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Noise and current-voltage characteristics of near-ballistic GaAs devices

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Title:
Noise and current-voltage characteristics of near-ballistic GaAs devices
Creator:
Schmidt, Robert Roy, 1952-
Publication Date:
Language:
English
Physical Description:
vi, 96 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Ballistics ( jstor )
Diodes ( jstor )
Electric potential ( jstor )
Electrons ( jstor )
Noise measurement ( jstor )
Noise spectra ( jstor )
Noise temperature ( jstor )
Thermal noise ( jstor )
Trucks ( jstor )
Velocity ( jstor )
Electronic noise -- Measurement ( lcsh )
Gallium arsenide ( lcsh )
Noise -- Measurement ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1983.
Bibliography:
Includes bibliographical references (leaves 93-95).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Robert Roy Schmidt.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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ACA4853 ( NOTIS )
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Full Text














NOISE AND CURRENT-VOLTAGE CHARACTERISTICS
OF NEAR-BALLISTIC GaAs DEVICES










By

ROBERT ROY SCHMIDT


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY





UNIVERSITY OF FLORIDA


1983


I
















ACKNOWLEDGMENTS


The author wishes to express his sincere gratitude to

Dr. C. M. Van Vliet and Dr. G. Bosman for their research guidance and

aid in improving drafts of the work, and to Dr. A. van der Ziel,

Dr. E. R. Chenette, and Dr. A. Sutherland for their helpful suggestions

and kind interest. He wishes to thank Mark Hollis for fabricating the

devices.

Finally, the author appreciates the help of his fellow students

in the Noise Research Laboratory, especially Bill Murray for drawing

most of the figures contained herein.


















TABLE OF CONTENTS


ACKNOWLEDGMENTS . .

ABSTRACT . .

CHAPTER

I INTRODUCTION . .

II THEORY OF VERY SMALL LAYERS IN GaAs .

III EXPERIMENTAL PROCEDURES AND MEASUREMENT CIRCUITS .

3.1 Current-Voltage Measurements .
3.2 The 0.24 Hz to 25 kHz Correlation System .
3.3 The 50 kHz to 32 MHz System .

IV MEASUREMENT RESULTS .

4.1 Current-Voltage Characteristics ..
4.1a The n+n-n+ Device .
4.1b The n+p-n+ Device .
4.2 The n+p-n+ Device Noise .
4.3 The n+n-n+ Device Noise .

V DISCUSSION OF EXPERIMENTAL RESULTS .

5.1 The n+n-n+ Device .
5.1a Current-Voltage Characteristic
and Impedance .
5.1b Excess 1/f Noise .
5.1c High-Frequency Noise .
5.2 The n+p-n+ Device .
5.2a Current-Voltage Characteristics
and Impedance .
5.2b Noise .

VI CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER WORK .

6.1 The n+n-n+ Device .
6.2 The n+p-n+ Device .


Page

. ii








3

S. 12

. 12
12
. 14
. 40

S. 46

. 46
. 46
. 49
. 56
. 63

. 75

. 75

. 75
. 76
. 79
. 80

. 80
. 82

. 84

. 84
. 85


1













Page

APPENDIX: COMPUTER PROGRAMS FOR THE HP 9825 ... 88

REFERENCES .. .. .. .. ..... 93

BIOGRAPHICAL SKETCH . 96




















































iv


1
















Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



NOISE AND CURRENT-VOLTAGE CHARACTERISTICS
OF NEAR-BALLISTIC GaAs DEVICES

By

Robert Roy Schmidt

April 1983

Chairperson: C. M. Van Vliet
Major Department: Electrical Engineering
+ + + +
Conduction processes in novel submicron n n n and n p n mesa

structures in GaAs are investigated. The widths of the n-type and p-

type layers are 0.4 pm and 0.47 im, respectively. These small devices

are of interest due to the possibility of ballistic (collision free)

transport which leads to picosecond switching times at femtojoule

power levels, since the velocities of the carriers may be much greater

than the collision-limited drift velocity.

Three noise measurement systems are described, covering the

frequency range 0.2 Hz to 64 MHz. The first system, for frequencies up

to 25 kHz, features a spectrum analyzer with a dual-channel fast-

Fourier transform algorithm. By exploiting the correlation feature,

the system,when combined with very low-noise preamplifiers, has a

background noise level for a low-impedance source of less than 0.3 ohms

noise resistance. The other two systems for radio frequencies use a











tuned step-up transformer to bring the noise of the very quiet n-typF

device to a level above that of the RF preamplifier. One is a conven-

tional single-channel system. The other uses RF mixers before the

dual-channel FFT analyzer to extend the range of the correlation system

to high frequencies, depending on the preamplifier frequency response.

Current-voltage measurements, DC, pulsed DC, and AC impedance,

are presented versus parameters bias level, temperature, and frequency,

as are noise measurements. The n-type device is linear and relatively

temperature independent exhibiting full thermal noise and, at low

frequencies, very low levels of excess 1/f noise five orders of magni-

tude less than for the bulk material. This suggests that collisions are

mainly absent from this device. The p-type device, conversely, is

nonlinear with large levels of excess low frequency noise. There is

a linear, noisy, and temperature-dependent low-bias region followed by

a transition to a less noisy temperature-independent high-bias regime

with large conductance. This may represent a transition from ambi-

polarly governed to near-ballistic transport.


1
















CHAPTER I
INTRODUCTION


There is great interest in very thin GaAs layers of lengths

less than 1 m. This is of the order of the mean free path lengths

of the dominant collision mechanisms for electrons. Under favorable

conditions, carriers may cross the layers undergoing few or no col-

lisions, which leads to very high velocities, much greater than the

collision-limited drift velocity. The resulting so-called ballistic or

near-ballistic transport leads to very fast-response devices for pico-

second switching at very low power or other new applications. These

new devices have been made possible by advances in fabrication tech-

niques such as electron beam lithography and, in particular, molecular

beam epitaxy.

Such small devices enable us to investigate physical mechanisms

on a small scale for which the traditional models no longer apply. In

addition, practical knowledge of the performance limitations of these

new devices is obtained.

Early theories of these devices treated the ballistic motion of

the carriers as the dominant mechanism affecting its characteristics.

Recent measurements and later theories have shown that boundary condi-

tions, in particular the velocity dispersion of the carriers, have

much greater influence.

The organization of the chapters follows. In Chapter II some

theories of small n n n and n p n layers in GaAs are surveyed and


__






2





predictions are noted. Measurement techniques and procedures are

presented in Chapter III. In Darticular, a crosscorrelation noise

measurement system with a noise resistance less than 0.3 ohms suitable
+ +
for low-impedance low-noise devices such as the n n n diode is dis-

cussed. In Chapter IV, experimental results of measurements on two
+ +
devices are described. One is an n n n structure of length 0.4 pm.

The other is an n p n device of length 0.47 um. Current-voltage charac-

teristics, DC, pulsed DC, and AC to 25 kHz are presented, as are

measurements from 0.2 Hz to 32 MHz. The measurements are repeated at

77 K to investigate the effect of temperature. In some cases, T is

reduced to 12 K. Chapter V contains a discussion of the results.

Finally, conclusions and recommendations for further work are presented

in Chapter VI. An HP (Hewlett Packard) 9825A computer program used

to automate the noise measurements is recorded in the Appendix.


1
















CHAPTER II
THEORY OF VERY SMALL LAYERS IN GaAs


In most semiconductor devices, the drift current is limited

by the rate of carrier collisions. However, for sufficiently thin

devices few or no collisions occur permitting ballistic or near-

ballistic transport. The resulting carrier velocities can be much

larger than in the collision-dominated case so that such parameters as

switching speed can be greatly increased. Advances in fabrication

techniques such as molecular beam epitaxy and electron lithography have

resulted in submicron gallium arsenide devices with dimensions the

same order of magnitude as the mean free paths of the dominant collision

mechanisms. Thus near-ballistic transport in these devices becomes

probable.

For polar optical phonon emission at room temperature, Eastman

et al. [1] report mean free path lengths in GaAs of about 0.1 pm for

electron energies near 0.05 eV to 0.2 um up to 0.5 eV. Phonon absorp-

tion is about four times less probable. They also report that the

change in the direction of motion due to a collision is small (5-10

degrees). Intervalley scattering becomes significant for higher elec-

tron energies than 0.5 eV. Shur and Eastman [2] calculated the mean

free path in the high-purity GaAs at 77 K to be 1.3 pm. Barker, Ferry,

and Grubin [3] have suggested some complicating factors even for

devices of length equal to the mean free path. First, in large area


1











devices, some carriers may move at angles to the length direction,

thereby increasing the distance traveled and the number of collisions.

Also, some of these carriers may collide with the device boundaries.

Third, space-charge effects may limit the current due to Coulomb

scattering. Finally, short devices may be dominated by contact effects

such as carrier reflection.

The theory for pure ballistic transport was revived from the

days of vacuum tubes by Shur and Eastman [2] based on a simple model.

They assumed a one-dimensional n-type device of length, L, neglecting

diffusion and electron scattering. Then the current density is


J = qnv (2.1)


where q is the electronic charge, n is the free electron density, and

the velocity, v, is found from


1 2
qVx m v (2.2)


where Vx is the voltage at x due to the applied potential, V, and m is

the effective mass. The initial velocity and the field at the injecting

contact are assumed to be zero. Poisson's equation


dV2
x 9 (n n) (2.3)
dx2 EOr 0


is then solved yielding Child's law












J / Er V3/2 (2.4)
/ m L


for large bias voltage and large consequent injected space charge. For

small applied voltages, the fixed charge due to ionized donors is

greater than that due to the injected carriers and an electron beam

drift region (J V1/2) results. The range of voltages for which

their model is valid is


kT AE
-< V < (2.5)
q q

where T is the lattice temperature and AE is the energy difference

between the main conduction band minimum and the secondary valley.

For very small voltages diffusion cannot be neglected. Intervalley

scattering is the cause of the upper voltage limit. Some measurements

[1,4] have been reported that appear to support this model, but their

interpretation has been questioned as will be discussed.

To take into account the effect of a few collisions, Shur [5]

and Shur and Eastman [6] have extended the theory. Frictional "drag

terms" to represent the collisions are added to the equations of

balance of energy and momentum. In general, the effective mass, m ,

and relaxation times, Tm and TE' are functions of energy so that the

balance equations are


dm*(E)v m*(E)v
dt qE --T (2.6)
dt T~M(E











and


dt qEv E (2.7)



Shur assumes a constant energy-independent effective mass and a single

momentum relaxation time, T, so that, neglecting (2.7), (2.6) is

replaced by


dv my
m -t = qE (2.8)


Equations (2.1), (2.3), and (2.8) along with the boundary conditions

for the ballistic case are then solved analytically. For T much longer

than the transit time and sufficient applied voltage to justify

neglecting no, Child's law is again obtained. Conversely T much less

than the transit time yields (for V= -iE where i is the mobility) the

Mott-Gurney law for collision-dominated space-charge limited current


9 V
J EEr (2.9)


In the second paper, values of m*(E), v(E), Tm(E), and TE(E) are

obtained from Monte Carlo calculations. Steady state is assumed and the

boundary conditions are taken to be the initial field (and velocity)

equal to zero. Again letting TE and Tm become large yields Child's

law. For the general case, computer solutions are presented for various

lengths and doping densities which show small deviations from their

previous (ballistic) results.












The interpretation that current proportional to V3/2 indicates

ballistic transport has been seriously questioned by some authors.

Rosenberg, Yoffa, and Nathan [7] have shown the crucial importance of

the boundary conditions. For the ballistic case, they assumed a

simple one-dimensional model neglecting collisions. For each electron,

they solve equations (2.1), (2.3), and the energy-velocity relationship


S(v2 v) = q(Vi V) (2.10)


where vi and V. are the initial velocity and potential, respectively. A

displaced Maxwellian distribution of initial velocities characterized by

a temperature and mean velocity is assumed. A set of nonlinear equa-

tions in V result which are solved numerically for specified electric

field, E, initial carrier density, ni, and initial (temperature normal-

ized) mean velocity, v. They demonstrate that a wide variety of current

versus potential curves can be generated for various choices of initial

conditions [7, Figure 3]. The authors agree with others [3,8] that,

conversely with Shur and Eastman, the current-voltage characteristic may

be used to infer the boundary conditions for a device that has already

been shown to be ballistic by other means (such as noise behavior).

Another significant effect in short devices is "spillover" of

carriers from high- to low-doped regions. The result is that the

effective length of the low-doped region is shortened and hence the

resistance is less than expected naively even for devices many Debye

lengths long [7]. Universal curves have been calculated by van der Ziel











et al. [9] from which the magnitude of the effect can be determined

for a particular device. The authors calculated the near-equilibrium

resistance for a 0.4 pm n-type device with doping densities Nd+ =
18 -3 15 -3
101 cm and Nd = 101 cm for liquid nitrogen and room temperatures.

They solve Poisson's equation including electrons and ionized donors

in the high- and low-doped regions, neglecting holes. They assume


n(x) = Nd+ exo fIq()l (2.11)


where Nd+ is the ionized donor density in the highly doped region and

,(x) is the potential. Reciprocal mobilities due to diffusion/drift

and thermionic emission are added to give an effective mobility which

agrees within 10/ of experimental results at room temperature. However,

a similar calculation for an n pn+ device is off by more than an order

of magnitude. They assumed that the electron spillover depletes the

holes in the p-region such that they (holes) may be neglected. This is

apparently not the case; the effects of the holes are important.

The simple theory of ballistic [2] and near-ballistic [5,6]

transport does not take into account the energy or velocity distribu-

tions of the carriers. Cook and Frey [10] have suggested the inclusion

of an electron temperature gradient term in the momentum balance

equation for this purpose. This may be sufficient for first order

effects such as the current-voltage characteristic. Their treatment

includes an average collective velocity dispersion term, in essence

treating the ensemble as a single particle for such effects as the











noise, a drawback they themselves point out in the simple theory.

For the noise, more adequate analysis should include a Langevin or

Monte Carlo approach based on the momentum and energy balance equations

which include terms necessary to account for the significant physical

mechanisms. Such a Monte Carlo calculation for a 0.25 pm n -i-n

diode at 77 K has been reported by Awano et al. [11]. Ionized impurity

and intervalley and intravalley phonon scattering is included in their

model but not nonparabolicity of the band edges. Particles with a

distribution of velocities in equilibrium with the lattice at 77 K are

injected at one contact. They find that a large proportion of the

particles are transported ballistically. Groups of particles cor-

responding to one or two collisions and some backscattering from the

collecting contact is observed. For potentials less than 0.5 volts,

the current-voltage characteristic is surprisingly close to Child's

vacuum diode law.

Holden and Debney [12] have presented a calculation based on

ideas from Fry's theory for thermionic values [13]. They assume injec-

tion from both n regions of carriers with Maxwell-Boltzmann velocity

distributions. Scattering is neglected. Following Fry, the free-

carrier charge density is found by an integration over the injected

carrier velocity distribution. A potential minimum due to injected

space-charge is assumed. They calculate current-voltage characteristics

for T = 77 K and lengths of 0.1, 0.2, and 0.5 um. The slopes at high

bias are less than 3/2 (L = 0.1 and 0.2 pm give ,1.3, L = 0.5 pm gives

<1.14). At low bias, the characteristics are somewhat more linear.











The authors conclude that ballistic effects do not lead to a particular

current-voltage power law and suggest that another method must be

found to determine ballistic effects.

The noise behavior of thin structures gives important informa-

tion on the extent of ballistic effects. Current-voltage character-

istics for these devices are dominated by events at the boundaries such

as the carrier velocity distribution. In addition, practical knowledge

of the performance limitations of the devices is obtained, as well as

useful insight into the physical mechanisms causing the various types

of noise.

The noise is naturally divided into two frequency regions. At

high frequencies the dominant mechanism, variously called thermal,

velocity fluctuation, or diffusion noise is the electrical result of

the Brownian motion of the carriers. While the theory is not yet

complete, preliminary calculations have been made by van der Ziel and

Bosman [14,15]. For the ballistic case (collisions neglected) at large

bias, the device is modeled similarly to a dual-cathode vacuum diode

with opposing thermionic emission currents across the potential mini-

mum. Then correlated fluctuations in the current due to fluctuations

in the minimum due to the space charge result in a lowered (64%) level

of noise compared to the low-bias value of thermal noise of the AC

conductance. For the collision-limited case, at sufficiently large

bias, the device is modeled as a space-charge-limited solid-state

diode and the noise becomes twice the thermal noise of the AC conduc-

tance. Thus ballistic effects should reduce the noise and











collision-dominated transport should increase it if sufficient bias

levels can be achieved.

At low frequencies, excess noise characterized by a spectral

density with a 1/f frequency dependence dominates. According to most

recent theories [16,17] such noise is thought to be caused by mobility

fluctuations which represent fluctuations in scattering cross-sections.

The usual method to describe the noise is using Hooge's empirical

formula [18]


2
aHl0
S-I(f) fN (2.12)


where I0 is the dc current, N is the number of carriers, f is the fre-

quency, and aH is Hooge's parameter.

This is accurate for a uniform sample. However, for a non-

homogeneous sample or a mesa structure, van der Ziel and Van Vliet [19]

have shown that the formula becomes


m 2 r L
maH 0 dx
S (f) H dx
S fAL2 nrx)
0


(2.13)


where m is the number of layers, L is the length of one layer, A is the

cross-sectional area, and n(x) is the electron density. The value of

aH was first thought to be constant (2 x 10-3). Later research has
found that aH depends on material variation, the dominant scattering

mechanism, and carrier heating [20] in Si. Thus the magnitude of aH

should give a good measure of the rate of collisions or lack of them.
















CHAPTER III
EXPERIMENTAL PROCEDURES AND MEASUREMENT CIRCUITS

+ +
The procedures used in the measurement of both devices, n n n
+ +
and n p n are similar, but the actual circuits depend on the device

characteristics which are quite different. The p-type near-ballistic

diode (NBD) exhibited an impedance about 100 times that of the n-type

device at small bias. It also generated excess noise at low frequencies

orders of magnitude larger than the n-type NBD. Consequently, the noise

of the n-NBD was the most challenging measurement requiring an equiva-

lent noise resistance of the measurement system of about 0.2 ohms.


