close

Вход

Забыли?

вход по аккаунту

?

Study of photoelectronic properties of semiconductors by the advanced method of transient microwave photoconductivity (AMTMP)

код для вставкиСкачать
INFORMATION TO USERS
This manuscript has been reproduced from the microfilm master. UMI
films the text directly from the original or copy submitted. Thus, some
thesis and dissertation copies are in typewriter face, while others may be
from any type o f computer printer.
T he quality of this reproduction is dependent upon the quality of the
copy submitted.
Broken or indistinct print, colored or poor quality
illustrations and photographs, print bleedthrough, substandard margins,
and improper alignment can adversely afreet reproduction.
In the unlikely event that the author did not send UMI a complete
manuscript and there are missing pages, these will be noted.
Also, if
unauthorized copyright material had to be removed, a note will indicate
the deletion.
Oversize materials (e.g., maps, drawings, charts) are reproduced by
sectioning the original, beginning at the upper left-hand comer and
continuing from left to right in equal sections with small overlaps. Each
original is also photographed in one exposure and is included in reduced
form at the back of the book.
Photographs included in the original manuscript have been reproduced
xerographically in this copy. Higher quality 6” x 9” black and white
photographic prints are available for any photographs or illustrations
appearing in this copy for an additional charge. Contact UMI directly to
order.
UMI
A Bell & Howell Information Company
300 North Zeeb Road, Ann Arbor MI 48106-1346 USA
313/761-4700 800/521-0600
of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
STUDY OF PHOTOELECTRONIC PROPERTIES OF SEMICONDUCTORS BY THE ADVANCED
METHOD OF TRANSIENT MICROWAVE PHOTOCONDUCTIVITY (AMTMP)
A Thesis
Presented to
The Faculty of G raduate Studies
of
The University of Guelph
by
SERGUEI GRABTCHAK
In partial fuflilment of requirements
for the degree of
Doctor of Philosophy
August, 1998
©Serguei G rabtchak, 1998
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1 * 1
National Library
of Canada
Bibfiotheque rationale
du Canada
Acquisitions and
Bibliographic Services
Acquisitions et
services bibliographiques
395 Wellington Street
Ottawa ON K1A0N4
Canada
395. rue Wellington
Ottawa ON K1A0N4
Canada
Your Ha Votro
Our So Notn relirmnca
The author has granted a non­
exclusive licence allowing the
National Library of Canada to
reproduce, loan, distribute or sell
copies o f this thesis in microform,
paper or electronic formats.
L’auteur a accorde une licence non
exclusive permettant a la
Bibliotheque nationale du Canada de
reproduire, preter, distribuer ou
vendre des copies de cette these sous
la forme de microfiche/film, de
reproduction sur papier ou sur format
electronique.
The author retains ownership of the
copyright in this thesis. Neither the
thesis nor substantial extracts from it
may be printed or otherwise
reproduced without the author’s
permission.
L’auteur conserve la propriete du
droit d’auteur qui protege cette these.
Ni la these ni des extraits substantiels
de celle-ci ne doivent etre imprimes
ou autrement reproduits sans son
autorisation.
0 612 33301-9
-
-
Canada
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
GRABTCHAK
UNIVERSITY
^ iI uelph
Serguei Yuryevich
928 013 450
GRADUATE PROGRAM SERVICES
Chemistry and Biochemistry
PhD
CERTIFICATE OF APPROVAL fDOCTORAL THESIS)
The Examination Committee has concluded that the thesis presented by the above-named candidate in partial
fulfilment of the requirements for the degree
Doctor of Philosophy
is worthy of acceptance and may now be formally submitted to the Dean of Graduate Studies.
Title:
" S t ud y o f p h o t o e l e c t r o n i c p r o D e r t i e s o f semi conduc tor s by the advanced method
o f t r a n s i e n t mi cr owave p h o t o c o n d u c t i v i t y
(AMTMP)"
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
'-* < Y Y T r* L f\r.
Chair. Doctoral Examination Committee
11.
External‘Examiner
111.
Gradua ;e Faculty Member
IV .
Advisory Committee
V.
Advisory Committee
r
Received by:
Date:
for Dean of Graduate Studies
rev. iv/96
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
AUG 2 8 1998
ABSTRACT
STUDY OF PHOTOELECTRONIC PROPERTIES OF SEMICONDUCTORS BY
THE ADVANCED METHOD OF TRANSIENT MICROWAVE
PHOTOCONDUCTIVITY (AMTMP)
Serguei Grabtchak
University of Guelph, 1998
Advisor:
Professor M. Cocivera
The current work has described the new experimental method, Advanced Method
of Transient Microwave Photoconductivity (AMTMP) and its capabilities to study the
photoelectronic properties of semiconductors. AMTMP measureo not only the effect
proportional to the excess conduction band electrons (“photoconductivity” itself, related
to the changes in the imaginary part of the complex dielectric constant), but also the
changes in the real part of the complex dielectric constant ( “photodielectric effect “). In a
rigorous treatment o f the complex dielectric constant changes in a microwave cavity
general expressions relating the changes in complex dielectric constant to two
experimentally measured quantities (change in the cavity quality factor and the shift of
the resonance frequency) were derived. Based on this method it was possible to
systematize basic types of excitations (free electrons, plasma, trapped electrons, excitons)
as bound/non-bound states.
To interpret the behavior of the kinetics in semiconductors having distributions of
the localized states in the band gap the simulation approach based on solving numerically
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
multiple trapping rate equations was developed. With this approach the major basic types
of distributions (rectangular, linear, exponential, Gaussian) were explored thoroughly.
We tested the method on various semiconductors: two types of polycrystalline
CdSe thin films (#1 and #2), semi-insulating (SI) GaAs, single crystal Si and porous Si.
We identified the distributions present in CdSe #1 as an exponential distribution and in
CdSe #2 as a Gaussian-like peaked distribution. The parameters of the distributions were
estimated. We showed that with respect to the distribution, the transient methods always
probe only the part of the real distribution which is present in a sample. The distribution
becomes distorted from the initial form by the Fermi function describing the occupational
probability and truncated at high energies by the demarcation level.
For SI GaAs we were able to separate the mobility changes from the
concentration changes in the photoconductivity decays. We showed that to observe
excess free electrons rather than trapped electrons dominate in the changes of the real part
of the dielectric constant the material has to possess high mobility carriers and relatively
low trap densities.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
This thesis is dedicated to my wife Elena and my daughter Arisha.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ACKNOWLEDGMENTS
I would like to thank Professor Michael Cocivera for his support and guidance
during this work. I would like to acknowledge Dr. Shixing Weng, a former postdoctoral
fellow in our group and Dr. Dolf Landheer of NRC, Canada for supplying CdSe samples.
Professor Guy Adriaenssens (Katholieke Universiteit Leuven) for helpful discussions
and Professor Charles Dunkl (University o f Virginia) for supplying the algorithm for
finding zeros o f high degree polynomials. Thanks is due to Steve Siedfried and Ian
Renaud for their assistance with electronics, Terry White for machining a resonance
cavity and many other pieces o f the equipment, Uwe Oehler for writing many programs,
Yves-Marie Savoret for the glass blowing.
Research funding was provided by the Natural Science and Engineering Research
Council of Canada through grants to Professor Michael Cocivera. The Ontario Graduate
Scholarship provided by the Ministry of Education, Canada, the Summer Fellowship
provided by the Electrochemical Society and the US Department of Energy, the
University of Guelph graduate scholarships and support from the Department of
Chemistry and Biochemistry are also gratefully acknowledged.
Finally, special thanks to my family, my wife Elena and my daughter Arisha.
Their constant support and patience have made this work possible.
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE OF CONTENTS
Acknowledgments.................................................................................................................... i
Table of Contents....................................................................................................................ii
List of Figures........................................................................................................................ vi
List of Tables.......................................................................................................................xvi
1. INTRODUCTION............................................................................................................... 1
2. PERTURBATION THEORY FOUNDATIONS OF AMTMP.......................................16
3. EXPERIMENTAL SETUP OF AM TM P....................................................................... 25
3.1. The Outline of the Experiment......................................................................................25
3.2. Setup 1............................................................................................................................ 29
3.3. Setup 2............................................................................................................................ 34
3.4. Time-Resolved Measurement Requirements................................................................38
3.5. Noise Reduction and the FFT........................................................................................43
4. HARMONIC OSCILLATOR BASED ANALYSIS OF BOUND/NON-BOUND
ELECTRON STATES.......................................................................................................... 46
4.1. Free Electron Effects...................................................................................................... 46
4.2. Plasma Effects.................................................................................................................48
4.3. Trapped Electron Effects............................................................................................... 54
4.4. Dominant Contributions for the AMTMP Measurements............................................56
ii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5. MULTIPLE TRAPPING RATE EQUATION SIMULATION FOR VARIOUS
FORMS OF DISTRIBUTION OF LOCALIZEDSTATES.............................................. 58
5.1. Exponential Distributions..............................................................................................59
5.1.1. Details of the Simulation...........................................................................................60
5.1.2. Behavior of An(t), An,(t) andPhotoabsorption Signal(PA((t))................................... 62
5.1.3. The Effect of the Width of Exponential Distributions: Dispersive/Non-Dispersive
Transport Transition.............................................................................................................. 69
5.1.4. The Effect of the Temperature, T............................................................................... 77
5.1.5. The Effect of Density of States, g0............................................................................. 77
5.1.6. The Effect of E,........................................................................................................... 81
5.1.7. Summary......................................................................................................................84
5.2. Rectangular and Linear Distributions......................................................................... 84
5.2.1. Details of the Simulation............................................................................................85
5.2.2. Behavior of An(t)........................................................................................................ 86
5.2.3. Effects of S, xr and An0 on the Behavior of An(t)..................................................... 95
5.2.4. The Effect of the Width of RectangularDistributions............................................... 97
5.2.5. Effect ofE ,.................................................................................................................100
5.2.6. Temperature Effects.................................................................................................. 102
5.2.7. Summary....................................................................................................................104
5.3. Gaussian Distributions............................................................................................... 108
5.3.1. Behavior of An(t) and 5 s'( / ) .................................................................................... 108
5.3.2. The Effect of the Width of Gaussian Distribution...................................................110
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5.3.3. The Effect O f The Peak Energy Position, E0............................................................112
5.3.4. The Relation Between Power-law Decays and Individual Levels In The
Distribution........................................................................................................................... 114
5.3.5. The Criterion O f The Continuity Of The Distribution............................................. 123
5.3.6. Summary.....................................................................................................................125
6. EXPERIMENTAL RESULTS AND ANALYSIS........................................................127
6.1. CdSe (#1)........................................................................................................................ 127
6.1.1. Experimental...............................................................................................................127
6.1.1. Analysis o f 5e' Transients.........................................................................................132
6.1.2. Kinetics Analysis....................................................................................................... 135
6.1.3. Summary.....................................................................................................................152
6.2. CdSe (#11)..................................................................................................................... 153
6.2.1. Experimental.............................................................................................................. 153
6.2.2. Behavior at 300 K...................................................................................................... 155
6.2.3. Behavior at 123 K...................................................................................................... 164
6.2.4. Behavior at 358 K...................................................................................................... 170
6.2.5. Temperature dependence of a ................................................................................... 172
6.2.6. Gaussian Distribution Based Model......................................................................... 175
6.2.7. Summary.....................................................................................................................196
6.3. Semi-insulating (SI) GaAs.......................................................................................... 197
6.3.1. Experimental.............................................................................................................197
6.3.2.. Approach 1.................................................................................................................198
iv
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6.3.3. Approach II............................................................................................................... 202
6.3.4. Approach III.............................................................................................................. 204
6.3.5. Trapped Electron Estimations.................................................................................. 205
6.3.6. Kinetics Analysis......................................................................................................208
6.3.7. Summary....................................................................................................................219
6.4. Si....................................................................................................................................220
6.5. Porous Si.......................................................................................................................224
7. CONCLUSIONS............................................................................................................. 225
8. REFERENCES................................................................................................................228
V
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
LIST OF FIGURES
Figure 1. Time dependence of the resonance curve and the reconstruction of the
difference signal from the transient decays.
Figure 2. The geometry o f a microwave cavity used. Inset: a field distribution inside the
cavity.
Figure 3. Schematic diagram of the experimental setup 1.
Figure 4. Detector calibration curve.
Figure 5. Typical measured and simulated Lorentz resonance curves for setup 1.
Figure 6. Schematic diagram of the experimental setup 2.
Figure 7. Typical measured and simulated Lorentz resonance curves for setup 2.
Figure 8. The tail of the photoresponse at the resonance frequency with different number
of averages: (top) no average; (middle) 16; (bottom) 256
Figure 9. Square-law equivalent circuit of detector voltage measurements.
Figure 10. Signal after 16 averages and 50% level FFT filtering.
Figure 11. FFT editing: (a) the frequency domain before editing; (b) the frequency
domain after zeroing channels above 162.08 Hz; (c) the time domain after zeroing.
Figure 12. Changes in the real and imaginary parts of the complex dielectric constant due
to excess free electrons as functions of mobility.
Figure 13. The plasma resonance frequency as a function of electron concentration for
various values of depolarization factor L.
vi
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 14. Changes in the real and imaginary parts of the complex dielectric constant for
a plasma as a function of time for various values of depolarization factor L. The change in
sign indicated for the real part is due to the plasma/free electron transition.
Figure 15. (a) Transformation of non-resonance absorption of free electrons to the
resonance absorption of bound electrons with increasing binding energy; (b) The change
in the imaginary part o f the dielectric constant for plasmas as function o f the relaxation
time and relative plasma frequency. The plasma effect occurs over the narrow region of
the peak.
Figure 16. The simulated decays o f free electrons (An(t)) and electrons localized in
selected traps (An^t)) for an exponential localized-state distribution.
Figure 17. The simulated decays o f free and total weighted trapped electrons for an
exponentially localized-state distribution.
Figure 18. The simulated decays An(t) and 5 e '(0 for an exponentially localized state
distribution with T0=400 K..
Figure 19. The simulated decays of An(t) for selected values of the characteristic
temperatures of the exponential distributions, T0.
Figure 20. The dependence of a on 1/T0 for the decays shown on Figure 19.
Figure 21. Parameter Ano extracted with Eq.(49) from decays shown on Figure 19.
Figure 22. Time constants extracted with Eq.(49) from decays shown on Figure 19.
Figure 23. The simulated decays of An(t) for selected values of the absolute temperature,
T.
Figure 24. The temperature dependence of a for decays shown on Figure 23.
vii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 25. Parameter Ano extracted with Eq.(49) from decays shown on Figure 23.
Figure 26. Time constants extracted with Eq.(49) from decays shown on Figure 23.
Figure 27. Simulated An(t) decays for selected values of g0.
Figure 28. An(t) and 5e'(f) decays simulated for selected values o f the offset Eh
Figure 29. Behavior o f simulated An(t) decays for selected values of g0 increasing in
powers of ten from 1014 cm’3 eV'1 (curve 1) to I022 cm’3 eV*1 (curve 9). Inset: same in
semi-log scale.
Figure 30. The time constant (ts ) of the exponential tail as a function of the density of
states ( g 0).
Figure 31. Decays of excess electrons (An(t)), electrons (An/t)) trapped in a localized level at
0.21 eV, photoadsorption signal (PA(t) ) and photodielectric signal ( 5 s '(/) ) for weak retrapping.
Figure 32. Decays of excess electrons (An(t)), electrons (An/t)) trapped in a localized level at
0.21 eV, photoadsorption signal (PA(t)) and photodielectric signal (5 e ' ( 0 ) for strong
retrapping.
Figure 33. The exponent o f the power law decay as a function o f g0.
Figure 34. The decay o f excess electrons (An(t)) for different values of capture crosssection (S ).
Figure 35. Decays of excess electrons (An(t)) and photodielectric signal ( 5 e '( 0 ) for
selected widths (mA) for rectangular DOS with weak retrapping.
Figure 36. Decays o f excess electrons (An(t)) and photodielectric signal ( 5 s '( 0 ) for
selected widths (mA) for rectangular DOS with strong retrapping.
viii
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
Figure 37. Decays o f excess electrons (An(t)), photodielectric signal (8s'(O X
photoadsorption signal (PA(t)) for selected offsets (£/) from the bottom of the conduction
band with weak retrapping.
Figure 38. Decays of excess electrons (An(t)) at selected temperatures for weak
retrapping.
Figure 39. Decays of excess electrons (An(t)) at selected temperatures for intermediate
retrapping.
Figure 40. Parameter a of the power law decay as a function of temperature for strong,
intermediate and weak retrapping.
Figure 41. An(f) and 5 s '(/) simulated for Gaussian distributions ,T0=800 K.
Figure 42. An(t) and 8e'(/) simulated for Gaussian distribution using selected values of
the characteristic temperature, T0.
Figure 43. Values of oci extracted from the power-law decay in the region before the knee
from the curves simulated for Gaussian distributions and selected values of T0.
Figure 44. Values of a 2 extracted from the power-law decay in the region after the knee
from the curves simulated for Gaussian distributions and selected values of Tn.
Figure 45. Kinetics o f An(t) simulated for the Gaussian distribution for two different
peak energy positions, E0.
Figure 46. Kinetics An(/) simulated for Gaussian distributions with selected values of the
offset energy, E,.
Figure 47. The total Gaussian distribution used for the simulation in Figure 46.
ix
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 48. Relation between individual trapping time, release time, recombination time,
average trapping time and trap energy for a Gaussian distribution with different offset
energies, Hj: a) E|=0.06 eV, b) E|=0.3 eV, c) E|=0.5 eV, d) Et=0.6 eV.
Figure 49. Kinetics of An(t) simulated for various values of the separation (A) between
levels at kT=0.0\06 eV.
Figure 50. The difference signal between the “dark” and the light induced resonance
curves obtained 2.1 psec after the laser pulse. The solid line represents the fit according
to Eq.(25).
Figure 51. Changes in the cavity quality factor
(upper curve) and the shift in the
resonance frequency 5/ 0 (lower curve) as function of time for light intensities
corresponding to I0 = 3.3 1020 electrons/cm3; T = 300 K.
Figure 52. Changes in the cavity quality factor 5(A/i/x) (upper curve) and the shift in the
resonance frequency 5/ 0(lower curve) as function of time for light intensities
corresponding to I0 = 1.8 1018 electrons/cm3; T = 300 K.
Figure 53. The changes in the complex dielectric constant for the I pm CdSe thin film
(# 1).
Figure 54. The simulation of the Clausius-Mossotti equation (Eq.(52)) for CdSe.
Figure 55. Loss tangent at two intensities, I0 = 3.3 1020 electrons/cm3, T = 300 K.
Figure 56. Intensity dependence of the maximum of 8 (A/j/2) associated with the fast
component prior the power-law dependence.
Figure 57. Intensity dependence of parameter a obtained for the power-law fit of the
time dependence of the cavity quality factor change.
x
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 58. Intensity dependence o f parameter a obtained for the power-law fit of the time
dependence of the shift of the resonance frequency.
Figure 59. Intensity dependence of the apparent bimolecular recombination time for the cavity
quality factor change.
Figure 60. Intensity dependence of the apparent bimolecular recombination time for the
shift o f the resonance frequency.
Figure 61. Intensity dependence of the extrapolated values of 5(A/I/2) and 5/0 based on
the power law decay
Figure 62. Changes in the cavity quality factor 6(Afl/2) (upper curve) and the shift in the
resonance frequency 5/0(lower curve) as function of time for light intensity
corresponding to I0 = 3.3 1020 electrons/cm3; T = 331 K.
Figure 63. Experimental and simulated curves of An(t) and 5e'(t) obtained for
exponential distribution o f localized states, T=300 K, I=I0.
Figure 64. Experimental and simulated curves of An(t) and 8s'(t) obtained for
exponential distribution o f localized states, T=300 K, 1=0.005 I0.
Figure 65. Experimental and simulated curves of An(t) and 5s' (t) obtained for
exponential distribution of localized states, T=331 K, I=I0.
Figure 66. Changes in the real and the imaginary parts of the dielectric constant for 1pm
CdSe (#2), T=300 K, I=I0. The inset: the same curves in a semi-log scale.
Figure 67. The decay o f the excess electron concentration for 1pm CdSe (#2), T=300 K,
I=I0.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 68. The kinetics o f the movement of the electron Fermi level, EFn, after excitation
for lpm CdSe (#2), T=300 K, I=I0.
Figure 69 The intensity dependence o f 5(AfI/2) for 1pm CdSe(#2), T=300 K.
Figure 70. The intensity dependence of -5f0 for 1pm CdSe(#2), T=300 K. Intensities are
the same as on Figure 68. Curves are scaled.
Figure 71. Intensity dependence o f the time constant of the fast exponential decay for the
bandwidth change, 1pm CdSe (#2), T=300 K.
Figure 72. Intensity dependence of the pre-factor of the initial exponential decay for the
bandwidth change, 1pm CdSe (#2), T=300 K.
Figure 73. Intensity dependence o f the parameter a from the power law decay in the
bandwidth change, 1pm CdSe (#2), T=300 K.
Figure 74. Intensity dependence o f the time constant of the slow exponential decay for
5(An/2) and 6f0, I pm CdSe (#2), T=300 K.
Figure 75. Changes in the real and the imaginary parts of the dielectric constant for 1pm
CdSe (#2), T=123 K, I=I0. The inset: the same curves in a semi-log scale.
Figure 76. The decay of the excess electron concentration for 1pm CdSe (#2), T=123 K,
I=IoFigure 77. Intensity dependence of the pre-factor o f the initial exponential decay for the
bandwidth change, lpm CdSe (#2), T=123 K.
Figure 78. Intensity dependence of the time constant of the fast exponential decay for the
bandwidth change, lpm CdSe (#2), T=123 K.
xii
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
Figure 79. Intensity dependence of the parameter a from the power law decay in the
bandwidth change, lpm CdSe (#2), T=123 K.
Figure 80. Intensity dependence of the time constant of the slow exponential decay for
S(Afl/2) and 5f0, lpm CdSe (#2), T=123 K.
Figure 81. Changes in the real and the imaginary parts o f the dielectric constant for lpm
CdSe (#2), T=358 K, I=I0.
Figure 82. The decay of the excess electron concentration for lpm CdSe (#2), T=358 K.
I=I0Figure 83. Temperature dependence of the parameter a extracted from the bandwidth
change, lpm CdSe (#2), I=I0.
Figure 84. Experimental and simulated with Gaussian DOS decays of An(t) and
5 e '(0 for lpm CdSe (#2), T=I23 K, I=I0.
Figure 85. Experimental and simulated with Gaussian DOS decays of An(t) and 5s '(t) for
lpm CdSe (#2), T=300 K., I=I„.
Figure 86. Parts of Gaussian distributions obtained from 123 K and 300 K kinetics using
the simulation approach, 1 pm CdSe (#2), I=I0.
Figure 87. Real Gaussian DOS, occupied states determined by the Fermi function, states
available and fitting to Gaussian DOS (T=300 K).
Figure 88. Curves from Figure 87 plotted in a semi-log scale (T=300 K).
Figure 89. Real Gaussian DOS, occupied states determined by the Fermi function, states
available and fitting to Gaussian DOS (T=123 K).
Figure 90. Curves from Figure 89 plotted in a semi-log scale (T=123 K).
xiii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 91. The energy of the last trap extracted according Eq.(62) from the terminal
exponential decay in 8f0. for lpm CdSe (#2).
Figure 92. Half-Gaussian distribution: real distribution, occupied states and available
states, T=123 K.
Figure 93. Half-Gaussian distribution: real distribution, occupied states and available
states, T=300 K.
Figure 94. The decay of excess electron concentration in SI GaAs, T=358 K, I=I0,
A.=1064 nm
Figure 95. The changes in the real part of the dielectric constant of SI GaAs, T=358 K.
I=I0, A=1064 nm
Figure 96. The relative increase in the mobility, Ap and the absolute value of the electron
mobility, p as a function of time in SI GaAs after excitation, T=358 K, I=I0, X=1064 nm.
Figure 97. Kinetics of the bandwidth change, 8(AfI/2) for selected temperatures in the
range +85°C - 0°C, I=I0.
Figure 98. Kinetics of the bandwidth change, 8(Af1/2) for selected temperatures in the
range 0°C - -60°C, I=I0.
Figure 99. Kinetics of the bandwidth change, S(AfI/2) at selected intensities for SI GaAs,
T=358K.
Figure 100. Intensity dependence of the time and the amplitude at the “break” point for
SI GaAs, T=300 K.
Figure 101. Temperature dependence of the time of the break point for selected
intensities, SI GaAs.
xiv
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
Figure 102. Temperature dependence of the time constant o f the dominant component for
selected intensities, SI GaAs.
Figure 103. Electron thermal release time as a function of trap depth (below conduction
band) for T = 213, 300, 358 K.
Figure 104. The bandwidth change and the shift of the resonance frequency for p-type Si
sample with e'=15, s"= 45, T=300 K.
Figure 105. The bandwidth change and the shift of the resonance frequency for p-type Si
sample with e'=15,e"=100 , T=300 K.
XV
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
LIST OF TABLES
Table 1. Simulated time constants obtained with methods #1 and #2.
Table 2. Release times for specific levels at selected temperatures.
Table 3. Estimated upper limit to trapped electron density.
xvi
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1. INTRODUCTION
The relatively short period which has elapsed since the invention of the bipolar
transistor has seen a dramatic and unprecedented industrial development based on the
electrical and optical properties of semiconducting materials. For activities related to
applications and fundamental understanding to flourish, it is necessary to prepare
materials with carefully specified properties, and this clearly requires that these properties
be accurately measured. Because the principal use of semiconductors is for devices which
rely on electronic properties for their operation, knowledge about those properties is of
vital importance. One of the main goals of the research undertaken in our laboratory is the
determination of the effect of dopants on the carrier transport properties of semiconductor
materials (commercially made as well as prepared in our laboratory). Trapping and
recombination phenomena can affect a performance of semiconductor based devices (i.e.
thin film transistor, solar cells, photovoltaic cells) in various ways depending on a
particular application. There are numerous experimental techniques suitable to extract
information about specific materials properties. Photoconductivity of solids is a wellknown effect that can be used for electrical characterization of semiconductors [1, 2].
Photoconductivity is based on the measurement of changes in a sample’s conductivity
after producing excess charge carriers in the sample by an external light source. In
contrast to luminescent methods, photoconductivity may detect both non-radiative and
radiative processes if both are accompanied by conductivity changes. Transient
photoconductivity (TP) is a dynamic method which allows one to follow the above
mentioned changes in a real time. Before discussing the capabilities of TP we would like
l
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
first to consider the phenomena which manifest themselves in TP measurements.
Consider a semiconductor which has an equilibrium concentration o f electrons and holes
denoted as n0 and p 0with corresponding mobilities p 0n and \iQp. The conductivity of this
sample can be written as
(0
The excitation of the material can produce an incremental change in concentration as well
as in the mobility. Changes in the sample’s conductivity Aci(t) can be described as the
following result:
ACT(/)=e(|io„+p0/,)(An(r)+A/7(O)+eAp(r)(n0+p0)+eAp(O(An(r)+Ap(O)
(2)
where Ap.(r)=Ap.„(f)+Ap (f)- This is a general photoconductivity expression which
serves as a basis for exploring photoconductivity phenomena. Eq.(2) is exact and no
assumptions have been made so far. In our subsequent discussion we consider only
positive increments. (Negative photoconductivity has been observed experimentally [2]
but lies out the scope of the present work.) The first step in a simplification of Eq.(2)
requires some extra knowledge about the semiconductor and we step back to Eq.(l) for
this purpose. When the contributions of both terms in Eq.(l) are comparable one deals
with the ambipolar conductivity. For example, this condition can be realized for intrinsic
materials when the values of the electron and hole mobilities are close. In the case o f
dark ambipolar conductivity, the full expression (Eq.2) should be used, and some
additional knowledge about the dominant light-induced changes is required. Next,
consider for simplicity n-type semiconductors where n(p,>p0. Even if the equilibrium
2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
mobilities o f electrons and holes are comparable only the first term in Eq.(l) will
determine the equilibrium conductivity value. One can see, that the condition n0» p n
alone does not bring the desired simplification into Eq.(2). For some semiconductors (for
example, GaAs, Si, a-SiC, GaSb, InP, InAs, InSb, ZnSe, CdS, CdTe, CdSe, ZnS, AgBr)
where the electron mobility is at least three times of the hole mobility [3] one can drop
the equilibrium hole mobility in Eq.(2). Therefore, when both conditions (n0» p 0 and
M-on>1*0/7 ) aie realized one can reduce Eq.(2) to the form
A
ct
( O » e H 0n( A w ( O + A / ? ( O ) + e A p ( O « 0 + e A ! * ( O ( A « ( O + A / ? ( O )
This is probably the greatest simplification one can made with some preliminary
knowledge of the equilibrium properties of the semiconductor. Usually, a verification of
the above conditions does not constitute any difficulties. Looking at Eq.(3) one can see
that a first problem in interpreting the conductivity changes comes from the fact that both
the concentration and mobility may undergo changes after illumination. However,
illumination does not allow us to consider only one type of carrier as a dominant source
for carrier and mobility changes. All subsequent simplifications o f Eq.(3) will require
knowledge of the relative behavior of excess electrons and holes in the particular
semiconductor. Therefore, we have ended up with the same problem with which we
started. To make progress in this problem, one has to obtain information on carrier
dependent mobility changes by other means and assume its validity for conditions o f the
experiment. The value of photoconductivity measurements can still be preserved. For
example, from transient cathodoluminescence experiments [4] it has been shown that in
all high quality n- and p-type GaAs samples free holes disappear essentially
3
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
instantaneously relative to electrons. One can then drop Ap(t) in Eq. (3) and consider
An(t) only. From photo-Hall experiments [5], information about the type and value of
the
mobility changes
in GaAs
enable
one to drop
the
App( 0
term
in
A|4.(0=A|i„(O+Ap p(0 • After those two steps Eq.(3) can be simplified further
A ct ( / ) * e p 0nA«(/)+«?Ap„(O/?0+ e A p n( / ) A « ( O
Under high intensity illumination
(4)
( A/z(r)»«0), the second term can be neglected.
Finally, when the mobility changes are less than the equilibrium mobility values one can
end up with the following widely used but not always justified expression
A ct ( O * e p 0„An(O •
(5)
At this point one can see the two main limitations of TP namely: a) the inability to
separate the effects coming from the oppositely charged carriers; b) the inability to
separate mobility changes from concentration changes in the photoconductivity signal. As
a result, photoconductivity method can not be used alone.
It requires background
information to separate the different contributions. Usually Eq.(5) is involved to draw
conclusions with respect to parameters from measurements [6, 7]. The time resolution
capabilities of TP (and Eq.(5)) determines the primary purpose for its application, i.e. to
extract the value of the lifetime, xn. In contrast to a loose use of this term its definition [6]
is the overall minority carrier lifetime (tn) related to the lifetimes o f radiative ( tr) and
non-radiative (Tnr) processes:
4
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The minority carrier lifetime can be considered a fundamental parameter. Because the
operation o f many electronic semiconductor devices is based primarily on minority
carriers, it becomes clear why its measurement is important. This is a so called “classical”
application of the technique. However, this definition does not preclude TP from
measuring the lifetime of the excess majority carriers when they can be seen more easily
in the photoconductivity signal due their larger mobility [8] (e.g. electrons in n-type
semiconductors, see discussion above.) Moreover, the value of the mobility often
determines the type of the carrier that dominate the TP signal. For example, electrons in
both p- and n-type GaAs will always be observed due to their larger mobility relative to
holes. This means that one would not be able to study directly the trapping/recombination
process involving holes in n-type GaAs. Clearly, this imposes a limitation on materials
which can be studied by TP in a “classical” way. When planning lifetime experiments
one also should consider the following: a) the lifetime can be obtained only if the low
intensity condition is satisfied, i.e., when an exponential decay due to monomolecular
recombination is observed; b) the presence of traps due trapping/retrapping processes can
distort the measurement of the lifetime and needs to be taken into account; and c) the
influence of surface on the measured lifetime, often presented in terms of a surface
recombination velocity, should be considered as well. As one can see the extraction of
data from TP measurements done in a proper way will often involve much more than
simply plotting the kinetics in semi-log scale to obtain time constants. Our concern in this
chapter is with measurement techniques rather than models of recombination. However,
5
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
an awareness of the [imitations o f the theoretical concepts is nevertheless essential to a
proper interpretation o f the resulting data.
A search for “photoconductivity” in titles of periodicals can convince any reader
that the scientific literature is literally flooded with photoconductivity related articles.
Therefore, our aim is not in giving a comprehensive review of this subject. Rather we will
attempt to outline of a spectrum of problems which can be treated with TP. First we
consider a contact TP, i.e. the method which requires making good ohmic contacts to the
sample under study. GaAs and related compounds were among the first targets of TP [922]. The earliest works [9,10] obtained values of minority and majority lifetimes in GaAs
using TP and PEM (photoelectromagnetic) techniques. The justification of results
involved a great deal of analysis involving radiative and Shockley-Read recombination.
The photoelectric memory effect observed in semi-insulating GaAs has been a subject of
numerous works [11, 14-16]. A siow-relaxation photoconductivity (up to tens of hours) at
low temperature was quenched optically and thermally to establish the nature of this
phenomenon. However, in spite of a large number of studies the mechanism of the
phenomenon is still not completely explained. Developments in the TP method focused
on measurements of diffusion coefficients of non-equilibrium carriers [12] and on
spatially resolved mapping of the entire surface o f wafers [13]. The well known effect
that the conductivity of SI GaAs can be increased by up to 5 orders o f magnitude served
as a basis for developing a concept of the semiconductor switch [17-19]. Those works
involved a great deal o f numerical simulation based on rate equations as well. Combining
TP and transient photo-Hall measurements authors [20] attempted to use potential
6
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
fluctuations in the band gap o f SI GaAs to explain the high degree of non-exponentiality
found in TP kinetics in SI GaAs and the variations in the mobility values after excitation.
The photoconductivity effect arising in GaAs based hetorstructures has received
increased attention [21, 22]. The growing interest is caused by the ability to control the
transport o f majority/minority carriers in sandwich-like structures.
Si is another material which has been studied intensively [23-27] by TP. In early
work [23, 24] trapping o f minority carriers in Si (both rt- and p-type) was studied. With
some approximations authors developed models based on trapping/recombination rate
equations which allowed an estimation of the trap density, energy levels, and capture
cross-sections of traps. These studies are good illustrations of the point made above that
varying the conditions of the experiment and detailed analysis are the only way to obtain
less ambiguous conclusions. They also show that TP depends to a large extent on some
assumptions and semi-quantitative approximations in order to extract quantitative data. In
[25] the dependence o f the rate of interband Auger recombination on the carrier density in
Si samples was experimentally investigated. Among the theoretical developments, we
would like to mention that this article [26] gave an analysis o f the interaction o f a laser
pulse with a silicon wafer. A comprehensive theory for determination of the bulk lifetime
and the surface recombination velocity was also presented. A separation of the surface
and the bulk lifetimes is the frequent theme of a number of studies since the silicon
surface and interfaces are among most frequently used in semiconductor devices [27,28].
There is a growing interest toward porous Si, and TP produces kinetics which often
puzzle the authors [29].
7
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The transient nature of TP stimulated further development in the statistics excess
carriers in semiconductors. Some variations of Shockley-Read recombination were
analyzed rigorously [30, 32]. The new methodology to separate Shockley-Read
recombination lifetime, trap-assisted Auger and band-to-band Auger recombination
coefficients was suggested [31]. Serious attention was given to the analysis of the lightbiased photoconductivity decay measurements [33].
Almost every known material or structure has been the subject of TP studies: InSb
[34], HgCdTe [35], InAs, InAsSb and InAsSb-InAlAsSb quantum wells [36], single
crystal and polycrystalline CdSe [37 - 44] etc. Polycrystalline and amorphous materials
are widely used in the semiconductor industry for fabrication of thin film transistors,
solar elements, flat panel displays, light emitting diodes,
laser diodes, and radiation
detectors. It is clear from this list, that these applications require an understanding of the
excess charge carrier behavior. Due to the structure of the samples, presence of grain
boundaries, large surface/volume ratio, varying degree of geometrical and/or energetical
disorder, the kinetics of TP are related to the materials parameters in a complex,
sometimes hardly predictable, way [45 - 53]. Nonetheless, this is a fast growing field
involving a large number of experimental and theoretical work.
From an experimental point of view, contact methods suffer some drawbacks: 1)
they require making contacts to the sample which can destroy the sample after use; 2) the
contacts can modify the sample's parameters; 3) some materials (free standing films,
powders, highly resistive samples) present a real challenge for making good ohmic
contacts. Therefore, the world is going contactless, i.e. microwave in particular.
8
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
When adding the microwave aspect to the photoconductivity method (transient
microwave photoconductivity, hence TMP) one has the following advantages:
•
no contacts are required, and the problem of contact potentials does not arise
•
any high impedance material can be studied
•
because of the very small (no more than tens of angstroms) displacemento f free
electrons during each half cycle of the microwave field, intergrain potentialsneed not
to be important
•
low mobility materials can be studied
TMP methods can be classified as waveguide and resonance (or cavity) based
methods. In waveguide TMP, the sample is placed in piece of waveguide connected to a
microwave source. Depending on the setup the reflected/transmitted microwave power is
measured. In resonance TMP, the sample is located in a resonance microwave cavity
which is equivalent to the resonance contour with distributed parameters. Excess carriers
produced in the sample by the illumination pulse begin to absorb the microwave energy
(Joule losses) when moving in the sample and dissipate the energy via various
collision/scattering processes involving the crystal lattice. I would like to discuss
waveguide TMP first.
