Title Page
 Table of Contents
 Computer-based measurements
 Charge transport in schottky-barrier...
 Theory and experiments of 1/F noise...
 Shot noise in schottky-barrier...
 Review of electrical properties...
 Excess noise anomalies observed...
 An analytical model for 1/F noise...
 Conclusions and suggestions for...
 Biographical sketch

Title: Charge transport and noise properties of Schottky barrier diodes and polycrystalline silicon thin films
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00082270/00001
 Material Information
Title: Charge transport and noise properties of Schottky barrier diodes and polycrystalline silicon thin films
Physical Description: viii, 245 leaves : ill. ; 28 cm.
Language: English
Creator: Luo, Min-Yih, 1958-
Publication Date: 1989
Subject: Diodes, Schottky-barrier   ( lcsh )
Thin films   ( lcsh )
Electronic noise   ( lcsh )
Silicon -- Electric properties   ( lcsh )
Polycrystalline smiconductors   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis (Ph. D.)--University of Florida, 1989.
Bibliography: Includes bibliographical references (leaves 241-244)
Statement of Responsibility: by Min-Yih Luo.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00082270
Volume ID: VID00001
Source Institution: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 001512969
oclc - 21969374
notis - AHC5960

Table of Contents
    Title Page
        Page i
        Page ii
        Page iii
    Table of Contents
        Page iv
        Page v
        Page vi
        Page vii
        Page viii
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    Computer-based measurements
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    Charge transport in schottky-barrier diodes
        Page 47
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    Theory and experiments of 1/F noise in schottky-barrier diodes operating in the thermionic-emission mode
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    Shot noise in schottky-barrier diodes
        Page 86
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    Review of electrical properties of polycrystalline silicon
        Page 97
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    Excess noise anomalies observed in phosphorus-doped polycrystalline silicon films
        Page 145
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    An analytical model for 1/F noise in polycrystalline silicon thin films
        Page 182
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    Conclusions and suggestions for further research
        Page 206
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    Biographical sketch
        Page 245
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Full Text








Many people have generously donated their time and

expertise during my graduate career. I would especially like

to thank Dr. G. Bosman, who in his role as my dissertation

advisor continuously offered his advice, help, and support to

my program. I sincerely express my gratitude to Dr. A. van

der Ziel and Dr. C. M. Van Vliet who provided many motivating

suggestions in my research. Great appreciation is also

extended to other members of the supervisory committee--

Dr. L. L. Hench, Dr. A. Neugroschel, and Dr. R. M. Fox for

their participation.

The graduate program has given me the opportunity to

discuss many topics with my colleagues. I would like to thank

Chris Whiteside, Arthur van Rheenen, Young-June Yu, Sheng-

Lyang Jang, and Horng-Jye Luo.

Finally I come to my wife, daughter and family. Without

moral and financial support offered by my parents and

parents-in-law this project would not have been possible.

There are not enough words to express my gratitude to my wife,

Ya-Ling. She has generously and lovingly given of herself,

allowing herself to be uprooted from her family and job to

start a life with me. Thanks also goes to my daughter,

Christine. Her mischiefs created many moments of comfort when

times were rough.

Although my grandmother passed away a few days before my

qualifying exam, she is deep in my memories. Her love led to

this gratifying conclusion.



ACKNOWLEDGMENTS ........................................... ii

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


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

II. COMPUTER-BASED MEASUREMENTS ...................... 6

2.1. Introduction ............................. 6
2.2. Noise Measurements Using the HP3582A
Spectrum Analyzer ...................... 9
2.2.1. Examination of Operational
States via HP-IB ............. 9
2.2.2. Modes of Data Transfer ........ 15
2.2.3. The Manipulation of Noise
Data .......................... 21
2.2.4. The Procedure of a Typical
Noise Measurement ........... 23
2.2.5. Brief Description of the
Control Program ............. 34
2.3. Current-Voltage Measurements Using the
HP4145B Parameter Analyzer ............ 36
2.3.1. Description of the Source/
Monitor Units (SMUs) ........ 37
2.3.2. Compliance Limit ............... 41
2.3.3. Access of the Status Byte
and Data Transfer ........... 42
2.3.4. Brief Description of the
Control Program ............. 45
2.4. Conclusions .... ......................... 45


3.1. Introduction ........... .................. 47
3.2. Charge Transport Theories ................ 49
3.2.1. Diffusion-Limited
Conduction .................. 49
3.2.2. Thermionic-Emission-Limited
Conduction ................... 51

3.3.3. Thermionic Emission-Diffusion-
Limited Conduction .......... 53
3.3. Electron Density in the Space-
Charge Region .......................... 54
3.4. Boundary Conditions ...................... 56
3.5. Junction Dynamic Resistance ............. 58
3.6. Conclusions ......... ........................ 62

THERMIONIC-EMISSION MODE ........................ 63

4.1. Introduction .............................. 63
4.2. Theory ...................................... 65
4.3. Device Description and
Experimental Results .................... 74
4.4. Discussion ................................. 79
4.5. Conclusions ............................... 83


5.1. Introduction .............................. 86
5.2. Noise Contribution from the
Interface Region ....................... 87
5.3. Noise Contribution from the
Space-Charge Region ..................... 90
5.4. Discussions and Conclusions ............... 95

POLYCRYSTALLINE SILICON ......................... 97

6.1. Introduction .............................. 97
6.2. Description of Grain-Boundary
Barrier Formation ......................... 99
6.2.1. Uniform Distribution
of Trapping States .......... 104
6.2.2. Monoenergetic Distribution
of Trapping States ............ 108
6.2.3. Exponential Distribution
of Trapping States ............ 110
6.2.4. U-Shaped Distribution
of Trapping States ............ 113
6.2.5. Gaussian Distribution
of Trapping States ............ 116
6.3. Factors Affecting the Barrier Height ...... 119
6.3.1. Doping Level ................... 119
6.3.2. Hydrogenation .................. 121
6.3.3. Dopant Segregation .............. 122
6.4. Modeling of Charge Transport in
Polycrystalline Silicon ................ 126
6.4.1. Dopant Segregation Model ........ 126
6.4.2. Carrier-Trapping Model .......... 128

6.4.3. Mixed Model ..................... 136
6.5. Nonuniform Properties .................... 139

SILICON FILMS .................................. .145

7.1. Introduction .............................. 145
7.2. Sample Description ........................ 146
7.3. Experiments ................................ 149
7.3.1. I-V Characteristics ............. 149
7.3.2. Noise Measurements .............. 158
7.4. Discussion ................................ 178
7.5. Conclusions ............................... 181

8.1. Introduction ............................. 182
8.2. Theory .. ....... ......................... 185
8.2.1. Grain-Bulk Region
1/f Noise ................... 185
8.2.2. Depletion-Barrier Region
1/f Noise .................... 191
8.3. Experiments .. .. .......................... 198
8.4. Discussion ............................... 201
8.5. Conclusions ......... ...................... 205

FURTHER RESEARCH ............................... 206
9.1. Charge Transport in Schottky-
Barrier Diodes ......................... 206
9.2. Shot Noise in Schottky-
Barrier Diodes ........................ 206
9.3. 1/f Noise in Schottky-
Barrier Diodes ........................ 207
9.4. Low-Frequency Excess Noise
in Polysilicon ......................... 208

SPECTRUM ANALYZER ....................... 211

PARAMETER ANALYZER ...................... 232

REFERENCES ................................................. 241

BIOGRAPHICAL SKETCH .................................... 245

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



Min-Yih Luo
August 1989
Chairperson: G. Bosman
Major Department: Electrical Engineering

A 1/f noise model for Schottky barrier diodes operating

in the thermionic-emission mode under forward conditions has

been developed. The model is based on mobility and

diffusivity fluctuations occurring in the space charge region

and accounts for the current limiting role of the

semiconductor interface. The bias dependence of the 1/f noise

spectral density calculated from this model is in excellent

agreement with our experiments, but is at variance with the

predictions of a model developed by Kleinpenning. From the

experimental data a value of 4.2x10"9 for the Hooge parameter

is derived, indicating that intervalley scattering is absent

in the barrier region.

Extensive experiments performed on phosphorus-doped

polysilicon thin films show that the excess noise spectra of


this material either have a form 1/f" where n is close or

equal to one, or consist of an ensemble of Lorentzian-shaped

and other time-scaled spectral components. These partial

spectra, having the forms A/(1+w2 2) and A/(1+474 ), are

observed simultaneously in our experiments. Also we found

that the measured excess noise density is not always

proportional to I2.

A large amount of pure 1/f noise is observed in low or

moderately doped n-type polysilicon. Noise calculations have

been performed for the quasi-neutral regions and the depletion

regions of the material using Hooge's empirical law.

Comparing the theoretical predictions with the experimental

results, we demonstrate that the 1/f noise in polysilicon is

depletion-region dominant. Our calculations are based on an

extension of our 1/f noise theory developed for Schottky

barrier diodes. The limiting role of grain boundaries and the

noise correlation between depletion regions on both sides of

a grain boundary are taken into account. The Hooge

parameter found from our model is 1.45x10"3. Finally, software

to computer control a noise measuring setup and a current-

voltage measuring setup was developed and tested.



This thesis describes experiments and theories developed

for barrier limited transport and low-frequency noise in metal

semiconductor systems and in polycrystalline material. In the

following we first address the metal semiconductor system, in

particular Schottky barriers formed on n-type silicon. The

second part of this thesis describes the results of our

efforts on polycrystalline material with special emphasis on


The diffusion barrier of a Schottky-barrier diode arises

because of the difference in work functions between the metal

and the semiconductor. When electrons travel, under forward

bias conditions, from an n-type bulk semiconductor to a metal

contact, they will experience scattering in the local field

of the space-charge region. Therefore the motion of electrons

in this region is governed by drift and diffusion. If the

electrons approach the interface and have enough energy, they

may be thermionically emitted over the top of the Schottky

barrier. If the electrons do not have sufficient energy to

surmount the barrier maximum, reflection or quantum-mechanical

tunneling through the barrier can occur.

Low-frequency 1/f noise has been observed in Schottky-

barrier diodes by many researchers. Several theories have

been developed to explain the measured noise data. Hsu [1][2]

assumes the presence of a uniform spatial and energy

distribution of traps over the space-charge region. In his

model even though the charge transport in Schottky-barrier

diodes is dominated by thermionic emission, a small amount of

current can tunnel through the barrier via trap states. The

thermionic emission process produces shot noise. However, the

multitunneling process gives rise to a large amount of 1/f

noise. Hsu's model has been criticized by Kleinpenning [3],

who argues that a uniform trap distribution over space and

energy is unrealistic for actual devices.