3.1 Current Voltage Measurements

Three I-V measurements were done on each diode: DC, pulsed DC,

and AC from 2.4 Hz to 25 kHz. A block diagram of the n-NBD circuit

is shown in Figure 3.1. A three-terminal measurement is required since

the resistance of the active region is comparable to that of the bonding

wire to the top surface of the diode (the devices were mounted in TO-5

cans). The resistance of the output wire, RL2, is negligible since it

is in series with the large input impedance of the oscilloscope. Then

V2 is the voltage across RX + RS which represents the series combina-

tion of substrate resistance and active region. The current through

the device is


Vl V2
I R R (3.1)
RB + RLl



























n-NBD


DC OR
PULSE
POWER
SUPPLY


Figure 3.1 Block diagram of n +current-voltage measurement
Figure 3.1 Block diagram of n n n diode current-voltage measurement


I











Pulsed measurements were done using an HP 214A pulse generator. The

pulse width was 1 psec and the repetition rate was 100 Hz.

The AC I-V measurement employed the HP 3582A spectrum analyzer.

The dual-channel feature was used to measure the transfer function as

a function of frequency from 2 Hz to 25 kHz and as a function of DC

bias. The circuit diagram of the AC resistance measurement is shown

in Figure 3.2. Taking into account the parallel resistances, RX is

found to be



V 1 + RS + R
V C R S RL S
C RB
R H= V- R-0R- L -(3.2)
X VO II RC1 (3.2
1 1 + T~\
VC t RB

The measurement for the p-type device is similar. Here, the

lead resistances may be neglected since the impedance is large; the

diode may be treated as a two-terminal device.

3.2 The 0.24 Hz to 25 kHz Correlation System

For the electrical noise measurement of low impedance devices

with resistances of the order of an ohm, the usual single-channel

method is not useful. Typical preamplifiers have noise resistances of

about 10 to 20 ohms. The device noise is masked by the preamplifier

noise. Therefore, the correlation method [21] is used as shown in Figure

3.3.

A Hewlett Packard 3582A spectrum analyzer featuring a dual-

channel Fast-Fourier transform is employed. Low-noise preamplifiers















n-NBD


RC' R,

Ll i RL2
B Rx
V Vo I MR


AC resistance measurement circuit for n-NBD


Figure 3.2


































- Vbatt

I


-------1
CHANNEL A


HP 3582A
SPECTRUM
ANALYZER

CHANNEL B

2
p2= ISAB
pA-SB
SA' SB


Correlation measurement setup


Figure 3.3











precede each channel from a common input. By measuring the coherence

(square of the correlation between the channels) the cross-spectra

can be calculated. The noise generated at the output of each channel

is uncorrelated with the other and therefore averages out in the final

reading. Only the noise generated at the input of each channel gives

a contribution to the final reading. Hence a better device noise

versus amplifier background noise ratio may be obtained.

The device under test (DUT) noise is compared to a known cali-

bration signal which is applied through a series resistor much greater

than the DUT resistance. The bias current is applied in a similar

manner. An equivalent circuit of the measurement setup is shown in

Figure 3.4. To characterize the noise of the system, we first neglect

the effect of the device noise. The series resistors to the bias and

calibration sources are R' and Rs. The calibration source, v al,

transforms to a Norton equivalent, ic, where


ci vocal (3.3)
c Rm


The source resistance is represented by RX. The preamplifiers are

assumed identical (with uncorrelated noise sources, vl, i1, v2, and i2)

with gain, G (assumed constant), and input resistance, Ri. The output

of the amplifiers, u1 and u2, are then multiplied together and averaged.

The measurement procedure is to compare the output for three con-

ditions. They are (1) shorted input (RX = 0) giving output reading M1,

(2) open input (RX = W, RS J R >> Ri) giving M2, and (3) open input







18
































IM













rQ-





+ "_






c.









C4-
r4a






t,
*- U













C)
a,
U,











LO







LL










with calibration signal applied (i 3 0) which gives M,. The output
c
for each case is easily found to be


M =0 (3.4)


R?
M2 G2 1(i* + i (3.5)
2 = l T4-

and

R
M3 G2 (ili i2 + ) (3.6)


In reality, M1 is not exactly zero. There is a residual back-
ground noise. This can be accounted for by introducing a correlated

background noise source, vb, to the input voltage sources so that

vI = vi + vb and v2 = v2 + vb. The primed portions are uncorrelated.

Then the analysis yields


Ml =G2 VbV (3.7)


S __ ___ R2-
=G2 v (i i* + i (8)
M2 = bVb ( + i2i) (3.8)

and


M3 = 2 vb + (i i* + 2 + (3.9)
3 1











Writing the mean square averages as spectral densities, i.e.,
i i SiVc
S = 2 f etc., and noting Sc = allows the current and
S
voltage noise to be calculated.


M M S
M3 M2 SIc
K- M S SI (3.10)
M2 N11 Il + 12


Thus


SV
SII + SI2 ,S (3.11)
KRP'
S

and


R2S2
SVb i VM (3.12)

2 M2 )



The current noise spectra of the two channels are SI and SI2 and the

background voltage noise spectrum is SVb'

A significant source of background noise, after such causes

as pickup of unwanted signal, ground loops, and power supply noise have

been eliminated, is the finite averaging time of the spectrum analyzer.

The sample coherence for shorted input versus number of averages is

shown in Figure 3.5. It is a large-side biased estimate inversely pro-

portional to the number of averages which is limited to 256. The

distribution of sample coherences is not easily described [22] for





















































1 10 100 10


NUMBER OF AVERAGES


Figure 3.5


HP 3582A sample coherence
of averages


for shorted input versus number


IT


LJ
z
Z
W
LT


0
w











small expected value. The residual nonzero value is interpreted as

background noise.

The system is used with two similar sets of preamplifiers

denoted PAl and PA2; PA2 features five parallel input stages like that

of PA1 but with larger bias current. They are shown in Figures 3.6

and 3.7; PA1 has larger voltage noise but extended low frequency

response suitable for excess 1/f noise measurement. The PA2 has been

designed for minimum voltage noise.

The PAl features a capacitively coupled common emitter transistor

first stage with shunt feedback. This reduces the input impedance but

does not affect the noise. The primary motivation for this circuit

is a practical one. The RC settling time of the base bias circuit is

substantially reduced compared to the usual configuration. The tran-

sistor (GE 82) is chosen for large and small series base resistance.

The second stage is a low-noise operational amplifier (Burr-Brown

OPA101BM) in noninverting configuration;12-volt automotive batteries

are used for the power supply.

An equivalent circuit of the input stage is shown in Figure

3.8. It is the low-frequency hybrid pi including source resistance,

feedback, and noise sources. The source resistance is RS. Thermal

noise of the series base resistance, rb, is vb. Shot noise of the

base and collector currents are denoted ib and ic, respectively. To

transform this circuit to a form similar to that in Figure 3.4, the

closed loop gain, the input resistance, and the noise sources must

be calculated. The closed loop gain is found by summing currents at
















-12 v




12K


487K

IlOOhF
SINPU GE 8,
INPU I I


+12V
9 or
oPA01BM


0.5, F


Figure 3.6. Model one (PAl) low-noise premplifier


OUTPUT


91K

















the input


v v. v.
i + o i 0, (3.13)
s RF 1 Rs


and at the output


V V. V
i. + 1 0- (3.14)
1 R R
F C


where R = r R, i = gmv and the noise sources have been neglected
vi
for now. Letting r = rb + r so i = the output equation gives i

in terms of v which, when substituted into the input equation allows i
ri
to be expressed as a function of v Then, for RF -, the closed

loop gain, A, is found to be


v -RF
A R- F (3.15)
RF + RC + r. + 1 +
S C 1 RF RSJ
1 +
BRC


The loop transmission is found by disconnecting the output current

generator, gmv and applying a test current generator, iT, there. The

output voltage, v is


v = iT [RC (RF + RS r)] (3.16)


The input voltage fed back is





27




r.RS (v v.)
v. RS 0 (3.17)
i r + R RF
1 S F


so that


ri RS
(r + RS) RF
i = vR (3.18)
ri + RS) RF


Then, noting that gmv, = gvi gives


g v -_Rc
1 RV r-BR
T RF + RS+ir ri + r(s
C F R + r R
S il( S)


The input resistance is RS in parallel with Ri where, for RF >> RC and

RF >> ri/,


RC + R
R. = r. (3.20)
1 + r R g
1+ Rcgm
I


The output resistance is R in parallel with RL where


r.R
R + r+ R-
R = r (3.21)
0 0 r.
B+ 1 +
RS


Except for the noise, the circuit of Figure 3.8 now becomes that of

Figure 3.9 where the voltage gain is
































































lE ----


C)











fR + R
G = A R 1 (3.22)
( RSRi


For the case that RF >> RC, ri, Ri >> rb, RS, and (riRF)/RS >> BRC,

then


-BRC
G -Fr (3.23)



The gain becomes open loop as the input is shorted by a small RS. To

transform the noise sources, the effect of feedback is neglected. Open

circuiting the inputs (RS = m) of Figures 3.8 and 3.9 and setting the

output voltages equal yields


i
S rb Vb + 9C
i =g rb m (3.24)
a b r + r rb + r,


Short circuiting (RS = 0) the inputs yields


i
a = Vb + + ibrb (3.25)
gm


Substituting numerical values for components with RS = 5.6 ohms and

assuming 8 350 yields about 1200 ohms for the input resistance. The

measured value for two PAls in parallel is 660 ohms as shown in Figure

3.10. The input resistance of two PA2s in parallel is shown in Figure

3.10. For the gain the calculated value is about 51 decibels. The

result including the second stage is 90 decibels. The measured value

is 89.5 decibels as shown in Figure 3.12.










O
0


o
0


0
0
oN

o
o 0I
O o
o LO >-
o C\W
ooz
0 oJ
0 LLj
0LOLL

0 m
U-
O
O
0
0


O
o 0
0
0 0
o
0

c j 0 0 d
(L>)30BNVISIS3 d IndNl














8




8








8
0
0


0 0 0
UL rno
(2) 30NViSJI S3]


N

C-)
z

O
O 0*
0
W0L


0










O
O




J O
0


0
O









II
0 t,
w=






r-




0 -
Q)

(8




0
-
0
=-r


O -
O -

0 0L
o co too P
(SP) N1V9











The noise resistance of PA1 and PA2 were measured to be about

35 and r2.5 ohms, respectively. The five in parallel input stage of

the PA2 gives that factor of reduction in the noise resistance.

Another factor of three reduction comes from increasing the input bias

currents by that amount since, to first order approximations, the

voltage noise sources are inversely proportional to bias [23].

Using the correlation setup gives about a factor of ten reduc-

tion in the voltage noise. This is seen in the ploLs of voltage noise

resistance and current noise conductance for the two setups shown in

Figures 3.13 through 3.16. The noise resistance, Rn, is defined by


SVb = 4kTRn (3.26)


Similarly, the noise conductance is defined by


S1 + SI2 = 4kTGn (3.27)


The current noise of the PA2 is much larger than that of the PA1

as would be expected from larger bias currents and five in parallel.

For the sensitive measurements, the current noise is shorted by the

small resistance of the DUT. This is the full current noise which

remains in the correlation calculation. Of the voltage noise, only the

background level remains.

The device noise measurement procedure is very similar to

that for the amplifier. Three conditions are recorded. They are

(1) DUT on, (2) DUT off, and (3) calibration signal applied with DUT

off. Any change in DUT resistance between the on and off state must be















0
0


0
0


" 0




D0

00
O
0
O


0
O
O


D

0


0


0


T 1 J CO 0D C j
(SIHO) 33NVISIS3d 3SION


S 4-
N 0
>-
o

LUL

40
D
Sr-
01





r-
.4-
o
0
4-


0
U-























N


z

0 :


O-=
00 0
o L

--

00


CY O
( u

o I

ou t

10 0 o
o 'o 'o -
(N B"I lS) -0NV1.I,.Nf NOO "SION
L-













0




O
S0O
O
0

0


)

O

0

O


0
0


O0
O
%


0%


(SINHO) 3DNVLSISld 3SICN 39V71iOA


,0
1 ro


I-

02
>-
O
- I
a
LiJ
LL


-0


I

















0
0
.. O
O















--O








o i


> --

LLJ

0






u

a>


V)


S-
;5



o
1
r-






- O (n) ONVI.onON0O o ,
-- 3SION IN3fnO ,

U-
.,-
1 ,!











taken into account (replacing the device by a resistance equal to its

"on" resistance for the latter two measurements). The equivalent circuit

of the input can be drawn as in Figure 3.17. The DUT is represented

by the parallel combination of Rx, the AC resistance of the device,

StLf, the thermal current noise generator, and V^-'f the excess
th x
current noise generator. The calibration signal is ^cal f. The ampli-

fier is characterized by Ra, /Sif and /VLf
A Af
For the three conditions, the outputs are


M = G2[(S + S + S)R2 + SI] (3.28)
1 x th A A


M2 = G2[(Sth + Si)R2 + SA] (3.29)


and


M3 G[(Sth + SA + Sal)R2 S] (3.30)


where G is the (constant) gain and R is the parallel resistance at the

input. Then, as before,


M1 M S
K 1 2 x (3.31)
M3 M2 Scal


so that the device noise current can be written


S = S +S = KS + (3.32)
DUT x th cal R
x










































L.LJ


Z
zI







a)
w ,






o






























S.-
4-,

'4-'
0)

4,
v-





C,










c,



r-,






CL
ar

C
a)
5-










+ + dV
Note that for the nonlinear n p n diode, Rx d- and depends on the

bias current applied. An HP 9825A controller can easily automate this

measurement and be used to average many data sets for increased

accuracy. A program which does this for eight logarithmically spaced

points per decade is presented in the Appendix.


3.3 The 50 kHz to 32 MHz System

At radio frequencies, the same three-measurement procedure as

before is used for the noise. The circuit is more conventional, as

shown in Figure 3.8. A step-up transformer consisting of four trans-

mission line transformers in cascade with capacitance-tuned input and

output is employed for the n-type device. See the Radio Amateur's

Handbook for details. A Micronetics KSD20LEE solid-state noise source

is used for the calibration signal. Frequency selection is done by an

HP 8557A spectrum analyzer. The IF output signal from the spectrum

analyzer is passed through a 21 MHz bandpass filter and detected with an

HP 431C power meter.

The preamplifier, designed and built by Christopher Whiteside

of the noise research group, is shown in Figure 3.19. It employs two

stages of low-noise FETs (2N4393) in cascade configuration to obtain

30 dB gain with a noise voltage resistance of about 70 ohms. This is

less than the stepped-up noise of the DUT.

The HP 3582A spectrum analyzer can be used to make noise

measurements in the radio frequency range. This is accomplished by

mixing down the RF signal just before the analyzer as shown in Figure

3.20. Only a single channel is shown, although the correlation method




























F-'1









0


O,


0 >
* If
-I-


J.
z
n*






43










C C







ULU


U-


Z4 -
0tD












LS-








tN
0 (n\
<








U-








C~1)


--LO---:












can be used just as easily. The bias and calibration signal are

applied as for the conventional RF system. The circuit for the n-type

device, using a tuned step-up transformer, is shown. The 60 dB radio

frequency preamplifier is actually two of the 30 dB preamplifiers

shown in Figure 3.19 in cascade. The large gain is needed to overcome

the poor noise performance of the mixer (Mini-Circuits ZAD-1). Using a

standard RF signal generator with output amplitude capability of

+7 dBm, the mixer has a frequency response of 100 kHz to 500 MHz.

The same signal generator can be used to supply both mixers

(correlation setup). Then two 60 dB preamplifiers are used and, if

series resistance or unwanted pickup of the step-up transformer is not

negligible, two of these also. Feedthrough of local oscillator har-

monics may overload the spectrum analyzer at some frequencies if a

bandpass filter similar to that shown in Figure 3.21 is not employed.

This simple filter had good (90 dB) attenuation to 100 MHz. The band-

width of a single FFT bin of the spectrum analyzer was extremely small

compared to the frequencies of interest, causing significant inaccuracy.

To reduce that, a program to average 80% of the bins to synthesize a

bandwidth of up to 20 kHz was written. It is shown in the Appendix.












































1 4

-I


O
















CHAPTER IV
MEASUREMENTS RESULTS


The near-ballistic diode (NBD) is a sandwiched mesa structure

of five lightly doped p or n layers, alternating with heavily doped n

layers. A not-to-scale sketch of the p-NBD is shown in Figure 4.1.
18 -3 +
The doping densities of the various regions are 10 cm for the n
15 -3
regions, approximately 2 x 10 cm for the n regions, and approxi-
14 -3
mately 6 x 101 cm for the p regions. The diameter of the mesas is

100 pm. The devices were manufactured by molecular beam epitaxy at

the Cornell University Submicron Research Facility by Mr. M. Hollis.

The mesas have very low-noise ohmic AuGe contacts.

Current-voltage measurements of three types-DC, pulsed DC,

and AC-are reported. Also presented are noise spectra versus parameters

bias, frequency, and temperature.


4.1 Current-Voltage Characteristics
+ +
4.1a The n n n Device
+ +
The DC current-voltage characteristics of the 0.4 m n n n device

at room temperature and 77 K are shown in Figure 4.2 In contrast to

the device reported in Eastman et al. [1], there is no vV region.

The characteristic is linear for both temperatures up to very large

bias. Pulse techniques are used for currents greater than 100 mA.

The resistance of the active region is 0.75 ohms at room temperature


















I00/Lm DIAMETER
18 -3
NJ 10 cm
15 -3
Np- r 10 cm



I V2


GND


Figure 4.1 P-type near-ballistic diode mesa structure





48










IOA




z IA
H-
(r

o 0 300 K
[] 77 K
100 mA




10mA




ImA-




0.1 mA '




0.OlmA
O.ImV ImV lOmV

VOLTAGE


Figure 4.2 DC I-V characteristic of n n n 0.4 nm device


ICOmV


I












and 0.68 ohms at 77 K. This is confirmed by the AC impedance

measurement at 300 K for IDC = 75 mA shown in Figure 4.3. Low-

frequency equivalent circuits of the device at room temperature and

77 K are shown in Figures 4.4 and 4.5. The series resistances of the

gold wires of the TO-5 can are significant and must be taken into

account.


