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Theoretical investigation of aluminum gallium nitride cathodes and their use in microminiature microwave triodes

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THEORETICAL INVESTIGATION OF
ALUM INUM GALLIUM NITRIDE CATHODES
A N D THEIR USE IN
M ICROM INIATURE MICROWAVE TRIODES
bv
C H R IS T O P H E R W ILLIAM H A T F IE L D
A dissertation subm itted to the G raduate Faculty of
N orth Carolina State University
in partial fulfillment of the
requirem ents for the Degree of
Doctor of Philosophy
E L E C T R IC A L E N G IN E E R IN G
R aleigh
1999
A P P R O V E D BY:
Cnair of Advisory C om m ittee
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UMI Number:
9922687
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Abstract
HATFIELD, CH RISTOPH ER WILLIAM. Theoretical Investigation of Aluminum
Gallium N itride Cathodes and Their Use in M icrom iniature Microwave Triodes
(Under the direction of Griff L. Bilbro.)
The purpose of the research was to theoretically investigate AlxG ax_xN cath­
odes and their use in microminiature microwave triodes. The use of AlxG ai_xN
as a cathode m aterial was investigated due to th a t m aterial’s unique combination
of physical properties. Research was directed toward the simulation of a variety
of AlxG ax_xN cathode structures operating at various tem peratures. The cathode
simulations were accomplished with semiconductor device theory. Poisson's equa­
tion was solved using a Runge-Kutta numerical m ethod to determine th e emis­
sion barrier.
The thermionic emission theory of metal-semiconductor junctions
was used to com pute the em itted current density. The results for n-type doped,
compositionally-graded AlxGa1_xN cathodes indicate th at very high current densi­
ties may be em itted from these cathodes at relatively low tem peratures, compared
to conventional thermionic cathodes. However, electron diffusion limits the current
density for certain cathode structures. M icrominiature microwave triodes utiliz­
ing high-current-density AlxG ax_xN cathodes were investigated using vacuum tube
theory and electron optics software. The results of the investigation into micro­
m iniature microwave triodes indicate that these triodes enjoy an enhanced cutoff
frequency compared to larger triodes with conventional thermionic cathodes. How­
ever, these devices do not appear to be com petitive with commercially available
solid state devices at room tem perature, for sim ilar device size and biasing condi­
tions. It is concluded in the study that these m icrom iniature triodes might find use
in harsh environments, involving high tem perature and radiation.
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Biography
Christopher William Hatfield was bora in Cheverly, M aryland on July 24, 1967.
His family lived in Sicklerville, New Jersey from 1971 to 1978. In 197S, his family
settled in Macungie, Pennsylvania. He graduated from Em m aus High School in
Emmaus, Pennsylvania in 1985. He enrolled in the Pennsylvania State University
and atten d ed classes at the Allentown Campus, located in Fogelsville, Pennsylvania,
from 1985 to 1987. While there, he won the Eric A. and Josephine S. Walker Award,
an annual award given to one student from each Pennsylvania S tate University com­
monwealth campus. In 1987, he transferred to Pennsylvania S tate University's main
campus in University Park, Pennsylvania. He was inducted into the Eta K appa Nu
electrical engineering honor society in 1988 and the Tau B eta Pi engineering honor
society in 1989. He completed his undergraduate studies at Pennsylvania S tate Uni­
versity in 1989. and received a Bachelor of Science in Electrical Engineering degree
with distinction. He enrolled in graduate school at Pennsylvania State University
in 1989. From 1989 to 1993, he held a series of research and teaching assistantships
at the Pennsylvania S tate University. He received a M aster of Science in Electrical
Engineering degree in 1991. The title of his m aster's thesis was The Development
o f Thin D iam ond Film Semiconductor Technology. He lived in S tate College, Penn­
sylvania from 1989 to 1993. Between 1989 and 1993. he co-authored seven journal
papers and four conference papers. In 1993, he moved to Raleigh, North Carolina.
He enrolled in graduate school at the North Carolina S tate University in 1994 to
pursue a doctorate. From 1994 to 1999, he held a series of research and teaching
assistantships at the North Carolina State University. His research at the North
Carolina S ta te University included AlxGai_xN and diam ond cathodes, silicon car­
bide sem iconductor devices, vacuum tubes and vacuum microelectronic devices, and
microwave devices and circuits. Between 1994 and 1999. he authored two journal
papers and two conference papers. He has been a m em ber of the IEEE Electron
Devices Society since 1989. He is also a member of the M aterials Research Society,
the American Vacuum Society, and the American In stitu te of Physics.
ii
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Acknowledgements
I th an k my graduate advisor, Dr. Griff Bilbro. for supporting m e academically
and em otionally and for chairing my graduate committee. His advice on every piece
of research I worked on from 1993 to 1999 was consistently helpful.
I thank my collaborator, Dr. Robert Nemanich. for help and encouragement with
my research on cathodes and vacuum microelectronics, and for being a member of
m y graduate com m ittee. His ideas, feedback, and data proved invaluable.
I thank Dr. Michael Steer for serving on my committee and encouraging me
to investigate the circuit analysis of vacuum microelectronic devices in his circuit
analysis class in 1995. I thank Dr. Jam es Mink for serving on my com m ittee and and
encouraging m e to investigate the microwave performance of field emission arrays
in his microwave circuits class in 1995. I thank Dr. Mehmet O zturk for encouraging
m e to research the history of vacuum tubes and vacuum microelectronics in his class
on the history of microelectronics in 1997.
I thank Dr.
electron optics.
A rthur Morris, who helped me learn about vacuum tubes and
I th an k Dr.
Peter Baumann for working w ith me on diamond
cathodes. I thank Mr. Brandon Ward for working with me on III-V nitride and
diam ond cathodes. I thank Dr. Robert Davis and his research group for providing
d a ta during my research on III-V' nitride cathodes. I also thank Mr. John Driscoll
for his support of my research on III-V nitride cathodes.
I thank the many graduate students who helped me with my research at North
Carolina S tate University. Particularly, I thank Mr. Real Pom erleau and Dr. Cagatay Tekmen for helping me with my preliminary oral exam ination.
I th an k the Office of Naval Research (Washington. DC) and the PTS Co.
(Raleigh, NC) for financially supporting my research. I thank Dr. William Herrm annsfeldt at Electron Optics Simulations (Los Altos. CA) who provided software
support.
I thank my parents, Mr. Deets Hatfield and Mrs. Ruth Hatfield, for providing
love and support during the m any years th a t I remained in school. I owe them
special thanks for helping me to make the transition from one graduate school to
another in 1993 and 1994. I thank my wife, Dr. Bo Jin Hatfield, for providing
iii
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the love and encouragement which I needed during my years at North Carolina
S tate University. W ithout her tireless and enthusiastic support. I would have never
completed this dissertation. Finally. I thank my daughter Jin n a Ju liette Hatfield
for providing a wonderful family life for me as I completed my dissertation.
iv
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C on ten ts
1
I n tr o d u c tio n
1.1 Historical B a c k g ro u n d ................................................................................
1.2 Motivation for the R e s e a r c h ......................................................................
1.3 Goals of the R esearch..............................................................
1.4 Original C o n trib u tio n s ................................................................................
1.5 P u b lic a tio n s...................................................................................................
1
1
2
4
4
5
2
L ite r a tu r e R e v ie w
2.1 Cathodes .......................................................................................................
2.1.1 Thermionic C a th o d e s .......................................................................
2.1.2 Field Emission C a th o d e s ................................................................
2.1.3 Wide Bandgap Sem iconductor C a th o d e s ...................................
2.2 Microwave T r i o d e s ......................................................................................
2.2.1 Microwave Triodes with Therm ionic C a th o d e s .........................
2.2.2 Microwave Triodes with Field Emission C a th o d e s ..................
2.2.3 Microwave Triodes with W ide Bandgap Semiconductor C ath ­
odes ....................................................................................................
7
7
7
10
12
IS
19
23
2S
3
AlxG ai_xN C a th o d e s
3.1 Cathode S tru c tu r e s ......................................................................................
3.2 Simulation of Cathode S tr u c tu r e s ...........................................................
3.3 Solution of Poisson’s E q u a tio n ..................................................................
3.3.1 Formulation of Poisson's E q u a tio n ................................................
3.3.2 Runge-K utta Numerical P r o c e d u r e ............................................
3.4 Com putation of Em itted Current D en sity ..............................................
30
30
32
34
34
39
41
4
M ic r o m in ia tu re M icro w ave T r io d e s
4.1 M icrominiature Microwave Triode S tru c tu re s .......................................
4.2 Simulation of M icrominiature Microwave Triode Unit Cell .............
4.2.1 Vacuum Triode T h e o ry ...................................................................
4.2.2 Electron O p t i c s ................................................................................
4.3 M icrominiature Microwave Triode A rra y s ..............................................
45
45
48
48
49
51
5
R e s u lts
5.1
AlxG ai_xN C a th o d e s ..................................................................................
5.1.1 Basic Cathode Structure O peration at 300 K ............................
5.1.2 Cathodes with a Thicker A lxG ai_xN L a y e r...............................
5.1.3 Cathodes with Non-Linear G r a d in g ............................................
5.1.4 Cathodes with M oderate and Low Donor Concentrations . .
56
56
56
65
66
68
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5.1.5 Basic Cathode Structure O peration at Elevated Tem peratures
5.2 M icrom iniature Microwave T r i o d e s ...........................................................
5.2.1 Sm aller Unit Cell with Cathode-to-G rid.............S p a c e rs ..........
5.2.2 Larger Unit Cell with Cathode-to-Grid and Grid-to-Anode
Spacers ..............................................................................................
5.2.3 Comparison with O ther Active D e v ic e s ..............
S3
6
C o n clu sio n s
6.1 Sum m ary of Results .....................................................................................
6.1.1 AlxGax_xN C a th o d e s ..................................................... ..................
6.1.2 M icrom iniature Microwave T r i o d e s .....................
91
6.2 Implications of the R e s e a rc h ........................................................................
6.2.1 AlxGax_xN C a th o d e s .......................................................................
6.2.2 M icrom iniature Microwave T r i o d e s .....................
94
6.3 Suggestions for Further R e s e a r c h ..............................................................
6.3.1 AlxGax-jcN C a th o d e s .......................................................................
6.3.2 M icrom iniature Microwave T r i o d e s .....................
96
R efe re n ce s
A
71
75
75
SO
89
S9
S9
93
93
95
95
99
Maple V In p u t F ile for C a th o d e S im u la tio n
110
B EGN2e In p u t F iles for M icrom in iatu re M icrow ave T riode U n it C ell
S im u la tio n
117
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L ist o f T ables
5.1
Results of EGN2e Simulations of Half of the Smaller Unit Cell . . .
5.2 Results of EGN2e Simulations of Half of the Larger Unit Cell
...
77
SI
5.3 Compaxison of a SiC JF E T with M icrominiature Microwave Triode
Arrays Operating at 773 K .........................................................................
85
5.4 Comparison of a SiC JF E T with Scaled M icrominiature Microwave
Triode Arrays O perating at 773 K ...........................................................
86
5.5
Increase of Microwave Triode Cutoff Frequency with T i m e ..............
88
6.1
Summary of Cathode Simulation R e s u l ts ...............................................
90
6.2
Summary of M icrom iniature Microwave Triode Array Simulation Re­
sults .................................................................................................................
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91
List o f F igu res
3.1
A generic cathode...........................................................................................
30
3.2
AlxGai_xN cathode stru ctu re......................................................................
31
3.3
Extrapolation of d ata for the electron affinity of AlxG ai_xN as a
function of A1 fraction x ...............................................................................
33
3.4
Simplified cathode structure assumed for one-dimensional simulation. 34
3.5
Two approximations for Si donor ionization energy in AlxG ai_xN as
a function of x. Lower curve is "best case"1; upper curve is "worst
case.”
3.6
.............................................................................................................
37
Two ways of defining the heterojunction conduction band offset for
AlxG aj_xN as a function of Al fraction x. Lower curve is "best case”:
upper curve is "worst c a s e . " ......................................................................
3.7
Qualitative energy band diagram for a metal-semiconductor junction
with an n-type sem iconductor.....................................................................
3.8
38
41
Qualitative energy band diagram for an n-type. graded AlxG ai_xN
cathode..............................................................................................................
42
3.9
Electron mobility in AlxG ai_xN as a function of x ................................
44
4.1
Electrode configuration of a conventional planar vacuum triode.
46
4.2
M icrominiature microwave triode array with control grid wires fabri­
. .
cated on cathode-to-grid spacers and the anode m ounted on a remote
spacer.................................................................................................................
4.3
46
Microminiature microwave triode array with control grid wires fab­
ricated on cathode-to-grid spacers and the anode m ounted on gridto-anode spacers..............................................................................................
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46
4.4
Graphical output for an electron optics simulation using th e EGN2e
com puter program............................................................................................
4.5
High-frequency small-signal model for an array of m icrom iniature
microwave triode unit cells..................................................................
51
4.6
Circuit diagram indicating how cutoff frequency is defined........
55
5.1
Basic cathode structure........................................................................
57
5.2
Potential vs. position in basic cathode structure at 300 K .........
57
5.3 Conduction band minimum energy' vs.
position in basic
cathode
stru ctu re at 300 K..................................................................................
5.4
59
Ionized donor concentration vs. position in basic cathode stru ctu re
at 300 K....................................................................................................
5.6
58
Electron concentration vs. position in basic cathode stru ctu re at 300
K.................................................................................................................
5.5
50
60
Total charge concentration vs. position in basic cathode stru ctu re at
300 K .........................................................................................................
60
5.7
Electric field vs. position in basic cathode structure at 300 K. . . .
5.8
Conduction band minimum energy vs.
61
position of basic cathode
stru ctu re at 300 K, assuming a sm aller ( “best case"1) heterojunction
conduction band offset...........................................................................
5.9
Conduction band minimum energy vs.
structure at 300 K, assuming lower
62
position of basic cathode
( “best case’’ ) donor ionization
energies......................................................................................................
63
5.10 Conduction band minimum energy vs. position for a cathode with
0.25-/zm-thick AlxG at_xN layer...........................................................
64
5.11 Conduction band minimum energy vs. position for a cathode with a
0.50-^m-thick AlxG at_xN layer...........................................................
64
5.12 Linear and quadratic grading profiles for the AlxG ai_xN layer. . . .
5.13 Conduction band minimum energy vs. position for a basic cathode
structure with a concave-up quadratic grading...............................
68
5.14 Conduction band minimum energy vs. position for a cathode with a
concave-down quadratic grading..........................................................
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69
67
5.15 Conduction band minim um energy vs. position for a basic cathode
stru ctu re with a m oderate donor concentration of 5 x 1017 cm -3 .
5.16 Conduction band m inim um energy vs. position for
.
70
a basic cathode
stru ctu re with a low donor concentration of 1 x 1017 cm -3......
70
5.17 Conduction band minimum energy vs. position (A,B) and Fermi level
(C,D) of basic cathode structure at 300 l\ (A.C) and 700 K (B.D ). .
72
5.18 Emission barrier of a basic cathode structure vs. tem perature. . . .
73
5.19 E m itted current density of basic cathode structure vs. tem perature.
73
5.20 Free electron concentration vs. position in basic cathode stru ctu re
a t 300 K and 700 K...............................................................................
74
5.21 G radient of electron concentration at cathode-vacuum interface vs.
tem p eratu re in basic cathode structure.....................................................
74
5.22 M inimum electron m obility required at cathode-vacuum interface
(where x = 0.75) to achieve barrier-limited current density vs. tem ­
p eratu re..............................................................................................................
75
5.23 Electrode configuration of the smaller unit cell.......................................
76
5.24 Graphical outp u t from an EGN2e simulation of the smaller unit cell.
76
5.25 Transconductance per unit area vs. bias voltages for an array of the
sm aller unit cell. The curves, from left to right, correspond to Vq =
0.00 V, -0.02 V. -0.04 V, -0.06 V. -0.08 V. -0.10 V. -0.12 V. and -0.14
V..........................................................................................................................
77
5.26 Cutoff frequency vs. bias voltages for an array of the sm aller unit
cell. The curves, from left to right, correspond to Vq = 0.00 V. -0.02
V. -0.04 V, -0.06 V, -0.08 V. -0.10 V. -0.12 V. and -0.14 V..........
7S
5.27 Anode current density vs. bias voltages for an array of the smaller
unit cell. The curves, from left to right, correspond to Vq = 0.00 V,
-0.02 V, -0.04 V, -0.06 V, -0.08 V. -0.10 V. -0.12 V, and -0.14 V.
. . 78
5.28 Anode DC power dissipation density vs. bias voltages for an array
of the sm aller unit cell. The curves, from left to right, correspond to
Vc = 0.00 V, -0.02 V, -0.04 V. -0.06 V, -0.08 V, -0.10 V. -0.12 V, and
-0.14 V................................................................................................................
x
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79
5.29 Anode resistance vs. bias voltages for a 1 cm 2 array of the smaller
unit cell. The curves, from left to right, correspond to Vq = 0.00 V,
-0.02 V, -0.04 V, -0.06 V, -0.08 V, -0.10 V, -0.12 V, and -0.14 V. . .
79
5.30 Electrode configuration of the larger unit cell........................................
SO
Graphical output from an EGN2e simulation of th e larger unit cell.
81
5.31
5.32 Transconductance per unit area vs. bias voltages for an array of the
larger unit cell. The curves, from left to right, correspond to Vq =
0.0 V, -0.2 V. -0.4 V, -0.6 V. -0.8 V and -1.0 V......................................
82
5.33 Cutoff frequency vs. bias voltages for an array of th e larger unit cell.
The curves, from left to right, correspond to Vg = 0.0 V, -0.2 V, -0.4
V, -0.6 V, -0.8 V and -1.0 V........................................................................
82
5.34 Anode current density vs. bias voltages for larger unit cell. The
curves, from left to right, correspond to I g = 0.0 V. -0.2 V. -0.4 V.
-0.6 V, -0.8 V. and -1.0 V.............................................................................
83
5.35 Anode DC power dissipation density for an array of larger unit cells
with anode spacers. The curves, from left to right, correspond to Vq
= 0.0 V, -0.2 V, -0.4 V, -0.6 V. -0.8 V. and -1.0 V ................................
84
5.36 Anode resistance for a I cm 2 array of larger unit cells with anode
spacers. The curves, from left to right, correspond to Vg = 0.0 V,
-0.2 V, -0.4 V. -0.6 V, -0.S V and -1.0 V...................................................
84
5.37 Increase of microwave triode cutoff frequency w ith tim e.....................
88
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C hapter 1
In trod u ction
1.1
Historical Background
Both thermionic and field emission were experimentally observed prior to 1900.
Fleming utilized a thermionic cathode when he invented the vacuum diode in 1904.
T he field of electronics was born in 1906 when Lee De Forest introduced a third
electrode into a vacuum diode to create a vacuum triode. This three-term inal device
was capable of signal generation and amplification and quickly found use in radio
communication, radio navigation, and audio circuitry.
During the first third of the tw entieth century, a variety of therm ionic cathodes,
including tungsten filaments and alkali metal oxide cathodes, becam e commercial­
ized products and were used in a variety of low-frequency, low-power vacuum tubes.
As tim e passed, electronic system s evolved and demanded more power at a higher
frequency. By the 1930's, engineers had designed smaller triodes which could op­
erate at microwave frequencies. Development of radar and com m unication systems
during VVWII led to the construction of high-power. high-frequency microwave tri­
odes.
After WWII, engineers developed a number of different schemes for the improve­
m ent of microwave tubes. One idea involved the replacement of hot thermionic
cathodes with cold cathodes. A second idea involved scaling the devices down such
th at the interelectrode spacings becam e microscopically small. However, neither of
these ideas became feasible until the 1960’s, when techniques were developed for
the fabrication of m icrom iniaturized field em itter cathodes and th e m anufacture of
1
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integrated circuits.
During the 1970’s, 1980’s, and 1990’s, researchers studied various m eans of devel­
oping cold cathodes and fabricating m icrom iniature vacuum tube structures. Most
of the research on cold cathodes has focussed on the development of a field emission
cathode, although many other types of cathodes have been proposed and researched.
Technology for th e manufacture of m icrom iniature vacuum tubes is not yet m ature,
but it is rapidly developing due to the recent commercialization of field emission
displays (F E D ’s).
1.2
M otivation for the Research
The convenient and economical generation, amplification, and processing of mi­
crowaves has become essential for applications such as comm unications (satellite
and terrestrial), radar (ground-based, airborne, and naval), electronic warfare (elec­
tronic counterm easures, decoys, and jam m ers), radionavigation (land, air. and sea),
remote sensing (telem etry and weather m onitoring), cooking, industrial heating,
medical diagnostics, surveillance, TV broadcasting (terrestrial UHF-TV and direct
broadcast satellites), radio astronomy, and space exploration.
As m ilitary and commercial microwave systems have evolved, they have be­
come increasingly high-power. high-frequency, efficient, compact, and lightweight.
All m ilitary branches want to mount com pact, lightweight, high-performance radar
and electronic counterm easure pods on wings or in small com partm ents. They want
active guided missiles with radionavigation subsystems of minimal size and weight.
They want to deploy phased array radars which consist of a densely packed array
of elements, w ith each element serviced by a compact, high-performance module.
Industry has sim ilar needs. Companies want to build inexpensive com m unication
systems in which com pact, lightweight, portable devices quickly, reliably, and ef­
ficiently receive and transm it voice channels, video channels, and com puter data.
Com m unication networks consisting of terrestrial base stations or satellites are lim­
ited by the perform ance, size, and weight required for generating and amplifying
microwaves in receivers and transm itters. Communication systems for spacecraft
have similar requirem ents. Furthermore, m ilitary, industrial, and research organi­
zations want to use these advanced microwave circuits and systems in environm ents
2
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involving extrem e tem peratures or high levels of radiation which can disable or
destroy most existing devices.
The performance of microwave circuits depends upon the performance of their
passive and active devices.
At higher power levels and higher frequencies, mi­
crowave circuits requiring active devices utilize microwave tubes. These tubes can
be classified into four categories, as follows: classical gridded electron tubes (tri­
odes and tetrodes), linear beam tubes (klystrons, reflex klystrons, traveling-wave
tubes, backward-wave oscillators, ubitrons. and peniotrons). crossed field tubes
(magnetrons and crossed-field amplifiers), and relativistic tubes (gyrotrons. gyromonotrons, gyroklystrons, gyro-traveling-wave tubes, and vircators). These tubes
have the ability to generate and/or amplify microwaves.
Due to their performance limitations, conventional microwave triodes are typi­
cally relegated to lower-power (< 100 VV), lower-frequency (< 4 GHz) circuits, but
these tubes do enjoy certain advantages over other microwave tubes. Triodes are
generally small in size and weight, making them advantageous for airborne and
m ilitary applications. They involve simpler operation with respect to biasing and
circuit design. They do not require a magnetic field to operate, so solenoids or per­
manent magnets are not required. Finally, due to their simple construction, they
are lower in cost. Hence, triodes cannot perform at the same level as other more
advanced types of microwave tubes, but they enjoy other advantages th at continue
to make them useful for certain applications.
All microwave tubes require a suitable cathode for successful operation. The
"ideal” cathode for microwave tube service would em it a high current density at
a reasonably low operating tem perature.
It must exhibit chemical and therm al
stability, and have a long operational lifetime. It should be resistant to both ion
bom bardm ent and residual gases (such as oxygen, carbon, and water vapor). The
cathode should consist of readily available m aterials and should have a stru ctu re
which can be fabricated with available techniques.
The cathodes which have been used in conventional microwave tubes are hightem perature thermionic cathodes. Such cathodes can be classified into five general
groups as follows: (1) pure materials. (2) oxide cathodes, (3) dispenser cathodes.
(4) cermet cathodes, and (5) thorium-based cathodes. All of these types of cathodes
3
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must be heated to at least 1100 K, and none can produce a maxim um continuous
current density above 10 A /cm 2. Higher current densities are possible from these
cathodes, but only in the form of very short pulses. T he relatively low continuous
current density of these thermionic cathodes, along with their heating requirements,
constrains th e compactness, efficiency, and reliability of microwave triodes.
The three most promising candidates for a cold cathode have been field emis­
sion cathodes, hot-electron semiconductor cathodes, and negative electron affinity
(NEA) cathodes. All three of these cathodes utilize standard m etals (e.g.. Mo, Ti.
W) or stan d ard semiconductors (Si, GaAs). In spite of extensive research, none of
them has led to a cathode suitable for high-performance microwave triodes.
1.3
Goals of the Research
Potential improvements in compact, lightweight microwave systems and circuits
were exam ined through the simulation of a new type of active microwave device.
This new device takes the form of an array of m icrom iniature microwave triode struc­
tures utilizing high-performance AlxC!al_xN cathodes. This research involved two
m ajor investigations. The first investigation sought to determ ine if an AlxGai_xN
cathode stru ctu re could be designed which could provide performance adequate for
m icrom iniature microwave triodes. The second investigation sought to determine
if a m icrom iniature triode structure could be designed which could provide useful
microwave performance, assuming that a AlxGa!_xN cathode is available.
1.4
Original Contributions
The original contributions of the research described in this dissertation are as fol­
lows:
• The first recognition of the possibility of thermionic emission from an AlxGai_xN
cathode.
• The first calculations indicating how the energy barrier which limits electron
emission is lowered by a combination of n-type doping and spatial grading of
the A1 fraction x in an AlxG ai_xN cathode.
4
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• T he first numerical calculations indicating th at the energy barrier limiting
electron emission for an AlxG a!_xN cathode could be in th e range of 0.4-0.6
eV, which is substantially lower than the values observed for conventional
thermionic cathodes.
• T he first recognition of the role which electron diffusion plays in limiting the
em itted current density of an AlxG at_xN cathode.
• T he first detailed designs for AlxG aj_xN cathodes which would be capable of
providing adequate performance for microwave tubes (such as triodes).
