HOTELECTRON NOISE IN GALLIUM ARSENIDE/ALUMINUM GALLIUM
ARSENIDE HETEROJUNCTION INTERFACES
By
CHRISTOPHER FRANCIS WHITESIDE
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1987
ACKNOWLEDGMENTS
The author wishes to express his sincere gratitude to Dr. G.
Bosman for his research guidance and many helpful discussions and also
to Dr. C.M. Van Vliet and Dr. A. van der Ziel for their support and
encouragement. The services of Dr. Morkoc at the University of Illinois
in supplying the heterostructures for the experiments are greatly appre
ciated. He also wishes to thank his fellow students in the Noise
Research Laboratory for their help and many interesting discussions and
Miss Katie Beard for the editing and typing of the manuscript.
Special thanks go to his wife, Susan, and his family, who have
supported and encouraged him over the years.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ................................................... i
ABSTRACT ........................................... ................ V
CHAPTER
I. INTRODUCTION ................ ...... ..................... 1
1.1. Band Structure of AlGaAs/GaAs ........................ 2
1.2. HotElectron Effects ................................. 4
1.3. Noise Characterization of HotElectron
Phenomena ........................................ 7
1.4. Device Applications of Heterojunctions ............... 14
II. MONTE CARLO MODELING OF HOTELECTRON TRANSPORT .............. 18
2.1. Description of Physical Model ........................ 19
2.2. Determination of rL Intervalley Coupling
Constant and its Relation to the
DiffusionField Characteristics .................... 32
2.3. Monte Carlo Spectral Analysis of
Velocity Fluctuations .............................. 39
2.4. Position Monitoring and Boundary Conditions
in Monte Carlo Programming ......................... 49
2.4.1. Program algorithm .......................... 52
2.4.2. Simulation results ......................... 57
III. EXPERIMENTS ON ALGAAS/GAAS INTERFACES ........................ 63
3.1. Description of Device Structures ..................... 63
3.2. Noise Temperature Measurement Setup and
Experimental Procedures ............................ 65
3.3. Experimental Results ................................. 71
3.4. Discussion of Results ................................ 83
IV. THE DC, AC AND NOISE CHARACTERIZATION OF THE
ALGAAS/GAAS MODFET CHANNEL ................................. 89
4.1. Impedance Field Modeling ............................. 90
4.1.1. Review of impedance field method ........... 90
4.1.2. Application of the impedance field
method to the MODFET ..................... 94
4.2. ChargeVoltage Dependence ............................ 99
4.3. Device Description ................................... 103
4.4. Measurement Procedure ............................... 103
4.5. Results and Discussion ............................... 107
4.6. Conclusions ..........................................115
iii
V. CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH ............. 118
5.1. Monte Carlo Transport Modeling ....................... 118
5.2. Experimental Characterization of
Heterostructures ................................... 119
5.3. MODFET Characterization .............................. 120
APPENDICES
A. MONTE CARLO ELECTRON TRANSPORT ALGORITHM ..................... 121
B. VELOCITY TIME SERIES ALGORITHM ............................... 122
C. POSITIONMONITORING ALGORITHM ............................... 123
D. MONTE CARLO COMPUTER PROGRAM ................................ 124
REFERENCES ............................................... .......... 141
BIOGRAPHICAL SKETCH.................. ............ .............. 144
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
HOTELECTRON NOISE IN GALLIUM ARSENIDE/ALUMINUM GALLIUM
ARSENIDE HETEROJUNCTION INTERFACES
By
Christopher Francis Whiteside
May 1987
Chairperson: G. Bosman
Major Department: Electrical Engineering
In recent years much attention has been paid to the study of semi
conductor heterojunction interfaces. An interest in the hotelectron
behavior of electron transport parallel to the interface has arisen.
In this dissertation the chargetransport noise in the direction
parallel with the GaAs/AlGaAs interface is studied. Monte Carlo calcu
lations of the electron transport properties of bulk GaAs are fitted to
recent experimental data of the fielddependent diffusion coefficient.
This method provides a better theoretical value of the rL intervalley
coupling constant. The effects of GaAs device length on the velocity
fluctuation spectrum are investigated using the Monte Carlo technique.
In addition, an experimental investigation of the velocity fluctuation
spectrum as a function of electric field and length for different
AlGaAs/GaAs heterojunctions is completed. Finally, the dc, ac and noise
properties of the AlGaAs/GaAs MODFET channel are investigated both
experimentally and theoretically using the impedance field method.
CHAPTER I
INTRODUCTION
In recent years much attention has been paid to the study of semi
conductor heterojunction interfaces. Due to increased processing capa
bilities, novel semiconductor heterojunctions of various material compo
sitions can be manufactured. This opens many new possibilities in the
development of existing and novel semiconductor devices and their
applications.
In order to take full advantage of these new heterojunctions, more
information on the chargetransport properties must be attained. In
this dissertation the chargetransport noise in the direction parallel
with the GaAs/AlGaAs interface is studied. Monte Carlo calculations of
the electron transport properties of bulk GaAs are fitted to recent
experimental data of the fielddependent diffusion coefficient. This
method provides a better theoretical value of the rL intervalley
coupling constant. The effects of GaAs device length on the velocity
fluctuation spectrum are investigated using the Monte Carlo technique.
In addition, an experimental investigation of the velocity fluctuation
spectrum as a function of electric field and length for different
AlGaAs/GaAs heterojunctions is completed. Finally, the de, ac and noise
properties of the AIGaAs/GaAs MODFET channel are investigated both
experimentally and analytically using the impedance field method.
In this introductory chapter the band structure of the AlGaAs/GaAs
system is reviewed, including energy band lineup and modulationdoping
techniques. The second section of this chapter outlines some of the
effects of hot electron behavior on charge transport, such as real
spacecharge transfer vs. intervalley transfer, in providing negative
differential mobility. The length of the active device region can have
a significant effect on the chargetransport properties of GaAs in the
hot electron regime, and this effect is discussed.
Noise measurements have been used for many years as a method for
investigating charge transport properties in materials and devices. A
review of both analytical and computational methods of noise modeling is
presented. Device applications of the AlGaAs/GaAs heterojunction, such
as the heterojunction bipolar transistor (HBT) [1] and the modulation
doped fieldeffect transistor (MODFET) [2], are discussed.
1.1. Band Structure of A1GaAs/GaAs
The alloy system AlxGal_xAs/GaAs is of great importance in high
speed electronic devices since it allows the possibility of bandgap
engineering. The lattice constants between the two materials are
closely matched. If properly grown, this small lattice constant
difference results in highquality interfaces between GaAs and AlGaAs
with an insignificant concentration of interface states.
The most important device parameter of interest in AlxGalxAs is
the energy bandgap dependence on the alloy composition. The energy gap
as a function of the mole fraction x can be expressed by [3,4]
E (x) = 1.424 + 1.247x(0 < x < .45) (1.la)
g
E (x) = 1.900 + 0.125x + 0.143x2(0.45 < x < 1.0) (l.lb)
g
and the units are in eV. For mole fractions less than approximately
0.45, the AlGaAs has a direct bandgap. For larger mole fractions, the
alloy bandgap is indirect, with the Xvalley having the lowest energy.
The Lvalley lies between the r and Xvalleys for mole fractions larger
than 0.45.
The sum of the valence and conductionband discontinuities must
equal the energy bandgap difference between the GaAs and AlGaAs.
Originally, it was believed that the conductionband discontinuity was
0.85 E (x) [5]. However, more recent measurements [6] have shown that
sixtyfive percent of the bandgap difference lies in the conduction band
for x ( 0.45. Then the conductionband discontinuity follows from
AE = 0.81x (eV) for x .0.45 (1.2)
From measurements of the valenceband discontinuity as a function of
mole fraction, it was determined that the maximum conductionband
discontinuity lies in the vicinity of x = 0.45 [7]. A further increase
of the mole fraction results in a decrease in the conductionband dis
continuity with a corresponding larger increase in the valenceband
discontinuity.
Of special interest in the A1GaAs/GaAs system is the method of
modulation doping. In this doping process, the GaAs layer is undoped
and the dopant atoms (usually silicon atoms for ntype) are deposited in
the AlGaAs layer. In equilibrium the Fermi level must be constant
across the interface. Consequently, the electrons from the donors near
the interface transfer to the lower energy GaAs conduction band. In
this way, the electrons become specially separated from the parent donor
atoms, causing an electric field normal to the interface. The electric
field leads to energy band bending in both the AlGaAs and the GaAs in
the vicinity of the heterojunction. This band bending forms a quasi
triangular potential well for the electrons in the GaAs. If the elec
tric field is large enough, the width of the triangular potential well
may be smaller than the carrier deBroglie wave length. The momentum
vector perpendicular to the interface then becomes quantized. Shown in
Figure 1.1 is an example of a heterojunction where the AlGaAs is doped
ntype and the GaAs is slightly ptype.
The modulationdoping process was developed to increase the low
field mobility of electrons in the direction parallel to the interface.
At room temperature (300K), the dominant scattering mechanism is polar
optical phonon scattering. At lower temperatures, however, the dominant
scattering mechanism becomes ionized impurity scattering. Because the
electrons are specially separated from the donor atoms, a reduction of
ionized impurity scattering is obtained, leading to a higher electron
mobility. The inclusion of a thin spacer layer of undoped AlGaAs
between the doped AlGaAs and GaAs reduces even further the coulombic
interaction between the donor atoms and free carriers [8].
1.2. HotElectron Effects
With microelectronic devices approaching submicron dimensions,
even moderate applied voltages result in very high electric fields.
These large fields cause the average carrier energy to increase sig
nificantly beyond the thermal equilibrium value. This increase in
energy leads to nonlinear charge transport (i.e., deviations from Ohm's
Law), also known as hotelectron transport. To improve the modeling and
performance of electron devices, highfield transport properties need a
more thorough understanding.
An interesting hotelectron effect is the occurrence of a negative
differential mobility regime in bulk GaAs. This phenomenon, commonly
GaAs
EC
EF
spacer
layer
I
I
AlGaAs/GaAs heterojunction at equilibrium.
AIGaAs
I
t
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
Fig. 1.1.
known as the Gunn effect [9], is due to the transfer of electrons from
the highmobility central rvalley to the lowmobility satellite L and X
valleys. At low fields the electron velocity increases in proportion to
the electric field. At higher field strengths, the electrons partially
occupy the lowmobility satellite valleys, and the average velocity is
lowered. This net decrease in velocity with increasing field gives rise
to the negative differential mobility in GaAs.
From the measurements by Ruch and Kino [10] on bulk GaAs, it was
found that the diffusion coefficient shows a sharp increase in the same
field range as the onset of transfer of electrons to the satellite
valleys. Since the diffusion coefficient is closely related to the
velocity fluctuations caused by random scattering, information about the
scattering process can be obtained. The increase in diffusion is very
sensitive to the intervalley coupling constant DFL. Thus, accurate
measurements of the field dependence of the diffusion coefficient can
provide a more accurate value for this constant.
Since the L valley is located .33 eV above the conductionband
minimum, it takes time for most electrons to gain sufficient energy
under an applied field to undergo intervalley transfer. If the device
length is short, few electrons will transfer to the satellite valleys
before being collected by the contact. The large changes in velocity
associated with intervalley transfer will not occur and less noise will
be produced in the external circuit. A description of a Monte Carlo
experiment to observe this effect is outlined in Chapter II.