4.1b The n p n Device
+ +
Conversely, the n p n structure is quite nonlinear. The DC

current-voltage characteristics of the 0.47 im n+p-n+ diode at 300 K

and 77 K are shown in Figure 4.6. The device is linear for both

temperatures up to about 100 mV. The characteristics then move through

transition to a temperature independent high bias regime in which the

slope falls off. The maximum slopes in the transition regions are

3 for the 300 K case and 4.5 at 77 K. The low-bias resistances at

300 K and 77 K are 90 ohms and 320 ohms, respectively. The measurements

of AC resistance for several bias currents at T = 300 K shown in Figure

4.7 and for T = 77 K shown in Figure 4.8, display no frequency

dependence.

To compare the large and small signal resistances more easily,

the exponent, B, can be examined where I = V Differentiating both

sides and solving for g gives


DC
S DC (4.1)
AC
















-, 0
0


0



0
O

0
O




o o




0 0
S-'
O Q








o 4-
0
0 -




0 +








0 -
O o
o L.)






c-
o
0r2




o 0 -


o00
0 4-



0
0 a)
C)










0 =U
0 vr


o 1j























S"RLI RL2




0.75 < R,

RSI


0.37

0.08 RS2














Figure 4.4 Equivalent circuit of n n n structure at 300 K showing
parasitic elements























R LI RL2

0.08 0.10


0.68 R

RSI


0.23

0.06 R2
RS2












+-+
Figure 4.5 Equivalent circuit of n n n structure at 77 K showing
parasitic elements




















0 300 K
E3 77K


ImV lOmV lOOmV IV lOV

VOLTAGE


Figure 4.6 DC I- characteristic of n- 0.47 m device
Figure 4.6 DC I-V characteristic of n p n 0.47 ,im device


IA -


1OmA



H-
SImA
LU




0.1mA


O.1mV














E E
< Ln <
O C

E7> O


S7[> 0


[ > O


Q o
Q 0
0 0
E 0

[ 0

D 0
a o

O o
E 0



0 O
E 0
So




7 0
E 0

S0


a o


0o
o( -


LD
Lu
IC

Lu
0
o
0
0


- 0


7 0> 0
E71> 0
1> 0O


ED>












E< E
-E li
00- ro'E

CBF 0




FDizK
DD7 0














ED7
CD[7 Q






oD 7 o



ED 7 0
LID7 c
cD 7 0








ED[7 0


(u)i) na 0o 0v


O
0
-0




-o


N


C)
z

Cy
LU
-0




-o


r o


I 0


r 0
I


1 I I


E
w E












This is plotted in Figure 4.9 for small bias up to 34 mA. At higher

bias, pulsed DC but not AC measurements were done. The resistance

falls dramatically at high bias. This is seen more clearly in Figure

4.10 which presents the DC and pulsed DC conductance versus bias

voltage for several temperatures down to 12 K. The large-bias

conductance in the temperature independent regime rises to a limiting

value of about 0.33 Siemens. In the low-bias regime, the conductance

quickly falls from its 300 K value by about a factor of three at 150 K

to 200 K and remains nearly constant for further decreases in tempera-

ture.

+ +
4.2 The n p n Device Noise

The n p n 0.47 ;m structure exhibited large levels of excess

low-frequency noise. The room temperature noise current spectrum

(less than 25 kHz) for several bias currents is shown in Figure 4.11.

The frequency dependence for all bias currents is about .75. In

Figure 4.12 is shown the extension to 32 MHz of the spectra for bias

currents of 100 pA and 1 mA. The 100 1A spectrum is approaching the

thermal level at the high-frequency end. The turnover frequency to

thermal noise occurs in the GHz range for larger bias. Figures 4.13

and 4.14 display the complete spectra versus frequency for bias currents

of 34 mA and 68 mA, respectively. Both show slopes of -0.77 over many

decades. No turnover frequency to a different slope is found. These

spectra represent measurements in the low-bias regime and transition

region. Measurement in the high-bias regime requires pulsed techniques.





















iCD


O


O
ro
0


1 I 1 -
qo d- to() CiJ C


Z
LU

:D

0
O







58








> =
L)








cc












Ln
-- |










9> >












L M
4--)
o F 0 a
u


*r

L-0
o V, I n c o









P +-1




















~C


00
0


0 0
0

O 0o
0
0
0 O o

Q 0 0 ..,
u-i- "G 0 0o,
^ -a^ So->
A2 I 0r !- 0 0
N-/ /" U -\ 0 [-


LJ-
LL.


k,)


0


rn-


0.


S34 mA
S13.6 mA
0 3.4 mA
0 I mA
0 400oA


V o
'I


SlooLA p7a



-20
10


f (H:)


Figure 4.11 Low-frequency p-NBD room temperature current noise spectra


-14
10 --





-15





-16
10 --
10'6-


<


WO

C),


D
77


D
Or7
00
































- 7
77v
77


7 ImA


D

IOOA E


17
'7
77n

v77
v77
v


I \


Figure 4.12


UIK lOOK IM IOM lOOM
FREQUENCY (Hz)

Current noise of 0.47 pm n pn+ NBD at T = 300 K for
I = 100 IA and 1 mA


10-!




IO-17


N

zJ`
7-


10-20
10


i ,,
















































I I
r0 0


2
0
-0


-0


N







1.
-o


0



-0


(ZH/VIs 10 10
Sz






















0



CO


CO



N-
-o

J =





Q
S-O >- o



















.--
o ac
G- 0 L
S| LL z
^r +


o r-







0 U-
-0 0 A











C o-
C .?'
I-











The room temperature current noise at 100 Hz is plotted

versus bias current in Figure 4.15. There is an 12 dependence at

lower currents up to a few mA. At higher bias the noise increases

less fast. The deviation from 12 behavior appears to coincide with

the transition region.

The low-frequency current noise spectra at 77 K for several

bias currents are shown in Figure 4.16. The levels of excess low-

frequency noise are again large. The slopes are somewhat steeper than

for the room temperature case, being approximately -0.85. Figure 4.17

displays the current noise at 100 Hz versus bias. Again, 12 dependence

in the low-bias regime with a fall-off in the transition region is

observed.

The temperature dependence of the noise in the low-bias regime

was investigated further. The low-frequency small bias (100 uA) noise

spectra were measured for several temperatures (300 K, 250 K, 200 K,

150 K, 77 K, and 12 K) down to 12 K as shown in Figure 4.18. The

magnitude increases with decreasing temperature down to about 200 K.

Thereafter, further lowering of the temperature does not affect the

magnitude very much, but the slope becomes more closely 1/f. The

other temperatures are not plotted since they only obscure the figure.

At 12 K, the slope is fully 1/f as shown in Figure 4.19.

+ +
4.3 The n n n Device Noise

The magnitudes of the noise levels of the 0.4 um n+n n struc-

ture, both thermal and excess low frequency, are very small. The

current noise spectra for several bias currents at room temperature is








64

















-14
10







-15
10 -
10





-16
10







-17
i.j



10 --










-19
-2I

10 10 I 10 o0o

I (mA)




Figure 4.15 Current noise at 100 Hz versus bias current for p-NBD
at T = 300 K















0
(



Go
ED


7
17
7
7
7
7
7


Y 10 0L A
VIOOfiA
, 400oA
E ImA
O 3.4mA
] 13.6mA
034mA


7 -


"V
'77


7


100


0O


0K2


FREQ UENCY (Hz)


Figure 4.16 Current noise of p-NBD at T = 77 K


I O13


N
I
c-
H
CO


-18-


IOK


lOOK



















10-13




1 x 71
x /





u -4
N xg






10-16 <-







10-17-


1OU A lOOA ImA 10mA IO0mA I A
CURRENT

Figure 4.17 Current noise versus current of the p-NBD for f = 100 Hz
and T = 77 K






































I I I i


100 IK
FREQUENCY (Hz)


Figure 4.18


+-+
Current noise of n+p n+ diode for I = 100 -A and three
temperatures


lq 300K
<0250K
0 12K


0-15-
10-


N
IO
-

c0
10-19-



o10-20-


IOK


lOOK











10-15--


-16
10





N
Ir-

cn

1 0 -19



,-20-



10 100 IK IOK lOOK
FREQUENCY(Hz)

Figure 4.19 Current noise of n p n diode at T = 12 K and IDC = 100 IA












shown in Figure 4.20. Thermal levels and excess low-frequency noise

for some levels of bias can be seen. The frequency dependence of the

low-frequency noise is 1/f. The spectra from 1 to 25 kHz with the

1/f levels subtracted is shown in Figure 4.21. The lower three bias

currents result in approximately thermal noise. There is still excess

noise for 75 mA, however. Therefore, a spot noise measurement at

500 kHz for that bias was done. Less than J- deviation from the

thermal level was found. Values at 10 Hz obtained from the straight

line approximations to the 1/f noise are plotted versus bias current

in Figure 4.22. The expected behavior for 1/f noise, SI 12 is well

satisfied.

Spectra for T = 77 K and the same bias currents are plotted in

Figure 4.23. Excess low-frequency noise is found for all current

levels. The noise in the thermal region is difficult to determine

accurately because the expected value for full thermal noise (%0.17 2)

is below the background noise of even the correlation setup. Also, for

some bias currents, the noise is not yet flat at these frequencies.

The 1/f noise at 10 Hz is plotted versus bias in Figure 4.24. Again,

the I2 dependence of the magnitude is found.



































017
10


-, 75 mA

5C mA


II mA





10 L--- -



-2C
-19 K) r-n r-








10 -I






-21
10
10 00 IK OK JO0K

f (H:)


Figure 4.20 Current noise spectra of n 0.4 m NBD at T 300 K
Figure 4.20 Current noise spectra of n n n 0.4 pm NBD at T = 300 K

































O 54 mA 2.5 x 1020
-20
S27 mA S. (,VG.) I KHz 25 Hz

10 o mA 2.1 x -20,






r -

c, ,




lo/V
C) ^ ^ ^ ^ ^------- -,

V THEFi.!AL ,'CISE-
CF 075f1


i0-
10 ';5_r


f (H:)


Figure 4.21 Thermal (-like) noise of n n n device













































-19
zz 0O


SLOPE 'L 2


!00


i (MA)


Figure 4.22 Excess 1/f noise of n+nn+ device versus current

















7 75mA
E] 50mA
O 27mA
0 IlmA


"7
7 7


25 2.5 25 250 2.5K 25K

FREQUENCY (Hz)


Figure 4.23 Current noise of 0.4 nm n-NBD at T = 77 K


o16
10


N


H
CJO


10-21
0




























N
I
C\J




10
















107
10 0OC

DC CURRENT(mA)


Figure 4.24 SI of n-NBD at 10 Hz versus bias current for T = 77 K
















CHAPTER V
DISCUSSION OF EXPERIMENTAL RESULTS



5.1 The n nn+ Device

5.la Current-Voltage Characteristic and Impedance

The I-V characteristic of the n-type device is seen from

Figure 4.2 to be linear for bias voltages up to about 1 volt and

currents up to about 1 amp for both room temperature and 77 K. An

attempt to apply pulses at higher bias resulted in melting the gold

bonding wire at the top of the mesa. Higher bias was desired since a

slight nonlinearity appears at currents greater than 1 amp. Neverthe-

less, 1 amp corresponds to a current density of 12,800 A/cm2. No V1/2
3/2
or V32 dependence is found which suggests that theories with these

results are not adequate.

Similarly, the collision-dominated Mott-Gurney theory which

predicts V2 current dependence at high currents does not apply.

Indeed, sublinear current dependence at high bias seems to be indicated

by the sparse data. The more realistic theory of Holden and Debney

[12] gives a high-bias current dependence of V14 for a 0.5 im device

where collisions are neglected. At lower bias, their result appears to

be somewhat sublinear, similar to the V /2 region of Shurand Eastman's [2]

theory. The theory of van der Ziel et al. [9] gives linear behavior

for small bias. They include spillover from the highly doped regions












and calculate separately the mobilities due to diffusion-drift and

thermionic emission. Setting the calculated and measured values of

resistance at room temperature equal and solving for the diameter of

the mesa gives a 96 pm diameter which is very close to the reported

value of 100 jm. At 77 K, the measured value decreases 10%, the cal-

culated value decreases 23%. The impedance is purely resistive at

these frequencies.


5.1b Excess 1/f Noise

In the 1/f noise region, we would like to apply Hooge's

empirical formula [18] which is Equation (2.12). For mesa structures

or nonhomogenous samples [19], Equation (2.13) replaces it and is

correct whether or not the transport is ballistic.

In a ballistic or near-ballistic device, many carriers do not

undergo any collisions at all. This is in contrast with a typical

semiconductor device in which every carrier collides many times.

Hooge's formula was developed for the second case, requiring that N,

the number of carriers in Equation (2.12) can be determined. In the

near-ballistic case, it is desired to exclude those carriers which are

transported ballistically, including only those that contribute to

the noise. This can be very difficult. Therefore, an alternative

expression to describe "noisiness," avoiding this problem, will also be

used. Noisiness is described as (S A(f)f)/I2 which is still dimension-

less unless the spectral slope is not -1. Substituting values gives,

at 300 K,











S i(f)f
S = 1.6 x 10-15 (5.1)


and at 77 K,

SI(f)f -14
2 1.4 x 104 (5.2)
I

If the slope is not -1, then just (S A(f)f)/I2 can be reported at a
specified frequency.
If n(x) = n is a reasonable approximation despite the complex
nature of n(x), then Equation (2.13) becomes

S A(f)fALmn
Hb = 2 (5.3)
I0

where the subscript, b, denotes validity for the ballistic case. If
the transport can be characterized by a constant mobility, i, the
measured resistance is

L
R m j n (5.4)
q4A n-5Tx)
0

Then, for the collision-dominated case, Hooge's constant becomes


S A(f)f(mL)2
Hc = 2 (5.5)
IoquR











2 2
Assuming that 300K = 0.74 mV- 14 mec A = 7.9 x 10 m2




aHc = 7.2 x 10-8 (5.6)


and


aHb = 5 x 10-8 (5.7)


At 77 K the values are


cHc = 3.7 x 10-8 (5.8)


aHb = 4.5 x 10-7 (5.9)


These values are five orders of magnitude smaller than the value

reported by Hooge et al. [16] for n-type bulk GaAs of 6 x 10-3. Thus

the number of collisions for this device is very small. The remaining

collisions involve polar optical phonon emission typically with a very

small deflection angle [1], 6. Handel's [24] quantum theory of 1/f noise
.2 1
indicates that the magnitude of the relative 1/f noise goes as sin2

so that the residual noise is very low.

Comparing the ballistic and collision-limited case temperature

dependence of aH suggests that the device is not purely ballistic at

either temperature since the low temperature ccHb is nine times larger

than the value at room temperature. Further, the diffusion-drift

resistance calculated by van der Ziel et al. [9] is larger than the












thermionic emission resistance for both temperatures. However, the

values are within a factor of two of each other. The very low value

of aH suggests near-ballistic transport.


5.1c High-Frequency Noise

The device exhibits nearly 100% thermal noise at all currents

to 75 mA if sufficiently high frequencies are attained. This current
2
corresponds to a current density of 960 A/cm2. There is no detailed

theory for the noise as yet developed for all applied bias, just the

preliminary calculations of van der Ziel and Bosman [14,15]. For the

collision-dominated space-charge-limited diode at low bias, the noise

is due to diffusion noise sources which transform via Einstein's

relation to 4kT/R At high bias for which the Mott-Gurney law
For a bal-
(I V2) applies, the noise becomes 8kT/R where R For a bal-
x x di.
listic device the noise is due to shot noise. At sufficiently high

bias where soace charge effects dominate, correlations between current

components due to fluctuations in the potential minimum caused by the

space charge lead to subthermal noise.

In order to differentiate between the two models, high bias

must be achieved. Our measurements have found no deviations from

thermal noise. One possible reason for this is that the current density

of 960 A/cm2 is insufficient. Another is that the device is operating

in between the two regimes as suggested by the 1/f measurements so that

extreme bias may be required to see which effect dominates.












5.2 The n p n Device

5.2a Current-Voltage Characteristic and Impedance
+ +
The n p n device shows substantial nonlinearity. There are

two regimes with a transition region in between as shown in Figure

4.10. At low bias, the device is linear with a DC conductance at room

temperature 100 times smaller than the n-type device. It decreases

with decreasing temperature to a limiting value a factor of 3 below its

room temperature value. It reaches this value near 150 K and remains

constant thereafter down to 12 K. The high bias regime is temperature

independent and is also linear with a large conductance 4 times less

than the n-type device.

Due to the thinness of p-regions and the large doping density

of the n regions, the spillover of electrons into the p-regions is

not very different than for the n-type diode case [9]. Then current

flow should be by nearly ballistic electron emission through the poten-

tial minimum. The resulting characteristic should be linear with a

large conductance and nearly temperature independent. This appears to

be a good model for the high bias regime. At low bias, the model

fails, however. One possible reason for this is that enough holes re-

main in the p-region to control the transport ambipolarly. The details

of such an effect are unclear, but a qualitative description by

Dr. C. M. Van Vliet follows. The motion is not strictly ambipolar,

since there is space charge, as indicated by the presence of the poten-

tial minimum [9], which even for V = 0 can be computed from Poisson's

equation. Therefore, near the potential minimum the excess electron












charge is small. Roughly speaking, only ambipolar pairs with energies

within kT of the potential minimum are able to cross the minimum.

If the injected carrier density An is less than p ata point (labeled x')

approximately kT greater in energy than the minimum on the injecting

side, then the current will be ambipolar (instead of ballistic).

Clearly, with decreasing T, p(x') decreases. Thus, with decreasing T

the ambipolar current decreases, and the transition to ballistic

behavior-which is independent of T-sets in at lower bias. The I/V

versus V characteristic is therefore as shown in Figure 4.10.

In any case, assuming that holes control the mobility for low

bias, and electrons at high bias gives a factor of 21 change in the con-

ductance at room temperature since typical values are pn 8,500 cm2
n V-sec
2
and p k 400 cm /V-sec [25]. There still remains to be explained the

factor of 4 difference between the n-device conductance and the high

bias p-device conductance. The potential barrier is larger in the

p-type device [26]. In the p- region, Poisson's equation is


2 q(n + N p)
a (5.10)
dx 0


where '(x) is the potential, n(x) the electron density, p(x) the hole

density, Na the acceptor density, and ee0 the dielectric constant. For

the n-device in the n region


d2 q(n Nd)
x Nd (5.11)
dx2 E0












If p(x) is assumed negligible due to electron spillover,


then


I 22 dp
p-


(5.12)


so that the n-device barrier is smaller. The above model of ambipolar

collision-dominated flow at low bias and near-ballistic electron

emission at high bias gives a qualitative explanation of the experi-

mental data.


