CHARGE TRANSPORT AND NOISE PROPERTIES OF SCHOTTKY
BARRIER DIODES AND POLYCRYSTALLINE SILICON THIN FILMS
BY
MINYIH LUO
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
1989
ACKNOWLEDGMENTS
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, YoungJune Yu, Sheng
Lyang Jang, and HorngJye Luo.
Finally I come to my wife, daughter and family. Without
moral and financial support offered by my parents and
parentsinlaw this project would not have been possible.
There are not enough words to express my gratitude to my wife,
YaLing. 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.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ........................................... ii
ABSTRACT ............................................. vi
CHAPTER
I. INTRODUCTION ............... .................. 1
II. COMPUTERBASED MEASUREMENTS ...................... 6
2.1. Introduction ............................. 6
2.2. Noise Measurements Using the HP3582A
Spectrum Analyzer ...................... 9
2.2.1. Examination of Operational
States via HPIB ............. 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. CurrentVoltage 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
III. CHARGE TRANSPORT IN SCHOTTKYBARRIER DIODES ...... 47
3.1. Introduction ........... .................. 47
3.2. Charge Transport Theories ................ 49
3.2.1. DiffusionLimited
Conduction .................. 49
3.2.2. ThermionicEmissionLimited
Conduction ................... 51
3.3.3. Thermionic EmissionDiffusion
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
IV. THEORY AND EXPERIMENTS OF 1/F NOISE IN
SCHOTTKYBARRIER DIODES OPERATING IN THE
THERMIONICEMISSION 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
V. SHOT NOISE IN SCHOTTKYBARRIER DIODES ............ 86
5.1. Introduction .............................. 86
5.2. Noise Contribution from the
Interface Region ....................... 87
5.3. Noise Contribution from the
SpaceCharge Region ..................... 90
5.4. Discussions and Conclusions ............... 95
VI. REVIEW OF ELECTRICAL PROPERTIES OF
POLYCRYSTALLINE SILICON ......................... 97
6.1. Introduction .............................. 97
6.2. Description of GrainBoundary
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. UShaped 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. CarrierTrapping Model .......... 128
6.4.3. Mixed Model ..................... 136
6.5. Nonuniform Properties .................... 139
VII. EXCESS NOISE ANOMALIES OBSERVED IN
PHOSPHORUSDOPED POLYCRYSTALLINE
SILICON FILMS .................................. .145
7.1. Introduction .............................. 145
7.2. Sample Description ........................ 146
7.3. Experiments ................................ 149
7.3.1. IV Characteristics ............. 149
7.3.2. Noise Measurements .............. 158
7.4. Discussion ................................ 178
7.5. Conclusions ............................... 181
VIII. AN ANALYTICAL MODEL FOR 1/F NOISE IN
POLYCRYSTALLINE SILICON THIN FILMS ............. 182
8.1. Introduction ............................. 182
8.2. Theory .. ....... ......................... 185
8.2.1. GrainBulk Region
1/f Noise ................... 185
8.2.2. DepletionBarrier Region
1/f Noise .................... 191
8.3. Experiments .. .. .......................... 198
8.4. Discussion ............................... 201
8.5. Conclusions ......... ...................... 205
IX. CONCLUSIONS AND SUGGESTIONS FOR
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. LowFrequency Excess Noise
in Polysilicon ......................... 208
APPENDIX A. THE CONTROL PROGRAM OF THE HP3582A
SPECTRUM ANALYZER ....................... 211
APPENDIX B. THE CONTROL PROGRAM OF THE HP4145B
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
CHARGE TRANSPORT AND NOISE PROPERTIES OF SCHOTTKY
BARRIER DIODES AND POLYCRYSTALLINE SILICON THIN FILMS
By
MinYih Luo
August 1989
Chairperson: G. Bosman
Major Department: Electrical Engineering
A 1/f noise model for Schottky barrier diodes operating
in the thermionicemission 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 phosphorusdoped
polysilicon thin films show that the excess noise spectra of
vii
this material either have a form 1/f" where n is close or
equal to one, or consist of an ensemble of Lorentzianshaped
and other timescaled 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 ntype polysilicon. Noise calculations have
been performed for the quasineutral 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
depletionregion 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.
viii
CHAPTER I
INTRODUCTION
This thesis describes experiments and theories developed
for barrier limited transport and lowfrequency 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 ntype silicon. The
second part of this thesis describes the results of our
efforts on polycrystalline material with special emphasis on
polysilicon.