In the AlGaAs/GaAs interface there are two mechanisms that can
produce negative differential mobility at high electric fields. The
first is the Gunn effect just outlined for bulk GaAs. The second
mechanism is called realspacecharge transfer, which stands for the
following physical process. Electrons in the highmobility GaAs gain
energy as they drift under an applied electric field parallel to the
interface. When the energy becomes comparable to the conductionband
difference, there is the possibility of transferring to the AlGaAs.
Because of the high doping concentration, which introduces a significant
amount of ionized impurity scattering in the A1GaAs layer, the electron
mobility in this layer is lower than in the GaAs layer. The increasing
percentage of electrons transferring to the AlGaAs layer with increasing
field causes the drift velocity to decrease, similar to the Gunn effect.
Experimentally, realspacecharge transfer has been shown to be the
cause of negative differential device conductance of specially made
heterostructures [11]. However, accurate modeling of the processes
involved is difficult, and even getting experimental verification of
negative conductance is rather involved. The length of the hetero
structure may also play a role in the transfer of carriers.
Investigation of the fielddependent diffusion coefficient in
conjunction with velocityfield measurements should provide information
on the hotelectron behavior of the GaAs/A1GaAs interface system. In
Chapter III these measurements will be presented for different interface
compositions and compared with bulk GaAs.
1.3. Noise Characterization of HotElectron Phenomena
Noise measurements are used to provide information on charge
transport processes in semiconductors. In this section the methods of
characterizing hotelectron effects by noise measurements are reviewed.
There are basically three types of noise in semiconductor devices:
1/f, generationrecombination (gr) and velocityfluctuation. At low
frequencies 1/f and gr noise, caused by fluctuations in the sample
resistance, can be observed by the passage of a current through the
sample. The 1/f noise mechanism has been attributed to mobility and
number fluctuations. Generationrecombination noise is caused by the
interaction of carriers with trapping states in the forbidden energy
band. The trapping process gives rise to fluctuations in the number of
free carriers available for conduction. Velocity fluctuations are a
result of carrier interactions with the scattering mechanisms associated
with the thermal vibrations of the host crystal. Since the mean inter
collision time of the carriers in highmobility semiconductors is very
small, the velocityfluctuation spectrum extends to very high frequen
cies. The emphasis in this dissertation is placed on the hotelectron
effects that are associated with the various scattering mechanisms.
Therefore, velocity fluctuation noise, also known as thermal or diffu
sion noise, is used as a tool for probing these effects.
Consider a oneport network biased by an arbitrary dc voltage V0
with a dc current 10 flowing through it. The smallsignal Thevenin and
Norton equivalent circuits, evaluated around the bias point, are
depicted in Figure 1.2. In general, the smallsignal impedance Z(VO,f)
and admittance Y(V0,f) are functions of bias and frequency. The voltage
and current noise generators represent the noise mechanisms in the net
work. The mean square voltage fluctuations AV2 can be expressed in
terms of the voltage spectral density SAV by
AV2 = S A(VO,f)df (1.3)
0
where f denotes frequency. A similar relation,
00
A = f S(Vf)df (1.4)
oI 0 AIO (1.4)
Z
Fig. 1.2.
Thevenin and Norton smallsignal
equivalent circuits.
c~
holds for mean square current fluctuations in terms of current noise
spectral density.
One can now define the concept of an ac noise temperature T (V0,f)
of the network in analogy with the Nyquist relation in the following
way:
S (V0,f) = 4kBTn 0,f)Re{Z(V0,f)} (1.5)
SAI(V,f) = 4kBTn(V0,f)Re{Y(Vo,f)} (1.6)
where kB is Boltzmann's constant and Re { } stands for the "real part
of." It should be noted that the noise temperature is an electrical
parameter of the network and has nothing to do with the electron tem
perature. By connecting a conjugately matched load to the network, the
maximum available power is delivered to the load. This maximum avail
able power has the value
P = k T (V ,f)Af (1.7)
av Bn O 17)
where Af is the bandwidth of the measuring system. Therefore, T has
physical meaning and can be measured. At high frequencies (f > 10 MHz)
measurements of T are preferred because it is much easier to measure
power flow than terminal voltages and currents.
The above definitions are valid for every oneport network whether
it is linear or nonlinear. The following discussion is restricted to
homogeneous semiconductor samples for which a onedimensional treatment
is warranted. The link between diffusion coefficient and velocity fluc
tuations is outlined. It should be noted that the quantum correction
factor for thermal noise is neglected [12].
Let the instantaneous velocity of a carrier i at time t be
vi(t) = vd(E) + Av(t) (1.8)
++
where vd(E) is the average drift velocity and E is the electric field.
The term Av (t) represents the fluctuations in the velocity about
+ +
vd(E), with the average Avi(t) = 0. By definition [12] the diffusion
d i
coefficient is related to the spectrum of velocity fluctuations by
S (E,f) ______
Av dT (1.9)
VD(E,f) = = Av(t)Av(t+T) e2f (1.9)
where the term Av(t)Av(t+T) is the autocorrelation function of the
velocity fluctuations. At low frequencies eq. (1.9) reduces to the
wellknown Einstein formula for diffusion
Ax2 = 2Dt (1.10)
for sufficiently long t.
Consider a semiconductor sample of length L and crosssectional
area A with ohmic contacts. An electron with velocity Av (t) gives rise
to a current Ai (t) in the external circuit such that
qAv (t)
Ai(t) = L (1.11)
i L
and the corresponding spectrum of current fluctuations is
2 2
SAi(f) = S (f) = 4 D(f) (1.12)
L L
If the electron gas in nondegenerate and there are nAL electrons in the
sample, then the total noise current spectral density becomes
2 nA
SA(f) = nAL Si(f) = 4q2 D(f) (1.13)
Using eq. (1.6), one obtains
kBTn(f)L
D(f) = Re(Y) (1.14)
q nA
Recognizing that Re(Y) = Re(u')qnA/L, one arrives at the generalized
Einstein relationship
kBT (E,f)
D(E,f) = Bn Re(u') (1.15)
q
where u' is the differential mobility. This equation is valid for all
cases in which the field remains uniform throughout the sample. As the
electric field approaches zero, the noise temperature becomes equal to
the lattice temperature, and the Einstein relation reduces to the
familiar form in equilibrium
kBT
D = B (1.16)
q O
If the electron gas is degenerate, as in heavily doped semi
conductors or metals, electronelectron interactions can no longer be
ignored. In this case, electrons cannot be treated as statistically
independent particles, and crosscorrelation terms must be included in
the spectrum. Van Vliet and van der Ziel [13] have extended the rela
tion for current spectral density using statistical mechanics and
derived
S = 2 nA D(f)kBT(3 log n) (1.17)
F T
The expression for diffusion in terms of mobility for degenerate semi
conductors then becomes
3 log n_
qD(f) ( a ) = Re(u') (1.18)
EF
Once the sources of noise in semiconductors have been determined,
it is possible to characterize the noise at the terminals of solidstate
devices. Three methods are used in hotelectron problems: the Langevin,
the impedance field, and the transfer impedance. First, in all three
methods the equations describing the device behavior are formulated.
Then each variable involved is set equal to Q = Q0 + AQ exp(jwt) The
zeroorder terms give the dc characteristics. The firstorder terms
give the ac equations.
In the Langevin method [14], the appropriate white noise source is
added to each ac equation. Auxiliary variables are then eliminated to
get a relationship between the ac current AI and the ac field AE.
Writing the solution of AE in terms of the other variables and integrat
ing over the device length, one gets the ac voltage across the termi
nals. By setting the noise sources to zero, the device impedance is
obtained. Conversely, when AI = 0, multiplying by the complex conju
gate AV results in the ac voltage noise around the bias point. An
extensive review of this method was given by Nicolet et al. [15], in
application to singleinjection diodes.
The impedancefield method was developed by Shockley et al. [16]
to describe diffusion noise in devices. In this method the transfer
function between the positiondependent ac current noise sources and the
ac voltage at the terminals is derived. Once the transfer function has
been obtained, the impedance and noise characteristics easily follow. A
general outline of this method applied to a MODFET is reviewed in Chap
ter IV.
When the variables used to describe the ac properties of a device
are written in terms of current AI and electric field AE, the most
general technique for calculating the impedence and noise properties is
the transfer impedance method. Van Vliet et al. [17] developed this
method to describe the noise behavior in spacecharge limitedcurrent
(SCLC) solidstate diodes. It was found that the transfer impedance
method is quite general and encompasses the impedancefield technique.
Its ability was recently utilized by Tehrani et al. [18] in SCLC silicon
carbide devices.
As device dimensions continue to shrink, traditional analytical
methods of characterizing solidstate devices become questionable.
Transient transport effects and boundary conditions will become increas
ingly important in device modeling. Computer methods for obtaining
device noise characteristics are beginning to emerge. In these methods
fewer approximations are made concerning carrier transport phenomena;
consequently, these methods are expected to lead to more accurate
results. In Chapter II the use of the Monte Carlo method to calculate
velocityfluctuation noise is outlined. The technique is then used in
modeling noise behavior of very short GaAs diodes.
1.4. Device Applications of Heterojunctions
In the following, two wellknown examples of devices based on
AlGaAs/GaAs heterojunction operation are discussed.
There are essentially two main parameters that influence the
commonemitter current gain 0 in bipolar transistors; they are the
emitter efficiency y and the base transport factor. The base transport
factor determines how many carriers injected from the emitter into the
base reach the collector before recombination occurs. With the short
base regions achievable with present processing capabilities, recombina
tion becomes negligible. Then the emitter efficiency, caused by back
injection of carriers from the base to the emitter, dominates the cur
rent gain. In homojunction technology, the emitter is heavily doped
with a lightly doped base region to decrease the back injection. How
ever, the high resistance of the lightly doped base severely limits the
high frequency and noise performance of bipolar devices. Doping the
base more heavily would lower the resistance but degrade emitter effi
ciency.
To circumvent these effects, it was proposed by Kroemer [19] that
a heterojunction at the emitterbase junction be used. Using a wide
bandgap emitter would allow the base region to be heavily doped, thus
lowering the base resistance while maintaining high emitter efficiency.
This is the basic premise of the heterojunction bipolar transistor
(HBT). Much research is currently being pursued on this interesting
device topic. Although the technology is available to make hetero
junction bipolar transistors today, the processing of integrated cir
cuits is difficult due to layout and interconnection problems.
Heterojunctions have also improved fieldeffect transistor tech
nology. The SiSiO2 interface has been used to make MOSFETs for
years. However, the interface is often degraded due to surface rough
ness and interface states. The AlGaAs/GaAs heterojunction does not have
these problems if properly grown. The modulationdoped fieldeffect
transistor (MODFET), also known as the high electron mobility transistor
(HEMT), was developed for highspeed applications. In this case the
carrier transport is parallel to the interface.
