• TABLE OF CONTENTS
HIDE
 Title Page
 Dedication
 Acknowledgement
 Table of Contents
 Abstract
 Introduction
 Quantum noise
 Physical model of schottky barrier...
 Noise of schottky barrier diodes...
 High-frequency noise of schottyky...
 Physical model of tunnel diode...
 Noise in tunnel diodes
 Conclusion and suggestions for...
 Appendix
 Reference
 Biographical sketch
 Copyright














Group Title: study of the noise of microwave Schottky barrier and tunnel diodes
Title: A study of the noise of microwave Schottky barrier and tunnel diodes
CITATION THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
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Permanent Link: http://ufdc.ufl.edu/UF00082429/00001
 Material Information
Title: A study of the noise of microwave Schottky barrier and tunnel diodes
Physical Description: vii, 177 leaves : ill. ; 28 cm.
Language: English
Creator: Trippe, Michael Wayne, 1957-
Publication Date: 1986
 Subjects
Subject: Tunnel diodes -- Noise -- Measurement   ( lcsh )
Diodes, Schottky-barrier -- Noise -- Measurement   ( lcsh )
Diodes, Semiconductor -- Noise -- Measurement   ( lcsh )
Electrical Engineering thesis Ph. D
Dissertations, Academic -- Electrical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (Ph. D.)--University of Florida, 1986.
Bibliography: Bibliography: leaves 173-176.
Statement of Responsibility: by Michael Wayne Trippe.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00082429
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 000957012
oclc - 17314050
notis - AER9695

Table of Contents
    Title Page
        Page i
    Dedication
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
        Page v
    Abstract
        Page vi
        Page vii
    Introduction
        Page 1
        Page 2
    Quantum noise
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
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    Physical model of schottky barrier diodes
        Page 31
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    Noise of schottky barrier diodes at low and intermediate frequencies
        Page 48
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    High-frequency noise of schottyky barrier diodes
        Page 66
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    Physical model of tunnel diodes
        Page 103
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        Page 128
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        Page 130
    Noise in tunnel diodes
        Page 131
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    Conclusion and suggestions for further research
        Page 152
        Page 153
        Page 154
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    Appendix
        Page 156
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    Reference
        Page 173
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    Biographical sketch
        Page 177
        Page 178
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    Copyright
        Copyright
Full Text














A STUDY OF THE NOISE OF MICROWAVE SCHOTTKY
BARRIER AND TUNNEL DIODES










By
MICHAEL WAYNE TRIPPE

























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


UNIVERSITY OF FLORIDA


1986







































TO MY FAMILY




















ACKNOWLEDGEMENTS


The author wishes to express his sincere thanks to Dr. A. van der

Ziel, who supplied many ideas and contributions concerning this

research. Special thanks go to Dr. G. Bosman for his daily guidance and

motivation. Without his help this work would not have been possible.

I also wish to thank the other members of the supervisory com-

mittee--Dr. E.R. Chenette, Dr. P. Kumar and Dr. C.M. Van Vliet--for

their participation. The author also wishes to acknowledge the

assistance of Dr. A.D. Sutherland, who passed away on May 8, 1984.

Finally, the patience of Katie Beard in typing the manuscript is

appreciated.

















TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS........................................................................... iii

ABSTRACT............................................ ........ ......... vi

CHAPTERS

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

II QUANTUM NOISE............................................. 3

Introduction......... ...................................... 3
Quantum Noise Theory.................................. ...... 4
Measurement Technique...................................... 11
Fabrication of MOM Tunnel Diodes............................ 18
Alternative Devices: Schottky Barrier
and p -n+ Tunnel Diodes.................................. 23
Conclusion.................... ...... ...... ............... 28

III PHYSICAL MODEL OF SCHOTTKY BARRIER DIODES .................... 31

Introduction........................... ........ .... 31
Physical Model.............................................. 32
I-V Characteristics of Schottky Barrier Diodes.............. 42
Summary ....................... ......................... 45

IV NOISE OF SCHOTTKY BARRIER DIODES AT LOW
AND INTERMEDIATE FREQUENCIES ............................... 48

Introduction................................................. 48
Intermediate-Frequency Theory.............................. 48
Noise Measurement Schemes................................... 51
Discussion of Noise Measurements at
Intermediate Frequencies.................................. 58
Discussion of Excess Noise Measurements...................... 61
Conclusion .......................................... 63

V HIGH-FREQUENCY NOISE OF SCHOTTKY BARRIER DIODES.............. 66

Introduction....... ..... ........... .................. ....... 66
Calculation of the Junction Admittance..................... 66
Calculation of the Spectral Intensities...................... 72
Calculation of the Noise Temperature......................... 81
High-Frequency Measurements................................ 88
Measurement Results and Discussion........................... 97
Conclusion............ ...................................... 100











VI PHYSICAL MODEL OF TUNNEL DIODES............................. 103

Introduction...................10........ 103
Physical Model............................................. 104
Heavy Doping Considerations................................. 109
Calculation of I-V Characteristic.......................... 121
Measured I-V Characteristic................................. 124
Conclusion................................................. 130


VII NOISE IN TUNNEL DIODES..................................... 131

Introduction.................. ............................ 131
Low-Frequency Excess Noise.................................. 132
High-Frequency Theory (Shot Noise).......................... 143
Microwave Noise Measurements................................ 143
Discussion of Microwave Measurements........................ 145
Conclusion........... .............................. 150


VIII CONCLUSION AND SUGGESTIONS FOR FURTHER RESEARCH............. 152

Quantum Noise............................................... 152
Schottky Barrier Diodes. ................................. 152
Tunnel Diodes...................................... ..... 154


APPENDICES

A COMPUTER PROGRAM TO CALCULATE THE HIGH-FREQUENCY
NOISE OF SCHOTTKY BARRIER DIODES............................. 157

B COMPUTER PROGRAM TO CALCULATE THE 1/f NOISE OF
SCHOTTKY BARRIER DIODES.................................. 163

C COMPUTER PROGRAM TO CONTROL THE HP 3478A
DIGITAL VOLTMETER............... ......................... 167

D COMPUTER PROGRAM TO CALCULATE THE I-V AND SHOT
NOISE OF TUNNEL DIODES..................................... 169

REFERENCES....................... ......................... ........... 173

BIOGRAPHICAL SKETCH......................... .................................. 177

















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


A STUDY OF THE NOISE OF MICROWAVE SCHOTTKY
BARRIER AND TUNNEL DIODES

By

Michael Wayne Trippe

May 1986


Chairman: A. van der Ziel
Cochairman: G. Bosman
Major Department: Electrical Engineering


A complete DC, AC and noise characterization of both Schottky

barrier and tunnel diodes is presented. Emphasis was placed on the

high-frequency noise properties of these devices.

For the Schottky barrier diodes, high-frequency noise measurements

were performed at 2.2, 12 and 97.5 GHz and are shown to agree well with

the theory. In the theoretical analysis the electrons participating in

the charge-transport process across the barrier are subdivided into four

groups based upon their initial velocity. The contribution of each

group to the device conductance, susceptance and current spectral

intensity was incorporated, including the effect of the transit time.

By taking each of these effects into account, an accurate model which

applies over a wide range of bias and frequency has been developed.

Although the emphasis of this model has been the high-frequency per-

formance, it also gives the correct result in the low-frequency limit.











Noise measurements were performed on germanium p-n+ tunnel diodes

at frequencies between 10 Hz and 8 GHz. The low-frequency excess noise

is found to be of the generation-recombination type in all bias

ranges. This indicates the presence of a large number of localized

energy levels within the bandgap.

At high frequencies, the noise was found to be the full shot noise

of the individual tunneling currents when the diode was operated at low

bias. For high-bias operation, less than full shot noise was observed.

This also indicates that the current transport process takes place

through the localized energy levels.

In addition, the effect of zero-point fluctuations and their con-

tribution to thermal noise was considered. Measurement techniques which

compare two thermal sources are shown to be ineffective in determining

whether zero-point fluctuations exist. The three types of devices

considered were metal-oxide-metal tunnel diodes, Schottky barrier diodes

and p+-n+ tunnel diodes. For various reasons, in each case it was

impossible to determine whether zero-point fluctuations exist. A

measurement system which can detect signals with an effective noise

temperature of 0.1 K was demonstrated. This is considered adequate

should suitable devices become available in the future.

















CHAPTER I
INTRODUCTION


This work was supported through the Solid State and Microstructures

Engineering Division of the National Science Foundation. Although the

grant title was "Quantum Noise at 90 GHz, and 2 K," the areas of pro-

posed research were considerably more broad. Among the original objec-

tives were 1) create a noise measurement system which operates in the 90

GHz range and which can detect the noise of a device with a noise tem-

perature of 2 K; 2) measure the quantum correction factor of thermal

noise at high frequencies (zero-point fluctuations); and 3) investigate

the high-frequency properties of MOM (metal-oxide-metal) diodes and SBDs

(Schottky barrier diodes). Of the original goals, at least the first

and third were satisfactorily completed.

A measurement system which operates at 97.5 GHz was developed and

tested. It can detect signals with an effective temperature as small as

0.1 K. This is remarkable considering the fact that the background

noise temperature of the measurement system is higher than 1000 K.

The quantum correction factor of thermal noise was never measured.

Detailed analysis of the proposed cross-correlation measurement scheme

showed that it would not respond to the quantum effect under investiga-

tion. Indeed, it will be seen that no detection system which requires a

comparison between two thermal noise sources will be able to determine

whether the zero-point fluctuations are present. A measurement tech-

nique which can detect the presence of zero-point fluctuations was











developed. This technique requires the use of an active device such as

an MOM diode. The effect of zero-point fluctuations is present in all

devices. However, certain difficulties arise in the operation of some

devices at high frequencies, obscuring the effects of zero-point energy

fluctuations, and ultimately the second goal was never achieved.

Chapter II discusses the limitations of the measurement systems and also

the consideration and fabrication of devices.

The device properties of Schottky barrier diodes were studied in

detail. This is the subject of Chapters III, IV and V. Noise measure-

ments were made as a function of bias for frequencies between 10 Hz and

97.5 GHz. The low-frequency noise measurements show that there are no

excess noise sources which extend into the microwave region. At inter-

mediate frequencies, the noise is found to be just the shot noise of the

DC current. It will be seen that transit-time effects are important

when considering the device operation at high frequencies.

Metal-oxide-metal tunnel diodes with oxide thicknesses of 23 A were

fabricated. This is still too large for the devices to give a suitably

low junction resistance. Instead, proprietary samples of microwave

p -n+ germanium tunnel diodes were obtained, and their noise properties

as a function of frequency and bias were measured from 10 Hz to 8 GHz.

For these devices an explanation of the noise properties follows from a

consideration of a two-step conduction process through localized energy

levels. To the best of our knowledge, this is the first observation of

this current transport mechanism through the use of noise measurements.

