The diffusion barrier of a Schottkybarrier diode arises
because of the difference in work functions between the metal
and the semiconductor. When electrons travel, under forward
bias conditions, from an ntype bulk semiconductor to a metal
contact, they will experience scattering in the local field
of the spacecharge 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 quantummechanical
tunneling through the barrier can occur.
2
Lowfrequency 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 spacecharge region. In his
model even though the charge transport in Schottkybarrier
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 TED 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
3
zero since then the diffusion velocity term is missing from
the current equation. This prediction is inconsistent with
our experimental findings on Schottkybarrier 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 chargetransport process in a Schottkybarrier
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.
4
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 thermionicemission 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
Schottkybarrier 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 currentvoltage measurements performed by the
HP4145B semiconductor parameter analyzer.
In Chapter III, the chargetransport properties of
Schottkybarrier 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 depletionregion impedance using a general
chargetransport model.
In Chapter IV, a general theory for shot noise in
Schottkybarrier diodes operating in TED mode is developed.
In Chapter V, we develop and test a new model for 1/f
noise in Schottkybarrier diodes operating in the TElimited
mode.
5
In Chapter VI, we extensively review the electrical
properties of polysilicon material, including the dependence
of the barrier height on the grainboundary trap distribution,
various chargetransport models, nonuniform properties and
more.
In Chapter VII, a comprehensive experimental study on
lowfrequency 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.
CHAPTER II
COMPUTERBASED MEASUREMENTS
2.1. Introduction
This chapter presents information about measurements of
noise and currentvoltage characteristics controlled by
computers. Personal computers can be used to carry out both
types of measurements efficiently. A noise measurement
performed manually is a timeconsuming 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 HPIB board have been set up.
In the case of a measurement of currentvoltage
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 HPIB board is an instrumentation interface which
primarily solves the compatibility problems between various
7
components and integrates the components into a system. The
board uses a 16line 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 HPIB 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 HPIB 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 8bit status word is
performed by sending the command "LST1" to the HP3582A via the
HPIB. 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 HPIB library is
8
"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 HPIB
subroutine "IOSPOLL". This subroutine covers the functions
of "LST1", "LSTO" ,and "rdb" or "IOENTERS" in an equivalent
manner.
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 HPIB 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
9
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 dualchannel spectrum
analyzer covering the frequency range of 0.02 Hz to 25.6 KHz.
These two components are interconnected via an HPIB. 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 HPIB
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 HPIB and the analyzer responds with an
8bit 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
XY plot Average Single sweep Time record
complete complete spectrum complete
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.
11
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 IFstatement
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
IBM PC
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.
13
"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 ASCIIformat 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.
15
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.
"LMK"
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
LMK
LDS
LFM
Fig. 2.3. Format of data transferred by commands LMK
and LDS.
17
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.
18
The data transfer using "LMK" via the HPIB allows a high
degree of accuracy to be obtained. However the marker tracing
is very timeconsuming.
"LDS"
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
I
19
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.
"LFM"
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 HPIB are in 8bit
byte, two bytes are required to transfer or receive a 16bit
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.
T=300K
Rbia=99.92K
 rd=3.31K
0: "LDS"
x: "LMK"
x /
4KT/rd
I l I IIIII I m I I I I I, ,I I.,, II  I 1, 1
100 101
102
104
105
f(Hz)
Fig. 2.4. Thermal current noise spectrum of a resistor of
3.31 Kn measured by two different programs. One
uses the datatransfer command LDS. The other
uses LMK.
