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Analysis of gallium arsenide field effect transistors

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Title:
Analysis of gallium arsenide field effect transistors a comparison of simulation tools
Creator:
Naukam, Michael Arthur
Publication Date:
Language:
English
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75 leaves : ; 28 cm

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Subjects / Keywords:
Gallium arsenide semiconductors ( lcsh )
Gallium arsenide semiconductors -- Mathematical models ( lcsh )
Gallium arsenide semiconductors ( fast )
Gallium arsenide semiconductors -- Mathematical models ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references (leaf 47).
General Note:
Department of Electrical Engineering
Statement of Responsibility:
by Michael Arthur Naukam.

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Source Institution:
|University of Colorado Denver
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|Auraria Library
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
40336928 ( OCLC )
ocm40336928
Classification:
LD1190.E54 1998m .N38 ( lcc )

Full Text
ANALYSIS OF GALLIUM ARSENIDE
FIELD EFFECT TRANSISTORS
A COMPARISON OF
SIMULATION TOOLS
B.S., University of Colorado at Denver, 1993
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Electrical Engineering
by
Michael Arthur Naukam
1998


This thesis for the Master of Science
degree by
Michael Arthur Naukam
has been approved
by
Hamid Z. Fardi
Jan T. Bialasiewicz
7-1 &
Date
ii


Naukam, Michael Arthur (M.S., Electrical Engineering)
Analysis of Gallium Arsenide Field Effect Transistors, A Comparison of Simulation
Tools
Thesis directed by Associate Professor Hamid Z. Fardi
ABSTRACT
This thesis is a study of the Analysis of Gallium Arsenide Field Effect Transistors and
a comparison of two GaAs transistor-modeling tools. In order to understand device
modeling, the device must be understood. Once the characteristic of the device is
understood then an evaluation of the tools is appropriate. The modeling tools used
will be GATES, from Gateway Modeling, Inc., and a FORTRAN based computer
simulation program designed, in-house, by Dr. Hamid Z. Fardi of the University of
Colorado at Denver. The engineering models include the type materials, process
model, concentration profiles, and formation of the current-voltage characteristic
curves and small signal analysis. How each model is compared to the general ideal
will be the focus. The importance of the modeling tools is reflected in the need to
find devices that could be used in many areas, which could include high frequency,
and high temperature applications.
This abstract accurately represents the content of the candidates thesis. I
recommend its publication.
SignedJ
111
Hamid Z. Fardi


CONTENTS
Chapter
1.0 Introduction ......................................................... 1
2.0 GaAs Mesfet Device Basics ............................................ 4
2.1 Electrical Characteristics .................................... 5
2.2 Small Signal Circuit Model .................................. 10
3.0 Simulation .......................................................... 16
3.1 Data Used .................................................... 29
3.2 Simulation Results ........................................... 31
3.2.1 GATES Simulation Results ............................. 31
3.2.2 FORTRAN Simulation Results ........................... 33
4.0 Analysis ............................................................ 35
4.1 Pinchoff Voltage ............................................. 38
4.2 Doping Profile ............................................... 39
4.3 Current ...................................................... 41
4.4 Mobilities ................................................... 42
4.5 Electron Concentration ....................................... 43
5.0 Conclusions ......................................................... 41
IV


References ..................................................... 43
Appendix
A. FORTRAN Program .......................................... 47
B. GATES Functions .......................................... 75
V


FIGURES
Figure
2.1 n-channel MESFET ...........................................................5
2.2 Small Signal Circuit Model ................................................10
2.3 Small Signal Circuit Model Superimposed Over MESFET .......................10
3.1 FET Active Region .........................................................18
3.2 I-V Characteristics, GATES, FET 1 .........................................31
3.3 I-V Characteristics, GATES, FET 2 ....................................... 32
3.4 I-V Characteristics, GATES, FET 3 .........................................32
3.5 I-V Characteristics, FORTRAN, FET 2 34
3.6 I-V Characteristics, FORTRAN, FET 3 34
VI


Tables
1.1 Principle Properties of GaAs and Si atC ............................. 2
3.1 GATES Simulation Parameters ......................................... 29
3.2 FORTRAN Simulation Parameters ....................................... 30
vii


1. Introduction
The Gallium Arsenide (GaAs) Metal Semiconductor Field Effect
Transistor (MESFET) is widely used in many applications such as amplifiers,
modulators, current limiters and oscillators [1], This widespread use is primarily due
to the requirement for faster devices that possess higher higher electron velocities and
mobilities. The higher electron velocities and mobilities increases are responsible
for smaller transit and faster response times. The need for ever increasing speed
demands the implementation of devices with superior properties. The use of GaAs
devices instead of Si based devices is required if the demand on the device is greater
speed and less power.
As one can see from the following table, the properties as exhibited by GaAs
as compared to those of silicon are far superior and are much more suited to high
frequency applications (GHz) [2],
l


Table 1.1
Principle Proper ties of GaAs and Si at 27C
Property Symbol Units GaAs Si
Low-field electron mobility Mb cm2/Vs 4,000-9,000 500-1,200
Max. electron drift velocity Vd cm/s 2(10)7 l(10)v
Critical field for high-field transport Ec V/cm 3(10)3 1(10)4
Low-field hole mobility Mp cm2/V*s 400 480
Relative permittivity 12.9 11.7
Band gap energy Eg eV 1.43 1.11
Intrinsic carrier cone. Di -3 cm 9(10)6 1.45(10)10
Maximum resistivity Q*cm l(10)a 1(10)5
Schottky- barrier heights Fabrication of the GaAs MESFET can be expensive to manufacture and is
time constrained in the manufacture. There are many defects that can effect the
performance of a GaAs device. Some of these defects include the level of compound
purity and gate leakage current. These problems as well as many others can be
2


resolved by using accurate simulations of devices, which would include device
fabrication, and circuit simulation.
3


2.
GaAs Mesfet Device Basics
The circuit is made on an undoped GaAs substrate. The gate is made by
placing the Schottky-barrier metal in direct contact with the n-type GaAs that forms
the channel of the MESFET at zero bias. The gate electrode length and width define
the channel length and width. The source and drain areas are made by implanting n+
type GaAs in direct contact with ohmic contact metal.
The surface channel is where the conduction takes place in which a Schottky
barrier is formed under that gate. When the voltage applied to the gate is zero, the
barrier voltage depletes some of the channel mobile carriers. With a positive bias
applied, the depletion region is reduced making the channel more conductive. With
an increasing negative bias applied to the gate, the depletion layer will increase until
the conductance of the channel is zero.
4


2.1 Electrical Characteristics
A pictorial view of a MESFET, is shown below.
Depletion
Region
Schottky-barrier metal gate
Drain
Channel
Semi-insulating GaAs substrate
Figure 2.1 n channel MESFET
5


The metal semiconductor field effect transistor, Shockley model, has a gate
contact that is used to vary n-channel conductivity. The source and drain regions are
connected by this n channel. The n channel electrons are removed by a voltage
applied to the gate. This removal of electrons provides for a depletion region in the
channel.
There are two modes of MESFET, the enhancement mode and the depletion
mode. The enhancement mode MESFET (normally off) n channel is completely
depleted of electrons by the positive gate voltage. In the depletion mode (normally
on) the gate voltage is negative.
Using an uniform n-type doping profile in the channel drain to source voltages
are considered, several assumptions are made. The electric field (E) is smaller and
proportional to the velocity saturation field, which is proportional to the electron drift
velocity. The channel is absent of charge with no fringe effects. Since the charge in
the channel varies with x, then the change in channel potential incrementally is
6


where,
(2.1)
dV=IdsdR=
qHnNDW[A-Ad{X)\
q = electron charge
Ids= channel current
dR = incremental channel resistance
A = channel thickness
Ad(x) = depletion layer thickness
W = gate width.
The depletion layer thickness Ad(x) is found using the following
Ad(x)
2jV(x)+VH
qND
where,
(2.2)
Vbi = the Schottky barrier voltage
s = dielectric permittivity
Vg = gate potential
Nd = donor concentration
Vx = potential of the neutral channel.
Using the previous two equations and integrating from x=0 to L the basic equation
for MESFETS are
7


Ids = gOTdc<
Vd-

(2.3)
3^
where the conductance of the undepleted doped channel (gOTdt) is
Scudc
_ qnNDWA
and
(2.4)
L is the gate length
Vd is the voltage drop in the channel under the gate.
The channel pinch off voltage is described as
V.
po
rqNDA2>
^ 2 s j
(2.5)
When the pinch off condition is described as Aa(L) = A, is reached, the saturation
voltage is = Vpo Vbi + VG.
8