Kleinpenning has published a mobility fluctuation theory

[3], wherein he calculates the output noise by considering

fluctuations in the "diffusion velocity", vd, which enters in

the TE-D terminal current equation.

Our objections to his treatment are threefold: First,

in his evaluation of the noise density, Kleinpenning

linearizes the terminal current expression instead of the

local current expression, which in most cases leads to

erroneous results [4]. Secondly, the diffusion velocity is

locally defined at the barrier maximum [5]-[7]; however, in

his work, the fluctuation in diffusion velocity is incorrectly

treated as spatially dependent. Thirdly, his model predicts

that for the TE mode of operation, the 1/f noise should go to

zero since then the diffusion velocity term is missing from

the current equation. This prediction is inconsistent with

our experimental findings on Schottky-barrier diodes operating

under this condition.

As mentioned before, the second part of this thesis deals

with barrier limited transport in polycrystalline silicon

material. Polysilicon material consists of small crystallites

or grains of silicon with a dimension of the order of 102 to

104 A. These grains are separated by thin regions of nearly

amorphous silicon, which are called grain boundaries.

The high density of dangling bonds and defects at the

grain boundaries is the origin of trapping states and

segregation sites around which carrier trapping and dopant

segregation models were developed. Researchers argue that the

charge transport mechanism across a grain boundary is somewhat

similar to the charge-transport process in a Schottky-barrier

diode. They modify the expressions for thermionic emission

and thermionic field emission derived for Schottky barriers

to describe DC current flow in polysilicon.

Based on this insight, de Graaff et al. [8] and Bisschop

[9] applied Kleinpenning's 1/f noise model for Schottky-

barrier diodes to polysilicon. The erroneous result given by

Kleinpenning's theory was adopted by de Graaff et al. and

Bisschop. Consequently, this led to a wrong interpretation

of the experimental data obtained for polysilicon material.


At the onset of this work we defined two goals. One was

to develop a more advanced experimental system, which would

allow us to carry out the noise measurement more efficiently

and increase the accuracy of measured data. The other goal

was to rederive and test the theory for 1/f noise in Schottky-

barrier diodes operating in the thermionic-emission limited

mode and then to apply this newly derived theory to

polysilicon. Both goals were reached and our study has shed

new light on the various aspects of charge transport in

Schottky-barrier diodes and polysilicon material.

The organization of this thesis is as follows; in Chapter

II, we present two automatic measurement systems. One is for

noise measurements performed by the HP3582A spectrum analyzer.

The other is for current-voltage measurements performed by the

HP4145B semiconductor parameter analyzer.

In Chapter III, the charge-transport properties of

Schottky-barrier diodes operating in thermionic emission,

diffusion, or mixed mode are derived and interpreted. Special

emphasis is placed on the derivation of the interface

impedance and the depletion-region impedance using a general

charge-transport model.

In Chapter IV, a general theory for shot noise in

Schottky-barrier diodes operating in TE-D mode is developed.

In Chapter V, we develop and test a new model for 1/f

noise in Schottky-barrier diodes operating in the TE-limited



In Chapter VI, we extensively review the electrical

properties of polysilicon material, including the dependence

of the barrier height on the grain-boundary trap distribution,

various charge-transport models, nonuniform properties and


In Chapter VII, a comprehensive experimental study on

low-frequency excess noise in polysilicon is presented. Our

experiments were carried out on samples with different doping

concentrations and dimensions, and under various bias

conditions and temperatures.

In Chapter VIII, we apply our 1/f noise theory initially

derived for Schottky barrier diodes to polysilicon resistors.

Finally, in Chapter IX conclusions and recommendations

for future research are presented.


2.1. Introduction

This chapter presents information about measurements of

noise and current-voltage characteristics controlled by

computers. Personal computers can be used to carry out both

types of measurements efficiently. A noise measurement

performed manually is a time-consuming task. The accuracy of

data obtained in this way can be degraded by human errors.

In addition, the treatment of the manually measured data, for

example, the input of data to a computer for a graphics plot,

is not efficient. To overcome these problems noise

measurement procedures using the HP3582A spectrum analyzer

controlled by an IBM PC via an HP-IB board have been set up.

In the case of a measurement of current-voltage

characteristics the IBM PC is also efficient. A control

program for the HP4145B parameter analyzer has been developed.

Automation greatly speeds up the measurement of the current-

voltage characteristics. Data retrieved by the PC have higher

accuracy than those displayed on the screen of the analyzer.

The HP-IB board is an instrumentation interface which

primarily solves the compatibility problems between various


components and integrates the components into a system. The

board uses a 16-line bus to connect up to 15 components,

including one controller and 14 instruments. Each component

on the bus is connected in parallel to the 16 lines of the

bus. Eight of the lines are used to transfer data and the

remaining eight are used for handshake (timing) and control.

Currently, the HP-IB system in the noise laboratory consists

of three devices, namely IBM PC, HP3582A, and HP4145B. The

system has the flexibility to accommodate more products.

The programs for the remote operation of HP3582A and

HP4145B were written in Microsoft GW Basic V3.2 which is an

interpreted Basic run on IBM PC/XT/AT and compatibles. To

work with the subroutines in the HP-IB library, the SETUP.BAS

has to act as a header of the programs, where the extension

BAS refers to the BASIC language. When different languages

are used, different setup files must be applied.

In remote operation, one of the most important tasks is

to check for the status which determines the next operation.

For the HP3582A, the transfer of an 8-bit status word is

performed by sending the command "LST1" to the HP3582A via the

HP-IB. The HP3582A places the status word on the data bus.

The computer executes the input command in the next operation

cycle to enter the status word. For different computers,

different inputs commands are used. The input command built

in the HP9825A programmable calculator is "rdb", whereas for

the PC the input command provided by the HP-IB library is


"IOENTERS". After the status word is entered, the computer

issues the command "LSTO" to reset the internal status word

of the HP3582A.

For the more advanced spectrum analyzer, for example,

HP3651A, the above sequential processes for fetching the

status word are implemented simply by calling the HP-IB

subroutine "IOSPOLL". This subroutine covers the functions

of "LST1", "LSTO" ,and "rdb" or "IOENTERS" in an equivalent


Likewise, the status of the HP4145B may be identified by

the PC by performing the same command "IOSPOLL". However, for

HP4145B we refer to 8 bits of status as a status byte instead

of the status word used for spectrum analyzers. It should be

noted that the status byte has a meaning different from the

status word. The former refers to the HP-IB status byte

representing states, for example, command syntax error and

data ready. The latter status word refers to the operational

states of the instrument like overloading, average complete

and others.

In the following sections, we focus first on the HP3582A

spectrum analyzer. We describe the remote operation and the

treatment of measured data by the computer. Then the setup

of noise measurement and the measurement procedures are

presented. A brief description of the control program of the

HP3582A is given. Then, we illustrate the details of

communication between the HP4145B parameter analyzer and the


computer. Finally an introduction of the control program for

this analyzer is also given.

2.2. Noise Measurements Using the HP3582A Spectrum Analyzer

The setup of the automatic noise measurement primarily

includes a computer and an HP3582 dual-channel spectrum

analyzer covering the frequency range of 0.02 Hz to 25.6 KHz.

These two components are interconnected via an HP-IB. The

major tasks in this automatic measurement are as follows: (i)

to check whether or not the spectrum analyzer is overloaded;

(ii) to check whether the averaging performed by the analyzer

is completed; (iii) to transfer data from the analyzer to the

computer; (iv) to perform noise calculations on the computer.

2.1.1. Examination of Operational States via HP-IB

In order to obtain accurate results in the noise

measurement, the overload condition has to be examined at any

time during the averaging sequence. When the averaging stops,

the information must be identified by the computer. To

accomplish these objectives, the computer issues the command

"LST1" through the HP-IB and the analyzer responds with an

8-bit status word shown in Table 2.1 on the data bus. When

the overload is detected, bit 2 or bit 3 is set, depending on

which channel is used. This flag acknowledges that the

computer will perform the following two tasks immediately.

One is to issue another command "LSTO" to reset the existing

status word in the analyzer so that new information may be

entered on the next machine cycle. The other is to inform the

Bit 7 Bit 6 Bit 5 Bit 4

X-Y plot Average Single sweep Time record
complete complete spectrum complete

Bit 3 Bit 2 Bit 1 Bit 0

Channel B Channel A Arm light is Diagnostic on
overload overload ON. screen

Table 2.1. The status word of the HP3582A spectrum

analyzer to restart the measurement. These processes proceed

only when the overload condition occurs.

After examining the overload condition, the computer goes

a step further to check for bit 6 which tells whether the

averaging sequence performed by the analyzer has ended.

Between these two steps, if the renewal of the status word is

again carried out as shown in Fig. 2.1, there will be delay

time due to the I/O access of the status word. During this

delay time, if overloading of the spectrum analyzer should

occur, the information is lost by the computer. The

overloaded data are therefore counted to the average value

being figured by the analyzer. This situation will affect the

accuracy of the measured data. In addition, when the status

word is obtained for the purpose of checking the overload, the

computer only takes a little time to execute an IF-statement

to identify bit 2 or bit 3. Within this short period of time,

the status word very likely remains unchanged. Therefor it

is not necessary to renew the status word before further

examining the completion of the averaging. When bit 6 is set,

it indicates that the averaging process is completed and the

measured data are ready for output.

Both the HP calculator (or PC) and the IBM PC use the

same command "LST1" or "LSTO" to request HP3582A for the

status word. However the ways that the word enters from the

data bus to the register of the computer are quite different.

The HP computer provides a pair of powerful commands "rdb" and


no need to
renew the
status word

Fig. 2.1. Flow diagram for an IBM PC to examine the
spectrum analyzer overload condition and the
averaging sequence procedure.


"bit". The former reads the status word into the computer in

a binary form, for example, 00100101; the latter provides the

capability to directly check a certain bit of the status word.

In comparison, the IBM PC does not have the above commands.

It therefore enters the status word in an ASCII character

form, for example, %(term.) by calling the subroutine

"IOENTERS", where % is an ASCII-format status word followed

by a terminator. The terminator is required for "IOENTERS".