The MODFET fabrication process begins with a semiinsulating GaAs
substrate on which an undoped buffer layer of GaAs is grown. A doped
AlGaAs layer is then deposited on top of the buffer layer. After ohmic
contacts are defined for the source and drain pads, the AlGaAs layer is
etched down to provide a Schottkytype gate. The depletion region of
the gate is made to overlap the depleted area in the AlGaAs adjacent to
the GaAs/AlGaAs interface. Careful control of the gate to interface
spacing determines the threshold voltage of the FET structure. A
typical MODFET conduction band diagram showing the overlapping depletion
regions is shown in Figure 1.3.
The MODFET has shown excellent gain and noise figure character
istics at high frequencies and will probably exceed conventional MESFET
capabilities into the millimeterwave region. In Chapter IV the de, ac
and noise properties of the MODFET channel are derived and experi
mentally verified.
AIGaAs
I
I
I
i
Doped ,< 
GaAs
UndoEF
Undoped
Fig. 1.3. MODFET conduction band diagram.
q4b
CHAPTER II
MONTE CARLO MODELING OF HOT ELECTRON TRANSPORT
The semiclassical Boltzmann transport equation (BTE) describes the
evolution of the distribution function in phase space. Once the distri
bution function is known, the pertinent transport parameters can be
obtained by taking the appropriate moments of this function. Solutions
of the integrodifferential Boltzmann equation can be difficult to
obtain analytically. In seeking solutions in the hotelectron regime,
drastic approximations have to be made for analytical results.
Monte Carlo techniques were first devised as a computational tool
for calculating difficult integral expressions. The general principles
have been applied to the solution of differential equations and many
other problems in the applied sciences. In this chapter the method of
Monte Carlo simulation of electron transport properties in GaAs is
examined. The method is very versatile since steadystate as well as
transient phenomena can be simulated in situations near as well as far
from equilibrium. A main disadvantage of the method is, however, that
it requires large amounts of computer time to obtain sufficient statis
tical accuracy. Therefore, the Monte Carlo technique is not always the
most efficient means of investigating a problem.
First, a description of the physical modeling of transport in
semiconductors and an explanation of how the Monte Carlo methods are
used to describe stochastic processes will be given. Subsequently, the
band structure of GaAs in space and the electronphonon scattering
band structure of GaAs in k space and the electronphonon scattering
mechanisms are reviewed. The sensitivity of the diffusionfield and
velocityfield characteristics on the rL intervalleyscattering
coupling constant is examined. The proper FL coupling constant is
found from fitting the calculated diffusionfield characteristics to
recently measured data on GaAs obtained from noise measurements.
Next, the techniques used to obtain the velocity fluctuation
spectrum from a velocity time series are described, and results obtained
for GaAs under highfield conditions are presented.
As the device length shortens, it is expected that transport
behavior becomes more dependent on the imposed boundary conditions. To
study the effects of boundary conditions, the Monte Carlo program was
modified in such a way that the active device length and initial elec
tron velocity could be adjusted. The effects of the length and boundary
conditions on the calculated velocity fluctuation spectrum are examined
and compared with bulk behavior.
2.1. Description of Physical Model
The Monte Carlo method can be applied to many physical systems
whose parameters are governed by probability distributions. The ability
to map simple pseudorandom distributions, available in most computers,
into more complex ones is very powerful. The mapping process begins by
equating the areas under the different distribution functions. Solving
the equations allows one to obtain the physical variable of interest
from the known, computergenerated distribution. In the example given
by Boardman [20], p(r) and p(p) are the respective probability densi
ties, where r is associated with the pseudorandom computer distribution
and $ is the physical quantity to be obtained from the mapping.
Equating the cumulative distributions
Sr
f p(4')d4' = f p(r')dr' (2.1)
0 0
and using a uniform distribution for p(r) = 1,
r = I p(')d$' (2.2)
0
Evaluating the integral of eq. 2.2, one obtains 0 in terms of r. In the
following Monte Carlo program, this method of obtaining random variables
is used to generate freeflight times, choose between scattering mechan
isms, and select the final kspace position after scattering. In
addition, the energy of the electrons injected into the active device
region is calculated using random numbers.
The program to be described is built upon the Fortran version
outlined by Boardman [20]. The original Boardman program only allowed
for a central valley and one type of satellite valley in the energy
wavevector dispersion relation E(k) for electrons in the conduction
band. Originally, it was believed that the ordering in energy of the
conductionband valleys was r X L for GaAs in increasing order of
electron energy. For this reason the original version included only
the r and X valleys, since the L valley population in this model could
be neglected. More recently, it was discovered that the ordering of the
valleys is r L X [21]. The program was rewritten to include all
three valleys in the appropriate order. The values for intervalley
scattering coupling constants and energy offsets between valleys were
taken from Pozhela and Reklaitis [22]. Figure 2.1 shows the energy
wavevector relationship for the GaAs conduction band. Each valley is
taken to be parabolic.
E(k)
0.52eV
Fig. 2.1. Energywavevector relation for GaAs.
[I11] 10003 o100o k
Electron motion is most easily described in k space. In simple
semiconductors the electrons are regarded as free particles with an
effective mass m* of the appropriate valley. The electron energy is
then given by
2+2
E(k) = (2.3)
2m
where k is the reduced wavevector of the electron.
To simulate electron motion, one first generates a random number
based on the scattering rates of the valley occupied by the electron.
This number is then used to calculate the flight time between colli
sions. The wavevector k changes during the collision freeflight time
in proportion to the applied electric field. If the electric field is
in the z direction, only the k component of wavevector increases
z
linearly with time during the free flight as indicated by
kzi() = kf + t (2.4)
The subscript i refers to the initial state before scattering and f
denotes the final state after the previous scattering event. During the
free flight the wavevector component kp perpendicular to the z axis does
not change. Upon scattering, however, both the kz and kp components may
change and obtain values determined by the particular scattering
mechanism involved. This process is shown pictorially in Fig. 2.2.
Having outlined the band structure and concept of electron motion
in GaAs, the electronphonon scattering mechanisms will be reviewed.
The program accounts for the following electronphonon interactions:
acoustic phonon (intravalley), polar optical phonon (intravalley),
S
I /
I
/t
kz
elec. field E
Electron motion in k space.
Electron motion in k space.
do
Fig. 2.2.
equivalent intervalley (L L or X X), and nonequivalent intervalley
(L X, etc.). Intravalley scattering means that the initial and final
states before and after scattering are in the same valley, and inter
valley scattering means that the two states are in different valleys.
Both types of intervalley scattering are via optical phonons since
acoustic and polar optical phonon scattering does not allow for large
changes in the wavevector.
For all intravalley scattering processes involving optical phonon
fields, the energy state after scattering must satisfy the relation
E(k') = E(k) Aw (2.5)
where m is the radian frequency of the lattice vibration, the plus sign
indicates absorption, and the minus sign emission of an optical
phonon. Acoustic phonon scattering, however, is treated as an elastic
+ +
process and therefore E(k') = E(k)
The energy of the electron is measured with respect to the minimum
of the valley it occupies. Therefore, when a nonequivalent intervalley
transition occurs, the energy difference between valleys must be
accounted for. When the transition is such that the final state is in a
valley with a minimum higher in energy than the initial valley, the
energy of the electron becomes
E(k') = E(k) I m A (2.6)
where A is the energy difference between the valley minima and mu is the
optical phonon energy. If the transition is to a valley with a lower
minima, then the energy difference is added to the final energy.
Each process that can scatter an electron at the end of a colli
+ +
sion free flight is characterized by a transition rate S (k,k'), which
is equal to the probability per unit time that an electron is scattered
+ +
from the state k to a state k'. The subscript n denotes a particular
+ +
scattering process. The scattering rate X (k) from state k to any
n
+ th
state k' due to the nth process is found by integrating over all possi
ble final states k'. Hence
+ +'+ +
X (k) = \S (k,k')dk' (2.7)
n n
The total scattering rate is then found from a summation over all
processes
N
X(k) = I \ (k) (2.8)
n=1
n=l
The scattering rates for each process are listed in Table 2.1, where the
rates are presented in terms of energy rather than in terms of wave
vector. The values of the physical constants used to fit experimental
data are listed in Table 2.2.
The scattering rates for the central (r) valley are depicted in
Fig. 2.3. Polar optical phonon absorption dominates over acoustic
phonon scattering at low energy levels. Once the electron energy
exceeds an energy of 0.035 eV, it becomes possible to emit a polar
optical phonon. The scattering rates for polar optical phonons become
smaller as the electron energy increases due to the coulombic nature of
the interaction. The dominance of polar optical phonon scattering in
GaAs is responsible for the polar runaway phenomena to be discussed
later. When the electron energy approaches the energy of the satel
lite valleys, intervalley transfer plays a role in the total scattering
rates. As seen in Fig. 2.3, polar optical phonons still dominate up to
TABLE 2.1
Scattering Rates
Mechanism Scattering Rates X(E)
3/2 2 1/2
Acoustic phonon (2m ) kT D E
(intravalley) a
4TAps 24
Polar optical Yq m w 1/2 +E1/2
phonon (log
(intravalley) 4K <0(2E)/2 0 E /E1'/2
Y = N absorption
= N0 + 1 emission
1
NO = [exp(%0/kBT) 1]
3/2
Equivalent intervalley (Gl)m 2D2 E1/2Y
(satellite satellite) e
/2 iipwe3
eG = 3 for X valley
G = 4 for L valley
Y = Ne absorption
= (N +1) emission
N = [exp( e/k T) 111
.3/ 2 1/2
Nonequivalent Gm D E'/2 Y
intervalley 3
n TPn G = 1 for r valley
G = 3 for X valley
G = 4 for L valley
Y = Nn absorption
= Nn+1 emission
Nn = [exp(%n/k T) 1]
p = material density
s = sound velocity
N = phonon occupation number
KO = permitivity of free space
S,e = high frequency, static dielectric constants
Da = acoustic deformation potential
De = equivalent intervalley coupling constant
Dij = nonequivalent intervalley coupling constant
E = energy of initial state
E' = energy of final state
G = parameter associated with the symmetry of the valleys
TABLE 2.2
Physical Constants
1. Material density (g/cm3) 5.37
2. Sound velocity (105cm/s) 5.22
3. Highfrequency dielectric constant 10.82
4. Lowfrequency dielectric constant 12.53
5. Optical phonon frequency (1013rad/sec) 5.37
6. Intervalley phonon frequency (1013rad/sec) 4.54
7. Acoustic deformation potential (eV) 7.0
8. Intervalley coupling constants (109eV/cm):
r L 0.325
rX 1.0
LX 0.1
LL 0.5
S X 1.0
9. Energy separation between valleys (eV):
r L .33
r X .52
10. Effective mass (m*/mo):
r .063
L .17
X .58
14 '1 k4
10' T=300 K x
(S') x c
13
10 
PO
012
L
AC
10
10 10
0 .2 .4 .6 .8 1.0
Energy (eV)
Fig. 2.3. Scattering rates for r valley.
about 0.5 eV. Above 0.5 eV the r to Xvalley transitions dominate.
The scattering rates in the L and X valleys are shown in Figs. 2.4 and
2.5, respectively.
