close

Вход

Забыли?

вход по аккаунту

?

MICROWAVE STUDIES OF AMORPHOUS SILICON (SPIN, RESONANCE, RELAXATION)

код для вставкиСкачать
INFORMATION TO USERS
This reproduction was made from a copy of a document sent to us for microfilming.
While the most advanced technology has been used to photograph and reproduce
this document, the quality of the reproduction is heavily dependent upon the
quality of the material submitted.
The following explanation of techniques is provided to help clarify markings or
notations which may appear on this reproduction.
1 The sign or "target" for pages apparently lacking from the document
photographed is "Missing Page(s)" If it was possible to obtain the missing
page(s) or section, they are spliced into the film along with adjacent pages This
may have necessitated cutting through an image and duplicating adjacent pages
to assure complete continuity
2 When an image on the film is obliterated with a round black mark, it is an
indication of either blurred copy because of movement during exposure,
duplicate copy, or copyrighted materials that should not have been filmed For
blurred pages, a good image of the page can be found in the adjacent frame If
copyrighted materials were deleted, a target note will appear listing the pages in
the adjacent frame.
3 When a map, drawing or chart, etc , is part of the material being photographed,
a definite method of "sectioning" the material has been followed It is
customary to begin filming at the upper left hand corner of a large sheet and to
continue from left to right in equal sections with small overlaps If necessary,
sectioning is continued again-beginning below the first row and continuing on
until complete
4 For illustrations that cannot be satisfactorily reproduced by xerographic
means, photographic prints can be purchased at additional cost and inserted
into your xerographic copy These prints are available upon request from the
Disseitations Customer Services Department
5 Some pages in any document may have indistinct print In all cases the best
available copy has been filmed
University
Microfilms
International
300 N Zeeb Road
Ann Arbor, Ml 48106
8502064
Askew, Thomas Rendall
MICROWAVE STUDIES OF AMORPHOUS SILICON
University of Illinois at Urbana-Champaign
University
Microfilms
I n t e r n S l t l O n c U 300 N Zeeb Road, Ann Arbor, Ml 48106
PH.D. 1984
(
MICROWAVE STUDIES OF
AMORPHOUS SILICON
BY
THOMAS RENDALL ASKEW
B.A., Gordon College, 1977
M.S., University of Illinois, 1982
THESIS
Sutnitted in partial fulfillment of the requirements
for the degree of Doctor of Philosophy in Physics
in the Graduate College of the
University of Illinois at Urbana-Champaign, 1984
Urbana, Illinois
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
THE GRADUATE COLLEGE
OCTOBER 1984
W E HEREBY RECOMMEND THAT T H E THESIS BY
THOMAS RENDALL ASKEW
ENTITLED
MICROWAVE STUDIES OF AMORPHOUS SILICON
BE ACCEPTED IN PARTIAL F U L F I L L M E N T OF THE REQUIREMENTS FOR
T H E DEGREE OF_
DOCTOR OF PHILOSOPHY
t JUtffA
Director of Thesis Researcli
Head of Department
Committee on Final Examination!
fl/^7
Chairperson
w§r£Tc
f Required for doctor's degree but not for master's
o 117
wnu<>uiwi«uiu«uinuMuiaii
twMwtwTiIMWUBammw »
iii
MICROWAVE STUDIES OF
AMORPHOUS SILICON
Thomas Rendall Askew, Ph.D.
Department of Physics
U n i v e r s i t y of I l l i n o i s a t Urbana-Champaign, 1984
t
Electron paramagnetic resonance and e l e c t r o n spin r e l a x a t i o n r a t e s of
the i n t r i n s i c paramagnetic c e n t e r (g=2.005?~) In amorphous s i l i c o n have
been studied in the 0.3 - 1.2 K temperature range.
Various sample
p r e p a r a t i o n techniques were u s e d , including ion i m p l a n t a t i o n ,
and vacuum evaporation.
sputtering,
The temperature dependence of t h e spin l a t t i c e
r e l a x a t i o n r a t e s depends somewhat on sample preparation but i s alwas very
c l o s e to simple Tn power laws.
i n t o two ranges:
The n values observed in t h i s study f a l l
2.09 - 2.36 and 3.26 - 3 . 4 7 .
Comparison of measurements
a t 9.3 GHz and 16.5 GHz i n d i c a t e s that the observed r a t e s a r e independent
or very nearly independent of microwave frequency and a p p l i e d magnetic
field.
Conventional one and two phonon spin l a t t i c e r e l a x a t i o n mechanisms
cannot account for the observed temperature dependences.
A theory
i n v o l v i n g spin l a t t i c e r e l a x a t i o n by coupling to a d i s t r i b u t i o n of two
l e v e l systems (TLS) i s p r e s e n t e d .
The t h e o r y i s adjusted so t h a t i t can
be applied in the relevant temperature range and i t s p r e d i c t i o n s are
compared to the experimental r e s u l t s .
An attempt i s made t o i d e n t i f y
TLS and the TLS-spin coupling u s i n g e l e c t r o n - n u c l e a r double resonance
(ENDOR), microwave frequency d i e l e c t r i c a b s o r p t i o n , and magnetization
studies.
The r e s u l t s of t h e s e experiments a r e discussed.
the
iv
ACKNOWLEDGMENTS
The author would l i k e to express s i n c e r e a p p r e c i a t i o n to his a d v i s o r ,
Prof. Harvey J . Stapleton, for i n s p i r a t i o n , invaluable guidance, and
nearly i n f i n i t e p a t i e n c e .
Prof. S t a p l e t o n ' s enthusiasm f o r research and
devotion to h i s students exemplify the h i g h e s t level of u n i v e r s i t y
professorship.
The a s s i s t a n c e of Dr. Keith L. Brower of Sandia Laboratories was
e s s e n t i a l and is g r e a t l y appreciated.
Among my fellow graduate students I would p a r t i c u l a r l y like to thank
Dr. Douglas G. Stinson, Dr. J . Trevor Colvin, and Mr. P h i l i p J . Muench
for c r e a t i n g a friendly, s t i m u l a t i n g , and occasionally mischievous
environment in which to work.
F i n a l l y , I would l i k e to thank my w i f e , Mary L. Askew, for providing
the support, patience, and encouragement I needed to complete this course
of study.
This research was supported by the U. S. Department of Energy,
Division of Materials S c i e n c e s , under c o n t r a c t no. DE-AC02-76ER01198.
V
TABLE OF CONTENTS
Ch. 1
INTRODUCTION
1
1.1
The N a t u r e of Amorphous S i l i c o n
5
1.2
M o t i v a t i o n and Scope of P r e s e n t S t u d y
19
Ch. 2
ONE PHONON SPIN LATTICE RELAXATION PROCESSES
21
2.1
2.2
2.3
2.4
Ch. 3
The D i r e c t P r o c e s s
Two Phonon P r o c e s s e s
The D i r e c t P r o c e s s w i t h a Phonon B o t t l e n e c k
R e l a x a t i o n by Modes A s s o c i a t e d w i t h Disorder
EXPERIMENTAL EQUIPMENT AND TECHNIQUES
23
28
30
34
37
EPR S p e c t r o m e t e r
T e m p e r a t u r e Measurement and C o n t r o l
Magnetization Studies
Sample P r e p a r a t i o n s
Ion I m p l a n t a t i o n
E l e c t r o n Beam E v a p o r a t i o n
Sputtering
37
44
48
48
49
55
56
RESULTS AND INTERPRETATION
58
3.1
3.2
3.3
3.4
3.4.1
3.4.2
3.4.3
Ch. 4
4.1
4.2
4.3
4.4
4.4.1
4.4.2
4.4.3
Ch. 5
5.1
5.2
5.3
Appendix A:
EPR S p e c t r o s c o p y
58
Spin L a t t i c e R e l a x a t i o n Measurements - - R e s u l t s
65
Theory of Spin L a t t i c e R e l a x a t i o n by Two L e v e l S y s t e m s . . . . 75
A u x i l i a r y Experiments
96
Measurement of M a g n e t i z a t i o n as a F u n c t i o n of T e m p e r a t u r e . 9 6
ENDOR Study
99
D i e l e c t r i c Response Measurements
99
CONCLUSIONS
C o n c l u s i o n s C o n c e r n i n g Spin L a t t i c e R e l a x a t i o n i n
Amorphous Si l i c o n
C o n c l u s i o n s C o n c e r n i n g t h e S t r u c t u r e of Amorphous
Silicon
Comments on Spin L a t t i c e R e l a x a t i o n by Coupling t o Two
L e v e l Systems
COMBINED He-3 CRYOSTAT AND MICROWAVE CAVITY
101
101
102
103
104
REFERENCES
128
VITA
136
1
CHAPTER 1
INTRODUCTION
As traditionally taught and practiced, solid s t a t e physics has meant
crystal physics,
The reasons for t h i s range from such inane statements as
"the normal condition of solid matter i s c r y s t a l l i n e " to the cogent
realization that most of the analytic techniques of solid state physics
assume the existence of a periodic array of atoms (Ashcroft and Mermin,
1976).
Many solids lack any vestige of long range order, however, and
interest in these has grown rapidly in recent years.
Amorphous silicon
(a-Si) is such a material.
The great technological interest in a l l forms of silicon has resulted
in experiments which have played a pivotal role in the developing theory
of amorphous solids.
The f i r s t significant progress in this area was
motivated by Electron Paramagnetic Resonance (EPR) studies of slightly
donor doped, very pure, crystalline silicon ( c - S i ) .
Fletcher and Feher
found clear evidence t h a t as the donor concentration is decreased, the
l a s t (5th) donor valence electron ceases to exist in a delooalized
impurity band and becomes localized near individual atoms (Feher, et a l . ,
1955).
P.W. Anderson offered a theoretical explanation of t h i s effect
(Anderson, 1958), now called a metal-insulator t r a n s i t i o n , which was
largely ignored due to i t s conceptual complexity and seemingly intractable
mathematics (Anderson, 1978).
2
For many years i t was thought t h a t only a s m a l l class of materials
could be prepared as amorphous s o l i d s and that t h i s property was therefore
unusual.
true.
During the 1960's i t became c l e a r that e x a c t l y the opposite i s
The modern viewpoint i s that " n e a r l y a l l m a t e r i a l s can, i f cooled
f a s t enough and far enough, be prepared as amorphous solids" (Turnbull,
1969).
The emphasis here i s on speed i n quenching from a m e l t .
Modern
o
quenching equipment can maintain c o o l i n g rates of up to 10 K/sec over
temperature ranges of roughly 1000 K.
Silicon i s one of a s e l e c t group of
m a t e r i a l s which cannot be made amorphous by rapid quenching.
Other
members of t h i s group are Ge, HO and c e r t a i n e l e m e n t a l metals, among
them Fe, Co, Au and Bi.
Except for Au, a l l of t h e s e materials can oe made
amorphous by condensation from the vapor phase.
In addition t o vapor
condensation, there are o t h e r methods of making a - S i which w i l l be
discussed l a t e r .
To d a t e , a l l a t t e m p t s to make amorphous Au h a v e
f a i l e d . ( Z a l l e n , 1983)
In the l a t e 1960's Cohen, F r i t z s c h e and Ovshinsky (1969) proposed
t h a t gradually i n c r e a s i n g the d i s o r d e r of a c r y s t a l l i n e semiconductor
would cause t a i l s to form a t each edge of the band gap.
They were led to
t h i s r e s u l t by the a p p l i c a t i o n of a p e r t u r b a t i o n t o conventional band
theory which r e s u l t e d in a small, randomly d i r e c t e d , nonconservation of
momentum.
As the p e r t u r b a t i o n was i n c r e a s e d , the band t a i l s grew in s i z e
and extent i n t o the gap u n t i l they overlapped in t h e middle.
I t i s now
g e n e r a l l y believed t h a t t h e s e s t a t e s i n the middle of the gap a r e highly
l o c a l i z e d , with a small associated m o b i l i t y , and t h a t conduction takes
p l a c e by a thermally a c t i v a t e d hopping process from one l o c a l i z e d state t o
3
another.
Mott was able to show (1967) t h a t t h i s process should give a
conductivity p r o p o r t i o n a l to exp(-BT
) , B depending on t h e density of
l o c a l i z e d s t a t e s and r a d i a l e x t e n t of t h e i r wavefunctions.
A-Si and a-Ge
provided dramatic confirmation of t h i s p r e d i c t i o n a s they f o l l o w the
hopping conductivity law over e i g h t orders of magnitude (Knotek, 1975).
The band t a i l s in these m a t e r i a l s completely obscure the band gap and in
a-Si the s t a t e s i n the gap c o n t r o l the c o n d u c t i v i t y , via t h e hopping
mechanism, from 40K to about 300K.
A c e r t a i n c l a s s of these s t a t e s in t h e
gap a r e paramagnetic, and t h i s study focuses upon them.
The
i n v e s t i g a t i o n s were c a r r i e d out a t temperatures well below 40K, where a - S i
i s e s s e n t i a l l y an i n s u l a t o r .
The conduction band t a i l , reaching down i n t o the band gap, provides
an environment where a m e t a l - i n s u l a t o r t r a n s i t i o n can occur.
The ideas of
Mott, Cohen, F r i t z s c h e and Ovshinsky have been l a r g e l y i n t e g r a t e d into a
model, best described by quoting Mott d i r e c t l y :
"We now understand t h a t in any n o n c r y s t a l l i n e system the
lowest s t a t e s in the conduction band are ' l o c a l i z e d ' , t h a t i s
t o say, t r a p s , and t h a t on t h e energy s c a l e t h e r e Is a c o n t i n uous range of such l o c a l i z e d s t a t e s leading from the bottom of
t h e band up to a c r i t i c a l energy, called t h e mobility edge,
where s t a t e s become nonlocalized or extended." (Mott, 1978)
When the Fermi energy of the m a t e r i a l c r o s s e s the mobility edge a
m e t a l - i n s u l a t o r t r a n s i t i o n occurs (Mott, e t a l . , 1975).
between l o c a l i z a t i o n , d i s o r d e r and the m e t a l - i n s u l a t o r
continues to be an area of i n t e n s e research.
The i n t e r p l a y
transition
I t i s i n t e r e s t i n g to note
t h a t almost 30 years a f t e r F e h e r ' s seminal work on s l i g h t l y donor doped
4
c-Si (Feher, et a l . ,
1955), people a r e again studying l o c a l i z a t i o n through
t h e magnetic resonance of t h i s m a t e r i a l (Paalanen, e t a l . , 1 9 8 4 ) .
A-Si has
not played a c e n t r a l role i n the study of l o c a l i z a t i o n because the Fermi
l e v e l is pinned a t mid-gap (Adler and Yoffa, 1976).
The c e n t r a l problem with the Cohen, F r i t z s c h e and Ovshinsky (CFO)
theory i s t h a t i t f a i l s miserably f o r the oxide and chalcogenide g l a s s e s .
These m a t e r i a l s lack any v e s t i g e of long range order and therefore t h e
theory p r e d i c t s huge band t a i l s and a completely obscured band gap.
The
chalcogenide g l a s s e s (such a s As_S_) are generally t r a n s p a r e n t in the
infrared while the oxide g l a s s e s (such as SIO ) are almost a l l
t r a n s p a r e n t in the v i s i b l e .
This s t a r k evidence of broad, well defined
band gaps d i r e c t l y c o n t r a d i c t s the key assumption of t h e CFO theory,
namely t h a t the degree of long range order governs the e x i s t e n c e of band
gaps.
Mott and coworkers were among t h e f i r s t to ask the simple and very
relevant q u e s t i o n :
How can g l a s s , w i t h a l l i t s compositional v a r i a t i o n
and contamination and generally disordered s t r u c t u r e be t r a n s p a r e n t ,
I s , have a band gap 7
that
The explanation is t h a t the band gap r e s u l t s from
each atom having t h e r i g h t number and type of neighbors so t h a t a l l , or
almost a l l , valence e l e c t r o n s are occupied in chemical bonds (Mott, 1969
and 1978).
This l o c a l bonding p i c t u r e of allowed s t a t e s i s , of course, a
very old idea and one never abandoned by the chemists.
5
Due largely to the influence of Mott (1978; Mott and Davis, 1979),
Adler (1978, 1981), Fritzsche (1981), and J.C. Phillips (1979, 1981), the
local bonding picture now dominates the analysis of amorphous materials.
The band tails of the early CFO theory are seen as a consequence of
departure from chemical bonding symmetry (short range order) as opposed to
departure from long range order.
In this view, the CFO theory is
successful in slightly defective crystalline materials only because In
that case the presence of long range order and short range order are
intimately connected.
The nature of a-Si will now be discussed in the
context of the local bonding picture.
1.1 The Nature of Amorphous Silicon
Covalently bonded amorphous solids are generally classified according
to their bond coordination.
A useful concept is that of average
coordination number m, defined by J.C. Phillips (1979) as follows:
Consider a binary alloy
N(B) respectively.
A
X B.,_ X
where A and B have valences N(A) and
The average coordination number m is then
m = xN(A) + (1-x)N(B).
The extension to three or more componants is
straightforward.
Figure 1.1 is a two dimensional representation of a chalcogenide
glass, such as As 2 S-, taken from an early paper by Zachariesen (1932).
Notice that there is a good balance between the spatial degrees of freedom
and the bonding constraints, allowing bond distances and angles to be
nearly identical from site to site. Also notice that the bond bending is
6
Figure 1.1
Two Dimensional Representation of a Chalcogenide Glass
from Zachariesen (1932)
Figure 1.2
Two Dimensional Representation of Amorphous Arsenic
7
greatest for the divalent species.
The flexibility of covalent bond
angles is greatest when the coordination number is much less than the
number of valence electrons.
Under these conditions, there exists a great
variety of possible atomic orbital admixtures which can form the covalent
bond.
Bond flexibility is thus greatest for divalent species such as 0
and S, and smallest for tetravalent species such as Si, where the
coordination number equals the number of valence electrons.
Almost all
covalently bonded materials which can be made amorphous by quenching from
a melt have m values between 2 and 3.
near m = 2.5.
Glass forming tendency is greatest
The structure shown in figure 1.1 has a m value of 2.H as
do all of the stoichiometric chalcogenide glasses.
The structure of
amorphous SiO ? is derivable from figure 1.1 by replacing the trivalent
species with Si tetrahedra and moving to three dimensions.
The material
has a m value of 2.67, and is easily quenched from a melt.
Figure 1.2 is a two dimensional representation of amorphous As.
This
structure can be derived in a heuristic sense by removing the divalent
species from figure 1.1 and allowing the remaining atoms to relax.
Notice
that in contrast to figure 1.1, the bond lengths and angles are now
severely distorted from their equilibrium values.
If the bond lengths and
angles are not sufficiently flexibile, correct coordination is impossible
and dangling bonds and strain relieving defect structures will develop.
When such structures develop, the system is said to be overconstrained.
The system shown in figure 1.2, with a ra value of 3, is probably
overconstrained in two dimensions.
exists.)
(No relevant experimental data
It is certainly a glass in three dimensions as amorphous As can
8
be made without dangling bonds and v o i d s .
the t h r e e dimensional analog of f i g u r e
A-Si, with a m value of 4, i s
1.2.
There is a tendency i n the l i t e r a t u r e to use the following words
interchangeably:
noncrystalline.
amorphous, glassy (or v i t r e o u s ) , disordered and
In an attempt to be more p r e c i s e , and to s t r e s s the
difference between m a t e r i a l s l i k e a - S i (figure 1.2 plus dangling bonds and
voids) and l i k e A3 2 S, (figure 1.1), F r i t z s c h e (1980) has proposed the
following scheme:
Glasses are m a t e r i a l s with 2 < m < 3 , and m a t e r i a l s
with 3 < m < 4 are called overconstrained amorphous.
Materials with m < 2
are c a l l e d undercrosslinked amorphous, i n d i c a t i n g t h e i r c l o s e a s s o c i a t i o n
with polymeric systems.
Materials with m > H a r e amorphous m e t a l s .
u n c e r t a i n t y must be allowed near the boundary values m = 2 , 3 and 4 .
Some
For
i n s t a n c e , a n o n c r y s t a l l i n e material which can be quenched from a melt, has
a band gap, and d o e s n ' t have dangling bonds would be c a l l e d a g l a s s even
if m = 3 . 2 .
The p r i n c i p a l a l t e r n a t i v e to t h i s scheme i s t o define g l a s s
as a supercooled l i q u i d with a v i s c o s i t y g r e a t e r than x, where x i s some
l a r g e , but measureable, v a l u e .
The J . C . P h i l l i p s theory implies t h a t a-Si, with a m value of 4, i s a
badly overconstrained m a t e r i a l .
This i s indeed the case.
The f i r s t
d i r e c t evidence t h a t a-Si contains I n t e r n a l microstructure came from EPR
s t u d i e s of a - S i by Brodsky and coworkers a t IBM, Yorktown Heights, N.Y.
(Brodsky and T i t l e , 1969; Brodsky, e t a l . , 1970; Crowder, e t a l . , 1970)
They saw a s i n g l e , u n s t r u c t u r e d , i s o t r o p i c EPR l i n e a t g = 2.0055 _+ .0005
with a Lorentzian l i n e shape and a width of about 5 gauss.
This s i g n a l
I
9
wa3 3een in samples prepared by radio frequency sputtering, by vacuum
condensation of Si vapor created by electron beam heating, and by ion
implantation of Si, P, and As into c-Si.
The observed EPR signal was
virtually identical to that observed on c-Si surfaces by Feher (1959) and
later, in more detailed studies, by Haneman (1968) and coworkers.
These
surface signals were attributed to dangling bonds and were seen on cleaved
faces, on mechanically damaged (polished) surfaces, and in crushed
samples.
Brodsky and coworkers were able to show for the deposited
samples (sputtering and evaporation) that the EPR signal strength was
proportional to the film thickness.
The detected resonance was thus a
bulk property of a-Si and not a surface phenomenon.
Brodsky proposed that
the detected signal was due to dangling bonds on the Internal surface of
voids which are dispersed throughout a-Si.
The IBM group found that the EPR signal amplitude could be decreased
by annealing at temperatures between 100 C and 400 C.
They also found a
dramatic correlation between the EPR signal strength, the room temperature
DC conductivity, and the optical absorption at 2 microns.
This indicates
that the states associated with the EPR signal are located In the band gap
and control the electrical and optical properties of a-Si.
Above 400 C
crystallization begins and the physical properties of a-Si begin to
resemble those of polycrystalline Si.
Once it was clear that the physical properties of a-Si were being
controlled by defect states in the gap, great efforts were made to
characterize the defects and to discover preparation and/or annealing
10
techniques t o eliminate them.
This led to refinements in vacuum
evaporation and s p u t t e r i n g techniques (for review see Townsend, e t a l . ,
1976, and Stewart, 1983), and to a new deposition p r o c e s s , glow discharge
decomposition of s i l a n e (SiHj.) gas ( C h i t t i c k , 1970; Le Comber, et a l . ,
1972).
Ion implantation into c-Si was a l s o studied e x t e n s i v e l y and t h i s
technique w i l l be reviewed in chapter t h r e e .
al.,
E l e c t r i c p l a t i n g (Tauc, e t
1970) and chemical vapor deposition (Hasegawa, e t a l . , 1979) have
received l i t t l e a t t e n t i o n a s a-Si p r e p a r a t i o n techniques.
Analysis of x-ray and e l e c t r o n d i f f r a c t i o n data from a v a r i e t y of
sources i n d i c a t e s t h a t in a - S i almost every atom i 3 surrounded by four
neighbors, each 2.35 8 away, j u s t a s in the c r y s t a l l i n e s t a t e .
Second
n e a r e s t neighbors show s l i g h t d e v i a t i o n s from c r y s t a l l i n e spacing,
corresponding to bond angle d e v i a t i o n s of about 10° (rms).
The spread
i n c r e a s e s u n t i l e s s e n t i a l l y random l o c a t i o n s ocoour a t about the 10th
neighbor (Moss and Graczyk, 1969).
Small angle s c a t t e r i n g of x - r a y s and
e l e c t r o n s , a s well as high r e s o l u t i o n e l e c t r o n microscopy, shows i n d i r e c t
evidence of heterogeneous composition with r e l e v a n t length scales i n the
5 - 40 8 range (Shevchik and Paul, 1974).
These r e s u l t s and s i m i l a r
work by o t h e r s have been s u b j e c t to a v a r i e t y of i n t e r p r e t a t i o n s .
Some
have argued for the e x i s t e n c e of roughly s p h e r i c a l voids with a
d i s t r i b u t i o n of s i z e s from s i n g l e vacancies up to about 40 8 diameters.
Others have argued for a m i c r o c r y s t a l l i n e model where the g r a i n s have a
5 - 40 A s i z e range.
Some have proposed a combination of t h e two
models where f u l l y coordinated g r a i n s are embedded in reduced d e n s i t y
m a t e r i a l t h a t contains voids,
(for review see Mott and Davis, 1979)
11
Similar grain structures have been observed directly in a-Ge using high
resolution electron microscopy (Donovan and Heinemann, 1971).
Artificially developed structural models of a-Si have played an
important role in the interpretation of experimental data.
The first of
these was constructed by Polk (1971), using the following criteria:
(1)
all interior atoms are properly coordinated; (2) bond length variations
are less than 1%; (3) bond angle distortions are under 20°; (4) each new
atom is added in such a way that local strain is minimized.
The resulting
model contained 440 atoms and appeared to be completely homogeneous and
capable of Infinite extension.
It seemed therefore that fully coordinated
a-Si structures were possible, and might be created under appropriate
experimental conditions.
After the atomic coordinates of the model were
relaxed using iterative numerical techniques, the bond length variation
dropped to less than 0.2% and the distribution of bond angle deviations
was found to match the experimental results of diffraction studies (Polk
and Boudreaux, 1973).
The density of the relaxed model was found to be
99% of the crystalline value.
The reported density values for a-Si range
from 3 to 15% below that of the crystal, and this large density defect is
usually attributed to voids,
(for review see Mott and Davis, 1979)
The
density defect and the spin density observed by EPR are not well
correlated in a-Si (Brodsky, et al., 1972).
The year 1975 was marked by two significant events in the area of
a-Si research.
Haneman retracted much of his earlier work on EPR signals
due to surface states of c-Si (Lemke and Haneman, 1975).
The previously
12
observed signals were apparently due to adsorbed gasses and not dangling
bonds.
I t Is now believed t h a t broken bonds r e c o n s t r u c t on the surface of
o-Si so t h a t the only remaining dangling bonds are in regions of atomic
mismatch such a s s t e p s and microcracks.
Haneman's e a r l i e r work was t h e
main motivation behind the b e l i e f t h a t dangling bonds in a - S i are on t h e
surface of voids.
More dramatic was the announcement by Spear and Le Comber (1975) t h a t
a-Si made by glow discharge decomposition of s i l a n e gas could be doped
both p and n type.
This s u r p r i s i n g p r o p e r t y , in contrast t o other types
of a - S i and chalcogenide g l a s s e s , r e s u l t s from the incorporation of l a r g e
amounts of H (3 - 20 at.%) i n t o the s t r u c t u r e .
Since much of the I n t e r e s t
in a-Si was motivated by t h e desire to make cheap s o l a r c e l l s , many
research e f f o r t s shifted to hydrogenated a-Si, usually w r i t t e n a-Si:H,
when i t s doping behavior became c l e a r .
In a d d i t i o n , a-Si:H displays
normal, thermally activated conductivity (instead of hopping); complete
absence of the dangling bond EPR s i g n a l ; photoluminescence; and
photoconductivity; a l l c h a r a c t e r i s t i c of an i n t r i n s i c semiconductor
(Spear, 1977).
