Title Page
 Table of Contents
 Monte carlo modeling of hot-electron...
 Experiments on algaas/gaas...
 The dc, ac and noise characterization...
 Conclusions and suggestions for...
 Biographical sketch

Group Title: Hot-electron noise in gallium arsenide/aluminum gallium arsenide heterojunction interfaces
Title: Hot-electron noise in gallium arsenidealuminum gallium arsenide heterojunction interfaces
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00082293/00001
 Material Information
Title: Hot-electron noise in gallium arsenidealuminum gallium arsenide heterojunction interfaces
Physical Description: v, 144 leaves : ill. ; 28 cm.
Language: English
Creator: Whiteside, Christopher Francis, 1959-
Publication Date: 1987
Subject: Gallium arsenide semiconductors -- Noise   ( lcsh )
Hot carriers   ( lcsh )
Electrical Engineering thesis Ph. D
Dissertations, Academic -- Electrical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis (Ph. D.)--University of Florida, 1987.
Bibliography: Bibliography: leaves 141-143.
Statement of Responsibility: by Christopher Francis Whiteside.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00082293
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 000947009
oclc - 16865103
notis - AEQ8989

Table of Contents
    Title Page
        Page i
        Page ii
    Table of Contents
        Page iii
        Page iv
        Page v
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        Page 11
        Page 12
        Page 13
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        Page 15
        Page 16
        Page 17
    Monte carlo modeling of hot-electron transport
        Page 18
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    Experiments on algaas/gaas interfaces
        Page 63
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    The dc, ac and noise characterization of the algaas/gaas modfet channel
        Page 89
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    Conclusions and suggestions for further research
        Page 118
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    Biographical sketch
        Page 144
        Page 145
        Page 146
Full Text









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.


ACKNOWLEDGMENTS ................................................... i

ABSTRACT ........................................... ................ V


I. INTRODUCTION ................ ...... ..................... 1

1.1. Band Structure of AlGaAs/GaAs ........................ 2
1.2. Hot-Electron Effects ................................. 4
1.3. Noise Characterization of Hot-Electron
Phenomena ........................................ 7
1.4. Device Applications of Heterojunctions ............... 14

2.1. Description of Physical Model ........................ 19
2.2. Determination of r-L Intervalley Coupling
Constant and its Relation to the
Diffusion-Field 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

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

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. Charge-Voltage Dependence ............................ 99
4.3. Device Description ................................... 103
4.4. Measurement Procedure ............................... 103
4.5. Results and Discussion ............................... 107
4.6. Conclusions ..........................................115



5.1. Monte Carlo Transport Modeling ....................... 118
5.2. Experimental Characterization of
Heterostructures ................................... 119
5.3. MODFET Characterization .............................. 120



B. VELOCITY TIME SERIES ALGORITHM ............................... 122

C. POSITION-MONITORING 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



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 hot-electron

behavior of electron transport parallel to the interface has arisen.

In this dissertation the charge-transport 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 field-dependent diffusion coefficient.

This method provides a better theoretical value of the r-L 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.


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


In order to take full advantage of these new heterojunctions, more

information on the charge-transport properties must be attained. In

this dissertation the charge-transport 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 field-dependent diffusion coefficient. This

method provides a better theoretical value of the r-L 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 modulation-doping

techniques. The second section of this chapter outlines some of the

effects of hot electron behavior on charge transport, such as real-

space-charge transfer vs. intervalley transfer, in providing negative

differential mobility. The length of the active device region can have

a significant effect on the charge-transport 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 field-effect 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 high-quality interfaces between GaAs and AlGaAs

with an insignificant concentration of interface states.

The most important device parameter of interest in AlxGal-xAs 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)

E (x) = 1.900 + 0.125x + 0.143x2(0.45 < x < 1.0) (l.lb)

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 X-valley having the lowest energy.

The L-valley lies between the r- and X-valleys for mole fractions larger

than 0.45.

The sum of the valence- and conduction-band discontinuities must

equal the energy bandgap difference between the GaAs and AlGaAs.

Originally, it was believed that the conduction-band discontinuity was

0.85 E (x) [5]. However, more recent measurements [6] have shown that

sixty-five percent of the bandgap difference lies in the conduction band

for x ( 0.45. Then the conduction-band discontinuity follows from

AE = 0.81x (eV) for x .0.45 (1.2)

From measurements of the valence-band discontinuity as a function of

mole fraction, it was determined that the maximum conduction-band

discontinuity lies in the vicinity of x = 0.45 [7]. A further increase

of the mole fraction results in a decrease in the conduction-band dis-

continuity with a corresponding larger increase in the valence-band


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 n-type) 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

n-type and the GaAs is slightly p-type.