CHAPTER II
QUANTUM NOISE


Introduction

There are two types of noise which are present in thermal equilib-

rium. Thermal noise, as its name implies, has thermal energy as its

origin. Quantum noise, although present in thermal equilibrium, comes

about from the noncommuting operators when solving the simple harmonic

oscillator problem. While a classical oscillator would have no kinetic

energy in its lowest state, a quantized oscillator still has some energy

even in its lowest allowed energy state. It is the properties of

thermal noise and especially quantum noise which were investigated. The

credit for first measuring thermal noise goes to J.B. Johnson for his

work of 1928 [1]. Nyquist also made a contribution in the same year by

giving a simple description of Johnson's results [2]. Einstein made a

theoretical treatment as early as 1906 using the theory of random

Brownian motion [3].

Thermal noise is derived from the thermal agitation of charge

carriers in the sample. Although, on the average, the charge carriers

do not have any net velocity in thermal equilibrium, their velocity does

fluctuate. Since the velocity fluctuations have thermal energy as their

origin, this source of noise may be reduced to arbitrarily close to zero

for very low temperatures. This is not true for the quantum noise

fluctuations. These fluctuations are only a function of the operating

frequency and do not depend on the thermodynamic temperature.










It should be noted that it will be tacitly assumed that the zero-

point energy fluctuations exist even though the details are still the

subject of some controversy [4]. (In this work the terms quantum noise

and zero-point energy fluctuations will be used interchangeably.)

The zero-point energy introduces additional fluctuations into the

system which must be taken into account. It will be seen that these

fluctuations dominate the noise behavior at very high frequencies and

set the absolute limit to the sensitivity of a measurement.

Quantum Noise Theory

The quantum mechanical problem to be considered is a quantized

simple harmonic oscillator. This is useful because it accurately models

a wide variety of physical systems. It is a direct analogy with the

classical simple harmonic oscillator. Equation (2.1) expresses the

allowed energy eigenvalues for this system.


En hf(n +2
En = hf(n +), n = 0, 1, . (2.1)


where n is the level number, h is Planck's constant and f is fre-

quency. The term hf/2 represents the zero-point energy (n = 0). It can

be seen from Figure 2.1 that the lowest possible energy level for this

system is E0 = hf/2. As a result, the lowest energy level is non-zero

[5].

In order to find the average energy of the system, E, a summation

over the distribution of photons in the energy levels must be made.

See, for example, Marcuse [6].

This leads to the result


= hf ( 1 -) (2.2)
ex(hf 2
e ) 1













V(x) = Cx2


E2


= 5/2 hf

= 3/2 hf

= 1/2 hf


Figure 2.1. Energy eigenvalues and potential well
for the simple harmonic oscillator problem.










where T is the thermodynamic temperature and k is Boltzmann's constant.

By making a Taylor series expansion of the exponential term, it is

simple to show that for hf << kT



E = kT. (2.3)



This is the well-known result for low frequencies.

It is useful in many cases to describe the noise of a system in

terms of its equivalent noise temperature. From equation (2.2) it

follows that


E hf/k hf
T hf/k + = T + T (2.4)
n k exphf- 2k B ZPE
exp(-k) 1


where TB represents the thermal contribution and TZPE is the effective

noise temperature due to the zero-point fluctuations. In Figure 2.2 Tn

and its various components are shown as a function of the thermodynamic

temperature. This illustrates that for high frequencies TB is less than

T, that the zero-point term is a constant with respect to temperature

and that the total noise temperature, Tn, is always greater than or

equal to the thermodynamic temperature.

The noise temperature as a function of the operating frequency is

presented in Figure 2.3. This illustrates that for low frequencies the

noise temperature is equal to the thermodynamic temperature. For higher

operating frequencies the thermal contribution falls off. In the same

frequency range where TB has begun to decrease, TZPE has increased to a

significant level. In this manner the total noise temperatures a con-

tinuously increasing function of the operating frequency.










10
9
8
7
6


(K)


TZPE


01 2 3 4 5 6 7 8 910


T (K)


Figure 2.2.


Noise temperature vs. T for an
operating frequency of 100 GHz.
















1000

Tn
(K)

100





10
-7

-3 T
-2
23
1 I -
108



























Figure 2.3.


10
f (Hz)
f (Hz)


Noise temperature as a function of frequency with
the thermodynamic temperature as a parameter.
Solid curves are the total noise temperature, Tn.
Dashed curves are the thermal contribution, T .
The solid line represents TZPE.










At any fixed operating frequency the difference between Tn and TB

is always given by TZPE. However, its relative importance is seen only

at low temperatures for an operating frequency of 100 GHz. At optical

frequencies the effect of the zero-point energy completely dominates the

noise behavior, even for high temperatures.

A main point of discussion concerning the zero-point energy is

whether or not it can be extracted from a system. The most general con-

clusion seems to be that it is not exchangeable, but yet has certain

properties which allow its presence to be substantiated by measurements.

In order to illustrate this point, consider an electron which is in

oscillatory motion.

It must be emphasized that the zero-point energy level correspond-

ing to Eg is the average energy of the particle. This is the sum of the

kinetic energy and potential energy. It is possible for the velocity to

fluctuate by converting kinetic energy to potential energy and vice

versa. In this way the fluctuations may be detected.

This is in close analogy to the case of two matched resistors, in

equilibrium, operated at low frequencies. In this situation the elec-

trons in the resistors give noise through velocity fluctuations, even

though they have zero net velocity.

The noise of one resistor creates a signal across the other and

represents the flow of power. This cannot continue unless the inverse

process is also taking place. Noise power must flow back from the

second resistor at an equal rate. This is merely another description of

the term thermal equilibrium.

In order for the zero-point energy to be extractable from a system,

there would need to be an energy level lower than E0. Since, by











definition, EO is the lowest possible energy level, there is no way for

the system to give up energy (on the average).

Historically, the presence of the zero-point energy was first seen

by Mulliken. The following quotation from French and Taylor describes

his work.

The study of molecular spectra led to the verification of
the existence of zero-point energy. In fact, the need to
describe the permitted vibrational energy levels by odd
multiples of 1/2 h . rather than by integral mul-
tiples of hw0 (whicR are the even multiples of 1/2 h )
0 0
was actually inferred from molecular spectra before wave
mechanics had been invented. This was done by R.S.
Mulliken in 1924 [see R.S. Mulliken, Nature 114, 350
(1924)]. Although the analysis was rather complicated in
details, it depended in essence on comparing the vibra-
tional energy levels for two molecules of the same chemi-
cal type but involving different isotopes. . Mulliken
was able to show that the inclusion of the zero-point
vibrated energy 1/2 hw at the base of the energy
structure leads to a good fit with the spectral data,
whereas without it there is a small but significant
discrepancy. This is a particularly interesting and
impressive piece of analysis: one might be tempted to
reason (incorrectly) that, since a particle in the ground
state can never radiate away its zero-point energy,
therefore the existence of the zero-point energy cannot be
verified by experiment. Mulliken's work showed that the
zero-point energy has observable consequences. [5, p. 173]

Another experimental verification came about through the work of

M.J. Sparnaay. These unusual measurements were on the force between two

closely spaced parallel metal plates at low temperatures. An electric

field between the two conductors induces eddy currents and causes a

force to exist on the plates. This is known as the Casimir effect. It

is the force due to the fluctuating zero-point field which Sparnaay

measured and which is described by Boyer [7].

At least one other observation of the zero-point energy has been

made. In 1981 Koch, Harlingen and Clarke reported on measurements using

a resistively shunted Josephson junction [8,9]. This technique made use











of the AC Josephson effect whereby the application of DC voltage to the

junction causes the production of a very high frequency oscillation (up

to 450 GHz). This high-frequency signal then mixes with the noise of

the resistive shunt and converts it down to the 100 kHz IF.

Their measurements clearly showed the influence of zero-point

fluctuations at high frequencies. At low frequencies the noise was just

the well-known thermal noise of the resistance.

Measurement Technique

It was the original intent of this project to measure the noise of

a matched termination at 2 K and 97 GHz using a very low noise (cross-

correlation) receiver system. When it became evident that this method

would not work [10], it was decided that the noise in MOM tunneling

junctions should be pursued instead. The proposed slow modulation

scheme for use with MOM diodes has been described in a paper by Van der

Ziel and Sutherland [11]. The several measurement schemes which were

considered will be briefly discussed.

The block diagram of the cross-correlation system is shown in

Figure 2.4. It is the resistive termination of the power divider which

causes the measurement system to fail in the determination of the zero-

point energy term.

As shown in equation (2) of the reference, the output reading, S,

of the system is


S = Tk[Ts T ]Af (2.5)


where T is the coupling between ports (= 1//2) Ts is the total noise

temperature of the device, TO is the total noise temperature of the

internal termination, k is Boltzmann's constant and Af is the



















ISOLATOR


DEWAR


I---1
MATCHED
TERMINATION


MIXER


Figure 2.4.


Simplified block diagram of the originally
proposed cross-correlation receiver.











measurement bandwidth. In the general case Ts and TO will consist of

the sum of two terms: a thermal contribution and the zero-point energy

contribution. This allows the total noise temperatures to be expressed

as


T = TB + TE (2.6)
S SB ZPE

T = TB + T ZpE. (2.7)
0 OB ZPE


It should again be emphasized that TZPE does not depend on the

thermodynamic temperature. It is strictly a function of the operating

frequency.

Now it is clear that the contributions from the zero-point energy

exactly cancel in equation (2.5) and thus the meter reading does not

depend on the value of TZPE. Therefore, the presence or absence of the

zero-point energy term cannot be detected by this method.

This is not to say that the system does not offer some advantages

over a single-channel receiver. It still responds with a better noise

performance than a standard receiver. Cross-correlation techniques are

used extensively in radio astronomy in order to separate the very faint

signals from the stars from the much larger background noise of the

detection system.

In a similar way it may be shown that the microwave noise-measuring

system of Chapter V also does not respond to the zero-point energy.

This method also involves the difference between two noise temperatures.

In a more general sense, any noise measurement scheme which involves

taking the difference (or making a comparison) between two noise

temperatures will fail to determine whether the zero-point energy term

is present.










The use of an MOM diode was considered to be a strong tool for

measuring the zero-point energy. The measurement technique makes use of

the fact that for large enough bias the noise should be the full shot

noise of the DC current. This provides a calibration directly from the

device itself and does not require a comparison to an external noise

source.

A simplistic band diagram and I-V characteristic of an MOM tunnel

diode is shown in Figures 2.5a and 2.5b respectively. The thickness of

the oxide region must be less than approximately 100 A in order for

significant tunneling to occur. At low bias voltages there are two

tunneling currents which flow. In equilibrium they are equal in magni-

tude but opposite in direction. Lecoy investigated the noise properties

of this type of device and found the noise to be full shot noise of the

individual currents [12]. In equilibrium the thermal noise of the

device may be shown to exactly equal the full shot noise of the two

tunneling currents.