The terminator may be any character, and has to be predefined

by calling the subroutine "IOMATCH" before "IOENTERS" is used.

Without the terminator, the execution of "IOENTERS" will not

terminate, and the PC appears to be hung up.

The status word read in the ASCII character form is still

unusable. The character must be concerted into its

corresponding ASCII code using the instruction "ASC", for

example, ASC (%)=37 where 37 is a decimal number. Next, the

computer performs the logic operation on the decimal number

so that it knows each bit value of the status word. For

example, if (status word AND 4)=4 is true, indicating that

overload on channel A has occurred, then bit 2 is set.

The HP and the IBM both read and analyze the status word

of the HP3582A. The difference in programming between the HP

computer and the IBM PC is shown in flow diagrams of Figs.

2.1 and 2.2. Since the HP computer can deal with the status

word in binary, it works more efficiently than the IBM PC.

However, the ASCII format used for data transmission is

HP controller

Fig. 2.2. Flow diagram for an HP calculator or PC to
examine the spectrum analyzer overload
condition and the averaging sequence procedure.

standard and can be recognized by all personal computers,

including the IBM and the HP computers.

2.2.2. Modes of Data Transfer

As the averaging performed by the spectrum analyzer is

completed, the measured data are temporarily stored in the

buffer. They can be outputted in following two ways: (1) byte

transmission and (2) block transmission. The HP3582A provides

three optional list commands to carry out the data transfer,

namely, "LMK", "LDS" and "LFM". The first two commands are

used to transfer the measured data which are formatted in an

ASCII character form for display, as shown in Fig. 2.3. The

third command "LFM" accesses the binary data directly from the

memory. "LMK" transfers data by byte, but "LDS" and "LFM"

transfer data by block. Although these commands are all

capable of fetching the measured data from the analyzer, their

functions are different. "LMK" is applied to read control

settings and "LDS" is used for data output from the display.

"LFM" working with "WTM" is designed for a professional user

who wishes to do something special. For example, "WTM" writes

the user's own data to the memory and "LFM" lists the results

from the memory after some operations performed by the

machine. These commands are described in detail below.


This command is used to list the amplitude and the

frequency at the position of the marker. For the single

trace, single channel mode of operation, a total of 256 data



Fig. 2.3. Format of data transferred by commands LMK
and LDS.


are measured and stored in the buffer. When the analyzer

receives the command "LMK" issued by the computer, we then

observe on the analyzer screen that the marker moves slowly

over 256 data points one by one. Meanwhile each corresponding

data value is sent from the analyzer to the computer. For the

dual trace in the dual channel mode of operation, the 256

measured data in the buffer are arranged in the way that the

first 128 data are measured from channel A while the following

128 data are measured from channel B. When the analyzer

receives "LMK", it again outputs the data one by one to the

computer. However, the data of channel A are transferred

first followed by those of channel B. In the dual channel

mode the marker first traces the display curve of channel A

and then switches to that of channel B.

The "LMK" instruction requires two variables to

accommodate each data value. One is for the amplitude of the

marker and the other is for the frequency where the marker is

located. The amplitude is formatted by NNNNENN. In other

words, the data obtained using "LMK" have an accuracy up to

four digits. The output unit in this case may be dBV,

dBV/(Hz)1/2, or Volt, which is determined by both instructions

"SC" and"MB". "SC" controls the scale for example linear,

10dB/Div or 2dB/Div, whereas "MB" determines whether the

amplitude is divided by the square root of the bandwith.


The data transfer using "LMK" via the HP-IB allows a high

degree of accuracy to be obtained. However the marker tracing

is very time-consuming.


Compared to the command "LMK" which only lists the data

at the position of the marker, the command "LDS" lists the

entire display graphics. According to a test, the transfer

of all 256 data using "LDS" takes less than five seconds in

either the single channel mode or the dual channel mode of

operation. However, the transmission of the same number of

data points by "LMK" requires three minutes. The transfer

rate of "LDS" is more than 36 times faster than that of "LMK".

The output unit when the command "LDS" is applied may be

either dBV or Volt and the instruction "MB" mentioned before

has no effect on the unit. Besides, the amplitude of data is

formatted by N.NNENN with the accuracy up to three digits.

Under this situation, when the measured data have a value

equal to 100.4 dBV on the log scale and 100.4 V (or mV, uV)

on the linear scale, they will be rounded off as 100 dBV and

100V. Both cases give a maximum error of approximately .4%,

on the linear scale. However, in a practical measurement,

the amplitude of noise data is statistically distributed.

Performing a number of averages will minimize the error to a

negligible level. To demonstrate this we used two different

programs developed in terms of "LMK" and "LDS" respectively

to measure the thermal noise of a resistor of 3.31 KM. In the

latter program the readings have been divided by the square

root of the bandwidth so that all data have the unit of

dB/(Hz) /. As shown in Fig. 2.4, both programs can give very

satisfactory results.


Like the command "LDS", the command "LFM" is capable of

sending data by block. However, several differences exist

between "LFM" and "LDS". For example "LFM" accesses memory

data directly and transfers the data to the computer in a

binary form. In addition "LFM" gives programmers the

flexibility to specify the length of output words, whereas

"LDS" transmits all data measured.

Since data transmitted through the HP-IB are in 8-bit

byte, two bytes are required to transfer or receive a 16-bit

word. The most significant byte is first. Normally, we do

not have to be concerned with this problem because whether

"LMK", "LDS" or "LFM" is used, both analyzer and computer

automatically take care of "packing" two bytes into a word.

It should.be noted that the HP9825A calculator does not have

the packing feature if "LFM" is used.

The program developed for the noise measurement is listed

in Appendix A and is run on IBM personal computers. The

program uses the data transfer command "LDS" and therefore

speeds up the measurement without losing significant accuracy.

,I li.

- rd=3.31K

0: "LDS"
x: "LMK"

x /

I l I IIIII I m I I I I I, ,I I.,, II | I 1, 1

100 101





Fig. 2.4. Thermal current noise spectrum of a resistor of
3.31 Kn measured by two different programs. One
uses the data-transfer command LDS. The other
uses LMK.










_ __ ____ I _____ _~~I


, , n n
. , n
, ,


2.2.3. The Manipulation of Noise Data

The HP3582A spectrum analyzer features different sweep

modes. 0-START refers to an analysis span that starts at DC

and SET START or SET CENTER refers to one that does not start

at DC. In addition, the 0-25 KHz mode allows a quick look at

the entire spectrum. The 0-START mode is utilized for the

control program of the analyzer. The following frequency

spans are measured: 25 KHz, 2.5 KHz, 250 Hz, 25 Hz and 2.5 Hz.

For any frequency span, the computer reads 256 data points in

the single channel mode and 128 data points in the dual

channel mode. For either mode, as illustrated in Fig. 2.5,

the computer is programmed to take eight independent small

frequency sections from each following span: 25 KHz, 2.5 KHz,

250 Hz and 25 Hz, but only three small sections from the span

of 2.5 Hz. The resulting 35 frequency sections contain

several data points ,respectively. Performing the

"horizontal" average over the data in each frequency section

gives rise to 35 representative data points on the overall

spectrum.. The value of each data point approximately

represents the noise spectral density at the central frequency

of each frequency section. These data points are uniformly

distributed on the log frequency scale.

It is important to note that the interval for each small

frequency section has to be properly chosen. An interval that

is too narrow will cause large fluctuations in resulting

spectra. The significance of the average performed is thereby


horizontal average

35 <
points\,i ,, ,



35 <
points\ l


. 2.5Hz E


horizontal average

n1 n1 n n nI


frequency section


I 2.5Hz

-*1 data point

Fig. 2.5. Chart illustrating the manipulation of measured
data. Horizontal average: averaging data over
each frequency section of five frequency spans.
Vertical average: repeating the measurement four
times and taking the average.








lost. If a wide interval is chosen, the data distribution will

be too smooth, greatly changing the noise nature of a DUT.

The proper arrangement of the frequency interval is listed in

Table 2.2.

The above data measurement and manipulation are

programmed to be repeated four times. As shown in Fig. 2.5,

the program further carries out the "vertical" average so that

the statistical error in the 35 representative data points is

reduced. These data are then stored in the computer for

further use.

2.2.4. The Procedure of a Typical Noise Measurement

In an actual noise measurement, several measurements have

to be performed to extract the actual noise spectral density

of a DUT. Fig. 2.6(a) shows the setup of the first

measurement, including 12-Volt wet-cell batteries, metal film

bias resistors, a Brookdeal 5004 ultra-low-noise preamplifier,

an HP3582A spectrum analyzer and a controller. The controller

used is either the HP9825A calculator or the IBM PC. If a

low-temperature measurement is made, a cooling system and a

vacuum pump system have to be incorporated into this setup.

The noise sources in the elements of Fig. 2.6(a) can be

represented by external noise generators connected to the

terminals of the noiseless equivalent circuit, as shown in

Fig. 2.6(b). For convenience of noise calculation, the output

current noise generator of the amplifier has been transferred

freq. section

0.9 -
1.4 -
1.9 -

2.1 -
2.1 -
4.6 -
5.6 -
7 -
9 -
14 -
19 -

26 -
36 -
46 -
62 -
70 -
90 -
140 -
190 -





2.6K -
3.6K -
4.6K -
5.6K -
7K -
9K -
14K -
19K -

Table 2.2. The arrangement of frequency sections.





260 340
360 440
460 540
560 640
700 900
900 1.1K
1.4K 1.6K
1.9K 2.1K





# of point

freq. (Hz)

r 1 Turbomoleculari
I cooling I & mechanical
I system pump
L J 1 J
r L2

r- M-
SiR b I ix rd 1 (b)


Fig. 2.6. Noise-measurement setup.
(a) the setup for measuring M2.
(b) the noise equivalent circuit.


into a voltage noise generator at the input. The measured

output noise spectral density based on this scheme is

generated by the bias resistor, the amplifier and the DUT and

is referred to as M12

M,2 = G2[Sveq + (Sia + Six + SiR) (Ri//rd//Rbias)2] (2.1)

where G2 is the squared voltage gain of the amplifier. Sv

and Sia are the noise sources associated with the amplifier,

whereas Six and SiR refer to the noise sources of the DUT and

the bias resistor, respectively. Ri is the input impedance of

the amplifier, rd the dynamic resistance of DUT and Rbias the

bias resistance. Practically, Ri is much larger than rd and

Rbias so that (2.1) becomes

M G2[Sveq + (Sia + Six + SiR) (r /Rbias)2] (2.2)

Note that Sveq, Sia and Six are unknown parameters. To solve for

the Six two more equations are required. In other words we

have to perform two other independent measurements.

