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A physics-based analytical model of an AlGaN/
GaN high electron mobility transistor
Jonathan Sippel
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A PHYSICS-BASED ANALYTICAL MODEL OF AN AIGaN/GaN
HIGH ELECTRON MOBILITY TRANSISTOR
by
Jonathan C. Sippel
A Thesis submitted in Partial Fulfillment of the
Requirements for the Degree of
MASTERS OF SCIENCE
In
Electrical Engineering
Approved by:
Professor
(Dr. Syed Islam - Advisor)
Professor
Dr. James Moon - Committee Member)
Professor
(Dr. Santosh Kurinec - Comminee Member)
Professor
(Dr. Robert Bowman - Department Head)
DEPARTMENT OF ELECTRICAL ENGINEERING
COLLEGE OF ENGINEERING
ROCHSTER INSTITUTE OF TECHNOLOGY
RCOHESTER, NEW YORK
MAY 2004
THESIS RELEASE PERMISSION
DEP ARTMENT OF ELECTRICAL ENGINEERING
COLLEGE OF ENGINEERING
ROCHESTER INSTITUTE OF TECHNOLOGY
ROCHESTER, NEW YORK
Title of Thesis:
A PHYSICS-BASED ANALYTICAL MODEL OF AN AIGaN/GaN HIGH
ELECTRON MOBILITY TRANSISTOR
I, Jonathan C. Sippel, hereby grant permission to Wallace Memorial Library of the
Rochester Institute of Technology to reproduce my thesis in whole or in part. Any
reproduction will not be for commercial use or profit.
-f-,(&-'--'--,I./..-C!<1\.L.,--'-Y_
Signature _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _Sl
Date
ACKNOWLEDGEMENTS
tenure at RIT has gone
My
by
so
sixth year as a matriculated student
have been
joined
a
brother
and
enrolled
fraternity,
get
have
degree. In short, I have
At this time I
guidance
pretty
of
experienced a great
helped
would also
professional
To
me over
on a
topic
If I had to do it
schedules
They
research process.
six
ending up in the
I
years,
right
one),
deal
an
during
the passing
influence
on
my time here
into the
person
I
on of
he
He
was
was
my life to
and would
am
to
originally shaky
again, I
with
some
like to
today.
support and
brained student,
to grind out a thesis
would still choose
be
part
offer
is
of
and
-
to study GaN HEMTs
Dr. James Moon for taking time
my thesis
second
committee.
fellow
researchers
played a pivotal role
mention
that
The knowledge
out
and
to none.
was outfitted with a computer and
alongside
Not to
to take a scatter
able
my father,
professor.
my research, I
the EE department
Shailesh Rai.
have had
the years to grow
all over
insight they have to
perform
experienced
like to thank Dr. Santosh Kurinec
hectic
(luckily
In those
like to thank my advisor, Dr. Syed Islam, for his
would
focus him
amazing.
their
fraternity,
all of whom
people,
beneath Dr. Islam. Thanks
I
a
throughout this thesis work.
myself, and
programs
program.
traveled across country, been graced with a second niece, had a
met numerous
who've
in the RIT engineering
in three engineering
married, disjoined from
thank those
fast it is hard to believe that I'm approaching my
Sankha
desk in the MEMs lab in
Mukherjee, Brett Klehn,
in my retaining any
sort of
and
sanity through the
every time I felt like trading in the thesis towel for
the graduate paper
towel, they
were able
to knock
some sense
into
me and
keep
me on
track.
I
would also
Certainly none
support
like to thank my
of
from my
Lastly, I'd like
family
who
always supported
everything I've done.
if I didn't have the
mental and emotional
has
this would have been possible
parents and siblings.
to thank the Gleason Foundation for
funding this
research.
in
ABSTRACT
Popular
semiconductors
currently
being
for RF
used
InP. The operating frequencies for HEMTs built
with
applications
these semiconductors
range
from 800 MHz to 100 GHz. Although high-speed
GaAs
or
where a
years
InP,
high breakdown field
ago
a solution to
as
and perhaps
challenge,
products, particularly
band gap,
material
and
high
parameter
and
power amplifiers.
values
researchers and
proposed
equations
device
to
approximated
Schrodinger
derivations
as
voltages.
a
The
triangular
equation.
as
used
The
and channel charge
and
viable
GaAs). GaN
to work on the
with
use of
fifteen
candidate
to
in RF
formed
equations
derivations for
such
On
modeling
near
as
carrier
a worldwide
of
GaN-based
future.
density, Fermi Level,
at
the
two eigenstates, both
control
possesses other
the Schrodinger and Poisson
sheet carrier
well
quantum
charge
a
about
the dominant semiconductor
commercially in the
between the
well
as
and piezoelectric coefficients.
is physics-based, making
establish relationships
terminal
be
using
This is due to its high breakdown field, large
experts continue
can
the
covers
attainable
Proposed
needed.
favorable to existing technologies
that are
industry
model
InP
is
emerged
conductivity (relative to InP
HEMT devices in hope that GaN
The
and
is
operation
and
considering high-power applications,
this problem, GaN has
overtake, GaAs
thermal
when
thermal conductivity
velocity, dielectric constant,
saturation
scale,
is limited
product performance
include GaAs
were
heterointerface
and
is
determined from the
carried
over
into the I/V
capacitance calculations.
IV
Spontaneous
the high
density
expression
heterointerface
and piezoelectric polarizations at the
of carriers
in the
channel and are accounted
for the device threshold
aluminum mole
fraction
of the
voltage.
Device
are responsible
for in the
performance
barrier layer. This is because the
model
is dictated
fraction
mole
for
in the
by
the
controls
the amount of polarization at the heterointerface and
consequently the 2DEG density.
I/V
region
equations were
model
modulation.
was
Device
results compared
To
address
to
high
derived
adopted
were
experimental
frequency
data
assumed
both drift
saturation
cutoff
region
diffusion
to account
components.
for
channel
A
two-
length
current expressions and
gathered.
expressions
for the
and
derived from the drain
device behavior,
frequency
Relationships between the
bias
for the
conductances were
expressions and cutoff
conditions
incorporating
parasitic gate capacitance
are
drain bias
frequency
of the
are shown and compared with published
data
derived
to
and presented.
simulate
device,
a
high
the length
reported
by other
of
(Cgs
and
High
power
voltage
scenario.
the gate, and
authors.
Cgd)
drain
Table
of
Contents
Acknowledgements
ii
Abstract
Table
of
iv
Contents
vi
List
of
Figures
List
of
Tables
List
of
Acronyms
xiii
List
of
Symbols
xiv
List
of
Publications
1. Chapter 1
-
ix
xii
Introduction
xv
and
Historical Review
1.1 Introduction
1
1.2HEMT:Whatisit?
2
1.3
Why GaN?
6
1
Contemporary Modeling Issues
8
Literature Review
9
1
.4
.5
12
1.6 Contributions
1
.7
13
Thesis Organization
2. Chapter 2
-
Theory / Model
Formulation
14
2.1 Introduction
2.2 Sheet Carrier Density, ns,
vs.
Fermi Level,
2.3 Sheet Carrier Density, ns,
vs.
Gate Voltage, Vgs
Ef
14
17
vi
2.4 Sheet Carrier Density, ns,
2.5 Threshold Voltage, Vth,
vs.
vs.
Barrier Thickness, dd
20
Doping Density, ND
2.6 Current-Voltage Characteristics
20
2.7
27
Transconductance,
2.8 Cgs
and
gm, and
Output Conductance, go
29
Cgd
2.9 Cutoff Frequency,//3.
Chapter 3
-
Materials
and
32
Apparatus
3.1 Hardware Data
4.
Chapter 4
-
Results
and
34
Discussion
4.1 Introduction
36
4.2 Sheet Carrier Density, ns,
vs.
Fermi Level,
4.3 Sheet Carrier Density, ns,
vs.
Gate
4.4 Sheet Carrier Density, ns,
vs.
Barrier Thickness,
4.5 Threshold Voltage, Vth,
vs.
37
Ef
39
Voltage, Vgs
dd
Doping Density, ND
4.6 Current-Voltage Characteristics
4.7 Transconductance, gm,
4.8
Cgs
and
and
Output Conductance, g0
Chapter 5
-
Conclusions
42
44
45
49
54
Cgd
57
4.9 Cutoff Frequency,//
5.
19
and
Future Research
5.1 Conclusion
60
5.2 Future Research
61
References
65
vn
Appendix I
Idlin Derivation
-
Appendix II
-
Vc(x) Derivation
68
7{
vin
4.1 1
go as
a
Curves
4.12
of
Vgs
with
for Vds
were plotted
Output conductance, go,
plotted
4.13
function
for Vgs
values of
-2
Vgs
Vds
and
of
V, 0 V,
-1
function
a
as
5 V, 10 V,
function
V,
Vgs
as a parameter.
values of
as a
Output Resistance, Ro,
parameter.
Vds
values used
for
with
and
1 V.
Vgs
Drain
from
-3
V to 2 V.
51
Curves
52
15 V.
and
V^
of
varied
as a parameter.
with
voltage
simulation are
the
Vgs
same as
as
a
those
in
53
Fig. 4.12.
4.14
Gate-to-source
capacitance as a
Geometry
parameter.
demonstrate "high
4.15
4.16
Cgd
as a
for
a
of
15
fr
fr as
a
of
function
and
Threshold
density.
linearly
ND
1 V
Vgs
with
device
5 V
Vds
as a parameter.
7VD
with
=
voltages used
55
to
simulations were
and
T
=
300K. Vds
done
56
values
of width
75
um and
length 1
57
to represent typical gate amplification voltages.
was used
of gate
All
2x10
Vds.fr calculated for device
length, L. Simulations done for drain biases
voltage as a
with
High drain
as a
were used.
5 V. Dimensions
Vth is
given on plot.
drain bias
capacitance values.
urn
of
device is
of
of gate voltage with
power"
of
and
function
VgS
10 V,
5.1
1
um x
V, 10 V.
as a
um.
4.17
75
function
function
of
device
function
observed
increasing
values at m values of
were
of mole
75
um
by
fraction
1
and
0.38
or
practical range
58
doping
62
um.
barrier layer
to decrease exponentially with
ND. The
15 V,
of
increasing
to consider
m
and
here is for
all
less.
xi
List
1.1 Semiconductor
3. 1 Device
4.1 Table
parameter comparison
parameters as reported
of constants used
4.2 AlGaN/GaN
of
for
by Wu
Tables
[4,5].
et al.
[23]
fraction. A
for
simulations.
results calculations with values and references.
material system equations used
aluminum mole
that were used
room
to
calculate results. //;
temperature of 300K was
is
assumed
the
for
all
calculations.
/n
List
of
Acronyms
Acronym
Meaning
2DEG
2-Demensional Electron Gas
Al
Aluminum
AlGaN
Aluminum Gallium Nitride
BJT
Bipolar Junction Transistor
CW
Continuous Wave
FET
Field Effect Transistor
GaAs
Gallium Arsenide
GaN
Gallium Nitride
HBT
He teroj unction Bipolar Transistor
HEMT
High Electron
Mobility Transistor
ID
Heteroj unction FET
Intentionally Doped
InP
Indium Phosphide
HFET
LHS
Left Hand Side
MESFET
Metal Semiconductor FET
MODFET
Modulation Doped FET
MOS
Metal Oxide Semiconductor
MOSFET
Metal Oxide Semiconductor FET
MS
Metal Semiconductor
RHS
Right Hand Side
SDFET
Selectively doped
FET
Si
Silicon
SiC
Silicon Carbide
Si02
Silicon Dioxide
TEGFET
Two Dimensional Electron Gas FET
UID
Un-Intentionally
Doped
VGA
Video Graphics
Array
Xlll
List
Symbol
AlGaN Dielectric
e(m)
1
g*
,
Basal Strain
;;
Sain
":
;
eGaN
.:.
MO.
,
heterointerface
Constant
Dielectric Constant
of
AIN
Dielectric Constant
of
GaN.
Low-Field
.
Mobility
Mobility
Pi
(.-
*.
.../
;..,-
constant
Field Dependent
ft{x)
V,
at
Permittivity
So
Symbols
Symbol
Meaning
*
'
**
of
IMeaotting
Boltzmann's Constant
V
I
Gate Length
L,
L2
Eds
Ecd
'
Schottky
<Pm(m)
aAicaiAm)
'
'
^w
'
M
AlGaN Electron
MeAlGaN
m-e.AIN
GaN Lattice Constant
Nd
AIN Lattice Constant
ns
Gate-to-Source Capacitance
ns0
Gate-to-Drain Capacitance
nsat
Thickness
Thickness
dd
4
of
of
of
2DEG
Layer
Spacer Layer
rest
Mass
AIN Electron Rest Mass
Barrier Layer
Pd
PpbJAlGaN)
PsponAAlGaN)
Psp,)nt(GaN)
AlGaN Piezoelectric Polarization
AlGaN Spontaneous Polarization
GaN Spontaneous Polarization
1
Electron Charge
Output Resistance
Channel
Rs
Parasitic Source Resistance
Critical Electric Field
R"
Parasitic Drain Resistance
T
Temperature
Discontinuity
Position Dependent Electric Field
Lowest Eigenstate in Channel
.
Doping Density
Density
Equilibrium Sheet Carrier Density
Saturation Sheet Carrier Density
Power Density
Sheet Carrier
Q
Ro
Conduction Band
Ac
of
Cap
Doped Barrier Layer
Thickness
(x)
fl
Drain Contact Length
Electron Rest Mass
AlGaN Lattice Constant
Effective Width
d cap
k
to
GaN Electron Effective Mass
'
'
High-Field Region
"
n
C
c^
of
Aluminum Mole Fraction
ma
Barrier Height
Low-Field Region
Length
Gate
m*
Density
of
Drain to Source Contact Length
Polarization-induced Sheet Charge
a{m)
Length
Second Lowest Eigenstate in
,
Channel Charge
'"
Ec
">S;
Conduction Band (In diagram
Ec
only)
Fermi Level
E/^y
Valence Band
Ev
': EfAlCaN
AlGaN
EgMtf'
'!!''
E^GaN
/r ;
tii
/man
',
<.,..
..
Max.
r
,
Energy Gap
AIN Energy Gap
GaN Energy Gap
Cutoff Frequency
Frequency of Oscillation
Transconductance
..
V
Volts
Channel Carrier
:V(X)
Drain-to-Source Bias
Gate-to-Source Bias
Vm
vH
vsa,
Velocity
Channel Potential
Vc(x)
vds
,-.
Voltage
,,
across
High-Field Region
Electron Saturation
,
Velocity
Thermal Voltage
So
Output Conductance
VT
V,h(m)
h
Planck's Constant
W
Gate Width
h2,
Forward Gain
X
Channel Location
.
,
'
l
'.
Current
U,Un
lis
Linear Region Drain Current
'ds'at
Saturation Region Drain Current
x'
y
,,;
Threshold Voltage
Modified Channel Location
AlGaN barrier depth location
Drain Current
XIV
PUBLICATIONS
1.
Jonathan C. Sippel, Syed S. Islam
GaN/AlGaN HEMT incorporating
presentation
2.
and
Conference,
Jonathan C. Sippel
p.
and
physics-based
analytical
model
of a
for
polarizatio
Syed S. Islam, "A
spontaneous and piezoelectric
3.
S. S. Mukherjee, "A
spontaneous
in IEEE Canadian Conference
Jonathan C. Sippel
New York
and
on
piezoelectric
and
Electrical
and
Computer Engineering, 2004.
charge control model
27th
polarizations,"
Proceedings
Accepted
for AlGaN/GaN HEMTs
incorporating
Annual EDS/CAS Activities in Western
18, November 2003.
Syed S. Islam, "A
physics-based
model
for AlGaN/GaN
HEMTs,"
To be
submitted.
xv
Chapter 1
Introduction
Historical Review
and
1.1 Introduction
1.2 HEMT: What is it?
1.3 Why GaN?
1.4
Contemporary Modeling
Issues
1.5 Literature Review
1.6 Contributions
1.7 Thesis Organization
1.1 Introduction
The
wireless
telecommunications
to smaller
ever-expanding demands
technologies capable
on
has
high- volume products to
demand for low-cost
technology has led
industry
of
of
circuit
such
such
integrated
as
graphics
video
demands.
transceivers.
These
array
supply ranges,
companies
(VGA)
and
camera
sensor
have been
and
The
rapid
increased
researchers
development
to
used
(InP)
are used
modules,
to
to
operating
semiconductor
technologies
implement
based
various
driver-stage amplifiers,
design
circuits
Each, however, has been
power applications
find
of such
features. The
product
wireless semiconductor
Indium Phosphide
wide-ranging functionality.
in its ability to handle high frequency, high
Contemporary
and
decade. This is due to the increased
at a rapid pace.
Contemporary
semiconductors
performance characteristics with
power
have led
Silicon (Si), Gallium Arsenide (GaAs),
products
be developed
sizes, lower
consumers
meeting
exploded over the past
with
shown
and
remarkable
to be
limited
at temperature extremes.
devices that have been implemented using the
aforementioned
semiconductors
include Metal Oxide Semiconductor Field Effect Transistors (MOSFETs), Metal Semiconductor Field
Effect Transistors (MESFETs), Bipolar Junction Transistors (BJTs), Heterojunction Bipolar Transistors
(HBTs),
and
High Electron
implement many different
which
do
introduce
temperature,
power,
device is HEMT.
Transistors (HEMTs). These devices have
types of products, though their
not require simultaneous
sections will
the
Mobility
of application
device technology
simultaneously.
The
capable of
operation.
operating in
proposed semiconductor
their capability to
is limited to
high-power, high-frequency, high-temperature
a semiconductor and
frequency)
field
proven
those products
The
following
all three areas
(high
is Gallium Nitride (GaN)
and
The
imply
following
otherwise:
variables all refer to
Vgs, Vlh, Vds, lds, IdMm lisa
DC
and
quantities and not
AC
quantities as their subscripts would
Vc(x).
1.2 HEMT: What is it?
HEMT is
an acronym
"aliases"
for High Electron
Mobility
Transistor. Like
Effect Transistor), HFET (Heterojunction Field Effect Transistor),
Electron Gas Field Effect
dimensional
Transistor) [1], Each
it. For example, from the
characteristic of
made
has
numerous
MODFET (Modulation Doped Field Effect Transistor), SDFET (Selectively Doped Field
such as
entirely
the HEMT
devices,
other
from
one
electron gas
entirety, complete
semiconductor
that
and
.
drain
describes
aliases provided
(2DEG). Figure 1 1
with source and
alias
current
shows
either
it
can
and
TEGFET (Two-Dimensional
the structure of the transistor or some
be
conduction
gathered
is
that the structure
carried
the epitaxial structure of a
out
through
a
is
not
two-
HEMT transistor in its
terminals.