The modulation-doping 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. Hot-Electron 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 hot-electron transport. To improve the modeling and

performance of electron devices, high-field transport properties need a

more thorough understanding.

An interesting hot-electron effect is the occurrence of a negative

differential mobility regime in bulk GaAs. This phenomenon, commonly







AlGaAs/GaAs heterojunction at equilibrium.





Fig. 1.1.

known as the Gunn effect [9], is due to the transfer of electrons from

the high-mobility central r-valley to the low-mobility 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 low-mobility 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 conduction-band

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 real-space-charge transfer, which stands for the

following physical process. Electrons in the high-mobility GaAs gain

energy as they drift under an applied electric field parallel to the

interface. When the energy becomes comparable to the conduction-band

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, real-space-charge 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 field-dependent diffusion coefficient in

conjunction with velocity-field measurements should provide information

on the hot-electron 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 Hot-Electron Phenomena

Noise measurements are used to provide information on charge-

transport processes in semiconductors. In this section the methods of

characterizing hot-electron effects by noise measurements are reviewed.

There are basically three types of noise in semiconductor devices:

1/f, generation-recombination (g-r) and velocity-fluctuation. At low

frequencies 1/f and g-r 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. Generation-recombination 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 high-mobility semiconductors is very

small, the velocity-fluctuation spectrum extends to very high frequen-

cies. The emphasis in this dissertation is placed on the hot-electron

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 one-port network biased by an arbitrary dc voltage V0

with a dc current 10 flowing through it. The small-signal Thevenin and

Norton equivalent circuits, evaluated around the bias point, are

depicted in Figure 1.2. In general, the small-signal 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)

where f denotes frequency. A similar relation,
A = f S(Vf)df (1.4)
oI 0 AIO (1.4)


Fig. 1.2.

Thevenin and Norton small-signal
equivalent circuits.


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


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 one-port network whether

it is linear or nonlinear. The following discussion is restricted to

homogeneous semiconductor samples for which a one-dimensional 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

well-known Einstein formula for diffusion

Ax2 = 2Dt (1.10)

for sufficiently long t.

Consider a semiconductor sample of length L and cross-sectional

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)

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

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)

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

D =-- B- (1.16)
q O

If the electron gas is degenerate, as in heavily doped semi-

conductors or metals, electron-electron interactions can no longer be

ignored. In this case, electrons cannot be treated as statistically

independent particles, and cross-correlation 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


S = 2 nA D(f)kBT(3 log n) (1.17)

The expression for diffusion in terms of mobility for degenerate semi-

conductors then becomes

3 log n_
qD(f) ( a ) = Re(u') (1.18)

Once the sources of noise in semiconductors have been determined,

it is possible to characterize the noise at the terminals of solid-state

devices. Three methods are used in hot-electron 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

zero-order terms give the dc characteristics. The first-order 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 single-injection diodes.

The impedance-field method was developed by Shockley et al. [16]

to describe diffusion noise in devices. In this method the transfer

function between the position-dependent 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 space-charge limited-current

(SCLC) solid-state diodes. It was found that the transfer impedance

method is quite general and encompasses the impedance-field 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 solid-state 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

velocity-fluctuation 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 well-known examples of devices based on

AlGaAs/GaAs heterojunction operation are discussed.

There are essentially two main parameters that influence the

common-emitter 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-


To circumvent these effects, it was proposed by Kroemer [19] that

a heterojunction at the emitter-base 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 field-effect transistor tech-

nology. The Si-SiO2 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 modulation-doped field-effect

transistor (MODFET), also known as the high electron mobility transistor

(HEMT), was developed for high-speed applications. In this case the

carrier transport is parallel to the interface.

The MODFET fabrication process begins with a semi-insulating 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 Schottky-type 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 millimeter-wave region. In Chapter IV the de, ac

and noise properties of the MODFET channel are derived and experi-

mentally verified.




Doped ,-<- -




Fig. 1.3. MODFET conduction band diagram.