Material with these p r o p e r t i e s can a l s o be made by
s p u t t e r i n g where H i s added to the Ar s p u t t e r i n g gas (Paul, e t a l . , 1976).
I t i s i n t e r e s t i n g to note how these r e s u l t s f i t into t h e
J.C. P h i l l i p s theory of covalent coordination.
Glasses, with m values
between 2 and 3 , and s t r u c t u r e s l i k e figure 1 . 1 , can not be doped because
they are underconstrained.
The addition of an atom of d i f f e r e n t valence
will merely cause the system to reconfigure so a l l atoms a r e properly
13
coordinated.
A-Si, on the o t h e r hand, i s overconstrained a t m = 4 and can
not be doped because i t has s t a t e s in t h e band gap which pin t h e Fermi
l e v e l (Adler and Yoffa, 1976).
Si.
A-Si:H, more p r o p e r l y written as
H , seems to have the r i g h t balance between bonding c o n s t r a i n t s
and s p a t i a l degrees of freedom to allow doping.
This corresponds t o a m
value of 4 - 3x or 3.85 for samples which are 5% H.
The t o p o l o g i c a l
i m p l i c a t i o n s of t h i s appear to be i n t e r e s t i n g ( J . C . P h i l l i p s ,
1979b).
Kaplan and coworkers (1978) discovered t h a t films of a-Si made by
evaporation under u l t r a high vacuum (UHV) conditions can be heated in a H
plasma so as to completely e l i m i n a t e t h e EPR dangling bond s i g n a l .
This
led to a s e r i e s of H evolution and passivation experiments where t h e ba3ic
goal was to r e l a t e t h e number of dangling bond spins t o the number of H
atoms necessary to p a s s i v a t e the dangling bonds.
Various measurements
i n d i c a t e d t h a t 30 to 100 H atoms a r e required for each detected spin to
achieve p a s s i v a t i o n .
This led to a v a r i e t y of arguements (Adler, 1978 and
1981; Brodsky and Kaplan, 1979; Brodsky, 1981; J.C. P h i l l i p s ,
1979b)
attempting to explain how H can convert a given assumed s t r u c t u r e of a-Si
i n t o a given s t r u c t u r e of a-Si:H and e f f e c t such a r a d i c a l change i n
physical p r o p e r t i e s .
I t has r e c e n t l y been discovered t h a t a-Si:H c o n t a i n s aggregates of
s o l i d H which do not p a r t i c i p a t e in the p a s s i v a t i o n of dangling bonds
(Lohneysen, e t a l . , 1984; Graebner, et a l . , 1984).
The presence of these
aggregates i n v a l i d a t e s e a r l i e r counting arguments based on H e v o l u t i o n and
p a s s i v a t i o n and implys that t h e
raicrostructure
of a-Si:H i s even more
14
complicated than t h a t of a - S i .
Recent i n f r a r e d absorption spectroscopy
measurements (Chabal and P a t e l , 1984) i n d i c a t e the presence of gaseous
molecular H~ a t very high p r e s s u r e s (~2000 atm ) .
From t h e s t a n d p o i n t
of s t r u c t u r a l c o n s i d e r a t i o n s , e l e c t r o n i c p r o p e r t i e s , and magnetic
resonance, a-Si:H i s an i n t r i n s i c a l l y d i f f e r e n t m a t e r i a l .
Although a-Si:H
has many i n t e r e s t i n g spin dependent p r o p e r t i e s (Voget-Grote, e t a l . , 1980;
Carlos and Taylor, 1980 and 1982; S t r e e t , 1982; Depinna, e t a l . , 1982;
B o u l i t r o p , 1983), t h e i r a n a l y s i s i s probably of limited u t i l i t y in
e l u c i d a t i n g t h e s t r u c t u r e of a - S i and the n a t u r e of i t s paramagnetic
states.
The f a c t t h a t H passivation e l i m i n a t e s the g = 2.0055 EPR s i g n a l
i s s i g n i f i c a n t , however, and supports the dangling bond concept as I t s
origin.
The 10 y e a r long EPR s t u d i e s of a-Si a t IBM form the core of useful
data on the s u b j e c t ,
(for review see Thomas, e t a l . , 1978)
Other
s i g n i f i c a n t work in t h i s area h a s been performed by Stuke and coworkers a t
the University of Marburg, F.R.G. (Stuke, 1977; Voget-Grote, e t a l . , 1976
and 1980; Movaghar and Schweitzer, 1977 and 1978; Bachus, e t a l . , 1979);
v a r i o u s people a t Kanazawa U n i v e r s i t y , Japan (Hasegawa, e t a l . , 1977 and
1979; Shimizu, e t a l . , 1979; I s h i i , e t a l . , 1981j Suzuki, e t a l . , 1982);
and Gourdon and coworkers a t U n i v e r s i t y P a u l - S a b a t i e r , France (Gourdon, e t
a l . , 1981).
These s t u d i e s employed a g r e a t v a r i e t y of p r e p a r a t i o n
techniques and deposition methods and found t h a t the g = 2.0055 dangling
bond s i g n a l was u n i v e r s a l l y p r e s e n t u n l e s s l a r g e amounts of H were
d e l i b e r a t e l y i n c o r p o r a t e d into the sample.
This has led to t h e g e n e r a l
b e l i e f t h a t pure (unhydrogenated) a-Si i s i n t r i n s i c a l l y overconstrained
15
and therefore cannot be made without dangling bonds.
Although t h i s b e l i e f
i s in agreement with the phenomenology of t h e J.C. P h i l l i p s theory, t h e r e
i s no compelling t h e o r e t i c a l reason why t h i 3 should be t r u e .
I t has never
been shown t h a t the fully coordinated a-Si models developed by Polk (1971)
and others could be f u r t h e r relaxed by the i n t r o d u c t i o n of dangling bonds
and/or v o i d s .
Is f u l l y coordinated a-Si thermodynamioally a c c e s s i b l e 7
i s the t r u e ground s t a t e of a-Si?
Is
These a r e open q u e s t i o n s .
There i s general agreement among the various e x p e r i m e n t a l i s t s t h a t
the g = 2.0055 EPR l i n e i s s t r u c t u r e l e s s and i s o t r o p i c .
Major
disagreements have a r i s e n concerning the spin d e n s i t y , l i n e w i d t h ,
line
shape, s a t u r a t i o n power l e v e l , and behavior of t h e s i g n a l a f t e r annealing.
In an effort to r e c o n c i l e a v a r i e t y of c o n f l i c t i n g r e s u l t s , t h e IBM group
undertook an exhaustive study of evaporated a-Si and the dependence of the
d a t a on suDstrate temperature, deposition r a t e , annealing temperature,
ambient contaminants i n vacuum during d e p o s i t i o n , and exposure of samples
to a i r (Thomas, e t a l . , 1978).
Dependences were discovered which c a s t a
p a l l of doubt over a l l previous work, including t h e i r own.
Using t i g h t l y
c o n t r o l l e d UHV c o n d i t i o n s , they obtained r e s u l t s t h a t a r e independent of
deposition r a t e and sample thickness over a wide range and show complete
symmetry between deposition a t a given s u b s t r a t e temperature and anneals
a t the same temperature of samples made on lower temperature s u b s t r a t e s .
The spin d e n s i t y for room temperature deposited samples was observed t o be
~5x10 19 cm - 3 .
This number declines monotonically as t h e s u b s t r a t e
temperature or annealing temperature i s i n c r e a s e d .
At 430 C t h e spin
d e n s i t y has dropped by a f a c t o r of 2 . 2 and c r y s t a l l i t e s begin t o form.
16
Crystallization
undetectable.
is complete by 630 C where the spin density i s
The IBM group concludes t h a t there must be a v a r i e t y of
defect s t a t e s a s s o c i a t e d with the unpaired e l e c t r o n spins and t h a t these
anneal out a t v a r i o u s temperatures.
The d i s t r i b u t i o n of types of d e f e c t s
must be nearly continuous, otherwise s t r u c t u r e would appear in the
r e l a t i o n s h i p between spin density and annealing temperature.
Thus, as the
d e n s i t y of d e t e c t a b l e s p i n s decreases on annealing (or increased s u b s t r a t e
temperature) t h e nature of the d e f e c t s associated with the s i g n a l probably
changes.
It i s i n t e r e s t i n g to note t h a t t h e spin d e n s i t y i s almost an
i n t r i n s i c p r o p e r t y of t h e m a t e r i a l , never varying by more than a f a c t o r of
2 from 3x10
cm
J
for UHV evaporated samples.
The line shape for a l l unannealed, room temperature deposited samples
i s b a s i c a l l y Lorentzian and symmetric.
As the spin density i s decreased
by annealing t h e linewidth measured a t 9 GHz narrows from 8 to 5 Gauss and
the l i n e takes on s l i g h t Gaussian c h a r a c t e r as well a s a l i t t l e asymetry.
The temperature and frequency dependence of the linewidth i s complicated
and has not been explained in a convincing way.
P.W. Anderson (1951) has
c a l c u l a t e d the linewidth and l i n e shape expected f o r d i p o l a r broadening
for a randomly d i s t r i b u t e d spin population with n e g l i g i b l e exchange
interactions.
He finds a Lorentzian l i n e where t h e width (HWHM in
r a d i a n s / s e c ) i s given by 3.8 t fiN, where N i s the s p i n d e n s i t y and K* i s
the gyromagnetic r a t i o .
Abragara (1961).
This and s i m i l a r c a l c u l a t i o n s are reviewed in
For a t y p i c a l spin d e n s i t y for a - S i ( 3x10 1 9 cm" 3 )
t h i s works out t o a peak-to-peak d e r i v a t i v e width of 2.5 Gauss which i s
l e s s than the observed l i n e w i d t h .
Also, t h e observed linewidth does not
17
s c a l e c o r r e c t l y with spin d e n s i t y and shows some frequency dependence.
I t i s b e l i e v e d , t h e r e f o r e , t h a t t h e observed l i n e i s inhomogeneously
broadened, t h a t i s , i t c o n t a i n s a d i s t r i b u t i o n of g f a c t o r s .
Lee and
Corbett (1973) have compiled EPR data on a g r e a t v a r i e t y of d e f e c t s seen
in irradiated c-Si.
I t i s p o s s i b l e (Thomas, e t a l , , 1978) to c o n s t r u c t an
average over various types of dangling bond d e f e c t s such t h a t the observed
g f a c t o r is 2.0055 and i s o t r o p i c and the linewidth i s about 7.5 Gauss a t 9
GHz.
In c o n t r a s t to t h i s approach, I s h i i and o t h e r s have used molecular
o r b i t a l theory to c a l c u l a t e g values for c l u s t e r s c o n t a i n i n g a dangling
bond ( I s h i i , e t a l . , 1981).
They find a range of g values which i s
c o n s i s t e n t with the experimental d a t a .
The presence of an inhomogeneous
l i n e i s c o n s i s t e n t with the annealing behavior of the spin density and the
d i s t r i b u t i o n of bond a n g l e s , both of which p r e d i c t a range of p o s s i b l e
paramagnetic s t a t e s .
I t Is not f e a s i b l e to c a l c u l a t e t h e a c t i o n of d i p o l a r broadening on
an inhomogeneous l i n e in a randomly d i s t r i b u t e d spin system.
This i s
because the d i p o l a r i n t e r a c t i o n contains spin f l i p terms which can
t r a n s f e r energy between v a r i o u s p a r t s of the inhomogeneously broadened
line.
This leads to a net narrowing of t h e l i n e , j u s t a s i n exchange
narrowing.
I t i s p o s s i b l e t h a t exchange i n t e r a c t i o n s a r e Important and
a l s o c o n t r i b u t e to l i n e narrowing.
The presence of two broadening
mechanisms, and one and maybe two narrowing mechanisms on a randomly
d i s t r i b u t e d spin system has prevented any s u b s t a n t i a l conclusions
concerning the observed l i n e w i d t h .
I t I s p o s s i b l e , of c o u r s e , t h a t t h e
18
spin system i s not randomly d i s t r i b u t e d and t h i s makes the problem even
worse.
Furthermore, Thomas notes t h a t linewidth measurements are
p a r t i c u l a r l y subject to contamination problems (Thomas, e t a l . , 1978).
I f l a r g e exchange i n t e r a c t i o n s are p r e s e n t , t h e s u s c e p t i b i l i t y must
show a deviation from the simple Curie Law a t temperatures n e a r J/k, where
J i s the exchange energy.
The unsaturated resonance signal amplitude was
found by the IBM group to vary with temperature according to a Curie Law
(1/T) between 5 and 120 K.
Other r e s e a r c h e r s have reported t h a t the
amplitude follows a Curie-Weiss Law, 1/(T+9), at and above He-4
temperatures with 0 values in the range of 1.1 - 5 K.
a l . , 1981 for review of data)
(See Khokhlov, e t
The IBM group put an upper l i m i t of 1K on
0 and points out t h a t data below 1.2 K are needed to c l a r i f y the i s s u e .
The following p i c t u r e t h e r e f o r e emerges:
A-Si i s an overconstrained
m a t e r i a l which c o n t a i n s dangling bonds t h a t r e s u l t i n an inhomogeneously
broadened EPR l i n e a t g = 2.0055.
These dangling bonds are a s s o c i a t e d
with compositional h e t e r o g e n e i t y of an unknown n a t u r e .
The s p a t i a l
d i s t r i b u t i o n of dangling bonds in a - S i i s therefore a l s o unknown.
The
dangling bonds are a s s o c i a t e d with s t a t e s In the band gap which have a
profound influence on the e l e c t r i c a l and o p t i c a l p r o p e r t i e s of a-SI.
M o t t ' s hopping c o n d u c t i v i t y law works well for a-Si i n the temperature
range of 50 - 300 K.
The r e l a t i o n of s i t e s associated with hopping
conductivity to the s t a t e s discussed above i s also n o t c l e a r .
19
1.2 Motivation and Scope of Present Study
This study attempts to f u r t h e r e l u c i d a t e the microscopic n a t u r e of
a - S i and i t s paramagnetic s t a t e s through t h e use of two experimental
c a p a b i l i t i e s not a v a i l a b l e to p r e v i o u s r e s e a r c h e r s .
The f i r s t i s the a b i l i t y to observe EPR a t temperatures down to
0.3 K.
I t i s indeed c u r i o u s t h a t such a complex system e x h i b i t s a s i n g l e ,
s t r u c t u r e l e s s EPR l i n e even a t 1.2 K.
Cooling the sample to He-3
temperatures might s i g n i f i c a n t l y reduce averaging and narrowing
which could obscure underlying s t r u c t u r e In t h e EPR l i n e .
effects
Furthermore,
the nature of the inhomogeneous broadening might change as c e r t a i n c l a s s e s
of paramagnetic e l e c t r o n s freeze onto t r a p s o r drop i n t o nonmagnetic
singlet states.
Any observed change in l i n e w i d t h , l i n e shape, or
s t r u c t u r e a s t h e temperature is reduced to 0 . 3 K would be useful
Information.
F i n a l l y , i n v e s t i g a t i o n of t h e adherence of the
s u s c e p t i b i l i t y to a Curie Law in t h i s low temperature region i s needed.
Measurements i n the He-3 temperature range should be capable of d e t e c t i n g
a Curie temperature 0 a s small a s 0.1 K and of determining i f t h e
temperature dependence i s t h a t of t h e Curie-Weiss Law or some o t h e r
behavior.
The second important experimental technique i s t h e a b i l i t y to
d i r e c t l y observe the recovery of t h e spin system from s a t u r a t i o n .
This
allows the d i r e c t measurement of T 1 , the l o n g i t u d i n a l r e l a x a t i o n time.
Useful p h y s i c a l information can be derived from the dependence of T. on
20
experimental c o n d i t i o n s .
This r e q u i r e s i d e n t i f i c a t i o n and t h e o r e t i c a l
understanding of the s p i n l a t t i c e r e l a x a t i o n (SLR) process in t h e
material.
Most of t h e SLR measurements In t h i s study were conducted in
t h e 0.3-3K r a n g e , where one phonon SLR mechanisms ( a s opposed t o two
phonon mechanisms) tend to dominate.
Various one phonon SLR mechanisms a r e discussed in chapter two along
with previously reported a-Si SLR measurements.
Chapter t h r e e , along with
an appendix, contains a d e s c r i p t i o n of experimental equipment and
techniques used in t h i s study.
The experimental r e s u l t s and t h e i r
t h e o r e t i c a l i n t e r p r e t a t i o n are t h e subject of c h a p t e r four.
a r e presented in c h a p t e r five.
Conclusions
21
CHAPTER 2
ONE PHONON SPIN LATTICE RELAXATION PROCESSES
A non-equilibrium system of paramagnetic e l e c t r o n s in an e x t e r n a l
magnetic f i e l d w i l l approach thermal e q u i l i b r i u m by exchanging energy with
the l a t t i c e e x c i t a t i o n s of the solid in which i t is imbedded.
This
phenomenon is c a l l e d spin l a t t i c e r e l a x a t i o n (SLR) and can involve a
number of d i f f e r e n t processes, s e v e r a l of which are reviewed in t h i s
thesis.
Consider a simple two level system where N. is t h e population of the
ground s t a t e , N2 i s the population of the upper s t a t e and N=N-+N2 i s
fixed.
The r e t u r n of the system to e q u i l i b r i u m is governed by t h e r a t e
equations:
N
(2.1)
1 = - N 1 W 12 +N 2 W 21
N2 = - N 2 W 2 1 + N l W l 2
where Wxy i s the t r a n s i t i o n r a t e from s t a t e x to s t a t e y .
r e w r i t t e n in terras of t h e population d i f f e r e n c e nsN.-N-:
(2.2)
n = (W 12 +W 21 ) N(W 21 -W 12 ) - n
W
12+W21
These can be
22
At equilibrium rt=0 and the population d i f f e r e n c e is given by
n =N(W 01 -W 10 )/(W 10 +W 01 ).
o
21 12
12 21
1/T.=W.2+W2..
Equation 2.2 then becomes n=(n - n ) / T , where
o
1
I f t h e t r a n s i t i o n r a t e s W a r e independent of N and n,
then the r e t u r n of t h e system to equilibrium i s characterized by an
exponential t r a n s i e n t with time c o n s t a n t T .
This simple two l e v e l system
c o r r e c t l y d e s c r i b e s the spin population of a magnetically d i l u t e , s p i n
1/2, paramagnetic s o l i d in an e x t e r n a l f i e l d ,
in t h a t case the energy
difference between l e v e l s 1 and 2 i s given by the Zeeman s p l i t t i n g
£=gBH
where g i s the s p l i t t i n g factor, B i s the Bohr magneton, and H i s t h e
applied magnetic f i e l d strength.
difference n
o
The thermal equilibrium population
i s defined by the Boltzmann f a c t o r N„/N,=exp(-i/kT) where T
i s the l a t t i c e temperature.
d
The quantity 1/T
\
i s r e f e r r e d to as t h e spin
l a t t i c e r e l a x a t i o n r a t e (SLR r a t e ) .
Experimental s t u d i e s of SLR t y p i c a l l y f o c u s on t h e dependence of 1/T.
on experimental c o n d i t i o n s .
This provides information about the m a t r i x
elements which couple the spins t o various l a t t i c e e x c i t a t i o n s and a l s o
information about the density of s t a t e s a s s o c i a t e d with these e x c i t a t i o n s .
Extraction of t h i s information r e q u i r e s t h e o r e t i c a l understanding of the
r e l e v a n t SLR processes in the m a t e r i a l .
now be reviewed.
S e v e r a l d i f f e r e n t processes will
23
2.1 The Direct Process
In a p e r f e c t l y ordered i n s u l a t i n g c r y s t a l an e l e c t r o n spin w i l l
g e n e r a l l y relax by i n e l a s t i c phonon s c a t t e r i n g .
This o c c u r s via phonon
modulation of the ligand e l e c t r o s t a t i c f i e l d ( c r y s t a l f i e l d ) which
perturbs the o r b i t a l e l e c t r o n i c s t a t e s of t h e system.
couples to the spins v i a the spin o r b i t i n t e r a c t i o n .
therefore be i n e f f e c t i v e for a pure S s t a t e e l e c t r o n .
This p e r t u r b a t i o n
This process w i l l
(Kronig, 1939; Van
Vleck, 1940)
The t r a n s i t i o n r a t e s W.- and W?1 may be calculated from Fermi's
golden r u l e which i s t o second o r d e r :
(2.3)
Wif = 2tf|<f lH0Lli>
'
+ y<f|H 0L |mXm|H 0L |i)J
m
E.-E
i m
2
£ D (E 1 -E f )
'
The i n i t i a l , i n t e r m e d i a t e , and f i n a l s t a t e s | i > , |m>, and |f> are each a
product of e l e c t r o n i c and phonon s t a t e s and f_ i s the Dirac d e l t a
function.
HQ
i s an o r b i t - l a t t i c e i n t e r a c t i o n Hamiltonlan which s p e c i f i e s
the coupling between t h e paramagnetic spins and the phonon spectrum.
complete c a l c u l a t i o n of HQT i s a formidable problem and i s reviewed by
Orbach and Stapleton (1972).
The essence of t h e i r c a l c u l a t i o n is
preserved in the following simplified phenomenological o r b i t - l a t t i c e
i n t e r a c t i o n Hamiltonian (Allen, e t a l . , 1982):
The
24
(2.4)
V
HOL = ^
= ^y
/ 2
(
b f f
V)v
q
is a tensor operating on electronic wavefunctions and has, for
notational simplicity, a single index q.
As V
interactions, it is even under time reversal.
arises from electrostatic
M Is the mass of the
crystal, v is the appropriately averaged velocity of sound and the R _ are
q<r
geometrical coupling constants.
The strain e has been expanded in terms
of creation and annihilation operators for phonons of frequency " and
index o- (Orbach and Tachiki, 1962).
In the direct process the spin is flipped by the absorption or
emission of a 3ingle phonon whose energy is equal to the Zeeman
splitting f.
Only the first order term in 2.3 contributes, giving the
transition rate:
(2.5)
W , = 2* 2I/W\
W
1 /2
<n<r+1 I h , - b r + 1 0 < - 1 R ^ | +>| 2 £D( E 2 -E 1 -tlft),)
2Mv 2
where the initial state |2> is |+>|...n^.. .> and the final state |1> is
I->I...n^+1.. .^ .
Note that |+> is a full electronic wavefunction with a
spin and orbital component.
The sum over phonon states is replaced by an
integral over energy with a Debye phonon spectrum p ( E ) assumed:
25
(2.6)
JJD(E)
=
3VE
2
:ff2n3v3
where V i3 the volume of the c r y s t a l and v i s t h e velocity of sound.
The
phonon occupation is given by the Bose-Einstein d i s t r i b u t i o n
p (ftu) = 1/(exp( , Rw/kT)-1).
Collapsing the various matrix elements between
t h e e l e c t r o n i c s t a t e s i n t o <C-|V|+>, the SLR r a t e 13 given b y :
1/T
(2.7)
1 d i r e c t = W21+W12 = R o ( 2 P 0 ( * W o ) + 1 ) = V o t h ( f % / 2 k T )
RQ = 3(1iw0)3/(2tfp-fl4v5)
ho
/<-IVl+>| 2
= f = gBH
where o i3 the d e n s i t y of the c r y s t a l .
The f a c t o r multiplying R i s t h e
sum of terms for stimulated a b s o r p t i o n , stimulated emission, and
spontaneous emission.
I t is often t r u e t h a t a t temperatures of i n t e r e s t
£<<2kT and coth(C/2kT) & 2kT/f
1/T1 .« T .
making the d i r e c t process SLR r a t e
In the work that follows, t h e temperature (T) and magnetic
f i e l d (H) dependence of 1/T. are measured.
If |<-lVl+>!
of H then equation 2.7 p r e d i c t s 1 / T 1 d * H2T when
i s Independent
f«2kT.
Kramers' theorem, which i s based on time r e v e r s a l symmetry, is
important in determining the magnetic f i e l d dependence of t h e matrix
element in equation 2 . 7 -
A system i s said to p o s s e s s time r e v e r s a l
symmetry i f i t s time development may be reversed, with a l l physical
processes running backwards and the i n i t i a l and f i n a l s t a t e s interchanged.
Symmetry between the two d i r e c t i o n s of time flow implies t h a t every
26
state V is related to a time reversed conjugate state Kt, called a Kramers
conjugate, and that the transformation represented by K preserves the
values of all physical observables.
For spatial wave functions, time can be reversed by taking the
complex conjugate.
K
For example:
? r 1 = ?+
K p K~1 = -p +
(2.8)
K L K~1 = -1*
As a r e s u l t , K i s an a n t l u n i t a r y operator so t h a t KaH' = a Kt and
(Kt,Kf) = ( W ) .
K a l s o reverses s p i n :
K S K ' = -S +
.
K S ± K"' = -S± + = -Ss
(2.9)
Consider the action of K on a one e l e c t r o n spin s t a t e s
SJ1/2,±1/2> = +1/2 |1/2, t 1/2>
z
(2.10)
S z K|1/2,±1/2> = T 1/2 K|1/2,t1/2>
K|1/2,+1/2> = exp(ik)|1/2,,:1/2>
where k i s r e a l .
Consider the a c t i o n of K on the same s t a t e :
27
K2|1/2,±1/2> = K2S±|1/2,T1/2>
= -KS T K|1/2,:p1/2>
(2.11)
= -KSTexp(ik)|1/2,±1/2>
= -Kexp(ik)|1/2,T1/2>
= -exp(-ik)exp(ik)|1/2,i1/2> = -1|1/2,±1/2>
2
K t h u s i n t r o d u c e s a f a c t o r of - 1 , r e m i n i s c e n t of t h e r e s u l t for
of a fermion through 360°.
rotation
(from Pake and E s t l e (1973) as modified by
Colvin (1984))
If Q i s a time even o p e r a t o r then Q+=KQK
.
Consider the matrix
element of Q between Kramers c o n j u g a t e s :
(Kf,Qt) = (K2V,KQt)* = -(KQt,T)
(2.12)
= -(Q + Kr,t0 = -(K?,Qt) = 0
The case where Q) is the identity operator indicates that Kramers
conjugates are orthogonal.
This implies that for a single electron state,
a time even Hamiltonlan will not connect Kramers conjugate states and each
energy level is therefore at least two-fold degenerate.
An important and
obvious corollary of Kramers' theorem is that the two-fold degeneracy is
not split by time even perturbations such as V. The matrix element in
equation 2.7 is therefore zero for the dangling bond electron In a-Si.
The degeneracy is removed by the applied magnetic field (B changes sign
under time reversal), creating a Zeeman doublet with splitting
f=gBH.
Note that it is conventional to write "H" in this equation even though the
28
c o r r e c t field i s a c t u a l l y "B".
The applied magnetic f i e l d H causes a s l i g h t admixture of excited
s t a t e wave functions i n t o the ground s t a t e doublet, breaking t h e time
r e v e r s a l symmetry.
The leading non-zero term in t h e matrix element
<1|V|2> is then p r o p o r t i o n a l to H.
Equation 2.7 then p r e d i c t s
1/T 1d * H 5 coth( S/2kT) which goes over to H4T when
£«2kT.
2 . 2 Two Phonon Processes
Two phonon SLR processes a r e p o s s i b l e and for Kramers i o n s are
derived from t h e second order term in equation 2 . 3 .