Barrier
Layer
Fig
1.1. HEMT
epitaxial structure.
made possible via a
Schottky
From top to bottom, charge control is
followed by a cap layer, charge
contact,
spacer layer, and buffer. The buffer is
(substrate) layer (not shown in the picture).
donor region,
nucleation
At first glance, the HEMT
the
HEMT is
from
a combination of
the channel
separated
structure pictured
from
by
an
the
MOSFET
in Fig. 1.1 is
and
insulator, allowing for
the channel
by
a
grown on
top
reminiscent of other
MESFET. In
a
of a
FET devices. In actuality,
MOSFET structure, the
charge control over
the channel. In a
barrier layer. The barrier layer is
made
of an
gate
is
HEMT,
alloy
separated
the gate
of the
is
buffer
is
in
figure
have three
sub
Layer, 2) Intentionally Doped (ID) Charge Donor Layer,
and
semiconductor and
layers
sub
shown
the
will expounded upon
later. The
differences between the MOSFET
MOSFET
contact.
only
by
occurs
The
Schottky
MOS
a
voltage of the
on
Assuming
is
substrate
on
conduction
few
the order of a
band
not
an
use
discontinuity for
electron
insulator
is
performed.
so current
flows
by
directly
region width
HEMT
There is
mediated
directly
beneath the
flowing
by
is
control
barrier height
the
of
This is
HEMT devices
concerns
and
heterointerface
on
The band
to Si. For the same
and the
reverse
beneath the
bias
applied
Si02 insulator
polarization
fields
and
HEMT is different from
HEMT employ
a
gate
drain terminals. The
a
into
contact
in the
and
The
interface
MOSFET,
Schottky
contact to
mediation
is
gate and channel
magnitude of current
the current path
device standpoint, this is
Si
a
gate and source terminals
region extends
the
1 eV, resulting in
than
insulation layer between the
between the
device. From
modulating the
controlling the
fundamentally
different
like the MOSFET.
MESFET,
biased to
or
source and
The depletion
difference between the HEMT
a
barrier
between the
gate
gate.
no
in
placed
of
carrier confinement.
discontinuity is less
reason a
limited
the carrier confinement
(eV) resulting in exemplary
relies
voltages,
The height
is
the semiconductor the gate
a
the case with the
not
the metal.
in
Schottky
through a
used over a wide range of gate
its
of the
that the charge control
the gate and channel.
Both the MESFET
through the
which operates more
another
is
volts
MESFET there is
beneath the
in the devices. In
channel and
a
the amount of reverse
volume of current
than a
and
charge
and
buffer layer. One
flow in the device. The difference between these devices is how the
For example, in
regulated
depletion
to a MESFET.
similar
current
mediate
lies in the fact
made of and
carrier confinement.
a worsened carrier confinement relative
the HEMT
be
the
MOSFET fabricated in Si, the energy difference between the
a
HEMT device does
is
is
is located in
HEMT the
a
is limited
Another difference between the MOSFET
channel.
in
insulator between
the type of metal the gate
3) UID Spacer Layer. The barrier layer
structures
capacitor can
contact whose gate voltage range
barrier depends
with.
HEMT
and
1) Un-intentionally doped (UID) Cap
layers:
current channel
capacitor whereas
difference is that
big
breakdown
the
MOS
via a
to
control
in the doped
a
and
Schottky
MESFET
contact
the amount of current
semiconductor.
So, if
is
structures other than
placed on
flowing in
top
of a
the device.
one wishes
to design
how the
doped
The
current
is
semiconductor
channel
is located
a structure capable of
higher
current
density,
problem with that
decrease
with
is that,
as the
be inclined
to
current
of
density
the carriers
its introduction
may be realized, the
were able
by
increase
the
doping density increases,
as more and more carriers are squeezed
increasing
mobility
one would
to maintain
its
into
doping density
semiconductor.
The
the mobility of the current-carrying electrons
a static-sized region.
So
although
magnitude of change will not
value.
the
of
This is
the problem the
Dr. Takashi Mimura in the late 1970's [2].
By
be
the overall goal of
as pronounced as
HEMT
employing
was proposed
a modulation
if the
to fix
doping
scheme, the HEMT physically separates the current channel from the doped semiconductor, allowing for
the carrier
density
reduced carrier
in the
channel to
mobility due to
A discussion
of the
impurity
band diagram
far. Fig. 1.2 depicts
mentioned thus
benefit from
t
Er
f
the
a
doped
semiconductor region without
suffering from
scattering.
of the
device may
help
device
to clarify some of the
band diagram corresponding to the
structure shown
\
F
aspects
in Fig. 1.1.
L
'
/'":
Buffer
*
Donor
*
'
Region
W,<-.:
\
E
f
r
i^~
F
v
o
CD
'
CO
/
Fig
1.2. HEMT Band diagram corresponding
to structure in
Fig. 1.1.
The band diagram has been appropriately labeled to
from left to right, the diagram
internal
electric
region
(MS)
contact
(donor region)
is doped n-type,
at
side of
the conduction
the
relationship between the figures.
how the energy bands
fields, band discontinuity,
metal-semiconductor
semiconductor
shows
show
and
depletion
the left side
the
junction
band, Ec, is
of
of
the semiconductor
regions created at
the
figure
results
and an associated
observed
the
in
barrier
a
bend
Progressing
to account
for
different interfaces. The
depletion
height,
ipm.
region
on
the
Since the donor
to slope downward as the distance from the
MS interface increases. The
layer causing the
spacer
is
phenomenon
barrier layers had the
had
a
defined alloy
discontinuity
base
same
layer
in
formulation
the
doping density
state
is
created
of
2) Polarization
the charge
in the
the
potential well and
that
the
absence of
the
be thin
must
and mobile carriers are
left in the region,
device to
is
the
such an extent
that
polarization
component
Spontaneous
component.
differences in lattice
is
composition
in
number of
into
parasitic
up
charges.
is
at the
dark
the
related
of
of
the
two
gate
The
between the buffer
and
degTade
semiconductor
created
mention
pertinent
from
tensile strain
at
add
not
and
overlap
presence
the performance of the
be
density
piezoelectric).
The
and
2)
resulting from
the
component,
the tensile strain
the
to
MS interface
the
1) Piezoelectric
the channel and raises the concentration of the
It is important to
to
1)
empty into
adding to the 2DEG
barrier layers. The tensile
amount of
carriers
may form in the barrier layer in the
channel will
electron
an equilibrium
is entirely depleted. If they do
sub-components:
is
the
well
all regions and all
At this time it is
other component
buffer layer
in the
to the thickness and
between
of
interface. In
two components:
on
The depleted
regions
layer to be
buildup
dimensionality
directly
gate.
buffer
interface. This
to the
adjacent
by
aligns
that the
the barrier
of
2DEG depends
depletion
MESFET
rendered unusable.
barrier layer increases the
induced
the
channel
piezoelectric component
constants
more charges
be
the
on either side.
below the
conducting
(assuming
made
The
field pulling
the
will
component
polarization
a
bias, Vds. This
it
of
reduced
below the
barriers
that
so
the semiconductor area
source
(indicated
component
directly
potential
enough
so
drain to
energy gap
bias. The Fermi level
barrier layer
by
the
density
The donor
external
any
heterointerface overlap
of an applied
to
was stated
making up the barrier layer
material
directly
interesting
an
rightward,
in the barrier layer. When the device is fabricated
region
become trapped
barrier layer
the
diffuse into the
region
band discontinuity, AEC,
(2DEG) forms
[1]. The
component.
donor
depleted from
mobile carriers are
type, but that the
refers
donor
Continuing
of a triangular potential well
the accumulated region
Donor component,
the
"two-dimensional"
beneath the Fermi Level);
inside
from
buffer layer. Earlier it
the
a conduction
this potential well, a two-dimensional electron gas
momentum
upwards.
composition causes
buffer layer resulting in
of the
meets
semiconductor
This alloy
composition.
results
undoped so electrons
band to bend
conduction
observed where the spacer
and
larger than that
layer is
spacer
strain creates an electric
2DEG.
Increasing
the alloy
interface and, consequently, the
here that the alloy composition, if too high,
can
cause
the
bonds to break
composition
alloy
full
strain).
The
at
the
is
so
interface, forming
high that bonds break,
optimal scenario
is
to
have full
piezoelectric polarization-induced charges.
from the
polarization,
results
spontaneous
polarizations
cation
of two
and
fields
can now return
to the band
diagram. As
band diagram is
observed to
flatten
discussing
understanding device behavior
effect on
bending
if
the
system.
in the barrier layer.
Vgs is
Any
band
discontinuity
conduction
band
and
relationships will
In
recent
years
which
presented
highest
amount of
having
right
Vgs
confine
the carriers to the
extending into the buffer
Fig. 1.2
section.
here that any
Level, Ef. It is
the
sheet carrier
the
2DEG
the
attention
layer,
the
level in
concentration
Fermi Level, Ef, is
pertinent
of
in
Vgs does
the entire
the
not change
barrier layer
distance between
bottom
the
2DEG
density,
concentration.
ns, and
In the
Ef and ns and
band
either
distance between the bottom
this
to
provides a visual representation of
change
energy bands
magnitude of
for
and the
thereby modulating
between the 2DEG
to
Both
carriers.
of mobile
been thoroughly discussed,
responsible
between
shifts the
Fermi
density
help
which
further
ultimately
note
high
a
create
constant as expected.
in the theory
dictates the
Gallium Nitride
better than the
and
(GaN) has
listed in Section 1.1
simultaneous
transistors on
GaAs,
Ef
Vgs is
positive,
and the
can
potential well
Vgs
the
(less than
relaxed
lattice [3]. Differences between the
the
heterointerface
become
in
to be
next
up,
of
the
of
the
chapter,
Vgs.
GaN?
semiconductors
require
be
change
down, if Vgs is
negative, or
Why
the
in
positions
layers
It is important to
said
alternate polarization sub-component, termed spontaneous
the relationship
presented
conduction
1.3
out and
interface is
degradation. If
performance
interface resulting in
strain at the
one progresses
be demonstrated in this thesis,
the potential well so
Vgs's
at
dislocations resulting in
the strain at the
anion
With the reasoning behind the
potential well.
will
The
adjacent
polarization effects create electric
As
misfit
GaN
high
have demonstrated
comparison.
a
viable
It is
clear
candidate
limited
to
replace
the
"wireless"
them to products that
power, high temperature operation. High Electron
aforementioned semiconductors.
InP for
as
whose performance capabilities
frequency, high
substrates
emerged
an
ability to handle the
Table 1 1 lists the
from
.
the table that
operational modes
material parameters
GaN collectively is
did
not
Mobility
collectively
for GaN, SiC, Si,
a
very appealing
semiconductor
temperature
technology in
capability,
breakdown field
and
high
with
bandgap
testament to its high power
5
at
second
any
only to SiC. The
device. For
in the device
instance, having
gate-to-source
Cgs,
and
capacitance values raises the cutoff
and/mav
are
figures
will
of merit used
to
GaN to
velocity is 1.45x1
intrinsic
the
helps to lower
operate at
cm/sec,
Lowering
the
frequency, fmax,
of
SiC
Si
GaAs
InP
5xl06
lxlO6
3xl05
4xl05
5xl05
1.43
1.344
8500
5400
Relative Dielectric Constant
10
9.66
11.7
12.5
12.5
cm/sec)
1.45
2.5
1
0.7
1.5
Conductivity (W/cm-K)
1.7
4.9
1.3
0.54
0.68
at elevated temperatures.
thermal conductivity, therefore
aspect of
GaN
2DEG
of the
concentration
has been
densities. The
a great
observed
most
to
deal. When
be
commonly
indicating
attractive
mentioned
heterointerface is Aluminum (Al). Most
simply AlGaN HEMTs
enhancing
it very
which makes
polarization effects
concentration
only to SiC, allowing GaN to
GaN HEMT devices have
in
grown on
is
that
it is
have
the barrier
device. /T
maintain performance
SiC
substrates to
a piezoelectric material.
to
benefit
times
These
larger
[6] leading
made of.
to
help
increased
create the
to raise the
Authors
lattice
the
2DEG
current and
strain at the
AlGaN/GaN HEMTs,
or
will sometimes opt not
to
referred to such structures as
layer is
effects
It, therefore,
GaAs-based HEMT structures,
alloy in the barrier layer to
authors
what
been
the previous section.
five
the
performance.
compared
more than
used
even
of the
[4,5].
GaN
second
that
adevice, respectively [1].
1.11
W/cm-K is
is
aforementioned
1400
.7
which
constant means
900
1
high
07
frequency response
speed and power gain of
material parameter comparison
conditions.
the values of the gate capacitance,
Cgd.
capacitance,
biasing
2.36
benefits from the
power
of
ability
SiC. The low dielectric
of
of an effect on
define the switching
severe
3.42
thermal conductivity of
Another
the
reveal
highest
the
material parameters are a
1100
Thermal
from the higher
These
and maximum oscillation
(xlO7
levels
to that
gate-to-drain
(V-cm1)
1.1, GaN has
Bandgap (eV)
Electron Mobility (cm2/V-sec)
vsa,
The
respectively.
velocity
constant
Parameter
Table
to
and the saturation
have less
frequency,/?-,
Table 1.1. Semiconductor
Breakdown Field
According
frequency
revealing its ability to handle
superior
lower dielectric
a
capacitance,
is 10
constant
high
-
3.42 eV,
and
mobility is
electron
parasitic capacitances
MV-cm"'
and carrier saturation
dielectric
relative
of
capability.
density, PD, capability
The mobility, dielectric constant,
frequencies. The
power
operation, high
everything"
it blends "the best
that
explicitly say
for the device
the acronym
1
and
with a
what the aluminum composition
with
and
fabricated
fT
along
with
the
being
The
and
for
considered
is
barrier layer, instead
(CW)
and
biasing
pure
fraction
50 GHz,
went as
high
as
techniques
is between 0
and
PD,
values,
itself
as
as a power
high
performance.
16.5 W/mm
as
a power
HEMT
device for
motor
In [8],
60 V revealing
The
for device
tested
and
respectively.
legitimizing
as a subscript to
AIN.
density
power
power amplifier applications.
growth
indicating it
called the aluminum mole
have been fabricated
up to 600 V
biasing
In [7], drain
the
1 corresponding to
100.9 GHz
as
steady improvement in
eagerly awaiting its
1.4
high
to withstand
and power applications.
in
GaN
revealed continuous wave
values as
and shown
effectiveness
pure
where m
report physical samples that
devices
of these
fmax
AlmGa]mN,
0 corresponding to
References [7,8,9]
Testing
as:
is in
the
was
drive
GaN-based HEMTs
results published
by
these authors,
for GaN, has many
researchers and companies
GaN-based HEMTs
are
commercial availability.
Contemporary Modeling Issues
modeling issues
growth and
which
has thus far kept it from
although more
is
associated with
GaN
and
wide-spread use commercially.
needed until mass production of
GaN
Steady
products can
progress
begin.
undoubtedly the
has been
reason
made each
issues
Contemporary
year,
under
scrutiny include:
1)
Modeling
2)
Correct incorporation
3)
Modeling
4)
I/V
of
thermal and
of growth
current
trapping
effects.
of polarization
process,
modeling
term
in device
threshold voltage equation.
and
incorporating
both diffusion
and
drift
current components
in linear
and
saturation regions of operation.
The
above
issues
are of utmost
importance if GaN-based HEMTs
effect"
wireless products.
up.
This
effect
phenomenon
is
is
The "thermal
observed as a
essential
refers
drop
refers
current with
for predicting device behavior
effect"
"trapping
in drain
to the mobility
to
electron
traps
are to
degradation
increasing
under
incorporated in the
be
used
to
develop
of the carriers as the
device heats
drain bias. Correct modeling
high bias, high temperature
structure
high-speed
during
of the
conditions.
material growth and
The
device
fabrication. The trapping
sensitivity,
and
gate-
effects
drain-lag
and
model them allows
correctly
include transconductance
for
accurate
circuit simulations that will
be
Although encouraging
has been
[6,1 1],
is
more work
voltage of
work
for
levels in both
the
linear
Such
parameters and provide
GaN
how the
elevated mole
diffusion
and
Many
I/V
usage
is
be
in that it
in this
to
sure
but none, to
chapter
different
the
provide
increase in the
near
future
how to
and
increases the reliability
polarization
have been
in the derivation
under
light
circuit
of
designs.
induced
charge
the heterointerface affect the threshold
models
could
collapse,
effects
incorporated into
are
components
valuable
current
trapping
modeling the
polarization effects at
fractions.
these
performed and
GaN-based HEMTs
insight into device behavior
outlined
be
simulations to
performed with regards to
a model would
The modeling issues
commercial use.
Understanding
and saturation regions of operation
have included both drift
current equations.
device
performed when
needed to model
the transistor
transients [10].
frequency dispersion,
the
used
biasing
of
to
to predict current
and saturation region
derive
small-signal
device
conditions.
reasoning why GaN has
as these
knowledge,
the author's
linear
the
of
be
best
reported
issues become
not
seen
wide
well understood
by
researchers.
1.5 Literature Review
Several existing models, equations,
later. All
however,
of the
adopted
future
In
in this thesis, why they
for why
chosen
because
were
in this thesis to
of
its
100%
proven
accurate.
adopted, and what their
produce
accuracy.
This
the results presented
This does
chapter
limitations
are.
certain parameters or expressions were used and
details
not
mean,
what models
This is done
to provide
to give
motives
for
research.
[1,12]
a
relationship is derived between the
the derivations differed slightly
assumed
the
equation
to
and
was
adopted models and/or equations are
reader an appreciation
the
of
borrowed literature
that all the
have been
and constants were used
quantum well at
generate
Ef was found,
the
from
sheet carrier
density,
ns, and the
one another, although the end result
the heterointerface to
quantum well eigenstates.
in both cases, to be [1,12],
be
Fermi Level, Ef. The flow
is the
same.
Both derivations
triangular and mentioned the use of the
The final
equation
detailing
the
Schrodinger
relationship between ns
E/~E0
ns
where
and
kB
D is
a constant equal
With only two
47rm*/h2
to
(h is Planck's
eigenstates considered, equation
accurate results.
reasonably
However
"weakness"
this
when
eigenstates are
highly
the
to
The relationship is
s(m) is the dielectric
Ad is the
region
shown
in
the
also
to
formulate
and
does
the equation's greatest weakness.
it is limited to
of
produce
Since
the
Most don't
certain regions.
region
in GaN).
lower
the
since
deemed
relating ns to
Vgs using
+Ad)
1
(1.2)
)
the electron energy,
dd is
the thickness of the undoped spacer region
2DEG, Vgs is
the applied gate
equation
a
bias,
(1.2) has been
and
V,h is
to
shown
the thickness
the threshold
be
quite
observed to
the transistor was
decrease
biased to
of
voltage.
in its ns
accurate
Vgs
When
values near
at an exponential rate
operate
the
in the barrier layer,
linear equation, its accuracy decreases for
has been experimentally
acceptable since
the Poisson
V^-V^m)*-
barrier layer, q is
[15]. Since (1.2) is
threshold and below where ns
weakness was
+ d.
dt is
inversion region,
concentration predictions
This
electron effective mass
(1.2),
{m)
=
in the barrier layer,
the strong
is
charge-control models
equation
constant of the
effective width of
biased in
is the
m*
(1.1) is relatively easy
two, the accuracy
linear
q{dd
donor
are the quantum well eigenstates,
device is biased in the strong inversion
provided
ns{m)
charge
E,
and
(l.l)
populated.