A long list compiled from selected publications starting from 1986 describes some
aspects of waveguide TMP [54-92] and shows the popularity of the method. There are a
number of reasons for this popularity. First, it is relatively easy to assemble the
microwave setup for this type of experiment. All microwave components/devices are
standard, and some effort is needed only to make adjustments in the waveguide to
9
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
accommodate the sample and provide a possibility for its illumination. Second, the
changes in the microwave power are directly related to conductivity changes. The amount
and validity o f information extracted from TMP experiments depend to a large degree on
the approach used by authors. In a large number o f papers [54, 57, 60, 61, 67, 78, 83] the
photoconductivity kinetics obtained under limited experimental conditions were reported,
and the authors tried to explain it qualitatively by relating the decays to known processes.
An extension of this approach is to perform a surface mapping o f whole wafers to obtain
information about the uniformity of electrical properties o f samples [55.56]. There are
many, commercially available, microwave setups which are mostly used to provide
qualitative material characterization by measuring the decay time. With these setups the
idea is that the material is better if the decay is longer. However, in a more rigorous
study one really needs to strive for an interpretation of the decay. In other articles [70, 76]
the goal was to establish the correlation between the measured effective lifetimes and the
donor/acceptor concentration in the samples. A lot of work [58, 62-64, 68, 71, 84, 85, 88,
91] has been devoted to an attempt to separate the bulk and the surface recombination
lifetimes in TMP measurements, primarily for Si samples. A closely related issue, the
ability to extract mobility values from TMP experiments, was given attention as well [59,
66, 81, 87], A growing technological interest in effects of various annealing/etching
procedures on sample parameters [65, 73, 79, 82] and probing the properties of real
devices and multi-interface layered structures [69, 77, 89, 90, 92] was reflected in
corresponding research works.
10
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
At this point I would like to draw a line under the discussion o f the contactless
waveguide TMP and make few comments which can be applied equally both to contact
and contactless waveguide TP. Summarizing all the above I want to stress that the TP
decay is only one experimentally measured parameter (while a number o f such decays
can be measured by changing the conditions of the experiment), and the subsequent
analysis is aimed at separating (or neglecting) mobility changes from the conduction band
excess carrier contribution in order to: a) extract the kinetics of these two quantities (if
both undergo changes after illumination); b) affect by variation of experimental
parameters (temperature, intensity, wavelength, doping etc.) the observed decay; and c)
extract useful information about carrier lifetimes, trap depths, density of traps,
recombination/trapping parameters using physical models and some correlation between
photoconductivity data and independently obtained data. Therefore, the complexity of the
studied materials and phenomena put a lot of pressure on the theoretical side of TP to
validate the interpretation. Developing a model to explain a single photoconductivity
decay can be considered as the initial step, and the subsequent verification of the model
must come from the experiments done when various parameters (i.e., intensity,
wavelength, temperature etc.) are changed. Sometimes, only few parameters are varied in
the experiment or the range o f the variation is too small. All those factors may raise
questions about uniqueness o f the model. This issue pertains equally to any experimental
approach.
The resonance TMP is also a useful technique for monitoring electrons in the
conduction band [93-133]. Two effects are known to occur in the experiment involving
11
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the resonance cavity. The first is known as photoconductivity which manifests itself in
the change of the cavity bandwidth and is usually (but not always) related to the
imaginary part of dielectric constant. The second is the photodielectric effect [2] which
manifests itself in a change (shift) o f resonance frequency and is related to the real part of
dielectric constant. Hartwig and Hinds [93-95] demonstrated that excess electrons in the
conduction band and traps cause changes, in the imaginary and real parts of the dielectric
constant, respectively at 4.2° K where the decay rates were slow and steady state
measurements could be made. Their work represented the first attempt to separate the
contributions from excess free and trapped electrons in microwave photoconductivity
experiments and was a step forward towards better understanding photoconductivity
phenomena. To extend this type of analysis to shorter time scales, subsequent approaches
employed automatic frequency control [99] or rapid scan [104] of only a small portion of
the resonance signal (near the peak maximum) after a short pulse of light. In the latter, the
entire light-induced signal was not determined, and the change in the imaginary part of
the dielectric constant could only be inferred from the decrease in the peak maximum.
Also the frequency scanning rate was not sufficiently rapid for studies below the
microsecond range.
A more direct method in which the time resolution is limited only by the time
constant of the cavity and detector diode, was suggested by the author and tested on silver
halide powders and crystals [118-120]. In this approach, light induced transients were
measured at a number of fixed frequencies near the “dark” resonance frequency, and a
partial curve representing the difference between the “dark” and “light” Lorentz
12
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
resonance signals was determined at various times during the decay.
From each
difference signal, the light induced frequency shift (related to the change in the real part
of the dielectric constant) was calculated using the “zero frequencies” approach, which
required a number of assumptions. The change in the imaginary part of the dielectric
constant was subsequently calculated from this result.
In 1993 we developed a modification of this microwave technique which we
called the Advanced Method o f Transient Microwave Photoconductivity (AMTMP).
AMTMP is based on a fit of the total difference signal to the difference between the
“dark” and “light” Lorentz resonance signals [121-131]. Because the total rather than the
partial difference signal was used, the fit could be accomplished without the simplifying
assumptions required for the zero frequencies analysis described above. Furthermore, it
avoids the limitations inherent in the rapid scan approach discussed above. Therefore,
AMTMP measures not only the effect proportional to the excess conduction band
electrons (“photoconductivity” itself, related to the changes in the imaginary part of the
complex dielectric constant),
but also the changes in the real part of the complex
dielectric constant ( “photodielectric effect”).
These last changes may result from
conduction band electrons and/or from trapped electrons. This method facilitates the
interpretation o f the observed kinetics. With this approach, one can
determine the
changes in the real and imaginary parts of the dielectric constant unambiguously. As a
result, one would be able to separate the contributions of the conduction and trapped
electrons.
13
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
It should be clear from the review of the techniques that most o f the previous
microwave studies o f transients in semiconductors have been done in a traditional
resonance configuration or in a waveguide configuration. In the resonance configuration
the transient was measured only at the resonance frequency, and the difference signal was
not determined. Consequently, it was not possible to separate the relative contributions
o f conduction and trapped electrons. In the waveguide method the measured signal is
proportional to conductivity changes as well.
On the other hand, AMTMP can be considered as a more universal and
comprehensive technique. This method could be of particular interest when studying thin
films.
Indeed, problems of contacts, intergrain barriers, low mobility become less
important. Because trapping and recombination in thin films often demonstrate
peculiarities due to large surface/volume ratios. AMTMP can be very useful for shedding
light on these processes. We should mention that other than the author’s work in this area
there have been only few sporadic attempts [113-115] to study light induced changes in
the complex dielectric constant by the resonance technique but those measurements were
done under steady-state conditions.
This thesis explores the major aspects of AMTMP. The main goal is to show the
advantages and the versatility of the method for extracting useful information about
photoelectronic properties of semiconductors. Chapter 2 shows our achievements in a
rigorous development of the cavity perturbation theory which provides the necessary
links between the experimentally measured parameters and the changes in the complex
dielectric constant. In Chapter 3 the experimental setup of AMTMP and related aspects
14
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
are presented. Chapter 4 is devoted to our classification of the basic electron states as
bound/non-bound electrons. The harmonic oscillator model was used to show what
peculiarities in the complex dielectric constant will appear as a result of various states.
Chapter 5 explores the multiple trapping model in semiconductors via numerical
simulation for various forms of the localized state distributions in the band gap. Using
this approach some unique features of the distributions which manifest themselves in
AMTMP transients have been traced and identified. Chapter 6 presents our experimental
results for two types of polycrystalline CdSe thin films, semi-insulating (SI) GaAs, single
crystalline Si and porous Si. Finally, conclusions are drawn in Chapter 7.
15
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2. PERTURBATION THEORY FOUNDATIONS OF AMTMP
Insertion of a semiconductor or dielectric material into a microwave cavity
(resonator) causes a change in the resonance frequency, f , and the cavity quality factor,
QL. Cavity perturbation theory, which relates these changes to real and imaginary parts of
the complex dielectric constant, s * = s ' - ye " , of the material, was developed by Slater
[134] and later used for various materials [94, 95, 111, 135-139]. This first perturbation
applies to a sample placed in an empty cavity, and it was used to determine absolute
values of the real and the imaginary parts of the dielectric constant. A second perturbation
occurs when a sample already in the cavity is subjected to photons, electric current,
temperature, X-rays etc. to produce excess electrons and cause a change in the complex
dielectric constant, 8s* = 8 s '- y'8s". This second perturbation is applied to AMTMP,
and below the term “perturbation'’ refers to the second perturbation unless otherwise
stated.
Irradiation creates excess carriers in an active volume of the sample, which is not
necessarily equal to the total volume. Diffusion can increase this active volume
significantly when the diffusion coefficient is large. In addition to size, the geometry of
this active volume determines the exact form of the final expressions for 5s' and for
5 s " , and these expressions are known for only a limited number of sample
geometries[135]. Initially [121-125] we used a long thin strip approximation for the
sample geometry but this approximation restricts the samples that can be studied by
AMTMP. Below general perturbation expressions are developed using a depolarization
factor L to include the dependence on the sample geometry.
16
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
When a homogeneous material is isotropic, the dielectric constant can be
expressed as a complex function
e *= e '- j z "= e
(7)
O)S0
in which e' is a normalized (relative) dielectric constant (real part), e" is the imaginary
part,
ct
is the material conductivity, e0 is the permittivity and
cd
is the real angular
frequency [135].
The change in the complex angular frequency gj \ is [135]
(lu - y i ; ) \ H xH2d V - ( $ ;
FxF2dV
=
^
(g)
K
in which ej and z\ are the complex dielectric constants of the sample before and after
illumination, Vs and Vc are the active sample and cavity volumes, and H and F are the
magnetic and electric fields. This basic cavity perturbation formula involves no
approximations. For a non-magnetic sample it reduces to
oco
®2
(9)
2 \ z l F xF1dV
l 'c
Note that F{ and F2 correspond to the field in the sample before and after illumination.
There are actually two terms in the denominator, and Eq.(9) can be written as :
( & - * : ) \ F xF2dV
5(0 ‘
y
—
“2
f
'
— •
2e'meJiua \ F}F2dV + 2e; \F xF2dV
Vc-V,
V,
17
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
(10)
The first integration is over the cavity volume and includes z m
' tdiumthe dielectric constant
o f the medium filling the cavity. Normally, the experiments are done either in air or
vacuum, so the dielectric constant equals 1. When the sample is sufficiently small and
does not change significantly the field in the cavity volume, F, = F,= F0 is a good
approximation, and F0 is the field in the cavity outside the sample. In the second term the
integration is over the sample and knowledge of fields F2 and F, is required. When F: =
F,= F0 one obtains the simplest expression:
o i)
2
1‘
which applies to a long thin strip placed into the cavity. Changes in the resonance
frequency / 0 and loaded cavity quality factor O, also can be related to the changes in the
complex angular frequency by [135]
5“_ l = & +y8
® 2 foi
^
in which 5/0 = f Q2
( 12)
and 5(1 / 2 0 , ) = (1 / 2QL2) -(1 /2 Q U) . The cavity quality
factor is
& = / o/A /i/2,
(13)
in which Aft/2 is the fiill bandwidth at half-maximum of the reflected power. Equating
E q s.(ll)
and
(12) and separating
the real
Kfio
and the imaginary
A20
18
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
parts
gives:
(14)
Vsf 2o
4
fA200
'
In practice, / 02 « / 0I and P"c. » 4 P ''s ' are good approximations. The relative magnitudes
of the real and the imaginary parts of the dark dielectric constant dictate which terms can
be neglected in Eqs. (14). For low conductivity samples, the terms containing e " can be
neglected, and Eqs. (14) are reduced to
5c, _
1 KVo _
2 V J 20
&/.
f 10G
(15)
5c>,
1 F;.5(A/1/2) _ 5(A/,/2)
4 Vsf 20
2 /,0G
These equations are almost the same as those derived previously [121] except z\ is
absent. Therefore, changes in the complex dielectric constant were overestimated
previously [121].
To treat more complicated possibilities, we return to Eq.(lO) and consider the case
in which the first term dominates in the denominator. According to Landau [140], the
field F2 inside the sample can be related to the external field F0 using a depolarization
factor L2:
(16)
19
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
which is valid when a sample with dielectric constant s 2 is placed in a vacuum. An
analogous expression applies to the field Fx. The same equations were used to derive the
corresponding relations for the first perturbation [136]. For the second perturbation,
substitution in the numerator and integration for a rectangular TE10I cavity gives
5co*
co
, .
(e;
1
1
(17)
+
in which G = 2VS / Vr . The simplest solution is obtained when the whole sample is
illuminated, and then L2 = L l =L. Equating Eq.(12) to Eq.(17) and separating the real and
the imaginary parts provides a system of two equations for the changes in the real and
imaginary parts of the dielectric constant. Maple V was used to solve this system
analytically for the following expressions:
5 s' = - ( L38 (A fV2)2e " 2 -4 G f02d(A /[l2)L2£" + 4Gf02Lb(A fm ) z " -4 G /02Z.25/0e " 2
+ 4 G / 0 2 Z.25 ( A / , , 2 ) s ' s "
-2 4 5 /02Z,V - 6 5 ( A
+ 1 2 5 / 0 2 L 4 e ' - 3 5 ( A / ‘i / 2 ) 2
/ ; /2 ) 2
iV
-
1 2 5 / 0 2 Z.4s ' 2 + 3 5 ( A / i / 2 ) 2 I 4s '
Z,4e '2 - 8G/025/0Is z ' - 8G/025/0L +4Gf028fQL2 + 4 / 02G5/0I 2s '2
+ 8 / 02G 5 / 0 I s ' + L 45 ( A / 1/2) 2 s " V
+
125f 2l2 + 4 e"25/02Z,V -4 5 /02L4e " 2 +45/02Z.
-125/02L2 +8 (Af V2)2L*s ' 3 + 35 (A/1/2)2 L V '2 +5 (A/„2)2L - 5 (A/1/2)2L*e "2
+4Gf025fQ+ 35(A/1/2)2J}z' - 5 (Af xn)2 L4- 45/02Z,4 + 35(Af v2 f L2
+4e"25/02/:3 + 12Z2s '5 / 02 + 4 Z ,4s ' 35 / 02 + 1 2 I
3s ' 25 / 02 -3 5 (A /1/2)2Z,2) /
M
20
permission of the copyright owner. Further reproduction prohibited without permission.
8s" = -(8 8 /0GZ.s'/02 - 2 5 (A /i/2)G/02 + 4 5 /02Z.V' + 2S(Af V2)2 L3z ”z ' + 6(4/;,, )2 ZV '
-8 8 /0L2G s r o2 +2d(AfV2)l} e ',2Gf02 - 2 5 (A /!/2)Z .V 2G/02 + 45(A f U2)L2z'Gf02
-48(A/-1/2)Zs'G /02-2& (AfV2)L2Gf02 + 46(A fV2)LG f02 +5(A/;/2)2 Z V '3
(I o)
+8 ( A/i/2)2 Z4s ' 2s " - 25 ( A/i/2)2Z3e " - 25 (A/;,, )2 Z4e's " + 45/0' Z4s ' 2e "
t4 5 /02Z2s " - 85/02Z4s's " + 8 (A/;/2)2Z2e " + 45/02Z4e " 3
+85/02Z2e 'Gs 7o2 + 85/02Z3e "s' - 85/02Z3e ") / AZ
M = 8G/025/0Z.2s ' + 8G/025/0Z + 4 / 02G2 - 8 G /025/0Z2 - 8 5 /02Z V
+5(A/;/2)2 Z4e '2 - 2 8 (A /i/2)2 Z4s ' +25(A /;/2) 2 ZV +5(A /;/2)2 Z4s "2
+45/02Z V 2 + 4G /025(A/;,2)Z2s " + 88/02Z V + 46/02Z V '2 + 5 (A /,,)2 Z4
-5 (A /;/2)2Z.3 + 5 (A/; , ) 2Z2 + 46/02Z4 - 8 8 /02Z3 +45/0: Z2.
As one can see, the final expressions are very complicated. However, because o f the
generality of these equations, a suitable choice of constants can be used to reduce them to
the simpler forms derived earlier for the first perturbation as well as those derived earlier
for simple geometries. The first perturbation expressions can be considered as a special
case and can be obtained by using z | = 1 and z[ = z ' - j z " (using the relative dielectric
constant). The resulting expressions are identical to those derived earlier except that the
frequency shift was defined to be positive [136]:
G /,05 /0 + 5 /02Z + ^-Z5(A/I/2)2
s ' - l = ------------------------± --------------( / 20G + 5 /0Z)2 +-(Z 5(A /;,,2))2
(19)
1
S
/ , 0G 5( a /;;2)
—
|
•
2 ( / 20g + s/ 0z )2 + - ( Z 5 ( a /;/2))2
21
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Eqs. (19) were used extensively to determine microwave properties of some materials
[137,138] and are not discussed here.
In addition to the first perturbation, the general second perturbation equations can
be used to obtain any o f the approximate relations derived earlier for various sample
geometries. For example, setting L = 0 gives Eqs. (15) for an infinitely long thin strip
inside the cavity. For a flat thin disk normal to the electric field direction, L = 1 [141],
and substitution into Eqs. (18) gives exact solutions in which 5 s' and 5s" contain both
5/0 and 8 (A f];2). For only the highly resistive material the expressions can be simplified
to:
5/0 _
G 5s'
( 20 )
5
(
1 ^
G5s"
The notation used in Eqs. (20) has been chosen for easy comparison with the
corresponding expressions derived earlier [94],
For many practical cases, it is not possible to use L = 0. For a long thin rod of
finite length / that is parallel to the electric field, the depolarization factor L is [140]:
(21 )
in which d is the diameter of the rod and / » d. This expression was used for a circular
wire [136] and a rolled sheet [138], and it also may apply to samples having a square
22
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
cross section. However, for thin films and thin strips o f wafers it is more reliable to use
the depolarization factor developed for a general ellipsoid [142]:
( 22 )
in which b, c, / are the semiaxes of the ellipsoid. For example, for the SI GaAs sample
used in our experiments (5 mm long, -1.5 mm wide, 0.52 mm thick), L obtained from
Eq.(21) was sufficiently small (0.007) so that Eq. (18) gave 5 s' and 5s" nearly identical
to the values obtained for L = 0. In other words for the sample dimensions used, the
approximation of an infinitely long thin strip is justified.
The final sample geometry to be discussed involves circular illumination of a
portion of a thin strip having length parallel to the electric field, which corresponds to
[140] determination of the electric field in an ellipsoid of dielectric constant s j placed in
a medium with a dielectric constant e \ . The resulting equation differs from Eq.( 16):
This derivation assumed an infinite surrounding medium in which the field F x is not
necessarily equal to the field in the cavity with the sample absent. For the thin flat disk
parallel to the electric field, the depolarization factor is:
(24)
in which r is the disk radius and 2c is the disk thickness. For a very thin disk the
depolarization factor is close to zero, and F2 approaches F x. Because F x for the thin strip
geometry is very close to the field in the cavity with the sample absent, the circular spot
23
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
is expected to give the same result as illuminating the whole strip (accounting for the
differences in the volumes).
In view of the strong dependence of L on geometry, it is not surprising that
sample
orientation
relative
to
the
microwave
field affects
photoconductivity
measurements. Anisotropy was observed in emulsions containing thin flat microcrystals
o f AgHal when the field was in the plane of the grain and perpendicular to it [111]. This
behavior was explained in terms o f the depolarization effects.
24
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3. EXPERIMENTAL SETUP OF AMTMP.
3.1 The Outline of the Experiment
Insertion of a semiconductor sample into a microwave cavity can change two
parameters of the resonance curve: the resonance frequency and the bandwidth at a half
maximum of amplitude. This behavior at various times after illumination is shown in
Figure I where to is the time before illumination. The short illumination pulse creates
extra carriers in the sample and changes the real and the imaginary parts o f dielectric
constant. This causes a perturbation of the resonance curve, which is manifested in a
shift of the resonance frequency and in the broadening of the resonance curve. The
resonance curve immediately after the illumination is denoted at time t,. During the decay
o f excess carriers due to recombination processes, the resonance signal eventually returns
to its to parameters. Two intermediate curves corresponding to instants t2 and t3 after the
excitation are shown, also. Thus, both the bandwidth and the frequency of the maximum
amplitude change with time. The shift of the resonance frequency usually is related to the
change in the real part of dielectric constant and is known as the photodielectric effect.
The change in the bandwidth is known as the photoconductivity and is usually ( but not
always) directly related to the change in the imaginary part o f dielectric constant. The
final goal is to extract above mentioned changes.
In their seminal papers, Hartwig and Hinds [93-95] demonstrated that excess
electrons in the conduction band and traps caused changes in the real and imaginary parts
of the dielectric constant at 4.2° K where the decay rates were slow. It allowed them to
monitor directly the resonance curve in a time scale of minutes. Extending this type of
25
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
photoconductivity ~8(A f1/2 ) ~ 8 e "
photodielectric effect ~ 5f0~8e'
f5 f4f3, f2 fi
frequency
time
time
time
frequency
time
time
Figure 1. Time dependence of the resonance curve and the reconstruction of the
difference signal from the transient decays.
26
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
analysis to systems in which rapid decay occurs requires measurement on a much shorter
time scale.
Such studies were reported for silver halide powders for which the time
domain was extended to transients having lifetimes in the 100 ms [99] and I ps range
[104], In the second paper, the time scale was achieved by scanning only a small portion
of the resonance curve (near the peak maximum) after a short pulse of light.
Consequently, the entire light-induced signal was not determined, and the change in the
imaginary part of the dielectric constant was inferred from the decrease in the peak
maximum.
In both these studies, the microwave system required equipment such as
automatic frequency control or a rapid frequency scanner, which can limit the time
resolution o f the measurement.
A more direct method in which the time resolution is limited only by the time
constant of the cavity and detector diode, was suggested by the author and tested on silver
halide powders and crystals [118-120]. In this approach, light induced transients were
measured at a number of fixed frequencies near the “dark” resonance frequency, and a
partial curve representing the difference between the “dark” and “light” Lorentz
resonance signals was determined at various times during the decay.
From each
difference signal, the light induced frequency shift was calculated using the “zero
frequencies” approach, which required a number of assumptions (i.e. on a relation
between “zero frequencies” and parameters of “light” Lorentz resonance curve, on a value
of the signal measured in a vicinity of the resonance frequency). After estimating the
contribution of the shift o f the resonance frequency to that measured near the resonance
27
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
signal the change in the imaginary part of the dielectric constant was subsequently
calculated by subtraction. The time resolution was reported to be 50 ns.
In contrast, AMTMP [121-131] involves a fit of the total difference signal to the
difference between the “dark” and “light” Lorentz resonance signals. Because the total
rather than the partial difference signal is used, the fit could be accomplished without the
simplifying assumptions required for the zero frequencies analysis described above.
Furthermore, it avoids the time scale limitations inherent in the rapid scan approach
discussed above.
In AMTMP, photoresponses are measured at a number of frequencies around the
initial resonance frequency. The typical time dependence of the signal at several
frequencies is shown at the bottom left of Figure 1. To reconstruct the difference signal
(dark vs. light), the values corresponding to the same time are taken from all transient
curves and plotted versus frequency. The bottom right of Figure 1 illustrates three such
curves for times, t,, t2 and t3 after the excitation. Analysis of each curve according to Eq.
(25) gives the value o f f 0 and A/1/2 at that specific time:
To(')
(25)
In this equation, / is the microwave frequency, y0 is the amplitude at the resonance
frequency/o and A/I/2 is the bandwidth at half-maximum amplitude for the curve before
illumination,
and
the
three
adjustable
parameters,
y^iO,
28
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
/o'(0=/o+S/o(0>
(4/1/2 XO > correspond to the resonance curve at time t after the
4/1/2
illumination. Thus, analysis according to the AMTMP method provides a separate time
dependence for the shift o f the resonance frequency and the change o f the bandwidth.
During my Ph.D. research work I developed two experimental designs of
AMTMP which are described below.
3.2 Setup I.
The sample was placed on a long (3x60x1 mm) quartz strip. Using a specially
designed micrometer sample holder this strip could be carefully positioned in the cavity
in a maximum of the electric field (Figure 2). The dark resonance and the light induced
transients were measured for this sample using the X-band microwave apparatus shown
schematically in Figure 3 [121]. The microwave source was a 1W klystron (8.2-9.6 GHz),
model A7273, controlled by a Narda 438 power supply.
The maximum frequency
deviation achieved was —5 kHz /30 min. If the experiment was run fast enough to be
completed in less than 30 minutes the minimum frequency shift which could be measured
was not less than 5 kHz.
A rectangular TEm cavity, constructed in our laboratory, had a loaded cavity
quality factor 0 L =1814, which corresponded to a time constant, t = 2Q, /at ~ 60 nsec,
co=27c/J).
The cavity, which had a volume Vc = 4.89 cm3, had two perpendicular
openings for sample insertion and illumination (Figure 2).
The sample cavity, which was placed in one of arm of a hybrid (magic) tee, was
mounted on a slide screw tuner used for impedance matching. This tuner and the one in
29
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5S3EKSEE5
SSSSSSSSSSS^I
Figure 2. A geometry of a microwave cavity used. Inset: a field distribution inside the
cavity.
RS 232
detector mount
oscilloscope
isolator
thermistor mount
power m eter
directional
couoler
PC AT 486
attenuator
slide tuner
short-circuit ^ 5 5 ?
® sample cavity
magic tee
nitrogen laser
w|weguide-coaxial frequency counter
klystron
klystron power supply
Figure 3. Schematic diagram of the experimental setup 1.
30
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the
reference arm were adjusted to assure that the dark resonance profile had the
appropriate Lorentz form. All microwave signals were measured using an IN23WE
detector diode. The time resolution in these experiments was limited by the rise time of
this diode in combination with Narda 510 detector mount (about 500 nsec). The signals
from the detector were adequately recorded by a 20-MHz sampling rate Nicolet 2090IIIA digital scope connected to an PC AT486 computer by an RS 232 interface. The
bandwidth of the scope was 10 MHz (100 nsec time constant, -200 nsec rise time). It had
an 8-bit-resolution plug-in unit and 100 mV per full scale which corresponded to 0.4
mV/point resolution. The input impedance was 1 MQ with 47 pF parallel capacity.
The curve from the scope was transferred into the computer and saved as ASCII
file. Each file was consisted of 4096 pairs of points what corresponded to - 64 kb. Such a
curve was measured at 22 frequencies and corresponded to ~1.4 Mb o f data. It took about
11 sec to transfer 50 kb from the scope into the computer using the maximum allowable
rate offered by RS232 C. It required about 4 minutes to transfer all 1.4 Mb to the
computer. The program to acquire the data was written using QUICK BASIC.
The
oscilloscope was triggered by an electric pulse produced by the laser power supply. A
known fraction of the microwave power was diverted from the detector diode to an HP
X486A thermistor mount and measured with an HP 43IB power meter for calibration of
the diode. The microwave frequency was measured by an HP 5340A frequency counter.
The light source used in these experiments was an LN 1000 nitrogen (337 nm)
pulsed laser (Photochemical Research Assoc. Inc.) with a pulse width of 600 ps. The
maximum incident laser power was 0.25 mJ/pulse (4 -1014photons/pulse), which
31
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
corresponds to 6 -1015photons/cm2 pulse. Laser power was measured using an ED-100A
joulemeter (Opticon Corp.).
Calibration of the diode detector (Figure 4) indicated that it produced a voltage
proportional to the rf power at low levels of incident microwave power (square law
behavior). In the milliwatt range, it produced a voltage proportional to the square root of
the microwave power (linear behavior). The microwave power used in our experiments
lay in the range at which the transition from square law to linear behavior occurred. A fit
to the sum of the linear and square law provided the calibration curve for subsequent data
processing.
The "dark" resonance profile of the cavity containing the un-illuminated sample
was measured as the power reflected from the cavity using a slow sawtooth modulation
of the klystron. The profile was often distorted from the pure Lorentz form because the
slide screw tuner was not properly adjusted. Adjustment to the proper setting required a
fitting procedure, which used only about 40 points from the profile in the early stages of
the fit. The experimental resonance curve and the fitted (smooth) curve are shown in
Figure 5. The final fit (according to Eq.(25)) was carried out over all 4096 experimental
points to determine y 0l, f Ql and Af ir2 which are the three adjustable parameters. As a
check, the fitting procedure was tested on a simulated difference signal which was a
difference between two Lorentz curves and was found to be unambiguous.
The analysis of experimental kinetics showed that the were two components of
noise present. The first one was the typical high frequency noise due to the broad band of
the oscilloscope’s input amplifier and to the laser induced noise. The second was low
32
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
c alib ra tio n c u rv e o f d io d e IN 23W E
fittin g : ci+ a » q rt(x i+ b
a-0 .2 0 9 0 3
b-M ).12472
c -a 0 0 0 4 7
•.
- I ------------- -------------- -------------- ------------- --------------------- --------0
100
200
300
400
500
600
power, mW
Figure 4. Detector calibration curve.
700 j
600
-
$00-j
J
300
-
200
-
too -
9.545
9.550
9 555
9560
95 6 5
9.570
9575
frequency, GHz
Figure 5. Typical measured and simulated Lorentz resonance curves for setup 1.
33
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
frequency noise due to 60 Hz and its harmonics at 120, 180, 240 Hz etc. This caused the
distortion of the background “zero” level of the signal. The main source of this noise was
the modulation of the high voltage reflector circuits of the klystron power supply by those
harmonics. This noise was also random because it was not synchronized with the laser
pulse (and as a result the oscilloscope triggering). This source introduced a relatively
small contribution to the noise in these experiments because it was masked by the highfrequency noise. The estimation o f the signal-to-nose ratio (SNR) gave the value of - 20.
As a result the tail region in the kinetics could not be measured accurately, and it did not
allow a clarification of the nature of charge carrier decays at long times.
3 3 Setup 2.
The schematic diagram of the microwave system given in Figure 6 [125]
represents a significant change from the equipment used in the previous study.
The
microwave source was a 20 mW Wiltron 68137B synthesized sweep generator (2.0-20.0
GHz) connected to a PC 486DX4 by an IEEE-488 interface. This sweep generator had the
following performance specifications: 1) the internal time base stability with aging
<5xlOlo/day, with temperature < 2 10'I0/°C; 2) an output power of 20 mW; 3) resolution
of 0.1 Hz; and 4) the presence of digital and analog sweep modes, phase-locked mode and
IEEE-488 interface. Because a three port circulator WFX-C (Microwave Resources Inc.)
replaced hybrid (magic) tee used previously, the tuning procedure to obtain the Lorentz
form o f the dark resonance profile was simplified greatly. A typical resonance curve is
shown in Figure 7. All microwave signals were measured using a DT8012 (Herotek Inc.)
34
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
E3
oscilloscope
detector mount
isolator
attenuator
slide tuner sample cavity
PC 486DX4
circulator
iwaveguide-coaxial
adapter
m
microwave synthesizer
Figure 6. Schematic diagram of the experimental setup 2.
025
0 20
i
0.15
I
0.10
0.05
0.00
8405
8410
6.415
&420
8.430
8.435
(raquancy. GHz
Figure 7. Typical measured and simulated Lorentz resonance curves for setup 2.
35
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
detector diode with a time constant corresponding to ~ 1.3 nsec. This time resolution
represents a substantial improvement from the 500 ns time constant of the previous
detector. The signal from the detector diode was recorded by a 100 MHz TDS 320
(Tektronix) oscilloscope connected to the computer by an IEEE-488 interface.
The
oscilloscope had 100 MHz bandwidth with 500 MS/sec sampling rate, 2 mV/division
input sensitivity, input impedance 1 MQ in parallel with 20 pF, IEEE-488 interface, a
“friendly” GUI (graphical user interface), an acquisition mode with ability to do up to
256 signal averaging scans. The oscilloscope was triggered by an electric pulse produced
by a photodiode, which received a small fraction of the intensity of the laser pulse. The
time resolution in the experiment was limited by the time constant of the cavity used
(about 60 nsec). Since it is possible to determine a time constant for a physical process
that is at least 2.2 times larger that the maximum instrumental time constant, the
minimum decay time constant that could be determined was 130 ns.
The SNR for this setup was about 80 compared to 20 in our previous study. The
SNR was increased in part by using ensemble averaging o f 16 transients at each
frequency and up to 256 when the frequency stability was sufficient (Figure 8).
(Additional SNR improvement was achieved using a digital filtering procedure that
involved averaging the last 50% of the channels in the frequency domain. See Chapter 3.5
for more details about noise reduction.) A program was written to automate data
acquisition to the extent that the computer controlled every operation except setting the
intensity o f the laser pulse. The program set the frequency o f the sweep generator, the
scale of the oscilloscope and repeated all the steps until, the chosen frequency range was
36
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Talc Run: lOOkS/s
Sample
ff-
■■ii
Acquisition |
c h i Wax
t2.4m v
XU
peak Detect
(> I0p$/d)v)
JU
Envelope
_TL
Average
MS00|1S
Ext
X -HOmV
■flnKftopAfterT
Buttonj
Ingle Seq lOOkS/s
ffEHRlngleSeq
1
CF—
S top A fter
Chi Max
!3.S4mV
Ms o a p s
Ext
RUN/STOP
b u tto n on ly
x -ic o m v
i Average
Mode qm hh
TekflBBsRglW^i oIokS/s
—
Stop A fter
c h t Max
t2.92mV
MSOOps
Ext
RUN/STOP
b u tto n o nly
X -l*9m V
Mode
Average
Figure 8. The tail o f the photoresponse at the resonance frequency with different number
of averages: (top) no average; (middle) 16; (bottom) 256
37
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
covered. To accurately determine the signal over the whole time domain ranging from
nanoseconds to milliseconds, each transient signal was measured at six different
overlapping time-scales. At each time scale, the transient signal was measured at the
highest vertical sensitivity automatically controlled by the program. For analysis, all files
were merged into one file.
In addition to the laser described in setup 1 the other laser used was a Nd:YAG
laser with the following parameters: 1064 nm (6.55 ns, 30 mJ), 532 nm (5.75 ns, 13.5
mJ), 355 nm (5.75 ns. 5 mJ), 266 nm (4.75 ns, 3 mJ). The circular laser spot was made
rectangular (1.7x5.5 mm) using a cylindrical lens. This shape was chosen to conform
more closely to the long narrow strip geometry needed for the analysis.
3.4. Time-Resolved Measurement Requirements
In an experiment of this kind the requirements of the time resolution play a key
role. The researcher must know the time constants of all devices involved in the
measurements. The time constant of any system (t )
is related to the rise time tr (the
time taken for the leading edge of a pulse to rise from its minimum to its maximum
values- typically measured from 10% to 90% of these values) as
/r * 2 .3 -T
(2 6 )
This expression allows determination o f the time constant of any device or process if the
rise time is known. Knowing the maximum time constant allows an estimate of the
minimum time constant of any physical process which can be resolved:
38
R e p ro du ced with permission o f the copyright owner. Further reproduction prohibited without permission.
where t mjn is the minimum time constant which can be resolved and t maxinst is the
maximum time constant among all devices used. In a typical AMTMP experimental setup
one can find three devices with their own time constants. The first is the scope with its
t s which usually can be determined from the minimum rise time t rs ( 3.5 ns) tabulated in
the manual. The second is the detector diode in its detector mount. The microwave
detector rise time tdt (< 5 ns) depends also on the parameters o f the load used :
'dr = 2.2
Rv + Rl
[CV+C L]
(28)
where Rv and RL are the detector diode resistance and the detector load resistance,
Cv and CL are the detector circuit bypass capacitor and the detector load capacitance
respectively (Figure 9). In our experiments the load was the oscilloscope.
The third is the resonator (or the cavity). The time constant of a cavity xc can be
determined using the value of a cavity quality factor Q and the value of an angular
frequency co :
2—
2
x„ = —
co
(29)
Ideally one can resolve the time constant of any physical processes (xp ) which is at least
2.2 times larger that the maximum instrumental time constant (xinst ~ 60 ns). The question
is: what is the error in determining the time constant which does not meet this
requirement? To answer this question one should consider the result (Vout) o f the
39
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
in
Figure 9. Square-law equivalent circuit of detector voltage measurements.
0.010
0.009 0.008 0.007 >
® 0.006 CO
c
i- 0.005 -
0)
|
0.004 -
Q.
0.003 0.002
-
0.001
-
0.000
-0.001
0.000
0.001
0.002
0.003
0.004
0.005
time, se c
Figure 10. Signal after 16 averages and 50% level FFT filtering.