The setup of the second measurement is presented in

Fig. 2.7(a). Compared to Fig. 2.6(a) a dummy metal film

resistor of the same value as the dynamic resistance of the

DUT is used to replace the DUT in this measurement.

Fig. 2.7(b) represents the corresponding noise circuit of

Fig. 2.7(a). The output spectral density obtained in this

step is written as

RbIa (dummy) o
\4 oscillo-
I scope



blas Sir d S Sa


Fig. 2.7. Noise-measurement setup.
(a) the setup for measuring M22 (with dummy).
(b) the noise equivalent circuit.

M2 = G [Seq + (Sia + Sir + SiR) (rd/Rbias) ] (2.3)

where Sir is the noise source of the dummy resistor and given


Sir =- (2.4)

The final measurement requires a calibration noise source

for which, in this case, we use the built-in noise generator

of the spectrum analyzer. Since the amplifier used has a

large gain of 60 dB, the magnified calibration noise is very

likely to overload the spectrum analyzer. Therefore an

attenuator is placed at the output of the noise generator to

reduce the calibration noise level, as shown in Fig. 2.8(a).

The output impedance of this attenuator Ro is much smaller

than the equivalent value of Rbias in parallel to rd. By

neglecting Ro the noise circuit for this measurement is

presented in Fig. 2.8(b). The output noise spectral density

according.to this circuit is as follows:

2 Svca l
M2 = G [Sveq + (SiR + Sir + Sia + -) ] (2.5)

where Svcat' is the calibration noise spectral density after

attenuation. From (2.1), (2.3), (2.4) and (2.5), we obtain

a general form for the noise spectral density of the DUT

Rblas rd (dummy) oscillo-


Sycogl attenuator Svcal


G---- _M3

-f SiRC RbIoa. Sir rd Sia J


Fig. 2.8. Noise-measurement setup.
(a) the setup for measuring M3 & Svcat (with
(b) the noise equivalent circuit.

Svcalt M M22 4KT
S = + ---- (2.6)
ix (Rbias//rd) M32 2 rd

The thermal current noise level of a DUT with a small dynamic

resistance is high. It is therefore safe to assume that the

spectral density of the DUT, Six, is larger than the current

noise source of the JFET amplifier, Sia at low and high

frequencies. Under this situation Sia can be neglected in the

noise calculation and the measurement setups of M22 and M32 may

be simplified. However there is no change in measuring Mi2

except that (2.2) becomes

M2 = G2[Sveq + (ix + SiR) (rd/Rbias)2] (2.7)

To measure M22 in the case of small rd, we shorten the

input of the amplifier, as shown in Fig. 2.9(a). The

equivalent noise circuit in Fig. 2.9(b) then gives M22 in the


M22 = G2Sveq (2.8)

Fig. 2.10(a) shows the circuit for the measurement of
M32. The calibration noise source and the attenuator in this

case are directly connected to the amplifier in series without

involving a dummy resistor. Again, from the corresponding

noise circuit in Fig. 2.10(b), we have

2 = G(Sveq + Svcal )




Fig. 2.9. Noise-measurement setup.
(a) the setup for measuring M22 (no dummy).
(b) the noise equivalent circuit.


M M3


Fig. 2.10. Noise-measurement setup.
(a) the setup for measuring M3 & SvcaL (no
(b) the noise equivalent circuit.

Equations (2.7)-(2.9) allow us to extract Six

S vcal M- M2 4KT
Six ( ias// bas
(Rbias//rd) 2 M32 M22 Rbias


The above expression only holds when rd is small. In the case

of large rd, the general form for Six given by (2.6) has to be


Note that some readings of Mi and Svcat' in (2.6) and

(2.10) are frequency-span dependent, while others are not.

This can be explained by considering the units used in the

measurement. To clarify this, a set of data is presented


Mi at 8 KHz:

frequency span

25 KHz

10 KHz

M3 at 8 KHz:

frequency span

25 KHz

10 KHz


100 Hz

40 Hz


100 Hz

40 Hz

in dBV



in dBV



in dBV/(Hz)1/2



in dBV/(Hz)1/2



If the measurement is performed using the unit dBV/(Hz) '2, the

readings M12 and M2z are frequency-span independent, whereas M32

and SvcaL vary with the frequency span used. If data are


measured in the unit dBV, M12 and M22 depend on frequency span,

but M32 and Svcal do not.

Since the unit dBV/(Hz)1/2 is very often used for Mi and

Svca' in (2.6) or (2.10), Mi2 and M22 are actually absolute
values, and M32 & Sca are relative values. Under these

circumstances (M32 M22) in (2.6) and (2.9) does not seem to

make sense. Fortunately, since M32 is much greater than M22 in

the practical case and Svcl '/M32 becomes frequency-span

independent, no mistakes occur in the expression for Six..

When the noise measurement is performed automatically,

the treatment of MI2, M22 and M32 by the computer follows from

the description presented in section 2.2.3. The computer then

calculates the noise spectral density of the DUT using either

(2.6) or (2.10).

The time required for the measurement of M12, M22, and M32

& Svcat (four times) using "LMK" and "LDS" respectively is

presented in Table 2.3. A complete noise measurement using

"LDS" is 2 hrs and 24 mins faster than using "LMK".

2.2.5. Brief Description of the Control Program

A brief description of the operation and the advantages

of the program are presented below:

1. The program is written in MS GW BASIC and may be stored

in either an ASCII or a binary form. When the binary form

is utilized, the PC only takes 1-2 seconds to load the

program to the BASIC interpreter. Then the program may be

run immediately.


M32 & SvcaL

measure. time

Table 2.3. Comparison of measurement time for M 2, M22
and M3 & SvcaL using different data-transfer

LMK 2 hrs 2 hrs 1 hr 5 hrs
(byte) 10 mins 5 mins 10 mins 25 mins

LDS 1 hr 1 hr 34 mins 3 hrs
(block) 16 mins 11 mins 1 min

time saved 2 hrs
by block 54 mins 54 mins 36 mins 24 mins

2. Basically, the program has been developed into a menu-

driven type of program. At the beginning the program

provides four options,namely, (a) measuring M12, M22, and M32

and Svcat in order, (b)skipping M12 to measure the others,

(c) only measuring M32 and Svca and (d) performing noise

calculations. The user may choose option (a) to measure

all quantities. The measured data may be stored on drives

A, B or C, or printed directly.

3. Since M22, M22 and Scal are not related to a DUT, they are

simply measured once. To only measure M,2, the user may

choose option (a). When the measurement of Mi2 is finished,

the user stores the data and then presses CTRL+BREAK to

terminate the program. The noise density of the DUT is

then calculated by running the program again and selecting

option (d) which allows the user to load M22, M32 and Svca

from any drive.

4. As mentioned, the program uses the data transfer command

"LDS" which transmits data by block. The measurement time

is greatly reduced without losing significant accuracy.

5. The program can be run on IBM PC/XT/AT and compatibles

without any modification.

2.3. Current-Voltage Measurements Using the HP4145B Parameter


The HP4145B parameter analyzer is a high performance,

programmable test instrument designed to measure, analyze, and

graphically display the DC characteristics of a wide range of


semiconductor devices. The HP4145B is equipped with eight

channels. Channels 1 through 4 are source/monitor units

(SMUs), channels 5 and 6 are voltage sources and channels 7

and 8 are voltage monitors. These channels are used for

device stimulation and characteristics measurement.

2.3.1. Description of the Source/Monitor Units (SMUs)

Since SMUs are the most commonly used units in I-V

measurement, their definitions are given below:

1. If the SMU is defined as a voltage source, it is also

a current monitor.

2. If the SMU is defined as a current source, it is also a

voltage monitor.

3. The SMU only works as a common. Voltage and current

outputs from the SMUs can be swept in a staircase manner.

The sweep,modes include main sweep (VAR1), subordinate

sweep (VAR2) and sweep synchronized with VAR1 (VAR1'),

where VAR1 and VAR2 are frequently used in I-V

measurements. Variations in the definition of SMUs and in

sweep modes will allow the user to perform all possible

I-V measurements on a DUT.

To provide better understanding of the applications of

SMUs, an example of measuring the collector-current and the

collector-voltage characteristics of an n-p-n bipolar

transistor biased in different base current is given below.

In this example, the emitter, the base and the collector are

hooked up to SMU1, SMU2 and SMU3 of the HP4145B, respectively


as illustrated in Fig. 2.11. To remotely define SMUs, the

computer sends the following codes via the HP-IB:

CODE1$="DE CH1,'VE','IE',3,3;CH2,'VBE','IB',2,2;CH3,'VCE',


In this command string, DE makes the HP4145B go to the channel

definition page and CH1, CH2 and CH3 simply correspond to

SMU1, SMU2 and SMU3, respectively. 'VE', 'IE','VBE', 'IB','V-

CE' and 'IC' are names of output data. The numbers in the

string have particular meanings. The first number of each

pair specifies the source mode which may be a voltage source,

a current source, or a common. The second number determines

the sweep mode, namely, VAR1, VAR2 or VAR1'. For example,

"3,3" indicates that SMU1 acts as a common and does not

operate in the sweep mode. "2,2" specifies SMU2 as a (base)

current source in the subordinate sweep mode. Finally, "1,1"

makes SMU3 function as a (collector) voltage source in the

main sweep mode.

The next step is to set up the voltage sweep and the

current sweep mode on the SOURCE SETUP (SS) page by entering

each field in the following command string:



(current source)


I --
L-----.. J




AIIB subordinate
IB2 sweep



Fig. 2.11. Diagram illustrating the measurement of
Ic-VcE characteristics of a bipolar
transistor using the HP4145B.