• T he first detailed analysis, based on electron optics calculations, of spacecharge-limited m icrom iniature microwave triodes.
• T he first comparisons of space-charge-limited m icrom iniature triodes to semi­
conductor devices operating at high tem perature.
1.5
Publications
Publications resulting from the work presented in this dissertation were as follows:
• C.VV. Hatfield and G.L. Bilbro, "Simulation of Room T em perature Thermionic
Emission from AlxG ai_xN Negative Electron Affinity C athodes.’’ accepted for
publication in Journal o f Vacuum Science and Technology B. M arch/April
1999 issue.
• C.W . Hatfield and G.L. Bilbro. "Simulation of Therm ionic Emission from
AlxG ai_xN Cathodes at Elevated Temperatures." subm itted for publication
to Journal o f Vacuum Science and Technology B on January S, 1999.
• C.W . Hatfield, G.L. Bilbro, S.T. Allen, and J.W . Palm our, “DC I-V Charac­
teristics and RF Performance of a 4H-SiC JF E T at 773 K .” IE E E Transactions
on Electron Devices, vol. 45, pp. 2072-2074, Septem ber 1998.
• C.W . Hatfield and G.L. Bilbro. "Modeling and Sim ulation of AlxG ai_xN Neg­
ative Electron Affinity Cathodes,” Technical Digest o f the Eleventh Interna­
tional Vacuum Microelectronics Conference. July 1998, pp. 289-290.
5
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• C.W . Hatfield, G.L. Bilbro, A.S. Morris, P.K. Baumann. B.L. Ward, and R.J.
Nemanich, ‘‘Investigation of an NEA Diamond Vacuum Microtriode Array."
Materials Research Society Symposium Proceedings Vol.
(III-Nitrides,
SiC, and Diamond fo r Electronic Applications), April 1996, pp. 33-38.
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C h ap ter 2
L iterature R eview
2.1
Cathodes
Many different types of cathodes have been studied and commercialized, includ­
ing therm ionic cathodes, field emission cathodes. Schottky cathodes, hot electron
semiconductor cathodes, and negative electron affinity cathodes.
Of these, only
therm ionic cathodes have found the most use in commercial microwave triodes.
The other type of cathode extensively tested in microwave triodes and micro­
m iniature triodes is the field emission cathode. These cathodes have promised, but
have not yet achieved, a level of performance which would allow them to be used
in commercialized microwave triode tubes.
W ide bandgap semiconductors exhibiting a low or negative electron affinity pro­
vide a new approach for the development of cathodes for microwave applications.
As a result, these materials have been extensively studied during the past few years.
Previous research efforts on thermionic cathodes, field emission cathodes, and
wide bandgap semiconductor cathodes are summarized and discussed below.
2.1.1
T herm ionic C athodes
Edison performed his first experiments on thermionic emission in 1882 and reported
his results in 1883 [1]. Around 1903. Wehnelt discovered th at a piece of m etal wire
coated with an alkali metal oxide could em it a large current when heated to a high
tem perature [1]. In 1914, Langmuir showed that thoriated tungsten emits signifi­
cantly more current than pure tungsten, and such cathodes soon found commercial
7
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use [1].
By the early 1920’s, the process of making “oxide cathodes” (VVehnelt’s cath­
odes) had become commercialized and widely used [1]. Around 1925. nickel became
the substrate of choice for oxide cathodes [2]. Oxide cathodes continued to improve
in quality in 1920’s, until by 1930 they could em it 4 A /cm 2 in pulsed operation [2].
Oxide cathodes also were greatly improved during YVWII. By 1943, oxide cath­
odes could produce 25 A /cm 2 in pulsed operation [2j. During th e war, oxide cath­
odes were dem onstrated which could produce 30 A /cm 2 peak density in production
units and 80 A /cm 2 peak density for several hundred hours of operation. Specially
constructed oxide cathodes could em it 140 A /cm 2 peak density for a short tim e [3].
After the war, oxide cathodes capable of 0.5 A /cm 2 continuous current density under
optim al conditions were dem onstrated [3]. The Bell Laboratories 1553 microwave
triode (later m anufactured as the Western Electric 416A triode) was developed in
the late 1940’s and utilized a sprayed oxide cathode on nickel which could em it 0.1S
A /cm 2 continuously [4].
In spite of the improvements in oxide cathode technology, there was still a need
for higher continuous emission densities and less susceptibility to dam age [5]. Dur­
ing the early 1950’s, Philips moved thermionic cathode technology forward with the
development of the L-cathode [6] and the B-cathode [6]. The L-cathode consisted
of a porous tungsten plug containing a reservoir of barium and strontium carbon­
ates. It achieved improved performance at the expense of reduced reliability due
to barium ’s high evaporation rate. The B-cathode consisted of a porous tungsten
plug “im pregnated” with barium calcium alum inate. Such a cathode can supply 1-2
A /cm 2 continuously at 1000-1050°C' and became the standard cathode in numerous
types of tubes for many years. Operation of a B-cathode up to 1100°C provides
several A /cm 2, but this limits the cathode lifetime to a few thousand hours [5].
In 1964, Figner et al. subm itted a patent on a cathode consisting of tungsten
and Ba3Sc40g, now called the pressed scandate cathode [7]. The cathode could
em it a pulsed current density of 5 A /cm 2 at a tem perature of 1030°C. The Mcathode was invented during the late 1960’s [5]. THe M -cathode is sim ilar to a
B-cathode, except that the top surface of the im pregnated plug has a thin (less
than 1 /zm) layer of osmium and/or ruthenium sputtered upon it. This ex tra layer
8
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reduces the work function, allowing the smae current density as a B-cathode but at
a tem perature about 100°C cooler. These cathodes were first m arketed by Philips
Metalonic around 1970.
During the 1970’s, a number of new thermionic cathodes were investigated.
Starting around 1975, research began on mixed metal m atrix (MMM) cathodes [5].
with plugs fabricated by sintering a m ixture of tungsten powder and iridium or
osmium powder. A nother type was the im pregnated scandate cathode [7], which
provided a pulsed current density of 20 A /cm 2 at 1030° C. Controlled porosity den­
sity (CPD ) m atrix cathodes made of Ir-BaO achieved a continuous current density
of 6 A /c m 2 in 1978 [6]. Van Oostrom and Augustus dem onstrated a pressed scan­
date cathode which could em it a pulsed current density of 10 A /cm 2 at 1030° C in
1979 [7]. T hat sam e year, a tungsten-iridium m atrix utilizing barium calcium aluminate dem onstrated continuous current densities of 4-8 A /cm 2 in the tem perature
range 1025-1100° C [6].
Mixed m atrix scandate cathodes were first fabricated in the early 19S0*s and
dem onstrated pulsed current densities in the range of 20-35 A /cm 2 [7]. During the
mid-1980’s, top-layer scandate cathodes were developed by Hasker et al. which could
emit a pulsed current density of 100 A /cm 2 [7]. The commercialized mixed metal
m atrix (M M M -cathode) was introduced around 1986 [8]. These MMM-cathodes
enjoy a higher resistance to poisoning, continuous current densities on the order
of 10 A /cm 2, and a lifetime over 100.000 hours. During the late L9S0’s, Varian's
Microwave Tube Division began developing a controlled doping (CD) cathode con­
sisting of a standard YV m atrix with an overcoating of tungsten alloyed with another
metal [9]. These CD cathodes have since been commercialized [10].
Currently, the therm ionic cathode receiving the most industrial research atten ­
tion is the scandate cathode [11]. In 1995. researchers at Philips dem onstrated a
top-layer scandate cathode fabricated using laser ablated deposition (LAD) which
could em it a pulsed current density of 400 A /cm 2 at 1030°C [7]. This is the largest
peak current density ever measured for a thermionic cathode. However, the util­
ity of scandate cathodes is limited by sensitivity to ion bom bardm ent (compared
to commercialized im pregnated cathodes), non-uniform emission, and by lack of
reproduciblity of the fabrication process.
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Thus, high-tem perature thermionic cathodes have been found to be the only
viable electron source for microwave triodes. However, m any problems exist con­
cerning the use of thermionic cathodes. Thermionic cathodes suffer from low contin­
uous current density (< 10 A /cm 2). Thermionic cathodes require heating circuitry
operating at 1100 K or higher (a current density of 10 A /cm 2 requires 1300 K or
higher) which consumes considerable power and results in therm al aging of the en­
tire vacuum device, resulting in reliablility problems. Because of the evaporation
of barium , conventional thermionic cathodes have a limited operational life. Also,
conventional thermionic cathodes are quite senstitive to ion bom bardm ent. Finally,
it is im portant to note th at there is no physical means of fabricating the micro­
m iniature triodes structures described in this dissertation on the relatively rough
surfaces of conventional tungsten matrix cathodes. The, the use of conventional,
high-tem perature thermionic cathodes is problematic for m icrom iniature microwave
triodes.
Recent work on thermionic cathodes has concentrated on extending operational
life [12], understanding the emission mechanisms [13], reducing the tim e required
for heating up the cathode from the ambient tem perature to the cathode operating
tem perature [14], and characterizing how the cathodes are poisoned by residual
gases [10].
2 .1 .2
Field E m ission C athodes
Field emission was extensively studied, well before WWII [15]. Field emission was
first observed by Wood in 1897. As early as 1920. attem pts were m ade to use field
emission in practical devices. Fowler and Nordheim used quantum mechanics to
explain the phenomenon in 1928.
Efforts to replace low-current-density thermionic cathodes in microwave tubes
w ith high-current-density field emission cathodes began soon after W W II, under the
direction of Dyke [15]. His group was never able to construct a useful microwave
tu b e because it was impossible at that time to fabricate a field em itter array (FEA)
w ith adequate microwave performance.
After Spindt invented the first thin-film field emission cathode in 1968 [16]. it
becam e feasible to put a field emission cathode in a microwave tube. At the Stanford
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Research In stitu te (SRI), Brodie and Spindt pursued this goal through the 1970’s,
further m iniaturizing the m olybdenum tips and improving th e emission properties
and reliability of the field em itter arrays [17]. In 1991, the SRI group reported FEA
cathodes with 1.5 x 107 tip s/cm 2 and 10 //A of current per tip [IS]. These values
imply a current density per unit area of 150 A /cm 2.
In 1993, Gammie et al. at Raytheon Company fabricated M o-tip FEA cathodes
and tested them at microwave frequencies [19], These arrays could em it 14 y A per
tip; hence, an array of 5000 tips produced a total current of 70 mA.
In certain respects. Si FEA tips are simpler than m etal tips to fabricate and
process, and the first Si field em itter arrays were reported in 1974 [20]. Researchers
at the Microelectronics Center of North Carolina (MCNC) reported large arrays of
Si tips for microwave tu b e cathodes in 1994 [21]. These arrays could achieve a total
current of 7 mA with a current density of 7 A /cm 2 for a period of about IS hours.
In 1995, this group reported a Si FEA cathode which could provide IS.5 mA total
current from a 0.00554 cm 2 cathode area with a current density of 3.3 A /cm 2 [22].
T he Vacuum Microelectronics Initiative of the Advanced Research Projects
Agency and the Naval Research Laboratory resulted in the testing of FEA cathodes
in a 10-GHz klystrode assembly in 1997 [23]. The goal was to achieve 50 VV output
power from the klystrode. The four FEA cathodes tested were a Mo-tip FEA from
SRI International, a Ni-coated Mo-tip FEA from MIT Lincoln Labs, a Si-tip FEA
from MCNC, and a GaAs thin film edge em itter FEA from the Varian Ginzton
Research Center (GRC). When characterized in a specially constructed test sta­
tion, all four cathodes achieved a current density of at least 0.4 A /cm 2, and the
MIT cathode achieved a peak current density of 2400 A /cm 2. However, when the
cathodes were tested inside the klystrode assembly, none of them could achieve a
peak current density greater than 35 A /cm 2, and the klystrode output power with
any of the four cathodes never exceeded 4 mVV.
NEC Corporation reported in 1997 that it had developed a reliable vertical
current lim iter (VECTL) Mo-tip FEA cathode [24]. This new VEC'TL FEA cathode
exhibited a gate-to-cathode breakdown at 120 V. a stable emission of 1 y A per tip
for 70 V on the gate, and a packing density of 5 x 10' tip s/c m 2, resulting in a total
current density of 50 A /cm 2.
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The most com m on form of cathode in experim ental m icrom iniature triode struc­
tures has been a field em itting cathode. Many m aterials have been used, including
molybdenum [33], silicon [34, 35. 36. 37. 38. 39]. tungsten [40. 41], gallium ar­
senide [42], polycrystalline silicon [43. 44, 46], and gold [47]. Various m ixtures of
m aterials, such as TiW [48], NiCr [40]. and titanium polycide [45] have been stu d ­
ied. These cathodes have been fabricated in th e form of conical tips [33, 39], pyra­
mids [34, 35, 36, 37], thin-film edge em itters [40. 48]. lateral tips [40. 41, 43, 44, 46],
inverted conical tips [38], deposited “supertips” (on standard field em itter tips) [47],
and wedges [42],
Thus, field emission has found use in field emission array (FEA) cathodes for ex­
perim ental klystrodes and travleing-wave tubes. Field emission is observed for both
metals (such as tungsten, molybdenum, and titanium ) and semiconductors (such as
silicon and gallium arsenide). These cathodes are a reliable, commercialized tech­
nology for field emission displays (FED's) [49]. However, reliable, commercialized,
high-current-density FEA ’s useful for microwave tubes have not yet been dem on­
strated. Of all th e field emission cathodes studied, the most promising appears to
be th e VECTL Mo-tip field emission array, developed by the NEC Corporation.
The use of new field-emitting materials such as HfC and ZrC is promising, but this
will require fu rth er development before they can be commercialized.
Research is currently still active concerning the development of FEA cathodes
for microwave applications. Many new m aterials are being researched. Among the
ones most recently studied include molybdenum silicide [25], hafnium carbide [26],
zirconium carbide [27], and carbon nitride [28]. Efforts are underway to produce
Si-tip F E A ’s on glass substrates [29] and protect m etal-tip FEA ’s from vacuum arc
damage [30]. T h e improvement of Si-tip arrays by reducing the effect of am bient
gases to achieve useful reliability and operating lifetim e is now being studied by the
MCNC group [31, 32].
2.1.3
W id e B andgap Sem iconductor C ath od es
The m aterials used to fabricate thermionic and field emission cathodes exhibit rela­
tively large work functions. Therefore the energy barrier for electrons attem p tin g to
escape out to vacuum is large. For example, both tungsten and m olybdenum both
12
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have work functions greater than 4 eVr. The use of these m etals as a thermionic or
field emission cathode requires high temperatures or high electric fields to achieve
reasonable emission density. The use of alkali metal oxide coatings on nickel (oxide
cathodes) results in a work function of about 1 eY\
For a semiconductor m aterials, the energy barrier for electron emission is de­
termined by the electron affinity at the surface. A certain class of cathodes known
as negative electron affinity (NEA) cathodes consist of a standard semiconductor
material (such as Si or GaAs) coated with cesium. T he cesium layer bends the
conduction band down in such a way that the external vacuum level lies below the
conduction band minimum energy deep in the bulk of the semiconductor. These
NEA cathodes have found lim ited commercial use.
There are certain wide bandgap semiconductor materials, such as diamond, sil­
icon carbide (SiC), boron nitride (BN), gallium nitride (GaN), alum inum nitride
(AIN), and aluminum gallium nitride (AlxG ai_xN). which have been studied as
cathode materials. Some of these, such as diamond. BN. and AIN. exhibit a nega­
tive electron affinity, without the use of cesium. The external vacuum energy level
lies below the conduction band minimum energyand the energy barrier for electrons
in the conduction band of th e semiconductor attem pting to escape out to vacuum
is negligible. AlxG ai_xN m aterials can also exhibit low or negative electron affinity
for larger x values. GaN and SiC [51] exhibit positive electron affinities on the order
of 3.5 eV and have also been studied as cathode materials. Cathodes made from
these materials, particularly those with very low or negative electron affinity have
the potential of being able to em it very high current densities at low extraction
voltages, while exhibiting robust chemical, structural, and therm al properties.
C arbon C athodes
In the late 1970’s, it was confirmed that diamond naturally exhibits a condition of
negative electron affinity on certain crystalline faces [52]. As a result, m any forms
of carbon have recently received a large amount of attention as an electron em itter.
A recent review of electron emission from diamond [53] classified diamond into
four categories, based on th e source of the film, as follows: (1) natural diamond,
(2) high-pressure high-tem perature diamond, (3) chemical vapor deposited (CVD)
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diam ond, find (4) shock-type nanodiamond (ND). There are numerous types of
CVD diamond, including crystalline diamond (c-D), amorphous diamond (a-D),
amorphous carbon (a-C), tetrahedral amorphous carbon (ta-C ), and diamond-like
carbon (DLC). All of these have been studied for use in cold cathodes.
Diamond has been studied as both a planar em itter m aterial and as a coating
on m etal or semiconductor field em itting tips. Much data has been collected, and
some general trends have been observed. Diamond with smaller crystallites and
higher defect density tends to em it more electrons [53]. G raphite inclusions tend to
enhance emission [53]. Exposure of a diamond surface to hydrogen tends to increase
emission; exposure to oxygen tends to reduce emission [53]. It has been postulated
th a t the electrical contact between the diamond film and the substrate is a crucial
factor in determining emission [53].
Some researchers have reported the incorporation of nitrogen [54] and phospho­
rus [55] into diamond films, and this seems to improve electron emission.
It is
unclear whether actual n-type doping is taking place, or if the inclusion of N or P
sim ply creates structural defects which lead to increased emission.
The best result so far obtained for a planar carbon em itter is a turn-on electric
field of 2 V /p m and a current density of 0.9 A/cm 2 at about 7 V /w n [56]. When
conventional metal and semiconductor field em itter tips are coated with carbon,
they tend to exhibit lower turn-on voltages. larger m axim um current densities,
b etter stability, longer operational life, and softer vacuum operation. Up to 100 n A
can be em itted per carbon-coated tip for
1 0 0 0
hours or more in vacuum pressures
of 10- 8 to 10- 6 torr [53]. Diamond-like carbon (DLC) is of particular interest, and
both DLC-coated Si tips [57] and DLC flat films [5S] have been used as cathodes in
field em itter triodes.
Thus, carbon-based cathodes seem promising, but the correct combination of
small crystallites, defect density, graphitic inclusions, hydrogen plasma treatm ent,
and im purities needed to create a carbon-based cathode with characteristics needed
for microwave performance has not yet been identified. Coating m etal or semicon­
ductor tips with somw form of carbon is also promising, but carbon-coated tips
supplying adequate performance have also not been dem onstrated. To summarize,
carbon-based cathodes are difficult to fabricate under repeatable conditions, there is
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a lack of understanding concerning the electron emission mechanisms from carbon,
and no optim al combination of design factors has been identified.
B oron N itr id e C ath od es
Cubic boron nitride (c-BN) has also received much attention as a possible cold cath­
ode m aterial. Ultraviolet photoemission spectroscopy (UPS) m easurem ents indicate
th at c-BN has a negative electron affinity at its surface [59]. S ubstantial electron
emission current densities (about 2 A /c m 2) have been measured from flat c-BN films
with an extraction field as low as 30 V / fim [60, 61]. It has been dem onstrated that
coating of m olybdenum tips with intrinsic BN [62] or coating Si tips w ith S-doped
BN [63] can significantly enhance the electron emission.
Recently, Ronning, et al. studied th e structural and electronic properties of
BN doped n-type with Si [64]. They found that the incorporation of the Si led
to the form ation of hexagonal BN (h-BN) instead of cubic BN (c-BN) for large Si
concentrations. The electron affinity of hexagonal BN is higher than th a t of cubic
BN.
Thus, cubic BN (c-BN) has also shown promise as both a tip coating m aterial
and as a flat film em itter. However, th e problems which prevent BN from being
widely used are sim ilar to those of diam ond - repeatable growth has not been
dem onstrated, emission is not well understood, and no optim al cathode structure
has been specified.
A lxG a i_ xN C a th o d es
In 1994, Benjamin et al. observed, through the use of ultraviolet photoemission
spectroscopy (UPS), that negative electron affinity exists on the surface of AIN
grown on SiC using molecular beam epitaxy [65]. This discovery im m ediately led
to speculation about the use of AIN for cold cathodes.
Nemanich et al. reported field emission measurements for AIN grown on SiC. and
observed a threshold field (the electric field at which 1 nA of emission is observed)
of 31 V//zm [6 6 , 67]. VVojak et al. and Zhirnov et al. reported field emission
m easurem ents for AIN grown on Si tips using reactive magnetron sputtering [6 8 , 69].
They found th a t the AIN coating resulted in a lower turn-on voltage and enhanced
15
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emission com pared to Si tips with no AIN coating.
Shaw et al. theoretically described a cold cathode containing graded AlxG ai_xN
m aterial in 1996 [70]. T he graded layers were either 50
were undoped.
A
or 400
A
thick and
Poisson's equation was solved to determ ine the position of the
conduction band and the electron concentration as functions of position. Schetzina
described [71] and patented [72] an electron em itter consisting of a graded layer of
n-type AlxGax_xN between a SiC substrate and a thick layer of AIN.
Underwood et al. reported a selective-area regrowth technique to fabricate GaN
pyram id structures in 1997 [73]. The pyramids were grown using a hexagonal dielec­
tric mask upon a layer of GaN which had been grown on sapphire using metalorganic
chemical vapor deposition (MOCVD). The pyramids were used as a field emission
array, and when the anode was positioned 0.5 mm from the array, a current of
0.8 fiA was extracted at a voltage of 2000 V. The pyramids had a base diam eter
of about 5 /im , a height of about 5 ^m, and a center-to-center spacing of 11 /im.
T he actual num ber of pyram ids involved in the emission was unknown, because the
growth was uneven across the sample, leaving some pyramids uncompleted with no
apex.
Forsythe et al. investigated the field emission properties of polycrystalline AIN
films in 1997 [74]. They studied both Ge-doped and undoped films grown with ion
beam assisted deposition (IBAD). They observed field emission from 10-nm-thick
undoped AIN films grown on n-type Si wafers using a 10-20 ^m -thick mica spacer
with a 2-mm -diam eter hole between the film and a C’u anode. They observed about
1
fiA of emission through a 2-mm-diameter hole at an anode voltage of 200 V.
Underwood et al. reported the fabrication and testing of vacuum diodes with
a cathode of GaN field em itting pyramids and an integrated air-bridge anode in
1997 [75]. W ith the anode suspended 2 fim above the pyramids, 0.15 fiA per tip was
m easured at an anode voltage of 570 V. They published these results in 1998 [76].
Christm an et al. and Sowers et al. fabricated and characterized AlxG ai_xN
cold cathodes in 1997 [77, 78]. One type of cathode utilized a layer of AIN grown
(using MOCVD) upon a SiC substrate. A second type of cathode utilized a layer
of Al;cGai_xN graded from x=0.05 to x=0.90 grown (using MOCVD) upon a SiC
substrate. It was found th at no current could be em itted unless the cathode struc16
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tures were chemically cleaned and exposed to a hydrogen plasma. Total emission
currents for a 5 x 5 array of 5 fim x 5 ^m emission holes were observed to range
from 10 to 100 nA.
Kozawa et al. reported the fabrication of GaN hexagonal pyram ids using metalorganic vapor phase epitaxy (MOVPE) in 199S [79]. The pyramids were about 10
fim in height, about 10 fim wide at the base, had a center-to-center spacing of 40
fim, and were found to have a tip radius less than 100 nm. When a 1-mm-diameter
Au ball-shaped collector was placed 1.5 fim from the array surface, 19 fiA could be
extracted with an anode voltage of 360 V.
Chung et al.
calculated the bulk states contribution to field emission from
GaN in 1998 [80]. They computed the field emission current from n-type GaN as a
function of the free electron concentration (from
1 0 15
to
1 0 19
cm-3 ) and the electron
affinity (from 1.8 to 2.4 eV).
Kang et al. studied how varying the thickness of magnetron sputtered AIN
on Mo field em itting tips affects electron emission in 199S [81].
They found a
consistent improvement in emission as the AIN thickness was increased up to 20
nm, but observed inconsistent results for film thicknesses above
2 0
nm.
Ward, et al. grew arrays of Si-doped GaN pyramids using m etallorganic vapor
phase epitaxy (MOVPE) [82]. The pyramids were 5 fim wide at the base, about 5
fim high, and were spaced 15 fim apart, and had a tip radius under 100 nm. They
studied the pyramids using field emission measurements and a variety of other
analysis techniques. Their results indicate that after a hydrogen plasm a treatm ent
a threshold field for electron emission was about T V / fim.
Thus, alum inum gallium nitride (AlxG ai_xN) seems to be a particularly promis­
ing m aterial for cold cathode development. This is because AlxG ai_xN enjoys four
key advantages. First, high-quality AIxG aj_xN material can be grown using a num­
ber of different techniques on a number of different substrates [83. 84]. Second, it
is possible to dope AlxG ai_xN material using Si as the n-type dopant [83, 84], and
this means th at an ohmic contact can be easily formed [85,
8 6
]. T hird, it is possible
to grow layers of AlxG ai_xN m aterial in which the Al fraction x is varied with po­
sition [77, 78]. Fourth, ultraviolet photoemission spectroscopy d a ta indicates that
AlxG ai_xN m aterial for which x > 0.75 exhibits zero or negative electron affinity
17
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at its surface [87,
8 8
].
Previous research has concentrated on several different ways of using A lxG ai_xN
m aterials to fabricate cold cathodes, but none were successful in achieving a cathode
capable of useful performance in a microwave triode. These approaches were as
follows:
• Using n-type GaN [73. 75. 76] is a reasonable approach, but it does not produce
an optim al cathode. The electron affinity of GaN is about 3.3 eV, which is
only 1 eV less than a conventional field em itting m etal, such as Mo.