10
22
10
23
10
N
I
<
24
25
10
10
_ __ ____ I _____ _~~I
I I
, , n n
,
. , n
, ,
103
2.2.3. The Manipulation of Noise Data
The HP3582A spectrum analyzer features different sweep
modes. 0START 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 025 KHz mode allows a quick look at
the entire spectrum. The 0START 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
oHz
I
horizontal average
35 <
points\,i ,, ,
I I I I I
oHz
/4
35 <
points\ l
25KHz
2.5KHz
250Hz
25H7
. 2.5Hz E
o
0)
L.
I_
horizontal average
n1 n1 n n nI
I I I I
frequency section
25KHz
2.5KHz
250Hz
25Hz
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.
L
I II
I
I
I
I I I I I
23
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 12Volt wetcell batteries, metal film
bias resistors, a Brookdeal 5004 ultralownoise preamplifier,
an HP3582A spectrum analyzer and a controller. The controller
used is either the HP9825A calculator or the IBM PC. If a
lowtemperature 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 
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
2.6K 
3.6K 
4.6K 
5.6K 
7K 
9K 
14K 
19K 
Table 2.2. The arrangement of frequency sections.
1
1.5
2
1.1
1.6
2.1
4.8
4.9
5.4
6.4
9
11
16
21
34
44
54
68
90
110
160
210
260 340
360 440
460 540
560 640
700 900
900 1.1K
1.4K 1.6K
1.9K 2.1K
30
40
50
65
80
100
150
200
300
400
500
600
800
1K
1.5K
2K
3K
4K
5K
6K
8K
10K
15K
20K
3.4K
4.4K
5.4K
6.4K
9K
11K
16K
21K
# of point
freq. (Hz)
r 1 Turbomoleculari
I cooling I & mechanical
I system pump
L J 1 J
(a)
Sveq
r L2
r M
SiR b I ix rd 1 (b)
(b)
Fig. 2.6. Noisemeasurement setup.
(a) the setup for measuring M2.
(b) the noise equivalent circuit.
26
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
(a)
Sveq
M2
blas Sir d S Sa
(b)
Fig. 2.7. Noisemeasurement 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
by
4KT
Sir = (2.4)
rd
The final measurement requires a calibration noise source
for which, in this case, we use the builtin 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:
S
2 Svca l
M2 = G [Sveq + (SiR + Sir + Sia + ) ] (2.5)
(Rbias//rd)
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
scope
Sycogl attenuator Svcal
(a)
Sveq
G _M3
f SiRC RbIoa. Sir rd Sia J
(b)
Fig. 2.8. Noisemeasurement setup.
(a) the setup for measuring M3 & Svcat (with
dummy).
(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
following:
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 )
(2.9)
(a)
(b)
Fig. 2.9. Noisemeasurement setup.
(a) the setup for measuring M22 (no dummy).
(b) the noise equivalent circuit.
Sveq
M M3
(b)
Fig. 2.10. Noisemeasurement setup.
(a) the setup for measuring M3 & SvcaL (no
dummy).
(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
(2.10)
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
utilized.
Note that some readings of Mi and Svcat' in (2.6) and
(2.10) are frequencyspan 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
below:
Mi at 8 KHz:
frequency span
used
25 KHz
10 KHz
M3 at 8 KHz:
frequency span
used
25 KHz
10 KHz
bandwidth
100 Hz
40 Hz
bandwidth
100 Hz
40 Hz
measured
in dBV
90.0
93.4
measured
in dBV
71.1
71.1
measured
in dBV/(Hz)1/2
109.2
109.2
measured
in dBV/(Hz)1/2
91.1
87.1
If the measurement is performed using the unit dBV/(Hz) '2, the
readings M12 and M2z are frequencyspan independent, whereas M32
and SvcaL vary with the frequency span used. If data are
34
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 frequencyspan
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 12 seconds to load the
program to the BASIC interpreter. Then the program may be
run immediately.
transfer
command
M32 & SvcaL
total
measure. time
Table 2.3. Comparison of measurement time for M 2, M22
and M3 & SvcaL using different datatransfer
commands.