5.2b Noise
+ +
The n p n device showed much larger levels of noise than the

n-type device. The frequency dependence of the spectra for all

measured currents is about (1/f)75 for all measured bias currents at

room temperature. Then Hooge's parameter is not well defined since it

is not dimensionless (unless the slope is -1) and depends on frequency.

The noisiness at 10 Hz and 100 pA is


SAI(f)f
Ix -9
2 = 6 x 10-9
I


(5.13)


This is about 4 x 106 times

slopes do not change in the

magnitude of the noise does

seen at low bias, however.

with the low bias transport


the n-type device value. The spectral

transition from low to high bias. The

fall off at high bias from the 12 dependence

This suggests that the noise is associated

mechanism and that the high bias mechanism






83





is much less noisy. That is in good agreement with the conjecture

of ambipolarly governed flow at low bias and near-ballistic flow at

high bias. The -0.75 slope is not common although van de Roer [27]

has also found spectra going slower than 1/f in 6 im p n p punch-

through diodes. The spectra become more closely 1/f as temperature

decreases. At 77 K the slope is -0.85 which is common for intermediate

temperatures down to 12 K. The magnitude of the noise is


= 3.3 x 10-8


(5.14)


at 10 Hz and 100 pA.

magnitude


S A(f)f
2
I


at 10 Hz and 100 yA.

slope is unclear.


Finally, at 12 K, the slope is fully -1 with


(5.15)


The explanation of this temperature-dependent


S (f)f
2
I


- 10
















CHAPTER VI
CONCLUSIONS AND RECOMMENDATIONS
FOR FURTHER WORK


6.1 The n+n n Device
+ +
The n n n device is nearly ballistic. Carrier transport is

by both thermionic emission and by collision-based diffusion-drift.

Neither process can be neglected in a physical model. For the I-V

characteristic, it is desired to calculate n(x), v(x), and J(V) for

any applied bias, expanding on the calculations of van der Ziel et al.

[9] for small V. Alternately, the model of Holden and Debney [12]

should be further developed to investigate the effect of adding col-

lisions to their model. Also the temperature-dependence and the

effects of replacing the Maxwell-Boltzmann velocity distribution by

a Fermi-Dirac distribution should be investigated.

Since nonlinearity in the I-V characteristic appears at high

bias, very fast (1l nsec) pulse neasurements at yet higher bias

would be very interesting. The mechanism causing this nonlinearity

should be identified.

For the noise, the measured value of aH found was extremely

low. This is the best confirmation yet that lattice phonon scattering

causes 1/f noise. It would be valuable to repeat the measurement on

other mesas. Perhaps in such small devices Handel's [24] fundamental

theory can be tested.












In the thermal noise region, the results were inconclusive,

neither subthermal nor greater than thermal noise was found. Perhaps

the bias level was not great enough. In that case pulsed noise

measurements could be attempted. This may be difficult since the noise

is already very low. A detailed and complete noise theory for these

devices is desirable, including collision effects and velocity dis-

persion. A Langevin equation based on the momentum and energy balance

equations of Shur and Eastman [5,6] but including the above effects

should give accurate results.

+ +
6.2 The n p n Device

The p-type device was very intriguing, beginning with the

I-V characteristic. At present it is believed that at low bias the

carrier transport is ambipolar and collision-limited, centering about

the potential minimum. At high bias, the injected electrons overrun

the holes and near-ballistic electron flow results similar to the n-

type device. The conductance is less because the potential barrier is

greater. This model deserves further detailed investigation.

The noise is very large and shows slopes becoming progressively

less 1/f as temperature is increased. This interesting characteristic

suggests that subtle and complex mechanisms may be taking place that

are not well understood yet. To develop a theory that explains these

effects would be a great step forward in our knowledge of very small

devices. The 1/f noise extends to the GHz range to frequencies greater

than our measurement system so that thermal noise measurements have still












to be done. In addition, only these low-bias measurements have

been done; pulsed noise measurements in the high-bias regime are

underway; they may yield valuable information.

Investigation of this device has led to another novel device
+ +
[28]. A p n p device with nonlinear characteristics is predicted to

have negative differential conductance. At low bias, electron-

controlled ambipolar flow should give a large conductance. At larger

bias, hole injection takes over with collision-limited flow resulting

in a low conductance. The transition region, therefore, should show

the negative differential conductance.



































APPENDIX
COMPUTER PROGRAMS FOR THE HP 9825
















APPENDIX
COMPUTER PROGRAMS FOR THE HP 9825


A.1 Correlation-System Three-Measurement Averaae

This program calculates equations for the current noise spectral

density of a device under test (DUT) using the three-measurement

technique. Actually, the magnitude of the calibration signal is also

recorded. The program calculates data for eight logarithmically equal-

spaced points covering one decade in frequency. It is used with the

HP 3582A spectrum analyzer which features a dual-channel Fast Fourier

transform algorithm for frequencies from 0.02 Hz to 25 kHz. Many data

sets can be averaged to increase the accuracy of the measurement which

is described in Chapter III.


0: "REPEATED-3 MEASUREMENT CORRELATION SYSTEM":
1: 0A-AB-C-D
2: dim A[24],B[24],C[24],F[8],M[8],S8][8] 8],Q$[1],LS[20]
3: dim MS[8,5];"MP12"-M$[1];"MP18"M$[2]
4: "MP25"-MS[3];"MP35"-M$[4];"MP50"-M$[5]
5: "MP70"-MS[6];"MP95"-*M$[7];"MP125"-M$[8]
6: 12S[l];18S[2];25S[3];35+S[4];5Q0S[5];70S[6];95-S[7];125-S[8]
7: ldf 1,A,B,C,D,A[*],B[*],C[*],F[*],M[*]
8: fxd 0;dsp A,B,C,D,F[8];beep;stp
9: fit 3
10: "n"-W$;ent "DUT ON (y/n)",QS;if Q$="y";gto "DUTON"
11: "n"Q$;ent "DUT OFF (y/n)",Q$;if Q$="y";gto "DUTOFF"
12: "n"-QS;ent "CAL ON (y/n)",QS;if Q$="y ;gto "CALON"
13: "n"-QS;ent "CAL MAG (y/n)",O$;if Q$="y";gto "CALMAG"
14: "n"-QS;ent "CALC NOISE (y/n)",Q$;if Q$="y";gto "NOISE"
15: "n"*Q$;ent "CLEAR DATA FILE (y/n)",Q$;if QS="y";gto "CLEAR"
16: "DONE":
17: rcf 1,A,B,C,D,A[*],B[*],C[*],F[*],M[*]
18: fxd O;dsp A,B,C,D;beep






89




19: Icl 7i1;end
21: "DUTON":
21: for I=l to 8;gsb "GET"
22: if A=0;X-A[I];Y*B[I];Z-C[I];next I
23: if A0O;jmp 4
24: (X+A*A[I])/(A+1)+A[I]
25: (Y+A*B[I])/(A+1)B[I]
26: (Z+A*C[I])/(A+1)*C[I];next I
27: A+1-A;gto "DONE"
28: "DUTOFF":
29: for I=1 to 8g:sb "GET"
30: I+8-J
31: if B=0;X-A[J];Y-B[J];Z-C[J];next I
32: if B=0:jmp 3
33: (X+B*A[J])/(B+1)*A[J];(Y+B*B[J])/(B+1)*B[J]
34: (Z+B*C[J])/(B+1)C[J];next I
35: B+1-B;gto "DONE"
36: "CALON":
37: for I=1 to 8;gsb "GET"
38: I+16-~1
39: if C=0;X-A[J];Y-B[J];Z-C[J];next I
40: if C=0;jmp 3
41: (X+C*A[J])/(C+1)-A[J];(Y+C*B[J])/(C+1)*B[J]
42: (Z+C*C[J])/(C+1)C[J];next I
43: C+1-C;gto "DONE"
44: "CALMAG":
45: wrt 711,"AAIMN1MB1"
46: for I=1 to 8
47: wrt 711, "MP",S[I],"LMK";red 711,U,V
48: if A+B+C+D=0;V-F[I]
49: if V#F[I];beep;dsp "FREQUENCY MISMATCH";stp
50: tnt(U/10)-U
51: if D=0;U-M[I];next I
52: uf D#0;(U+D*M[I])/(D+1)4M[I];next I
53: wrt 711,"AA0";D+1-D;gto "DONE"
54: "NOISE":
55: ent "TEtFP",T;ent "DUT RESISTANCE",R
56: ent "CAL RESISTANCE",S;ent "DC CURRENT",U
57: gsb "LABEL"
58: for I=1 to 8;spc
59: v/(A[I]*B[I]*C[I])-V;Y/(A[I+8]*B[I+8]*C[I+8])*W
60: v(A[I+16]*B[I+16]*C[I+16])X
61: (V-W)/(X-W)-Y;M[I]/S/S-Z
62: Y*Z+4*1.38e-23*T/R-N
63: prt "SI=",N;fxd 1;prt "FREQ=",F[I];flt 3
64: next I
65: spc ; spc ;gto "DONE"
66: "CLEAR":
67: 0A-+B-tC-D
68: for I=l to 8;0F[I]-1M[I];next I











69: for I=1 to 24;0-*A[I]-B[I]C[I];next I
70: gto "DONE"
71: "GET":
72: gsb "DATA"
73: if A+B+C+D=0;V-F[I]
74: if V#F[I];beep;dsp "FREQUENCY MISMATCH";stp
75: asb "COH"
76: tnt(X/10)X;tnt(Y/10)Y;ret
77: "DATA":
78: wrt 711,"AAlMNlMB1"
79: wrt 711,MS[I]
80: wrt 711,"LMK";red 711,X,V
81: wrt 711,"AA0AB1LMK";red 711,Y,V
82: wrt 711,"ABO";ret
83: "COH":
84: wrt 711,"LFM,",76000+dto(4*S[I]),",4";red 711
85: for J=1 to 4;rdb(731)-U;rdb(731)-Z
86: ior(rot(U,8),Z)-X[J];next J
87: wrt 711,"LFM,",77000+dtoS[I],",1";red 711
88: rdb(731)-X[2];rdb(731)-X[4]
89: wrt 711,"LFM,",75000+dtoS[I],",1";red 711
90: ior(rot(rdb(731),8),rdb(731))-X[5]
91: wrt 711,"LFM,",75200+dtoS[I],",1";red 711
92: ior(rot(rdb(731),8),rdb(731))-X[6]
93: wrt 711,"LFM,",77200+dtoS[I],",1";red 711
94: rdb(731)-X[7];rdb(731)X[8]
95: X[1]*2t(X[2]-15)X[1]
96: X[3]*2+(X[4]-15)X[3]
97: X[5]*2+(X[7]-15)X[5]
98: X[6]*2+(X[8]-15)-+X[6]
99:(X[5]+2+X[6]+2)/X[1]/X[3]+Z
100: ret
101: "LABEL":
102: ent "Label",L$;spc ;prt L$
103: prt "DCI=",U;prt "TEMP=",T
104: prt "DUT RES=",R;prt "CAL RES=",S
105: prt "# SETS=",A
106: ret
*19711


A.2 Wide-Band Filter Synthesizer

This program is written for the HP 9825 computer in conjunction

with the HP 3582A spectrum analyzer. The high-frequency 4/5 of the

FFT bins are averaged to synthesize a wide-band filter to use with the












radio frequency FFT system which utilizes a mixer just before the

spectrum analyzer. Without the use of this program, the accuracy

of a measurement is poor since the bandwidth is then very small com-

pared with the frequency of interest. A bandwidth of 20 kHz can be

generated by setting the frequency span on the 3582A to 25 kHz,

although the program displays the filter output in normalized dBV//Vz.



0: "Last 4/5'ths of display averager:
1: dim B[10],Y[256]
2: wrt 711,"LFM,77454,5"
3: red 711
4: for I=1 to 10;rdb*711)-~[I]
5: next I
6: wrt 711,"LSP";red 711,S
7: if B[3]>0;sfg 1
8: B[2]-H
9: if H>127:H-128-H4
10: if H>63;H-64-H
11: if H>31;H-32-H
12: if H>15;H-16-H
13: if H>7;H-8-H
14: if H=0;250-B
15: if H>1;68.87+B
16: if H>3;166.6667-B
17: B[4]-H
18: if H>127;H-128-H
19: if H>63;H-56.-H
20: if H>31;H-32-4
21: if H>15;H-16-*H
22: if H>7;H-8-4H
23: if H>3;H-4-+H
24: if H>2;sfg 2
25: 128-N;if flgl;256-N
26: if not flgl;B/2-B
27: if flg2;256-N
28: wrt 711,"LDS"
29: red 711
30: for I=1 to N;red 731,Y[I]
31: next I
32: Icl 711
33: if flg2;128+N
34: 0-A;fxd 4
35: gsb "calc"






92




36: if flg2;XYY;0+X;128-A;gsb "calc"
37: if flg2;prt 20*log(Y/.8N),201og(X/.8N);end
38: prt 201og(X/.8N);end
39: "calc":
40: for I=A+N/5 to A+N;X+tnt(Y[I]/20)//(S/B)-X;next I
41: ret
*30119
















REFERENCES


1. L. F. Eastman, R. Stall, D. Woodard,
M. S. Shur, and K. Board, "Ballistic
Room Temperature," Electron. Letters


N. Dandekar, C. E. C. Wood,
Electron Motion in GaAs at
16, 524 (1980).


2. M. S. Shur and L. F. Eastman, "Ballistic Transport in Semiconduc-
tor at Low Temperatures for Low-Power High-Speed Logic," IEEE
Trans. Electron Devices ED-26, 1677 (1979).

3. J. R. Barker, D. K. Ferry, and H. L. Grubin, "On the Nature of
Ballistic Transport in Short-Channel Semiconductor Devices," IEEE
Electron Device Letters EDL-1, 209 (1980).

4. R. Zuleeg, "Possible Ballistic Effects in GaAs Current Limiters,"
IEEE Electron Device Letters EDL-1, 234 (1980).

5. M. S. Shur, "Ballistic Transport in a Semiconductor with Col-
lisions," IEEE Trans. Electron Devices ED-28, 1120 (1981).

6. M. S. Shur and L. F. Eastman, "Near Ballistic Electron Transport
in GaAs Devices at 770K," Solid-State Electron. 24, 11 (1981).

7. J. J. Rosenberg, E. J. Yoffa, and M. I. Nathan, "Importance of
Boundary Conditions to Conduction in Short Samples," IEEE Trans.
Electron Devices ED-28, 941 (1981).

8. W. R. Frensley, "High-Frequency Effects of Ballistic Electron
Transport in Semiconductors," IEEE Electron Device Letters EDL-1,
137 (1980).


9. A van der Ziel, M. S. Shur, K. Lee,
"Carrier Distribution and Low-Field
and n+-p--n+ Structures," submitted


T. Chen, and K. Amberiadis,
Resistance in Short n+-n--n+
to Solid-State Electron.


10. R. K. Cook and J. Frey, "Diffusion Effects and 'Ballistic Trans-
port'," IEEE Trans. Electron Devices ED-28, 951 (1981).


11. Y. Awano, K. Tomizawa, N.
Carlo Particle Simulation
Electron. Letters 18, 133


Hashizume, and M. Kawashima, "Monte
of GaAs Submicron n+-i-n+ Diode,"
(1982).


12. A. J. Holden and B. T. Debney, "Improved Theory of Ballistic Trans-
port in One Dimension," Electron. Letters 18, 558 (1982).












13. T. C. Fry, "The Thermionic Current between Parallel Plane Elec-
trodes; Velocities of Emission Distributed according to Maxwell's
Law," Phys. Rev. 17, 441 (1921).

14. A. van derZiel and G. Bosman, "Collision-Free Limit of Near-
Thermal Noise in Short Solid-State Diodes," Physica Status Solidi
a) 73, K93 (1982).

15. A. van derZiel and G. Bosman, "Collision-Dominated Limit of Near-
Thermal Noise in Short Solid-State Diodes," Physica Status Solidi
(a), 73, K87 (1982).

16. F. N. Hooge, T. J. G. Kleinpenning, and L. K. van Damme, "Experi-
mental Studies on 1/f Noise," Reports Progress in Physics 44,
479 (1971).

17. A. van der Ziel, "Flicker Noise in Electronic Devices," Advances
in Electronics and Electron Physics 49, 225 (1979).

18. F. N. Hooge, "1/f Noise Is No Surface Effect," Phys. Letters A29,
139 (1969).

19. A. van der Ziel and C. M. Van Vliet, "Mobility-Fluctuation 1/f
Noise in Nonuniform Nonlinear Samples and in Mesa Structures,"
Physica Status Solidi (a) 72, K53 (1982).

20. G. Bosman, R. J. J. Zijlstra, and A. D. van Rheenen, "Flicker
Noise of Hot Electrons in Silicon at T = 78 K," Phys. Letters A78,
385 (1980).

21. T. M. Chen and A. van der Ziel, "Hanbury Brown-Twiss Type Circuit
for Measuring Small Noise Signals," Proc. IEEE 53, 395 (1965).

22. L. D. Enochson and R. K. Otnes, Programming and Analysis for
Digital Time Series Data, Shock and Vibration Center, Naval
Research Laboratory, Washington, D.C., 201 (1968).

23. J. Kilmer, A. van der Ziel, and G. Bosman, "Presence of Mobility-
Fluctuation 1/f Noise Identified in Silicon P+NP Transistors,"
Solid-State Electron. 26, 71 (1983).

24. P. H. Handel, "Quantum Approach to 1/f Noise," Phys. Rev. A22,
745 (1980).

25. S. M. Sze, Physics of Semiconductor Devices, 2nd ed., Wiley-
Interscience, New York, 851 (1981).

26. R. R. Schmidt, G. Bosman, A. van der Ziel, C. M. Van Vliet, and
M. Hollis, "Nonlinear Characteristic of Short n+p-n+ GaAs Devices,"
submitted to Solid-State Electron.