We find that the new drain to source saturation current is
(Vu~VB)i
(Ids)sat Scudc*
po
+ -
-V*+VG\
3^2
pi
(2.6)
The linear region transconductance is
§m
Au)
A^)
, where VD is constant, results in
(2.7)
9


(VI,+Vl,-V(!)i-(VM-Va)2
§m Scudc<
1
V 2
po
> .
(2.8)
2.2 Small Signal Circuit Model
The key to understanding the operation of a MESFET is to properly model the
characteristics of the device at the circuit level. This section will present the
equivalent circuit model representation of a MESFET and present a two dimensional
view of a Gallium Arsenide MESFET with the equivalent circuit elements placed in
the representative areas of the device[3].
Small Signal Circuit Model Small Signal Circuit Model Superimposed
over MESFET
Figure 2.2
Figure 2.3
10


As one can see by the diagram, there is a relationship between all
parameters[l]. The gain of the transistor is a voltage controlled current source with
transconductance (gm), Drain Source Resistance (R^), and Transit Delay Time (t) as
integral parts. Transconductance(gm) is defined as the ratio of drain conduction
current to gate source voltage represented by
The Drain Source Resistance(RdS) is the ratio of the drain source voltage to the
conduction current at a constant gate source voltage represented by
(2.9)
R^ = where
(2.10)
(2.11)
11


The total gate charge Qg is the magnitude of the electric field on the gate. Using
Gausss Law a general relationship is stated
a
Qg= eW\Ey{x,y)dx. (2.12)
o
The electric field is calculated over both the saturation and linear regions
using the x and y components of the electric field. The gate source capacitance(Cgs)
is the rate of change of the charge on the gate electrode with the gate source voltage
Vgs with Vds being constant as stated in the following equation
Cgs = Cgs0 + Cgsi (2.13)
= (space charge under gate + gate source capacitance)
12


(2.14)
CgSo

Vds
Cgsl =
CQSin 1
n
2
v\
\V*-Vrj
; Vgs > Vpo
(2.15)
;Vgs < Vpo
The domain capacitance (Cdc) is the variation of the accumulated charge in the
channel and is represented by
Cdc-
ftQ*'
ds y
. (accumulation charge)
(2.16)
The Gate-Drain Capacitance(Cgd) is the static capacitance between the gate and drain
and is represented by
Cgd Cgdo + C
gdl
(2.17)
13


CodO

V &
3nd Qgo Qg Qdo:
(2.18)
The drain source capacitance (0^) is the static capacitance between the drain and
source. The following formula states it best
Cds -

(2.19)
Qd is the charge on the drain electrode. The transit time t is the time it takes for
carriers to cross the channel and can be represented by the following formula
x =
r dx
Jo|vM
where v is the electron velocity.
(2.20)
14


In the saturation mode
t
L
v.
(2.21)
The charging resistor py is the charging time of the gate source capacitance
f
R i=K
V
(2.21)
where K is a proportionality factor.
15


Simulation
j.
Simulation tools are used to predict analyses at the circuit level. The
physical model is very complex and can not be solved in a purely analytical closed
form so solutions are obtained by the use of simulation tools. An accurate model
requires solutions of semiconductor device equations. These equations include the
current, continuity, and Poisson equations. A simulation tool/model uses a
numerical method to solve the above equations.
The simulation tools studied are GATES [4], a circuit level simulation tool,
which takes an empirical approach to the analysis and FORTRAN [5], which solves
simplified but real physical equations.
The GATES [4] simulation tool has two main modeling aspects: process and
device. In the process model for the GaAs MESFET, GATES uses an ion
implantation profile. This profile allows multiple acceptor and donor implantations
into a seminsulating GaAs substrate. Accurate Secondary-Ion Mass Spectroscopy
(SIMS) measurements accumulated on Si, Se and Be implanted in GaAs is what the
profiles are based on. The GATES simulation program has two different activation
models for the donors in GaAs. The first one is a constant activation model which
16


include values of between 30% (for low energy, high dose implants) to 90% (for high
energy low dose implants). The second is an activation saturation model that gives
an electron profile concentrations of 5(10)17 to 2(10)18cm-3.
The control of charge, due to effect of the field on the gate, is determined by
doping, composition and the thickness of the layers between the gate and the channel.
The solution to the charge control problem is best represented by solving the Poisson
equation, in one dimension, for electron and hole concentration profiles and by using
the correct mobility model (constant or dependent on electron concentrations) and
electron saturated velocity.
GATES device modeling uses the empirical approach physical aspects of the
MESFET. Implanted characteristics such as resistivities, saturated current, average
mobility and saturated velocity, as well as contacted resistances, are used in GATES.
Other characteristics of the MESFET such as source and drain resistance, gate metal
resistance, gate Schottky-diode characteristics and implanted resistor values are
represented. GATES calculates threshold voltages, computes Ids for selected Vg and
Vds as well as computes Vg and other device characteristics for selected Ids and Yds.
17


The equivalent device circuit parameters (gm, Rds, Cgs, Cgd, Cds, Ri and x) are use
to appropriately model the MESFET.
The FORTRAN [5] simulation tool relies on the physical device model for the
GaAs MESFET. This device model is predicated on finding solutions for the device
equations of potential, electric field and finally conduction current for the area of the
channel directly under the gate. The geometry of the active region of a FET as well
as the basic device equations is illustrated below.
Figure 3.1- FET Active Region
18


The basic device equations are
A1'Â¥ = -^[N{y)-n{x,y)\}
(3-1)
J = -qnv + qD0Vn (3.2)
c r & VJ = q H a (3.3).
T X SE Jt = J+ s a (3.4)
E = -V'P (3.5)
where,
E = the electric field
Â¥ = electrostatic potential
N = an arbitrary channel donor concentration
n = the free electron density
v = electron velocity
Do = the diffusion constant
J = the conduction current density
Jt = the total current density
Velocity is a function of the electric field. The magnitude of the electric field
allows control of the device between pinch-off and saturation. Since velocity
19


saturation, in GaAs MESFETs, occurs before channel pinch-off, current saturation
occurs before pinch-off. The continuous free electron density (n) will vary from zero
to that of the doping concentration. The function used for the doping profile is
shown below
n(x,y) = (1 +y(x-L))T (d(x),y)N(y)
(3.6)
T(d(x),y) = 1-
_____1_____
l+exp(rf^)
(3.7)
N(y) = donor electrons
d(x) = effective depletion layer width
(1+Y(x-L)) = charge accumulation/depletion in the channel
T(d(x),y) = transition region smoothing function.
The transition area is that area between the depletion region and channel
where n would vary between 5% and 95% of N. If a device has a uniformly doped
channel in which N(y) varies slowly with y than n, the spread of charge is the balance
between diffusion and drift currents.
20


The solution to the potential (equation 3.1) can be defined as the combination
of two factors, the Laplacian potential of voltage on the electrodes, vj/0, and space
charge on the channel, \}/i, to include the appropriate boundary conditions. The
Laplacian potential of voltage on the electrodes vj/0 and space charge on the channel
vj/j to include the appropriate boundary conditions are shown below.
The Laplacian potential of voltage on the electrodes is
V2vF0 = 0
(3.8)
using the boundary conditions
Vo(0,a) = 0
(3.8a)
i|/0(L,a) = V0
(3.8a)
(3.8b)
v|/0(x,0) = 0
(3.8c)
21


The space charges on the channel equations are
A2*?, = -[N -].
s
(3.9)
using boundary conditions
M/,(0,a) = 0
(3.9a)
Vi(L,a) = V,
(3.9b)
dy
{x,a)= 0
(3.9c)
v|/i(x,0) = Vgs-Vbi
(3.9d)
where
Vbi = built in voltage
Vgs = gate source voltage.
22


So the solution to the space charges on the channel is
% (*.?)= J }[! T (d1,y")]N (y )dy "dy + ^-x
y y
+ --/(* L )J ][T(d1,y'')]N(y")dy "d
£ y y'
(3.10)
The Solution for the Electric field
E = -VT (3.11)
arises from its dependence on the Laplacian potential due to impressed voltages on
the electrodes vj/0 and space charge on the channel i|/i. The linear and saturation
region, with respect to the y component, are illustrated by the following equations:
23


The Linear region
Ey(x,y) = | [1 T(d{x), y')]N (y')dy £ sinh
nx I I Try
---- cos
\2a J 2a
Vn
(3.12)
with,
£ =
7tL
2a
sinh
2a
(3.13)
The Saturation Region
r 7cc\ (
Ey(x,y) = --f[1 -T(d(x),y)]N(y)dy -£sinh| ~ Icosj ^
cJ \2aJ \la)
_o
L
+ --rt- i)| ]{T(dy'')}N(y')dy 'd
y y
(3.14)
24