Sanchez et al. have given the zero-bias resistance of MOM diodes as

[13]


S exp(S) (2.8)
R = S l (2.8)
R0 324 00A


where


S = 1.025 L1/2

A = junction area in um2

0 = barrier height in eV

and


L = oxide thickness in A.





















METAL OXIDE METAL



(a)


(b)


Figure 2.5. Important characteristics of MOM diodes.
a) Band diagram. b) Current-voltage
characteristic


__ L [4-











If a diode resistance of 450 ohms is desired (approximately the

characteristic impedance of WR-10 waveguide at 100 GHz), then
-12 *
for (0 = 1 eV, L = 10 A, a junction area of 2 x 10 m2 is required.

Since the MOM diode operates by quantum mechanical tunneling, the

spectrum of the shot noise should be white. This is due to the very

short transit time of a tunneling electron.

The experiment may be described with the aid of Figure 2.6. This

shows the total spectral intensity which is measured as a function of

the bias voltage. The technique requires that the bias voltage of the

diode be very slowly swept from positive to negative values. This gives

a time-varying signal at the output of the system, which may be

detected. In order to distinguish the quantum effects, it is necessary

for the bias to change by an amount on the order of V P hf/q. For an

operating frequency of 1011 Hz this is approximately 1 mV.

The signal at the output of the system will consist of two compo-

nents: a DC contribution, SO, due to the background noise of the

measurement system, and an AC component, AS(t), due to the modulated

noise of the diode. By using a very large series blocking capacitor,

the DC component may be eliminated and only the desired AC signal will

remain.

In order to test the validity of the slow modulation technique, the

measurement system of Figure 2.7a was implemented. The measurement

system works in the following manner. The isolator at the input reduces

the possibility of reflected receiver noise. A SSB filter is used to

select only the upper sideband response of the mixer. The mixer, local

oscillator and first IF amplifier are all supplied as a custom design

from Hughes. The thermistor and power meter respond to the 2-4 GHz IF.











SHOT NOISE
ASYMPTOTE
S




THERMAL
THERMALNO ZPE ,/ AS(t)
NOISE
WITH \ /
ZPE So
SO
e ; RECEIVER
'( BACKGROUND


-1 0 1
V (mV)


Figure 2.6. Spectral intensities for the MOM measurement.











The recorder output signal from the power meter is passed through a DC

block and low-pass filter, and the 0.2 Hz signal is finally detected

using the HP 3582A FFT spectrum analyzer. Convenient units of measure

for the spectrum analyzer are dBV. That is, a voltage proportional to

power (noise temperature) is produced by the 432B power meter.

In order to verify that the system had the required sensitivity, a

test signal using the arrangement of Figure 2.7b was measured. The

waveguide termination is kept at temperature T, which for the example

shown is 77 K. The ferrite modulator is an electrically controllable

switch which has two states. In the low-loss state the modulator passes

signals with only a small attenuation (~ 2 dB). The high-loss state

gives an attenuation of greater than 20 dB. In the high-loss state the

noise temperature out of the system is equal to the ambient temperature

of the modulator. Both the ferrite modulator and calibrated attenuator

are at room temperature. The calibrated attenuator is used to set the

signal level.

Results of the measurements are plotted in Figure 2.8 and show that

the system is able to detect signals on the order of 0.1 K. The system

is very linear for signals of 0.5 K and larger. These measurements

indicate that the system is able to accurately detect very small changes

in noise temperature, even though the receiver noise temperature is on

the order of 1000 K. This sensitivity is due to the large bandwidth

(" 2 GHz) and long integration time (m 1000 seconds).

Fabrication of MOM Tunnel Diodes

Many sources have reported the fabrication of MOM-type structures

[14,15]. The techniques are similar to those used to construct

Josephson junctions [16,17]. In most early cases the device geometry













AMPLICA


DC
432B BLOCK


ISOLATOR


2-4 GHz


94 GHz


3582A


0.2 Hz


REF
PLANE



II

CALIBRATEDI
ATTENUATOR



(b)


Figure 2.7.


Millimeter wave-noise measurement system.
a) Block diagram of the receiver channel.
b) Block diagram of the system used to create
a test signal in order to measure
the receiver sensitivity.


REF
PLANE
I
I


FILTER


zo
AT T










-50


-60



-70 -
METER
READING
-80 -
(dBV)

-90



-100


-110


0.1


1.0
ATeff


Figure 2 8.


Output meter reading vs. effective input
noise temperature for the noise measurement
system of Figure 2.7a.


10.0


(K)











was quite large, and the zero-bias impedance was often 106 ohms or

greater [18]. This indicates the need to reduce the oxide thickness of

the device as well as the junction area. It is necessary to stay within

a factor of seven of the characteristic impedance of the waveguide in

order to be able to match to the device.

For the patterning of the required fine metal lines, the lift-off

technique was chosen [19,20]. The processing steps are outlined below.

1) Clean the substrates. Boil for 5 minutes in TCE, Acetone and

Methanol.

2) Rinse in DI water.

3) Spin dry.

4) Dry bake at 1800C for 20 minutes.

5) Spin on photoresist at 2000 rpm for 20 seconds if using

AZ 1470.

6) Softbake the photoresist for 20 minutes at 950C.

7) Allow substrates to cool for approximately 5 minutes.

8) Expose the photoresist (9 seconds for the Karl Suss mask

aligner).

9) Develop the photoresist pattern by immersing the sample for 60

seconds in MP-315 developer.

10) Rinse in DI water.

11) Spin dry.

12) Place samples in evaporator and evacuate to 10-5 Torr.

13) Coat samples with aluminum to 2000 A thickness. Evaporation

should be carried out in as near normal incidence as possible

to aid lift-off.

14) Remove slides and soak in acetone to remove the unwanted metal.











The sequence of steps is repeated for the counter electrode after

the growth of the oxide has been completed. Several methods were inves-

tigated for the control of oxide growth.

Plasma oxidation in a commercial rf photoresist stripper always

yielded very thick oxides regardless of the oxidation time [18]. This

was unsuitable for our application. In fact, the main problem was to

limit the growth of oxide. Even the thermally grown oxide which

resulted from exposure to atmospheric air during processing for the

counter electrode was sufficiently thick to give a high impedance

junction.

A DC sputtering system was constructed to try and remove the

natural oxide just prior to evaporating the counter electrode. This

still did not provide low-resistance junctions, possibly due to poor

electrical connection to the devices. Next the evaporation system was

modified to perform RF sputtering. This method normally works even for

sputtering dielectrics and does not require an electrical connection to

the device. Even RF sputtering did not yield low-resistance devices.

One possible explanation is that the background pressure of oxygen may

have been too high. Argon was used as the main sputtering gas.

According to Chapman the monolayer formation time for a gas at a

pressure of 10-6 Torr is approximately 1 second [21]. A contaminant gas

at this pressure would be able to form a monolayer in this time. A

typical value of the best vacuum achieved in our system was 2 x 10-6

Torr. This is near the specification of the evaporation system and is

adequate for most metal depositions. It is not sufficient for use in a

sputtering system when control of the oxide thickness must be within a

monolayer.










The best fabrication method resulted in junctions with a zero-bias

resistance of 350 kS. In this technique the first electrode was pro-

cessed in the usual manner. The photoresist was prepared as described

for the second electrode, but immediately prior to placing the samples

into the evaporator they were immersed in a dilute (3:1) solution of

aluminum etchant. This etchant removes any oxide and also the first

layers of aluminum, and should give a clean, oxide-free surface. The

samples were loaded and the system pumped down in the shortest possible

time. The I-V characteristic which resulted is shown in Figure 2.9.

It is seen that the characteristic has the desired shape except for

the high value of zero-biased resistance. The characteristic is linear

near zero bias and antisymmetrical. Using equation (2.8) and assuming

i0 = 1 eV, the oxide thickness is determined to be 23 A. More typical

overall values of oxide thickness were on the order of 100 A.

Even the best device which was fabricated is not good enough for

use in the 450-ohm waveguide system since it had such a large

impedance. The impedance mismatch in this case would only allow, at

most, 0.5% of the noise signal from the device to be transferred to the

system. This is unacceptable.

Alternative Devices: Schottky Barrier and p -n+ Tunnel Diodes

The lack of adequate devices was by this time a major obstruction

to progress. Two other devices, p+-n+ tunnel diodes and Schottky

barrier diodes, were selected as possible alternatives to use in the

100 GHz measurements.

Commercial Schottky barrier detector diodes (Hughes model

47316H-1111) were available already mounted in a waveguide holder.

These are considered to be the finest detectors available for 100 GHz

operation. The mount structure is illustrated in Figure 2.10.















4

3

2


I(pA) 1
0

-1

-2

-3


-4

-5


-5-4-3 -2 -1 0 1 2 3 4 5

S-4)
V (10 V)


Figure 2.9. Measured I-V characteristic of the MOM
diode with the thinnest oxide barrier.


p
0
*



" *


*
-* RO== 350 kQ

- e

- *

- *
II
nS
n0

B0

0 RO 35Okfl






SI I I II I I I
















FULL
HEIGHT
INPUT


LINEAR TAPER
TO 1/4 HEIGHT


DIODE


TUNABLE
SHORT


~I


SSMA
CONNECTOR


Figure 2.10. Hughes Schottky barrier detector mount.











Noise measurements were made at 97.5 GHz on this device and gave

the surprising result that the noise temperature did not change as a

function of bias. This result was unusual since at low frequencies the

noise temperature is found to quickly drop to mT/2 where m is the non-

ideality factor of the diode and T is the ambient temperature. Further

study led to the explanation of this result through a transit-time model

of the device. The complete theoretical model and its verification

through noise measurements are presented in Chapters III through V.

The result of attempts to apply Schottky barrier diodes to the

problem of measuring the zero-point energy was the conclusion that the

device was unsuitable This is due to the dominance of the group of

returning electrons for low bias, as will be shown.

As a final consideration p-na tunnel diodes were measured. The

main advantage of p -n semiconductor tunnel diodes is that the zero-

bias resistance is near the characteristic impedance of the waveguide

system. Typically, RO is in the range 50 150 ohms.

A four-step Tchebycheff waveguide transformer was designed follow-

ing the work of Cohn [22]. The transformer dimensions are given in

Figure 2.11. The waveguide transformer was made from gold-plated brass

in split-block style.

Even though the transformer was fabricated with the greatest care,

it did not operate satisfactorily at 97.5 GHz. Losses due to the

increased surface roughness from machining were found to be on the order

of 1.5 dB.

A better solution was to use the commercial Hughes detector mount

and replace the Schottky barrier diode with the tunnel diode chip.
































LENGTH = 38.1


Four-step Tchebycheff transformer for use with
p+-n+ tunnel diodes. All dimensions are in mils.


7.0
14.9

9.7

2.4

S15.8


Figure 2.11.