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O p^|^|§jill


NOISE AND CURRENT-VOLTAGE CHARACTERISTICS
OF NEAR-BALLISTIC GaAs DEVICES
By
ROBERT ROY SCHMIDT
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1983

ACKNOWLEDGMENTS
The author wishes to express his sincere gratitude to
Dr. C. M. Van Vliet and Dr. G. Bosman for their research guidance and
aid in improving drafts of the work, and to Dr. A. van der Ziel,
Dr. E. R. Chenette, and Dr. A. Sutherland for their helpful suggestions
and kind interest. He wishes to thank Mark Hollis for fabricating the
devices.
Finally, the author appreciates the help of his fellow students
in the Noise Research Laboratory, especially Bill Murray for drawing
most of the figures contained herein.

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
ABSTRACT v
CHAPTER
I INTRODUCTION 1
II THEORY OF VERY SMALL LAYERS IN GaAs 3
III EXPERIMENTAL PROCEDURES AND MEASUREMENT CIRCUITS 12
3.1 Current-Voltage Measurements 12
3.2 The 0.24 Hz to 25 kHz Correlation System .... 14
3.3 The 50 kHz to 32 MHz System 40
IV MEASUREMENT RESULTS 46
4.1 Current-Voltage Characteristics 46
4.1a The n+n_n+ Device 46
4.1b The n+p-n+ Device 49
4.2 The n+p_n+ Device Noise 56
4.3 The n+n_n+ Device Noise 63
V DISCUSSION OF EXPERIMENTAL RESULTS 75
5.1 The n+n_n+ Device 75
5.1a Current-Voltage Characteristic
and Impedance 75
5.1b Excess 1/f Noise 76
5.1c High-Frequency Noise 79
5.2 The n+p_n+ Device 80
5.2a Current-Voltage Characteristics
and Impedance 80
5.2b Noise 82
VI CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER WORK 84
6.1 The n+n~n+ Device 84
6.2 The n+p_n+ Device 85
i i i

Page
APPENDIX: COMPUTER PROGRAMS FOR THE HP 9825 88
REFERENCES 93
BIOGRAPHICAL SKETCH 96

Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
NOISE AND CURRENT-VOLTAGE CHARACTERISTICS
OF NEAR-BALLISTIC GaAs DEVICES
By
Robert Roy Schmidt
April 1983
Chairperson: C. M. Van Vliet
Major Department: Electrical Engineering
Conduction processes in novel submicron n+n"n and n+p”n+ mesa
structures in GaAs are investigated. The widths of the n-type and p-
type layers are 0.4 pm and 0.47 ym, respectively. These small devices
are of interest due to the possibility of ballistic (collision free)
transport which leads to picosecond switching times at femtojoule
power levels, since the velocities of the carriers may be much greater
than the collision-limited drift velocity.
Three noise measurement systems are described, covering the
frequency range 0.2 Hz to 64 MHz. The first system, for frequencies up
to 25 kHz, features a spectrum analyzer with a dual-channel fast-
Fourier transform algorithm. By exploiting the correlation feature,
the system, when combined with very low-noise preamplifiers, has a
background noise level for a low-impedance source of less than 0.3 ohms
noise resistance. The other two systems for radio frequencies use a
v

tuned step-up transformer to bring the noise of the very quiet n-typr
device to a level above that of the RF preamplifier. One is a conven¬
tional single-channel system. The other uses RF mixers before the
dual-channel FFT analyzer to extend the range of the correlation system
to high frequencies, depending on the preamplifier frequency response.
Current-voltage measurements, DC, pulsed DC, and AC impedance,
are presented versus parameters bias level, temperature, and frequency,
as are noise measurements. The n-type device is linear and relatively
temperature independent exhibiting full thermal noise and, at low
frequencies, very low levels of excess 1/f noise five orders of magni¬
tude less than for the bulk material. This suggests that collisions are
mainly absent from this device. The p-type device, conversely, is
nonlinear with large levels of excess low frequency noise. There is
a linear, noisy, and temperature-dependent low-bias region followed by
a transition to a less noisy temperature-independent high-bias regime
with large conductance. This may represent a transition from ambi-
polarly governed to near-ballistic transport.
vi

CHAPTER I
INTRODUCTION
There is great interest in very thin GaAs layers of lengths
less than 1 ym. This is of the order of the mean free path lengths
of the dominant collision mechanisms for electrons. Under favorable
conditions, carriers may cross the layers undergoing few or no col¬
lisions, which leads to very high velocities, much greater than the
collision-limited drift velocity. The resulting so-called ballistic or
near-ballistic transport leads to very fast-response devices for pico¬
second switching at very low power or other new applications. These
new devices have been made possible by advances in fabrication tech¬
niques such as electron beam lithography and, in particular, molecular
beam epitaxy.
Such small devices enable us to investigate physical mechanisms
on a small scale for which the traditional models no longer apply. In
addition, practical knowledge of the performance limitations of these
new devices is obtained.
Early theories of these devices treated the ballistic motion of
the carriers as the dominant mechanism affecting its characteristics.
Recent measurements and later theories have shown that boundary condi¬
tions, in particular the velocity dispersion of the carriers, have
much greater influence.
The organization of the chapters follows. In Chapter II some
theories of small n+n n+ and n+p"n+ layers in GaAs are surveyed and
1

2
predictions are noted. Measurement techniques and procedures are
presented in Chapter III. In Darticular, a crosscorrelation noise
measurement system with a noise resistance less than 0.3 ohms suitable
for low-impedance low-noise devices such as the n+n"n+ diode is dis¬
cussed. In Chapter IV, experimental results of measurements on two
devices are described. One is an n+n"n+ structure of length 0.4 ym.
The other is an n+p n+ device of length 0.47 ym. Current-voltage charac¬
teristics, DC, pulsed DC, and AC to 25 kHz are presented, as are
measurements from 0.2 Hz to 32 MHz. The measurements are repeated at
77 K to investigate the effect of temperature. In some cases, T is
reduced to 12 K. Chapter V contains a discussion of the results.
Finally, conclusions and recommendations for further work are presented
in Chapter VI. An HP (Hewlett Packard) 9825A computer program used
to automate the noise measurements is recorded in the Appendix.

CHAPTER II
THEORY OF VERY SMALL LAYERS IN GaAs
In most semiconductor devices, the drift current is limited
by the rate of carrier collisions. However, for sufficiently thin
devices few or no collisions occur permitting ballistic or near-
ballistic transport. The resulting carrier velocities can be much
larger than in the collision-dominated case so that such parameters as
switching speed can be greatly increased. Advances in fabrication
techniques such as molecular beam epitaxy and electron lithography have
resulted in submicron gallium arsenide devices with dimensions the
same order of magnitude as the mean free paths of the dominant collision
mechanisms. Thus near-ballistic transport in these devices becomes
probable.
For polar optical phonon emission at room temperature, Eastman
et al. [1] report mean free path lengths in GaAs of about 0.1 pm for
electron energies near 0.05 eV to 0.2 pm up to 0.5 eV. Phonon absorp¬
tion is about four times less probable. They also report that the
change in the direction of motion due to a collision is small (5-10
degrees). Intervalley scattering becomes significant for higher elec¬
tron energies than 0.5 eV. Shur and Eastman [2] calculated the mean
free path in the high-purity GaAs at 77 K to be 1.3 pm. Barker, Ferry,
and Grubin [3] have suggested some complicating factors even for
devices of length equal to the mean free path. First, in large area
3

4
devices, some carriers may move at angles to the length direction,
thereby increasing the distance traveled and the number of collisions.
Also, some of these carriers may collide with the device boundaries.
Third, space-charge effects may limit the current due to Coulomb
scattering. Finally, short devices may be dominated by contact effects
such as carrier reflection.
The theory for pure ballistic transport was revived from the
days of vacuum tubes by Shur and Eastman [2] based on a simple model.
They assumed a one-dimensional n-type device of length, L, neglecting
diffusion and electron scattering. Then the current density is
J = qnv (2.1)
where q is the electronic charge, n is the free electron density, and
the velocity, v, is found from
qVx = I m*v2 (2-2)
•k
where V is the voltage at x due to the applied potential, V, and m is
X
the effective mass. The initial velocity and the field at the injecting
contact are assumed to be zero. Poisson's equation
q
£ £
0 r
n)
(2.3)
is then solved yielding Child's law

5
£0£r v3/2
(2.4)
for large bias voltage and large consequent injected space charge. For
small applied voltages, the fixed charge due to ionized donors is
greater than that due to the injected carriers and an electron beam
1 /2
drift region (J 'v V ) results. The range of voltages for which
their model is valid is
kT
q
V
(2.5!
where T is the lattice temperature and AE is the energy difference
between the main conduction band minimum and the secondary valley.
For very small voltages diffusion cannot be neglected. Intervalley
scattering is the cause of the upper voltage limit. Some measurements
[1,4] have been reported that appear to support this model, but their
interpretation has been questioned as will be discussed.
To take into account the effect of a few collisions, Shur [5]
and Shur and Eastman [6] have extended the theory. Frictional "drag
terms" to represent the collisions are added to the equations of
balance of energy and momentum. In general, the effective mass, m ,
and relaxation times, x and x~, are functions of energy so that the
m E
balance equations are
dm*(E)v _ m*(E)v
—dt qE ‘ TTeT
ITT
(2.6)

6
and
(2.7)
Shur assumes a constant energy-independent effective mass and a single
momentum relaxation time, x, so that, neglecting (2.7), (2.6) is
replaced by
Equations (2.1), (2.3), and (2.8) along with the boundary conditions
for the ballistic case are then solved analytically. For t much longer
than the transit time and sufficient apolied voltage to justify
neglecting n^, Child's law is again obtained. Conversely t much less
than the transit time yields (forV= -yE where y is the mobility) the
Mott-Gurney law for collision-dominated space-charge limited current
(2.9)
In the second paper, values of m*(E), v(E), xm(E), and x^(E) are
obtained from Monte Carlo calculations. Steady state is assumed and the
boundary conditions are taken to be the initial field (and velocity)
equal to zero. Again letting x^ and xm become large yields Child's
law. For the general case, computer solutions are presented for various
lengths and doping densities which show small deviations from their
previous (ballistic) results.

7
3/2
The interpretation that current proportional to V ' indicates
ballistic transport has been seriously questioned by some authors.
Rosenberg, Yoffa, and Nathan [7] have shown the crucial importance of
the boundary conditions. For the ballistic case, they assumed a
simple one-dimensional model neglecting collisions. For each electron,
they solve equations (2.1), (2.3), and the energy-velocity relationship
\ (v2 - v*) = q(V. - V) (2.10)
where v. and V. are the initial velocity and potential, respectively. A
displaced Maxwellian distribution of initial velocities characterized by
a temperature and mean velocity is assumed. A set of nonlinear equa¬
tions in V result which are solved numerically for specified electric
field, E, initial carrier density, n^ , and initial (temperature normal¬
ized) mean velocity, v. They demonstrate that a wide variety of current
versus potential curves can be generated for various choices of initial
conditions [7, Figure 3], The authors agree with others [3,8] that,
conversely with Shur and Eastman, the current-voltage characteristic may
be used to infer the boundary conditions for a device that has already
been shown to be ballistic by other means (such as noise behavior).
Another significant effect in short devices is "spillover" of
carriers from high- to low-doped regions. The result is that the
effective length of the low-doped region is shortened and hence the
resistance is less than expected naively even for devices many Debye
lengths long [7], Universal curves have been calculated by van der Ziel

et al. [9] from which the magnitude of the effect can be determined
for a particular device. The authors calculated the near-equilibrium
resistance for a 0.4 ym n-type device with doping densities Nd+ =
10 cm’ and = 10 cm’ for liquid nitrogen and room temperatures.
They solve Poisson's equation including electrons and ionized donors
in the high- and low-doped regions, neglecting holes. They assume
n(x) = Nd+ exD (2.11)
where N^+ is the ionized donor density in the highly doped region and
ik(x) is the potential. Reciprocal mobilities due to diffusion/drift
and thermionic emission are added to give an effective mobility which
agrees within 1 Of of experimental results at room temperature. However,
a similar calculation for an n+p’n+ device is off by more than an order
of magnitude. They assumed that the electron spillover depletes the
holes in the p-region such that they (holes) may be neglected. This is
apparently not the case; the effects of the holes are important.
The simple theory of ballistic [2] and near-ballistic [5,6]
transport does not take into account the energy or velocity distribu¬
tions of the carriers. Cook and Frey [10] have suggested the inclusion
of an electron temperature gradient term in the momentum balance
equation for this purpose. This may be sufficient for first order
effects such as the current-voltage character!-stic. Their treatment
includes an average collective velocity dispersion term,, in essence
treating the ensemble as a single particle for such effects as the

9
noise, a drawback they themselves point out in the simple theory.
For the noise, more adequate analysis should include a Langevin or
Monte Carlo approach based on the momentum and energy balance equations
which include terms necessary to account for the significant physical
mechanisms. Such a Monte Carlo calculation for a 0.25 pm n+-i-n+
diode at 77 K has been reported by Awano et al. [11]. Ionized impurity
and intervalley and intravalley phonon scattering is included in their
model but not nonparabolicity of the band edges. Particles with a
distribution of velocities in equilibrium with the lattice at 77 K are
injected at one contact. They find that a large proportion of the
particles are transported ball istically. Groups of particles cor¬
responding to one or two collisions and some backscattering from the
collecting contact is observed. For potentials less than 0.5 volts,
the current-voltage characteristic is surprisingly close to Child's
vacuum diode law.
Holden and Debney [12] have presented a calculation based on
ideas from Fry's theory for thermionic values [13]. They assume injec¬
tion from both n+ regions of carriers with Maxwell-Boltzmann velocity
distributions. Scattering is neglected. Following Fry, the free-
carrier charge density is found by an integration over the injected
carrier velocity distribution. A potential minimum due to injected
space-charge is assumed. They calculate current-voltage characteristics
for T = 77 K and lengths of 0.1, 0.2, and 0.5 pm. The slopes at high
bias are less than 3/2 (L = 0.1 and 0.2 pm give r^1.3, L = 0.5 pm gives
^1.14). At low bias, the characteristics are somewhat more linear.

10
The authors conclude that ballistic effects do not lead to a particular
current-voltage power law and suggest that another method must be
found to determine ballistic effects.
The noise behavior of thin structures gives important informa¬
tion on the extent of ballistic effects. Current-voltage character¬
istics for these devices are dominated by events at the boundaries such
as the carrier velocity distribution. In addition, practical knowledge
of the performance limitations of the devices is obtained, as well as
useful insight into the physical mechanisms causing the various types
of noise.
The noise is naturally divided into two frequency regions. At
high frequencies the dominant mechanism, variously called thermal,
velocity fluctuation, or diffusion noise is the electrical result of
the Brownian motion of the carriers. While the theory is not yet
complete, preliminary calculations have been made by van der Ziel and
Bosman [14,15]. For the ballistic case (collisions neglected) at large
bias, the device is modeled similarly to a dual-cathode vacuum diode
with opposing thermionic emission currents across the potential mini¬
mum. Then correlated fluctuations in the current due to fluctuations
in the minimum due to the space charge result in a lowered (64%) level
of noise compared to the low-bias value of thermal noise of the AC
conductance. For the collision-limited case, at sufficiently large
bias, the device is modeled as a space-charge-1imited solid-state
diode and the noise becomes twice the thermal noise of the AC conduc¬
tance. Thus ballistic effects should reduce the noise and

11
collision-dominated transport should increase it if sufficient bias
levels can be achieved.
At low frequencies, excess noise characterized by a spectral
density with a 1/f frequency dependence dominates. According to most
recent theories [16,17] such noise is thought to be caused by mobility
fluctuations which represent fluctuations in scattering cross-sections.
The usual method to describe the noise is using Hooge's empirical
formula [18]
SiI=-FT <2-12
where Iq is the dc current, N is the number of carriers, f is the fre¬
quency, and is Hooge's parameter.
This is accurate for a uniform sample. However, for a non-
homogeneous sample or a mesa structure, van der Ziel and Van VIiet [19]
have shown that the formula becomes
SM(f>
m .2 r L
maH 0
2
fAl/
dx
T
(2.13)
where m is the number of layers, L is the length of one layer, A is the
cross-sectional area, and n(x) is the electron density. The value of
_3
was first thought to be constant (2 x 10 ). Later research has
found that depends on material variation, the dominant scattering
mechanism, and carrier heatinq [20] in Si. Thus the maqnitude of au
should give a good measure of the rate of collisions or lack of them.

