A STUDY OF THE NOISE OF MICROWAVE SCHOTTKY
BARRIER AND TUNNEL DIODES
By
MICHAEL WAYNE TRIPPE
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1986
TO MY FAMILY
ACKNOWLEDGEMENTS
The author wishes to express his sincere thanks to Dr. A. van der
Ziel, who supplied many ideas and contributions concerning this
research. Special thanks go to Dr. G. Bosman for his daily guidance and
motivation. Without his help this work would not have been possible.
I also wish to thank the other members of the supervisory com
mitteeDr. E.R. Chenette, Dr. P. Kumar and Dr. C.M. Van Vlietfor
their participation. The author also wishes to acknowledge the
assistance of Dr. A.D. Sutherland, who passed away on May 8, 1984.
Finally, the patience of Katie Beard in typing the manuscript is
appreciated.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS........................................................................... iii
ABSTRACT............................................ ........ ......... vi
CHAPTERS
I INTRODUCTION................................................ 1
II QUANTUM NOISE............................................. 3
Introduction......... ...................................... 3
Quantum Noise Theory.................................. ...... 4
Measurement Technique...................................... 11
Fabrication of MOM Tunnel Diodes............................ 18
Alternative Devices: Schottky Barrier
and p n+ Tunnel Diodes.................................. 23
Conclusion.................... ...... ...... ............... 28
III PHYSICAL MODEL OF SCHOTTKY BARRIER DIODES .................... 31
Introduction........................... ........ .... 31
Physical Model.............................................. 32
IV Characteristics of Schottky Barrier Diodes.............. 42
Summary ....................... ......................... 45
IV NOISE OF SCHOTTKY BARRIER DIODES AT LOW
AND INTERMEDIATE FREQUENCIES ............................... 48
Introduction................................................. 48
IntermediateFrequency Theory.............................. 48
Noise Measurement Schemes................................... 51
Discussion of Noise Measurements at
Intermediate Frequencies.................................. 58
Discussion of Excess Noise Measurements...................... 61
Conclusion .......................................... 63
V HIGHFREQUENCY NOISE OF SCHOTTKY BARRIER DIODES.............. 66
Introduction....... ..... ........... .................. ....... 66
Calculation of the Junction Admittance..................... 66
Calculation of the Spectral Intensities...................... 72
Calculation of the Noise Temperature......................... 81
HighFrequency Measurements................................ 88
Measurement Results and Discussion........................... 97
Conclusion............ ...................................... 100
VI PHYSICAL MODEL OF TUNNEL DIODES............................. 103
Introduction...................10........ 103
Physical Model............................................. 104
Heavy Doping Considerations................................. 109
Calculation of IV Characteristic.......................... 121
Measured IV Characteristic................................. 124
Conclusion................................................. 130
VII NOISE IN TUNNEL DIODES..................................... 131
Introduction.................. ............................ 131
LowFrequency Excess Noise.................................. 132
HighFrequency Theory (Shot Noise).......................... 143
Microwave Noise Measurements................................ 143
Discussion of Microwave Measurements........................ 145
Conclusion........... .............................. 150
VIII CONCLUSION AND SUGGESTIONS FOR FURTHER RESEARCH............. 152
Quantum Noise............................................... 152
Schottky Barrier Diodes. ................................. 152
Tunnel Diodes...................................... ..... 154
APPENDICES
A COMPUTER PROGRAM TO CALCULATE THE HIGHFREQUENCY
NOISE OF SCHOTTKY BARRIER DIODES............................. 157
B COMPUTER PROGRAM TO CALCULATE THE 1/f NOISE OF
SCHOTTKY BARRIER DIODES.................................. 163
C COMPUTER PROGRAM TO CONTROL THE HP 3478A
DIGITAL VOLTMETER............... ......................... 167
D COMPUTER PROGRAM TO CALCULATE THE IV AND SHOT
NOISE OF TUNNEL DIODES..................................... 169
REFERENCES....................... ......................... ........... 173
BIOGRAPHICAL SKETCH......................... .................................. 177
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
A STUDY OF THE NOISE OF MICROWAVE SCHOTTKY
BARRIER AND TUNNEL DIODES
By
Michael Wayne Trippe
May 1986
Chairman: A. van der Ziel
Cochairman: G. Bosman
Major Department: Electrical Engineering
A complete DC, AC and noise characterization of both Schottky
barrier and tunnel diodes is presented. Emphasis was placed on the
highfrequency noise properties of these devices.
For the Schottky barrier diodes, highfrequency noise measurements
were performed at 2.2, 12 and 97.5 GHz and are shown to agree well with
the theory. In the theoretical analysis the electrons participating in
the chargetransport process across the barrier are subdivided into four
groups based upon their initial velocity. The contribution of each
group to the device conductance, susceptance and current spectral
intensity was incorporated, including the effect of the transit time.
By taking each of these effects into account, an accurate model which
applies over a wide range of bias and frequency has been developed.
Although the emphasis of this model has been the highfrequency per
formance, it also gives the correct result in the lowfrequency limit.
Noise measurements were performed on germanium pn+ tunnel diodes
at frequencies between 10 Hz and 8 GHz. The lowfrequency excess noise
is found to be of the generationrecombination type in all bias
ranges. This indicates the presence of a large number of localized
energy levels within the bandgap.
At high frequencies, the noise was found to be the full shot noise
of the individual tunneling currents when the diode was operated at low
bias. For highbias operation, less than full shot noise was observed.
This also indicates that the current transport process takes place
through the localized energy levels.
In addition, the effect of zeropoint fluctuations and their con
tribution to thermal noise was considered. Measurement techniques which
compare two thermal sources are shown to be ineffective in determining
whether zeropoint fluctuations exist. The three types of devices
considered were metaloxidemetal tunnel diodes, Schottky barrier diodes
and p+n+ tunnel diodes. For various reasons, in each case it was
impossible to determine whether zeropoint fluctuations exist. A
measurement system which can detect signals with an effective noise
temperature of 0.1 K was demonstrated. This is considered adequate
should suitable devices become available in the future.
CHAPTER I
INTRODUCTION
This work was supported through the Solid State and Microstructures
Engineering Division of the National Science Foundation. Although the
grant title was "Quantum Noise at 90 GHz, and 2 K," the areas of pro
posed research were considerably more broad. Among the original objec
tives were 1) create a noise measurement system which operates in the 90
GHz range and which can detect the noise of a device with a noise tem
perature of 2 K; 2) measure the quantum correction factor of thermal
noise at high frequencies (zeropoint fluctuations); and 3) investigate
the highfrequency properties of MOM (metaloxidemetal) diodes and SBDs
(Schottky barrier diodes). Of the original goals, at least the first
and third were satisfactorily completed.
A measurement system which operates at 97.5 GHz was developed and
tested. It can detect signals with an effective temperature as small as
0.1 K. This is remarkable considering the fact that the background
noise temperature of the measurement system is higher than 1000 K.
The quantum correction factor of thermal noise was never measured.
Detailed analysis of the proposed crosscorrelation measurement scheme
showed that it would not respond to the quantum effect under investiga
tion. Indeed, it will be seen that no detection system which requires a
comparison between two thermal noise sources will be able to determine
whether the zeropoint fluctuations are present. A measurement tech
nique which can detect the presence of zeropoint fluctuations was
developed. This technique requires the use of an active device such as
an MOM diode. The effect of zeropoint fluctuations is present in all
devices. However, certain difficulties arise in the operation of some
devices at high frequencies, obscuring the effects of zeropoint energy
fluctuations, and ultimately the second goal was never achieved.
Chapter II discusses the limitations of the measurement systems and also
the consideration and fabrication of devices.
The device properties of Schottky barrier diodes were studied in
detail. This is the subject of Chapters III, IV and V. Noise measure
ments were made as a function of bias for frequencies between 10 Hz and
97.5 GHz. The lowfrequency noise measurements show that there are no
excess noise sources which extend into the microwave region. At inter
mediate frequencies, the noise is found to be just the shot noise of the
DC current. It will be seen that transittime effects are important
when considering the device operation at high frequencies.
Metaloxidemetal tunnel diodes with oxide thicknesses of 23 A were
fabricated. This is still too large for the devices to give a suitably
low junction resistance. Instead, proprietary samples of microwave
p n+ germanium tunnel diodes were obtained, and their noise properties
as a function of frequency and bias were measured from 10 Hz to 8 GHz.
For these devices an explanation of the noise properties follows from a
consideration of a twostep conduction process through localized energy
levels. To the best of our knowledge, this is the first observation of
this current transport mechanism through the use of noise measurements.
CHAPTER II
QUANTUM NOISE
Introduction
There are two types of noise which are present in thermal equilib
rium. Thermal noise, as its name implies, has thermal energy as its
origin. Quantum noise, although present in thermal equilibrium, comes
about from the noncommuting operators when solving the simple harmonic
oscillator problem. While a classical oscillator would have no kinetic
energy in its lowest state, a quantized oscillator still has some energy
even in its lowest allowed energy state. It is the properties of
thermal noise and especially quantum noise which were investigated. The
credit for first measuring thermal noise goes to J.B. Johnson for his
work of 1928 [1]. Nyquist also made a contribution in the same year by
giving a simple description of Johnson's results [2]. Einstein made a
theoretical treatment as early as 1906 using the theory of random
Brownian motion [3].
Thermal noise is derived from the thermal agitation of charge
carriers in the sample. Although, on the average, the charge carriers
do not have any net velocity in thermal equilibrium, their velocity does
fluctuate. Since the velocity fluctuations have thermal energy as their
origin, this source of noise may be reduced to arbitrarily close to zero
for very low temperatures. This is not true for the quantum noise
fluctuations. These fluctuations are only a function of the operating
frequency and do not depend on the thermodynamic temperature.