As mentioned previously, one must generate collision freeflight
times from the scattering rates. If p(t) is the probability per unit
time that an electron has a flight, of duration t, and subsequently
scatters, then the flight time t is found from
t
r = f p(t')dt' (2.9)
0
where r is the uniformly distributed random number. As shown in
Boardman [20], eq. (2.9) can be written as
t
r = 1 exp{ f X(k)dt'} (2.10)
0
The integral cannot be evaluated analytically and thus requires numeri
cal evaluation. This involves a significant amount of computer CPU time
for each flight. To circumvent this problem, Boardman et al. [23]
developed the concept of virtual scatterings. In a virtual scattering
event, the state of the electron does not change. The scattering rate
for the electron, including virtual scattering, becomes equal to
GAMMA = XT(k) = X(k) + V(k) (2.11)
where GAMMA is the nomenclature used by Boardman, and V is the virtual
scattering rate. Since GAMMA is a constant, the integral in eq. (2.10)
can be easily evaluated, and one finds for t
t = log(r)/GAMMA (2.12)
The value of GAMMA is usually taken to be equal to the maximum electron
real scattering rate evaluated over the possible electron energy range.
i14
1014 
ST=300 K
AC
(SI) 'TC
1013
PO
EQ
rX
IOI
10 10
[_10  '
0 .2 .4 .6 .8 1.0
Energy (eV)
Scattering rates for L valley.
Fig. 2.4.
(s1) EQ
013 PO
___po
PO
1012 AC
r^
L
1010
0 .2 .4 .6 .8 1.0
Energy (eV)
Fig. 2.5. Scattering rates for X valley.
At the end of a collision free flight, the electron is scattered
by one of the real processes or by a virtual process. The scattering
rate for each process is evaluated and normalized to unity by dividing
by GAMMA. A random number is generated uniformly between 0 and 1 to
choose between scattering mechanisms. If a virtual scattering mechanism
is chosen, the wavevector k remains the same and the program proceeds to
generate a new flight time. However, if a real scattering mechanism is
chosen, a new wavevector must be stochastically determined before a new
flight time can be produced. A block diagram of the electron transport
simulation is outlined in Appendix A.
At the end of a sufficiently long time interval to obtain conver
gence, the quantities of interest such as the mean velocity, the diffu
sion coefficient, etc., are evaluated and outputed. More details can be
found in Boardman [20].
2.2. Determination of FL Intervalley Coupling Constant and its
Relation to the DiffusionField Characteristics
Recently obtained experimental data [24,25] of the fielddependent
diffusion coefficient of GaAs at room temperature can be combined with
Monte Carlo calculations to investigate transport parameters. Input
variables to the program such as effective masses, valley separations in
energy, deformation potentials, etc., can be adjusted to give a proper
fit to experimental data. The most uncertain of these transport param
eters are the intervalley scattering coupling constants, which represent
the strength of the electron transfer mechanism resulting in intervalley
transitions via a deformation potential. According to recent literature
[5], these constants have values that range from 108 to 109 eV/cm for
GaAs.
At low electric fields the intervalley transitions play a minor
role in transport since most of the electrons stay in the central (r)
valley. With larger fields present, the electrons move to higher ener
gies in the conduction band, thus enabling the intervalley transfer from
r to L valleys. This intervalley transfer to the lowmobility satellite
valleys is the cause of the negative differential mobility regime in
bulk GaAs.
Shown in Fig. 2.6 is the variation of the velocityfield charac
teristic as a function of the rL intervalley coupling constant. The
minimum and maximum values of the constant were obtained from other
Monte Carlo simulations of bulk GaAs [20,22]. It can be seen that the
maximum sensitivity of the velocityfield relationship lies in the 3 to
5 kV/cm range. This is the same range where Gunn oscillations of bulk
GaAs make accurate measurements almost impossible.
Figure 2.7 unveils the effect of changing the rL intervalley
coupling constant on the fielddependent, lowfrequency diffusion
coefficient of GaAs. Measurements, utilizing noise techniques, of the
diffusion coefficient of GaAs done by Bareikis et al. [24] and Gasquet
et al. [25] are included in the figure. Accurate noise measurements can
be done below 3 kV/cm without the problems associated with Gunn domain
formation. There is a strong peak in the diffusion coefficient around 3
kV/cm that is very sensitive (much more so than the velocity) to the rL
intervalley coupling strength. The value of 0.325 x 109 eV/cm gives the
best fit to the experiments.
The increase in the diffusion coefficient with electric field in
bulk GaAs has been attributed to two mechanisms, intervalley transfer
2.0
V
(107cm/s) 
1.0
0
0
Fig. 2.6.
2 4 6 E(kV/cm)8
Velocityfield characteristics of bulk GaAs
vs4, rL intervalley coupling constant.
5
4
3
2
O
Fig. 2.7.
D
Do
2 4 6 8
E(kV/cm)
Normalized diffusionfield characteristics of
bulk GaAs vs. FL intervalley coupling constant.
Measured values indicated by circles and squares
are from Bareikis et al. [24] and Gasquet et al.
[25], respectively.
and polar runaway. In both of these mechanisms the intervalley coupling
strength has an effect on the diffusion coefficient.
Electrons undergoing intervalley transfer experience large changes
in velocity due to the randomizing nature of the intervalley process and
due to the change in effective mass. These large fluctuations in
velocity might be responsible for the increased diffusion coefficient.
As can be seen in Fig. 2.8, the population of the L valley starts to
increase, due to intervalley transfer, in the same field range where the
diffusion coefficient increases.
Polar runaway is a term used to describe a process attributed to
semiconductors in which polar optical phonon scattering dominates [26].
Since polar optical phonons emphasize smallangle scattering, the elec
trons heat up fast. In addition, the process becomes less efficient
with increasing electron energy. Consequently, fast electrons move even
faster. This causes the velocity distribution to widen, corresponding
to an increase in Av Assuming that the correlation time of velocity
fluctuations is not significantly altered in this process, the low
frequency diffusion coefficient (plateau value) will increase with elec
tric field (see eq. 1.9).
Figure 2.9 shows the results of a Monte Carlo simulation of the
frequency dependence of the diffusion spectrum for different electric
field strengths. In this simulation all electrons were confined to the
central valley as no intervalley transfer was allowed. As Fig. 2.3
indicates, polar optical phonon scattering dominates in this case. The
spectral shape remains essentially unaltered while the plateau value
increases with field strength. This confirms the fact that the velocity
distribution spreads (Av2 increases), while the time dependence of the
100
OOr
pop(%)
80
60
40
20
O
0o
0
2 4 6 8
E (kV/cm)
Fig. 2.8. Average L and X valley population of bulk GaAs
vs. rL intervalley coupling constant.
105
D(f)
(cm2/s)
102
10 L
10
Fig. 2.9.
f(GHz)
Diffusion spectra vs. electric field for electrons
in r valley (polar runaway).
velocity autocorrelation function experiences little change. Sec
tion 2.3 will explain how the calculations of the spectrum are made.
It is expected that when all three valleys are included in the
simulation, a decrease of the rL intervalley coupling constant will
cause the electrons in the central valley to have a smaller probability
of scattering to the L valley. Since the probability of intervalley
scattering is smaller, the electrons spend more time, on the average, in
the central valley where polar optical phonon scattering dominates. If
polar runaway is the cause of increased diffusion, then decreasing the
rL intervalley coupling strengths would result in an increase in the
diffusion coefficient as indicated in Fig. 2.7. However, when one
observes that the decreased coupling constant gives rise to a larger
average electron population in the L valleys, the effects of polar run
away are not as clear. As a result of the decreased coupling strength,
once the electron scatters to the L valley there is a small probability
of returning to the central valley. This is the cause of the average L
valley population increase.
2.3. Monte Carlo Spectral Analysis of Velocity Fluctuations
Since the diffusion coefficient is a frequencydependent parameter
as shown by eq. (1.9), it is interesting to investigate the velocity
fluctuation spectrum to gain insight into transport processes. In this
section the process of how to generate the velocity time series in the
Monte Carlo program is described, including the selection of sampling
rate. The definition of spectral density is reviewed, and the calcula
tion of the spectrum from discretely sampled signals is explained.
Results for bulk GaAs are presented, and they indicate the effects of a
dominant scattering process on the spectrum.
The Nyquist sampling theorem states that the sampling rate
f = 1/At must be at least twice the highest frequency component of the
s
waveform being sampled. Here, At is the length of time between samples.
If the sampling rate is less than twice the highest frequency present,
then aliasing will occur. In other words, the sampling time At used in
the Monte Carlo program must be much smaller than the intercollision
time. After considering the scattering rates in GaAs (see Figs. 2.3 
2.5), the sampling time of 5 x 10 15sec was chosen, reducing aliasing to
a negligible effect. However, after observing the spectral character
14
istics of GaAs, the sampling time was increased to 2.5 x 10 sec to
reduce the timeseries length.
To obtain the velocity time series, the program first generates
the freeflight time of the electron between scattering events, as
described in section 2.1. The collision freeflight time is then broken
up into smaller subflights of duration At. The subflights are treated
similarly to the virtual scattering process. At the end of each
subflight, the velocity of the electron is calculated from the knowledge
of the wavevector component kz in the direction of the applied field.
In principle one could also sample the velocity components normal to the
field direction. This would allow characterization of the transverse
diffusion coefficients. The velocity of the electron in one dimension
is given by
+ %k
1 aE(k) z
vz = k W (2.13)
z m
These subflights continue until the end of the free flight or the time
window length T (= NAt) has been reached.
s
The actual algorithm of the process is described below and out
lined in the flow chart of Appendix B. Since scattering events occur,
in general, between the sampling times, the computer must keep track of
when it last took a sample. The program labels the time passed since
the last sample was taken as TIMEX. When the freeflight time between
scattering events (TIME) is generated, the algorithm first determines
whether the generated time (TIME) plus the time remaining since the last
sample (TIMEX) is long enough to reach the next sampling time. If it is
not, the flight time is added to TIMEX to become the new TIMEX, and the
program proceeds to the next scattering selection without taking a
sample of the velocity. When the generated freeflight time plus TIMEX
is greater than the sampling time At, the program advances to the next
sampling time by the time labeled TIMEV = At TIMEX. The velocity of
the electron is then calculated using eq. (2.13) and stored in the time
series. Figure 2.10 pictorially represents the relationship of the
times in the sampling process.
The number of samples in the time series that can be taken in the
remaining freeflight time is designated by NN. The value of NN is an
integer number and is determined by dividing the remaining flight time
by At and truncating any fractional quantity. When NN is equal to zero,
the remaining time is not long enough to reach the next sampling time,
so the remaining time becomes TIMEX. The program then goes to the next
scattering selection. For NN greater than zero, the algorithm loops
through NN subflights, generating additional timeseries data. At the
end of the NN loop the new value of TIMEX is updated by the remaining
time before the loop minus the loop time. The program then proceeds to
the next scattering selection.
TIME
1
H
TIME
I
NNxAt
55
TIMEX
4
scattering
event
scattering
event
Fig. 2.10. Representation of times used to generate
velocity time series.
, !
I
IC
I
The number of samples N in the time series is monitored at each
sampling time. When the number reaches T /At, the program stops
s
sampling and proceeds to the FFT algorithm.