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
transfer
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. CurrentVoltage Measurements Using the HP4145B Parameter
Analyzer
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
37
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 IV
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 IV
measurements. Variations in the definition of SMUs and in
sweep modes will allow the user to perform all possible
IV measurements on a DUT.
To provide better understanding of the applications of
SMUs, an example of measuring the collectorcurrent and the
collectorvoltage characteristics of an npn 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
38
as illustrated in Fig. 2.11. To remotely define SMUs, the
computer sends the following codes via the HPIB:
CODE1$="DE CH1,'VE','IE',3,3;CH2,'VBE','IB',2,2;CH3,'VCE',
'IC',1,1"
In this command string, DE makes the HP4145B go to the channel
definition page and CH1, CH2 and CH3 simply correspond to
SMU1, SMU2 and SMU3, respectively. 'VE', 'IE','VBE', 'IB','V
CE' and 'IC' are names of output data. The numbers in the
string have particular meanings. The first number of each
pair specifies the source mode which may be a voltage source,
a current source, or a common. The second number determines
the sweep mode, namely, VAR1, VAR2 or VAR1'. For example,
"3,3" indicates that SMU1 acts as a common and does not
operate in the sweep mode. "2,2" specifies SMU2 as a (base)
current source in the subordinate sweep mode. Finally, "1,1"
makes SMU3 function as a (collector) voltage source in the
main sweep mode.
The next step is to set up the voltage sweep and the
current sweep mode on the SOURCE SETUP (SS) page by entering
each field in the following command string:
CODE2$="SS VR1, STARTVCE,STOPVCE,STEPVCE,MAXIC;IP STARTIB,
NO.OF IB, MAXVB"
SMU1
(current source)
source)
I 
I I
L.. J
I
L______.
ISMU2
(common)
I(common)
IC NO. OF IB
AIIB subordinate
IB2 sweep
STEPVCE STARTIB
STARTVCE main sweep STOPVCE
STARTVCE main sweep STOPVCE
Fig. 2.11. Diagram illustrating the measurement of
IcVcE characteristics of a bipolar
transistor using the HP4145B.
I
'7 I
40
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 IV
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$, the
subordinate sweep becomes voltage.
The third step is to set up the display graphics on the
MEAS & DISP MODE SETUP (SM) page by choosing the axis
variables and specifying their ranges. In our example, the
41
Xvariable is 'VCE' and the Yvariable is 'IC'. The program
codes for these settings are as follows:
CODE3$="SM DM1 XN 'VCE',1,STARTVCE,STOPVCE;YA 'IC',1,MINIC,
MAXIC"
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 IV measurement of the npn 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 userdefined
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 HPIB. 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 HPIB subroutine
"IOSPOLL".
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 HPIB. 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
analyzer.
Bit 7 Bit 6 Bit 5 Bit 4
SelfTEST
Emergency RQS Fail Busy
DRO
(data not ready)
bit0
data ready^ No
Yes
define
output channel
"DO xx"
4
enter data
Fig. 2.12. Flow diagram describing the procedure of
accessing data measured by the HP4145B.
IOSPOLL
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 menudriven 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
below:
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
IV characteristics of two and threeterminal devices
including diodes, BJTs (npn/pnp), JFETs (p/n channels),
MOSFETs (p/n channels in depletion/enhancement modes).
3. The polarities of all IV 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
directly.
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
46
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.
CHAPTER III
CHARGE TRANSPORT IN SCHOTTKYBARRIER DIODES
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 metalsemiconductor
interface region, the spacecharge 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 ntype Schottky
barrier diodes is determined by the passage of electrons from
the bulk semiconductor through the spacecharge 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^
I^^
q(x) q(vEv)
TE
SEC
semiconductor
1w x
0
Fig. 3.1. Band diagram of an ntype Schottky barrier
diode operating in TE, D, and TED modes,
respectively.