40
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
convolution of the exponential decay input signal Via using the RC-circuit impulse
response function h(t) :
-mo
(30)
—03
where
h{t) = -^—e '" RC= — *r'/T«» at / > 0
RC
(31)
Vin= T0-e'"T‘’ at f > 0
One can get the final expression for the output voltage:
We used Eq.(32) to simulate the response as a function of Tp/xinst. The time constant o f
Voul was extracted from the simulated curve by fitting the selected part to an exponential
decay. Two methods of such extraction were compared. The first one corresponded to a
more ideal case. In this method to exclude the effect of the instrumental time constant, we
removed from the simulated curve the part of the signal over the period corresponding to
five instrumental time constants and used two subsequent orders o f magnitude in Fout for
fitting. A deviation of fitted time constant from the actual time constant was used to
calculate the percentage error. The resulting time constants are listed in a Table 1 (under
#1). In the case where the physical time constant is smaller than the instrumental time
constant the error is more than 100% and it makes no sense even to attempt to resolve
something. When the process time constant becomes larger than instrumental
41
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
Table I. Simulated time constants obtained with methods #/ and #2.
fitted time constant,
error, %
#1
fitted time constant,
error, %
#2
1.2
1.336
11.30
1.505
25.43
1.5
1.597
6.46
1.709
13.95
1.7
1.742
2.47
1.898
11.65
2.0
2.031
1.55
2.166
8.31
2.2
2.521
0.84
2.351
6.87
2.639
5.38
2.5
3.0
-
-
3.117
3.89
3.5
-
-
3.584
2.41
4.0
-
“
4.077
1.92
4.5
-
•
4.571
1.58
5.0
-
5.070
1.4
5.5
-
5.566
1.21
6.0
-
6.073
1.21
6.5
•
6.554
0.84
“
42
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
time constant by a factor o f 1.2 the error is 11.3%. One can see from the table that at the
ratio 2.2 the error would be about 1%. This finding is further justification of the use of
Eq.(27).
In the second method we analyzed the whole curve starting from a maximum and
using 4-5 orders o f magnitude as a dynamic range. This was closer to experimental
conditions. The results are listed in the Table 1 under #2. One can see that in this case one
gets more severe deviations. Thus when the actual time constant is 1.2 of instrumental
time constant the error jumps to ~ 25% (compare with ~11% for method #1). To reach the
same 1% deviation one has to have the time constant o f the process 6 times bigger than
the instrumental time constant!
In real experiments the process time constant is unknown, the dynamic range is
often limited and, therefore, both methods can not be used. But method #2 is more useful
because it gives a better estimation of errors when one interprets the measured time
constants which differ from the instrumental time constant by factors 1 to 3.
3.5. Noise Reduction And FFT.
This section describes briefly the use of fast Fourier transformations (FFT) to
reduce noise further. For this objective we used the Table-Curve 2D (Jandel Scientific)
program which employed FFT filtering and FFT editing. FFT filtering is an automated
Fourier domain smoothing routine with zeroing the high frequency components. A 10%
smoothing level zeroes the upper 1/2 of the frequency channels, a value of 20% zeroes
the upper 3/4 of the channels, a value of 50% zeroes the upper 9/10 channels. Figure 10
43
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
shows the experimental signal after 16 averages and 50% o f FFT smoothing. Comparing
it with Figure 8 (middle) one can see the improvement in the low frequency components
after the high frequency noise is removed. (Significant distortion of the signal will occur
when zeroed regions starts to include the signal frequencies.)
The ideal FFT smoothing consists of finding the optimum threshold for zeroing
channels so as to preserve all of the signal and hopefully to zero most of the noise. Rather
than experiment with various FFT % smoothing levels, it is often much easier to directly
edit FFT. This offers a means to zero one or more of the frequencies present in the data (
e.g. 60 Hz and its harmonics). The result of such an editing is shown at Figure 11 ( afrequency domain of a signal after 16 averages during the experiment, b - frequency
domain with zeroing all channels after 162.08 Hz, c- time domain after zeroing). One can
see that direct FFT editing gives almost the same SNR in present example as FFT
smoothing
but
allows
more
flexibility
in
the
frequency
44
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
domain.
500
1000
1500
fraquancycttarvM l
(a)
loaoooo
1.0000
|
0.1000
E
0.0100
aooio
0.0001
0
500
1500
1000
frv^jancycttnH
2000
(b)
0.009 -
0.007 >
0.006 -
I
0.003 0.002
-
0.001
-
0.000
On*. MC
x
(c)
Figure 11. FFT editing: (a) the frequency domain before editing; (b) the frequency
domain after zeroing channels above 162.08 Hz; (c) the time domain after zeroing.
45
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4. HARMONIC OSCILLATOR MODEL ANALYSIS OF BOUND/NON-BOUND
ELECTRON STATES.
A simple physical model of dynamic dielectric polarization treats the motion of
electrons in the microwave field as harmonic oscillation [94.143]. Benedict and Shockley
[144] used this model to determine the effective mass of free electrons in germanium.
Recent work [145] showed that this model adequately reproduces the dielectric constant
o f a semiconductor for excitation levels up to 1020 cm*3. In the context of AMTMP. this
model provides an adequate description o f free and trapped electrons including plasma
effects. Furthermore, it can be extended to include other types of bound electrons i.e.
electron-hole droplets and excitons.
4.1. Free Electron Effects
According to the Drude-Zener theory, when a reactive force proportional to the
carrier displacement in the electromagnetic field is neglected in the equation of motion
and the relaxation time is independent of energy, the free carrier effects on the real and
imaginary parts of the dielectric constant are:
5ey>«(0= -An(/)e*V / e 0m'[l + (cox)2],
= An(r)e2t /cos0/n’[l + (G)x)2],
(33)
(34)
volume concentration o f freecarriers,m is the
in which
An(t)
is thechange inthe
effective
mass,
xisthe momentum relaxation time and to themicrowave angular
46
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10-11
co = 5.5x10 1° rad/s.
-5
e
'
10-15
co
slope = 1„
co
C
O
co
T3
C
(0
slope = 2
"co
coI
10-22
1 0 -2 4
10*
10-5
10-4
10-3
10-2
10-1
100
101
mobility (m2V*1s-1)
Figure 12. Changes in the real and imaginary parts of the complex dielectric constant due
to excess free electrons as functions o f mobility.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
frequency. The momentum relaxation time becomes time dependent when the carrier
mobility p. varies with time:
x(t) = m’(\LQ+b\x(t))/e.
(35)
In this equation p0 is the “dark” equilibrium mobility and Ap(t) is the change in the
mobility value after excitation. From Eq.(33) one can see that free (conduction band)
electrons cause a negative change in the real part of the dielectric constant. Because the
frequency is very nearly constant during AMTMP measurements, it is informative to
analyze Eqs. (33) and (34) for various mobility values keeping the frequency constant. In
Figure 12, -S e'an d 8e" are plotted as a function of the mobility for an angular
frequency corresponding to microwave measurements (5.5 1010 rad/s). The values have
been scaled to permit comparison of the plots. Thus, the change in the imaginary part is
positive and proportional to the first power of the mobility, whereas the change in the real
part of the dielectric constant is negative and proportional to the square o f the mobility.
This behavior will be discussed in regard to the experimental results for low-mobility (~
10° m2/(Vs)) CdSe thin films and high-mobility (~1 m2/(Vs)) single crystal SI GaAs.
4.2. Plasma Effects
Plasma depolarization occurs when the concentration of excess carriers is
sufficiently large to cause a harmonic restoring force due to confinement of mobile
carriers in a small sample volume [143]. Including this force in the equation of motion
and using the relations between the complex dielectric constant and the complex
conductivity gives:
48
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Se'ptema(0=-A «( O eV (l-<o 2P/co 2)/s0/n*[l+((l-a>2 /to 2)g)t ]f ],
6e';/anM(/)=A«(Oe2T/e0w*co(l+((I-o)2 /to2)g>t Jf ],
cd p = J — ----- — ---------,
F Vm e 0(l+ £(e{ -1))
(36)
(37)
(38)
in which co ^ is the resonance plasma frequency that includes plasma depolarization
effects via the depolarization factor L introduced in Chapter 3, sj is the real part of the
dielectric constant before excitation; and n is the concentration of free electrons
responsible for plasma effects (which in fact is the sum of the equilibrium and the excess
electron concentrations).
According to Eq.(38), plasma effects depend on the sample shape and are
unimportant only for the infinitely long thin strip (L=0) when the plasma frequency is
zero, and Eqs.(36) and (37) transform to the corresponding equations for free electrons.
For a long thin strip of finite length parallel to the electric field the depolarization effects
can be reduced but not completely avoided by choosing dimensions to give co,, « co .
Figure 13 shows that the plasma frequency can be reduced by no more than one order of
magnitude using appropriate dimensions. Plasma depolarization may affect AMTMP
measurements, and the transition from plasma to free electron effects can be observed if
Aw(r) » nQ and An{t) ~ 10l° - 1015 cm"3 . At this level of excitation the plasma frequency
is time-dependent also, and how this can affect the kinetics expressed by Eqs.(36) and
(37) is shown below.
49
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10«
n (cm -3)
Figure 13. The plasma resonance frequency as a function of electron concentration for
various values of depolarization factor L.
— Se” (all L)
L=1
-
5e‘. L=0.04
- - 8e'. L=0.007
plasm >0
0.0
5 e’
plasm
<0
2 .0 x 1 0 -5
1 .0 x 1 0 -5
time (s)
Figure 14. Changes in the real and imaginary parts of the complex dielectric constant fo
plasma as a function of time for various values of depolarization factor L. The change in
sign indicated for the real part is due to the plasma/free electron transition.
50
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
To simulate the kinetics the excess electron concentration An(r) was assumed to
decay exponentially in time with an initial value o f 1015 cm'3 and lps time constant. The
time-dependent plasma frequency was calculated according to Eq.(38) for three values of
L (1, 0.04, 0.007) using t = 10'13 s, to = 5.5xl010 rad/s. In addition, the dark parameters
o f SI GaAs ( s | =12.85, m =0.063m0) were used as a specific example. Shown in
Figure 14 is the time dependence calculated according to Eqs.(36) and (37). The most
interesting feature is that 5s "(f) follows the exponential decay of the excess electrons
exactly while 5s '(0 follows An(t) only at longer times. In fact, during the decay o f A/z(/),
5 s ' (r) exhibits two exponential regions and undergoes a change in the sign at a time that
increases as L increases. As a result for low electron density or appropriate sample
geometry, co,, < co , and the plasma effects are unimportant; free electron behavior is
observed, giving a negative value to 5 s'(()
In this case all three parameters ( An (t),
8 e ’(t), 5s"(t) ) decay with the same time constant as indicated by the middle part and
tail region of the plots in Figure 14. At shorter times when the electron density is large,
oo P > 03, and 5s '(0 changes sign to a positive value while the behavior of 5s "(f) remains
almost unchanged. Because the numerator of Eq.(36) depends on the square of the plasma
frequency, the time constant for 5s '(f) in this region is one half the value in the free
electron region. As Figure 14 illustrates, the crossover between these regions shifts to
shorter times as L decreases.
While the plasma effect on 5s '(f) can be identified unambiguously by the sign
change during the decay, the effect on 5s "(0 is harder to discern. In Figure 15 (b), the
51
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
a
&
<w
a.
k.
©
ca
J©
es
(a)
(b)
Figure 15. (a) transformation of non-resonance absorption of free electrons to the
resonance absorption of bound electrons with increasing binding energy; (b) the change
in the imaginary part of the dielectric constant for plasma as function of the relaxation
time and relative plasma frequency. The plasma effect occurs over the narrow region of
the peak.
52
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
plot of 8s"(f) as a function of co ,,/co illustrates the small region of frequency over
which the effect is manifested. Furthermore this effect decreases with decreasing x, and at
small x the spectrum o f 5e"is identical to the free electron spectrum. In practice the
kinetic measurements are made when cox « 1 . Consequently the free and plasma
electrons cause the same time dependence for 5e "(f) and cannot be distinguished by this
parameter alone.
Behavior similar to that shown in Figure 14 was observed experimentally for Ge
at 1.14 K [96] as a result of the dissociation o f electron-hole droplets. It corresponded to
the dissociation of the electron-hole plasma into free electrons plus excitons. In general,
to observe the change in the sign of 5s'(f) during the decay, a transition from a bound to
an unbound state is required (or, vice versa ). The exciton itself is a bound state for the
electron and dissociation of excitons to free electrons/holes would cause changes
analogous to those discussed above for 5e '( f ) . If the binding energy of excitons is the
same as the threshold energy for free electrons calculated in the harmonic oscillator
model, the exciton will make the same contribution to 5e'(f) as the free electron, i.e. a
negative rather a positive value. This threshold energy calculated for CdSe [121] is about
3 meV, and values of few meV are expected for other materials. The exciton binding in
Ge and GaAs is about 4 meV. Consequently, no sign change would be observed for the
transition from excitons to free electrons, and this transition cannot be detected during the
decay. The results for Ge support this conclusion since a mixture of free electrons and
excitons produced only negative changes in the dielectric constant [96].
53
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
As was shown in [96] the sign o f x ' can change as the concentration of electronhole droplets is decreased. Electron-hole droplets, which have properties o f a degenerate
Fermi gas of electrons and holes, can be formed by condensation of excitons in Ge at low
temperatures. Droplets are made from the electron-hole pairs, and these dipoles increase
the real part of the complex dielectric constant and may also exhibit resonance absorption
under appropriate conditions. It was shown that the Drude-Zener formalism can be used
to describe the positive changes in the real part of the susceptibility caused by the
droplets. According to Eqs. (36) and (38), the inner region of the droplet can be
characterized by its own plasma resonance frequency and
go
8e' is positive when
« co . As free electrons produced by intrinsic and surface Auger recombinations
overcome the droplet contribution, negative values of 8 s' are detected. The kinetics of
X' observed for Ge exhibited the major features of the simulated decay shown at Figure
14 except that the behavior was found to be strongly non-exponential in all regions. The
infrared resonance adsorption of electron-hole droplets in Ge showed that the plasma
resonance frequency is consistent with the Drude formalism [146] . Extending this
formalism to a system containing excitons and free electrons (excluding electron-hole
droplets) results in an analogous change in the sign of 8e' at the exciton/free electron
transition.
4 3 . T rapped Electron Effects
The harmonic oscillator model for a trapped electron involves a restoring force
proportional to the electron displacement and related to the oscillator binding energy Eb
(i.e., a trap depth). The characteristic frequency of the oscillator oo0 is
54
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
co5=(2/m-)[(47t80)2/ e4]£*\
(39)
The corresponding equations for the changes in the real and imaginary parts of the
complex dielectric constant are [94,95]:
5e ’lr(0 = Anlr(t)e2((ol -c o 2) / e 0/w*[(to2 - to 2)2 + (to / t ) 2],
(40)
5 s"(/) = Anlr(t)e2( o / s 0m \[((a 20 - to 2)2 + (to /x )2],
(41)
in which A/z,r(0 is the density o f the excess trapped electrons. In contrast to free
electrons, trapped electrons make a positive contribution to 5 s'(0 • When to0 « t o ,
Eq.(40) reduces to Eq.(36), which corresponds to the free electron condition. Note that
the sign of 5s '(0 changes from positive to negative for a transition from plasma to free
electrons, but it changes from negative to positive for a transition from free to trapped
electrons. This behavior facilitates interpretation of observed sign changes because each
transition occurs at different points as the concentration of excess electrons decreases.
When to 0 » o) and to I » co / x for the microwave frequency employed, Eqs.(40) and
(41) can be simplified to
8e'lr(t) = &nlr(t)e2/ e 0m'a)l,
(42)
8 e "(0 = An,r(t)e2(o / e 0/n‘xto4.
(43)
The contribution of trapped electrons to 8s " has the same resonance character as
plasma electrons (Figure 15 (a)). The plot (b) in Figure 15 willcorrespond to 5s" if
log(to P Uo )is replaced by log(to0 / to) . Consequently, the effect of trappedelectrons on
8s" occurs only over a small frequency range about to0. Because to0> to at 5.5xl010 rad/s
even for shallow traps, trapped electrons do not make a significant contribution to 8s " .
55
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
As seen in Eq.(39), o>0 increases as the trap depth increases. Consequently Eq.(40)
indicates that electrons in shallow traps make a bigger contribution to 8e'(f) than those
in deep traps at equal concentrations. However, because 5 s'(/) depends on the
concentration of trapped electrons as well significant accumulation o f electrons in deep
traps can make those deep levels ‘‘visible”. The sensitivity of the photodielectric effect to
certain types of traps was verified experimentally [95] using CdS:Ag, which exhibited
electron traps at 0.007, 0.17 and 0.35 eV.
4.4. Dominant Contributions In AMTMP Measurements
For to = 5.5x10 10 rad/s, the relative contributions of the various excited species to
the complex dielectric constant can be compared using the equations developed above
and:
(44)
(45)
Because the plasma state occurs only at high carrier density, it will always precede the
free electron state, and the only coexisting states will be plasma/trapped electrons and
free electrons/trapped electrons. In case of plasma/trapped electrons, their presence
cannot be distinguished because each makes a positive contribution to 5 e'(t). On the
other hand the negative contribution from the free electrons can be distinguished from the
positive contribution from the traps having energies within 0.003-0.1 eV. When the
mobility is large, free carriers dominate. Trapped electrons dominate when the mobility is
small -0.001 m2/(V s). This analysis is based on the equations presented above using the
56
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
following ranges of parameters:
A/i = 109 - 1 0 15 cm'3),
cd0
t= 1 0 13- 1 0 " 15
s,
co P =3*109-3 -1 0 12 Hz
(for
= 1012 -1 0 15 Hz (corresponding to the trap depth range 0.001 -
I eV). The major contributions to 5e"(0 would be from the free electrons only or from
the electron plasma at high concentrations. Because these two states do not coexist, either
one or the other will contribute to 8e "(/). These estimations apply at room temperature
and will change at sufficiently low temperatures when
cot >
1.
57
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5 . MULTIPLE TRAPPING RATE EQUATION SIMULATION FOR VARIOUS
FORMS OF DISTRIBUTION OF LOCALIZED STATES
Very often the time dependence of photoconductivity transients can be reproduced
by a number of models. To distinguish between models, it is necessary to compare their
predictions with experimental transients obtained when a variety o f parameters are altered
(i.e., temperature, light intensity, donor density etc.). The present chapter is devoted to the
analysis of the effect various types of the distributions o f localized states (rectangular,
linear, exponential, Gaussian) on photoconductivity transients simulated within a
framework of multiple trapping (MT). The major approximations and previously
overlooked features o f the distributions are described.
The multiple trapping transport model was initially developed by Schmidlin [147]
to describe the photocurrent transients observed in time-of-flight experiments (TOF). The
rate equations included a diffusion term specific to TOF phenomena. Alternatively, a
statistical interpretation o f TOF data related a waiting-time distribution function to the
photocurrent transient [148]. In both approaches, the power-law behavior o f the
photocurrent observed experimentally could be reproduced using a distribution o f levels
with a density g(E). Because Monte Carlo simulations using a waiting distribution
function showed [149] that a number of distributions (exponential, rectangular, linear,
Gaussian-peaked) could produce very similar power law behavior, it seemed questionable
that photocurrent could be used to identify localized state distributions.
58
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A better understanding o f multiple trapping can be obtained if both free and
trapped electron transients can be observed and analyzed. Photoconductivity is related to
free electrons. Direct information about trapped electrons can be obtained by
photoadsorption (PA) [150] and the advanced method o f transient microwave
photoconductivity (AMTMP) measurements [121-133].
5.1. Exponential Distribution.
Since late 70’s a theoretical study of dispersive transport in amorphous materials
in terms of a multiple trapping (MT) model has been the focus of many researchers
[45.46,147,149,151.153-159]. Featureless exponential distributions o f localized states has
been given particular attention because it is believed to be a hallmark of amorphous
semiconductors. It has been accepted that the time dependence o f the photocurrent
observed in transient photocurrent (TPC) studies should be identical to the so-called ‘pretransit’ regime in time-of-flight (TOF) experiments [156]
The model describing the
phenomena observed in TPC was thoroughly developed by Orenstein and Kastner
[45,46,155] (referred to as OK below). The essential feature of the OK approach was a
joint analysis of decays of excess conduction band and trapped carriers. To confirm the
model experimentally, OK used photoabsorption (PA) experiments and related the
observed kinetics to trapped carriers. Using a thermalization approach, OK predicted that
retrapping by deeper levels would deplete shallow traps so fast that a thermal equilibrium
between electrons in shallow traps and the conduction band would not be observed. The
difference in transient behavior observed for PA and TPC supported this conclusion. The
59
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
OK model has two weaknesses. First, it does not provide a general spectroscopic
interpretation of transients, since it was developed for an exponential distribution only.
Also only those transients having a power-law behavior can be analyzed by this model.
Second, the approximations used in this model for the analysis of carrier thermalization
(in particular, the concept of a demarcation level) has been shown to be unreliable. Our
goal in this section is to obtain information about major features of the exponential
distribution which manifest themselves in phototransients without the OK assumptions
using simulation of MT equations. Below we show the following: (1) the form of the
transients peculiar to photoconductivity can be easily obtained as a solution of the MT
rate equations and does not need the more complicated theory developed by OK; (2)
detailed analysis of these rate equations gives the transient decay of the total
concentration of trapped electrons which follows the relation developed by OK and can
describe PA transients; (3) use of OK relation can lead to erroneous results; (4) by
varying temperature, density of states, and width of the distribution, some of the known
features of the exponential distribution will be confirmed and new features outlined.
5.1.1. Details O f T he Simulation
Values of the free and trapped electron densities based on MT and a continuous
density of states (DOS) were simulated numerically using the discrete approach o f Ref.
[157, 158]. The continuous exponential DOS function was replaced by a set of m discrete
levels separated by spacing A so that the energy of z'th level is 6£,=-£,-A z (where El is
the distance between the bottom of the conduction band and the first level in the
60
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
distribution), and the concentration o f corresponding traps is Nt = g0 exp(5£, / tcT0) A
MT rate equations (Eqs.39) were as follows:
dAn(t)
^
'£ riAni(t)-A n (t)Jja - A n (t)k r
;a|
/^1
(46)
- ^ - ~=An(t)(o-An, (t)r,
at
where An(t) is the concentration of excess free electrons, Art/t) is the concentration of
electrons trapped in the z'th level, xr is a time constant for monomolecular recombination,
r, is the release probability and o)j is the capture probability. The capture and release
probabilities are:
ri= v 0exp (6E,/kT)
in which S is the capture cross-section, v=107 cm/s is the thermal velocity, and v0=1012 s 1
is the attempt-to-escape frequency.
Values of An(t) and Ant(t) were simulated between I O'12 and 10'3 s, which was the
time range involved in most of our measurements. For these simulations, the initial
concentration of excess free electrons was An0, and all traps were empty at t = 0. Other
parameters used in the simulations were: v0=1012 s’1, S=10’15 cm2, T0=6QO K, v=107 cm/s,
g0=1021 cm’3 eV'1, T=300 K, An0=1019 cm'3, t= 1 0 ’9 s, £ /= 0. By making A sufficiently
small, the number o f levels m was sufficiently large to ensure the discrete approach
provided a very good approximation to the continuous DOS. Values of A=0.02 eV and
m=40 were sufficient because the discrete nature of the DOS manifested itself (i.e., in an
appearance of the final mth exponent) only after 10’1 s, well outside the range o f interest.
61
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Furthermore, the addition o f extra levels did not affect the form of the kinetics in the time
range o f interest.
/I*+|
Solutions o f Eqs. (46) have the form, ^C„exp(.y,/), j= \jn ¥ \. To find eigenvalues
/= !
5j a matrix obtained from Eqs. (46) was used to find the roots of the corresponding
polynomial. A set of initial conditions was used to find the coefficients C,y. The interested
reader can find more details about this linear algebra approach in Ref. [157, 158] . The
whole algorithm was developed on Maple V R4 except a faster algorithm written in
Quick C (supplied by Professor C. Dunkl) was used to solve the high degree polynomial
equation. In the discrete approach, m+ 1 kinetics (the free electron decay and decays of
trapped electrons at m levels) were obtained, and the temporal behavior of electrons
trapped at levels within 0.02-0.8 eV range was examined.
5.1.2. Behavior of An(t), An{(t) and the Photoabsorption Signal (PA(t)).
The decays of free electron concentration A n ( t) , and the decays of localized
electrons An,(t) in selected traps from 0.1 to 0.8 eV are shown on Figure 16. First, we
will discuss some features of the curve corresponding to An(t) , which exhibits a
“classical” transition from t<I<l) to f (1+a>. This result suggests a complete equivalence
between TOF and TPC curves, which often exhibit this transition. Subsequent
redistribution among deeper traps due to thermal release and retrapping gradually lowers
the density of free carriers which corresponds to the pre-transition range. Flowever, the
two experiments differ in the mechanism for ultimate removal of carriers - by collection
62
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
miJ
ii ii iiJ
imittl
i i i i i J
iimij m J
i t m t J
iijiJ
i l l
40 levels, 0.02 eV spacing, T=300 K, Tg=600 K
t
=10'9 s
CO
I
E
o
.0.4 eV
c
<3
\ \
0.6 e V ^
c
O
)
o
0.8 eV
i i iiiu] inniq l ining i iimi^ i iiuu^ 1 1 iiiiu^ 11'linii^ iinui^ limn
-12 -11 -10
-9
-8
-7
-6
-5
-4
log (time [s])
Figure 16. The simulated decays of free electrons (Ari(t)) and electrons localized
in selected traps (An((t)) for exponential localized-state distribution.
63
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
at an electrode in TOF and by recombination in TPC. To our knowledge, only one earlier
paper [152] obtained such behavior of An(t) by solving the MT rate equations
numerically, but this phenomenon was not analyzed in detail. In general, rate equations
including a diffusion term were used with some approximations to describe the
photocurrent obtained in TOF experiments. The thermalization/recombination model
developed by OK was used to describe the photocurrent in cells with coplanar contacts.
In this analysis, the recombination time was extracted from a log-log plot of the A n{t)
curve as an intersection of two straight lines corresponding to f (l"a) and f <I+a>. For the
corresponding curve in Figure 16, this intersection is located inside a wide transition
range and gives a value about 3x1 O’7 s. According to Marshall and co-workers, one
difficulty with this method is that the two linear regions of the curve occur only at very
short and long times. Our simulation reveals another important feature— the intersection
does not give the recombination time even satisfying the above mentioned condition. The
recombination time used in this simulation is 10*9 s, which is two orders of magnitude
smaller than the value obtained by the intersection method. As a result, using this method
can lead to a large error in the estimation of the recombination time. It turns out that the
recombination time occurs when the curve starts to deviate from the initial f (1'ot) slope, as
can be seen in Figure 16.
The time dependence of electrons in selected traps in Figure 16 is characterized by
a rise to a m axim um and a subsequent decay that eventually matches the decay rate of
free electrons, indicating thermal equilibrium has been attained. The trapping time is
distributed over 10 orders of magnitude due to a spread in the densities of the levels, and
64
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
Iiiij
I i i m i j
■ I iiiiiJ
I iiihJ
Iinim l
I i m i j
i i ni m i i i i m
i uni
40 levels, 0.02 eV spacing, T=300 K, Tq=600 K
x =1O'1* s
§1 '~Snj(t)/(-5Ej)
2~ 1/(1+(t/x )0-5), xr=3x10'7 s
S-Znj^-SEj)
1 4~ 1/(1+(t/Tr)0 -5), Tr=10‘9 s
i iimij i i imuj i i iiiiuj i i imij iiimij lining i i iiiii^1rmni^ inn
-12 -11 -10
-9
-8
-7
-6
-5
-4
log (time [s])
Figure 17. The simulated decays of free and total weighted trapped electons
for exponential localized-state distribution (see param. on the plot).
65
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
the dynamics of trapping and release are such that the maxima correspond roughly but
not exactly to the release times, indicated by the arrows in Figure 16. As was pointed out
by Marshal and Main for a simple two discrete trap level model, thermal equilibrium is
indeed established but not at the time corresponding to the release time for the particular
level. This behavior is also indicated in Figure 16 in which these times occur when free
electrons and trapped electrons decay in the same manner. In fact for a particular trap
level, we found that approximately five release times (typically, between 5.0 and 5.5)
must pass before the same decay was found for free and trapped electrons. A possible
explanation is a competition between this process and a faster one such as the fast
trapping to shallower levels and recombination. As in transient circuit analysis, a level
characterized by a particular time constant would take about five time constants to
complete 99.33 % of the transient process when reacting to an external perturbation
having a substantially shorter time constant. Thus, before reaching thermal equilibrium, a
deep level must respond to processes faster than its release time.
The sum of An,(0 can be related to experimentally measured quantities in both
PA and AMTMP. As was discussed previously the photodielectric effect 5 s ' is measured
in AMTMP, and the contribution of trapped electrons to 5 s ' is inversely proportional to
the third power o f the trap depth. Therefore, in the case of an exponential distribution,
m
5s'(0°c^A «,(f)/|5£,|3, and the resultant curve is shown on Figure 17 (curve 1). The
i=l
decay of free electrons An(t) is shown in Figure 17 for comparison (all curves were
scaled). Up to about 10'7s, the two curves exhibit the same time dependence. Although
66
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
this correspondence appears to indicate thermal equilibration, Figure 16 shows that it
would occur only for the more heavily weighted shallow traps. In fact at later times, the
two curves deviate because of the larger contributions of the deeper traps that have not
reached equilibrium. Intuitively, these two curves would match only for a relatively
narrow distribution with negligible density of deep traps. This was confirmed by the
simulation using T0=400 K (instead of 600 K as before). The results are shown on Figure
18. One can see that the curves continue to follow one another until almost 10‘2 s. This
plot also shows that the transition from one power-law decay to another occurs at longer
-6
-9
times because the recombination time constant was chosen to be 10 s instead of 10* s.
Therefore, by varying the value of the recombination time one can shift the time of
appearance of the transition to longer times.
According to theory [150], the PA signal is proportional to a product of the
density o f adsorbing centers and the adsorption cross section, which is believed to
decrease as the trap depth increased. However, OK after two assumptions (a demarcation
energy and a constant occupation number for all traps) arrived at an equal weighting for
all levels in an exponential DOS.
They developed the following expression for the
averaged trapped electron decay, which was subsequently used to fit decays of
photoinduced adsorption in PA experiments:
m
-
N
<«>
in which a=7/T0. Curve 2 in Figure 17 is a plot of Eq. 48 using a=0.5 and Tr=3xl0 ' 7 s.
To reproduce the form of this curve, the values Art;(t) from the simulation were used to
67
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
19
~m=4Q, S=10~15 cm2. T0=400 K, T=150 K, A=0.02 eV, g0=1020 cm'3 e V 1
18
17
16
15
14
13
12
An(t)
11
10
9
-12 -11 -10 -9
-8
-7
-6
-5
-4
-3
-2
-1
log(t)
Figure 18. The simulated decays An(t) and Se '(t) for exponential localized state
distribution with T0=400 K. (see other param. on the plot).
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
calculate ^ A n l(t)/\d)El\w in which w = 0 , 1 or 2 , and the denominator is a weighting
i=i
factor to explore the effect o f an energy dependent absorption cross section. The best
results were obtained using w = 0 (curve 3 on Figure 17), and w = I and 2 showed a
completely different behavior. In other words, the best fit of the simulated data was
obtained with all trap levels weighted equally as was assumed in the OK model. An
important outcome o f this analysis is that a fit o f “experimental” data according to the
OK equation gives an “apparent” recombination time (3x1 O'7 s) that is two orders of
magnitude larger than the correct value (10‘9 s). For comparison, curve 4 in Figure 17,
which is based on Eq.(48) and the correct recombination time (10‘9 s), clearly does not
conform to the simulated “real” data. Consequently, using Eq.(48) in the OK model to fit
a PA response would give the correct form of the time dependence but significantly
overestimated the time constant o f recombination.
5.1.3. The Effect of the Width for an Exponential Distribution: Dispersive/NonDispersive Transport Transition.
Next, we analyzed the effect of the width of the distribution on the slope of the
power-law decay in An(t). It was shown in a number of publications that the parameter a
can be related to the width of the distribution as a=T/T0 in a particular temperature range
when T<Tn. The behavior o f the kinetics at T >T0 was another objective in this thesis
since it was not given much attention in previous work. We explored these phenomena by
keeping all parameters of the simulation constant and varying the characteristic
69
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
m in i
I I IHid I Illial
L IIllnJ
I
linilJ
I
lin J
|
lllld
L L1UIj
I I lUlfll
I
I IlLuJ
I
I Mini
m=40, S=10*15 cm2, A=0.02 eV, g0=1021 cm'3 eV~1, T=300 K
n0=1019 cm"3, t =10-9 s, E,=0
.1500
T0=200 K
2000
log (tim e),s
Figure 19. The simulated decays of An(t) for selected values of the characteristic
temperatures o f exponential distribution, T0.
70
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
temperature from 200 K to 2500 K when T=300 K. It corresponds to 0.017 eV and 0.22
eV on the energy scale, respectively. The simulated curves of An(t) are shown on Figure
19. The interesting feature o f the curves for which the condition T>T0 holds (i.e. 200,
250, 300 K) is the absence o f the initial power-law decay. Instead an exponential decay
(t» 10'9s at 7>=200 K) appears in the beginning of the kinetics which gradually transforms
to the power-law decay with increasing the width of the distribution. This time constant is
very close to the time constant of recombination. When the width of the distribution is
relatively small, the total density of states is small as well, and as a result the trapping
time is relatively large. The recombination time is the smallest parameter, and once the
electrons are released after first trapping, they recombine almost instantly. Therefore, the
electrons will not be involved in the processes of retrapping and redistribution between
various traps. The electron transport is no longer dispersive. In fact, in [160] the condition
T>T0 was considered, in analyzing the behavior of the mobility in time-of-flight (TOF)
experiments. It was shown that, indeed, the mobility is constant under those conditions
and the dispersion cancels. Therefore, our simulation provides the first consideration of
the dispersive-non dispersive transport condition for transient photoconductivity
experiments. With increasing width of the distribution (T<Tn) the density o f states
increases as well, leading to a decreased trapping time below the recombination time.
Now electrons have a chance to go down the ladder of localized states until the
probability o f subsequent retrapping will be equal to probability of recombination. At that
moment the released electrons will predominantly recombine. Extending the length o f the
first power-law decay to longer times in Figure 19 corresponds to the fact that the
71
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
boundary condition for recombination will be met for the same density of states, but those
states are now much deeper in energy.
To extract the parameters from the simulated curves, we fitted power-law parts
before and after the transition (the knee) in An(t) kinetics using the following expression:
A«(O=An0/(l-Kt/T)a-1)
(49)
where An0 provides the information about the excess electron concentration at t=0 and is
expected to be close to An0 =10 19 cm ’3 used in the simulation. The parameter t is related
to the time constant of the process which is responsible for the power-law decays before
and after the knee regions. Clearly, this process does not have to be the same in both
cases. Finally, the parameter a is expected to be same in both regions and related to T/Tn .
The results of the simulation confirms, in general, our expectations. Tne
dependence of a on 1/T0 is shown on Figure 20 for r=300 K. The straight line
corresponds to expected dependence T/T0 in the interval of T0 used . White circles denote
the values of a extracted from the region before the knee, and black circles correspond to
the region after the knee. For the region before the knee, a continues to follow the linear
dependence until the proximity of T=T0 region where it starts to deviate from the linear
dependence. The values extracted from the region after the knee deviate about 13% from
the linear dependence under the condition of no dispersion (T0= 200 K). Note, the linear
dependence predicts a > l for T>T0, as was confirmed by our simulation. Values continue
to follow the linear dependence very closely. The deviation which is evident at high
values of Tn can be a result of the diminishing range where the second power-law was
observed.
It
is
closely
related
to
the
previously
mentioned
72
R eprod u ced with permission o f the copyright owner. Further reproduction prohibited without permission.
fact
that
the
before the knee
after the knee
a=T/Tni T=300 K
1
T=T.
0
0.000
0.001
0.002
0.003
0.004
0.005
0.006
1rrQ, K-1
Figure 20. The dependence of a on 1/T0 for the decays shown on Figure 19.
73
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
true value of a is achieved only when far away from the knee region. We believe that by
extending the time domain of the simulation the true values of a could be obtained.
Figure 21 shows the dependence of the parameter An0 extracted from the both
regions. The points are very scattered probably due to
complications in the fitting
procedure when working with very high values. Nonetheless, the average of those values
was close to the 1019 cm '3 value used in the simulation. The plot gives a measure o f the
error expected if this approach is used to estimate the initial concentration in real
experiments.
The dependence of the time constants according to Eq.(49) is shown in Figure 22.
It is very clear that the time constant extracted from the region after the knee is very close
to the recombination time constant ( 10'9 s) used in the simulation. It has a clear physical
basis since this second power-law decay is governed by recombination. The time constant
extracted from the power-law decay before the knee has an average value between 10'12
and 10"13 s. The scatter we believe is a result of working with very small values. (In
addition, the parameters n0 and t are not completely independent in the region before the
knee.) Clearly, these values do not provide the recombination time constant (10"9 s) but
they are very close to the average trapping time. As one may recall from the previous
results, the recombination does not start until 10'9 s. The region before the knee
corresponds to the high density of localized states involved in dispersive transport, the
trapping time is very small and is comparable with 10‘13 s. Therefore, from the fitting in
this recombination free region one can get a good estimation of the average trapping
time.
74
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
parameter An0from the power law decay, cm
1.e+20
1.e+19
1.e+18
before the knee
after the knee
1.e+17
0.000
0.001
0.002
0.003
0.004
0.005
0.006
i r r 0, k -1
Figure 21. Parameter Ano extracted with Eq.(49) from decays shown on Figure 19.