'-7 I


The fields before the semicolon are used to define the main

sweep (VAR1), while the following fields define the

subordinate sweep (VAR2). Therefore VR in CODE2$ gives an

indication that the (collector) voltage source operates in the

main sweep mode. In addition, "1" following VR further

specifies that this collector voltage source is swept

linearly. Also, STARTVCE and STOPVCE determine the range of

the applied collector bias voltage and STEPVCE is the voltage

step during the sweep. On the other hand, IP means that the

current source is in the subordinate sweep mode. STARTIB

gives the starting bias base current, and NO. OF IB allows for

different base currents being applied so that multiple I-V

curves will be displayed on the scope. MAXIC and MAXVB in

CODE2$ refer to the compliance limit which is discussed in the

following section.

In the current example, the main sweep is voltage and

linear and the subordinate sweep is current. However, other

cases are allowed. For example, the substitution of VR1 with

IR3 in CODE2$ indicates that the current source is in the main

sweep mode and sweeps logarithmically at 25 steps per decade.

In addition, if VP is applied to replace IP in CODE2$, the

subordinate sweep becomes voltage.

The third step is to set up the display graphics on the

MEAS & DISP MODE SETUP (SM) page by choosing the axis

variables and specifying their ranges. In our example, the


X-variable is 'VCE' and the Y-variable is 'IC'. The program

codes for these settings are as follows:



where DM1 displays the measured graphics on the scope of the

analyzer. If DM1 is replaced by DM2, then only a data list

is shown. XN and YA indicate the X and Y a*is, respectively.

Now the setup of all pages is completed. When the

computer gives the measurement code "MD ME1", the HP4145B

starts the I-V measurement of the n-p-n transistor. The VCE-

ICE characteristics obtained according to the above setup are

illustrated in Fig. 2.11.

2.3.2. Compliance Limit

Each SMU can be programmed to output DC voltages from 0

V to 100 V over three ranges, namely, 0 V to 20 V, 20 V to

40 V and 40 V to 100 V with the maximum corresponding

output current 100 mA, 50 mA and 20 mA. When the measured

current exceeds the above ranges, the current is automatically

limited by the machine and an error message is displayed.

In practical use, even though the current does not reach

the limits of the machine, it could be beyond the maximum

tolerance of the device under test and so burn out the device.

To avoid this, several levels of output protection, referred

to as compliance, can be incorporated into the HP4145B. When

the machine is remotely controlled, the user-defined

compliance limits can be directly inputted through the

program. For example, the user may specify MAXIC and MAXVB in

CODE2$ in the previous example when the program is run.

2.3.3. Access of the Status Byte and Data Transfer

The data transfer of the HP4145B is primarily controlled

by the status byte shown in Table 2.4. As long as the HP4145B

is polled by the computer, the HP4145B places the status byte

on the HP-IB. This byte has to be converted into an ASCII

character before the computer reads it. Later, the computer

converts this ASCII character back to its corresponding ASCII

code and then assigns the value to an integer variable.

Meanwhile, the status byte in the analyzer is reset by the

poll so that new information can enter again. These

complicated steps are performed by the single HP-IB subroutine


When all measured data are ready for output onto the HP-

IB, the bit 1 of the status byte is set. Then the command

"DO xx" is executed to perform the following two tasks: one

is to specify what data is delivered; namely, the output data

"xx" can be VCE, IC, VBE, IB and others. The other is to put

the specified data on the HP-IB. Next the computer executes

the instruction "IOENTERA" to enter data by block and assign

them to a real array. The flow diagram describing the data

transfer between the HP4145B and the computer is presented in

Fig. 2.12.

Table. 2.4 The status byte of the HP4145B parameter

Bit 7 Bit 6 Bit 5 Bit 4

Emergency RQS Fail Busy

(data not ready)


data ready^ No


output channel
"DO xx"


enter data

Fig. 2.12. Flow diagram describing the procedure of
accessing data measured by the HP4145B.

1.enter status byte
(ASCII character)
Z.reset status byte
3.convert ASCII
char. to its code

2.3.4. Brief Description of the Control Program

The control program of the HP4145B listed in Appendix B

is also developed into a menu-driven one using the MS GW BASIC

language. It can be run on the IBM PC/XT/AT and compatibles

without any change. The features of the program are presented


1. A guide for the measurement setup of a DUT is shown at the

beginning when the program is run. The user can easily

understand how to plug the DUT in the test fixture.

2. The program is capable of measuring almost all major

I-V characteristics of two- and three-terminal devices

including diodes, BJTs (npn/pnp), JFETs (p/n channels),

MOSFETs (p/n channels in depletion/enhancement modes).

3. The polarities of all I-V quantities associated with a

device are tutorial, avoiding confusion due to current and

voltage polarities.

4. The measured data can be stored on any drive or printed


5. Data obtained by the computer have a higher accuracy than

data displayed on the screen of the analyzer.

2.4. Conclusions

The status word of the HP3582A may be accessed using a

sequence of commands, namely, LST1, LSTO and IOENTERS (or

rdb). In the noise measurement, the status word presents the

important information about the data overload condition and


the completion of the averaging routine. This information

determines whether the measured data are available for output.

The HP3582A provides three commands for data transfer

namely "LMK", "LDS" and "LFM". Each command has its special

application. The control program of the HP3582A takes

advantage of "LDS" because of its fast data transmission. In

addition according to a program test, the accuracy of the

measured data is very satisfactory.

The noise measurement setup depends on the dynamic

resistance of the DUT, rd. In the case of large rd, a dummy

resistor is required in the measurement of M22 and M32, whereas

for the case of small rd, the dummy resistor is not necessary

so that the measurement can be simplified.

When the computer accesses the status word of the

HP3582A, three steps are required. To access the status byte

of the HP4145B, however, the computer simply executes the

command "IOSPOLL". This command performs the equivalent

functions of LST1, LSTO and IOENTERS (or rdb) for the HP3582A.


3.1. Introduction

For the purpose of modeling charge transport in Schottky-

barrier diodes, it is helpful to divide the diode into three

different regions. These are: the metal-semiconductor

interface region, the space-charge region, and the quasi-

neutral bulk semiconductor region. For practical doping

concentrations the bulk region has little or no effect on

charge transport.

Charge transport in forward biased n-type Schottky

barrier diodes is determined by the passage of electrons from

the bulk semiconductor through the space-charge region into

the metal contact. During their passage through the space-

charge region, electrons experience a number of scatterings.

Therefore their motion in this region is governed by drift and

diffusion in the local electric field. When electrons arrive

at the interface region, charge transport is controlled by

random emission of electrons over the top of the barrier. As

shown in Fig. 3.1, these two processes occur in series. The

current flow is essentially determined bywhich process is the

larger impediment to the transport of electrons.

TE diffusion & drift
/ \H^

q(x) q(vE-v)


1w x

Fig. 3.1. Band diagram of an n-type Schottky barrier
diode operating in TE, D, and TE-D modes,



When current flow through the diode is limited by the

conductivity of the space-charge region, charge transport is

dominated by drift and diffusion mechanisms. Then the

so-called diffusion (D) model developed by Schottky and Spenke

[10] applies. When current flow is limited by the

properties of the metal-semiconductor interface however,

charge transport is dominated by thermionic emission (TE).

In this case the TE theory developed by Bethe [11] is valid.

When charge transport is controlled by both the space-charge

region and the metal-semiconductor interface, the thermionic

emission-diffusion (TE-D) model proposed by Schultz [12] is

used to calculate the charge transport properties of the

Schottky diode.

In this chapter a quantitative description of charge

transport in n-type forward biased Schottky barrier diodes is

presented. We base our description on D, TE and TE-D models.

Then we derive a general form for the electron density in the

space-charge region from the local current. Finally, we

discuss the boundary conditions at the metal-semiconductor

interface and the junction dynamic resistance of Schottky

barrier diodes.

3.2. Charge Transport Theories

3.2.1. Diffusion-Limited Conduction

The current density in the space-charge region of

Schottky barrier diodes can be expressed as

J = qgn(x)E(x) + qD- (3.1a)

dEF (x)
= pn(x) (3. b)

where g is the electron mobility, E(x) is the electric field,

D is the electron diffusion constant, and n(x) is the electron

density given by

EFn(x) Ec(X)
n(x) = Ncexp( ) (3.2)

In (3.2) Nc is the effective density of states in the

conduction band and Ec(x) represents the conduction band edge.

From (3.1b) and (3.2) we obtain after some algebra the

following expression for the current in the space-charge


-qBO qV EFn (0)
J, = qvdNcexp(-- )[exp(-- ) exp( )] (3.3)

where OBO is the barrier height, V is the applied bias voltage,

and vd is the effective diffusion velocity defined as [13]

vd = (3.4)
w -q0o(x)
Sexp(- )dx

In practical cases vd can be approximated as IEmax [13], where

Emax is the maximum electric field at the metal-
semiconductor interface.

According to the diffusion theory for charge transport

in Schottky barrier diodes, we ignore the limiting role of the

interface. This is equivalent to assuming that the electron

collection velocity at the interface is infinite [5].

Therefore, the electron density at the interface remains at

its equilibrium value. This indicates that for a forward

biased Schottky barrier diode the electron Fermi level drops

throughout the space-charge region, and is pinned at the Fermi

level in the metal. The spatially dependent Fermi level is

shown in Fig. 3.1.

By substituting the condition that EFn(O) z 0 in (3.3),

we obtain the following expression for the diffusion-limited

current density

-qOBO qV
JD = qVdNcexp( )[exp(- ) 1] (3.5)

3.2.2. Thermionic-Emission-Limited Conduction

The thermionic current flowing over the barrier maximum

mainly depends on the number of electrons with enough energy

to surmount the barrier. The current density can be generally

written as [5]:

JE = q(n(0) nth)vr (3.6)

where vr is the effective recombination velocity, equal to

(KT/2rm*)1/2. The electron density n(0) with bias voltage

applied is given by

EFn(O) q0BO
n(0) = Nexp( ) (3.7)

The equilibrium electron density nth at the interface follows


nth = Ncexp( ) (3.8)

Incorporating (3.7) and (3.8) into (3.6) leads to the

following form for the interface TE current density

-qOBO EFn(0)
JTE = qrNcexp( )[exp( ) 1] (3.9)

Suppose that Schottky barrier diodes operate in the TE-

limited mode. The diffusion effect in the space-charge region

is less important than the limiting role of the interface.