In the Raman
r e l a x a t i o n mechanism, a phonon i s absorbed, the paramagnetic spin is
f l i p p e d , and a phonon i s emitted t h a t is s h i f t e d in energy from the
i n i t i a l phonon by the Zeeman s p l i t t i n g .
Van Vleck proposed t h i s mechanism
0 Q
(1940) and calculated for Kramers ions 1/T, D
« H T when T<<0 n , .
1 Raman
Debye
2
Being a two phonon p r o c e s s , the r a t e dependence must approach T in t h e
classical limit.
number of systems.
Van V l e c k ' s p r e d i c t i o n s have been verified i n a g r e a t
Orbach has described and v e r i f i e d experimentally a
r e l a x a t i o n process which occurs when an e x c i t e d s t a t e e x i s t s within an
energy kOD .
of the ground s t a t e doublet (Finn, e t a l . , 1961). For
3
-1
Kramers ions the r a t e dependence i s 1/T.Q .
i s the excited s t a t e energy.
.<* H (exp(E/kT)-1)
where E
Kronig (1939) proposed a Raman process for
Kramers ions t h a t involves excited s t a t e admixtures, just as i n the d i r e c t
2 7
p r o c e s s . The predicted H T r a t e i s apparently very weak, and has been
observed only i n rare i n s t a n c e s (Marchand and S t a p l e t o n , 1974).
An
29
extensive literature exists on the theory and experimental results for
these various SLR processes.
Major reviews have been written by Stevens
(1967), Abragam and Bleaney (1970), and Orbach and Stapleton (1972).
Spin lattice relaxation in ionic solids tends to be dominated by the
direct process at temperatures of about 1K or less.
In this temperature
range the phonon spectrum peaks at an energy close to the Zeeman splitting
for commonly used microwave fequencies (9-20 GHz). This creates optimal
conditions for the direct process because only those phonons within a
small width of the Zeeman splitting can contribute to the spin relaxation.
In contrast, the Raman process can use the entire phonon spectrum above
the Zeeman splitting.
At 1K this population is generally not large enough
to cause a second order process rate significant in comparison to that of
the direct process.
By 4K the peak in the phonon spectrum will have moved
out to higher energies and the Raman process will begin to dominate.
By
10K the contribution of the direct process to the total rate will
generally be negligible.
The radically different temperature dependences of the direct and two
phonon processes make them easily distinguishable.
The temperature
dependences of the SLR rates in this study show no evidence of two phonon
processes and they will not be considered further.
The remainder of this
chapter is devoted to two variations of the direct process, one of which
will be proposed as an explanation of the experimental results.
2.3 The Direct Process with a Phonon Bottleneck
The direct SLR process involves only phonons whose energy falls in a
narrow band about the Zeeman splitting i.
These are called the resonant
phonons, and their bandwidth 4««> is on the order of the homogeneous EPR
linewidth.
In the previous calculation it is assumed that the resonant
phonons are always in thermal equilibrium with the remainder of the phonon
spectrum and the sample and associated hardware being held at a constant
temperature T.
The sample, the hardware, and the nonresonant part of the
phonon spectrum represent a constant temperature thermal reservoir, often
called the "bath". Van Vleck (1941) pointed out that under certain
conditions the rate of energy transfer from the spin system to the
resonant phonons might exceed the rate of energy transfer between the
resonant phonons and the bath.
Energy will then accumulate in the
resonant phonon spectrum, causing the occupation numbers p(<o) to exceed
their thermal equilibrium values at the bath temperature T.
This effect,
called a phonon bottleneck, was first observed in 1962 and has since been
seen in a variety of systems (Scott and Jefries, 1962; Ruby, et al., 1962;
Mlkkelson and Stapleton, 1965; Muench, et al., 1984).
The net result of a
bottleneck is a spin recovery which is inhibited from its normal, direct
process rate and is nonexponential in character.
A useful parameter, called the bottleneck factor, has been defined by
Scott and Jeffries (1962):
31
(2.13)
cr=ET.
z ph
T
=NRT.
o ph
1d E P h
tanh(£/2kT) 5
htanh(£/2kT)
u
?«W
The bottleneck factor is the energy of the spin system, E , times the SLR
z
rate to the resonant phonons, 1/T-. (see 2.7)> divided by the energy of
the resonant phonon system, E . = (p(£) + 1/2)j>(S)JttA<«>, times the relaxation
rate of the resonant phonons to the bath, T
.
As before, p(f) is the
Bose-Einstein distribution evaluated at the Zeeman energy.
the spin system is E
The energy of
= n£ = (1/2)N£tanh(£/2kT) (see 2.2 and associated
z
text).
A variety of evidence indicates that the predominant mechanism for
thermalization of the resonant phonons is inelastic scattering at the
surface of the crystal (Scott and Jeffries, 1962; Giordmaine and Nash,
1965).
At cryogenic temperatures T , and thus t\_ are nearly independent
of T.
Complete analysis of the phonon bottleneck requires the solution of
two coupled, nonlinear rate equations—one for the spins, and the other
for the resonant phonons.
General, closed form solutions do not exist but
various techniques have been used to solve the problem within certain
limits.
(Faughnam and Strandberg, 1961; Scott and Jeffries, 1962; Abragam
and Bleaney, 1970)
Under intermediate and full bottleneck conditions
(cr£>1), the latter stage of the recovery of the spin system to the bath
temperature is nearly exponential, with a time constant T.. ...
(cr+DT. ,. The observed (spin-bath) relaxation rate is then:
.=
32
(2 1i|)
'
1/T
1 b o t t l e n e c k = V°<*<S/2kT)
1+ tanh(£/2kT)
In the l i m i t of a strong bottleneck (<r>M) t h i s becomes:
(2.15)
1/T 1b = j,(J)M<o
NT
eoth 2 (£/2kT)
ph
In the l i m i t of a weak b o t t l e n e c k (o-«1) 1/T 1h approaches t h e ordinary
d i r e c t process r a t e 1/T. . given by equation 2 . 7 .
The f i r s t d i r e c t measurements of SLR in a-Si were performed by
Gourdon, e t a l . , (1981) and are shown in figure 2 . 1 , curves 1-4.
Curves
1-4 i n d i c a t e data for f o u r d i f f e r e n t samples whose spin d e n s i t i e s are
respectively:
1.5x10 2 0 C m' 3 ,
1.1x10 20 cm" 3 ,
8x10 19 cm" 3 , and 6x10 1 9 cm" 3 .
The samples were made by evaporation with e l e c t r o n beam h e a t i n g , a
preparation a l s o used in t h i s study.
For the two h i g h e s t d e n s i t y samples
p
(1 and 2) the observed r a t e s were p r o p o r t i o n a l to T between 4.2K and 10K.
Above 10K the temperature dependence g r a d u a l l y changes over t o r a t e s
p r o p o r t i o n a l to T, the T dependence holding from ~30K t o 100K.
Above
100K another process e n t e r s which the authors claim i s r e l a t e d to M o t t ' s
hopping c o n d u c t i v i t y process (Mott, 1969).
Gourdon, e t a l . , (1981) propose the phonon bottleneck a s an
explanation for t h e i r r e s u l t s below 100K.
f o r two r e a s o n s .
This i s u n l i k e l y to be c o r r e c t
F i r s t , the temperature dependence of <r (see 2.13) i s
such t h a t t r i p l i n g T w i l l not take <T from the b o t t l e n e c k l i m i t (<r»D
to
33
108
— I — I I I Mil |
1—I I I 1111 |
I 11
IM
& > © Gourdon eta!.
(§)
Stutzmann etal.
© © Askew etal.
(extrapolation)
10'
V1 a T
id 1
ia
10
I
I
I I I Mill
LJ
' »' ""I
10
100
Temperature (K)
2.1
i
'
I I INI
500
Spin Lattice Relaxation Rates for Amorphous Si
Data are from various sources: Curves 1-4 are from
Gourdon, et al., 1981, and correspond to.spin
densities of 1.5, 1.1, 0.8, and 0.6 x 10
cm"
respectively. Curve 5 is from Stutzmann, et al.,
1983. Curves 6 and 7 are from the current study.
34
the unbottlenecked limit (o-«1).
al., (9.4GHz),
At the frequency used by Gourdon, et
o- is reduced by slightly less than a factor of 3 in
moving from 10K to 30K.
Secondly, the observed g values (^2.0055) are so
close to the free electron value that the spin orbit interaction («L«S)
will be extremely weak.
This makes the strong spin-phonon coupling
required for a bottleneck very unlikely.
A phonon bottleneck has never
been observed for an EPR line so close to the free electron g value.
2.4 Relaxation by Modes Associated with Disorder
In imperfect crystals and amorphous materials new excitations are
possible which can relax electron spins.
In imperfect crystals
interstitial and substitutional impurities can give rise to local
vibrational modes.
A simple rate equation argument shows that the spin
relaxation by a local mode at frequency
to
dependence of 1/T..,
This dependence has been observed
1
u expt-fiM/kT).
will have a temperature
for hydrogen and deuterium impurities in CaF ? (Feldman, et al., 1965).
In
some instances an interstitial impurity can tunnel between various minima
in the crystal potential.
The effects of such tunneling behavior are
commonly observed in the alkali halides (Narayanamurti and Pohl, 1970).
The first observation of SLR by tunneling modes was by Feldman, et
al., (1966) who studied the effect of atomic hydrogen in fused silica.
Murphy (1966) was able to explain the data by postulating that the
hydrogen existed in a double well potential.
He derived an SLR rate
temperature dependence of 1/T. = 2W csch(E/kT) where W is temperature
35
independent and E Is an energy splitting which characterizes tunneling in
the double well potential.
data when E/k = 16K.
This result provided a good fit to Feldman's
Similar experimental results have been obtained for
relaxation due to color centers in Li doped KBr (Ohkura, et al., 1975) and
Li doped CaO (Stinson and Stapleton, 1983).
More relevant to the study of a-Si are cases where there appears to
be a continuous distribution of tunneling state energies E, rather than
just one or two individual modes.
Kurtz and Stapleton (1979, 1980) have
observed SLR rates due to color centers in Li, Na, and K beta-alumina
which are proportional to T^ where n = 2 to 4.
They have proposed a
theory which involves spin relaxation by coupling to a continuous
distribution of tunneling states and predicts 1/T. «* T n , with n»4.
4+
Sirailar behavior has been seen in the SLR of v* T ions in amorphous V-O,- by
Deville, et al., (1983) but n was about 2, rather than 4.
The
Kurtz-Stapleton theory predicts that the T° dependence should hold even at
temperatures low enough that 2kT<£.
Neither of the experiments mentioned
above used temperatures low enough to test that limit.
A principal motivation of this study is to test the hypothesis that
SLR In a-Si is caused by excitations related to disorder and is not the
phonon bottlenecked direct process.
A clear distinction between the
phonon bottleneck with 1/T « coth (£/2kT) (see 2.15) and the disorder
2
related process with 1/T- °< T will be possible only at temperatures low
enough that 2kT<£.
range (0.3-1.1K).
At 16.5 GHz this requires temperatures in the He-3
The equipment required to maintain these temperatures
36
is reviewed in the appendix.
Chapter three contains a discussion of
various experimental details, including sample preparations, temperature
calibration, and operation of the microwave spectrometer.
The
experimental results and their theoretical interpretation is the subject
of chapter four.
Conclusions are presented in chapter five.
37
CHAPTER 3
EXPERIMENTAL EQUIPMENT AND TECHNIQUES
3.1 EPR Spectrometer
In this study SLR rates were measured using the pulse saturation and
recovery technique.
The cycle begins with a high power microwave pulse
which is of sufficient amplitude and duration to completely saturate the
EPR signal.
At the end of the pulse the microwave power drops to a much
lower monitor power level and the spin population begins its recovery back
to thermal equilibrium.
The increase in the EPR absorption is monitored
as a function of time.
Since the absorption is proportional to the spin
population difference n, the increase of n from 0 to n
monitored.
Is being
If the spin recovery is characterized by a single time
constant, the increase in the EPR absorption will be exponential with the
baseline defined by the monitor power level.
This power level should be
sufficiently low that there is no observable deviation of the spin
population from Its thermal equilibrium values.
A schematic diagram of the main spectrometer used in this study is
shown in figure 3.1.
The frequency of the microwave source (a klystron)
is controlled using a Pound stabilizer circuit (Poole, 1967a) which locks
the source frequency to a resonant frequency of an external reference
cavity.
The resonant frequency of the reference cavity is set manually to
38
SAMPLE
CAVITY
/MAGNET
POUND
STABILIZER
«
FABRITEK
SIGNAL AVG
# +
ISOLATOR VARIABLE MAGIC
ATTENUATOR
T
y
VARIABLE
PHASE
SHIFTER
Figure 3.1
COAX
CABLE
-JS.48.
<
\
1
DC SHIFT AND
LOW PASS FILTER
e
IN 78 A
CROSS
CRYSTAL
GUIDE
nrTFCTOR
COUPLER • " ^ " • • O H
u
. _ u c Dn
" f ™ E
LOAD
wv
^
©
MICROWAVE
WAVEGUIDE V A - 9 2
KLYSTRON
MANUAL
WAVEGUIDE
SWITCH
Ku Band (16,5 GHz) Pulse Saturation Electron
Paramagnetic Resonance Spectrometer
39
match t h a t of the sample c a v i t y .
This makes i t p o s s i b l e to hold the
microwave frequency on t h e sample c a v i t y resonance even though t h e
microwave power l e v e l s a t the sample c a v i t y are f l u c t u a t i n g r a p i d l y .
The
achievable frequency s t a b i l i t y i s a few p a r t s in 10' over a 15 minute time
span.
The sample c a v i t y Q is h i g h enough (~20,000) t h a t changes l a r g e r
than t h i s will produce an observable (and unwanted) deviation in t h e
d e t e c t e d output.
The waveguide path between the source k l y s t r o n and t h e sample c a v i t y
c o n t a i n s a 90dB microwave switch, labeled "main switch" i n figure 3 . 1 .
This switch c o n s i s t s of two MD-20K15D u n i t s (Micro Dynamics, I n c .
Mass.)
in s e r i e s with an i s o l a t o r in between.
Woburn,
The main switch i s
bypassed by a waveguide shunt c o n s i s t i n g of two c r o s s guide c o u p l e r s , a
v a r i a b l e phase s h i f t e r , and a v a r i a b l e a t t e n u a t o r .
During the p u l s e phase
t h e main switch i s open and the a t t e n u a t e d k l y s t r o n s i g n a l can t r a v e l
d i r e c t l y to the sample c a v i t y .
During the monitor phase t h e main switch
i s closed and e s s e n t i a l l y a l l of the microwave power reaching t h e sample
c a v i t y t r a v e l s through the shunt.
This allows the monitor power to be
c o n t r o l l e d independently ( v i a t h e shunt a t t e n u a t o r ) and minimizes t h e
i n f l u e n c e of the main switch t r a n s i e n t on t h e monitor power l e v e l .
The r e f l e c t e d s i g n a l from t h e sample c a v i t y i s observed through a
10dB broadwall d i r e c t i o n a l c o u p l e r .
The reasons f o r using t h i s d e v i c e , as
well a s the v a r i a b l e phase s h i f t e r next to i t , are given in the appendix.
The sample cavity and associated hardware a r e also discussed in t h e
appendix.
If the 10dB power l o s s in t h e d i r e c t i o n a l coupler has a
40
s i g n i f i c a n t influence on the s i g n a l / n o i s e r a t i o ( i t u s u a l l y does not) then
a c i r c u l a t o r can be used i n s t e a d .
The problems a s s o c i a t e d with t h i s
s u b s t i t u t i o n are a l s o discussed In the appendix.
The d e t e c t o r must have broad bandwidth OlOOKHz) and high s e n s i t i v i t y
(min. d e t e c t a b l e signal <1pW), n e c e s s i t a t i n g t h e use of a superheterodyne
receiver.
The f i r s t s t a g e of t h e r e c e i v e r i s a MP015/2A (RHG E l e c t r o n i c s ,
I n c . , Deer Park, N.Y.)
balanced mixer with an I n t e g r a l preamplifier.
It
operates a t a 60MHz i n t e r m e d i a t e frequency with a 10MHz bandwidth and a
microwave to i . f .
gain of 27dB.
The u n i t ' s wide dynamic range, -"60dB, i s
p a r t i c u l a r l y important In t h i s a p p l i c a t i o n because high power microwave
p u l s e s are allowed to e n t e r the d e t e c t o r .
Since the monitor power i s
t y p i c a l l y 30-50dB down from the pulse power, both power l e v e l s can be f i t
i n t o the dynamic range.
I t i s e s s e n t i a l t h a t mixer s a t u r a t i o n be avoided
during the p u l s e , as t h e recovery time c o n s t a n t from s a t u r a t i o n i s a few
msec.
The microwave a t t e n u a t o r ahead of the mixer i s used to position t h e
p u l s e power j u s t below the s a t u r a t i o n p o i n t of t h e mixer.
Adding a t t e n u a t i o n between the sample and r e c i e v e r I s , of course,
throwing away s e n s i t i v i t y .
This can be avoided by p l a c i n g a waveguide
switch d i r e c t l y i n front of t h e mixer and running i t o u t of phase with t h e
main switch.
The mixer switch i s off during t h e main p u l s e , s h i e l d i n g t h e
r e c e i v e r and preventing s a t u r a t i o n .
This arrangement has been used
s u c c e s s f u l l y in t h e past ( S t i n s o n , 1981) and was t r i e d in t h i s case u s i n g
an MD-20K15D u n i t (the same a s t h e main s w i t c h ) .
Unfortunately,
the
s i g n a l t r a n s i e n t a s s o c i a t e d with the opening of the mixer switch was
41
larger than the signal due to spin relaxation.
17
(~10
A larger spin population
15
instead of 10
problem.
) or a faster microwave switch would have solved the
As neither was readily available, the indicated arrangement was
used instead.
The ndxer-preamp output is fed to an EST6010LD (RHG Electronics, Deer
Park, N.Y.)
linear i.f. amplifier and matched video detector.
This unit
provides 80dB of i.f. gain (60MHz center frequency, 10MHz bandwidth)
before rectification at the video detector.
The detector provides linear
i.f. to DC conversion with a dynamic range (±0.5dB) of 30dB.
stage of the detector includes 10dB of DC gain.
The output
Under normal operating
conditions 50 to 70dB of the available 90dB of gain was used.
i.f. amplifier was allowed to saturate during the pulse phase.
The
This was
not a problem because the unit is designed to have a rapid (~1//sec) and
clean (no overshoot or ringing) recovery from saturation.
The output of the superheterodyne is DC shifted, amplified 10dB and'
then digitized in a Fabrltek 1072 signal averager.
The system bandwidth
is controlled with passive low pass filtering at the point of final
amplification and also just before the digitizer.
The low pass time
constants were always less than 1/40 of the measured 1/T. and less than
the dwell time for digitization.
The signal averager contains a memory
with 1K words, 18 bits deep, where the averaged signal accumulates.
The signal averager memory can be Interrogated, converted back to
analog, and fed to a logarithmic amplifier which Is described elsewhere
42
(Herrick, 1976).
An exponential recovery will generate a straight line at
the output of the logarithmic amplifier, the time derivative of which is
proportional to the recovery rate 1/T... SLR rates reported in this study
were computed from measurements of these slopes.
scheme was checked in two ways.
The accuracy of this
First, analog RC transients of known time
constant were generated and then checked with the slope measurement
procedure.
constant.
The measured values were always within 2% of the true time
Second, the signal averager is interfaced to an LSI-11/2
microcomputer so that digitized transients can be stored and analyzed by
various curve fitting programs.
Computed curve fits to digitized
recoveries were always consistent with the measured rates, although they
generally showed more scatter.
This increased scatter is a consequence of
the nonrandom nature of the system noise, most of which seems to be
related to structural vibrations of the building and microphonic effects.
Even after averaging 1000 transients, individual spikes and glitches
appear in the data.
An experienced observer can ignore these points in
determining a rate while the curve fitting routines were written to treat
all points equivalently.
The signal averager was also interfaced to a pulse generator which is
described elsewhere (Herrick, 1976).
The pulse generator operates the
microwave switch and allows for independent adjustment of the microwave
pulse duration and the delay until the next pulse.
The system shown in figure 3.1 can be converted to a microwave
reflectometer by changing the position of the two manual waveguide
43
switches.
An EPR spectrometer is designed to accurately measure changes
of a few percent or less in the sample cavity reflection coefficient.
A
reflectometer, on the other hand, is designed to measure the reflection
coefficient over a wide range of values, typically two orders of
magnitude.
This reflectometer feature can be used to measure t h e low
temperature d i e l e c t r i c response of the sample a t the cavity resonant
frequency ("'^GHz).
The change in the cavity reflection coefficient is
directly related to the imaginary part of the sample d i e l e c t r i c constant
while the resonant frequency (measured by the frequency counter) i s
related to the real part (see Sucher and Fox, 1963).
The reflectometer functions through careful comparison of t h e output
voltages of the two crystal detectors shown in figure 3 . 1 .
(These are not
to be confused with the two crystal detectors In the Pound s t a b i l i z e r . )
The crystal connected to the main switch shunt functions as an incident
power monitor while the crystal that replaces the superheterodyne measures
reflected power.
To the extent that these detectors are well matched,
variations in the signal klystron power with time or frequency w i l l not
affect the measurement.
The attenuators are adjusted u n t i l the c r y s t a l
outputs are balanced and then the attenuator readings are used to compute
the reflection coefficient.
All measurements are calibrated against a
waveguide short (reflection coefficient = 1) which can be inserted in
place of the sample cavity.
The main switch can be used to modulate the
microwaves so that a lock-in amplifier can be used to detect the balance
(voltage null) condition.
This scheme can be used only a t low microwave-
power where d i e l e c t r i c saturation in the sample will be negligible.
At
44
i n c i d e n t microwave powers above 10/W, the n u l l can be adequately detected
with a DC microvoltmeter (such a s a Boonton E l e c t r o n i c s Corp. Model 95A,
Parsippany, N . J . )
and modulation i s not necessary.
SLR measurements a t 9.3 GHz were made on a spectrometer which I s
described elsewhere ( A l l e n , et a l . , 1982).
I t s configuration and
o p e r a t i o n were s i m i l a r t o t h a t of the spectrometer j u s t described.
3.2 Temperature Measurement and Control
All of the sample c a v i t i e s used in t h i s study have germanium
r e s i s t a n c e thermometers attached f o r temperature measurement and c o n t r o l
( s e e figure A.2 i n appendix).
The r e s i s t a n c e of t h e s e sensors i s measured
w i t h a 19Hz AC bridge whose error signal i s fed to a lock-in
anplifier.
This system i s capable of 4 to 5 p l a c e accuracy in the r e s i s t a n c e while
h o l d i n g power d i s s i p a t i o n in the sensor under 1^«W.
The output of t h e
l o c k - i n a m p l i f i e r i s sent to a temperature control c i r c u i t which g e n e r a t e s
a feedback signal t h a t c o n t a i n s components p r o p o r t i o n a l to t h e d e r i v a t i v e ,
t h e v a l u e , and the I n t e g r a l of t h e error s i g n a l .
This feedback signal i s
applied to a low r e s i s t a n c e (~30A) heater winding which surrounds each of
t h e sample c a v i t i e s .
The r e s u l t i s a system that can measure and c o n t r o l
temperatures to an accuracy of s e v e r a l mK.
Temperature measurements above 1.4K were made with previously
c a l i b r a t e d sensors (see Herrick, 1976).
The r e s i s t a n c e thermometer in t h e
He-3 c r y o s t a t and microwave cavity assembly (required for temperatures
45
below 1.2K) was calibrated by the author using equipment which will now be
described.
The He-3 cryostat and microwave cavity assembly is described in
detail in the appendix.
The upper surface of the microwave cavity (see
figure A.2) is the bottom of a He-3 evaporation pot and is in good thermal
contact with the temperature sensor.
The remaining walls of the cavity
are in contact with the liquid He-4 bath in which the entire assembly is
immersed.
The temperature of the He-4 bath can be regulated, via pumping
line constriction, at values between 1.3K and 4.2K.
During He-3
refrigeration the He-4 bath was typically at 1.2K, the lowest value
attainable with available pumping capacity.
After an equilibration period
of two hours this minimum He-4 temperature was very stable, changing by no
more than a few mK over the next 10 hours.
It is unusual to have two independently controlled temperatures
available in the same microwave cavity, and the sensor calibration
procedure makes good use of it.
Two samples of a known Curie law
paramagnet were used, one attached to the He-3 pot and the other on the
cavity bottom at 1.2K.
The EPR 3ignal strength («to magnetization) for
the two samples is measured and the ratio specifies the ratio of the two
sample temperatures.
The He-3 pot temperature can then be computed from
the He-4 bath temperature, which is measured directly from the pressure
over the bath.
An oil manometer was constructed for making this pressure
measurement and filled with dibutyl phthalate.
This liquid is unusual in
-4
that, like Hg, both its vapor pressure and viscosity are low (4x10
torr
46
and 14 centistokes respectively, at NTP).
At 1.043, lt3 density is 1/13
of that of Hg, allowing for much more accurate pressure readings in the
few mmHg pressure range.
It also works well as a rotary vacuum pump
lubricant, should it happen to get sucked into one. The oil manometer was
also used for direct calibration of the temperature sensor in the 1.2K to
2.1K range.
Nd doped lanthanum magnesium nitrate (La2Mg_(N0_)12x24H2O,
abbreviated LMN:Nd) was used as the Curie law paramagnet.
The magnetic
behavior of the lanthanide double nitrate salts has been studied
extensively (see Abragam and Bleaney, 1970b, for review) and certain
members of this family are known to exhibit simple Curie law paramagnetism
into the millikelvin range, even when magnetically concentrated (Fisher,
et al., 1973).
The samples were made with 1 at.^ Nd doping to insure
magnetic dilution and simple Curie law behavior.
One sample was tilted
slightly so that the two EPR lines could be resolved and measured
independently.
This works because the g factor is anisotropic:
gx=2.702,
g„ =0.362.
One rather subtle complication occurred.
It is conventional in EPR
spectroscopic work to apply a small AC magnetic field in addition to H and
thereby observe the derivative of the absorption rather than the
absorption itself.
Observation of absorption requires DC gain and
detection while the derivative technique makes use of a lock-in amplifier
and achieves much better signal/noise.
In this study the magnetic field
modulation colls were placed outside the microwave cavity, as is
47
conventional.
As a r e s u l t , the modulation f i e l d must propagate through
t h e cavity w a l l s .
These w a l l s have a skin depth that depends on, among
o t h e r things, t h e r e s i s t i v i t y of the metal.
The r e s i s t i v i t y of s t a n d a r d
m i l l grade copper drops w i t h decreasing temperature u n t i l about 20K, where
impurity and d e f e c t s c a t t e r i n g mechanisms begin t o dominate.
By 10K t h e
phonon s c a t t e r i n g c o n t r i b u t i o n is n e g l i g a b l e and the r e s i s t i v i t y I s
e s s e n t i a l l y independent of T (de Haas, e t a l . , 1934).
The amplitude of
t h e field modulation r e a c h i n g the samples in a standard copper c a v i t y w i l l
t h e r e f o r e be independent of T below ~10K.
As noted in t h e appendix,
the
microwave c a v i t y on the He-3 c r y o s t a t was constructed from very p u r e ,
vacuum refined copper.
The r e s i s t i v i t y of t h i s m a t e r i a l i s apparently
temperature dependent even a t 1K.
f i e l d modulation a t the samples.