• Using AIN in the form of either a flat film [74] or a tip coating [6 8 . 69] may
lead to a cathode with a NEA surface. However, AIN is an insulating m aterial,
and no proven m ethod of doping it has been dem onstrated. W ith no n-type
doping, there are no electrons in the conduction band to be em itted.
• Using a layer of undoped AlxGa!_xN in which the value of x changes from 0
to
1
over a distance of 50
A
or 400
A
[70] will result in a NEA surface, but
the lack of doping will result in few electrons for emission.
• Using a layer of n-type doped AlxG ai_xN in which the value of x changes from
0.00 to 0.90 [77, 78] will result in a cathode with a NEA surface. However,
the use of m aterial for which 0.75 < x < 0.90 is unwise, because Si does not
serve as an effective n-type dopant in such m aterial [89].
2.2
Microwave Triodes
Vacuum triodes date back to the beginning of this century, and they have gone
through many changes and improvements. They have been replaced by solid state
devices in many applications, but they continue to find use in certain types of
m ilitary, commercial, and scientific systems.
The area of research known as “vacuum microelectronics” got its s ta rt around
1960, when Shoulders proposed the development of microm iniature vacuum tri­
odes [90]. At th at tim e. Shoulders’ motivation for developing these devices was to
create ultra-high-speed digital circuits, and he believed that field emission cath­
odes would be the best choice for the electron source. Forty years of research have
IS
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resulted in a m ultitude of vertical and lateral geometries and various types of cath­
odes, grids, and anodes. During the 1970’s the idea of using m icrominiature triodes
for microwave service was developed by a research group at the Stanford Research
In stitu te (SRI).
Previous research efforts on microwave triodes with various types of cathodes
are summarized and discussed below.
2 .2 .1
M icrow ave T riodes w ith T herm ionic C ath od es
Electrical engineers realized in the 1920’s that the use of higher-frequency elec­
trom agnetic waves would lead to highly directive beams th at could be used for
transm itting large am ounts of information (e.g.. television) and for radar [91]. As
of late 1927, commercial vacuum tubes could not oscillate above 97 MHz and even
high-performance experim ental tubes could not exceed 286 MHz [91]. Japanese en­
gineers reported in 1930 how they had to use seven tubes in parallel to communicate
at a frequency of 600 MHz [91].
In 1933, Thom pson an d Rose developed new. m iniaturized vacuum tubes which
were capable of higher frequency operation [92]. The tu b e enclosures were about
the the size of an acorn, and the interelectrode spacings were reduced to just a
few thousandths of an inch. Using these tubes, they were able to construct RF
receivers operational at 300 MHz and 400 MHz. They built a feedback oscillator
which operated at
1
GHz with a plate voltage of 115 V and a plate current of 3
mA. The RCA Radiotron Company later manufactured and m arketed these “acorn”
tubes.
Fay and Samuel reported experim ental tubes which could generate
MHz and produce a m easurable signal at
1
6
VV at 500
GHz in 1935 [91]. Samuel later developed
larger vacuum tubes capable of producing 40 VV at 400 MHz and smaller tubes
capable of generating 1 VV at 1.7 GHz in 1937 [93]. He achieved this improvement
by bringing two leads from each electrode out from either end of the tube, thus
reducing the inductance of the leads. These tubes were later commercialized and
known as ‘‘door knob” tubes.
During WVVII, microwave electronics progressed rapidly, due to the development
of high-frequency com m unication and radar systems. The technology for fabricating
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th e cathodes, grids, and anodes of conventional transm itting and receiving tubes
was improved [3], and the lighthouse tube was developed.
Lighthouse tubes utilized three technological advances [3]: (1) the use of disk
seals to reduce lead inductance, (2 ) miniaturization and the use of a parallel-plane
structure to minimize interelectrode capacitances, and (3) a tube package which fa­
cilitated the tu b e ’s insertion into microwave circuits with waveguides and cavities.
Much im portant research on such tubes was done at the MIT Radiation Labora­
tory [94]. One type of M IT triode mounted in an oscillator achieved an output
power of
1
W, a gain of 20 dB. a 6 -MHz bandwidth. 9 dB NF. and an efficiency of
25% at 3 GHz. Another tube operated up to 10 GHz with a 10 dB gain. 18 dB NF.
and an efficiency of 1 %.
Improvements of triode tubes during the war had a significant im pact [144].
Before VVWII, no single triode could produce more than 100 VV at 350 MHz; after
VVWII, a single triode could produce 100 VV at 700 MHz. Before VVWII, even the
best triode could not produce more than 1 or 2 VV CVV above 1 GHz; after VVWII.
a single triode could supply at least I VV CVV at frequencies greater than 4 GHz.
T he 2C40 planar triode could provide 50 mW of CVV power at 3.37 GHz [3]. The
2C43 triode could provide a pulse power of 750 VV at 3.37 GHz [3]. Higher-power
planar triodes included the 3C22, which could provide 25 W CW at 1.4 GHz [3],
and the 2C38, which could provide 10 W CVV at 2.5 GHz [3].
During the late 1940’s, Bell Laboratories developed a new high-frequency triode
for use in a 4-GHz microwave relay system [4]. The tube utilized grid wires just
0.0003 inch in diam eter wound 1000 turns per inch, a cathode-to-grid spacing of
0.0006 inch, and a plate-to-grid spacing of 0.012 inch. These dimensions resulted
in an amplification factor of 350. Such a tube typically operated with 250 V on the
anode and -0.3 V on the grid with a DC anode current of about 25 mA (resulting
in a DC plate dissipation of 6.25 VV). Typical performance for the tube was a
transconductance of 50 mS and a plate resistance of 7 kfi. The 1553 triode was
later manufactured and m arketed by Western Electric as the 416A triode. The
416A triode was used in 4-GHz grounded-grid amplifier circuits and provided 9 dB
power gain (for Pout less than 20 mW ) and a bandwidth of 100 MHz [96]. The gain
of the tube decreased with increasing output power, and the gain was about 5 dB
20
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for an output power of 0.5 W.
During the 1950’s, lighthouse tubes such as the 2C39 and the 2C43 became
quite popular, and British tubes such as m iniature CV90 and the ACT23 (capable
of delivering 200 VV at
1
GHz) were in use [97]. By the early 1960’s, lighthouse
tubes (such as the 2C49) were improved to the point th at the tube could supply
several watts of o utput power up to 3 GHz [98].
2C39’s were used in 1965 to
transm it photographic d a ta from lunar probes taking pictures of the moon. In this
application, the 2C39’s supplied 60 VV of output power at 960 MHz [98]. Larger diskseal tubes could provide 5-10 kVV of output power in the 700-900 MHz range [98].
Improved higher-frequency triodes (such as the GL-7391) could provide 0.1 VV CVV
power at
6
GHz [98].
In the mid-1960’s, research and development of power triode tubes continued to
extend their usefulness to higher powers and higher frequencies. The usefulness of
such triodes was increased by using oxide cathodes in titanium -ceram ic structures,
which increased the m axim um allowable em itted current density to 2 A /c m 2 [98].
W ith this increased current density, which was ten times th at of the conventional
2C39, an order of m agnitude improvement in transconductance and power output
was achieved. At General Electric, experim ental triodes. designated L-65 and L-64.
achieved transconductances of 0.3 S and 0.7 S [99. 98]. The L-65 typically operated
with 2100 V on the anode and about 0.7 A anode current (resulting in a DC anode
power of 1470 VV). The L-65 could output 1000 VV at 1.3 GHz with an input power
of 50 VV and a plate efficiency of 67% (note th at it took about 20 VV of power to
heat the cathode, so th e overall efficiency was actually 65%). GE also developed
power triodes for use above 10 GHz. Two of these tubes could operate in the range
of 11.4 to 12.3 GHz, providing a few watts of output power with an efficiency of
4-6%. A third tube with a m iniaturized cavity could provide 0.1 VV at IS GHz.
Microminiature triode structures with thermionic cathodes were first fabricated
by Geppert in 1972 [1 0 0 ]. In this work, the cathodes consisted of flat films of tita ­
nium and tungsten for the grids and anodes and triple carbonate for th e cathodes.
Lynn, et al. continued this work, and in 1985 reported on thermionic integrated
circuits constructed with a cathode made of a standard oxide cathode coating on
m etal [1 0 1 ].
21
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As of th e mid-1980’s, state-of-the-art commercialized planar triodes operating at
4 GHz could provide an average power of
1
VV or a peak pulse power of 10 kVV [102].
For such tubes, the efficiency was about 60% at
1
GHz and just a few percent at 5
GHz.
E astm an compared the perform ance of vacuum and sem iconductor devices which
both have th e same physical dimensions in 1989 [103]. He analyzed a space-chargelim ited m icrom iniature microwave triode array with an anode-to-cathode spacing
of 1.6 fim, using basic vacuum tu b e theory. He concluded th at if such a device can
operate with 64 A /cm 2 current density, then the unity-current-gain frequency of
th e device would be 11.5 GHz. He concluded that such arrays would be capable
of low-current, high-voltage, high-frequency performance if current densities on the
order of 100’s-1000’s A /cm 2 could be achieved.
New types of m icrom iniature triode structures were constructed with thermionic
cathodes in the 1990’s. One type used a thin filament of polycrystalline silicon [104].
A nother type used a suspended filament of pure tungsten [105]. Recent work in
th e Republic of Belarus has developed thermionic vacuum microelectronics by way
of thin emissive coatings on anodic alum ina [106. 107]. However, none of these
structures exhibited microwave performance.
G ridded vacuum tubes continue to find use for TV and radio broadcasting,
shortwave radio, m ilitary radio, commercial radio, am ateur radio, industrial heat­
ing, scientific instrum entation, medical applications, radar, and electronic warfare.
Planar triodes such as the 2G'39, 2C'43. 5675. and 5893 still find use in am ateur
radio. T he 3CX100A5 triode and the ceramic-sealed 7289 triode are particularly
popular, and a triode such as the Eimac YU-129 can provide more th an 200 VV CVV
o u tp u t at 1.296 GHz [108].
Thus, conventional microwave triodes are limited in their high-frequency per­
formance by a number of factors. The most significant factor, obviously, is the
transconductance, which m ust be maximized to achieve the best possible perfor­
mance. O ther factors include im pedance limitations and transit tim e lim itations.
T he im pedance problems are the result of the tube’s interelectrode capacitances
and th e leads’ inductances. Specifically, when these are too large, they tend to load
th e circuit at high frequencies, causing a mism atch with the source and load and
22
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dim inishing gain. Transit tim e refers to the tim e it takes for an electron to move
from the cathode to the anode. If this tim e is long in comparison to the period of
the operating frequency, the triode will not operate properly and gain and efficiency
will be drastically reduced. In spite of all of the above limitations, certain actions
can be taken to reduce their effect. The transconductance of a tube can be in­
creased by increasing the current density em itted by the thermionic cathode. Tube
interelectrode capacitances and lead inductances vary proportionally with th e phys­
ical dimensions of the tube and its leads, so microminiaturization can help reduce
this effect. Finally, the transit time of a tube is decreased by reducing the spacing
between the electrodes, so m icrominiaturization should also help to alleviate this
problem.
2.2 .2
M icrowave Triodes w ith Field E m ission C a th o d es
Efforts to replace low-current-density thermionic cathodes in microwave tubes with
high-current-density field emission cathodes began soon after VVWII. under the
direction of Dyke. His research group at the Linfield Research Institute continued
this work into the early 1960’s, but they were never able to construct a useful
microwave tube [15]. It was impossible at that time to construct a field em itter
array (FEA) with the necessary geometry and physical properties for adequate
performance.
Buck and Shoulders first began discussing the possibility of microm iniature vac­
uum tubes in 1958 [109] and Feynman delivered his famous lecture on m icrom inia­
turization in 1959 [110]. Shoulders proposed m icrominiature triode devices in an
extensive paper published in Advances in Computers in 1961 [90]. He specified th at
these devices would have a vacuum gap micrometers in size and a field emission
cathode. He envisioned both vertical configurations and lateral configurations. Af­
ter Spindt (at SRI) invented the first thin-film field emission cathode in 1968 [16].
it became feasible to build microwave triodes with field emission cathodes. T he SRI
group continued to work on the applications of field emission into the 1970’s, and
proposed th at their field em itter arrays be used in either conventional microwave
tubes or in microminiature vacuum tube structures [17].
At the 1986 International Electron Device Meeting (IEDM), Gray, et al. pre23
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sented experimental d ata for a lateral vacuum m icrotriode fabricated with silicon
field em itting tips [34]. An oscillograph showed the 4-V peak-to-peak modulation
of anode voltage corresponding to a 2-V peak-to-peak gate voltage signal, at very
low frequency.
Kosmahl obtained a patent in 198S entitled “Microwave Integrated Distributed
Amplifier with Field Emission Triodes” [ ill] . He published a paper entitled "A
W ide-Bandwidth High-Gain Small-Size Distributed Amplifier with Field-Emission
Triodes (FETRO D E’s) for the 10 to 300 GHz Frequency Range” in 1989 [112]. Kos­
mahl calculated transconductance, interelectrode capacitances, transit tim e effects,
and efficiency for the “FETR O D E’s.”
Anderson analyzed the gain-bandwidth product of microm iniature microwave
triodes with field emission cathodes [113]. He assumed a device in which the anode
is ju st 0.75 fim above the gate plane. His calculations indicate th at with such a
geometry, the Mo-tip FEA ’s available at that tim e could achieve no better than 2.5
GHz cutoff frequency, and that the achievement of cutoff frequencies on the order of
250 GHz could only be achieved with a tip material with a work function of 3.0 eV
or less and applied voltages of the order of 240 V. He concludes th at microminiature
field emission triodes cannot achieve the same level of performance as solid state
devices for similar dimensions.
Lally, et al. at Teledyne MEC proposed a design for an X-band tuned amplifier
in the form of a microwave triode with a FEA cathode in 1989 [114]. They predicted
th at FEA amplifiers operating at 10 GHz with an output power of 10 VV, a power
gain of 10 dB, and a bandwidth of 50 MHz are feasible.
Neidert, et al. constructed a triode using a field emission array cathode and
reported DC and AC experim ental results in 1991 [115]. The array used exhibited a
total transconductance of 38 ftS, and the cutoff frequency of the triode amplifier was
400 kHz. That same year, Holland, et al. at SRI International published the results
of sim ilar experiments involving triodes with field emission cathodes [33].
The
triodes were also constructed with Mo-tip FEA cathodes. The first experim ent used
a stainless steel tube anode located about 5 mm from the gate. A transconductance
of 5 fiS per tip and a cutoff frequency of 0.S MHz were measured. In a second
experim ent, an anode was m ounted to the TO-5 header and was 1.5 mm from the
24
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gate. This experim ent yielded a cutoff frequency of 0.14 MHz.
Asano published simulations of the effects of geometry on the electrical char­
acteristics of a microm iniature field emission triode in 1991 [116]. Specifically, he
exam ined the effects of a misalignment of the tip with the gate opening a n d /o r a
variation in the gate aperture width. Transconductance and the interelectrode ca­
pacitances were com puted. Two types of structures (one with the grid close to the
tip, one with the gate further away) were compared by examining their transcon­
ductance, capacitance, and anode resistance values. He concluded th at use of a low
work function m aterial is necessary and th at the spacing between the gate an d the
em itter should be minimized.
By 1992, SRI had developed a new type of low-capacitance triode stru c tu re in
which th e overlap of the gate metallization and the cathode m etallization were m in­
imized [117]. They published experim ental d a ta for a triode structure in 1993 [1 1 S,
119]. They dem onstrated gain for the structure with oscilloscope traces up to 1
GHz and measured the S-parameters in the frequency range 0.5-1.0 GHz.
Akinwande, et al. described lateral vacuum microelectronic triodes utilizing
thin-film-edge TiVV em itters at the 1992 IEDM [48]. These edges achieved up to 10
/zA//zm current emission density. Using their experim ental results, they developed
an equivalent circuit model to predict the microwave performance of a triode am ­
plifier. These com putations of maximum available gain indicated cutoff frequencies
in the range of
1
to 3 GHz.
A 1993 paper by Calame, et al. presented calculations for microwave amplifiers
constructed with arrays of microminiature microwave triodes with field emission
cathodes [120]. These authors developed a complete circuit model for th e vac­
uum m icrotriode array and they designed resonant and nonresonant networks for
im pedance m atching the array. They com puted power gain and bandw idth a t 10
GHz. O utput power and power added efficiency were presented for m icroelectronic
triodes which were designed with an additional acceleration electrode added above
the extraction electrode. Finally, they performed a therm al analysis for triodes in
which beryllium oxide and/or diamond are used for heat sinking. They conclude
th at microwave amplifiers capable of delivering 50-100 W output power at 20 GHz
with octave bandw idth and 10 dB gain are possible if FEA cathodes can be ob25
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tained which em it 10-20 /zA per tip and exhibit a transconductance of 10 /zS per
tip. Such performance could be obtained from a FEA cathode which has tips with
50-A radius, 2000-A gate aperture, and tip m aterial w ith a work function of 3.5 eV.
Zaidm an presented a set of simulations for a m icrom iniature microwave triode
with a field emission cathode in 1993 [121]. The sim ulations were done using a ver­
sion of MAGIC, a two-and-a-half dimensional particle-in-cell (PIC ) code. To model
the field emission, a special field emission module was created and inserted into the
MAGIC code. Collected current, em itted current, field enhancement factor, and
transconductance were all computed. He derived a com plete set of DC TV char­
acteristics with an appropriate load line for amplifier operation. Power dissipation
in th e anode (in w atts per square meter) versus tip radius was computed. For an
anode-to-gate spacing of
radius of
1 0 0
1 0 0
/zm. he concluded that using a field emission tip with
A provides a transconductance of 7.75 fj.S. which, when combined with
gate capacitance of 0.193 fF, provides a cutoff frequency of 6.39 GHz.
Gammie, et al. at Raytheon Company fabricated m etal-tip FEA cathodes and
tested them in 1993 [19]. A 5000-tip array successfully em itted a total current of
70 mA and the triode formed by bring a molybdenum tube close to the cathode
exhibited a transconductance of nearly 10 mS. They used a vector network analyzer
to m easure the S-param eters of the triode up to 1.00 GHz and used these values to
com pute capacitances.
Nicolaescu investigated the modeling of cone- and wedge-shaped field em itter
vacuum microdiodes and microtriodes in 1994 [122]. He used software designed to
model field emission in order to to com pute transconductance, plate resistance, gain,
capacitance, and cutoff frequency as functions of the geometrical factors, the work
function, and the electrode potentials. He found th at cutoff frequency is increased
by shrinking the gate aperture, reducing the grid height, reducing the em itter work
function, increasing the em itter height, and decreasing the tip-to-anode spacing.
Kopka and Erm ert modeled electron trajectories in vacuum to characterize field
emission vacuum m icrotriodes [123]. These devices had an anode-to-tip distance of
2 /zm and a gate-to-tip distance of 0.6 /zm. They conclude th a t the work function
of a 10-nm-radius tip m ust be reduced to 3.5 eV to achieve a cutoff frequency of 1
GHz, down to 3 eV to achieve 10 GHz, and down to 2.5 eV to achieve 100 GHz.
26
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Palm er, et al. performed RF measurements on a microwave triode constructed
with a Si FEA cathode and an anode [124]. T he DC bias of the triode was set at 120
volts on th e gate and 1.45 mA total anode current. They presented an oscilloscope
trace showing th at at I GHz a swing of 2 V on the gate produced a swing of 75 fi.\
at the anode.
Phillips, et al. constructed microwave triode amplifiers using Mo thin-film fieldemission cathodes in 1995 [125]. The DC I-V characteristics of the cathode were
m easured, m atching circuits were designed and utilized for the RF measurem ents,
and th e S-param eters of the amplifier were measured. An amplifier with 250 Mo
tips, a total DC current of 4.8 mA, and a to tal transconductance of 840 fiS achieved
an intrinsic power gain of 7 dB at 1.1 GHz.
These researchers concluded from
their study th a t fully integrated, realistic microstrip matching circuits would have
resulted in an amplifier displaying a gain of
6
dB at 1.5 GHz. They predict future
microwave amplifiers utilizing field emission arrays which provide a few dB of gain
up to 20 GHz.
Nicolaescu, et al. modeled a m icrominiature microwave triode with a field em it­
ting tip coated with porous silicon in 1996 [126]. For a structure with an anode-to-tip
spacing of ju st 0.4 fim and a tip radius of 100 nm. they predict cutoff frequencies
in the range of 1-10 GHz for gate voltages between 78 and 98 V'.
Qin, et al. presented a high-frequency equivalent circuit model for a triode with
field emission cathode at the 1996 International Vacuum Microelectronics Confer­
ence (IVMC) [127]. Expressions were derived for the triode’s interelectrode capac­
itances, series resistances, transconductance, Y param eters, high frequency short
circuit current gain, maximum stable power gain, and cutoff frequency. Calcula­
tions are presented for the power gain of the triode versus operating frequency.
They com pared their calculations to the experim ental values reported by Neidert.
et al. They indicated that further modeling would be needed for wire inductance
and other parasitic effects.
Yokoo and Ishizuka reviewed the radio frequency (RF) applications of FE A ’s at
the 1996 IVMC [128]. They discussed FE cathodes, vacuum microelectronic triodes,
linear beam devices, pre-bunched beam devices, high energy beam applications, and
beam diagnostics. They predict a cutoff frequency for a field emission microminia27
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tu re microwave triode of roughly 20 GHz, based on a transconductance of 10 fiS
and a capacitance of 0.1 fF.
Sazonov and Rum yantsev described a means of extending the performance of
vacuum microtriodes to higher frequencies by introducing three additional elec­
trodes at the 1997 IVMC [129], The inclusion of the ex tra electrodes effectively
increases the transconductance of the device and reduces the parasitic capacitances,
at the expense of increased fabrication complexity.
Nam, et al. constructed a field emission triode using a Si-tip FEA and performed
DC and AC m easurem ents in 1997 [130], The measurements were performed using
cables fed through the wall of a vacuum chamber, and the large parasitic capac­
itances of this m easurem ent set-up resulted in a cut-off frequency of just
6
kHz.
instead of the expected value of 43 kHz. They published these results in 1998 [131].
Thus, an extensive am ount of research has been conducted on triodes with field
emission cathodes. Some simulations of these devices predict lower cutoff frequen­
cies of 6.39 GHz [121], 10 GHz [126]. and 20 GHz [128]. O ther researchers predict
th a t these devices will have cutoff frequencies as high as 100 GHz [123] or even 250
GHz [113]. These very high cutoff frequencies assume th at field emission tips are
available which have a very low work function m aterial and a very small tip radius,
both of which are presently unavailable. Some of the m easured values of cutoff
frequency for these field emission triodes are as low as
6
kHz [130], 140 kHz [33].
400 kHz [115], and 800 kHz [33]. In the early 1990s, m odulation of current was
dem onstrated up to
1
GHz [117, 124] and calculations performed on the extrapola­
tion of experim ental d ata taken from certain groups [48. 125] indicate operation in
the range of 1-2 GHz. More recently, current modulation and small but detectable
amplification were dem onstrated at 10 GHz for FEA's installed in a klystrode test
station [132]. Hence, field emission cathodes have found successful use in field emis­
sion displays, but have not yet shown a performance useful for microwave tubes.
2 .2 .3
M icrow ave T riodes w ith W ide B andgap Sem iconduc­
tor C a th o d es
Only very recently have wide bandgap semiconductor m aterials been used to con­
stru ct triode structures. AIN, graded AlxGai_xN. and diamond-like carbon (DLC)
28
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have all been utilized.
Christm an, et al, and Sowers, et al fabricated and characterized AlxGax_xN
cold cathodes in 1997 [77, 78]. O ne type of cathode utilized a layer of AIN grown
(using MOCVD) upon a SiC substrate. A second type of cathode utilized a layer
of AlxG ai_xN graded from x=0.05 to x=0.90 grown (using MOCVD) upon a SiC
substrate.
In order to draw out th e electrons, an extraction grid stru ctu re was
created with 0.25-/tm-thick Al electrodes on top of I-/zm-thick Si0
2
- G rid voltages
were varied from 20 to 110 V during the measurements. A triode stru ctu re was
achieved by way of a rem ote probe biased up to 650 V. It was found th a t no current
could be em itted unless the cathode structures were chemically cleaned and exposed
to a hydrogen plasma. Total emission currents for a 5 x 5 array of 5 fim x 5 /im
emission holes ranged from 10 to 100 n.A.
Ko, et al. fabricated a gated FEA cathode using thin films of DLC and reported
their results in 1997 [58]. A total emission current of 0.5 fiA was observed when
100 V was applied to the gate. Lee, et al. fabricated a triode using a FEA cathode
with DLC-coated Si tips [57]. Using a total of 3600 tips, a total anode current of
about 15 fiA was measured with a gate voltage of about 95 V.
The few results for large-sized, low-frequency triodes m ade w ith wide bandgap
semiconductor cathodes are prelim inary and inconclusive. Obviously, further work
needs to be done. So far. no one has constructed a m icrom iniaturized triode using
a wide bandgap semiconductor cathode. The emphasis of this dissertation is to
identify a promising way of developing microminiature microwave triodes with wide
bandgap semiconductor cathodes.
29
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C hapter 3
AlxG ai_ xN C ath od es
3.1
Cathode Structures
A cathode is an electrode in an electron tube through which a stream of electrons
enters the interelectrode space. A "generic cathode" is shown in Figure 3.1. As
indicated in the figure, in order for electrons to be em itted into vacuum, they m ust
move from ground, through an electrical contact, through the cathode structure,
and exit the cathode surface.
As explained in Chapter 2, AlxG al_xN has a number of physical properties th at
make it suitable for the development of a cathode. The focus of the research in this
dissertation is on AlxG ai_xN cathodes which will provide suitable performance for
a microminiature microwave triode.