The proprietary tunnel diode samples were supplied through TRW and

are designated as type TD-153. These are 15 x 15 mil. chips of

germanium and have gold-plated contact areas.

The measured noise data are presented in Figure 2.12. It is seen

that the measurements do not agree with the theory. This is most

probably due to the large junction capacitance of these diodes. The

heavy doping and narrow space-charge region required to produce

tunneling cause the junction capacitance to be large. This capacitance

was approximately 0.2 pF for the diodes which were measured. At an

operating frequency of 100 GHz this gives a junction reactance

of 8 a and effectively shorts out the junction.

Conclusion

A theoretical overview of the quantum effect which it was desired

to measure has been presented. For various reasons it was not possible

to actually measure this effect.

It has been shown that any comparative type of system cannot detect

the presence of zero-point fluctuations. An alternative method sug-

gested by Van der Ziel has merits but requires an active device. The

three types of devices considered were MOM tunnel diodes, p -n semi-

conductor tunnel diodes and Schottky barrier diodes. Each had the

promise of allowing the measurement of the quantum noise, but none were

actually suitable. This made clear, especially for the p+-n+ diodes and

Schottky barrier diodes, that the device operation and description for

high frequencies was not well understood.

The physical models, theories and measurements which were developed

for Schottky barriers and tunnel diodes are the subject of the remainder

of this work. It is hoped that this will prove to be a useful











400-


DUT

(K)
300





200



o
-30


-20 -10


0 10 20 30


BIAS VOLTAGE (mV)









Figure 2.12. Measured and theoretical noise
temperatures for the p+n tunnel
diodes at a frequency of 97.5 GHz.







30



contribution to the description of the high-frequency behavior of these

devices.

A measurement system which can detect changes in noise temperature

on the order of 0.1 K was constructed and its operation at 97.5 GHz

demonstrated. This technique is considered adequate should suitable

devices become available in the future.

















CHAPTER III
PHYSICAL MODEL OF SCHOTTKY BARRIER DIODES


Introduction

Schottky barrier diodes have become increasingly important due to

their excellent high-frequency properties. They are in widespread use

in the mixing and direct detection of signals at frequencies up to

several hundred gigahertz [23]. Since these devices are being operated

at such high frequencies, it is necessary to have a model which is

capable of describing the device behavior on time scales on the order of

the transit time of electrons across the junction. This dissertation

presents such a model for silicon Schottky barrier diodes and compares

the theory with experimental results.

Chapter III presents background information and introduces the

physical description necessary to create a complete device model. The

main emphasis is to include transit time effects. In addition, the

measured DC characteristics are discussed.

Chapter IV deals with the noise behavior of the device at low and

intermediate frequencies. In order to show that the results of Chap-

ter V are due to transit time effects, it was necessary to show that the

diode behaves in the well-known manner at intermediate frequencies.

For the lowest frequencies the excess noise is dominant. The

excess noise is important since it can be upconverted to microwave

frequencies. This may limit the usefulness of some diodes when operated

as a mixer with a low-frequency IF.











At intermediate frequencies, above the excess noise corner

frequency, the spectral intensity is equal to the shot noise of the DC

current. The well-known shot noise theory is adequate in this range.

Chapter V gives a complete model of the Schottky barrier diode

which includes transit-time effects. This is necessary in order to

describe the properties of the diode at microwave frequencies. Very

good agreement was found between theory and experiment for the high-

frequency case.

Physical Model

In their basic form Schottky barrier diodes have a very simple

configuration. This is depicted in Figure 3.1. A small metal anode

contact (e.g. gold) is deposited on top of a semiconducting wafer (e.g.

Si or GaAs). A large-area ohmic back contact is used in order to lower

the series resistance of the bulk semiconductor.

In spite of the simple configuration of Schottky barrier diodes,

the physical mechanisms which govern their operation are complex. The

importance of each mechanism may also vary with frequency and bias.

There are several current transport processes which may be present

for the electrons traveling from the semiconductor to the metal. It is

necessary to distinguish between the possibilities of tunneling,

thermionic emission, diffusion and recombination of electrons in the

space-charge region. Justification will be given for the neglect or

consideration of each form of current transport.

The device description starts with a consideration of the band

diagram, Figure 3.2. The metal on the n-semiconductor gives an energy

barrier of q ms. The value of q'ms is often calculated for metal-

semiconductor systems by considering the work functions, electron














METAL


7, A


ANODE
DEPLETION
REGION


CONTACT


Figure 3.1.


Configuration of a simple Schottky
barrier diode configuration


n-SEMICONDUCTOR


BACK


/ 0/////////////////












- -- - -


n-SE
n-SE


Figure 3.2.


Q

0

0


qVdif


-------- EC
EF

MICONDUCTOR














Band diagram of the metal-
semiconductor contact.


Ims

q~ms


METAL










affinities and doping densities of the materials. This calculated value

may be of little use as the actual barrier height also depends on the

number of interface states, and this number may not be well controlled.

In practice the actual barrier height is determined from I-V measure-

ments, once the reverse saturation current is known. This eliminates

the need for assumptions concerning the metal-semiconductor barrier

height. It will be assumed that the Schottky barrier height remains

constant with bias.

The difference in energy between the conduction band and Fermi

level (far from the junction region) is qVn. This is governed by the

doping density in the semiconductor material and may be found from [24]


N
Vn = VT n(C) (3.1)



where VT is the thermal voltage (= kT/q), NC is the effective density of

states of the conduction band and ND is the doping density.

Once qOms and qVn have been determined, then the diffusion

potential, Vdif, may easily be found:



V = V (3.2)
dif ms n


This will be a convenient quantity for many calculations.

In this work the depletion approximation is applied such that in

the depletion region the space-charge density is the same as the doping

density of the semiconductor. This is illustrated in Figure 3.3a. Once

established that the space-charge density, p, is constant in the junc-

tion region, then it follows immediately from Poisson's law that the

electric field varies linearly:














qND -


-EMAX


Figure 3.3.


Device model in the junction region.
a) Fixed space-charge density as a
function of position;
b) Electric field variation with
position.











d (eE) = p (3.3)
dx


By imposing the boundary condition that the field is zero in the

semiconductor, then equation 3.3 is solved to yield



E(x) = E 1) (3.4)



This may be integrated once to find the potential. In the junction

region the potential is parabolic and, thus, so is the bending of the

conduction band, as already indicated in Figure 3.2.

In order to describe the junction behavior, the current transport

properties must be identified. Although many exceptions are possible,

some general guidelines can be given.

For extremely high doping the semiconductor will become degenerate,

and the depleted region will become thin enough for quantum mechanical

tunneling to occur. This process is not believed to be significant in

the diodes which were measured, as was discerned from the I-V character-

istics and noise properties. Diodes which have significant tunneling

show a large non-ideality factor. For the diode under consideration the

non-ideality factor was 1.09, and this will be discussed in detail in

the section on I-V measurements. More importantly, since tunneling is a

very fast phenomenon, there should be little variation of the device

parameters and noise with frequency. The measured results show a

significant change of both the noise and admittance as the frequency is

increased.

In the case of very low doping the current transport is dominated

by diffusion processes. Since low doping causes the depletion region of











the semiconductor to be large, the possibility of an electron scattering

in the junction region is high. Once the number of electron collisions

becomes large, then the time required for the current to respond to an

applied signal increases. A device operating in this mode would be

unsuitable for use at microwave frequencies. The silicon Schottky

barrier which was used in this work is a commercially available detector

which operates to 110 GHz. Therefore, the diffusion limited process is

discounted.

Another possible current mechanism is the recombination of

electrons in the space-charge region. This mechanism is characterized by

a non-ideality factor of two. As previously mentioned, a non-ideality

factor of 1.09 was observed. This allows one to neglect the possibility

of recombination current for the bias range involved.

Lastly, there is the possibility of thermionic emission of

electrons over the metal-semiconductor barrier. Since it is believed

that this mechanism controls the device behavior, it will be examined in

detail. In order for thermionic emission to dominate the DC current,

the depletion region width must lie within a range of values. That is,

the doping density should not be so low that significant scattering

could occur in the space-charge region. In this work it will be assumed

that electrons can cross the space-charge region without suffering any

scatterings. Conversely, the doping should not be high enough to allow

the tunneling of electrons across the resultant narrow space-charge

region.

According to Sze [24], the DC current flow will be due to

thermionic emission if the electric field is between 104 V/cm and 105

V/cm in silicon diodes. This is the case for the assumed doping density

of 1017 cm-3.










The process of thermionic emission is the thermal excitation of

charge carriers to a sufficient kinetic energy such that they may cross

the energy barrier of the junction. Since the bottom of the conduction

band represents zero kinetic energy, electrons in the semiconductor must

have at least an energy of qVdif in the negative x-direction in order to

reach the metal contact and be collected.

The electrons of the semiconductor participating in the charge

transport can be subdivided into three groups. See Figure 3.2. The

physical mechanisms involved will be presented in this section, but a

detailed mathematical description will be delayed until Chapters IV and

V.

Electrons of group 1 have insufficient kinetic energy in the

negative x-direction (KE < qVdif). See Figure 3.2. These electrons

always return to the neutral semiconductor and give no contribution to

the DC current. It will be shown in Chapter V that these electrons have

the same characteristic time of flight regardless of their initial

velocity. At low frequencies these electrons give no contribution to

the device conductance. As the operating frequency increases, the

conductance due to these electrons rises as w2 and dominates the overall

device conductance. It will be shown that this group of electrons

controls the high-frequency behavior of the device at low biases. The

conductance and noise of this group of electrons does not change

significantly with bias. Most electrons of the semiconductor occupy

states near the bottom of the conduction band. A change in the barrier

height due to an applied voltage will not cause any appreciable change

in the number of electrons in this group.

The electrons of group 3 have sufficient kinetic energy to just

reach the metal contact (KE = qVdif). If no signal were applied, then











these electrons would always be collected at the metal contact. With an

applied small-signal voltage some electrons of this group will be

collected while others will return to the neutral semiconductor. For

this group of electrons dI/dV is large. At intermediate frequencies

this group of electrons dominates the device conductance, even though

only a fraction of the DC current is carried by them. The conductance

due to this group of electrons is constant up to frequencies on the

order of the reciprocal transit time and decreases rapidly at higher

frequencies. The intermediate frequency conductance of this group of

electrons is given by the slope of the I-V characteristic.

The electrons of the semiconductor which have sufficient velocity

(kinetic energy) to always pass the energy barrier are designated as

belonging to group 2, as shown in Figure 3.2. Since the electrons of

this group are always collected, they give rise to the DC current. With

an applied signal the velocity of the electrons during the transit time

is slightly modulated. This is a small effect and the quantity dI/dV is

approximately zero for this group. The contribution to the conductance

from this group of electrons may be neglected for all frequencies of

interest.