This r e s u l t s i n a temperature dependent
S i n c e a l l temperature c a l i b r a t i o n
measurements a r e based on r a t i o s of t h e two l i n e a m p l i t u d e s , one might
e x p e c t t h i s e f f e c t to c a n c e l out.
The two samples do not s i t in
equivalent p o s i t i o n s in t h e cavity (one i s surrounded by an annular g a p
and s i t s on a warmer piece of copper) so the c a n c e l l a t i o n I s not complete.
As d e r i v a t i v e spectroscopy produced an erroneous temperature
c a l i b r a t i o n , absorption spectroscopy was used i n s t e a d .
Signal/noise
problems were a l l e v i a t e d somewhat by t h e use of optimally l a r g e spin
populations (~10
17
).
Extremely low microwave power l e v e l s (<50pW) were
r e q u i r e d to avoid measurable s a t u r a t i o n e f f e c t s .
The magnetoresistance of
t h e temperature sensor was a l s o measured, and a small c o r r e c t i o n f a c t o r
(<15mK) was added to account for i t .
The net r e s u l t was a temperature
c a l i b r a t i o n a c c u r a t e to a few mK in t h e 0.3K to 2.OK r a n g e .
48
3.3 Magnetization Studies
The temperature dependence of the magnetization of a sample can be
checked in a manner s i m i l a r t o the temperature sensor c a l i b r a t i o n
procedure.
The EPR signal s t r e n g t h of t h e sample i s compared to t h a t of
LMN:Nd a t various temperatures.
Any change in the r a t i o of the two
signals i n d i c a t e s a temperature dependent l i n e shape or a deviation from
simple Curie law behavior.
The sample under study and the LMN:Nd standard
are put in equivalent p o s i t i o n s on the He-3 evaporation pot 30 t h a t
d e r i v a t i v e spectroscopy (with e x t e r n a l f i e l d modulation) can be used.
If
the l i n e shape remains constant the area under the absorption curve, the
height of t h e absorption curve, and the peak-to-peak d e r i v a t i v e h e i g h t
(measured i n d e r i v a t i v e spectroscopy) w i l l a l l be p r o p o r t i o n a l .
This
p r o p o r t i o n a l i t y can be checked by computing the f i r s t moment of the
d e r i v a t i v e curve which is equal to the area under t h e absorption curve. A
mechanical Amsler I n t e g r a t o r was used for t h i s purpose and i t indicated
temperature independent l i n e shapes.
All changes i n the signal r a t i o s
therefore i n d i c a t e deviations from simple Curie law behavior.
3.4 Sample Preparations
All of the sample preparations used In this study started with .025
In. thick, high purity, c-Si wafers which were vacuum float zone grown, p
type with room temperature resistivities above 1000 ohm-ora, and had (110)
faces.
The wafers were cut to fit the sample cavity and then etched to
remove saw damage.
The broad faces were polished with 30 micron diamond
49
g r i t and then 50 microns waa removed by e t c h i n g .
The samples were
o r i e n t e d in the c a v i t y such t h a t the q u a n t i t y H X H . (H = s t a t i c applied
magnetic f i e l d , H.. = microwave frequency magnetic f i e l d ) did not vary by
more than a f a c t o r of 3 over t h e volume of t h e sample and H was p a r a l l e l
to £110].
Each side of the wafer was amorphized s e p a r a t e l y i n successive
production runs under i d e n t i c a l c o n d i t i o n s .
The r e s u l t was an amorphous
region with a surface area of 3.2 cm and a depth between 36005 and
13,0008, depending on the p r o c e s s .
Three d i f f e r e n t processes were used
to prepare samples for t h i s study:
various types of ion implantation i n t o
c - S i , vacuum condensation of Si vapor created by e l e c t r o n beam heating,
and s p u t t e r i n g .
Note t h a t in the l a s t two cases the wafers discussed
above function only as s u b s t r a t e s for the amorphous f i l m s .
All of the
samples were prepared a t Sandia Laboratories under t h e supervision of Dr.
Keith Brower.
Each of the preparation processes w i l l now be reviewed.
3.4.1 Ion Implantation
Between 1965 and 1975 ion implantation progressed from being the
s u b j e c t of numerous fundamental i n v e s t i g a t i o n s to becoming a useful and
widely applied t o o l for research and technology (for review see Mayer, e t
a l . , 1970j Townsend, e t a l . , 1976).
In t h i s study ion implantation is
used a s a tool for producing very pure a-Si with c o n s i s t e n t sample
c h a r a c t e r i s t i c s and minimal H, C, and 0 contamination.
Ion implantation
i s unique among a-Si p r e p a r a t i o n techniques in t h a t i t i s not a deposition
process.
I t converts c-Si to a-Si by producing c l u s t e r s of h i g h l y
disordered Si which coalesce to form a continuous random phase if the dose
50
is high enough (Mazey, et al., 1968).
The process by which this occurs is
not understood on the microscopic scale.
In general, the distribution of
damage produced in o-Si will depend on the substrate temperature and
orientation (with respect to the ion beam) as well as the projectile
energy, mass, and fluence.
It is possible, with careful manipulation of
these variables, to produce a thick and continuous amorphous layer with a
relatively small volume of clustered and defect ridden c-Si.
Semlemperical theories have been developed to codify the various
experimental data and to assist the experimentalist (and englneeer) in the
proper specification of these variables.
A heavy charged particle moving through the substrate at KeV energies
loses energy through screened Coulomb Interactions with the substrate
atoms.
It is conventional to partition the energy loss Into two
components:
(1) nuclear collisions which result in translational motion
of the target atom and (2) electronic collisions which result in the
excitation or ejection of an atomic electron.
The former process results
in cascades of off-lattice atoms and significant deflections of the
trajectory, and predominates at low projectile energies (<40KeV) and high
projectile masses (>50amu).
The latter process dominates at high
projectile energies (>100KeV) and low projectile masses (<20amu) and
produces few off-lattice atoms and negligible deflection.
that one 40KeV
ln + ion incident on room temperature c-Si will produce
about 3000 off-lattice atoms.
and
The result is
Under the same conditions a
Si + ,
4 +
He will produce about 1000, 10, and zero off-lattice atoms
respectively.
These results were obtained from channeled He
B+,
51
b a c k s c a t t e r i n g (Mayer, e t a l . , 1968), a technique which takes advantage of
the I n a b i l i t y of He to c r e a t e l a t t i c e damage.
Channeling e f f e c t s are
s i g n i f i c a n t throughout the KeV energy range and must be accounted for if
the ion beam i s o r i e n t e d along d i r e c t i o n s of high symmetry.
In t h i s work
the ion beam was oriented 5° off the s t r o n g e s t channeling d i r e c t i o n in the
diamond l a t t i c e , <110>.
Simpler, non-channeling c a l c u l a t i o n s can then be
used and these have the added advantage of applying to a-Si a s w e l l .
Methods for c a l c u l a t i n g t h e d i s t r i b u t i o n of energy d e p o s i t i o n into
atomic displacement have been developed by Brice (1970) and Winterbon
(1972).
These methods are derived from the range-energy r e l a t i o n s of
Lindhard and o t h e r s (Lindhard and Scharff, 1961) which are reviewed in
Gibbons (1968).
The c a l c u l a t i o n s of Brice include the c o n t r i b u t i o n s of
r e c o i l i n g t a r g e t atoms (Brice, 1975a), are a v a i l a b l e in t a b u l a r form
( B r i c e , 1975b), and were used in t h i s study.
Figure 3.2 shows t h e
d i s t r i b u t i o n for t h e ion implantation c o n d i t i o n s used for sample no. 1:
40
240KeV
+
Ar
1 5 - 2
a t a fluence of 4x10
cm
on a room temperature s u b s t r a t e .
As t h e ions produce the most atomic displacement when t h e i r energy i s
below 50KeV, the damage in the c r y s t a l i s concentrated a t a depth somewhat
l e s s than the p r o j e c t e d range which In t h i s case i s 24908. Various data
i n d i c a t e that the atomic displacement energy d e n s i t y threshold for
production of a - S i i n room temperature c-Si s u b s t r a t e s i s about 2x10 24
eV/cra3 or 2 eV/8 3 ( S t e i n , e t a l . , 1971; Crowder and T i t l e , 1971; Vook,
1973).
This t h r e s h o l d is indicated in figure 3.2 and can be used to
p r e d i c t the depth of the amorphous region a s shown.
As the l a t t i c e damage
b u i l d s up during implantation, the threshold w i l l be crossed f i r s t
at
52
cu
'
1
1
1
i
1
'
1
••
£(A 15 —
—
-
C
-
o
Displacement Energy
Q
-
-
u
E
'
o
Threshold for Amorphous State
<
n
Figure 3.2
I
1
1000
i
1
1
2000
3000
Depth Into Crystal (&)
_ \^
-
i
4000
Displacement Energy Distribution
Computed according to Brice (1975a) for nonchanneled 240 KeV Ar ion implantation at a
15
-2
fluence of 4x10
cm
onto room temp. c-Si.
53
about 18008.
As t h e dose c o n t i n u e s , the amorphous region spreads inward
and outward to the c r y s t a l s u r f a c e .
In o r d e r to minimize t h e volume of
the damaged but non-amorphous r e g i o n , s u f f i c i e n t fluence was used to b r i n g
t h e amorphous region o u t to the s u r f a c e .
Figure 3.2 i s a b i t misleading in t h a t t h e atomic displacement energy
does not continue t o grow a f t e r c r o s s i n g t h e amorphization t h r e s h o l d .
i s apparently i m p o s s i b l e to ion damage amorphous s i l i c o n .
It
Evidently t h e
system has enough topological f l e x i b i l i t y t h a t displaced atoms i m e d i a t e l y
r e t u r n to qua s i - e q u i l i b r i u m p o s i t i o n s .
The r e s u l t is t h a t damage beyond
t h e threshold " s t i r s t h e pot but d o e s n ' t change the c o n s i s t e n c y of t h e
stew."
The IBM group has found t h a t a-Si made by ion implantation i s
i n d i s t i n g u i s h a b l e ( i n terms of measured p r o p e r t i e s ) from t h a t made by
conventional means such as evaporation and s p u t t e r i n g (Crowder, et a l . ,
1970; Crowder and T i t l e , 1971).
This is s u r p r i s i n g in l i g h t of the
compositional complexity of a-Si and i t s a p p a r e n t heterogeneous s t r u c t u r e .
Six different samples were prepared by i o n implantation and are
described in table 3 . 1 .
All implantations were performed a t room
temperature with t h e ion beam misoriented from [110] by 5 ° .
The d e p t h s of
the amorphous r e g i o n s were c a l c u l a t e d from t h e Brice t h e o r y as o u t l i n e d
above.
A3 the mass of the implanted atom i s decreased, h i g h e r and h i g h e r
fluence ( a t the same implantation energy) must be used to make the sample
amorphous out to t h e surface.
The r e s u l t i n g atomic displacement energy
d i s t r i b u t i o n s have i n c r e a s i n g l y l e s s amplitude and more d e p t h than t h e one
shown In figure 3 . 2 .
Two r e s u l t s of t h i s a r e a larger volume of damaged
54
Table 3.1
Preparation Conditions for Samples made by Ion Implantation
Fluence
(cm"2)
Sample
Implanted
No.
Ion
Implantation
Energy (KeV)
1
40 A +
Ar
240
4x10
2
40. ++
Ar
250
5x10
3
28
Si+
250
5x10
4
20
Ne++
240
1.4x10
16
o+
250
2.0x10
IV
240
2.3x10
5
6
Calculated Depth of
Amorphous Region (8)
15
3600
15
7200
15
4500
16
13000
16
7900
16
8100
55
but non-amorphous sample, and a less d i s t i n c t t r a n s i t i o n from amorphous to
crystalline regions.
This trend is observed in t h e data which w i l l be
discussed in the next c h a p t e r .
Sample t h r e e is n o t amorphous a l l t h e way
out to the surface and a h i g h e r fluence should have been used.
The Brice
theory p r e d i c t s an amorphous region from 658 to 46008.
3.4.2 Electron Beam Evaporation
The most widely used p r o c e s s for c r e a t i n g a - S i i s vacuum condensation
of Si vapor.
All of the EPR research groups mentioned in c h a p t e r one used
t h i s type of sample at one time or a n o t h e r .
S i l i c o n melts a t 1680K but
must be heated to about 1800K to achieve a s i g n i f i c a n t vapor p r e s s u r e .
In
t h i s s t a t e Si i s extremely r e a c t i v e and w i l l i n t e r a c t to some degree with
a l l known c r u c i b l e m a t e r i a l s .
The standard solution to t h i s problem I s t o
use e l e c t r o n beam heating of a water cooled t a r g e t .
The technology and
advantages a s s o c i a t e d with t h i s technique are reviewed in Stewart (1983).
Sample 7 was created under standard deposition c o n d i t i o n s :
pressure - 4x10" t o r r , s u b s t r a t e temperature -
base
25K, e l e c t r o n beam - 2.6mA
a t 10KV, deposition r a t e - 428/sec, film thickness - 700o8.
Film
adherence to t h e c-Si s u b s t r a t e was good and there was no v i s u a l evidence
of s p a l l i n g or cracking.
Thomas, e t a l . , (1978) found contamination
e f f e c t s (probably due to H, C, and 0) i n samples made by evaporation a t
-8
p r e s s u r e s above 3x10" t o r r .
This i s one of the reasons for t h e use of ion
implantation as the p r i n c i p a l preparation method i n t h i s s t u d y .
56
3 . 4 . 3 Sputtering
S p u t t e r i n g has been described as the process of painting a wall by
p l a c i n g open p a i n t cans near the wall and throwing rocks into them.
Ar+
wall i s the s u b s t r a t e , t h e p a i n t i s the t a r g e t , and t h e rocks a r e
ions.
The
The g r e a t u t i l i t y of s p u t t e r i n g i s t h a t i t works for a l l known
s o l i d m a t e r i a l s and a l l reasonable combinations of t a r g e t and s u b s t r a t e
materials.
I t s great disadvantage is t h a t i t is a plasma d e p o s i t i o n
process and i s therefore hard to analyze from f i r s t p r i n c i p l e s .
Another
disadvantage i s that some of the s p u t t e r i n g gas ( u s u a l l y Ar) i s
incorporated into the deposited film — u s u a l l y at t h e 1 or 2 a t . $ level.
A good review of the l o r e of s p u t t e r i n g can be found i n Stuart (1983).
Much of the early IBM work on a-Si used sputtered samples (Brodsky
and T i t l e , 1969; Brodsky, e t a l . , 1970).
Since then t h e desire t o have
b e t t e r control over experimental c o n d i t i o n s has caused most r e s e a r c h e r s t o
deemphesize s p u t t e r i n g i n favor of ion Implantation o r evaporation.
In
t h i s study the samples made by Ar ion implantation behaved d i f f e r e n t l y
than t h e o t h e r s .
The main motivation for making a sputtered sample was t o
see i f the presence of Ar in the amorphous region was associated with t h i s
difference.
Sample 8 was prepared with a Varian (Palo Alto, CA) "S" Type Sputter
Gun run a t 980VDC and 1.0A.
The base p r e s s u r e was 1x10" t o r r and the Ar
gas pressure during s p u t t e r i n g was 3.5x10
J
torr.
The r e s u l t i n g film was
50008 t h i c k (computed from deposition r a t e monitoring), adhered well to
57
the c-Si substrate, and showed no visible evidence of spalling or
cracking.
58
CHAPTER 4
RESULTS AND INTERPRETATION
4.1 EPR Spectroscopy
EPR spectra were obtained a t ~2K and ~0.38K for a l l the samples
l i s t e d in t a b l e 3 . 1 .
In a l l cases the s t r u c t u r e of the g=2.0055±»0005
l i n e remained the same a t t h e two t e m p e r a t u r e s .
There i s no s t r u c t u r a l
evidence for changes in motional narrowing or t r a p p i n g mechanisms of any
kind a t temperatures down t o 0.38K.
A d e r i v a t i v e spectrum for sample 3 ( S i + implant) I s shown in f i g u r e
4.1.
The spectrum i s within a few percent of being a symmetric Lorentzian
l i n e except for a small i n t e r f e r i n g l i n e a t g=2.0101.
In figure 4.1 a
Lorentzian f i t to t h e g=2.0059 l i n e has been superimposed in t h e r e g i o n
n e a r g=2.0101 to make the i n t e r f e r i n g l i n e more obvious.
This I n t e r f e r i n g
l i n e is not i s o t r o p i c and i s t h e r e f o r e a s s o c i a t e d with c l u s t e r i n g and/or
damage in t h e c-Si region.
I t is almost c e r t a i n l y due to the S1-P3 c e n t e r
(a -Clio} planar four-vacancy) which has been i d e n t i f i e d by Brower (1971).
Brower finds (1977) that t h i s c e n t e r e x h i b i t s a s t r o n g l i n e a t
g=2.0102±.0002 with H p a r a l l e l to £110],
The spectrum in figure 4.1 I s
very s i m i l a r to t h o s e measured for samples 1, 2, and 4, which were Ar + ,
Ar
, and Ne
implants r e s p e c t i v e l y .
The i n t e r f e r i n g l i n e was s m a l l e r for
t h e Ar implants and l a r g e r for the Ne implant.
This i s c o n s i s t e n t with
Magnetic Field (gauss)
c
H
01
5845
*•
o
o
o
s:
(0
<
<
<
ID
at
H
CD
t)
:»
C
(t)
CO
P
ns
o
rt
«
I-
1
c
3
o
rt
o
3;
N
CO
a>
3
(0
3
O
•
"O
I—
5855
—T~
5865
—1
5875
I
5885
T
5895
5905
—1
60
the increasing fractional volume of the damaged but non-amorphous region
as the mas3 of the implanted species decreases.
A derivative spectrum for sample 5 (0 + implant) is shown in figure
4.2.
Notice that the interfering line at g=2.0101 (5876 gauss in figure
4.2) has become much larger and that there are other superimposed lines as
well.
The superimposed lines are all anisotropic and at least one of them
is sitting very close to the center of the underlying amorphous signal for
the orientation (H parallel to [110]) shown in figure 4.2.
This is the
reason why the apparent g factor of the amorphous line is high at 2.0061.
The line at 5876 gauss indicates that a large fractional volume of the
sample is damaged but not amorphous.
The spectrum in figure 4.2 is very
similar to one obtained by Brower (1977) for c-Si damaged by l60KeV 0 +
13 -2
ions at a fluence of 2x10 "cm . This fluence is definitely below the
threshold for amorphization.
The spectroscopic evidence therefore implies
that a large fraction of the paramagnetic response of sample 5 is not due
to a-Si.
This is true even at the center of the line.
These conclusions are consistent with those of Brosious (1977), who
conducted an extensive investigation of 0 + implanted Si.
He concluded
that the atomic displacement energy density threshold for amorphization is
about 6 eV/8
for 0 + implantation at room temperature.
Brosious used the
Brice (1975a) method, as this study did, to calculate the distribution of
energy deposition into atomic displacement.
Unfortunately sample 5 (0 +
implant) was made assuming a threshold of 2 eV/8
larger ions.
which works well for
If the Brosious threshold is used, the Brice method would
61
g = 2.0061
Figure 4.2
D e r i v a t i v e EPR Spectrum for Sample 5
microwave frequency = 16.5292 GHz
(0
implant)
temp. = 0.360 K
62
p r e d i c t an amorphous region from ~12008 to ~65008 i n t o the surface and
explain t h e l a r g e non-amorphous componant i n t h e spectrum.
Sample 6 (N+ implant) was made under t h e same assumptions.
The
spectrum f o r sample 6 i s almost i d e n t i c a l t o t h a t shown In f i g u r e 4.2 and
i n d i c a t e s t h a t the observed f e a t u r e s are not a s s o c i a t e d with oxide or
n i t r i d e complexes in t h e S i .
Such complexes would be expected s i n c e the
high fluence used works out to about 1/2 a t . ? in the "amorphous" region.
Although samples 5 and 6 have helped to c l a r i f y t h e r o l e of t h e
g=2.0101 l i n e , they are of l i t t l e use in t h e a n a l y s i s of the a-SI
resonance.
The spectrum for sample 7 (e-beam evaporation) i s Lorentzian and
shows no evidence of i n t e r f e r i n g l i n e s .
With i t s s l i g h t asymmetry taken
i n t o account, (the high f i e l d peak exceeds t h e low f i e l d peak by 4.658) the
l i n e is within 2% of being a p e r f e c t Lorentzian.
The peak-to-peak
linewidth i s 10 gauss ( a t 16.5 GHz) which i s considerably g r e a t e r than t h e
«6 gauss l l n e w i d t h s seen for samples 1-6 and expected on t h e b a s i s of
previous r e s u l t s (see Thomas, e t a l . , 1978).
Thomas, e t a l . , did see
llnewidths of about 10 g a u s s , but only for evaporated films deposited onto
77K s u b s t r a t e s .
For room temperature deposition or room temperature
anneals of films made a t lower temperatures, Thomas, et a l . , saw t h e
normal 6 gauss l i n e w i d t h .
but not unprecedented.
The 10 gauss linewidth i s t h e r e f o r e unexpected,
63
The spectrum of sample 8 ( s p u t t e r e d ) is q u i t e d i f f e r e n t from the
o t h e r s in t h a t the l i n e shape i s i n t e r m e d i a t e between Gaussian and
Lorentzian, the linewidth i s g r e a t e r than the o t h e r s a t 13.8 gauss
(peak-to-peak a t 16.5 GHz), and t h e r e i s s i g n i f i c a n t asymmetry p r e s e n t
(the high f i e l d peak exceeds the low f i e l d peak by 2 2 ? ) .
Previous s t u d i e s
have shown t h a t these t h r e e p r o p e r t i e s a r e influenced by contamination
(Thomas, e t a l . , 1978).
I t t h e r e f o r e appears t h a t t h e s p u t t e r e d sample i s
of l e s s e r q u a l i t y than the o t h e r s .
Spectroscopic data on samples 1-8 i s summarized in t a b l e 4 . 1 .
measurements in t a b l e 4.1 were made by Dr.
The
Keith Brower a t Sandia
Laboratories on samples prepared a t the same time and under the same
c o n d i t i o n s a s samples 1-8.
The g f a c t o r s and l i n e widths for samples 1-8
were checked by the author and found the be c o n s i s t e n t with Brower's
r e s u l t s , although not as a c c u r a t e .
The llnewidths measured by t h e author
a t 16.5 GHz and ~4K were about 5% l e s s than Brower 1 s r e s u l t s which were
obtained a t 20.3 GHz and 20K (H was p a r a l l e l t o L1103 in both c a s e s ) .
The
llnewidths are not temperature dependent below 20K b u t do have a s l i g h t
frequency dependence due to the inhomogeneous broadening of t h e l i n e .
The
5% d i f f e r e n c e observed between 16.5GHz and 20.3 GHz values i s c o n s i s t e n t
with s i m i l a r observations made by Thomas, e t a l . , (1978).
The spin density v a l u e s reported in t a b l e 4.1 a r e a l s o due to Brower.
The sample EPR s i g n a l s were compared to t h a t of a w e l l c h a r a c t e r i z e d
sample of p type c-Si obtained from E.A. Gere of Bell L a b o r a t o r i e s , Murray
H i l l , NJ.
The r e s u l t s were adjusted for changes in t h e c a v i t y Q and
64
Table 4.1
Sample
No.
Spectroscopic Data for Amorphous Silicon
Preparation
g factor
(+0.0001)
linewidth
(+0.1 gauss)
spin density
(cm-3)
1
40. + .
Ar imp.
2.005S
6.1
19
4.2xl0xv
2
40. ++.
Ar imp.
2.0058
6.3
2.2x10
3
28„.+ .
Si imp.
2.0059
6.3
19
1.7x10 *
4
20 M ++.
Ne imp.
2.0058
6.0
19
l.lxl0x*
19
5
0
imp.
2.0061
6.2
8.4xl018
6
N
imp.
2.0058
6.0
2.4xl018
7
evaporated
2.0057
10.2
8
sputtered
2.0058
13.8
19
19
l.lxl0xv
Note: All measurements made at 20.3 GHz and 20K with H parallel to [110]
65
f i l l i n g f a c t o r and are based on numerical i n t e g r a t i o n of a Lorentzian f i t
to the d e r i v a t i v e s p e c t r a .
The spin d e n s i t y values for samples 1, 2 , 3,
7, and 8 are about a f a c t o r of 4 below those reported by Gourdon, e t a l . ,
(1981); about 305& below those reported by Thomas, e t a l . , C1978) y and a
f a c t o r of 2 above those r e c e n t l y reported by Waddell, e t a l . , (1984).
This order of magnitude range in t h e reported values i n d i c a t e s the
d i f f i c u l t y In making accurate spin density measurements.
contamination may also have been a f a c t o r ,
reported by Waddell and co-workers.
Sample
p a r t i c u l a r l y for the low values
In s p i t e of t h e disagreement o v e r
a b s o l u t e magnitude, a l l of the mentioned groups find about a factor of 3
range in spin density a s the p r e p a r a t i o n c o n d i t i o n s are v a r i e d .
The spin
d e n s i t i e s for samples 4 and 8 are low in t h i s r e s p e c t , being a f a c t o r of 4
below the maximum reported d e n s i t y in t a b l e 4 . 1 .
samples 5 and 6 (
0
and
The problems with
N r e s p e c t i v e l y ) have a l r e a d y been discussed.
I t i s p o s s i b l e t h a t Ne i s l i g h t enough at mass=20 to cause s i m i l a r
problems during i m p l a n t a t i o n .
Sample 4 has a normal EPR spectrum ( a s
compared to samples 1, 2 , and 3) however, i n d i c a t i n g t h a t t h e f r a c t i o n a l
volume of non-amorphous but damaged Si i s small.
4.2 Spin L a t t i c e Relaxation Measurements — Results
In addition to T and H, the observed SLR r a t e may depend on o t h e r
v a r i a b l e s such as the p u l s e d u r a t i o n and power l e v e l or the monitor power
level.
(These terms, and t h e i r connection with t h e experimental
equipment, are discussed in chapter t h r e e . )
Such dependences are
i n d i c a t i v e of nonldeal experimental c o n d i t i o n s and are p o s s i b l e sources of
66
systematic e r r o r s .
For i n s t a n c e , dependence of t h e observed r a t e on t h e
p u l s e duration or power level sometimes i n d i c a t e s t h a t t h e sample i s being
heated by the microwave p u l s e .
Pulse duration and power l e v e l were chosen
so t h a t t h e i r influence on the measured SLR r a t e s was minimized or
eliminated.
The v a r i o u s samples were analyzed under the following
standard c o n d i t i o n s :
microwave pulse duration - 100/Msec,
difference
between monitor power and pulse power - 40dB, time delay between p u l s e s 0.6 s e c .
I t was observed t h a t the optimal power l e v e l s ( t h o s e l e v e l s
where the observed r a t e was almost independent of power l e v e l and
s i g n a l / n o i s e was adequate) were roughly p r o p o r t i o n a l to T.
Approximately
1/tW of p u l s e power was used a t 1K and the power a t other temperatures was
scaled from t h i s l i n e a r l y in T.
Any dependence of t h e measured r a t e s on
p u l s e duration was unobservable for d u r a t i o n s between 50/tsec and 400/sec.
The same was t r u e for ±3dB changes i n power l e v e l s and ±6dB changes In t h e
d i f f e r e n c e between pulse and monitor powers.
Measured SLR r a t e s for sample 3 (
4.3.