It should be noted that it will be tacitly assumed that the zero
point energy fluctuations exist even though the details are still the
subject of some controversy [4]. (In this work the terms quantum noise
and zeropoint energy fluctuations will be used interchangeably.)
The zeropoint energy introduces additional fluctuations into the
system which must be taken into account. It will be seen that these
fluctuations dominate the noise behavior at very high frequencies and
set the absolute limit to the sensitivity of a measurement.
Quantum Noise Theory
The quantum mechanical problem to be considered is a quantized
simple harmonic oscillator. This is useful because it accurately models
a wide variety of physical systems. It is a direct analogy with the
classical simple harmonic oscillator. Equation (2.1) expresses the
allowed energy eigenvalues for this system.
En hf(n +2
En = hf(n +), n = 0, 1, . (2.1)
where n is the level number, h is Planck's constant and f is fre
quency. The term hf/2 represents the zeropoint energy (n = 0). It can
be seen from Figure 2.1 that the lowest possible energy level for this
system is E0 = hf/2. As a result, the lowest energy level is nonzero
[5].
In order to find the average energy of the system, E, a summation
over the distribution of photons in the energy levels must be made.
See, for example, Marcuse [6].
This leads to the result
= hf ( 1 ) (2.2)
ex(hf 2
e ) 1
V(x) = Cx2
E2
= 5/2 hf
= 3/2 hf
= 1/2 hf
Figure 2.1. Energy eigenvalues and potential well
for the simple harmonic oscillator problem.
where T is the thermodynamic temperature and k is Boltzmann's constant.
By making a Taylor series expansion of the exponential term, it is
simple to show that for hf << kT
E = kT. (2.3)
This is the wellknown result for low frequencies.
It is useful in many cases to describe the noise of a system in
terms of its equivalent noise temperature. From equation (2.2) it
follows that
E hf/k hf
T hf/k + = T + T (2.4)
n k exphf 2k B ZPE
exp(k) 1
where TB represents the thermal contribution and TZPE is the effective
noise temperature due to the zeropoint fluctuations. In Figure 2.2 Tn
and its various components are shown as a function of the thermodynamic
temperature. This illustrates that for high frequencies TB is less than
T, that the zeropoint term is a constant with respect to temperature
and that the total noise temperature, Tn, is always greater than or
equal to the thermodynamic temperature.
The noise temperature as a function of the operating frequency is
presented in Figure 2.3. This illustrates that for low frequencies the
noise temperature is equal to the thermodynamic temperature. For higher
operating frequencies the thermal contribution falls off. In the same
frequency range where TB has begun to decrease, TZPE has increased to a
significant level. In this manner the total noise temperatures a con
tinuously increasing function of the operating frequency.
10
9
8
7
6
(K)
TZPE
01 2 3 4 5 6 7 8 910
T (K)
Figure 2.2.
Noise temperature vs. T for an
operating frequency of 100 GHz.
1000
Tn
(K)
100
10
7
3 T
2
23
1 I 
108
Figure 2.3.
10
f (Hz)
f (Hz)
Noise temperature as a function of frequency with
the thermodynamic temperature as a parameter.
Solid curves are the total noise temperature, Tn.
Dashed curves are the thermal contribution, T .
The solid line represents TZPE.
At any fixed operating frequency the difference between Tn and TB
is always given by TZPE. However, its relative importance is seen only
at low temperatures for an operating frequency of 100 GHz. At optical
frequencies the effect of the zeropoint energy completely dominates the
noise behavior, even for high temperatures.
A main point of discussion concerning the zeropoint energy is
whether or not it can be extracted from a system. The most general con
clusion seems to be that it is not exchangeable, but yet has certain
properties which allow its presence to be substantiated by measurements.
In order to illustrate this point, consider an electron which is in
oscillatory motion.
It must be emphasized that the zeropoint energy level correspond
ing to Eg is the average energy of the particle. This is the sum of the
kinetic energy and potential energy. It is possible for the velocity to
fluctuate by converting kinetic energy to potential energy and vice
versa. In this way the fluctuations may be detected.
This is in close analogy to the case of two matched resistors, in
equilibrium, operated at low frequencies. In this situation the elec
trons in the resistors give noise through velocity fluctuations, even
though they have zero net velocity.
The noise of one resistor creates a signal across the other and
represents the flow of power. This cannot continue unless the inverse
process is also taking place. Noise power must flow back from the
second resistor at an equal rate. This is merely another description of
the term thermal equilibrium.
In order for the zeropoint energy to be extractable from a system,
there would need to be an energy level lower than E0. Since, by
definition, EO is the lowest possible energy level, there is no way for
the system to give up energy (on the average).
Historically, the presence of the zeropoint energy was first seen
by Mulliken. The following quotation from French and Taylor describes
his work.
The study of molecular spectra led to the verification of
the existence of zeropoint energy. In fact, the need to
describe the permitted vibrational energy levels by odd
multiples of 1/2 h . rather than by integral mul
tiples of hw0 (whicR are the even multiples of 1/2 h )
0 0
was actually inferred from molecular spectra before wave
mechanics had been invented. This was done by R.S.
Mulliken in 1924 [see R.S. Mulliken, Nature 114, 350
(1924)]. Although the analysis was rather complicated in
details, it depended in essence on comparing the vibra
tional energy levels for two molecules of the same chemi
cal type but involving different isotopes. . Mulliken
was able to show that the inclusion of the zeropoint
vibrated energy 1/2 hw at the base of the energy
structure leads to a good fit with the spectral data,
whereas without it there is a small but significant
discrepancy. This is a particularly interesting and
impressive piece of analysis: one might be tempted to
reason (incorrectly) that, since a particle in the ground
state can never radiate away its zeropoint energy,
therefore the existence of the zeropoint energy cannot be
verified by experiment. Mulliken's work showed that the
zeropoint energy has observable consequences. [5, p. 173]
Another experimental verification came about through the work of
M.J. Sparnaay. These unusual measurements were on the force between two
closely spaced parallel metal plates at low temperatures. An electric
field between the two conductors induces eddy currents and causes a
force to exist on the plates. This is known as the Casimir effect. It
is the force due to the fluctuating zeropoint field which Sparnaay
measured and which is described by Boyer [7].
At least one other observation of the zeropoint energy has been
made. In 1981 Koch, Harlingen and Clarke reported on measurements using
a resistively shunted Josephson junction [8,9]. This technique made use
of the AC Josephson effect whereby the application of DC voltage to the
junction causes the production of a very high frequency oscillation (up
to 450 GHz). This highfrequency signal then mixes with the noise of
the resistive shunt and converts it down to the 100 kHz IF.
Their measurements clearly showed the influence of zeropoint
fluctuations at high frequencies. At low frequencies the noise was just
the wellknown thermal noise of the resistance.
Measurement Technique
It was the original intent of this project to measure the noise of
a matched termination at 2 K and 97 GHz using a very low noise (cross
correlation) receiver system. When it became evident that this method
would not work [10], it was decided that the noise in MOM tunneling
junctions should be pursued instead. The proposed slow modulation
scheme for use with MOM diodes has been described in a paper by Van der
Ziel and Sutherland [11]. The several measurement schemes which were
considered will be briefly discussed.
The block diagram of the crosscorrelation system is shown in
Figure 2.4. It is the resistive termination of the power divider which
causes the measurement system to fail in the determination of the zero
point energy term.
As shown in equation (2) of the reference, the output reading, S,
of the system is
S = Tk[Ts T ]Af (2.5)
where T is the coupling between ports (= 1//2) Ts is the total noise
temperature of the device, TO is the total noise temperature of the
internal termination, k is Boltzmann's constant and Af is the
ISOLATOR
DEWAR
I1
MATCHED
TERMINATION
MIXER
Figure 2.4.
Simplified block diagram of the originally
proposed crosscorrelation receiver.
measurement bandwidth. In the general case Ts and TO will consist of
the sum of two terms: a thermal contribution and the zeropoint energy
contribution. This allows the total noise temperatures to be expressed
as
T = TB + TE (2.6)
S SB ZPE
T = TB + T ZpE. (2.7)
0 OB ZPE
It should again be emphasized that TZPE does not depend on the
thermodynamic temperature. It is strictly a function of the operating
frequency.
Now it is clear that the contributions from the zeropoint energy
exactly cancel in equation (2.5) and thus the meter reading does not
depend on the value of TZPE. Therefore, the presence or absence of the
zeropoint energy term cannot be detected by this method.
This is not to say that the system does not offer some advantages
over a singlechannel receiver. It still responds with a better noise
performance than a standard receiver. Crosscorrelation techniques are
used extensively in radio astronomy in order to separate the very faint
signals from the stars from the much larger background noise of the
detection system.
In a similar way it may be shown that the microwave noisemeasuring
system of Chapter V also does not respond to the zeropoint energy.
This method also involves the difference between two noise temperatures.
In a more general sense, any noise measurement scheme which involves
taking the difference (or making a comparison) between two noise
temperatures will fail to determine whether the zeropoint energy term
is present.
The use of an MOM diode was considered to be a strong tool for
measuring the zeropoint energy. The measurement technique makes use of
the fact that for large enough bias the noise should be the full shot
noise of the DC current. This provides a calibration directly from the
device itself and does not require a comparison to an external noise
source.
A simplistic band diagram and IV characteristic of an MOM tunnel
diode is shown in Figures 2.5a and 2.5b respectively. The thickness of
the oxide region must be less than approximately 100 A in order for
significant tunneling to occur. At low bias voltages there are two
tunneling currents which flow. In equilibrium they are equal in magni
tude but opposite in direction. Lecoy investigated the noise properties
of this type of device and found the noise to be full shot noise of the
individual currents [12]. In equilibrium the thermal noise of the
device may be shown to exactly equal the full shot noise of the two
tunneling currents.