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 integro-differential Boltzmann equation can be difficult to

obtain analytically. In seeking solutions in the hot-electron 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 steady-state 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 electron-phonon scattering
band structure of GaAs in k space and the electron-phonon scattering

mechanisms are reviewed. The sensitivity of the diffusion-field and

velocity-field characteristics on the r-L intervalley-scattering

coupling constant is examined. The proper F-L coupling constant is

found from fitting the calculated diffusion-field 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 high-field 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 pseudo-random 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, computer-generated 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 pseudo-random computer distribution

and $ is the physical quantity to be obtained from the mapping.

Equating the cumulative distributions

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)

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 free-flight times, choose between scattering mechan-

isms, and select the final k-space 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

conduction-band 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.



Fig. 2.1. Energy-wavevector 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

E(k) = (2.3)

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 free-flight time

in proportion to the applied electric field. If the electric field is

in the -z direction, only the k component of wavevector increases
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 electron-phonon scattering mechanisms will be reviewed.

The program accounts for the following electron-phonon interactions:

acoustic phonon (intravalley), polar optical phonon (intravalley),


I /




elec. field E

Electron motion in k space.
Electron motion in k space.


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
+ 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

X(k) = I \ (k) (2.8)

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

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
NO = [exp(%0/kBT) 1]-

Equivalent intervalley (G-l)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) 11-1

.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

Physical Constants

1. Material density (g/cm3) 5.37

2. Sound velocity (105cm/s) 5.22

3. High-frequency dielectric constant 10.82

4. Low-frequency 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
r-X 1.0
L-X 0.1
L-L 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
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 X-valley 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 free-flight

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

r = f p(t')dt' (2.9)

where r is the uniformly distributed random number. As shown in

Boardman [20], eq. (2.9) can be written as

r = 1 exp{- f X(k)dt'} (2.10)

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.

1014 -
ST=300 K

(S-I) 'TC





10 10
[_10 -- '
0 .2 .4 .6 .8 1.0
Energy (eV)

Scattering rates for L valley.

Fig. 2.4.

(s-1) EQ

013 PO


1012 AC


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 F-L Intervalley Coupling Constant and its

Relation to the Diffusion-Field Characteristics

Recently obtained experimental data [24,25] of the field-dependent

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


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 low-mobility satellite

valleys is the cause of the negative differential mobility regime in

bulk GaAs.

Shown in Fig. 2.6 is the variation of the velocity-field charac-

teristic as a function of the r-L 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 velocity-field 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 r-L intervalley

coupling constant on the field-dependent, low-frequency 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 r-L

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

(107cm/s) -



Fig. 2.6.

2 4 6 E(kV/cm)8

Velocity-field characteristics of bulk GaAs
vs4, r-L intervalley coupling constant.






Fig. 2.7.


2 4 6 8

Normalized diffusion-field characteristics of
bulk GaAs vs. F-L 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 small-angle 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






2 4 6 8
E (kV/cm)

Fig. 2.8. Average L and X valley population of bulk GaAs
vs. r-L intervalley coupling constant.



10 -L

Fig. 2.9.


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 r-L 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

r-L 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 frequency-dependent 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
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-
istics of GaAs, the sampling time was increased to 2.5 x 10 sec to

reduce the time-series length.

To obtain the velocity time series, the program first generates

the free-flight time of the electron between scattering events, as

described in section 2.1. The collision free-flight 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.

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 free-flight 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 free-flight 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 free-flight 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 time-series 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.








Fig. 2.10. Representation of times used to generate
velocity time series.

, !




The number of samples N in the time series is monitored at each

sampling time. When the number reaches T /At, the program stops
sampling and proceeds to the FFT algorithm.

The methods of calculating the velocity-fluctuation 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)

where fK = (K = 0, 1;, 2 ...) and aK is the Fourier coefficient of
x(t) at fK. The discrete Fourier coefficients aK are defined as

aK =NA At x(nAt) e-j2KfnAt (2.15)

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 + a-Ke (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

N-1 N-1
XK = 2 aKaK = 2 1 1 (At)2 x(n)x(m) exp(j2fK(m-n)At) .
K K K (NAt)2 n=0 m=0 K

For a stationary process and setting s = m-n, 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)

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)

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)

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. Time-series

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 single-precision

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:

X(-) = s A(J) ej27KJ/N (2.24)
s J=0

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

velocity-fluctuation spectral density is then calculated from

S () = X(~ )X (* ] (2.25)
s N s s

In order to obtain the diffusion coefficient, one simply divides

the velocity-fluctuation 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 velocity-fluctuation

spectral density of bulk GaAs at room temperature (300 K). Calculations

of the low-frequency diffusion coefficient D(E,0) as a function of elec-

tric field, using the time-of-flight and spectral-density methods, show

good agreement.