References [1,13,14] have
where
kj
DkBT In l + e
constant and
this strength
has been limited
number of eigenstates
equation.
+
Boltzmann's constant, 7 is temperature in Kelvin, E0
is
consider
kT
l+e
-DkBTln
[16].
in the strong inversion
region only.
The I/V
that this
model
that was adopted closely
model was
[17],
accounted
[18],
and
for
the
best to
channel
incorporated
start since
length
in [17] is
that
that reported
it incorporated drift
modulation
a reliable carrier
current model reported
followed
mobility
it does
in the
and
by
Rashmi
diffusion
saturation region
model reported
not account
for
by
by
Ruden
channel
et al.
[17,18]. It
components
was
in the linear
using the 2D Poisson
et al.
length
decided
[19]. The
modulation
region
equation
problem with the
in the
saturation
10
region, assuming drain current to remain constant
reported
in
[18] is
papers reported good
current
components,
The threshold
be dependent
it
that
neglects to
results,
with
incorporate the diffusion
be
a revised model should
and channel
length
drain bias. The
increasing
Although these
current component altogether.
made
the model
problem with
that accounts
for both drift
and
Rashmi
[15]. In [15],
V,h is
diffusion
modulation.
voltage model that was used was proposed
on the magnitudes of the polarization
by
fields induced
at the
et al.
given
heterointerface as,
V>)=f.M-ABI-^-<^W,+rf,)
2e(m)
where
<pm is the
is the
doping density,
Schottky barrier height, AEC
theoretically be
difficult to
conduction
o{m) is the polarization-induced
and
calculated
is the
for
device
a
calculate reasonable
Vlh
of specified
values
using
adopted
for a{m)
was reported
and experimental
point.
content was
Further
may have tried to
The
models
predictions
The
inability
found to be
research
for
is
a
huge
needed
to
the
biasing
model
of
(1.3)
point
[6]
to reliably
be
in
and
used
mole
calculate
Using (1.3), V,h
fraction. The
Vlh
V,h
and
values
scope of
this thesis
for
one of
2) They
can
Attempts
that was
model
between
0.15,
calculated
were made
to
locate
a(m) but very little has been done
for devices
the model to a
two
excess of
The
can
found it
author
fractions in
comparison.
the polarization effect on the
considered, or
hetero-interface, ND
the
density.
of mole
for data
at
and showed excellent agreement
in that it limits the
were used
sheet charge
relationship between
understand
conditions
starting
et al.
discontinuity
polarization model's credibility.
obtain a
setback
fully
just discussed
provided an excellent
have done to
Ambacher
results, substantiating the
other authors who
up to this
by
band
(1.3) for devices
the number of experimental samples that could
(13)
{m)
geometry
limiting
to
high
with
narrow range of
device threshold
reasons:
be improved
for understanding device behavior along
aluminum
1) They
upon.
In
devices.
voltage.
provide accurate
either
case,
they
with what past researchers
that behavior.
1.6 Contributions
The
contributions of
this thesis are as follows:
11
1)
A relationship between the
is
quantum well eigenstates
2)
A relationship between the
was
relationship
Vgs affects
3)
4)
discussion to
included in both
conditions
and
by
Predicted
region and
conditions.
high power, high
An
expression
established
are
model.
gate-to-source
Low-Field
for
density is
on
Ef.
presented.
The
of
how
understanding
doping density
aspects of
the
derived. Drift
has
barrier layer
were
will change
components
been incorporated in the
measured
provide
derived using the
data.
for
various
insight into the
results
have been derived. Both
discussion to
diffusion
and
also
experimentally
are presented
are plotted
current equations so
both
incorporated.
derived. When the device
and gate-to-drain capacitance are
channel charge was calculated
region).
The device
to
using
a
two-region model (High-
capacitances are plotted against applicable
interpret the
results and provide
was
insight into
the
bias
device's
capabilities.
the intrinsic cutoff
between the
and the
modulation
with
Conductances
provided
frequency
V,h
current equations are
coupled
region, the
Discussion is
and
how changing
Channel length
components are
saturation
of
and output conductance
derived
diffusion
Expressions for
Field
and the sheet carrier
results are compared with
plots
whose
the
drain
regions.
Device transconductance
biased in the
7)
understanding
and saturation region
calculated
6)
provide
Linear
drift
dependence
show ns's
equation and aims to provide an
barrier thickness
and the
performance.
bias
Poisson
the
Vgs,
visually
two
ns.
saturation region.
5)
are given to
applied gate voltage,
device
are
Plots
level, Ef, incorporating
ns, and Fermi
density,
carrier
shown.
derived from
Relationships between ns
with
sheet
cutoff
frequency
frequency
of
of
the device
the transistor,
length
is
presented and relationships are
of
the gate, and the
drain bias.
1.7 Thesis Organization
This thesis has been
Model Formulation,
Work. Each
of
these
partitioned
3)
Materials
into five
and
chapters:
1)
Introduction
Apparatus, 4) Results
chapters provides
information
and
pertinent
and
Historical Review,
Discussion,
to the
and
2) Theory /
5) Conclusions
understanding
of
and
Future
the proposed material
12
with what
along
the
future
larger chapters,
chapters
just
Chapter 1
a
favorable
such
-
MOSFET
that were
Other
of what a
derivation
saved
for
formal discussion
data is
each
details the derivational
lengthy
detailing
all materials used
along
with
the
insight is
future
in
of
process
for
important
semiconductor and
why it is
motivation
up into
a
also
the
"Materials
for the thesis along
discussed. Chapter 2
be
Some
intermediate
in the
for
paper.
complete analysis
In Chapter 5
brief summary
-
highlighting
weaknesses are again mentioned
is
directly.
steps of the
Apparatus"
reference used
-
equations
referenced
and
reported
points.
a
structures,
most equations used.
in Chapter 2. A
form
other
the sources can
-
to
needed
to
compares
equation with
Chapter 3
each section
wrapped
Major
The
the device that were
offered to reinforce
the results of the thesis are
the proposed model.
for
GaN
to complete this thesis. The
dimensions
presents results
how it
engineering field is
derivations listed only the final
weaknesses of
research
about the
other authors were not re-derived since
and additional
Work"
and
wireless semiconductors.
given to the electrical
Discussion"
and
figure
Future
leading
the appendix attached at the end.
also mentioned
"Results
have been
borrowed from
background information
provides all
given
included in
are
detailing their contents.
Review"
MESFET. Background is
and
had
brief description
further. Subsections
topics are presented and discussed. The
important
HEMT device is, how it operates,
Formulation"
equations that
made to extend the work
with a
Historical
and
when compared with other
/ Model
be
this one, where many
listed below
"Introduction
with what contributions
"Theory
as
mentioned are
basic understanding
notably
modifications can
is
a
semi-
experimental
Chapter 4
provided
"Conclusions
-
for
and
the strengths and
to provide a
foundation for
this area.
13
Chapter 2
Theory / Model Formulation
2.1 Introduction
2.2 Sheet Carrier Density, ns,
2.3 Sheet Carrier Density, ns,
2.4 Sheet Carrier Density, ns,
Fermi Level, Ef
Gate Voltage, Vgs
vs.
vs.
vs. Barrier Thickness, dd
2.5 Threshold Voltage, V,h, vs. Doping Density, ND
2.6 Current-Voltage Characteristics
2.7
Transconductance,
2.8
Cgs
gm, and Output
Conductance,
g0
Cgd
Cutoff Frequency, fT
2.9
and
2.1 Introduction
This
the
two models that are capable of
chapter presents
2DEG
such as
2DEG
is later
models
used
derive drain
to
predicting 2DEG
to the Fermi
concentration
Of the two
level using Fermi-Dirac
derived
transistor I/V characteristic curve equations are
next
models
statistics
bias using the Poisson
to the applied gate-to-source
levels. One
of
current equations and associated small-signal parameters
transconductance and parasitic gate capacitances.
concentration
carrier concentration
and
presented, the
by
a
relates
the second relates the
equation.
followed
first
The high
electron
detailed derivation
of
the
2DEG
mobility
the
small-
signal parameters.
2.2 Sheet Carrier Density, ns,
The
performance of
must
E,
the HEMT
be developed that
density-of-state
an equation
by
for
vs.
can
Fermi
device is
accurately
the probability of
the
Level, Ef
centered on
and
reliably
the concentration of the 2DEG.
predict
occupancy function for
channel electron
density
n,
can
=
the
finding
2DEG
Therefore
concentration.
a model
Multiplying
an electron at a particular
the
energy level
be [1],
\D{E)f{E)dE
(2.1)
o
where
D(E) is
particular
the density-of-state
energy E.
function
Employing Fermi-Dirac
and
f{E) is
the
statistics, f(E)
probability function for
finding
an electron at a
becomes [1],
14
/()=
(2.2)
k"T
l+e
where
kB is Boltzmann's
band energy level. Fermi-Dirac
conduction
distinguish
constant, T is the temperature in
one electron
from another,
to as the Pauli exclusion principle [1].
with respect
to momentum, the
density-of-state function
quantum well,
E0
and
can
and
statistics account
(2) No
Since the
be
can
valley
be defined
as
minimum
follows
and rn
lowest
is
the
subband
/j-
first
4nm*/h2
two
Using
subband
Therefore the
only the two lowest energy levels in the
E
the second-lowest subband
in GaN.
Plugging (2.2)
AX7t%K
.
and
kT
K'
=
DkBT In l + e
E-E.
E>l +
Schrodinger's
energy levels,
E0
kT
equation and
and
'0
Eh
=
were
+
i
by
a
yields
[1],
-dE
ekT
density,
DkBT In l + e
assuming
found
(2.3) into (2.1)
J"
Ei
energy level, h is Planck's
1
.-r
+
,
(2.3)
the integration yields an equation relating the channel carrier
nK
the
<
E,'
imdE+Vm
=^-dE
"l + e
=
spherical
E0<<,
r-m
1
J
o,
D
m*.
referred
*
m
electron effective mass
4K
Performing
mass,
to
E<En
4k
level, E\ is
energy
n*=vm
where
a single effective
(considering
h2
constant,
by
the
(1) It is impossible
energy valley in GaN, the Y valley, is
characterized
2
the
two restrictions:
Ec is
and
/),
D(E):
E0 is
for
is the Fermi level,
Ef
two electrons may occupy the same energy state,
0
where
Kelvin,
n to the Fermi
level, Ef,
k"T
(2.4)
triangular potential well, expressions
Rashmi
et al.
[15]
to
for
be,
2.123xlO"12(nJ2/3)
,=3.734xl(T12(ni2/3)
15
Fig. 2.1
shows
a
E0
eigenenergies,
close
and
E,,
the
of
up
triangular
in
are shown
quantum
the quantum well
well
at
with
along
heterointerface. The incorporated
the
the
Fermi level.
\
V
\\
Ec
-X
F
t
2.1.
Fig
Conduction
Vgs filling
If the Fermi level
increases,
E0
eigenstates
Conversely, E0
concentration.
or
and
band
showing the discrete
and down with applied
up
y
emptying the energy levels with electrons.
eigenenergy levels E0
Et
be
can
Ej
and
profile
,.
and
shifts
can
expected
be
to
expected to
be
fill
with electrons
raising the 2DEG
their electrons if the Fermi
emptied of
level
drops.
Analysis
of equation
(2.4)
relationship between
reveals an exponential
the arguments of the exponential terms to evaluate to
values of
eigenenergy
from
,
106
cm"2,
and
when
region
to
Ef approaches E0
in
n,
values of
the AlGaN/GaN heterostructure
To explicitly find
iteratively
equilibrium sheet carrier
would
into
hand
be
used
the right
side
for ns in
hand
(LHS)
of
side
the
for
E0
(RHS)
region
density
a given
density,
the
magnitude
forcing
region and
for
In this range,
the range
for higher
n,
Ef is
for ns in this
values of
that cause
Ef values
below both
well
region
is
anywhere
Ef. A linear dependence between
the exponential term to evaluate to
is
the target
have been found to be
biasing
region
on the order of
for
or
device for
the
10"~1013
unity
cm"2
using
system.
the sheet carrier
convergence
until
in this
indicates
zero.
cm"2
strong inversion
as the
values
research
1010
Ef,
is known
amplifier applications,
Et. Previous
and
for very low
Ef begins
larger. This
E0
less than
Ef and
ns0,
and
E,
This
function
Fermi level
Ef would
be
expressions.
of equation
equation.
as a
(2.4)
new value
so
of
the
to zero
The resulting
that a
equation
For example, if it
value.
set equal
Fermi Level,
E0
in
equation
and
E,
re-substituted
(2.4)
can
into
be
solved
desired to find the
and an
initial
values would then
numerical answer can
for rc, is then
was
(2.4)
be
value
plugged
be determined for the left
the
E0
and
E,
expressions.
16
The
new
solved
E0
and
E,
for. This iterative
happens, it
can
be
Having
shown
can also
section, a
plot will
light
process
on their
be
how
assumes a
equation
be
made
detailing
relationship
triangular
approximation.
can
and
vs.
how they
the
be
If it is
obtained
by
the
aluminum mole
2DEG, q is
is
the
<pm is the
to
are
E,
and
for ns for
an
ns
the value
longer
values no
for ns for
find
and
for
a given
a specified range of
and
ultimately dependent
Rvalue,
will
be
When this
eV.
one can see
In the
Ef values.
Efand insight
on
0
LHS is
the
change.
the given Rvalue of
value
for
given
results
to shed
Vgs.
the sheet carrier
only the first two
in
Fermi level, Ef,
ns, and the
(E,
quantum states
equation
interface depletion
occurs
density,
and
conjunction
region and gate
E0)
the total
with
depletion
in the AlGaN barrier layer, simplifying
occupied.
region
and
and
Another
depletion
one
overlap
solving Poisson
[15],
the
polarization-dependent
where
used
using the Poisson
assumed that the
doped AlGaN barrier layer,
of
E0,
(2.4)
Gate Voltage, Vgs
Vgs-V!h(m)-
=
q(dd
is
of equation
relationship between ns
the
potential well with
ns\m)
where m
be
can
relationship between
another, and that total depletion
equation yields
(2.4)
used to obtain a series of values
shows the
for ns
RHS
repeated until the ns,
equation
2.3 Sheet Carrier Density, ns,
Equation (2.4)
is
the
said that the equation converged on a value
how (2.4)
some
into
values are again plugged
d, is
d,
+
e(m) is the dielectric
charge,
Vgs is
threshold voltage given as
barrier height,
and
Ad)
(2.5)
<7
constant of
AlGaN, dd is
the thickness of the undoped AlGaN spacer
electron
Schottky
doping density,
fraction,
+
a(m) is the
AEC is
the thickness of the
layer, Ad is
the effective width
the applied gate-to-source voltage, and
the
[15, 17, 18],
the conduction
band
discontinuity
polarization-induced sheet charge
density
at
the
given as
hetero-interface, ND
[20],
\<?H K, {AlmGa,_mN)-Psponl {GaN)+ Ppiezo{AlmGa,_mN]
=
V,h{m) is
(2.7)
17
Pipom(AlmGa,.mN)
where
and
Pspnnl(GaN)
spontaneous polarization of the
pspm, {AlmGa,_mN)
Pspmu{GaN)
=
=
the spontaneous polarization of the AlGaN barrier
GaN buffer layer, respectively,
-0.052m
and can
be
expressed as
layer
and
[20],
0.029
-
-0.029
PPjezo(AlmGai_mN)
and
are
is
the piezoelectric polarization of the AlGaN
barrier layer
and can
be
expressed as
[21],
Pp,eZ0 {AlmGa,_mN) <w \e{m)] + (l
=
-
m)PN
[e{m)]
where
PJfa [e(m)]
*
= -1
PGaN [e{m)]
e
is
the
basal
=
-
.808f
-0.91
%e
strain and
is
7.888f
+
9.54
n
If*2
related to the
lattice
constants of the
GaN
substrate and
AlGaN barrier
that,
such
t(m)=aGaN~aAfN^
aAlGaN\m)
where
Law
the
as
aAlmGai
The
the
lattice constants, aGaN
in Coulombs
GaN buffer layer, Ppiew(GaN), is
The
done
it
were
absolute value
and models
curve
experimental
data using Vegard's
0.31986-0.00891m
=
units of each polarization are
linear
interpolated from
were
m/v(m)
inside the
was
aAiCaN(m),
[21],
the barrier layer. If
0V)
and
the total
regression
a
included, it
brackets
would
density
can
best fit line to
analysis,
to
be
of equation
amount of charge
sheet carrier
by finding
assumed
per square meter
be
the
a second-order
from
(2.7). Equation
depleted from
linear
the
negligible since
subtracted
be determined
(C/m ). The
the
as a
portion of
best-fit line
(2.5)
buffer layer is
AlGaN
the
piezoelectric polarization of
much
thicker than
piezoelectric polarization term
assumes no
drain-to-source bias (Vds
=
barrier layer.
function
the ,, vs.
of
Vgs according
to equation (2.5).
Ef curve discussed in
was obtained
for
the
linear
Section 2.2.
portion of the
ns
This
Using
vs.
Ef
as,
18
nj=(7xl014)j+(9xl0l3)/+3xl012
Solving
for ns
the above equation
function
as a
of gate
for
plugging into
Ef and
Vgs
g
qid.+d.+M)
Solving
an expression
be
that can
solved
7.5x10"
-V,h{m) +
+
*-(-
v
7xl07
140
'
for n
*7
ns
(2.5) leaves
Vgs,
voltage,
{m)
ns{m)--
equation
(m)
=
d Vgs-Vlh+a
+
b
-
1
db
(49d2b2
2MaY'
+4c +
2%dVp
2MV,,
-
,
+
(2.8)
where
V
140
b
V-cm
=
70000000
l/cmz
c
=
F/C-cm"
q{dd +di+Ad)
Equation
since a
(2.8) is
the
best-fit line
predicting
observed
resulting linear
was used
channel
between ns
for the linear
concentrations
and
Vgs. This
2.4 Sheet Carrier Density, ns,
Equation
Vgi
and
(2.5)
Vds
was used
were
fixed
the threshold voltage,
at
vs.
in
will
portion of
the strong
0V
V,h,
material parameters and
zero
since
magnitude
the ns vs.
inversion
is
last
device
unknown
between
be
left in
region
equation
only,
where
should
(2.8)
will
be
be
noted
that
accurate
for
linear dependence is
a
results section.
compared
sheet carrier
to
density
and
performed under zero
equation
geometries provided
negligible
aforementioned assumptions and
Ef curve,
V^. It
of
Barrier Thickness, dd
so that the analysis could
as the
for ns in terms
be discussed further in the
to show the relationship
GaN
its
charge control equation
in
the
(2.5)
which was
barrier thickness. Both
bias
readily
the subsequent chapter.
other
terms
in
conditions.
calculated
Ef was
equation
simplifications, n was calculated over a specified range
That left
using
assumed to
(2.6).