CHAPTER III
EXPERIMENTAL PROCEDURES AND MEASUREMENT CIRCUITS
-j" — "t"
The procedures used in the measurement of both devices, n n n
and n+p n+, are similar, but the actual circuits depend on the device
characteristics which are quite different. The p-type near-ballistic
diode (NED) exhibited an impedance about 100 times that of the n-type
device at small bias. It also generated excess noise at low frequencies
orders of magnitude larger than the n-type NBD. Consequently, the noise
of the n-NBD was the most challenging measurement requiring an equiva¬
lent noise resistance of the measurement system of about 0.2 ohms.
3,1 Current Voltage Measurements
Three I-V measurements were done on each diode: DC, pulsed DC,
and AC from 2.4 Hz to 25 kHz. A block diagram of the n-NBD circuit
is shown in Figure 3.1. A three-terminal measurement is required since
the resistance of the active region is comparable to that of the bonding
wire to the top surface of the diode (the devices were mounted in T0-5
cans). The resistance of the output wire, R^, is negligible since it
is in series with the large inDut impedance of the oscilloscope. Then
V2 is the voltage across R^ + R<- which represents the series combina¬
tion of substrate resistance and active region. The current through
the device is
I
(3.1)
12

13
n-NBD
Figure 3.1
Block diagram of n rfn
diode current-voltage measurement

14
Pulsed measurements were done using an HP 214A pulse generator. The
pulse width was 1 ysec and the repetition rate was 100 Hz.
The AC I-V measurement employed the HP 3582A spectrum analyzer.
The dual-channel feature was used to measure the transfer function as
a function of frequency from 2 Hz to 25 kHz and as a function of DC
bias. The circuit diagram of the AC resistance measurement is shown
in Figure 3.2. Taking into account the parallel resistances, is
found to be
Rv =
vn Í
V7 lRc ¡] +
RS + RL1
+ Rs + rli
0 I Rrl
:3.2'
The measurement for the p-type device is similar. Here, the
lead resistances may be neglected since the impedance is large; the
diode may be treated as a two-terminal device.
3.2 The 0.24 Hz to 25 kHz Correlation System
For the electrical noise measurement of low impedance devices
with resistances of the order of an ohm, the usual single-channel
method is not useful. Typical preamplifiers have noise resistances of
about 10 to 20 ohms. The device noise is masked by the preamplifier
noise. Therefore, the correlation method [21] is used as shown in Figure
3.3.
A Hewlett Packard 3582A spectrum analyzer featuring a dual¬
channel Fast-Fourier transform is employed. Low-noise preamolifiers

15
n-NBD
Mil
Figure 3.2 AC resistance measurement circuit for n-NBD

16
Figure 3.3 Correlation measurement setup

17
precede each channel from a common input. By measuring the coherence
(square of the correlation between the channels) the cross-spectra
can be calculated. The noise generated at the output of each channel
is uncorrelated with the other and therfore averages out in the final
reading. Only the noise generated at the input of each channel gives
a contribution to the final reading. Hence a better device noise
versus amplifier background noise ratio may be obtained.
The device under test (DUT) noise is compared to a known cali¬
bration signal which is applied through a series resistor much greater
than the DUT resistance. The bias current is applied in a similar
manner. An equivalent circuit of the measurement setup is shown in
Figure 3.4. To characterize the noise of the system, we first neglect
the effect of the device noise. The series resistors to the bias and
calibration sources are Ri and Rc. The calibration source, v ,,
transforms to a Norton equivalent, i , where
i = V (3<3)
c Rs
The source resistance is represented by R The preamplifiers are
assumed identical (with uncorrelated noise sources, v-|, i-,, v^, and i^)
with gain, G (assumed constant), and input resistance, R.. The output
of the amplifiers, u-| and u^, are then multiplied together and averaged.
The measurement procedure is to compare the output for three con¬
ditions. They are (1) shorted input (R^, = 0) giving output reading M,,
(2) open input (R^, = °°, R<. ]! R¿. >> R^.) giving , and (3) open input

Figure 3.4 Measurement setup equivalent circuit

19
with calibration signal applied (i ? 0) which gives M„. The output
for each case is easily found to be
M2 = G
1111 + V2'
R?
i
4
(3.5)
and
M =
3
(V*
i *
2 2
__ R2
\ l
1 1 / —r-
c c' 4
(3.6)
In reality, is not exactly zero. There is a residual back¬
ground noise. This can be accounted for by introducing a correlated
background noise source, v^, to the input voltage sources so that
V1 = ví + vb and v2 = v2 + V The Pnmed portions are uncorrelated.
Then the analysis yields
M, -
G vtvb
(3.7)
r
G j vbvb + (1111 + 1212y 4
(3.8)
and
IU = G‘
. R2
V,V* + (i 1 if + iol'S + i.i*) 4-
1'1 2 2 c c' T
(3.9)

20
Writing the mean square averages as spectral
_ Sy c
3I(
= 2 9.-c , etc., and noting Sj
R1
voltage noise to be calculated.
S
densities, i.e.,
allows the current and
K =
M, -
3 ¿
Ic
!I1
+ S
12
(3.10)
Thus
sn +s
Vc
12
KR
and
2
RTS,,
i Vc
KR
fM
2
- 1
(3.11)
(3.12)
The current noise spectra of the two channels are and and the
background voltage noise spectrum is S^.
A significant source of background noise, after such causes
as pickup of unwanted signal, ground loops, and power supply noise have
been eliminated, is the finite averaging time of the spectrum analyzer.
The samóle coherence for shorted input versus number of averages is
shown in Figure 3.5. It is a large-side biased estimate inversely pro¬
portional to the number of averages which is limited to 256. The
distribution of sample coherences is not easily described [22] for

COHERENCE
21
O
G
O
O
G
G
G
10 100
NUMBER OF AVERAGES
Figure 3.5 HP 3582A sample coherence for shorted input versus number
of averages

22
small expected value. The residual nonzero value is interpreted as
background noise.
The system is used with two similar sets of preamplifiers
denoted PA1 and PA2; PA2 features five parallel input stages like that
of PA1 but with larger bias current. They are shown in Figures 3.6
and 3.7; PA1 has larger voltage noise but extended low frequency
response suitable for excess 1/f noise measurement. The PA2 has been
designed for minimum voltage noise.
The PA1 features a capacitively coupled common emitter transistor
first stage with shunt feedback. This reduces the input impedance but
does not affect the noise. The primary motivation for this circuit
is a practical one. The RC settling time of the base bias circuit is
substantially reduced compared to the usual configuration. The tran¬
sistor (GE 82) is chosen for large 3 and small series base resistance.
The second stage is a low-noise operational amplifier (Burr-Brown
OPA101BM) in noninverting configuration; 12-volt automotive batteries
are used for the power supply.
An equivalent circuit of the input stage is shown in Figure
3.8. It is the low-frequency hybrid pi including source resistance,
feedback, and noise sources. The source resistance is R<~. Thermal
noise of the series base resistance, r^, is v^. Shot noise of the
base and collector currents are denoted i^ and i , respectively. To
transform this circuit to a form similar to that in Figure 3.4, the
closed loop gain, the input resistance, and the noise sources must
be calculated. The closed loop gain is found by summing currents at

23
INPU
-12 v
Figure 3.6. Model one (PA1) low-noise premplifier

O OI/)F
-12 V
3.6K 9
—wv
4 7 0/4 F
I50K
rAAA/—
INPUT
i e v
4 7 0/4 F
3 6K
15 OK
16 V
I50K
i—VA—[
4 70/4 F I
HI !•'
I 6 V
3-6K
3.6K
150 K
47 0/4 F I
-r
I 6 V
4 70/4 F
3.6 K
150 K
r^VW-
rn
I6V
r/7
600V
0 01/^F
6 0 0 V
0 OI^F
600 V
0 0l/ 600 V
0 0l;(F
600 V
Figure 3.7 Model two (PA2) low-noise preamplifier
OUTPUT

Figure 3.8 PA 1 input stage equivalent circuit

26
the input
i
s
- v,
and at the output
(3.13)
V - V . V
Bii + ~VF~ + = 0 (3.14)
where R^, = rQ | R^5 = and the noise sources have been neglected
vi
for now. Letting r. = r^ + r^ so i.¡ = -¡^r- , the output equation gives i..
in terms of vQ which, when substituted into the input equation allows i
ri
to be expressed as a function of v . Then, for Rr » the closed
o F 6
Iood gain, A, is found to be
The loop transmission is found by disconnecting the output current
generator, g^^, and applying a test current generator, i-, there. The
output voltage, v , is
vo = V [RC 11 (RF + RS !'ri)] (3.16)
The input voltage fed back is

27
v
i
r. Rc
i S
r. + Rc
i S
Rn
r
so that
v.
i
v
o
ri RS
i +
ri RS
+1rs) rf
(3.17)
(3.18)
Then, noting that g v
a am i
91vi gives
T =
g v
ir
-3Rr
RC + RF + R
Rrr.
S i
S + ri
(3.19)
1 + íf
RS
The input resistance is R<. in parallel with R.. where, for Rp » Rp and
Rp » r./6,
RC + RF
R. = r. t— (3.20)
i i r ' '
The output resistance is Rq in parallel with R^ where
R
o
Rc + r.
F i
. R
R
r.
6 + 1 + r1
RS
(3.21)
Except for the noise, the circuit of Figure 3.8 now becomes that of
Figure 3.9 where the voltage gain is

Figure 3.9 Transformed equivalent input circut of PA1

29
(3.22)
For the case that Rp >> R^, r., FT >> r^, R^, and (r^ Rp)/R<. » gR^,
then
-gR
C
G =
r.
(3.23)
The gain becomes open loop as the input is shorted by a small R<.. io
transform the noise sources, the effect of feedback is neglected. Open
circuiting the inputs (R^ = °°) of Figures 3.8 and 3.9 and setting the
output voltages equal yields
(3.24)
Short circuiting (R^ = 0) the inputs yields
(3.25)
Substituting numerical values for components with R<. = 5.6 ohms and
assuming 8 n, 350 yields about 1200 ohms for the input resistance. The
measured value for two PAls in parallel is 660 ohms as shown in Figure
3.10. The input resistance of two PA2s in parallel is shown in Figure
3.10. For the gain the calculated value is about 51 decibels. The
result including the second stage is 90 decibels. The measured value
is 89.5 decibels as shown in Figure 3.12.

INPUT RESISTANCE(KQ)
1.2
1.0
0-8(5
OOoOqOq 0o000o0 0oooo°o000°0°0 Oq
U-b” Oq
0.4
o
o
o
0.2-
0' 1 f 1 —i \-
.25 25 25 250 2.5K 25K
FREQUENCY (Hz)
Figure 3.10 PA1 input resistance

iPUT RESISTANCE (il)
Figure 3.11 PA2 system input resistance

GAIN (dB)
Üíz)
Figure 3,12 PA1 single-channel gain

33
The noise resistance of PA1 and PA2 were measured to be about
35 and r^2.5 ohms, respectively. The five in parallel input stage of
the PA2 gives that factor of reduction in the noise resistance.
Another factor of three reduction comes from increasing the input bias
currents by that amount since, to first order approximations, the
voltage noise sources are inversely proportional to bias [23].
Using the correlation setup gives about a factor of ten reduc¬
tion in the voltage noise. This is seen in the plobs of voltage noise
resistance and current noise conductance for the two setups shown in
Figures 3.13 through 3.16. The noise resistance, R , is defined by
S
Vb
4kTR
n
(3.26)
Similarly, the noise conductance is defined by
S + S
5I1 ^12
4kTG
n
(3.27)
The current noise of the PA2 is much larger than that of the PA1
as would be expected from larger bias currents and five in parallel.
For the sensitive measurements, the current noise is shorted by the
small resistance of the DUT. This is the full current noise which
remains in the correlation calculation. Of the voltage noise, only the
background level remains.
The device noise measurement procedure is very similar to
that for the amolifier. Three conditions are recorded. They are
(1) DUT on, (2) DUT off, and (3) calibration signal applied with DUT
off. Any change in DUT resistance between the on and off state must be

Figure 3.13 Correlation system voltage noise with PA1

NOISE CONDUCTANCE (SIEMEN)
Figure 3.14 Correlation system current noise with PA1

VOLTAGE NOISE RESISTANCE (OHMS)
05
0.4-
03
02
01
o
O
CP
0
O
o °o
oO
o
° oOO ° d>
° o °
o
o
0
-4-
10
[-
100 IK
FREQUENCY (Hz)
-f F
10K 30K
OO
04
Figure 3.15 Correlation system voltage noise with PA2

CURRENT NOISE
CONDUCTANCE (V)
Figure 3.16 Correlation system current noise with PA2

38
taken into account (replacing the device by a resistance equal to its
"on" resistance for the latter two measurements). The equivalent circuit
of the input can be drawn as in Figure 3.17. The DUT is represented
by the parallel combination of R , the AC resistance of the device,
/S^Af, the thermal current noise generator, and /S Af the excess
current noise generator. The calibration signal is /S .jAf. The ampli¬
fier is characterized by Rfl, ysiAf and
A A
For the three conditions, the outputs are
G2[(Sv + Sth + s\)R2 + S¡¡]
(3.28)
G2[(Sth + S¡)R2 + S)(]
(3.29)
and
l,3 = G2[
where G is the (constant) gain and R is the parallel resistance at the
input. Then, as before,
1 - M
1 2
S ,
2 cal
(3.31)
so that the device noise current can be written
DUT
S + S,, = K S . +
x th cal
4kT
(3.32)

r
CO
Figure 3.17 Noise equivalent circuit of the measuring setup

40
Note that for the nonlinear n+p n+ diode, R = -A- and depends on the
bias current applied. An HP 9825A controller can easily automate this
measurement and be used to average many data sets for increased
accuracy. A program which does this for eight logarithmically spaced
points per decade is presented in the Appendix.
3.3 The 50 kHz to 32 MHz System
At radio frequencies, the same three-measurement procedure as
before is used for the noise. The circuit is more conventional, as
shown in Figure 3.8. A step-up transformer consisting of four trans¬
mission line transformers in cascade with capacitance-tuned input and
output is employed for the n-type device. See the Radio Amateur's
Handbook for details. A Micronetics KSD20LEE solid-state noise source
is used for the calibration signal. Frequency selection is done by an
HP 8557A spectrum analyzer. The IF output signal from the spectrum
analyzer is passed through a 21 MHz bandpass filter and detected with an
HP 431C power meter.
The preamplifier, designed and built by Christopher Whiteside
of the noise research group, is shown in Figure 3.19. It employs two
stages of low-noise FETs (2N4393) in cascade configuration to obtain
30 dB gain with a noise voltage resistance of about 70 ohms. This is
less than the stepped-up noise of the DUT.
The HP 3582A spectrum analyzer can be used to make noise
measurements in the radio frequency range. This is accomplished by
mixing down the RF signal just before the analyzer as shown in Figure
3.20. Only a single channel is shown, although the correlation method

CFW-IOI
30\
IF
OUT
HP43IC
POWER
METER
HP8557A
SPECTRUM
ANALYZER
Figure 3.18 Radio frequency noise measurement block diagram

OUTPUT
ro
Figure 3.19 Radio frequency low-noise preamplifier

-p*
OJ
Figure 3.20 Radio frequency noise measurement setup using FFT spectrum analyzer

44
can be used just as easily. The bias and calibration signal are
applied as for the conventional RF system. The circuit for the n-type
device, using a tuned step-up transformer, is shown. The 60 dB radio
frequency preamplifier is actually two of the 30 dB preamplifiers
shown in Figure 3.19 in cascade. The large gain is needed to overcome
the poor noise performance of the mixer (Mini-Circuits ZAD-1). Using a
standard RF signal generator with output amplitude capability of
+7 dBm, the mixer has a frequency response of 100 kHz to 500 MHz.
The same signal generator can be used to supply both mixers
(correlation setup). Then two 60 dB preamplifiers are used and, if
series resistance or unwanted pickup of the step-up transformer is not
negligible, two of these also. Feedthrough of local oscillator har¬
monics may overload the spectrum analyzer at some frequencies if a
bandpass filter similar to that shown in Figure 3.21 is not employed.
This simple filter had good (90 dB) attenuation to 100 MHz. The band¬
width of a single FFT bin of the spectrum analyzer was extremely small
compared to the frequencies of interest, causing significant inaccuracy.
To reduce that, a program to average 80% of the bins to synthesize a
bandwidth of up to 20 kHz was written. It is shown in the Appendix.

IN o
IOQmF 120-íl 120-0.
-AAA/-
I20H<
-JWV
- 0.035,:
Smf
15Hz 38KHz
HIPASS LOPASS
Figure 3.21 Bandpass filter for RF-FFT system
120-0.
AAA/—
OUT
o

CHAPTER IV
MEASUREMENTS RESULTS
The near-ballistic diode (NBD) is a sandwiched mesa structure
of five lightly doped p or n layers, alternating with heavily doped n
layers. A not-to-scale sketch of the p-NBD is shown in Figure 4.1.
18-3 +
The doping densities of the various regions are 10 ^ cm for the n
15 3
regions, approximately 2 x 10 0 crrfJ for the n regions, and approxi-
Id _3
mately 6 x 10 cm for the p regions. The diameter of the mesas is
100 pm. The devices were manufactured by molecular beam epitaxy at
the Cornell University Submicron Research Facility by Mr. M. Hollis.
The mesas have very low-noise ohmic AuGe contacts.
Current-voltage measurements of three types—DC, pulsed DC,
and AC—are reported. Also presented are noise spectra versus parameters
bias, frequency, and temperature.
4.1 Current-Voltage Characteristics
4.1a The n+n~n+ Device
The DC current-voltage characteristics of the 0.4 pm n+n”n' device
at room temperature and 77 K are shown in Figure 4.2 In contrast to
the device reported in Eastman et al. [1], there is no AÍ region.
The characteristic is linear for both temperatures up to very large
bias. Pulse techniques are used for currents greater than 100 mA.
The resistance of the active region is 0.75 ohms at room temperature
46

47
Figure 4.1 P-type near-ballistic diode mesa structure

CURRENT
48
I 0 A
I A
100 mA
10 mA
I mA
O.i mA''
0 -01 mA
0
Figure 4.2
O 300 K
â–¡ 77 K
lm V
Im V
10 mV
ICO mV
V0LTAGÍ
+ - +
DC I - V characteristic of n n n 0.4 jam device

49
and 0.68 ohms at 77 K. This is confirmed by the AC impedance
measurement at 300 K for 1^ = 75 mA shown in Figure 4.3. Low-
frequency equivalent circuits of the device at room temperature and
77 K are shown in Figures 4.4 and 4.5. The series resistances of the
gold wires of the TO-5 can are significant and must be taken into
account.
4.1b The n+p~n+ Device
Conversely, the n+p n+ structure is quite nonlinear. The DC
current-voltage characteristics of the 0.47 urn n+p"n+ diode at 300 K
and 77 K are shown in Figure 4.6. The device is linear for both
temperatures up to about 100 mV. The characteristics then move through
transition to a temperature independent high bias regime in which the
slope falls off. The maximum slopes in the transition regions are
3 for the 300 K case and 4.5 at 77 K. The low-bias resistances at
300 K and 77 K are 90 ohms and 320 ohms, respectively. The measurements
of AC resistance for several bias currents at T = 300 K shown in Figure
4.7 and for T = 77 K shown in Figure 4.8, display no frequency
dependence.
To compare the large and small signal resistances more easily,
a
the exponent, 8, can be examined where I = V. Differentiating both
sides and solving for 8 gives
(4.1)

OF DU
iu
o I
iii
rt
o
o
09
O
o o
G O
o o oo o o
o o o o
o o
G
G
G
G G G
G
G
G
G
G
G
O
10
—I
100
FREQUENCY (HZ)
I K
I0K
I00K
1X1
O
Figure 4.3
AC resistance of n+n n
device versus frequency at room temperature

51
Figure 4.4 Equivalent circuit of n+n nT structure at 300 K showing
parasitic elements

52
Figure 4.5 Equivalent circuit of n+n n structure at 77 K showing
parasitic elements

CURRENT
53
Figure 4.6 DC I-V characteristic of n+p n+ 0.47 urn device

IOO
El
g
go Fi
El
E] El E]
El
El El El
El
El El El
0 A
V
V
vv V
V
VE E/
V
vv E7
V
V V V
1.5 mA
o
O
ooo
o
oo o
O
oo O
o
oo o
3.4 mA
o
.3V in
/
â–¡
â–¡
â–¡ â–¡ n
n
nn â–¡
â–¡
â–¡ â–¡ â–¡
â–¡
â–¡ â–¡ â–¡
13.6 mA
Q
rr
i
O
o
ooo
o
ooo
o
ooo
O
ooo
34 mA
1
10
100
IK
I0K
100 K
FREQUENCY (Hz)
Figure 4.7 AC resistance of 0.47 pm n+p~n+ NBD at T = 300 K

(u)ina p

56
This is plotted in Figure 4.9 for small bias up to 34 mA. At higher
bias, pulsed DC but not AC measurements were done. The resistance
falls dramatically at high bias. This is seen more clearly in Figure
4.10 which presents the DC and pulsed DC conductance versus bias
voltage for several temperatures down to 12 K. The large-bias
conductance in the temperature independent regime rises to a limiting
value of about 0.33 Siemens. In the low-bias regime, the conductance
quickly falls from its 300 K value by about a factor of three at 150 K
to 200 K and remains nearly constant for further decreases in tempera¬
ture .
4.2 The n+p~n+ Device Noise
The n+p n+ 0.47 pm structure exhibited large levels of excess
low-frequency noise. The room temperature noise current spectrum
(less than 25 kHz) for several bias currents is shown in Figure 4.11.
The frequency dependence for all bias currents is about
In
Figure 4.12 is shown the extension to 32 MHz of the spectra for bias
currents of 100 uA and 1 mA. The 100 pA spectrum is approaching the
thermal level at the high-frequency end. The turnover frequency to
thermal noise occurs in the GHz range for larger bias. Figures 4.13
and 4.14 display the complete spectra versus frequency for bias currents
of 34 mA and 68 mA, respectively. Both show slopes of -0.77 over many
decades. No turnover frequency to a different slope is found. These
spectra represent measurements in the low-bias regime and transition
region. Measurement in the high-bias regime requires pulsed techniques.