Sanchez et al. have given the zerobias resistance of MOM diodes as
[13]
S exp(S) (2.8)
R = S l (2.8)
R0 324 00A
where
S = 1.025 L1/2
A = junction area in um2
0 = barrier height in eV
and
L = oxide thickness in A.
METAL OXIDE METAL
(a)
(b)
Figure 2.5. Important characteristics of MOM diodes.
a) Band diagram. b) Currentvoltage
characteristic
__ L [4
If a diode resistance of 450 ohms is desired (approximately the
characteristic impedance of WR10 waveguide at 100 GHz), then
12 *
for (0 = 1 eV, L = 10 A, a junction area of 2 x 10 m2 is required.
Since the MOM diode operates by quantum mechanical tunneling, the
spectrum of the shot noise should be white. This is due to the very
short transit time of a tunneling electron.
The experiment may be described with the aid of Figure 2.6. This
shows the total spectral intensity which is measured as a function of
the bias voltage. The technique requires that the bias voltage of the
diode be very slowly swept from positive to negative values. This gives
a timevarying signal at the output of the system, which may be
detected. In order to distinguish the quantum effects, it is necessary
for the bias to change by an amount on the order of V P hf/q. For an
operating frequency of 1011 Hz this is approximately 1 mV.
The signal at the output of the system will consist of two compo
nents: a DC contribution, SO, due to the background noise of the
measurement system, and an AC component, AS(t), due to the modulated
noise of the diode. By using a very large series blocking capacitor,
the DC component may be eliminated and only the desired AC signal will
remain.
In order to test the validity of the slow modulation technique, the
measurement system of Figure 2.7a was implemented. The measurement
system works in the following manner. The isolator at the input reduces
the possibility of reflected receiver noise. A SSB filter is used to
select only the upper sideband response of the mixer. The mixer, local
oscillator and first IF amplifier are all supplied as a custom design
from Hughes. The thermistor and power meter respond to the 24 GHz IF.
SHOT NOISE
ASYMPTOTE
S
THERMAL
THERMALNO ZPE ,/ AS(t)
NOISE
WITH \ /
ZPE So
SO
e ; RECEIVER
'( BACKGROUND
1 0 1
V (mV)
Figure 2.6. Spectral intensities for the MOM measurement.
The recorder output signal from the power meter is passed through a DC
block and lowpass filter, and the 0.2 Hz signal is finally detected
using the HP 3582A FFT spectrum analyzer. Convenient units of measure
for the spectrum analyzer are dBV. That is, a voltage proportional to
power (noise temperature) is produced by the 432B power meter.
In order to verify that the system had the required sensitivity, a
test signal using the arrangement of Figure 2.7b was measured. The
waveguide termination is kept at temperature T, which for the example
shown is 77 K. The ferrite modulator is an electrically controllable
switch which has two states. In the lowloss state the modulator passes
signals with only a small attenuation (~ 2 dB). The highloss state
gives an attenuation of greater than 20 dB. In the highloss state the
noise temperature out of the system is equal to the ambient temperature
of the modulator. Both the ferrite modulator and calibrated attenuator
are at room temperature. The calibrated attenuator is used to set the
signal level.
Results of the measurements are plotted in Figure 2.8 and show that
the system is able to detect signals on the order of 0.1 K. The system
is very linear for signals of 0.5 K and larger. These measurements
indicate that the system is able to accurately detect very small changes
in noise temperature, even though the receiver noise temperature is on
the order of 1000 K. This sensitivity is due to the large bandwidth
(" 2 GHz) and long integration time (m 1000 seconds).
Fabrication of MOM Tunnel Diodes
Many sources have reported the fabrication of MOMtype structures
[14,15]. The techniques are similar to those used to construct
Josephson junctions [16,17]. In most early cases the device geometry
AMPLICA
DC
432B BLOCK
ISOLATOR
24 GHz
94 GHz
3582A
0.2 Hz
REF
PLANE
II
CALIBRATEDI
ATTENUATOR
(b)
Figure 2.7.
Millimeter wavenoise measurement system.
a) Block diagram of the receiver channel.
b) Block diagram of the system used to create
a test signal in order to measure
the receiver sensitivity.
REF
PLANE
I
I
FILTER
zo
AT T
50
60
70 
METER
READING
80 
(dBV)
90
100
110
0.1
1.0
ATeff
Figure 2 8.
Output meter reading vs. effective input
noise temperature for the noise measurement
system of Figure 2.7a.
10.0
(K)
was quite large, and the zerobias impedance was often 106 ohms or
greater [18]. This indicates the need to reduce the oxide thickness of
the device as well as the junction area. It is necessary to stay within
a factor of seven of the characteristic impedance of the waveguide in
order to be able to match to the device.
For the patterning of the required fine metal lines, the liftoff
technique was chosen [19,20]. The processing steps are outlined below.
1) Clean the substrates. Boil for 5 minutes in TCE, Acetone and
Methanol.
2) Rinse in DI water.
3) Spin dry.
4) Dry bake at 1800C for 20 minutes.
5) Spin on photoresist at 2000 rpm for 20 seconds if using
AZ 1470.
6) Softbake the photoresist for 20 minutes at 950C.
7) Allow substrates to cool for approximately 5 minutes.
8) Expose the photoresist (9 seconds for the Karl Suss mask
aligner).
9) Develop the photoresist pattern by immersing the sample for 60
seconds in MP315 developer.
10) Rinse in DI water.
11) Spin dry.
12) Place samples in evaporator and evacuate to 105 Torr.
13) Coat samples with aluminum to 2000 A thickness. Evaporation
should be carried out in as near normal incidence as possible
to aid liftoff.
14) Remove slides and soak in acetone to remove the unwanted metal.
The sequence of steps is repeated for the counter electrode after
the growth of the oxide has been completed. Several methods were inves
tigated for the control of oxide growth.
Plasma oxidation in a commercial rf photoresist stripper always
yielded very thick oxides regardless of the oxidation time [18]. This
was unsuitable for our application. In fact, the main problem was to
limit the growth of oxide. Even the thermally grown oxide which
resulted from exposure to atmospheric air during processing for the
counter electrode was sufficiently thick to give a high impedance
junction.
A DC sputtering system was constructed to try and remove the
natural oxide just prior to evaporating the counter electrode. This
still did not provide lowresistance junctions, possibly due to poor
electrical connection to the devices. Next the evaporation system was
modified to perform RF sputtering. This method normally works even for
sputtering dielectrics and does not require an electrical connection to
the device. Even RF sputtering did not yield lowresistance devices.
One possible explanation is that the background pressure of oxygen may
have been too high. Argon was used as the main sputtering gas.
According to Chapman the monolayer formation time for a gas at a
pressure of 106 Torr is approximately 1 second [21]. A contaminant gas
at this pressure would be able to form a monolayer in this time. A
typical value of the best vacuum achieved in our system was 2 x 106
Torr. This is near the specification of the evaporation system and is
adequate for most metal depositions. It is not sufficient for use in a
sputtering system when control of the oxide thickness must be within a
monolayer.
The best fabrication method resulted in junctions with a zerobias
resistance of 350 kS. In this technique the first electrode was pro
cessed in the usual manner. The photoresist was prepared as described
for the second electrode, but immediately prior to placing the samples
into the evaporator they were immersed in a dilute (3:1) solution of
aluminum etchant. This etchant removes any oxide and also the first
layers of aluminum, and should give a clean, oxidefree surface. The
samples were loaded and the system pumped down in the shortest possible
time. The IV characteristic which resulted is shown in Figure 2.9.
It is seen that the characteristic has the desired shape except for
the high value of zerobiased resistance. The characteristic is linear
near zero bias and antisymmetrical. Using equation (2.8) and assuming
i0 = 1 eV, the oxide thickness is determined to be 23 A. More typical
overall values of oxide thickness were on the order of 100 A.
Even the best device which was fabricated is not good enough for
use in the 450ohm waveguide system since it had such a large
impedance. The impedance mismatch in this case would only allow, at
most, 0.5% of the noise signal from the device to be transferred to the
system. This is unacceptable.
Alternative Devices: Schottky Barrier and p n+ Tunnel Diodes
The lack of adequate devices was by this time a major obstruction
to progress. Two other devices, p+n+ tunnel diodes and Schottky
barrier diodes, were selected as possible alternatives to use in the
100 GHz measurements.
Commercial Schottky barrier detector diodes (Hughes model
47316H1111) were available already mounted in a waveguide holder.
These are considered to be the finest detectors available for 100 GHz
operation. The mount structure is illustrated in Figure 2.10.
4
3
2
I(pA) 1
0
1
2
3
4
5
543 2 1 0 1 2 3 4 5
S4)
V (10 V)
Figure 2.9. Measured IV characteristic of the MOM
diode with the thinnest oxide barrier.
p
0
*
" *
*
* RO== 350 kQ
 e
 *
 *
II
nS
n0
B0
0 RO 35Okfl
SI I I II I I I
FULL
HEIGHT
INPUT
LINEAR TAPER
TO 1/4 HEIGHT
DIODE
TUNABLE
SHORT
~I
SSMA
CONNECTOR
Figure 2.10. Hughes Schottky barrier detector mount.
Noise measurements were made at 97.5 GHz on this device and gave
the surprising result that the noise temperature did not change as a
function of bias. This result was unusual since at low frequencies the
noise temperature is found to quickly drop to mT/2 where m is the non
ideality factor of the diode and T is the ambient temperature. Further
study led to the explanation of this result through a transittime model
of the device. The complete theoretical model and its verification
through noise measurements are presented in Chapters III through V.