The equation that represents the electric field in the x direction for both the linear and
saturation regions are as follows:
Ex(*,y)
\o
y
1
- % cosh
fxx_"
K2a;
sinh
2a) L
(3.15)
The source and drain conduction currents are as follows:
The Source Current is represented by
IS = AV1 + B
(3-16)
where
A
' qW '
, L ,
[F,(do ) -
(3.17)
B =
r qW_
, L

(3.18)
and
25


M
/=1
p*=f4&P>yi))
' i-i
-1
, s
i-l
M 1
i = 1,2,3,.-,M
and
F3 (d) |T(d, y) 1
0
a J
N (y)dy
(3-19)
(3.20)
(3.21)
(3.22)
(3.23)
F 4(d) S jr (d ,y)sin ()>)<&
(3.24)
26


The Drain Current is represented by
Id = CV1 = D
(3.25)
C =
-if
[1 +
r(L-L^
F3(d,) £ cosh
tiL n
27>
FM1)
(3.26)
ud qWD o/Fsid^)
(3.27)
where
F5(d) = \T(d,y)N(y)dy
0
(3.28)
27



(3.29)
M* = M(E(L,yt))
(3.30)
28


3.1 Data Used
The GATES parameters used in the study are as follows:
Table 3.1 GATES Simu ation Parameters
Variable 1 2 3 Unit
Frequenc y 5.5 10 10 GHz
L 1.0 0.42 0.56 pm
W 2.5 1.0 1.2 mm
EP 2.50 3.75 3.75 kV/cm
vs 1.57(10)7 1.57(10)7 1.57(10)7 cm/s
Ho 6000 4000 4000 cm2/Vs
D 10 10 10 cm2/s
vhi 0.78 0.78 0.78 V
X 0.1 0.1 0.1 pm
a 2.0 2.0 2.0 -
Rgg .8 .573 .311 Q
Rj! .5 .62 1.3 n
dd 2.2 1.83 1.57 Q
Lee .02 .05 .08 pH
Lss .05 .02 .08 pH
Ldd .015 .05 .08 pH
29


The FORTRAN parameters used in the study are as follows:
Table 3.2 FORTRAN Simulation Parameters
FET1 FET 2 FET 3 Unit
Length Kioy4 ,42(10)'4 56O0)-4
Area 35(10y6 4.2(10)'4 .72(10)^
Width .25 .1 12
Ec 2.5(10)3 3.75C10)3 3.75(10)3
Vo 6000 4000 4000
Do 10 10 10
vs 1.57(10)7 1.57(10)* 1.57(10)7
Es 12.9 12.9 12.9
Vgs 0 0 0
Vbi .78 .78 .78
X Ol(lO)-4 .Ol(lO)-4 .Ol(lO)-4
a 2 2 2
Ilimit 16 16 16
Tol i(i oy1 i(io)1 Kioy1
Vds ,i .1 ,i
VO .2 .2 .2
Vlimit 10 10 10
dVgs -0 -0 -0
VgsL -0 -0 -0
The differences in the parameters used are due to the level of input parameters
required by the simulation tools.
30


3.2 Simulation Results
3.2.1 GATES Simulated Results
The GATES results for FET 1,2, 3 were predictable. The Ids-Vds curve
shows the linear and saturation regions. The dividing line between the two regions is
described by the formula Vds=Vgs-Vpo.
GATES
Uds Figure 3.2 I-V Characteristics of GATES, FET 1
31


Figure 3.3 I-V Characteristics of GATES, FET 2
Figure 3.4 I-V Characteristics of GATES, FET 3
32


3.2.2 FORTRAN Simulation Results
The FORTRAN simulation program does much of the same modeling but
there are several basic differences to that of the GATES simulation program. These
differences do present themselves during the presentation of variables as well as the
results obtained from the modeling. The similarities as well as the differences are
presented during the analysis, conclusions and recommendations in this thesis. Due
to some constraints within the FORTRAN simulation program, adjustment had to be
made. This was due to the modeling ranges designed within the program. Further,
due to the range simulation constraints, the FET 1 parameters would not yield any
results.
FET 1 not able to acquire data.
The ranges allowed within the FORTRAN simulation program would not allow for
the device parameters to work.
33


FET 2
8.00E-02
6.00E-02
2 4.00E-02
2.00E-02
O.OOE^OO
O^C0i0h*-0>^-;C0iq
r (\i ri t in N 00
VDS
Series 1
Figure 3.5 I-V Characteristics of FORTRAN, FET 2
The results as shown in Figure 3.5 shows a typical 1^ curve with Vgs =0
As you can see from the results that the 1^ curve results gained are typical.
Figure 3.6 I-V Characteristics of FORTRAN, FET 3
34


4. Analysis
There is a level of sophistication difference between simulation tools. The
different levels of sophistication are important but not germane to this discussion.
This analysis will focus on the basic building blocks of each tool. It will bring to the
forefront differences and similarities.
GATES was modeled to incorporate most of the MESFET design process. It
starts with the assumption that we have p type material to start. Into this p-type
material n+ and n- ions are implanted. A concentration profile is calculated. The
Vth as well as the Ids is calculated and presented graphically. Additionally, the
following small signal representations are calculated and graphically presented IDS,
Qg> Cgs, Cgd, Gm, and Gd.
The FORTRAN program uses a set of FET parameters into which preliminary
calculations are made. The size of the channel is then figured. Depending on the
size of the channel, Id and Is are then calculated. As soon as there are enough
calculations for and Is the calculations are stopped and the results are placed in a
35


file. Both GATES and FORTRAN have the user supply the implant and the doping
information.
The GATES simulation program was based on a set of 17 modeled structures
that would allow the changing of the device parameters within a selected structure.
These structures have physical and electrical characteristics that could be changed to
effect the MESFET under design. The process would start by using a series of.
command sequences (Appendix B) to give the MESFET the desired characteristics.
These characteristics could be reviewed at will, further processed or saved for future
use.
The FORTRAN simulation program was a UNIX based system that depended
on an input file in which to set specific parameters and an output file that would
contain the final results. The results would be processed and presented in graphical
form using a spreadsheet program. The GATES tool was somewhat easy to use. This
tool depended on a specific sequence of processes happening. If this sequence
werent followed GATES would crash. The FORTRAN tool was very easy to use,
and, as long as the input parameters were in the correct order, the program would run.
36


The GATES simulation software produced an IV curves that were as
expected. The FORTRAN code produced the typical IV curve. However, that Vgs
=0, IV curve was not of the same value as that of the due to the softwares inability to
handle values of numbers outside a certain range. Those values were proportionally
adjusted so as to find the limits of where it would work. It seemed that the splin
routine in the FORTRAN program would only allow certain values to be used
(otherwise the messages out of splin range or no convergence would present).
The GATES program has many functions in which information can be derived
as well as presented in alphanumeric or graphical form. This would include a
graphical representation of the IV Curve as well as a sideview representation of the
MESFET. The functional aspects of the two simulations have been covered and that
will leave the comparison of the profile assumptions to be covered.
GATES was developed in cooperation with Stanford Universitys GaAs
SUPREM3.5 process modeling program and that methodology has been incorporated
into GATES thus the type of ion, ion energy, dose, ...etc. has already been
incorporated into the MESFET development process. In that light several aspects of
what profiles an MESFET model will be covered which include but will not be
37


limited to: implanted layer characteristics, source and drain resistances, gate metal
resistances, threshold voltages, 1^ computation, equivalent circuit parameters, gm,
Rds, Cgs, etc., electron and hole concentration, etching, mobility, electron velocity,
and Gate Schottky diode characteristics.
In the preceding sections there was a discussion of the basis for calculation of
the various quantities such as Vtp, Vpo, Nd, etc. This section will give a comparison
of the various calculations.
4.1 Pinch Off Voltage
The Pinch off voltage is defined as the voltage on the gate when the depletion
layer extends through the entire channel and the channel conductance goes to zero.
The GATES pinchoff calculations are based on a linear function of the effect of the
depletion region where the FORTRAN calculation is based on the area of the
depletion region. The effects of the donor concentration are more evident in the
FORTRAN calculation than it is in GATES.
38


GATES
(4.1)
FORTRAN
Vpo=j Vbi-^\\N(y")dy"dy'
(4.2)
4.2 Doping Profile
The doping profiles do not have much difference in there basic assumptions.
A minor difference is in the flexibility of data entry between the simulation tools. In
GATES, the parameter can be entered through a menu as compared to FORTRAN
where the parameter is a constant. Otherwise there is not much difference.
39


GATES
n(x)=pk exp
(x-Rpf
2CT2
(4.3)
FORTRAN
N(y)=2(10)nexp
1 >- 0.307' 2
2 .110
(4.4)
for FET1.
N(y)=2.1(10)nexp
1 y 0.075' 2"
2 .093
(4.5)
for FET2.
N(y)=1.8(10)17 (4.6)
forFET3.
40