Using
be
the
for dd.
19
2.5 Threshold
Voltage, Vlh,
Doping Density, ND
vs.
Equation (2.6)
was used to plot the
doping density,
ND. The
were set equal
OV,
to
relationship between
same assumptions were used
and
here
device
the
as
threshold voltage,
in Section 2.4,
which
is
V,h,
say that
to
and
Vds
barrier
and
Vgs
Eflo OeV.
2.6 Current- Voltage Characteristics
The HEMT
be
channel current can
modeled
using the
f
Ids=Wq/l(x)
where x
Vc(x) is
is the location in
n
(m,x)
current
dVc (x)
density
equation
kBT
dn, (m, x)
q
dx
+
dx
the channel with origin at the source side (refer to
is
the channel potential, and^(jt)
the
field-dependent mobility
//(*)
[15,17],
given
x
(2.9)
Fig. 2.2), W is the
gate
width,
in [19] as,
(2.10)
=
1
1
dVc{x)
E,
dx
+
with
P
Ec vsat
_
MoEc-Vsa,
where
Ec is
mobility.
second
field,
The first term in the
term
equations,
under
the critical electric
for
(2.8)
drain bias
the diffusion
or
(2.5),
can
conditions.
be
The
vM, is the
saturation
parentheses of equation
component
in
used
of
drift velocity
(2.9)
the current.
equation
(2.9) for
charge control model
to
be
of
accounts
Neither
and
for the drift
of the
ns because
used
electrons,
iu0 is the low-field
component and the
two previous charge control
neither considers
here has been
reported
the charge
in
[1,15,13]
density
as,
(2.11)
q{dd+di+Ad)
which
includes the
channel
Plugging (2.10) into (2.9)
and
potential
term and can
be derived in
the same manner as equation
(2.5).
cross-multiplying yields,
20
1
The
boundary
account
for
conditions
of the channel
into
EcVsat
V
V
dV(x)
MoEc-vsl
+
directly
equation
beneath
making the
J
dx
(2.12)
the gate.
boundary
=
can
ns(m,x)
be formed
In this case,
conditions as
dVc(x)
WqjU0
kBT dns(m,x)
+
by knowing
parasitic source and
(2.12)
dx
dx
the
intrinsic
drain
voltages at either end
resistances
have been taken
follows:
K(x)U=Vds-IdsRd
where
Rs
and
respect to the
Rd are the parasitic source and drain resistances respectively. Integrating
Vc(x) from source to drain and solving for Ids yields
'
*d,lin
equation
(2.12)
j3-jj32-4ay
with
(2.13)
~
2a
where
a
=
{Rd+R5)
M0Ec-vsa
W{l0(m)
EcVsa,
2d
\
(Rl-Rl)
p=m^vA-m!*{RD+RtY-L.
M0Ec-vsa!
VJs
EcVsat
r-
Wju0(m).w
V V
Wfi0(m)v2
-
Vgsyds
d=d,,+ d,
+
ds
2d
Ad
k T
gs
=V
gs
-VAm)
The intermediate
the
current model
operation
^-
steps of the
for
derivation
the linear
of equation
region of operation.
(2.13)
The
are shown
condition
in Appendix I. Equation (2.13)
that was used to
determine linear
gives
region
was,
dldjin
1
f;.
>
3V*
dl dsal
,
dVds
v>v,
gs
th
21
where
To
!&,
refers to the saturation
get the saturation region
is
channel
high-field
separated
be
drain
of
drift
considered
toward the source, the channel
From this
diffusion
current components act.
components
in the
point
voltage will
The
approach was used.
and
(2) A High-Field Region [18]. The
through the region.
stressed"
by
is in
V^
the applied
low-field
the
carriers are
the carriers will
channel
become velocity
velocity
position, x, is
drain
equal
to
from
in Fig.
for
drain
un-
[18]. Here, both drift
and
and
diffusion
the current.
dominating
current equation
Lj,
the
eventually becomes
and
follows includes the drift
saturated and the
a saturation region
Progressing
region
appropriate regions of the channel to obtain an expression
low-field region, L;. When the
In saturation, the
across this pinched off region and all current
saturation current model that
is due to drift. The first step in establishing
value and
Region,
field
to the source the channel
In the high-field region, the
the
different
be dropped
electric
becomes "less
pinched.
next.
side where the channel experiences pinch-off as shown
due to the
current
be derived
A Low-Field
(1)
drain
drain
the applied
current to
current equation a
regions:
region occurs close to the
2.2. The majority
can
into two
drain
the electric
current component
is to find the length
field
will reach
its
of
critical
saturated.
<r
-}
Metal Gate
/N
AlGaN
v=v
E=E,
M/
2DEG
-L2-
-L1-
GaN
Figure 2.2. HEMT device
channel
i-
is
at
showing separation of
high-field region, L2. Point
structure
into low-field region, L,,
and
the onset of saturation where the electric
critical value and
field
assumes
its
velocity reaches its saturated value.
[18].
the origin for the offset axis y and
the
carrier
x'
Point
5 also acts as
22
The
boundary
potential
equation
condition
and electric
(2.9) from 0
b7,
t
.
2Vc2(x)
+
just
field in
to
x and
low-field
the
is
,
.
utilized once an expression
region.
An
expression
is
for the
obtained
for Vc(x) is
channel
by integrating
obtained
shown as,
\_
(aIdsal-bVgsy.(x) +
.
be
mentioned can
(
bVIdsat R
gs
b
,
s
+ */..
dsat
-
aI2R.
s
dsat
-
V
-
^
,
II
dsat
^
R2
=
2
0
(2.14)
j
where
_fi0Ec-vsl
EcVsat
b
W/i0(m)
=
d
The details
found for
of
this derivation as
well as
the
field
the channel potential, the electric
position, x, and,
since
higher potential,
the
integration is starting
a negative sign
is
E(x)
=
placed
for Vc(x)
solution
be found
in Appendix II. With
by taking
the derivative
the source and progressing toward the
at
in front
can
are shown
of
an expression
with respect
drain
which
at a
the derivative as,
'*"
-d^=
(2-15)
.
dX
is
to
J(aIdmr-bVj-2b(a0-/30)
where
a0=xIdsal+bVgsIdaitRs
A
=
When
x
=
L,, E(x)
=
L,
-Ec, and
A =^ +
2b
With
an expression obtained
high-field
continuity.
region
(L/
< x <
can
solved
^
2Idsat
for
L)
be
the
+
The 2-dimensional Poisson
the
(2.16)
LaA+^^-<-bVgsRs-^
2
2bEc
length
where
for using (2.15),
of
the low-field region,
Poisson
equation must
equation near
be
attention can now
used
to account
the drain end of the channel
is
be focused
for
on the
current and
given as
field
[18],
23
d2Vc(x',y)
dx'2
d2Vc(x',y)
^-T^
+
|
dy2
=
(2.17)
-^~ND(y)
(m)
x'
is defined
where
follows [18]
as
0<jc'<L2=L-L,;
x'-
It is
x-L^
assumed that there
conditions
is
concerning the
a uniform
doping
=
0,
0<y<d,
ND(y)
=
ND,
di<y<d
(2.17) is
profile
=
following boundary
{dd+di)
following boundary
subject to the
in the doped AlGaN layer. The
concentration result,
ND(y)
Equation
doping
conditions
[18]:
\
K (0, y)
=
(Vgs
-
(pm{m))
+
-M
f
Nd{y)dyLdsat
(m)\-b
qV
qwvsal
(d-y)
j
(2.18a)
z(dy(Nd(y)dy
(m
avc(o,o)
-Ec
(2.18b)
(Vgs-<pm{m))
(2.18c)
dx
Vc(x',d)
=
Ll
i
=
dy
In
equation
The
did
(2.18a)
net result of
not come
is found
by
the second term
the
expression
from the
n-doped
in the
inside the
{m)
parentheses accounts
parentheses equals
AlGaN layer.
summing together the ID Poisson
(218d)
_^_i2L
Following
for
the number of electrons
the solution outlined
equation solution and
V{x\y)
=
in
the channel.
in the
channel that
charge compensation
in [18], the
solution to
(2.17)
the 2D Laplace equation solution as,
V{y) + <S>{x\y)
(2.19)
where
24
a2^(v)
q
dy2
{m)
"M
d_j3Hx\y)_ 9^(x\Z)_
,
dx'2
Using
dy2
the
boundary
conditions
listed in (2.18),
a solution
for
the
ID Poisson
equation
in (2.19) is
shown to
be,
^y) ygs-<PM+^UNd(y)dy~n)(d-y)-~f-(dy(Nd(y)dy
=
Combining
equations
(2.18), (2.19),
and
(2.20) lead
to the
0>(0,y)
=
boundary
a$(y,o)
ay
Laplace
equation can
readily be
obtained as
=
=
together equations
(2.20)
Vc(x',y)-
and
+
(2.21c)
o
(2.21d)
conditions above, the solution
for
the 2D
[18],
COS
(
K
the
2D
channel potential
N
{2d
in
the
high-field region,
\
I n
COS
(2.22)
y
y~2d
J
yields
.
\
( K
ir
-sinh
K
0
-sinh
(2.22)
2dE^
(2.21b)
boundary
K
Adding
(2.21a)
2dE,
0(X',J):
equation
F
{x',d)
the method of separation of variables and the
for the 2D Laplace
0
33>(0,0)
Using
conditions
(2.20)
v
2d
\U
J
(d-y)
T-J(NAy)dy-J^1-
2-fdy[Nd(y)dy
(2.23)
25
Since y
equals
0
the
at
hetero-interface,
v
(
Vc{x
^
equation
(2.23)
1
2dEc
=
+-^\
2d
by evaluating
(2.24),
equation
be
an expression can
(2.24)
difference. The resulting
%dy^Nd(y)dy
v
2dE,
.
-sinh
K
To
obtain an expression
voltage across
high-field
the
voltage
for
voltage
dropped
represented
region.
The
following
the high-field region
L2,
and
finding
the
can represent
the
0)
-
and
as,
kL2
~2d
the saturation current, another equation must
high-field
across
x'
=
region
is
region
(2.24)
(jc'
high field
high-field
the
for the
obtained
at either end of the
voltage across
^s2-
(Nd(y)dy
{m)
equation
^
K
u
to
-sinh
,y)
n
Using
simplifies
equation
illustrates
be formed that
an alternate
way for
finding
the
drop:
v=v(wL-vewU
The first term in the
by
=
VH equation is
(2.16) into (2.14)
plugging
re-arranging
0
above
and
some variables results
solving for Vc(x).
in
the
I2Jl-a2Ec-2abE2Rs
2
the drain-to-source
Equating
-b2E2Rs)+
saturation current
be
realized.
for
a given
Bringing both
solved
Vgs
and
negative
the second term can
be found
region voltage equations and
Idsal(2bE2L +
explicitly for
Vds. First,
2abVgsE2
+2b%RsE2)
Vlls-V-Idsa,Rd+^
ldsa,
equation
\\
1
I,
2dE,'A
71
be
both high-field
n
4Wi-A
above equation can not
voltage and
following ldm, expression,
-Ka)2-
The
bias
a +
-
(2.25)
Jc
so an alternate approach
(2.25)
must
be
is
needed
re-arranged so
) ))
to obtain the
that a solution can
terms to the other side of the equal sign gives
26
Ka):
,2
.
+
-
4bdI,F
dsat
c
K
2d
K
<L (l
Now that
function
be
-
*2Ec
~
Vgi, Vdi,
and
used to generate an
In the
Idi
transconductance
drain
held
current
constant.
gm
of a
gm
can
a
be
and
an
expression
between
point
(2.25)
and
The transconductance
given
2abVgsE2
+
sides can
be
the two curves
(2.13)
plotted
provides
)
individually
as a
the values that can
change
viewed as a
are
to derive device
used
small-signal
and parasitic gate capacitances.
the performance
of
the greater the gain
in
mathematical equation can
be found
the
of
by taking
transistor.
voltage
in
the
while
current
derivative
of
the
for
the
voltage
a
drain
delivering,
the
larger the
when all other
as the magnitude of change
keeping
relating the input
output
In general,
capable of
any transistor is defined
parameter
change
of a
(amplification) it is
gate-to-source
sensitivity
greater
(Vgs in
format, both
J
2b%RsE2
Output Conductance, g0
higher the gm value, the
voltage
=
intersection
a+-
,
Vds plot.
device,
constant.
for
the
sections, equations
gm, is
Transconductance,
are
vs.
and
Idsat (2bE2L +
n"
"m
to an
dsat
R +
T
1dsatlKd
-I
gs
'c
b2E2Rs)+
-
V
transconductance, drain conductance,
Transconductance,
factors
Idmh
following
characteristics such as
2.7
2abE2eRt
(2.25) has been formatted
of
V
Vds
drain-to-source
in
voltage
to the output current; the
change
in input
The
voltage.
current with respect
to the
input
this case),
(2.26)
dV.
k*
The
transconductance was
linear
region
derived for both
regions of
operation, linear
and saturation.
The
result
for
the
is,
dl djin
> m,lin
(2.27)
dV.
2cddXm+P
gs
where
a,
fi,
a, and
transconductance
in
b have
the same values as
the saturation region was
they did in
the
linear
region
drain
current equation.
The
found to be
27
3/*
_
5+2bldsatEca,
dV.
4bdE2a
c"3
2hsaM+U2
K
W
+
2bldsatEcaA R,
a +
(2.28)
-
Ec JJ
where
or, =X-a2E)-2abE)Rs
a2
=
2bE2L +
or3
=
sinh
2abV'E2
-b2E2Rs
+2b2V'RtE2
K
a.
2dE
-1/2
^
K
a4
<*s
1+
=
a*
2dc J
hsat
=
i^abE2
2b2R,E2
+
+
af,=V,-V0-I,Rl
dsat
d
gs
The
output
in
change
while
as possible
output
current,
the
for
depend
output
dsat
2b2E2Vgs
a +
-
conductance, g0, of a transistor
output current
keeping
)-
)
is
similar
to
observed with respect
gate-to-source voltage constant.
g0.
on
is
Jc
an
voltage, and
exclusively.
input
The
voltage
incremental
change
amplifier circuits where
output conductance can
be
incremental
in the drain-to-source
Unlike gm, however, it is desirable to have
This is important for high-power
the input
to the transconductance except that the
as
low
voltage
a value
it is desirable to have the
represented
in terms
of
the output
as,
_^
,0
(2.29)
dV
,/s
go was derived for both
regions of
) ojin
operation,
linear
and saturation.
The
result
for
the
dl dJ,n
Id,J"-bRd) + b(vds-Vgs)
dVds
2aldXm+p
linear
region
is
(2.30)
28
The drain
conductance
=
for the
is
saturation region
dldsa,
2bEcIdiata4
2Idsa,ai+(x2-
(2.31)
4bdE2a
+
-
2bEJdsa,Rda4-2E(.Idsala,
a
+
-
K
2.8 Cgs
and
Of
the parasitic capacitances
all
frequency
response
Either
terminals.
when an
Cgd
are
(2.32)
be
one can
incremental
Equations
arguably
change
(2.33)
and
inherent
those that
modeled
in
device,
to a
lie between the
by determining
potential occurs
show
how both
capacitances can
C
Q is
length
of
channel and
[22]. Since the two
also
in
regions
before the
the channel
source, and the
gate
gate and source or gate and
drain
and
in the
drain
channel
terminals.
be found mathematically,
(2.32)
dQ
Kd
(2.33)
dvds
be found
by
result
regions of operation under
be found for both
amount of charge
multiplying the
and
high-
a transistor's
dQ
dV
=
the amount of channel charge and can
the
gate
detrimental to
the amount of charge that changes
between the
C~
where
the two most
the
width of
scrutiny
capacitance
in the linear region,
by integrating
in
are
the channel and
linear
and
(2.34)
carrier
by
must
density
over the
the electron charge, q
saturation, the
each region can
equation
the sheet
channel charge must
be determined. To find the total
be used,
L
-Qun
Equation
(2.11) is
used
necessary integration
for ns(mj)
since
and multiplication,
~U.Hr,
d
it includes the
the total
Vgt
,
channel
channel charge
W(m)
W(m)L
~
(2.34)
=qW$ns(m,x)dx
potential
in the linear
term.
After performing the
region was
found
to
be,
l
{B-CEf2-
AE +
3C
3C
(2.35)
29
where
A_Ks~aId,n
(ah,u -bVj-2b2VgJdl,nRs +
B
2abRJ2dMn+{bIdMnRi)2
b1
C
2/,
=
-
b
Vgt-Vgs-V,h{m)
Taking
the
capacitance,
C gsjin
derivative
Cgs, in
the
with
to
respect
the
32,
We(m)L
W(m)
dV
d
d
)
(B-CL)
dv.j
dB
dA
T
2
dC
v3^
+
3V,
To find
voltage
yields
the
for
expression
the
gate-to-source
linear region,
\.5C{B-CL)
L
gate
3/2
dC
L5CBU2^--B"2
3^
the gate-to-source capacitance in the saturation region, the total channel charge must
To do this,
the channel must once again
each region must
be
summed together
to
be
(2.36)
c1
c-
3
dV..
dV..
2
dC
split
into low-field
determine
and
the total amount
high-field
in
first be found.
regions and the charge
the channel
for
[22],
U
Qsm
The first
term accounts
approximation
(pinched-off)
still
for the
holds
region.
If the
and
=
qW
\ns (m, x)dx + qW \ns (m, x)dx
total amount of charge
the second term accounts
current
density
equation
ld.sat
then an
carrier
expression
for
the
in the low-field
carrier
density in
in the
for
(2.37)
region where
the gradual-channel
the total amount of charge
saturation region
is
represented
in
the saturated
as,
-qWvsalnsal(m,x)
the
saturation region can
be
obtained.
Substituting
in for the
densities in both integrals,
30
and
the
performing
necessary integration
and multiplication yields the total channel charge
in
the saturation
region as
td.satJL-L,)
WL](m)Vgl
w( m)
Vsa,
d
d
-Q
Taking
the
derivative
of
(2.38)
(2.38)
3C
3C
voltage,
"
dQ
Cgs.sal
with respect to the gate
(5-Ci,)3/2-
AL,+
8mL
SmL\
o m
dV
V
V.
V
+
ld.sa, dLj
V
*a(
sal
dV
w r
gs
WL,(m)
+
+
J
W(m)Vg, dL,
dV,
ss
J
W(m)
V
(2.39)
^gs
^
where
dB
\.5C{B-CLlf
3C
B'=
1.5CB
?>CZ
ii
dV
dB
channel charge equations
instead
to
Vds
for
in
~
gs
the gate-to-drain capacitance,
Cgd,
the same approach can
derivative
Cgs,
except
taking
accordance with
(2.33).