The result of attempts to apply Schottky barrier diodes to the
problem of measuring the zeropoint energy was the conclusion that the
device was unsuitable This is due to the dominance of the group of
returning electrons for low bias, as will be shown.
As a final consideration pna tunnel diodes were measured. The
main advantage of p n semiconductor tunnel diodes is that the zero
bias resistance is near the characteristic impedance of the waveguide
system. Typically, RO is in the range 50 150 ohms.
A fourstep Tchebycheff waveguide transformer was designed follow
ing the work of Cohn [22]. The transformer dimensions are given in
Figure 2.11. The waveguide transformer was made from goldplated brass
in splitblock style.
Even though the transformer was fabricated with the greatest care,
it did not operate satisfactorily at 97.5 GHz. Losses due to the
increased surface roughness from machining were found to be on the order
of 1.5 dB.
A better solution was to use the commercial Hughes detector mount
and replace the Schottky barrier diode with the tunnel diode chip.
LENGTH = 38.1
Fourstep Tchebycheff transformer for use with
p+n+ tunnel diodes. All dimensions are in mils.
7.0
14.9
9.7
2.4
S15.8
Figure 2.11.
The proprietary tunnel diode samples were supplied through TRW and
are designated as type TD153. These are 15 x 15 mil. chips of
germanium and have goldplated contact areas.
The measured noise data are presented in Figure 2.12. It is seen
that the measurements do not agree with the theory. This is most
probably due to the large junction capacitance of these diodes. The
heavy doping and narrow spacecharge region required to produce
tunneling cause the junction capacitance to be large. This capacitance
was approximately 0.2 pF for the diodes which were measured. At an
operating frequency of 100 GHz this gives a junction reactance
of 8 a and effectively shorts out the junction.
Conclusion
A theoretical overview of the quantum effect which it was desired
to measure has been presented. For various reasons it was not possible
to actually measure this effect.
It has been shown that any comparative type of system cannot detect
the presence of zeropoint fluctuations. An alternative method sug
gested by Van der Ziel has merits but requires an active device. The
three types of devices considered were MOM tunnel diodes, p n semi
conductor tunnel diodes and Schottky barrier diodes. Each had the
promise of allowing the measurement of the quantum noise, but none were
actually suitable. This made clear, especially for the p+n+ diodes and
Schottky barrier diodes, that the device operation and description for
high frequencies was not well understood.
The physical models, theories and measurements which were developed
for Schottky barriers and tunnel diodes are the subject of the remainder
of this work. It is hoped that this will prove to be a useful
400
DUT
(K)
300
200
o
30
20 10
0 10 20 30
BIAS VOLTAGE (mV)
Figure 2.12. Measured and theoretical noise
temperatures for the p+n tunnel
diodes at a frequency of 97.5 GHz.
30
contribution to the description of the highfrequency behavior of these
devices.
A measurement system which can detect changes in noise temperature
on the order of 0.1 K was constructed and its operation at 97.5 GHz
demonstrated. This technique is considered adequate should suitable
devices become available in the future.
CHAPTER III
PHYSICAL MODEL OF SCHOTTKY BARRIER DIODES
Introduction
Schottky barrier diodes have become increasingly important due to
their excellent highfrequency properties. They are in widespread use
in the mixing and direct detection of signals at frequencies up to
several hundred gigahertz [23]. Since these devices are being operated
at such high frequencies, it is necessary to have a model which is
capable of describing the device behavior on time scales on the order of
the transit time of electrons across the junction. This dissertation
presents such a model for silicon Schottky barrier diodes and compares
the theory with experimental results.
Chapter III presents background information and introduces the
physical description necessary to create a complete device model. The
main emphasis is to include transit time effects. In addition, the
measured DC characteristics are discussed.
Chapter IV deals with the noise behavior of the device at low and
intermediate frequencies. In order to show that the results of Chap
ter V are due to transit time effects, it was necessary to show that the
diode behaves in the wellknown manner at intermediate frequencies.
For the lowest frequencies the excess noise is dominant. The
excess noise is important since it can be upconverted to microwave
frequencies. This may limit the usefulness of some diodes when operated
as a mixer with a lowfrequency IF.
At intermediate frequencies, above the excess noise corner
frequency, the spectral intensity is equal to the shot noise of the DC
current. The wellknown shot noise theory is adequate in this range.
Chapter V gives a complete model of the Schottky barrier diode
which includes transittime effects. This is necessary in order to
describe the properties of the diode at microwave frequencies. Very
good agreement was found between theory and experiment for the high
frequency case.
Physical Model
In their basic form Schottky barrier diodes have a very simple
configuration. This is depicted in Figure 3.1. A small metal anode
contact (e.g. gold) is deposited on top of a semiconducting wafer (e.g.
Si or GaAs). A largearea ohmic back contact is used in order to lower
the series resistance of the bulk semiconductor.
In spite of the simple configuration of Schottky barrier diodes,
the physical mechanisms which govern their operation are complex. The
importance of each mechanism may also vary with frequency and bias.
There are several current transport processes which may be present
for the electrons traveling from the semiconductor to the metal. It is
necessary to distinguish between the possibilities of tunneling,
thermionic emission, diffusion and recombination of electrons in the
spacecharge region. Justification will be given for the neglect or
consideration of each form of current transport.
The device description starts with a consideration of the band
diagram, Figure 3.2. The metal on the nsemiconductor gives an energy
barrier of q ms. The value of q'ms is often calculated for metal
semiconductor systems by considering the work functions, electron
METAL
7, A
ANODE
DEPLETION
REGION
CONTACT
Figure 3.1.
Configuration of a simple Schottky
barrier diode configuration
nSEMICONDUCTOR
BACK
/ 0/////////////////
   
nSE
nSE
Figure 3.2.
Q
0
0
qVdif
 EC
EF
MICONDUCTOR
Band diagram of the metal
semiconductor contact.
Ims
q~ms
METAL
affinities and doping densities of the materials. This calculated value
may be of little use as the actual barrier height also depends on the
number of interface states, and this number may not be well controlled.
In practice the actual barrier height is determined from IV measure
ments, once the reverse saturation current is known. This eliminates
the need for assumptions concerning the metalsemiconductor barrier
height. It will be assumed that the Schottky barrier height remains
constant with bias.
The difference in energy between the conduction band and Fermi
level (far from the junction region) is qVn. This is governed by the
doping density in the semiconductor material and may be found from [24]
N
Vn = VT n(C) (3.1)
where VT is the thermal voltage (= kT/q), NC is the effective density of
states of the conduction band and ND is the doping density.
Once qOms and qVn have been determined, then the diffusion
potential, Vdif, may easily be found:
V = V (3.2)
dif ms n
This will be a convenient quantity for many calculations.
In this work the depletion approximation is applied such that in
the depletion region the spacecharge density is the same as the doping
density of the semiconductor. This is illustrated in Figure 3.3a. Once
established that the spacecharge density, p, is constant in the junc
tion region, then it follows immediately from Poisson's law that the
electric field varies linearly:
qND 
EMAX
Figure 3.3.
Device model in the junction region.
a) Fixed spacecharge density as a
function of position;
b) Electric field variation with
position.
d (eE) = p (3.3)
dx
By imposing the boundary condition that the field is zero in the
semiconductor, then equation 3.3 is solved to yield
E(x) = E 1) (3.4)
This may be integrated once to find the potential. In the junction
region the potential is parabolic and, thus, so is the bending of the
conduction band, as already indicated in Figure 3.2.
In order to describe the junction behavior, the current transport
properties must be identified. Although many exceptions are possible,
some general guidelines can be given.
For extremely high doping the semiconductor will become degenerate,
and the depleted region will become thin enough for quantum mechanical
tunneling to occur. This process is not believed to be significant in
the diodes which were measured, as was discerned from the IV character
istics and noise properties. Diodes which have significant tunneling
show a large nonideality factor. For the diode under consideration the
nonideality factor was 1.09, and this will be discussed in detail in
the section on IV measurements. More importantly, since tunneling is a
very fast phenomenon, there should be little variation of the device
parameters and noise with frequency. The measured results show a
significant change of both the noise and admittance as the frequency is
increased.
In the case of very low doping the current transport is dominated
by diffusion processes. Since low doping causes the depletion region of
the semiconductor to be large, the possibility of an electron scattering
in the junction region is high. Once the number of electron collisions
becomes large, then the time required for the current to respond to an
applied signal increases. A device operating in this mode would be
unsuitable for use at microwave frequencies. The silicon Schottky
barrier which was used in this work is a commercially available detector
which operates to 110 GHz. Therefore, the diffusion limited process is
discounted.
Another possible current mechanism is the recombination of
electrons in the spacecharge region. This mechanism is characterized by
a nonideality factor of two. As previously mentioned, a nonideality
factor of 1.09 was observed. This allows one to neglect the possibility
of recombination current for the bias range involved.
Lastly, there is the possibility of thermionic emission of
electrons over the metalsemiconductor barrier. Since it is believed
that this mechanism controls the device behavior, it will be examined in
detail. In order for thermionic emission to dominate the DC current,
the depletion region width must lie within a range of values. That is,
the doping density should not be so low that significant scattering
could occur in the spacecharge region. In this work it will be assumed
that electrons can cross the spacecharge region without suffering any
scatterings. Conversely, the doping should not be high enough to allow
the tunneling of electrons across the resultant narrow spacecharge
region.
According to Sze [24], the DC current flow will be due to
thermionic emission if the electric field is between 104 V/cm and 105
V/cm in silicon diodes. This is the case for the assumed doping density
of 1017 cm3.
The process of thermionic emission is the thermal excitation of
charge carriers to a sufficient kinetic energy such that they may cross
the energy barrier of the junction. Since the bottom of the conduction
band represents zero kinetic energy, electrons in the semiconductor must
have at least an energy of qVdif in the negative xdirection in order to
reach the metal contact and be collected.