T he fabrication of a useful cathode imposes certain requirements. One of these
©
©
©
t t t
emitted
electrons
cathode
back
contact
ground
Figure 3.1: A generic cathode.
30
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©
©
t
t
ohmic contact
AIGaN graded layer
GaN
AIN
SiC
Figure 3.2: AlxG ai_xN cathode structure.
requirem ents is the ability to grow a high-quality film upon an available substrate.
A lxG ai_xN can be grown on a num ber of different substrates (such as sapphire
an d SiC) by a number of different techniques (such MOCVD and MBE) [S3, 84].
A nother requirement is the ability to perform n-type doping, in order to supply an
adequate concentration of free electrons. Silicon can be used as an n-type dopant
in AlxG ai_xN [83, 84]. A final requirem ent is the formation of an ohmic contact
somewhere on the cathode structure. Because GaN material can be n-type doped,
forming low-resistivity ohmic contacts is not difficult [85.
8 6
]. It is shown in this
dissertation that the ability to spatially grade the Al fraction of AlxG ai_ xN. which
has also been dem onstrated [77. 78], can also be used to construct a useful cathode.
A cathode structure taking the above considerations into account and utilizing
A lxG a!_xN m aterial is shown in Figure 3.2. The n-type. compositionally-graded
A lxG a!_xN is grown on top of a layer of n-type GaN. To obtain high-quality growth,
a thin buffer layer of AIN is grown on th e SiC surface before the GaN and AIxG ai_xN
layers are grown [133]. An ohmic contact is formed on the n-type GaN [85,
8 6
].
Specifications for the cathode structure include the thickness of th e GaN layer,
th e thickness of the AlxG ai_xN layer, the spatial grading of the Al fraction x in the
A lxG at _xN material, and the profile of n-type Si dopant in the GaN and AlxG a!_xN
layers.
The thickness of the AlxG ai_xN represents a tradeoff between film grow th lim ita­
tions and cathode performance. The lattice mismatch between AlxG a j_ xN and GaN
31
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for large x values (greater than 0.5) is significant for thick AlxG ai_xN layers [133].
Some researchers have reported cracking in th e form of grooves in AlxG aj_xN grown
on GaN [133]. T he doping profile of the GaN and AlxG ai_xN layers also represents
a tradeoff between film growth limitations and cathode performance. T he GaN and
AlxG ai_xN layers should be doped with Si to as high a concentration as possible to
provide a large free electron concentration. Unfortunately, studies indicate th a t an
excessive Si concentration in such materials m ay also lead to cracking [134].
T he grading profile of the Al fraction (x) in the AlxG ai_xN is based upon ex­
perim ental d ata which indicates how electron affinity varies with x. The electron
affinity describes the energy difference between the external vacuum level and the
conduction band of the semiconductor at th e surface. Hence, for adequate emis­
sion, the value of the electron affinity should be minimized. The electron affinity of
AlxG ai_xN can be determ ined experim entally by way of ultraviolet photoemission
spectroscopy (UPS). Benjamin et al. [8 8 ] and Nemanich et al. [6 6 . 87] used UPS to
determ ine the electron affinity ( \) of GaN. Alo.1 3 Gao.s7 N, Alo.5 5 Gao.4 5 N. and AIN.
They found th a t \ = 3.3 eV for x=0.00, \ = 2.9 eV for x=0.13. and \ = 1.0 eV
for x=0.55. A dditional d ata indicates th at AIN (x = 1.00) has a negative electron
affinity, but they could not specify a numerical value. When the first th ree data
points are extrapolated to higher x values, it is found th a t a zero electron affinity is
indicated for x somewhere around 0.75. as shown in Figure 3.3. Since zero electron
affinity corresponds to zero energy barrier for electrons at the surface residing in the
conduction band, it is desirable to have a value of x at the em itting surface equal
to or greater th an 0.75 if possible.
3.2
Simulation of Cathode Structures
To sim ulate the cathode structure, the energy barrier for electrons attem p tin g to
escape to vacuum is determ ined and the current density em itted by the cathode is
com puted for each operating tem perature. In order to perform the sim ulation in one
dimension, the cathode structure shown in Figure 3.2 is simplified to the cathode
stru ctu re shown in Figure 3.4. The variable y is the vertical position in the cathode
stru ctu re, and y = 0 corresponds to the GaN-AlxG ai_xN interface. Only a small
error is introduced by reducing the structure shown in Figure 3.2 to th a t shown
32
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0
0.2
0.4
0.6
0.8
Al fraction x
1
Figure 3.3: Extrapolation of d a ta for the electron affinity of AlxG a!_xN as a function
of Al fraction x.
in Figure 3.4, because the heavily-doped n-type GaN layer and the ohmic contact
both exhibit very low series resistance and ordinarily no appreciable potential drop
would appear across them .
First, the charge neutrality condition is imposed at the back of the n-type GaN
layer. This consists of setting the free electron concentration equal to the ionized
donor concentration at the tem perature of interest and determ ining the location of
the Fermi level [135]. This determ ination can be performed through the use of a
simple bisection numerical routine. This constant Fermi level is utilized throughout
the rem aining calculations. Semiconductor device theory indicates th at the move­
m ent of current through a semiconductor structure corresponds to the change of the
Fermi level with position [135], but in this case, only a small error is introduced by
keeping the Fermi level constant throughout the cathode structure. This is because
the current densities typically em itted by the cathode (1000 A /cm 2 or less) are just
a m inute fraction of the m axim um diffusion and drift currents (which are on the
order of 1014 A /cm 2) in the cathode structure. Hence, the movement of the small
current densities typically associated with the cathode’s emission into vacuum are
merely a slight perturbation of the semiconductor's therm al equilibrium condition.
The simulation is performed by solving Poisson’s equation in the GaN and
AlxG ai_xN layers to obtain the potential distribution.
This potential distribu-
33
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vacuum
AlGaN graded layer
GaN
ohmic contact
Figure 3.4: Simplified cathode structure assumed for one-dimensional sim ulation.
tion provides the location of the conduction band minimum energy, which in turn
provides the emission barrier. The solution of Poisson’s equation is accomplished
with a fourth-order Runge-Kutta numerical procedure. The em itted current den­
sity is estim ated using the theory of metal-semiconductor junctions, specifically,
therm ionic emission theory. After the barrier-lim ited current density is com puted,
the gradient of the electron concentration at the cathode-vacuum interface is com­
puted. From these calculations, a determ ination can be made as to w hether the
emission is barrier-limited or diffusion-limited.
3.3
Solution of Poisson’s Equation
3.3.1
Form ulation o f P o isso n ’s E quation
Electromagnetic theory [136] asserts th at the electric flux density D(y) is related to
charge density p(y) by the equation V • D(y) — p{y). Electric flux density is related
to electric field £(y) by the constitutive relation D(y) = t{y)£{y), where t{y) is the
perm ittivity of the medium. Hence, in one dimension,
V • D(y) = V • (e(y)£(y)) = ^ S ( y ) +
= p(y),
(3.1)
where y indicates position. The electric field is related to scalar electric potential
*p (y) in one dimension by
£(*) =
(3.2)
Combining the above equations gives
dV(y) =
dy2
p(y)
u{y)
i
«»(y)
34
d<p{ y ) de s{y)
dy
dy
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3.3)
Thus, the electric potential in th e cathode depends upon the charge density and
the perm ittivity.
In order to form ulate and solve Poisson’s equation, the charge density m ust be
expressed as a function of position. In semiconductor m aterials, doping leads to a
situation in which electrons, holes, ionized acceptor atom s, and ionized donor atom s
may all be present simultaneously. Hence, the charge density in a sem iconductor is
given by [135]
p(y) = q{p{y) - n ( y) + -VdO/) - -Y l(y)).
(3.4)
where q is the electron charge. p(y) is the hole concentration. n(y) is the electron
concentration, N p i y ) is the concentration of ionized donors, and .V j(y) is the con­
centration of ionized acceptors.
Semiconductor theory asserts th at the concentration of electrons n(y) at a given
point in a sem iconductor is [135]
/ \
at t \
( E c {y) — E F \
I,
n(y) = Nc ( y) ex p i ---------—
(3.o)
where Nc( y) is th e effective density of states in the conduction band, Ec{y) is the
conduction band m inim um energy. E F is the Fermi level, k is Boltzmann's constant,
and T is tem perature. Similarly, th e concentration of holes p{y) at a given point in
a semiconductor is [135]
.
p{y) = N v {y)exp
,3.6,
where Nv{y) is th e effective density of states in the valence band and Ey{y) is the
valence band m axim um energy.
The values of Nc[y) and Nv ( y ) are determ ined from the effective mass of elec­
trons in the conduction band and the effective mass of holes in the valence band.
The effective density of states in the conduction band is given by [135]
MC.
*»<») = 2
(3.7)
where m e(y) is th e density of states effective mass for electrons, h is Planck’s con­
stan t, and Me is the num ber of equivalent m inim a in the conduction band. Note
th a t the above equation implies th a t the value of Nc{y) m ust be recom puted for
35
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each tem perature of interest. The only experimentally determ ined value for the
effective mass of electrons in AlxG ai_xN m aterials reported in the literature is that
for GaN, which has been reported as 0.19mo [137]. The effective mass for all values
of A1 fraction x was assumed to be 0.19mo. For materials w ith a direct energy band
structure, such as AlxG ai_xN materials, the number of equivalent m inim a in the
conduction band Me is 1 [138]. Thus, it is found that .Vc=2.084 x 1018 cm -3 .
The AlxGax_xN is doped n-type with Si. and it is assumed th at no acceptor
atom s are present. Semiconductor theory asserts that the donor atoms will ionize
according to [135]
M£{y) = X D(y)
1+2exp(* ^ > k > )
-i
(3-8)
where Np^ y ) is the concentration of ionized donor atoms. No( y ) is the concentration
of donor atom s, and Eo( y) is the energy level of the donor atom .
The donor ionization energy, E cd = Ec — E d - for Si in AlxG ai_xN. as a func­
tion of A1 fraction x, is a complex issue. No precise values for the ionization energy
have been calculated or measured. Electrical measurements of Si-doped AlxG ai_xN
films [84] (which indicate that Si forms a relatively shallow n-type dopant in GaN)
and quantum molecular dynamics calculations [89] and other available informa­
tion [134] allow for a rough approximation of the energies. Information compiled
by Bremser, et al. [84] indicate th at Si is a relatively shallow donor in GaN (with
an ionization energy on the order of 15-30 meV) and that the energy rises as the Al
fraction x increases. Boguslawski and Bernholc's quantum molecular dynam ics cal­
culations [89] indicate that Si is a shallow donor for x < 0.6 and th at the ionization
energy of Si DX centers in AlxGa!_xN rises from 0.0 eV for x=0.60 to about 0.5 eV
at x=0.75 to about 1.5 eV for x=1.00. Since more ionization results in a larger free
electron concentration, a lower ionization energy is desired. These two possibilities
are illustrated in Figure 3.5. with the lower curve being the ubest case” (less energy
needed to ionize) and the upper curve being the "worst case" (more energy needed
to ionize).
At room tem perature, the energy bandgap (Ec — Ev) of GaN is 3.45 eV and
the energy bandgap of AIN is 6.2 eV [133]. It has been experim entally confirmed
th a t the energy bandgap of AlxG ai_xN varies approximately linearly with the value
36
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0.6
12 0.4
“
03
-
0.1
0
0.2
0.6
0.4
Al fraction (x)
0.8
1
Figure 3.5: Two approximations for Si donor ionization energy in AixG ai_xN as a
function of x. Lower curve is “best case’’; upper curve is “worst case."’
of x [133]. Because the cathode is doped n-type. and the energy bandgap of the
AlxG ai_xN m aterial is so large, we can assume th at the quantity Ep — Ev { y) will
be very large compared to the the quantity Ec{y) — E f and so it is valid to assum e
th at the cathode is a m ajority carrier device and that there are virtually no holes
(p(y) = 0)- Hence, the charge concentration at each point in the cathode is given
by
Pit)
= « ( ‘V S ( 0 ) - « < » ) ) =
qN oit)i
| —
)
-g jV cex p
(3-9)
In order to solve Poisson’s equation, it must be put into a form such th at the po­
tential ip appears on both sides. This can be done by relating the electric potential to
the conduction band minimum energy through the relation E c = —qp>—A E c , where
A E c is the heterojunction conduction band offset. The heterojunction conduction
band offset describes how the conduction bands of two dissimilar sem iconductor
m aterials will line up if one is grown on top of the other.
Two estim ates for the heterojunction conduction baud offset were considered in
this work. A simple energy band analysis which assumes that the Fermi level lies
exactly in the middle of the energy gap for an undoped AlxGai_xN layer results
37
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4>
1.4
0.6
0.4
0.2
0.4
0.6
Al fraction (x)
0.8
Figure 3.6: Two ways of defining the heterojunction conduction band offset for
AlxG ai_xN as a function of Al fraction x. Lower curve is '‘best case’’; upper curve
is “worst case.”
in a band offset which starts at 1.03 eV at x=0.00 and falls to zero for x=0.75.
However, the heterojunction conduction band offset can also be constructed from
the conduction band offset between GaN (x = 0.00) and AIN (x = 1.00). A value
of 1.92 eV has been calculated using a first-principles total-energy pseudopotential
m ethod [139]. Since it is desired to minimize the energy difference between the
minimum conduction band energy in the GaN and th e minimum conduction band
energy at the cathode-vacuum interface, a minimum offset is desired. These two
ways of defining heterojuction conduction band offset are illustrated in Figure 3.6.
The lower curve is th e “best case” (minimum offset) which results from the energy
band analysis and th e upper curve is the “worst case” (maximum offset) which
results from a first-principles analysis.
Using Equation 3.9, Poisson's equation takes on th e form
d 2v { y ) _
qi^Npj
p ( yy)) ( .l +9exp f E F+q <p ( y ) + AEc(y) + E Cp ( y ) \ \ ~L
dy2
tsiy)
qNc
kT
\
).
1 d y { y ) d t s[y)
( E F + g<p(y) + A£c(t/)\ ________________
(3.10)
M y ) exp V
kT
)
£*(y) dy
dy
The dielectric p erm ittiv ity of GaN has been reported to lie somewhere between
S.9e0 and 9.5e<, [133] (one reference specifies it as 9.0eo [140]) and the perm ittivity
38
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of AIN has been reported to be S.Se* to within an error of ±0.2e„ [133]. Because
of this relatively minor variation in relative perm ittivity, Poisson's equation can be
approxim ated as
q^Di y) ( , ,
# V (y )
=
~
( E F + qip(y) + A E c (y) + E CD ( y ) \ \ ~ l
[1+ 2expl
. ^ e x p
Vf
JJ
( E r + (i r t y ) + ± E c ( y ) \
(:3 n )
e.
V
kT
)
by assum ing a relative perm ittivity. es. of 8.75 for all values of Alfraction x. Unless
otherw ise stated,
this approximation of Poisson's equation was used for all the
sim ulations.
3 .3 .2
R u n ge-K u tta N u m erical P roced ure
Poisson’s equation was solved using numerical analysis on a com puter. The fourthorder Runge-K utta technique is com putationally efficient and accurate [141], and
was chosen as the numerical analysis m ethod for solving Equation 3.11.
E quation 3.11 is a second-order differential equation which can be split into two
first-order differential equations. One equation is of the form
Ty =
(3 1 2 )
where p denotes the slope of potential ( not the hole concentration). The second
equation is of the form
^ =
dy
where / is a function that represents the right hand side of Equation 3.11.
(3.13)
Given th a t that the variable y represents position, the GaN layer is on th e left
(y < 0), the GaN-AlxG ai_xN interface is at y = 0, and the AlxG ai_xN layer is
on th e right (y > 0). The one-dimensional cathode structure is subdivided into
N segments, each of width h (here h denotes the width of the Runge-K utta step.
not Planck’s constant). The initial value of potential <p0 and the initial slope of
potential p0 are specified at the left hand boundary, at the back of the GaN layer.
T he com putation then proceeds from left to right.
39
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T he fourth-order Runge-Kutta procedure requires a total of ten com putations
for each successive step, as follows:
^2
k i = hpn
(3.14)
li = h f { y n.*pn)
(3.15)
= h(pn + 0.5/i)
(3.16)
h = h f ( y n + 0.5h. <pn + 0.5£j)
(3.17)
^3 = h(pn + O.5 /2 )
(3.IS)
k = hf { y n + 0.5 h.tpn + 0.ok2)
(3.19)
k4 = k(pn + I3 )
(3.20)
U — h f ( y n + h,<pn + k$)
(3.21)
’n+i = 'Pn + ~x(ki + - k 2 + —Z-'3 + k4)
(3.22)
0
(3.23)
Pn+l — Pn + 7-(/l + 2/2 + 2 / 3 + U)
b
where the potential and the slope of potential for the (n + l)th step is determ ined
from the previous position (/„, potential ~pn and slope of potential pn. This RungeK u tta numerical procedure was programmed using the Maple V m athem atical soft­
ware package [142]. A Maple V input file used for simulating an AlxG ai_xN cathode
is shown in Appendix A.
It is im portant to note th at Equation 3.10 has an infinite number of possible
solutions, but only one corresponds to the physical situation inside the cathode.
Semiconductor theory asserts th at, in the absence of externally applied fields, a
semiconductor structure should be completely charge neutral [135]. A charge neu­
tral solution is indicated for the cathode when the electric field at both ends of
the structure (the back of the GaN layer and the cathode-vacuum interface) are
approxim ately zero. Therefore, when the R unge-K utta procedure was used, the
left hand boundary pQ was adjusted until a solution was found which provided a
charge-neutral solution.
After the Runge-Kutta numerical procedure has been used to solve for the poten­
tial throughout the GaN layer and the graded AlxG ai_xN layer, a series of com puta­
tions are done to evaluate all im portant physical entities at each location inside the
40
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E
’c J q V n
metal
semiconductor
E
v
Figure 3.7: Qualitative energy band diagram for a metal-sem iconductor junction
with an n-type semiconductor.
cathode. The second derivative of potential d 2^p/dy2 is the left-hand side of Poisson’s equation, and was used to verify that Poisson's equation was solved correctly.
The location of conduction band minimum energy Ec { y) is used later to compute
the emission barrier. Finally, the free electron concentration n(y) describes the elec­
tron diffusion mechanism which may limit the current density. Maple V code used
to com pute these quantities is included in the input file shown in Appendix A.
3.4
Computation of Emitted Current Density
The energy barrier seen by electrons attem pting to escape out into vacuum can
be defined in a way which is similar to the way the energy barrier is defined for
metal-semiconductor junctions [135]. Figure 3.7 shows a qualitative energy-band
diagram for a metal-semiconductor junction, assuming an n-type semiconductor and
no surface states. A qualitative energy band diagram indicating the situation inside
the AlxGax_xN cathode is shown in Figure 3.S. The sim ilarity of the two energyband diagrams, with respect to the conduction band, implies th at the calculations
which describe the electron flow in a metal-semiconductor junction with an n-type
semiconductor should also describe the electron flow in the AlxG ai_xN cathode.
In th e metal-semiconductor junction, qVf, is the energy difference between the
conduction band m inim um energy at the metal-semiconductor interface and the
conduction band m inim um energy deep in the semiconductor. In the AlxG ai_xN
41
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GaN
E
AIGaN
vacuum
v
Figure 3.8: Qualitative energy band diagram for an n-type, graded AlxG ai_xN
cathode.
cathode, the quantity qVb is the energy difference between the vacuum level at the
cathode surface and the conduction band deep in the n-type GaN layer. In the
m etal-sem iconductor junction, qVn is the energy difference between the conduction
band and the Fermi level deep in th e semiconductor. In the AlxG ai_xN cathode,
the quantity qVn is the energy difference between the conduction band and the
Fermi level deep in the n-type GaN layer. In both cases, the effective barrier is
qq> = qVb + qVn.
A num ber of different theories have been developed to describe the current trans­
port in metal-semiconductor junctions [135]. A theory based purely on thermionic
emission (the transport of electrons over an energy barrier) is appropriate for highm obility semiconductors with m oderate doping levels operating at m oderate tem ­
peratures [135]. A theory based on diffusion (movement of electrons from regions
of high electron concentration to regions of low concentration) is appropriate for
Iow-mobility semiconductors with m oderate doping operating at m oderate tem per­
atu re [135]. A generalized theory which combines both thermionic emission and
diffusion has been devised, which is more accurate than either one alone [135]. In
the work described in this dissertation, it was found th at both barrier-lim iting and
diffusion-limiting are possible for certain cathode structures operating at certain
tem peratures.
Three additional physical mechanisms are possible in m etal-sem iconductor junc­
tions. One is carrier recombination. Recombination seems extrem ely unlikely in
42
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these cathode structures, since the bandgap of these materials is very large (be­
tween 3.45 and 6.2 eV) and the hole concentration is vanishingly small. A nother
possible mechanism is hole injection. Based on the very large bandgap of these
m aterials, the energy barrier for holes attem pting to enter the cathode’s surface
is very large (on the order of 2 eV). T he third possible mechanism is tunneling.
Q uantum -m echanical tunneling is a significant transport process if the semiconduc­
tor is heavily doped or the junction is operated at low temperatures. Since the
doping in these cathode structures is not unusually high ( 1 0 1 8 cm - 3 or less) and
the operating tem perature of the cathode in a microminiature microwave triode will
probably be well over 300 K, tunneling does not seem to be a significant electron
transport mechanism.
Based on th e theory of m etal-sem iconductor junctions, the current density ex­
pected due to thermionic emission is [135]
J = , r r 2exP ( ^ ? ) .
(3.24)
where J is current density. .4“ is the Richardson constant, and o is the energy
barrier. The Richardson constant is given by [135]
Airqm.k2
A' = S
- .
(3.25)
If the effective mass of electrons is assumed to be the same as that of GaN (0.19mo).
then A “ = 2.28 x 10s A /m 2 -K2.
The therm ionic emission current density given by Equation 3.24 will only be
attained if an adequate diffusion current density can supply the cathode-vacuum
interface.
After each simulation, it is im portant to check the diffusion current
density at the cathode-vacuum interface to see if it is adequate. If not, the em it­
ted current density will be the diffusion current density instead of the calculated
therm ionic emission current density.
The gradient of electron concentration d n (y )/d y is automatically com puted after
Poisson’s equation is solved, as shown in the Maple V code in Appendix A. This
quantity provides for the com putation of the diffusion current density, given by [135]
•/diffusion = kTp.n— .
dy
43
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(.3.26)
250
200
M
I
£
S
u 150
SB
£
0
0.1
0.2
0.4
Al Fraction (x)
0.6
Figure 3.9: Electron mobility in AlxG at_xN as a function of x.
Estim ation of diffusion current density requires knowledge of the electron mobil­
ity in the graded layer at each point. Experimental data for electron mobility in
AlxG ai_xN which was heavily doped in the range of 4.5 x 101 8 to 1.2 x 102° exists.
This d ata which was reported in 1998 [84] and is plotted in Figure 3.9. Very heavy
doping tends to reduce mobility, hence, the mobility for a lower Si concentrations
would be expected to be higher. In addition, as film growth techniques improve,
the mobility may be increased. Data is not vet available for material for which
x > 0.58, but it could be roughly ascertained from the data and the above com­
ments th at an electron mobility of about 30 cm 2 /V -s for x = 0.75 material would not
be unreasonable. This value of electron mobility is used later for diffusion current
calculations.
44
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C h ap ter 4
M icrom in iatu re M icrow ave
T riodes
4.1
Microminiature Microwave Triode Structures
A vacuum triode is an electron tube in which three electrodes exist inside a vacuum
enclosure. One electrode is the cathode, which em its electrons. A nother electrode
is th e anode, which collects the electrons. The third electrode is the control grid,
usually a multi-element electrode, which is placed between the cathode and the
anode in order to control the flow of electrons from cathode to anode. T he electrode
configuration of a conventional planar vacuum triode is shown in Figure 4.1.
Two types of microminiature microwave triode structures were exam ined. The
first type is shown in Figure 4.2, in which the control grid is fabricated on the
cathode using dielectric spacers and th e anode is placed near the cathode by way of
a rem ote spacer. The second type is shown in Figure 4.3. in which the control grid
is fabricated on the cathode using cathode-to-grid spacers and the anode is held in
place with additional grid-to-anode spacers.
For the structures examined in this research, the cathode was assumed to be
a heated AlxG ai_xN cathode as described in the previous chapter. Since cathode
tem peratures well under the conventional 1100 K are required, the cathode heater
can presum ably be simplified from th e conventional tungsten spiral assembly to a
sm aller, sim pler resistive tungsten m etallization in the form of a serpentine pattern
on th e back of the SiC wafer upon which the cathode structure is fabricated.
In conventional microwave triodes, ceramic m aterial is used to provide insulation
45
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ANODE
CONTROLGRID
CATHODE
Figure 4.1: Electrode configuration of a conventional planar vacuum triode.
ANODE
remote spacer
CONTROL
GRID
dielectric
spacers
CATHODE
Figure 4.2: M icrom iniature microwave triode array with control grid wires fabri­
cated on cathode-to-grid spacers and the anode m ounted on a remote spacer.
ANODE
didectnc
C O N T R O L G RID
CA T H O D E
Figure 4.3: M icrom iniature microwave triode array with control grid wires fabri­
cated on cathode-to-grid spacers and the anode m ounted on grid-to-anode spacers.
46
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
between the electrodes and provide strength and durability for the tube assembly.
T he spacer material for the microminiature microwave triodes should be a good
insulator and have as low a relative perm ittivity as possible, to minimize interelec­
trode capacitance. Furtherm ore, the dielectric should be able to w ithstand harsh
conditions (e.g., heat and radiation). The most commonly encountered insulator in
semiconductor device processing is silicon dioxide (Si 0
2
). which has a relative per­
m ittiv ity of 3.9 [135]. Silicon nitride (Si3 N4) is also widely available as an insulator
in semiconductor device processing, but it has a relative perm ittivity of 7.5 [135].