5
4
O 300K
â–¡ 77K
01
10* A
00/íA ImA
CURRENT
Figure 4.9 Beta versus current at 300 K and 77 K
â–¡
â–¡
O
O
en
IOmA

5)
d
UJ
o
<
i-
C 0
u
Q
6
O
u
Q
0
0 01
0 30 OK
0250 K
V200K
© 150 K
â–¡ 77 K
0 12 K
000!
0 0—O—©-©O 0-©-© O—q—©iO©
in to o
01 mV
-i
I mV
10 mV lOOrnV
VOLTAGE
V
-i-
10V
cn
CO
Figure 4.10 DC conductance versus bias voltage for 0.47 pm
+ -
n p
n device at several temperatures

(A /Hz)
59
-i4 ,
io —
10
-16
10 â– 
ló' ' -
-18
10 -
-19
10
10
-20
O
O
G
o
G
o
J G,
'on
&
u
G G
Oo
ccc©
Q Gc-><
G q
U . ,
o v
G
G
••o —i
qG.
o
o
o
CvQ0
o
0
Oo
â–¡ r
O O ^ 0 o>
G Gm ° n
O
o.
0
0
On
0
o
v
G
0^
°o
gg ^Gn
G G
G G
G
G G
O
G
O
G
G G
0
G
Gr
“V
Oo
v
G
"0
G
G
0
G
o
G 3 4mA
G 13.6mA
O 3.4 m A
â–¡ I mA
O 400/xA
G 10 0 ¿A A
^G
o
o
G
Oo
G
G
G
G
V<
10
100 IK
f (Hz)
I0K
Figure 4.11 Low-frequency p-NBD room temperature current noise
I0CK
spectra

60
-i
10
64-
JGU
10'
■¡7-
4ti
10
-18-
â–¡
â–¡
â–¡
77
n
M
CM
<
cn
V
â–¡ ImA
W
V.
V
V
L_J
V
n
10
,-20-
77 lOO^A
V
â–¡
V
n
V
77
10
-21-
V
v
T7
10
c22
V
T7
IK
I0K
I00K iM
FREQUENCY (Hz)
10 M
+ - +
Figure 4.12 Current noise of 0.47 pm n p n NBD at T - 300 K for
I = 100 uA and 1 mA
IOOM

SJA/Hz)

FREQUENCY (Hz)
Figure 4.14 Current noise of 0.47 ym n+p n+ NBD at T = 300 K for I = 68 niA
CT>
rv>

63
The room temperature current noise at 100 Hz is plotted
2
versus bias current in Figure 4.15. There is an I dependence at
lower currents up to a few mA. At higher bias the noise increases
2
less fast. The deviation from I behavior appears to coincide with
the transition region.
The low-frequency current noise spectra at 77 K for several
bias currents are shown in Figure 4.16. The levels of excess low-
frequency noise are again large. The slopes are somewhat steeper than
for the room temperature case, being approximately -0.85. Figure 4.17
2
displays the current noise at 100 Hz versus bias. Again, I dependence
in the low-bias regime with a fall-off in the transition region is
observed.
The temperature dependence of the noise in the low-bias regime
was investigated further. The low-frequency small bias (100 yA) noise
spectra were measured for several temperatures (300 K, 250 K, 200 K,
150 K, 77 K, and 12 K) down to 12 K as shown in Figure 4.18. The
magnitude increases with decreasing temperature down to about 200 K.
Thereafter, further lowering of the temperature does not affect the
magnitude very much, but the slooe becomes more closely 1/f. The
other temperatures are not plotted since they only obscure the figure.
At 12 K, the slope is fully 1/f as shown in Figure 4.19.
4.3 The n+n~n+ Device Noise
The magnitudes of the noise levels of the 0.4 ym n+n"n+ struc¬
ture, both thermal and excess low frequency, are very small. The
current noise spectra for several bias currents at room temperature is

(A /Hz)
64
I 1mA)
Figure 4.15 Current noise at 100 Hz versus bias current for p-NBD
at T = 300 K

(AVHz)
65
FREQUENCY (Hz)
Figure 4.16 Current noise of p-NBD at T = 77 K

66
CURREN l
Figure 4.17 Current noise versus current of the p-NBD for f = 100 Hz
and T = 77 K

67
temperatures

(ZHAV) S
68

69
shown in Figure 4.20. Thermal levels and excess low-frequency noise
for some levels of bias can be seen. The frequency dependence of the
low-frequency noise is 1/f. The spectra from 1 to 25 kHz with the
1/f levels subtracted is shown in Figure 4.21. The lower three bias
currents result in approximately thermal noise. There is still excess
noise for 75 mA, however. Therefore, a spot noise measurement at
500 kHz for that bias was done. Less than 4% deviation from the
thermal level was found. Values at 10 Hz obtained from the straight
line approximations to the 1/f noise are plotted versus bias current
2
in Figure 4.22. The expected behavior for 1/f noise, o I is well
satisfied.
Spectra for T = 77 K and the same bias currents are plotted in
Figure 4.23. Excess low-frequency noise is found for all current
levels. The noise in the thermal region is difficult to determine
accurately because the expected value for full thermal noise (00.17 Q)
is below the background noise of even the correlation setup. Also, for
some bias currents, the noise is not yet flat at these frequencies.
The 1/f noise at 10 Hz is plotted versus bias in Figure 4.24. Again,
2
the I dependence of the magnitude is found.

(a7hj)
70
Figure 4.20 Current noise spectra of n+n n+.
0.4 ym NBD at T = 300 K

(A /Hz)
71
¡o
G i ~
_
( D m A
3.15 x ¡0
1 | 3-+ mA
2.5 x 10
G 27 mA
2.3 x 10
0 II mA
2.1 x 10
j
!01
20 !
>T (AVG.) 7 KH: ANO A5GVE
S. (AVG.) ! KHz - 25 KHz
g
G
O g)
i_j
o
G G G
0 l— i—'
CP
V
v
V
)
THERMAL NOISE'"'
CF 0,7 5 i)
_pn
10 “
f (Hz)
Figure 4.21 Thermal (-like) noise of n+n n+ device

(A /Hz)
72

(AV Hz)
73
Figure 4.23 Current noise of 0.4 pm n-NBD at T = 77 K

74
DC CURRENT (mA)
Figure 4.24 Sj of n-NBD at 10 Hz versus bias current for T = 77 K

CHAPTER V
DISCUSSION OF EXPERIMENTAL RESULTS
5,1 The n+n~n+ Device
5.1a Current-Voltage Characteristic and Impedance
The I-V characteristic of the n-type device is seen from
Figure 4.2 to be linear for bias voltages up to about 1 volt and
currents up to about 1 amp for both room temperature and 77 K. An
attempt to apply pulses at higher bias resulted in melting the gold
bonding wire at the too of the mesa. Higher bias was desired since a
slight nonlinearity appears at currents greater than 1 amp. Neverthe-
2 1/2
less, 1 amp corresponds to a current density of 12,800 A/cm . No V
3/2
or V dependence is found which suggests that theories with these
results are not adequate.
Similarly, the collision-dominated Mott-Gurney theory which
2
predicts V current dependence at high currents does not apply.
Indeed, sublinear current dependence at high bias seems to be indicated
by the sparse data. The more realistic theory of Holden and Debney
[12] gives a high-bias current dependence of v"'’^ for a 0.5 pm device
where collisions are neglected. At lower bias, their result appears to
be somewhat sublinear, similar to the V region of Shur and Eastman' s [2]
theory. The theory of van der Ziel et al. [9] gives linear behavior
for small bias. They include spillover from the highly doped regions
75

76
and calculate separately the mobilities due to diffusion-drift and
thermionic emission. Setting the calculated and measured values of
resistance at room temperature equal and solving for the diameter of
the mesa gives a 96 ym diameter which is very close to the reported
value of 100 ym. At 77 K, the measured value decreases 10%, the cal¬
culated value decreases 23%. The impedance is purely resistive at
these frequencies.
5.1b Excess 1/f Noise
In the 1/f noise region, we would like to apply Hooge's
empirical formula [18] which is Equation (2.12). For mesa structures
or nonhomogenous samples [19], Equation (2.13) replaces it and is
correct whether or not the transport is ballistic.
In a ballistic or near-ballistic device, many carriers do not
undergo any collisions at all. This is in contrast with a typical
semiconductor device in which every carrier collides many times.
Hooge's formula was developed for the second case, requiring that N,
the number of carriers in Equation (2.12) can be determined. In the
near-ballistic case, it is desired to exclude those carriers which are
transported ballistically, including only those that contribute to
the noise. This can be very difficult. Therefore, an alternative
expression to describe "noisiness," avoiding this problem, will also be
used. Noisiness is described as (S T(f)f)/I which is still dimension-
A1
less unless the spectral slope is not -1. Substituting values gives,
at 300 K,

77
sAÍ(f)f
1.6 x 10
-15
(5.1)
and at 77 K,
sAi(f)f
1.4 x 10
â– 14
(5.2)
If the slope is not -1, then just (S^j(f)f)/I^ can be reported at a
specified frequency.
If n(x) - n is a reasonable approximation despite the complex
nature of n(x), then Equation (2.13) becomes
S A j(f)fALmn
aHb 72 (5.3)
0
where the subscript, b, denotes validity for the ballistic case. If
the transport can be characterized by a constant mobility, y, the
measured resistance is
R =
m
qyA
(5.4)
Then, for the collision-dominated case, Hooge's constant becomes
a
He
SM(f)f(mL)2
igOyR
(5.5)

78
Assuming that u300K = 0.74 u?7K = 14 A = 7.9 x 109 m2,
and n = 2 x Id"'8 cm 3 gives at 300 K,
au =7.2x10
He
(5.6)
and
aHb = 5 x 10“8 (5.7)
At 77 K the values are
cl, = 3.7 x 10
He
aHb ~ x
These values are five orders of magnitude smaller than the value
reported by Hooge et al. [16] for n-type bulk GaAs of 6 x 10”3. Thus
the number of collisions for this device is very small. The remaining
collisions involve polar optical phonon emission typically with a very
small deflection angle [1], 6. Handel's [24] quantum theory of 1/f noise
2 1
indicates that the magnitude of the relative 1/f noise goes as sin
so that the residual noise is very low.
Comparing the ballistic and collision-limited case temperature
dependence of ct^ suggests that the device is not purely ballistic at
either temperature since the low temperature is nine times larger
than the value at room temperature. Further, the diffusion-drift
resistance calculated by van der Ziel et al. [9] is larger than the
(5.8)
(5.9)

79
thermionic emission resistance for both temperatures. However, the
values are within a factor of two of each other. The very low value
of ot„ suggests near-ballistic transport.
5.1c High-Frequency Noise
The device exhibits nearly 100% thermal noise at all currents
to 75 mA if sufficiently high frequencies are attained. This current
?
corresponds to a current density of 960 A/cm . There is no detailed
theory for the noise as yet developed for all applied bias, just the
preliminary calculations of van der Ziel and Bosnian [14,15], For the
collision-dominated space-charge-limited diode at low bias, the noise
is due to diffusion noise sources which transform via Einstein's
relation to 4kT/Rx- At high bias for which the Mott-Gurney law
2 d V
(I 'v V ) applies, the noise becomes 8kT/R where R, = For a bal-
X X U i .
listic device the noise is due to shot noise. At sufficiently high
bias where soace charge effects dominate, correlations between current
components due to fluctuations in the potential minimum caused by the
space charge lead to subtnermal noise.
In order to differentiate between the two models, high bias
must be achieved. Our measurements have found no deviations from
thermal noise. One possible reason for this is that the current density
2
of 960 A/cm is insufficient. Another is that the device is operating
in between the two regimes as suggested by the 1/f measurements so that
extreme bias may be required to see which effect dominates.

80
5.2 The n+p~n+ Device
5.2a Current-Voltage Characteristic and Impedance
The n+p n‘ device shows substantial nonlinearity. There are
two regimes with a transition region in between as shown in Figure
4.10. At low bias, the device is linear with a DC conductance at room
temperature 100 times smaller than the n-type device. It decreases
with decreasing temperature to a limiting value a factor of 3 below its
room temperature value. It reaches this value near 150 K and remains
constant thereafter down to 12 K. The high bias regime is temperature
independent and is also linear with a large conductance 4 times less
than the n-type device.
Due to the thinness of p-regions and the large doping density
of the n+ regions, the spillover of electrons into the p-regions is
not very different than for the n-type diode case [9], Then current
flow should be by nearly ballistic electron emission through the poten¬
tial minimum. The resulting characteristic should be linear with a
large conductance and nearly temperature independent. This appears to
be a good model for the high bias regime. At low bias, the model
fails, however. One possible reason for this is that enough holes re¬
main in the p-region to control the transport ambipolarly. The details
of such an effect are unclear, but a qualitative description by
Dr. C. M. Van VIiet follows. The motion is not strictly ambipolar,
since there is space charge, as indicated by the presence of the poten¬
tial minimum [9], which even for V = 0 can be computed from Poisson's
equation. Therefore, near the potential minimum the excess electron

81
charge is small. Roughly speaking, only ambipolar pairs with energies
within kT of the potential minimum are able to cross the minimum.
If the injected carrier density An is less than p at a point (labeled x')
approximately kT greater in energy than the minimum on the injecting
side, then the current will be ambipolar (instead of ballistic).
Clearly, with decreasing T, p(x') decreases. Thus, with decreasing T
the ambipolar current decreases, and the transition to ballistic
behavior—which is independent of T—sets in at lower bias. The I/V
versus V characteristic is therefore as shown in Figure 4.10.
In any case, assuming that holes control the mobility for low
bias, and electrons at high bias gives a factor of 21 change in the con-
cm2
ductance at room temperature since typical values are u n, 8,500 tt——
K Hn ’ V-sec
2
and pip ^ 400 cm /V-sec [25], There still remains to be explained the
factor of 4 difference between the n-device conductance and the high
bias p-device conductance. The potential barrier is larger in the
p-type device [26], In the p” region, Poisson's equation is
,2 Q (n + N - p)
d c _ a
dx2 " e£0
(5.10)
where ip(x) is the potential, n(x) the electron density, p(x) the hole
density, Na the acceptor density, and ccq the dielectric constant. For
the n-device in the n" region
2,
d v
dx
q(n - Nd)
(5.11)

82
then
If p(x) is assumed negligible due to electron spillover,
A
r.2.1
d \i>
\ dx
JP-
i dx
(5.12)
so that the n-device barrier is smaller. The above model of ambipolar
collision-dominated flow at low bias and near-ballistic electron
emission at high bias gives a qualitative explanation of the experi¬
mental data.
5.2b Noise
The n+p n+ device showed much larger levels of noise than the
n-type device. The frequency dependence of the spectra for all
75
measured currents is about (1/f)' for all measured bias currents at
room temperature. Then Hooge's parameter is not well defined since it
is not dimensionless (unless the slope is -1) and depends on frequency.
The noisiness at 10 Hz and 100 pA is
(f)f
2
6 x 10
(5.13)
This is about 4 x 10 times the n-type device value. The spectral
slopes do not change in the transition from low to high bias. The
2
magnitude of the noise does fall off at high bias from the I dependence
seen at low bias, however. This suggests that the noise is associated
with the low bias transport mechanism and that the high bias mechanism

83
is much less noisy. That is in good agreement with the conjecture
of ambipolarly governed flow at low bias and near-bal1istic flow at
high bias. The -0.75 slooe is not common although van de Roer [27]
has also found spectra going slower than 1/f in 6 ym p+n~p+ punch-
through diodes. The spectra become more closely 1/f as temperature
decreases. At 77 K the slope is -0.85 which is common for intermediate
temperatures down to 12 K. The magnitude of the noise is
(f)f
2
= 3.3 x 10'8
(5.14)
at 10 Hz and 100 yA. Finally, at 12 K, the slope is fully -1 with
magnitude
S
Al
(f)f
(5.15)
at 10 Hz and 100 yA. The explanation of this temperature-dependent
slope is unclear.