28 +
Si implant) are shown in figure
Also shown are the t h e o r e t i c a l l y predicted temperature dependences
for the d i r e c t process (see equation 2.7) and the phonon b o t t l e n e c k (see
equation 2 . 1 5 ) , scaled to meet t h e d a t a a t 4K.
Both s e t s of data ( 9 . 3 GHz
and 16.5 GHz) are c l e a r l y well f i t by a simple Tn power law which i s
165 T
.
The s e p a r a t e power law f i t s for 9.3 GHz data and 16.5 GHz data
are (167±11) T 2 * 3 4 ± 0 * 0 8 and (169±12)
2 3
T
- 9±°-°9
re3pectively.
The
combined f i t a t (165±3) T 2 , 3 6 ± 0 , 0 2 i s well within t h e e r r o r s for t h e
s e p a r a t e f i t s , i n d i c a t i n g that t h e observed r a t e i s independent of
frequency and applied magnetic f i e l d H.
The SLR process i s c l e a r l y not
10
Direct Process
coth (0.396/T)
a
Phonon Bottleneck
coth2 (0.396/T)
165 T
2.36
10 2
X -
9.3 GHz (X band) data
+ - 16.5 GHz (Ku band) data
1
10
•
J
0.3
I
I
I
0.4 0.5 0.6
I
I
I I
1
TEMPERATURE
ure 4.3
2
3
4
(K)
Electron Spin Lattice Relaxation Rates for Amorphous Silicon
Made by 28g£+ i o n implantation
(sample 3)
68
the phonon bottlenecked d i r e c t process proposed by Gourdon, e t a l . ,
(1981).
Experimental r e s u l t s for samples 1 and 2 (
Ar+ implant and
Ar ++
implant r e s p e c t i v e l y ) along with the f i t from sample 3 a r e shown in figure
4.4.
In c o n t r a s t to t h e simple power law behavior of sample 3 , these
samples show two d i f f e r e n t power law dependences with a sharp t r a n s i t i o n
between the two a t ~ 1 . 2 K .
This behavior Is very unusual.
When a spin
system changes from one SLR process to another there i s o r d i n a r i l y a small
range in temperature where t h e two process compete.
This causes a gradual
t r a n s i t i o n from one temperature dependence to t h e other.
The sharp
t r a n s i t i o n s in figure 4.4 i n d i c a t e t h a t a d i s t i n c t change in t h e SLR
process i s occurring a t ~1,2K.
The d a t a for sample 1 a r e best f i t by
(396±24) T 3 , i , 0 ; t 0 ' 1 0 below ~1.2K and (497±23) T 2 * 0 9 * 0 , 0 8 above t h a t
temperature.
The i n t e r s e c t i o n of t h e s e two f i t s occurs a t 1.19±0.08 K.
The data for sample 2 are b e s t f i t by (277±14)
(349±14) i 2 , 1 2 * 0 , 0 * 1 above t h a t temperature.
16.5 GHz and 9.3 GHz d a t a .
(355±22)
2
T
-10±0-06
(345±26) T 2 , 1 5 ± * .
whlle
T 3.26±0.O8 b e l Q w
K2K
an(J
The l a t t e r f i t includes both
The f i t for the 9.3 GHz data alone Is
the
f i t for t h e 16.3 GHz data i s
The SLR r a t e appears to be independent of frequency
and H, j u s t as i t was for sample 3.
The high and low temperature f i t s
for
sample 2 i n t e r s e c t a t 1.23±0.06 K, a value very close t o the i n t e r s e c t i o n
for sample 1.
SLR data for samples 4, 7 , and 8 a r e shown in f i g u r e 4.5 along with
the power law f i t from sample 3 .
The d a t a are well f i t by simple power
69
o
<u
<n
0.3
0.4 0.5 0.6
1
TEMPERATURE
Figure 4.4
2
4
5
(K)
Electron Spin Lattice Relaxation Rates for Amorphous Silicon
Made by * Ar + and *°Ar ++ Ion Implantation
(samples 1 and 2
respectively)
70
10:
10 2
•
10'
0.3
0.4
0.5 0.6
1
TEMPERATURE
Figure 4.5
(K)
E l e c t r o n S p i n L a t t i c e R e l a x a t i o n R a t e s f o r Amorphous S i l i c o n
Made by V a r i o u s P r e p a r a t i o n s ( f r e q u e n c y = 1 6 . b GHz)
71
laws and there is no evidence of the d i r e c t process or the phonon
bottlenecked d i r e c t p r o c e s s .
The sample made by s p u t t e r i n g (sample 8)
3hows a power law dependence of (118t10) T 2 * 3 4 ± 0 , 0 5 which I s s i m i l a r to
28 +
t h e best f i t for sample 3 ( Si Implant). Apparently the presence of 1
or 2 at./£ Ar in the sputtered sample and the less than i d e a l sample
p r e p a r a t i o n (as evidenced by the non-standard EPR spectrum) have n o t had a
major influence on the SLR.
Since the amorphous region of sample 8
c o n t a i n s more Ar than t h a t of sample 1 or 2 (1 or 2 at.Js v s . ~0.2 at.56),
i t seems u n l i k e l y t h a t Ar Is associated with the sharp t r a n s i t i o n a t 1.2K
for samples 1 and 2 .
The frequency and H dependence of t h e SLR r a t e of
sample 8 were checked a t 1.50K.
302.9*3.7 1/sec.
The measured r a t e a t 9.3 GHz was
The best f i t for the 16.5 GHz data p r e d i c t s 304.8 1/sec
for the SLR r a t e a t 1.50K.
Samples 2, 3 , and 8 thus i n d i c a t e t h a t the
observed SLR r a t e s are independent of frequency and H.
The best f i t s for sample 4 (
are r e s p e c t i v e l y :
NeT+ implant) and sample 7 (evaporated)
(466*19) T 3 , 2 7 * 0 , 0 8 and (740±43) T 3 , 4 7 * 0 , 0 9 .
I t is
i n t e r e s t i n g to note t h a t the exponents for a l l six samples f a l l in two
narrow r a n g e s :
2.09-2.36 and 3 . 2 6 - 3 . 4 7 .
The data in figure 4.5 shows a
s t r o n g c o r r e l a t i o n between the SLR r a t e p r e f a c t o r and the amount by which
the exponent n exceeds 2.
This r e l a t i o n s h i p i s shown in figure 4.6 where
X=n-2 i s p l o t t e d against the r a t e p r e f a c t o r s .
The b e s t f i t to the data is
shown which corresponds to a r a t e p r e f a c t o r of 419A.
All of the d a t a in
figure 4.5 can be f i t to within 18% by the expression 419A T 2 "^.
72
sample 7
O
O
n..
h
o
o
i
^
sample 4
O
Q
a
a
«..
sample 3
sample 8
0.0
Figure 4.6
0.2
a4
0.6
0.8
1.0
1.2
1.4
1.6
Correlation of Electron Spin Lattice Relaxation Rates With A
( Rate«<Tn, X = n - 2 )
73
The microwave s i g n a l r e c o v e r i e s for samples 1-4, 7» and 8 were not
exponential over t h e i r e n t i r e range in amplitude.
The SLR r a t e was
determined from the l a s t portion of the recovery where t h e s i g n a l recovery
appeared exponential within an accuracy of s e v e r a l p e r c e n t .
t h i s exponential region varied somewhat, depending on T.
The range of
At M).4K t h e
r e c o v e r i e s were exponential over the l a s t 3556 of t h e t o t a l recovery
amplitude.
This percentage g r a d u a l l y drops with I n c r e a s i n g temperature,
reaching about 2056 for the highest r a t e s measured.
This behavior i s r e l a t e d to the fact t h a t the EPR l i n e s a r e
inhomogeneously broadened.
During the p u l s e s a t u r a t i o n experiments only
the spin packet a t the c e n t e r of the EPR l i n e is r e s o n a n t with the
microwaves.
The high power microwave pulse s a t u r a t e s only the c e n t r a l
spin packet in the l i n e , reducing i t s EPR s i g n a l amplitude to zero.
This
process i s referred to as "burning a hole" i n the EPR l i n e and for systems
with s u f f i c i e n t l y slow r e l a x a t i o n , the hole can be seen d i r e c t l y in t h e
EPR spectrum.
After s a t u r a t i o n t h e spins i n the c e n t r a l spin packet can
relax i n two d i f f e r e n t ways:
(1) coupling to l a t t i c e e x c i t a t i o n s which i s
the mechanism discussed in chapter 2 , i s r e f e r r e d to as l o n g i t u d i n a l
r e l a x a t i o n , and i s c h a r a c t e r i z e d by 1/T , and (2) exchanging energy with
nearby unsaturated spin packets through s p i n - s p i n i n t e r a c t i o n s .
l a t t e r process i s c a l l e d s p e c t r a l d i f f u s i o n .
This
The same mutual spin f l i p
processes t h a t cause exchange narrowing by r e d i s t r i b u t i n g energy within an
EPR l i n e (see chapter 1) w i l l cause s p e c t r a l diffusion i n t o a s a t u r a t e d
spin packet.
The EPR l i n e s in t h i s study a r e inhomogeneously broadened
and exchange narrowed; "hole burning" and s p e c t r a l diffusion are therefore
74
expected.
The idea of spin packets and inhomogeneous broadening d a t e s
baok to papers by P o r t i s (1953, 1956).
A short review of s p e c t r a l
diffusion and "hole burning" can be found in a r e c e n t paper by Anderson
(1978).
The e a r l y p a r t of the spin recovery f o r samples 1-4, 7, and 8 i s
nonexponential and dominated by s p e c t r a l d i f f u s i o n .
This process i s
a p p a r e n t l y f a s t enough t h a t s p e c t r a l diffusion into t h e c e n t e r of t h e l i n e
i s e s s e n t i a l l y over in a time short compared to T...
The l a t t e r p a r t of
t h e recovery of the c e n t r a l spin packet i s then dominated by l o n g i t u d i n a l
r e l a x a t i o n which produces ( i n t h i s case) an exponential recovery.
Two other measurements of SLR i n a-Si have been made (Gourdon, e t
al.,
1981; Stutzmann and Biegelsen,
1983), both by i n d i r e c t methods.
d a t a of Gourdon and co-workers i s shown in figure 2 . 1 , curves 1-4.
The
These
r e s u l t s are c o n s i s t e n t with the data from the p r e s e n t study which are
represented by curves 6 and 7.
sample 8 (118 T
3
)
3 47
sample 7 (740 T J * ' ) .
Curve 6 i s an e x t r a p o l a t i o n of the f i t
while curve 7 i s an e x t r a p o l a t i o n of t h e f i t
for
for
The data of Stutzmann and Biegelsen (curve 5) are
not in good agreement with the o t h e r d a t a .
I t i s not c l e a r t h a t t h e
experimental technique employed by Stutzmann and Biegelsen properly
accounts for s p e c t r a l d i f f u s i o n and t h i s i s probably responsible for the
disagreement.
Spin recoveries were obtained for samples 5 and 6 and were found to
be nonexponential a t temperatures above 0.45K.
Even when the l a t t e r
75
p o r t i o n s of the recovery were e x p o n e n t i a l , the observed T. was s t r o n g l y
dependent on p r e c i s e l y where in the c e n t e r of the l i n e the resonance was
tuned.
This behavior is c o n s i s t e n t with the a n a l y s i s of t h e EPR s p e c t r a
for these samples (see section 4 . 1 ) .
The S1-P3 c e n t e r seen by Brower
(1971, 1977) has s e v e r a l c l o s e l y spaced l i n e s which are superimposed on
the c e n t e r of the a - S i l i n e in the sample o r i e n t a t i o n used in t h e s e
experiments (H p a r a l l e l t o (1103).
The dependence of T. on the e x a c t
p o s i t i o n in the c e n t e r of the l i n e i n d i c a t e s i n t e r f e r e n c e from t h e s e
centers.
Samples 5 and 6 are not v a l i d examples of a-Si and will not be
discussed
further.
The simple Tn temperature dependence seen in t h e SLR r a t e s f o r
samples 1-4, 7, and 8 w i l l now be discussed in t h e context of a theory
developed by Kurtz and Stapleton (1979, 1980).
4.3 Theory of Spin L a t t i c e Relaxation by Two Level Systems
The low temperature p r o p e r t i e s of amorphous m a t e r i a l s have been t h e
subject of numerous i n v e s t i g a t i o n s over t h e l a s t 15 y e a r s ,
see W.A. P h i l l i p s ,
1981)
(for review
Two s u r p r i s i n g observations can be made
concerning the thermal and a c o u s t i c p r o p e r t i e s of amorphous i n s u l a t o r s a t
temperatures below 1K:
(1) Data for a wide v a r i e t y of t h e s e materials
f a l l s into a very narrow range, i n d i c a t i n g the presence of c h a r a c t e r i s t i c
magnitudes and temperature dependences,
(2) This c h a r a c t e r i s t i c behavior
i s markedly d i f f e r e n t from that seen In c r y s t a l l i n e m a t e r i a l s .
These
r e s u l t s are s u r p r i s i n g because on the length scale of 1K thermal phonons
76
(~1000 fi) both crystals and amorphous materials should be well
approximated by elastic continua and thus exhibit similar behavior.
Experimental evidence is to the contrary, indicating that glasses,
undercrossllnked amorphous materials, and many polymers behave universally
as a group and anomalously as compared to crystals.
The theoretical
principles which underlie this apparently universal behavior are largely
unknown.
A useful model has emerged, however, and is widely used in the
Interpretation of experimental results,
1981b)
(for review see W.A. Phillips,
This theory proposes that amorphous solids possess a set of extra
excitations which are strongly coupled to the normal Debye phonons, and
are not found in perfect crystals.
These excitations are attributed to
entities that have a ground state and one excited state that are well
separated in energy from any higher excited states.
Since the theory is
concerned only with transitions between the two lowest states, these
entities are referred to as two level systems or two level states (TLS
either way).
The original form of this theory was proposed independently by
Anderson, Halperin, and Varma (1971), and by W.A. Phillips (1972).
The
authors suggested that the entities responsible for the extra excitations
might be atoms or configurations of atoms which tunnel between the minima
of a multi-well potential.
Such an entity can be represented as a mass m
in a one dimensional, asymmetric, double well potential like the one shown
in figure 4.7.
These are usually called tunneling states (TS).
77
V,
r
2d
Figure 4.7
A schematic representation of a tunneling state (TS).
Potential energy is plotted vertically versus spatial
coordinate. V 0 is the barrier height, £ is the
asymmetry and 2d is the well separation.
78
The a n a l y s i s begins with basis s t a t e s t h a t are t h e ground s t a t e
wavefunctions of each i n d i v i d u a l well and each well i s an harmonic
oscillator potential.
Assuming e i s n e g l i g i b l e , the ground s t a t e energy
for each well is V /X and the overlap energy between t h e two w e l l s i s
A = (2mV ) 1 / 2 d / h .
o
(V / A ) e " * S A/2 where
o
The system i s normally
considered under c o n d i t i o n s where t u n n e l i n g through the b a r r i e r i s
important but hopping over i t i s n o t .
This r e q u i r e s a small but f i n i t e
overlap (often defined as A>4) and temperatures low enough t h a t kT<<V .
o
Note that under these conditions the first excited state of the individual
wells, with energy 3V /A, will not be significantly occupied.
In the
basis of the individual well ground states, hereafter called the spatial
basi3, the TS Hamiltonian can be written as follows:
(4.1 )
H ' = 1/2
u -«.
In the diagonal (unprimed) basis the Hamiltonian becomes:
(4.2)
H = 1/2
o
/E
0
0
-E
2
2 1/2
where the energy s p l i t t i n g E i s given by E = (€ +A )
.
The TS thus has
the e s s e n t i a l c h a r a c t e r of a TLS — a ground s t a t e and one excited
state
( s p l i t by E) which a r e well separated in energy from any higher s t a t e s .
I t I s generally assumed t h a t in amorphous m a t e r i a l s t h e r e e x i s t s a
d i s t r i b u t i o n of TS p r o p e r t i e s V ,
d i s t r i b u t i o n in the values of E,
e , and d.
A, and
This generates a
£, only two of which are
79
independent.
I t i s often assumed t h a t the TS d i s t r i b u t i o n can be
described by:
(4.3)
j3(E,A)
j»(E)
Ad-tA/E) 2 ?' 2
where
p(E) is weakly dependent on E up to some cutoff E
drops r a p i d l y to zero ( J a c k i e , 1972).
, where i t
Physical p r o p e r t i e s of the system
are then computed a s an average over the d i s t r i b u t i o n .
The TS-phonon i n t e r a c t i o n i s found by expanding the TS Hamiltonian
(4.1) in s t r a i n .
A s t r a i n wave of amplitude e causes a f i r s t
order
c o r r e c t i o n H Tp ' given by
(4.4)
where e. . i s a l o c a l s t r a i n t e n s o r .
Using the s c a l a r approximation for
e and transforming to the diagonal b a s i s gives t h e following Hamiltonian:
H
(4.5)
"=,/2(M - l ) e
D = e i ^ + A M.
E ie
E £e
M
= AAL " «!£.
E *e
Eie
Measurements of a c o u s t i c p r o p e r t i e s are used to determine the magnitudes
80
of D and M.
A variety of data indicate that the magnitudes are roughly
comparable, at least when averaged over a TS distribution (Hunkllnger and
Arnold, 1976).
The development just outlined follows that of Hunkllnger and Arnold
(1976).
More rigorous treatments have been given (Jackie, et al., 1976),
and the theory can be cast in numerous mathematical forms, some of which
are reviewed by W.A. Phillips (1981b).
Although the TS theory has been widely used in the interpretation of
experimental data for amorphous solids, actual identification of the
tunneling species and potential is usually nonexistent or highly
speculative.
It is unlikely that all amorphous materials possess similar
distributions of asymmetric, double well potentials with appropriate
tunneling units.
(See Pohl, 1981 for discussion)
To the extent that a
large density of TS seems unlikely in a given amorphous material the
theory become pure phenomenology.
In the other limit are a few unusual
materials where TS seem to be a good physical description of the local
properties of the solid.
One of these is the crystalline superionic
conductor beta-alumina.
This family of materials are layered crystals, consisting of plates
2of alumina bonded together by 0
+
ions.
Other ions, in particular Li ,
Na + , and K + , can be diffused into the region between the plates where
their mobility is very high.
When beta-alumina is cooled to He-4
temperatures the mobile ions freeze out into a disordered array in the
81
planes between the alumina p l a t e s .
As t h i s d i s o r d e r e d l a y e r i s only one
atom t h i c k , the system i s e f f e c t i v e l y a two dimensional g l a s s in intimate
c o n t a c t with the face of a t h r e e dimensional c r y s t a l .
The low temperature
thermal p r o p e r t i e s of beta-alumina e x h i b i t the same c h a r a c t e r i s t i c
behavior seen in t h r e e dimensional g l a s s e s and undercrossllnked amorphous
m a t e r i a l s (Anthony and Anderson, 1976 and 1977).
A wide v a r i e t y of
experimental d a t a , including s t r u c t u r a l information r e l a t e d to t h e
presence of c r y s t a l l i n e o r d e r , i n d i c a t e t h a t t h e a l k a l i i o n s tunnel
between v a r i o u s i n e q u i v a l e n t p o t e n t i a l minima (Strom, e t a l . , 1978;
Anthony and Anderson, 1979; Kurtz, e t a l . , 1981).
Kurtz and Stapleton (1979, 1980) were a b l e to c r e a t e paramagnetic
c o l o r c e n t e r s in t h e d i s o r d e r planes of v a r i o u s beta-aluminas and thereby
observe EPR i n t h i s family of m a t e r i a l s .
Their measurements of SLR
i n d i c a t e d behavior i n c o n s i s t e n t with known r e l a x a t i o n p r o c e s s e s .
The
r e s u l t s were a t t r i b u t e d to a new process - e l e c t r o n SLR by coupling to a
d i s t r i b u t i o n of TS.
Drawing on a v a r i e t y of s t r u c t u r a l information,
Kurtz
and Stapleton found a s i g n i f i c a n t hyperfine i n t e r a c t i o n between the
nucleus of t h e tunneling s p e c i e s ( L i , Na, or K) and the paramagnetic color
center electrons.
The motion of t h e c a t i o n s i n the asymmetric double well
p o t e n t i a l modulates t h i s i n t e r a c t i o n , giving t h e following TS-spin
i n t e r a c t i o n Hamiltonian in the s p a t i a l b a s i s :
82
O1
/1
HL« «. AI-S
(4.6)
Tfa
±1
R VO
-1
A Fermi contact interaction i3 assumed, and R i s the distance to the color
center.
When transformed to the diagonal basis, the result i s :
(4.7)
H_,_ <* A?-S d 1 [
)
Using HTp (4.5) and Upg (4.7) in Fermi's Golden Rule for a mixed,
second order process,
(4.8)
Wfl = 2g
•R
£
<f|HTp|m><m|HTSli> 2 (
/ V
i
m
Kurtz and Stapleton were able to calculate the SLR rate.
The calculation
which follows is a generalization of that due to Kurtz and Stapleton
(1980).
As there is no particular reason to believe there are TS in a-Si, the
more general TLS (two level system) theory will be used instead.
In this
theory there is no reference to asymmetric, double well potentials or the
spatial basis associated with them.
The states are characterized only by
their energy splitting E (no A or €) and are otherwise undefined.
form of H
p
The
is given by equation 4.5, but unlike the TS theory there is no
83
particular connection between the values of D and M.
The relative merit
of the TLS theory, and its relation to the TS theory, is a matter of
controversy at present,
(for review see W.A. Phillips, 1981)
Lacking any specific knowledge of TLS in a-Si, It would be
speculative to specify a particular HT_.
The following general form is
used, where B is an unspecified operator at present and C and N are real
numbers.
(4.9)
C
N
N
-C
H T g = B(S + + S_)
The spin operators a c t on e l e c t r o n s , the matrix a c t s on the TLS, and
p o s s i b i l i t i e s for B w i l l be considered l a t e r .
More general i n t e r a c t i o n s
are p o s s i b l e , but the concern here i s with the component of such
interactions that f l i p s electron spins.
A schematic diagram of t h e energy l e v e l s and product wavefunctions
for t h e coupled TLS (<f±) - e l e c t r o n spin ( ± ) system i s shown in figure
4.8.
The v e r t i c a l l i n e s represent t r a n s i t i o n s which c o n t r i b u t e to the
r e l a x a t i o n of the TLS via the TLS-phonon coupling ( 4 . 5 ) .
I t i s assumed
t h a t t h i s i n t e r a c t i o n Is strong enough to keep the TLS in thermal
equilibrium with the phonon system.
The diagonal t r a n s i t i o n s r e p r e s e n t
the slower combined e l e c t r o n spin-TLS r e l a x a t i o n process, the r a t e of
which i s being c a l c u l a t e d .
The observed r a t e w i l l b e :
84
state a
l+,t|/+>—*
state b
|
-.</ / + > 7 r
E/2+B/2
E/2-8/2
V
A
/
\
state c
state d
l-,u>_:
Figure 4.8
E/2 + 8/2
E/2-8/2
Schematic diagram of the energy levels and product wave
functions for the coupled TLS (Y±) - electron spin (±)
system. The vertical lines represent transitions which
relax the TLS via TLS-phonon coupling. The diagonal
transitions represent the slower combined electron spinTLS relaxation process. E is the TLS energy splitting
and S is the Zeeman splitting.
85
(4.10)
1 / T l = Wad + Wda
+
Wbc
+
Wcb
W . i s c a l c u l a t e d using equation 4 . 8 .
The i n i t i a l s t a t e i s | + , %., n«>,
the f i n a l s t a t e i s l - , f - , n^+1>, and | + > , I fV>, and |n w > r e p r e s e n t the
e l e c t r o n s p i n , TLS, and phonon s t a t e s r e s p e c t i v e l y .
<-,4'-|H T S l
•R
,^><^,n<t+1|HTp|f4,not>
+
**•
+ <-,^|H TS U,f,X > P->^^|H Tp |^,n < >
E - fie***.
+ <*-,n«+1|HTp|*.t,rv)<-,%IHTSl + , ' ^ )
(4.11)
_
+ <y-,n<(+1|HTp|t.,nM><-,Y-|HTS|4,^>
2
E +$
ync^-(E+t))
T^ETkl
In t h i s equation £ i s the Zeeman energy s p l i t t i n g and £.. i s the Dirac
FVlrT
delta function.
The term ( H e * ' * 1 )
TLS in the upper s t a t e i n i t i a l l y .
1
i s the p r o b a b i l i t y of finding the
S u b s t i t u t i n g f o r t h e matrix elements
from equations 4.5 and 4 . 8 , t h e r e s u l t i s :
1.12)
W
aa
= 2ff 4BC /CM - ND\
fi
{ S E+S/
% l<n*+1 | e | n « > r £,
"*
—„ E/kT
1+e
where B is the matrix element of B(S +S ) between appropriate spin 1/2
states.
Note that the density of TLS, js(E), has not been included so W g d
is the rate per TLS with energy E. The strain matrix element is evaluated
as before (see equations 2.4 and 2.5), a Debye phonon spectrum is assumed,
and the sum over phonon states is changed to an integral over E and done
in the usual way ( s e e equation 2 . 6 ) .
w
(4.13)
A = 2?
*
4B2
/ 2 ! - ND\2 3(E+g) 3
U
E +S J
4Tf2fTi 3 v 5
N«+1
1+e E/kT
N. = < e E + f / k T r 1
Here o i s the d e n s i t y of t h e solid and v the v e l o c i t y of sound.
The
equation for W. i s j u s t l i k e 4.13 except t h e l a s t term i s N M /(1+e~
The equations for W,
replaced by E-f.
and W . are l i k e the f i r s t two except t h a t E+S i s
Assuming t h a t Eȣ so t h a t E+S s*E-S ^ E , t h e sum of the
four r a t e s i s :
(4.14)
W. . = A B2 /CM
/CM - NDV
ND"^2
U
tot
V * E~7
E/
Aa
21Y
Hi
E3
sinh(E/kT)
6
tfjpti3v5
The observed spin l a t t i c e r e l a x a t i o n r a t e (1/T..) i s approximated by the
average of W. , (E) over t h e d i s t r i b u t i o n of TLS in energy.
E
^max
(
"'
15
< 1 / T 1> =
>
).
?™ w t o t ( E )
J min
E
/'max
' j»(E) dE
E
min
dE
87
Since 1/T, depends on E for t h e p a r t i c u l a r tunneling s t a t e involved,
one might expect a d i s t r i b u t i o n of r e l a x a t i o n r a t e s and therefore a n e t
spin recovery which is nonexponential.
The same spin-spin
interactions
which cause the observed exchange narrowing and s p e c t r a l diffusion
will
tend to narrow t h e d i s t r i b u t i o n of 1/T.. values by exchanging energy
throughout the s p i n system.
If t h i s spin diffusion i s f a s t enough, t h e
spin system will recover w i t h a s i n g l e time c o n s t a n t (which i s observed)
and equation 4.15 w i l l be v a l i d .
^max/k
must correspond to a low enough
temperature that t h e phonons are e f f e c t i v e l y i n the long wavelength l i m i t ,
or the Debye approximation w i l l not h o l d .
than
E .
must be much g r e a t e r
£ or the thermodynamic functions will n o t sum to s i n h " (E/kT).
Without more knowledge about t h e matrix elements i t I s impossible to
determine which term, if any, dominates in (CM/f - ND/E) .
Kurtz and
S t a p l e t o n conclude (1980), in the c o n t e x t of t h e TS theory, t h a t the CM/S
term dominates.