Therefore, taking
dQu
the
the
be done using the
with respect to
derivative
of
the drain voltage
(2.35)
with respect
dV
W{m)
Vgs
dC
dB
L
gs
gives
-gdjin
,
dV,
sj;
dC
just derived for
of the gate voltage
dC
dV
dVr
obtain expressions
(^-CL,)
3 V,
9V
g3/2
g*
To
,/2
ci+Ai
l.5C{B-CLf
dA
2
+
dVdi
-
3
K^ds
3V*,
c1
3/2
(B-CL)
dC
1.5C5
dv.ds
3
,2
dB
dC_
dV,
dv,.
c2
(2.40)
31
cU
a
-v*
b
1
dB
o,//n
1
=
7T[2k,;,
~62
dC
-KlaSoMn)-2b2VgsRsgoMn+4abRJdMngoMn+2b2R2IdJmgoMn]
2
"**
3V
Taking
the
odin
derivative
of
(2.38)
Vds
with respect to
gives
the gate-to-drain capacitance
in
the saturation
region,
LSo..
-Q
C gd,sat
W{m)Vgl 3L,
dV
dV.
W{m)
AdL^ +
d
dVds
^_
,,.
y
(2.41)
dVds
where
'c^^sc"
dB
l.5C{B-CLi)>'2
A"=3C1
svds
B"--
1.5CB
12
3C2
2.9 Cutoff
frequency
cutoff
the
frequency
Though this
forward
analog
converted to
though
dC
.12
dVds
dC
dV.
Frequency,//-
the transistor.
which
3/2
dV,r
The
at
dB
dVds J )
\B-CIj
transistor
|h2i|
of
the
To accurately
method can also
a
figure
of merit used
to characterize the
power
circuits, it is
intrinsic device becomes unity [1]
calculate
h-parameters. The h-parameters
less accurate,
is
to digital circuitry than RF
pertains more
gain
circuits.
of a
be
are
used
this, the
which
y-parameters must
is
switching
also the
be found for
frequency. As
frequency
applicable to high-
the
then used to calculate the /t- of the transistor.
to calculate the cutoff
speed of
shown
device
and
A simplified,
in [1],
dividing
32
the
intrinsic
can
be determined. Equation (2.42)
transconductance of the
device
gives the
fT
The
"2tc"
excluded
to
keep
is included in
from
this
functions
of
the
denominator
by
L
and
the resulting
(Cgs
plus
Cgd), fr
(2-42)
r^
-
S
2K(Cgs+Cgd)
to yield a result
in
it is usually
crude simplification as accurate as
and
capacitance
equation,
the expression altogether since
V^
the total of the parasitic gate
possible,
relationships will
terms of
hertz
and not radians.
orders of magnitudes
Cgd is kept in.
be
less
Equation
presented and
than
(2.42) is
Often
Cgd can be
Cgs. In
an attempt
used
to
plot/7-
as
discussed.
33
Chaptery
Materials
Apparatus
and
3. 1 Hardware Data
3.1 Hardware Data
Published
Results
reported
by
Wu
model proposed
confirm
be
et al.
[23]
with calculated values.
I/V
HEMT
experimental results on numerous
[23]
by Chang
et al.
with
experimental
[22]
As
the experimental data that
largely
was used
the validity of the model.
compared
provided
and
will
data,
samples were collected and analyzed
be
because the
Rashmi
shown
multiple
used
[18] in
which
in
[23]
were
for
was used as a
the subsequent chapter,
references
this thesis
comparison.
comparison
in Chapter 2 closely follows
proposed model
et al.
in
is
for
obtained
for any
benchmark to
results
to verify that
an
that couldn't
both
the curve
profiles and curve magnitudes were okay.
The
physical structure of
Deposition (MOCVD)
labeled
on
the sample fabricated
C-plane
a
in
[23]
substrate.
sapphire
was grown
Fig. 3.1
by
Metal Organic Chemical Vapor
shows
the
device
with
appropriately
regions.
DRAIN
-
GATE-
SOURCE-
u
I
T"l
UID AlGaN
"cap
Cap
Layer
Iaa
':. ;
n+
/;,,,
ID AlGaN
.
Charge Donor Region
i.L:.
1
d,
UID AlGaN Spacer Layer
X
Ad:
...,
Fig
the
A 200
A
thick
GaN
3.1. HEMT
results used
nucleation
GaN.
structure reported
for
layer
by
comparison with
was grown on
barrier
insulation (buffer) layer. The Alo.sGaossN
Wu
et al.
[23]
which produced
the proposed model.
top
of the substrate
grown
followed
by
between the GaN buffer
a
2
and
thick
GaN
Schottky
gate
u.m
34
consisted of a
30
layer. The 220
A charge
were annealed at
of
250
3um,
900 C for 30
was grown
and gate-drain
leakage
currents
by
the
doping densities
interface, Ad,
to
spacing
and
spacer
was silicon
layer, 220 A
doped to
Transfer
have
of
1
was assumed to
Table 3.1. Device
in
be 40
The
source and
gate
length
was grown
barrier height [24,25].
peak
respectively.
in
for
drain
be in the
of
1 um,
and
Table 3.1 lists
all
in
help
the
cap
proportions
spacing
Q-
of
suppress gate
figure
necessary device
results.
A
0.5 to 0.7
source-drain
"ID"
"UID"
150
ohmic contacts
range
the structure to
the subsequent chapter to obtain
A
and a
Titanium/Aluminum/Nickel/Gold in
75 um,
The cap layer
intentionally doped,
that were used
cm"3
contact resistance was measured to
a gate width of
um.
donor layer,
n-doped charge
2xl018
seconds and were made of
raising the HEMT
unintentionally doped
and
donor layer
A/2000 A/400 A/450 A.
The device
mm.
A unintentionally doped
stand
for
geometries
The 2DEG distance from
all calculations.
parameters as reported
by
Wu
et al.
[23]
that were used
for
simulations.
Structure Parameter
AlGaN
complete
Layer Thickness,
A
A
A
A
150
d^p
AlGaN Charge Donor Layer Thickness, dd
220
AlGaN Spacer Layer Thickness, d,
30
2DEG Electron Cloud Distance from Interface, Ad
40
GaN Buffer Layer Thickness
2
Doping
This
Cap
Value
Concentration
of
Charge Donor Region, ND
urn
2x1
018
Gate Width, W
75
Gate Length, L
1
um
Drain-to-Source Spacing,
3
um
Gate-to-Drain Spacing,
1
um
chapter aimed
this thesis work.
to
give
Since
against reported experimental
no
the
LDS
LGD
reader an
hardware tests
understanding
were
of
cm"3
urn
the materials/resources necessary to
performed, all simulation results were compared
data.
35
Chapter 4
Results
and
Discussion
4.1 Introduction
4.2 Sheet Carrier
Density,
ns,
vs.
Fermi Level, E,
4.3 Sheet Carrier Density, ns, vs. Gate
Voltage, Vgs
4.4 Sheet Carrier Density, ns, vs. Barrier
Thickness,
4.5 Threshold Voltage, Vm, vs.
Density,
Doping
4.6 Current-Voltage Characteristics
4.7 Transconductance, gm, and Output
4.8
Cas
ygs
and
4.9 Cutoff
dd
ND
Conductance,
g0
C,gd
Frequency, fT
4.1 Introduction
All
results shown
Formulation
been
in this
section were obtained
and are compared with experimental
collected
from
published results to
verify the
to substantiate any graphs that could not
follows
by
the chapter
given
be
calculations were
finally
data
equations
derived in Chapter 2
when possible.
model.
Numerous
In
charge control of
the
-
Theory
/ Model
this work, experimental
data has
data
are used
sources
citing
reinforced with experimental results.
2 layout closely in that the
the current-voltage results, and
All
using the
2DEG
will
simulated
The layout
be
analyzed
of this chapter
first, followed
small-signal parameter results.
done using
the material constants given
in Table 4.1
and material equations
in Table 4.2.
Table 4.1. Table
of constants used
for
results calculations with values and references.
Constant Description
Value
Electron Rest Mass, m0
AIN Electron Mass, m,,AIN
9.1094xl0"31kg
0.22m0 kg
0.33m0 kg
Permittivity
8.8541 8xlO"14F/cm
Reference
GaN Dielectric constant, eGaN
10e0F/cm
AIN Dielectric constant, eMN
8.5co F/cm
[26]
[20]
[20]
[26]
[27]
[27]
Electron charge, q
1.602xl0"19C
[26]
1.38066xlO"23J/K
[26]
Planck constant, h
6.62607xlO"34J-sec
GaN Lattice constant, aCaN
3.189xlO"'m
[26]
[20]
GaN Electron Mass,
m*
constant, s0
Boltzmann constant,
kB
3.112x10"'
AIN Lattice Constant, aMN
GaN Band Gap,
AIN Band
EeGaN
Gap, EeAiN
m
[20]
3.42
eV
[20]
6.13
eV
[20]
36
For
constants that are given
AlGaN
material where the mole
Vegard's Law due to the
(such
4.2
for GaN
as the
fraction lies between 0
formed using Vegard's Law
and provides references
the
be formed using
material parameter and the mole
necessary
for
fraction
Table
equations.
for those
that were
aluminum
mole
up.
Table 4.2. AlGaN/GaN
material
fraction. Room temperature (300K)
Schottky
equations used to
system
was assumed
Equation Description
for
calculate
results,
<pm(m)
=
0.91
me,AiGan(m)
AlGaN Dielectric
e(m)
constant
-
(10
+
2.44m
(0.22
-
[16]
eV
0. 1
+
Vegard's Law
\.5m)e0F/cm
-
(3.189
10"'
Conduction Band
Thermal Voltage
VT
constant
aucatlm)
Energy Gap
Discontinuity
In Section 2.1,
relationship between
presented on
where
used and
how to
the
obtain a plot
sheet carrier
Ef was
varied
for ns
density
from
-6VT
~
Fermi
is
to
=
channel
density
the
Fermi level
and
(2.4)
function
produce
was used
of
was
derived
and
a procedure
to produce the results shown
the Fermi
level. A temperature
of
in Fig.
300K
was
results.
-
-
300K
=
m
=
0.15".
c
2.123x10"12(ntf3
=
1E+13
8E+12
,.....
3.734x10"12(n,)2/3
E,
'55
g
[20]
-
Ef. Equation
vs.
r
1.2E+13
[20]
Level, Ef
3 VT to
1.4E+13
m(l-m)
eV
kBTlq
plotted as a
E
2-
-
-
-
Density, ns,
Vegard's Law
m
-
=
4.2 Sheet Carrier
vs.
=
Vegard's Law
\m)m0 kg
0.077m) x
6.13m
+
EeAiGaN(m)
3.42(l-m)
*
AEC 0.7 [EeAiCaN EeCaN]
a
is the
Reference
AlGaN Effective Electron Mass
AlGaN
m
all calculations.
Equation
Barrier Height
AlGaN Lattice
4.1
an equation
equations couldn't
so references were obtained that contained the
shows the equations that were
looked
1. A few
and
to extrapolate
was used
relationship between the
non-linear
AlGaN band gap),
AIN, Vegard's Law
and
=
-
i-
*
6E+12
<0
O
4E+1^
|
(O
2E+12
-
-0.15
-0.1
Fermi
Fig 4.1.
Ef
at
Cfc
The
for T
=
0.05
0
-0.05
0.1
Voltage, Ef (V)
channel sheet carrier
300K. E, varied from
density
-6VT
as a
function
of
to 3 VT
37
It is
from the figure that there
clear
relationship depends
V,
0.025
when
on
which
section
ns is observed to have
-0.1
dependence is
V due to the
in Fig. 4.2
is
being
dependence
of
on
n,
Ef
on
as a
though the type of
voltages
which turns to a
/is difficult
to the
function
Ef,
and
At Fermi
analyzed.
small relative
very
ns is plotted
where
being
dependence
exponential
magnitude of ns
more evident
the curve
of
an exponential
yis increased from there. The
less than
direct relationship between ns
exists a
of
to see at
Fermi
logarithmic
a
than
linear dependence
plotted magnitude.
Efon
less
voltages
This type
of
scale.
o
O
T
5.
c
i
-
300K
=
m
=
0.15
E0
=
E,
=
2.123x10'l2(ns)M
3.734x1
^^
0-12(ns)2/3
yS
0 1
0.01
U.UU1
i
-0.2
...
-0.15
-0.1
Fermi
function
The
exponential
-0.025
curve.
V
after which
This "slope
covered
curves
dependence
in
more
positively
the curve
ns
Fermi
on
yis
references
begins to "slope
indicates the
detail in the
logarithmically
have been
Although
gathered
use of equation
that
by
the
linear
The
in Section 2.1,
a
provide
ns
which was
with
visual
portion of
Vgs
and
ns
Fig. 4.1
which will
Fig. 4.1
the
the
and
the premise that as the
interpretation to
be
what
obtained
equation
Fermi
(2.4)
for comparison,
nrEf behavior [1,12,28]. The
cited
be
Fig. 4.2
electrons, raising the value of the
values could not
similar
between
profile and magnitude of
become filled
(2.4) for predicting
slope of the curve prior to Ef~
corresponding to the linear
experimental
report
as a
over"
subsequent section.
quantum well eigenstates
0.1
dependence type.
point where charge control occurs
reinforce what was established
mathematically.
legitimize the
in Fig. 4.2
evident
density. The curves, furthermore,
providing
plotted
voltage to emphasize
over"
level increases, the
carrier
of
of
0.05
Voltage, Ef (V)
4.2. Sheet Carrier Density
Fig
.
0
-0.05
sheet
was
several
references
help
values.
38
Fig. 4.2, in
values.
reading
2
cm
In
V),
showing the ns dependence
the exponential region (Ef<
Ef increased in
as
-0.025
addition to
from
value
-0.15
V to
V),
-0.025
-0.0258
effectively than Fig. 4.1, is
more
ns increased from
ns is observed to have less response to increases in
for
used
1.47xl012
1.435xl010
V. As Renters the "linear
better
also
cm"2
to
region"
voltage range
(>>
1.42xl013
Ef, reaching
a maximum value of
cm"2
for
an
Ef of 0.0776 V.
A relationship between the 2DEG concentration,
n
section.
once
For
less than
Ef values
Ef exceeds
determined from
The
-VT.
a plot of
next section shows
ns
equation
it is
discussion is
the
in the
discontinuity
in
to
potential
voltages
shifts
channel
channel
and
is
expected
density
A, d:
=
of
30
be
close
direct linear relationship
(2.6)
A,
to
be
can
seen
When
Ad
well
for
region
for
30
A, ND
and
the
is very low
of
This
region
also shown on
in Fig. 4.3,
and
Vgs.
and
values)
Fermi level
of
(106
the
all
at
the
band
difference
changes.
At
gate
the quantum well and the
1010
cm"2
-
complete
is known
Looking
between
the well and the charge
has
some
applied, the conduction
Vgs
that
charge
as the
cm"2).
As
Vgs
density
in
the
control over the
strong inversion
region
amplifier applications.
2xl018
=
is
below the bottom
to the bottom
is
Ef
between ns, Ef,
Using
voltage.
results shown
negative
discontinuity
eigenstates
results.
to the gate
relationship.
a gate voltage
was used to calculate the threshold voltage,
=
be
gate capacitance can
that the Fermi level aligns
values, up
band
level lies
relationships
to become large. Above threshold, the gate
and a
how
and
Before analyzing the
Vgs
positive
the conduction
expected
is the desirable operating
220
in Chapter 1, it
occupying the discrete
Ef is
linear dependence
that turns to a
density
sheet carrier
understanding to the
than threshold, the Fermi
Equation
=
the gate voltage.
accordingly (down for
threshold,
Vgs
on
to visually show the
was created
system under equilibrium conditions.
concentration of electrons
approaches
derived relating the
on
provided
between the bottom
less
how both depend
this
Gate Voltage, Vgs
provide some
energy band diagram
materials
was
dependent
also
given
vs.
(2.5), Fig. 4.3
equation and equation
graph since
(2.8)
dependence
an exponential
in
established
V^.
vs.
4.3 Sheet Carrier Density, ns,
In Section 2.1,
nt exhibits
-VT,
Fermi level, Ef, has been
and the
cm"3)
which was
V,h, for
found to be
the given sample
-3.218
V. This
was
(m
=
0.15, dd
the calculated
39
threshold voltage
treated as
value
fitting
a
but
by
Vg,
=
0 V
in Fig. 4.3
Fermi level is
concentration
well
is very
Ef begins
inherently
be
-0.0569
further
If
a
Vgs
order
few tenths
be
small and
of
of a volt
were
of
to
Increasing Vgs
equation
V>
increase
2DEG
by
depending
=
ability
of
The
further
even
band
the
Schottky
was calculated
Vgi
to
is
dependence
in ns
be
location
on
Vgs
and
V,
n,
cm"2
and
figure {Vgs
=
0.72 V),
Naturally,
so small
107. As
a
to
it
Vgs
be 4.96x1
0"
in
can not
n,
be
approaches
enough
to be
on
Vgs
(2.5) is
cm2
Eflo
Increasing Vgs
7.53xl012
an
V),
2DEG
the
and
>to be 0.0259 V.
results
this
-3.5
linear dependence
cm"2
value of
more, ns would eventually attain a value on the order of
concentration
values (<
expected since equation
was calculated
6.21xl012
Vgs
ns becomes large
Ef both having
and
is
about
Fermi level [1]. The
At low
on the order of
assumption
vary slightly
of the
concentration
This behavior is
gate.
= -3
voltage will
(an
atoms
edge and eigenstates.
2DEG
calculated value
results
the
on
impurity
more understanding.
the magnitude of the
relating ns to Vgs. At
0 V, ns
bit
of the conduction
plot.
donor
the
of
0 V, the threshold
observed with a
in fact
the
90% ionization
a
not equal
to the maximum value shown on the
considered
10"
V. At
Vgs does
below the bottom
charge-control
linear
assuming
to show signs of an exponential
noticed on the graph.
showing the
If
can now
determined from inspection
threshold,
and
parameter).
no more than a
results shown
the
at
1013
cm"2.
It is this high
that makes the AlGaN/GaN heterostructure system very attractive and,
most, superior to its AlGaAs/GaAs counterpart where concentration values rarely get above
cm"2
0.2
T
0
-'2
>
Uj
-0.4
>
-0.6
-0.8
-1.2
-1.4
-3
-4
-1
-2
Gate Voltage,
Fig
4.3. Fermi level
gate voltage.