BO
#BO
metal
49
When current flow through the diode is limited by the
conductivity of the spacecharge region, charge transport is
dominated by drift and diffusion mechanisms. Then the
socalled diffusion (D) model developed by Schottky and Spenke
[10] applies. When current flow is limited by the
properties of the metalsemiconductor 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 spacecharge
region and the metalsemiconductor interface, the thermionic
emissiondiffusion (TED) 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 ntype forward biased Schottky barrier diodes is
presented. We base our description on D, TE and TED models.
Then we derive a general form for the electron density in the
spacecharge region from the local current. Finally, we
discuss the boundary conditions at the metalsemiconductor
interface and the junction dynamic resistance of Schottky
barrier diodes.
3.2. Charge Transport Theories
3.2.1. DiffusionLimited Conduction
The current density in the spacecharge region of
Schottky barrier diodes can be expressed as
dn(x)
J = qgn(x)E(x) + qD (3.1a)
dx
dEF (x)
= pn(x) (3. b)
dx
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)
KT
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 spacecharge
region
qBO qV EFn (0)
J, = qvdNcexp( )[exp( ) exp( )] (3.3)
KT KT KT
where OBO is the barrier height, V is the applied bias voltage,
and vd is the effective diffusion velocity defined as [13]
D
vd = (3.4)
w q0o(x)
Sexp( )dx
JO KT
In practical cases vd can be approximated as IEmax [13], where
Emax is the maximum electric field at the metal
semiconductor interface.
51
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 spacecharge 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 diffusionlimited
current density
qOBO qV
JD = qVdNcexp( )[exp( ) 1] (3.5)
KT KT
3.2.2. ThermionicEmissionLimited 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)
KT
The equilibrium electron density nth at the interface follows
from
q'BO
nth = Ncexp( ) (3.8)
KT
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)
KT KT
Suppose that Schottky barrier diodes operate in the TE
limited mode. The diffusion effect in the spacecharge 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 spacecharge 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 TElimited current density
qsBO qV
JTE = qvrNcexp( )[exp() 1] (3.10)
KT KT
It should be noted that electron scattering could still occur
in the spacecharge region without invalidating the TE theory
[14].
3.2.3. Thermionic EmissionDiffusionLimited Conduction
The TED 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 spacecharge 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 TED mode of transport the current density in the
spacecharge 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 TED 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 diffusionlimited
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 SpaceCharge Region
In this section we study the electron density in the
spacecharge region of Schottky barrier diodes based on
different chargetransport models. Since the spacecharge
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 spacecharge 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 TED models,
a variation in the Fermi level complicates the
characterization of the electron density in the spacecharge
region.
We calculate the electron density in the spacecharge
region based on the TED 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 spacecharge region is then obtained as
q (VV) qo0(x) J q o(x)
n(x) = NDexp( ).exp( )  exp( ).
KT KT qD KT
w q0o(x')1
exp( ) dx' (3.13)
Sx KT
where VD is the diffusion potential. J can be either Jo, JTE
or JTED* Suppose that Schottky barrier diodes operate in the
TElimited 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 righthand 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
(3.14)
Since for the TElimited case vr/vd << 1, the second term on
the RHS of (3.13) is negligible. With this condition the
electron density in the spacecharge region reduces to
q (VDV) qo (x)
n(x) z NDexp( ).exp( ) (3.15)
KT KT
The above expression shows a pure exponential form. This
agrees with the condition in the TElimited mode that the
56
Fermi level is nearly flat in the spacecharge region with a
parabolic band structure.
When either the diffusion or the TED 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 spacecharge 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,(wx) KT
(3.16)
so that (3.13) becomes
q(VDV) qo0(x) J EKT
n(x) = NDexp( ).exp( ) (3.17)
KT KT qD q2ND(WX)
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
spacecharge region. In this section our study focuses on the
electron density at the metalsemiconductor interface. This
57
boundary electron density determines the thermionic current
density in this region.