This implies that the Fermi level in this case is nearly flat

in the space-charge region, as shown in Fig. 3.1. By imposing

the condition that EFn(O) V upon (3.9), we obtain the

following expression for the TE-limited current density

-qsBO qV
JTE = qvrNcexp(-- )[exp(-) 1] (3.10)

It should be noted that electron scattering could still occur

in the space-charge region without invalidating the TE theory


3.2.3. Thermionic Emission-Diffusion-Limited Conduction

The TE-D theory is a synthesis of diffusion and TE

approaches, accounting for the effect of drift and diffusion

in the space charge region and that of thermionic emission at

the interface. Both mechanisms occur in series and

effectively determine the behavior of the Fermi level over

the space-charge region. Therefore the Fermi level in this

case lies somewhere between those Fermi levels based on

diffusion and TE theories. The Fermi level as a function of

position is shown in Fig. 3.1.

For the TE-D mode of transport the current density in the

space-charge region is given by (3.3), whereas the TE current

density at the semiconductor interface is described by (3.9).

According to the rule of current conservation, these two

currents are equal. Therefore the current expression based

on the TE-D model can be derived by equating (3.3) to (3.9).

It follows that

vrvd -qBO qV
JTED qNc exp(-- )[exp(-) 1] (3.11)
vr+vd KT KT

If vd << Vr, (3.11) reduces to (3.5) and the diffusion-limited

theory applies. If vr << Vd, (3.11) reduces to (3.10) and TE-

limited theory has to be used to characterize the charge

transport in Schottky barrier diodes.

3.3. Electron Density in the Space-Charge Region

In this section we study the electron density in the

space-charge region of Schottky barrier diodes based on

different charge-transport models. Since the space-charge

region has a parabolic band structure for homogeneous doping,

the electron density shows a spatial dependence. This spatial

dependence becomes even stronger if the electron Fermi level

is not flat in the space-charge region.

For the TE model the Fermi level is nearly flat and the

electron density simply decays exponentially in the direction

of the semiconductor surface. For diffusion and TE-D models,

a variation in the Fermi level complicates the

characterization of the electron density in the space-charge


We calculate the electron density in the space-charge

region based on the TE-D model from (3.1a). Multiplying both

sides of (3.1a) by a factor exp(-qo0(x)/KT) and rearranging

the equation, result in

J -q0o(x) d -q0((x)
exp( ) = [n(x)exp( ] (3.12)
qD KT dx KT

Next we perform the integration on both sides of (3.12) from

x to w. A general form describing the free electron density

in the space-charge region is then obtained as

-q (V-V) qo0(x) J q o(x)
n(x) = NDexp( ).exp( ) -- exp( ).

w -q0o(x')1
exp( ) dx' (3.13)

where VD is the diffusion potential. J can be either Jo, JTE

or JTED* Suppose that Schottky barrier diodes operate in the

TE-limited mode. We substitute JTE given by (3.10) in (3.13)

and evaluate the ratio of the second term to the first term

on the right-hand side (RHS) of (3.13). It is found that

the second term vr w -q0o(x')
exp( ) dx'
the first term D Jx KT

vr [w -qo0(x') vr
< -- exp( ) dx' -
D J0 KT vd


Since for the TE-limited case vr/vd << 1, the second term on

the RHS of (3.13) is negligible. With this condition the

electron density in the space-charge region reduces to

-q (VD-V) qo (x)
n(x) z NDexp( ).exp( ) (3.15)

The above expression shows a pure exponential form. This

agrees with the condition in the TE-limited mode that the

Fermi level is nearly flat in the space-charge region with a

parabolic band structure.

When either the diffusion or the TE-D model applies, the

ratio Vr/Vd in (3.14) is no longer much smaller than one.

Therefore the second term on the RHS of (3.13) cannot be

neglected. This term is attributed to spatial variations in

the Fermi level over the space-charge region.

In order to simplify the integral part of the mentioned

second term, we apply the depletion approximation. This

approximation is sufficiently accurate if the Schottky barrier

diodes have a large barrier [15]. We therefore get [16]

w -q0o(x') EKT -qo (x)
I exp( ) dx'-- exp(
Jx KT q2N,(w-x) KT

so that (3.13) becomes

-q(VD-V) qo0(x) J EKT
n(x) = NDexp( ).exp( ) -(3.17)
KT KT qD q2ND(W-X)

3.4. Boundary Conditions

As mentioned, the general charge transport in Schottky

barrier diodes is the synthesis of electron thermionic

emission at the interface, and diffusion and drift in the

space-charge region. In this section our study focuses on the

electron density at the metal-semiconductor interface. This

boundary electron density determines the thermionic current

density in this region.

For the diffusion-limited mode the boundary electron

density can be obtained by substituting JD (eqn.(3.5)) in the

general n(x) expression (eqn.(3.13)) with x=0. With the help

of (3.4) we write n(0) as

-q(VD-V) J,
n(0) = NDexp( ) -
KT qvd

= NDexp( ) (3.18)

The above boundary electron density is independent of the

applied bias voltage. This fully agrees with the assumption

made in the diffusion theory that the interface remains at

thermal equilibrium under any bias condition. The excess

electron density n(0)-nth at the interface in this case is

zero. This indicates that at the interface the thermionic-

emission process does not occur.

To evaluate n(0) for the TE-limited case, we follow the

same procedure by combining JTE (eqn.(3.10)) and the general

form of n(x), or simply set 0o(0) = 0 in (3.15). It follows


n(0) = NDexp( ) (3.19)

Compared to (3.18), (3.19) exponentially depends on the

applied bias voltage. The forward bias voltage may greatly

increase the boundary electron density and hence the TE

current density at the interface.

Finally, for the TE-D mode of transport, n(0) is obtained

from (3.11) and (3.13) as

-qV, Vd qV vr
n(0) = NDexp(- ) [ exp(- ) + -- ] (3.20)
KT vd+vr KT vd+Vr

The above boundary electron density is between those electron

densities given by (3.18) and (3.19) under the same bias

condition. Compared to (3.19) the bias dependence of the

boundary electron density of (3.20) is weakened by the term

of v~ (Vd+vr). Therefore for the TE-D model the TE current

density at the interface does not entirely follow the applied

bias voltage.

3.5. Junction Dynamic Resistance

The dynamic resistance of Schottky barrier diodes

consists of a junction resistance and a series resistance.

The junction resistance includes the dynamic resistance of the

metal-semiconductor interface Rint and of the space-charge

region RSCR. The series resistance is associated with the

neutral region, substrate and the ohmic contact at the back

side. All the above dynamic resistance components are

arranged in series. At the practical doping levels of

ND > 1016 cm"3, and for low to moderate bias conditions, the

series resistance is small compared to the junction

resistance and can be neglected.

Suppose that Schottky barrier diodes operate in the

TE-D mode. We evaluate for RSCR and Rint, respectively. As

shown in Fig 3.2, RscR may be obtained using the following


dISCR dITED Rint = 0 EFn(0) = 0

By substituting (3.11) in (3.21), we write

RscR -- (3.22)
ITED Rint = 0 EFn(0) = 0

where VT = KT/q. In the TE-D mode of operation the expression

describing the diffusion current in the space-charge region

is given by (3.3). This is equal to the terminal current ITED*

Combining (3.3) and (3.22) yields

vr VT
RSCR = ( ) (3.23)

Similarly Rint can be derived using the following definition:





Fig. 3.2. Equivalent dynamic resistance circuit
associated with interface and space-charge
regions of Schottky barrier diodes.

dVint dV
Rint = =
dIint dITED SC = 0 EFn(O) = qV


CITED SCR = 0 EFn(O) = qV


Again, in the TE-D mode the form for the thermionic-emission

current flowing over the interface barrier is given by (3.9).

Utilizing (3.9) and (3.24), we find that

Rint = ( )- (3.25)
Vr + vd CITED

The total junction dynamic resistance of Schottky barrier

diodes is calculated by summing up RscR and Rint. Thus

R = RSCR + Rint


which is equal to dV/dITED. In addition we evaluate the

resistance ratio of the space-charge region to the interface


Rint Vd

For the diffusion-limited model, vr > vd and hence RSC >> Rint

indicating that Rj RSCR. For the TE-limited model, vr < vd

so that Rj = Rin,. Finally, for the TE-D model, vr vd and

consequently both interface and space-charge regions

contribute significantly to the device dynamic resistance,

resulting in Rj = RscR + Rnt.

3.6. Conclusions

Under low to moderate bias conditions, the charge

transport in Schottky barrier diodes is determined by the

interface region or the space-charge region. The bulk effect

is negligible. If the device resistance is dominated by the

space-charge region, the charge transport is limited by this

region and the diffusion model applies. If the device

resistance is mainly due to the semiconductor interface, the

charge transport is limited by this narrow region. Therefore

the TE model is valid. Also, if both interface and space-

charge regions contribute significantly to the device

resistance, the charge transport is controlled by both

regions. In this case the TE-D model has to be applied to

calculate the electrical properties of Schottky barrier



4.1. 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.

The noise properties of these devices at high frequencies were

successfully explained by Trippe et al.. [17] in terms of shot

noise and transit time effects. At low frequencies, excess

noise is dominant. A good understanding of this excess noise

is important since, for example, it can be up-converted to

microwave frequencies limiting the usefulness of some diodes

when operated as a mixer with a low intermediate frequency.

In this chapter our study is concerned with low-

frequency noise observed at low to moderate forward-bias

levels. For these regimes, the effect of the quasi-neutral

semiconductor bulk region on the current-voltage

characteristic can be neglected, and we will assume in our

discussion of the noise that this region acts as a perfect

ohmic contact supplying electrons to the space-charge region.


Several noise theories have been developed to explain the

low-frequency 1/f noise observed in Schottky barrier diodes.

Hsu [2] presented a number fluctuation theory based on the

fluctuating charge occupancy of traps in the space-charge

region. This fluctuating occupancy in turn would modulate the

barrier height and induce current fluctuations. However,

Hsu's model has been criticized by Kleinpenning [3], who

argues that the trap density and trap energy distribution

required to get 1/f noise are unrealistic for actual devices.

In 1979, Kleinpenning gave a "rough estimate" of the

magnitude of the 1/f noise based on the novel idea that

mobility and diffusivity fluctuations can lead to low-

frequency 1/f noise in Schottky barrier diodes. He calculated

the output current noise by considering fluctuations in the

diffusion velocity vd, which enters into the TE-D terminal

current equation [5]-[7].