It may eventually be possible to avoid grid-to-cathode dielectric spacers by using
air-bridge fabrication techniques [75. 76].
In conventional microwave triodes, control grids are usually constructed from a
mesh of tungsten, m olybdenum, or gold wires [8 ]. Conducting materials commonly
used in semiconductor processing include tungsten, gold, aluminum, polycrystalline
silicon, platinum , and titanium . For the purposes of producing a m icrom iniature
microwave triode which can be used in harsh conditions, metals suitable for use
at high tem peratures, such as tungsten or titanium , would seem to be best. The
control grid, cathode-to-grid spacers, and grid-to-anode spacers would be produced
using reactive ion etching (RIE) techniques to achieve straight sidewalls and a large
aspect ratio [143].
Conventional triodes typically have a copper anode, because copper has a very
high therm al conductivity [8 ]. Thus, copper would probably be used as the anode
m etal for these triode structures. The anode in the first type of m icrom iniature
triode structure would be a flat piece of copper mounted onto the cathode base
using a thin spacer m aterial placed remotely outside the active area of the cathode.
Currently, a technique of this type may only perm it a minimum cathode-anode
distance of a 0.001-0.002 inches (25-50 fim) or more, instead of the extremely small
cathode-to-anode spacing (less than 2 /zm) described in this work. The anode in
the second type of m icrom iniature microwave triode could simply be a flat piece of
copper laid upon the grid-to-anode spacers. For large power dissipation densities,
some form of anode cooling would need to be used, which might take the form of a
heat sink or a system which flows air or water.
47
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4.2
Simulation of M icrominiature Microwave Tri­
ode U nit Cell
4 .2 .1
Vacuum T riode T h eory
Given an electrode configuration for a triode. the DC I-V characteristics of the
device can be simulated. Assuming th at the AlxG at_xiN^ cathode can produce a
substantial current density, the current in the vacuum device will be space-charge
limited. The DC I-V characteristics for space-charge lim ited devices are described
by the Child-Langmuir law [144]. According to this law. the current of any spacecharge-limited device is related to voltage by way of a th ree -h a lve s-p o w e r law [144].
Thus, for any triode, the total cathode current is [144]
Ic =
g
(
vg
+ 1j J ' .
(4.1)
where I c is the total cathode current, G is a constant referred to as the perveance
(with dimensions A /V 3/2), Vx is the anode voltage. Vg is the control grid voltage,
and p is a constant referred to as the amplification factor.
When the control grid of the triode has a negative potential, all of the cathode
current goes to the anode, and there is virtually no grid current, and I \ = IcW hen the control grid has a positive potential, a fraction of the cathode current
goes to the grid and a fraction of it goes to the anode. T he m anner by which the
current is divided between the anode and the control grid when the grid is positive
is described by the c u rre n t-d iv isio n f a c to r [144] 8. where the anode current is
-i
(4.2)
I a = Ic 1 + (^8\JVx/Vc^j
and the grid current is
-i
Ic
=
Ic
1 + 8\JV x /V g
(4.3)
When electrons are moving through a vacuum region. Poisson's equation [136]
VV = - L
(4.4)
can be used to relate charge density p and perm ittivity e0 to potential. T he tra­
jectories of the electrons in th e region are determined by solving the Lorentz force
4S
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
equation [8]
F
=
^
= ^ + f
xB,
(4.5)
where F is the force on the electron, m is the electron mass, v is the velocity, £
is the electric field, and B is the magnetic flux density. Research concerning the
formation and m anipulation of electron beams by way of electric and m agnetic fields
is called “electron optics.”
For conventional triodes with cylindrical grid wires, a m ature theory exists which
describes triode characteristics [144]. However, for structures such as m icrom inia­
tu re microwave triodes which have non-cylindrical control grid wires or spacers m ade
of dielectric m aterials, simple vacuum tube theory cannot be used. For these cases,
the DC I-V characteristics should be com puted using numerical analysis by way of
electron optics software [8 ].
4 .2 .2
E lectro n O ptics
Because of the num erical difficulties involved in solving the above equations, various
m athem atical techniques have been developed in order to do these calculations on
a computer. T h e two most widely used techniques are the finite element m ethod
(FEM ) and the finite difference method (FDM) [8 ]. The region of interest is either
broken into variable-sized triangular subregions (FEM ) or defined by a mesh of
equally-spaced nodes (FDM). The computer program must solve the differential
equations at th e boundaries between adjacent subregions (FEM) or at each node
(FDM ). Finite difference and finite element m ethods both currently find extensive
use. FEM techniques tend to be preferred for problems involving m agnetic fields,
electrode geom etries which involve small protrusions which which require a very fine
mesh (the triangular regions can be made dense at the location of interest), and
for electrode boundaries th at are curved. FDM techniques tend to be preferred for
problems involving boundaries which are straight lines and for problems requiring
accurate num erical derivatives (such as problems involving the motion of charge
through a vacuum region). Thus, for the purposes of this research, electron optics
calculations for th e microminiature microwave triode structures were perform ed
using FDM software.
49
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
20
RCWOX.KCl
0
40
60
80
100
NZA TR IC D Z SIMULATION - - WITH Q X 1D Z
1 -2 -9 6
120
CWH
140
160
Figure 4.4: Graphical output for an electron optics sim ulation using the EGN2e
com puter program.
The electron optics FDM software used for this work was developed at the Stan­
ford Linear Accelerator Center (SLAC) and is called EGN2e [145]. In order to run
the program, the user generates an input file which specifies the two-dimensional
electrode geometry in either rectangular or cylindrical coordinates, assigns a poten­
tial to each electrode, and imposes a grid of equally-spaced nodes upon the struc­
ture. The input files for EGN2e which were used for sim ulating the microminiature
microwave triode structures for this work are listed in Appendix B.
Typical EGN2e graphical output (corresponding to the second type of micro­
m iniature triode structure studied in this work, with both cathode-to-grid and gridto-anode spacers) is shown in Figure 4.4. The cathode is on the left and emits
current past a control grid to an anode on the right. T he dielectric spacer regions
between the cathode and grid and grid and anode are shown. The numbers appear­
ing along the axes correspond to mesh units (the spacing between nodes, which is
specified by the user) which in this case are 20 nm in length. The simulation is
performed by EGN2e by assuming that the depth of the structure (into the paper)
equals one mesh unit, or 20 nm. W ith this mesh unit, the simulation involves 150 x
25 = 3750 nodes. The vertical lines between the grid and anode describe the poten­
tial distribution (each line is an equipotential). The program uses a Runge-Kutta
m ethod to compute the space-charge distribution. These calculations are presented
graphically as charge trajectories, shown in Figure 4.4 as curved lines proceeding
from left to right. Each line represents a quantity of negative charge.
In order to utilize EGN2e to sim ulate the m icrom iniature microwave triode struc­
tures, a “unit cell” approach was used. In this approach, the triode structures were
viewed as periodic structures which can be specified with a repetitive, single unit
50
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C
ga
o A
Go
ccT
c
Figure 4.5: High-frequency small-signal model for an array of m icrom iniature mi­
crowave triode unit cells.
cell of width u?uc, height h uc, and d ep th d uc. For com putational efficiency, only h a lf
of the unit cell was simulated, as indicated in Figure 4.4. For this sim ulation, due
to sym m etry arguments, the boundaries representing vacuum can be implemented
using Neumann boundary conditions, i.e.. the derivative of the potential normal to
the vacuum boundary is zero.
4.3
Microminiature Microwave Triode Arrays
In order to predict the microwave perform ance of an array of m icrom iniature mi­
crowave triode unit cells, a circuit m odel was constructed. A small-signal model
for a triode operating at high frequencies [147] is shown in Figure 4.5. where g m
is the transconductance, r . 4 is the anode resistance, C g c is
grid-to-cathode
capacitance, C .\c is the anode-to-cathode capacitance, C g a is the grid-to-anode
capacitance, and vx is the voltage between the grid and cathode.
At a given operating point
Vg —
VgO
1
1 .4
=
1 .4 0 . I .l.u c =
I a o .u c ,
the amplification factor \i is given by [147]
51
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the transconductance of a single unit cell gm,uc is given by [147]
gm.uc = j ±
= \ i ‘G U V a + Vi l v ) ' l 1,
(4.7)
and th e anode resistance of a single unit cell r^.uc is given by [147]
= j j j = j/-G - '(V c + Ki
(4.S)
T he above three param eters can be related at a given operating point using [147]
— 9m,uc^A.uc-
(4-9)
T he capacitance between the three electrodes of a triode are im portant in de­
term ining its high-frequency lim itations. The most im portant capacitance is the
gate-to-cathode capacitance C g c • It is expected that the capacitance C g c lies
between one of two extrem e values. The first value for C g c can be com puted by as­
sum ing th a t each grid wire forms a distinct parallel-plate capacitor w ith th e cathode
and th e total capacitance is just the sum m ation of these individual capacitances.
In these cases, only the region directly under the grid wire is taken into account
when the capacitance is com puted. The dielectric material between th e grid and
cathode are taken into account by m ultiplying the capacitance corresponding to
no oxide being present by a weighting factor determined from th e proportion of
volume taken up by the dielectric and the relative perm ittivity of th e dielectric.
For example, if one-fourth of the volume under the grid wire is filled with silicon
dioxide with a perm ittivity of 3.9. then the parallel-plate capacitance calculated
w ith no oxide present is multiplied by the factor W = 0.75 + (3.9 x 0.25) = 1.725.
This value represents the “best case." since it ignores all fringing between th e grid
wires. A second value value for C g c would computed by simply assum ing that
two m etallic parallel plates exist, one located at the cathode surface and a second
p late located at the lower surface of the grid wires. Again, the dielectric m aterial
between the grid and cathode are taken into account by m ultiplying th e capacitance
corresponding to no oxide being present by a weighting factor determ ined from the
proportion of volume taken up by the dielectric and the relative p erm ittiv ity of
the dielectric. This approach for com puting C g c produces a “worst case” value,
52
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
because it assumes th at th e maximum possible fringing will occur in the regions
between th e grid wires.
For th e purposes of estim ating the high-frequency lim itations of the triode, the
“worst case” capacitance value of C g c was used, since the extent of fringing is un­
known and additional capacitance may arise elsewhere in the triode packaging. The
interelectrode capacitance C .\ g is computed in a sim ilar m anner as C gc. assuming
parallel plates at the anode plane and at the top surface of the grid wires, with
appropriate weighting due to spacers, if they are present.
According to vacuum tu b e theory [148]. the anode-to-cathode capacitance C'a c
is given by
C’a c
= —
(4.10)
due to the fact that the grid provides electrostatic shielding between the anode and
cathode. Hence, in this work, the interelectrode capacitance C.-tc was com puted by
taking the non-weighted gate-to-cathode capacitance and dividing by the amplifica­
tion factor [148]. This should provide a value consistent with the fact that, between
the grid wires, the triode unit cell looks like a classical vacuum tube.
The expressions for g m ,uci r A,uc- C'g c .uc C-g a .uc- and C .ac .uc, describe the unit
cell, but they must be converted to a form useful for describing entire arrays of unit
cells. This was done for each unit cell by deriving expressions which provide (1) DC
current density, (2) anode DC power dissipation density. (3) transconductance. (4)
interelectrode capacitance, and (5) anode resistance. By putting the equations in
this form, they become useful for choosing an appropriate array area. The unit cell
width wuc and unit cell depth duc are used to perform the conversion. As explained
earlier, the unit cell depth corresponds to one EGN2e mesh unit. The area of a
single unit cell is u?uc</uc.
The DC operating conditions of an array can be described as follows.
The
current density at the anode is
Ji a ,d c
=
C“y D( cv H J. l A ) 3/2 •
wucduc \
/i /
,1
m
(4-11)
Note th at the emission density at the active part of the cathode must about twice
the density at the anode, because only half of the unit cell's bottom surface is
53
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
em itting. The DC power dissipation per unit area at the anode is
C
5-
Pa ,dc = J a ,dc Va= VA
Wucd-uc
(
V \ 3^2
[Vc + —
•
V
V- )
(4.12)
The transconductance per unit area is
3 G ac /
VA'
9m = ^ — 7 - *fe + —
2 w ucduc \
ft I
1 /2
.
(4.13)
The anode resistance of an array of unit cells is
rA
=
2 wucduc f
VA V /2
I •'c H----- •
3 G lic \
fi
(4.14)
T he interelectrode capacitances per unit area are given by
C gc
---------------------------------------------------------- (4 -1 0 )
C gc
Cat =
(4.16)
wucduc
C a C.uc
,. ,
CAc = ----- 7 —
(4.10
^UC“UC
In order to roughly estim ate th e maximum frequency at which the triode arrays
can be used, a figure of merit can be defined. As indicated in Figure 4.6, the figure
of m erit is derived by calculating the short-circuit current gain (often referred to as
the hybrid param eter A2 1 [149]) of a simplified small-signal model of the triode array.
The figure
of merit is called the “cutoff frequency”’ (denoted
f r ) and is defined as
the frequency at which the m agnitude of the input current *,•„equals the magnitude
of the output current iout. Circuit analysis indicates that
h
=
I itC gc
(4.18)
This definition of cutoff frequency is commonly used to describe the high-frequency
lim itations of solid-state devices (where C g c is replaced by th e gate-to-source ca­
pacitance C g s ) [135] and high-frequency field-emission triodes [8 ].
54
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Ltn
—
v
G
m
GC
Lout
Figure 4.6: Circuit diagram indicating how cutoff frequency is defined.
55
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h ap ter 5
R esu lts
5.1
5 .1 .1
A lxG a i _ xN
Cathodes
B asic C a th o d e Structure O p eration at 300 K
The m ost basic AlxG aj_xN cathode design assumed in this work is a structure of
the form shown in Figure 5.1, in which an AlxG at_xN layer sits upon a GaN layer.
The AlxG ai_xN layer is compositionally graded from x = 0.00 to x = 0.75 over a
thickness of 75 nm. The grading is linear with position, i.e. Al fraction x varies
with y as 10' y, where y is the position in meters. The cathode structure is doped
uniformly n-type with Si at a concentration of 10IS cm -3. At 300 K. the Fermi
level in the GaN m aterial with 1018 cm-3 Si doping is located 0.041 eV below the
conduction band. For this basic cathode structure at 300 K, all of the relevant
physical quantities as function of position will be plotted, in order to illustrate how
the cathode operates. For all other cathode structures, only the physical quantities
needed to compare the performance of one structure to the performance of another
stru ctu re will be plotted.
T he distribution of potential in the basic cathode structure is shown in Fig­
ure 5.2. The coordinate system is defined such th at the origin (y = 0) is placed at
the interface between the GaN and the graded AlxG at_xN layer. The back of the
cathode structure (y = —50 nm) is grounded, so the potential is zero at the back of
the GaN layer. The potential rises up to a value of 0.92 V at the cathode-vacuum
interface (y = 75 nm).
T he conduction band minim um energy as a function of position in the basic
56
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*50 0____ 75
y (nm)
Figure 5.1: Basic cathode structure.
0.8
> 0.6
cs
s
“o 0.4
a.
0.2
-100
-50
0
50
100
Position (nm)
Figure 5.2: Potential vs. position in basic cathode structure at 300 K.
57
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-LI
AlGaN
vacuum -
-1_3
-1.4
-100
GaN
-50
0
50
100
150
Position (nm)
Figure 5.3: Conduction band minimum energy' vs. position in basic cathode stru c­
tu re at 300 K.
cathode structure is shown in Figure 5.3. The energy values are referenced to th e
back of the GaN layer, at which an ohmic contact is grounded. This ground is
taken as both the zero energy and zero potential references. The conduction band
minim um energy at th e back of the GaN layer starts at -1.44 eV. which corresponds
to the heterojunction conduction band offset of the GaN material. The band dips
down to about -1.46 eV at the origin. It rises up to -0.92 eV at the cathode vacuum
interface, corresponding to the value of potential there.
The Fermi level {E p) is located at -1.4S1 eV. as shown in Figure 5.3. The en­
ergy barrier for electrons attem pting to escape into vacuum is the energy difference
between the conduction band minimum energy and the Fermi level at the cathodevacuum interface, and this value is 0.560 eV. At 300 K, thermionic emission theory
indicates an emission density of 7.92 A /m 2, or 792 /iA /cm 2.
The free electron concentration as a function of position in the basic cathode
structure is shown in Figure 5.4. The concentration of electrons at the back of the
th e GaN layer is 4.31 x 101' cm -3. The electron concentration rises up to about 1.4
x 1018 cm-3 at the origin and then rapidly drops off into the AlxGat_xN layer. This
electron distribution results in a diffusion current at the cathode-vacuum interface
th a t supplies electrons during emission. The gradient of electron concentration
at the interface is 8.90 x 1015 cm-4. W ith this gradient, the electron m obility
58
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
le+18
8e+17
6e+17
4e+17
2e+17
-100
0
-SO
50
100
Position (nm)
Figure 5.4: Electron concentration vs. position in basic cathode structure at 300
K.
required at the cathode-vacuum interface to achieve the emission density indicated
by therm ionic emission theory is 21.3 cm2/V-s. Based on the mobility data for
AlxG ai_xN materials (see Figure 3.9) indicating that /x„ « 30 cm2/V -s for the
x = 0.75 m aterial, the current will probably be barrier-limited, not diffusion-limited.
T he ionized donor concentration as a function of position in the basic cathode
stru ctu re is shown in Figure 5.5. The concentration of ionized donors at the back
of the the GaN layer is 4.31 x 101‘ cm -3. The concentration dips down to about
1.8 x 101, cm -3 at the origin, rises back up to about 2.1 x 101‘ cm -3 at y = 4 nm.
dips down to about 5 x 1016 cm -3 at y = 38 nm. and rises to a maximum of about
8.5 x 101' cm -3 at the cathode-vacuum interface.
T he total charge concentration as a function of position in the basic cathode
stru ctu re is shown in Figure 5.6. The back of the GaN layer is charge neutral. A
“spike” of negative charge exists at the origin which has a peak concentration of
about 1.28 x 1018 cm -3. The point within the AlxGax_xN layer at which zero net
charge exists occurs at about y = 11 nm. Past this point, only net positive charge
exists, rising up to m axim um value of about 8.5 x I0ir cm -3 at the cathode-vacuum
interface. At therm al equilibrium , the total negative charge in the cathode structure
m ust equal the total positive charge.
59
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
le+18
<
E
8e+17 -
s©
6e+17 e©
oe
o
U
u©
4e+17 -
s
©
Q
■8
~e
c
2e+17 -
100
Position (nm)
Figure 5.5: Ionized donor concentration vs. position in basic cathode structu re at
300 K.
g
le+18
S
5e+17
SJ
2
s1)
os
e
U
a*
2es? -5e+17
-C
•100
-50
0
50
100
Position (nm)
Figure 5.6: Total charge concentration vs. position in basic cathode structure at
300 K.
60
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-8e+06
-1.6e+07
-100
-50
0
50
100
Position (nm)
Figure 5.7: Electric field vs. position in basic cathode stru ctu re at 300 K.
T he electric field distribution in the basic cathode structure corresponding to
the total charge distribution shown in Figure 5.6 is shown in Figure 5.7. The electric
field is zero at each end of th e cathode structure. The field reaches a peak absolute
value of about 1.5 x 107 V /m a t about ij = 11 nm. corresponding to the point where
there is zero net charge. This electric field distribution corresponds to the potential
distribution shown in Figure 5.2.
R e su lt o f A ssu m ing a S m a ller H etero ju n ctio n C o n d u ctio n B a n d O ffset
As explained in Chapter 3. the heterojunction conduction band offset can vary
based on what assumptions are made. The baseline simulation previously presented
assumed a "worst case" m axim um band offset. The emission barrier resulting from
the “best case” offset was com puted. The conduction band m inim um energy as a
function of position in the basic cathode structure, assuming th e sm allest reasonable
heteroj unction conduction band offset (the “best case” ), is shown in Figure 5.S.
Note th a t this simulation was performed with the more complex form of Poisson’s
equation (equation 3.10), w ith relative perm ittivity varying w ith A1 fraction x as
9.0-0.5x. The conduction band at the back of the GaN layer starts at -1.03 eV.
which corresponds to the sm aller assumed heteroj unction conduction band offset of
the GaN material. The band dips down to about -1.05 eV at th e origin. It rises up
61
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
0.6
« -0.7
M
-
0.8
-
1.1
-150
-100
-50
0
50
100
150
Position (nm)
Figure 5.8: Conduction band m inimum energy vs. position of basic cathode struc­
tu re at 300 K, assum ing a sm aller ( “best case” ) heteroj unction conduction band
offset.
to -0.66 eV at th e cathode vacuum interface.
T he Fermi level is located at -1.071 eV. The energy barrier for electrons a t­
tem pting to escape into vacuum is 0.411 eV. down from 0.560 eV observed with
th e larger heterojunction conduction band offset. At 300 K, thermionic emission
theory indicates an emission density of 0.26 A /cm 2, which is much higher th a n the
792 fiA /c m 2 predicted for the larger band offset.
This sim ulation was performed by assuming a 100-nm-thick GaN layer below
th e 75-nm-thick AlxGa!_xN layer, instead of a 50-nm-thick GaN layer. Inspection
of Figure 5.8 indicates th a t virtually all of the space charge (indicated by bending
of th e conduction band) in the GaN layer exists within 15-20 nm of the origin, and
placem ent of the back of the GaN layer at y = —100 nm instead of at y = —50 nm
makes no difference.
R e s u lt o f A ssu m in g a L ow er D on or Ion ization E n ergy
As explained in C hapter 3, the value of the donor ionization energy can vary, based
on what assumptions are m ade. The baseline sim ulation previously presented as­
sum ed uworst case” m axim um ionization energies. T he emission barrier resulting
from the “best case” ionization energies was also com puted. The conduction band
62
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~
>
-1.05
^
- 1.1
12
ueu -1.15
w
-12
|
-1.25
e
1
1
-1.35
-
-1.4
68
e
-1.45
o
U
-100
-50
0
50
100
Position (nm)
Figure 5.9: Conduction band minimum energy vs. position of basic cathode struc­
tu re at 300 K, assuming lower ( ubest case’’) donor ionization energies.
m inim um energy as a function of position in the basic cathode structure, assuming
the smallest reasonable ionization energies (the “best case” ), is shown in Figure 5.9.
Note th at this simulation was also performed with the slightly m ore complex form
of Poisson’s equation (equation 3.10). with relative perm ittivity varying with A1
fraction x as 9.0-0.5x. The conduction band at the back of the GaN layer starts at
-1.44 eV, which corresponds to the (larger) heteroj unction conduction band offset
of the GaN material. The band dips down to about -1.47 eV at the origin. It rises
up to -1.104 eV at the cathode vacuum interface.
The Fermi level is located at -1.476 eV. The energy barrier for electrons at­
tem pting to escape into vacuum is 0.360 eV. down from 0.560 eV observed with the
larger donor ionization energies. At 300 K. thermionic emission theory indicates
an emission density of 1.79 A /cm 2, which is much higher th an the 792 /j A /cm 2
predicted for the larger ionization energies.
Again, it should be noted th at this simulation was performed by assuming a
100-nm-thick GaN below the 75-nm-thick AlxG ai_xN layer, instead of a 50-nmthick GaN layer. However, the placement of the back of the GaN layer at y = —100
nm instead of y = —50 nm does not change the charge distribution in the GaN
layer.
63
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
>
S
-i
to
u»
w
-1.1
E
I
i
■B
a -13
aa
s
_o
y3
•O
B
-1.4
•100 -50
0
50
100 150 200 250 300
Position (nm)
Figure 5.10: Conduction band minimum energy vs. position for a cathode with
0.25-/zm-thick AlxGat_xN layer.
>
-1
i
'
i
i
■" r
i
-1.05
>»
to
u4> -1.1
S
w -1.15
/
—
I
e
_
/
3
s -13
*e
-1.25 S
•3 -13 S
—
y
/
-
C
5
sa -135
g
.2
u -1.4
3
-a
so -1.45
U
-15
-100
_
'r
_____ 1_____ 1_____ 1_____ 1_____L
0
100
200
300
400
500
Position (nm)
Figure 5.11: Conduction band minimum energy vs. position for a cathode with a
0.50-/tm-thick AlxG ai_xN layer.
64
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5 .1 .2
C ath od es w ith a T hicker AlxGai_xN Layer
A lxGax_xN graded layers can be grown with varying thicknesses.
The effect of
m aking the layer thicker was studied by simulating cathodes with graded layers
with linear grading which were 0.25 /im thick and 0.50 y m thick.
C a th o d e w ith 0.25-^ m -T h ick AlxGai_xN Layer
The conduction band m inim um energy as a function of position for a cathode struc­
ture with a 0.25-/zm-thick AlxG ai_ xN layer is shown in Figure 5.10. The 0.25-/zmthick layer is graded linearly w ith position, i.e. the A1 fraction varies as 3 x 10by.
where y is given in meters.
T he conduction band rises up to -1.006 eV at the cathode-vacuum interface.
T he Fermi level is located at -1.4S1 eV. The energy barrier for electrons attem pting
to escape into vacuum is 0.474 eV. down from 0.560 eVr observed with the 75-nmthick AlxG ai_xN layer. At 300 K. therm ionic emission theory indicates an emission
density of 221 A /m 2 , or 22.1 m A /cm 2, which is much higher than the 792 /iA /cm 2
predicted for the 75-nm-thick AlxG ai_xN layer.
T he gradient of electron concentration at the cathode-vacuum interface in this
case is 6.31 x 1016 cm -4. W ith this gradient, the electron mobility required at the
cathode-vacuum interface to achieve the emission density indicated by therm ionic
emission theory is 34.6 cm2/V -s. Based on the mobility d ata for AlxG ai_xN layers
(see Figure 3.9) indicating th at y n % 30 cm2/V-s for the x = 0.75 m aterial, the
current density for this structure will probably be diffusion-limited, not barrierlim ited. The expected value of emission current density based on this lower mobility
value would be 7.8 m A /cm 2.