For the diffusionlimited 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(VDV) J,
n(0) = NDexp( ) 
KT qvd
qVD
= NDexp( ) (3.18)
KT
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 TElimited 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
that
q(VDV)
n(0) = NDexp( ) (3.19)
KT
Compared to (3.18), (3.19) exponentially depends on the
58
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 TED 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 TED 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
metalsemiconductor interface Rint and of the spacecharge
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
TED mode. We evaluate for RSCR and Rint, respectively. As
shown in Fig 3.2, RscR may be obtained using the following
definition
dVSCR dV
RSCR
dISCR dITED Rint = 0 EFn(0) = 0
(3.21)
By substituting (3.11) in (3.21), we write
VT
RscR  (3.22)
ITED Rint = 0 EFn(0) = 0
where VT = KT/q. In the TED mode of operation the expression
describing the diffusion current in the spacecharge 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)
Vr + Vd CITED
Similarly Rint can be derived using the following definition:
EF,(O)/q
11
CITED
Rint
RSCR
Fig. 3.2. Equivalent dynamic resistance circuit
associated with interface and spacecharge
regions of Schottky barrier diodes.
dVint dV
Rint
Rint = =
dIint dITED SC = 0 EFn(O) = qV
VT
CITED SCR = 0 EFn(O) = qV
(3.24)
Again, in the TED mode the form for the thermionicemission
current flowing over the interface barrier is given by (3.9).
Utilizing (3.9) and (3.24), we find that
Vd VT
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
VT
(3.26)
TED
which is equal to dV/dITED. In addition we evaluate the
resistance ratio of the spacecharge region to the interface
region
RSCR Vr
(3.27)
Rint Vd
For the diffusionlimited model, vr > vd and hence RSC >> Rint
indicating that Rj RSCR. For the TElimited model, vr < vd
so that Rj = Rin,. Finally, for the TED model, vr vd and
consequently both interface and spacecharge 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 spacecharge region. The bulk effect
is negligible. If the device resistance is dominated by the
spacecharge 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 TED model has to be applied to
calculate the electrical properties of Schottky barrier
diodes.
CHAPTER IV
THEORY AND EXPERIMENTS OF 1/F NOISE IN SCHOTTKY
BARRIER DIODES OPERATING IN THE
THERMIONICEMISSION MODE
4.1. Introduction
Schottky barrier diodes have become increasingly
important due to their excellent highfrequency 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 upconverted 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 forwardbias
levels. For these regimes, the effect of the quasineutral
semiconductor bulk region on the currentvoltage
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 spacecharge region.
64
Several noise theories have been developed to explain the
lowfrequency 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 spacecharge
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 TED 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 positiondependent 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
65
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 TElimited mode and is based on
local mobility and diffusivity fluctuations occurring in the
spacecharge region of the diode. The thermionic emission
process at the interface does not produce any lowfrequency
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 lowfrequency 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 ntype Schottky barrier
diode operating in the TElimited forwardbias 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
dn(x)
ITE = A[qn(x)lE(x) + qD ] (4.1)
dx
66
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)
dx
and
dAn(x)
AITE = A[qgoAn(x)0oEo(x) + qno(x)0OAE(x) + qD0 ] + H,(x,t)
dx
(4.2b)
with
dno(x)
H,(x,t) = A[qno(x)Eo(x)At(x) + qAD(x) ] (4.2c)
dx
The suffix zero indicates steady state. Equation (4.2c)
represents a noise term stemming from mobility and diffusivity
fluctuations. Using the Einstein relation
Do AD KT
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 TElimited case, the
imref in the spacecharge 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)
KT KT
where VD is the buildin potential and 0o(x) is the potential
defined in Fig. 3.1. Substituting (4.5) into (4.2b) and using
do0(x)
E0(x) = (4.6)
dx
we obtain
qpo(x) d qoo(x)
AITE(t)exp( )= qDoA[ (exp( )An(x))
KT dx KT
qND q(VDV)
+ exp( )AE(x) ]
KT KT
q0 (x)
+ H (x,t)exp( ) (4.7)
KT
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 (VV)
AITE(t) [exp( )An(w) An(0)
Ic KT
(1) (2)
qND q(V,V) w
+ exp( ) w AE(x)dx
KT KT JO
(3)
1 w qo0(x)
+ H,(x,t)exp( ) dx (4.8)
I J 0 KT
(4)
with
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 spacecharge 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
lowfrequency noise problem. Furthermore, since
exp(q(VDV)/KT) << 1 for VDV >3KT/q, the product of An(w) and
exp(q(VDV)/KT) in (4.8) can safely by neglected. In
addition, under ac shortcircuited conditions
w AE(x) dx = 0 (4.10)
Jo
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
(4.11)
The first term on the RHS of (4.11) explicitly accounts for
the effect of the metalsemiconductor boundary on the noise
69
output, while the second term relates the noise output to the
local mobility and diffusivity fluctuations in the space
charge region.