The purpose of the present chapter is to give an exact

expression for the magnitude of the 1/f noise in Schottky

barrier diodes by taking the details of the charge transport

mechanism properly into account. As we will show in the

following sections, the main difference between the

Kleinpenning approach and ours is that our calculations reveal

the presence of a position-dependent integrating or weight

function that couples the local 1/f noise current fluctuations

to the external circuit. Kleinpenning does not consider this

weight function since he linearizes the terminal current

expression instead of the local current expression (see, for

example, [4]).

The theory given in the present chapter predicts low-

frequency 1/f noise in the TE-limited mode and is based on

local mobility and diffusivity fluctuations occurring in the

space-charge region of the diode. The thermionic emission

process at the interface does not produce any low-frequency

1/f noise, but limits current flow and therefore strongly

affects the noise output observed at the terminals. The

theory is applicable to most commercially available Schottky

diodes since the relatively high doping density (= 1016 cm'3)

used in these diodes to reduce the series resistance

guarantees TE operation. In addition, the theory applies to

the description of low-frequency 1/f noise in polycrystalline

silicon films, for which de Graaff and Huybers [8] and

Bisschop [9] generalized Schottky barrier noise models to

calculate the 1/f noise spectral density.

*4.2. Theory

In this section, we consider an n-type Schottky barrier

diode operating in the TE-limited forward-bias mode. The band

diagram of the diode is presented in Fig. 3.1. The local

current between the barrier maximum at x=0 and the edge of the

electron depletion layer at x=w is given by

ITE = A[qn(x)lE(x) + qD-- ] (4.1)


where A is the junction area of the diode, n(x) is the free-

electron density, and p and D are the electron mobility and

diffusivity. Expressions for the current fluctuation AI and

the dc current IT0 can be obtained by linearizing (4.1).

Neglecting the second and higher order terms, we get

o dn (x)
ITE = A[qno(x)AoEO(x) + qD ] (4.2a)


AITE = A[qgoAn(x)0oEo(x) + qno(x)0OAE(x) + qD0 ] + H,(x,t)



H,(x,t) = A[qno(x)Eo(x)At(x) + qAD(x) ] (4.2c)

The suffix zero indicates steady state. Equation (4.2c)

represents a noise term stemming from mobility and diffusivity

fluctuations. Using the Einstein relation

S- = (4.3)
o0 Ag q

this equation can be rewritten as

A L(x ,t)
H (x,t) = A (x,) (4.4)

It is generally assumed that, in the TE-limited case, the

imref in the space-charge region is independent of position

[5][7 [16] and then the electron density no(x) is given by

-q(V,-V) q0o(x)
no(x) = NDexp( )exp( ) (4.5)

where VD is the build-in potential and 0o(x) is the potential

defined in Fig. 3.1. Substituting (4.5) into (4.2b) and using

E0(x) = (4.6)

we obtain

-qpo(x) d -qoo(x)
AITE(t)exp( )= qDoA[ -(exp( )An(x))
KT dx KT

qND -q(VD-V)
+ exp( )AE(x) ]

-q0 (x)
+ H (x,t)exp( ) (4.7)

where we introduced the integrating factor exp(-q#0(x)/KT).

Equation (4.7) can be integrated from x=0 to x=w and we obtain

qDoA -q (V-V)
AITE(t) [exp( )An(w) An(0)
(1) (2)

qND -q(V,-V) w
+ exp( ) w AE(x)dx

1 w -qo0(x)
+-- H,(x,t)exp( ) dx (4.8)
I J 0 KT

w -q0o(x)
I, = exp( ) (4.9)
J 0 KT

Since the electron density in the bulk of the semiconductor

is much higher than in the space-charge region, we assume that

the dielectric relaxation time of excess electrons in the bulk

is short enough to make An(w) small on the time scale of our

low-frequency noise problem. Furthermore, since

exp(-q(VD-V)/KT) << 1 for VD-V >3KT/q, the product of An(w) and

exp(-q(VD-V)/KT) in (4.8) can safely by neglected. In

addition, under ac short-circuited conditions

w AE(x) dx = 0 (4.10)

which makes the third term of (4.8) equal to zero. Hence, the

expression for the noise current output of a Schottky barrier

diode operating in the TE mode reduces to

-qDA 1 Fw -qo0 (x)
AITE(t) = An(0) + H (x,t)exp( --) dx
I, I J0 KT


The first term on the RHS of (4.11) explicitly accounts for

the effect of the metal-semiconductor boundary on the noise


output, while the second term relates the noise output to the

local mobility and diffusivity fluctuations in the space-

charge region.

At the metal-semiconductor interface, the current is pure

thermionic-emission current and is given by equation (3.6) of

chapter 3
ITE = qA(no(0) nth)v, (4.12)

where the electron density no(0) with bias voltage applied can

be expressed as

q (V-0,o)
no(0) = Ncexp( ) (4.13)

and the quasi-equilibrium electron density nth at the interface

follows from equation (3.8)

nth = Ncexp(-qB0o/KT) (4.14)

Incorporating (4.13) and (4.14) into (4.12), we obtain for

TE-limited current flow

o-qBO qV
ITE = qANcvexp(-- ) [exp(-) 1] (4.15)

As can be seen from (4.12), fluctuations An(0) in the number

of electrons that have enough energy to surmount the barrier

maximum induce fluctuations AITE in the thermionic current flow

across the metal-semiconductor interface, i.e.

AITE = qAAn(0)vr


In terms of the number fluctuations, we then find for the

boundary condition An(0)

An(0) no(0) (4.17)

Substituting this equation and (4.13) and (4.15) into (4.11)

yields the following expression for the noise current:

Vr 1 fw -q0o(x)
AITEt) (-) H(x,t) exp( )
vd Ic JO KT

where we set Do/Ic equal to vd. The expression for the

corresponding noise density becomes

v 2 1 [w [w
S,(f) (- )2 Sy(x,x',f).
Vd Ic Jo J0

-q0o(x) -q#o(x')
exp( )exp( ) dxdx' (4.19)

The cross-correlation spectrum S,(x, x', f) of H,(x,t) is

obtained from (4.4)

(ITEO) 2S (X, X' ,f) H a(ITEO)2
SH(X,x',f) = 2( ')
o0 fno(x)A

where Su(x,x',f) is given by Hooge's empirical relation [18].

Note that (4.19) and (4.20) show that the 1/f noise is


generated in the space-charge region and is associated with

mobility and diffusivity fluctuations caused by electron

lattice scattering.

Next, attention will be focused on the evaluation of

(4.19). As we mentioned before, our discussion is limited to

low to moderate forward-bias levels. Then the width w of the

space-charge region is significantly larger than the electron

Debye length Lb given by

Lb = ( )1/2 (4.21)

As a consequence, the depletion approximation can be used and

the potential 0o(x) can be expressed as [13]

qND x2
k0(x) (wx -) (4.22)
E 2


W = [ ]e1/2 (4.23)

By substituting (4.22) and (4.23) into (4.5), we find for the

electron density

-q2ND (w-x)2
no(x) = NDexp[ ] (4.24)
2 eKT

Upon further substitution of (4.20), (4.22), and (24) into

(4.19), we derive

Vr 2 H (ITE ) 22EKT q2NDw2
S,(f) = ( )( ) exp (() .
vd fANIc2 3q2N0 2EKT


exp(-z2) exp(s2) ds (4.25)

This equation involves a Dawson function with

2 3 q2NW 2


2 3 q2N 2
s (w-x)
2 eKT

Since typically z >> 1, it holds that [19]

2w 1
exp(-z2) exp(s2) ds (4.26)
Jo 2z

Moreover, the evaluation of Ic in (4.9) also involves a

Dawson function with

Z2 2 2
2 = q---
2 KT


q2 2
s2 (w-x)2
2 KT

The result is found to be

Ic 2 (4.27)

Upon combining (4.26) and (4.27) with (4.25), we obtain the

following closed form for Si(f):

q2WH (ITE02 q2 w2
S,(f) =(-) exp( ) (4.28)
Vd 3EKTfA 2eKT

For convenience, we rewrite (4.28) using

-q0BO -qV,
Ncexp( ) = Nexp(-- ) (4.29)

and obtain

Vr q 3aHITE0 ND(VD-V) 1/2
S,(f) (-) [ ] (4.30)
Vd 3f qErKTm*

The result for SI(f) obtained by Kleinpenning [3], who

starts with the terminal current equation instead of the local

current equation and therefore does not take the integrating

factor exp[-q~0(x)/KT] into account, is

cHITE KT q m*e
S,(f) =- [ 5/2 2 ( 1/2 (4.31)
167rf q(VD-V) pm* 7N,

To illustrate the difference between (4.30) and (4.31), we

calculate the ratio 7 of these two expressions

S ref[3] 3 KT
7 = [ ]2 (4.32)
Sthis work 4 q(V,-V)

It is important to note that 7 depends on bias and that for

typical values of V z 0.1 V, T=300 K, and VD z 0.5 V, the 1/f

noise density calculated from (4.31) is approximately two

decades lower than the result of (4.30).

4.3. Device Description and Experimental Results

To verify the validity of the theory outlined in the

previous section, we measured the noise of a Hewlett-Packard

(HP5082-2305-634A) n-type silicon Schottky barrier diode at

frequencies between 5 Hz and 25kHz as a function of bias.

Figs. 4.1 and 4.2 show the scanning electron micrographs of

the measured diode. The junction area was found to be

1.0 x 10-9 m2. The measured I-V characteristic is shown in

Fig. 4.3. The linear behavior of the semilog I-V

characteristic indicates that the effect of the series

resistance can be neglected for forward-bias voltages smaller

than 0.3 V. From the sublinear region we calculated that the

series resistance is approximately equal to 11 n. The C-V

characteristic of the Schottky barrier diode under reverse-

bias condition was measured using an HP4280A 1-MHz capacitance

meter and is shown in Fig. 4.4. The inset of Fig. 4.4. shows

the doping density ND as a function of the metal-semiconductor

interface distance. ND was calculated using [13]

N = (4.33)
2 dCd

Fig. 4.1. Scanning electron micrographs (SEM) of an HP
5082-2305-634A mesh Schottky-barrier diode.


Fig. 4.1. Scanning electron micrographs (SEM) of an HP
5082-2305-634A mesh Schottky-barrier diode.






0.1 0.2 0.3


Fig. 4.3.