75
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
parameter x from the power-law decay, s
1.e-8
1.e-9
I I » * 1 ■ » t I I I 1 l. .± I t_i i i_L I i L. -I I_I_I
x=10-9 s
1.e-10
1.e-11
O
•
1.e-12
before the knee
after the knee
O O
oo
trapping
1.e-13
o o
o
1.e-14 -i— r— i— i— [— i— i— i— i— |— i— i— i— i— |— i— i— i— i— [— i— i— i— i— |— i— i— i— r
0.000
0.001
0.002
0.003
0.004 0.005
0.006
1/T0. K-1
Figure 22. Time constants extracted with Eq.(49) from decays shown on Figure 19.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
5.1.4. The Effect of the Temperature, T.
From the form of the expression a=T/T0 one would assume a complete
reproducibility o f the major results when keeping T0 constant and varying T. For this
simulation we used T0 = 400 K and xr =10‘6 s. The range of temperature used was 120 450 K. The other parameters of the simulation were unchanged. The simulated kinetics
are shown in Figure 23. The major features from the previous simulation are the same.
For the temperature range T>T0 there is no dispersion. The fast exponential decay with a
time constant (3x1 O'6 s) close to the recombination time is evident at r=450 K. Below
this temperature until ~ 330 K, this decay gradually transforms to the power-law decay
pertinent to thermalization/dispersion. The dependence of the parameter a extracted from
both power-law decays is shown on Figure 24. The values follow closely a linear
dependence. The extracted values of An0 shown in Figure 25 demonstrate the same
scattering around the 1019 cm'3 value which was used in the simulation. The results for the
time constants (Figure 26) shows that before knee region, the extracted time constants are
related to the recombination time constant (ICf6 s), and after the knee region the time is
close to the trapping time (10‘12 - 10' 13 s). Therefore, the complete equivalence o f effects
of T and T0 on a was confirmed as might be expected.
5.1.5. The Effect of Density of States, goAnother simulation parameter which can affect the dispersive/non-dispersive
transport transition is g0. This is just another way of varying the density of states. The
kinetics obtained with the following parameters: m=40, 5=10 ' 15 cm2, T0=400 K, 7K300
77
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
«m rJ
19
t ihum
t_ n m d
i nuJ
n m d__t_n»j
in irJ
.ttinJ
■ hi ii J t m a d
iiim j
i m tJ
in u J
itm J
im m
m=40, S=10'15 cm2, T0=400 K. A=0.02 eV. g0=1021cm-3 e V 1
AnQ=10^ cm"^, t r=10“® s
18
17
T=270 K
16
200 K
log (An(t))
15
14
13
400 K
12
150 K I
360 K
11
330 K
10
300 K
9
120 K
8
i runi^ m ii^ -rnna miua i niiiq •>tin^
- 1 2 - 1 1 - 1 0 -9
-8
-7 -6 -5 -4 -3 -2
i ini^-riniitf nrniq rrntaf
-1
0
1
2
3
log (t)
Figure 23. The simulated decays of An(t) for selected values of the absolute temperature,
T.
78
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
before the knee
after the knee
a=T/T„, Tn=400 K
1
0
100
150
200
250
300
350
400
450
500
T.K
Figure 24. The temperature dependence of a for decays shown in Figure 23.
79
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.e+20
before the knee
after the knee
>.
S
*o
©
1
o
1.e+19
5
An„=1018 cm-3 !
<D
<o
E
1 .e + 1 8
r
i~ r '
1
100
—
150
' i
............
200
i
1
|
1 '
250
i i
30 0
1 i- ' - - ' - '
350
■ : 1
400
'
1 i—
■—
450
500
T. K
Figure 25. Parameter Aiiq extracted with Eq.(49) from decays shown in Figure 23.
p -—
4
l.e-5 -j
«
—— ------—1— .................... i
I
.
2
f
•
P
3T 1 ®-€ -i-------------------• ---------- — ---------- • — * • —
8
1.e-7
©
1
„ •“-*
|
i
r =10*$
-j3
--------------r
*
I
r
^ before the knee §r
O
F
* after the knee |-
a.
4
0 1.e-9 -j
§ 1.e-1oj
£
1
l.e-11-J
r
1
1.e-1i|
f.e-12-J
6
o
O
1.e -15j
O
O
O
C
p
1 -0-14 | i i > i ■ ■ i - - i i - - ■ r - i ' 1 i ■ * i > ; 1 * ■— i ■
100
ISO
200
250
300
3S0
400
i
450
500
T .K
Figure 26. Time constants extracted with Eq.(49) from decays shown in Figure 23.
80
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
K, Aw0=lO 19 cm"3, xr—10"**s and lO^-lO 22 cm^eV' 1 as a range for g0 are shown on Figure
27. The transition from the non-dispersive transport to the dispersive one starts when the
density of localized states is high enough to provide a small value o f the trapping time
constant which could compete successfully with the recombination time constant value
(10"6 s). Above go=1020 the appearance of the power-law decay due to thermalization/
dispersion is evident. Subsequent increase in g0shifts the position o f knee to longer times
without affecting the slopes.
5.1.6. The Effect of E t.
Finally, we would like to examine the behavior of both An(t) a n d 5 e'(0 when the
distribution is truncated between the conduction band and some value Et below it. States
above Et are simply cut from the distribution. Because we used same value o f g0 for all Et
the form of the distributions should still be the same and is characterized by the same g0
and T„. This truncation not only alters the range of energies involved but also decreases
the total density o f the levels due to the nature of the exponential DOS. Therefore, the
trapping time will increase. This will prevent the rapid equilibration between various
levels mainly because of larger trapping times. As a result,8s'(0 behavior will deviate
from An(t) behavior with increasing E,. This result is shown on Figure 28 with simulation
parameters on the plot. To compensate the decrease in the total DOS with increasing Et
we increased the value of g0 to have approximately the same trapping time as for the case
ErO (not shown). It caused an appearance of the fast initial exponential decay due to fast
trapping followed with a plateau region both in 8 e '(0 a n d An(t). This plateau lasts until
81
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
20
m aJ
i m J
. »m«J
r »niW
i m J
i m n^
i m nJ
im n J
>mmJ
m
J
i mwJ
im n J
i m nJ
m=40, S=10'15cm2, TQ=400 K, A=0.02 eV. T=300 K, AnQ=1019an-3. t ^ K
t tu n *
T 6 s
19
18
17
16
= 1022
15
14
13
12
11
10
9 0=10 16
i i mia
i mini i inni i mur i i nin^—i muq- nnuq i rrnu^ lining i iiXii^ lining i m ii^ i liiii^ i mm
-1 3 -12-11-10 -9 -8 -7 -6 -5 -4 -3 - 2 - 1 0
log(t)
Figure 27. Simulated An(t) decays for selected values o f g0.
82
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
1.e+19
An(t) and 8e'(t)(scaled), arb.units
1.e+18
1.e+17
1.e+16
x
\
E,=0
E,=0.1 eV
E,=0.2 eV
E,=0.3 eV
An(t)
1.e+15
1.e+14
1.e+13
1.e+12
1.e+11
m =40, S = 10
1.e+10
c m , TQ= 4 0 0 K. A = 0 .0 2 e V . T = 3 0 0 K
AnQ= 1 0 1 9 c m '3 . g Q= 1 0 2 1 c m '3 eV " 1 . t ^ I C F 6 s
1.e+9
12 -11 -10 -9
-8
-7
-6
-5
-4
-3
-2
-1
0
log (time), s
Figure 28. An(t) and 5e '(/) decays simulated for selected values of the offset Et.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the time is equal to the release time from the shallowest level in the distribution. After
the plateau the usual power-law decays will start. Therefore, the result will be a simple
shift o f the power-law behavior to longer times in contrast to the truncated distribution as
discussed above.
5.1.7. Summary
Summarizing the results of this section, we were able to simulate all features
peculiar to photocurrent decay in the presence of an exponential distribution of localized
states using MT rate equations without making any of the assumptions used in the OK
model. We showed that the recombination time extracted according to the OK analysis
gives erroneous results. Therefore, we have reason to believe that the approach outlined
in this chapter can provide a more realistic analysis of transients from TPC. PA and
AMTMP measurements. For the first time, we showed the features arising in the kinetics
when moving from non-dispersive to dispersive transport. The relation between the real
and those extracted from the fits was discussed. The accuracy and range o f information
extracted from the experimental kinetics were explored.
5.2. Rectangular and Linear Distributions
The present section investigates multiple trapping transport in the presence of a
rectangular/linear distribution of localized states. First, in contrast with results of the
Monte Carlo simulation, this distribution does not generally provide a power law decay.
84
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Power law behavior occurs only under very limited conditions. Furthermore when it does
occur, it is only over a limited time range, and unlike experimental observation it does not
exhibit a transition (so-called knee) to a second steeper power law at longer times. On the
other hand, the exponential distribution does provide a photoresponse that undergoes a
transition from one power-law dependence to another. Second, in place of the second
power law behavior the rectangular distribution causes an exponential decay related to the
deepest level in the distribution. Third, this behavior occurs for weak retrapping. Fourth,
as retrapping becomes stronger the power law region shortens and eventually disappears.
Although the exponential region remains at longer times, the time constant is a composite
of several time constants. Fifth, the transition of the temporal behavior from weak to
strong retrapping depends on the values of the recombination time, density of localized
states and capture cross-section. The linear distribution demonstrated similar major
features, i.e. a limited range of conditions where the power law decay can be observed
and the exponential decay in the tail region. Unlike the results of the Monte Carlo
simulations, these results identify clear differences between the photoresponses expected
for the rectangular/(linear) and exponential density of states in the multiple retrapping
model and, hopefully, will provide guidance when analyzing experimental data.
5.2.1. Details of the Simulation
The analysis of photoconductivity transients in the presence o f rectangular DOS
employed multiple trapping equations with a term for recombination [130]. The
continuous DOS was replaced by a set of m discrete levels with spacing A so that the
85
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
energy of the /th level is 5£, = —E, - A/', and £, is the distance from the bottom o f the
conduction band. The resulting concentration of traps was calculated as N t = g0A . The
rate equations were the same as those used in section 5.1 (Eqs.(46)). Simulations using
several values of A (all less than k T ) indicated that the decays were identical for values of
0.02 eV or less in the time range of interest (1 O' 12 - 1O'2 s).
Solution of Eq. (46) gives m+1 decays: one for tsn(t) plus m for the various Ant(t).
The An(t) transients can be compared directly with the photoconductivity transients
obtained from contact and contactless (AMTMP) methods. The sum o f An/t) can be
compared with transientsfrom photoadsorption (PA)
microwavephotoconductivity
(AMTMP).
As
and the advancedmethod of
was shown
in section
5.1
the
photodielectric effect 5s '( t) measured in AMTMP is related to this sum by:
S e 'f r ^ A / i ^ O ^ I 3
(50)
/= !
It was also shown in section 5.1 that the signal PA(t) measured in PA experiments is
related linearly to this sum:
PA(t )oc]T A^ (t )/|8 E, j°
(51)
1=1
5.2.2. Behavior of An(t)
Although Monte Carlo simulations using a rectangular DOS [149, 151] indicated
that Aw(if) did not exhibit power-law behavior over a large time scale, the authors
nonetheless were able to fit limited regions to power-law decays and analyzed the
86
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
temperature dependence o f the exponents a . In those simulations, the dynamic range was
limited to no more than I order of magnitude [149]. To establish unambiguously the form
o f the kinetics, we have extended the dynamic range significantly. To analyze the effect
on An(t) decays caused by the density of localized states, Nt , values of g0 were varied
over a wide range. The results of the simulation and the parameters are shown in Figure
29. The plots on the log-Iog scale reveal three regions: an initial fast exponential decay,
power-law decay in the middle (which disappears at high g0) and a slow exponential
decay at the tail. The exponential tail is clearly indicated in semi-log plot in the inset of
Figure 29. Most likely, a semi-log plot of the Monte Carlo simulations would reveal that
the tail section is an exponential decay rather than a power law decay. Exponential DOS
is known [45,46,147,149,151,153-159] to provide two straight lines with different slopes
in a log-log scale, a so called "knee’’. Thus in contrast to the exponential DOS, the
rectangular DOS does not provide two power law regions over the period of the decay.
Furthermore as gBincreases, power-law behavior becomes less prominent and eventually
disappears when the plot becomes curved. The plateau that develops with increasing g0
(curves 8 and 9) is caused by the increasing time constant of the exponential tail.
First we will consider the effect o f g„ on the time constant, 1 5, of this exponential
decay. This time constant was determined by an exponential fit to the tail region, and the
results are plotted in Figure 30. For convenience, subsequent discussion uses a trapping
time constant, t ; . = 1/to,- , and a release time constant release, t Ti = 1 / r ,. Figure 30
illustrates two regions for t , . In the region I t T is independent o f g„ and calculation
87
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
17
1=10, AnQ=1017cm'3 , S=10_13cm2 , A=0.02eV
E,=0.01 eV, Tr=10-12 s, T=300 K
c
<
O
)
o
14
r
cn
t ns
-12
11
-10
•9
log (t)
-8
7
-6
Figure 29. Behavior of simulated An(t) decays for selected values o f gO increasing in
power of ten from 1014 c m 3 eV' 1 (curve 1) to 10"~ c m 3 eV 1 (curve 9). Inset: same in
semi-log scale.
88
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
IIIIIJ'
-6
■7
O
)
o
8
9
14
15
16
17
18
19
20
21
22
log 9o
Figure 30. The time constant (rs ) of the exponential tail as a function of the density of
states ( g 0).
89
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
shows that : , = x T)0, the release time from the I Oth or deepest level at 0.21 eV,. In the
region II it increases superlinearly with g0 , and we will show that x, = x rx tI0 / x iI0.
These results are obtained if the tail region is assumed to be governed solely by
trapping/release processes for the deepest level. In this case, Eq.(46) transforms into a
second order differential equation with a general solution having a sum of two
exponential terms. Analytical solutions o f the quadratic equation for three regions (I, II.
m ) are based on relative contributions ofxr,xTl0,xi|0:
I. x r « x Tl0,x i|0. Recombination is the fastest process. Once the carrier is released from
the trap it recombines almost instantly. This is the weak retrapping condition. The two
time constants are x r for the fast exponential and x Tl0 for the slow exponential of the
solution. Indeed, each fit of the initial fast exponential decay of curves 1 to 5 in Figure 29
gives 10‘12 s, and fits of the slow exponential tail section gives 3.36x10‘9 s. These values
are identical to those used in the simulations in Figure 29.
II.
X ;i0
« x r ,x t|0. Trapping is the fastest process. This is the strong retrapping
condition. Because recombination is weak, almost all electrons eventually accumulate in
the deepest trap. The short and long time constants are x il0 and xrxT]0 / xil0 , respectively.
Values for xi]0andxTl0 were calculated using Eq.(47). Note, that the density of the first and
last levels is half that o f the other levels i.e. N t = g 0A / 2. The trapping process is very
fast (~10‘15 s) and does not appear on our time scale. The solid line corresponding to
x r = x rx tl0 / x il0 plotted in Figure 30 (region II) agrees well with the “data” points
90
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
obtained by a fit of the exponential tail. Therefore, the transition from region I to II
shown in Figure 30 corresponds to a transition from weak to strong retrapping.
III. t Tl0 « r r ,r ll0. The solution gives r Tt0 and t r as the fast and slow time constants.
This result is presented only for completeness, and will not be discussed here. It might
occur in samples having only very shallow traps with repulsive barriers.
The change in the behavior o f An(t),8e'(t) and PA(t) for the change from weak to
strong retrapping can be seen in Figures 31 and 32, respectively, for the case when the
initial excess electron concentration was 1017 cm’3. Values o f 5 s'( 0 and PA(t) were
scaled to present them with An(t) and An/t) (and i = 10). For weak retrapping one might
conclude all these transients exhibit power law behavior part of the time; however, none
have the same slope. Also, equilibration between free electrons and those trapped in the
deepest level is established only when release from this level becomes rate determining.
Likewise release from this level is the reason that the behavior of An(t), §£'(/) and PA(t)
becomes similar in the exponential tail. For strong retrapping (Figure 32), none o f the
parameters exhibit power law behavior and electrons ultimately accumulate in the deepest
level. Analytical expressions could not be established for these transients.
As discussed above the rectangular DOS does not generally provide power law
behavior. Furthermore the magnitude of the exponent of the power-law was a limited
function of g„ as illustrated in Figure 33. To avoid confusion the transients were fit to
r (,_a> decay as was done in [149] rather than simply to
. Notice for low values o f g0
(weak retrapping), the decay is almost hyperbolic (/’;), which is also the time dependence
expected for bimolecular recombination. However, in the present case it is due to
91
R eprod u ced with permission o f the copyright owner. Further reproduction prohibited without permission.
I (weak retrapping)
-2
® ^A nn= 1017 cm'3. t = 1 0 ' 12 s. m=10, A=0 02 <
3 rE pO -01 eV. g0= l0 15 cm'3 e V 1. S=10*13 cm2
2 Sr
■ ■ i ■m i l
'
■ ' '" " I
'
" I___ i i i m i l l
-12
Figure 31. Decays o f excess electrons (An(t)\ electrons (An/t)) trapped in a localized level at
0.21 eV, photoadsorption signal (PAft)) and photodielectric signal (5 s '(f) ) for weak retrapping.
18
s i i i iiuij
i
111nit]
i’ituiuj
i
11nirtj
n niiuj
> ■huh
PA(t) (scaled)
17
16
tCL 15
'^ A n ^ t ) at 0.21 eV
a Hq= 1 0 17
cm '3 , rr= 1 0 '12 s . m =10. 4= 0 .0 2 eV
£ .1 4
c
<
Ej=0.01 eV , g 0 = 1 0 15 cm '3 e V 1 . S = 1 0 '13 cm 2
And)
=
II (strong retrapping)
S> 12
6e'(t) (scaled)
I m ill
i I I m ill
I I i m ill
I I m in i
I I I m ill
L I IIIU I
Figure 32. Decays o f excess electrons (An(t) ), electrons (An/t)) trapped in a localized level at
0.21 eV, photoadsorption signal {PA(t)) and photodielectric signal ( 8 s ' ( ( ) ) for strong
retrapping.
92
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
monomolecular recombination of electrons released from a rectangular DOS. As a result
a t'1 dependence is not necessarily indicative of bimolecular recombination. Power-law
decay was not observed in the strong retrapping region. But a evolved very rapidly in a
transition region (5xl0 18 - l.5 x l0 19 cm '3 eV*1 ) going from 0 to ~ 0.3. The maximum
value was cc=0.3 because the dynamic range diminished (<1.5 orders) and the curvature
of the decay increased to eliminate the linear region. Furthermore values larger than ~ 0.3
could not be obtained by variation of other parameters as discussed below. As a result any
interpretation of these a values for the rectangular DOS is questionable. In contrast, the
exponential DOS provides well defined a values over the whole range (see above in
section 5.1).
The linear DOS demonstrated the same major features. We used the following
parameters for this simulation: m=9, A=0.02 eV, A/io=l019 cm'3, S -1 0 15 cm " , T=300 K,
t =10'M s. The effect of g0 (I0 19 - 1022 cm° e V 1) on An(t) was in changing the slope of
the power low decay from /’u at 1019 cm° eV' 1 to approximately
75 in the intermediate
region (1021 cm "3 eV"1). At 1022 cm "3 eV”1 the curve in a double-log scale began to bend
showing no power law decay. The slow exponential decay was always present in the tail
region as for rectangular DOS. The kinetics of 5 e '(0 coincided with An(t) in the
exponential tail region as well. All this features can be understood in terms of the model
developed above for the rectangular DOS.
93
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
o>
Ql
? 0.9
weak retrapping
strong
retrapping
0.8
0.7
0.6
14
15
16
18
17
g0 ,cm'3 e V 1
19
20
21
Figure 33. The exponent of the power law decay as a function of g0.
94
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
5.23. Effects o f S, xr and An0 on Behavior of An(t)
As might be expected for rectangular DOS, the transition from weak to strong
retrapping can be effected by increasing the capture cross-section, 5 ( Figure 34). Weak
retrapping occurs with 5=10'17cm2 (x iI5 = I0~8s » x r = 10'n s). The slow exponential
part gives exactly the release time (1.61xl0‘7s for 0.31 eV trap), and the power-law decay
produces t {. The second value o f the capture cross-section, 5=1 O'15 cm2 corresponds to
the intermediate retrapping region (x iI5 = 10"105 > x r = 10""j) giving a power law slope
close to 0.7 and x 5 = 2.0x1 O’7 s , which is still close to the true release time (deviation
about 25 %). Strong retrapping is obtained for 5=1 O'13 cm2 ( x ;|5 = 10~l2s »
x r = 1 0 '" 5 ), and x t is a combination of three time constants and equals to 6x10"6 s,
which is about two orders o f magnitude higher than the release time. A similar transition
from weak to strong retrapping occurs as the recombination time is increased. The only
difference is that there is a maximum value beyond which the curvature of An ft) does not
change. Subsequent increase in xr increases the plateau region and the time constant of the
slow exponential decay.
The value o f An0 does not affect either the transition from slow to fast retrapping
or the slope of the power-law decay. Increasing An„ causes a linear increase in the value
of An(t) and x 5 remains unchanged.
We studied the effect of recombination time (10’7 - I O’" s) on An(t) for linear
DOS. The kinetics changed its behavior in the way similar for the rectangular DOS.
Because of these similarities we did not do any further investigations of the linear DOS.
95
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
18
17
16
15
S = 10*15 cm'
14
13
12
S = 10'13 cm;
11
10
9
8
7
6
m=15, AnQ=10^7 cm*3 , gg=1020 cm*3eV*1
5
Ei=0.01 eV, A=0.02 eV, Tr= 10'11 s
4
3
2
-12
11
-10
•9
8
log (t)
7
-6
5
-4
Figure 34. The decay of excess electrons (An(t)) for different values of capture crosssection (S ).
96
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5.2.4 The Effect of the Width of Rectangular Distribution
It is well known that the width o f an exponential DOS determines the value o f a
for the power law decay of the photoconductivity in the pre-transit time or during the
thermalization period. This section shows that the rectangular DOS does not cause a
similar response. Below, the influence o f the width of the rectangular DOS on n(t) and
5s '(0 is described relative to weak, strong and intermediate retrapping.
Simulated decays for weak retrapping are shown on Figure 35 for selected widths
mA of rectangular DOS. For n(t), the exponent of the power-law decay is not affected by
the width changes and continues to follow t x. The main effect is to extend power law
behavior to longer times as well as to increase the time constant o f the exponential tail
due to the release time from the deepest trap. The decay of 8 e '(0 follows And) only in
the slow exponential tail region. In the central part, the And) and 5s '(0 transients differ
because 5e'(r)is calculated from Eq.(50), which gives a higher weighting factor for the
shallower traps. For strong retrapping (Figure 36), the central part of the And) decay
remains curved in the Iog-log scale as the width increases, indicating no power law
behavior. This suggests that a more complicated analytical expression with parameters
related to DOS is required to extract a meaningful information from this part. Because
the time constants of the exponential tails are a combination of the tree time constants
( t , = t rxT. It*') and an increase in the width of DOS increases /, the value of t s increases
due to release timeTf . For the intermediate retrapping the power-law decays follow f 0'7
(as a limiting slope) and are not affected by the width changes. The exponential tails
follow
the
same
behavior
as
the
strong
retrapping
97
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
condition.
AriQ=10'® cm*3 , gg=1016 cm '3eV_1, E|=0.01 eV
A=0.02 eV, S=10*15 cm2 , T=300 K, Tr=10'1 1 s
14
An(t)
U3
tO
mA=0.1 eV
cn
0.2 eV
0.3 eV
0.4 eV
r~
-12
11
-10
9
8
•7
-6
5
-4
-3
log (t)
Figure 35. Decays of excess electrons (An(t)) and photodielectric signal (5s ' (f)) for
selected widths (mA) of rectangular DOS for weak retrapping.
98
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
17
Ang=101^ cm"^, gQ=1021 cm'^eV-1 , Ej=0.01eV
16
A=0.02 eV, S=10-15 c m 2 , r= 300 K, t r= 1 0 '11 s
15
14
log (An(t), 6e'(t))
13
12
11
10
0.2 eV
9
0.3 e ’
8
0.4 eV
7
6
5
4
-12
11
-10
9
-8
-7
-6
5
■A
3
log (t)
Figure 36. Decays of excess electrons (An(t)) and photodielectric signal ( 8 e '( 0 ) for
selected widths (mA) o f rectangular DOS for strong retrapping.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5.2.5. Effect of Et
The analysis above used a starting trap level (E, = 0.01 eV) sufficiently small to
locate the first level of the distribution very close to the conduction band. By increasing
E,
while keeping the width unchanged, the distribution is moved farther from the
conduction band. The effect of this shift on the kinetics of An(t) and 5 e '(0 can be seen in
Figure 37 for weak retrapping. One can see that the power-law decay, which follows t'1.
is unchanged when E, changes from 0.01 to 0.3 eV. An interesting feature o f the 0.3 offset
is the development of a plateau region in An(t) after the fast exponential decay and prior
to the power-law decay. This plateau region decreases as E, decreases and cannot be
observed at 0.01 eV. Because dhn(t)/dt = 0 in this plateau region, the rate of release from
the traps must equal the recombination rate plus the trapping rate according to Eq.(46).
Because retrapping is weak one obtains that the rate of release from the traps is equal the
m
recombination rate. In other words, the decay rate of ^ A n ((0 illustrated in Figure 37
<=i
equals the recombination rate. The time at which the plateau transforms to the r'1 decay is
equal the release time from shallowest level (0.3 eV) in the distribution. In addition the
slow exponential tail is shifted to correspond to the release time of the deepest level (0.6
eV) in this distribution. Thus the effect of the offset is to shift the onset o f the power law
behavior to longer times for the weak retrapping. This “delayed” release is manifested
also in 5e'(t) kinetics illustrated Figure 37. As has been observed in steady-state
photoconductivity experiments, these plateaus can extend to long times, depending on the
offset of the distribution. As might be expected, the extent of the power law decay is
100
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A=0.02 eV. S=10"15 cm2 , T=300 K, t ,=10‘
Ang=10 1a cm
, gQ=10 1° cm"'3 eV "1, m=15
PA(t), E,= 0.3 eV
- n(t)
—
£
Se'(t)
12
Ei=0.3 eV
r
Ei=0.1 eV
-12
11
-10
9
8
7
-6
•5
-4
3
•2
log (t)
Figure 37. Decays of excess electrons (An(t)), photodielectric signal (8 s'(r)),
photoadsorption signal (PA(t)) for selected offsets (£/) from the bottom of the conduction
band for weak retrapping.
101
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
diminished when the width of the rectangular distribution is shortened, but the extent o f
the plateau is unaffected for weak retrapping. For intermediate and strong retrapping,
plateau regions are also developed by these offsets for the distribution.
5.2.6 Temperature Effects
In the case o f an exponential DOS, the temperature dependence of the parameter
a in the power-law decay provides information about the width of the distribution in
terms o f a characteristic temperature 7*0, i-e. a = T / T0 .O n the other hand, it was shown
[149] that a linear increase of a with temperature was not conclusive evidence for an
exponential DOS because other distributions (rectangular, linear, Gaussian) also gave a
similar temperature dependence. In this section, we will show that the temperature
dependence of a depends the nature of the retrapping. Furthermore, the temperature
dependence of a is not always linear. When a linear dependence does obtain, it does not
provide direct information about the width of the rectangular distribution.
First, consider weak retrapping. The simulated transients of An(t) in Figure 38
illustrate that the hyperbolic decay does not change with temperature, indicating a is not
temperature dependent. As might be expected the exponential tail moves to longer times
as the temperature decreases because it depends on the release time o f the deepest trap.
For intermediate retrapping illustrated in Figure 39, the power-law part of An(t) becomes
steeper with decreasing temperature, indicating a follows the linear relation, a = 77 T0
with T0= 1622 K as illustrated in Figure 40. In terms o f energy units it would correspond
to 0.14 eV. The temperature dependence of a for strong retrapping is also shown in
102
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
m=15, Ang=10^ cnrf^, gg=10^® cm'^eV'l
A=0.02 eV, E|=0.01 eV, S=10*15 cm2 , xr=10
16 r
15
14
<.
10
o
o
100 K
400 K
350 K
-12
11
-10
9
8
\ 300 K
■7
-6
200 K
5
-4
•3
log (t)
Figure 38. Decays o f excess electrons (An(t)) at selected temperatures for weak
retrapping.
103
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 40 and T0= 1250 K, equivalent to 0.108 eV. The relation of those energies to the
width of the rectangular DOS (0.3 eV) is not clear. But one trend is obvious; a decreases
as retrapping progresses from strong through intermediate to weak at which a diminishes
to 0 (Figure 40). For strong retrapping, 1 1= x rx Tl0 / t iI0 for the exponential tail of An(t),
and temperature dependence follows t Tiq since the other parameters are defined to be
temperature independent in this analysis. The trapping process would be temperature
activated if the centers were repulsive or neutral.
5.2.7. Summary
In this chapter we provided a physical picture of photoconductivity behavior for
rectangular/linear DOS based on simulation results using multiple trapping equations. We
showed that due to finite width of the rectangular/linear DOS (in contrast to exponential
DOS) the last deepest level will always manifest itself as an exponential decay (unless the
last level is not reached within the time scale of the experiment, but this would
correspond a quasi-continuous uniform distribution within the band gap). The conditions
for a direct relation between the time constant of this slow exponent and the release time
from the deepest trap was developed. The transition from weak to strong retrapping could
be achieved by changing density of states, the capture cross-section and the
recombination time. This transition affects the part of the kinetics exhibiting power-law
decays in a predictable way. It was shown that rectangular DOS imposes some limitations
on values of a used to describe power-law decays. The insensitivity of a to the changes
in the density width and the distance from the bottom of the conduction band was
104
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
18
17
m=15, Ang=101 7 cm'3, g g = 10^ cm'3eV_1, E|=0.01eV
16
A=0.02 eV, S =10"15 cm2, T(=10-11 s
15
14
log (An(t))
13
12
11
200 K
10
350 K
9
8
100 K
7
6
-12
300 K
400 K
11
-10
9
7
-8
-6
5
-4
3
log(t)
Figure 39. Decays of excess electrons (An(t)) at selected temperatures for intermediate
retrapping.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.35
0.30
strong retrapping
0.25
Tn= 1250 K
0.20
intermediate retrapping Tn= 1622 K
3
0.15
0.10
0.05
no retrapping, a=0 ( Tg= infinity? )
0.00
0
100
200
300
400
500
T.K
Figure 40. Parameter a of the power law decay as a function of temperature for strong,
intermediate and weak retrapping.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
demonstrated. The temperature dependence of a was shown did not provide the direct
relation to the width o f the distribution.
Based on the effects of g0 and xr on An(t) and Se'(t) for the linear DOS we can
conclude that this distribution possesses features very similar to rectangular distribution.
We did not study the other effects for the linear DOS, but it is possible that for some
conditions (i.e. temperature) the behavior will not be the same.
Nonetheless, our simulations based on other various types of distributions
(exponential, Gaussian, linear) confirmed an important point of [149] that there are
certain values of a which can be obtained for more than one distribution. Therefore, it
supports the conclusion made in [149] that a so called inverse 'spectroscopic’ problem
(given An(t), of calculating g(E)) does not always provide a unique solution. The use of
Laplace transformations, for example [152] to extract the form g(E) from the
experimentally measured photocurrent decay could provide reliable results only for the
kinetics which are known to be fingerprints of specific distributions. In contrast, in this
chapter we offered a more comprehensive approach, based on analysis of a complex of
parameters and conditions. Using this approach, some unique features of the distributions
can indeed be traced and identified.
It is understood that the model presented in the paper must be adjusted for
particular experimental conditions. For example, we did not included bimolecular and
Auger recombination terms. It was also assumed that all traps were empty at the
beginning and that the distribution of traps is located well above the Fermi level.
107
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5 3 . Gaussian Distribution.
This
form
of
DOS
can
be
described
by
the
following
analytical
expression: gi=g0e x p (-((E -E 0)/kT0)2), where E0 corresponds to the energy location of
the peak of the DOS. As one can see this a peaked distribution which can be located
anywhere within the bandgap. This form of DOS is very different from all types
considered so far (rectangular, linear, exponential). For rectangular, linear, exponential
distributions the density o f states decays or stays constant with increasing energy. For the
Gaussian (or any other peaked) DOS there is a region where the density of states
increases with increasing energy. This feature adds only one more parameter to the
simulation but it increases the complexity of the behavior of Gaussian distribution by a
large extent. In contrast to the previously studied distributions where we were able to
establish solid relations between various parameters and the final appearance of the
kinetics the behavior of the kinetics for Gaussian DOS is not so well predictable.
5.3.1. Behavior of An(t) and 5s ' ( f ) .
The shape of the Gaussian distribution can be considered to be intermediate
between rectangular and exponential distributions in terms of the degree of closeness
between An(t) and 5s ‘(t) kinetics. One may recall that in general the rectangular
distribution gives the kinetics which coincide only at the tail region when the last level is
seen for An(t) and 5s '(t). The exponential distribution in contrast provides the kinetics
which coincide at short times only and begin to diverge at longer times when electrons in
deeper levels dominate in 5s ‘(t). The Gaussian distribution in general does not provide
108
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
19
m=30, S=1(r15 cm2. T=300 K, a=0.02 eV, g0=10 21 cm"3 e\A1
An0=1019 cm"3. tr=10-11 s. Eq=0.3 eV, E,=0. T0=800 K
18
17
log(An(t), 8e'(t))
16
15
An(t)
14
13
12
11
10
9
8
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
log (t)
Figure 41. An(t) and 5s'(t) simulated for the Gaussian distribution , r o=800 K (see
param. on the plot).
109
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
any region at all where both kinetics will decay in the similar way. It is illustrated by
Figure 41 where simulated kinetics of An(t) and 6s ‘(t) are shown. The nature of the
divergence in the tail region (starting from ~ 10‘J s) is the same as in exponential
distribution, i.e. due to increasing domination o f deeper levels. Therefore, in general
5s ‘(0 will always decay more slowly than An(t). This is true for the “infinite” distribution
when the release from the last level of the distribution can not be detected within the time
scale o f the experiment However, if within the range of the detection the last level
manifests itself both in An(t) and 5s '(t), the exponential decay (compare with the
rectangular distribution) following the power law decay will be seen. This truncation
situation will be later considered when explaining experimental results in Chapter 6.2.
53.2. The Effect of the Width of Gaussian Distribution.
Because we are mostly concerned with power-law decays in And) we will
concentrate on the corresponding parts in the kinetics. It has been shown in Monte Carlo
simulations [149, 151] of TOF kinetics that this distribution can produce a transition from
one power-law to another in simulated photocurrents. It was also shown that parameter a
extracted from
slopes has different values in two regions and does not necessarily
increase with temperature with a linear dependence (accT/T0). Our first goal was to
explore this behavior with our MT model, i.e. with a recombination rather than a
diffusion term. The simulated for selected T0 curves are shown on Figure 42. One can see
that the width of the distribution affects the slopes in An(t) in a predicted way. Another
trend is evident: the region where the first power-law can be detected shrinks as T0 is
110
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
19
m=30, S=1CT15 cm2, T=300 K, a =0.02 eV, g0=10 21 cm*3 e\A1
An0=1019cm*3, Tr=10*11 s, Eq=0.3 eV, E|=0
18
5s’(t)
17
log(An(t), 8e'(t))
16
An(t)
15
14
13
12
11
10
9
8
-12
-11
-10
9
8
7
6
5
-4
3
log (t)
Figure 42. An(t) and 8e'(t) simulated for Gaussian distribution for selected values o f the
characteristic temperature, T0.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
decreased. Therefore, one may expect difficulties in extraction of a for narrow widths of
distribution. For 5s '(t) the changes in T0 manifest themselves mostly in the tail regioa
The parameters a! (before the knee) and a 2 (after the knee) extracted from power law
decays by fitting to the same expression as used in Chapter 5.2 are shown on Figure 43
and Figure 44 correspondingly. Our data show that a linear dependence of a can be found
in a limited range only. As compare with [149] where simulations were done in -0.1 - 0.5
7/To range our values are somewhat lower in a high value limit (for example, at T/Tn=0.5
we have a^O .5 and [149] provides a,«0.8). Our simulations were done in a larger
dynamic range of An(t) than those in [149]. This would make our values more reliable
than those in [149] since the extraction of the true power-law slopes can be obscured by
the proximity o f the transition region from one power-law to the other. Since covering a
larger range of
T/Tn would require extending significantly the time range of the
simulation (to get enough decades for establishing the slope) we did not pursue this study
in the current work. From the discussed above figures another conclusion can be made:
the values obtained from the region after the knee can significantly exceed unity and.
therefore, are different from those the before the knee. Those slopes were established over
more than several orders of magnitude of dynamic range and provide a strong support for
this conclusion.
5 3 3 . The Effect of the Peak Energy Position, Eg.
Going back to Figure 42 one can see that a change in temperature causes changes
in the slopes, but the time range where power-law decays exist is limited. In other words,
112
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
0.55
T=300 K
a=a (T/T0)+ b. a=0.35 , b=0.34
0.50 -
0.45 -
a
0.40 -
0.35 -
0.30 -
0.25
0.10 0.15 0.20
0.25 0.30 0.35 0.40 0.45 0.50 0.55
Figure 43. Values of a t extracted from the power-law decay in the region before the knee
from the curves simulated for Gaussian distribution for selected values of T0.