T
=
Vgs (V)
and sheet carrier
300K,
m
=
1 5%,
Vd%
0
density
=
plotted as a
function
of
0 V.
40
The behavior
these references reveals
curve
is
observed to
exponential
in
linearly
very
relationship between ns
considered
slightly different
relationship is
the exponential
visible
and
Vgs is
relationship it is
the
quantum
concentration values to the
eigenstates
to be
need
concentration
in Fig. 4.3 has been demonstrated
of the curve
levels
capacitance
increases
the
the ns
slope of
as
line, multiplying by
on and more
This
shown
to
channel.
negative
to
section extended
on
=
meet at the
references
[15]
[19]
and
the
To improve the
eigenstates
not need to
the
below. The
is due to the fact that only two
of
closer
look
4.3,
at
the ns
At this point, the
x-axis.
values at threshold and
be
able
being
not
exponential
reason
why
eigenstates were
in predicting 2DEG
model
precision
the model,
of
is important only if
be included here
more
subthreshold
since the
device is
to extrapolate the capacitance of the
saturates when
low
visible at
Vgs. Therefore it
to
gate capacitance
for the
could
be
Vgs becomes
W,
Vgs
values,
that the
said
device's
than V,h.
greater
and gate
its behavior
Taking
length, L,
the
is
directly
barrier thickness
smaller.
One
to the
distance between
increased,
one could expect
related
were
could also expect
extra electrons
the threshold voltage to
in the barrier, making it
easier
to turn the
off.
the relationship
Vgs
In
A
{l.602xW-,9ch5um)(lum)^-
Therefore, if
account
Vgs. For
they
as well.
to threshold. In Fig.
the electron charge, q, the gate width,
by becoming
difficult to turn it
be dependent
did
Despite ns
eventually
228.55 fF. The
the slope of the curve to change
device
states
[15,19]
values close
effectiveness
increase in
respect
and
=qWL-^
revealed a gate capacitance of
increasingly
this model
others
all current-voltage calculations and small-signal calculations.
with
Vgs increases
the gate and the conducting
Vgs
charge vs. voltage, one would
exponential
C
become
for
curve at a given gate voltage.
determined to be
for
region only.
to be known. More
Vg5
where
This limits the
though the
region
at
values of ns.
more pronounced
strong inversion
Since Fig. 4.3 is plotting
was
except
for very low
well.
biased in the strong inversion
device from the
Vgs
with
not visible with
included,
need
behavior
curve
by
values at
established
in Section 2.1
threshold and
below,
to
ns has
include Vgs. Both island ns
an exponential
are
dependence that
41
turns
linear for Vg,
values at and above
capacitance under zero
4.4 Sheet Carrier
Equation (2.5)
drain bias
Density, ns, vs.
Barrier
shows the aforementioned plot
touched
upon
that as
previous section
Intuitively,
dd increases,
density. From
assuming Vgs
concerning
the
larger dopant
a quantitative
is
area
Ef=
Vds
vs.
contain
more electrons when
to
increase
at
a
A
The 350
biased to do
decreasing
calculation of
the gate
to
increase
electrons to the
of
thickness
This
rate
briefly
density
barrier
carrier
for dopant
more room
concept
is
eventually stabilizing
increasing
the channel
at
,
of thickness
say,
simple reason
IxlO19
doped
350
will not
A doped
electrons and can therefore
Fig. 4.4. The
by
reinforced
and the
increases for the
channel,
A
250
barrier,
electrons as a
barrier has
so.
dd
as
graph shows what was
"donating"
the same number of
the same concentration.
to a
AlGaN barrier layer thickness, dd.
0 V. The
=
"depletable"
physically
lead
can
relationship between the
barrier layer
a
standpoint,
=
density,
density
one would expect the carrier
a
Vgs
vs.
Thickness, dd
was used to produce a plot of sheet carrier
in the
n,
conditions.
Fig. 4.4
thickness.
Plotting
threshold.
carrier
by increasing linearly
as
density is
the
at
donate
observed
barrier thickness
increases.
o
Na
10
2x10'8
cm?
=
1.5x1018
"o
ND
W0
8
=
1x10'8cm?
\.\<f\^\^-^^'
-
c
0.5x1018cm"3
D
J^\^-
X^^S^^^'
T-
X
cm^
=
.
=
\
\y^\y^^^^^
'3S
*"
6
ND
-
=
0
~__
cm'3
S?/&^^*-
\
Q
y^^^^^
T
0)
CO
4
Jjp'
-
O
(Br
CD
0)
."..
5
'
'/;.
*"
AlGaN Barrier Thickness,
Fig
4.4. 2DEG density
500x1
would
increase the 2DEG
as a
function
indicate,
as a parameter.
0"8
2xl018
and
As Section 4.3
=
0.15
30x10'8
of
cm
30x10"*
cm
=
>',<
5.5
4.5
dd
(xlO"6
cm)
barrier thickness
with
100x10"
doping density
and
d,
3.5
2.5
1.5
0.5
=
Ad
^
*J
300K
=
m
cm.
Doping
0.5xl018
cm"3
in
dd
varied
concentration varied
increments.
era"3
this type of 2DEG behavior
density by increasing
the
between
is
V^
cm
between 0
=
Vgs
=
expected.
cm'3
0V.
A larger barrier thickness
amount of electrons available
to the
would
2DEG. Fig. 4.4
also
42
indicates
that the channel
example,
at
yields a
higher 2DEG
concept, the curve
dd
is
concentration than other
to
corresponding
ND
350x10s
=
ND
cm whereas the
350x10
cm.
This
barrier
more sensitive to the
barrier thickness between
given
any
density
lOOxlO"8
500xl0'8
2xl018
with
lower
doping
8.1428xl012
yields a sheet carrier
lxlO18
density
curve yields a carrier
higher
makes sense since a
density
doping
at
cm"2
at
of
higher
concentration will yield a
cm"2
of
6.8174xl012
cm"3
=
doped. For
To quantify this
concentrations.
cm"3
=
highly
cm, the higher doped barrier
cm and
barriers
if it is
thickness
dd
=
volume of available
electrons.
An
interesting
doped. This
are
run
can
be
for example,
consider
be less
which
"B"
which
300x10s
its
cm and
cm"2
This
provides
the
"B,"
"B."
frequency
response compared
to
barrier layer. Conversely, device
up
the
more vertical space on the wafer.
freedom to
capacitance
As
"B"
choose
the
tradeoff with the
a
dd increases,
also
,
the
"A,"
he/she
This type
of
shown
space
free
but
for
electrons
the
density
densities
of
response
et al.
drive
as
degraded high
but
would
gives the
The
take
designer
foregoing
[15].
dependence
in the barrier
these
be higher due to the thinner
application.
Rashmi
"A"
is
fabricating
when
frequency response,
briefly by
carrier
doping
experience a
frequency
on
much
thickness
same amount of current
"B,"
depending
discussed
detailing
as the number of
device
improved high
tradeoff,
was
enjoy the
density,
its barrier
yield
freedom
gate capacitance will
device geometry
barrier thickness
to
of
[29].
For comparison,
cm"3.
that
and
undoped
Consider device
lxl018
of
Fig. 4.4, both
degree
would
wafer compared
would provide an
optimum
increases
the certain
with
as well.
except
to
with
the channel
"A"
device
According
This is because the
relationship has been
In this section,
the same as
fabricate
electron mobilities
device
the
to
fact, devices
can affect
doping density
cm and a
designer
conserve vertical space on
response of
250xl0"8
devices. Should he/she decide to build device
device
doping density
and
0.5xl018
is
higher
noisy, and to yield
those curves that
same as
has led many
In
results.
encouraging
high-frequency
is in every way
doping density
amongst researchers and
with
both barrier thickness
has barrier thickness
behaves the
the undoped curve
hot topic
leaky, less
their relationship to the
device
5.73xl012
observed to
established that
said of
a current
barrier devices [29,30]
tests on undoped
Having
is
phenomenon
barriers have been
here is how
result to note
on
barrier thickness.
goes up.
43
4.5 Threshold
Near the
Voltage, Vlh,
end of
Doping Density, ND
vs.
Section 4.3 it
magnitude of the threshold voltage.
is therefore
lead to
an
devices
A larger
quite clear that either the
increasingly
negative
increasing
so
contemporary digital circuitry
try
the
same with
number of electrons affects
doping density
or
ND
would
make
any GaN-based
Fig. 4.5
GaN-based HEMTs. Enhancement
Five
circuit
was constructed
a parameter.
V,h is
240x1
0s
in
using
plotted as a
the simulation).
cm, and 280x1
in
the
even
affects the
inherently
turn
a
off.
finite
GaN
some
would
depletion-mode
them
which require
This has led
Most
gate-to-
researchers
devices have been fabricated [31,32]
considered
It
more negative.
barrier layer thickness
difficult to
FET devices
mode
being
directly
V,h by making it
or the
more
and source.
barrier
commercially viable,
as
is
and
the case
design.
(2.6)
equation
function
curves were made, each with a
considered
it
connecting drain
studied, although the fabrication process is far from
with
barrier layer
the
of
employs enhancement-mode
source voltage to create a current channel
to
number of electrons
threshold voltage. Most fabricated HEMTs are
dd
either
how the
was stated
ND
of
to show the
which
is
V,h dependence
varied
between
different barrier thickness
The five thicknesses
used were
dd
(only
ND
on
=
0
ND
with
barrier thickness
3.5xl018
cm"3
to
ND
as
cm"3
=
thicknesses that are practical were
120x10s
=
160xl0"8
cm,
cm, 200x1
0s
cm,
0"8
cm.
o
D)
CO
o
>
a
o
c
f
CO
<a
m
=
=
d,=
300K
0.15
30x10
cm
2
15
1
0.5
Doping Density, ND
Fig
ND,
The
graph shows
given
that
V,h
4.5. Threshold
with
becomes
doping density, V,h is
V,h,
dd, as
voltage,
barrier thickness,
more negative as
more negative
for
as a
2.5
function
a parameter.
the
thicker
3
(x1018
cm"3)
of
V&
doping density,
Vgs 0 V.
=
=
doping density is
increased. Furthermore, for
barrier layers. For example,
a
barrier
of
a
thickness
44
160xl0"8
yields a
V,h
of
doping density
As
reported
are
Before
\Q\,
what
density
by
of
2DEG
Rashmi
into
designer has
\Q\
bias
device
electrons
he/she
to choose which technique
280x10s
in Fig. 4.5
cm
of
that the number of
be. Barrier
will
free
threshold voltage. The results shown
and
current-voltage results, some
results.
Equations
6V
to
obtain
1E-12
thickness and
in the barrier layer.
will use
to obtain a
are similar to those
&
Gate
\ix),
versus
(2.38)
and
density, Q,
voltages of
-2
V,
channel charge
drain
were used
as a
-1
voltage
will
be
given
to
to observe the relationship
function
V, 0 V,
concentration,
of
and
the applied drain-to-
1 V
were used and
V&
the results shown.
i
9E-13
Si
(2.35)
shows the channel charge
with gate voltage as a parameter.
to
background concerning the
and channel carrier velocity,
the I/V
Vdv Fig. 4.6
from 0 V
was varied
freedom
is
concept which
overlying
the threshold voltage of the
the
barrier
whereas a
Characteristics
insight into
and
V
-2.05
in [15],
et al.
drain voltage, VA,
provide some
reinforce the
of
parameters used to control the number of
level
versus
source
dictate
a
delving
between
to
V,h
yields a
further
fabrication
Current- Voltage
4.6
help
cm"3
of
results
in Section 4.4,
stated
specified
V. These
-4.7
in the barrier
electrons
2xl018
doping density
cm at a
~\^^
^~*~--^^
8E-13
T
=
W
t.
^^
7E-13
300K
=
=
1
75
[im
urn
y
=
TV
V*
=
V
6E-13
J
<>
^""^-^"V~--
5E-13
___
4E-13
CD
C
^
CO
.C
Vfl.=-1V
__
3E-13
2E-13
>/gs=-2V
O
1E-13
0
-
Drain
5
4
3
2
1
()
Voltage, Vds (V)
V,
4.6. Channel charge as a function of VA. Gate voltages of
V, 0 V, and 1 V were used for the curves. Width and length of
Fig
-1
-2
device
Each
in
curve
charge
is
is
and temperature used
observed
to decrease
expected
by
linearly
inspection
and
in
simulation are shown on
eventually flatten
of equation
(2.11)
which
out
the graph.
progressing to the right. The decrease
has the
sheet carrier
density,
ns,
being
45
related to the channel voltage.
negatively
device
Vc(x) is directly
and
The
increases.
"flattening?"
FET
devices,
becomes
This is
is, if
\Q\
decreases
device reaching
the
pinched-off and the carriers are
evident
by
drain
of
the slope that
is
be
In the linear
constant.
carrier
the channel
increases rapidly
velocity
not
velocity increases rapidly
for
the decrease
(2.13)
The
and
(2.25)
results resemble
eventually flattens
gate voltages of
(Wu
currents of
-1
results of
-2
et al.
V,
-1
out with
V, 0 V,
[23]). All
and
-2
V,
characteristics
increasing
and
for
increasing
lateral
regards
and
electric
to
Vds
68
of
8%
within
and
65
6 V. These
mA/mm at
Vds
lds
linearly
their
current
in the
with
continues to
field. The velocity
isn't
of
channel
linear
a
increase
of the carriers
increase in
in
carrier
the drain current experiences an
with more understanding.
shown
in Fig. 4.7. At first
initial linear increase in
curves were obtained
found between the
of
channel
not pinched-off nor are
decreases
done in Fig. 4.6,
measured values at all
closely to the
for
predicted and measured
mA/mm were calculated
match
=
is
transistors where an
agreement was
V^
both the velocity
to the drain current, this
was
the
drain
the
amount of charge
be interpreted
most
as
short-channel
Vds because
drain biases,
channel charge and
for
189 mA/mm,
respectively, at a
mA/mm, 350 mA/mm, 200 mA/mm,
channel charge
the
each curve experience a
the curves. With
constant, the
drain bias. As
1 V. Good
mA/mm,
decrease
of
of channel-length modulation.
to plot the I/V characteristic curves
curves were predicted
537 mA/mm, 350
V,
the I/V
in
because
part of
the I-V curves can now
were used
should
FETs,
the case since the channel
the
with
more than compensates
magnitude.
short-channel
"flat"
being
observed
increases. With
glance, the
V, 0 V,
is
Vds
Equations
values
in the
as
increase in
current
\Q\
why does
of
In
saturated.
linear region, the
increase in Vc(x). Despite the decrease in
\Q\
"independent"
becomes
channel
this
that
clear
then
the saturation region
in the
region
the carriers velocity saturated. In the
because the
in
still apparent
the carriers and the amount of current
must also
Vds, it is
to ns through the dimensions
the saturation region of operation. Like other
velocity
voltage
related
Vdt increases,
as
the current in the saturation region
completely independent
directly
related to the magnitude of
question now
This is due to
Since Q is
for
times.
Drain
gate voltages of
experimental values of
1
500
6 V.
46
2
3
4
Drain Voltage,
4.7. HEMT drain
Fig
current as a
Current
voltage as a parameter.
0,
and
1 V. Maximum
of
537
mA/mm.
inspection
closer
Solid lines
the experimental
in the data does
peak point
found to
of
peak at
drain
channel
temperature to
results
in
would
be
a
drain
a
between the
change
length
in
channel charge and
magnitude.
of the
linear
The
region
the transistor will remain
results
in
there
being
an
dissipates
the
change
is
drain
in the linear
increase in
current
positive
is determined
effects were
in the future.
research
by
1 V
-1
-2,
with an
,
ldi
predictions; symbols
for
gate voltages of
1 V
6 V. Instead,
is
voltage
goes up.
Prior to saturation,
drain
charge,
leading
voltages.
for
a
to the
current
that the
for
the
The decrease in mobility
the current model and
and
4.7,
the
relationship
all curves experience a
for the
Increasing
current
this
flows in the
of power causes
linear
charge curves.
The
the gate voltage means
larger Vdt. The higher
higher
reveals
possible reason
more
Figures 4.6
0 V
experimental current was
This dissipation
the current curves and negative
the gate and
and
however, included in
Comparing
clear.
increases,
the carrier mobility.
not,
region of operation
channel
values of
=
voltage of
As the drain
the transistor
decrease. Thermal
topic to
for
voltage with gate
Vgs
Vgs
decrease slightly thereafter. One
and
thermal effects.
for
data [23].
increase, subsequently decreasing
current
valuable
4 V
drain
correspond to model
highest drain
the
voltages around
channel and the amount of power
of
data corresponding to
not occur at
behavior is due to
observed
function
curves plotted
current predicted was
correspond to experimental
A
Vds (V)
gate voltage also
levels.
47
1
09
0.8
07
06
05
0.4
o
0.3
~o
c
c
(0
0.2
.c
O
12
3
Drain
Fig
of
4.8. Drain
drain
curve
current and channel charge
Plot
voltage.
its
and
4
Voltage, Vds (V)
used
both
plotted as a
function
to show relationship between each charge
corresponding
current
curve
linear
in
both
be
appreciated visually.
and
saturation regions of operation.
Fig. 4.8
plots
readability
both
of the
Fig. 4.9
Ids
Q
and
figure,
Vgs
against
so the
data
the experimental
shows the
relationship
were not
relationship between the
can
To increase the
included.
gate voltage and the
drain
current while
keeping V&
constant.
800
m=0-15
700
.Exp..
T
E
^
600
W
E
~
L
500
300K
=
75
=
=
"t
[23]
1
um
pm
A
Vds=3V
y
OO
OO
CO
OO
TO
-*
OO
>
1
4-3-2-10
Gate Voltage,
Fig
4.9. Drain
Temperature
Maximum
current
was
/^
of
set
as
equal
671
2
Vgs (V)
a
function
to
mA/mm
300K
of
and
predicted
gate
V^
voltage.
to
3 V.
against
604
mA/mm measured.
As V
increases from left to the right, the transistor is
operation around
Vgs
-
0 V (the
transition can
be
seen on
initially
in saturation, eventually reaching linear
the graph as there
is
a slight
discontinuity
on the
48
curve around
The largest
that
less
it
=
-0.2
discrepancy
V). Excellent
occurred at
of an
will
between
issue is the fact
be biased in the
the calculated
Vg,
=
2 V
10%
still within
be
that the transistor will
measured
values
in
imitate
the
differed from
mostly for
saturation
{Vgs
region
current source.
0 V)
<
was
by 65
discrepancy
the
amplifier applications.
high-impedance
a
experimental
Making
of the measured value.
used
values.
the calculated and measured
the calculated value
where
saturation region so as to
and
found between
agreement was
This difference is large but is
mA/mm.
even
Vgs
This
means
Agreement
found
be
to
exceptional.
4.7
Transconductance, gm,
Equations (2.27)
4.10. The drain
-4
V
and
leveling
(2.28)
were used
to calculate the
voltage was
kept fixed
at
and
ending
off and
at
2 V.
declining
immediately
of operation
Vgs
~
-0.7
2 V
Progressing from
the
while
transconductance,
left to right, the
Vgs becomes
after
greater
is
curve
than Vlh.
to the linear
saturation region
gm.