I n the TS theory D and M a r e roughly comparable, a t l e a s t
when averaged over a d i s t r i b u t i o n (Hunkllnger and Arnold, 1976).
i s t r u e of C and N for TS.
The r e l a t i v e s i z e of the energy denominators
then causes the f i r s t term to dominate,
Stinson and S t a p l e t o n , 1983)
(see also D e v i l l e , e t a l . , 1983;
In c o n t r a s t , Lyo and Orbaoh (1980) have
developed a theory where t h e ND/E term dominates.
developed to e x p l a i n the T
The same
This theory was
dependence of homogeneous o p t i c a l linewldths
for fluorescent c e n t e r s i n g l a s s e s .
Their a p p l i c a t i o n of t h i s theory to
magnetic resonance remains unpublished, but p a r a l l e l s the development of
Kurtz and Stapleton (1980).
If I t should happen t h a t
C/E W i e « A/E iC/Ae (see equation 4 . 5 ) then M<<D and the ND/E term would
88
dominate.
Although t h i s might be t r u e for a few p a r t i c u l a r TS, i t
probably would not be t r u e for an e n t i r e d i s t r i b u t i o n .
I t i s p o s s i b l e , of
c o u r s e , that CM/J**ND/E for a s i g n i f i c a n t p a r t of t h e TLS d i s t r i b u t i o n .
This would r e s u l t in a very complicated expression for 1/T...
two limiting c a s e s w i l l be considered in what follows:
Only the
the
Kurtz-Stapleton (K-S) case where CM/J dominates, and the Lyo-Orbach (L-0)
case where ND/E dominates.
Since the matrix elements are unknown, only
the magnetic f i e l d (H) and temperature (T) dependence w i l l be kept in the
following c a l c u l a t i o n s .
I t i s assumed, as i n the TS theory, t h a t j>(E) i s
weakly dependent on E and can be approximated as p ( E ) * E
up to E
where i t drops rapidly to zero ( J a c k i e , 1972).
The predicted T and H dependence for t h e L-0 case i s :
E
/-max
{
E p ( E ) dE
sinh(E/kT)
•^ E min
E__../kT
<1/T..)oiB2
(4.16)
2.2+A
.
(p(E)*E X )
r^-HA
dx
sinh(x)
^ B'T'
(x=E/kT)
J WkT
If
^ax^
kT
and
E
min^ k T
then
e s s e n t i a l l y independent of T.
of H T
or H T
the
value
of
tne
inte
Sral
wil1
be
This r e s u l t s in a predicted r a t e dependence
, depending on whether or not the matrix element B i s
proportional t o the applied magnetic f i e l d H.
observed SLR r a t e dependence — H° T
.
This theory p r e d i c t s the
At high temperatures (kT>E
)
max
89
p
the i n t e g r a l w i l l cause the r a t e dependence to change over to B T, as i t
must for the high temperature l i m i t of a one phonon p r o c e s s .
S i m i l a r a n a l y s i s a p p l i e s to t h e K-S case where the p r e d i c t e d r a t e
dependence is H"
T
or H T
+
, a g a i n depending on whether H appears In
the matrix element.
Although t h i s theory c o r r e c t l y p r e d i c t s t h e observed r a t e dependence
(H T
+
) , there are problems a s s o c i a t e d with i t s use a t t h e low
temperatures employed in t h i s study.
I t has been assumed t h a t E . » £
t h a t E . < kT and in t h e s e experiments £=kT a t 0.792K.
min ~
a p p l i c a t i o n of the theory a t temperatures where
revisions.
and
Correct
JsskT w i l l r e q u i r e two
In the a n a l y s i s that follows only t h e L-0 case (ND/E dominates
i n equation 4.14) w i l l be considered a s I t alone p r e d i c t s the observed
r a t e dependences.
The f i r s t r e v i s i o n i s the e l i m i n a t i o n of t h e assumption t h a t E . » £ .
This means t h a t E+S t E t E-i and t h e thermodynamic functions for t h e
v a r i o u s terms w i l l no longer sum to s i n h " (E/kT).
The second r e v i s i o n involves a change in t h e t r a n s i t i o n r a t e
computation for W.
and W . i n equation 4.10.
For the t r a n s i t i o n s
computed in equation 4.10 the d e e x c l t a t i o n ( e x c i t a t i o n ) of a TLS i s always
connected with the c r e a t i o n ( a n n i h i l a t i o n ) of a phonon.
For s t a t e s a and
d t h i s t r a n s i t i o n i s p o s s i b l e for any value of t h e TLS energy E.
s t a t e s c and d are connected by t h i s t r a n s i t i o n only when E>f.
The
When E<£ a
90
different t r a n s i t i o n i s possible between s t a t e s c and d, one where t h e
deexcltation ( e x c i t a t i o n ) of a TLS i s connected with the a n n i h i l a t i o n
(creation) of a phonon.
The t r a n s i t i o n r a t e W • for the E<£ case i s
shown below:
V = &2'r J
<-,4'.JHTSl + ,4'-><4>.,n.<+1|Hrp|y.,n«>
•<
-h<oM
+ < - , % | H T S l + ,4>><S4,n M +1|H T p |^.,n^>
-h<A, - E
(4.17)
+ <f+ > n t ,+1|H Tp |4>- > n BL ><-,4 J -IH TS l + ,%>
_
+ < ^ > n < t + 1 | H T p | ^ , n < > < - , % l H T S l + ,S-'.>
i - E
2
fD(hwu+E-f)
1+e-E/kT
Evaluating t h e matrix elements as b e f o r e (see equations 4.10-4.13) and
keeping only t h e L-0 t e r m s , the r e s u l t i s :
W
(4.18)
Q h'
= A_B 2 /ND_\ 2 (£-E) 3
2 \i-E/
N« +1
l+e"E/kT
i
N = fe f - E / k T -l)- 1
W. . i s s t i l l t h e sum of four terms (equation 4 . 1 0 ) but W . + Wu must be
replaced by W Qb ' + Wbc' when E<S.
The algebraic expressions for WQb + Wfeo
and W ' + Wfa ' turn o u t to be the same function of E-S.
terms, the r e s u l t i s :
Keeping the L-0
91
W
tot
(4.19)
2
A B2 (ND)
2
h
sinh(E/kT)
E+£)(eE/kT - e i / k T ) + ( E - Q ( e E / k T - e ' i / k T )
[< e
1
2 sinh(E/kT) L
E+i/kT
eE"S/kT - 1
- 1
Proceeding a s before ( s e e equations 4.14-4.16), the predicted
temperature and f i e l d dependence of the r a t e i 3 :
. •-max
<(l/T ) <* B 2 T 2+A /
x* dx
1
sinh x
(4.20)
2x + (x+y)(e x -e y ) + ( x - y ) ( e x - e " y )
ex+y _ 1
e x-y _ 1
J Emln/kT
where x = E/kT and y = £/kT.
When E=S an electron spin can relax d i r e c t l y
to a nearby TLS with no phonon i n t e r a c t i o n s .
This f i r s t order c r o s s
r e l a x a t i o n process has been considered by Kurtz and Stapleton (1980).
the TLS remain in thermal equilibrium w i t h the l a t t i c e , and
If
sufficiently
rapid spin diffusion occurs, t h e r e s u l t i n g r e l a x a t i o n process i s
temperature independent.
Kurtz and Stapleton saw evidence of such a
process in t h e i r lowest temperature data for SLR r a t e s in beta-alumina
(~2K).
There is no evidence of such a process i n the SLR r a t e s from t h i s
study.
This is not s u r p r i s i n g because such resonant p a i r s of spins and
TLS w i l l be very r a r e .
Since E
i s normally i n the 30-150K range (Kurtz and S t a p l e t o n ,
1980; Deville, et a l . , 1983) E
/kT i s large even a t the highest
IQclX
temperatures in t h i s study.
The upper l i m i t of t h e i n t e g r a l in 4.20 was
92
therefore approximated a s i n f i n i t e and does n o t influence the temperature
dependence.
The i n t e g r a l was evaluated numerically u s i n g Simpson's r u l e
and A and E . /k were adjusted to f i t the v a r i o u s r e l a x a t i o n data.
Equation 4.20 will p r e d i c t SLR r a t e temperature dependences which a r e very
c l o s e to s t r i c t T11 power laws for c e r t a i n combinations of E .
and K
The
best f i t for sample 8 ( s p u t t e r e d ) i s shown by curve 2 i n figure 4 . 9 . This
2 34
f i t corresponds to E . / k = 0.58K, A = 0.29 and deviates from a T
power
law by no more than 158 between 0.3K and 3K.
Such power law dependence i s
not a general property of equation 4.20 however, and c u r v e s 1 and 3 i n
figure 4.9 demonstrate t h i s .
Curve 1 is the same f i t a s curve 2 except
the lower l i m i t has been raised to E , /k = 1.2K.
Likewise, curve 3 i s
the same f i t except t h e lower l i m i t i s E . / k = 0.
r
min
The b e s t f i t s for samples 3 , 4 , 7, and 8 are summarized in t a b l e 4 . 2 .
The best simple Tn power law f i t s a r e shown along with parameters f o r t h e
best f i t s from the extended Kurtz-Stapleton theory ( e q u a t i o n 4.20).
The
extended theory f i t s i n d i c a t e the e x i s t e n c e of a non-zero E . which i s
mln
g r e a t e r than £ for samples 4 and 7 , and l e s s than £ f o r samples 3 and 8.
The data for samples 3 and 8 follow a s t r i c t T 0 power law very c l o s e l y
(see f i g u r e s 4.3 and 4 . 5 ) .
extended theory f i t ,
The r e s u l t is t h a t 2+A, where A i s from t h e
i s very c l o s e to n from t h e simple power law f i t .
This c o r r e l a t i o n is not a s strong f o r samples 4 and 7 because the 'data
show some curvature on a log-log p l o t (see f i g u r e 4.5) and are t h e r e f o r e
b e t t e r f i t by the extended theory ( s e e curves 4 and 5 i n figure 4 . 9 ) .
93
10!
H
10s
Preparation
Evaporated
Ne
imp.
Sputtered
10'
0.3
0.4
0.5 a 6
TEMPERATURE
Figure 4 . 9
1
(K)
Electron Spin L a t t i c e Relaxation Rates for Amorphous S i l i c o n
Made by Various P r e p a r a t i o n s ( frequency = 1 6 . 5 GHz, data
f i t s are from the Extended Kurtz-Stapleton Theory )
2
Analysis of Spin L a t t i c e Relaxation Rates for Amorphous S i l i c o n
Preparation
Power Law F i t
Extended Kurtz-Stapleton Theory
Prefactor
A
E . /k (K)
min
28
(165+3)T 2 " 3 6 : f c 0 - 0 2
16.8
0.32±0.08
O.59±0.10
°Ne + + imp.
(466+19)T3,27±0"°8
27.0
1.10*0.13
1.10*0.12
evaporated
(740+43)T3,47±0'°9
38.1
1.22±0.14
1.23±0.14
sputtered
(118+10)T2'34:t0'05
12.2
0.29±0.08
0.58*0.11
2
S i + imp.
v
'
95
Equation 4.20 must be multiplied by the p r e f a c t o r s l i s t e d in t a b l e
4.2 to produce the c o r r e c t r a t e p r e f a c t o r s (assuming B
data.
= 1) seen in the
If the extended Kurtz-Stapleton theory c o r r e c t l y predicted t h e
observed r a t e p r e f a c t o r s , the p r e f a c t o r s l i s t e d in t a b l e 4.2 would a l l be
the same.
The f a c t t h a t they are not even s i m i l a r i n d i c a t e s t h a t t h e
theory i s not p r e d i c t i n g the observed c o r r e l a t i o n of t h e p r e f a c t o r with
(see figure 4 . 6 ) .
The theory cannot explain the unusual behavior of t h e Ar implanted
samples (samples 1 and 2, see figure 4.4) unless one allows for
discontinuous changes in A and E . / k a t 1.2K.
mln
If the matrix element B i s independent of H, the theory c o r r e c t l y
p r e d i c t s the lack of observed frequency and H dependence in the SLR data.
Equation 4.20 p r e d i c t s a s l i g h t frequency and H dependence in the SLR r a t e
which becomes s i g n i f i c a n t only below 1K for the frequencies employed in
t h i s study (9.3 GHz and 16.5 GHz).
Using E
/k and A from the sample 3
max
fit, equation 4.20 predicts a 9.3 GHz rate which is 0.5? below the 16.5
GHz rate at 2.8K.
This percentage grows as the temperature is reduced,
reaching 2% at 1.4K, 4$ at 1.0K, and 23% at 0.38K.
data were not available below 1.4K.
Unfortunately 9.3 GHz
Above 1.4K experimental resolution is
insufficient to detect such a small frequency dependence in the rate.
The ability of the Kurtz-Stapleton theory to explain the observed SLR
rates raises two important questions:
is the TLS-spin coupling?
(1) what are the TLS?
and (2) what
The lack of observed frequency and H dependence
96
in the measured r a t e s i n d i c a t e s t h a t the matrix element B i s n o t
p r o p o r t i o n a l t o H.
Since Kramers' theorem a p p l i e s , t h i s r u l e s out
modulation of t h e e l e c t r o s t a t i c f i e l d as t h e dominant r e l a x a t i o n mechanism
(see section 2.1 ).
Several a u x i l i a r y experiments were conducted i n an attempt to
e l u c i d a t e the nature of t h e TLS and the TLS-spin coupling.
These are
described in t h e following s e c t i o n .
4 . 4 Auxiliary Experiments
4 . 4 . 1 Measurement of Magnetization a s a Function of Temperature
Over the p a s t eight years s e v e r a l research groups have reported
that
the EPR signal strength i n a-Si shows small d e v i a t i o n s from a s t r i c t C u r i e
law ( t a n h
£/2kT) temperature dependence.
The r e p o r t e d d e v i a t i o n s have
been described in terms of the Curie-Weiss law ( t a n h
£/2k(T+Q)).
F r i t z s c h e and Hudgens (1976) r e p o r t a value for 0 of 5K based on bulk
s u s c e p t i b i l i t y measurements of a sputtered sample.
Pawlik, e t a l . , (1976)
r e p o r t a 0 value of 1.1K, also from bulk s u s c e p t i b i l i t y measurements on
s p u t t e r e d samples.
The most r e c e n t measurements from the IBM group
(Thomas, et a l . , 1978) show no deviation from a s t r i c t Curie law
dependence.
These r e s u l t s were obtained from EPR s i g n a l amplitude
measurements between 5 and 120K on evaporated and i o n implanted samples.
Unfortunately t h e thermometry was r e l a t i v e l y i n a c c u r a t e (±1K) and the
region below 5K was not probed.
I n c o n t r a s t to a l l previous r e s u l t s ,
97
Khokhlov, e t a l . , (1981) report ferromagnetic ordering In a - S i with
t r a n s i t i o n temperatures in the 12-50K range.
In an attempt t o resolve t h i s controversy and to i n v e s t i g a t e a
p o s s i b l e TLS-spin i n t e r a c t i o n (modulation of exchange c o u p l i n g ) , the
temperature dependence of the EPR s i g n a l s t r e n g t h for samples 1 (Ar+
implant) and 3 (Si + implant) were measured.
described in section 3 . 3 .
The experimental procedure i s
Detailed comparisons were made between the EPR
signal amplitude of t h e sample ( a - S i ) and the standard (LMN:Nd) at 1.250K,
0.700K, and 0.400K.
The r e s u l t s are reported a s temperature dependent
signal amplitude r a t i o s R(T):
(4.21)
R(T)
-
LMN:Nd (T)
a-Si (T)
x
a-Si (1.250K)
LMN:Nd (1.250K)
The r e s u l t s for sample 1 (Ar + implant) are R(0.700K) = 1.075 ± 0.016
and R(0.400) = 1.152 * 0.013.
These r a t i o s were measured a t 150 pW
microwave power and were independent of microwave power between 50 pW and
500 pW.
Each r a t i o r e p r e s e n t s roughly 100 i n d i v i d u a l amplitude
comparisons and the d i s t r i b u t i o n of experimental r e s u l t s appeared to be
Gaussian (1o* error ranges are r e p o r t e d ) .
The reported r a t i o s are
c o n s i s t e n t with a 0 v a l u e of 0.22 ± 0.02 K.
The r e s u l t s can a l s o be
described in terms of an exchange i n t e r a c t i o n between spin p a i r s ,
-J S»S-.
The reported r a t i o s are c o n s i s t e n t with a J/k value of
-0.52 ± 0.02 K where t h e minus s i g n i n d i c a t e s a n t i f e r r o m a g n e t i c coupling.
I
98
The results for sample 3 (Si+ Implant) indicated slight deviations
from a strict Curie law magnetization but these deviations were strongly
dependent on the microwave power used.
Reduction in microwave power
reduced the deviation from the Curie law and this trend continued all the
way to the lowest operating power which was 50 pW.
At this power level
R(0.700) = 1.00 ± 0.03 and R(0.400) = 1.02 * 0.02.
As before, each ratio
represents about 100 individual amplitude comparisons and the distribution
of results appeared to be Gaussian (1o* errors are reported).
The observed
trend in the power law dependence indicates that R(0,400) will converge to
1.00 and indicate strict Curie law dependence at lower operating powers.
The observed R(0.400) is consistent with a 0 range of 0 < 0 <; 0.03 K and
a J/k range of -0.25 £ J/k £ 0.
It is expected that both values would
converge to 0 at lower operating power.
The absence of a significant exchange interaction in sample 3
indicates that a TLS-spin coupling involving the exchange interaction is
unlikely.
The significant exchange interaction in sample 1 may be
responsible for the unusual SLR rate temperature dependence seen in the Ar
implanted samples (see figure 4.4). It thus appears that exchange
interactions are not crucial to the SLR process but can have a significant
effect on it if they are strong enough.
99
4.4.2 ENDOR Study
Kurtz and Stapleton (1979, 1980) found t h a t the TLS-spin i n t e r a c t i o n
in Li, Na, and K beta alumina involves moduation of a hyperfine (I»S)
interaction.
In a-Si a hyperfine i n t e r a c t i o n i s possible between t h e
pa7
Si
n u c l e i (1=1/2, n a t u r a l abundance 4.7%) and the paramagnetic e l e c t r o n s .
E l e c t r o n - n u c l e a r double resonance (ENDOR) was employed t o search for
evidence of this i n t e r a c t i o n .
None was found.
The theory of double
resonance experiments i s discussed in S l i c h t e r (1978) and Pake and E s t l e
(1973a).
Kurtz (1980) describes the equipment and experimental techniques
used for t h i s i n v e s t i g a t i o n .
4.4.3 D i e l e c t r i c Response Measurements
If the TLS d i s t r i b u t i o n c o n s i s t s of charged units which are involved
in t r a n s l a t i o n a l tunneling the low temperature d l e l e o t r l c response of the
sample w i l l be affected.
The microwave frequency d i e l e c t r i c response of
charged TS increases with decreasing temperature.
As t h e sample
temperature is reduced, the other c o n t r i b u t i o n s to e = e ' + i e "
gradually freeze out u n t i l t h e TS response dominates.
will
The d i e l e c t r i c
response w i l l reach a minimum and then begin to increase again as t h e
temperature is lowered f u r t h e r .
al.,
This e f f e c t was discovered by Strom, et
(1978), who observed i t in Na b e t a alumina.
LI_N by Baumann, et a l . , (1980).
I t has a l s o been seen in
The microwave frequency d i e l e c t r i c response of sample 3 (Si
implant)
was checked in the 1.2-20K range using techniques described in s e c t i o n
3.1.
Both components of
reduced to 1.2K.
C declined monotonically a s the temperature was
The s e n s i t i v i t y of the experimental apparatus was
checked by r e p e a t i n g the experiment of Strom, e t a l . , on Na beta alumina.
I f the TLS in a-Si are s i m i l a r to t h e charged TS in Na beta alumina they
would have been d e t e c t a b l e in the a p p a r a t u s used a t a density of about
1 Q
10
o
cm" .
Since the charged TS d i e l e c t r i c response signature was not
observed in sample 3 , t h i s number i s a rough upper l i m i t on the density of
s i n g l y charged TS i n a - S l .
CHAPTER 5
CONCLUSIONS
5.1 Conclusions Concerning Spin L a t t i c e Relaxation in Amorphous
Silicon
Conventional one and two phonon SLR mechanisms can not account f o r
the observed SLR r a t e s in a - S i a t temperatures in t h e 0.3-4.0 K range.
Comparison of SLR r a t e s measured a t 9 . 3 GHz and 16.5 GHz i n d i c a t e s t h a t
the r a t e s are independent or very n e a r l y independent of microwave
frequency and applied magnetic f i e l d .
The temperature dependence of t h e
r a t e s depends somewhat on sample preparation b u t is always very close t o a
T power law.
The n values observed i n this study a r e in the range
2.09 - 2.36 or 3.26 - 3.47.
A theory of SLR by coupling to a d i s t r i b u t i o n of TLS has been
presented.
The theory p r e d i c t s SLR r a t e temperature dependences which a r e
c l o s e to simple Tn power laws.
The theory i n d i c a t e s t h e p o s s i b i l i t y of
SLR r a t e s which are nearly Independent of microwave frequency and applied
magnetic f i e l d .
The theory does not p r e d i c t t h e observed c o r r e l a t i o n
between the r a t e p r e f a c t o r s and n-2 ( s e e figure 4 . 6 ) .
The t h e o r y also
does not p r e d i c t sharp t r a n s i t i o n s in t h e SLR r a t e temperature dependence.
Such t r a n s i t i o n s were observed in two Ar implanted samples.
I t has been
u n i v e r s a l l y observed t h a t the low temperature thermal and a c o u s t i c
102
p r o p e r t i e s of amorphous m a t e r i a l s are controlled by d i s o r d e r r e l a t e d
e x c i t a t i o n s r a t h e r than Debye phonons.
I t i s extremely l i k e l y that such
e x c i t a t i o n s exist in a-Si In s i g n i f i c a n t numbers.
It is certainly
p l a u s i b l e that some of these e x c i t a t i o n s contribute to electron SLR.
Recent thermal conductivity measurements on a-Ge in t h e 0.03-5K range show
evidence of such disorder r e l a t e d e x c i t a t i o n s (Graebner and Allen, 1983).
The TLS theory c o r r e c t l y describes the observed r e s u l t s (Allen and
Graebner, 1984).
5.2 Conclusions Concerning the S t r u c t u r e of Amorphous S i l i c o n
The development of a comprehensive microscopic d e s c r i p t i o n of the
s t r u c t u r e of a-Si has been and continues to be a vexatious problem.
It
was hoped that EPR spectra taken a t 0.3K might show a d d i t i o n a l s t r u c t u r e
and thereby y i e l d information about the paramagnetic electron wave
functions.
This did not happen.
I t appears that SLR in a-Si occurs via coupling to a d i s t r i b u t i o n of
TLS.
I d e n t i f i c a t i o n of the TLS and the TLS-spin i n t e r a c t i o n would help to
c h a r a c t e r i z e the wave functions of i n t e r e s t .
Unfortunately, attempts to
make these i d e n t i f i c a t i o n s were unsuccessful.
The conclusions from the EPR s p e c t r a l measurements are i d e n t i c a l t o
those of Thomas, et a l . , (1978) which are based on measurements a t higher
temperatures (4-300K).
The s i n g l e , unstructured EPR l i n e is b a s i c a l l y
Lorentzian and inhomogeneously broadened; the l i n e width and spin density
103
depend on sample preparation while t h e g f a c t o r does n o t ; the exchange
i n t e r a c t i o n , i f present a t a l l , does not s i g n i f i c a n t l y affect t h e
p r o p e r t i e s of the EPR l i n e .
These conclusions have now been v e r i f i e d
at
temperatures down to 0.3K.
5.3 Comments on Spin L a t t i c e Relaxation by Coupling to Two Level
Systems
There a r e now t h r e e cases where unusual e l e c t r o n SLR has been
observed and a t t r i b u t e d to a d i s t r i b u t i o n of TLS.
D e v i l l e , e t a l . , (1983)
have observed SLR r a t e s s i m i l a r to those of t h i s study in amorphous
Vp05 and conclude t h a t the TLS-spin i n t e r a c t i o n involves modulation of
the c r y s t a l f i e l d .
Kurtz and Stapleton (1979, 1980) observed s i m i l a r
r e s u l t s in Li, Na, and K beta alumina and concluded t h a t the TLS-spin
i n t e r a c t i o n involves modulation of a hyperfine i n t e r a c t i o n .
The TLS-spin
i n t e r a c t i o n i s unknown in t h e present case but i s u n l i k e l y to be e i t h e r of
the two j u s t mentioned.
I t i s s u r p r i s i n g t h a t a wide v a r i e t y of amorphous
m a t e r i a l s show s i m i l a r thermal and a c o u s t i c p r o p e r t i e s .
of electron SLR in these m a t e r i a l s ?
I s the same t r u e
I s the TLS concept a valid
d e s c r i p t i o n of the physical system or j u s t useful phenomenology?
The
answer to these questions awaits a more comprehensive understanding of
d i s o r d e r r e l a t e d e x c i t a t i o n s i n amorphous m a t e r i a l s .
APPENDIX A
COMBINED He-3 CRYOSTAT AND MICROWAVE CAVITY
A.1 Introduction
This appendix discusses He-3 cryogenic equipment used in this study
to obtain temperatures in the 0.28 to 1,2 K range.
The combined He-3
refrigerator and microwave cavity is a reconstructed version of a device
designed and built by Bohan and Stapleton and described elsewhere (Bohan
and Stapleton, 1968).
This appendix Is intended as a supplement to their
original description.
As such, it focuses mainly on various modifications
made by the author and certain details omitted in the original article.
After a brief description of the system and associated figures, the
modifications are discussed.
Following that is a detailed list of
operating instructions which functions as an instruction manual for the
modified system.
Finally, the appendix closes with suggestions for
further improvements in the design.
A.2 Brief Description of the System
An outline of the combined He-3 cryostat and microwave cavity along
with its He-4 dewar vessel is shown In figure A.I.
Not shown is a
vertical control rod which engages a variable microwave coupler near the
point labeled "E". This rod continues up through the dewar head so that
105
TO SPECfl^pMEtER\ \
TO DIFF.
PUMP
Figure A . l
\
\
TO DIFF.
.PUMP
Combined He-3 Cryostat and Microwave Cavity
A--Control knob for He-3 t h r o t t l e valve
B--Return l i n e for He-3 gas
C--Line for evacuating cavity and exchange gas region
D--Top of thermal isolation column and location of
He-3 throttle valve
E--Microwave coupling hole, sealed with .005 in.
mylar sheet
F--He-3 Evaporation Pot
G--ESR Sample
H--0uter wall of thermal isolation region
(figure from Bohan, 1968)
106
the microwave coupling to the resonant cavity can be varied under
cryogenic conditions.
Figure A.2 Is a scale drawing of the lower portion of the cryostat.
The microwaves travel In the waveguide structure shown on the left with
its H plane in the plane of the paper.
The microwaves enter the
cylindrical resonant cavity via a round, .230 in. diameter coupling hole
shown in cross section at the lower end of the teflon plunger "I".
The
cavity resonates at about 16.5 GHz in the cylindrical T E 0 1 1 mode,
commonly called the wavemeter mode.
In the original design, shown in
figure A.2, the lower portion of the waveguide is narrowed so that
frequencies below 18 GHz will be beyond the cutoff wavelength and
attenuate exponentially.