4 -f-
}
T
T=300 K
a=a (T/T.)+ b. a=7.6 , b=-0.7
■j
i
3 -j
tT 2 -*
0.20
<1 1 r '
0.25
1 1 -r i ' 1 1 i- 1' 1 1 1 i 1 ' ■ .............
0 .3 0 0.35 0.40
0.45
0.50
i
0.55
T/T-
Figure 44. Values of a 2 extracted from the power-law decay in the region after the knee
from the curves simulated for Gaussian distribution for selected values of T0.
113
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
changes in the temperature do not move the knee position to higher/lower times. But this
can be achieved by moving the peak energy position, E0 (Figure 45). The two curves
shown on Figure 45 have identical slopes. The only difference in parameters of
simulation is that E0 moved to higher energies for the dashed curve. (To ensure that we
will not see the release from the last level in a original 30-level distribution when
extending the time range we increased the number of levels up to 40. This step does not
affect the slopes o f the simulated curves.)
5.3.4. The Relation Between Power-Law Decays and Individual Levels in The
Distribution.
Eventually we were able to assign specific parts of An(t) decay to particular levels
in the Gaussian distribution. For this purpose we varied the offset value. Et from 0.06 to
0.65 eV. It shifted the position of the first level in the distribution to higher energies. The
distribution was peaked around 0.5 eV and contained total number of 22 levels with the
step 0.03 eV. It provided the energy range between 0.06 eV and 0.73 eV. We limited the
time range of the simulation to 1 s which corresponded to the release time from 0.73 eV
level. Therefore, the upper boundary for the level which could be accessed by our
simulated curves was set firmly. The resulting curves along with the simulation
parameters are shown on Figure 46. The transient remained virtually unchanged when Et
was changed from 0.06 to 0.3 eV. Therefore, those levels remained invisible in the
simulated data. The An(t) curve corresponding to those values consisted of the initial fast
exponential decay followed by a small plateau region and then by two subsequent power-
114
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.e+19
1.e+18
iiimI
iiiiiiJ
Iimnj I i III J Iimml imiml iiiiinj I Illlld—I IJIlul— I I llinl—I I1HB1—I III1B,
=10'15 cm2 , T=300 K, T0=800 K, A=0.02 eV, g0=1021cm*3e V 1
AnQ=1019 cm*3. xr=10"11 s, E,=0
1.e+17
1.e+16
m=30, E0=0.3 eV
1.e+15
1.e+14
\ 1.e+13
S
m=40, E0=0.5 eV
1e+12
¥ 1.e+11
1.e+10
1.e+9
1.e+8
1.e+7
1.e+6
1.e+5
mrn|—i rmrn—n nm^—i nun—i 11 mu(—i uiiu^—n nin^—rrra^ iniiiu| iiimi^ i i
-12 -11 -10 -9 -8
-7
-6
-5
-4
-3
-2
n
-1
0
log (t)
Figure 45. Kinetics of An(t) simulated for the Gaussian distribution for two different
peak energy positions, E0.
115
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
law decays. The time constant of the fast component remained the same during these
variations which indicated that the rate of the initial removal of electrons from the
conduction band was unchanged when changing E{ in this range. The time constant of the
recombination used in this simulation was 10 s. We fitted the initial decay of this
simulation to the exponential decay and obtained a perfect fit with the time constant
7.6x10‘10 s and the pre-exponential factor 1019 which is identical to the initial excess
electron concentration used in the simulation. The calculation of the average trapping
22
time as r i =l/(5’v ^vVJ) gave 8xlO'10 s which is very close to the value obtained by
r= 1
fitting.. Therefore, the fast exponential decay corresponds to trapping.
When changing E{ one should also consider how it can affect the total
concentration of states available for trapping. The Figure 47. which depicts the total
distribution, can help to follow this idea. As was mentioned, states from 0.3 to 0.73 eV
are visible in this simulation and this area is shaded. The total area between these two
values gives the total number of states that are active in terms of trapping, and this
number determines the trapping time constant. One can see that adding extra levels when
the offset scanned 0.06 - 0.3 eV range almost did not change the total number of states
(reflected in the constant trapping time as above) because the density of the added centers
was much lower than the present one. Because dAn(t)/dt = 0 in the plateau region, the rate
o f release from the traps must equal the recombination rate plus the trapping rate. The
equilibrium condition breaks down when electrons begin to be released intensively from
0.3 eV level. The plateau region ended at a time corresponding to the release from the
116
R e p ro du ced with permission o f the copyright owner. Further reproduction prohibited without permission.
19
I m ild
I -U ilid
in>mj | m n j
i m tn J
l m ud
i i
. hi J
iitind
■i n J
...m d
. i .i u d
■■■■
18
\
17
16
06) eV
15
^
14
°
12 i
E,= 0.4 eV.
E.=0.5 eV
'
r
11-1
f
10 i
r
9 1
F
8 1
r
: An
=1019 cm*3, t=10^
Ano=1019
t^K T 8 s
s
7
:
: m=22, S=1CT15
S=10-15 cm2, T=300
T=3CX) K, T0=800 K. A=0.03 eV. g0=1018 cm-3
cm*3 e v
Vfl:
1}
ri —
iii’
mii^^mnuq
tilinlginimnigmimii
i ii m
iiiiiq
i-M
nni^"i
i i ii^ n
iurqn—
iu^ri—
iinn^
11mu
u—
----ri
—
i ini'iiu
iiijhi iiiii
i^mim
mnyi iiirn
qi irinnqrm
inmq—
i—iiim
-12 -11 -10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
log (t)
Figure 46. Kinetics An(t) simulated for Gaussian distribution for selected values o f the
offset energy, Et.
seen before
the knee
seen after
the knee
i
not seen at all
not seen at au
5*
n 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
E,eV
Figure 47. The total Gaussian distribution used for simulation in Figure 46.
117
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.3 eV level and this was the point where the power-law decay started. We started to
observe changes in the kinetics at £,=0.4 eV. As one can see it affected only the middle
part o f the kinetics by extending the plateau region. The time of trapping obtained from
the initial exponential was the same indicating that cutting out few (4) levels from the
distribution has no effect on trapping because o f a relatively low density o f states in this
range. Starting from £,=£0=0.5 eV the time constant of the fast exponential decay starts
to increase. This means that cutting down the levels which produce high concentrations of
the traps slow down the trapping process. To be more rigorous we should mention that
the addition of the new deeper levels (above 0.73 eV) occurs simultaneously with the
removal o f shallower levels. Recall that we are just moving the energy frame while
keeping the same number of levels in between. Although these levels have release times
too long to observe the release from them, nothing prohibits these levels from trapping
the electrons. The fact that we see a progressive decrease means that the added levels do
not increase noticeably the total number of centers compare with the removal process.
One can see that Er o .55 eV corresponds approximately to the knee position. Subsequent
increasing of the offset has two effects: freezing the time constant of the fast exponential
decay and moving the curve down the second power-law part in An(t). Fitting this fast
decay now gives 10’8 s as a time constant which is the recombination time. It becomes
clear that the fast exponential decay always corresponds to the fastest process involved.
Initially, it is the trapping which is the fastest process. As the total concentration of levels
decreases the trapping time increases and matches the recombination time (10
s) at
Ef=0.6 eV and continues to increase. Beyond this point recombination is the fastest
118
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
process and, therefore, the time constant of the fast component is not affected any longer
by changes in the total density of centers. Therefore, we have shown that when the
transition from one power-law to another is observed the levels with energies < E0 will
manifest themselves in the power-law decay before the knee and the levels with energies
> E0 will be responsible for the second power-law decay after the knee. This situation is
depicted in Figure 47 where the total shaded area in the distribution is the range seen in
the simulated “experimental” data. The light gray area corresponds to the region before
the knee and the dark gray area to the region after the knee. This can help us understand
the phenomena of shifting the whole transient to the larger times when increasing En
which was discussed previously.
In the particular example discussed above the peak energy of the Gaussian DOS is
manifested at the knee point o f excess electron kinetics. In general, it is not the peak
energy location which determines the range where appropriate power-laws will be
observed. A relation between the recombination time and the trapping time to individual
levels in the distribution (individual trapping time) will dictate the time range for the
power-law decays. This is illustrated by Figure 48 where the recombination time, the
release time from the individual traps (■ct .=(l/v0)exp(5£^/£7,)), the individual trapping
tim e (ti(=l/(5v//;))
m
and the average trapping time (x iav= \/(Sv'^N l))
for four
1=1
distributions examined at Figure 46 are shown: a) E|=0.06 eV, b) E|=0.3 eV, c) Et=0.5 eV,
d) E|=0.6 eV. In all four simulations the recombination time was 10'8 s, the average
trapping time was calculated by summation of all traps as was discussed above, the
individual trapping time was calculated in the same way but using the specific levels only
119
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
and the release time was calculated for every level in the distribution for T=300 K. Figure
48 (a) illustrates the point made in the preceding paragraph that levels can be seen in the
excess electron decay only starting from some particular value. For this offset energy
(0.06 eV) the first level in the distribution is 0.09 eV. But the intersection between two
lines corresponding to the release time and the individual trapping time will dictate the
first level seen. One can see that it will correspond to ~ 0.34 eV and will be observed
starting from ~ 4x10‘7 s. Below this energy the release time is faster than the individual
trapping time, and the concentration in the levels < 0.32 eV will be too low to contribute
to the total decay of excess electrons, An(t). Note, that these levels are not empty as one
may think. They equilibrate very fast with conduction band electrons and the
concentration of electrons trapped at these levels will follow the decay of conduction
band electrons. In this picture the line corresponding to the average trapping time is
located below the line corresponding to the recombination time and this means that the
fastest process (average trapping in this case) will be responsible for the initial fast
exponential decay in hjn(t). The next important intersection is the first intersection
between the line corresponding to the recombination time and the line corresponding to
the individual trapping time which occurs for 0.44 eV level (release time ~ 2x10° s). At
this point (2x10° s) An(t) approaches the first power law-decay and continues to follow it
until time determined by the second intersection between these two lines which occurs for
~ 0.56 eV level (at about 3x10'3 s). Therefore, the first power law occurs between the two
times corresponding to the two intersection points between the recombination time and
120
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10 i
[individual trapping time
individual trapping time!
release time
time
-2 -j.
- 3 ~i
- 4 -j
-5 i
-6
05
-4 -i
-5 -i
-j.
-7 n
-6 -j —
-7
-8 -j —
•9 i —
average trapping time:
recombination time
average trappino time
-10 J —
0 .0
0 .1
0 .2 0 .3
0 .4
0 .5
0 .6
0 .7
0 .2 0 . 3 0 . 4 0 .5 0 .6 0 . 7
0 .8
m
3
i| iiii]
0 .8 0 .9
1 .0
E,eV
E.eV
10
13 -
9
12
8
7
i i I i i r i | i iT T | r r i i I i i i i | i
6
5
4
3
-
10 9 8-
individual trapping time
release time
6
2
1
-
ti)
3
•
3 _
I
individual trapping time
release time
2—
05
O 0-
0
-1
-2
-3
-4
-5
-2
-
-3 -
average trapping time
-5 -
-6
-7
recombination time
-8
average trapping time
-7 -
re c o m b in a tio n tim e
-9
0 .4 0 .5 0 .6 0 . 7
0 .8
0 .9
1 .0
1 .1
1 .2
0 .5
0 .6
0 .7
0 .8
0 .9
1 .0
1 .1
1 .2
1 .3
Figure 48. Relation between individual trapping time, release time, recombination time,
average trapping time and trap energy for Gaussian distribution with different offset
energies, E(: a) E(=0.06 eV, b) E(=0.3 eV, c) E|=0.5 eV, d) E|=0.6 eV.
121
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the individual trapping time. Starting from the second intersection point, tsn(t) changes
its slope to the second power-law decay. One can monitor these features by simply
comparing two Figures, 46 and 48.
Figure 48 (b) (E|=0.3 eV) predicts that in spite of starting the distribution from
0.33 eV the first level seen will be still determined by the intersection of the release time
curve and the individual trapping time curve. One can see that there are no changes
(compare with case (a)) in this first level which is observed in An(t). The two
intersections between the recombination time curve and the individual trapping time
curve gives the same time range for the power-law decay as in case (a). One new feature
appears on Figure 48 (b) - the second intersection between the release time curve and the
individual trapping time curve. Beyond this intersection the energy of the trap has a
release time that is smaller than the trapping time. This means that starting from this trap
energy there will be no electron accumulation in the traps and the dominant release from
these traps will be observed.
Figure 48 (c) (E|=0.5 eV) shows that the average trapping time moves closer to
the recombination time due to cutting the levels in the DOS which supply high density of
traps. Because the first intersection point between the release time curve and the
individual trapping time curve was passed, the first level observed in Hm(t) will be
determined by the first level (0.53 eV) in this part of the distribution. The first
intersection point between the recombination time curve and the individual trapping time
curve was passed as well. Therefore, the first power law decay will be observed in the
very narrow range determined by the time corresponding to the release from the first level
122
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(~ 1O'3 s) and the time corresponding to the intersection o f the recombination time and the
individual trapping time curves (~ 3xl0‘3s). The second power-law decay starts from
3xl0‘3s.
Figure 48 (d) (El=0.6 eV) shows that now the recombination time curve is the
lowest line and therefore, the initial fast exponential decay will be the determined by the
recombination. The first level in the distribution (0.63 eV) is seen in the second powerlaw decay staring from - 3x10‘2 s (as determined by the release time from this level).
Therefore, one can see that this type of analysis can reveal all major peculiarities
observed in An(t) kinetics.
5.3.5. The Criterion of the Continuity of the Distribution.
Finally, we would like to address the issue of the continuity of the distribution.
This final discussion has a universal character and is valid for the other types of
distributions discussed previously. We will illustrate it with the Gaussian distribution in
these closing remarks. The “discrete” distribution means that the individual levels can be
pinpointed in their appearance in the observed phenomena, e.g. photoconductivity
kinetics in this particular case. In this case the transient will reveal exponential
components. In case of continuous distribution it is more appropriate to talk about the
collective behavior o f the levels since any particular level can not be sectioned from the
observed kinetics. In this case the transient will have a form of the power-law decay. To
discriminate between discrete/continuous distributions the separation between the levels
can be compared with kT. Figure 49 shows the kinetics of An(t) obtained in the
123
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.e-3
LJJilll
» »m nu
I
LIIIUI 1
♦11■»»« 1 11tm» 1 1•im»
N A=0.05 eV
1.e-4
A=0.01 eV
1.e-5
A=0.02 eV
< 1.e-6
o5
_o
A =0.03eV ^x
Vv
1.e-7
1.e-8
1.e-9
tt 7tt|—
-9
i i i iiiii| — i i i Tinij — i i 1 1 uiij — r i i iiiiij— i i i iiiii| — i i 11 inrj— i i i n m |
-8
-7
- 6 - 5 - 4
-3
r
-2
log(t)
Figure 49. Kinetics of An(t) simulated for various values of the separation (A) between
levels using £7=0.0106 eV.
124
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
simulation for 7=123 K (£7=0.0106 eV)
with varying the energy step A in the
distribution (all other parameters are irrelevant for the present discussion). One can see
that when A=0.01 eV the curve follows a smooth power-law decay, and the separation
between levels corresponds to continuous distribution. When the separation is doubled
(A=0.02 eV), very tiny waves begin to appear on the curve, i.e. the discrete nature starts
to manifest itself. When it is tripled (A=0.03 eV) the exponential components due to the
individual levels can be resolved easily. Eventually, when A=0.05 eV only few exponents
are left. With the MT model simulation we have shown that the power-law decay due to
thermalization of electrons down the ladder of bandgap states is a combination of
exponential decays due to individual release times from particular levels. Therefore, the
degree of continuity can be judged from the appearance of the measured kinetics.
Observing "non-modulated1" power-law decays (without individual exponentials) would
mean that the separation between levels is comparable to kT.
53.6. Summary.
We have shown that the Gaussian DOS provides generally An(t) and 5e'(/)th at
do not decay in the same way over the time range. The parameter a is related to the width
of the distribution but the linear dependence can be observed in a limited temperature
range only. It has been shown that when the transition from one power-law to another is
observed the levels with energies < E0 will manifest themselves in the power-law decay
before the knee and the levels with energies > E0 will be responsible for the power-law
decay after the knee. Finally, we proved with our simulations that any distribution for
125
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
which the separation between the levels is less/equal kT will behave as a continuous
distribution marked by the “collective” behavior of participating levels.
126
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6. EXPERIMENTAL RESULTS AND ANALYSIS
6.1. CdSe (#1)
6.1.1 Experimental
The sample of 1pm thick polycrystalline CdSe film was deposited on a thin quartz
strip (3x60xlmm3) by conversion of CdO to CdSe as described in [162]. The conversion
process involves two steps: 1) spray pyrolysis of an aqueous cadmium nitrate solution to
form thin-film cadmium oxide; and 2) reaction of selenium vapor with cadmium oxide to
form cadmium selenide. During the spray pyrolysis the temperature of the substrate was
kept at 300°C. The subsequent treatment in the Se vapor was done at 500°C for one hour.
Composition data indicated that CdO was completely converted to CdSe. The ratio of
atomic concentrations Se/Cd was 1.18. The sample was n-type with the following
equilibrium parameters: «0=2.7xlOI0 cm'3, p0“ 14 cm2 V '1s*1.
This is the only sample which was studied on both experimental setups described
in Chapter 3. Below we would like to outline major features o f the experiment done on
setup I which stimulated our work on its improvement ( the setup 2). Due to the use of
the tee bridge the adjustment procedure to obtain the proper Lorentz resonance curve was
very time consuming and not always completely successful. The typical experimental
resonance curve (Figure 5) deviated from the Lorentz form. The slow sawtooth
modulation of the klystron which was used to obtain the dark resonance curve allowed
measurement of only initial and final frequencies within the modulation range. This
limitation was a drawback of the analog modulation which caused some uncertainty in
127
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
the dark resonance frequency position. Collecting transients in setup I was done in a
manual mode. The microwave frequency of the klystron generator was set by hand every
time and measured separately by the frequency meter. Therefore, the ability of setting the
desired frequency was limited by a mechanical vernier. Photoresponces were collected at
individual frequencies which were measured with 5 kHz accuracy. The major problem
arose when establishing the correspondence between these individual frequencies and
those which were calculated from the dark resonance curve measurements. The signal-tonoise ratio (20) was not high enough to establish the true form of decays unambiguously.
Therefore, we will present only data obtained with setup 2.
For setup 2 all parameters in the experiment were controlled by the computer. The
major advantage was gained by using the microwave sweep synthesizer (described in
section 3.3). The dark resonance curve was measured in a digital sweep mode which
allowed us to measure any frequency with high precision (1Hz). The same precision was
preserved when collecting photoresponces at individual frequencies. The use of the
circulator instead of the magic tee simplified the tuning procedure to get the Lorentz
form. As a result, it allowed us to obtain a high degree of fit of the experimental curve
with the Lorentz form (see Figure 7). The signal-to-noise ratio (80) increased the
reliability of the fitting results for the collected kinetics.
Before illumination, the resonance curve parameters of the loaded cavity
containing
the
CdSe
sample
were:
y0 =0.425 m W ,
f0 = 9,628.061 MHz,
AfI/2 =3372 MHz [125]. The frequency resolution was 7 kHz, which was determined
using the frequency range and the number of channels in the oscilloscope. The light-
128
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
induced transients, which were measured in the range 9623.0-9630.6 MHz, were
determined at 20 frequencies with a step of 0.4 MHz. Figure 50 illustrates a typical plot
(open circles) o f a reconstructed difference signal corresponding to 1.00 psec after the
laser excitation. The corresponding fit according to Eq.(25) is illustrated as the solid
curve in Figure 50. Fits like this were obtained at 100 ns intervals from 100 ns to 1.0
ms. These fits provide the light-induced values of y02, / 02, Af(a at each time after the
laser pulse.
The corresponding time dependence of the light-induced shift of the
resonance frequency (8/0) and the change the bandwidth 5(A /[/2) are presented as the
solid curves in Figure 51 on a logarithmic scale to cover the dynamic range o f the data.
The light source used was an LN 1000 nitrogen (337 nm) laser (described in sections 3.2
and 3.3) The penetration depth at 337 nm for CdSe is to be 3.5xl0'5 cm. Based on the
light intensity and the depth of penetration, these curves were obtained for an initial
photogenerated electron concentration
- 3.3 1020 /cm3 which corresponded to the
maximum laser intensity. The laser intensity was varied by 3 orders of magnitude. The
18
3
•
corresponding curves obtained at the electron density - 1.8 10 /cm are shown on Figure
52.
The measurements were done at two temperatures: 300 K and 331 K. Higher
temperatures were not accessible for this experiment. The temperature was controlled by
blowing
hot
air
through
the
quartz
tube
with
the
129
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
sample
inside.
0.03
2.1 p sec
0.02
-
0.01
-
I
E
o>
0 .0 0 -
w
c
o
Q.
c/i
(D
O
• •
-0 .0 1
-
o
sz
CL
-
0.02
-
-0.03 -
-0.04 9.622 9.623 9.624 9.625 9.626 9.627 9.628 9.629 9.630 9.631
frequency, GHz
Figure 50. The difference signal between the “dark” and the light induced resonance
curves obtained 2.1 psec after the laser pulse. The solid line represents the fit according
to Eq.(25).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.a-2
- e x p ( - t / r ), r = 9 0 n s
1 .0 -3 -
N
oX
a = 0.69
■a
c
a
a = 0.67
1.e-S -
1. 0 -7
1 . 0-6
1.0-5
time, s
Figure 51. Changes in the cavity quality factor SfA/J/j) (upper curve) and the shift in the
resonance frequency 8/0(lower curve) as function of time for light intensity
corresponding to I0 = 3.3 1020 electrons/cm3; T = 300 K.
1.0-3
a *0.67
M
a * 0.67
o|Xo
«?
■eo
a
e
VS
1.e-5 -
l.ft-7
1.e-5
t
time, s
Figure 52. Changes in the cavity quality factor 5(A/I/2) (upper curve) and the shift in the
resonance frequency 8/0(lower curve) as function o f time for light intensity
corresponding to I0 = 1.8 1018 electrons/cm3; T = 300 K.
131
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6.1.1. Analysis O f 5 s' Transients.
Figure 53 illustrates the changes in the real and imaginary parts o f the complex
dielectric constant calculated according to Eq. (18) using 5(A /1/2) and (8 /0) kinetics.
Note, that we use the normalized dielectric constant, i.e. divided by e0. As a result, both
real and imaginary part changes are in relative units. As indicated in Figure 53, the
change in the real part o f the dielectric constant has a maximum value of -2 2 (giving the
absolute value about 32). Although this value is large, we believe that it is not unrealistic
because the intensity dependence results, which will be discussed later, indicate that all
the detectable traps appear to be saturated. Thus, the value of 5s '( 0 was saturated over
almost 3 orders of the magnitude of light intensity, indicating all traps that can be
detected in this time scale ( <0.4 eV) must be saturated. Due to the high polarizability of
traps in this energy range, a dramatic increase in the dielectric constant is expected
according to the Clausius-Mossotti relation:
(5 2 )
3s,
in which sj is the real part of the dielectric constant of the host material, e'2 is the
increased dielectric constant due to the presence of N donors (electrons in shallow traps),
a is the polarisability of donors. This equation was used to estimate the electron densities
in the shallow traps assuming equal polarisability of all trapped electrons. The
03
polarizability of all traps was taken to be a £VW«6.8-105A [121]. This value is similar to
03
the corresponding polarizability values of shallow centers in Si (6.42-10 A ) [162] and
132
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100.0
1 nm CdSe film (#1)
10=3.3x1020 el./cm3, T=300 K
CO
10.0
'
)
CU
O
40
I
i
1.0
0.1
1.e-7
1.e-6
1.e-5
time, s
1.e-4
1.e-3
Figure 53. The changes in the complex dielectric constant for I pm CdSe thin film (#1).
300
Clausius-Mossotti eq., CdSe, a=6.8x10-19cm3 .
200
100
-200
-300
1015
1016
1017
1018
1019
1020
1021
1Q22
N, cm-3
Figure 54. The simulation o f Clausius-Mossotti equation (Eq.(52)) for CdSe
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
AgCI (2-105A ) [104],
It was calculated for CdSe using the effective mass
approximation (EMA) as in [162,163]
In this equation, a „ is the polarisability of atomic hydrogen, e' = 10 is the real part of
dielectric constant, and the effective electron mass m = 0.13m for CdSe. Based on the
changes in the real part of the dielectric constant and Eq.(52), the average electron density
in the detected traps was 2.0x10
|Q
1
c m ', which is in the region o f the asymptotic value
(= 3.5* 101S cm'3) where large changes in the dielectric constant occur.
A plot of Eq.(52) as a function of trapped electron concentration (Figure 54)
indicates the increase in the dielectric constant may reach values ~ 200 near the critical
concentration, 3.5xl018 cm'J, which is the critical density of shallow traps for CdSe.
Nonetheless, due to some uncertainty in the active sample volume resulting from nonhomogeneous excitation, the error in this experimental value could be as large as 40%.
In the absence of optical excitation, a similar change is expected when the donor
concentration approaches the region of the metal-insulator transition or when the shallow
traps are all populated at low temperatures. A number of authors observed giant dielectric
constants, up to 700 in this region at low temperatures [164,165]. For example, the static
dielectric constant of single crystal germanium, increased from 16 to 107 when shallow
ionized donor states were filled at liquid helium temperatures [164], The impurity
concentration was 1017 to 1018 cm’3. Values as high as 35 to 40 were found for antimony
and arsenic doped single crystal silicon at low temperatures [166]. The impurity densities
134
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
were (2-3) x 1018 cm'3. These results were supported by the low temperature metalinsulator transition photoinduced in AlGaAs [167], Also, they are consistent with Mott
transition theory for heavily doped semiconductors [168-170]. Eventually, the dielectric
constant diverges in the metal-insulator transition region [169,170,172,173], As discussed
in [169,171] the Clausius-Mossotti relation can provide satisfactory numerical results
near the Mott transition. (A more sophisticated treatment o f the divergence of the
dielectric constant uses scaling theories [165].) The transient microwave results for CdSe
at room temperature demonstrate that the detection of large changes in the dielectric
constant is not restricted to steady state conditions at low temperatures and that it is not
necessary to heavily dope the sample. Thus, the photoinduced concentrations of electrons
in naturally existing shallow traps are sufficient to effect large changes in the dielectric
constant on a short time scale. These results indicate that AMTMP could be a very useful
tool to study light induced Mott transitions because both parts of the complex dielectric
constant undergo large changes at the transition [165]. Optical generation of this
transition with a subsequent AMTMP detection would be a logical extension of the work
done in [165] that detected it in the dark using various doping levels in a resonant cavity.
6.1.2 Kinetics Analysis.
This section discusses the form of the observed kinetics (Figure 51). The
complexity o f these curves illustrates that more than one decay process has occurred. For
5(A /1/2), there are three regions: a rapid exponential decay, a linear region characterized
by a power law decay, and finally a break followed by a more rapid decay. Although
135
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(8 /0) did not exhibit the initial fast exponential decay, it had almost the same time
dependence as 5(A /1/2) in the other two regions. This can be seen more readily by the
plot of the loss tangent
s s : s A (4 k )
3c’
2S/0
versus log t given in Figure 55 using the data from Figures 51 and 52. The almost
constant value following the initial fast drop indicates that that the ratio 5(A /!/2)/(5 /0)
remained almost unchanged over nearly four orders of magnitude in time.
For the
reasons discussed previously (sections 4.1 and 4.3), the negative value of (5 /0) indicates
that the frequency shift is due to electrons in shallow traps, and 5(A /I/2) is due to free
electrons in the conduction band.
Consequently, the region of constant 5(A/l/2)/(5 /0)
indicates that the free electrons are in equilibrium with the electrons in shallow traps
weighted more heavily (according to chapter 5) during the power law decay.
At short times the loss tangent is not constant, and there is a rapid drop to the
constant value. As indicated in Figures 51 and 52, this change is due mainly to the
exponential change in 8(A/1/2) that precedes the quasi-equilibrium region. Because the
time constant for the exponential fit (dashed lines in Figures 51 and 52) to this pre­
equilibrium region is only 90 ns, which is smaller than 130 ns (2.2 times the time
constant of the cavity), the relaxation time cannot be resolved for the process responsible
for the change in S(Af m) in this region. The longer time constant calculated for the
lower light intensity data in Figure 52 may be an over-estimate because the contribution
136
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
3
1
I
i
0J
i
1------------------------------------------------------
10-'
10*
10*
10“*
10*
•
time, s
Figure 55. Loss tangent at two intensities, I0 = 3.3 1020 electrons/cm3, T = 300 K.
X
o
i.e-3 -j
c©
C
o
Q.
E
8
w
3
1.e-4
©
o
D
"<
O
3
a.
1.e-5
0.001
0.010
0.100
1.000
normalized intensity (l/l0), rel. units
Figure 56. Intensity dependence of the maximum of 5(A/l/2) associated with the fast
component prior the power-law dependence.
137
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
of the exponential term is small. At any rate, it is apparent that there is a fast process that
cannot be resolved by our equipment On the other hand, although the process cannot be
resolved, 5(A /1/2) provides a measure of the total density of electrons involved in this
process because the RC circuit of the microwave system acts to integrate the effect of
these electrons.
As a result at about 100 ns after the pulse, the total value of 5(A /I/2) can be
considered to have two contributions: an integrated term associated with a process that
was too fast to be resolved, and a term due to a slower, time-resolved process observed at
later times when quasi-equilibration occurred. Figure 56 illustrates that the maximum in
the integrated portion of 5(A /1/2) follows a square root dependence on the light intensity.
Furthermore, it decreased to almost zero at the lowest light intensity. On the other hand
as discussed below, the maximum in the time resolved portion of 5(A /I/2) remains
constant except at the lowest light intensity. The same was true for (S/0). These results
indicate that all the traps detected by the microwave technique are saturated at all light
intensities except perhaps, the lowest one. Furthermore in the time resolved region, the
free and trapped electrons are in quasi-equilibrium. Consequently, the exponential region
is due to free electron density in excess of the value needed for equilibration with the
saturated traps.
The square root dependence presented in Figure 56. indicates a
bimolecular relaxation process for these electrons. Recombination of free electrons with
free holes would be consistent with these results.
The power law time dependence can be analyzed with models involving multiple
trapping o f excess free carriers in spatially localized subband gap states having a density
138
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
function varying in energy (i.e. exponential, Gaussian, linear, etc.). The lack of deviation
o f trapped electron decay from the free electron decay at long times suggests that the
exponential DOS can be responsible for this similarity, and that deep traps were present
in densities small enough to ensure the similar decay of the bandwidth change and the
shift of the resonance frequency (based on Chapter 5.2. results). Initially we analyzed our
experimental data based on the OK model [45,46.155]. It showed some inconsistencies
with the OK model and led us to develop the MT simulation approach. The discussion
starts with the OK model, and the simulation treatment will follow.
Because the OK model was too complicated to fit the total experimental transient,
we attempted to find regions in our kinetics which could follow the analytical expressions
of this model. We found what appeared to be a power-Iaw decay and a more complicated
function at the tail of the kinetics in accordance of OK model. The pertinent equations to
describe the time dependence of both bandwidth change and shift of the resonance
frequency over the whole time domain are :
(54)
(55)
in which t^ . is the bimolecular recombination time involving mobile electrons and
trapped holes (a justification of this process is below), and the exponential term, which
was discussed above, was not derived from the OK model. As one can see from Eqs.
(54,55) we observed
decay but we did not find any convincing evidence for the
139
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
transition to the r-0’1 decay according to OK. Indeed, using an intersection method applied
in the OK model, it was hard to find a second straight line in a double-log scale of our
plots. The transients look curved at the tail as one can see in Figure 51. Therefore, we
invoked another mechanism developed by OK as well which resulted in a logarithmic
term in Eqs.(54.55). The model predicted that at high concentrations when the
bimolecular recombination dominates over the monomolecuiar recombination the
logarithmic time dependence for t > r^. (shown in Eqs.(54,55)) is required for a nonhomogeneously illuminated sample. Because the penetration depth of the 337 nm light
was substantially shorter than the thickness of the film in our experiments, there was a
gradient in the photo-generated carrier density. We considered it as a justification for
using the logarithmic term. The fit of the experimental data to these equations is
presented as the dashed lines in Figures 51, 52 which also give the values for a and Trcc .
The value of a was the same for S (A /1/2) and 5 /0, and also it was also the same for
both light intensities. Figures 57 and 58 indicate that all but the lowest light intensity
gave the same value of a . We attempted to explain the intensity independence o f a with
the OK model. According to the OK model, a increases with decreasing intensity in a
range of bimolecular recombination until the monomolecuiar recombination regime is
reached where a is intensity independent. The predicted changes in a were confirmed
experimentally in [46], Our experimental data suggested that in our experiments the
effect of saturation rather than monomolecuiar recombination was responsible for an
intensity independent a . On the other hand, because the intensities used were very high it
was unlikely that we avoided bimolecular recombination. The condition of saturation for
140
R e p ro du ced with permission o f the copyright owner. Further reproduction prohibited without permission.
0.70
0.68
-
0.66
-
0.64 0.62 -
0.56
0.56 0.54 0.52 0.50
1.000
0.010
0.100
n o c m a S z ed in ten sity (I/Iq ), re t. u n its
0.001
Figure 57. Intensity dependence of parameter a obtained for the power-law fit of the
time dependence o f the cavity quality factor change.
0.58 ------0 001
—
0.010
0.100
1 000
n o rm a liz e d intensity (1/1o ). rd .u n its
Figure 58. Intensity dependence of parameter a obtained for the power-law fit o f the
time dependence o f the shift o f the resonance frequency.
141
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
bimolecular recombination was not considered in the OK model. Supporting the
saturation concept, the 8(A/1/2) and 8 /0 transients gave approximately the same
within experimental error (8 psec and 5 psec respectively), and Figures 59 and 60
indicate that the value remains relatively constant over the intensity range employed,
although it might increase at the lowest intensities. The lack o f an intensity dependent
behavior observed for CdSe indicates the initial density of electrons involved in the
dispersive transport process was constant at all but, perhaps, the lowest light intensity
employed. Support for this conclusion is provided by extrapolation o f the power law data
to give the initial values of 5(A /1/2) and S/0 after the light pulse for each light intensity.
Figure 61 illustrates that the extrapolated values of 5(A/I/2) and 5 / 0, respectively, were
not dependent on intensity, except for the lowest intensity. Consequently, at the start of
the power law decay the total electron density involved in multiple trapping and
dispersive transport was about the same for all but the lowest light intensity because the
variation in light intensity was manifested only in the rapid non-resolved (apparently)
bimolecular recombination process involving mobile holes and mobile electrons. The
presence of the bimolecular recombination from the beginning of the kinetics could not
justify completely the use of the logarithmic term in Eq.(54,55) leaving the nature and the
form of the decay appearing after the knee point open. Unfortunately, the OK model did
not provided the relation between a and the width of the exponential distribution in the
case of bimolecular recombination. Because the OK model was developed solely for
exponential distribution it did not allow us to make an unambiguous conclusion about the
form of the distribution.
142
permission of the copyright owner. Further reproduction prohibited without permission.
=
10-5 -
0.001
0.100
0.010
1.000
normalized intensity (I/Iq). rel. units
Figure 59. Intensity dependence o f the apparent bimolecular recombination time for the
cavity quality factor change.
C
O
3O)'
O)
10-5
(cO
oco
e
o
4=
(A
<0
w
O
10-6
0.001
0.010
0.100
1.000
normalized intensity (I/y, rel. units
Figure 60. Intensity dependence o f the apparent bimolecular recombination time for the
shift of the resonance frequency.
143
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
N
5(Afi/2)
X
0
c>o%
so
*o
«T
0w) 10-6
0$Q.
I
1
i
/)
o
*
L
)
4
I
I
i
<!p
i
1
-SfQ
CO
1
*o
c
CO
"5
<,
60
w
£
0CO)
^3
10-7
CO
>
u
0}
k.
JO
CL
10-8
0.001
0.010
0.100
.000
normalized intensity (I/Iq). rel.units
Figure 61. Intensity dependence o f the extrapolated values of
) and 5/0 based
on the power law decay
144
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
As was mentioned earlier, the negative shift of the resonance frequency was
caused by trapped electrons, and the change of a cavity quality factor is proportional to
the conductivity of conduction band electrons. When the time dependence of the total
concentration of photogenerated carriers is determined, the drift mobility pd and the total
excess carrier density ANtot can be time dependent. If the motion of only conduction band
electrons is probed directly and the mobility does not change after illumination only the
free carrier density An is time dependent and p0 is the Hall mobility:
cr(f) = ANlol(t)iid(t)e = An(/)p0e .
(56)
Because the observed conductivity cr in microwave experiments is always due to
conduction band electrons, the time dependence of An can be followed. The resolution of
this time dependence is restricted by the instrumental time constant to - 100 ns as
discussed above. By this time equilibration has been achieved between conduction band
electrons and electrons in traps to a depth of 0.3 eV. This depth was estimated using the
relation between the release time t(E) and trap depth £:
T (£)=v '1exp(£/&7,)
(57)
in which v is the attempt-to-escape rate (-1012 Hz). By the end o f measurement (~ 10‘3
sec) all electrons in traps < 0.5 eV will be involved in the thermal equilibrium. For aA s2S3 the population of traps with depths > 0.4 eV was followed by the photoabsorption
technique [46], and it was implied that the population o f shallow traps was depleted very
rapidly. Our results indicated that (3 meV- 0.5 eV) traps in polycrystalline CdSe reached
thermal equilibrium at times faster than 60 nsec and remained populated at least until 1
ms.