In order to calculate the noise temperature of the device, the

spectral intensity of the current pulse due to each electron must first

be found. From the equations of motion the velocity of an electron as a

function of time may be found for an electron of any group. This

velocity relates directly to the current flowing,



i(t) v(t) (3.5)
d











Once the current flowing as a function of time has been determined.

it is only necessary to take the Fourier transform in order to find the

spectral intensity of the current as a function of frequency. See

Chapter V.

For the electrons of group 1 the spectral intensity will be seen to

start at zero for zero frequency (no DC current) and then to increase

as w2. This spectral intensity is not a strong function of bias since

the number of electrons in group 1 does not change significantly with

bias.

Group 3 is a special case of group 2 under the assumption of no

applied signal. The same description will hold for both in calculating

the spectral intensity of the noise. These electrons carry the DC

current and, since the electrons are emitted at random times, give an

intermediate frequency spectral intensity equal to the shot noise of the

DC current. The spectral intensity is white up to frequencies on the

order of the reciprocal transit time and drops steeply at higher fre-

quencies. This noise component varies linearly with the device current

(exponentially with voltage). At high bias the passing electrons will

dominate.

One additional current must be included in order to form a complete

model of the device. This is the reverse saturation current. The

reverse saturation current consists of those electrons in the metal

which have sufficient kinetic energy in the positive x-direction to

cross the energy barrier of q ms Once the electrons have crossed the

energy barrier, they are swept by the electric field to the neutral

semiconductor. Since the electric field causes the electrons to be

accelerated to a very high velocity, their transit time is extremely











short. For this reason the spectral intensity due to these electrons is

considered white and equal to the shot noise of the DC current. The

energy barrier qOms is not a function of bias in this model. This

allows the magnitude of the reverse saturation current to be considered

constant.

I-V Characteristics of Schottky Barrier Diodes

The current transport mechanism which will be considered is the

thermionic emission of electrons. The results of this model have been

known for many years [25]. Some background information will be

presented in order to show the assumptions invoked, but the reader is

referred to the references for complete details.

The calculation of the I-V characteristic is simplified by the

realization that only electrons of groups 2 and 3 contribute to the DC

current. The reverse saturation current is considered constant in the

present model. Thus, it is only necessary to find the number of

electrons traveling from the semiconductor to the metal per-unit time.

This current is a function of the applied bias voltage. The description

to be used is that of Van der Ziel [26], although the notation has been

changed slightly.

As already mentioned, only electrons with a kinetic energy of

greater than qVdif directed in the negative x-direction can escape the

semiconductor. The electrons lose kinetic energy in the amount of qVdif

in doing work against the built-in electric field. Conservation of

energy requires that



Sv-2 + 1 mv2 (3.6)
2 x dif 2 x











where the unprimed velocity is the velocity of the electron upon arrival

at the metal contact, and the primed velocity is the electron velocity

upon departure from the semiconductor.

In addition to the conservation of energy, it is also necessary to

describe the distribution of electrons in velocity. By summing over all

possible electron velocities, the total current may be obtained. The

number of electrons with velocities that lie in the velocity space

between v' and v' + Av', v' and v' + Av', and v' and v' + Av' is given
x x x y y y z z z
by ref. [26]:


22 E m2 1 mv2 -1 ,I-2 qV
2m2 F 2 x -my -qV
AN = exp( x 2 2 )AvAvA
h3 kT x y z
(3.7)

It is assumed that the semiconductor is non-degenerate so that Maxwell-

Boltzmann statistics apply.

The actual number of electrons which arrive at the edge of the

space-charge region is found as follows. Consider a volume which has a

cross-sectional area of 1 m2 in the y-z plane. All electrons which

have a velocity of v' will arrive at the surface in one unit of time
x
(one second). Thus the number of arriving electrons per unit time per

square meter which have a velocity in the prescribed range is given by



An = v'AN (3.8)
x


Since each electron carries a charge of q, the DC current which is

flowing is


AId = -qAn(Area) .


(3.9)











The total current Id is found by summation over all possible velocity

intervals. Carrying out the integration yields the familiar Richardson

equation,


V s
Id = Area(4hqmkr)T2 exp( ms) (3.10)
h3 T


The expression in parentheses is the Richardson constant, A, which

assumes an electron mass equal to the free mass. For semiconductors the

actual mass should be replaced by the effective electron mass, and this

revised constant is termed A For free electrons the Richardson

constant is 120 A/cm2/K2. The effective Richardson constant may also be

a function of the electric field strength. This is designated by A .

A detailed calculation by Andrews and Lepselter [27] shows that an

average value of effective Richardson constant A** equal to

110 A/cm2/K2 is appropriate for silicon doped to 1016 cm-3. This value

of effective Richardson constant was used in all calculations reported

here.

An effect which has been omitted in the calculations is the image

lowering due to the Schottky effect. The image lowering causes a

reduction of the barrier height as a function of the applied electric

field (voltages). The barrier height may no longer be considered

constant, but its variation with bias must also be taken into account.

The variations from the ideal case are adequately modeled by

including a non-ideality factor m in the expression for the diode

current. A simplified expression of the diode current is



= I(exp(- 1) (3.11)
T











where

-8
IO = (Area)A**T2 exp( ) (3.12)
0 V VT


Note that the constant reverse saturation current has now been included.

Schneider has reported a model in which the non-ideality factor influ-

ences the reverse saturation current [28].

The measured I-V characteristic of the Hughes model 47316H-1111

silicon Schottky barrier diode detector is shown in Figure 3.4. All

measurements were performed at 300K. The slope of the line indicates a

non-ideality factor m equal to 1.09. By extrapolating to zero bias the
-8
reverse saturation current is found to equal 3 x 10 A. Assuming a

junction diameter of 2 jm, the actual Schottky barrier height is deter-

mined to be 0.585 eV.

The saturation current and Schottky barrier height are important

parameters in the device model. Another important parameter which is a

parasitic effect is the series resistance. This plays an important role

at high-bias currents for both the admittance and noise. The value of

the series resistance is determined from the deviation of the measured

I-V curve from a true exponential curve at high bias. For a current of

2 mA the deviation is 20 mV. This indicates that the series resistance

is 10 ohms.

The device parameters determined in this section will be used in

the description of the noise properties. Unless otherwise stated, the

measured values were used without modification.

Summary

A physical model of the operation of Schottky barrier diodes has

been presented. Equations have purposely been kept to a minimum.


























I(A)


0 100 200 300 400 500 600
V (mV)





Figure 3.4. I-V characteristic of Hughes
diode at 300K.






47



It has been pointed out that for forward bias the three groups of

electrons traveling from the semiconductor towards the metal are

important. The role of each group of electrons varies with frequency

and bias.
















CHAPTER IV
NOISE OF SCHOTTKY BARRIER DIODES AT LOW
AND INTERMEDIATE FREQUENCIES


Introduction

This chapter gives a brief review of the well-known intermediate-

frequency theory of the noise in Schottky barrier diodes. Results of

measurements at intermediate frequency are presented, and these are seen

to agree with the standard theory.

Intermediate-Frequency Theory

The intermediate-frequency noise theory has been developed for some

time [29]. The basic starting point of an analysis is the DC current,

which one assumes can be described in the form


Id = I exp(-Vdif/mVT) exp(Vd IdRs/mVT) (4.1)


where Id is the diode current, I is the saturation current, Vd is the

applied diode voltage, Vdif is the diffusion potential, VT = kT/q is the

thermal voltage and m is the diode non-ideality factor. At low bias it

is common to neglect the effect of the series resistance, R This

equation is valid if the bias voltage Vd is approximately five times

larger than mVT, in order that the reverse saturation current may be

neglected. It should be noted that in this work Is does not contain the

term exp(-Vdif/mVT), as it does in many other representations, and thus

Is is much greater than Id.

Assuming that the current Id gives full shot noise and that the

device conductance is given by the derivative of the I-V relationship










when evaluated at the operating point, it follows directly that the

noise temperature is


SI(f) = 4kTnd = 2qld (4.2)


Tn = 2q Id/4k gd = mT/2 (4.3)


This is the well-known intermediate-frequency result and is commonly

used. Note that it does not depend on bias under the stated assump-

tions.

The spectral intensity of the shot noise (intermediate-frequency

noise) which is measured is the thermal noise from an equivalent

resistor of equal value. It is a common misconception that these

devices will show both thermal noise and shot noise when forward

biased. In fact, the shot noise is the actual measured quantity. Even

though the device has a known impedance, it is not a passive resistor at

300K, but rather it is an active device. Since the device is not in

thermal equilibrium, it is not necessary for it to give thermal noise.

Now consider the addition of a series resistance to the device

model as shown in Figure 4.1.

In a measurement only Seq can be determined and this has a value

of


Idr2 4kT r
S = 2q + ---- (4.4)
eq (rs+rj)2 (rsr )2


This may be expressed in terms of a noise temperature with


4kT /(r + r.) = S (4.5)
eq


which gives















v4kTAf/RS


Figure 4.1. Intermediate-frequency model of a
Schottky barrier diode.


EQ











2qld r T r
T i (4.6)
n 4k r +r r +rj
s j s j


Noise Measurement Schemes

The noise measurements were carried out using two standard tech-

niques. At low frequencies (10 Hz 25 kHz) the system depicted in

Figure 4.2 was used. At slightly higher frequencies (40 kHz 1.5 MHz)

the same system was used, but the FFT spectrum analyzer was replaced

with an HP Wave Analyzer (Model 310A, 10 kHz 1500 kHz). Above 30 MHz

a different scheme was used which will be described later.

Three measurements are required in order to determine the device

noise. Both Rbias and Rcal are chosen to be >>rdum and are also

selected to be wirewound or of thin film (for low excess noise).

First measurement: The device under test is mounted in place and

biased to the proper current.

Second measurement. The device under test is removed and replaced

with a dummy resistor which has the same resistance as the small-signal

resistance of the biased device.

Third measurement: The dummy load remains in place, but the noise

source is turned on in order to give a calibration reading. The value

of Rcal must be chosen large enough to convert the voltage noise source

into an equivalent current source.

Thus, there are three equations (see Figure 4.3),


M2 = G2.B{e2 + [i2 + i2](R Rca .iasr )2} (4.7)
1 n dut n calbias dut

M2 = G2.B{e2 + [i2 + i2](R calIR r )2} (4.8)
2 n dum n cal ias dum

M2 = G2. B2 + [i2 + 12 + i2](R R ir )21 (4.9)
3 n cal dum n cal Tias dum


















-VTOT


Rbias


CFW-101


Figure 4.2.


Noise measurement system for low-
frequency measurements.


