CHAPTER VI
CONCLUSIONS AND RECOMMENDATIONS
FOR FURTHER WORK
6.1 The n+n~n+ Device
— -j-
The n n n device is nearly ballistic. Carrier transport is
by both thermionic emission and by collision-based diffusion-drift.
Neither process can be neglected in a physical model. For the I-V
characteristic, it is desired to calculate n(x), ^(x), and J(V) for
any applied bias, expanding on the calculations of van der Ziel et al.
[9] for small V. Alternately, the model of Holden and Debney [12]
should be further developed to investigate the effect of adding col¬
lisions to their model. Also the temperature-dependence and the
effects of replacing the Maxwell-Boltzmann velocity distribution by
a Fermi-Dirac distribution should be investigated.
Since nonlinearity in the I-V characteristic appears at high
bias, very fast (M nsec) pulse neasurements at yet higher bias
would be very interesting. The mechanism causing this nonlinearity
should be identified.
For the noise, the measured value of found was extremely
low. This is the best confirmation yet that lattice phonon scattering
causes 1/f noise. It would be valuable to repeat the measurement on
other mesas. Perhaps in such small devices Handel's [24] fundamental
theory can be tested.
84

85
In the thermal noise region, the results were inconclusive,
neither subthermal nor greater than thermal noise was found. Perhaps
the bias level was not great enough. In that case pulsed noise
measurements could be attempted. This may be difficult since the noise
is already very low. A detailed and complete noise theory for these
devices is desirable, including collision effects and velocity dis¬
persion. A Langevin equation based on the momentum and energy balance
equations of Shur and Eastman [5,6] but including the above effects
should give accurate results.
6,2 The n+p~n+ Device
The p-type device was very intriguing, beginning with the
I-V characteristic. At present it is believed that at low bias the
carrier transport is ambipolar and collision-limited, centering about
the potential minimum. At high bias, the injected electrons overrun
the holes and near-ballistic electron flow results similar to the n-
type device. The conductance is less because the potential barrier is
greater. This model deserves further detailed investigation.
The noise is very large and shows slopes becoming progressively
less 1/f as temperature is increased. This interesting characteristic
suggests that subtle and complex mechanisms may be taking place that
are not well understood yet. To develop a theory that explains these
effects would be a great step forward in our knowledge of very small
devices. The 1/f noise extends to the GHz range to frequencies greater
than our measurment system so that thermal noise measurements have still

86
to be done. In addition, only these low-bias measurements have
been done; pulsed noise measurements in the high-bias regime are
underway: they may yield valuable information.
Investigation of this device has led to another novel device
[28], A p+n p+ device with nonlinear characteristics is predicted to
have negative differential conductance. At low bias, electron-
controlled ambipolar flow should give a large conductance. At larger
bias, hole injection takes over with collision-limited flow resulting
in a low conductance. The transition region, therefore, should show
the negative differential conductance.

APPENDIX
COMPUTER PROGRAMS FOR THE HP 9825

APPENDIX
COMPUTER PROGRAMS FOR THE HP 9825
A.1 Correlation-System Three-Measurement Average
This program calculates equations for the current noise spectral
density of a device under test (DUT) using the three-measurement
technique. Actually, the magnitude of the calibration signal is also
recorded. The program calculates data for eight logarithmically equal¬
spaced points covering one decade in frequency. It is used with the
HP 3582A spectrum analyzer which features a dual-channel Fast Fourier
transform algorithm for frequencies from 0.02 Hz to 25 kHz. Many data
sets can be averaged to increase the accuracy of the measurement which
is described in Chapter III.
0:
1
i:
2:
3:
4:
5:
6:
"7 .
/ .
3:
9:
10
11
12
13
14
15
16
17
18
"REPEATED-3 MEASUREMENT CORRELATION SYSTEM":
dim A[24],B[24],C[24],F[8],M[8],S[8],X[8],OS[1],LS[20]
dim MS[8,5] ;"MP12"->M$[1 ] ;"MP18"-HM$[2]
"MP25"-»M$[3];"MP35"-»M$[4];"MP50"+M$[5]
"MP70"-*MS[6] ;"MP95"->M$[7]; "MP125"->M$[8]
12+S[1];18-S[2];25+S[3];35+S[4];50+S[5];70+S[6];95+S[7];125+S[8]
Idf 1,A,B,C,D,A[*],B[*],C[*],F[*],M[*]
fxd 0;dsp A,B,C,D,F[8];beep;stp
fl t 3
"n"-K3$ :ent "DUT ON (y/n)",Q$;if Q$-"y":oto "DUT0N"
"n"-K3S;ent "DUT OFF (y/n)",Q$;if QS="y";gto "DUTOFF"
"n"->0$;ent "CAL ON (y/n)" ,QS ;if 0$="y ;gto "CALON"
"n"^Q$;ent "CAL MAG (y/n)",0S;if QS="y";gto "CALMAG"
"n"->OS ;ent "CALC NOISE (y/n)" ,Q$;if QS="y";gto "NOISE"
"n"->0$ ;ent "CLEAR DATA FILE (y/n)" ,Q$ ;i f Q$="y";gto "CLEAR"
"DONE":
ref 1,A,B,C,D,A[*],B[*],C[*],F[*],M[*]
fxd 0;dsp A,B,C,D;beep
88

89
19: lcl 711;end
23: "DUTON":
21: for 1=1 to 8;asb "GET"
22: if A=0;X->A[I j ;Y-»-B[I] ;Z->C[I] ;next I
23: if A^CUjmp 4
24: (X+A*A[I])/(A+l )->A[I]
25: (Y+A*8[I]) / (A+l )-H3[ I]
26: (Z+A*C[I])/(A+1 )-*C[l] ;next I
27: A+l->A;qto "DONE"
28: "DUTOFF":
29: for 1=1 to 8:asb "GET"
30: I+8-hJ
31: if B=0;X+A[J];Y^B[J];Z+C[J];next I
32: if B=0:jmp 3
33: (X+B*A[.Jl)/(B+l )+A[J] ; (Y+B*B[J])/(B+1 )+B[J]
34: (Z+B*C[J])/(B+lK[J];next I
35: B+1-+B:ato "DONE"
36: "CALON":
37: for 1=1 to 8;gsb "GET"
38: I+16-nl
39: if C=0;X+A[J];Y-*B[J];Z+C[J];next I
40: if C=0;jmp 3
41: (X+C*A[J])/(C+1)->A[J];(Y+C*B[J])/(C+1)+B[J]
42: (Z+C*C[J])/(C+l)-*C[j];next I
43: C+l-Kl;gto "DONE"
44: "CALMAG":
45: wrt 711 , "AA1MN1MB1"
46: for 1=1 to 8
47: wrt 711, "MP" ,S[I], "LMK";red 711,U,V
48: if A+B+C+D=0;V->F[I]
49: if V#F[I];beep;dsp "FREQUENCY MISMATCH";stp
50: tnt(U/10)->U
51: if D=0;U->M[I] ;next I
52: uf D#0;(U+D*M[I])/(D+lhM[l];next I
53: wrt 711 ,"AA0" ;D+l->D;gto "DONE"
54: "NOISE":
55: ent "TEMP",T;ent "DUT RESISTANCE" ,R
56: ent "CAL RESISTANCE",S;ent "DC CURRENT",U
57: gsb "LABEL"
58: for 1=1 to 8:spc
59: /(A[I]*B[I]*C[l])^V;v/(A[l+8]*B[I+8]*C[I+8])^W
60: /(A[I+16]*B[I+16]*C[I+16])-X
61: (V-W)/(X-W)—Y ;M[I]/S/S-^Z
62: Y*Z+4*1.38e-23*T/R->N
63: prt "SI = ",N;fxd 1 ;prt "FREQ=",F[I];f11 3
64: next I
65: spc ; spc ;gto "DONE"
66: "CLEAR":
67: 0-»-A->B->C->D
68: for 1 = 1 to 8;0-*F[I]-hM[I] ;next I

90
69: for 1 = 1 to 24;0-*-A[I]+B[l]-*C[I] ;next I
70: gto "DONE"
71: "GET":
72: gsb "DATA"
73: if A+B+C+D=0;V+F[I]
74: if V#F[I];beep;dsp "FREQUENCY MISMATCH" ;stp
75: gsb "COH"
76: tnf(X/10)-*X;tnf(Y/10)+Y;ret
77: "DATA":
78: wrt 711,"AA1MN1MB1"
79: wrt 711 ,M$[I]
80: wrt 711,"LMK";red 711 ,X,V
81: wrt 711,"AA0AB1LMK";red 711 ,Y,V
82: wrt 711,"AB0";ret
33: "COH":
84: wrt 711 ,"LFM,",76000+dto(4*S[I])," ,4";red 711
85: for J=1 to 4;rdb(731 )+U;rdb(731)+Z
86: ior(rot(U ,8) ,Z)-*X[J] ;next J
87: wrt 711 ,"LFM," ,77000+dtoS[I] ,",1";red 711
88: rdb(731)-X[2];rdb(731)-X[4]
89: wrt 711,"LFM," ,75000+dtoS[I],", 111;red 711
90: ior(rot(rdb(731) ,8) ,rdb(731 ))->X[5]
91: wrt 711 , "LFM," ,75200+dtoS[I],",1";red 711
92: ior(rot(rdb(731),8),rdb(731))+X[6]
93: wrt 711 ,"LFM,",77200+dtoS[I],",1";red 711
94: rdb(731)->X[7];rdb(73lH[8]
95: X[l]*2t(X[2]-15)->X[l]
96: X[3]*2T(X[4]-15)+X[3]
97: X[5]*2f(X[7]-l5)->X[5]
98: X[6]*2f(X[8]-15)+X[6]
99:(X[5]+2+X[6]+2)/X[l]/X[3]-*-Z
100
101
102
103
104
105
106
TG "C
"LABEL":
ent "Label" ,L$;spc ;prt L$
'DCI =" ,U;prt "TEMP=" ,T
'DUT RES=",R;prt "CAL RES='
'# SETS = " ,A
prt
prt
prt
ret
H 9711
,S
A.2 Wide-Band Filter Synthesizer
This program is written for the HP 9825 computer
with the HP 3582A spectrum analyzer. The high-frequency
in conjunction
4/5 of the
FFT bins are averaged to synthesize a wide-band filter to use with the

91
radio frequency FFT system which utilizes a mixer just before the
spectrum analyzer. Without the use of this program, the accuracy
of a measurement is poor since the bandwidth is then very small com¬
pared with the frequency of interest. A bandwidth of 20 kHz can be
generated by setting the frequency span on the 3582A to 25 kHz,
although the program displays the filter output in normalized dBV/v'Hz.
0: "Last 4/5’ths of display averacer":
1: dim B[10] , Y[256]
2: wrt 711 ,"LFM,77454,5"
3: red 711
4: for 1 = 1 to 10;rdb*711 )-H3[l]
51 next I
6: wrt 711 ,"LSP" :red 711 ,S
7: if B[3]>0 ;sfg 1
8: B[2>H
9: if H>127 ;H-128-HH
10: if H>63 ;H-64->-H
11 : if H>31 ;H-32-HH
12: if H>15 ;H- 16-HH
13: if H>7;H-8->H
14: if H=0;25O+6
15: if H>1;68.87+B
16: if H>3;166.6667-B
17: B[4]-*H
18: if H>127 ;H-128->H
19: if H>63;H-64->H
20: if H>31 ;H-32->H
21 : if H>15 ;H-16-HH
22: if H>7;H-8->H
23: if H>3;H-4-Hi
24: if H>2;sfg 2
25: 128-*N;if flgl;256^N
26: if not flgl ;B/2->B
27: if f 1 g2:256->N
28: wrt 711 ,"LDS"
29: red 711
30: for 1 = 1 to N;red 731 ,Y[I]
31: next I
32: 1c1 711
33: if flg2; 128-*N
34: 0->A;fxd 4
35: gsb "calc"

92
36: if flg2;X+Y;0+X;128->A;gsb "calc"
37: if flg2;prt 20*1og(Y/.8N),201og(X/.8N);end
38: prt 201oq(X/.8N) ;end
39: "calc":
40: for I=A+N/5 to A+N;X+tn + (Y[I]/20)//(S/BH ;next I
41: ret
*30119

REFERENCES
1. L. F. Eastman, R. Stall, D. Woodard, N. Dandekar, C. E. C. Wood,
M. S. Shur, and K. Board, "Ballistic Electron Motion in GaAs at
Room Temperature," Electron. Letters 16_, 524 (1980).
2. M. S. Shur and L. F. Eastman, "Ballistic Transport in Semiconduc¬
tor at Low Temperatures for Low-Power High-Speed Logic," IEEE
Trans. Electron Devices ED-26_, 1677 (1979).
3. J. R. Barker, D. K. Ferry, and H. L. Grubin, "On the Nature of
Ballistic Transport in Short-Channel Semiconductor Devices," IEEE
Electron Device Letters EDL-J_, 209 (1980).
4. R. Zuleeg, "Possible Ballistic Effects in GaAs Current Limiters,"
IEEE Electron Device Letters EDL-J_, 234 (1980).
5. M. S. Shur, "Ballistic Transport in a Semi conductor with Col¬
lisions," IEEE Trans. Electron Devices ED-28, 1120 (1981).
6. M. S. Shur and L. F. Eastman, "Near Ballistic Electron Transport
in GaAs Devices at 77°K," Solid-State Electron. 24, 11 (1981).
7. J. J. Rosenberg, E. J. Yoffa, and M. I. Nathan, "Importance of
Boundary Conditions to Conduction in Short Samples," IEEE Trans.
Electron Devices ED-28, 941 (1981).
8. W. R. Frensley, "High-Frequency Effects of Ballistic Electron
Transport in Semiconductors," IEEE Electron Device Letters EDL-1,
137 (1980).
9. A van der Ziel, M. S. Shur, K. Lee, T. Chen, and K. Amberiadis,
"Carrier Distribution and Low-Field Resistance in Short n+-n~-n+
and n+-p--n^ Structures," submitted to Solid-State Electron.
10. R. K. Cook and J. Frey, "Diffusion Effects and 'Ballistic Trans¬
port'," IEEE Trans. Electron Devices ED-28, 951 (1981).
11. Y. Awano, K. Tomizawa, N. Hashizume, and M. Kawashima, "Monte
Carlo Particle Simulation of GaAs Submicron n+-i-n+ Diode,"
Electron. Letters J_8, 133 (1982).
12. A. J. Holden and B. T. Debney, "Improved Theory of Ballistic Trans¬
port in One Dimension," Electron. Letters J_8, 558 (1982).
93

94
13. T. C. Fry, "The Thermionic Current between Parallel Plane Elec¬
trodes; Velocities of Emission Distributed accordinq to Maxwell's
Law," Phys. Rev. 1_7, 441 (1921 ).
14. A. vanderZiel and G. Bosman, "Collision-Free Limit of Near-
Thermal Noise in Short Solid-State Diodes," Physica Status Solidi
a) 73, K93 (1982).
15. A. vanderZiel and G. Bosman, "Collision-Dominated Limit of Near-
Thermal Noise in Short Solid-State Diodes," Physica Status Solidi
(a), 73, K87 (1982).
16. F. N. Hooge, T. J. G. Kleinpenning, and L. K. van Damme, "Experi¬
mental Studies on 1/f Noise," Reports Progress in Physics 44,
479 (1971). —
17. A. van der Ziel, "Flicker Noise in Electronic Devices," Advances
in Electronics and Electron Physics 49_, 225 ( 1979).
18. F. N. Hooqe, "1/f Noise Is No Surface Effect," Phys. Letters A29,
139 (1969). —
19. A. van der Ziel and C. M. Van Vliet, "Mobility-Fluctuation 1/f
Noise in Nonuniform Nonlinear Samples and in Mesa Structures,"
Physica Status Solidi (a) 72_, K53 (1982).
20. G. Bosman, R. J. J. Zijlstra, and A. D. van Rheenen, "Flicker
Noise of Hot Electrons in Silicon at T = 78 K," Phys. Letters A78,
385 (1980). —
21. T. M. Chen and A. van der Ziel, "Hanbury Brown-Twiss Type Circuit
for Measuring Small Noise Signals," Proc. IEEE _53, 395 (1965).
22. L. D. Enochson and R. K. Otnes, Programming and Analysis for
Digital Time Series Data, Shock and Vibration Center, Naval
Research Laboratory, Washington, D.C., 201 (1968).
23. J. Kilmer, A. van der Ziel, and G. Bosman, "Presence of Mobility-
Fluctuation 1/f Noise Identified in Silicon P+NP Transistors,"
Solid-State Electron. 26_, 71 (1983).
24. P. H. Handel, "Quantum Approach to 1/f Noise," Phys. Rev. A22,
745 (1980). ~
25. S. M. Sze, Physics of Semiconductor Devices, 2nd ed., Wiley-
Interscience, New York, 851 (1981).
26. R. R. Schmidt, G. Bosman, A. van der Ziel, C. M. Van Vliet, and
M. Hollis, "Nonlinear Characteristic of Short n+p_n+ GaAs Devices,"
submitted to Solid-State Electron.

95
27. Th. G. van de Roer, "1/f Noise and Velocity Saturation in Punch-
Through Diodes," Solid-State Electron. 23_, 695 (1980).
28. A. van der Ziel, G. Bosman, and C. M. Van Vliet, "Predicted
Nonlinear Characteristic of Short p+-n_-p+ GaAs Devices," internal
report, Noise Research Laboratory, Electrical Engineering Depart¬
ment, University of Florida, Gainesville.

BIOGRAPHICAL SKETCH
Robert Roy Schriidt was born on November 1 1 , 1952 , in Edmonton,
Alberta, Canada. Moving to the United States at the age of four, he
lived in California and Washington, graduating from Redmond High School
in 1970, He then attended the University of Washington, in Seattle,
graduating in 1974 with a double major in physics and mathematics. In
order to support his interest in music, for the next four years he
worked as a professional bass guitarist, TV technician, residential
carpenter, and electrician. Deciding to continue his education, he
came to the University of Florida in January 1979 and began graduate
study in electrical engineering. During this time he was employed in
the Noise Research Laboratory as a graduate research assistant. He
received the Master of Science in March 1981, writing a thesis on
various transistor noise measurement systems from 0.06 Hz to 32 MHz.
He is a member of Phi Kappa Phi Honor Society.
96

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
C. M. Van Vliet, Chairperson
Professor of Electrical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
E. R. Chenette, Cochairperson
Professor of Electrical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
G. Bosman
Assistant Professor of Electrical
Engineering

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
A. D. Sutherland
Professor of Electrical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
—
T. T. Bowman
Associate Professor of Mathematics
This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate Council, and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
April 1983
I'liJhi'fCt.'
Dean, College of Engineering
Dean for Graduate Studies and Research

O p^|^|§jill

O p^|^|§jill