The methods of calculating the velocityfluctuation spectral
density from the time series will now be reviewed. Here we follow the
outline given by Tehrani [27]. Let a random signal x(t) in the interval
0 < t < Ts be defined in terms of a Fourier series
CO j2rfKt
x(t) = I aKe (2.14)
K=o
K
where fK = (K = 0, 1;, 2 ...) and aK is the Fourier coefficient of
s
x(t) at fK. The discrete Fourier coefficients aK are defined as
N1
aK =NA At x(nAt) ej2KfnAt (2.15)
n=0
where x(nAt) is the sampled time data, Af is the frequency spacing
defined as Af = 1/NAt, and K denotes the frequency component f = K
k NAt
= KAf. The Fourier component XK of x(t), having a frequency fK, is
given by
j2rf t j2tf t
XK = aKe + aKe (2.16)
The ensemble average of XK is found to be
2 i
XK = 2 aKaK (2.17)
since the Fourier coefficients have an arbitrary phase resulting in
2 2
a = aK = 0, and since for a real signal x(t)
a_K = aK (2.18)
Writing the ensemble average in terms of the discrete Fourier transform
results in
N1 N1
XK = 2 aKaK = 2 1 1 (At)2 x(n)x(m) exp(j2fK(mn)At) .
K K K (NAt)2 n=0 m=0 K
(2.19)
For a stationary process and setting s = mn, the two summations can be
decoupled as shown in [27], resulting in
12 2 ___
X =N At x(n)x(n+s) exp(j2fK sAt) (2.20)
s=M
Since x(n)x(n+s) = 0 for s>M and if N>X>, the limits of summation change
such that
2 2
X = NA At x(n)x(n+s) exp(j2nfKsAt) (2.21)
s=m
The spectral density of x(t) can be defined by the discretized Wiener
Khintchine theorem,
S (fK) = 2 1 At x(n)x(n+s) exp(j2fK sAt) (2.22)
s=0
which is essentially the Fourier transform of the discretized auto
correlation function x(n)x(n+s) of the process x(t). Since Af is the
frequency interval between adjacent fk's, the spectral density can be
written as
2 *
XK 2 aKaK
Sx(f ) = K (2.23)
Equations (2.22) and (2.23) show that there are two routes that
can be taken to calculate the spectral density of a signal. One can
either generate the discretized autocorrelation function from the time
series and then compute the Fourier transform, or one can calculate the
Fourier coefficients aK directly by fast Fourier transform (FFT) and
average each spectrum. The latter method was preferred in this case
since it involves calling only one of the standard library subroutines
available on the Harris 800 computer system. Computing the spectrum
from the autocorrelation function would require calling two subroutines,
one to compute the autocorrelation function and another to take the
Fourier transform. This method would be more time consuming.
The Harris subroutine used is named FFTRC. This subroutine com
putes the fast Fourier transform of a real valued sequence. Timeseries
lengths up to N = 20,000 have been transformed by the subroutine very
quickly and without any problems. All variables transferred to the sub
routine must be defined in single precision. Since many of the varia
bles used in the Monte Carlo program are implemented in double precision
for accuracy, the data to be passed on are stored in singleprecision
variables before calling the subroutine.
The computer program proceeds as follows. First, the program com
putes the average value of the velocity time series and subtracts it
from the data. This new time series is then stored in the single
precision real vector A(G). A running average of the average velocity
is made for each time series. The vector A(G) is the data to be trans
formed by the FFT. The output is expressed in the complex vector X(G).
The FFT algorithm computes the following summation:
NI JT
X() = s A(J) ej27KJ/N (2.24)
s J=0
K
Therefore, the spectral density at frequency f of the time series is
K s X K
found by multiplying X(T) by its complex conjugate X (T), then multi
s S
plying the result by 2T/N2 and averaging over many time series. The
velocityfluctuation spectral density is then calculated from
2T
S () = X(~ )X (* ] (2.25)
s N s s
In order to obtain the diffusion coefficient, one simply divides
the velocityfluctuation spectral density by 4, as indicated in eq.
(1.9). The average diffusion spectrum is stored in the vector AV(G).
The constant 2Ts/4N2 is lumped into a single number to improve computa
tion time. Although the first N/2 + 1 coefficients of the Fourier
transform are available, only the first 400 frequencies were outputed
for our purposes. The time window Ts was chosen to be 100 ps, giving a
frequency resolution Af(= 1/T ) of 10 GHz.
We are now in a position to calculate the velocityfluctuation
spectral density of bulk GaAs at room temperature (300 K). Calculations
of the lowfrequency diffusion coefficient D(E,0) as a function of elec
tric field, using the timeofflight and spectraldensity methods, show
good agreement.
At each field strength the spectra have been normalized to their
lowfrequency plateau level so that the relative spectral shapes can be
compared, as shown in Figs. 2.11 and 2.12. For the lowfield range,
10
D(f) T=300 K
D (0)
D(O)
"I 3 kV/cm '\
5 kV/cm
.0 1 I I ,
10 102 f(GHz) 103
Fig. 2.11. Normalized diffusion coefficient spectral
density for 3 and 5 kV/cm.
I0
ST=300 K
D(f) 
D(O)
7 kV/cm
 10 kV/cm
S. 20 kV/cm
.0 1 i 1 i l I I ,
10 102 f(GHz) 103
Fig. 2.12. Normalized diffusion coefficient spectral
density for 7, 10, and 20 kV/cm.
between 1 and 3 kV/cm, the spectrum has a Lorentzian shape with a half
power bandwidth on the order of 500 GHz. At 5 kV/cm a peak shows up in
the spectral characteristics around 300 GHz. As indicated in Fig. 2.12,
the peak in the spectra moves gradually to higher frequencies as the
field increases. The presence of a peak in the spectrum was first
observed in Monte Carlo simulations of InP and explained by Hill et al.
[28]. The peak has also been observed in simulations of GaAs done by
Fauquembergue et al. [29] and Grondin et al. [30].
The observed peak in the spectrum has been attributed [28] to a
strong scattering cycle of electrons from a satellite valley back to the
central valley with a velocity in the direction negative to the motion
under the applied field. The electrons are then accelerated under the
applied field through the valley minima and travel almost ballistically
until they scatter once again to the satellite valley. This scattering
process is pictorially shown in Fig. 2.13. The ballistic motion is
characterized by an almost sawtoothlike velocity waveform as indicated
in Fig. 2.14, where the associated Fourier spectrum of this waveform is
also presented. As the electric field increases, the motion through the
central valley becomes faster, causing the peak in the spectrum to move
higher in frequency as displayed in Fig. 2.12.
2.4. Position Monitoring and Boundary Conditions
in Monte Carlo Programming
It is expected that as device dimensions shrink to submicron
levels, the boundary conditions and the length of the active region will
have significant effects on the chargetransport characteristics of
these devices. To investigate these phenomena, the Monte Carlo program
 ~
E(k)
Scattering process responsible for spectral peak.
Fig. 2.13.
sA
2 4 6
Fig. 2.14.
8 10
Dominant scattering process: a) velocity waveform;
b) spectral characteristics.
Y(E)
was rewritten to monitor the electron position in real space. Electrons
can be injected from a cathode at some welldefined position (z = 0) and
removed at the anode after drifting some fixed length L. The injected
electron energy at the cathode can be tailored to any probability dis
tribution desired.
The program still contains the velocity timeseries capability so
that the velocity spectral density information of shortchannel devices
is obtained. This method is physically satisfying, in that it is analo
gous to the measurement of noise in actual devices. The program does
not, however, account for noise associated with the injection process.
Also, the effects of space charge are neglected by assuming a uniform
electric field throughout the active device region.
An outline of the algorithm to monitor the position of the elec
tron in real space is given below. This will include a description of
the back scattering of electrons from the active region into the
cathode. Although the backscattered electrons do not contribute to the
dc characteristics, they do have an effect on the velocity spectrum.
Thus, it is important to include the back scattering in the simulation.
When an electron reaches the anode or returns to the cathode, a new
electron is injected. The injected electron velocity is derived from a
modified Maxwellian velocity distribution [31].
After explaining how the program functions, results for different
length GaAs devices will be presented. The experimental data of Andrian
[32] on 1.1 pm GaAs diodes will then be compared with the simulations.
2.4.1. Program algorithm
The equations describing electron motion, in one direction, during
the freeflight time between collisions can be expressed most simply in
terms of a classical velocity
V = v + at (2.26)
z 0
and position
1 2
z = + vt + at (2.27)
where t is time, v0 is the initial velocity, z0 is the initial position,
and the acceleration a is given by qlEl/m*. At the beginning of the
simulation, an electron is injected with some positive initial velocity,
derived from the modified Maxwellian velocity distribution, at the posi
tion z = 0. The Monte Carlo program then generates the collision free
flight time according to the standard procedure as outlined in sec
tion 2.1.
The first thing that has to be determined at the beginning of each
collision free flight is whether at any time during the flight the elec
tron ever goes back into the cathode. In other words, a check is made
to see if the electron position becomes negative (z < 0). The position
of the electron obeys a quadratic equation (2.27) in time and depends on
the direction of the initial velocity. If the initial velocity is
positive, then the electron position only increases with time. However,
when the initial velocity is negative, the position first decreases and
then increases given that the flight time is long enough. Therefore, to
determine if the electron ever goes back into the cathode, the program
only has to check those flights in which the initial velocity is less
than zero.
With a negative initial velocity, the program first calculates the
time in which the electron velocity vz is equal to zero. This time,
labeled TMIN, corresponds to the minimum position of the particle during
the flight if the flight time is at least that long. The value of TMIN
is determined by
v
TMIN = a (2.28)
A check is made to determine whether the flight time (TIME) is less than
TMIN. If it is less than TMIN, the minimum position is determined by
the flight time. The minimum position ZMIN, during the electron flight,
is now calculated from (2.27) using TMIN or TIME, whichever is smaller.
When ZMIN is less than zero (ZMIN < 0), the program calculates at
what time during the flight ZMIN = 0. Solving (2.27) for the time when
z = 0, one obtains
v [v0 2az0]1/2
t = (2.29)
a
since the smaller of the two roots of the quadratic equation corresponds
to the first time the electron crosses the boundary. This new calcu
lated time (2.29) is the actual flight time of the electron in the
active region. Now the velocity time series can be updated for this
flight.
A flag in the program, labeled IFLAG, is set equal to 1 each time
that ZMIN is found to be less than zero. This flag causes the program
to reinject another electron from the cathode after the time series has
been updated. The value of 1 is assigned to the flag to keep track of
how many electrons scatter back into the cathode during the simulation.
The number of backscattered electrons is stored in the variable ILTO.
When the initial velocity is positive or the minimum position is
found to be still in the active region, the program determines the final
position from (2.27), using the freeflight time. This final position
is compared to device length L to see if the electron was collected by
the anode. When z < L, the time series is updated during the free
flight and then proceeds to the next scattering selection. If the final
position is greater than L, the exact time that the electron passed the
anode boundary must be calculated. This new flight time is
2 1/2
v + [v2 2a(z0L)]1/2
t = (2.30)
a
and is used in updating the time series. As before, the flag is set
when the electron reaches the anode, indicating injection of another
electron from the cathode. This time, though, the flag value is equal
to 2 so as not to interfere with keeping track of the number of back
scattering. A block diagram of the positionmonitoring algorithm is
given in Appendix C.
During the process of evaluating the velocity time series, the
counter N is monitored. When the appropriate time window length
T (= NAt) is reached, the program proceeds with the spectral density
calculations as outlined in section 2.3.