4.3 Current
There is some difference is the way the currents are presented. The GATES
relies on the linear nature of the device where the FORTRAN tool adds in more detail
for charge accumulation and the transition of the doping concentration. This change
makes the FORTRAN calculation appear to closely resemble the device
characteristics in the depletion region.
GATES
(4.7)
FORTRAN
a
Is = -qWpsJ T(d0,y)N(y)Ex(0,y)dy
(4.8)
o
41


4.4 Mobilities
There is a difference in the GATES mobility access or at least the treatment.
The GATES offers a selection of various levels where the FORTRAN offers mobility
based on weighted position levels within the region.
GATES
Constant
Caughey-Thomas parameterization
MBE-Data of Anderson
FORTRAN
M

/=!
(4.9)
M
i=I
Us. = P(E(0,yi))
(4.10)
/-I
(4.11)
a
Ug = weighted average mobility
42


(0; = weights
(E(0,yi)) = electric field magnitude
yi = position
4.5 Electron Concentration
There is a difference between the calculations for electron concentration.
The GATES offers the basics with respect to a basic location in the channel.
However, the FORTRAN electron concentration offers specific remedies for charge
accumulation as well as the movement of the charge transitioning through the
channel.
GATES
n(x) = Pk exp -
(4.12)
Pk = concentration in cm'3
Rp = peak position
ct = distribution width
43


FORTRAN
n(x,y) = [l+y(x-L)] T(d(x),y) N(y)
(4.13)
N(y) = donor electrons
y = sign factor(accumulation/depletion)
T(d(x),y) = 1-
1
1 + exp ~y-d(x)
l
(4.14)
N(y) = donor concentration
[1+Y(x-L)J = charge accumulation
A, = Debye length
44


5.
Conclusions
There are many differences between the GATES and FORTRAN simulation
tools. The greatest difference is in the tools themselves. The GATES tool is menu
driven and has a multitude of choices that can be used to simulate a MESFET. The
FORTRAN tool is very limited with respect to number of parameters that can be
accessed (See Appendix A). However, the FORTRAN tool is a better representation
of the MESFET in several ways. The Pinchofif voltage calculation is a better
representation of the donor concentration in the channel. There wasnt too much
difference in the doping profiles except that there was less flexibility in the way the
doping was determined. The representation of the mobilities, offered by the
FORTRAN based program, was more appropriate due to the weighted average
technique that was used. The FORTRAN tool offered a better representation of the
electron concentration, which is important to the current developed within the
channel region. The GATES tool might have been better served by concerning
themselves with representing charge accumulation within their simulation tool.
Additionally, the GATES software should be more concerned with the transitional
movement of charge in the channel and accounted for it in their calculations. And
lastly, the representations of the currents showed a basic assumption difference. It
45


appears that the GATES current calculation does not include a representation for
mobility of electron transitions through the channel. The effect is a much smaller
electron simulation transition.
The FORTRAN based program was not the most elegant of the simulation
tools nor did it address all the parameters of the MESFET. However, in the
comparison of the tools, the FORTRAN based tool, physical characteristic
assumptions were more representative of the MESFET.
The FORTRAN based program should be expanded. It should be menu
driven with a variety of inputs available to the user. Its range should be expanded in
order to handle devices of vaiying lengths, maintaining the standard and thoroughness
of how it calculated its results.
4
46


REFERENCES
1. Wang, Cheng T., Introduction to Semiconductor Technology: GaAs and
Related Compounds, John Wiley and Sons, 1990
2. Hodges, D. A., Analysis and Design of Digital Integrated Circuits, 2nd ed.,
McGraw-Hill, Inc., 1988
3. Khatibzadeh, M. A., Large-Signal Modeling of Gallium Arsenide Field Effect
Transistors, Ph.D. Thesis, North Carolina State University, 1987
4. Gateway Modeling,. Inc., GATES A Process and Device Modeling Program
for Gallium Arsenide (GaAs) Mesfet Technology, Gateway Modeling, Inc.,
1993
5. Fardi, H., FORTRAN GaAs MESFET FORTRAN based simulation program,
University of Colorado at Denver, 1998
6. Sedra, A. S./Smith, K. C., Microelectronic Circuits, 3rd ed., Saunders
College Publishing, 1991
47


APPENDIX A
FORTRAN Simulation Program
Date: Wed, 12 Feb 1997 15:01:08 -0700
From: Hamid Fardi
To: manaukam@ouray.cudenver.edu
SUBROUTINE SPLIN(N)
implicit double precision (a-h,o-z)
COMMON/DAT AI/C1 (61 ),C2(61 ),C3(61 ),C4(61)
COMMON/XDIR/ADIS(61)
COMMON/DATAO/D(61 ),DIAG(61)
DIAG(1)=1.0D0
D(1)=0.D0
NP1=N+1
DO 10 M=2,NP1
D(M)=ADIS(M)-ADIS(M-1)
10 DIAG(M)=(C 1 (M)-C 1 (M-1 ))/D(M)
DO 20 M=2,N
C2(M)=3.0*(D(M)*DIAG(M+1 )+D(M+1 )*DIAG(M))
20 DIAG(M)=2.0*(D(M)+D(M+1))
DO 30 M=2,N
G=-D(M+1 )/DIAG(M-1)
DIAG(M)=DIAG(M)+G*D(M-1)
30 C2(M)=C2(M)+G*C2(M-1)
NJ=NP1
DO 40 M=2,N
NJ=NJ-1
40 C2(NJ)=(C2(NJ)-D(NJ)*C2(NJ-1 ))/DIAG(NJ)
DO 50 M=1,N
DX=ADIS(M+1 )-ADIS(M)
DIVDF1 =(C 1 (M+1 )-C 1 (M))/DX
DIVDF3=C2(M)-C2(M+1 )-2.0*DIVDF 1
C3(M)=(DIVDF 1 -C2(M)-DIVDF3 )/DX
48


50 C4(M)=DIVDF3/DX/DX
RETURN
END
SUBROUTINE PL 1SP1 (NSPLIN,NVAL1)
implicit double precision (a-h,o-z)
COMMON/DATAI/F VAL(61 ),C2(61 ),C3(61 ),C4(61)
COMMON/XDIR/ADIS(61)
COMMON/DATAO/SPLX(61 ),SPLY(61)
CALL SPLIN(NVAL1)
DX=( ADIS(NVAL1 )-ADIS (1 ))/FLO AT (N SPLIN)
SPLX( 1 )=ADIS( 1)
DO 3000 I=1,NSPLIN
XBAR=SPLX(I)
CALL PCUBIC(XBAR, YBAR,DIFF,NVAL 1)
SPL Y (I)=YB AR
IF(I.NE.NSPLIN)SPLX(I+1 )=SPLX(I)+DX
3000 CONTINUE
RETURN
END
SUBROUTINE PCUBI C(XB AR, YB AR,DIFF,N)
implicit double precision (a-h,o-z)
COMMON/DATAI/C1 (61 ),C2(61 ),C3(61 ),C4(61)
COMMON/XDIR/ADIS(61)
1=1
DDX=0.
DX=XBAR-ADIS( 1)
IF(DX) 10,30,20
10 IF(I.EQ.l)GOTO 30
1=1-1
DX=XBAR-ADIS(I)
IF(DX) 10,30,30
19 1=1+1
DX=DDX
20 IF(I.EQ.N)GOTO 30
DDX=XBAR-ADIS(I+1)
IF(DDX)30,19,19
30 YBAR=C 1 (I)+DX*(C2(I)+DX*(C3(I)+DX*C4(I)))
49


nnnn
DIFF=C2(I)+(2.*C3(I)+3.*C4(I)*DX)*DX
RETURN
END
FUNCTION TRANSI(TD,NY)
implicit double precision (a-h,o-z)
COMMON/XYBOX/X( 101), Y( 101 ),elec(2,101 ),si 1 (2,101)
COMMON/ONE/GAMA,DLAMDA
temp=(y(ny)-td)/dlamda
if(dabs(temp).gt. 17.3D0)then
if(temp.gt.0.0D0)then
transi=1.0D0
elseif(temp. It. 0.0D0)then
transi=0.D0
endif
else
TRANSI=1.-1 ./(l .+DEXP(temp))
endif
RETURN
END
subroutine poten(index)
implicit double precision (a-h,o-z)
external fl,f5
common/xybox/x( 101 ),y( 101 ),elec(2,101 ),si 1 (2,101)
common/one/gama,dlamda
common/two/a,dl,dl 1
common/three/vO,es,q,ec,w,dO
common/five/vl ,vds,vgs,vbi
common/six/dxO,dx 1
nxl=101
nyl=101
do 100 j=l,nyl
temp=psi(indexj)
50