At each field strength the spectra have been normalized to their

low-frequency plateau level so that the relative spectral shapes can be

compared, as shown in Figs. 2.11 and 2.12. For the low-field range,

D(f) T=300 K

D (0)

"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.

ST=300 K
D(f) -

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 sawtooth-like 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 charge-transport characteristics of

these devices. To investigate these phenomena, the Monte Carlo program

- ~---


Scattering process responsible for spectral peak.

Fig. 2.13.


2 4 6

Fig. 2.14.

8 10

Dominant scattering process: a) velocity waveform;
b) spectral characteristics.


was rewritten to monitor the electron position in real space. Electrons

can be injected from a cathode at some well-defined 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 time-series capability so

that the velocity spectral density information of short-channel 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 back-scattered 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 free-flight 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

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)

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


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 back-scattered 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 free-flight 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(z0-L)]1/2
t = (2.30)

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 position-monitoring 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, central-valley 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

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


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 x-y 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.

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 steady-state 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

Simulation Results for Short-Length 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 r-L 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 r-L 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 low-frequency 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 space-charge 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 low-field 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 -n-n GaAs mesa structure, performed by Andrian [32]. The donor con-

centration in the 1.1 um active n-layer was 1015 cm-3. The Monte Carlo

data support the measurements, showing that the relative increase in the

current-noise spectral density measured in short GaAs regions is not as

large as for bulk GaAs.








Fig. 2.15..

Fraction of time electrons spent in L valleys
at 3 kV/cm as a function of active region length.





-0 ,


T=300 K

---- bulk,4pm
--- 2pm

I pm

o I.Ijpm meas.

) I 2 3 E(kV/cm)

Fig. 2.16.

Relative increase in GaAs low-frequency
velocity spectral density versus electric
field for several active region lengths.





Again, the mechanisms causing the increase in the low-frequency

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.


In this chapter, we will discuss the experiments which are done to

determine the de, ac and noise properties in the hot-electron regime of

AlGaAs/GaAs heterojunction interfaces. Two device structures with dif-

ferent characteristics, such as length, fraction of aluminum content,

sheet-carrier 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 modulation-doped 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 device-mounting 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 semi-insulating (SI)

GaAs substrate starting with an undoped GaAs buffer layer. This buffer

layer is grown to smooth out any defect properties associated with the


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 silicon-doped AlGaAs layer is grown next. The doping

level, as well as the spacer layer thickness and conduction-band dif-

ference, determine the quasi-two-dimensional 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

field-dependent 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 g-r 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.

Device Structure Parameters

#1483 #1885

1. Cap layer thickness (A) 50 50

2. Cap layer doping level (cm-3) 2.5x1018 2.5x1018

3. Doped A1GaAs thickness (A) 600 350

4. AlGaAs doping level (cm-3) 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 broad-band low-noise 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 broad-band 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


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

short-circuit termination is placed. The reference signal is totally



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 low-noise 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

T c (3.8)
a M2 Ml

and the gain bandwidth of the system is given by


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 hot-electron 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 pin-diode 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 25-mil.-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 current-voltage characteristic for this

4 um heterostructure is depicted in Figure 3.3. The low-field

equilibrium resistance is found to be 31 ohms. As can be seen in the

figure, the I-V characteristic begins to deviate from Ohm's law around

600 mV. This nonlinear behavior is an indication of hot-electron

effects in the channel. Changing the polarity of the voltage had no


100 V(V)

Current-voltage characteristic of #1483.


Fig. 3.3.

effect on the I-V relationship, indicating that the contacts were indeed

ohmic and had no rectifying properties.

From the I-V 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

(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 [37-39]. 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 g-r 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.

1 2


Fig. 3.4.


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],
right-side-up triangles indicate Masselink et al.
[38], and squares indicate van Welzenis et al. [39].

0 0 0 O



2- 1483 T=300 K
7- o
3- 0 0 0

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 I-V 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)

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







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-
side-up triangles indicate Bareikis et al. [24].

3 A
0 A V
----o ----0-- 0-400ooP---




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 different-length

devices were available for measurement. In this way the contact resis-

tance could be determined more accurately. The low-field 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 current-voltage characteristics of #1885 for

lengths of 2 and 8 um are shown in Figs. 3.8 and 3.9, respectively.