145
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The function (a ta' 1) was used to fit the power-law decay. Fitting was based on
arbitrary cutting the curves to exclude the initial exponential decay and the final
logarithmic decay. One can see from Figure 51, 52 that what was left covered less than
one order of magnitude on a vertical scale. This allowed some variation in the slope o f
the power-law decay when choosing various truncated intervals.
Figure 62 shows the time dependence o f the cavity quality factor (bandwidth) and
the change of the resonance frequency obtained at +58° C. The behavior o f the cavity
quality factor and the frequency shift at this higher temperature was similar to but not
exactly the same as the lower temperature behavior.
In both cases, the difference
occurred after the break in the power law (/“'*) dependence at longer time. In this region
instead of the f lIn t/xrec dependence used for both parameters at the lower temperature,
we found the decays followed a t^ A dependence using the intersection method.
According to the OK model it would mean that bimolecular recombination transformed
to monomolecuiar with increasing temperature.
Summarizing the results of use of the OK model, we found the following: 1) the
OK model could not explain the same time dependence of the bandwidth change and the
shift o f the resonance frequency observed in our experiments and the issue of an
exponential distribution remained open; 2) the intensity dependence of the constant a
indicated the saturation regime under bimolecular recombination conditions; and 3) due
to this saturation
the OK model was not well suited for these experiments and no
estimation of the parameters of DOS could be done. As a result we developed our MT
simulation approach and the results are interpreted on this basis.
146
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
1.e-3 -
x
~ t H~ \ a = 0.65
1.e-4 -
- 1
a = 0.6
1.e-5 -
1.e-6
1.e-8
1.e-7
1.e-6
1.e-5
1.e-4
time, s
Figure 62. Changes in the cavity quality factor 5(A/1/2) (upper curve) and the shift in the
resonance frequency 8/0(lower curve) as function of time for light intensity
corresponding to I0 = 3.3 1020 electrons/cmJ; T = 331 K.
147
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
As was clearly shown in section 5.2 the same time dependence of both free and
trapped electron decays observed in the AMTMP experiments would require: 1) an
exponential distribution (ruling out all other forms) and 2) a narrow energy distribution
o f traps. The time range where similar behavior is observed will dictate the width o f the
distribution. Therefore, we removed some uncertainties inherent in the OK model.
However, due to computational difficulties, our simulations were restricted to the
monomolecuiar recombination between the free electron and the trapped hole. Even so.
we still could not obtain the desired relations between a and the parameters o f the
established exponential distribution. Since it has been shown previously [46] that at high
intensities (identified in the OK model as the bimolecular recombination regime) the
decay of excess electrons still shows two power-law regions, and the region before the
knee exhibits the intensity dependent slope, we believe that the addition o f the
bimolecular recombination term to the rate equations would lead to the same form for the
kinetics but with an intensity dependent a . Therefore, this term would not change the
form o f the distribution but only slopes of the transients.
The experimental data with simulation curves are shown on Figure 63 (7=300 K.
/=3.3xl020 cm'3), Figure 64 (7=300 K, /=1.8xl018 cm'3), Figure 65 (7=331 K, 7=3.3xl020
cm°). One can see from the parameters from the plots that all analyzed curves (high and
intermediate intensity at 300 K and high intensity at 331 K) could be described by a very
narrow energy distribution with less than 10% deviation using the MT model with a
monomolecuiar recombination term. The width o f the DOS varies from 400 K to 440 K
(0.034 to 0.038 eV in the energy scale). The simulated curves reproduce the experimental
148
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.e-2
An(t)
An(t) and 6e*(t), arb. units
1.e-3
\ MT simulation: T0=420 K (a=0.71), Tr=4x10-° s
1 e-4
■ MT simulation: Tg=440 K (a=0.68), t r=9x10‘
1.e-5
1.e-6
1.e-7
1.e-6
1.e-5
1.e-4
1.e-3
time, s
Figure 63. Experimental and simulated curves o f An(t) and 5s' (t) obtained for
exponential distribution of localized states, T=300 K, I=I0-
149
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
» ■ i i- i i i 11
.
iiiii
1.e-3
1 pm CdSe, 300 K, 1=0.005 l0
• MT simulation: Tq=430 K (a=0.7), t r=1.5x10"^ s
tn
exponential DOS
=
xi
w
(0
1.e-4
“ec©
o
■Co
(0
MT simulation: Tq=430 K (a=0.7), t^IxIO"® s
f 1-e-5
1.e-6
i i i1
1
1
1.e-7
1.e-6
i ■r i T ' r m , 1
1.e-5
1.e-4
1.e-3
time, s
Figure 64. Experimental and simulated curves of A/j(r) and 5s'(t) obtained for
exponential distribution of localized states, T=300 K, 1=0.005 I0.
150
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.e-2
1pm C dSe, 331 K, l=l
\ An(t)
An(t) and 8e'(t), arb. units
1.e-3
ITsimulation: Tq=400 K(a=0.83), Tr=6x1 O'
exponential DOS
1.e-4
1.e-5
MT simulation: Tg=410 K (a=0.81), Tr=2.15x10"' s
1.e-6
1.e-8
1.e-7
1.e-5
1.e-6
1.e-4
1.e-3
time, s
Figure 65. Experimental and simulated curves of An{t) and 5s'(t) obtained for
exponential distribution of localized states, T=331 K, I=I0.
151
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ones well once one makes allowances for the small dynamic range and for the
discontinuities o f the curves obtained as a result of merging data measured at different
oscilloscope scales. The subtraction o f the base line when it was comparable with the
signal in the tail region led to the systematic deviation in the tail as can be seen clearly in
Figure 63, 64. Because a should increase with decreasing intensity, reaching an intensity
independence under monomolecular recombination, we can estimate the upper limit of
the width of the distribution. Using the linear relation, a=T/T0 and the average value
obtained in our experiments T0 « 410 K, we can predict that the real distribution must be
narrower than 0.035 eV. Addressing the issue of the apparent decrease o f a observed at
the lowest intensity (Figures 57. 58) we would like to mention that the error (-20%)
associated with this value brings the upper limit to values close to 0.62, i.e. to intensity
independent values.
6.1.3. Summary
Using second perturbation theory relations we reported high values of the real part
o f the dielectric constant of CdSe after the laser excitation. Since all detectable traps
appeared to be saturated those values are consistent with values obtained from the
Clausius-Mossotti relation in the vicinity of metal-insulator transition. Light intensity
studies indicated that the all detectable traps were saturated so that the average density of
these traps could be determined using the polarisability estimations. The analysis of the
experimental data for this sample o f CdSe was done in two ways: the OK model and our
MT simulation approach. The OK model could not explain the similar decay observed for
152
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the real and imaginary parts of the complex dielectric constant. In our MT simulation
approach, this behavior was related to the form of the distribution. While in both
approaches the effect of the saturation under conditions of bimolecular recombination
could not be confirmed quantitatively, the MT simulation allowed us to determine the
form of the distribution present in the sample as an exponential. It also allowed us to
estimate the upper limit o f the real distribution as 0.035 eV. Therefore, the MT simulation
provided a more rigorous treatment of both bandwidth changes and the shift o f the
resonance frequency obtained by AMTMP. In general, the present results for thin film
CdSe indicate that dispersive transport occurred for the photogenerated carriers.
According to our knowledge this phenomenon has been never reported before.
6.2. CdSe (#2)
6.2.1 Experimental
The sample consisted of a thin (1pm) n-type film (2x10mm2) of CdSe deposited
on a strip (3x60x1 mm3) of quartz. The sample was prepared by CVD technique with
substrate temperature of 70° C. After deposition, the sample was annealed for 4 hours at
350°C in nitrogen. The details of a fabrication of the film can be found elsewhere [174].
The Hall measurements gave the following values for the dark mobility of the sample and
the equilibrium electron concentration: p 0=27.8 cm2V '‘s'1 and n0 = 7x1012 cm'3 .
The sample was studied with setup 2. The sample was illuminated with Nd:YAG
laser at A=355 nm with a pulse duration ~ 6 ns. The incident laser power was ~4 mJ/pulse
( 1016photons/pulse). Taking the penetration depth at 355 nm for CdSe (3.5x10° cm) and
153
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the rectangular profile of the laser beam (0.17x0.55 cm2), the active sample volume was
calculated to be 3.27x1 O'6 cm3. Considering the quantum yield for photocarriers
generation to be 1 this would give a very high initial excess electron concentration (~ 10
cm'3). As is shown below in this section, this concentration dropped very fast during the
non-resolved process by about 5 orders of magnitude. Therefore, the typical electron
concentration we worked with most likely did not exceed IO17 cm"3.
The cavity with a piece of waveguide was pumped out to allow low temperature
measurements. A temperature of the cavity itself was stabilized by a water jacket and kept
at 22±0.4°C for all measurements. For low temperature measurements, evaporated liquid
nitrogen passed through temperature controlled heater into a specially designed double­
wall quartz tube which accommodated the sample. The space between tube walls was
evacuated to avoid condensation on outer walls. This tube was inserted into the
microwave cavity. The temperature controller was a 1600 series microprocessor based
controller by LOVE Controls Corp. This arrangement provided temperatures between
123±0.1 K. and 300±0.1 K. The temperature step was chosen to be 10 K. High
temperature measurements were done by heating a large copper block to desired
temperature and blowing dried, compressed air through it and, subsequently, into the
quartz tube. The highest temperature used was 358±0.1 K. The light induced changes in
the resonance curve were obtained as discussed earlier, and the final kinetics of the cavity
quality factor changes and the shift of the resonance frequency were extracted by fitting
the constructed, light-induced difference signal to a difference between two Lorentz
curves. The parameters o f the dark Lorentz resonance curve were measured prior the light
154
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
exposure. Light induced changes in the resonance curve were extracted by the fitting
program using Levenberg-Marquardt 3-parameter method. The experiment and the raw
data processing were computerized.
O
I
The kinetics were measured in 123-358 K temperature range and on 10 - 10’ s
time scale. Intensity measurements were done at 298 K and 123 K. The intensity was
varied by 2 orders of magnitude with neutral density filters. Below, we discuss the
behavior of the bandwidth changes and the shift of the resonance frequency at three
selected temperatures: 300 K, 123 K, 358 K.
6.2.2. Behavior at 300 K.
The measured kinetics of the bandwidth changes and the shift of the resonance
frequency were used as input data to calculate changes in the complex dielectric constant
according to Eq.( 18) from Chapter 2. The results of the calculation corresponding to the
m axim um laser intensity and 300 K are shown in Figure 66. We found the following
regions in the8s "(0 decay: a fast initial exponential decay with a time constant close to
the time constant of the cavity (~ 80 ns) followed by a power law decay
(a=0.41)
which was terminated by what appears to be a slow exponential decay (2.1 ms) buried in
the noise. T he8e'(0 decayed more slowly but revealed a similar exponential decay (2.6
ms) at the tail region. The parameters were obtained by fitting the entire region except the
ta ilo f5 s " (/) to
uexp(—f/x)+c/(l+(60“-1)
155
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
(57)
100.0000
i
L.J
m
ill
—
I
I
» i n
i li_
I
1
. j L I III
I
.l. ■ m i l l
1 |im CdSe (#2)
300 K, max.int.
5s"
10.0000
B
3
‘c
1.0000
q5
35
0.1000
CM
TJ
c
CO
£
-M
oo
CM
0.0100
25s'
TJ
C
re
“to
CO
03
O
0.0010
.lB -4
0.000
0.0001
0.004
i i . i i i i ----------- 1— i
0.008
0.012
■ i ■ 1111
1
Figure 66. Changes in the real and the imaginary parts o f the dielectric constant for 1pm
CdSe (#2), T=300 K, I=I0. The inset: the same curves in a semi-log scale.
156
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
with TableCurve (Jandel). The tail of 5s "(t) as well as 8s '(t) were fitted to exponential
decays. What appears to be a sharp transition for 5 s'( 0 at ~ 3 ms is a result of some
merging problems when parts o f the decay measured at various oscilloscope scales were
spliced into a single file. It also explains some minor steps frequently observed on some
other decays. The inset on Figure 66 shows the same kinetics on a semi-log scale.
Using the relation between 8e" and 5cr from Eq. (7) and assuming that the
mobility of excess electrons equals the mobility o f equilibrium electrons p0, we obtained
the plot of excess electron concentration shown in Figure 67. Recalling that the maximum
initial concentration could be as high as 1022 cm'3, one can see that the fast unresolved
process which is indicated by the initial fast exponential decay, is responsible for
reducing drastically the initial electron concentration down to ~ 1016 cm'3 . For example,
it could involve a direct free electron-hole or Auger recombination. The dark equilibrium
concentration is also shown on Figure 67 as a straight line. Therefore, the lower detection
limit of the excess electron concentration can be expressed as A/zmin~n0. The excess
electron concentration from Figure 67 was used to calculate the dynamics of the Fermi
level movement after the excitation according to
(58)
Eh, = - k T \ n ^ where EFn is the electron Fermi level, Nc= l.17x10
to
1
cm' is the density of states in the
conduction band edge. The resulting kinetics along with the dark position of equilibrium
Fermi level are shown in Figure 68. One can expect from this figure that at times « 1 0 '7 s
the Fermi level could move to the conduction band leading to degeneracy o f the
157
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
l.e+17
1 pm CdSe (#2)
300 K. max.int
1 e+16
c!
<
i.e+14
equilibrium concentration
l.e+13
i.e+12
le - 6
i.e-7 ’
1 e-5
1 e-4
1 e-2
ts
Figure 67. The decay of the excess electron concentration for 1nm CdSe (#2), T=300 K,
I= Io -
0.40
1 Mm CdSe (#2)
300 K. maxint
0.35
equilibrium electron Fermi level position
0.30
0.25
0.20
0.15
0.10
0.05
1 e-7
l.e-4
1.8-5
t. s
Figure 68. The kinetics of the movement of the electron Fermi level, EFn, after excitation
for l(im CdSe (#2), T=300 K, I=I0.
158
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
sample. The resolved movement o f the Fermi level covers ~ 0.2 eV, and these dynamics
will be used later when discussing results.
To explore the behavior of the observed bandwidth change and the shift of the
resonance frequency, we did intensity dependence measurements at 300 K. The intensity
was varied with neutral density filters with a minimum intensity corresponding to 0.01 I0.
The results for the bandwidth change are shown in Figure 69 on a double-log scale. The
results for the shift are shown in Figure 70 on a double-log scale. The resultant
dependence of the time constant for the fast exponential decay of the bandwidth change is
shown in Figure 71. As was expected the processes responsible for this component was
too fast to be resolved but it slowed down with decreasing intensity. It might be due to
fast direct electron-hole recombination which is known to decrease (resulting in increase
of the time constant) with decreasing intensity. In the absence o f the sufficient time
resolution, the intensity dependence of the pre-factor of the fast exponential decay would
give an indication about the order of the process. This dependence shown on Figure 72
demonstrates a dependence ~ (I/T0)°63 supporting the suggestion that the bimolecular
electron-hole recombination might be consistent with this behavior. Instead of the
hyperbolic decay characterized by the fast lifetime (which is known to be a feature of
bimolecular recombination) we observed rather a product of the convolution of this
hyperbolic decay with instrumental response function which resulted in the fast
exponential decay.
One o f points o f the major interest in the intensity dependence measurements was
the behavior o f a from the power low decay and the time constant o f the final slow
159
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.e-3
1 pm C d S e (# 2 ). T = 3 00 K
.e-4 IM
X
o
c*
<
1= 1,
50
e-5 -
1= 0 . 0 1 1,
e-6
1.e-S
1 .e-7
1.e-6
1.e-5
1.e-4
1.e-3
1.e-2
t s
Figure 69. The intensity dependence of 8(Af[/2) for 1pm CdSe(#2), T=300 K.
1e+2 -r
i.e + i ->
i.e+0
! 1iim CdSe (#2). Ts300 K
Figure 70. The intensity dependence of -5f0 for 1pm CdSe(#2), T=300 K. Intensities are
the same as on Figure 68. Curves are scaled.
160
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
normalizedintensity(l/l„), rel.units
Figure 71. Intensity dependence of the time constant of the fast exponential decay for the
bandwidth change, 1pm CdSe (#2), T=300 K.
1 .e-2
2
1 .e-3
e -4
1
1 .e-5 i —
0.01
0.10
1.00
norm alized intensity (1 /y . rel.units
Figure 72. Intensity dependence of the pre-factor of the initial exponential decay for the
bandwidth change, 1pm CdSe (#2), T=300 K.
161
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
exponential decays in the bandwidth change and frequency shift. The former is shown in
Figure 73. One can see from this plot that there is a gradual increase of a with decreasing
the intensity. The very slow increase in a at high intensities can be a result of the
saturation effect. This event is quite possible if one recalls that the initial concentration of
excess electrons was very high at all studied intensities.
We were able to resolve the slow exponential decay at three intensities for the
bandwidth and at four intensities for the frequency shift. The level of noise prevented
determination of the exponential part at other intensities. The time constants extracted
from these decays are shown in Figure 74. One can see that the corresponding time
constants are very close to one another and are almost temperature independent. The
averaged time constant is 2.4 ms. It is also interesting to note that the time of appearance
of this exponential decay also falls into the range between 2 and 3 ms. The pre-factor of
the slow exponential decay for the bandwidth was intensity insensitive and the one for the
shift decreased by a factor of 3 when the intensity decreased by one order of magnitude.
We are not going to discuss to the intensity behavior of two other parameters
from the power law decay : c which would give the concentration of the carriers at the
instant t=0 and b which would be interpreted as the time constant for the processes
involved in the power law decay. We found that those parameters are not completely
independent in fitting because both affect the total amplitude o f the power law decay. A
more detailed interpretation of these parameters would require a lower noise level and
preferably
a
larger
dynamic
range
of
the
power
162
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
law
decay.
normaized intensity (l/Ig), rel. units
Figure 73. Intensity dependence o f the parameter a from the power law decay in the
bandwidth change, 1pm CdSe (#2), T=300 K.
0I
C
c
©
oa. -
S
5
To5
£
C
8
<D
E
*
O
0.001
bandwidth change
shift of the resonance frequency
—
0.1
10
normafized intensity (I/1q). rel units
Figure 74. Intensity dependence of the time constant of the slow exponential decay for
5(An/2) and 8f0,1pm CdSe (#2), T=300 K.
163
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6.23. Behavior at 123 K.
The kinetics of 5 s "(0 and 5 s '( 0 obtained at 123 K in the same way as discussed
at 300 K are shown in Figure 75. Except for some increase in the absolute amplitude of
kinetics the same features as at 300 K were observed: 1) an initial exponential decay
followed by the power law decay eventually truncated with an exponential decay for
5 s ”(t) ; 2) a much slower power-law decay of 5 s '(/) revealing the same exponential
decay at the tail region. The slow exponential decays have time constants (4.1 ms for
5s "(f) and 5.4 ms for 5 s '(i )) which are comparable with the 2.4 ms decays at 300 K. To
estimate An(t) we had to use the value o f the mobility at 300 K to calculate this value due
to the lack of experimental values for 123 K. Normally one would expect an increase of
the mobility by approximately factor of 2-3 at low temperature due to decrease in
scattering. The values of An(t) shown in Figure 76 indicate a slight increase in the
concentration of excess electrons compare with 300 K. Since the initial intensity was kept
the same in both cases, this increase can be attributed to a decreasing rate o f processes
which are responsible for the ultimate removal of excess electrons from the conduction
band. The increased time constant (220 ns) of the initial fast exponential decay for
Se "(f) and the decreased pre-factor (2x103 at 123 K and 4x103 at 300 K) are consistent
with a decreasing rate. While the process was still too fast to be completely resolved, an
increased time constant points at the direct electron-hole recombination as a possible
candidate for this process. Indeed, it was shown in [184] that the luminescence observed
for CdSe crystals decays more slowly and has a lower intensity at low temperatures. We
would like to stress again that the fast exponential decay is a product of the convolution
164
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T=123 K, 1 fim CdSe (#2)
100.000
10.000
1.000
to
CM
0.100
00
28e‘
CM
o©
0.010
0.000
0.005
0.010
time, s
0.001
1.e-7
1.e-6
1.e-5
0.015
0.020
______________
1.e-4
t, s
1.e-3
1 . 6-2
1.6-1
Figure 75. Changes in the real and the imaginary parts of the dielectric constant for 1|im
CdSe (#2), T=123 K, I=I0. The inset: the same curves in a semi-log scale.
165
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T=123 K, 1 nm CdSe (#2)
.e+17
.e+16
1
1.e+15
o
c
<
.e+14
1.e+13
1.e+12
1.e-8
1.e-7
1.e-6
1.e-5
1.e-4
1.e-3
1.e-2
1.e-1
t, s
Figure 76. The decay o f the excess electron concentration for 1fjm CdSe (#2), T=123 K,
I=Io-
166
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
o f the assumed hyperbolic decay (due to bimolecular recombination) and the exponential
instrumental response function.
The intensity dependence of the pre-factor for the fast exponential decay for
8s " (0 is shown in Figure 77. The dependence follows a square root law very closely
which is in agreement with the 300 K. data. It indicates that this fast process does not
change its nature with temperature. The intensity dependence of the time constant o f this
fast exponential decay is shown in Figure 78. The time constant indicates that the process
is close to being resolved. Due to scattering it is hard to conclude unambiguously about a
direction in which the time constant changes with intensity. Another problem is that the
contribution o f the fast exponential decay to the whole signal was very small at the two
lowest intensities.
The intensity dependence of a is shown in Figure 79. Comparing it with the
corresponding dependence at 300 K one can see that the trend is the same: an increase of
a with a decrease in intensity. One can observe the same relatively slow increase at high
intensities. A difference is that the values o f a are bigger than those at corresponding
intensities at 300 K.
The intensity dependence of the time constant of the slow exponential decay for
the bandwidth and the shift is shown in Figure 80. One can see that all values are very
close and intensity independent. The average time constant is close to 4 ms (compare
with 2.4 ms at 300 K.). Again due to noise limitations we were able to trace the bandwidth
only over one order of magnitude in intensity change. The shift was less noisy and we
followed the slow exponential decay over two orders of magnitude in intensity change.
167
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T=123 K. 1nm CdSe (#2)
■i
0.01
0.10
1.00
normalized intensity (l/y. rel.units
Figure 77. Intensity dependence of the pre-factor of the initial exponential decay for the
bandwidth change, 1pm CdSe (#2), T=123 K.
320
1=123 K. 1pm CdSe (#2)
300
280 -j
260 -i
r
a 240
220 A
200
•
0.01
0.10
1.00
normalized intensity (t/lg), rel.units
Figure 78. Intensity dependence of the time constant o f the fast exponential decay for the
bandwidth change, 1pm CdSe (#2), T=123 K.
168
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.01
0.10
1.00
normalized intensity (1/1,), rel.units
Figure 79. Intensity dependence o f the parameter a from the power law decay in the
bandwidth change. 1pm CdSe (#2), T=123 K.
0.010
T=123 K. 1iim C dSe (#2) j
1!
a
I
X
o
i>
©
5
9
i
©
E
•
O
bandwidtti change
shift of the resonance frequency
0.001
0.01
0.10
1.00
nonnalized intensity (1/1,). rel.units
Figure 80. Intensity dependence o f the time constant of the slow exponential decay for
8(AfI/2) and Sf0, 1pm CdSe (#2), T=123 K.
169
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6.2.4. Behavior at 358 K.
The kinetics of 5e"(t) and 8c'(t) for the highest temperature in the temperature
range used, 358 K are shown in Figure 81. One can see that 5 e "(0 still follows the
power law (~ 0.46) decay following the fast exponential decay. The total amplitude o f the
signal decreased slightly but the relative contribtrtion o f the fast process increased
significantly compared with the 300 K results. This indicates that the process which
reduces the concentration of excess electrons proceeds slightly more effectively at higher
temperatures. The time constant o f the fast exponential decay is close to the instrumental
x indicating that the process is not going slower (as at 123 K) but may be faster.
Therefore, the power-law part o f 5 e"(0 starts at lower electron concentration reducing
the dynamic range o f the observation. Due to this fact and to the higher level of noise, the
slow exponential component could not be observed at the tail. It seems that extending the
time range of measurements to ~I ms, we would be able to get the final part o f the 6s '(0
decay (Figure 81). The curve of 6 s '(/) starts to bend down indicating a possible presence
of the slow exponential decay. The projected exponential behavior is shown in Figure 81
by the dashed line. The estimation of the excess electron concentration for 358 K was
done in the same way as at 300 K. and 123 K. The results are plotted in Figure 82. Indeed,
15
i
the starting electron concentration for the power law decay is less than 10 cm which
explains the smaller dynamic range.
170
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
“ ■«*
t
» » « ■«»■*
»
»
I
«
*
. . . .
.m l
.
. . .....I
1 nm CdSe (#2),T= 358 K
10.000
-
1.000
c
3
*Co
(0
0.100
0.010
0.001
time, s
Figure 81. Changes in the real and the imaginary parts of the dielectric constant for 1pm
CdSe (#2), T=358 K, 1=I0.
1nm CdSe (#2). T=358 K
1.e+16 -
5 1.e+14 -
1.e+13 -
1.e+12
1.e-7
1.e-5
1.e-6
1.e-4
1.e-3
I s
Figure 82. The decay o f the excess electron concentration for 1pm CdSe (#2), T=358 K,
Mo171
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
6.2.5. Temperature dependence of a.
Combining values of a from all temperatures for the maximum intensity resulted
in the following plot (Figure 83). The temperature dependence of a shows an initial
decrease with increasing temperature (123-200 K) with a plateau region (210 - 270 K.)
followed by a subsequent increase (280 - 358 K).
The explanation of experimental data is based on the fact that we are dealing with
some form of distribution o f localized states involved in a multiple trapping process. We
did extensive simulation using various forms of distributions described in Chapter 5. This
allowed us to identify some unique features of the distributions which were manifested
both in An(t) and 8s '( f ) . First we would like to concentrate on the power law behavior
and address the final slow exponential decay below. The forms of experimental decays of
An(t) and 5e'(f) allowed us to rule out exponential, rectangular and linear distributions of
localized states as possible distributions in the sample. Indeed, based on results presented
in Chapter 5 rectangular and linear distributions would produce power law decays only at
limited conditions giving values of a not exceeding 0.3 at 300 K. Our values of a were in
0.38 - 0.54 range at the maximum intensity. On the other hand, exponential DOS seems
to produce any value of a depending on the characteristic temperature T0. However, An(t)
and 8e'(f) decayed differently starting from the beginning of the kinetics, a possibility of
having a narrow exponential DOS was ruled out. Recall from the previous, that the
change in the real part of the dielectric constant is proportional to the trapped electron
concentration. For a relatively wide distribution, electrons accumulated in deeper traps
would cause8s'(f) to diverge from An(t) at longer times allowing an initial similar decay
172
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.56
1pm C dSe (#2), l=l
0.54
0.52
0.50
0.48
a 0.46
0.44
0.42
0.40
0.38
0.36
100
150
200
250
300
350
400
T, K
Figure 83. Temperature dependence of the parameter a extracted from the bandwidth
change, 1pm CdSe (#2), I=I0.
173
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
in both kinetics. We did not see even a small region where both kinetics followed the
same time dependence. Also, lowering the temperature should broaden the time range o f
the similar decays due to freezing out contributions from deeply trapped electrons. This
should facilitate the observation of a similar decay for the broad distribution. However,
we did not see this behavior at low temperatures.
Now we would like to direct attention to the second process which terminated the
power law decay, the slow exponential decay. For continuous types of DOS one will
always have the power law decay if the distribution is '‘infinite” on the time scale of the
experiment, i.e. the release from the last level in the distribution is not observed within
experimental time frame. (In other words, the termination of the DOS will not manifest
itself in the experiment). Our simulations showed that in the exponential DOS one could
not terminate the power law decay by an exponential decay at about 2 ms at 300 K. The
reason is the following. For an exponential DOS in order to get this termination, the knee
o f transition from t a~x to / ‘1_a must be moved to - 2 ms which requires increasing the
recombination time to values about 10° s. Truncating the DOS with the last level having
the release time of about 3 ms will not result in a power law/exponential decay transition
but in a power law/constant transition. Indeed, when the recombination time is bigger
than the release time from the last level, the excess electrons released from the last level
will stay in the conduction band without decaying until the recombination occurs.
Therefore, while it is possible in general to obtain the observed value of a with an
exponential DOS it is not possible: 1) to obtain the behavior observed for8e'(/) and An(t);
2) to get the desired truncation of the power law decay at a specific time for An(t).
174
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
6.2.6. Gaussian distribution based modeL
A peaked Gaussian-like distribution gave very close agreement between the
experimental transients and the simulation results. In contrast to the exponential DOS
there is no severe limitation on allowable values of the recombination time due to the
peaked shape of Gaussian DOS. Our best results based on visual inspection, are shown
for 5 e '(0 and And) in Figure 84 for 123 K. The parameters of the simulation are shown
in the figure as well. One can see that the simulated curves follow the experimental ones
very closely. Because the experimental kinetics of And) and 5 e '(0 can be obtained from
the bandwidth change and the shift of resonance frequency, respectively, it is possible to
match not only the shape but the absolute amplitude by choosing the appropriate
parameters of the simulation. However, this step would require more justification for the
choice o f such parameters as the electron capture cross-section (S'), the pre-factor g0 in
the DOS expression. Therefore, at the present stage we restrict ourselves to the analysis
of the form of the decays only.
The parameters of simulation show that the finite number of levels (m=12) located
between 0.13 and 0.24 eV with a step 0.01 eV is enough to reproduce the decays. This
restriction is imposed by the finite time range of the experiment along with the
temperature. In our experiments the time range was approximately 10"-10 ~ s at all
temperatures. As was shown in Chapter 5 only the levels which have their release times
falling into the time range of experiment will manifest themselves in the kinetics. The
results o f the calculation of the first (Etl), last (E ^) levels and the effective width probed
175
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
■ i i 11i n i
i
i ■m i l l
i i ' n i'il
...................'I____ i i i
mul—
i i i m d ------ 1 i 11 m il—
-3 -
5e'(t)
log (A n ( t) , Sc'(t)) (arb. units)
i_ i_
Vi=12, A=0.01eV, E|=0.12 eV, Eq=0.165 eV, Tq=400 K, T=123 K
\ g^loV* cm*3 eV"1, An0=1013 cm'3, S=10- '5 cm2. tr=10'7 s
(scaled)
-4 MT simulation
-5 -
"6 1 T
-
1.e+17
-
1.e+16
-7 -4
0.16
E eV
0.20
-8
Figure 84. Experimental and simulated with Gaussian DOS decays of An(t) and
8 s'(/) for 1pm CdSe (#2), T=123 K, Mo-
176
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(E^-E^) are summarized in a Table 6.2.1 for selected temperatures. One can see that with
increasing temperature the width extends and the trap levels move to deeper energies.
Table 2. Release times for specific levels at selected temperatures
T, K
kT, eV
Eh, eV
Et2 , eV
Etj-Ej,, eV
123
0.0106
0.122
0.244
0.122
203
0.0175
0.201
0.402
0.201
300
0.0258
0.297
0.594
0.297
358
0.0308
0.354
0.709
0.354
This clearly shows that while working in the same time range only a limited part
o f the real distribution is probed at a particular temperature. Keeping this in mind, one
would expect that once the parameters of the distribution at 123 K (Figure
established from comparison to simulation
( g o = 1 0 18
cm*J eV'1,
T 0= 4 0 0
K,
84)
E 0= 0 .1 6 5
are
eV)
the measurements at other temperatures would confirm those parameters with some
refinement by probing missed parts of the distribution. (Those parameters are the only
parameters which determine the slope of the power law decay. For Gaussian DOS the
characteristic temperature T0 is related to the parameter a (a ac T/T0) but not directly as
for an exponential DOS (a =77T0).) But this expectation turned out to not be the case for
real measurements as will be shown below.
The simulation results for 300 K are shown in Figure 85. The degree of
closeness for the shift is not as good as at 123 K. The simulation could provide better
results if we were able to fit rather than manually adjust parameters. However, we believe
177
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
» i Lt i m i l
i
i i i m il
i
i
i i nul
i
i i m ill
i
t i n ml
_l i
ill
mi
1 I i
l llll
A=0.03 eV, g0=1018 eV'1 cnrr3, An0=1019 cirr3, t =10'8 s'
m=8, Tq=1700 K, Eq=0.35 eV, Ep0.3 eV, S=1Q-15 cm2
-2 -
An(t)x10
MT simulation
Gaussian DOS
oj
1.e+18
-6
0.30
-8
0.35
0.40 0.45
E, eV
0.50
1.e+17
0.55
— I— 1~ I I I . 111------- 1— 1 I T I 111|-------- 1— I I I 11 l lj---------- 1— I I I I 11 i |
-9
-8
-7
-6
-5
I
rT T T T T T j
-4
I
TTTTTT7]
-3
I
T T T T T T T J1
-2
log (t)
Figure 85. Experimental and simulated with Gaussian DOS decays o f An(t) and 8s'(t) for
lp m CdSe (#2), T=300 K, I=I0.
178
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
that this is only part of the problem. One can see from the parameters listed on Figure 85.
that the distribution parameters for 300 K ( T0=1700 K, E0=0.35 eV) are not even close to
the corresponding values for 123 K (T0=400 K. E0=0.165 eV). For comparison, the DOS
obtained for 300 K is plotted along with the DOS obtained for 123 K in Figure 86. One
can see that with increasing temperature the peak becomes wider and shifts to larger
energies. Below we explain why measurements at various temperatures will always give
different parameters for Gaussian-Iike peaked DOS.
Consider the Gaussian DOS (g(E)) with the following parameters: go=1018 cm'3
eV'1, To=1700 K, E0=0.35 eV. Because we suggest a priori that this DOS well describes
the material, we call it the real DOS. This DOS describes thestates in the bandgap. but
generally speaking not all those states will be available for multipletrapping processes as
described by our model in Chapter 5. At a specific temperature (say, 300 K.) some states
will be occupied at equilibrium (i.e. before illumination of the sample). The density of
those occupied at the equilibrium levels (goc(E)) is determined as a product o f the density
of localized states (g(E)) and the occupation probability/f£y:
g„c (£ )=&oexP(- (( E -E 0)/kT)2) f ( E )
(59)
where /(£ )= l/(e x p (-£ + £ /„)A t7 + l) is the Fermi function describing the occupation
probability. The density of
states (g^E)) available for trapping electrons can be
calculated as,
g „ ( E) =g( E) - goc(E)
(60)
All three curves (g(E), goc(E),gav(E)) are shown on Figure 87. For calculations, we used
7=300 K and EFn=0.31 eV (which is the equilibrium electron Fermi level position at 300
179
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.e+18 -
300 K
123 K
m
>0
«? 1.e+17 H
E
o
d)
9
-e
1.e+16 0.1
0.2
0.3
0.4
0.5
0.6
E, eV
Figure 86. Parts of Gaussian distributions obtained from 123 K. and 300 K kinetics using
the simulation approach, 1 pm CdSe (#2), I=I0.
180
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
K obtained from the dark equilibrium electron concentration). The conclusion is quite
obvious: the density o f available states is not even close to the real DOS. Moreover, it can
not be even described as a Gaussian DOS. It was confirmed by fitting g^ E ) to the
Gaussian DOS. The fitted curve is also shown in Figure 87 with parameters on the plot.
The parameters (go= 6 x l0 17 cm'3 eV'1, T0=1008 K, E0=0-26 eV) are far away from those
for the real DOS. The shape of gm(E) suggests that the left (low-energy) side continues to
follow the low-energy side of g(E) while the right (high-energy) part is truncated
smoothly and distorted from the Gaussian DOS by occupied states at the specific T and
the specific
To get an idea what would be the deviation in the range o f interest which
was probed in our experiments at 300 K, we chose the energy range 0.3 - 0.6 eV and
plotted the same curves in a semi-log scale in Figure 88. One can see a constant deviation
of gaviE) from the fitted curve. It implies that one should expect to have the same sort of
deviation either in the bandwidth change or shift of the resonance frequency when
attempting to approximate them with An(t) or 5 s'(0 using a Gaussian form for DOS. In
our experiments the deviation between the shift of the resonance frequency and simulated
5e '( 0 at 300 K was appreciable.
To determine the temperature behavior of these deviations we repeated the same
calculations for 123 K, the lowest temperature used in our experiments. We used the
same parameters of the "real” Gaussian DOS. Because the position of the electron Fermi
level was unknown for 123 K we assumed that it moved closer to the conduction band
(which can be considered as a general trend in semiconductors [3]) to the position
Ef„=0.2
eV.
At
high
temperatures
most
semiconductors
181
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
move
real Gaussian DOS: gg=10^®, E g = 0 . 3 5 eV, T g = 1 7 0 0 K
1.e+18 - E p n = 0 .3 1 eV, T = 3 0 0
fitted to Gaussian DOS: g0=6x1017
E0=0.26 eV, T0=1008 K
real DOS
8.e+17 -
"o 6.e+17 -
occupied states at T and Epn
cn
E
o
d)
states available
4.e+17 fitted to Gaussian DOS
2.e+17 -
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
E, eV
Figure 87. Real Gaussian DOS, occupied states determined by the Fermi function, states
available and fitting to Gaussian DOS (T=300 K).