Every time the electron arrives at the anode or scatters back to
the cathode, a new, centralvalley electron is injected into the active
region to maintain constant space charge. The electrons injected in the
z direction obey a modified Maxwellian distribution [31]. The proba
bility that an electron is emitted with a velocity between vz and vz
+ Av is
z
1 2 1 2
 my T my
AP(v ) = exp( 2 z ) 2 (2.31)
where T is the lattice temperature. The modified Maxwellian distribu
tion can be generated from a uniform distribution with the techniques of
section 2.1. Here the variable EZ, associated with the energy due to
the vz component, is randomly generated from r, using
EZ = kBT log(r) (2.32)
The average value of this distribution is kBT.
The wavevector (or velocity) component is found directly from EZ
by
1/2
k = (2m E (2.33)
z 2
Only the positive root need be taken since negative values signify elec
tron motion back into the cathode.
The emitted electrons also have velocity components perpendicular
to the field direction. These electrons obey a Maxwellian distribution
in each direction. The energy distribution associated with the perpen
dicular velocity components can be expressed in a form similar to
(2.31), so the wavevector component kp is determined in much the same
way using (2.32) and (2.33). The actual magnitudes of the two perpen
dicular components kx and ky are not needed since the program simulation
considers the xy dimensions to be infinite. However, the two compo
nents can be determined by generating a random phase angle between 0 and
2 once the magnitude of the kp wavevector is known. This would need to
be done to gain information on the transverse diffusion coefficient.
1
Both perpendicular components have an average energy equal to k T.
2 B
Therefore, the total average energy of injected electrons into the
active region is 2 kBT [31].
An integer variable named IINJ is used to keep track of how many
electrons are injected from the cathode. The difference between IINJ
and the number of electrons returning to the cathode ILTO gives the
number of electrons traversing the entire length of the active region to
the anode. A copy of the entire program is listed in Appendix D.
2.4.2. Simulation results
The program was run for GaAs at room temperature (300 K) with the
same intervalley coupling parameters used to fit the bulk experiments in
section 2.2. The length of the active region was varied from 0.25 to
4 um, and the values at the electric field were chosen between 1 and 3
kV/cm. A number of interesting effects on transport behavior versus
device length can be observed in the results presented in Table 2.3.
First, when the electrons are injected into the active region,
they are rapidly accelerated by the electric field to very high veloci
ties. If the device length is short, the electron velocity has insuffi
cient time to relax to the steadystate bulk velocity before being
collected by the anode. This transient velocity phenomenon is termed
velocity overshoot [33]. As shown in Table 2.3, for a given field value
as the device length is reduced, the average velocity throughout the
active region increases. This increase in average velocity signifies
the occurrence of velocity overshoot near the cathode.
Second, the amount of intervalley transfer, from r to L valleys,
is reduced in short devices. In Fig. 2.15 the average fraction of time
spent in the L valley during the simulation of 3 kV/cm as a function of
TABLE 2.3
Simulation Results for ShortLength GaAs
Length Electric Field v Sv(0)/4
(um) (kV/cm) (107 cm/s) (cm2/s)
.25 1 1.59 113
2 2.16 124
3 2.55 128
.5 1 1.33 162
2 1.96 172
3 2.45 181
1.0 1 1.16 208
2 1.84 219
3 2.46 258
2.0 1 1.07 238
2 1.80 270
3 2.40 384
3.0 1 1.03 248
2 1.80 293
3 2.21 425
4.0 1 1.02 260
2 1.77 300
3 2.09 500
Bulk 1 0.96 250
2 1.73 326
3 1.82 508
active region length is presented and compared to the bulk GaAs value.
Injected electrons with high energies, near the rL energy offset of
0.33 eV, can undergo intervalley transfer after traveling over short
distances. However, since most of the injected electrons have low
initial energy (Eav = 2 kBT), they must travel farther into the active
region before gaining sufficient energy for rL intervalley transitions
to commence [34]. Therefore, as the length of the active region
decreases, a higher fraction of the injected electrons are swept out at
the anode before transferring to the L valley.
The velocity fluctuation spectrum is also determined as a function
of device length. Table 2.3 gives the lowfrequency plateau value of
the spectrum for each length and field. As mentioned before, shot noise
behavior associated with the random injection of electrons from the
cathode is not taken into account. Often in real device structures the
injection noise levels are reduced due to spacecharge suppression [35].
Only the effects of the velocity spectrum in the active region are
calculated and presented.
The spectrum plateau levels are normalized to the lowfield value
at each length and depicted in Fig. 2.16. This figure shows the rela
tive increase in the noise characteristics versus field strength for
various device lengths, as well as for a bulk device. Also included in
the figure are the normalized diffusion coefficient measurements of a
+ +
n nn GaAs mesa structure, performed by Andrian [32]. The donor con
centration in the 1.1 um active nlayer was 1015 cm3. The Monte Carlo
data support the measurements, showing that the relative increase in the
currentnoise spectral density measured in short GaAs regions is not as
large as for bulk GaAs.
30
20
I0
10
0
2
3
4
L(Um)
Fig. 2.15..
Fraction of time electrons spent in L valleys
at 3 kV/cm as a function of active region length.
bulk
0
0
0
0 ,
0
T=300 K
/
/
 bulk,4pm
 2pm
I pm
.5pm
o I.Ijpm meas.
) I 2 3 E(kV/cm)
Fig. 2.16.
Relative increase in GaAs lowfrequency
velocity spectral density versus electric
field for several active region lengths.
Sv
Svo
I
I
62
Again, the mechanisms causing the increase in the lowfrequency
velocity spectrum with increasing field can be due to intervalley
transfer or polar runaway. In very short active regions, the lower
noise produced could be attributed to the decrease in intervalley
transfer. It can also be argued that for the short regions there is
insufficient scattering before the electrons are removed at the anode,
so the spread in the velocity distribution attributed to polar runaway
cannot be attained.
CHAPTER III
EXPERIMENTS ON AlGaAs/GaAs INTERFACES
In this chapter, we will discuss the experiments which are done to
determine the de, ac and noise properties in the hotelectron regime of
AlGaAs/GaAs heterojunction interfaces. Two device structures with dif
ferent characteristics, such as length, fraction of aluminum content,
sheetcarrier concentration, etc., are used in the experiments.
In this dissertation the emphasis is on noise characterization, so
first a review of the methods of measuring the device ac noise tempera
ture T is given. With the use of noise temperature data, the diffusion
coefficient can be determined as a function of electric field for trans
port parallel to the AlGaAs/GaAs interface. The differences in the
experimental results between the two interfaces are examined and com
pared to bulk GaAs behavior.
3.1. Description of Device Structures
The devices used in the experiments were modulationdoped field
effect transistor structures without the gate metalization. The advan
tage of these structures was that they were readily suitable for high
frequency measurements and devicemounting procedures.
A diagram of the gateless MODFET device structure is presented in
Fig. 3.1. These MODFET structures were fabricated by Dr. Morkoc of the
University of Illinois. The devices are grown on a semiinsulating (SI)
GaAs substrate starting with an undoped GaAs buffer layer. This buffer
layer is grown to smooth out any defect properties associated with the
GaAs
Diagram of AlGaAs/GaAs structures.
Fig. 3.1.
surface of the SI substrate and to provide a relatively pure GaAs region
for electron transport. It also provides a means for obtaining an
uninterrupted growth cycle at the AlGaAs/GaAs interface. An undoped
AlGaAs spacer layer is incorporated to provide greater separation
between the parent donor atoms and the free electrons at the
interface. A silicondoped AlGaAs layer is grown next. The doping
level, as well as the spacer layer thickness and conductionband dif
ference, determine the quasitwodimensional electron sheet carrier con
centration ns at the interface. A thin, highly doped n+ cap layer of
GaAs is grown to facilitate ohmic contact formation. Source and drain
regions are defined by photolithography, and gold is deposited for con
tact pads.
Details of each structure are found in Table 3.1 for the two
wafers numbered 1483 and 1885. Also included are the geometrical width
w and length L of each device. The contact resistance Rc associated
with the ohmic contact to the active device region is given because of
its importance in interpretation of the experimental data.
3.2. Noise Temperature Measurement Setup and Experimental Procedures
Since our investigation is concerned with the determination of the
fielddependent diffusion coefficient, the noise component associated
with the velocity fluctuations needs to be measured. To measure the
velocity fluctuation spectrum, experiments have to be done at frequen
cies high enough to avoid the gr and 1/f noise contributions. At high
frequencies it becomes difficult to measure the actual terminal voltages
and/or currents, so it is easier to measure the available power from the
network. This available noise power is related to the noise temperature
Tn as described in Chapter I.
TABLE 3.1
Device Structure Parameters
#1483 #1885
1. Cap layer thickness (A) 50 50
2. Cap layer doping level (cm3) 2.5x1018 2.5x1018
3. Doped A1GaAs thickness (A) 600 350
4. AlGaAs doping level (cm3) 2.5x1018 2.5x1018
5. Aluminum mole fraction x .28 .23
6. Undoped spacer thickness (A) 30 15
7. GaAs buffer thickness (pm) 1 2
8. Contact resistance Rc (n) 8 11
9. Width w (pm) 145 75
10. Length L (pm) 4 2,8
The technique used to measure the noise temperature of the device
under test (DUT) is similar to the method developed by Gasquet et
al. [36]. The main advantage of this particular scheme is that it
allows measurement of the noise temperature without the need to match
the DUT to the characteristic impedance (50 n) at each bias and fre
quency. Not only can matching be a tedious task, but the stub tuners
used in matching can have different resistive losses depending on the
particular stub settings. The variability of these losses degrades
measurement accuracy.
The experimental setup to measure DUT noise temperature over the
frequency range 500 MHz to 12 GHz is depicted in Fig. 3.2. This broad
range of frequency coverage is obtained by using circulators with octave
bandwidths and broadband lownoise amplifiers. Frequency selection is
attained with the spectrum analyzer (HP8559A), which is capable of
receiving input frequencies from 10 MHz to 22 GHz. The noise source
(HP346B) is also broadband covering 10 MHz to 18 GHz with an effective
noise ratio of 15 dB (T = 9170 K). Amplification and power detection
are performed at the intermediate frequency (21.4 MHz) of the spectrum
analyzer.
The experimental procedure consists of four measurements to deter
mine the noise temperature Tn of the DUT. These measurements also make
available the power reflection coefficient rl2 at port 2 of the circu
lator, the noise temperature Ta of the measuring system, and the gain
bandwidth product GB of the system.
The first measurement M1 consists of a reference temperature sig
nal Tc flowing from port 1 in the preferred direction to port 2, where a
shortcircuit termination is placed. The reference signal is totally
SWITCH DET
SYNCH
Fig. 3.2. Noise temperature measurement setup.
reflected by this short circuit and proceeds to the amplifier stages at
port 3. The amplifier system provides proper impedance termination at
port 3. The measured power is then proportional to
M1 = k GB(T + T ) (3.1)
B a c
where kB is the Boltzmann constant. The second measurement M2 is essen
tially the same except that the reference temperature is now Th, where
Th > Tc, giving
M2 = kBGB(Ta + T) (3.2)
Now, the short circuit at port 2 is replaced by the DUT which is
biased to the dc voltage of interest. The reflection coefficient
between the DUT and circulator is defined in terms of the DUT impedance
ZDUT and the characteristic impedance Z00 by the relation
ZDUT 00 (33)
F = (3.3)
ZDUT + Z00
Since the DUT has, in general, an impedance different from the charac
teristic impedance of the circulator, the available noise power from the
DUT is reduced by the factor (1 Irl ) This property is exploited in
the next two measurements.