At the metalsemiconductor interface, the current is pure
thermionicemission current and is given by equation (3.6) of
chapter 3
0
ITE = qA(no(0) nth)v, (4.12)
where the electron density no(0) with bias voltage applied can
be expressed as
q (V0,o)
no(0) = Ncexp( ) (4.13)
KT
and the quasiequilibrium 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
TElimited current flow
oqBO qV
ITE = qANcvexp( ) [exp() 1] (4.15)
KT KT
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 metalsemiconductor interface, i.e.
AITE = qAAn(0)vr
(4.16)
In terms of the number fluctuations, we then find for the
boundary condition An(0)
AIT
An(0) no(0) (4.17)
ITE
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
(4.18)
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)
KT KT
The crosscorrelation 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
(4.20)
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
71
generated in the spacecharge 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 forwardbias levels. Then the width w of the
spacecharge region is significantly larger than the electron
Debye length Lb given by
EKT
Lb = ( )1/2 (4.21)
q2N
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
with
2E(VDV)
W = [ ]e1/2 (4.23)
qND
By substituting (4.22) and (4.23) into (4.5), we find for the
electron density
q2ND (wx)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
z
exp(z2) exp(s2) ds (4.25)
Jo
This equation involves a Dawson function with
2 3 q2NW 2
2 EKT
and
2 3 q2N 2
s (wx)
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
Sq2NDw2
Z2 2 2
2 = q
2 KT
and
q2 2
s2 (wx)2
2 KT
The result is found to be
EKT
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)
KT KT
and obtain
Vr q 3aHITE0 ND(VDV) 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(VDV) 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 HewlettPackard
(HP50822305634A) ntype 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 109 m2. The measured IV characteristic is shown in
Fig. 4.3. The linear behavior of the semilog IV
characteristic indicates that the effect of the series
resistance can be neglected for forwardbias voltages smaller
than 0.3 V. From the sublinear region we calculated that the
series resistance is approximately equal to 11 n. The CV
characteristic of the Schottky barrier diode under reverse
bias condition was measured using an HP4280A 1MHz 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 metalsemiconductor
interface distance. ND was calculated using [13]
2
N = (4.33)
2 dCd
qeA
dVR
Fig. 4.1. Scanning electron micrographs (SEM) of an HP
50822305634A mesh Schottkybarrier diode.
I
Fig. 4.1. Scanning electron micrographs (SEM) of an HP
50822305634A mesh Schottkybarrier diode.
o..*of
102
103
105
106
0.1 0.2 0.3
0.4
Fig. 4.3.
Currentvoltage characteristic of the HP5082
2305634A diode. The solid line is plotted
based on (4.15). The dotted curve represents
the experimental data.
10
T=298K
measured:*
theoret. TE
current: 
I(A)
108
oe10
0.5
V (volts)
20
16
CI 2)
Cd2
12
41
10o8
lo17
1 d6
ND
(cm3)1 01
10 .2 .3 4 .5
I 1 t I I i I I
2
3 4
5
VR (volts)
Fig. 4.4.