non
if(index.eq.l)then
call dint2(fl,dx0,sumj)
si 1 (index j)=vbi-(q/es)*sum+temp
elseif(index. eq. 2)then
call dint2(fl,dxl,sumlj)
call dint2(f5,dxl,sum2j)
c sil(indexj)=vbi-(q/es)*suml+vl+(q/es)*sum2*gama*(dl-dll)+temp
tl=-(q/es)*suml
t2=(q/es)*sum2*gama*(dl-dl 1)
si 1 (index j)=vbi+t 1 +v 1 +t2+temp
c write(*,444)vbi,tl,vl,t2,temp
c444 format(4D9.2)
endif
c vbi is introduced only for conv. comparison with PISCES
100 continue
return
end
subroutine carrier(index)
implicit double precision(a-h,o-z)
COMMON/XYBOX/X( 101),Y( 101 ),elec(2,101),si 1 (2,101)
COMMON/ONE/GAMA,DLAMDA
COMMON/TWO/A,DL,DL 1
COMMON/THREE/VO,ES,Q,EC,W,DO
COMMON/FIVE/V 1, VDS, VGS, VBI
COMMON/SIX/DXO,DX 1
nxl=101
nyl=101
if(index.eq.l)then
td=dx0
tx=x(l)
elseif(index.eq.2)then
td=dxl
tx=x(nxl)
endif
51


onnnn nnno
do 100 j=l,nyl
ELEC(indexj)=(l.+GAMA*(tX-DLl))*TRANSI(tdj)*DONOR(j)
100 continue
RETURN
END
FUNCTION DONORb(NY)
implicit double precision (a-h,o-z)
COMMON/XYBOX/X( 101), Y( 101 ),elec(2,101 ),si 1 (2,101)
DATA Cl/2. ID 17/, Y0/.075D-4/,DELY/0.093D-4/
DONORb=C 1 *DEXP(-. 5 *(( Y(NY)-Y 0)/DEL Y)* *2)
RETURN
END
FUNCTION DONORO(NY)
implicit double precision (a-h,o-z)
COMMON/XYBOX/X( 101), Y( 101 ),elec(2,101),si 1 (2,101)
IF(NY.LE. 101 )THEN
DONORO=7.5D16
ELSE
DONORO=O.DO
ENDIF
RETURN
END
function donor4(ny)
implicit double precision (a-h,o-z)
common/xybox/x( 101 ),y( 101 ),elec(2,101),si 1 (2,101)
if(y(ny).lt..263D-4)then
tl=. 5D17*dexp(-.5*((y(ny)-0.09D-4)/0.025D-4)**2)
t2=. 9D17*dexp(-. 5 *((y(ny)-0.133D-4)/0.14D-4)**2)
52


donor=tl+t2
else
t3=. 6D17*dexp(-.5*((y(ny)-0.248D-4)/0.038D-4)**2)
t3=. 5D16*dexp(-.5*((y(ny)-0.263D-4)/0.16D-4)**2)
donor=l .D15+t3+t4
endif
return
end
function donor(ny)
implicit double precision (a-h,o-z)
common/xybox/x( 101 ),y( 101 ),elec(2,101 ),si 1 (2,101)
if(y(ny).le.0.185D-4)then
donor=1.80dl7
else
tl=l .69D17*dexp(-((y(ny)-0.185D-4)/0.018D-4)**2)
t2=l. ID 17*dexp(-((y(ny)-0.185D-4)/0.122D-4)**2)
donor=tl+t2
endif
return
end
C
C
FUNCTION DONORA(NY)
implicit double precision (a-h,o-z)
COMMON/XYBOX/X( 101),Y( 101 ),elec(2,101 ),si 1 (2,101)
DATA C 1/2.D 17/, Y0/.037D-4/,DELY/0.110D-4/
DONORA=C 1 *DEXP(-.5*((Y(NY)-Y0)/DELY)**2)
RETURN
END
C
C
c
c
c
FUNCTION PSI(index,NY)
implicit double precision (a-h,o-z)
53


COMMON/XYBOX/X( 101), Y( 101 ),elec(2,101),si 1 (2,101)
COMMON/TWO/A,DL,DL 1
COMMON/THREE/VO,ES,Q,EC,W,DO
nxl=101
PI=4.*DATAN(1.D0)
PI=PI/(2.*A)
if(index.eq.l)then
tx=x(l)
elseif(index.eq.2)then
tx=x(nxl)
endif
PSI=(VO/DSINH(PI*DL))*DSINH(PI*tX)*DSIN(PI*Y(NY))
RETURN
END
C
C
c
c
c
c
SUBROUTINE DINT2(Fl,TD,SUM,nylow)
implicit double precision (a-h,o-z)
EXTERNAL FI
COMMON/ONE/GAMA,DLAMDA
COMMON/T WO/A,DL,DL 1
COMMON/XYBOX/X( 101), Y( 101 ),elec(2,101),si 1 (2,101)
SUM=0.
NY=101
DEL Y=A/FLOAT (NY-1)
i=nylow
100 G=0.
DO 10 J=I,NY-1
G=G+F 1 (TD,J)+F1 (TD, J+1)
10 CONTINUE
G=G*DELY/2.
IF((I. EQ. nylo w). OR. (I. EQ. NY))THEN
PRE=1.
ELSE
54


o n
PRE=2.
ENDIF
SUM=SUM+PRE*G
1=1+1
IF(I.Le.NY)GOTO 100
SUM=SUM*DELY/2.
RETURN
END
SUBROUTINE PINCH(VPO)
implicit double precision (a-h,o-z)
COMMON/ONE/GAMA,DLAMDA
COMMON/TWO/A,DL,DL 1
COMMON/THREE/VO,ES,Q,EC,W,DO
COMMON/FIVE/Vl,VDS,VGS,VBI
COMMON/XYBOX/X( 101), Y( 101 ),elec(2,101), si 1 (2,101)
SUM=0.
NY=101
DEL Y=A/FLOAT(NY-1)
1=1
100 G=0.
DO 10 J=I,NY-1
G=G+DONOR(J)+DONOR(J+1)
10 CONTINUE
G=G*DELY/2.
IF((I.EQ. l).OR.(I.EQ.NY-l))THEN
PRE=1.
ELSE
PRE=2.
ENDEF
SUM=SUM+PRE*G
1=1+1
IF(I.LT.NY)GOTO 100
SUM=SUM*DEL Y/2.
VPO=VBI-Q*SUM/ES
RETURN
END
55


uuuuuu uuou uuuu
FUNCTION F1(TD,J)
implicit double precision (a-h,o-z)
COMMON/XYBOX/X( 101), Y( 101 ),elec(2,101 ),si 1 (2,101)
COMMON/ONE/GAMA,DLAMDA
F1 =( 1 .-TRANSI(TD,J))*DONOR(J)
RETURN
END
FUNCTION DIFF2(TD,NY)
implicit double precision (a-h,o-z)
COMMON/ONE/GAMA,DLAMDA
COMMON/X YBOX/X( 101),Y(101 ),elec(2,101),si 1 (2,101)
T=TRANSI(TD,NY)
DIFF2=-T*( 1 .-T)*DONOR(NY)/DLAMDA
RETURN
END
FUNCTION F3(TD, NY)
implicit double precision (a-h,o-z)
COMMON/ONE/GAMA,DLAMDA
COMMON/X YBOX/X( 101),Y( 101 ),elec(2,101 ),si 1 (2,101)
COMMON/TWO/A,DL,DL 1
F3=TRANSI(TD,NY)*( 1 ,-(Y(NY)/A-1 .)**2)*DONOR(NY)
RETURN
END
56


uou uoou uuuu
FUNCTION F4(TD,NY)
implicit double precision (a-h,o-z)
COMMON/XYBOX/X( 101), Y( 101 ),elec(2,101 ),si 1 (2,101)
COMMON/ONE/GAMA,DLAMDA
C0MM0N/TW0/A,DL,DL1
PI=4. *DATAN( 1 .DO)
PI=PF(2.*A)
F4=TRANSI(TD,NY)*DSIN(PI*Y(NY))*DONOR(NY)
RETURN
END
FUNCTION F5(TD,NY)
implicit double precision (a-h,o-z)
COMMON/ONE/GAMA,DLAMDA
COMMON/XYBOX/X( 101), Y( 101 ),elec(2,101 ),si 1 (2,101)
F 5=TRAN SI(TD,NY)*DONOR(N Y)
RETURN
END
SUBROUTINE DINTl(F,TD,SUM,NLOW)
implicit double precision (a-h,o-z)
EXTERNAL F
COMMON/XYBOX/X( 101), Y( 101 ),elec(2,101 ),si 1 (2,101)
COMMON/ONE/GAMA,DLAMDA
COMMON/TWO/A,DL,DL 1
NY=101
DEL Y=A/FLO AT (NY-1)
SUM=0.
DO 10 J=NLOW,NY-l
57