.2
2 'dut
'cal Rca
Rcal


Calibration


DUT


AMPLIFIER


Source

















Figure 4.3. Equivalent circuit of the low-
frequency measurement system.











so that

__- M2 M2
"1 j2 _4kT
12 = -- ) i2 + -4k (4.10)
dut 2 cal r(4.10)
M^ M dum
3 2

This is the method used in all measurements below 1.5 MHz. The meter

readings are Mi, M2 and M3 and have the units of voltage. The gain of

the amplifier is G and the measurement bandwidth is B. The equivalent

voltage noise source of the amplifier is en and the current noise source

is in. For simplicity the amplifier is assumed to have infinite input

impedance, although non-infinite impedance would not affect the final

result. The values of the calibration resistor and bias resistor are

Rcal and Rbias respectively. The small-signal resistance of the device

under test is rdut and the dummy resistor is chosen to have an equal

value. The noise currents are subscripted in accordance with their

respective origins.

For measuring the noise at higher frequencies, it is more conven-

ient to use microwave techniques in order to eliminate parasitic effects

(particularly the capacitance of the device mount and interconnecting

cable). The system used is shown in Figure 4.4. It uses a double stub

tuner to obtain an impedance match to the device under test.

A typical measurement using the system of Figure 4.4 would proceed

as follows:

Connect the DUT to the bias tee and bias it, using a large series

resistor, to the proper operating point. The connector of the bias tee

will be the reference plane.

Turn off all digital voltmeters.

Turn on the noise source (6). The signal from this source is

coupled to the main transmission line via the directional coupler (3).











NOISE
SOURCE
(AMPLIFIER)


NOISE
SOURCE


LEVEL
SET.
ATTEN


Figure 4.4. Noise measurement system for 30-150 MHz.




Equipment list of Figure 4.4

1 Bias Tee Model 5575, 10 kHz to 12 GHz; Picosecond Pulse Labs,
Boulder, Colorado.
2 Double Stub Tuner Model DS 109H, 40 MHz to 400 MHz; Weinschel
Engineering, Gaithersburg, Maryland.
3 20 dB Directional Coupler Model Ch-132, 1 1000 MHz; ANZAC.
4 Low-Noise Amplifier Model W1G2H, 1.5 dB NF, 5-1000 MHz; Trontech.
5 Amplifier Model 8447F, 0.1 1300 MHz, Hewlett-Packard.
6 Amplifier (noise source for matching) Model 462A to 150 MHz;
Hewlett-Packard.
7 Spectrum Analyzer Model 8558B, 0.1 1500 MHz; Hewlett-Packard.
8 Step Attenuator Model 8841B11, to 100 MHz; Shall Co., Inc.
9 Power Sensor 8484A and Power Meter 435B, 10 MHz to 18 GHz;
Hewlett-Packard.
10 Low-Pass Filter, RC time variable; in-house design.
11 Digital Voltmeter Model 3466A; Hewlett-Packard.
12 Calibrated Noise Source Model SKTU 1-1000 MHz; Rhode & Schwarz.










For most frequencies the signal will be reflected by the stub tuner and

will give a large signal at the input of the LNA. At those frequencies

where a match is obtained, the signal returned to the LNA will be very

small.

Adjust the stubs and observe the spectrum analyzer display until

a match is found. The match should typically be better than 20 dB over

a 1 MHz bandwidth.

Turn off the noise source (6). Its only purpose is to provide a

wideband signal for matching.

Change the analyzer to zero-span mode. Select a bandwidth of

approximately one-half the 20 dB match bandwidth. A smaller bandwidth

decreases the statistical accuracy while a larger bandwidth may

introduce systematic errors in the measured noise temperature.

During this procedure the output step attenuator should have been

set to at least 50 dB. This is to protect the sensitive power meter

from the large signals used during matching. Now that an actual noise

measurement is ready to proceed, it should be changed to 0 dB.

Record the meter reading. This will be Mdut. Make sure that the

system is linear by changing the input attenuator of the analyzer by 10

dB. The meter reading of the power detector should change by a factor

of 10. Note that this only checks the linearity from the analyzer input

and following. It is possible that the low-noise amplifier or power

amplifier is saturated, but this must be checked with the calibrated

signal source.

Return the output step attenuator to 50 dB in order to protect

the detector.

Remove the device and connect in its place the Rhode & Schwarz

noise generator (12). Set the output of (12) to zero.











Use the tuning stubs in conjunction with the spectrum analyzer

(7) and matching source (6) to match to the Rhode & Schwarz.

Turn off the noise source used for matching and return to the

same conditions under which Mdut was measured.

Make at least two meter readings corresponding to different

settings of the calibrated noise source. These will be referred to as

MC and MH. A third reading would allow a check of the total system

linearity.

From these noise measurements it is possible to determine three

quantities, namely, Tamp, the background noise temperature; Tdut, the

device under test noise temperature; and K.G2*B, the gain-bandwidth

product. A constant of proportionality, K, exists which converts the

detected power into the measured voltage.

The three meter readings are


Md = K.G2B(Td + T ) (4.11)
dut dut amp

MC = K.G2B(T + T amp) (4.12)
C C amp

M = KG2B(T + Tamp) (4.13)
H H amp


The results are as follows:

T -YT M
H C H
T = where Y =-- (4.14)
amp Y 1 MC


This is a standard Y-factor measurement:


M ) ( -M
T14 dut C) (TH-TC) + TC (4.15)
dt MH-MCC











Once the noise temperature of the device under test is known,

equation 4.2 may be used to convert to the spectral intensity of the

current fluctuations, using the device conductance.

This was done for the measurements between 30 MHz and 150 MHz. The

diode was assumed to have a small-signal resistance equal to that

obtained from the DC I-V measurements.

Discussion of Noise Measurements at Intermediate Frequencies

The results of the noise measurements between 10 Hz and 150 MHz are

presented in Figure 4.5. It is seen that the Hughes diode exhibits

excess noise at low frequencies and then reaches a constant noise level

for higher frequencies. The low-frequency noise goes approximately as

1/f. The constant value which is reached corresponds closely with the

calculated values for the shot noise of the bias current when the effect

of the series resistance is included. The calculated values are indi-

cated by the solid lines, and these correspond to a noise temperature of

mT/2. The series resistance is assumed to be 10 ohms as determined by

the I-V measurements. For the purpose of establishing a reference, the

thermal noise of an equivalent resistor is also shown by using the

broken lines. This corresponds to a noise temperature of 300K.

Above the excess noise corner frequency, the theoretical shot noise

level of the DC current coincides well with the measured noise level.

See Figure 4.6. This is true for the several bias currents which were

used during the noise measurements. These shot noise measurements

indicate that the device behaves in the well-known manner at inter-

mediate frequencies.










-16
10


SI(F)

-1
10

(A2/HZ)


101 102 103 104 105 106 107 108
F (HZ)


Figure 4.5.


Spectral intensity of current fluctuations
vs. frequency for the Hughes Schottky
barrier diode.


109








-20
10


SI(F)



10

(A2/HZ)


10


Figure 4.6.


-5 -4 -3
10 10 10
I (A)


Shot noise plateau values vs. DC bias current.
The solid line corresponds to full shot noise
(= 2ql)and the dashed curve indicates the noise
after correctingfor the series resistance.


-2
10






61



Discussion of Excess Noise Measurements

As may be seen in Figure 4.5, the Schottky barrier diode shows

excess noise at low frequencies. This excess noise shows some charac-

teristic g-r type Lorenzians but has the general character of 1/f.

In Figure 4.7 the excess noise at a frequency of 1 kHz is plotted

vs. the bias current on a double logarithmic scale. It is seen that a

power law of 13/2 is obtained. This is not the usual case as it is more

common for the noise to go as 12

According to Handel [30], the 1/f noise may be expressed as


QH
S (f) = 12 (4.16)
1 fN


In this case I is the current due to a stream of monoenergetic elec-

trons, f is the frequency, N is the number of electrons in the sample

and


a (A) (4.17)
aH 3w c


-+
Here a is the fine structure constant (= 1/137), Av is the vector

change in the velocity of electrons and c is the speed of light.

In a Schottky barrier diode operating in the thermionic mode, the

current flow is not due to a monoenergetic beam of electrons but.

rather, there is a distribution of energies. By subdividing the

velocities of passing electrons into small intervals, Avx, it is

possible to calculate the 1/f noise due to the electrons which have a

velocity in this interval. The total 1/f noise is found by applying

Handel's formalism outlined in equation (4.16) and summing over all

possible velocity intervals.


















Sl(f)

-20
10

(A2/Hz)

-21
10



-22
10



-23
10



-24
10


aly
Y = 1.5


MEASURED


23 7
S i l


I-6
10


1-5
10


, ,I I


a, a l


-4
10 (A)
I (A)


Figure 4.7. Excess noise at 1 kHz vs. bias current I.


1-2
10


I, I I I













S (f) = 1 S (f) = (AI )2 f- (4.18)
I i i i


Now AI represents the DC current flowing due to electrons in the

velocity interval Av The quantity ANi is known from a calculation of

the transit time. See equation (4.19). The transit time and Ali are

defined in detail in Chapter V. The computer program which was used to

carry out the evaluation may be found in the appendices.


AI T
AiN (4.19)
i q


The measured and theoretical values of the 1/f noise at 1 kHz are

plotted as a function of the bias current in Figure 4.8. It is seen

that the measured values are considerably larger than the theoretical

ones and indicate that the theoretical limit of excess noise has not

been reached. At least two possible explanations exist: 1) There may

be another "non-fundamental" 1/f noise source which is present; and

2) the excess noise may actually be of the g-r type (McWhorter's model).

If Handel's formalism is correct, then there exists the possibility

of considerable improvement in the 1/f noise properties of these

devices. This is an important indication for the use of these diodes as

detectors and mixers.

Conclusion

At frequencies up to 150 MHz the well-known shot-noise theory is

sufficient to explain the measured results for the regions where there

is no low-frequency excess noise. The device conductance is equal to

the derivative of the I-V characteristic, and the value of the current













118




1-19
10

sl(f)

-20
10

(A2/Hz)

-21

102


-22
10


-23
10


-24
10


alv
Y= 1.5


MEASURED


THEORETICAL


V = 0.84


-3
-223 7i
*3 *


I I


-6 1-5
) ~10


1-4
10(A)
I (A)


I I


1-3
10


Figure 4.8.


Excess noise at 1 kHz vs. bias current.
Solid line measurements, dashed line theory.


1-2
10


I I


* *I











spectral intensity is given by assuming full shot noise in the DC

current.

The low-frequency measurements show that there are no excess noise

sources which extend into the microwave frequency region.

A theoretical model of the 1/f noise was developed using Handel's

formalism and taking a distribution of the electron velocities into

account. This leads to a theoretical value of the 1/f noise which is

below the measured data and indicates that considerable improvement

should be possible.

















CHAPTER V
HIGH-FREQUENCY NOISE OF SCHOTTKY BARRIER DIODES


Introduction

Van der Ziel has provided the theoretical background for analyzing

the effect of the transit time on Schottky barrier diodes operating in

the thermionic mode [31]. The case to be considered is for a uniformly

doped semiconductor (linearly varying electric field). The physical

model was presented in Chapter III.