Capacitancevoltage characteristic of the HP5082
2305634A diode. Cd is the capacitance associated
with the spacecharge 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.
8
,,rd
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 metalsemiconductor interface a
value of 2.4 x 102m"3. The value of VD was determined from
both the IV characteristic and the CV measurement. We found
VD = 0.46V.
The very low 1/f noise levels of the HP diode were
measured using a Brookdeal 5004 ultralownoise 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 aH4.2xlO
1020
A2
Si(Rz)
1022
1023
1024
i24
10725 0
10
0: 232.OpA +:16.4pA
X: 111.08pA C:7.06pA
A:47.27pA *:3.22uA
:( X>OCxx
I..
0
T=298K
I
I............I. ............... I 35.3.3
I I I I I l I I I I I 1 15t I a 1 
101
102
f(Hz)
10J
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
+:16.4puA
a:7.06pA
*:3.22,uA
.. I
10
f(Hz)
Fig. 4.6. Excess noise spectra as a function of bias.
1021
S 2 (
Ai
1024
1025100
100
, ,0
10"
tr
3a
I
measured:o
theory: this work
......ref.1 33
102
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.
101
leq (A)
103
104
106
83
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 Schottkybarrier
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 109 calculated by Kousik et al. [20] for normal
electronphonon scattering in ntype 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
Schottkybarrier diodes operating in the thermionicemission
limited forwardbias mode. The model is based on mobility and
diffusivity fluctuations occurring in the spacecharge 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
10
leq(I)
leq (3.22MA)
101
10 1 , ,
106 105 104
S(A)
Fig. 4.8. Normalized equivalent noise current versus
bias current.
103
10
85
agreement with our experiments. The experimental value of the
Hooge parameter found from our measurements is 4.2 x 109.
CHAPTER V
SHOT NOISE IN SCHOTTKYBARRIER DIODES
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 spacecharge 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 TED modes. For the TED mode
of transport we calculate the output noise spectral density
of a diode by accounting for noise contributions from the
the metalsemiconductor interface and the spacecharge region.
The noise density of the interface region is obtained by
considering fluctuations in the recombination process. The
noise density of the spacecharge 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 TED mode of transport.
5.2. Noise Contribution from the Interface Region
Consider an ntype 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
by
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 TED mode. According to the principle of
current conservation, the terminal TED 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
88
Under a forward bias condition, we may consider that the
current flowing through the metalsemiconductor 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)
Vd
where
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)
as
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 spacecharge
"lint
ITED 
Rint RSCR
(a)
SCR
SITED
Rint
(b)
Fig. 5.1.
Equivalent noise circuit of
(a) the metalsemiconductor interface
(b) the spacecharge region
RSCR
90
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
Rint
AITED = A int. ( ) (5.8a)
Rint + RSCR
From (3.27) AITED may be rewritten as
vd
AITED (AInt (5.8b)
Vr + Vd
The corresponding output noise spectral density is given by
SITED t(f) = 2qITEDr2 + 4qIr (5.9a)
vd
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 SpaceCharge Region
Even in the TED 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
91
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 FokkerPlanck moments which depend on
the transition and scattering rates [22]. In general we have
SH(x,x',f) = 4q2Dn(x)6(xx') (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
dn(x)
ITED = A(qn(x)pE(x) + qD ) + H(x,t) (5.11)
dx
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 shotnoise 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)
dx
and the fluctuation current
dAn(x)
AITED = A(qAn(x)t0Eo0(x) + qno(x)AioAE(x) +qD ) + H(x,t)
dx
(5.12b)
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 TElimited
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 TED 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 (VDV)
AITED = qAvd[An(w)exp( ) An(0) + exp().
KT KT KT
(1) (2) (3)
w JTED w AE(x) q0o(x)
AE(x) dx + exp( ) dx
o qD J 0 E0(x) KT
(4)
1 [w q0o(x)
+ H(x,t)exp( ) dx (5.13)
Ic J KT
(5)
w qo(x)
where Ic =j exp( ) dx (5.14)
Jo KT