C a th o d e w ith 0.5 0 -/im -th ick AlxGat_xN Layer
T he conduction band minimum energy as a function of position for a cathode struc­
tu re w ith a 0.50-/zm-thick AlxG a!_x.\’ layer is shown in Figure 5.11. The 0.50-^mthick layer is graded linearly w ith position, i.e. the A1 fraction varies as 1.5 x 106y,
where y is given in meters.
T he conduction band rises up to -1.051 eV at the cathode-vacuum interface.
T he Fermi level is located at -1.481 eV. The energy barrier for electrons attem pting
65
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
to escape into vacuum is 0.430 eV. down from 0.560 eV observed with the 75-nmthick AlxG ai_xN layer. At 300 K, thermionic emission theory indicates an emission
density of 1221 A /m 2 , or 0.1221 m A /cm 2. which is much higher than the 792
/zA /cm 2 predicted for the 75-nm-thick AlxG ai_xN layer.
T he gradient of electron concentration at th e cathode-vacuum interface in this
case is 1.46 x 101' cm -4. With this gradient, th e electron mobility required at th e
cathode-vacuum interface to achieve the emission density indicated by therm ionic
emission theory is 201.8 cm2/V-s. Based on the m obility d ata for AlxG at _xN layers
(see Figure 3.9) indicating that y n % 30 cm2/V -s for the x = 0.75 m aterial, the
current density for this structure will definitely be diffusion-limited, not barrierlim ited. The expected value of emission current density, based on this lower m obility
value, would be 18.1 m A /cm 2.
5 .1 .3
C a th o d es w ith N on-L inear G rading
To investigate the effect of using various types of grading profiles, two non-linear
grading profiles in the AlxGai_xN layer were studied.
The first profile was a
quadratic function which had x = 0.00 at the GaN-AlxG a!_xN interface, x = 0.75
at the cathode-vacuum interface, and was concave up. i.e.. 1.333 x 10l4y2 where y is
given in m eters. The second profile was a quadratic function which had x = 0.00 at
th e GaN-AlxG ai_ xN interface, x = 0.75 at the cathode-vacuum interface, and was
concave down, i.e.. —1.333 x 1014y2 + 2 x 10‘y where y is given in meters. These two
non-linear profiles, along with the basic linear profile, are plotted in Figure 5.12.
C a th o d e w ith C oncave-U p Q uadratic G rad in g
T he conduction band minimum energy as a function of position for a cathode struc­
tu re with the concave-up quadratic grading profile in the AlxG at_xN layer is shown
in Figure 5.13. The shape of the conduction band curve is quite different for this
grading than the linear grading case (see Figure 5.3). There is no sharp downward
dip at the origin, and the curve rises with upward concavity to -0.827 eV.
The Fermi level is located at -1.481 eV. The energy barrier for electrons a t­
tem pting to escape into vacuum is 0.654 eV. up from 0.560 eV observed with the
linear grading. At 300 K, thermionic emission theory indicates an emission density
66
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.7
0.6
* 0.5
eo
£
2
bm
0.4
— 03
<
0.1
0
10
20
30
40
50
60
70
Position (nm)
Figure 5.J.2: Linear and quadratic grading profiles for the AlxG ai_xN layer.
of 0.2137 A /m 2 , or 21.37 /xA/cm2. which is lower than the 792 /xA/cm2 predicted
for the linear grading.
The gradient of electron concentration at the cathode-vacuum interface in this
case is 7.40 x 1014 cm -4. W ith this gradient, the electron m obility required at the
cathode-vacuum interface to achieve the emission density indicated by thermionic
emission theory is 6.98 cm2/V-s. For this structure, based on the AlxG ai_xN mobil­
ity d a ta (see Figure 3.9) indicating that /xn ss 30 cm2/V-s for the x = 0.75 material,
the current is most likely barrier-limited, rather than diffusion-limited.
C a th o d e w ith C on cave-D ow n Q uadratic Grading
The conduction band minimum energy as a function of position for a cathode struc­
ture with the concave-down quadratic grading profile in the AlxG a1_xN layer is
shown in Figure 5.14. The shape of the conduction band curve is quite different for
this grading than the linear grading case (see Figure 5.3). The dip at the origin is
more pronounced, and the curve rises up with downward concavity to -1.003 eV.
The Fermi level is located at -1.481 eV. The energy barrier for electrons at­
tem pting to escape into vacuum is 0.478 eV, down from 0.560 eV observed with the
linear grading. At 300 K, thermionic emission theory indicates am emission density
of 191.4 A /m 2 , or 19.14 m A /cm 2, which is higher than the 792 /xA/cm2 predicted
67
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
=5
-
0.8
-
1.1
£o»
Ci
U
S
=5
£
M
l
e
•mt
S
"O
aa
sa
e
.2
s
•o
e
«
U
-1-3
-1.4
-100
-50
50
100
Position (nm)
Figure 5.13: Conduction band minimum energy vs. position for a basic cathode
structure with a concave-up quadratic grading.
for the linear grading.
The gradient of electron concentration at the cathode-vacuum interface in this
case is 1.S9 x 10IS cm -4. W ith this gradient, the electron m obility required at the
cathode-vacuum interface to achieve the emission density indicated by thermionic
emission theory is 2444 cm 2/V-s. For this structure, based on the AlxG ai_xN mobil­
ity d ata (see Figure 3.9) indicating th at fin % 30 cm 2/V -s for the x = 0.75 m aterial,
the current will definitely be diffusion-limited. The expected value of emission cur­
rent density, based on this lower mobility value, would be 235 /iA /cm 2.
5 .1 .4
C a th o d es w ith M od erate and Low D o n o r C oncentra­
tion s
The baseline cathode sim ulations were performed for structures in which the donor
concentration was 1 x 1018 cm -3 (a "high" concentration). To create a substantial
free electron concentration, the donor concentration should be kept high. However.
AlxG at_xN film growth studies indicate th at excess Si im purities in such films lead
to cracking [134]. Therefore, it is im portant to study th e emission properties of
cathode structures with varying donor concentrations.
Basic cathode structures
containing 5 x 1017 cm -3 (a “moderate"1 concentration) and 1 x 101, cm -3 (a “low"
68
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-O -1.5
e
©
°
- 1.6
Position (nm)
Figure 5.14: Conduction band minim um energy vs. position for a cathode with a
concave-down quadratic grading.
concentration) were simulated. Note th at, in order to perform these simulations,
th e position of th e Fermi level in the n-GaN layer had to be recalculated each time.
C a th o d e S tru ctu re w ith M o d era te D o n o r C on cen tration
T h e conduction band minimum energy as a function of position for a cathode struc­
tu re w ith m oderate donor concentration (5 x 101, cm -3 ) is shown in Figure 5.15.
T h e conduction band rises up with upward concavity to -0.886 eV. T h e Fermi level
is located at -1.493 eV. The energy barrier for electrons attem pting to escape into
vacuum is 0.607 eV, up from 0.560 eV observed with the higher donor concentra­
tion. At 300 K, thermionic emission theory indicates an emission density of 1.315
A /m 2 , or 131.5 /zA/cm2, which is lower than the 792 fiA /c m 2 predicted for the
high donor concentration.
T h e gradient of electron concentration at the cathode-vacuum interface in this
case is 1.481 x 1015 cm-4. W ith this gradient, the electron mobility required at the
cathode-vacuum interface to achieve the emission density indicated by thermionic
emission theory is 21.47 cm2/V-s.
For this structure, based on th e AlxG ai_xN
m obility d ata (see Figure 3.9) indicating th at fin sa 30 cm2/V -s for th e x = 0.75
m aterial, th e current density will probably be barrier-lim ited, not diffusion-limited.
69
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
„
>41
-0.8
uV
S
-1
W
I
1
-L1
|
-a
§
ea
1-3
o
-1.4
U
-
1.6
•100
-50
0
100
Position (nm)
Figure 5.15: Conduction band minimum energy vs. position for a basic cathode
structure with a m oderate donor concentration of 5 x 1017 cm -3.
100
Position (nm)
Figure 5.16: Conduction band minimum energy vs. position for a basic cathode
structure with a low donor concentration of 1 x 101' cm -3.
70
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C a th o d e Stru ctu re w ith Low D o n o r C oncentration
T he conduction band minimum energy as a function of position for a cathode struc­
tu re with a low donor concentration (1 x 101' cm -3) is shown in Figure 5.16. The
conduction band rises with upward concavity to -0.588 eV. T he Fermi level is located
at -1.524 eV. The energy barrier for electrons attem pting to escape into vacuum is
0.936 eV, which is much larger th a n the 0.560 eV observed with th e high concen­
tration. At 300 K, thermionic emission theory indicates an emission density of 3.89
x 10-6 A /m 2, or 389 pA /cm 2, which is much lower than the 792 /zA/cm 2 pre­
dicted for the high concentration. Clearly, reducing the donor concentration to this
level makes the emission barrier unacceptably high, comparable to th at observed
for conventional thermionic cathodes.
The gradient of electron concentration at the cathode-vacuum interface in this
case is 4.35 x 109 cm -4. W ith th is gradient, the electron m obility required at the
cathode-vacuum interface to achieve the emission density indicated by thermionic
emission theory is 21.66 cm2/V -s.
For this structure, based on the AlxG ax_xN
mobility d ata (see Figure 3.9) indicating th at /zn % 30 cm 2/V -s for the x = 0.75
m aterial, the current would probably be barrier-limited, not diffusion-limited.
5.1.5
B asic C ath od e S tru ctu re O peration at E lev a ted T em ­
peratures
The previous results indicate th at A lxG ax_xN cathode structures should have low
thermionic emission barriers. However, the current densities obtained with these
barriers at 300 K do not reach the 10 A /cm 2 which can be achieved w ith conventional
thermionic emitters. Hence, the operation of the basic cathode stru ctu re at elevated
tem peratures was investigated, in th e tem perature range 300-700 K.
Results of solving Poisson’s equation to find the location of the conduction band
at 300 K and 700 K are shown in Figure 5.17. Curve A corresponds to the location
of the conduction band at 300 K and Curve C indicates the location of the Fermi
level at 300 K. Curve B corresponds to the location of the conduction band at 700
K and Curve D indicates the location of the Fermi level at 700 K. T he energy band
orientation at 700 K results in an energy barrier of 0.569 eV.
The energy barrier at the cathode-vacuum interface is plotted as a function of
71
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-0.8
-0.9
-1
_
-
>
~
1.1
-L 2
S -13
e
W
-1.4
-13
C
1.6
D
-
-1.7
-100
0
-50
50
100
Position (nm)
Figure 5.17: C onduction band minimum energy vs. position (A,B) and Fermi level
(C,D) of basic cathode structure at 300 K (A,C) and 700 K (B,D).
tem perature in Figure 5.18. It is apparent th a t the barrier is changing very little
with tem perature. Thermionic emission current density is plotted as a function of
tem perature in Figure 5.19. The current density is 1 A /cm 2 at about 430 K. 10
A /cm 2 at about 490 K. and 100 A /cm 2 at about 580 K.
The free electron concentration as a function of position in the basic cathode
structure is p lotted in Figure 5.20 for 300 K and 700 K. The gradient of th e elec­
tron concentration at the cathode-vacuum interface (y = 75 nm where x = 0.75)
increases by several orders of magnitude as the tem perature is increased from 300 K
to 700 K, as shown in Figure 5.21. The m inim um required electron m obility needed
at the cathode-vacuum interface to achieve the current density predicted by the
thermionic em ission theory is plotted as a function of tem perature in Figure 5.22.
Hence, based on th e AlxGa!_xN mobility d a ta (see Figure 3.9) indicating th a t fin %
30 cm2/V -s for th e x = 0.75 material, it appears th a t the basic cathode stru ctu re
would be barrier-lim ited for T < 450 K and diffusion-limited for T > 450 K.
72
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
t—
i— i— i— i— i— i— i— r
0.568
0.566
0.564
250 300 350 400 450 500 550 600 650 700 750
Temperature (K)
Figure 5.18: Emission barrier of a basic cathode structure vs. tem perature.
4
3
2
1
0
-1
-2
-3
-4
250 300 350 400 450 500 550 600 650 700 750
Temperature (K)
Figure 5.19: Em itted current density of basic cathode structure vs. tem perature.
73
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
rri
18
700 K
<
E
16
B
V 14
300 K
U
s 12
b
uu
5 10
so
e
8
L -
-100
0
-50
50
100
Position (nm)
Figure 5.20: Free electron concentration vs. position in basic cathode stru ctu re at
300 K and 700 K.
22
21
20
19
18
17
16
15
250 300 350 400 450 500 550 600 650 700 750
Temperature (K)
Figure 5.21: Gradient of electron concentration at cathode-vacuum interface vs.
tem perature in basic cathode structure.
74
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
45
40
e 35
s 30
W 25
250 300 350 400 450 500 550 600 650 700 750
Temperature (K)
Figure 5.22: M inimum electron mobility required at cathode-vacuum interface
(where x = 0.75) to achieve barrier-lim ited current density vs. tem perature.
5.2
5 .2.1
M icrominiature Microwave Triodes
Sm aller U n it C ell w ith C ath od e-to-G rid Spacers
In 1989, L.F. Eastm an presented a paper to the Second International Vacuum Mi­
croelectronics Conference [103]. In th at paper. Eastm an described a m icrom iniature
triode geometry. T he electrode configuration of E astm an’s triode is shown in Fig­
ure 5.23.
The design has square control grid wires fabricated on dielectric supports. The
control grid wires have a 0.3 x 0.3 /im 2 cross section. For the purposes of this
research, the dielectric supports are assumed to have a 0.15 x 0.3 fim 2 cross section.
The anode is located exactly 1 fim above the top of the control grid wires, and is
supported there by a rem ote dielectric spacer. T he unit cell corresponding to this
structure wiil be referred to in the rest of this dissertation as the "smaller unit cell.”
The perveance and amplification factor of one-half of the smaller unit cell were
com puted using the electron optics code EGN2e. as shown in Figure 5.24. T he unit
cell had a height huc of 1.6 /im , a width wuc of 0.6 ^m , and a depth of 5 nm. Three of
the bias points com puted are shown in Table 5.1. The EGN2e simulations indicated
a n of 675 and a unit cell perveance G uc of 20.9 n A /V 3^2. The current-division factor
75
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Anode
[.0
electron
trajectories
u. m
Grid Electrodes
metal
dielectric
Cathode
Figure 5.23: Electrode configuration of the sm aller unit cell.
ECU: 2.708
C1TCLE- 40
1 1
■
0
E A S T -S
SO
100
ISO
EASTHJUT DEVICE — SCLC
2S0
3S0
CW
H
Figure 5.24: Graphical output from an EGNSe simulation of the smaller unit cell.
8 was estim ated to be 0.216.
Based on the capacitance calculation technique described in Chapter 4, the “best
case” (no fringing) value for
C'g c
is 3.61 nF/cm 2 and the “best case" value for
C'g a
is 0.443 n F /c m 2. The "worst case” capacitance values (m axim um fringing) used to
construct the circuit model for an array of the smaller unit cell, are
n F /c m 2,
C ga
C gc
= 5.09
= 0.885 n F /cm 2, and Cac = 4.37 pF /cm 2.
Figure 5.25 indicates how the transconductance per unit area varies with anode
and grid voltages for the smaller unit cell. Figure 5.26 shows how the cutoff fre­
quency of the sm aller unit cell varies with anode and grid voltages. It can be seen
th at if th e grid voltage is kept negative and the anode voltage is kept below 100 V,
the transconductance will not exceed 13 GHz.
76
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
K4 (V)
40.0
40.0
80.0
VG (V)
1.0
1.2
1.0
l c (nA)
22.746
29.483
24.513
I a (nA)
13.445
17.394
18.060
I c (nA)
9.301
12.089
6.453
Table 5.1: Results of EGN2e Simulations of Half of the Smaller Unit Cell
<: 4 0 0
SJ
52. 350
|
300
£
250
£
200
S§u
3
150
■o
100
1
50
3U
H
0
0
20
40
60
Anode Voltage (V)
80
100
Figure 5.25: Transconductance per unit area vs. bias voltages for an array of the
smaller unit cell. The curves, from left to right, correspond to VG = 0.00 V, -0.02
V, -0.04 V, -0.06 V, -0.08 V, -0.10 V. -0.12 V, and -0.14 V.
77
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
N
le+10
s
8e+09
u> 6e+09
U
fc
s
s 4e+09
u
0
20
40
60
Anode Voltage (V)
80
100
Figure 5.26: Cutoff frequency vs. bias voltages for an array of the smaller unit cell.
The curves, from left to right, correspond to Vo = 0.00 V, -0.02 V'. -0.04 V. -0.06
V, -0.08 V, -0.10 V, -0.12 V, and -0.14 V.
20
40
60
Anode Voltage (V)
80
100
Figure 5.27: Anode current density vs. bias voltages for an array of the smaller
unit cell. The curves, from left to right, correspond to Vq — 0.00 V, -0.02 V, -0.04
V, -0.06 V, -0.08 V, -0.10 V, -0.12 V. and -0.14 V.
78
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
£
3500
>t
0
20
40
60
Anode Voltage (V)
80
100
Figure 5.28: Anode DC power dissipation density vs. bias voltages for an array of
th e sm aller unit cell. The curves, from left to right, correspond to Vg = 0.00 V,
-0.02 V, -0.04 V, -0.06 V, -0.08 V, -0.10 V, -0.12 V, and -0.14 V.
10
8
E
O
u-J
s
e*
6
.2
8
T43>
06
4
e
c
<
2
0
0
20
60
40
Anode Voltage (V)
80
100
Figure 5.29: Anode resistance vs. bias voltages for a 1 cm2 array of th e sm aller unit
cell. T he curves, from left to right, correspond to Vg = 0.00 V, -0.02 V, -0.04 V,
-0.06 V, -0.08 V, -0.10 V, -0.12 V, and -0.14 V.
79
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Anode
metal
MA
dielectric
1.0 |im
Grid
Substrate
Figure 5.30: Electrode configuration of the larger unit cell.
Figure 5.27 shows how the current density at the anode varies with anode and
grid voltages.
Figure 5.2S shows how the DC power dissipation density at the
anode varies with anode and grid voltages. It is seen th at if the anode voltage is
kept below 100 V and the grid is kept negative, the current density at the anode
never exceeds 40 A /c m 2 and the power dissipation at the anode never exceeds 4000
YV/cm2. Figure 5.29 indicates the anode resistance of a 1 cm2 array of the smaller
unit cell.
5 .2 .2
Larger U n it C ell w ith C ath od e-to-G rid and G rid-toA n od e Spacers
In the mid-1990’s, a research group at the North Carolina State University designed
a microminiature microwave triode. This design was presented at the 1996 Spring
Meeting of the M aterials Research Society [150]. T he electrode configuration of
th at triode is shown in Figure 5.30.
The design used rectangular control grid wires fabricated on dielectric supports
and an anode m ounted upon grid-to-anode spacers. T he rectangular control grid
wires have a 0.5 x 0.1 /zm2 cross section. The cathode-to-grid spacers have a 0.5
x 0.45 fim 2 cross section. The grid-to-anode spacers have a 0.5 x 2.45 //m2 cross
section. This stru ctu re will be referred to in the rest of this dissertation at the
"larger unit cell.”
Electron optics sim ulations were used to compute the amplification factor n and
SO
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0
20
40
60
80
100
120
NOraX.EGS NZATRICDE SZMDUtnoia -- WITHOXIDE 4-2-96 CVH
140
160
Figure 5.31: Graphical o u tput from an EGN2e sim ulation of the larger unit cell.
VA (V)
40.0
40.0
80.0
VG (V)
1.0
1.2
1.0
tc (nA)
103.16
125.83
151.61
I a (nA)
86.122
9S.357
136.62
I g (nA)
17.038
27.473
14.991
Table 5.2: Results of EGN2e Simulations of Half of the Larger Unit Cell
perveance Guc of one-half of the larger unit cell, as shown in Figure 5.31. The unit
cell in this case had a height of 3.0 /ira. a width of 1.0 /im . and a depth of 20 nm.
T hree of the com puted bias points are shown in Table 5.2. The amplification factor
fi is 97 and the unit cell perveance G uc is 61.5 nA /Y 3/2. T he current-division factor
8 is 0.799.
Based on the capacitance calculation technique described in C hapter 4, the ubest
case” (no fringing) value for
C gc
is 3.84 n F /c m 2 and the "best case” value for
C qa
is 0.704 n F /cm 2. The "worst case” capacitance values (maxim um fringing) used
to construct the circuit model for an array of the larger unit cell are
n F /c m 2,
C ga
= 0.885 n F /c m 2, and
C'a c
C gc
= 4.82
= 40.6 p F /cm 2.
Figure 5.32 indicates how the transconductance per unit area varies with anode
and grid voltages for the larger unit cell. Figure 5.33 shows how the cutoff frequency
varies with anode and grid voltages. It can be seen th at if the grid voltage is kept
negative and the anode voltage is kept below 100 V, the cutoff frequency remains
below 16 GHz.
Figure 5.34 shows how th e current density at the anode varies with anode and
grid voltages.
Figure 5.35 shows how the DC power dissipation density at the
anode varies with anode and grid voltages. It can be seen th a t if the anode voltage
is kept below 100 V and the grid is kept negative, the anode current density will
81
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^
500
|
450
IC5
400
|
350
|
300
fa
V
cu 250
0 200
1 150
1
100
Ia
50
fa
H
0
0
20
60
40
Anode Voltage (V)
80
100
Figure 5.32: Transconductance per unit area vs. bias voltages for an array of the
larger unit cell. The curves, from left to right, correspond to Vg = 0.0 V. -0.2 V.
-0.4 V, -0.6 V, -0.8 V and -1.0 V.
1.6e+10
N
s
>
*
W
s&
>
3
1.2e+10
le+10
2e+09
0
20
60
40
Anode Voltage IV)
80
100
Figure 5.33: Cutoff frequency vs. bias voltages for an array of the larger unit cell.
The curves, from left to right, correspond to Vg = 0.0 V, -0.2 V, -0.4 V. -0.6 V, -0.S
V and -1.0 V.
82
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350
?
S
300
< 250
1
200
Q
|
150
U
u
U 100
1
.
0
20
40
60
Anode Voltage (V)
80
100
Figure 5.34: Anode current density vs. bias voltages for larger unit cell. The curves,
from left to right, correspond to Vg = 0.0 V, -0.2 V. -0.4 V, -0.6 V, -0.S V, and -1.0
V.
not exceed 350 A /cm 2 and the power dissipation at the anode will not exceed 35
kW /cm 2. Figure 5.36 indicates the anode resistance of a 1 cm2 array of the smaller
unit cell.
5 .2.3
C om parison w ith O ther A ctiv e D evices
C om p arison to a SiC JFE T at 773 K
Because one of the most im portant intended uses of microm iniature microwave triodes is for microwave systems and circuits operating in harsh environm ents, it is
instructive to com pare the results for the two microminiature microwave triode
arrays discussed in this chapter with a com peting solid state device for high tem ­
perature microwave operation. The results of the comparison are shown in Table
5.3 and are discussed below.
The device chosen for comparison is a SiC JF E T , for which sim ulations have
been previously performed [151]. The operating tem perature of the JF E T is 773 K.
The basic Uunit cell” of the SiC JF E T is a region 5.8 /im x 1 mm upon a SiC wafer.
The gate-to-source capacitance of the JF E T is 0.42 pF. It was found th at when
such a device was operated with a drain voltage of 45 V and a Class A bias point
S3
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35000
?
30000
^
25000
>)
i<u
20000
jj
15000
W
10000
u
■a
os
<
5000
Q
Bo.
0
20
40
60
Anode Voltage (V)
80
100
Figure 5.35: Anode DC power dissipation density for an array of larger unit cells
with anode spacers. T he curves, from left to right, correspond to l c = 0.0 V'. -0.2
V. -0.4 V, -0.6 V. -0.S V, and -1.0 V.
1.8
1.6
s
14
]a>u
1
0.6
<
0.4
0.2
0
20
40
60
Anode Voltage (V)
80
100
Figure 5.36: Anode resistance for a 1 cm2 array of larger unit cells with anode
spacers. The curves, from left to right, correspond to Vg = 0.0 V. -0.2 V, -0.4 V,
-0.6 V, -0.8 V and -1.0 V.
84
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anode/drain voltage (V)
total device area (m m 2)
grid/gate bias (V)
DC anode/drain current (mA)
DC anode/drain power (W)
Cg c / C gs ( p F)
transconductance (mS)
cutoff frequency (GHz)
anode/drain resistance (kff)
SiC JF E T
45
0.0058
-5
77
3.47
0.42
16
6.1
O.S
Sm aller U. C.
45
0.0058
-0.0247
0.348
0.0157
0.295
12.4
6.7
54.6
Larger U. C.
45
0.0058
-0.170
2.84
0.128
0.285
14.3
8.0
15.24
Table 5.3: Comparison of a SiC JF E T with M icrom iniature Microwave Triode Ar­
rays Operating at 773 K
of Vg = -5 V, a transconductance of 16 mS and a cutoff frequency of 6.1 GHz were
achieved. The drain resistance was about S00 ft. This operating point corresponds
to a total DC current of 77 mA and total DC power dissipation of 3.47 YV.
For the smaller unit cell, if it is assumed th a t the area of the triode array is the
sam e as the area of the SiC JF E T structure (5.S /im x 1 m m ), the total gate-tocathode capacitance will be 0.295 pF. It is found th at an anode voltage of 45 V and
a grid voltage of -0.0247 V will provide a total transconductance of 12.4 mS and
a cutoff frequency of 6.7 GHz. This operating point corresponds to a DC current
density of 6 A /cm 2 and a total DC anode current of 348 fiA. Note that the fact
th at half of the unit cell cathode region is em itting an d the fact that a Class A
biasing scheme will drive the anode current to about 1.5 tim es its DC value implies
th at the peak current density at the cathode will be approxim ately IS A /cm 2. The
anode resistance at this operating point is 54.6 kfi. T he power dissipation density
at the anode is 270 YV/cm2, and the total available anode DC power is 15.66 mVV.