Calculation of the Junction Admittance

In Van der Ziel's work the initial velocities of the electrons are

divided into three cases for the purpose of calculating the small-signal

admittance of the diode at high frequencies. It is assumed that all of

the electrons are emitted from the interface between the depleted/non-

depleted region of the semiconductor.

Electrons which belong to group one have insufficient kinetic

energy to cross the barrier and return to the undepleted semiconductor

region. During their time of flight, the current which flows has the

time dependence illustrated in Figure 5.1. This is shown for the case

of a single electron, but in practice the amplitude must be multiplied

by the number of electrons injected with the same initial velocity.

The electrons of group 1, regardless of the initial velocity

of an electron upon entering the space-charge region, all require

the same characteristic time to return to the neutral semiconductor

region. Only the amplitude of the current pulse changes due to the
























StR




\\0




















Figure 5.1. The solid curve indicates the current pulse
due to a single electron of group 1 with an
initial velocity of v (0). The dashed curve
is for an electron with higher initial velocity.
This figure shows that all electrons of group 1
have the same characteristic time of flight.











different possible initial velocities. The characteristic time for an

electron to return is



TR = w/a (5.1)



where a is the plasma frequency and is equal to (eEmax/m*d)1/2. Both

Emax and d are a function of the applied bias. The effective mass of an

electron is m*.

These electrons carry no DC current and at low frequencies do not

give any significant contribution to the device admittance. The

characteristic frequency associated with the current pulses is

fR = 1/2 lT R. When operated at high frequencies, the frequency of the

applied signal may be of a frequency on the same order of magnitude as

this characteristic frequency. These current pulses thus give a contri-

bution to the device admittance. This component of the device conduc-

tance increases as w2 for low frequencies and reaches a maximum when the

frequency of operation corresponds to 1.37 times fR. The maximum may be

found by differentiating the expression for the conductance. The

conductance due to these electrons is given in [31] as


I I
s d 1 c (1 cosw(l-c) 1 cosw(-c
11 if V 2 1- c2 1 -c 1 + c
(5.2)

where the parameter c is the normalized angular frequency defined as

w/a where w is the frequency of operation. In this notation Is is very

much larger than Id for normal operation. This conductance dominates at

high frequencies when the diode is operated at low bias currents. It










should be recalled that most electrons have energies near the bottom of

the conduction band edge and thus belong to group 1.

If the series resistance of the device is zero, then it is possible

to calculate the effective noise temperature of the device with no

knowledge of the junction susceptance. However, for most diodes the

series resistance is not negligible, especially for high bias

currents. The combined effect of the series resistance and junction

susceptance will be discussed in the section on the theoretical noise

temperature.

The component of junction susceptance due to those electrons which

return can be found to equal


I I
Ss d C1 sin(l-c) + sin(l+c)7r (
11 Vdif d 2 1 c 1 + c


The electrons with sufficient kinetic energy to always reach the

metal contact are those of group 2. The current pulses due to two

possible initial velocities are illustrated in Figure 5.2. The device

admittance is found by applying a small AC voltage and calculating the

response of the electron velocity. The transit time is also changed due

to the applied signal.

Since the electrons from this group are always collected, they

carry the DC current and are not much affected by an applied small-

signal voltage (dI/dV is approximately zero for this group of

electrons). Nonetheless, an applied signal does influence the velocity

of electrons during the transit time, and the contribution to the

conductance may be calculated as

















i (t)


'
Q

NO


t


Figure 5.2.


The current which flows as a function
of time for an electron belonging to
group 2. The amplitude and transit time
both depend on the initial velocity of
the electron.






71




I 0 eV' eV'
g22- Vd f exp -( T xd T c 2 (sin sin co
dif d 0 c


+(1 cos(l-c))_ (1 cos(l+c)>))
1 -c 1 + c


(5.4)


where = aT eV' is the energy of the arriving electrons and T is the

DC transit time.

Again, the component of device susceptance may be calculated and is

equal to


I d eV eV'
b22 = dif exp (-)d(-I-) x1 c [ + sin cos c
22 Vdif Vd kT kT 2 2


+ (sin(1-c)) + sin(1+c))]
1 c 1 + c


(5


It should be noted that for most cases the effect of electr

this group may be ignored in the admittance calculation. Howev

the calculation of the spectral intensity of the current fluctu

the electrons of group 2 play an important role.

The final group of electrons which must be considered are

with barely sufficient energy to reach the metal contact.

electrons dominate the device conductance at low frequencies.

contribution to the junction conductance is


(1 2c sin 1- c c2 cos cw)
33 g ( -2 2
933 0 (1 c2)2


.5)



ons of

er, in

nations


those

These

Their


(5.6)


where go is the low-frequency small-signal conductance.










The contribution to the device susceptance is


(c2 sin re 2c cos 2-)
b33 = g0 (5.7)
3 (1 c2)2



The total junction admittance is found from a superposition of the

effects of the three groups of electrons. At low frequencies the

admittance has only a real part. This conductance is due to the

electrons of group three and is the same result as obtained from the

derivative of the I-V relationship. The calculation of the derivative

is analagous to evaluating the change in the number of electrons which

cross the barrier when a signal is applied. Since only the number of

electrons in group three is modulated, they give the low-frequency

junction conductance.

Calculation of the Spectral Intensities

In order to calculate the noise temperature of the diode, the

spectral intensities of the current fluctuations due to each group of

electrons must be calculated. The total spectral intensity is found by

summing the contribution of each electron. It should be noted that the

calculation of the spectral intensity can proceed by considering only

two cases: electrons which are able to cross the space-charge region

and those which can not. The electrons of group three always cross the

barrier in the absence of an applied signal. Since the spectral

intensity is calculated for the case of an AC short circuit, groups two

and three may be combined.

The calculation of the spectral intensities follows along straight-

forward lines but involves lengthy expressions. The procedure is as

follows: 1) Calculate the current which flows as a function of time for










a given initial velocity of an electron; 2) take the Fourier transform

of the current pulse due to electrons with this initial velocity;

3) calculate the number of electrons flowing with this velocity;

4) apply Carson's theorem using results 1 3 in order to find the

spectral intensity for a particular initial velocity; 5) sum over all

possible initial velocities in order to find the total spectral

intensity.

The first case which will be considered is for electrons which

reach the metal contact. From the equation of motion for a

single electron one finds


v (t) = v (0)cos at (5.8)



and the time of flight TF is found when the electron reaches the metal

anode.



TF = 1/a Arcsin (ad/vx(0)) (5.9)


Since the velocity of the electron as a function of time is known,

the current which is flowing is also known [32]. The Fourier transform

of this current pulse is necessary in order to calculate the current

spectral intensity.


S-qv (0)
d cos at e tdt

C 1 -j(1+c)aTF j(-c)aT
c (e 1 c (-
(5.10)


where C1 = qvx(0)/2d.










The spectral intensity due to the electrons which cross with this

initial velocity is found by using Carson's theorem.



AS 22(f) = 2X*0 (5.11)
22


where A is the average number of electrons being emitted per unit time

with a given velocity. This spectral intensity is that due to those

electrons in a small velocity interval centered about v (0).

A calculation of lambda follows from the DC current. Since the

electrons which reach the metal contact are currently being considered,

the calculations follow the familiar lines of thermionic emission

theory. Numerically, this requires evaluating the function F(K) over

the velocity interval of interest [26].


Smv.(0,K)
2 xJL
F(K) = A v (0,K) exp( kT ) (5.12)
x k

*kT
where A = 4qm kT2/h3 and K is the Kth point under consideration. The

function F describes the current density due to the electrons of a par-

ticular initial velocity crossing the barrier (per unit velocity). This

function must be integrated in order to find the DC current. This is

done numerically by discretizing the function and applying Simpson's

Rule. The number K is an integer used as an index to keep track of the

velocity interval under consideration. See Figure 5.3.

For the case of passing electrons, F(K) was evaluated for 100 vx

values using a spacing in vx of 2500 m/sec. This gives 27 points for

every kT unit of energy, which allows sufficient accuracy in the

integration procedure.
















































Figure 5.3.


SVX


Vx(K)

-- vx(1)

(Vdif Vd) (



F


















Method of discretizing the function F(K)
to allow for numerical integration.










It is simple to calculate the spectral intensity due to a particu-

lar initial velocity once F(K) has been evaluated. The current in a

particular velocity interval is



AI22 = (Area)Av (F(K) + F(K+1))/2 (5.13)



and X(K) for this interval follows directly:



X(K) = AI22/q. (5.14)



Now that all of the necessary quantities have been evaluated, the

spectral intensity is calculated using Carson's rule. This yields


2X(K)C2 2
S 1 1 2 1(1 cos(1-c)a (K)) 2c
2 K a2 (1+c)(1-c) (I -) c

( 2c
(1 cos(l+c)aTF(K)) +


+ (1 cos 2aTF(K))} (5.15)



where the index K again refers to a particular small velocity interval.

It should be kept in mind that T (K) is the average transit time of an

electron in the small velocity interval under consideration.

Since the case being considered is for the electrons which are able

to cross the junction, the summation should start from v00 and go to

infinity. The value of v00 is found from the kinetic energy required to

cross the barrier of the junction. This is obtained from equation

5.16. In practice the summation only needs to be carried out over a few

kT of energy,












q(Vdif Vd) = -m (5.16)



The second case, electrons which have insufficient kinetic energy

in order to cross the junction, is considered in the same manner.

Again the calculation begins by taking the Fourier transform of a

single current pulse. The results of the previous calculation may be

used if it is kept in mind that F should be replaced everywhere by TR'

The resulting expression may be simplified using trigonometric

identities to the following form:


C2
1 8c2
S= ---(1 + cos cr). (5.17)
a2 (1-c2)2


As in the first case, F(K) must be evaluated in order to determine

the number of electrons injected with a particular initial velocity.

See equation (5.12). Only the limits of evaluation need to be changed

for this group of electrons. Electrons at the bottom of the conduction

band have only potential energy and no kinetic energy. Thus, the lower

limit of evaluation is vx(0) = 0. Electrons which almost reach the

metal contact, yet fail, have an initial velocity equal to v00, as

defined by equation (5.16).

In the numerical evaluation for the returning electrons, F(K) was

evaluated for 100 values of vx regardless of the actual barrier

height. Since typical values of Vdif Vd are less than one eV, this

resulted in a sufficiently fine spacing in vx to give accurate results

from a numerical integration.

The total spectral intensity of those electrons which do not reach

the metal contact and then return to the neutral semiconductor is











2X(K)C2 2
S (f) = AS = 1 8c2 (1 + cos cW). (5.18)
11 K 11 K a2 (1-c2)2


The method of calculating the spectral intensities has now been

outlined. The actual computer program which was used may be found in

the appendices.