Microwaves near 16.5 GHz can propagate In the
teflon however, due to its higher dielectric constant.
Moving the teflon
plunger up and down changes the amplitude and phase of the microwaves at
the coupling hole, thereby adjusting the coupling into the resonant
cavity.
The sample is normally attached to the upper wall of the
microwave cavity which is the bottom of a He-3 evaporation pot.
Access to
the cavity is obtained through the lower wall which has a removable plug
and an indium O-ring seal.
The opening, but not the plug, is shown in
figure A.2. The He-3 evaporation pot is connected via .010 In. wall CuNi
tubing to a large capacity vacuum pumping system.
system is condensed above needle valve "G".
The exhaust of this
When valve "G" is opened,
condensed He-3 flows down capillary tube "C" and into the evaporation pot,
thus completing the refrigeration cycle.
The microwave cavity as well as
the region above the evaporation pot is kept evacuated by pumping line
107
F/
Figure A.2
Lower Portion of Combined He-3 Cryostat and Microwave Cavity
A--Thermometer resistor
F--Groove for indium 0-ring
B--Heater winding
G--BeCu needle valve
C--Capillary tube feedline
H--Line for evacuating cavity
for liquid He-3
and exchange gas region
D--Evaporation pot
I--Teflon plunger for variable
E--Nylon spacer
microwave coupler
Note: Outer wall of waveguide is shown cutaway only in the
lower half of drawing. Holes in the teflon plunger are for
connecting the external control rod. (figure from Bohan,1968)
108
"H".
A .030 in. wide annular gap at the periphery of the evaporation pot
maintains thermal Isolation between the pot and the other structures shown
that are Immersed in superfluid He-4 at 1.2K.
In the original design the
gap was maintained by a star shaped nylon spacer.
Temperature measurement
is obtained by monitoring the value of thermometer resistor "A" with an
external AC bridge.
The temperature is controlled by applying a variable
voltage to the heater winding "B".
A schematic diagram of the three external pumping systems is shown in
figure A.3. At the top of the figure is a high throughput pumping system
whose sole purpose is to keep the He-4 bath at 1.2 K.
At the lower left
of the figure is the He-3 pumping system which forms a complete circuit,
beginning and ending at the He-3 evaporation pot.
On the lower right is
an auxiliary vacuum system used for servicing various parts of the
rerigerator.
It is therefore referred to as the service vacuum system.
When the He-3 refrigerator is in operation this system is used for
maintaining the vacuum around the He-3 evaporation pot.
When the
refrigerator is not in use, the service vacuum system keeps the entire
system evacuated except for He-3 storage areas.
The purpose of the
various vacuum connections will be made clear in the section on operating
procedures.
A.3 Modifications to the Original Design
The microwave cavity, coupler, and waveguide have all been modified
from the original design.
These changes were made to solve, or at least
109
140 cfm
ROTARY PUMP
F^ I
MANOSTAT
-«(402 B
IROTARYPUMPJ
Be-Cu
NEEDLE
VALVE
SERVICE
"CONNECTION
TC2
-62
F
10 -10 torr
DISCHARGE
GAUGE
>TC4
THERMOCOUPLE
GAUGES
>PUMPING"VERNIERS"
COLD
TRAP
32®0-800mm {>
33 — ' 7
23?)
31
9 DIAL GAUGE
30
He-3
RETURlN
LINE
150 l/s
DIFF. - * T - I
PUMP
'1
18®
DIFE
PUMP
i 1
198
4®
8
CYLINDER
—®-|20
I402K
He-3 PUMP
F i g u r e A.3
9
AUXILIARY
PUMP
J402B
SERV PUMP
Schematic Diagram of He-3 and He-4 Pumping
Systems
110
minimize, three separate problems:
First, the cavity Q was low (3000 to
4000) and was affected by simultaneous coupling to two slightly
nondegenerate modes (Bohan, 1968).
The degree of coupling to one mode or
the other depended on the coupler position as well as the type and
location of samples in the cavity.
Useful cavity coupling was not
attainable for the a-Si samples used in this study.
Secondly, the empty
microwave cavity contained a spurious EPR signal near g = 2 which was
about three times the amplitude of the signal of interest at g = 2.0059.
Finally, there were difficulties associated with the presence of
superfluid He-4 in the long vertical waveguide section shown in figure
A.1.
Solutions to each of these problems will now be discussed.
The cavity mode and coupler problems were a consequence of the fact
that the T E 0 1 1 mode is intrinsically degenerate with another mode,
TM.^.
The wall currents for the TE mode are entirely azimuthal and
therefore do not cross the annular gap surrounding the He-3 evaporation
pot.
The undesired TM mode does excite currents across the annular gap
and thus couples microwave power into the region above the He-3
evaporation pot.
not.
The TE mode has axial symmetry while the TM mode does
Any departure from axial symmetry in the cavity (such as the
presence of a sample) will generate a slight frequency spliting and tend
to enforce the undesired mode.
The coupling orifice and the annular gap
also contribute to this frequency splitting between the two modes.
Various solutions to this problem have been employed in wavemeter designs
(Montgomery, 1947).
The presence of a symmetry destroying sample in this
case limits the utility of these earlier designs.
The solution employed
Ill
in this case results from the observation that the wall currents for the
two modes are orthogonal at the mirror plane perpendicular to the
cylindrical axis of the cavity (Aron, 1967).
A vertical slot at this
location will preferentially intercept the azimuthal T E Q 1 1 wall currents
while the usual round coupling hole will intercept both mode currents
equivalently.
When a circular coupling hole is used, one relys on the
waveguide field polarization to enforce the desired mode and also on the
annular gap, which greatly reduces the Q for the undesired mode.
With
this arrangement, and with a lossy sample in the cavity, approximately one
third of the microwave power was coupled into the undesired TM . 1 mode.
Switching to a slot reduced this to an undetectable level, certainly less
than 1%.
For unknown reasons, modified Gordan couplers (Gordan, 1961) such as
the one shown in figure A.2 did not work well with a coupling slot.
It
was discovered by trial and error that a tapered sliding short in the
waveguide will function adequately as a variable coupler.
When the short
has a 45 degree taper, it efffectively forms a 90 degree mitered H plane
waveguide bend (Moreno, 1958).
This is an oversimplification because the
coupling slot does not present the same impedance as the waveguide.
In
fact, a taper angle of 30 degrees with respect to the axis of the cavity
works much better than one of 45 degrees.
Spatial constraints required
that the long dimension of the waveguide cross section be reduced from the
standard .622 in. for Ku band to .500 in.
This increases the cutoff
frequency from the standard 9.5 GHz to 11.6 GHz.
cavity has the following properties:
The cylindrical resonant
diameter - .990 in., height - .755
112
i n . , resonant frequency (empty) - 16.5 GHz, wall t h i c k n e s s a t coupling
o r f i c e - .030 i n .
Under these c o n d i t i o n s , and with a 30 degree taper
s l i d i n g s h o r t , i t was found t h a t a coupling s l o t .031 i n . wide and .201
i n . long performed well under a v a r i e t y of loading c o n d i t i o n s and a t
frequencies from 14.5 to 16.5 GHz.
The tapered s l i d i n g s h o r t was t r i e d
with a round coupling hole and the performance was very poor.
As mentioned e a r l i e r , the o r i g i n a l c a v i t y contained a spurious
resonance near g = 2 which was s e v e r a l times as l a r g e a s t h e s i g n a l of
interest.
The s i d e s and bottom of t h e o r i g i n a l c a v i t y were constructed of
f r e e c u t t i n g b r a s s (61.5? Cu, 35.5? Zn, 3.0? Pb).
I t was determined t h a t
the spurious s i g n a l was o r i g i n a t i n g i n the b r a s s .
Great e f f o r t was
expended to e l i m i n a t e i t using v a r i o u s combinations of e l e c t r o p o l i s h i n g ,
copper s t r i k i n g , and s i l v e r p l a t i n g on the i n t e r i o r of the c a v i t y .
was unsuccessful.
This
The problem was resolved by c o n s t r u c t i n g the c a v i t y
with vacuum refined OFE (oxygen free e l e c t r o n i c ) grade copper.
The
m a t e r i a l used met ASTM s p e c i f i c a t i o n F-68-77 and i s a v a i l a b l e from Hitachi
Cable Ltd., Tokyo, Japan (a domestic r e p r e s e n t a t i v e Is Copper and Brass
Sales, Inc.
Schaumberg, I L ) .
Use of t h i s copper reduced t h e amplitude of
the spurious s i g n a l by two o r d e r s of magnitude.
Although r e a d i l y
observable, t h i s was much smaller than the s i g n a l s of i n t e r e s t .
Some of
the remaining spurious s i g n a l Is probably due to the He-3 evaporation pot
which Is made of OFHC (oxygen free high conductivity) grade copper of
unknown o r i g i n .
113
The g r e a t disadvantages of high p u r i t y vacuum refined copper are that
i t is very soft, d i f f i c u l t to machine, and almost impossible to t a p .
The
tapping problem was resolved by using blocks of TeCu a l l o y (.5? Te, Copper
Development Association Alloy Code 145) wherever threaded holes were
needed.
These blocks were attached to t h e c a v i t y by vacuum brazing, which
precluded the use of copper a l l o y s containing z i n c .
Although d i f f i c u l t to
f a b r i c a t e , the high conductivity of OFE copper pays off with an e x c e l l e n t
cavity Q.
At 1.2 K, loaded cavity Q's were in the 17,000 to 20,000 range.
One f i n a l change was made to t h e c a v i t y design.
In t h e i n i t i a l
arrangement, shown in figure A2, t h e nylon spacer "E" s a t j u s t above the
annular gap on a small f l a n g e .
Contact with the c y l i n d r i c a l wall occurred
only a t four sharp points on the spacer i n order to reduce thermal
conduction from t h e evaporation pot to t h e o u t e r wall.
The problem was
t h a t small s u p e r f l u i d l e a k s i n t o t h e microwave cavity were always present
and the nylon spacer r a p i d l y acquired a superfluid film from them.
caused a severe thermal s h o r t .
tried.
This
Operation without any s p a c e r a t a l l was
This produced e x c e l l e n t thermal i s o l a t i o n but u n f o r t u n a t e l y
the
cavity Q was s t r o n g l y modulated by mechanical v i b r a t i o n s , making the
arrangement unworkable.
The spacer assembly was moved up one inch so t h a t
i t a t t a c h e s to t h e evaporation pot pumpline r a t h e r than t h e pot i t s e l f .
d e l r i n c o l l a r was f i t t e d into which three nylon screws with pointed ends
are threaded.
The pointed ends contact t h e o u t e r wall a t t h r e e p l a c e s ,
equally spaced around a c i r c l e .
I t is very important t h a t a l l t h r e e
points c o n t a c t the outer wall simultaneously or mechanical v i b r a t i o n
problems w i l l r e s u l t .
The c y l i n d r i c a l o u t e r wall is i n s t a l l e d over t h e
A
114
evaporation pot and spacer by using f i n g e r pressure to c r e a t e a t r i a n g u l a r
d i s t o r t i o n in the c y l i n d e r c r o s s s e c t i o n .
The d i s t o r t e d c y l i n d e r i s s l i d
over the pot and the p o i n t s of the spacer, which form an e q u i l a t e r a l
triangle.
If the p o i n t s are adjusted c o r r e c t l y , t h e c y l i n d r i c a l wall w i l l
c o n t a c t a l l three when i t r e l a x e s back to a c i r c u l a r cross s e c t i o n , but
without s u f f i c i e n t pressure to crack the nylon a t cryogenic temperatures.
In t h e higher p o s i t i o n the spacer assembly i s in a warmer region of the
He-3 c r y o s t a t and p r e s e n t s a much longer thermal path to the evaporation
pot.
The superfluid film problem can be further minimized by precharging
the c a v i t y region with a small amount of He-3 gas.
This w i l l mix with t h e
incoming He-4 and g r e a t l y reduce the lambda t r a n s i t i o n temperature.
The
He-4 concentration w i l l grow in time and eventually a film w i l l form.
Although mylar is porous to He a t room temperature, i t i s g e n e r a l l y
believed t h a t below 80 K no diffusion of He can occur in t h i s m a t e r i a l
(Moss, 1963).
(Mochel, 1984).
Recent high s e n s i t i v i t y measurements have confirmed
this
I t i s therefore u n l i k e l y that the mylar window i s the
source of t h e superfluid l e a k .
As mentioned e a r l i e r , problems were caused by the presence of
superfluid He-4 i n s i d e the long v e r t i c a l waveguide section of f i g u r e A1.
As t h e superfluid He-4 l e v e l in the dewar core drops, the l i q u i d - g a s
i n t e r f a c e in the waveguide a l s o moves.
This causes the c a v i t y response
s i g n a l to have a time dependent phase s h i f t and a l s o introduces a time
dependent spurious r e f l e c t i o n a t the l i q u i d - g a s i n t e r f a c e .
The "thumper
115
tube" e f f e c t (Gaffney and Clement, 1955) operating in the waveguide and
surface o s c i l l a t i o n s a t the l i q u i d - g a s i n t e r f a c e in the dewar core
modulate the unwanted effect in the 3 - 300 Hz range.
These e f f e c t s were
not completely eliminated, but were reduced to a t r a c t a b l e l e v e l by
s e v e r a l modificatons.
The long v e r t i c a l waveguide section was loaded with a styrofoam c o r e .
The ideal s i t u a t i o n would be to completely d i s p l a c e the He-4 i n s i d e the
waveguide with a sealed, low l o s s d i e l e c t r i c .
As no s u i t a b l e way was
found to s e a l the styrofoam core into t h e waveguide, He-4 was allowed to
creep up between the foam core and the waveguide w a l l s .
Various types of
closed c e l l expanded polystyrene foam were t e s t e d for microwave l o s s and
s t a b i l i t y under repeated cryogenic c y c l i n g .
"Formula R I n s u l a t i o n " , a
product of the UC I n d u s t r i e s Div. U.S. Gypsum Co., was found to cycle well
and to have a microwave loss under .2 dB/ft.
Small vent holes were
d r i l l e d a t various places in t h e waveguide walls to allow for pressure
equalization.
Efforts to c o n t r o l surface o s c i l l a t i o n s a t the l i q u i d - g a s i n t e r f a c e
in t h e dewar core were less s u c c e s s f u l .
Several l a r g e b a f f l e s were
i n s t a l l e d in the dewar core in an attempt to damp the o s c i l l a t i o n s .
These
were constructed from very c o a r s e grained open c e l l polystyrene foam in an
attempt to maximize the e f f e c t i v e surface area.
Some improvement was
noted with the b a f f l e s , but u n c o n t r o l l a b l e o s c i l l a t i o n s continued to occur
on r a r e occasions.
116
The r e s i d u a l e f f e c t s due to changing He-4 l e v e l s were f u r t h e r reduced
by c o n s t r u c t i n g the a s s o c i a t e d EPR spectrometer in such a way t h a t i t was
r e l a t i v e l y i n s e n s i t i v e to phase changes i n the c a v i t y s i g n a l .
As only one
waveguide l e a d s to the sample c a v i t y , a nonreciprocal device I s required
to s e p a r a t e the incident s i g n a l from the r e t u r n i n g 3 i g n a l .
Spectrometers
constructed before the mid 1960's tended to use a magic tee f o r t h i s
purpose while l a t e r designs have u s u a l l y employed a c i r c u l a t o r (Poole,
1967).
Both of t h e s e devices are prone to having phase dependent
p r o p e r t i e s when a high standing wave component i s p r e s e n t .
A high
d i r e c t i v i t y , multihole, broadwall d i r e c t i o n a l coupler was used in t h i s
case (Hewlett-Packard model P752C).
Even with t h i s h i g h l y r e f i n e d device,
phase dependent e f f e c t s were s t i l l observable.
A phase s h i f t e r was
i n s t a l l e d between the d i r e c t i o n a l coupler and the resonant c a v i t y so t h a t
a s e t t i n g could be s e l e c t e d that minimized t h i s problem.
Generally t h i s
could be accomplished by a d j u s t i n g the phase s h i f t e r to produce an
extremum in the detected microwave s i g n a l .
Two plumbing modifications were made to the c r y o s t a t .
gas pumpline "H" in figure A2 was enlarged from .125 i n .
in.
diameter.
The exchange
diameter to .250
In the molecular flow regime p i p e conductance i s
p r o p o r t i o n a l to the cube of the diameter.
This change reduced the
exchange gas pumpout time from about 60 min.
to about 8 min.
A condensation bulb was added j u s t above the needle valve "G" In
figure A.2.
This enlargement in t h e He-3 feed l i n e has an i n t e r n a l volume
of 3 ml and is connected to a cold f i n g e r which extends down t o the dewar
117
tail.
The bulb allows the use of a much l a r g e r He-3 charge (1.5 1-atm
i n s t e a d of 0.5 1-atm) and a l s o allows condensation of He-3 even when the
He-4 level In the dewar core is very low.
The r e s u l t is a longer run time
for each charge of He-3 and also a longer t o t a l run time.
Several plumbing changes were made in t h e He-3 pumping system.
c u r r e n t arrangement i s shown in f i g u r e A.3.
(Sergeant-Welch Co.
Skokle, 111.)
The
The 1402 K r o t a r y vacuum pump
exhausts large q u a n t i t i e s of o i l vapor
and d r o p l e t s under c e r t a i n unavoidable operating c o n d i t i o n s .
In t h e
o r i g i n a l design t h i s o i l had a tendency to freeze up and block t h e He-3
feed l i n e a t the needle v a l v e .
The cold t r a p is designed t o prevent t h i s
but occasionally became blocked with frozen o i l i t s e l f .
This freeze up
problem was solved by i n s t a l l i n g an o i l separator i n the He-3 exhaust
line.
This consisted of 5 f e e t of water cooled 3/16 in. ID tubing wound
in a h e l i x (pitch angle 45 degrees) and located above the 1402 K pump.
This worked well and had the useful feature that t h e condensed o i l ran
back into t h e pump o i l r e s e r v o i r .
The entrance to t h e s e p a r a t o r was
f i t t e d with several splash b a f f l e s t o d e f l e c t large drops of o i l back Into
the r e s e r v o i r d i r e c t l y .
A pressure equalization l i n e was i n s t a l l e d
between the intake and exhaust p o r t s of t h e 1402 K rotary pump.
Valve 21
( f i g u r e A3) i s opened b r i e f l y a f t e r the pump is switched off to prevent
o i l from being sucked back to the i n t a k e s i d e .
Thermocouple gauges TC1
and TC3 were added to monitor the intake and exhaust pressures of t h e
diffusion pump.
F i n a l l y , i t was convenient to a t t a c h a s e p a r a t e r o t a r y
vacuum pump where TC3 i s located.
This pump ( l a b e l e d " a u x i l i a r y pump" in
f i g u r e A.3) i s used to keep the diffusion pump evacuated and allows i t to
118
be i s o l a t e d from the r e s t of t h e system.
The diffusion pump can then be
kept hot and ready to use while various operations a r e performed on the
remainder of the system.
Those are described in d e t a i l in the next
section.
A.4 Operating Proceedures
This s e c t i o n functions a s an i n s t r u c t i o n manual for t h e modified
system.
Various device l a b e l s are used and these a r e shown In figure A3.
The valve nomenclature is organized as follows:
Valves 1-5 are p a r t of
the service vacuum system which is s e p a r a t e from t h e He-3 r e f r i g e r a t i o n
loop.
Valves 10-20 a r e part of the low pressure s i d e of t h e He-3
r e f r i g e r a t o r while 21-25 complete the loop on the high pressure s i d e .
Valves 30-33 are a s s o c i a t e d with storage of the He-3 charge.
In most
c a s e s the second d i g i t of the valve l a b e l i n c r e a s e s in the d i r e c t i o n of
gas flow.
The i n s t r u c t i o n s a r e organized into f i v e roughly chronological
sections:
i n i t i a l evacuation and He-3 charging, system s t a r t u p ,
r e f r i g e r a t i o n cycle-condensation phase, r e f r i g e r a t i o n
cycle-evaporation
phase, and system shutdown.
A.4.1 I n i t i a l Evacuation and He-3 Charging
All of t h e following assumes t h a t the s e r v i c e vacuum system i s a b l e
to reach a pressure of 5x10"
t o r r blanked off.
This i s measured on t h e
discharge gauge with 1,2, and 4 closed; 3 and 5 open.
119
Prepare the r e f r i g e r a t o r by opening 10-34 and completely evacuating
the system via the s e r v i c e vacuum system (2 and 4 c l o s e d ; 1,3, and 5
open).
The 1402 K r o t a r y pump should be run in b r i e f b u r s t s of s e v e r a l
seconds to help outgas i t s o i l .
The 1402 K cannot be run for more than an
hour or so with both i t s intake and exhaust p o r t s evacuated.
The pump
w i l l overheat and burn i t s o i l in a few hours i f i t i s operated without a
gas load and with the exhaust evacuated.
be about 1x10
-4
torr.
The system base pressure should
This i s t h e room temperature vapor pressure of
the r o t a r y pump o i l (Sergeant-Welch Duo-Seal O i l ) which eventually
migrates throughout the whole system.
I t takes about 72 hours t o outgas
the o i l in the 1402 K pump, assuming t h a t i t has been exposed to a i r .
O r d i n a r i l y t h i s would happen only when the r o t a r y pump o i l is changed.
If
the pressure a t TC3 i s under 100 microns Hg, 14 and 17 can be closed and
the 150 l i t e r / s e c diffusion pump s t a r t e d .
The l i m i t i n g forepressure of
t h i s pump is a c t u a l l y 300 microns Hg (when charged with Dow Corning 702
fluid),
but i t i s best not to o p e r a t e a t high f o r e p r e s s u r e s for extended
periods.
The He-3 charging procedure begins as follows:
pump by closing 14,17 and 18.
Isolate the
diffusion
Close 20 and 21 and s t a r t the 1402 K pump.
Watch the discharge gauge pressure as the 1402 K pump warms up.
The
pressure will probably climb above 1x10 - 4 t o r r a s the pump becomes hot
and outgasses t h e l a s t b i t of dissolved a i r from i t s o i l .
min.
After about 30
the pump case w i l l be g e t t i n g too hot to touch for more t h a n a
second or so and the p r e s s u r e should be f a l l i n g below 1x10 -4 t o r r again.
When t h i s happens close 1,12,13,23 and 3 1 .
Valves 17,18,20 and 21 should
120
already be closed, 1 9 , 3 0 , 3 2 , 3 3 , and 34 s t i l l open.
the valve leading to t h e He-3 supply c y l i n d e r .
Open 2 0 .
Crack open
The 0-800 t o r r dial gauge
w i l l show a r i s i n g pressure as the 1402 K pump gradually evacuates t h e
supply c y l i n d e r .
The normal charge for t h e system i s 1.0 1-atm and t h i s
w i l l cause about 200 t o r r on the d i a l gauge.
About 3/4 of t h i s charge
w i l l be stored in the 3 l i t e r He-3 r e s e r v o i r and the remainder in the 1402
K pump case which ha3 about 1 l i t e r of f r e e volume on the high pressure
side.
The system can be operated with a l a r g e r charge of He-3, probably up
to about 2 1-atm.
The danger in t h i s can be seen by considering the
previous operation b u t with 32 closed.
Now the 3 l i t e r r e s e r v o i r i s
i s o l a t e d and the only s i g n i f i c a n t volume on the high p r e s s u r e side i s i n
the pump c a s e .
The 1 1-atra charge w i l l now cause a reading of about 800
t o r r on the d i a l gauge.
A l a r g e r charge w i l l cause a s i g n i f i c a n t
overpressure with r e s p e c t to the atmosphere which can damage the 1402 K
pump shaft seal and exhaust bellows.
If t h e system i s operated with a
charge of more than 1 1-atm considerable care must be taken to insure t h a t
t h i s does not ooour.
After charging with the a p p r o p r i a t e amount of He-3, t h e system w i l l
be in a configuration
i d e n t i c a l to t h a t following a r e f r i g e r a t i o n r u n .
I n s t r u c t i o n s for s h u t t i n g down the system are l i s t e d in t h e f i n a l s e c t i o n .
A.4.2 System Startup
The normal storage c o n f i g u r a t i o n for the He-3 system i s as f o l l o w s :
The He-3 feed l i n e is capped off between 24 and 25 and the He-3 vacuum
l i n e I s capped upstream of valve 1.
2,4,14,17,19-22 and 32.
The following valves a r e closed:
The o t h e r valves are open.
The 150 l i t e r / s e c
diffusion pump, h e r e a f t e r called t h e diffusion pump, is warm with a b o u t 90
V AC applied to i t s h e a t e r .
I t i s kept evacuated by the a u x i l i a r y vacuum
pump which i s attached to the system along with TC3.
If an a u x i l i a r y pump
i s not used, 19 must be l e f t open the keep the diffusion pump evacuated.
The combined He-3 c r y o s t a t and resonant c a v i t y assembly i s connected
to t h e r e f r i g e r a t o r plumbing as follows:
The c r y o s t a t and c a v i t y assembly
should have i t s various connections capped off and a l l plumbing
p r e s s u r i z e d to 1.1 atra with He-4.
r e f r i g e r a t o r with 1.1 atm He-4.
Close 3 and 33 and p r e s s u r i z e the
The f a c t that 14,17,19-22 and 32 are
closed prevents the He-4 from contaminating the He-3 storage areas of
pumps.
I n s t a l l t h e c r y o s t a t and c a v i t y assembly i n t o the room temperature
dewar vessel and make t h e various vacuum c o n n e c t i o n s .
I t i s important t o
minimize the entrance of a i r into any of the vacuum plumbing as t h i s w i l l
g r e a t l y lengthen the pumpdown time, p a r t i c u l a r l y I n humid weather.
Shut
off t h e He-4 g a s supply and pump out the He-4 by closing 5 and opening 4
and 2 .
Close 25 and open 33.
After about 10 min. the p r e s s u r e at TC2
should be droping below 50 microns Hg and the s e r v i c e vacuum diffusion
pump can be engaged by c l o s i n g 4 and opening 5, t h e n 3.
n i t r o g e n to the dewar j a c k e t .
Add liquid
Within an hour t h e discharge gauge p r e s s u r e
122
should be below 10 microns Hg. Within 12 hours t h e pressure should be
-4
below 10
t o r r and the c r y o s t a t temperature w i l l be approaching 77 K.
The system can now be prepared for the t r a n s f e r of l i q u i d He-4.
Close 2 and p r e s s u r i z e the exchange gas area to about 50 t o r r with He-3.
Transfer l i q u i d He-4 into the dewar c o r e .
The p r e s s u r e observed a t the
discharge gauge should drop s t e a d i l y from about 8x10 J t o r r to
_5
1-2x10
t o r r . After the t r a n s f e r i s completed c l o s e valve 1.
open valve 2 and pump out He-3 u n t i l TC2 reads 35 microns Hg.
2.
Crack
Then close
This l e a s e s a small amount of He-3 in the exchange gas a r e a to
"poison" t h e He-4 which will g r a d u a l l y leak in and e v e n t u a l l y form a
superfluid
film.
Engage the He-4 pumping system and as the temperature I s dropping t h e
He-3 pumping system can be prepared a s follows:
closed.
Check to see t h a t 1 is
(This w i l l prevent a c c i d e n t a l exhaust of t h e He-3 c h a r g e . )
Increase t h e diffusion pump h e a t e r v o l t a g e to 120 V AC, i t s normal
operating v o l t a g e .
Close 18 and valve off the a u x i l i a r y vacuum pump.