The noise source temperature is again set to Tc for the third
measurement. Since the lownoise amplifier sees a constant impedance
looking into port 3 regardless of the impedance change at port 2, the
system noise temperature Ta remains the same. Also, the reference tem
perature is partially reflected at port 2 because of the mismatch, so
the measured power is proportional to
M3 = k GB(Ta + T (1 rl 2)+ Tcir2) (3.4)
A final reading is done with the reference temperature at Th providing
M4 = kBGB(Ta + Tn(1 rl2) + Thl 2) (3.5)
Manipulating these four measurements, one obtains the unknown device
noise temperature
T (M3 Ml) + T (M2 M4)
T c (3.6)
n M3 M1 + M2 M4
and the power reflection coefficient as
S12 M4 M3
I =M2 Ml (3.7)
It should be pointed out that the parameters determined by (3.6) and
(3.7) are associated with the network connected at the reference plane
of port 2 of the circulator. If there are no resistive or radiative
losses between the actual DUT and the circulator port, then T is the
actual device noise temperature. Any known losses between the DUT and
circulator can be easily corrected. Also, the loss between ports 1 and
2 only affect the values of the reference temperatures Tc and Th. Cor
respondingly, the loss between ports 2 and 3 affect only the system
noise temperature Ta.
The system noise temperature can be determined from
MITh M2T
T c (3.8)
a M2 Ml
and the gain bandwidth of the system is given by
71
1 M2 Ml
GB =  (3.9)
kB Th Tc
B h c
Another advantage of this measuring technique is that it allows
the determination of device noise temperature using either a continuous
or pulse bias. When the device is biased into the hotelectron regime,
significant Joule heating of the lattice occurs. A pulse bias at low
duty cycle is then required to keep the average power dissipation to a
minimum. A pindiode RF switch is used in the IF section to make sure
that only the noise power produced during the bias time is detected.
The actual length of bias pulse time is determined by the time constants
associated with the bias tee network and the DUT impedance. The bias
tee provides the necessary dc and RF isolations.
3.3. Experimental Results
Both device structures used for the measurements were mounted in a
500 microstrip transmission line test fixture. The microstrip line was
made from a 25mil.thick alumina substrate and attached to SMA coaxial
connectors. All measurements were made with a covered mount to keep the
device in the dark.
The first device to be measured was wafer #1483. Measurements
were done with a pulse bias time of 4 ms and a 3% duty cycle at room
temperature (300 K). The dc currentvoltage characteristic for this
4 um heterostructure is depicted in Figure 3.3. The lowfield
equilibrium resistance is found to be 31 ohms. As can be seen in the
figure, the IV characteristic begins to deviate from Ohm's law around
600 mV. This nonlinear behavior is an indication of hotelectron
effects in the channel. Changing the polarity of the voltage had no
I(A)
100 V(V)
Currentvoltage characteristic of #1483.
101
Fig. 3.3.
effect on the IV relationship, indicating that the contacts were indeed
ohmic and had no rectifying properties.
From the IV characteristic one can obtain the dc mobility as a
function of electric field. The electric field in the channel is
assumed to be uniform and found by taking the voltage drop across the
active region and dividing it by the length. The definition of dc
mobility is given by
v(E)
(E) = (3.10)
Figure 3.4 shows the dc mobility as a function of electric field, nor
malized to its equilibrium value U0 after correcting for a contact
resistance of 8 ohms. The advantage of normalization is that a better
comparison with other heterostructures can be made since it removes any
differences in the actual thermal equilibrium mobility values. Also
included in the figure are other published experimental results on
similar heterostructures [3739]. Since no other means of obtaining the
contact resistance for this structure was available, the value of 8Q was
derived by lining up the data with the previously published results in
the figure.
Next the noise temperature of #1483 was measured in the frequency
range 500 MHz to 1 GHz. No frequency dependence in the noise tempera
ture as a function of bias was observed, indicating the absence of any
1/f or gr noise components. Therefore, the noise temperature was
associated with velocity fluctuations. The actual noise temperature of
the active region measured between .5 and 1 GHz and as a function of
electric field is displayed in Figure 3.5. It can be seen here how
quickly the noise temperature increases for fields far from equilibrium.
3
1 2
Mo
100
F310
Fig. 3.4.
23
2 6 104
E (V/cm)
Normalized dc mobility as a function of electric
field. Circles indicate ungated MODFET #1483,
inverted triangles indicate Tsubaki et al. [37],
rightsideup triangles indicate Masselink et al.
[38], and squares indicate van Welzenis et al. [39].
0 0 0 O
V
02
3
2 1483 T=300 K
Tn(K)
1030
7 o
0
0
3 0 0 0
2
102
102 2 3 7103 2 3 7104
E (V/cm)
Noise temperature vs. electric field for #1483.
Fig. 3.5.
The differential mobility is obtained from a measurement of the
admittance of the DUT. First, the conductance or Re(Y) can be calcu
lated from the derivative of the IV characteristic. This method works
well as long as the Re(Y) .is not frequency dependent in the range of
interest. The admittance can also be measured with the use of an S
parameter test set (HP 8410). The drawback of this measurement tech
nique is that it has to be done with a continuous bias. Finally, the
magnitude of the reflection coefficient is determined during the noise
temperature measurement (3.7). When the susceptive elements are negli
gible, the Re(Y) can be found from the magnitude of the reflection
coefficient. Whenever possible, all three methods are combined. For
#1483 all methods showed good agreement in the Re(Y) since the parasitic
susceptive elements were small compared to the channel conductance.
The diffusion coefficient in the active region of #1483 can now be
determined from the generalized Einstein relation
k T (E)
D(E) = n Re(P') (3.11)
q
Normalizing the diffusion coefficient to the equilibrium value DO, one
obtains from the measured data
D(E) T (E) Re(Y)
E (3.12)
DO T Re(Y0)
where T is the lattice temperature and YO is the equilibrium admit
tance. A constant carrier concentration in the channel is assumed. The
normalized diffusion coefficient as a function of electric field for
#1483 is presented in Figure 3.6. Also included in the figure are the
7
D0
Do
100
7
3
1o'
102
2 3
25 7104
E (V/cm)
Fig. 3.6. Normalized diffusion coefficient for #1483 as a func
tion of electric field. Circles indicate ungated
MODFET, squares indicate Ruch and Kino [10], inverted
triangles indicate Gasquet et al. [25], and the right
sideup triangles indicate Bareikis et al. [24].
3 A
A
0 A V
o 0 0400ooP
I I I I I I
2
I
normalized diffusion coefficient measurements of bulk GaAs [10,24,25].
It can be seen that the diffusion coefficient of the heterostructure
does not increase as significantly as the bulk GaAs results.
Device structure #1885 was measured next. One of the main advan
tages of this structure was that on the same wafer differentlength
devices were available for measurement. In this way the contact resis
tance could be determined more accurately. The lowfield ohmic resis
tance as a function of device length is plotted in Fig. 3.7. The
circles indicate the data obtained by using the wafer probe station and
the triangles indicate the actual wire bonded values. The discrepancy
between the two different measurements is attributed to the contacting
problems associated with the wafer probes. However, both sets of data
extrapolate to the same value at the origin (L = 0), giving a contact
resistance of 11. The currentvoltage characteristics of #1885 for
lengths of 2 and 8 um are shown in Figs. 3.8 and 3.9, respectively.
Both devices showed hotelectron effects at high bias.
Having the IV characteristics and the contact resistance, it is
again possible to find the de mobility versus electric field. The nor
malized dc mobility of #1885 is given in Figure 3.10. There was no
difference found in the dc mobility for the two lengths measured. Also,
the dc mobility behavior of #1885 is very similar to that of #1483.
Obtaining the Re(Y) from measurements at high frequencies was
difficult for this structure. The measured admittance data at low
frequencies (f < 500 MHz) showed reasonable behavior, but became quite
difficult to model at high frequencies (f > 1 GHz). Because of the
large wafer size associated with this structure, very long bonding wires
had to be used to make connections to the actual DUT. These long
160
R(n )
120
80
40
0
10
Resistance vs. length for #1885 to determine
contact resistance. Circles indicate wafer
probed values whereas triangles indicate wire
bond values.
2 4 6 8
L(pm)
Fig. 3.7.
 T=300 K
1885 2pum
ooo"
//
0
0/
,o
O/
O
0/
/
I I I I 11111 I
I( 1
SI I 11111I
I I I I 1 I
V (V)
100
Currentvoltage characteristic of #1885
(L = 2 nm).
I(A)
106
6o
Fig. 3.8.
01
1885 8pm
I(A) T=300K /
5 /
/0
0/
o 0
/
0'
10c 3 ' 1 I I I Il ll I l lIlln
10' i00 V(V) 101
Fig. 3.9. Currentvoltage characteristic of #1885
(L = 8 um).
.I
E(V/cm)
Fig. 3.10. Normalized de mobility vs. electric field
for #1885.
103
1885 T=300 K
I r I I 1111 I I f 1 111
102
bonding wires made it difficult to obtain a good highfrequency ground,
and the large wafer may introduce other unaccounted for parasitics. As
a result, the Re(Y) of the DUT was determined from dI/dV, and we assumed
that it is frequency independent in the range of interest.
The noise temperature of #1885 was measured from .5 to 12 GHz.
After accounting for all known losses between the circulator and DUT,
the noise temperature showed a slight decrease for frequencies greater
than 2 GHz (see Fig. 3.11). This slight decrease was associated with
losses in the parasitics that could not be well defined. At other bias
values, similar behavior of Tn versus frequency was observed.
Taking the data between .5 and 1 GHz to be accurate, the noise
temperature of the active region versus electric field for #1885 is
depicted in Fig. 3.12. Again it can be seen that the noise temperature
increases with electric field, but not as rapidly as that of #1483.
The normalized diffusion coefficient as a function of electric
field is given in Fig. 3.13 for both 2 and 8 Pm structures. The Re(Y)
is determined from dI/dV. Both device lengths show the same decrease in
the diffusion coefficient with field, indicating that there is no
noticeable dependence on device length in this range.
3.4. Discussion of Results
In this section we will discuss the hotelectron behavior of the
AlGaAs/GaAs interfaces and compare the results to those of bulk GaAs.
Only qualitative explanations can be given due to the lack of suffi
ciently developed analytical models for parallel transport in the
heterojunctions or the availability of complex Monte Carlo programs.
Clearly, the first observation that can be made is the similarity
in the dc characteristics of different heterointerfaces. The dc
T=300 K
1885 2 pm
V=600mV
0 0
00O
i I I Iiiilll
108
I I I I 1 1 111
f (Hz)
Noise temperature vs. frequency for #1885.
Tn(K)
1010
I I I 1 I i I I I . .
Fig. 3.11.
Tn(K) T=300K
1885
103 
0 
0
o00 
00
000
0 0 0 0000 0
10 2 1' i,,
102 103 E(V/cm) 1(
Fig. 3.12. Noise temperature vs. electric field for #1885.