The electrons of the semiconductor participating in the charge
transport can be subdivided into three groups. See Figure 3.2. The
physical mechanisms involved will be presented in this section, but a
detailed mathematical description will be delayed until Chapters IV and
V.
Electrons of group 1 have insufficient kinetic energy in the
negative xdirection (KE < qVdif). See Figure 3.2. These electrons
always return to the neutral semiconductor and give no contribution to
the DC current. It will be shown in Chapter V that these electrons have
the same characteristic time of flight regardless of their initial
velocity. At low frequencies these electrons give no contribution to
the device conductance. As the operating frequency increases, the
conductance due to these electrons rises as w2 and dominates the overall
device conductance. It will be shown that this group of electrons
controls the highfrequency behavior of the device at low biases. The
conductance and noise of this group of electrons does not change
significantly with bias. Most electrons of the semiconductor occupy
states near the bottom of the conduction band. A change in the barrier
height due to an applied voltage will not cause any appreciable change
in the number of electrons in this group.
The electrons of group 3 have sufficient kinetic energy to just
reach the metal contact (KE = qVdif). If no signal were applied, then
these electrons would always be collected at the metal contact. With an
applied smallsignal voltage some electrons of this group will be
collected while others will return to the neutral semiconductor. For
this group of electrons dI/dV is large. At intermediate frequencies
this group of electrons dominates the device conductance, even though
only a fraction of the DC current is carried by them. The conductance
due to this group of electrons is constant up to frequencies on the
order of the reciprocal transit time and decreases rapidly at higher
frequencies. The intermediate frequency conductance of this group of
electrons is given by the slope of the IV characteristic.
The electrons of the semiconductor which have sufficient velocity
(kinetic energy) to always pass the energy barrier are designated as
belonging to group 2, as shown in Figure 3.2. Since the electrons of
this group are always collected, they give rise to the DC current. With
an applied signal the velocity of the electrons during the transit time
is slightly modulated. This is a small effect and the quantity dI/dV is
approximately zero for this group. The contribution to the conductance
from this group of electrons may be neglected for all frequencies of
interest.
In order to calculate the noise temperature of the device, the
spectral intensity of the current pulse due to each electron must first
be found. From the equations of motion the velocity of an electron as a
function of time may be found for an electron of any group. This
velocity relates directly to the current flowing,
i(t) v(t) (3.5)
d
Once the current flowing as a function of time has been determined.
it is only necessary to take the Fourier transform in order to find the
spectral intensity of the current as a function of frequency. See
Chapter V.
For the electrons of group 1 the spectral intensity will be seen to
start at zero for zero frequency (no DC current) and then to increase
as w2. This spectral intensity is not a strong function of bias since
the number of electrons in group 1 does not change significantly with
bias.
Group 3 is a special case of group 2 under the assumption of no
applied signal. The same description will hold for both in calculating
the spectral intensity of the noise. These electrons carry the DC
current and, since the electrons are emitted at random times, give an
intermediate frequency spectral intensity equal to the shot noise of the
DC current. The spectral intensity is white up to frequencies on the
order of the reciprocal transit time and drops steeply at higher fre
quencies. This noise component varies linearly with the device current
(exponentially with voltage). At high bias the passing electrons will
dominate.
One additional current must be included in order to form a complete
model of the device. This is the reverse saturation current. The
reverse saturation current consists of those electrons in the metal
which have sufficient kinetic energy in the positive xdirection to
cross the energy barrier of q ms Once the electrons have crossed the
energy barrier, they are swept by the electric field to the neutral
semiconductor. Since the electric field causes the electrons to be
accelerated to a very high velocity, their transit time is extremely
short. For this reason the spectral intensity due to these electrons is
considered white and equal to the shot noise of the DC current. The
energy barrier qOms is not a function of bias in this model. This
allows the magnitude of the reverse saturation current to be considered
constant.
IV Characteristics of Schottky Barrier Diodes
The current transport mechanism which will be considered is the
thermionic emission of electrons. The results of this model have been
known for many years [25]. Some background information will be
presented in order to show the assumptions invoked, but the reader is
referred to the references for complete details.
The calculation of the IV characteristic is simplified by the
realization that only electrons of groups 2 and 3 contribute to the DC
current. The reverse saturation current is considered constant in the
present model. Thus, it is only necessary to find the number of
electrons traveling from the semiconductor to the metal perunit time.
This current is a function of the applied bias voltage. The description
to be used is that of Van der Ziel [26], although the notation has been
changed slightly.
As already mentioned, only electrons with a kinetic energy of
greater than qVdif directed in the negative xdirection can escape the
semiconductor. The electrons lose kinetic energy in the amount of qVdif
in doing work against the builtin electric field. Conservation of
energy requires that
Sv2 + 1 mv2 (3.6)
2 x dif 2 x
where the unprimed velocity is the velocity of the electron upon arrival
at the metal contact, and the primed velocity is the electron velocity
upon departure from the semiconductor.
In addition to the conservation of energy, it is also necessary to
describe the distribution of electrons in velocity. By summing over all
possible electron velocities, the total current may be obtained. The
number of electrons with velocities that lie in the velocity space
between v' and v' + Av', v' and v' + Av', and v' and v' + Av' is given
x x x y y y z z z
by ref. [26]:
22 E m2 1 mv2 1 ,I2 qV
2m2 F 2 x my qV
AN = exp( x 2 2 )AvAvA
h3 kT x y z
(3.7)
It is assumed that the semiconductor is nondegenerate so that Maxwell
Boltzmann statistics apply.
The actual number of electrons which arrive at the edge of the
spacecharge region is found as follows. Consider a volume which has a
crosssectional area of 1 m2 in the yz plane. All electrons which
have a velocity of v' will arrive at the surface in one unit of time
x
(one second). Thus the number of arriving electrons per unit time per
square meter which have a velocity in the prescribed range is given by
An = v'AN (3.8)
x
Since each electron carries a charge of q, the DC current which is
flowing is
AId = qAn(Area) .
(3.9)
The total current Id is found by summation over all possible velocity
intervals. Carrying out the integration yields the familiar Richardson
equation,
V s
Id = Area(4hqmkr)T2 exp( ms) (3.10)
h3 T
The expression in parentheses is the Richardson constant, A, which
assumes an electron mass equal to the free mass. For semiconductors the
actual mass should be replaced by the effective electron mass, and this
revised constant is termed A For free electrons the Richardson
constant is 120 A/cm2/K2. The effective Richardson constant may also be
a function of the electric field strength. This is designated by A .
A detailed calculation by Andrews and Lepselter [27] shows that an
average value of effective Richardson constant A** equal to
110 A/cm2/K2 is appropriate for silicon doped to 1016 cm3. This value
of effective Richardson constant was used in all calculations reported
here.
An effect which has been omitted in the calculations is the image
lowering due to the Schottky effect. The image lowering causes a
reduction of the barrier height as a function of the applied electric
field (voltages). The barrier height may no longer be considered
constant, but its variation with bias must also be taken into account.
The variations from the ideal case are adequately modeled by
including a nonideality factor m in the expression for the diode
current. A simplified expression of the diode current is
= I(exp( 1) (3.11)
T
where
8
IO = (Area)A**T2 exp( ) (3.12)
0 V VT
Note that the constant reverse saturation current has now been included.
Schneider has reported a model in which the nonideality factor influ
ences the reverse saturation current [28].
The measured IV characteristic of the Hughes model 47316H1111
silicon Schottky barrier diode detector is shown in Figure 3.4. All
measurements were performed at 300K. The slope of the line indicates a
nonideality factor m equal to 1.09. By extrapolating to zero bias the
8
reverse saturation current is found to equal 3 x 10 A. Assuming a
junction diameter of 2 jm, the actual Schottky barrier height is deter
mined to be 0.585 eV.
The saturation current and Schottky barrier height are important
parameters in the device model. Another important parameter which is a
parasitic effect is the series resistance. This plays an important role
at highbias currents for both the admittance and noise. The value of
the series resistance is determined from the deviation of the measured
IV curve from a true exponential curve at high bias. For a current of
2 mA the deviation is 20 mV. This indicates that the series resistance
is 10 ohms.
The device parameters determined in this section will be used in
the description of the noise properties. Unless otherwise stated, the
measured values were used without modification.
Summary
A physical model of the operation of Schottky barrier diodes has
been presented. Equations have purposely been kept to a minimum.
I(A)
0 100 200 300 400 500 600
V (mV)
Figure 3.4. IV characteristic of Hughes
diode at 300K.
47
It has been pointed out that for forward bias the three groups of
electrons traveling from the semiconductor towards the metal are
important. The role of each group of electrons varies with frequency
and bias.
CHAPTER IV
NOISE OF SCHOTTKY BARRIER DIODES AT LOW
AND INTERMEDIATE FREQUENCIES
Introduction
This chapter gives a brief review of the wellknown intermediate
frequency theory of the noise in Schottky barrier diodes. Results of
measurements at intermediate frequency are presented, and these are seen
to agree with the standard theory.
IntermediateFrequency Theory
The intermediatefrequency noise theory has been developed for some
time [29]. The basic starting point of an analysis is the DC current,
which one assumes can be described in the form
Id = I exp(Vdif/mVT) exp(Vd IdRs/mVT) (4.1)
where Id is the diode current, I is the saturation current, Vd is the
applied diode voltage, Vdif is the diffusion potential, VT = kT/q is the
thermal voltage and m is the diode nonideality factor. At low bias it
is common to neglect the effect of the series resistance, R This
equation is valid if the bias voltage Vd is approximately five times
larger than mVT, in order that the reverse saturation current may be
neglected. It should be noted that in this work Is does not contain the
term exp(Vdif/mVT), as it does in many other representations, and thus
Is is much greater than Id.