uuuu
SUM=SUM+F(TD,J)+F(TD,J+1)
10 CONTINUE
SUM=SUM*DELY/2.
RETURN
END
SUBROUTINE INITIAL(VPO)
implicit double precision (a-h,o-z)
EXTERNAL F1,F3,F4,F5
COMMON/DATAI/C1 (61 ),C2(61 ),C3(61 ),C4(61)
COMMON/DATAO/SPLX(61 ),SPLY(61)
COMMON/XYBOX/X( 101), Y( 101 ),elec(2,101),si 1 (2,101)
COMMON/ONE/GAMA,DLAMDA
COMMON/THREE/VO,ES,Q,EC,W,DO
COMMON/FIVE/Vl,VDS,VGS,VBI
COMMON/TWO/A,DL,DL 1
COMMON/TABLE/CS1 (5,61 ),CS2(5,61 ),CS3(5,61 ),CS4(5,61)
COMMON/XDIR/ADIS(61)
NSPLIN=60
ADIS(1)=0.
DT=(2. *A)/FLOAT(NSPLIN)
DO 10 J=1,NSPLIN
ADIS(J+1 )=ADIS(J)+DT
10 CONTINUE
DO 20 J=1,NSPLIN+1
TD=ADIS(J)
CALL DINT2(F 1 ,TD,SUM, 1)
CS1(1,J)=SUM
CALL DINT1(F3,TD,SUM,1)
CS1(3,J)=SUM
CALL DENT1(F4,TD,SUM,1)
CS1(4,J)=SUM
CALL DINT 1(F5,TD,SUM, 1)
CS1(5,J)=SUM
20 CONTINUE
58


DO 99 J=1,NSPLIN
CS1 (2, J)=-(ES/Q)*(VPO-VBI)-CS 1 (1, J)
99 CONTINUE
DO 100 K=l,5
DO 25 I=1,NSPLIN+1
C1(I)=CS1(K,I)
25 CONTINUE
C2( 1 )=(C 1 (2)-C 1(1 ))/DT
C2(NSPLEM+1 )=(C 1 (NSPLIN+1 )-C 1 (NSPLIN))/DT
CALL SPLIN(NSPLIN)
DO 26 I=1,NSPLIN
CS2(K,I)=C2(I)
CS3(K,I)=C3(I)
CS4(K,I)=C4(I)
26 CONTINUE
100 CONTINUE
c DO 777 K=l,5
c DO 777 J=1,NSPLIN
c WRITE(7,77)ADIS(J),CS 1(K,J)
c777 CONTINUE
c77 FORMAT(2D10.3)
RETURN
END
C
C
C
c
SUBROUTINE DEPTH
implicit double precision (a-h,o-z)
LOGICAL MA,MB,MC
COMMON/MOD/MA,MB,MC
COMMON/ONE/GAMA,DLAMDA
COMMON/XDIR/D(61)
COMMON/X YBOX/X( 101), Y( 101 ),elec(2,101), si 1 (2,101)
COMMON/T WO/A,DL,DL 1
COMMON/THREE/VO,ES,Q,EC,W,DO
COMMON/FTVE/V1 ,VDS,VGS,VBI
COMMON/SIX/DXO,DX 1
59


COMMON/DATAI/C1 (61 ),C2(61 ),C3(61 ),C4(61)
COMMON/DATAO/SPLX(61 ),SPL Y(61)
COMMON/TABLE/CS1 (5,61 ),CS2(5,61 ),CS3(5,61 ),CS4(5,61)
common/out/volt(51 ),curr(51)
NSPLIN=60
DO 5 J=1,NSPLIN
C1(J)=CS1(1,J)
C2(J)=CS2(1,J)
C3(J)=CS3(1,J)
C4(J)=CS4(1,J)
5 CONTINUE
PI=4. *DATAN( 1 .DO)
PI=PI/(2.*A)
IF(MA)THEN
DL1=DL
ELSEIF(MB)THEN
ETA=( 1 ./(DL*PI))*((EC*DL-V1)/V0)*DSINH(PI*DL)
DLl=DLOG(ETA+DSQRT(ETA**2-1 .))/PI
ELSEIF(MC)THEN
DL1=0.
ENDIF
CALL ROOT(0.0,DX0)
IF(MB)THEN
CALL ROOT(DL1 ,DX 1)
ELSEff(MA)THEN
CALL ROOT(DL,DXl)
ELSEIF(MC)THEN
DX1=DX0
ENDIF
IF(MA)THEN
GAMA=0.
ELSEIF(MB.OR.MC)THEN
DO 6 J=1,NSPLIN
C1(J)=CS1(2,J)
C2(J)=CS2(2,J)
C3(J)=CS3(2,J)
C4(J)=CS4(2,J)
60


uuuu
6 CONTINUE
CALL PCUBIC(DX 1, YB AR,DIF,NSPLIN)
GAMA=-ES* V1/(Q*DL* YBAR)
ENDIF
RETURN
END
FUNCTION EY(NX,NY)
implicit double precision (a-h,o-z)
EXTERNAL FI,F5
LOGICAL MA,MB,MC
COMMON/MOD/MA,MB,MC
COMMON/XYBOX/X(101),Y(101),elec(2,101),sil(2,101)
COMMON/ONE/GAMA,DLAMDA
COMMON/T WO/A,DL,DL 1
COMMON/THREE/VO,ES,Q,EC,W,DO
COMMON/FrVE/Vl,VDS,VGS,VBI
COMMON/SIX/DXO,DX 1
C NX=1 ORNX=101
NY1=101
NX1=T01
IF(NY.EQ.NY1)THEN
EY=0.
RETURN
ENDIF
PI=4. *DATAN( 1 .DO)
PI=PI/(2.*A)
DLP=DL*PI
XI=DLP/DSINH(DLP)
XI=XI*DSINH(PI*X(NX))*DCOS(PI*Y(NY))*VO/DL
IF(NX.EQ. 1 )THEN
CALL DINT 1 (F1 ,DX0,SUM,NY)
EY=-SUM*Q/ES-XI
ELSEIF(NX.EQ.NX 1 )THEN
CALL DINT1(F1,DX1,SUM1,NY)
61


ouuo u uuuu
CALL DINT1(F5,DX1,SUM2,NY)
E Y=-(Q/ES)*(SUM1 -GAMA*(X(NX)-DL 1 )*SUM2)-XI
ENDIF
RETURN
END
FUNCTION EX(NX,NY)
implicit double precision (a-h,o-z)
COMMON/XYBOX/X( 101), Y( 101 ),elec(2,101),si 1 (2,101)
COMMON/T WO/A,DL,DL 1
COMMONAHREE/VO,ES,Q,EC,W,DO
COMMON/FIVE/V 1, VDS, VGS, VBI
PI=4. *DATAN( 1 .DO)
PI=PI/(2.*A)
DLP=DL*PI
XI=(DLP)/DSINH(DLP)
EX=-( V1/DL)*( 1. -(Y (NY)/A-1. )**2)
1-XI*DCOSH(PI*X(NX))*DSIN(PI*Y(NY))*VO/DL
RETURN
END
FUNCTION DMUE(NX,NY)
implicit double precision (a-h,o-z)
LOGICAL MA,MB,MC
COMMON/TWO/A,DL,DL 1
COMMON/THREE/V 0,ES,Q,EC, W,D0
COMMON/FIVE/V1,VDS,VGS,VBI
COMMON/SIX/DXO,DX 1
62


noon
c
c
c
c
COMMON/SEVEN/DMUO,VS
COMMON/MOD/MA,MB,MC
COMMON/XYBOX/X( 101), Y( 101 ),elec(2,101),si 1 (2,101)
ESET=DSQRT(EX(NX,NY)**2+EY(NX,NY)**2)
ELIM=2.*EC
IF(ESET.LE.ELIM)THEN
V=DMU0*ESET-(VS/(4.*(EC**2)))*(ESET**2)
ELSEIF(ESET.GT.ELIM)THEN
V=VS
ENDIF
DMUE=V/ESET
tl=eset/ec
t2=tl*tl
t4=t2*t2
dmue=(dmu0+(vs/eset)*t4)/( 1 +t4)
RETURN
END
SUBROUTINE MODE
implicit double precision (a-h,o-z)
LOGICAL MA, MB, MC
COMMON/MOD/MA,MB,MC
COMMON/TWO/A,DL,DL 1
COMMON/THREE/VO,ES,Q,EC,W,DO
COMMON/FIVE/V 1 ,VDS, VGS,VBI
MA=.FALSE.
MB=.FALSE.
MC=. FALSE.
PI=4. *DATAN( 1 .DO)
PI=PI*DL/(2. *A)
XI=PI/DSINH(PI)
ES0=(V1+XI*V0)/DL
ED=( V1+XI* VO *DCOSH(PI))/DL
IF((ED.LT.EC).AND.(ED.GT.ESO))THEN
MA=.TRUE.
63