The spectral intensity of current fluctuations due to electrons of

group 1 is shown in Figure 5.4. The noise of these electrons which

return is zero for zero frequency, since these electrons carry no DC

current. The spectral intensity is seen to rise initially as f2.

The lower curve is for a bias voltage of five times VT and the

upper curve is for a bias voltage of ten times VT. This illustrates

that the spectral intensity due to this group of electrons is a very

slow function of the bias.

Since all electrons of this group have the same characteristic time

of flight, the spectral intensity shows several sharp nulls. This would

not be the case if the time of flight took on a distribution of values.

The spectral intensity due to the passing electrons, groups 2 and

3, is illustrated in Figure 5.5 by the solid curves and is equal to the

shot noise of the DC current at low frequencies. At frequencies on the

order of the reciprocal transit time the spectral intensity falls off

steeply. The lower solid curve is for a bias voltage of five times VT

and the upper curve is for a bias voltage of ten times VT. It is clear

that this component of the spectral intensity is a very strong function

of the applied bias.

For a given frequency of operation, e.g. 12 GHz, it may now be seen

that there are two distinct regimes of operation. At low bias the noise













10-"1





10-20

S1(f)

CA2/Hz]

10-21





10-22





10-23


10-24
7

3
2
mu -S


/I

I,







/ /






///
/I/
/I/












/./
I3


//
I/








S3 ?, .. .


100








Figure 5.4.


10' 102
FREQUENCY CGHz)


-





turning
ctrons


I
'I
i,



!! *
I i
SII



i


I
i


103


104


Spectral intensity due to returning electrons.
Lower curve is for Vd = 5 VT;
Upper curve is for Vd = 10 VT.


I
















































FREQUENCY [GHz3


Figure 5.5.


Spectral intensities vs. frequency.
Lower curves are for Vd = 5 VT;
Upper curves are for Vd = 10 VT.


10-41
SX(f)

EA2/H z]


104










of the returning electrons is dominant while at high bias the noise of

the passing electrons becomes dominant. Furthermore, the higher the

frequency of operation, the higher the bias must be in order for the

passing electrons to dominate the noise.

Calculation of the Noise Temperature

The noise temperature is calculated as a function of both bias and

operating frequency by combining the various components of device con-

ductance and spectral intensity.



4kTng = S (f) (5.19)


where gj is the total junction conductance from all groups of electrons

and Sl (f) is the spectral intensity of current fluctuations due to all

groups of electrons.

Consider the case of a junction with non-ideality factor of unity

and no parasitic resistances or capacitances. If only the passing

electrons are considered, then the noise temperature would be equal to

half the ambient temperature. This is the standard result for inter-

mediate frequencies, where the returning electrons may be neglected.

For low-bias operation at high frequencies the returning electrons

may not be neglected. This group of electrons has a noise temperature

associated with it which is equal to the ambient temperature. As the

bias is increased, the passing electrons begin to dominate the device

behavior, and the noise temperature again drops to half the ambient

temperature.

The description of the junction presented so far has only been

concerned with an ideal exponential junction. In practice this is not

sufficient as the effects of the series resistance, diode non-ideality










factor and reverse saturation current must also be considered in an

actual device.

In most cases the series resistance is of the greatest concern.

This resistance degrades the performance of the device in every manner.

Due to the series resistance rectification properties are reduced,

matching to the actual junction becomes difficult and in general the

effective noise temperature of the diode is increased. The series

resistance can be appreciable in microwave diodes due to the small metal

contact area which is used (typically 2 pu diameter).

At very high bias levels the series resistance may be the source of

excess noise. At high frequencies the effective series resistance may

be greater than that obtained from I-V measurements due to the skin

effect. It is difficult to accurately model the series resistance for

high-bias and high-frequency operation [33]. In this work it will be

assumed that the series resistance is constant and equal to its low-

frequency value. In addition, it will be assumed that the noise tem-

perature of the series resistance is equal to the ambient temperature.

The diode non-ideality factor was assumed to be equal to unity in

these calculations. For the diode which was measured this is a

reasonable assumption since it was found experimentally that m = 1.09.

At worst, this gives an error of 9 percent in the noise temperature.

The non-ideality factor could be taken into account by recalculating the

Schottky barrier height for each bias point. This would lead to a

different value of the minimum velocity required for an electron to

reach the metal contact.

At very low values of bias (Vd < 5mkT/q) it is important to include

the reverse saturation current in the device model. If this current is











omitted, then the calculated current for zero applied voltage would be

non-zero, and the noise temperature would not match the ambient

temperature as it should.

The reverse saturation current is comprised of electrons escaping

the metal and flowing towards the semiconductor. These electrons are

traveling in a direction opposite to the electric field of the

junction. Since electrons are negatively charged, they will be

accelerated as they cross the junction. Their transit time will be much

less than for electrons traveling from semiconductor to metal. It is

concluded that the spectral intensity of this component of current may

be considered to be white over the frequency range of interest.

The complete results of the theoretical calculations are presented

in Figure 5.6. The Schottky barrier height of 0.585 eV and series

resistance of 10 ohms were obtained from DC I-V measurements. The diode

was assumed to have an anode diameter of 2 im and a doping density of

101 cm3. The substrate material is silicon.

The curves of Figure 5.6 indicate that for each frequency of opera-

tion there is an optimum bias. It is at this bias that the lowest noise

temperature of the diode will be obtained. The lowest noise temperature

which can be achieved increases with the operating frequency. At high

frequencies the lowest noise temperature is not much reduced from the

ambient temperature.

At low frequencies, with no series resistance, the effective noise

temperature is 295 K at zero bias and drops sharply to 147.5 K. This is

the well-known result for intermediate frequencies. By including the

series resistance of 10 ohms, it can be seen that for high biases the

series resistance dominates over the junction resistance and the noise


















300


Tn [K]






200










100

0


V/VT


Figure 5.6.


Theoretical noise temperature
vs. bias voltage.











temperature again rises to the effective temperature of the series

resistance.

When operated at high frequencies, the noise temperature does not

begin to drop until a higher bias is reached. This is due to the fact

that at low bias the noise and conductance of the device are dominated

by the returning electrons. At high frequencies the noise temperature

of the diode remains nearly constant due to the returning electrons. At

97.5 GHz there is almost no decrease in the noise temperature.

The roll-off from the ambient temperature towards a noise

temperature of mT/2 occurs when the spectral density and conductance of

the passing electrons dominate. The bias at which the roll-off begins

is a function of frequency. If there were no series resistance, then

the noise temperature would always reach a value of mT/2. This is not

the case.

At 2.2 GHz the high-bias portion of the curve is seen to be

dominated by the series resistance. It should be recalled that at low

bias the returning electrons cause both the junction conductance and

susceptance to be large compared to their low-frequency values. This

allows the series resistance to play a role even at the lower biases.

The series resistance, in conjunction with the junction susceptance,

causes the effective conductance at the device terminals to be

increased. The circuit transformation which is made is indicated in

Figure 5.7. It is the transformed quantities which are measured.

Since the numerical analysis also gives the theoretical junction

admittance, the device reflection coefficient as a function of bias may

be calculated. The results of these calculations are presented in

Figure 5.8a. In this case the series resistance was 10 ohms.
































































Figure 5.7. Circuit transformation due to the combined
effect of the series resistance and
junction susceptance.








1.0
2.2 GHz
0.8

Ir12 0.6 12 GHz
0.4 -
0.4 97.5 GHz

0.2

0.0
0 5 10 15
V/VT

(a)



1.0

0.8 2.2 GHz

Ir12 0.6 12 GHz

0.4

0.2 97.5 GHz


0 5 10 15
V/VT


(b)



Figure 5.8. Theoretical magnitude squared of
the reflection coefficient.
a) A series resistance of 10 Q
is assumed;
b) A series resistance of 15 a
is assumed.











Figure 5.8b presents the results of the same calculations, but for a

series resistance of 15 ohms. It will be seen that the results of

Figure 5.8b more closely approximate the measured results which will be

presented later.

The magnitude of the reflection coefficient depends more critically

on the value of the series resistance than does the noise temperature.

For this reason the noise temperature versus bias was presented for only

one case.

High-Frequency Measurements

The noise measurements were performed on a Hughes silicon diode

detector (model 47316H-1111) using the circulator method of Gasquet et

al. [34]. Since four measurements are performed, there are also four

quantities which can be determined. It is possible to determine Ta, the

system background noise temperature; K*G*B, the gain-bandwidth product

(K is a constant of proportionality); TDUT, the device under test noise

temperature; and Ir 2, the magnitude of the reflection coefficient

squared. Usually, only the latter two quantities are of interest.

Microwave measurements were performed at 2.2, 12 and 97.5 GHz. The

various front ends of the receivers are shown in Figures 5.9 through

5.11. The down conversion and detection were done using the setup shown

in Figure 5.12. This portion of the circuit was the same for all of the

measurements, although an HP desk-top computer was used to record the

voltmeter readings during 97.5 GHz noise measurements. Use of the com-

puter allows for long averaging times to obtain the required statistical

accuracy.

Measurements proceed as follows:

1) A short circuit is placed at the reference plane and the noise

source with a noise temperature of TN1 is applied. This gives a meter













HP
NOISE
SOURCE

AMPLICA


REF
PLANE


2-4 GHz


2.2 GHz


Figure 5.9. Measurement front end for 2.2 GHz.


bias


_T-

















NOISE SOURCE


STEP ATTEN


20 dB


8-12 GHz


f = 12 GHz
BW =1%


I


Noise measurement front end for 12 GHz.


REF
PLANE
I


CIRC


10 dB


Figure 5.10.













94 GHz
SOURCE


CIRC ISOLATOR
TN1 ---0 LN

REF
PLANE

TN2 DUT


Rbias


-I-
















Figure 5.11. Noise measurement front end
for 97.5 GHz.














21.4
MHz


DET


PWR
METER


SPECTRUM
ANALYZER


LEVEL
SET


Figure 5.12. Detection system for noise measurements.
Desk-top computer was needed only during
97.5 GHz measurements. See the appendix
for the required program.











reading of


M1 = K.G*B[Ta + TN]


(5.20)


2) The short circuit remains in place and a noise source of

equivalent temperature TN2 is applied. This gives a second meter

reading equal to


M = K.G.B[T + TN2]


(5.21)


3) The device under test is connected at the reference plane and

biased to the proper operating point. By applying the noise source of

temperature TN1 the meter reading becomes


M3 = KG.B[Ta + TDUT(1 Ir2) + TNlIr12] .


(5.22)


4) The device remains biased and in position, but the noise source

of TN2 is again applied. The meter reading for this case is


M = KGB[Ta + TDUT(1 Ir 2) + TN2 r2] .


(5.23)


These equations can be solved to yield all four of the unknown

quantities.


TN Y TN
T N2 N1
T =
a Y 1


M
H2
,where Y = ;
M1


Z T T M M1
T = Z 1 where Z= M3 M
DUT Z-1 M -M2


(5.24)



(5.25)




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