Assuming that the current Id gives full shot noise and that the
device conductance is given by the derivative of the IV relationship
when evaluated at the operating point, it follows directly that the
noise temperature is
SI(f) = 4kTnd = 2qld (4.2)
Tn = 2q Id/4k gd = mT/2 (4.3)
This is the wellknown intermediatefrequency result and is commonly
used. Note that it does not depend on bias under the stated assump
tions.
The spectral intensity of the shot noise (intermediatefrequency
noise) which is measured is the thermal noise from an equivalent
resistor of equal value. It is a common misconception that these
devices will show both thermal noise and shot noise when forward
biased. In fact, the shot noise is the actual measured quantity. Even
though the device has a known impedance, it is not a passive resistor at
300K, but rather it is an active device. Since the device is not in
thermal equilibrium, it is not necessary for it to give thermal noise.
Now consider the addition of a series resistance to the device
model as shown in Figure 4.1.
In a measurement only Seq can be determined and this has a value
of
Idr2 4kT r
S = 2q +  (4.4)
eq (rs+rj)2 (rsr )2
This may be expressed in terms of a noise temperature with
4kT /(r + r.) = S (4.5)
eq
which gives
v4kTAf/RS
Figure 4.1. Intermediatefrequency model of a
Schottky barrier diode.
EQ
2qld r T r
T i (4.6)
n 4k r +r r +rj
s j s j
Noise Measurement Schemes
The noise measurements were carried out using two standard tech
niques. At low frequencies (10 Hz 25 kHz) the system depicted in
Figure 4.2 was used. At slightly higher frequencies (40 kHz 1.5 MHz)
the same system was used, but the FFT spectrum analyzer was replaced
with an HP Wave Analyzer (Model 310A, 10 kHz 1500 kHz). Above 30 MHz
a different scheme was used which will be described later.
Three measurements are required in order to determine the device
noise. Both Rbias and Rcal are chosen to be >>rdum and are also
selected to be wirewound or of thin film (for low excess noise).
First measurement: The device under test is mounted in place and
biased to the proper current.
Second measurement. The device under test is removed and replaced
with a dummy resistor which has the same resistance as the smallsignal
resistance of the biased device.
Third measurement: The dummy load remains in place, but the noise
source is turned on in order to give a calibration reading. The value
of Rcal must be chosen large enough to convert the voltage noise source
into an equivalent current source.
Thus, there are three equations (see Figure 4.3),
M2 = G2.B{e2 + [i2 + i2](R Rca .iasr )2} (4.7)
1 n dut n calbias dut
M2 = G2.B{e2 + [i2 + i2](R calIR r )2} (4.8)
2 n dum n cal ias dum
M2 = G2. B2 + [i2 + 12 + i2](R R ir )21 (4.9)
3 n cal dum n cal Tias dum
VTOT
Rbias
CFW101
Figure 4.2.
Noise measurement system for low
frequency measurements.
.2
2 'dut
'cal Rca
Rcal
Calibration
DUT
AMPLIFIER
Source
Figure 4.3. Equivalent circuit of the low
frequency measurement system.
so that
__ M2 M2
"1 j2 _4kT
12 =  ) i2 + 4k (4.10)
dut 2 cal r(4.10)
M^ M dum
3 2
This is the method used in all measurements below 1.5 MHz. The meter
readings are Mi, M2 and M3 and have the units of voltage. The gain of
the amplifier is G and the measurement bandwidth is B. The equivalent
voltage noise source of the amplifier is en and the current noise source
is in. For simplicity the amplifier is assumed to have infinite input
impedance, although noninfinite impedance would not affect the final
result. The values of the calibration resistor and bias resistor are
Rcal and Rbias respectively. The smallsignal resistance of the device
under test is rdut and the dummy resistor is chosen to have an equal
value. The noise currents are subscripted in accordance with their
respective origins.
For measuring the noise at higher frequencies, it is more conven
ient to use microwave techniques in order to eliminate parasitic effects
(particularly the capacitance of the device mount and interconnecting
cable). The system used is shown in Figure 4.4. It uses a double stub
tuner to obtain an impedance match to the device under test.
A typical measurement using the system of Figure 4.4 would proceed
as follows:
Connect the DUT to the bias tee and bias it, using a large series
resistor, to the proper operating point. The connector of the bias tee
will be the reference plane.
Turn off all digital voltmeters.
Turn on the noise source (6). The signal from this source is
coupled to the main transmission line via the directional coupler (3).
NOISE
SOURCE
(AMPLIFIER)
NOISE
SOURCE
LEVEL
SET.
ATTEN
Figure 4.4. Noise measurement system for 30150 MHz.
Equipment list of Figure 4.4
1 Bias Tee Model 5575, 10 kHz to 12 GHz; Picosecond Pulse Labs,
Boulder, Colorado.
2 Double Stub Tuner Model DS 109H, 40 MHz to 400 MHz; Weinschel
Engineering, Gaithersburg, Maryland.
3 20 dB Directional Coupler Model Ch132, 1 1000 MHz; ANZAC.
4 LowNoise Amplifier Model W1G2H, 1.5 dB NF, 51000 MHz; Trontech.
5 Amplifier Model 8447F, 0.1 1300 MHz, HewlettPackard.
6 Amplifier (noise source for matching) Model 462A to 150 MHz;
HewlettPackard.
7 Spectrum Analyzer Model 8558B, 0.1 1500 MHz; HewlettPackard.
8 Step Attenuator Model 8841B11, to 100 MHz; Shall Co., Inc.
9 Power Sensor 8484A and Power Meter 435B, 10 MHz to 18 GHz;
HewlettPackard.
10 LowPass Filter, RC time variable; inhouse design.
11 Digital Voltmeter Model 3466A; HewlettPackard.
12 Calibrated Noise Source Model SKTU 11000 MHz; Rhode & Schwarz.
For most frequencies the signal will be reflected by the stub tuner and
will give a large signal at the input of the LNA. At those frequencies
where a match is obtained, the signal returned to the LNA will be very
small.
Adjust the stubs and observe the spectrum analyzer display until
a match is found. The match should typically be better than 20 dB over
a 1 MHz bandwidth.
Turn off the noise source (6). Its only purpose is to provide a
wideband signal for matching.
Change the analyzer to zerospan mode. Select a bandwidth of
approximately onehalf the 20 dB match bandwidth. A smaller bandwidth
decreases the statistical accuracy while a larger bandwidth may
introduce systematic errors in the measured noise temperature.
During this procedure the output step attenuator should have been
set to at least 50 dB. This is to protect the sensitive power meter
from the large signals used during matching. Now that an actual noise
measurement is ready to proceed, it should be changed to 0 dB.
Record the meter reading. This will be Mdut. Make sure that the
system is linear by changing the input attenuator of the analyzer by 10
dB. The meter reading of the power detector should change by a factor
of 10. Note that this only checks the linearity from the analyzer input
and following. It is possible that the lownoise amplifier or power
amplifier is saturated, but this must be checked with the calibrated
signal source.
Return the output step attenuator to 50 dB in order to protect
the detector.
Remove the device and connect in its place the Rhode & Schwarz
noise generator (12). Set the output of (12) to zero.
Use the tuning stubs in conjunction with the spectrum analyzer
(7) and matching source (6) to match to the Rhode & Schwarz.
Turn off the noise source used for matching and return to the
same conditions under which Mdut was measured.
Make at least two meter readings corresponding to different
settings of the calibrated noise source. These will be referred to as
MC and MH. A third reading would allow a check of the total system
linearity.
From these noise measurements it is possible to determine three
quantities, namely, Tamp, the background noise temperature; Tdut, the
device under test noise temperature; and K.G2*B, the gainbandwidth
product. A constant of proportionality, K, exists which converts the
detected power into the measured voltage.
The three meter readings are
Md = K.G2B(Td + T ) (4.11)
dut dut amp
MC = K.G2B(T + T amp) (4.12)
C C amp
M = KG2B(T + Tamp) (4.13)
H H amp
The results are as follows:
T YT M
H C H
T = where Y = (4.14)
amp Y 1 MC
This is a standard Yfactor measurement:
M ) ( M
T14 dut C) (THTC) + TC (4.15)
dt MHMCC
Once the noise temperature of the device under test is known,
equation 4.2 may be used to convert to the spectral intensity of the
current fluctuations, using the device conductance.
This was done for the measurements between 30 MHz and 150 MHz. The
diode was assumed to have a smallsignal resistance equal to that
obtained from the DC IV measurements.
Discussion of Noise Measurements at Intermediate Frequencies
The results of the noise measurements between 10 Hz and 150 MHz are
presented in Figure 4.5. It is seen that the Hughes diode exhibits
excess noise at low frequencies and then reaches a constant noise level
for higher frequencies. The lowfrequency noise goes approximately as
1/f. The constant value which is reached corresponds closely with the
calculated values for the shot noise of the bias current when the effect
of the series resistance is included. The calculated values are indi
cated by the solid lines, and these correspond to a noise temperature of
mT/2. The series resistance is assumed to be 10 ohms as determined by
the IV measurements. For the purpose of establishing a reference, the
thermal noise of an equivalent resistor is also shown by using the
broken lines. This corresponds to a noise temperature of 300K.
Above the excess noise corner frequency, the theoretical shot noise
level of the DC current coincides well with the measured noise level.
See Figure 4.6. This is true for the several bias currents which were
used during the noise measurements. These shot noise measurements
indicate that the device behaves in the wellknown manner at inter
mediate frequencies.
16
10
SI(F)
1
10
(A2/HZ)
101 102 103 104 105 106 107 108
F (HZ)
Figure 4.5.
Spectral intensity of current fluctuations
vs. frequency for the Hughes Schottky
barrier diode.