182
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
i-A® Ewo '6
.e+18
.e+17
.e+16
.e+15
.e+14
.e+13
.e+12
.e+11
.e+10
.e+9
.e+8
.e+7
.e+6
.e+5
.e+4
.e+3
.e+2
.e+1
i
1
real DOS
_j occupied states at T and Epn
“i
•-
" i
“1
1
“1
“1
1
1
1
T
states available
j
^ ,
fitted to Gaussian DOS
EFn=0.31 eV, T=300 K
real Gaussian DOS: gQ=101®, Eg=0.35 eV, Tg=1700 K
fitted to Gaussian DOS: go=6x1017
i En= 0.2 6 eV ,T = 1 0 0 8 K
1
■
0.3
0.4
0.5
0.6
E, eV
Figure 88. Curves from Figure 87 plotted in a semi-log scale (T—300 K).
183
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
closer to the intrinsic range where the position of the Fermi level is in the middle of the
bandgap. As the temperature is lowered the ionization of impurities begins to play a
major role in determining the electrical properties o f semiconductor. The Fermi level
moves closer to the donor level (we consider n-type), i.e. it shifts up to the conduction
band. Therefore, it justifies our assumption on a direction of Fermi level changes. On the
other hand there are also very peculiar temperature dependencies of the equilibrium
electron concentration observed in polycrystalline semiconductors and for CdSe in
particular [181]. This anomalous temperature dependence shows up as a sudden drop in
the concentration within 200-300 K. After a drop by as much as 3 orders of magnitude,
the concentration increased again with subsequent temperature increase. Our 300 K
experimental data would correspond to the minimum in the observed [181] temperature
dependence and the move in any direction would increase the electron concentration
moving the Fermi level closer to the conduction band (according to our calculations
based on data from [181]). Consequently, we expect a similar behavior in our CdSe thin
film. We were not able to do temperature dependent Hall measurements at the time.
Therefore, with the position of the Fermi level at 0.21, eV calculated curves (
(g(E), gocffl.gavfE) and the fitted curve) are shown in Figure 89. As might be expected
the curve corresponding to the density of available states narrowed and moved to lower
energies (go=2xl017 cm'3 eV'1, T0=582 K, E0=0.17 eV). We plotted the same curves in a
semi-log scale in the energy range probed at 123 K (0.1-0.24 eV) (Figure 90). The degree
o f closeness of gm(E) to a Gaussian DOS is better than at 300 K. Indeed, because the
energy range probed at 123 K (-0.12 eV) is narrower than the one at 300 K (-0.3 eV) it
184
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
real G aussian DOS: g0=10 , Eq=0.35 eV, T q=1700 K
1.e+18 - Epn=0.2 eV, T=123 tO->.
/
\ fitted to G aussian DOS: go=2x1017
:0=0.17 eV, Tq=582 K
real DO:
8.e+17
Id 6.e+17
CO
I
E
o
bi
4.e+17
I occupied states at T and Ei
itates available
2.e+17
fitted to Gaussian DOS
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
E, eV
Figure 89. Real Gaussian DOS, occupied states determined by the Fermi function, states
available and fitting to Gaussian DOS (T=123 K).
185
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
.e+18
.e+17
.e+16
.e+15
.e+ 14
.e+13
.e+12
.e+11
>
1.6+10
1.e+9
« 1.e+8
03 1.e+7
.e+6
.e+5
.e+4
.e+3
.e+2
.e+1
.e+0
states available
i
i,
-j
i
-j
i
-I
-j
fitted to Gaussian DOS
occupied states at T and Epn
real Gaussian DOS: gQ=101®, Eq=0.35 eV, Tq= 1700
EFn=0.2 eV, T=123 K
fitted to Gaussian DOS: g0=2x1017
E0=0.17 eV, T0=582 K
i
-j
i
-j
-j
-j
1-------- '-------- ■—
-—” 1------- r-
0.1
0.2
E, eV
Figure 90. Curves from Figure 89 plotted in a semi-log scale (T=123
186
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
is easier to get better fits at 123 K. This is supported by the experimental fact that we
obtained a higher degree of closeness between experimental and simulated data at 123 K
rather than at 300 K. Therefore, the temperature behavior of parameters characterizing the
Gaussian DOS is consistent with a reasoning that at a specific temperature an effective
density of states is probed which can deviate from the original DOS by a large extent as
well as from the Gaussian form.
Now one can take a new look at the temperature dependence of a (Figure 83).
Because it was shown above that the width of the effective DOS changes (i.e. increases)
with temperature, the temperature behavior of a will have the form axT+AT/T0+AT0 in
which AT0 is associated with the width of the effective DOS. When the increment ATn is
bigger than the increment AT, the parameter a will decline with increasing absolute
temperature. Considering the dynamics for movement of the effective DOS along g(E)
with temperature, one would expect the increment in Tn to vary
in contrast to the
constant increment in T. The increment would be biggest in a low temperature range.
With increasing temperature the Fermi level moves down towards the middle o f the band
gap leaving more states from the real distribution unoccupied, i.e. available for trapping
after excitation. This increases the degree of the correspondence between the real and the
effective DOS. When the position of the maximum in the effective DOS is very close to
those of the real DOS increasing the temperature farther will cause only minor changes in
the width since the degree of closeness between the real DOS and the effective DOS at
this temperature will be already high. Therefore, in this range one might expect very
minor changes in the effective peak width (since it is very close to the real width).
187
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Applying this explanation to Figure 83 one can conclude that in the low temperature
region (123-200 K) AT0 >AT and a decays with temperature. In the intermediate region
(210 - 270 K) both increments are approximately equal. At high temperature region (280
-358 K) the width and the central energy of the effective DOS are very close to those of
the real DOS and the temperature increase in a is governed solely by an increase in T .
Experimentally a number of authors observed decreasing or constant values of a
with increasing temperature [for example, 48], This observation contradicted the expected
behavior o f a , i.e. should increase with an increase with temperature. Indeed, for most (if
not for all) types of DOS (linear, rectangular, exponential, Gaussian) this parameter of the
power law decay is directly proportional to temperature via accT/T0. With this relation
alone it is difficult to explain the decrease of a with temperature. To explain these
phenomena some authors [48] used a concept of electronic doping which implies a
transformation of traps to recombination centers with different capture cross-sections for
electrons and holes as the temperature is increased. This explanation is based on a
temperature movement of a demarcation level and on a presence of two types of
recombination centers [2]. One type is a recombination centre with equal capture crosssections for electrons and holes. Addition or removing this type of center would affect the
lifetimes of electrons and holes in the same way. For the second type, electrons and holes
have different capture cross-sections. Addition of centers o f this type would increase the
life time of one type of carrier and work as a recombination center for the other type of
carrier. The transition of these centers from traps to recombination centers (or vise versa)
with temperature will be accompanied by a noticeable change in the recombination time
188
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
for a specific carrier [2]. While this can be the case, our explanation sheds some light on
other possible mechanisms involving demarcation levels which have not been considered
so far. Extending our mechanism to an exponential DOS, one would expect less
"‘abnormal” temperature behavior for a . Indeed, the vast majority of the levels in the
exponential DOS are located at lower energies and distorting the tail with the occupation
probability function f(E) would cause less severe effects.
Now we turn our attention to the explanation o f why we see truncation o f the
power law decay by the slow exponential decay. In the language o f DOS, this would
mean that in fact we are dealing with a continuous DOS when the release from the last
level of the DOS is seen at a specific time. But because the release from the trap is
temperature dependent, one would see that release starting within a time frame o f the
experiment only when temperature is high enough for thermal activation. Lowering the
temperature should demonstrate only the power law decay simply because the release
from the last level will start at times which are beyond the observation range. A striking
fact is that we observed the exponential decay at the tail at almost all temperatures where
we were able to resolve it! Even distortion o f the high energy tail with the Fermi function
would still cause a continuous (not abrupt) decay on the energy scale. The concept o f the
demarcation level can explain this abrupt tail.
The demarcation level for electrons D„ is defined as follows:
D ,= £ F„ + t n n ^ PS P
189
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(61)
where n and p are the total concentration o f electrons and holes including both
equilibrium and excess concentrations. Sn and Sp are the capture cross-sections for
electrons and holes, respectively. Different symbols assume that the centers with different
capture-cross sections are present (recall two types of centers in the electronic doping
model). The demarcation level physically separates the levels which are traps for
electrons (levels above the demarcation level) from levels that are recombination centers
for electrons [2]. When n » p
the demarcation level will be below the corresponding
Fermi level (this is valid for holes as well). Physically this means that there is quite a
sharp separation between the levels from which electrons can be released and observed in
the conduction band (i.e. are available for multiple trapping process) and the levels which
would led to a final termination o f the electron. The fact that crossing the demarcation
level could drastically change the electronic properties o f the material is well known
(electronic doping, optical and thermal quenching [2]). In our experiments the
demarcation level would determine the last level at a particular temperature which serves
as a trap. Levels deeper that the last level would be invisible in the experiments.
Therefore, adding the demarcation level to the formalism described above can explain
why we see a release from the apparent last level in the continuous Gaussian DOS at
every temperature. In other words, we just monitor the movement of the demarcation
level as the temperature changes. We obtained time constants of the final exponential
decay at all temperatures to get the depth of the last level according to
£=-fc7Tn—!—
v oTt
190
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(62)
where x . denotes the release time from a level with energy E below the conduction band.
The resulting curve is shown in Figure 91. One can see that the temperature dependence
of the trap depth extracted from the slow exponential decay follows the expected linear
temperature dependence. This means that a major contribution to the position of D„
comes from the second term in Eq.(61) allowing for some increment in EFn- When the
total concentrations of electrons and holes are known Eq.(61) can provide an estimation
of the ratio SJSp. We can estimate the concentration of electrons but not holes in our
experiments. The best what we can do is to estimate nS„/pSp. Taking into account the
electron Fermi level movement after excitation (Figure 68) the estimated Fermi level
position when the slow exponential decay starts to develop is ~ 0.25 eV. Using this value
and D„=0.55 eV obtained from Figure 91 at 300 K the value of the ratio is ~ 10\ Knowing
that the concentration of excess electrons was ~1014 cm° when the exponential
components started to develop we can only guess what would be the concentration of free
holes at that time. Because we are dealing with the n-type semiconductor, it is reasonable
to suggest that n » p . (The light creates equal concentrations of excess electrons and
holes but by the time we start measurements holes are trapped). Taking p=\0~2n (which
can be considered as a mild estimation) would give 5„/5; =10'\ The point we would like to
make is that with our very modest estimation the sample definitely should have the
recombination centers with different capture cross-sections for electrons and holes.
Using the approach developed above, in principle one should be able to extract the
information about the real DOS. Using our simulation results (section 5.3) one can
develop an empirical relation between a and T . Applying our simulation/fitting approach
191
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
0.65
0.60
0.55
0.50
> 0.45
<o
Ui
0.40
0.35
0.30
0.25
0.20
100
150
250
200
300
350
T,K
Figure 91. The energy of the last trap extracted according Eq.(62) from the terminal
exponential decay in 5f0. for 1pm CdSe (#2).
192
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
for the “high temperature” region (where the degree o f closeness between the real and
the probed DOS is very high) one can obtain the parameters £„ and T0 which are close to
those o f the real distribution. This approach will work for any peaked DOS which does
not have to be Gaussian. In any case developing the procedure which can fit both the
bandwidth change and the shift of the resonance frequency using a set of discrete levels
with the appropriate density will facilitate the process for the extraction of parameters by
a large extent.
To use this approach one must be certain that the experimental conditions favor
monomolecular recombination (MR) rather than bimolecular recombination (BR). One of
the manifestations of BR would be an intensity dependent a while for MR a does not
change with intensity. This was shown in the OK model developed for an exponential
DOS. Our data show a very gradual increase in a at all temperatures. As was discussed
above the effect of the saturation combined with BR rather than MR is responsible for
this slow increase. Unfortunately introducing the bimolecular term in our rate equations
complicated significantly the equations, increasing computation time enormously. With
the present facilities we were not able to do extensive simulations for BR and determine
the relation between a and the width of the DOS. Therefore, we can only estimate the
upper limit of the width of the real DOS from our BR affected data. We assumed from
Figure 73 that the value of a (0.6) obtained at the lowest intensity is close enough to the
corresponding value for MR. The value o f £„=0.35 eV obtained at 300 K is also expected
to be close to the real one. The DOS which gives a=0.6 for An(t) at 300 K with £,,=0.35
eV is expected to be the upper limit of the real DOS. In fact, the real DOS width should
193
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
not be bigger that the value obtained from the simulation with above mentioned
parameters. With a help of Figure 43 one can determine that a 7^5 5 0 K will correspond
to a=0.6. On the energy scale it will correspond to a 0.047 eV width around a 0.35 eV
central energy. Comparing this value (7 ^ 5 5 0 K) with the corresponding value (Tff* 1700
K) obtained from fitting of the maximum intensity data (Figure 85) at 300 K, one can see
the way that BR can affect the credibility o f the data. Indeed, because values of a for BR
are smaller than those for MR (i.e., slope of the power-law decay is less for BR), the
interpretation of Tn extracted from them will produce fictitious (bigger) widths for DOS.
The real width obtained from MR kinetics can be much smaller.
Finally, we would like to discuss the fact that the both simulations o f the
experimental curves (Fig. 86) produced what appeared to be half-Gaussian distributions.
Indeed, the forms suggested that the peak energy of the distributions is not observed at
both temperatures (123 K. and 300 K). While it might be indeed the case, it does not rule
out the possibility of having a real half-Gaussian DOS rather than the full Gaussian DOS.
In such a case the peak energy of the distribution is located at the edge of the conduction
band. This case can be analyzed within the same formalism of occupied and available
states as above. The results for two temperatures (123 K and 300 K) are shown in Figures
92 and 93, respectively. The general tendency is the same: the probed distribution is
distorted from the real distribution and it becomes narrower with decreasing temperature.
194
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
•
«— 1
t
I
I
I
I
.
I
I
.
1_ I— I— I
1_ I
real half-G auss: Eq= 0 , go= 10 18, Tq= 1 7 0 0
123 K: available
9002000001010000010200019001020002480000000200010200020201000102010001
0
15
O) 1 3
o 12
0.1
0.2
0 .3
0 .4
0 .5
0 .6
E, eV
Figure 92. Half-Gaussian distribution: real distribution, occupied states and available
states, T=123 K.
20
19
real half-Gauss: Eo=0, go=1018»Tq=1700
18
300 K: available
17
16
co
15
14
3
13
o 12
11
10
9
8
0.1
0.2
0.3
0.4
0.5
0.6
E, eV
Figure 93. Half-Gaussian distribution: real distribution, occupied states and available
states, T=300 K.
195
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6.2.7. Summary
The following summarizes the findings for this sample: 1) using the MT model
we have shown that the observed kinetics of the bandwidth change and the shift o f the
resonance frequency are consistent with the Gaussian-like distribution of localized states;
2) the developed mechanism involving the distortion with a smooth truncation o f the real
distribution by moving the Fermi level can explain the variation of parameters o f the
Gaussian DOS with temperature, which manifests itself in the different behavior o f a
with temperature; 3)
invoking the demarcation energy concept, we explained the
appearance of the final exponential decay truncating the power law decay o f the
continuous infinite DOS; 4) it was shown that the sample very likely contained
recombination centers with quite different capture cross-sections for electrons and holes;
5) we showed the limitations of the present level of knowledge about the interpretation of
the experimental results under conditions of bimolecular recombination; 6) we obtained
an upper estimation of the real DOS width (0.047 eV) and the peak energy position (0.35
eV); 7) by comparing two different forms of distributions observed in two CdSe thin
films (exponential in #1 and Gaussian-like in #2) we showed how two different
preparation techniques can result in not only different levels o f introduced defects but in
the different shapes of the distribution of these levels; and 8) finally, this type o f analysis
can not be done using the traditional TMP, therefore the advantages of AMTMP are
evident.
196
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6 3. Semi-insulating (SI) GaAs
63.1. Experimental
The tested sample was a strip (5x1x0.52 mm3.) of undoped n-type SI GaAs
(MACOM) cut from a wafer. Equilibrium concentrations of carriers were as follows: n0 =
8.3xl07 cm"3 and p 0 = 3.8xl04 cm’3. The dark equilibrium mobility was |4=7300 cm^/V
sec, which was reported by the supplier and confirmed by our Hall measurements. The
sample was measured on the setup #2. Excess photocarriers were generated by a NdrYAG
laser at X = 1064 nm (subband gap excitation) and X = 532 ran (band gap excitation) with
~5 ns pulse width. Kinetics of the change of the cavity quality factor and the shift of the
resonance frequency were measured at intensity ranges corresponding to the following
initial excess concentrations: ~ 1011-1013 cm’3 at 1064 nm and ~1015 - 1019 c m 3 at 532
nm. After excitation the "active” volume was defined by the rectangular laser beam (4x 1
mm), and the absorption coefficient measured at 1064 nm was accurate within 15%. For
X = 1064 nm, the temperature was varied from 213 to 358 K in 8 steps, and the light
intensity varied over three orders o f magnitude using neutral density filters. Temperature
was controlled in the same way as discussed in section 6.2. The intensity effect was
measured in great detail at 298 K. For X = 532 nm only room temperature intensity
experiments were done. In total, the kinetics of more than 50 independent transients were
analyzed.
Over the whole range of temperatures (213-358 K), intensities and wavelengths
used for SI GaAs, the change in the real part o f the dielectric constant As '(t) was
197
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
negative. Recall that in both CdSe samples discussed so far the real part of dielectric
constant has increased after illumination. According to the harmonic oscillator treatment
(Chapter 4) this decrease is due to the dominant influence of free carriers on the dielectric
constant rather than the trapped carriers. Below we will discuss three approaches [129]
based on the harmonic oscillator model to analyze the experimental data in order of
increasing complexity based on the number of effects involved. This treatment not only
highlighted the major features, but also provided quantitative information that the
conductivity decay for our SI GaAs sample depended on the decay o f the excess electron
concentration only. Mobility changes were shown to provide negligible contribution to
the conductivity decay.
6 3.2 Approach I
As
was
shown
in
Introduction
for
w-type
semiconductor
transient
photoconductivity may include both mobility and carrier concentration changes
(Eq.4).Therefore, the relative importance of mobility and electron density changes must
be considered in the interpretation of conductivity changes.
Under steady-state illumination, the mobility in SI GaAs increases up to 100%
relative to the dark value [175]. A similar increase was observed recently for transient
measurements of Hall mobility after pulsed illumination [20]. During much of the time
required for the decay of the electron density, the mobility remained almost constant.
Since the same period was used for the AMTMP measurement, the mobility was assumed
to increase by -100% after illum ination and to stay constant during the decay. Even if the
mobility were to change during the decay, it would have a small effect on the kinetic
an aly sis because the excess electron density would be corrected by no more than a factor
198
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
SI GaAs
lg = 1 0 10 el./cm , T=358 K
o
10a
O
approach
approach I
5x10*5
10-4
1x10^
2x 1 0 ^
time (s)
Figure 94. The decay of excess electron concentration in SI GaAs, T=358 K, I-I0,
A.=1064 nm.
199
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
of -2.5, which is minor in comparison to the several orders of magnitude change
observed during the transient decay. This conclusion is based on a calculation o f the
electron density using p=0.73 m^/(V sec) at 300 K, determined by Hall effect
measurements.
A test of this conclusion can be made by comparing the value of 8e '( 0 obtained
from the frequency change 5/0 (Eq.(15)) with the one calculated from S(A/u2) using the
following approach. Over the whole range of temperatures (213-358 K.) and intensities
used for SI GaAs. the change in the real part of the dielectric constant was negative,
indicating that free electrons provided the main contribution to 8e'(/), and plasma effects
can be excluded. As a result, only free electrons contribute to Se"(0- If changes in the
mobility are neglected An(t) can be obtained from S(Af V2) using Eq.(15) and the relation
between 8e"an d Act from Eq.(7). According to Eqs.(33) and (34), both 8 e '(0 and
8e"(0 must decay with the same time dependence as An(t). Values of An(t) obtained
from 8(A /1;2) are given on Figure 94. Using Eq.(33), these values were used to calculate
5 e '(0 (solid line in Figure 95), which is compared with 8 e '(0 (dashed line) determined
directly from the frequency shift according to the perturbation theory. The two curves
show not only the same time dependence but also they almost coincide numerically. For
these calculations, the mobility was adjusted to 1.8 m^/fV sec) to obtain the best match
between the two curves in Figure 95. These results support the conclusion that mobility
changes can be neglected during the decay of the carrier density. Furthermore, the DrudeZener theory adequately describes the laser induced behavior observed for SI GaAs.
200
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
SI GaAs
ln = 1015 el./cm3 , T=358 K
—
experiment (from the shift)
harm. osc. mod. (from An(t)), appr. I
time (s)
Figure 95. The changes in the real part of the dielectric constant o f SI GaAs, T=358 K,
I=I0, /.=1064 nm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6 3 3 . Approach II
This approach considers the mobility as a time dependent parameter (Eq.(35)).
Exclusion of An(t) is accomplished by dividing Eq.(33) on Eq.(34) to give the absolute
value of the electron mobility after excitation. p,:
M O 55 Ho + Ap(0 =
os (t) m co
(63>
Changes in the real and imaginary parts of the dielectric constant were calculated
according to perturbation theory as described in Approach I. The results plotted in Figure
96 (solid line) show the mobility stays almost constant from -3x10"' to ~ 4x10° s at a
value ~2 m2/(V s), which is very close to the constant value used in Approach I. These
results are consistent with the two-fold increase observed for several SI GaAs crystals by
the Hall method [20]. Furthermore, the constant region starts at about the same time, but
the range can not be compared because of the noise associated with our technique. The
initial apparent fast component is spurious because the corresponding fast components in
8 e '( 0 and 5 e"(0 were not completely resolved. The absolute value of mobility obtained
by this method is estimated to be accurate within 50%, which is sufficient to verify the
mobility is nearly time independent. Taking into account the time dependence o f the
mobility, the decay of the concentration of excess electrons is plotted on Figure 94, which
almost coincides with the corresponding curve obtained assuming constant mobility
(Approach I).
202
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10.0
Iq=10
-
SI GaAs
el./cm^, T=358 K
co
CM
e,
n o t resolved
=L
<
-a
c
CD
Mo = 0.73 rrrlfy s) at 300 K1
approach III (Ap(t))
approach II (ml, (t))
10-4
time (s)
Figure 96. The relative increase in the mobility, Ajj. and the absolute value of the electron
mobility, p. as a function o f time in SI GaAs after excitation. T=358 K, I=I0, A=1064 nm.
203
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6.3.4. Approach III
This approach is the most rigorous treatment. In contrast to Approaches I and I
which considered sample to be infinitely long (Z,=0) Approach III uses real dimensions of
the sample which may cause depolarization effects. In this case, the finite length o f the
sample results in a non-zero value of L =0.007, which was obtained from Eq.(22). As was
shown in section 4.2 introduction of the final sample’s length results immediately in the
necessity to consider a possibility of plasma effects. While from the forms o f the kinetics
we ruled out a possibility of plasma effects it could not justify the approximation of the
infinitely long sample. Therefore, this Approach not only confirms that the plasma effect
is unimportant for the present measurements but shows also that a finite sample’s length
results in minor changes. In place of Eqs.(33) and (34), Eq.(36) is divided by Eq.(37):
8 s '(Q
8e " ( / )
(Po+ApfOVn*,/, g>2/
e
Vco' y
-------------------------- CO 1-------- -T
(64)
In this equation, the ratio is not independent of An(t) because the plasma frequency is
concentration dependent. Maple V was used to solve Eqs.(4) and (64) for Ap(/) and An(t),
and the analytical expression for Ap(/) is:
Ap (f)= ^-co2/»*£0p 0 —res0o3 —T0eco2e 0 -r-e2n^Ll\iQ+ eL,A<s(t) + V z ^ 2 e 0co "m*
-2(a2m’s Q\ile2nQLi - 2 c o 2m 'e 0p 0eZtACT(r) + r 2e : £^co2
+ 2 r e 2e;;coJT 0 -2 r e lz (fan^Li\i0 -
2re2e 0o)LiA a ( t)
+ T 2e 2co4£o + 2 t 0e 3co 2e q/IqZ(p 0 + 2x 0e 2co 2s o Z, A ct (f)
+ e 4n 2Z2p^ + 2 e 5«0Z2p 0A a ( / ) ,
204
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
(65)
in which r = de’(t)/8 e " (t) and Z, = Z / 1+ Z(e, - 1 ) . Comparison o f Approaches II and
III in Figure 96 shows that the use of a finite sample length has only a minor effect on the
mobility change, and it is essentially constant over the time range where changes can be
resolved. Therefore, plasma effects are unimportant over this time range, and the three
approaches verify that the conductivity decay for our SI GaAs sample depended on the
decay of the excess electron concentration only.
63.5. Trapped electron estimations.
The presence o f shallow traps is well established in SI GaAs [175.176] although
there is some question about the exact density o f these traps [177,178]. As discussed
above, the positive shift of the resonance frequency indicates free electrons make the
dominant contribution to the real part of the dielectric constant, despite the good
possibility that shallow traps are filled in SI GaAs after illumination over the time range
studied. In contrast, thin-film, polycrystalline CdSe exhibited a negative shift of the
frequency, indicating electrons in shallow traps made the dominant contribution to the
real part (see sections 6.1, 6.2). The range of temperatures and times used for SI GaAs
was similar to that for CdSe. This difference between CdSe and SI GaAs in regard to
5 e '( 0 is related to the magnitude of the mobility for each material as discussed below.
Because of the difference in sign of the contributions made by free and trapped
electrons to changes in the real part o f dielectric constant, the relative concentrations at
which these effects cancel depend on the magnitude of the dark mobility. For CdSe and
SI GaAs, the values are 0.0014 and 0.73 m2/(V s), respectively. In the case of SI GaAs,
205
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the free electrons dominate at a density o f 2x10U cm’3 or lower (see Figure 94), and this
value was used to calculate the trapped electron densities at which the contribution of the
free electrons to 5 s'(/) is a factor o f 100 larger than the value due to trapped electrons.
Above this trapped electron density, the free electrons begin to lose their domination of
S e '( 0 . This ratio and Eqs.(33) and (40) were used to calculate Anlr for various binding
energies listed in Table I. The characteristic frequencies of the oscillator were calculated
according to Eq.(24), and the relaxation time was calculated according to Eq.(35). For
CdSe this time was calculated to be I O’15 s using m*=0A3 m0. For SI GaAs the following
*
values were used: m =0.063 mQ, x= 3.6x10
-13
sec. For a low mobility material like
polycrystalline CdSe, it can be seen that density of trapped electrons must be very low
before free electrons can dominate the change in the real part of the dielectric constant.
Indeed, the positive change in the real part observed for this material indicates that the
actual concentrations o f trapped electrons exceeds upper limits given in Table I. On the
other hand for SI GaAs, Table I indicates that 2x10
II
cm
-3
free electrons still dominate
when the electron density in the 0.1 eV trap is as high as 1x10
12
-3
cm . Consequently the
upper limit for SI GaAs is substantially larger than that for CdSe, and this difference is
due to the large difference in the mobility values. It is understood that a ratio is an
arbitrary choice; however, these conclusions will not change if a smaller value is chosen.
Thus it can be concluded that the effect on 5 e '(0 was dominated by free electrons in SI
GaAs, but trapped electrons dominated in CdSe because it has a substantially smaller
mobility.
206
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 3. Estimated upper limit to trapped electron density.
Trap depth, AE (eV)
Concentration of trapped electrons
in CdSe, cm
Concentration of trapped
electrons in SI GaAs, cm"3
0.001
30
lxlO6
0.01
2x104
2x109
0.1
3x10?
lxlO 12
0.3
8x 108
4 x l0 13
Abovethisdensity,thecontributionoftrappedelectronsto '{t)beginstodominate
overtheoneforfreeelectrons.Thisestimateisbasedonthefreeelectrondensitiesgiven
inthetext.
5s
a
This approach sets a more realistic upper limit than the polarizability method
[104] used to estimate the trapped electron density in CdSe (see also Chapter 6.1).
Assuming equal polarizability for all traps normally detected by photodielectric effect (in
the energy range, 0.003-0.3 eV), the concentration of electrons in shallow traps after
illumination was estimated to be less than 1011 cm"3 in SI GaAs, which is substantially
lower than the sum of the values in Table I. This approach suffers from the assumption
that all traps within 0.003-0.3 eV range have the same polarizability. Because this
condition applies only to a few tens of meV depth for the hydrogen-like centre, the
concentration of electrons in traps deeper than 10-20 meV was underestimated. In fact,
semi-empirical calculations from polarizability measurements (i.e. second order Stark
207
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
effect) made for AgCl showed that polarizability of electrons in hydrogen-like states was
about one order of magnitude bigger than the value for free electrons and about four
orders o f magnitude bigger than that for deep traps [104].
63.6. Kinetics analysis
Therefore, we have shown what type of analysis can be done if using all
advantages offered by AMTMP. Once An(t) has been extracted the following analysis
will be a more routine (but not necessarily simple) kinetics analysis. When we decided to
choose SI GaAs to demonstrate the abilities of AMTMP we expected to get relatively
simple “model” kinetics since the sample was not intentionally doped. But it turned out to
be a quite complicated material to analyze. Due to autocompensation [177] SI GaAs can
possesses quite a few levels in its band gap. Studying the literature [182] about the
impurities found in SI GaAs led us to a conclusion that almost any energy within the
band gap could be attributed to some defect. Therefore, the interpretation of the kinetics
in this sample has not been completed yet.
Figure 94 indicates that there are several regions of time over which the decay
appears to follow an exponential time dependence. This figure illustrates that the decay at
358 K is very complicated with at least four regions which appear to be exponential
components. Furthermore this complexity was preserved at all temperatures and light
intensities (Figure 97 and Figure 98). Although it is relatively easy to find an analytical
expression that fits for a limited set of conditions, determination of a model that applies
to a variety of experimental conditions (i.e., temperature, intensity, wavelength) can be
208
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
SI G aA s. 1=1,
1.e-2
1.e-3
1.e-4
A-
N
X
e-5
o
cs
.e-6
OO
.e-7
.e-8
.e-9
1.e-7
1 .e-6
1 .e-5
1.e-4
1.e-3
1.e-2
1.e-1
tim e.s
Figure 97. Kinetics of the bandwidth change, 8(Afl/2) for selected temperatures in the
range +85°C -5- 0°C, I=I0.
1.e-2
1.e-3
1 e-4
. 20 C
O
1e-5
2
§
1e-6
1.e-7
1 e-8
1.e-7
1.e-6
1.e~4
1.e-5
1.e-3
1.e-2
time.s
Figure 98. Kinetics of the bandwidth change, 8(AfI/2) for selected temperatures in the
range 0°C
-60°C, I=I0.
209
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
difficult. Due to the complexity of the observed phenomena and the large amount of
experimental data obtained under various conditions, the work on incorporating all
experimental results into one non-contradictory model is still in progress . Therefore, the
topic o f the following discussion is limited merely to outlining some major features and
giving qualitative explanations of the some obtained results.
We would like to start the analysis o f the kinetics by discussing some features
shown on Figure 99. The figure shows the behavior of the bandwidth change at four
selected intensities measured at 385 K. The laser wavelength was 1064 nm (-1.13 eV)
which provided subband gap excitation of GaAs (energy gap 1.424 eV at 300 K).
Therefore, one might expect that the major process would be defect ionization and
subsequent trapping on those centers. The semi-log plot clearly shows a major
exponential part in the middle of the kinetics. We called it the '‘dominant” component.
With straight lines we identify the other exponential components. There is one feature
which was preserved at all intensities and all temperatures. Almost every kinetics
demonstrated a region of faster decay after what appears to be a small transition region or
a part of another slower exponential decay. The detailed analysis of every kinetics in a
semi-log scale revealed that this “break point” always appears at the same value of
6(A/'l/-,)close to 10° GHz but at different times. Processing 5(A/1/2) according to the
second cavity perturbation formalism and with assumption of the constant mobility
resulted in A/i(r)«8.8xl08cm"3corresponding to the break point. This value is one order of
magnitude higher that the dark equilibrium concentration n0 (8.3x107 cm"3). Those values
will be discussed later in this section. The more detailed intensity dependence
210
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
I i ■»_
1.e-2
1.e~3
»
»
I
i
«
t
»
»
»
i
«
i - 1—
I—
SI GaAs, T=358 K
"dominanr component
8(Af1/2), GHz
1.e-4
"break point" level
1.e-5
1.e-6
1.e-7
0.11 I
0.0023 I
1.e-8
0.00000
0.00004
0.00008
0.00012
time, s
Figure 99. Kinetics of the bandwidth change, 8(AfI/2) at selected intensities for SI GaAs,
T=358 K.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
•
tn
1.e-3
O
time of break
amplitude of break
E
3
.d
m
c
o
CL
(0
as
.O
(U
£
(0
as
"O
3
"q .
L_
1.e-4
E 1.e-5
as
■a
c
as
as
E
1.e-6
0.0001
0.0010
0.0100
0.1000
1.0000
normalized intensity (l/l0), rel. units
Figure 100. Intensity dependence o f the time and the amplitude at the “break” point for
SI GaAs, T=300 K.
212
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
measurements done at 300 K show the behavior of the amplitude and the time location of
the break point (Figure 100). One can see that the amplitude at the break point is constant
and very close to 10° GHz.. The time of the appearance o f the break point shifts to longer
times with increasing intensity. We want to stress that the break point identification did
not involved any assumptions on the forms of the components and was done by a visual
analysis at the appropriate scale of every kinetics. The temperature dependence o f the
time of the break point at four intensities is shown on Figure 101. One can notice three
regions at all intensities except the smallest one. The break time increases with decreasing
temperature from 358 K to 300 K. At 300 K. it reaches a maximum. With subsequent
temperature decrease from 300 K to ~ 273 K it fails down (by larger degree at high
intensities) and starts to increase again up to 213 K.. At the smallest intensity the
maximum disappears and there is what can be considered as a monotonic increase in the
break time with temperature. We will return to the discussion of this plot after showing
the analogous features observed for the dominant component.
The analysis of this so called "dominant " component with temperature and
intensity changes was complicated by the variation of the contribution made by a second
component as conditions were changed. At 358 K, when the contribution of this dominant
exponential component was relatively large, the second component acted as a constant
rather an exponential decay (the part which we suggested earlier to be the transition
region or another exponential decay, see Figure 99). With the sole aim to isolate the
dominant component, the sum of an exponential decay and a constant was used for this
region. While this approach was justified at higher temperatures, the time constant o f the
213
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.e-3
1.e-4
1.e-5
1.e-6
0.0025
intensity lg
intensity 0.11 lQ
intensity 0.0210
intensity 0.0023 I,
0.0030
0.0050
0.0045
0.0040
0.0035
1/T, K'1
Figure 101. Temperature dependence of the time of the break point for selected
intensities. SI GaAs.
SI GaAs
1.e-3 -i
<D
I
1-e-H
8
3
C
(0
e
■§ 1.6-5 ~
c
CD
tSJ
I
I
1 .e-6-]
0.11 l0
0-02 !{,
0.002 L
1.e-7
■ 1 1 -|
0.0025
0.0030
.
0.0035
0.0040
1/T, K-1
0.0045
0.0050
Figure 102. Temperature dependence of the time constant of the dominant component for
selected intensities, SI GaAs.
214
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
dominant component was clearly overestimated at other temperatures because the
subsequent exponential decay could not be considered as the constant any more but as
true exponential. At these other temperatures for which the contributions of the
components were comparable, the double-exponential decay was more appropriate, and
underestimated time constants were obtained for this region. A temperature dependence
of the time constant o f the dominant component assuming the form of an exponential
decay plus a constant is shown at Figure 102. One can see now the amazing similarity
between the temperature dependence of the break time and the time constant of the
dominant component. (Recall, that the break time was discovered on all kinetics simply
looking for the transition point in a semi-log scale, i.e. without any assumptions on the
forms of components.) First, it proves that the phenomenon indeed exists. Second, it leads
us to invoke some quenching mechanism to explain this behavior.
The similar kinetic behavior can be obtained in the Schokley-Read recombination
model for a large defect density [183). According to this model the break point would
occur when An(t)/n0= l. Accepting it as an explanation of our data we still have one
order difference between the dark equilibrium concentration obtained from the Hall
measurements and our proposed concentration from the break point. A reason tor this
might be that the Hall measurements could provide a lower value due to a depleted
surface layer in SI GaAs. One can recall that in Hall measurements the average
concentration over the sample is measured, and it is attributed to the total sample volume.
The depletion which is known to be the case in SI GaAs could leave the true
concentration in the middle of the sample only. However, this is only a speculation, and it
215
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
can be explored further working with other samples which are known to have different
degree o f a depletion or with samples having different Fermi level positions.
The apparent activation energy o f the high temperature branch (358 - 293 K) was
about 0.75 eV. The second approach using a double-exponential decay changed the slope
to 0.55 eV but did not affect the total shape o f the whole temperature dependence. So. we
expect a true activation energy to be close to 0.65 eV.
A study of the photoconductivity of SI GaAs using the conventional technique
involving contacts and a constant applied voltage obtained a time dependence that was
very similar to our results [20]. The temperature dependence of an exponential
component similar to our dominant component gave an activation energy o f 0.61 eV in
the same temperature range (420 - 300 K). It is reassuring that these two different
techniques for measurement of the photoconductivity provide the same results for
undoped SI GaAs. From our data in Figure 102 one can see that below 300 K there is a
change in the behavior of the time constant i.e. it starts to decrease. It may explain why
the authors in [20] did not extend their temperature range down to 200 K.