Hence, although the cutoff frequency of the smaller unit cell is comparable to th at
of the JFE T , the DC power available from the device is 222 times smaller.
For the larger unit cell, if it is assumed th at the area of the triode array is the
sam e as the area of the SiC JF E T structure (5.S fim x 1 m m ), the total gate-tocathode capacitance will be 0.285 pF. It is found th a t an anode voltage of 45 V
and a grid voltage of -0.170 V will provide a total transconductance of 14.3 mS and
85
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anode/drain voltage (V)
total device area (mm2)
g rid /g a te bias (V)
DC anode/drain current (mA)
DC anode/drain power (VV)
C g c / C g s (pF)
transconductance (S)
cutoff frequency (GHz)
anode/drain resistance (fi)
SiC JF E T
45
0.0058
-5
77
3.47
0.42
0.016
6.1
800
Smaller U. C.
45
1.29
-0.0247
77
3.47
64.9
2.73
6.7
248
Larger U. G'.
45
0.157
-0.170
77
3.47
7.70
0.386
S.O
564
Table 5-4: Comparison of a SiC JF E T with Scaled Microminiature Microwave Triode Arrays O perating at 773 K
a cutoff frequency of S.O GHz. This operating point corresponds to a DC current
density of 49 A /cm 2 and a total DC anode current of 2.84 mA. Again, th e fact th at
half of the unit cell cathode region is em itting and the fact that a Class A biasing
scheme will drive the anode current to about 1.5 times its DC value im ply th a t the
peak current density at the cathode will be approxim ately 147 A /cm 2. T he anode
resistance at this operating point is 15.2 kQ. The DC power dissipation density
at th e anode is 2200 W /cm 2 and the total available anode DC power is 0.128 W.
Hence, for an array of larger unit cells, the cutoff frequency is slightly higher, but
th e available DC power has dropped by a factor of 27.
It can be ascertained from the above analysis th at in order for the m icrom iniature
microwave triode axrays to compete at the sam e power level as the SiC J F E T device
w ithout utilizing a higher anode voltage, the area of the arrays would need to be
scaled up accordingly. The smaller unit cell array area would need to be increased
by a factor of 222 and the larger unit cell array area would need to be increased
by a factor of 27. The characteristics of the scaied-up arrays compared to the SiC
JF E T are com pared in Table 5.4. If the arrays are scaled in this way. the cutoff
frequency is preserved and the available DC power is made equal to th at of the SiC
JF E T . Note th a t scaling the arrays in this way brings the anode resistances of the
arrays down below th at of SiC JFE T . closer to the characteristic impedances found
in typical microwave circuits.
86
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The continuous current densities calculated in this comparison are quite large in
comparison to those observed for conventional thermionic cathodes, and therefore
the use of AlxGa!_xN cathodes would seem to be required. The peak current density
of about 18 A /cm 2 needed for the smaller unit cell array and the peak current density
of about 147 A /cm 2 needed for the larger unit cell array are not achievable with
any commercially available thermionic cathodes.
In the comparison above, the microminiature microwave triode arrays are op­
erating at 773 K. This means that the AlxG ai_xN cathode will be heated to 773
K. For the basic cathode structure, examining Figure 5. IS. the emission barrier
at 773 K can be estim ated to be about 0.570 eY. Based on this barrier and this
tem perature, the barrier-lim ited thermionic emission current density would be 26*20
A /cm 2, hence barrier lim itations will not prevent the triode from working prop­
erly. Examining Figure 5.22, the mobility required to achieve the barrier-lim ited
current density should be approximately 43 cm2/V-s. Hence, the m obility required
in the Alo. 75Gcio.25N material at 773 l\ to achieve a continuous current density of
147 A /cm 2 would be only about 2.4 cm2/V-s. Based upon the AlxG ai_ xN mobility
d ata (see Figure 3.9) indicating that the electron mobility for the x = 0.75 material
would be about 30 cm 2/V -s at 300 K. 2.4 cmJ/V-s appears to be a reasonably small
mobility value, hence, both triode arrays should operate at 773 K as indicated in
the electron optics simulations.
C om p arison to O ther M icrowave Triodes
As described in the literature review [4. 99. 92]. microwave triodes steadily im­
proved over time. This improvement can be traced using the cutoff frequency as
a param eter. The typical transconductance for a typical bias point, the grid-tocathode capacitance, and the cutoff frequency (as defined in this work) of several
historically im portant microwave triodes are given in Table 5.5, along with the cut­
off frequency values com puted for the two microminiature microwave triode arrays
in this study. The rise in cutoff frequency of microwave triodes with tim e is shown
in Figure 5.37. The anode voltages of these older triodes were 250 V or higher.
The last two points on Figure 5.37 are theoretical calculations, corresponding to
the years in which the microminiature microwave triode arrays were proposed. It
S7
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acorn tube
2C43
1553/416A
L-65
smaller unit cell
larger unit cell
Year
1933
1945
1949
1965
1989
1995
9m (S)
C gc (pF)
0.00155
0.008
0.05
0.3
2.73
0.386
0.7
2.8
10
27
64.9
7.7
h
(GHz)
0.352
0.455
0.796
1.77
6.7
8.0
Reference
[92]
[1S2]
W
[99]
Table 5.5: Increase of Microwave Triode Cutoff Frequency with Time
10
O-
8
6
4
2
0 L_x----- 1-----1------ 1------ 1------ 1------ 1-----1930
1940 1950 1960 1970 1980 1990 2000
Year
Figure 5.37: Increase of microwave triode cutoff frequency with time.
is seen th at the cutoff frequency for the structures analyzed in this dissertation are
five tim es larger than for any previous microwave triode, with an anode voltage at
least five times smaller.
88
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C h a p ter 6
C on clu sion s
6.1
6 .1 .1
Summary of Results
AlxGai_xN C athodes
A sum m ary of the results for the cathode sim ulations performed in this stu d y are
shown in Table 6.1. The current values refer to the am ount of current which would
be em itted by a 1 cm 2 cathode. The diffusion-limited current density of cathode
structures which have a current density limited by clifusion were com puted by as­
suming th a t the electron mobility in the x = 0.75 m aterial is 30 cm2/V -s.
"B-L
C u rr.” refers to barrier-limited current. "D-L C u rr.“ refers to diffusion-limited cur­
rent, and “Exp. C urr.” refers to the current which would be expected when barrierand diffusion-limiting are taken into account. T he results for AlxG at_xN cathode
structures lead to many im portant guidelines concerning the design and operation
of AlxG ai_ xN cathodes.
T he use of n-type doping and compositional grading appears to be an effec­
tive means of creating an AlxGai_xN cathode with a very low thermionic emission
barrier. T he superpositioning of the n-type doping with the grading results in a
charge distribution which brings the conduction band at the surface down towards
the Fermi level, resulting in a very low thermionic emission barrier.
T he basic cathode structure, which utilized a 75-nm-thick AlxG ai_xN layer with
its A1 fraction (x) varied linearly with position and a constant doping of 1018cm -3 ,
exhibited a barrier of just 0.560 eV at 300 K. This emission barrier is much lower
than th at generally observed for commercially available thermionic cathodes ( i eV
89
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S tru c tu re
basic (300 K)
basic (450 K)
basic (700 K)
0.25 /zm (300 K)
0.50 /zm (300 K)
concave up (300 K)
concave down (300 K)
m oderate N q (300 I\)
low N q (300 K)
B a rrie r
0.560 eV
0.562 eV
0.569 eV
0.474 eV
0.430 eV
0.654 eV
0.478 eV
0.607 eV
0.936 eV
B -L C u r r .
792 /zA
2.34 A
S94 A
22.1 mA
122 mA
21.4 /zA
19.1 mA
132 /zA
3S9 pA
D -L C u r r .
1.13 mA
2.34 A
638 A
7.8 mA
18.1 mA
91.7 /zA
235 /zA
189 /zA
530 /zA
E xp. C u rr.
792 /zA
2.34 A
638 A
7.8 mA
18.1 mA
21.4 /zA
235 /zA
132 /zA
389 pA
Table 6.1: Summary of Cathode Simulation Results
or more) [8].
Because d ata is scarce on the ionization energy of Si donors and the heterojunc­
tion conduction band offset in AlxG ai_xN. reasonable assumptions had to be m ade
for these quantities. T he simulations indicated th at even the “worst case" values
for these param eters still indicated an emission barrier at or below 0.560 eV at 300
K.
Simulations were performed for AlxG at_xN layers of thickness 0.25 /zm and
0.50 /zm. These simulations clearly indicate th at increasing the thickness of the
AlxG ai_xN layer substantially reduces the emission barrier. The emission from these
thicker structures is diffusion-limited, but the em itted current density at 300 K is
more than an order of magnitude larger than th at of the 0.075-/zm-thick structure.
Hence, increasing the layer thickness is a useful technique for increasing current
density. Unfortunately, there is a limit to how thick the AlxGax_xN layer can be.
based on film growth considerations.
Simulations done for Si doping levels below 1018cm~3 indicate that the emission
barrier is a strong function of doping. Reducing the doping by a factor of 2 increased
the barrier by about 8%. which lowers the room tem perature current density to 17%
of its original value. Reducing the doping by a factor of 10 increased the barrier
by more th an 67%, which decreases the room tem perature emission by a factor of
more than 106. Obviously, excessively reducing the doping level to decrease the
probability of film cracking can have serious electronic ramifications.
90
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P a r a m e te r
anode voltage (V)
grid voltage (V)
anode current density (A /cm 2)
anode power dissipation (W /cm 2)
transconductance (S /cm 2)
C cc (n F /cm 2)
cutoff frequency (GHz)
S m a lle r U . C .
45
-0.0247
6
270
210
5.09
6.7
L a rg e r U . C .
45
-0.170
50
2200
250
4.82
S.O
Table 6.2: Sum m ary of M icrom iniature Microwave Triode Array Simulation Results
Simulations for non-linear grading profiles produced interesting results. A cath­
ode with a concave-down quadratic profile has a reduced emission barrier, but also
is diffusion-limited. A cathode with a concave-up quadratic profile has a large emis­
sion barrier, but also has enhanced electron diffusion. As a result, neither non-linear
grading profile provided a net increase in current density over the density produced
by the linear profile at 300 K. It seems unclear if optim ization, by way of a tailored
grading profile, could be accomplished.
T he basic cathode structure's operation at elevated tem peratures was investi­
gated. These sim ulations indicate th at the emission barrier changes very little with
tem perature. T he results indicate th a t the current density will be very large and
m ay exceed 600 A /cm 2 (even with diffusion lim itations taken into account) at or
below 700 K.
6 .1 .2
M icrom iniature M icrow ave T riodes
A sum m ary of the microm iniature microwave triode array simulations is given in
Table 6.2. The values for anode current density, anode power dissipation, transcon­
ductance, and cutoff frequency correspond to the DC bias point defined by the
anode and grid voltages. The sim ulation results for the m icrom iniature microwave
triode arrays describe how these devices work and how their performance compares
to th a t of both traditional microwave triodes and th at of solid state devices with
sim ilar bias conditions and size at elevated tem peratures.
According to the electron optics calculations done in this work, the smaller
91
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unit cell, originally discussed by Eastm an [103], operates quite differently th an the
way Eastm an originally described. Eastm an stated in his conference paper th at
the am plification factor of such a structure would be 23. but the detailed electron
optics calculations done in this work indicate an am plification factor of 675. which is
considerably larger. This larger value is suggested by the formulas derived in basic
triode theory for amplification factor as a function of electrode geom etry [144].
T he sim ulations indicate that for an anode bias of 45 V and an appropriate
Class A grid bias, the anode DC current density will be about 6 A /cm 2, th e DC
power dissipation density at the anode will be on the order of 270 W /c m 2, the
transconductance per unit area will be about 210 S /cm 2, and the cutoff frequency
will be about 6.7 GHz.
Based on these simulations, the larger unit cell, designed at NCSU [150], appears
to exhibit b etter performance compared to the sm aller unit cell. The sim ulations
indicate th a t an anode bias of 45 V and an appropriate Class A grid bias with
result in an anode DC current density of about 50 A /cm 2, a power DC dissipation
density at the anode on the order of 2200 W /cm 2. a transconductance per unit area
of about 250 S /cm 2, and the cutoff frequency will be about S GHz.
In an attem p t to assess the competitiveness of the triode arrays in harsh en­
vironments, the performance of the arrays was com pared to that of a SiC JF E T
operating at elevated tem perature (773 K). To make the comparison m eaningful, it
was assum ed th at the anode or drain bias of all three devices was the sam e (45 V).
It was found th at the smaller unit cell array could achieve about the sam e cutoff
frequency and power level as the SiC JF E T . but only with an array area 222 times
larger th an th at of the JF E T . However, an array of the larger unit cell perform ed
better. It was found th at an array of the larger unit cell could achieve a slightly
higher cutoff frequency than the SiC JF E T at the same power level, but with an
area 27 tim es larger than the JFET.
A comparison was also made between the cutoff frequency of these m iniature
microwave triodes and that of larger, conventional microwave triodes. Conventional
microwave triodes operate with anode voltages of 250 V or higher and have a cutoff
frequency below 2 GHz. It is seen that, at least theoretically, it should be possible
for these m icrom iniature microwave triode arrays to achieve a cutoff frequency 3-4
92
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tim es larger than those for conventional microwave triodes [99], while operating the
array with an anode voltage at least 5 times smaller than the anode voltage of a
conventional microwave triode [4, 152].
6.2
Implications of the Research
6.2.1
AlxGai_xN C ath od es
The sim ulations presented in this dissertation describe a new type of AlxG ai_xN
cathode which has th e potential for revolutionizing therm ionic cathode technology.
It appears that these cathodes will emit relatively high continuous current densities
(100 A /cm 2 or even more) at relatively low operating tem peratures (under 700 K).
Conventional therm ionic em itters can emit no more than 10 A /cm 2 continu­
ously, and they operate at tem peratures exceeding 1100 K. They suffer from lim ited
operational life, sensitivity to ion bombardment, and excessive power consumption.
The cathode heater heats the surrounding components, causing reliability problems.
The AlxG ai_xN cathodes described in this work, if realized, would be high-currentdensity, low er-tem perature, robust cathodes which enjoy much longer operational
life, consume much less power, and heat the tube assembly much less. Hence, these
cathodes would be an attractive replacement for conventional thermionic cathodes.
M anufacturers of all types of vacuum tubes, particularly high-power, highfrequency microwave tubes, would be interested in using these AlxG ai_xN cath­
odes. Such tubes would include traveling-wave tubes (T W T ’s) and klystrons and
also more exotic vacuum devices, such as electron-bombarded semiconductor (EBS)
tubes. The higher current density would allow higher power at higher frequencies,
more compact designs, and superior performance. The lower operating tem perature
will result in much lower heater power and better tube reliability. These AlxG ai_xN
cathodes would also be of interest to those constructing electron guns, which are
used to supply a stream of electrons in many types of vacuum systems. The com­
bination of higher current density at a lower operating tem perature will result in
electron guns which are smaller, operate at a lower tem perature, axe more reliable,
and consume less power.
Finally, it is also im portant to note that these new planar AlxG ai_xN cathodes
93
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operating at lower tem peratures enable the technology of space-charge-limited mi­
crom iniature microwave triodes. Conventional thermionic cathodes do not have a
planar surface (they are made from a granular tungsten m atrix), so microminia­
ture structures cannot be fabricated on their surfaces. Also, because conventional
therm ionic cathodes operate at such a high tem perature, the cathode surface is too
hot to allow for any material to rest on the cathode surface, and the high tem pera­
ture causes a massive deformation of all materials near the cathode surface. Finally,
and most importantly, conventional thermionic cathodes cannot supply high contin­
uous current densities. Thus, th e development of a new planar, lower-temperature,
low-emission-barrier thermionic cathode, such as an AlxGa!_xN cathode, is needed
for the realization of m icrom iniature microwave triode arrays.
6 .2.2
M icrom iniature M icrow ave Triodes
The simulations presented in this dissertation of the m icrom iniature microwave
triode arrays describe a new type of active microwave device. This device may
prove useful for certain types of microwave systems and circuits operating in certain
environments.
The arrays do not exhibit performance which could be considered competitive
with solid state devices at room tem perature in normal operating environments.
Commercially available M ESFET's and PHEMT s operating at room tem perature
can now easily achieve cutoff frequencies in the range of 22 to 55 GHz with drain
voltages no higher than 10 V [153]. Based on the results of the simulations of the
arrays, it seems unlikely that the arrays could compete with these devices, assuming
that the anode voltage is kept below 10 V and that both devices are of comparable
size.
However, for systems and circuits operating in conditions of higher tem perature
and high levels of radiation, these arrays might provide suitable performance. Stan­
dard semiconductor devices do not operate above 500 K. and if they are used in an
environment with radiation they must be shielded and/or fabricated with special
(and expensive) processing techniques to achieve radiation hardness. Examples of
harsh environments are those involving satellites, aircraft, engines, nuclear reac­
tors, particle accelerators, fusion reactors, well logging, spacecraft, power plants,
94
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
sensors, radar, and other applications. It is also im portant to recognize th at ordi­
nary solid s ta te devices are not resistant to high power microwaves (HPM ) or to
electrom agnetic pulses (EM P) which can exist in a m ilitary environm ent. Micro­
miniature microwave triode arrays may be able to fulfill the need for electronics
with this kind of resistance. Furthermore, as the am ount of electronics in satellites,
spacecraft, an d space stations increases over tim e, m icrom iniature microwave triode
arrays may provide a reasonable, low-cost three-term inal device for high frequency
circuits.
Much work has been done in recent years on the development of wide bandgap
semiconductor devices for high-power. high-frequency, and high-tem perature appli­
cations. Two m aterials which have received a considerable am ount of attention are
SiC and GaN. T he results presented in this dissertation comparing the microminia­
ture microwave triode arrays with a SiC JF E T operating at 773 K seem to indicate
that these triode arrays might be com petitive with with wide bandgap semicon­
ductor devices for applications involving harsh environments, but the outlook is
uncertain.
6.3
Suggestions for Further Research
6.3.1
AlxGai_xN C ath od es
The cathode sim ulations presented in this dissertation provide a strong motivation
for the commercial development of AlxG ai_xN cathodes. This developm ent will
require further work on simulation, fabrication, and characterization.
In order to obtain more accurate results for the basic sim ulations, the values
for the m aterial param eters of the AlxG ai_xN m aterials must be more accurately
determined. T h e key m aterial param eters for cathode operation are electron affin­
ity, donor ionization energy, heterojunction conduction band offset, and electron
mobility.
There are a num ber of factors which were left out of the sim ulations, which
deserve further consideration. The possibility of surface states or surface recon­
structions were not considered. The effects of having something other than Al.
Ga, or N on th e em itting surface (e.g.. 0 or C) was not exam ined. More experi95
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
m ental work is needed to investigate these surface conditions and include them in
sim ulations, if necessary.
T he R unge-K utta technique implemented in Maple V software on a Sun SPARCstation 4 was adequate for the purposes of this research. However, further sim u­
lations of AlxGax_xN cathode structures, particularly any attem pts involving opti­
m ization (through m aniupulation of doping, grading, and thickness), may require
faster software on a faster computer to keep the com putation tim e reasonable.
As work on AlxG ai_xN materials continues. AixG ai_xN material quality will
improve and the m aterial will be much better characterized.
Improved growth
techniques should result in higher-quality semiconductor m aterial and lead to higher
electron mobility. Enhanced electron mobility should ensure th at the cathodes are
barrier-lim ited rath er than diffusion-limited. Also im portant is development of the
ability to create sm ooth spatial grading of the A1 fraction x.
It would be useful to install these cathode structures in microwave tubes (or an
equivalent testing environm ent) and observe the cathodes’ reliability and operating
lifetime. Finally, because one of the most promising uses for these cathodes is in
microwave tubes operating with high levels of radiation, further characterization of
the radiation hardness of AlxG ai_xN materials would be helpful in determining the
suitability of these cathodes for such environments.
6 .3 .2
M icrom iniatu re Microwave Triodes
T he m icrom iniature microwave triode array simulations presented in this disserta­
tion suggest th at the development of these arrays m ight be worthwhile for harsh
operating environm ents involving high tem peratures and radiation. Some of the
work th at would be needed to achieve this development are discussed below.
The electron optics simulations in this work make certain assumptions that may
not be found in practice. First, the simple form of the equations describing the triode
assumes th a t the initial velocity of electrons coming from the cathode is zero. This
is usually not true - the electrons have a distribution of initial velocities. Second,
the calculations do not take into account the potential differences between the three
electrodes, due to the work function differences between the three materials. Both of
these effects will cause a deviation from the simple three-halves-power law equations
96
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used for these simulations.
T he interelectrode capacitances assumed in this work were "*worst case"1 values,
and are not particularly accurate. More detailed capacitance calculations (perhaps
using appropriate computer software) would provide more accurate results. It is
also interesting to note th a t the assumption in this work th a t the interelectrode
capcitances remain constant when space charge exists between the electrodes is not
necessarily true.
Due to the very small interelectrode spacings and reasonably large electrode
voltages of these triodes, it seems unlikely that transit tim e from cathode to grid or
from grid to anode could lim it these devices below the limit imposed by transcon­
ductance and capacitance. However, future analysis should investigate these effects.
These calculations only present the expected performance for small-signal, linear
conditions. Hence, strictly speaking, the results are only applicable to highly linear,
small-signal amplifier circuits. However, the research done here provides a triode
array model containing information complete enough to perform many types of
meaningful circuit simulations, and the use of these devices in power amplifiers,
oscillators, and other types of microwave circuits could be analyzed.
T he techniques needed to fabricate such devices are not m ature, although such
techniques are currently developing in the manufacture of field emission displays.
For the structures described in this work, the chief concern would probably be
the anodes and how they might be cooled. Cooling systems which use circulating
water are commercially available which can remove 1 k W / cm2 from an anode, and
more advanced, specially-built water-cooling systems have been constructed which
can remove 10 kW /cm 2 [154]. If more complex anode structures were allowed, then
depressed-collector techniques (in which the anode voltage is stepped down through
successive stages to reduce heat generation at the collector and increase efficiency)
would be very useful.
Packaging issues are also im portant, particularly with respect to vacuum sealing
and microwave circuit issues. The achievement of a high-quality vacuum seal is
one of the key problems currently faced by field emission display manufacturers.
Designing the small package containing the array in such a way th at it is properly
m atched to its circuit environment at the appropriate frequency is an im portant
97
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design problem th at must be solved. Experience with traditional triodes suggests
th at lead inductances can have a deleterious effect at high frequencies, so the design
of the device leads will be im portant, as well.
DC and R F testing of the devices at room tem perature should be performed,
and the devices should be tested under conditions of heat and radiation, as well.
One phenomenon im portant for microwave circuit performance is noise, and this
will need to be investigated.
It would be interesting to observe at how high a
tem perature the devices can survive and up to what level of radiation they will
be able to w ithstand.
From these tests, the suitability of these arrays in harsh
environments and their commercial viability would be determ ined.
98
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109
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A p p en d ix A
Maple V In p u t File for C ath od e
Sim ulation
Below is a sample input file for Maple V which was utilized to simulate AlxG at_xN
cathodes. The code does the following:
1. solves Poisson’s equation by way of a R unge-K utta numerical procedure to
obtain potential as a function of position
2. computes the derivative of potential with respect to position
3. computes the second derivative of potential with respect to position
4. evaluates the right-hand side of Poisson's equation to verify that Poisson's
equation was solved correctly
5. computes the location of the conduction band minimum energy as a function
of position
6. computes the free electron concentration as a function of position
7. computes the ionized donor concentration as a function of position
S. computes the total charge concentration as a function of position
9. computes diffusion current density as a function of position (by way of com­
puting the derivative of electron concentration with respect to position)
110
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10. c o m p u te s th e drift current d ensity as a fun ctio n of position (b y way of c o m ­
p u tin g t h e derivative of th e c o n d u ctio n ban d m in im u m e n e r g y w ith resp ect
to positio n)
N ote t h a t t h e lines preceded by the sy m b o l
are c o m m e n t lines.