109
20
10
SI(F)
10
(A2/HZ)
10
Figure 4.6.
5 4 3
10 10 10
I (A)
Shot noise plateau values vs. DC bias current.
The solid line corresponds to full shot noise
(= 2ql)and the dashed curve indicates the noise
after correctingfor the series resistance.
2
10
61
Discussion of Excess Noise Measurements
As may be seen in Figure 4.5, the Schottky barrier diode shows
excess noise at low frequencies. This excess noise shows some charac
teristic gr type Lorenzians but has the general character of 1/f.
In Figure 4.7 the excess noise at a frequency of 1 kHz is plotted
vs. the bias current on a double logarithmic scale. It is seen that a
power law of 13/2 is obtained. This is not the usual case as it is more
common for the noise to go as 12
According to Handel [30], the 1/f noise may be expressed as
QH
S (f) = 12 (4.16)
1 fN
In this case I is the current due to a stream of monoenergetic elec
trons, f is the frequency, N is the number of electrons in the sample
and
a (A) (4.17)
aH 3w c
+
Here a is the fine structure constant (= 1/137), Av is the vector
change in the velocity of electrons and c is the speed of light.
In a Schottky barrier diode operating in the thermionic mode, the
current flow is not due to a monoenergetic beam of electrons but.
rather, there is a distribution of energies. By subdividing the
velocities of passing electrons into small intervals, Avx, it is
possible to calculate the 1/f noise due to the electrons which have a
velocity in this interval. The total 1/f noise is found by applying
Handel's formalism outlined in equation (4.16) and summing over all
possible velocity intervals.
Sl(f)
20
10
(A2/Hz)
21
10
22
10
23
10
24
10
aly
Y = 1.5
MEASURED
23 7
S i l
I6
10
15
10
, ,I I
a, a l
4
10 (A)
I (A)
Figure 4.7. Excess noise at 1 kHz vs. bias current I.
12
10
I, I I I
S (f) = 1 S (f) = (AI )2 f (4.18)
I i i i
Now AI represents the DC current flowing due to electrons in the
velocity interval Av The quantity ANi is known from a calculation of
the transit time. See equation (4.19). The transit time and Ali are
defined in detail in Chapter V. The computer program which was used to
carry out the evaluation may be found in the appendices.
AI T
AiN (4.19)
i q
The measured and theoretical values of the 1/f noise at 1 kHz are
plotted as a function of the bias current in Figure 4.8. It is seen
that the measured values are considerably larger than the theoretical
ones and indicate that the theoretical limit of excess noise has not
been reached. At least two possible explanations exist: 1) There may
be another "nonfundamental" 1/f noise source which is present; and
2) the excess noise may actually be of the gr type (McWhorter's model).
If Handel's formalism is correct, then there exists the possibility
of considerable improvement in the 1/f noise properties of these
devices. This is an important indication for the use of these diodes as
detectors and mixers.
Conclusion
At frequencies up to 150 MHz the wellknown shotnoise theory is
sufficient to explain the measured results for the regions where there
is no lowfrequency excess noise. The device conductance is equal to
the derivative of the IV characteristic, and the value of the current
118
119
10
sl(f)
20
10
(A2/Hz)
21
102
22
10
23
10
24
10
alv
Y= 1.5
MEASURED
THEORETICAL
V = 0.84
3
223 7i
*3 *
I I
6 15
) ~10
14
10(A)
I (A)
I I
13
10
Figure 4.8.
Excess noise at 1 kHz vs. bias current.
Solid line measurements, dashed line theory.
12
10
I I
* *I
spectral intensity is given by assuming full shot noise in the DC
current.
The lowfrequency measurements show that there are no excess noise
sources which extend into the microwave frequency region.
A theoretical model of the 1/f noise was developed using Handel's
formalism and taking a distribution of the electron velocities into
account. This leads to a theoretical value of the 1/f noise which is
below the measured data and indicates that considerable improvement
should be possible.
CHAPTER V
HIGHFREQUENCY NOISE OF SCHOTTKY BARRIER DIODES
Introduction
Van der Ziel has provided the theoretical background for analyzing
the effect of the transit time on Schottky barrier diodes operating in
the thermionic mode [31]. The case to be considered is for a uniformly
doped semiconductor (linearly varying electric field). The physical
model was presented in Chapter III.
Calculation of the Junction Admittance
In Van der Ziel's work the initial velocities of the electrons are
divided into three cases for the purpose of calculating the smallsignal
admittance of the diode at high frequencies. It is assumed that all of
the electrons are emitted from the interface between the depleted/non
depleted region of the semiconductor.
Electrons which belong to group one have insufficient kinetic
energy to cross the barrier and return to the undepleted semiconductor
region. During their time of flight, the current which flows has the
time dependence illustrated in Figure 5.1. This is shown for the case
of a single electron, but in practice the amplitude must be multiplied
by the number of electrons injected with the same initial velocity.
The electrons of group 1, regardless of the initial velocity
of an electron upon entering the spacecharge region, all require
the same characteristic time to return to the neutral semiconductor
region. Only the amplitude of the current pulse changes due to the
StR
\\0
Figure 5.1. The solid curve indicates the current pulse
due to a single electron of group 1 with an
initial velocity of v (0). The dashed curve
is for an electron with higher initial velocity.
This figure shows that all electrons of group 1
have the same characteristic time of flight.
different possible initial velocities. The characteristic time for an
electron to return is
TR = w/a (5.1)
where a is the plasma frequency and is equal to (eEmax/m*d)1/2. Both
Emax and d are a function of the applied bias. The effective mass of an
electron is m*.
These electrons carry no DC current and at low frequencies do not
give any significant contribution to the device admittance. The
characteristic frequency associated with the current pulses is
fR = 1/2 lT R. When operated at high frequencies, the frequency of the
applied signal may be of a frequency on the same order of magnitude as
this characteristic frequency. These current pulses thus give a contri
bution to the device admittance. This component of the device conduc
tance increases as w2 for low frequencies and reaches a maximum when the
frequency of operation corresponds to 1.37 times fR. The maximum may be
found by differentiating the expression for the conductance. The
conductance due to these electrons is given in [31] as
I I
s d 1 c (1 cosw(lc) 1 cosw(c
11 if V 2 1 c2 1 c 1 + c
(5.2)
where the parameter c is the normalized angular frequency defined as
w/a where w is the frequency of operation. In this notation Is is very
much larger than Id for normal operation. This conductance dominates at
high frequencies when the diode is operated at low bias currents. It
should be recalled that most electrons have energies near the bottom of
the conduction band edge and thus belong to group 1.
If the series resistance of the device is zero, then it is possible
to calculate the effective noise temperature of the device with no
knowledge of the junction susceptance. However, for most diodes the
series resistance is not negligible, especially for high bias
currents. The combined effect of the series resistance and junction
susceptance will be discussed in the section on the theoretical noise
temperature.
The component of junction susceptance due to those electrons which
return can be found to equal
I I
Ss d C1 sin(lc) + sin(l+c)7r (
11 Vdif d 2 1 c 1 + c
The electrons with sufficient kinetic energy to always reach the
metal contact are those of group 2. The current pulses due to two
possible initial velocities are illustrated in Figure 5.2. The device
admittance is found by applying a small AC voltage and calculating the
response of the electron velocity. The transit time is also changed due
to the applied signal.
Since the electrons from this group are always collected, they
carry the DC current and are not much affected by an applied small
signal voltage (dI/dV is approximately zero for this group of
electrons). Nonetheless, an applied signal does influence the velocity
of electrons during the transit time, and the contribution to the
conductance may be calculated as
i (t)
'
Q
NO
t
Figure 5.2.
The current which flows as a function
of time for an electron belonging to
group 2. The amplitude and transit time
both depend on the initial velocity of
the electron.
71
I 0 eV' eV'
g22 Vd f exp ( T xd T c 2 (sin sin co
dif d 0 c
+(1 cos(lc))_ (1 cos(l+c)>))
1 c 1 + c
(5.4)
where = aT eV' is the energy of the arriving electrons and T is the
DC transit time.
Again, the component of device susceptance may be calculated and is
equal to
I d eV eV'
b22 = dif exp ()d(I) x1 c [ + sin cos c
22 Vdif Vd kT kT 2 2
+ (sin(1c)) + sin(1+c))]
1 c 1 + c
(5
It should be noted that for most cases the effect of electr
this group may be ignored in the admittance calculation. Howev
the calculation of the spectral intensity of the current fluctu
the electrons of group 2 play an important role.
The final group of electrons which must be considered are
with barely sufficient energy to reach the metal contact.
electrons dominate the device conductance at low frequencies.
contribution to the junction conductance is
(1 2c sin 1 c c2 cos cw)
33 g ( 2 2
933 0 (1 c2)2
.5)
ons of
er, in
nations
those
These
Their
(5.6)
where go is the lowfrequency smallsignal conductance.
The contribution to the device susceptance is
(c2 sin re 2c cos 2)
b33 = g0 (5.7)
3 (1 c2)2
The total junction admittance is found from a superposition of the
effects of the three groups of electrons. At low frequencies the
admittance has only a real part. This conductance is due to the
electrons of group three and is the same result as obtained from the
derivative of the IV relationship. The calculation of the derivative
is analagous to evaluating the change in the number of electrons which
cross the barrier when a signal is applied. Since only the number of
electrons in group three is modulated, they give the lowfrequency
junction conductance.
Calculation of the Spectral Intensities
In order to calculate the noise temperature of the diode, the
spectral intensities of the current fluctuations due to each group of
electrons must be calculated. The total spectral intensity is found by
summing the contribution of each electron. It should be noted that the
calculation of the spectral intensity can proceed by considering only
two cases: electrons which are able to cross the spacecharge region
and those which can not. The electrons of group three always cross the
barrier in the absence of an applied signal. Since the spectral
intensity is calculated for the case of an AC short circuit, groups two
and three may be combined.