E (V/cm)
Fig. 3.13.
Normalized diffusion coefficient vs. electric
field for #1885. Squares and circles indicate
2 and 8 um data, respectively.
D
Do
.I
102
1885 T=300 K
7 0 0 1 000 000
0000
Q 0
I o
I I I I I It I I I I I !
mobility of both #1483 and #1885 decreases with increasing field
strength in the hotelectron regime, which also agrees with the
previously published results.
However, there does seem to be differences in the noise behavior
between different heterointerface compositions. For device #1483 the
diffusion coefficient (or velocity fluctuation spectral density) remains
nearly constant with increasing field, whereas #1885 shows a slight
decrease in the hotelectron regime. This difference in diffusion
coefficients results mainly from the lower noise temperature measured in
#1885.
Both heterojunction interface structures show a clearly different
noise behavior than bulk GaAs. In bulk, the increase in the diffusion
coefficient with field was attributed to polar runaway and intervalley
transfer (sec. 2.2). A decrease in the importance of one or both of
these mechanisms in the heterointerfaces might be responsible for the
observed D(E) dependence. Differences due to device length are not
suspected since no noticeable length dependence of D(E) is observed in
the diffusion coefficient measurements of the 2 and 8 um channels of
#1885 presented in Fig. 3.13.
Yokoyama and Hess [40] calculate the twodimensional scattering
rates for electrons in the first five subbands of a quasitriangular
potential well at the AlGaAs/GaAs interface. Their results show lower
scattering rates for polar optical phonons as compared to the rates for
bulk GaAs at room temperature. Since the polaroptical phonon scatter
ing rates are reduced in twodimensional systems, the effects of the
polar runaway phenomenon may be less significant.
The second contributing factor to the diffusion coefficient in
bulk GaAs is intervalley transfer. In the case of interfaces, this
process is difficult to model because of the realspacecharge transfer
from the GaAs to the AlGaAs. An electron might cross the energy barrier
at the interface before gaining enough energy to undergo intervalley
transfer. Indeed, the conductionband difference at the interface is
smaller for device #1885 by 40 meV, which is the structure that shows
the decreasing diffusion coefficient with electric field. Since there
is a lack of experimental data on the AlGaAs system, it is very diffi
cult to model or otherwise evaluate hotelectron properties in this
region or its effect on realspacecharge transfer.
The only analytical support for the diffusion coefficient behavior
in the heterointerfaces is from the Monte Carlo model of van Rheenen and
Bosman [41]. In their model they use an infinitely high, square poten
tial well to simulate the twodimensional transport behavior of a two
valley GaAs channel. The diffusion coefficient in this simulation shows
a decrease with increasing electric field as opposed to the increase in
diffusion observed in their bulk simulation. Therefore, the decrease in
the diffusion coefficient with increasing field in the heterostructures
is possibly linked to the twodimensionality of the electron gas.
CHAPTER IV
THE DC, AC AND NOISE CHARACTERIZATION OF THE
ALGAAS/GAAS MODFET CHANNEL
In recent years much attention has been paid to AlGaAs/GaAs
modulationdoped fieldeffect transistors (MODFETs) for potential use in
highspeed logic circuits. The very high transconductance gm and high
cutoff frequencies fT also make them of interest for lownoise micro
wave amplification. Excellent articles by Solomon and Morkoc [2] and
Drummond et al. [42] have been written reviewing the characteristics of
these new transistors.
Since the first report of the noise figure of these devices in the
microwave frequency range, an interest in the noise behavior has devel
oped. The noise figures of various MODFETs have been reported recently
and show improvements over conventional GaAs MESFETs of comparable gate
lengths.
Up to now only noisefigure measurements have been reported in the
microwave frequency range. In this chapter we will not focus on the
noise figure, but instead report on the noise characteristics of the FET
channel. At intermediate frequencies (0.5 < f < 10 GHz) the channel
noise is due to fluctuations of the freecarrier velocity and is the
major contributor to the overall device noise. Measurements of the
thermal noise (i.e. velocity fluctuation noise) as a function of bias
are discussed in this chapter.
In section 4.1 we will outline the theory of the impedance field
method, which is used to obtain the ac and noise properties of the
MODFET channel. Section 4.2 explains the methods of obtaining the
chargevoltage relationship for the devices used in our experiments.
Some of the methods of obtaining the chargevoltage relationship involve
only lowbias data while other methods involve highbias data. Compar
ing the results of the different methods can help determine the presence
or absence of realspacecharge transfer. Section 4.3 describes the
MODFET structures to be considered. Measurement procedures are dis
cussed in section 4.4. The experimental results will then be presented
and discussed in section 4.5, followed by conclusions in section 4.6.
4.1. Impedance Field Modeling
In this section the procedure for obtaining the positiondependent
ac channel voltage in terms of the Green's function for a MODFET channel
is discussed. It will be shown how the Green's function is related to
the impedance field [16,17]. Once the impedance field is obtained, the
ac and noise properties can be easily calculated. Van Vliet [43] and
Nougier [44] have outlined this method for the case of the junction
fieldeffect transistor (JFET). In this chapter the impedance field for
a MODFET is calculated using the proper transport equations, and the ac
and noise properties of the MODFET are derived.
4.1.1. Review of impedance field method
The procedure begins by considering the device transport equa
tions. Smallsignal variations around the steadystate values of all of
the variables are introduced. Having done this, and neglecting second
order and higher terms, the ac and dc equations can be separated. The
dc equation can be used to obtain the steadystate currentvoltage
characteristic of the device.
The ac equation has some interesting properties. Generally, the
ac equation involves the positiondependent steadystate parameters.
This equation can be written as follows:
HAV(x) = AI(x)
(4.1)
where H is a linear operator and AV and AI are the smallsignal ac
channel voltage and current respectively. By letting z(x,x',f) be the
Green's function of H, i.e.
A
Hz(x,x',f) = S(xx'), (4.2)
where 6(xx') is the Dirac delta function and f denotes frequency, the
total ac voltage at position x can be calculated from
L
AVT(x) = f z(x,x',f)AI(x')dx' (4.3)
0
The integration is taken over the entire length of the device. The
total ac voltage at x given by eq. (4.3) is simply the summation over
all of the smallsignal current sources properly weighted by the terms
z(x,x',f).dx'. Depending on the charge transport mechanisms involved,
some of the terms z(x,x',f) might be zero. This point will be
illustrated when the equations are developed for the MODFET. Of course,
one is mainly interested in the values of the smallsignal quantities at
the device terminals, since these can be measured. The total small
signal voltage at the device terminal (x = L) is
L
AVT(L) = f z(L,x',f)AI(x')dx' (4.4)
0
The onedimensional device shown in Fig. 4.1 is grounded at x = 0
and has an arbitrary steadystate dc bias applied at x = L. Suppose a
current of value AI(x) is introduced at position x + Ax and extracted at
x. AI(x) will produce an opencircuit voltage response AV(L) at the
terminal (x = L). If the ac impedance between position x and ground
(x = 0) is given by Z(x), then the voltage response at L can be ex
pressed as
AI(x)
10o
AV(
x=O AX x=L
Fig. 4.1. A small signal current AI(x) produces a voltage
response AV(L) at the terminal x = L. Steady
dc current 10 and voltage V0 are indicated.
Vo
;L)
AV(L) = [Z(x+Ax,f) Z(x,f)]AI(x) (4.5)
For small Ax
Z(x+Ax,f) = Z(x,f) + dZ(x,f) Ax (4.6)
dx
and one obtains
AV(L) = VZ(x,f)AI(x)Ax (4.7)
The term VZ(x,f) is known as the impedance field and was first intro
duced by Shockley et al. [16]. The impedance field relates the ac
current inside the device to the voltage response at the terminals. In
the limit Ax + dx the total ac voltage at L is given by
L
AVT(L) = f VZ(x',f)AI(x')dx' (4.8)
0
Comparing eqs. (4.8) and (4.4), one sees that
VZ(x',f) = z(L,x',f) (4.9)
To find the total device impedance at the terminals, one makes use of
the fact that the ac current is conserved. Then AI(x) = AI, and con
sequently
AV (L) L L
Z(L) = AI = J VZ(x',f)dx' = L z(L,x',f)dx' (4.10)
0 0
Using the impedance field, one can express the noise in terms of
spectral densities. The spectral density of the opencircuit voltage
fluctuations measured at the terminals is given by [16]
L
SAV = I f K(x')IVZ(x',f)j2dx'dydz (4.11)
z y 0
where K(x') is the spectral density of the current fluctuations in
volume dx'dydz, and the integration is carried out over the entire
volume of the device. Using eq. (4.11), the spectral density of 1/f
noise, generationrecombination (gr) noise, and velocityfluctuation
noise can be calculated if the proper source term K(x) is inserted.
In this chapter the focus is on velocityfluctuation noise only.
It has been shown in Chapter I that the spectral density of velocity
fluctuations is directly related to the diffusion coefficient D(E),
which may be field dependent. Taking this effect into account, Nougier
shows that the spectral density of the voltage fluctuations due to
velocity fluctuations in a onedimensional treatment becomes [44]
L
SAV = f A(x')4q2D[E(x')]n(x')IVZ(x',f)12dx' (4.12)
0
where A(x') is the crosssectional area, n(x') is the carrier density
and q is the electron charge. The equivalent currentnoise spectral
density SI can be calculated from
S AV (4.13)
AI JZ(L)12
4.1.2. Application of the impedance field method to the MODFET
In the following a onedimensional, collisiondominated transport
model is used to obtain simple analytical expressions for the impedance
and noise of the device. The advantage of this approach is that it pro
vides physical insight into the ac and noise behavior of the channel.
Clearly this treatment breaks down for very short submicron devices
(L < .5 um) since in that case the usual concept of mobility and diffu
sion needs to be generalized (see Constant [45]). Assuming no leakage
current through the gate and neglecting both diffusion and displacement
currents, the charge transport equation is given by
I = qwns V(x)]v[E(x)], (4.14)
where w is the gate width and v[E(x)] is the fielddependent carrier
velocity. The sign convention is as follows. The source is chosen at
x = 0, the drain at x = L > 0, q > 0, V(x) > 0, E(x) < 0, v[E(x)] > 0,
and I > 0. The twodimensional sheet carrier concentration ns[V(x)] is
assumed to be only a function of the local electrical potential under
the gate. Velocity saturation will cause accumulation and/or depletion
of the sheet carrier concentration in the highfield region under the
gate, making eq. (4.14) invalid. For this reason the model we employ
only describes the linear and triode regimes of the currentvoltage
characteristic. At T = 300 K the velocityfield characteristic of the
twodimensional electron gas is assumed to be identical to the one of
bulk GaAs [46]. Consequently,
v(E) = EE (4.15)
c
where U0 is the lowfield mobility taken to be 8000 cm2/V sec at room
temperature, and the critical field Ec is chosen to be 11.4 kV/cm. When
the electric field exceeds 3.5 kV/cm, the model [eq. (4.14)] no longer
holds due to the saturation effects mentioned above. The large critical
field Ec is chosen to provide the proper curvature of the velocity
characteristic at low electric fields. Using the bulk GaAs velocity
field characteristic as a first attempt is justified since in the high
field region under the gate the reduced sheet carrier concentration