no non
ELSEIF((EC.LT.ED).AND.(EC.GT.ESO))THEN
MB=.TRUE.
ELSEIF((ESO.LT.ED).AND.(ESO.GT.EC))THEN
MC=. TRUE,
c elseif(es0.gt.ed)then
c ma=.true.
END IF
RETURN
END
FUNCTION DMU(INDEX)
implicit double precision (a-h,o-z)
EXTERNAL FI,F5
LOGICAL MA,MB,MC
COMMON/MOD/MA,MB,MC
COMMON/SIX/DXO,DX 1
COMMON/SEVEN/DMUO,VS
COMMON/TWO/A,DL,DL 1
COMMON/FOUR/ALFA
COMMON/FIVE/V 1, VDS, VGS, VBI
COMMON/XYBOX/X( 101),Y( 101 ),elec(2,101),si 1 (2,101)
COMMON/THREE/VO,ES,Q,EC,W,DO
DMU=0.
WISUM=0.
M=ll
NY=101
DO 101=1,M
IY=(I-1)*(NY-1)
IY=1+IY/(M-1)
WI=(FLOAT (I-1 )/FLO AT (M-1 ))* ALFA
WISUM=WISUM+WI
INDEX=1, MU(E(X=0,IY))
INDEX=101, MU(E(X=L,IY))
DMU=DMU+DMUE(INDEX,IY)*WI
10 CONTINUE
DMU=DMU/WISUM
64


uuuu
RETURN
END
FUNCTION DIFMU(INDEX)
implicit double precision (a-h,o-z)
EXTERNAL F5,DIFF2
LOGICAL MA,MB,MC
COMMON/MOD/MA,MB,MC
COMMON/SIX/DXO,DX 1
COMMON/SEVEN/DMUO,VS
COMMON/TW0/A,DL,DL 1
COMMON/FOUR/ALFA
COMMON/FIVE/Vl,VDS,VGS,VBI
COMMON/TFIREE/VO,ES,Q,EC,W,DO
COMMON/XYBOX/X( 101), Y( 101 ),elec(2,101 ),si 1 (2,101)
COMMON/ONE/GAMA,DLAMDA
COMMON/ELE V/DIFGM,DIFL 1 ,DIFD 1
PI=4. *DATAN( 1 .DO)
PI=PI/(2.*A)
DPL=PI*DL
XI=DPL/DSINH(DPL)
DIFMU=0.D0
WISUM=0.D0
M=ll
NX=101
NY=101
DO 101=1,M
IY=(I-1)*(NY-1)
IY=1+IY/(M-1)
WI=(FLO AT (I-1 )/FLOAT (M-1 ))* ALFA
WISUM=WISUM+WI
EX 1 =EX(INDEX,I Y)
E Y1 =E Y(INDEX,IY)
E=DSQRT (EX 1**2+EY1**2)
TEMP=PI*Y(IY)
65


nnnnn
DEX=-( 1.-(Y(I Y)/A-1. )**2)/DL+
1 (XI/DL)*DCOSH(PI*X(INDEX))*DSIN(TEMP)
IF((INDEX.EQ. 1). OR. (I Y.EQ.NY))THEN
DEY=0.
ELSEIF(INDEX.EQ.NX)THEN
CALL DENT1(DIFF2,DX1,T1,IY)
T2=XI*DSINH(DPL)*DCOS(TEMP)/DL
T3=+(Q/ES)*T1 *DIFD1+T2
CALL DINT1(F5,DX1,SUM,IY)
DEY=T3+(Q/ES)*(+(DL-DL1 )*SUM*DIFGM
1 -GAMA* SUM*DIFL 1 +GAMA*(DL-DL 1) *T 1 *DIFD 1)
ENDIF
DE=(EX1/E)*DEX+(EY1/E)*DEY
c ELIM=2.*EC
c IF(E.LE.ELIM)THEN
c DMUSI=-VS/(4. *(EC**2))
c ELSEIF(E. GT.ELIM)THEN
c DMUSI=-VS/(E**2)
c ENDIF
tl=e/ec
t2=tl*tl
t4=t2*t2
tv=l.+t4
tw=(dmu0+(vs/e)*t4)/tv
dmusi=(t4/tv)*((3. *vs/(e**2))-4. *tw/e)
DIFMU=DIFMU+DMUSI*DE*WI
10 CONTINUE
DffMU=DIFMU/WISUM
RETURN
END

SUBROUTINE ROOT(XBAR,DNEW)
implicit double precision (a-h,o-z)
COMMON/T WO/A,DL,DL 1
66


COMMON/THREE/VO,ES,Q,EC,W,DO
C0MM0N/FIVE/V1,VDS,VGS,VBI
COMMON/DATAI/C1 (61 ),C2(61 ),C3(61 ),C4(61)
COMMON/XDIR/D(61)
common/out/volt(51 ),curr(51)
ITL=0
ILIMIT=20
NSPLIN=60
FX=ES*(-VGS+VBI+V1 *XBAR/DL)/Q
DO 10 I=1,NSPLIN
IF(dabs((FX-C 1 (I))/fx).le. 1 .e-4)THEN
DNEW=D(I)
RETURN
ELSEIF(dabs((FX-C 1 (1+1 ))/fx).le. 1 ,e-4)THEN
DNEW=D(I+1)
RETURN
ELSEIF((FX. GT. C1 (I)).AND.(FX.LT. C1 (1+1 )))THEN
GOTO 20
ENDIF
10 CONTINUE
WRITE(7,40)FX
40 FORMAT(E10.3,' IS NOT IN THE GIVEN SPLINE RANGE')
GOTO 50
20 DNEW=(D(I)+D(I+1 ))/2.
30 CALL PCUBIC(DNEW,YBAR,DIF,NSPLIN)
F=YBAR-FX
DOLD=DNEW
DNEW=DOLD-F/DIF
ERROR=DABS(DNEW-DOLD)/DABS(DNEW)
IF(ERROR.LT.(l -D-6))RETURN
ITL=ITL+1
IF(ITL.LT.ILIMIT)GOTO 30
WRITE(7,11)
11 FORMATCNO CONVERGENCE FOR FINDING D(X)')
50 do60k=l,51
write(7,*)volt(k),curr(k)
60 continue
STOP
67


END
C
C
c
c
c
SUBROUTINE DSPLINE
implicit double precision (a-h,o-z)
COMMON/NINE/DIF(5),CF(5),CFO(2)
COMMON/DATAI/C1 (61 ),C2(61 ),C3(61 ),C4(61)
COMMON/XDIR/D(61)
COMMON/TABLE/CS1 (5,61 ),CS2(5,61 ),CS3(5,61 ),CS4(5,61)
COMMON/SIX/DXO,DX 1
NSPLIN=60
DO 30 K=l,5
DO 15 J=1,NSPLIN
C1(J)=CS1(K,J)
C2(J)=CS2(K,J)
C3(J)=CS3(K,J)
C4(J)=CS4(K,J)
15 CONTINUE
C CF0(1)=F3(D0), CF0(2)=F4(D0)
C CF(I)=FI(D 1 ),I= 1,5
IF((K.EQ. 3). OR. (K.EQ. 4))THEN
CALL PCUBIC(DXO,YBAR,DIFF,NSPLIN)
CF0(K-2)=YBAR
ENDIF
CALL PCUBIC(DX 1, YBAR,DIFF,NSPLIN)
CF(K)=YBAR
DIF(K)=DIFF
30 CONTINUE
RETURN
END
C
C
c
c
c
68