In the contact photoconductivity study [20], the observed activation energy was
attributed to recombination via the EL2 level (0.75 eV) with bandgap shrinkage due to
potential fluctuations in the vicinity of recombination center. Another possibility is that
the recombination via the deep EL2 center involves thermal activation of a trap. One of
the approaches to analyze the activation energies in photoconductivity experiments is to
relate the observed activation energy directly to the trap depth which governs the process.
In this approach, the activation energy corresponds to the trap depth due to a thermal
216
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
release of the electron (if electrons are detected). The time of the thermal release is related
to the trap depth by Eq.(52). In Eq.(62), x represents the earliest time when the electron
released from the trap can be seen in the conduction band. In steady-state experiments
when thermal equilibrium with deep traps has been established, it is quite appropriate to
attribute the observed high activation energy’ to the thermal release from the deep traps. In
transient experiments, the time o f thermal release dictates when the appropriate trap effect
might be observed. For the quick evaluation we plotted Eq.(62) at three chosen
temperatures (Figure 103). It can be seen that a thermal release from the trap depth of 0.6
eV can be seen only starting from 10“3 s at room temperature. The simple analysis of
plots of the present experimental data shows that the dominant component appeared much
sooner. Therefore, it can not be a thermal process related to a 0.6 eV trap. Based on this
analysis, the time constant of our dominant component and the time of its appearance in
the range 358 - 298 K would be consistent with thermal release from a trap with a depth
in the 0.45-0.5 eV range and with a temperature dependent capture cross section having
an activation energy of 0.19-0.14 eV. The depth (0.45-0.5 eV) will give the desired time
constant of the "dominant” component and the temperature dependent capture cross
section (with activation energy o f 0.19-0.14 eV) will give the energy needed to add to the
trap depth to obtain the experimentally observed activation energy (~ 0.65 eV).
Furthermore, the contribution o f this component changes with temperature. Thus, the
time of its appearance at 213 K (a lowest temperature in our experiments) would be
around 10"3-l 0 '1 seconds, which indicates that the dominant component has a different
nature below 300 K.
217
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
t mil mil mil mil mil mil mil
?L h9^ u9 J N
9 ) 9^ Qo« *o No )
release time (sec)
111111111 p r r r r
nrrr
i i 111 i i 11 i i i i i i i i i 1111111111 i 11 11 iTi 11 i y n i i i t pITTTTTrT'H
J
=-243-K-
T = 30CT
E
:
r-=-958-K
:
1 0 -6 4
;
10-7 4
10-6 4
:
:
10-10 -I
10-11 4
10-12 4
1n -13
0.0
-
:
-
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
trap depth below the c.b. (eV)
Figure 103. Electron thermal release time as a function of trap depth (below conduction
band) for T = 213, 300, 358 K.
218
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
To our knowledge, the temperature behavior in Figure 102 has not been reported
for SI GaAs (although parts o f this complicated dependence were observed by others
[20]). It may be caused by the interrelation between the various localized states in this
material. The situation is complicated further by the fact that temperature dependencies of
the components occurring after the breakpoint (~ 4.5x10*5 sec) exhibited a maximum in
the same temperature range, but it was less pronounced. Similar behavior was observed
for InSb [34] and the authors had to invoke a two center model to explain qualitatively
the temperature variations. Our detailed investigation of the intensity dependence at 298
K. revealed a supralinear intensity dependence of the time constant of the dominant
component. This may suggest a two center model with sensitizing centers [2] as an
explanation for the observed phenomena in our experiments. With the two center model
one can obtain the effect of temperature quenching when varying the temperature. The
temperature quenching appears when one of the centers crosses the position of the
demarcation level and changes its nature from a recombination center to the trapping
center (or vice versa). But this type of quenching would provide only one extreme in a
temperature dependence of the electron lifetime. To interpret two extremes temperature
dependence more work is needed.
63.7. Summary
We have shown that the harmonic oscillator model was found to be consistent
with the experimental results obtained for SI GaAs. The dominant contribution to the
changes in the real part of the complex dielectric constant came from the free (conduction
219
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
band) electrons. The changes in the conductivity were shown to depend on the decay of
the excess electron concentration only. Mobility changes provided a negligible
contribution to the conductivity decay. The results obtained for the finite sample length
were very similar to those obtained on the assumption of the infinitely long sample. A
rigorous approach was developed to include the depolarization effects due to finite
sample length. The different effects on the real part of the dielectric constant found for
CdSe and SI GaAs could be ascribed to the large difference in their mobility values. The
analysis of the dominant component ruled out a possibility that the electron release was
from EL2 level found in SI GaAs. Instead, the temperature activated capture cross-section
was suggested for 0.4-0.45 eV levels. Finally, the temperature behavior of the dominant
component suggested a Schokley-Read recombination model for a large defect density.
6.4. Si
Two samples of p-Si (~0.3 ohm"1 cm"1 with s '=15, e"=45 and -0.7 ohm '1 cm '1
with e '=15, e"=105) were evaluated with AMTMP [123]. Both were irradiated by a
laser pulse (337 nm, 0.6 nsec) which created the initial carrier density up to 1020
electrons/cm3.
In contrast to the previously described samples Si samples can be
considered as highly conductive. This would mean that the condition £ "> s' must be used
to get preliminary estimations o f the final look of the second perturbation theory
expressions for this sample.
•
According to our estimations, one would expect a very complex behavior of AMTMP
transients because under these conditions one could not obtain the simple expressions
220
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
having forms 5 /0oo5e' and 8 ( A / i/j)q c 5 s" as was a case in previous samples. Instead,
one would have both parameters contributed to the changes of the complex dielectric
constant, i.e.8/0oc(8s\5s") and S lA /^ o c ^ s", 8 e ') . At some point there might be
even a complete inversion, i.e. 8/0<x8e" and SCA/^jooSe'.
•
Besides, another complication could arise also because of the high mobility electrons
in Si. It will cause negative changes in e ' due to free electron contribution (as in
GaAs). It will result in opposite signs of 5e ' and 8s" in perturbation theory
expressions making negative values due to subtraction procedure.
•
The extra complication can be brought about by the high equilibrium electron
concentration. Estimations showed that it more easier to realize plasma conditions for
Si rather than for SI GaAs. It would mean that the regions with different signs will be
present in the kinetics (as in Chapter 4.2).
With all these possibilities the behavior of the frequency shift and the bandwidth
change shown on Figure 104 and Figure 105 for two Si samples is not very surprising.
Figure 104 shows that the bandwidth change is negative which means that the losses in
the cavity were decreased and the energy was released rather than absorbed! Of course,
this is possible if the change of the cavity quality factor was proportional to the imaginary
part of the complex dielectric constant. Electrodynamics allows in principle an increase in
the cavity quality factor after pumping the cavity with extra particles and increasing the
stored energy in the cavity. But it would be a very short burst of generation followed by a
relatively long dissipative process. Another known and practical way o f generating the
energy is a parametric amplifier which could be realized with the microwave cavity by
221
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
0.0020 -p-type Si,
e '= 1 5 ,
e"=45
0 .0 0 1 5 -
N
X
o
0.0010
-
eo
TcJ
CD
CM
0 .0 0 0 5 -
0.0000
-0 .0 0 0 5
Ve-9
1e-6
l.e - 7
l.e - 5
time.s
Figure 104. The bandwidth change and the shift of the resonance frequency for p-type Si
sample with e‘=15, e“=45, T=300 K.
0.018
p-type Si, e - 1
5 , e"= 100
0.016
0.014
0.012
N
X
0.010
TCD
0.008
CO
S
'1/2'
0.006
<
0.004
0.002
0.000
-
0.002
1.e-9
1.e-8
1.e-6
1.e-7
1.e-5
1.e-4
time, s
Figure 105. The bandwidth change and the shift of the resonance frequency for p-type Si
sample with s ‘=15, e“=100, T=300 K.
222
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
altering its parameters in an appropriate way. We doubt that we had parametric conditions
in our cavity. Therefore, the observed increase in the cavity quality factor can not be
related to the true generation of energy but most likely to “cross-talk” conditions when
both the real and the imaginary changes make contributions to the bandwidth changes.
With this reservation the interpretation of the double changes in the sign of the shift o f
the resonance frequency (Figure 104) will not be simple as well. While we are not ruling
out a possibility of plasma-free-trapped electron transitions (which would explain the
sequence in the sign changes), the contribution from the both terms as above have to be
considered. It is important to mention that the photoresponse measured at the resonance
frequency does not contain these peculiarities making its interpretation non complete
without considering all factors as above.
The situation remains complicated for a more conductive sample having
approximately £“/£' = 7 (Figure 105). Now the bandwidth change exhibits a different
behavior: it changes its sign twice as did the shift for the less conductive sample.
Apparently, due to higher conductivity this sample was closer to the complete inversion
when 8/„oo5£"and 5(A /I/2)<x5£'. Such effective negative values of dielectric constants
were reported for highly conductive silicon spheres in [179].
That is almost everything we can say about this complex behavior. This
interesting phenomena must be studied more to obtain a clear picture.
223
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6.5. Porous Si
Porous Si samples were kindly supplied by Dr. P. Fauchet, University of
Rochester [180]. These films were known for extremely long luminescence decays, and
we looked for a correlation between photoconductivity and luminescence measurements.
Unfortunately, poor response was obtained for all samples except the one having the
lowest porosity, which was the least efficient in producing the luminescence. For this
sample, both bfA/j^) and 8f 0 exhibited very short transients, comparable to the
instrumental time constant. The sign of 5fQ was negative indicating trapped electron
effects. Therefore, one possible explanation is that the free electrons quickly fill shallow
traps, then electrons thermalize very fast down to very deep trap levels which can not be
detected by our method. The very long luminescence observed in these samples could
support the speculation that it originates from such deep traps.
224
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
7. CONCLUSIONS
The current work has described the new method, Advanced Method o f Transient
Microwave Photoconductivity (AMTMP) and its abilities to study the photoelectronic
properties o f semiconductors. AMTMP measures not only the effect proportional to the
excess conduction band electrons ("photoconductivity” itself, related to the changes in the
imaginary part of the complex dielectric constant), but also the changes in the real part of
the complex dielectric constant ( “photodielectric effect “). In a rigorous treatment of the
complex dielectric constant changes in a microwave cavity, the basic perturbation theory
was extended to the second perturbation theory. It allowed us to obtain general
expressions relating the changes in complex dielectric constant to two experimentally
measured quantities; change in the cavity quality factor and the shift o f the resonance
frequency. It was demonstrated that any known relations for the simple geometries as
well as the first perturbation expressions can be obtained from our expressions. Based on
this method it was possible to systematize basic types of excitations (free electrons,
plasma, trapped electrons, excitons) as bound/non-bound states. This clearly showed what
peculiarities can be observed in the changes of the real and the imaginary parts of the
complex dielectric constant at those excitations.
To interpret the behavior of the kinetics in semiconductors having distributions of
the localized states in the band gap we developed the simulation approach based on
solving numerically multiple trapping rate equations. With this approach the major basic
types of distributions (rectangular, linear, exponential, Gaussian) were explored
thoroughly. We extracted the major features of the distribution manifested in the excess
225
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
electron decays and the changes in the real part of the dielectric constant related to the
trapped electron concentrations. It was shown that to get unambiguous conclusions about
the type of the distribution the excess electron decay as well as the decay related to
trapped electrons must be measured. During the simulations we obtained new features of
the distributions. We made some corrections to the theory widely used to explain multiple
trapping phenomena for the exponential distribution showing that the extraction from the
kinetics of the recombination time done in a traditional way gives a two orders of
magnitude error.
We tested our experimental method along with practical developments in the
perturbation theory, the harmonic oscillator model and the multiple trapping model on
various semiconductors: two types of polycrystalline CdSe thin films (#1 and #2), semiinsulating (SI) GaAs, single crystal Si and porous Si. We showed that polycrystalline
samples can be successfully treated with a multiple trapping model previously used
mostly for amorphous materials. We identified the distributions present in CdSe #1 as an
exponential distribution and in CdSe #2 as a Gaussian-Iike peaked distribution. The
parameters of the distributions were estimated. This type of information can not be
obtained by traditional methods, only data provided by AMTMP can do this. We showed
that in the peaked distribution transient methods always probe only the part of the real
distribution. This effective distribution is distorted from the original form by the Fermi
function describing the occupational probability and truncated at high energies by the
demarcation level.
226
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
For SI GaAs we demonstrated that the harmonic oscillator based analysis can
provide not only qualitative but quantitative information. Using this analysis we were
able to separate the mobility changes from the concentration changes in the
photoconductivity decays. We showed that to observe excess free electrons rather than
trapped electrons dominating in the changes of the real part of the dielectric constant the
material has to possess high mobility carriers and relatively low trap densities.
We reported very peculiar kinetics detected by AMTMP in highly conductive
single crystal Si samples. At the present stage we were not able to discriminate between
cross-talk conditions appearing in the second perturbation theory expressions and
sequential transitions between various bound/non-bound electron states. The observed
phenomena are challenging and require a more detailed study.
Testing AMTMP on porous Si revealed some limitations of the method when
applying it to the materials with very short lifetimes and high density of deep traps.
Overall, we would like to conclude that the theoretical and experimental
developments described in this work can result in many versatile applications.
227
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
8. REFERENCES
1. R.H. Bube, Photoconductivity o f Solids ( John Wiley & Sons. New York, I960).
2. R.H. Bube, Photoelectronic Properites o f Solids (Campridge University Press,
Cambridge, 1992)
3. S. M. Sze, Physics o f Semiconductor Devices , 2nd edition (John Wiley & Sons, New
York, 1981)
4. D. Bimberg, H. Munzel. A. Steckenbom. J. Christen, Phys. Rev. B 31, 7788 (1985)
5.
R. H. Bube in Semiconductors and Semimetals, ed. R . K. Willardson and A. C. Beer
(Academic Press, New York, 1967), 461.
6. J. W. Orton and P. Blood, The Electrical Characterization o f Semiconductors:
Measurment o f Minority Carrier Properties (Academic Press, London. 1990)
7. D. K. Schroder, Semiconductor Material and Device Characterization (John Wiley &
Sons. New York, 1990)
8. J. Reichman, Appl. Phys. Lett. 59, 1221 (1991)
9. B.R.Holeman and C.Hilsum, J.Phys.Chem.Solids 22, 19 (1961)
10. T.Kinzel and I.Kudman, Sol.-St. Electron. 8, 797 (1965)
11. G.P. Peka, V.A. Brodovoi, I.I. Mishova. and L.Z. Mirests,Sov.Phys.Semicond.12,
540(1978)
12. B.Tot, V.I. Korol’kov and A.A. Yakovenko, Sov.Phys.Semicond. 12, 111 (1978)
13. Y. Mita, J.Appl.Phys. 57, 1391 (1985)
14. T. Hariu, T.Sato, H. Komori. and K. Matsushita, J.Appl.Phys. 61, 1068 (1987)
15. S. Nojima and H.Asahi, T.Ikoma, J.Appl.Phys. 61, 1073 (1987)
228
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
16. J.Jimenez, P.Hemandez, and J.A. de Saja, J.Bonnafe, Phys.Rev.B 35. 3832 (1987)
17. K.H. Schoenbach.V.K. Lakdawala, R.Germer, and S.T. Ko, J.AppI.Phys. 63, 2460
(1988)
18. S.T. Ko, V.K. Lakdawala, K.H. Schoenbach, and M.S. Mazzola. J.Appl.Phys. 67.
1124(1990)
19. K.H. Schoenbach,R.P. Joshi, and F. Peterkin, R.L.Druce. J.Appl.Phys. 77. 5208
(1995)
20. V.Kazukauskas, J. Storasta and J.-V. Vaitkus, J. Appl. Phys. 80,2269 (1996)
21. Y.Beaullieu and J.B. Webb, J.L. Brebner. J. Appl. Phys.79. 1772 (1996)
22. S.Ehret, H.Schneider. C. Schonbein, G.Bihlmann. and J.Fleissner. Appl.Phys.Lett. 69.
931 (1996)
23. J. A. Hombeck and J. R. Haynes, Phys. Rev. 97, 3 11 (1955)
24. J. R. Haynes and J. A. Hombeck, Phys. Rev. 100, 606 (1955)
25. Yu. Vaitkus and V. Grivitskas, Sov. Phys. Semicond. 15. 1102 (1981)
26. K. L. Luke and L. Cheng, J.Appl.Phys. 61, 2282 (1987)
27. J. Ch. Fontaine, S. Bartle, J.P. Ponpon, J.P. Schunk and P.SifFert. Meas. Sci. Technol.
5, 47(1994)
28. E. Gaubas and J. Vanhellemont, J. Appl. Phys. 80. 6293 (1996)
29. R. Sedlacik, F.Karel, J.Oswald. A.Fejfar, I.Pelant, J. Kocka, Thin Sol.Films 255,269
(1995)
30. G. K. Wertheim, Phys. Rev. 109,1086 (1958)
31. S.K. Pang and A. Rohatgi, J. Appl.Phys. 74, 5554 (1993)
229
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
32. H. Kuwano, M.Abe and S. Okubo, J. Appl. Phys., 51,437 (1980)
33. A.G. Aberle and J. Schmidt, R. Brendel, J. Appl. Phys. 79, 1492 (1996)
34. R. N. Zitter, A.J. Strauss, and A.E. Attard, Phys. Rev. 115, 266 (1959)
35. R. Fastov and Y.Nemirovsky, Appl Phys.Lett. 55, 1882 (1989)
36. J. R. Lindle, J. R. Meyer, C. A. Hoffman, and F.J. Bartoli. G. W. Turner and H.K.
Choi, Appl. Phys.Lett. 67. 3153 (1995)
37. R.H. Bube, Proc.IRE, 1836 (1955)
38. M. Buragohain and K. Barua, Indian J. Phys. 61 A, 559 (1987)
39. C. Capon. J.De Baets, A.M. De Cubber, H. De Smet. A. Van Calster, and J.
Vanfletem, Phys. Stat. Sol. (a) 142, 127 (1994)
40. D. Samanta, B.Samanta, A.K. Chaudhuri, S. Ghorai and U. Pal, Sem.Sci.Techn. 11.
548(1996)
41. A.V. Lyubchenko, B.M. Bulakh, I.A. Gurina, and S.M. Krasikova, Sov. Phys.
Semicond. 10, 545 (1976)
42. V.A. Smyntyna, Sov. Phys. Semicond.17. 424 (1983)
43. C. Manfredotti, R. Murri, E.Pepe, and D. Semisa. Phys. Stat. Sol. (a) 20.477 (1973)
44. N.V. Joshi and S. Swanson, Solid-St. Electron. 30, 105 (1987)
45. J. Orenstein and M. A. Kastner, Phys. Rev. Lett. 46, 1421 (1981)
46. J. Orenstein, M. A. Kastner and V. Vaninov, Phil. Mag. B 46, (23) 1982
47. R. Pandya and B. A. Khan, J. Appl. Phys. 62, 3244 (1987)
48. G. J. Adriaenssens and P.Nagels, J. Non-Cryst. Sol. 114, 100 (1989)
49. T. E. Dyer, J . M. Marshall and J. F. Davies, Phil. Mag. 69, 509 (1994)
230
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
50. L. H Chu, Y. F. Chen, D. C. Chang and C. Y. Chang, J. Phys.: Condens. Matter 7.
4525 (1995)
51. C. H. Qiu, W. Melton, M. W. Leksono, J. I. Pankove, B. P. Keller, and S. P.
DenBaars, Appl. Phys. Lett. 69, 1282 (1996)
52. B. Dulieu, J. Bullot, J.Wery, M. Richard and L. Brohan, Phys. Rev. B 53. 10641
(1996)
53. M. S. lovu, E. P. Kolomeiko, and S.D. Shutov, Semicond. 31, 710 (1997)
54. A. Werner, M. Kunst, G. Beck, J. Christen, G. Weimann, Phys. Rev. B. 34. 5918
(1986)
55. K. D. Cummings, S. J. Pearton, and G.P. Vella-Coleiro, J. Appl. Phys. 60. 1676
(1986)
56. Y. Fujisaki and Y. Takano, J. Appl. Phys. 61, 2910 (1987)
57. M. C. Chen, Appl. Phys. 64, 946 (1988)
58. K. L. Luke and L. Cheng, J. Electrochem. Soc.135, 957 (1988)
59. W. Jantz, Th. Frey, and K. H. Bachem, Appl. Phys.A 45, 225 (1988)
60. R. J. Gutmann, C.S. Campbell, M. C. Heimlich, J. M. Borrego, D. Bliss. J.
Electrochem. Soc. 135, 3795 (1989)
61. M. S. Wang and J. M. Borrego, J. Electrochem. Soc.137, 3648 (1990)
62. A. Buczkowski, Z. M. Radzimski, G.A. Rozgonyi, and F. Shimura, J. Appl. Phys. 69.
6495 (1991)
63. A. Buczkowski, K. Katayama, G.A. Rozgonyi, and F. Shimura, Appl. Phys. Lett. 60.
1229(1992)
231
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
64. H. Daio and F. Shimura, Jpn. J. Appl. Phys. 32, L 1792 (1993)
65. H .C. Neitzert, W. Hirsch, and M. Kunst. J. Appl. Phys. 73, 7446 (1993)
66. A.W. Stephens and M. A. Green, J. Appl. Phys. 74, 6212 (1993)
67. X. Le Cleac’h, Sol. St. Commun. 85. 799 (1993)
68. A. Buczkowski, G.A. Rozgonyi, and F. Shimura, K. Mishra, J. Electrochem. Soc.
140, 3240 (1993)
69. F. Wunsch, G. Schlichthorl and H. Tributsch, J. Phys. D: Appl. Phys. 26, 2041 (1993)
70. P. A. Borodowskii. A. F. Buldygin, and V. G. Remeznik. Semicond. 28, 1159 (1994)
71. A.W. Stephens, A. G. Aberle and M. A. Green, J. Appl. Phys. 76, 363 (1994)
72. C. Swiatkowski, A. Sanders, K. D. Buhre, and M. Kunst, J. Appl. Phys. 78, 1763
(1995)
73. H .C. Neitzert, W. Hirsch, M. Kunst and M. E. A. Nell, Appl. Optics 34, 676 (1995)
74. M. Schofthaler and R. Brendel, J. Appl. Phys. 77,3162 (1995)
75. J. A. Eikelboom, C. Leguijt, C. F. A. Frumau, A. R. Burgers, Sol. Energ. Mat. Sol.
Cells 36, 169(1995)
76. A. Kaniava, A. L . P. Rotondaro, and J. Vanhellemont, U. Menczigar, E. Gaubas.
Appl. Phys. Lett. 67, 3930 (1995)
77. H .C. Neitzert and M. Kunst, J. Vac. Sci. Technol. A 13, 2753 (1995)
78. C. H. Ling, H. T. Teoh, W. K. Choi, T. Q. Zhou and L. K. Ah, Mater. Sci. Forum
173-174, 255 (1995)
79. H .C. Neitzert and W. Hirsch, Phys. Stat. Sol. (a) 151, 371 (1995)
232
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
80. Y. Kitagawara, T. Yoshida, T. Hamaguchi, and T. Takenaka, J. Electrochem. Soc.
142, 3505 (1995)
81. C. Swiatkowski, M. Kunst, Appl. Phys. A 61, 623 (1995)
82. W. K. Choi, L. K. Ah, and Y. M. Chan, A. Raman. J. Electrochem. Soc. 142, 1651
(1995)
83. Y. L. Khong, J. Electrochem. Soc. 142, L 74 (1995)
84. A. Cuevas, P. A. Basore, G. Giroult-Matlakowski, C. Dubois, J. Appl. Phys. 80, 3370
(1996)
85. E. Gaubas and J. Vanhellemont, J. Appl. Phys. 80, 6293 (1996)
86. M. Ichimura, H. Yoshida, and A. Usami, J. Electr. Mater. 25, 1088 (1996)
87. A.W. Stephens and M. A. Green, J. Appl. Phys. 80, 3897 (1996)
88. E. Gaubas, A. Kaniavaand J. Vaitkus, Semicond. Sci. Technoi. 12, I (1997)
89. M. Ichimura, T. Makino, H. Asakura, A. Usami, E. Morita and E. Arai, Jpn. J. Appl.
Phys. 36, L 839 (1997)
90. J. R. Elmiger, R. Schieck, and M. Kunst, J. Vac. Sci. Technoi. A 15, 2418 (1997)
91. E. Gaubas, A. Kaniava and J. Vaitkus, Semicond. Sci. Technoi. 12, 1 (1997)
92. M. Ichimura, H. Tajiri, E. Morita, N. Yamada, and A. Usami, Appl. Phys. Lett. 70,
1745(1997)
93. G. D. Arndt, W. H. Hartwig, and J. L. Stone, J. Apll. Phys. 39, 2653 (1968)
94. W. H. Hartwig and J. J. Hinds, J. Appl. Phys., 40, 2020 (1969)
95. J. J. Hinds and W. H. Hartwig, J. Appl. Phys., 42, 170 (1971)
233
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
96. J.C Hensel, and T. G. Phillips, Proceedings o f the Twelfth International Conference
on the Physics o f Semiconductors, Stuttgart, 1974, ed. M. H. Pilkuhn, Teubner,
Stuttgart, 1974, 51
97. J. Beutel, J. Appl. Phys. 46,4649 (1975)
98. S. S. Collier, A .K. Weiss, and R.F. Reithel, Photogr. Sci. Eng. 20, 54 (1976)
99. J.P. Spoonhower, Photogr. Sci. Eng. 24, 130 (1980)
100.R.J. Deri and J.P. Spoonhower, Photogr. Sci. Eng. 25, 89 (1981)
101.R.J. Deri and J.P. Spoonhower, J. Phys. D: Appl. Phys.15, 1071 (1982)
102.R.J. Deri and J.P. Spoonhower, J. Phys. D: Appl. Phys.15, 1059 (1982)
103.R. S. Eachus and J.P. Spoonhower, Photogr. Sci. Eng. 26,20 (1982)
104.R.J. Deri and J.P. Spoonhower, Phys. Rev. B 25,2821 (1982)
105.R.J. Deri and J.P. Spoonhower, Appl. Phys. Lett. 43,65 (1983)
106.R.J. Deri and J.P. Spoonhower, Rev. Sci. Instrum. 55, 1343 (1983)
107.R.J. Deri, J.P. Spoonhower, and J. F. Hamilton, J. Appl. Phys. 57, 1968 (1985)
108.R.J. Deri and J.P. Spoonhower, Photogr. Sci. Eng. 28, 92 (1984)
109.R.J. Deri and J.P. Spoonhower, J. Appl. Phys. 57,2806 (1985)
110.J.P. Spoonhower and R.J. Deri, Phys. Stat. Sol. (a) 133, 301 (1985)
111. J. P. Spoonhower and R.J Deri, J. Imag. Sci. 31, 141 (1987)
112.S. Damaskinos, A. E. Dixon, G. D. Roberts, and I. R. Dagg, J.Appl. Phys. 60, 1681
(1986)
113.N. S. Allen, Y. S. Chow, F. Thompson, T. S. Jewitt, M. R. Homby and L.A.
Simpson, J. Photochem. Photobiol. A : Chem. 60,369 (1991)
234
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
114.V. N. Semenenko, G. V. Belokopytov, and D. E. Ryabov, Phys. Solid State 35. 889
(1993)
115.V. Subramanian, V. R. K. Murthy and J. Sobhanadri, Pramana - J. Phys. 44, 19
(1995)
116.G. F. Novikov and B.I. Golovanov, J. Imag. Sci. Technoi. 39, 520 (1995)
117.5. A. Buldygin, S. V. Golod, V. V. Preobrazhenskii, B. P. Semyagin, E. M. Skok,
D.G. Tarlo, Mat. Sci. Eng. B 44,223 (1997)
118.5. Grabtchak. and G. Novikov, Zhoumal Nauchnoi i Prikladnoi Fotografii I
Kinematografii (Soviet Journal of Scientific and Applied Photography and
Cinematography) (in Russian) 33, 371 (1988)
119.G. Novikov, S. Grabtchak, and M. Alfimov, Zhoumal Nauchnoi i Prikladnoi
Fotografii I Kinematografii (Soviet Journal of Scientific and Applied Photography
and Cinematography) (in Russian) 35, 18 (1990)
120.5. Grabtchak, G. Novikov, L. Moiseeva, M. Lubovskii, and M. Alfimov. Zhoumal
Nauchnoi i Prikladnoi Fotografii I Kinematografii (Soviet Journal of Scientific and
Applied Photography and Cinematography) (in Russian) 35, 134 (1990)
121.5. Grabtchak and M. Cocivera, Phys.Rev.B. 50, 18219 (1994)
122.5. Grabtchak and M. Cocivera, Proceedings o f the Symposium on Diagnostic
techniques fo r semiconductor materials, 186th Meeting of the Electrochemical
Society, Miami Beach, Florida October 9-14, 1994. Proceedings Volume 94-33,171
(The Electrochemical Society, 1994)
permission of the copyright owner. Further reproduction prohibited without permission.
123.5. Grabtchak. and M. Cocivera, M. (1995), Progress in Surface Science. 50, Nos 1-4.
305 (1995)
124.5. Grabtchak, Interface, 5, N o.4,57 (1996)
125.5. Grabtchak and M. Cocivera, J. Appl. Phys. 79, 786 (1996)
126.5. Grabtchak and M .Cocivera, Proceedings o f the Symposium on Diagnostic
techniques fo r semiconductor materials, 191st Meeting of the Electrochemical
Society, Montreal, Quebec, May 4-9, Proceedings Volume 97-12, 235 (The
Electrochemical Society, 1997) (1997)
127.Grabtchak, S. and Cocivera, M. Eighth Canadian Semiconductor Technology
Conference (Ottawa, Aug. 12 - 15, 1997), Abstracts, 123
128.5. Grabtchak and M. Cocivera, be published in Proceedings o f the Symposium on
Electrically Based Microstructural Characterization II, Fall Meeting of the Material
Research Society, December 1-5, Boston, 1997
129.5. Grabtchak and M. Cocivera, Phys. Rev. B 58, issue 7 (1998)
130.5. Grabtchak and M. Cocivera, (accepted) Phil. Mag. B (1998)
131.5. Grabtchak and M. Cocivera, be published in Phys. Rev. B (1998)
132.5. Grabtchak, M. Cocivera, D. Landheer, workshop Materials Technology '98,
University of Waterloo, 13 June 1998
133. S. Grabtchak and M. Cocivera, workshop Materials Technology ’98, University o f
Waterloo, 13 June 1998
134. J.C. Slater, Rev. Mod. Phys.18,441 (1946)
236
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
135. H.M. Altshuller in Handbook o f Microwave Measurements, volume 2, ed. by
M.Suchner, and J. Fox, Interscience Publishers, New York (1963), chapter 9
136.L. J. Buravov and I. F. Shchegolev, Prib. Tek. Eksp., 2 171 (1971)
137. S. K. Khanna, E. Ehrenfreund, A. F. Garito, and A. J. Heeger, Phys. Rev.B 10, 2205
(1974)
138.B. Lehndorff, Meas. Sci. Technoi. 3, 822 (1992)
139. R. Hutcheon, M. de Jong, and F.Adams, J. Microw. Pow. Electromagn. En.
27.
87
(1992)
140. L. D. Landau and E. M . Lifshitz, Electrodynamics o f Continuous Media, 2nd
edition, Pergamon Press, 1984
141.C. Kittel, Introduction to Solid State Physics, 5th edition, J. Wiley & Sons. 1976
142.J. A. Osborn, Phys. Rev. 6 7 , 351 (1945)
143.G. Dresselhaus, A. F. Kip, and C. Kittel, Phys. Rev. 100, 618 (1955)
144.T.S. Benedict and W.Shockley, Phys. Rev. 89, 1152 (1953)
145.E. N. Glezer, Y. Siegel, L. Huang, and E. Mazur, Phys. Rev.B 51, 6959 (1995)
146. V. S. Vavilov, V. A. Zayatz, and V. N. Murzin, Eksitony v Poluprovodnifcah
[Excitons in Semiconductors] (Nauka, 1971, Moscow), 32.
147.F. Schmidlin, Phys. Rev. B 16, 2362 (1977)
148.H. Scherand E. W. Montroll, Phys. Rev. B, 12,2455 (1975)
149. J.M. Marshall, H. Michiel, and G. J. Adriaenssens, Phil. Mag. B 4 7 , 211 (1983)
150. J. Tauc, Semiconductors and Semimetals, edited by J. I. Pankove (Academic Press,
Orlando,1984), 21 B, Chapter 9
237
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
151. J.M. Marshall, Phil. Mag. 36, 959 (1977)
152. H. Naito, T. Nagase, T. Ishii, M. Okuda, T. Kawaguchi, and S. Maruno, J. noncrystalline Solids, 198-200, 363 (1996)
153.J. Nolandi, Phys. Rev. B 16, (4466) 1977
154.G. Pfister and H. Scher, Adv. Phys. 27, (747) 1978
155.J. Orenstein and M. A. Kastner, Sol. St. Comm. 40, (85) 1981
156.J. M. Marshall and C. Main, Phil. Mag. B 47, (471) 1983
157.H. Michiel and G. J. Adriaenssens, Phil. Mag. B 51, (27) 1985
158.G. Seynhaeve, G. J. Adriaenssens and H. Michiel, Sol. St. Comm. 56, (323) 1985
159.V. Vasudevan and J. Vasi, J. Appl. Phys. 74, (3224) 1993
160.T. Tiedje, Semiconductors and Semimetals, edited by J. I. Pankove (Academic Press.
Orlando. 1984), 21 C, Chapter 6
161.S. Weng and M. Cocivera, J. Electrochem. Soc. 139, 3220 (1992)
162.J. Bethin, T. G. Castner, and N. K. Lee, Sol. St. Comm. 14, 1321 (1974)
163. G. W. Castellan and F. Seitz, in Semiconducting Materials, ed. by H. K. Henisch
(Buttersworth, London, 1951), 8.
164. F. A. D’Altroy and H. Y. Fan, Phys. Rev. 103, 1671 (1956)
165.H. F. Hess, K. DeConde, T. F. Rosenbaum, G. A. Thomas, Phys. Rev. B 25, 5578
(1982)
166.T. G. Castner, N. K. Lee, G. S. Cieloszyk and G. L. Salinger, Phys. Rev.Lett. 34,
1627(1975)
238
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
167. S. Katsumoto, in Anderson Localization, Proceedings o f the International
Symposium, Tokyo, Japan, 1987 (Springer-Verlag, Berlin, 1988), 45.
168. D. G. Frood, Proc. Phys. Soc. 75, 1269 (1968)
169. N. Mott, Conduction in Non-Crystalline Materials, (Clarendon Press. Oxford.
1993), 52
170. H. Kamimura and H. Aoki, The Physics o f Interacting Electrons in Disordered
Systems (Clarendon, Oxford, 1989)
171.K.F. Berggren, J. Chem. Phys. 60. 3399 (1974)
172.M. Caprizzi, G. A. Thomas. F. DeRora. R. N. Bhatt. and T. M. Rice. Phys. Rev. Lett.
44, 1019(1980)
173.P. Edwards, M. J. Sienko, Intern. Rev. Phys.Chem. 3, 83 (1983)
174. S. Belkouch, D. Landheer, D.P. Masson, S.R. Das, T. Quance, L. LeBrun, and S.L.
Rolfe, J. Vac. Sci. Technoi. A. 16, 860 (1998); D. Landheer, D.P. Masson. S.
Belkouch, S.R. Das, L. LeBrun, and J.E. Hulse, J. Vac. Sci. Technoi. A. 16, 834
(1998)
175.R. H. Bube and H. E. MacDonald, Phys. Rev. 128,2062, 2071 (1962)
176.M. R. Burd and R. Braunstein, J. Phys. Chem. Solids 49, 731 (1988)
177.G. M. Martin, J. P. Farges, G. Jacob, J. P. Hallais, G. Poiblaud, J. Appl.Phys. 51,
2840(1980)
178.M.L. Sadowski and M. Grynberg, J. Phys. Chem. Solids 56,619 (1995)
179. K. S. Champlin and R. R. Crongard, IRE Trans. Microwave Theory Tech. MTT-9.
545 (1961)
239
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
180. J. von Behren, P. M. Fauchet, E. M. Chimowitz and C. T. Lira, Mat. Res. Soc. Proc.
452, 565 (1997)
181.M. Alibzhonov, M. A. Karimov, M. S. Saidov, and N .Kh. Yuldashev, Semicond.
30, 827(1996)
182. D. C. Look, in Semiconductors and Semimetals 19 , ed. by R. K Willardson and
Albert C. Neer ( Academic Press. New Y ork, 1983), 75-170
183.J.S.Blakemore, Semiconductor Statistics (Dover Publications, Inc., New York. 1987)
184.G.L. Belen’kii, A.V. Lyubchenko, and M.K. Sheinkman, Sov. Phys.- Semicon. 2.
445(1969)
240
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
IMAGE EVALUATION
TEST TARGET (Q A -3 )
150mm
IM/4GEE. In c
1653 East Main Street
Rochester. NY 14609 USA
Phone: 716/462-0300
Fax: 716/288-5989
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Документ
Категория
Без категории
Просмотров
0
Размер файла
9 042 Кб
Теги
sdewsdweddes
1/--страниц
Пожаловаться на содержимое документа