Current-voltage characteristic of the HP5082-
2305-634A diode. The solid line is plotted
based on (4.15). The dotted curve represents
the experimental data.



theoret. TE
current: -





V (volts)



CI -2)




1 d6

(cm-3)1 01

10 .2 .3 4 .5
I 1 t I I i I I


-3 -4


VR (volts)

Fig. 4.4.

Capacitance-voltage characteristic of the HP5082-
2305-634A diode. Cd is the capacitance associated
with the space-charge region. The inset shows the
doping concentration N as a function of interface
distance. The variations for x > 0.35 jm are
caused by numerical instabilities in the differen-
tiation of Cd"2 with respect to VR.



The variations in ND for x > 0.35 gm result from numerical

instabilities in the differentiation of Cd"2 with respect to

VR. From this graph we extrapolated for the doping

concentration ND close to the metal-semiconductor interface a

value of 2.4 x 102m"3. The value of VD was determined from

both the I-V characteristic and the C-V measurement. We found

VD = 0.46V.

The very low 1/f noise levels of the HP diode were

measured using a Brookdeal 5004 ultra-low-noise preamplifier

in conjunction with an HP3582A spectrum analyzer. Standard

procedures were used to correct for the noise of the

amplifier. Fig. 4.5 shows the measured current spectral

density at T=298 K for currents between 3.22 and 232.0 IA.

The spectra show the presence of excess noise at low

frequencies, whereas at high frequencies full shot noise is

observed. To determine more accurately the nature of the

excess noise, we subtracted the full shot noise levels from

the measured data. These results are presented in Fig. 4.6

and indicate that the excess noise is pure 1/f noise.

4.4. Discussion

Since Fig. 4.6 clearly demonstrates that the measured

excess noise is pure 1/f noise, an interpretation of the noise

data in terms of mobility and diffusivity fluctuations seems

warranted. In Fig. 4.7, we compare the measured noise data

at 40 Hz with theoretical values of the equivalent noise

current calculated from (4.30) and (4.31) using a,=4.2x10"9.
current calculated from (4.30) and (4.31) using aH-4.2xlO-






10725 0

0: 232.OpA +:16.4pA
X: 111.08pA C:7.06pA
A:47.27pA *:3.22uA

:-( -X->OC-x-x





-I............I. ............... I 35.3.3

I I I I I l I I I I I 1 15t I a 1 -





Fig. 4.5.

Measured spectral current density as a function
of bias.

0: 232.0,pA
X: 111.80 pA
A:4 7.27uA

S I I I iI I I I


.. I



Fig. 4.6. Excess noise spectra as a function of bias.


S 2 (



, ,0





theory: this work
......ref.1 33


I (A)

Fig. 4.7.

Equivalent noise current versus bias current.
Solid and dashed lines are plotted according to
(4.30) and (4.31), respectively, using f=40 Hz and
aH=4.2x10 The circles are .experimental data.


leq (A)





It should be noted that (4.30), which is based on the local

current fluctuation model, correctly predicts the bias

dependence of the measured noise, whereas the result of the

rough estimate, see (4.31), derived by Kleinpenning and based

on terminal current fluctuations, leads to a slightly steeper

current dependence than is observed experimentally. This

point is more clearly illustrated in Fig. 4.8, where we

plotted normalized equivalent noise currents versus bias

current. In our opinion, Fig. 4.7 demonstrates that

Kleinpenning's estimate for 1/f noise in a Schottky-barrier

diode is not quite accurate.

The low value of the Hooge parameter that follows from

(4.30) is in good agreement with the theoretical value of

3.4 x 10-9 calculated by Kousik et al. [20] for normal

electron-phonon scattering in n-type silicon and seems to

indicate that intervally scattering is absent in the space-

charge region of the diode since the latter process would

result in a much higher value of a, [20].

4.5. Conclusions

In this chapter we presented a 1/f noise model for

Schottky-barrier diodes operating in the thermionic-emission

limited forward-bias mode. The model is based on mobility and

diffusivity fluctuations occurring in the space-charge region

and accounts for the current limiting role of the metal-

semiconductor interface. The bias dependence of the 1/f noise

spectral density calculated from this model is in excellent

measured: o
theory: this work
...... ref.C 31
leq (3.22MA)


10 1 , ,
10-6 10-5 10-4


Fig. 4.8. Normalized equivalent noise current versus
bias current.


agreement with our experiments. The experimental value of the

Hooge parameter found from our measurements is 4.2 x 109.


5.1. Introduction

A theory of shot noise in Schottky barrier diodes based

on the diffusion model was first developed by van der Ziel

[21]. His theory was derived starting from distributed

diffusion noise sources in the space-charge region. The

result given by van der Ziel's derivation predicts full shot

noise in the diffusion mode of operation.

In this chapter we discuss shot noise in Schottky barrier

diodes operating in TE, D and TE-D modes. For the TE-D mode

of transport we calculate the output noise spectral density

of a diode by accounting for noise contributions from the

the metal-semiconductor interface and the space-charge region.

The noise density of the interface region is obtained by

considering fluctuations in the recombination process. The

noise density of the space-charge region is calculated using

the diffusion noise source. These two noise components are

essentially uncorrelated. This allows us to evaluate the

overall output noise density by adding individual noise

outputs. The result obtained in this way results in full shot

noise in the TE-D mode of transport.

5.2. Noise Contribution from the Interface Region

Consider an n-type Schottky barrier diode under zero

bias. Transport at the interface is balanced by two equal

but opposite thermionic electron currents over the barrier

maximum. These two current components are proportional to the

equilibrium electron density at the interface nth, and given


Jth = qAvrnth (5.1)

where A is the cross section and vr is the effective

recombination velocity at the interface.

Assume that the Schottky barrier diode is forward biased

and operates in the TE-D mode. According to the principle of

current conservation, the terminal TE-D current should be

equal to the thermionic excess electron flow over the

interface barrier. Therefore

ITED = qAno'vr (5.2)

where the supscript and the subscript zero denote the DC

quantities. no'(0) is the excess electron density equal to

no(0)-nth; n0(0) and nth are defined by (3.7) and (3.8),

respectively. We derive for n'(0) by equating (5.2) to (3.11)

the following result

VdND -qVD qV
no'(0) = .exp(-- )(exp(-) 1) (5.3)
Vd + vr KT KT


Under a forward bias condition, we may consider that the

current flowing through the metal-semiconductor interface

consists of three components: Two of them are given by (5.1)

and are macroscopically balanced by each other, whereas the

third current component is described by (5.2). Since each

current component gives full shot noise, the noise spectral

density due to the interface region, Sm(f), is found as

Sim(f) = 2q2vA(n' (0) + 2nth) (5.4)

By substituting (5.3) and (3.8) in (5.4), we have

Vr + Vd
Sim(f) = 2qITED + 4q(- ) .Io (5.5)


vrvd -qVD
10 =qA( ) .N0exp (- ) (5.6)
vr + Vd KT

To calculate the output noise spectral density measured

in the external circuit due to the noise source Sm(f), we

present an equivalent noise circuit shown in Fig. 5.1(a). In

this figure the noise current generator is found from (5.5)


A int = (SIm(f)Af)1/2 (5.7)

In addition, Rint and RSCR in Fig. 5.1(a) represent dynamic

resistances associated with the interface and the space-charge






Fig. 5.1.

Equivalent noise circuit of
(a) the metal-semiconductor interface
(b) the space-charge region


region, respectively. Rint and RScR are defined by (3.23) and

(3.25) in chapter three. According to simple network

considerations, the output noise current AITED coming from the

interface region is given by

AITED = A int. ( ) (5.8a)
Rint + RSCR

From (3.27) AITED may be rewritten as

AITED (AInt (5.8b)
Vr + Vd

The corresponding output noise spectral density is given by

SITED t(f) = 2qITEDr2 + 4qIr (5.9a)

and r m (5.9b)
vr + Vd

Note that SITEDint(f) is obtained from the thermionic emission

current occurring at the interface. This current is commonly

treated as a surface recombination current, and expressed in

terms of an effective recombination velocity. This seems to

indicate that the physical basis of (5.9a) is fluctuations

occurring in this recombination process.

5.3. Noise Contribution from the Space-Charge Region

Even in the TE-D mode, charge transport in the space

charge region is governed by drift and diffusion. We thereby

expect that the noise source in this region might be

associated with these mechanisms. Van Vliet [22] argued that

although charge transport is due to drift and/or diffusion,

the noise source is only related to the diffusion process.

The drift flow may affect the observed spectral response even

though no noise source is associated with drift. The

diffusion noise can be derived from a generalized formalism,

involving generalized Fokker-Planck moments which depend on

the transition and scattering rates [22]. In general we have

SH(x,x',f) = 4q2Dn(x)6(x-x') (5.10)

We calculate the response at the terminal of the device due

to SH(x,x',f) as follows.

A stochastic Langevin equation for the local current in

the space charge region is expressed as

ITED = A(qn(x)pE(x) + qD ) + H(x,t) (5.11)

where H(x,t) signifies the Langevin noise term. Suppose that

fluctuations in J and D are negligible on the short time scale

of our shot-noise study. We use the DC quantities AI and Dg

to replace M and D in (5.11). Then linearizing the electron

density n(x) and the electric field E(x) of (5.11), we get the

DC current

0 dn (x)
ITED = A(qno(x)A0oEo(x) + qD0 ) (5.12a)

and the fluctuation current

AITED = A(qAn(x)t0Eo0(x) + qno(x)AioAE(x) +qD- ) + H(x,t)


To further manipulate the noise current expression of (5.12b),

we follow procedures similar to those applied in the treatment

of 1/f noise presented in chapter IV. In that TE-limited

case, the expression for the electron density we used, eq.

(4.5), was based on the assumption of a flat imref. However
in the present TE-D mode of operation, the general expression
for no(x) in (3.17) has to be applied to (5.12b) because of a

nonflat imref. By multiplying both sides of (5.12b) by a

factor exp(-q,0(x)/KT) and performing the integration over the

range from x=O to x=w, we obtain

-q (V,-V) qN, -q (VD-V)
AITED = qAvd[An(w)exp( ) An(0) + exp(-).

(1) (2) (3)
w JTED w AE(x) -q0o(x)
|AE(x) dx + exp( ) dx
o qD J 0 E0(x) KT

1 [w -q0o(x)
+ H(x,t)exp( ) dx (5.13)


w -qo(x)
where Ic =j exp(-- ) dx (5.14)

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