# M aple V c o d e f o r s o l v i n g P o i s s o n ' s e q u a t i o n i n s i d e
# of a s e m ic o n d u c to r cath o d e.
D i g i t s := 1 8 :
#
#
#
#
#
#
#
#
#
#
#
#
#
#
e x l = r i g h t - h a n d s i d e o f P o i s s o n ' s e q u a t i o n ( i n V/mA2)
q = e l e c t r o n i c c h a r g e ( i n C)
k = B o l t z m a n n 's c o n s t a n t ( i n
J /K )
T = t e m p e r a t u r e ( i n K)
nd = d o n o r c o n c e n t r a t i o n ( i n mA-3)
n c = c o n d u c t i o n b a n d e f f e c t i v e d e n s i t y o f s t a t e s ( i n m A -3)
p h i = p o t e n t i a l ( i n V)
e f = F erm i l e v e l ( i n J)
y = p o s i t i o n ( i n m)
f x l =f u n c t i o n d e s c r i b i n g t h e A1 f r a c t i o n
x as a f u n c tio n of p o s itio n y
e c d =d o n o r i o n i z a t i o n e n e r g y ( i n J )
d e c =h e t e r o j u n c t i o n c o n d u c t i o n b a n d o f f s e t
(in J)
perm = d i e l e c t r i c p e r m i t t i v i t y ( i n F/m)
d i f f p h i = d e r i v a t i v e o f p o t e n t i a l ( i n V/m)
# D e f in it io n o f r ig h t- h a n d s id e o f P o i s s o n 's e q u a tio n .
e x l := p r o c ( p h i , y )
lo c a l q ,p e rm ,n d ,e c d ,k T ,n c ,fx l,d e c ;
q := 1 .6 0 2 e -1 9 ;
k T : = 0 . 0 2 5 8 5 * 1 . 6 0 2 e - 19;
n d := l.0 e 2 4 ;
n c := 2 .084e24;
p e r m : = 8 . 7 5 * 8 . 8 5 e - 12;
f x l : = p r o c ( y ) i f y<0 t h e n 0 e l s e 1 . 0 e 7 * y f i e n d ;
e c d : = ( ( 0 . 6 2 6 6 6 6 6 6 6 * f x l ( y ) ) + 0 .0 3 0 ) * 1 . 6 0 2 e - 19;
d e c : = ( 1 . 4 4 - 1 . 9 2 * f x l ( y ) ) * 1 . 6 0 2 e - 19 ;
- ( q / p erm ) * ( ( n d / ( 1 . 0 + 2 . 0*exp ( ( e f + q * p h i + d e c (y ) +ecd ( y ) ) / k T ) )
- (n c * e x p ( ( e f + q * p h i + d e c (y) ) /k T ) ) ) ) ;
end:
111
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
h = R u n g e - K u t t a s t e p s i z e ( i n m)
n = num ber o f R u n g e -K u tta s t e p s
# S te p s i z e and num ber o f s t e p s f o r R u n g e -K u tta p ro c e d u re
# a re chosen.
h := l.O e -9 :
n := 1 2 5 :
#
#
#
#
yk =
p h ik
pk =
ef =
p o s i t i o n ( i n m)
= p o t e n t i a l ( i n V)
s l o p e o f p o t e n t i a l ( i n V/m)
F erm i l e v e l ( i n J)
# I n i t i a l c o n d itio n s a t l e f t - h a n d boundary o f c a th o d e
# s tr u c tu r e are d e fin e d .
y k := -5 0 e -9 :
y k s to r e := -50e-9:
p h i k : = 0 .0 0 :
e f : = - 1 .4 8 0 7 3 7 7 6 7 2 2 4 3 7 * 1 . 6 0 2 e - 1 9 :
p k := 1 1 6 .3 5 3 8 :
# e x p t s = a r r a y f o r s t o r i n g a l l R u n g e -K u tta c o m p u ta tio n s
#
( p o s itio n , p o t e n t i a l , s lo p e of p o te n tia l )
# p e x p t s = a r r a y c o n t a i n i n g R u n g e - K u tt a r e s u l t s f o r p u r p o s e s
#
o f p lo t tin g g rap h s ( p o s itio n , p o te n tia l )
# S to ra g e a rra y s a re i n i t i a l i z e d .
e x p t s : = a r r a y (0. . n) :
p e x p t s := a r r a y ( 0 . .n ) :
# L e f t- h a n d b o undary c o n d i t i o n s a re p la c e d i n t o s to r a g e a r r a y s .
e x p t s [0 ] : = [ y k , p h i k , p k ] :
p e x p t s CO] : = [ y k , p h i k ] :
# F o u r th - o r d e r R u n g e-K u tta n u m e ric a l t e c h n i q u e .
P o te n tia l is p lo tte d .
f o r i from 1 to n
do
112
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
k l := h * p k ;
ll:= h * e x l(p h ik ,y k );
k 2 : = h * ( p k + 0 .5 *1 1) ;
1 2 : = h * e x l (p h ik + O . 5 * k l , y k + 0 . 5 * h ) ;
k 3 : = h * ( p k + 0 . 5*12) ;
1 3 : = h * e x l ( p h i k + 0 . 5 * k 2 ,y k + 0 . 5 * h .) ;
k 4 := h * ( p k + 1 3 ) ;
1 4 : = h * e x l ( p h ik + k 3 , yk+h) ;
p h i k : = p h i k + ( ( l . 0 ) / ( 6 . 0 ) ) * (k l+ 2 * k 2 + 2 * k 3 + k 4 ) ;
p k : = pk+ ( (1 .0 )/(6 .0 ))* (1 1 + 2 * 1 2 + 2 * 1 3 + 1 4 );
y k :=yk+h;
e x p t s [ i ] : = [ y k , p h i k , p k ] ,*
p e x p t s [ i ] := [y k , p h ik ] ;
od:
p lo t(p e x p ts ) ;
# D e riv a tiv e of p o te n tia l ( d if f p h i) is c a lc u la te d and p lo t te d .
d i f f p h i : = a r r a y ( l . .n ) :
f o r g f r o m 1 t o n do
d i f f p h i [g ] : = [ y k s t o r e + h * g - 0 . 5 * h , ( e x p t s [g] [2] - e x p t s [ g - l ] [2] ) / h ] ;
od:
p lo t(d iffp h i);
# Second d e r iv a tiv e of p o t e n t i a l (d iff2 p h i) is c a lc u la te d
# and p l o t t e d .
d i f f 2 p h i : = a r r a y ( l . .n -1 ) :
f o r g f r o m 1 t o n - 1 do
d i f f 2 p h i [ g ] : = [ y k s t o r e + h * g , ( d i f f p h i Cg+1] [ 2 ] - d i f f p h i [g ] [ 2 ] ) / h ]
od:
p lo t(d iff2 p h i) ;
;
# R ig h t-h a n d s id e o f P o i s s o n 's e q u a tio n (R ig h tS id e ) i s e v a l u a te d ,
# u s i n g t h e co m puted s o l u t i o n f o r p o t e n t i a l , i n o r d e r t o v e r i f y
# t h a t P o i s s o n ' s e q u a t i o n w as s u c c e s s f u l l y s o l v e d .
R i g h t S i d e : = a r r a y ( 0 . .n ) :
f o r g f r o m 0 t o n do
R i g h t S i d e [g] : = [ y k s t o r e + h * g , e x l ( e x p t s [g] [2] , y k s t o r e + h * g ) ] ;
od:
p lo t (R ig h tS id e );
113
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
# L o c a t i o n o f t h e c o n d u c t i o n b and minimum e n e r g y ( E c d i r e c t ) a s a
# fu n c tio n of p o s itio n is c a lc u la te d .
E c d ir e c t:= array ( 0 . . n ) :
e x l8 := p ro c (t)
lo c a l fx l.y ;
y := y k sto re + t* h ;
f x l : = p r o c ( y ) i f y<0 t h e n 0 e l s e 1 .0 e 7 * y f i e n d ;
C1 . 4 4 - 1 . 9 2 * f x l ( y ) ) * 1 . 6 0 2 e - 1 9 ;
end:
f o r t fro m 0 t o n do
E c d i r e c t [ t] : = [ y k s t o r e + t * h ,- l .6 0 2 e -1 9 * p e x p ts [t] [ 2 ] - e x l 8 ( t ) ] :
od:
# The c o n d u c t i o n b a n d d a t a p o i n t s a r e c o n v e r t e d t o u n i t s o f eV
# and p l o t t e d .
E c d i r e c t g r a p h : = a r r a y ( 0 . .n ) :
f o r t fro m 0 t o n do
E c d ire c tg ra p h [t] := [y k s to re + t* h ,E c d ire c t[t] [ 2 ] /l.6 0 2 e - 1 9 ] :
od:
p l o t (E c d ire c tg ra p h ) ;
# The f r e e e l e c t r o n c o n c e n t r a t i o n ( e l e c t r o n s ) a s a f u n c t i o n
# of p o sitio n is c a lc u la te d .
e le c tro n s:= a rra y (0 . . n ) :
f o r t fro m 0 t o n do
n c := 2 .084e24:
k T : = 0 .0 2 5 8 5 * 1 . 6 0 2 e - 1 9 :
e l e c t r o n s [ t ] := [ y k s to r e + t* h ,n c * e x p ( ( e f - E c d ir e c t [ t ] [ 2 ] ) /k T ) ] ;
od:
# The e l e c t r o n c o n c e n t r a t i o n d a t a p o i n t s a r e c o n v e r t e d t o u n i t s
# o f cmA-3 and p l o t t e d .
e le c tro n sg ra p h := a rra y (0 . . n ) :
f o r t fro m 0 t o n do
e le c tro n s g ra p h [t] := [ y k s to r e + t* h ,e le c tr o n s [ t] [ 2 ] /le 6 ] ;
od:
p lo t(e le c tro n sg ra p h );
114
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# T he i o n i z e d d o n o r c o n c e n t r a t i o n ( n d p l u s ) a s a f u n c t i o n o f
# p o s itio n is c a lc u la te d .
e x n d p := p r o c ( p h i , y )
lo c a l q ,n d ,e c d ,k T ,fx l,d e c ;
f x l : = p r o c ( y ) i f y<0 t h e n 0 e l s e 1 . 0 e 7 * y f i e n d ;
e c d := ( ( 0 . 6 2 6 6 6 6 6 6 6 * f x l ( y ) ) + 0 . 0 3 0 ) * 1 . 6 0 2 e - 1 9 ;
q : = l . 602 e-1 9 ;
k T : = 0 . 0 2 5 8 5 * 1 . 6 0 2 e - 19;
n d := 1 .0 e 2 4 ;
d e c : = ( 1 . 4 4 - 1 . 9 2 * f x l ( y ) ) * 1 . 6 0 2 e - 19;
n d /(1 .0 + 2 .0 * e x p ((e f+ q * p h i+ d e c + e c d (y ))/k T ));
end:
n d p l u s : = a r r a y ( 0 . .n ) :
f o r t f r o m 0 t o n do
n d p lu s [t]:= [y k s to re + t* h ,e x n d p (e x p ts [t] [ 2 ] ,y k s to r e + t* h ) ] ;
od:
# The i o n i z e d d o n o r d a t a p o i n t s a x e c o n v e r t e d t o u n i t s o f cmA-3
# and p l o t t e d .
n d p g ra p h : = a r r a y ( l .. n) :
f o r t f r o m 1 t o n do
n d p g ra p h [t] := [y k s to re + t* h ,n d p lu s [t] [2 ]/le 6 ] ;
od:
p lo t(n d p g ra p h );
# The t o t a l , c h a r g e c o n c e n t r a t i o n ( t o t a l c h a r g e ) a s a f u n c t i o n o f p o s i t
# is c a lc u la te d .
t o t a l c h a r g e : = a r r a y (0 . . n ) :
f o r t f r o m 0 t o n do
t o t a l c h a r g e [ t ] := [ y k s t o r e + t * h , n d p l u s [ t ] [ 2 ] - e l e c t r o n s [ t ] [ 2 ] ] ;
od:
# The t o t a l , c h a r g e d a t a p o i n t s a x e c o n v e r t e d t o u n i t s o f cmA-3
# and p l o t t e d .
t o t a l c h a r g e g r a p h : = a r r a y ( 0 . .n ) :
f o r t f r o m 0 t o n do
to ta lc h a x g e g r a p h [ t] := [y k s to re + t* h , t o t a l charge [ t] [ 2 ] / l e 6 ] ;
115
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od:
p lo t (to ta lc h a rg e g ra p h ) ;
# The d i f f u s i o n c u r r e n t ( d i f f c u r r ) a s a f u n c t i o n o f p o s i t i o n i s c a l c u l a t e d
# from t h e d e r i v a t i v e o f e l e c t r o n c o n c e n t r a t i o n ( d e l t a n ) .
d e l ta n := a r r a y ( 0 . .n -1 ) :
f o r t f r o m 0 t o n - 1 do
d e l t a n [ t ] : = [ t , ( e l e c t r o n s [ t + l ] [2] - e l e c t r o n s [ t ] [ 2 ] ) / h ] ;
od:
d iffc u rr:= a rra y (0 . .n -1 ):
ex 4 := p r o c ( t )
lo c a l f x l,y ,c h i;
y := y k s t o r e + t * h + 0 . 5 * h ;
f x l : = p r o c ( y ) i f y<0 t h e n 0 e l s e 1 . 0 e 7 * y f i e n d :
( 4 . 1 4 1 1 7 e - 2 1 ) * 1 . O e - 4 * ( 1 4 . 0+ exp( 5 . 2 2 6 - 5 . 5 4 * e v a l f ( f x l ( y ) ) ) ) ;
end:
f o r v f r o m 0 t o n - 1 do
d i f f c u r r [ v ] : = [v * h + y k s to re + 0 .5 * h ,e x 4 (v )* d e lta n [v ] [2 ]] ;
od:
p lo t(d iffc u rr);
# T he d r i f t c u r r e n t ( e c d r i f t ) i s c o m p u te d a s a f u n c t i o n o f p o s i t i o n
#
from t h e d e r i v a t i v e of th e c o n d u c tio n band en erg y ( d e l t a e c ) .
d e lta e c := a rra y (0 . .n -1 ):
f o r t f r o m 0 t o n - 1 do
d e l t a e c [ t ] : = [ y k s t o r e + t * h + 0 . 5 * h , ( E c d i r e c t [ t + 1 ] [2] - E c d i r e c t [ t ] [2] ) / h ] ;
od:
ex 5 := p r o c ( t )
lo c a l fx l,y ;
y := y k sto re + t* h + 0 . 5*h;
f x l : = p r o c ( y ) i f y<0 t h e n 0 e l s e 1 . 0 e 7 * y f i e n d ;
1 . O e-4 * ( 1 4 . 0+exp( 5 .2 2 6 - 5 .54 * ev alf ( f x l ( y ) ) ) ) ;
end:
e c d rift:= a rra y (0 . .n -1 ):
f o r v f r o m 0 t o n - 1 do
e c d r i f t [ v ] : = [ y k s t o r e + v * h + 0 . 5 * h , e x 5 ( v ) * 0 . 5 * ( e l e c t r o n s [ v + 1 ] [2]
+ e l e c t r o n s [v ] [2] ) * d e l t a e c [ v ] [ 2 ] ] ;
od:
p lo t(e c d rift);
116
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A p p e n d ix B
EGN2e Input F iles for
M icrom in iatu re M icrow ave T riode
U n it C ell Sim ulation
This chapter contains the input files for EGN2e which were utilized to sim ulate each
m icrom iniature microwave triode unit cell. An explanation of what each input file
contains is given below [146].
The section denoted INPUTl provides param eters for solving Poisson's equa­
tion. T he lines between the sections IN P U T l and INPUT5 specify the boundary
conditions. The section denoted INPUTS provides parameters used for incorporat­
ing space charge.
Here is an explanation of each param eter:
• RLIM - width of unit cell (in mesh units)
• ZLIM - height of unit cell (in mesh units)
• PO TN - specifies the total number of potentials
• PO T (i) - assigns each potential -pi its value
• CSYS
- specifies either rectangular coordinates or cylindrical sym m etry
• PASS
- num ber of times program passes to solveLaplace's equation at the
beginning of the simulation
117
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• XR - a value called the “spectral radius” which influences the convergence of
the Poisson’s equation solution
• ERROR - an error limit multiplier
• POIS - used to specify use of computer m emory when storing arrays
• START - specifies the starting conditions for the simulation
• UNIT - one mesh unit, given in meters
• DENS - m axim um allowable emission in am peres per square centim eter
• NS - num ber of iterations
• AV - chooses num ber of iterations over which space charge is averaged
• PE - initial energy of an electron at the cathode in eV
• RC - coordinate of lower end of starting surface
• ZC - specifies location of the cathode surface
• CL - maxim um length of starting surface (in mesh units)
EGN2e has a routine for constructing a com plete boundary from a sequential
list of points. The lines containing numbers in five columns specify the boundaries.
Each line contains PO T , R, Z, DELTAR, and DELTAZ. PO T is the potential num ­
ber.
R is the num ber of mesh units to the right of the origin.
Zis
the num ber of
mesh units above the origin. DELTAR describes th e horizontal offset of the actual
boundary away from the node defined by R.Z. DELTAZ describes the the verti­
cal offset of the boundary away from the node defined by R.Z. Placem ent of the
boundary directly on the grid point specifies a N eum ann boundary: offsetting the
boundary from the grid point specifies a Dirichlet boundary. Note also th a t th e use
of a number greater than 1.0 as an offset tells the program that the boundary is
perfectly vertical or perfectly horizontal. The use of a num ber greater than P O T N
at the beginning of the line indicates the end of the boundary data.
EGN2e has a special routine for incorporating dielectric materials into the sim u­
lation. This is done by letting the number of potentials be greater than 101. W hen
IIS
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i of PO T (i) is greater than 101, what is actually being defined by PO T(i) is not a
potential value, but rather th e relative perm ittivity of a dielectric. W hen a bound­
ary corresponding to a potential with i > 101, a region of dielectric is specified with
a relative perm ittivty value given by PO T(i).
Here is the EGNSe file for the smaller unit cell.
SMALLER UNIT CELL
&INPUT1 RLIM = 6 1 , ZLIM = 3 2 1 , POTN = 102,
P 0 T ( 1 ) = 0 . 0 , P 0 T ( 4 ) = 0 . 0 , P 0 T ( 3 ) = 1 .0 0 0 ,
P 0 T ( 2 ) = 8 0 . 0 , PQT(10) = 1 0 0 0 0 . 0 ,
P 0 T ( 1 0 1 ) = 3 . 9 , P 0 T ( 1 0 2 ) = 3 . 9 , CSYS = 0 ,
PASS = 4 , XR = . 9 9 5 0 , ERROR = .1 0 0 0 E -0 4 ,
POIS = 7 , &END
1 0 0 0.0000000 -.0 0 1 0 0 0 0
1 15 0 2 .0 0 0 0 0 0 0 - . 0 0 1 0 0 0 0
1 30 0 0 .0 0 0 0 0 0 0 - . 0 0 1 0 0 0 0
4 31 0 0 .0 0 0 0 0 0 0 - . 0 0 1 0 0 0 0
4 40 0 2 .0 0 0 0 0 0 0 - .0 0 1 0 0 0 0
4 58 0 0 .0 0 0 0 0 0 0 - . 0 0 1 0 0 0 0
0 58 1 2 .0 0 0 0 0 0 0 2 .0 0 0 0 0 0 0
0 50 1 2 .0 0 0 0 0 0 0 2 .0 0 0 0 0 0 0
101 45 1 2 .0 0 0 0 0 0 0 2 .0 0 0 0 0 0 0
101 45 30 2 .0 0 0 0 0 0 0 2 .0 0 0 0 0 0 0
101 45 59 2 .0 0 0 0 0 0 0 2 .0 0 0 0 0 0 0
0 52 59 2 .0 0 0 0 0 0 0 2 .0 0 0 0 0 0 0
102 59 59 2 .0 0 0 0 0 0 0 2 .0 0 0 0 0 0 0
102 59 30 2 .0 0 0 0 0 0 0 2 .0 0 0 0 0 0 0
102 59 1 2 .0 0 0 0 0 0 0 2 .0 0 0 0 0 0 0
4 59 0 0 . 0 0 0 0 0 0 0 - .0 0 1 0 0 0 0
4 60 0 0 . 0 0 0 0 0 0 0 - .0 0 1 0 0 0 0
0 60 30 2 .0 0 0 0 0 0 0 2 .0 0 0 0 0 0 0
3 60 60 .0 0 0 0 0 0 0 . 0010000
3 59 60 2 .0 0 0 0 0 0 0 .0 0 1 0 0 0 0
3 31 60 2 .0 0 0 0 0 0 0 .0 0 1 0 0 0 0
3 30 60 2 .0 0 0 0 0 0 0 .0 0 1 0 0 0 0
3 29 61 .9 9 9 0 0 0 0 2 .0 0 0 0 0 0 0
3 29 119 .999 00 00 2 .0 0 0 0 0 0 0
3 29 120 .9990000 2 .0 0 0 0 0 0 0
3 30 121 2 .0 0 0 0 0 0 0 - .9 9 9 0 0 0 0
3 58 121 2 .0 0 0 0 0 0 0 - .9 9 9 0 0 0 0
3 59 121 2 .0 0 0 0 0 0 0 - .9 9 9 0 0 0 0
3 60 121 .0 000000 - . 9 9 9 0 0 0 0
119
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0 60 200 2 .0 0 0 0 0 0 0 2 .0 0 0 0 0 0 0
2 60 320 .0 0 0 0 0 0 0 .0010000
2 59 320 2 . 0 0 0 0 0 0 0 .0010000
2 2 320 2 .0 0 0 0 0 0 0 .0010000
2 1 320 2 .0 0 0 0 0 0 0 .0010000
2 0 320 .0 0 0 0 0 0 0 .0010000
0 0 319 .0 0 0 0 0 0 0 2 .0 0 0 0 0 0 0
0 0 3 .0 0 0 0 0 0 0 2 .0 0 0 0 0 0 0
0 0 2 .0 0 0 0 0 0 0 2 .0 0 0 0 0 0 0
0 0 1 .0 0 0 0 0 0 0 2 .0 0 0 0 0 0 0
888
&INPUT5 START = ' GENERAL' ,P E = 0 .0 0 1 , NS=40, AV=35,
RC=0. 0 , ZC=2. 0 , CL=30. 0 , DENS=1000000. 0 ,
UNIT = 0 .0 0 0 0 0 0 0 0 5 , &END
Here is the EGN2e file for the larger unit cell.
LARGER UNIT CELL
&INPUTl
LSTP0T=0, RLIM=60, ZLIM=250, P0TN=104,
POT( 1 ) = 0 . 0 , POT( 2 ) = 4 0 . 0 , POT(101 ) = 3 . 9 , POT( 1 0 2 ) = 3 . 9 ,
P 0 T ( 2 4 ) = 1 . 0 , POT ( 4 ) = 0 . 0 , POT(lO)= 3 0 0 0 . 0 ,
POT (1 4 ) = 4 0 . 0 , CSYS=0, MI=3, TYME=300 . 0 ,
XR=0. 9 9 5 0 ,ERR0R=1E-4,P0IS=7,
&END
1 0 0 .0000000 -.0010000
1 1 0 2.0000000 -.0010000
1 6 0 2.0000000 -.0010000
1 12 0 0 .5 0 0 0 0 0 0 -.0 0 1 0 0 0 0
4 13 0 - 0 . 5 0 0 0 0 0 0 - .0 0 1 0 0 0 0
4 20 0 2 . 0 0 0 0 0 0 0 - .0 0 1 0 0 0 0
4 23 0 0 .0 0 0 0 0 0 0 - .0 0 1 0 0 0 0
0 23 1 2 . 0 2 . 0
0 19 1 2 . 0 2 . 0
101 12 1 2 . 0 2 . 0
101 12 10 2 . 0 2 . 0
101 12 22 2 . 0 2 . 0
0 17 22 2 . 0 2 . 0
102 24 22 2 . 0 2 . 0
102 24 10 2 . 0 2 . 0
102 24 1 2 . 0 2 . 0
4 24 0 0 .0 0 0 0 0 0 0 - .0 0 1 0 0 0 0
120
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4 25 0 0 .0 0 0 0 0 0 0 - . 0 0 1 0 0 0 0
0 25 10 2 . 0 2 . 0
2 4 25 23 .0 0 0 0 0 0 0 - . 5 0 0 0 0
24 24 23 .0 0 0 0 0 0 0 - . 5 0 0 0 0
24 19 23 2 . 0 - . 5 0 0 0 0
24 13 23 2 . 0 - . 5 0 0 0 0
24 12 23 0 . 0 0 0 0 0 0 - . 5 0 0 0 0
24 12 24 0 . 0 0 0 0 0 0 2 . 0
24 12 25 0 . 0 0 0 0 0 0 2 . 0
24 12 26 0 .0 0 0 0 0 0 2 . 0
24 12 27 0 .0 0 0 0 0 0 . 5 0 0 0 0
24 13 27 2 . 0 .5 0 0 0 0
24 18 27 2 . 0 .5 0 0 0 0
24 23 27 .0 0 0 0 .5 0 0 0 0 0 0 0
0 23 28 2 . 0 2 . 0
0 22 28 2 . 0 2 . 0
0 19 28 2 . 0 2 . 0
101 12 28 2 . 0 2 . 0
101 12 100 2 . 0 2 . 0
101 12 149 2 . 0 2 . 0
0 19 149 2 . 0 2 . 0
102 24 149 2 . 0 2 . 0
102 24 100 2 . 0 2 . 0
102 24 28 2 . 0 2 . 0
3 24 27 2 .0 0 0 0 0 0 0 .5 0 0 0 0
3 25 27 .0 00 000 0 0 . 5 0 0 0 0
0 25 100 .0 0 0 0 0 0 0 2 .0 0 0 0 0 0 0
0 25 125 .0 0 0 0 0 0 0 2 . 0 0 0 0 0 0 0
0 25 149 .0 0 0 0 0 0 0 2 .0 0 0 0 0 0 0
14 25 150 .0 0 0 0 0 0 0 .0 0 1 0 0 0 0
14 14 150 2 .0 0 0 0 0 0 0 .0 0 1 0 0 0 0
14 2 150 2 .0 0 0 0 0 0 0 .0 0 1 0 0 0 0
14 1 150 2 .0 0 0 0 0 0 0 .0 0 1 0 0 0 0
14 0 150 .0 0 0 0 0 0 0 . 0 0 1 0 0 0 0
0 0 149 .0 000 00 0 2 . 0 0 0 0 0 0 0
0 0 23 .0 000000 2 . 0 0 0 0 0 0 0
0 0 3 .0000000 2 . 0 0 0 0 0 0 0
0 0 2 .0000000 2 . 0 0 0 0 0 0 0
0 0 1 .0000000 2 . 0 0 0 0 0 0 0
999
&INPUT5
START=»GENERAL' , UNIT=0.0 0 0 0 0 0 0 2 ,DENS=10000000. 0 ,
121
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N S= 40, AV=35, P E=0.0 0 1 ,
RC=0. 0 , ZC=2. 0 , CL=12. 0 ,
&END
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IMAGE EVALUATION
TEST TARGET (Q A -3 )
<&■
150mm
IIV U G E . I n c
1653 E ast Main S treet
R ochester, NY 14609 USA
Phone: 716/482-0300
Fax: 716/288-5989
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