Both devices showed hot-electron effects at high bias.

Having the I-V 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


R(n )






Resistance vs. length for #1885 to determine
contact resistance. Circles indicate wafer-
probed values whereas triangles indicate wire-
bond values.

2 4 6 8

Fig. 3.7.

- T=300 K

1885 2pum






I I I I 11111 I

I( 1

SI I 11111I

I I I I 1 I

V (V)


Current-voltage characteristic of #1885
(L = 2 nm).



Fig. 3.8.


1885 8pm

I(A) T=300K /

5- /

o- 0


10c 3 -' 1 I I I Il ll I l lIlln

10' i00 V(V) 101

Fig. 3.9. Current-voltage characteristic of #1885
(L = 8 um).



Fig. 3.10. Normalized de mobility vs. electric field
for #1885.


1885 T=300 K

I r I I 1111 I I f 1 111


bonding wires made it difficult to obtain a good high-frequency 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 hot-electron 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


-0 0


i I I Iiiilll


I I I I 1 1 111

f (Hz)

Noise temperature vs. frequency for #1885.



I I I 1 I i I I I . .

Fig. 3.11.

Tn(K) T=300K
103 -
0 -
o00 -
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.




-1885 T=300 K

7- 0 0 1 000 000

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 hot-electron 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 hot-electron regime. This difference in diffusion

coefficients results mainly from the lower noise temperature measured in


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 two-dimensional scattering

rates for electrons in the first five subbands of a quasi-triangular

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 polar-optical phonon scatter-

ing rates are reduced in two-dimensional 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 real-space-charge 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 conduction-band 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 hot-electron properties in this

region or its effect on real-space-charge 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 two-dimensional 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 two-dimensionality of the electron gas.


In recent years much attention has been paid to AlGaAs/GaAs

modulation-doped field-effect transistors (MODFETs) for potential use in

high-speed logic circuits. The very high transconductance gm and high

cut-off frequencies fT also make them of interest for low-noise 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


Up to now only noise-figure 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 free-carrier 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

charge-voltage relationship for the devices used in our experiments.

Some of the methods of obtaining the charge-voltage relationship involve

only low-bias data while other methods involve high-bias data. Compar-

ing the results of the different methods can help determine the presence

or absence of real-space-charge 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 position-dependent

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

field-effect 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. Small-signal variations around the steady-state 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 steady-state current-voltage

characteristic of the device.

The ac equation has some interesting properties. Generally, the

ac equation involves the position-dependent steady-state parameters.

This equation can be written as follows:

HAV(x) = AI(x)


where H is a linear operator and AV and AI are the small-signal ac

channel voltage and current respectively. By letting z(x,x',f) be the

Green's function of H, i.e.

Hz(x,x',f) = S(x-x'), (4.2)

where 6(x-x') is the Dirac delta function and f denotes frequency, the

total ac voltage at position x can be calculated from

AVT(x) = f z(x,x',f)AI(x')dx' (4.3)

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 small-signal 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 small-signal quantities at

the device terminals, since these can be measured. The total small

signal voltage at the device terminal (x = L) is

AVT(L) = f z(L,x',f)AI(x')dx' (4.4)

The one-dimensional device shown in Fig. 4.1 is grounded at x = 0

and has an arbitrary steady-state 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 open-circuit 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




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.



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)

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

AVT(L) = f VZ(x',f)AI(x')dx' (4.8)

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-


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 open-circuit voltage

fluctuations measured at the terminals is given by [16]

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, generation-recombination (g-r) noise, and velocity-fluctuation

noise can be calculated if the proper source term K(x) is inserted.

In this chapter the focus is on velocity-fluctuation 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 one-dimensional treatment becomes [44]

SAV = f A(x')4q2D[E(x')]n(x')IVZ(x',f)12dx' (4.12)

where A(x') is the cross-sectional area, n(x') is the carrier density

and -q is the electron charge. The equivalent current-noise 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 one-dimensional, collision-dominated 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 field-dependent 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 two-dimensional 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 high-field 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 current-voltage

characteristic. At T = 300 K the velocity-field characteristic of the

two-dimensional electron gas is assumed to be identical to the one of

bulk GaAs [46]. Consequently,

v(E) = EE (4.15)

where U0 is the low-field 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

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