The
results are shown
gate voltage was swept over a
linearly. The sharp increase is due to
eventually transitions from the
seen around
Output Conductance, g0
and
observed to
the transistor
As
Vgs
range
starting
at
increase sharply before
entering the
continues to
resulting in
region
6V
in Fig.
saturation region
increase,
a peak
in the
the transistor
curve
(this is
V).
-p
180
E
160
E
140
-
-
-
E
120
'/
/
o
60
-
m
f
/
T3
40-
Vds
/
20
CO
=
[23]
ii
0.15
=
300K
ND=2x10"5cm"3
/
U
fl
Expt.
/
^
CD
O
^^^-
-
O
3
.
W
=
=
2V
75
vm
0-
4-3-21012
Gate Voltage,
4.10. HEMT
Fig
voltage.
300K.
As V
linear
increases further, the
region
of operation.
Drain
Vgs
transconductance
voltage was
was varied
from
transconductance
Reasonable
kept fixed
-4
is
Vgs (V)
as
at
a
2 V
function
and
of
gate
temperature at
V to 2 V.
observed
agreement
was
to
decrease
found between
as the transistor extends
predicted
and measured
into
the
values.
49
here
although the results
occur
gm
in the
are not as
middle of the graph where the
value was
found to be 137
HEMT
will
greater
interest. The
be biased to
predicted values
in the
accurately
predicts gm
Ideally,
when a
be independent
should
source with
current
is
infinite
expected
To determine how
what
the
output
help
is biased in
Vgs
151
of the
left half
values.
As
peak-calculated
mS/mm.
Since the
is
the curve
of
Vgl becomes larger,
between
of
the
is deficient
proposed model
agreement
Improvements in
range.
as
The
saturation near the
better
this problem and obtain
correct
V,h)
-
high
revealing that the
measured values,
measured values throughout the specified
control model could
accuracy
for low Vg, (Vgs
values when the transistor
be done to
research can
and saturation regions come together.
saturation region, the
begin to deviate from the
for predicting transconductance
More
linear
mS/mm whereas the measured values went as
operate
model
in Section 4.6. The largest discrepancies
as those obtained
encouraging
linear
region.
predicted
and
the mobility model or the charge
to alleviate this problem.
is biased in
transistor
drain
of the
voltage.
Since
output resistance.
to increase to
the notion of
infinite
output conductance of the transistor
is. Equations
in Fig. 4.1 1,
shown
and
output resistance
increase in Vds
with an
in
the output current will change with a change
conductance, g0, results
drain
This is because in saturation, the transistor behaves
degree
some
the saturation region, the magnitude of the
while
output
(2.30)
and
voltage, it is
(2.31)
as a current
is impractical,
the transistor
current
is in
the
drain
saturation.
imperative to know
were used
to produce the
4.12.
?6-
E
m =0.15
5
7
E
=
vSBl
o
0) 4
s
O
3
=
1
V*
07
.35x1
cm/s
=
5 V
v
-
\i0
<l)
O
c
300K
=
370 cm2/(V*s)
Vds
-
V*
2-
=
=
10
Vv/
15Vv/
\
V
O
o
T"
-2.5
-3 .5
Gate
Fig
4.11. g0
varied
from
V, 10 V,
as a
-3
and
function
0.5
-0.5
-1.5
1.5
Voltage, Vgs (V)
of
Vgs
V to 2 V. Curves
with
V^
as a parameter.
were plotted
for
Vds
Vgs
values of
5
15 V.
50
from left to right,
Progressing
into
an exponential
transistor will
drain
are close
in
another as
reach the
of
increase (most
be in the
initially
transistor
each curve
in Fig. 4.11 is
obvious
for Vgs
they
approach the
values
linear
linear region; higher drain
the curves reaching the
linear
V&
the
saturation region
current resistant to changes
magnitude
in
in the drain
less than 0 V. As
regime
for
the
Vgs
by
and
Vgs
is
value
bias
of
5 V
needed
drain-to-source bias
the others,
its
the linear
of
mS/mm
(V^
=
It is
function
of
shows the
into
10V),
and
also useful
Vds
makes
the
linear
increase
0.35
mS/mm
analyze
region.
relationship between g0
faster
at a
faster
in
{Vds
=
an
15 V),
high
curve
other
with
in
is
order
because
the
rate with respect
each
for
Vds
g0
value
as
device
as all curves
separate
from
one
in
none
increased further
since
increase the fastest,
Vds decreases,
under a
linear
device
region at
values of
5.63
Vgi
=
lower
0 V, g0
5 V, 10
mS/mm
a
lower
drain-to-source
than the same
to Vgs. At
to
making the
scenarios, resulting
observed to
increasing Vgs
figure
begin to
was not
words, a
will reach
the
high)
the
the plot, none of the curves
power
Vgs
be
expected as
V,
(V^
under a
Vgs
values of
and
-
than
15 V,
5V), 1.21
respectively.
output conductance relates
discussed in the
and
In
increase in
behavior is
from
the curves
V"
5
mS/mm were calculated
how the
concepts
=
evident
considered;
increase in this
curves
the
to 2 V resulted
0.68
to
and
This is
range shown
lower drain bias device
the
output conductance will
Increasing Vgs
The
region of operation
10 V. Since
1.67 mS/mm, 0.58 mS/mm,
respectively.
curves.
to get the device
will reach
Vgs
"Vds
of
the output resistance will
Vgs increases,
voltages
increase linearly, eventually turning
This type
voltages were used to represent
followed
15 V
curve).
voltage.
(note: for the
region
practically be that high.). The
10 V
5 V
(meaning
gate voltages will not
the
=
observed to
to the drain voltage.
previous paragraph easier
Plotting
g0
to understand. Fig.
as a
4.12
Vds.
51
0.35
-,
'
\
/V'ss=-2V
\
V \/
0.25
p
L
;:
..
0.30
\/V
0.20
1 pm
-
W
=
m
=0.15
N0
<;V
um
2x1018cm"3
=
di.=
yvp)=-iv
75
220x1
O^pm"3
55
"^
0.15
0.10
-
/ "^
X/\
/Y
\
v
=
/V<"'!'IV
>C\/\
0.05
0
12
3
Drain
Fig
4.12. Output
parameter.
Fig. 4.12 closely
eventually
to the transistor
off and
being
for
plotted
Vgs
Fig. 4.6 in that both
resembles
leveling
conductance, g0,
Curves
becoming
biased in
the
fact
nature of
a current
because
Vds,
a given
a
higher
conductance.
At
Vgs
and
with
conductance at
values
of
will result
a
Vgs
V^ values
in
transitioning g0
point where
output resistance,
As
source
Vgs
=
6.4
of
V,
-2
-2
-1
V
=
and
6 V,
all
28.4
g0
Ro is
into the
observed
The
trends;
a
linear decrease
is due
the transistor approaches
have
to
and
not
like
showing little deviation
and
1 V
a
larger
yielded
first to
a
resistor.
with respect
output conductance.
g0
values of
higher
In
to
This is
channel
0.1 1 S/mm, 0.19
reach saturation at a
Vds
value of
g0"
is the
"transitioning
at
V^
The
values
values were calculated
of
as a
1 V.
similar
saturation
regions meet.
mS/mm
function
Vgs
and
Vds increases,
in
curve was the
linear
with
concentration and a subsequent
V, 0 V,
mS/mm.
was plotted as a
in Fig. 4.13.
conditions extend
higher 2DEG
V^
V, 0 V,
concomitantly decreases further, reinforcing
are observed
the saturation and
bias, Vds
R0,
a
values of
0.31 S/mm. The
At full drain
results are shown
biasing
and
higher
14.2 mS/mm, 24.9 mS/mm,
respectively.
The
the
value
drain bias,
no
S/mm, 0.26 S/mm,
1.05 V
curves with
6
the curves prior to reaching saturation
saturation, the curves are observed to behave as a current source
V&. For
of
-1
following
show curves
region of operation.
behave like
that the transistor will
V,
-2
the saturation region of operation and the output conductance
the
5
function
as a
values of
flat. The linear
linear
4
Voltage, Vds (V)
to
Vds by taking
other
of
of the
"transitioning
1.85 V, 2.5 V,
be 0.002
the
value
mS/mm or
inverse
of
to increase to very large magnitudes
output
go's'1
and
had
3.25 V,
less.
g0 in Fig. 4.12. The
of resistance as
the
saturation region of operation.
52
L
60000
1 um
=
W = 75pm
_
./
_
3,
m
50000
.J?
8
<*
0.15
=
.
i/'
ND=2x1018cm"3
rfd
40000
d,
=
220x 1
/^
"I
0"6
^^
cm
0"8
cm
=30x1
ooo
1 V
20000
Vss=0V
>^
.
"
-
3
a.
10000
3
-
O
0
12
3
Drain
Fig
4.13. Output Resistance, R0,
VSJ
as a parameter.
those
a given
the
2DEG is lower consequently
drain bias (Vds
at
Vgs
value, resistances
6 V)
=
linear
of each
region
Analysis
of
the
R0 did
=
-2
not remain
V, R0
varied
varied
~
this
curve
Ro in
chapter.
though the
from 1 16 Q.
the
all
and
decreased
linear
to
as a
function
for
40 Q
and
for
of
Drain
voltage with
Vgs
simulation are the same as
R0
was
V^
found
magnitude
magnitude of
be
not
became
fl,
Multiple
profiles
in
references
which
to what
by
increasing
is
closer
data that
has been
inspection
for the
Fig. 4.13.
of
insight into the linear
at the calculated
and
data
region
revealed
for
gate
Vgs
700 Q. For
=
0 V
and
voltage, the
Vgs
=
1 V,
help
reinforce
was compared
shown
here, in
-1
R0
variation
to the ideal (ideal scenario
have been found to
the experimental
similar
was also calculated
the points were below
With
63621 2
and
that
to the saturation region values. For
coming below 400 Q;
constant,
R0
get some
Looking
comparison
most of
respectively.
more
(shown in Fig. 4.12). At full
appreciated
done to
region was
concentration of
be 6566 Q, 1 1060 Q, 21685 Q,
to
The
can
variation pales
350
higher because the
values are
be deduced from Fig. 4.13.
1200 Q though
~
lower
the conductance of the channel
the linear
which reported
report curve
to
respectively.
with most points
region).
[23]
V,
-2
constant, though the
Besides
[17,18,33,34]
and
can not
they
550 Q
region resistance
constant
V,
-1
from 70 fi to 830 Q
between 50 Q.
decreasing
plotted point values
resistance value since
6
Voltage, Vds (V)
values used
corresponding
the calculated value of
1 V, 0 V,
values of
5
in Fig. 4.12.
For
Vds
V^
and
4
Vgs
V, R0
varied
in linear
would
the results shown
be
in
to the present model,
comparable magnitude
ranges.
53
This
and the
section
terminal
helped to
biasing
output conductance and
similar to other
Using
voltages.
(2.36)
equations
V to 2 V. The
varied
and
lower than
using three different
biasing
-3.1
V
conditions
contemporary
capacitance
that
The
power
values
would
proposed model
for
the
similar
rate,
to
a
voltage
curve.
-
better
that the transistor will operate
high
the transistor as a
type of
C/V
density,
ns(mj),
being directly
can
the curves
values.
be
response
said that
in Fig. 4.14
It is
clear
lower n5(m,x)
from
is
the
device
device
to have high
was shown
behaving
behaving
as a resistor and
showed
very low
is
output
as a current source.
in
a
(2.11)
lower
high
values as
drop
The
"Vds
5
and
5
V)
over a
=
by
as
V"
curve
analysis of
limit
15 V
is
shown
in higher
at
shows
Vgs
device
the
range of
could
Vgs
be
was not
subthreshold region and would provide
the subthreshold
since
it is
were used
is
off, each curve
voltage will result
given
to
Cgs
show
to
can
voltages.
high-power
increase
have the highest
It
of
representative
observed
values.
higher drain
to
region
Cgs
at
a
value of
be implied, then,
This
characteristic
device.
analyzing
and
equation
negatively
related
Vgs
to
Vds
for the
(2.11). This
related
as well.
in
channel
Vds
equation shows the
to Vc(x). Since
Vc(x) is directly
This fact helps to
value, the curves with
that an increase
value
in the
any in-depth
initial
is negatively
V, 10 V,
Fig. 4.14
such that the capacitance of the
an
Vgs
15
=
was obtained.
used as the upper
was
Vds
power
to
why, for a
equation
value results
(Vds
higher frequencies
expected
related
ns{m^c)
and
at
frequency
This
V&, it
device
Cgs
a plot of
values
not provide
values.
the three curves, revealing that a lower drain
to
device
device
representative of real-world amplifier circuits.
does
device. After
y
be
here. 2 V
avoided
supply
Vds
were chosen
x"2
favors
of a
that would place the transistor
since
is why it has been
decreasing
in saturation,
when
indicative
curve
region, the
consistent with a
(2.39) derived in Section 2.7,
under circumstances
un-reliable results.
which
Conversely,
conductance parameters of the
Cgd
and
analyzed
nature.
steep transconductance
results that were calculated
-3.1
between the
When operating in the linear
low transconductance. This is
FET devices in
conductance and a
4.8 Cgs
establish relationships
lower
will result
in
a
V&
related
explain the profile of
values yield
lower
2DEG
value
for
higher Cgs
ns(mj.).
A
charge, Q.
54
0 10
L
0.185
1pm
=
\s"^^
W
0.18
2x1018cm'J
WD
T
0.175
75um
=
=
=
300K
a.
"
0.165
O
v*
yy^^\.
0.16
1 //s*\
V*
\s7s
0.155
0.15
0.145
5 v
=
"*=10V
=
'5V
0 11
-3.5
-2.5
-1.5
-0
5
Gate Voltage, Vgs
4.14. Gate-to-source
Fig
voltage with
drain bias
given on plot.
capacitance
(V)
function
a
as
Geometry
as a parameter.
High drain
1.5
0.5
gate
of
device is
of
demonstrate "high
voltages used to
power"
capacitance values.
Since ns(m^c) is
more sensitive
higher for lower
Vds
and
5
"Vds
the
V"
is
curve
amplification
applications,
data from
figure
and
pF
5 V,
(Vds
the
15
This type
decrease
Vds
Equations
and
(2.40)
for Vgs
between
-16
V
=
also
(2.41)
1 V
and
small
to increase
Cgd
values
dramatically
between 0 V
the type of
Vgs
model
and
to
(fF)
at
large
and reach values
2 V
0.1747
curve
for
is
on the
the curves
assumed
pF
Cgs for
both, Cgs is
has been
the plot
just
for Cgd
Vds
here.
is
bottom
In
small).
Extrapolating
15 V, 10 V,
values of
each curve
shown
shown
to values of
to
increase
with
in Fig. 4.14. These
Vgd
vs.
Vgd
shown
conditions considered
mentioned.
the transistor could
in
V"
0.1758
5 V).
=
that
biasing
values
negative
15
=
be
capacitance will
Vgs but
results are
small-signal parameter prediction.
obtain
between the
(V*
[35,36]. In
by
1 V. The
of
pF
response
for
and
values, the
magnitudes of
and the
to 2 V increased
0.1861
reported
were used
V, 0 V,
-1
difference in the
the
Vds
why the "V^,
explains
0.167 pF, 0.1696 pF,
and
been
to legitimize the
and
This
Increasing Vgs
10 V),
is precisely
values of
and
top (though
values of
0 V.
-
(Vds
value.
lower
gate voltage at
will assume a value
Cgs
has
the
Vgs
"amplification
so that the
have very
pF
which
help
on
Vgs
of response
with
encouraging
plotted
V), 0.179
a given
Vgs
reveals
respectively, at
=
for
values
to changes in the
values.
the pF range.
be
The
Vgd
This type
were
here involved varying
Vgd
conditions were once again chosen
analyzed.
Once
in Fig. 4.15. Curves
The figure
exceeds
of
-3
V,
behavior
shows each curve
each curve
is
to
observed
once again supports
the
55
argument that a
can
be
seen
GaN HEMT is
from the figure that
Vds
as
rises,
frequency
smaller of the two parasitic gate-to-channel
capacitances
as possible
only helps to further the high
voltages are of concern
the plot reveals
a
Vgd
of
-3
Cgd
here, Vgd
values of
frequency
larger than
values
6.7 fF, 9.9fF,
and
V. For higher drain voltages, both
0.725 fF, 0.66 fF,
and
curves similar to what
0.55 fF
has been
Vgs
at
more
response of the
(Cgs being
negative,
-3
V
the other),
ignored in
are
17.2 fF for
Vgs
values of
1 V, 0 V,
and
V,
-1
V, 0 V,
At
smaller.
Cgd becomes
decreasing its
the analysis.
1
applications.
device. Even though
performance of the transistor.
Vgd and Cgd become
values
Vgd
respectively.
smaller
Cgd is
the
value as much
Since only high drain
Pulling
and
=
It
-10
-1
V,
from
numbers
respectively, at
V, Cgd had
References
values of
[36,37]
show
here for Cgd.
shown
0.12
high-power, high-frequency
essentially making Vgd
desirable bi-product for the intrinsic
which is a
for
an exceptional candidate
-i
L
\im
=1
W
=
ND
75|jm
Vas=-1V
J
cm
2x10"
=
r = 300K
0.08
Q.
0.06
-
O
0.04
0
-I
-13
Gate-Drain Voltage,
Fig
4.15. Cgd
simulations
2xl018
function
as a
were
cm"3
and
done for
T
=
300K.
of
a
Vgs
V*
V&
with
75um
x
Vgd (V)
as a parameter.
l|im device
values of
with
15 V, 10 V,
All
ND
and
=
5 V
were used.
results
for both
Cgd
and
Cgs help
how it is
affected
by
the
drain bias
The
and
provided an
increase in
performance of
Vds helps
to
provide some
and
gate
to decrease both
Cgs
insight into
length.
and
Cgd
the
frequency
Clearly from
which act as
the
response of
discussion
limiting agents
the
device
and results
for the
just
frequency
the device.
4.9 Cutoff Frequency,/r
56
argument that a
be
can
which
from
seen
is
a
GaN HEMT is
the
figure
Vds
that as
rises, essentially making
desirable bi-product for the intrinsic
frequency
smaller of the two parasitic gate-to-channel capacitances
as possible
only helps to further the high
voltages are of concern
the
a
Cgd
plot reveals
Vgd of
-3
here, Vgd
values of
frequency
larger than
values
6.7 fF, 9.9fF,
and
V. For higher drain voltages, both
0.725 fF, 0.66 fF,
and
curves similar to what
0.55 fF
has been
for high-power,
an exceptional candidate
Vgs
at
values
Vgd
the
decreasing its
the other),
V
ignored in
are
17.2 fF for
Vgs
values of
Vgd and Cgd become
1 V, 0 V,
and
V, 0 V,
1
smaller.
V,
-1
the analysis.
At
Vgd
respectively.
the
value as much
Since only high drain
Pulling
and
-10
-
It
smaller
Cgd is
device. Even though
performance of the transistor.