Close 24 and 30, open 22 and s t a r t t h e 1402 K pump.
Crack open 32 while watching t h e p r e s s u r e at TC3.
Open 19 and 20.
Try to keep the pump
i n l e t pressure (TC3 reading) a t about 1000 microns Hg.
r e s e r v o i r i s empty c l o s e 31 and 32, and open 30.
t h e high p r e s s u r e s i d e of the 1402 K pump.
When the He-3
The d i a l gauge now reads
This value should be in the
700-800 t o r r range and will climb s l i g h t l y as the 1402 K pump warms up and
outgasses dissolved He-3 from i t s o i l .
traps.
Add l i q u i d nitrogen t o the cold
When the He-4 in the dewar core i s pumped down to 1.25 K the He-3
cycle can begin.
A.4.3 Refrigeration Cycle - Condensation Phase
Check that 25 is closed and open 24. The pressure on the dia] gauge
should start to drop immediately, indicating that ga3 is condensing.
pressure at TC3 may climb slightly as valve 25 leaks a bit.
The
When the dial
gauge reading drops to 30-40 torr the entire charge will be condensed and
sitting above the closed needle valve 25.
evaporation phase of the cycle.
take about 3 min.
The system is now ready for the
The condensation process will normally
When the He-4 level in the dewar core gets very low the
dial gauge pressure will not drop to the 30-40 torr range and the
condensation process will take longer.
When this happens, the system can
be cycled once more but then should be shut down.
A.4.4 Refrigeration Cycle - Evaporation Phase
Close 19 and open 18 so that accumulated gas in the diffusion pump is
exhausted.
Close 24 and open 25 about 2 turns.
The pressure at TCI
should jump immediately, indicating that He-3 is entering the evaporation
pot.
Wait about 90 sec while the condensed He-3 runs into the pot.
18 and open 19.
This allows the 1402 K pump to start pumping on the
evaporation pot.
Various valve combinations will now be required to
establish thermal equilibrium In the 0.3 - 1.2 K range.
summarized in table A.1.
Close
These are
In general, the pressure reading at TC1 will be
slightly less than the He-3 vapor pressure at the control temperature.
Table A.l
Control
Temperature
(K)
Equilibrium Operating Conditions for He-3 Refrigerator
Pressure
at T.C.I
(microns)
Valve Configuration
Pressure
at T.C.2
Closed
Open
Partially open
(microns)
<.45
<15
<20
19
17 18
.48
30
20
19
18
17-2 turns
.51
50
25
19
18
17-1/2 turn
.54
70
30
17,18
19 13
.57
200
35
11,13
10 12, 19
.61
1000
40
11,13
12 19
10-3 turns
.66
<<1000
45
ii
n
10-1 and 1/2 turn
.72
off scale
50
n
II
10-3/4 turn
.77
55
II
II
10-1/2 turn
.83
60
n
II
10-1/3 turn
.92
70
n
n
1.00
80
II
II
10-10 degrees
1.10
90
it
II
10-5 degrees
10-1/4 turn
125
The pressure a t TC3 i s a good gas flow i n d i c a t o r , and should be kept as
low as p o s s i b l e while maintaining temperature c o n t r o l .
If the diffusion
pump i s r e q u i r e d , the 1402 K pump must be used alone to reduce t h e
pressure a t TC1 to 100 microns Hg.
then 17 opened.
Then 18 can be opened, 19 closed and
The lower pumping " v e r n i e r " in f i g u r e A3 was n o t found to
be useful and thus 14 is closed in a l l of what f o l l o w s .
In g e n e r a l , the values l i s t e d for TC3 w i l l be required to e s t a b l i s h
s t a b l e temperature c o n t r o l .
sacrificed,
I f some temperature s t a b i l i t y can be
lower values should be used for TC3.
This w i l l r e q u i r e more
t h r o t t l i n g than shown in the t a b l e , and w i l l use t h e He-3 charge more
effeciently.
Using higher v a l u e s for TC3 w i l l cause l i t t l e i n c r e a s e in
temperature s t a b i l i t y and w i l l exhaust t h e He-3 charge more r a p i d l y .
The
amount of He-3 remaining in t h e evaporation pot can be computed from the
d i a l gauge reading which w i l l climb as t h e charge i s evaporated.
When t h e
He-3 charge i s n e a r l y a l l evaporated the condensation phase (A.4.3) can be
started.
Temperature control can be d i f f i c u l t while condensing He-3
because the He-4 bath warms s l i g h t l y and the needle valve 25 l e a k s
slightly.
This i s not a problem a t temperatures above .6 K but becomes
one below . 5 K.
After about 3 hours of running valve 2 should be opened.
This will
allow the s e r v i c e vacuum system to remove some of t h e He-4 which has
leaked into the c a v i t y region.
I t w i l l a l s o tend to remove the He-3
"poison" p r e v i o u s l y put in the c a v i t y region.
This arrangement of leaving
35 microns Hg of He-3 in a t t h e 3 t a r t and then w a i t i n g three hours to pump
126
seems to be about optimal for preventing superfluid film formation.
The
He-4 superfluid film will become a problem after 7-9 hours, but operation
in the .6 - 1.2 K temperature range will still be possible.
The liquid
He-4 in the dewar core will last about 9-14 hours, depending on operating
conditions.
A.4.5 System Shutdown
Close 24,17 and 18.
Open 25,13,19, and 32 (some valves may already
be in the correct position).
Turn up the evaporation pot heater voltage
to boil out any remaining He-3. When the pressure on TC1 drops below 20
microns Hg close 32 and 22.
Open 31.
In this phase the remaining He-3
gas is evacuated and deposited in the 1402 K pump case.
When the pressure
on TC1 drops below 20 microns Hg close 20 and stop the 1402 K pump.
1.
Open 21 momentarily and then close it again.
This equalizes pressure
on both sides of the pump rotors so oil suckback does not occur.
and open the valve to the auxiliary pump.
Hg open 18.
Open
Close 19
When TC3 is down to 50 microns
Cut back the diffusion pump heater voltage to about 90 V AC.
This will keep the oil warm and outgassed, but reduce the fluid
evaporation that occurs over long periods at 120 VAC operation.
If the system will be used In the next day or two the cold traps
should be kept filled with liquid nitrogen.
If the system Is to be warmed
up, it is important that the cold traps not be warmed up until the
cryostat is at room temperature.
Otherwise the cryopumping action of the
He-3 evaporation pot and associated pipe will recondense the oil and
127
contaminants which were o r i g i n a l l y in the cold t r a p s .
After t h e c r y o s t a t
i s a t room temperature, the cold t r a p s should be baked out with a heat gun
to remove condensed rotary pump o i l .
A.5 Suggestions for Improvement
The basic design of Bohan and Stapleton (1968) was and continues t o
be an optimal arrangement for a combined He-3 c r y o s t a t and microwave
cavity.
In a d d i t i o n to the modifications a l r e a d y described, t h r e e other
changes a r e suggested.
The lower pumping " v e r n i e r " in figure A.3 (valves
14-16 and a s s o c i a t e d pipe) i s not u s e f u l and should be d e l e t e d .
In the
pressure range where the diffusion pump i s u s e f u l an ordinary vacuum v a l v e
(such a s 17) w i l l function w e l l as a t h r o t t l e .
Secondly, an a u x i l i a r y
vacuum connection to the He-3 system cold t r a p would be very u s e f u l .
This
would allow the t r a p to be i s o l a t e d from the r e s t of the system and baked
out, e l i m i n a t i n g any chance of the contaminant recondensatlon discussed
earlier.
F i n a l l y , the microwave system performance would be g r e a t l y
enhanced i f l i q u i d He-4 were eliminated from i n s i d e the long v e r t i c a l
waveguide s e c t i o n .
This could be accomplished by enclosing t h e e n t i r e
waveguide and coupler assembly in a superfluid t i g h t envelope.
A l t e r n a t i v e l y , a mylar window could be placed a c r o s s the guide j u s t above
the coupler assembly.
This would allow He-4 around the coupler but
exclude i t from t h e upper p o r t i o n of t h e guide, where i t is in f a c t most
troublesome.
JL28
REFERENCES
Abragam, A., 1961, Principles of Nuclear Magnetism (Clarendon Press,
Oxford), p. 126.
Abragam, A., and B. Bleaney, 1970a, Electron Paramagnetic Resonance of
Transition Ions (Clarendon Press, Oxford), oh. 10.
Abragam, A., and B. Bleaney, 1970b, ibid., p. 320.
Adler, D., and E. J. Yoffa, 1976, Phys. Rev. Lett. 36, 1197.
Adler, D., 1978, Phys. Rev. Lett. 41, 1755. See also M. Kastner, D. Adler
and H. Fritzsche, 1976, Phys. Rev. Lett. 37, 1504.
Adler, D., 1981, "Defects in Covalent Amorphous Semiconductors" in
Fundamental Physics of Amorphous Semiconductors, F. Yonezawa ed.
(Springer Verlag, New York), p. 14.
Allen, J. P., J. T. Colvin, D. G. Stinson, C. P. Flynn, and H. J.
Stapleton, 1982, Biophys. J. 38, 299.
Allen, L. C., and J. E. Graebner, 1984, Bull. Amer. Phys. Soc. 29, 508.
Anderson, P. W., 1951, C. R. Acad. Sci. 82_, 342.
Anderson, P. W., 1958, Phys. Rev. 109, 1492.
Anderson, P. W., B. I. Halperin and C. M. Varma, 1971, Philos. Mag. 25, 1.
Anderson, P. W., 1978, Rev. Mod. Phys. 50, 191.
Anthony, P. J., and A. C. Anderson, 1976, Phys. Rev. B 14, 5198.
Anthony, P. J., and A. C. Anderson, 1977, Phys. Rev. B 16, 3827.
Anthony, P. J., and A. C. Anderson, 1979, Phys. Rev. B 19, 5310.
Aron, C. P., 1967, Proc. Inst. Elect. Engineers (London) 1J_4, 1030.
Ashcroft, N. W., and N. D. Mermin, 1976, Solid State Physics (Holt,
Rinehart, and Winston, New York), p. 62.
Bachus, R., B. Movaghar, L. Schweitzer, and U. Voget-Grote, 1979, Philos.
Mag. B_39, 27.
Baumann, T., M. v. Schiokus, S. Hunkllnger, and J. Jackie, 1980, Solid
State Commun. 35.» 587.
Bohan, T. L., 1968, Ph. D. Thesis, Univ. of Illinois (unpublished), p.
101, p. 108.
129
Bohan, T. L., and H. J. Stapleton, 1968, Rev. Sci. Instrum. _39, 1707.
Boulitrop, F., 1983, Phys. Rev. B 28, 6192.
Brice, D. K., 1970, Radiat. Eff. 6, 77.
Brice, D. K., 1975a, J. Appl. Phys. 46, 3385.
Brice, D. K., 1975b, Ion Implantation Range and Energy Deposition
Distributions, Vol. 1 (Plenum, New York).
Brodsky, M. H., and R. S. Title, 1969, Phys. Rev. Lett. 23, 581.
Brodsky, M. H., R. S. Title, K. Weiser, and G. D. Pettit, 1970, Phys. Rev.
B ±, 2632.
Brodsky, M. H., D. Kaplan, and J. F. Ziegler, 1972, Appl. Phys. Lett. 21.,
305.
Brodsky, M. H., and D. Kaplan, 1979, J . Non-Cryst. Solids 32, 431.
Brodsky, M. H., 1981, "The Effect of H and Other Additives on the
E l e c t r o n i c P r o p e r t i e s of Amorphous Si" in Fundamental Physics of
Amorphous Semiconductors, F. Yonezawa, ed. (Springer-Verlag, New York),
p . 56.
Brosious, P. R., 1977, Ion Implantation in Semiconductors 1976, F. Chernow
e t a l . eds. (Plenum P r e s s , New York), p. 417.
Brower, K. L., 1971, Radiat. Eff. 8., 213.
Brower, K. L., 1977, Ion Implantation in Semiconductors 1976, F. Chernow
e t a l . eds. (Plenum Press, New York), p. 427.
Carlos, W. E., and P. C. Taylor, 1980, Phys. Rev. L e t t . 45, 358.
Carlos, W. E., and P. C. Taylor, 1982, Phys. Rev. B 26, 3605.
Chabal, Y. J . , and C. K. N. P a t e l , 1984, Phys. Rev. L e t t . 5 3 , 210.
C h i t t i c k , R. C., 1970, J . Non-Cryst. Solids 3., 255.
Cohen, M. H., H. F r i t z s c h e , and S. R. Ovshinsky, 1969, Phys. Rev. Lett.
22, 1065.
Colvin, J . T., 1984, Ph. D. Thesis, Univ. of I l l i n o i s (unpublished), p. 9.
Crowder, B. L., R. S. T i t l e , M. H. Brodsky, and G. D. P e t t i t ,
Phys. L e t t . 16, 205.
1970, Appl.
130
Crowder, B. L. and R. S. Title, 1971, in Ion Implantation, F. H. Eisen and
L. T. Chadderton, eds. (Gordon and Breach, London), p. 87.
de Haas, W. J., J. H. de Boer, and G. J. Berg, 1934, Physica II, 1115.
Depinna, S., B. C. Cavenett, T. M. Searle, M. J. Thompson, J. Allison and
P. G. Le Comber, 1982, Philos. Mag. B 46, 473.
Deville, A., B. Gaillard, and C. Blanchard, 1983, J. Physique 44, 77.
Donovan, T. M., K. Heinemann, 1971, Phys. Rev. Lett. 27, 1794.
Faughnam, B. W. , and M. W. P. Strandberg, 1961, Phys. Chem. Solids 1_9,
155.
Feher, G., R. C. Fletcher, and E. A. Gere, 1955, Phys. Rev. 100, 1784. see
also G. Feher, 1959, Phys. Rev. n 4 , 1219. and G. Feher and E. A. Gere,
1959, Phys. Rev. 1J4, 1245.
Feher, G., 1959, Phys. Rev. 1_1_4, 1219. see also G. K. Walters and T. L.
Estle, 1961, J. Appl. Phys. 32, 1845.
Feldman, D. W., J. G. Castle, and J. Murphy, 1965, Phys. Rev. 1_3_8, 1208.
Feldman, D. W., J. G. Castle, and G. R. Wagner, 1966, Phys. Rev. 145, 237.
Finn, C. B. P., R. Orbach, and W. P. Wolf, 1961, Proc. Phys. Soc. 77, 261.
Fisher, R. A., E. W. Hornung, G. E. Brodale, and W. R. Giauque, 1973, J.
Chem. Phys. 58, 5584.
Fritzsche, H., and S. J. Hudgens, 1976, in Proc. of the Sixth Int. Conf.
on Amorphous Semiconductors, Leningrad, 1975 (Ioffe Institute,
Leningrad), p. 6. see also S. J. Hudgens, Phys. Rev. B 14, 1547.
Fritzsche, H., 1981, "What are Non-Crystalline Semiconductors" in
Fundamental Physics of Amorphous Semiconductors, F. Yonezawa, ed.
(Springer-Verlag, New York), p. 1.
Gaffney, J., and J. R. Clement, 1955, Rev. Sci. Instrum. 26, 620.
Gibbons, J. F., 1968, Proc. IEEE 56, 295.
Giordmaine, J. A., and F. R„ Nash, 1965, Phys. Rev. 1_3_8, A1510.
Gordan, J. D., 1961, Rev. Sci. Instrum. 32., 658.
Gourdon, J., P. Fretier, and J. Pescia, 1981, J. Physique Lett. 42, L21.
131
Graebner, J. E., and L. C. Allen, 1983, Phys. Rev. Lett. 51, 1566.
Graebner, J. E., B. Golding, L. C. Allen, D. K. Biegelsen, and M.
Stutzmann, 1984, Phys. Rev. Lett. 52, 553.
Haneman, D., 1968, Phys. Rev. 170, 705. see also D. Hanemwn, M. F. Chung,
and A. Taloni, 1968, Phys. Rev. 170, 719.
Hasegawa, S., and S. Yazaki, 1977, Solid State Commun. 23, 41.
Hasegawa, S., T. Kasajima, and T. Shimizu, 1979, Solid State Commun. 29,
13.
Herrick, R. C ,
p. 82-98.
1976, Ph. D. Thesis, University of Illinois (unpublished),
Hunkllnger, S., and W. A. Arnold, 1976, in Phsioal Acoustics, W. P. Mason
and R. N. Thurston, eds. (Academic, New York), p. 155.
Ishii, N., M. Kumeda, and T. Shimizu, 1981, Jpn. J. Appl. Phy3. 20, L673.
Jackie, J., 1972, Z. Phys. 257, 212.
Jackie, J, L. Piohe, W. Arnold, and S. Hunkllnger, 1976, J. Non-Cryst.
Solids 20, 365.
Khokhlov, A. F., A. I. Mashin, and A. M. Satanin, 1981, Phys. Status
Solidi B 1^5, 129. Note: This reference uses non-standard terminology
for Curie and Neel temperatures.
Knotek, M. L., 1975, Solid State Commun. 17, 1431.
Kronig, R. de L., 1939, Physica 6,, 33.
Kurtz, S. R., and H. J. Stapleton, 1979, Phys. Rev. Lett. 42, 1773.
Kurtz, S. R., and H. J. Stapleton, 1980, Phys. Rev. B 22, 2195.
Kurtz, S. R., D. G. Stinson, H. J. Stapleton, and M. M. Abraham, 1981,
Phys. Rev. B 24, 4983.
Le Comber, P. G., A. Madan, and W. E. Spear, 1972, J. Non-Cryst. Solids
li, 219.
Lemke, B. P., and D. Haneman, 1975, Phys. Rev. Lett. 35, 1379.
Lindhard, J., and M. Schraff, 1961, Phys. Rev. 124, 128.
132
Lohneysen, H. v., H. J. Schlnk, and W. Beyer, 1984, Phys. Rev. Lett. 52,
549.
Marchand, R. L., and H. J. Stapleton, 1974, Phys. Rev. B 9., 14.
Mayer, J. W., L. Eriksson, S. T. Picraux, and J. A. Davles, 1968, Can. J.
Physics 46, 663.
Mayer, J. W., L. Eriksson, and J, A. Davies, 1970, Ion Implantation in
Semiconductors: Silicon and Germanium (Academic, New York).
Mazey, D. J., R. S. Nelson, and R. S. Barnes, 1968, Philos. Mag. 17, 1145.
see also R. S. Nelson and D. J, Mazey, 1968, Can. J. Physics 46, 689.
Mlkkelson, R. C., and H. J. Stapleton, 1965, Phys. Rev. 1_40, A1968.
Mochel, J. M., 1984, (private communication, manuscript in preparation).
Montgomery, C. G., 1947, Technique of Microwave Measurements, M.I.T.
Radiation Laboratory Series Vol. 11 (McGraw-Hill, New York), p. 323.
Moreno, T., 1958, Microwave Transmission Design Data (Dover, New York), p.
164.
Moss, T. H., C. F. Kellers, and A. J. Bearden, 1963, Rev. Sci. Instrum.
34, 1267.
Moss, S. C , and J. F. Graczyk, 1969, Phys. Rev. Lett. 23, 1167.
Mott, N. F., 1967, Adv. Phys. 16, 49.
Mott, N. F., 1969, Philos. Mag. 1j), 835.
Mott, N. F., M. Pepper, S. Pollitt, R. H. Wallis, and C. J. Adkins, 1975,
Proc. R. Soc. London A 345, 169.
Mott, N. F., 1978, Rev. Mod. Phys. 50, 203.
Mott, N. F., and E. A. Davies, 1979, Electronic Processes in
Non-Crystalline Materials (Clarendon Press, Oxford), p. 320.
Movaghar, B., and L. Schweitzer, 1977, Phys. Status Solidi B 80, 491.
Movaghar, B., and L. Schweitzer, 1978, Philos. Mag. B 37, 683.
Muench, P. J., T. R. Askew, J. T. Colvin, and H. J. Stapleton, 1984, J.
Chem. Phys. (to be published).
133
Murphy, J . ,
1966, Phys. Rev. 145, 241.
Narayanamurti, V., and R. 0. Pohl, 1970, Rev. Mod. Phys. 42, 201.
Ohkura, H., M. Matsuoka, and Y. Mori, 1975, J . Phys. Soc. Jpn. 39, 547.
see a l s o Y. Mori and H. Ohkura, 1981, J . Phys. Soc. Jpn. J5.0, 1616.
Orbach, R., and H. J . S t a p l e t o n , 1972, i n Electron Paramagnetic Resonance,
S. Geschwind, ed. (Plenum, New York), ch. 2.
Paalanen, M. A., G. A. Thomas, and A. E. Ruckenstein, 1984, B u l l . Amer.
Phys. Soc. 29, v. 3 , p. 342.
Pake, G. E., and T. L. E s t l e , 1973, The Phsioal P r i n c i p l e s of Electron
Paramagnetic Resonance, 2nd ed. (Benjamin, Reading, MA), p. 86.
Pake, G. E., and T. L. E s t l e , 1973a, i b i d . , ch. 1 1 .
Paul, W., A. J . Lewis, G. A. N. Connel, and T. D. Moustakis, 1976, Solid
S t a t e Commun. 20, 969.
Pawlik, J . R., G. A. N. Connel, and D. Prober, 1976, in Proc. of t h e Sixth
I n t . Conf. on Amorphous Semiconductors. Leningrad, 1975 (Ioffe
I n s t i t u t e , Leningrad), p. 304.
Phillips, J. C ,
1979, J. Non-Cryst. S o l i d s , 34, 153.
Phillips, J. C ,
1979b, Phys. Rev. L e t t . 42, 1151.
Phillips, J. C ,
1981, J. Non-Cryst. S o l i d s , 43, 3 7 .
P h i l l i p s , W. A., 1972, J. Low Temp. Phys. ]_, 351.
P h i l l i p s , W. A., 1981, Amorphous S o l i d s : Low Temperature P r o p e r t i e s , W. A.
P h i l l i p s , ed. (Springer-Verlag, New York).
P h i l l i p s , W. A., 1981b, i b i d . , p. 1.
Pohl, R. 0 . , in Amorphous S o l i d s : Low Temperature P r o p e r t i e s . W. A.
P h i l l i p s , ed. (Springer-Verlag, New York), p. 2 7 .
Polk, D. E., 1971, J . Non-Cryst. Solids 5_, 365.
Polk, D. E., and D. S. Boudreaux, 1973, Phys. Rev. L e t t . 3J_, 92.
Poole, C. P . , 1967, Electron Spin Resonance ( I n t e r s c i e n c e , New York), p.
229-237.
Poole, C. P . , 1967a, i b i d . , p. 199.
P o r t i s , A. M., 1953, Phys. Rev. 9J_, 1071.
P o r t i s , A. M., 1956, Phys. Rev. 104, 584.
Ruby, R. H., H. Benoit, and C. D. J e f f r i e s ,
S c o t t , P. L., and C. D. J e f f r i e s ,
1962, Phys. Rev. 1_27, 5 1 .
1962, Phys. Rev. 127, 32.
Shevchik, N. J . , and W. P a u l , 1974, J . Non-Cryst. S o l i d s 1_6, 55.
Shimizu, T., M. Kumeda, I . Watanabe, and K. Kamond, 1979, Jpn. J . Appl.
Phys. 18, 1923.
S l i c h t e r , C. P . , 1978, P r i n c i p l e s of Magnetic Resonance 2nd ed.
( S p r i n g e r - V e r l a g ) , ch. 7«
Spear, W. E., and P. G. Le Comber, 1975, Solid S t a t e Commun. 17, 1193.
Spear, W. E., 1977, Adv. i n Phys. 26, 8 1 1 .
S t e i n , H. J . , F. L. Vook, D. K. Brice, J . A. Borders, and S. T. Picraux,
1971, in Ion Implantation, F. H. Eisen and L. T. Chadderton, e d s .
(Gordon and Breach, London), p. 17.
Stevens, K. W. H., 1967, Rep. Prog. Phys. 30, 189.
Stinson, D. G., 1981, Ph.D. Thesis, Universiy of I l l i n o i s
p. 15.
(unpublished),
Stinson, D. G., and H. J . S t a p l e t o n , 1983, Phys. Rev. B 27, 5386.
S t r e e t , R. A., 1982, Phys. Rev. B 26, 3588.
Strom, U., M. v. Sohickfus, and S. Hunkllnger, Phys. Rev. L e t t . 4 1 , 910.
S t u a r t , R. V., 1983, Vacuum Technology, This Films, and S p u t t e r i n g
(Academic, New York).
Stuke, J . , 1977, Proo. Seventh I n t . Conf. on Amorphous Semiconductors.
Edinburgh, W. E. Spear ed. (Univ. of Edinburgh P r e s s , Edinburgh), p.
406.
Stutzmann, N. and D. K. B i e g e l s e n , 1983, Phys. Rev. B 28, 6256.
Sucher, M., and J . Fox, 1963, Handbook of Microwave Measurements, 3rd ed.
(Wiley, New York), vol. 2 , ch. 8, 9.
»
135
Suzuki, M., M. Suzuki, M. Kanada, and Y. Kakimoto, 1982, Jpn. J. Appl.
Phys. 21, L89.
Tauc, J., A. Abraham, R. Zallen and M. Slade, 1970, J. Non-Cryst. Solids
4, 279.
Thomas, P. A., M. H. Brodsky, D. Kaplan, and D. Lepine, 1978, Phys. Rev. B
1_8, 3059.
Townsend, P. D., J. C. Kelley, and N. E. W. Hartley, 1976, Ion
Implantation, Sputtering, and Their Applications (Academic, New York).
Turnbull, D., 1969, Contemp. Phys. 10, 473.
Van Vleck, J. H., 1940, Phys. Rev. 57, 426.
Van Vleck, J. H., 1941, Phys. Rev. 59, 724 and 730.
Voget-Grote, U., J. Stuke, and H. Wagner, 1976, Conf. on the Structure and
Excitations of Amorphous Solids, Williamsburg, VA, G. Lucovsky, ed.
(A.I.P. Confrence Proceedings no. 31, New York), p. 91.
Voget-Grote, U., W. Kummerle, R. Fischer, and J. Stuke, 1980, Philos. Mag.
H , 127.
Vook, F. L., in Radiation Damage and Defects in Semiconductors (Institute
of Phsics, London), p. 60.
Waddell, C. N., W. G. Spitzer, J. E. Fredrickson, G. K. Hubler, and T. A.
Kennedy, 1984, J. Appl. Phys. 55, 4361.
Winterbon, K. B., 1972, Radiat. Eff. ]3,
215.
Zachariesen, W. H., 1932, J. Am. Chem. Soc. 54, 3841.
Zallen, R., 1983, The Physics of Amorphous Solids (Wiley, New York), p. 3.
136
VITA
Thomas Rendall Askew was born i n Geneva, I l l i n o i s on June 11, 1955.
He attended public schools in Wheaton, I l l i n o i s and Hamilton, Mass. and
graduated from the Hamilton-Wenham Regional High School in 1973.
That
year he entered Gordon College, Wenham, Mass., where he majored in
physics and math.
He received the Bachelor of Arts degree from that
i n s t i t u t i o n in 1977.
Since then he has studied physics a t Princeton
University and the University of I l l i n o i s , r e c e i v i n g the Master of Science
degree from the l a t t e r i n s t i t u t i o n i n 1982.
He i s a member of the
American Physical Society and the Materials Research Society.
Документ
Категория
Без категории
Просмотров
0
Размер файла
4 520 Кб
Теги
sdewsdweddes
1/--страниц
Пожаловаться на содержимое документа