SUBROUTINE ELEMENT(CIS,CID,DV1)
implicit double precision (a-h,o-z)
LOGICAL MA,MB,MC
COMMON/MOD/MA,MB ,MC
COMMON/ONE/GAMA,DLAMDA
COMMON/SEVEN/DMUO, V S
COMMON/DAT AI/CI 1(61 ),C2(61 ),C3(61 ),C4(61)
COMMON/XYBOX/X( 101), Y( 101 ),elec(2,101 ),si 1 (2,101)
COMMON/TABLE/CS1 (5,61 ),CS2(5,61 ),CS3(5,61 ),CS4(5,61)
COMMON/THREEA^O,ES,Q,EC,WJDO
COMMON/TWO/A,DL,DL 1
COMMON/FOUR/ALFA
COMMON/FIVE/V 1, VDS, VGS, VBI
COMMON/XDER/D(61)
COMMON/SIX/DXO,DX 1
COMMON/NINE/DIF(5),CF(5),CFO(2)
COMMON/ELEV/DIFGM,DIFL 1 ,DIFD 1
CALL DSPLINE
PI=4. *DAT AN( 1 .DO)
PL=PI*DL/(2. A)
XI=PL/DSINH(PL)
PL 1 =PI*DL 1 /(2. A)
if(ma.or.mc)then
difll=0.D0
else
difl 1 =dl *(xi *cosh(pl 1)-1. )/(pl *xi *sinh(pl 1 )*vO)
endif
difdl=(es/(q*dif(l)))*(dll/dl+(vl/dl)*difll)
if(ma)then
difgm=0.D0
else
difgm=-(es/(q*dl*cf(2)))*( 1 ,-(dif(2)/cf(2))*difdl *vl)
endif
TEMP=Q*W/DL
T2=DMU(1)
DT2=DIFMU( 1)
11 =CF0( 1 )-XI*CF0(2)
t4=XI*VDS*CF0(2)
69


if(ma)then
al=t2*temp*tl
bl=t2*temp*t4
dal=al*(dt2/t2)
dbl=bl*(dt2/t2)
else
t3=temp*( 1. -gama*dl 1)
dt3=-temp*(difgm*dl 1 +gama*difl 1)
tauxi=dt2/t2+dt3/t3
al=tl*t2*t3
bl=t2*t3*t4
dal=al*tauxi
dbl=bl*tauxi
endif
CIS=A1*V1+B1
Tl=CF(3)-XI*DCOSH(PL)*CF(4)
T2=DMU(101)
T3=TEMP*( 1 ,+GAMA*(DL-DL 1))
T4=XI*DCOSH(PL)*CF(4)*VDS
T5=Q*W*D0*GAMA*CF(5)
DT1 =(DIF(3)-XI*DCOSH(PL)*DIF(4))*DIFD 1
DT2=DI
FMU(lOl)
DT3=TEMP*(DEFGM*(DL-DL 1 )-GAMA*DIFL 1)
DT4=XI*DCOSH(PL)*DIF(4)*VDS*DIFD 1
DT5=Q*W*D0*(DIFGM*CF(5)+GAMA*DIF(5)*DIFD 1)
IF(MB.OR.MC)THEN
C1=T1*T2*T3
D1 =T2 *T3*T4-T5
DC1=DT1*T2*T3+DT2*T1*T3+T1*T2*DT3
DD1 =DT2 *T3 *T4+T2 *DT3 *T4+T2 *T3 *DT4-DT5
ELSEEF(MA)THEN
C1 =TEMP*T 1 *T2
D1=TEMP*T2*T4
DC 1 =TEMP*(T 1 *DT2+DT 1 *T2)
DD1=TEMP*(DT2*T4+T2*DT4)
ENDIF
70


n n n n n n
CID=C1*V1+D1
RHS=(A1-C1)*V1+(B1-D1)
DV1 =RHS/((DA 1 -DC 1 )* VI +(A1 -C1 )+(DB 1 -DD1))
c WRITE(7,44)TMU 1 ,DTMU 1 ,T2,DT2, V1 ,DV1 ,CIS,CID
c44 FORMAT(8D8.1)
RETURN
END
PROGRAM FETIV
implicit double precision (a-h,o-z)
real qdom(51)
external f5
LOGICAL MA,MB,MC
COMMON/MOD/MA,MB,MC
COMMON/ONE/GAMA,DLAMDA
COMMON/TWO/A,DL,DL 1
COMMON/THREE/VO,ES,Q,EC,W,DO
COMMON/FOUR/ALFA
COMMON/FIVEA^ 1, VDS, VGS, VBI
COMMON/S1X/DXO,DX 1
COMMON/SEVEN/DMUO,VS
COMMON/XYBOX/X( 101 ),Y( 101 ),elec(2,101 ),si 1 (2,101)
COMMON/XDIR/ADIS(61)
COMMON/DATAI/C1 (61 ),C2(61 ),C3(61 ),C4(61)
COMMON/DATAO/SPLX(61 ),SPL Y(61)
COMMON/TABLE/CS1 (5,61 ),CS2(5,61 ),CS3(5,61 ),CS4(5,61)
COMMON/NINE/DIF(5),CF(5),CFO(2)
COMMON/OUT/VOLT (51 ),CURR(51)
COMMON/ELE V/DIFGM,DIFL 1 ,DIFD 1
open(unit=4,file-data',status-old')
open(unit=7,file='output',status-new')
NSPLIN=60
DO 5 I=1,NSPLIN+1
71


DO 5 J=l,5
CS1(J,I)=0.0
5 CONTINUE
MA=FALSE.
MB=.FALSE.
MC=:FALSE.
ERROR=0.
Q=1.6D-19
READ(4,*)DL,A,W
READ(4,*)EC,DMU0,D0,vs
c VS=DMU0*EC
READ(4,*)ES
ES=ES*8.854D-14
READ(4,*)VGS,VBI,DLAMDA,ALFA
READ(4,*)ILIMT,TOL,VDSO,DVO,VLIMIT
READ(4,*)DVGS,VGSL
nxl=101
nyl=101
XT=DL/FLOAT (nx 1 -1)
YT=A/FLOAT (ny 1 -1)
X(1)=0.D0
Y(1)=0.D0
DO 10 I=l,nxl-1
X(I+1)=X(I)+XT
10 continue
do 20 i=l,nyl-l
Y (1+1 )=Y (I)+YT
20 CONTINUE
CALL PINCH(VPO)
CALL INITIAL(VPO)
500 Vl=O.0D0
VDS=VDS0
ITL=0
N=1
V OLT(N)=0.0D0
CURR(N)=0.0D0
qdom(n)=0.0d0
1000 V0=(-V1+VDS)
72


I
CALL MODE
CALL DEPTH
c write(*,*)dxO,dxl,vl,ma,mb,mc
CALL ELEMENT(CIS,CID,DV1)
C V1 =V 1 +DABS(DV 1)
V1=V1-DV1
ERROR=DABS(DV 1)
c WRITE(7,*)ERROR,Vl
IF(ERROR.LT.TOL)GOTO 2000
ITL=ITL+1
IF(ITL.LE.ILIMIT)GOTO 1000
GOTO 3000
2000 N=N+1
VOLT(N)=VDS
CURR(N)=. 5 *(CIS+CID)
call dintl(f5,dxl,sumq,l)
qdom(n)=-w*(q/2.)*gama*((dl-dl 1 )**2)*sumq
write(*,*)volt(n),curr(n),qdom(n)
VDS=VDS+DV0
ITL=0
C V1=0.
IF(VDS.Lt.VLIMIT)GOTO 1000
3000 WRITE(7,12)VGS,VPO
DO 501=1,N
WRITE(7,13) VOLT (I),CURR(I),qdom(i)
50 CONTINUE
IF(ITL.GT.ILIMIT)THEN
WRITE(7,11)
GOTO 4000
ENDIF
VGS=VGS+DVGS
IF(DABS(VGS).LT.DABS(VGSL))GOTO 500
11 FORMATCNO CONVERGENCE')
12 FORMATC VGS= ',F6.2,/,' PINCH-OFF VOLTAGE = ,F6.2)
13 FORMAT(F5.2,E 12.5, e 12.5)
c4000 write(*, *)v 1 ,v0,vds
c do 6000 m=l,2
c call poten(m)
73


c call carrier(m)
c6000 continue
c do 5000 j=l,101
c write(7,*)y(j)* 1 .e+4,si 1 (1 j),elec( 1 j),si 1(2 j),elec(2 j)
c5000 continue
4000 close(unit=4)
close(unit=7)
STOP
END
74


APPENDIX B
GATES Functions
GALLIUM ARSENIDE TRANSISTOR ENGINEERING MODELS,
GATES COMMANDS
PROCESSING DEVICES OUTPUT
IMPLANT VTP (POISSON SOLUTION) PLOT
ACTIVATION RS (SHEET RESIST) IVPLOT
DOSE GEOMETRY PISCES FILES
BOX PROFILES RG (GATE RESIST) DATA
ELECTRONPROFILES DIODE
DIFFUSION PROFILES IV REVFET INFO
ETCH THERMAL EFFECTS CLR
SC(SHORT CHAN) MODELS SAVE/RETR
PIEZOELECTRIC SPARAMETERS SET (TEMP,VBI)
ANNEAL PROFILE NOISE PARAMETERS
BULK IMPURITIES VTH BATCH
CV SIMULATIONS MONTE-CARLO
FAT-FET SYS/RUN
DBASE REND END RESISTANCES QUIET
CREATE BASE HELP
DEPOSIT DATA STOP
PRINT DATA JFET THRESHOLD VOLTAGE EXIT
SIDEWALL CAP (JFET)
75