-3
applications.
Cgd becomes
more negative,
response of
(Cgs being
high-frequency
-1
V,
numbers
from
respectively, at
V, Cgd had
References
values of
[36,37]
show
here for Cgd.
shown
0.12
L
=
W
0.1
1 |jm
=
75|jm
Vgs
W0
r
0.08
=
300K
V^
-0V
Vgs
a.
V
= "1
-1
V
0.06
o*
0.04
0.02
0
-13
Gate-Drain
Fig
4.15. Cgd
simulations
2xl018
function
as a
were
cm"1
and
done for
T
=
300K.
of
a
Voltage, Vgd (V)
Vgs
VA
Vd<
with
75um
x
as a parameter.
lum device
values of
with
15 V, 10 V,
All
ND
and
=
5 V
were used.
results
for both
Cgd
how it is
affected
by
increase in
Vds
The
and
provided an
performance of
and
Cgs help
the drain
to
bias
provide some
and
gate
helps to decrease both
Cgs
insight into the
length.
and
Cgd
Clearly
frequency
response of
from the discussion
which act as
limiting
agents
the
device
and results
for the
just
frequency
the device.
4.9 Cutoff Frequency,/r
56
Using
equation
the cutoff
(2.42)
and
varying
frequency. Fig. 4. 1 6
shows
from 0 V to 15 V
was varied
reinforce what was established
and
fT
Vgs
flattens
out
for Vds
"saturating"
the curve
increase in
are
is
in
values
encouraging for high
Cgd
and
Cgs
4 V
Cgd
and
Cgs
higher. The
region of
interest lies to the
Vdg equal to
3 V (with
V^
right of
Vds
=
Vds
Vgs),
-
Cgd
figures
were obtained
for
fT
function
L.
Vds
vs.
as a
improves
and
with
will
can
be
Cgs
also
with an
seen
the
increase in
from
flatten
drain
out.
Prior to
4 V. This
a magnitude of
voltage
figure
the
increase in Vdi. These
the
typically have
is
since this
of
V*. Fig. 4.16 helps to
reaches a value of about
decreasing
4 V
fT
response
both
that
until
Vdg (Vds
shows
increase in Vds. In
the
are
two
the plot of
frequency
indicating
power applications where
=
with
Fig. 4. 1 7
and
1 V for
device
the
-
increase quickly
because
the value of fT occurs
Vds
of
was set equal to
excess of
observed to
drain voltage, Vds,
and
function
as a
in Section 4.8
Vds. This is due to the decrease in both
that/7-
length, L,
the gate
rapid
results
3 V
or
that would make
1 V).
"n"
x
12
st
10
&
8
3-
6
1
I
3
0
n
/
um
75
urn
1 V
vgs
=
r
300K
=
Fig 4.16./r as
a
function
um and
15
10
5
75
1
=
/_
Drain Voltage,
width
=
W
/
4-
2
L
/
0)
2
;
^
of
length 1
Vds (V)
V^./, calculated for device
Vgs of 1 V was used
um.
of
to
represent typical gate amplification voltages.
Pulling
of
numbers
4 V, 10 V,
device
cut-off
from the
and
15
V,
frequency
curve yielded
respectively.
has
also
been
fT
values of
Similar
11.19 GHz, 12.69 GHz,
curve profiles
plotted as a
function
have
of
also
drain
been
and
13.15 GHz for
reported
by [38,39]
Vds
values
where
the
voltage.
57
L
=
X
W
o
Vgs
T
1 [im
75
=
=
=
Vds= 10V
[im
1 V
Vds
-
=
5V
300K
oO
^>>>to^
^^
o
3
o
0.1
1
Gate Length, L
(um)
Fig 4.17. fT as a function of gate length, L.
drain biases
75
Fig. 4.17
plotted
The
with
for drain biases
the
since a
frequency
values.
larger L
and
length
11.71
GHz,
increased the
similar
results shown
predict
device
the
of
have
an
and
increasing
and
magnitude of
the
of the
147.64 GHz
cut-off
are quite close
4.17 have
drain
of
to
were
the gate
length, L. Curves
the
"V^,
the gate
device for
(Vds
=
with
decrease in gm
=
a
-
V"
15
higher
5 V
and
length
all
Vds
as/7-
were
1 V.
steadily decreases
yielded
by
one
et al.
to
decreases
fT values
of
values of
[23]
expected
have
a
the
Cgd
to
199.6 GHz
and
Ambacher
with
the gate
here, revealing
higher fT
Cgs
and
13.15 GHz, 12.69
order of magnitude
its relationship
shown
is
this
since channel resistance
observed
value
drain biases to
and
has been
is
curve
5 V). Rashmi
frequency
length
the gate
logarithmic type). Intuitively,
to
15 V, 10 V,
what
and
values and a
in Fig. 4.16
Decreasing
frequency
frequency
the
Cgd and Cgs
length, L,
findings regarding the
cut-off
device
curves were plotted with a constant gate voltage of
axes were converted
lum, drain biases
10 V),
of
frequency, fT,
inverse relationship
same reason as
cutoff
Simulations done for
5 V. Dimensions
cutoff
5 V. All
in larger
respectively.
in [10,40]
Figures 4.16
either
=
and
a given gate
for the
a gate
V), 183.34 GHz (Vds
reported
to
value would result
the others
For
drastically
by
shown
increase in L (note that both
value than
GHz,
is
and
um.
15 V, 10 V,
of
become larger. For
would
1
relationship between the
shows the
cutoff
by
um
15 V, 10 V,
of
0.1
(V^
et al.
pm
=
15
have
length. The
the model's
ability to
values.
shown
that the
voltage or
gate-to-source
decreasing
the gate
GaN HEMT is
most
easily improved
length. This is because both
methods reduce
frequency response
of a
and gate-to-drain capacitances.
Although the
process
for
fabricating
58
GaN HEMT devices has
continuing
research
not yet achieved a mature
in the
area of
stage, these
results
HEMT modeling for high-power
certainly
provide a good reason
high-frequency
for
circuits.
59
Chapter 5
Conclusions
5.1
and
Future Research
Conclusions
5.2 Future Research
5.1 Conclusion
A
physics-based model
equation was used
carrier
density,
in
has been
conjunction with
ns, to the
Poisson
version of
the
which was
later
used
in predicting ns
to
derive I/V
found
the calculated and
their
for
Most
to
Most
values
results
was
concentrations to
linear
The linear
Vgs
measured
lacked
by
experimental
comparison
in that
the results were very encouraging
much
better than that for the
the results never
calculated
for
the
Ids
all
vs.
Vds
and
became larger than 10% for any
other curves since
they
by
of
limiting
and
included both
other authors.
the
simplified
Vgs
current
Discrepancies
performance-predicting ability
data to be
compared
to experimental
fell
In reality,
measured values were available).
equations
sheet
transconductance values supports the argument that
be done to improve the accuracy
the other calculated results
I/V
considered,
relationship between ns
charge-control
and saturation
were
relating the
below threshold. A
values
the conductance calculations and all of the capacitance and
(when
between
be
all
well eigenstates
upon previous results reported
profiles and magnitudes were validated
most of
not
of
improve
experimentally
more research and work needs
model.
to
equations.
Transistors. The Schrodinger
Mobility
statistics to generate an equation
quantum
equation was used to obtain a
components which was
between
Fermi-Dirac
Fermi level, Ef. Two
effectiveness of the model
High Electron
proposed to model
within
the agreement
Id,
vs.
curve at
were not able
to
Vgs
GaAs data. This
frequency
22%
of
the
which case
was
experimentally
the case
measured
simulated and measured
curves where
the percent
condition.
be explicitly
the
calculations.
between
any bias
against, in
of
Percent
deviation
errors could
compared with experimental
measurements.
The
most notable weakness
and around the transitional
considering that
in
area
the model
is its
between the
inability
saturation
peak transconductance occurs when
the
to accurately predict transconductance values
and
linear
regions.
This is
device transitions from
a serious
in
limitation
the saturation region to the
60
linear
Since gm is derived from
region.
looking
bump
at
Fig.
4.8,
found between
good agreement was
occurring in the
region or when
although
thermal effects degraded the
in
small
the current curves,
magnified and more pronounced with respect to the conductance parameters
data
experimental
were
found
the case
their
could not
be
obtained
for
provided simulation results that
with
the remainder of results that
accuracy
be
could not
validated.
however, did
not
have
the same
The ns
than their magnitudes.
having
and
"success."
vs.
closely
matched
were presented.
It is
between Figures 4.1, 4.12, 4.15, 4.16,
so
profile comparison
The
Vgs
discrepancy
negligible magnitudes
for
could not
with
Vgs
values
relationship,
with
ns
decreases to
near
threshold, ns begins to decrease asymptotically
in Fig. 4.1
Vgs
where
rc, is
range specified.
with
increasing Vgs
plotted against
Ef). In Fig, 4.14,
Although this is generally
what
be
g0.
Unfortunately,
References that
in Chapter 4. This
to
compared
references.
respective
pictured
relationship
and g0).
conductance,
the results presented
They
their
and
output
(gm
become
Cgs is
Figures 4.3
these curves
involved their
in Fig. 4.3
shows
less
than threshold.
with respect
shown
found
happens, Cgs
should rise
at a
faster
and
4.14,
profiles more
linear
near-perfect
In actuality,
to the x-axis
increase
to
a
was
real samples so
to mention that excellent agreement was
worthwhile
4.17
for
a slight
only
The largest differences between
regions meet.
in the linear
device. These differences,
of the
performance
either
predicted and measured results with
linear
curves where the saturation and
the measured and predicted values came
current
be found there. When
the current curves, an explanation may
[15]
decreasing
when
Vgs
(much like
rate
and stabilize
for the
in
value
[36].
5.2 Future Research
The
GaN
list
following
growth process and
1)
Although
work.
to
has been
until
and
the
is limited
offered
reasoning for future
has been
achieved
improvement has been
junction profiles,
refined
provide
major progress
Steady
however,
of statements
increasing
the
quality
of
recent
in
the
fact that there
exist
years, the
decreasing
GaN
the model and of the
growth process still needs
only
a
yield.
handful
values,
idealizing
progress
is needed,
contact resistance
the grown semiconductor.
process can produce acceptable
by
weaknesses of
research:
in
achieved
to detail the
The
of
More
rate at which
the process can
fabrication facilities
be
worldwide that
61
the
following Vlh discrepancy
and
-4.2
with
an
V
were reported
m
of
thickness
30
of
information
clear
of
be
for
used
and
the
V,h
fraction
as
3)
The
proposed model
V)
[23],
had
a
could
large
part
be
not
be
fractions
focused only
on
in
15%
of
the strong
or
to
be
than
values
resolved
15%. It
if
Nonetheless
was
unnecessary for the
subthreshold
region
amplification
device.
In Fig. 4.7, the
drain bias
experimental
continued
Clearly, if
a
since
the
Vgs
GaN-based
data
plotted
to increase. Thermal
device heats up, the ability
expected
was
proposed
(2.6)
Using
were
V,h
results would
and
the
[23] for
does
device
V,h
value
experimental
not
With that said, future
would not
V
the
calculated
the calculated
voltage
vary
with
research
limit the applicability
leading
included,
between
of
HEMT
is
to poor subthreshold ns
the
V,h
to
only two
the Poisson equation
experimentally
vs. m problem would still need
accurately
being
and
predicted and
desired for devices
model
increased,
fractions higher
of mole
predict
considered
values
ns
as
a
in the
high-power
peaked, and then decreased slightly as the
effects were offered as
of
layer
equation was used to obtain
in
and
spacer
-6.72
hardware
equation
inversion region,
be
between ns
other
less.
excellent agreement could
agreement
which
V
results, nullifying the
behind using
more eigenstates were
the subthreshold region.
and
-7
the
and
AlGaN
V
found between
in that it
Certainly if
ns
Using
suggest.
quantum well eigenstates.
measured
A,
350
donor layer.
other
device threshold
This is because Schrodinger's
simplified,
an
-10.14
all
deciding
used.
would
of great value
had
of
values of
The difference between
the motivation
Fig. 5.1
and
dd
charge
of
with
good agreement was
which was
would
with mole
values
discrepancies
to
and
samples
in the
concentration predictions.
was not
4)
which
(2.6)
equation
devices
to
model
directly
25%
[16], V,h
et al.
cm"3
(2.6), V,h
measured results would suggest, the
modeling their relationship
the
2xl018
of
with the measured values.
et al.
2.8
(
In Garrido
Both
respectively.
equation
discovery
and
in Wu
reported
As
and
lead
This
comparison
comparison.
mole
in [16]
the model.
A,
the author.
one with an m of
doping density
would
parameters reported
and
Si
280
of
disagreement
Vlh
measured
accuracy
A
by
for two samples,
dd
and
provided
calculated, in
and
22%
found
was
the carriers to move
the explanation
(mobility) in
for
this
behavior.
the presence of
an
63
electric
field
will
decrease. This
reasoning behind the decrease in
model
did
into account,
no
increasing
Attempts have been
to alleviate this problem
[41]). Although thermal
high drain biases (Vds
substrates are used.
and
the
>
20V),
A future
trapping
performance of
model
effects
relationship between the threshold
model
to
the
model
would
be
decrease
growing GaN
in
circuits
W-cm"1
as opposed
device
pronounced compared
that
it
could
to
on
in
top
the
to
more
'
for
to degrade at
when
effects such as
predict
SiC
of
0.35 Wcm
performance
includes performance-degrading
useful
proposed
observed
was
the
experimental and calculated values.
effects still cause
degradation is less
that
drain biases. Since the
is
sapphire
thermal
accurately the
device.
improvements just listed,
the proposed
by
have high thermal conductivity (around 4-5
sapphire substrates
effects
the
a negative manner which
associated
deviation between
calculated results, therefore
made
in
current observed at elevated
take these effects
not
substrates that
The
affects the channel current
devices
with serve as motivation
voltage and mole
of mole
fraction 15%
for future
fraction is
or
most
research
important
in this
area.
Modeling
since equation
the
(2.6) limits
less.
64
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67
Appendix I
Appendix I
Equation
(2.12)
IdUn Derivation
-
shows the current
1
+
'/J0EC
density
I
,
V
equation
M0EC
,
A
-v,,
^cVsat
V
(2.1
1) for
plugging in the mobility
(_
^dVc(x)
ns(mj)
(2.10)),
(2.12)
-
dx
dx
,
(equation
,
+
n,(m,x)-
=
model
kBTdn,(m>x)^
)dVc(x))_u,_
(_
Wq/U0
-v10,
Ecvwl
Substituting
equation after
dx
q
yields
A
".<>
.Ww>[f<=)(l,,_v,W)^>+*i.(^_v,w)
qd
dx
qd
q
dx
J
dxy
v.
(2.13a)
where
d=dd+dl+ Ad
Vg,=Vgs-Vlh{m)
Distributing
through on the right
,+
V.
hand
side
(RHS)
of
(2.13a),
dx
dVc (x)
|/fp______
V
Ecvsal
j
e{m),
=
dx
WqM0
[
,
s'
,\dVc{x)
c{X)l
qd
dx
kjfe{m)dVc{x)
q
{
qd
dx
(2.13b)
Simplifying
the
RHS
and
dx +
multiplying both
MqEc-v,,
Ev
The
boundary
dV(x)
by dx yields,
WAm)
V
Vc(x)-^]dVc(x)
q
J
sat
conditions represent
sides
the
intrinsic
voltages at either end of
the
(2.13c)
)
channel and are
listed here for
reference:
68
VM)\x=L=Vds-IdsRd
Since the intrinsic
=
L to
voltages are
obtain an expression
IA
known
at either end of
for the drain
current
\dx + a\dVc(x)
=
b
the channel,
(2.13c)
can
be integrated from
x
=
0
to x
in the linear region,
kj
Vg,~:
\dVc{x)-b\Vc{x)dVc{x)
(2.13d)
where
U0Ec-vsa
ErVsa,
b
=
Wfl0{m)
the integration yields,
Performing
kBT
x=L
Ids(L + aVc(x)\^)=b V.
a
Plugging in
IdsL +
the
boundary
'dSVdsa
'
J
vMZ-?M
(2.13e)
x=0
conditions and simplifying,
< (Rd + R,) bVgsVds bVgJds (Rd + R, )
-
=
-|(VJ. ~2VdsIdsRd
+l2ds{R]
-R;)) (2.i3f)
where
V
gs
=v
-^
s'
q
By combining
the
coefficients of
the
lds
terms,
o
=
equation
ai2ds
+
(2.13f)
pids
+
can
r
be
prepared
for the
quadratic
formula,
(213g)
and
'
(2.13)
d.lin
2a
where
69
a
=
P
=
a(Rd+Rsytl(R2_R2)
bVdsRd-bVgs{Rd+RsyL-aVd
r=bvgsvds-^v2
70
Appendix II
Appendix II
This
appendix
region
Starting
in
derivation
incorporation
here is later
expression obtained
Field
the
outlines
pertinent to the model's
Vc(x) Derivation
-
used
of the
potential,
length
modulation
of channel
to
Vc(x).
channel
form the bias-dependent
Obtaining
occurring in
expression
this
expression
the saturation region.
for L,, the length
was
The
of the Low-
the channel.
with equation
(2.9)
and
plugging in
equations
(2.10)
and
(2.1
1)
yields,
^dV(x)^
1 +
Ecvsal
v
V
j
^K
WqjU0
dx
-v,
qd
-v.ui
M)^+i^f^(v,
dx
q
ax\
qd
(2.14a)
Simplifying (2.14a)
gives the same expression as
'MoEc
1
-VsM
.
2. dV
dx+ ^-^
E v
Since the
-
0
can
channel
be
used.
is
pinched off and
Integrating (2.14b)
is
(2.13c)
and
is
repeated
here for reference,
,.]_Wf40e{mY
(2.14b)
(x)
of unknown
from 0 to
x
length, only
the
boundary
condition
corresponding to
x
gives,
fx
1
ds
[dx + a
\dVc (x)
=
b(vgl
V
Vo
-^-))dVc(x)-b)vc(x)dVc(x)
1
(2.14c)
JS
After integrating,
'A*+vmz)Av*-~YMZ--v^T,:
q j
Plugging
in
the
boundary
conditions
yields,
/<_(^ + a(VcU)-/(_^))
Distributing
>*
=
fcVff(VcW-/^I)-^(ve2W-/^)
the coefficients through and combining terms with
like
orders of
Vc(x)
simplifies
(2.14e)
(2.14e) to,
71
x<ds+bVgJd5Rs-aI2dsRs--I2sR2
=
oK2M+(/*-*v>cM+
The
quadratic
formula is
used to obtain a solution to
0
(2.14)
(2.14),
, (l)=z_______E
V
2a
where
6
a
=
2
/3
=
7=
aIds-bVigs
*hs
+bVJdR
-aI2R
ds Jvi
J
rfi-1
-
72