The calculation of the spectral intensities follows along straight
forward lines but involves lengthy expressions. The procedure is as
follows: 1) Calculate the current which flows as a function of time for
a given initial velocity of an electron; 2) take the Fourier transform
of the current pulse due to electrons with this initial velocity;
3) calculate the number of electrons flowing with this velocity;
4) apply Carson's theorem using results 1 3 in order to find the
spectral intensity for a particular initial velocity; 5) sum over all
possible initial velocities in order to find the total spectral
intensity.
The first case which will be considered is for electrons which
reach the metal contact. From the equation of motion for a
single electron one finds
v (t) = v (0)cos at (5.8)
and the time of flight TF is found when the electron reaches the metal
anode.
TF = 1/a Arcsin (ad/vx(0)) (5.9)
Since the velocity of the electron as a function of time is known,
the current which is flowing is also known [32]. The Fourier transform
of this current pulse is necessary in order to calculate the current
spectral intensity.
Sqv (0)
d cos at e tdt
C 1 j(1+c)aTF j(c)aT
c (e 1 c (
(5.10)
where C1 = qvx(0)/2d.
The spectral intensity due to the electrons which cross with this
initial velocity is found by using Carson's theorem.
AS 22(f) = 2X*0 (5.11)
22
where A is the average number of electrons being emitted per unit time
with a given velocity. This spectral intensity is that due to those
electrons in a small velocity interval centered about v (0).
A calculation of lambda follows from the DC current. Since the
electrons which reach the metal contact are currently being considered,
the calculations follow the familiar lines of thermionic emission
theory. Numerically, this requires evaluating the function F(K) over
the velocity interval of interest [26].
Smv.(0,K)
2 xJL
F(K) = A v (0,K) exp( kT ) (5.12)
x k
*kT
where A = 4qm kT2/h3 and K is the Kth point under consideration. The
function F describes the current density due to the electrons of a par
ticular initial velocity crossing the barrier (per unit velocity). This
function must be integrated in order to find the DC current. This is
done numerically by discretizing the function and applying Simpson's
Rule. The number K is an integer used as an index to keep track of the
velocity interval under consideration. See Figure 5.3.
For the case of passing electrons, F(K) was evaluated for 100 vx
values using a spacing in vx of 2500 m/sec. This gives 27 points for
every kT unit of energy, which allows sufficient accuracy in the
integration procedure.
Figure 5.3.
SVX
Vx(K)
 vx(1)
(Vdif Vd) (
F
Method of discretizing the function F(K)
to allow for numerical integration.
It is simple to calculate the spectral intensity due to a particu
lar initial velocity once F(K) has been evaluated. The current in a
particular velocity interval is
AI22 = (Area)Av (F(K) + F(K+1))/2 (5.13)
and X(K) for this interval follows directly:
X(K) = AI22/q. (5.14)
Now that all of the necessary quantities have been evaluated, the
spectral intensity is calculated using Carson's rule. This yields
2X(K)C2 2
S 1 1 2 1(1 cos(1c)a (K)) 2c
2 K a2 (1+c)(1c) (I ) c
( 2c
(1 cos(l+c)aTF(K)) +
+ (1 cos 2aTF(K))} (5.15)
where the index K again refers to a particular small velocity interval.
It should be kept in mind that T (K) is the average transit time of an
electron in the small velocity interval under consideration.
Since the case being considered is for the electrons which are able
to cross the junction, the summation should start from v00 and go to
infinity. The value of v00 is found from the kinetic energy required to
cross the barrier of the junction. This is obtained from equation
5.16. In practice the summation only needs to be carried out over a few
kT of energy,
q(Vdif Vd) = m (5.16)
The second case, electrons which have insufficient kinetic energy
in order to cross the junction, is considered in the same manner.
Again the calculation begins by taking the Fourier transform of a
single current pulse. The results of the previous calculation may be
used if it is kept in mind that F should be replaced everywhere by TR'
The resulting expression may be simplified using trigonometric
identities to the following form:
C2
1 8c2
S= (1 + cos cr). (5.17)
a2 (1c2)2
As in the first case, F(K) must be evaluated in order to determine
the number of electrons injected with a particular initial velocity.
See equation (5.12). Only the limits of evaluation need to be changed
for this group of electrons. Electrons at the bottom of the conduction
band have only potential energy and no kinetic energy. Thus, the lower
limit of evaluation is vx(0) = 0. Electrons which almost reach the
metal contact, yet fail, have an initial velocity equal to v00, as
defined by equation (5.16).
In the numerical evaluation for the returning electrons, F(K) was
evaluated for 100 values of vx regardless of the actual barrier
height. Since typical values of Vdif Vd are less than one eV, this
resulted in a sufficiently fine spacing in vx to give accurate results
from a numerical integration.
The total spectral intensity of those electrons which do not reach
the metal contact and then return to the neutral semiconductor is
2X(K)C2 2
S (f) = AS = 1 8c2 (1 + cos cW). (5.18)
11 K 11 K a2 (1c2)2
The method of calculating the spectral intensities has now been
outlined. The actual computer program which was used may be found in
the appendices.
The spectral intensity of current fluctuations due to electrons of
group 1 is shown in Figure 5.4. The noise of these electrons which
return is zero for zero frequency, since these electrons carry no DC
current. The spectral intensity is seen to rise initially as f2.
The lower curve is for a bias voltage of five times VT and the
upper curve is for a bias voltage of ten times VT. This illustrates
that the spectral intensity due to this group of electrons is a very
slow function of the bias.
Since all electrons of this group have the same characteristic time
of flight, the spectral intensity shows several sharp nulls. This would
not be the case if the time of flight took on a distribution of values.
The spectral intensity due to the passing electrons, groups 2 and
3, is illustrated in Figure 5.5 by the solid curves and is equal to the
shot noise of the DC current at low frequencies. At frequencies on the
order of the reciprocal transit time the spectral intensity falls off
steeply. The lower solid curve is for a bias voltage of five times VT
and the upper curve is for a bias voltage of ten times VT. It is clear
that this component of the spectral intensity is a very strong function
of the applied bias.
For a given frequency of operation, e.g. 12 GHz, it may now be seen
that there are two distinct regimes of operation. At low bias the noise
10"1
1020
S1(f)
CA2/Hz]
1021
1022
1023
1024
7
3
2
mu S
/I
I,
/ /
///
/I/
/I/
/./
I3
//
I/
S3 ?, .. .
100
Figure 5.4.
10' 102
FREQUENCY CGHz)

turning
ctrons
I
'I
i,
!! *
I i
SII
i
I
i
103
104
Spectral intensity due to returning electrons.
Lower curve is for Vd = 5 VT;
Upper curve is for Vd = 10 VT.
I
FREQUENCY [GHz3
Figure 5.5.
Spectral intensities vs. frequency.
Lower curves are for Vd = 5 VT;
Upper curves are for Vd = 10 VT.
1041
SX(f)
EA2/H z]
104
of the returning electrons is dominant while at high bias the noise of
the passing electrons becomes dominant. Furthermore, the higher the
frequency of operation, the higher the bias must be in order for the
passing electrons to dominate the noise.
Calculation of the Noise Temperature
The noise temperature is calculated as a function of both bias and
operating frequency by combining the various components of device con
ductance and spectral intensity.
4kTng = S (f) (5.19)
where gj is the total junction conductance from all groups of electrons
and Sl (f) is the spectral intensity of current fluctuations due to all
groups of electrons.
Consider the case of a junction with nonideality factor of unity
and no parasitic resistances or capacitances. If only the passing
electrons are considered, then the noise temperature would be equal to
half the ambient temperature. This is the standard result for inter
mediate frequencies, where the returning electrons may be neglected.
For lowbias operation at high frequencies the returning electrons
may not be neglected. This group of electrons has a noise temperature
associated with it which is equal to the ambient temperature. As the
bias is increased, the passing electrons begin to dominate the device
behavior, and the noise temperature again drops to half the ambient
temperature.
The description of the junction presented so far has only been
concerned with an ideal exponential junction. In practice this is not
sufficient as the effects of the series resistance, diode nonideality
factor and reverse saturation current must also be considered in an
actual device.
In most cases the series resistance is of the greatest concern.
This resistance degrades the performance of the device in every manner.
Due to the series resistance rectification properties are reduced,
matching to the actual junction becomes difficult and in general the
effective noise temperature of the diode is increased. The series
resistance can be appreciable in microwave diodes due to the small metal
contact area which is used (typically 2 pu diameter).
At very high bias levels the series resistance may be the source of
excess noise. At high frequencies the effective series resistance may
be greater than that obtained from IV measurements due to the skin
effect. It is difficult to accurately model the series resistance for
highbias and highfrequency operation [33]. In this work it will be
assumed that the series resistance is constant and equal to its low
frequency value. In addition, it will be assumed that the noise tem
perature of the series resistance is equal to the ambient temperature.
The diode nonideality factor was assumed to be equal to unity in
these calculations. For the diode which was measured this is a
reasonable assumption since it was found experimentally that m = 1.09.
At worst, this gives an error of 9 percent in the noise temperature.
The nonideality factor could be taken into account by recalculating the
Schottky barrier height for each bias point. This would lead to a
different value of the minimum velocity required for an electron to
reach the metal contact.
At very low values of bias (Vd < 5mkT/q) it is important to include
the reverse saturation current in the device model. If this current is
omitted, then the calculated current for zero applied voltage would be
nonzero, and the noise temperature would not match the ambient
temperature as it should.
The reverse saturation current is comprised of electrons escaping
the metal and flowing towards the semiconductor. These electrons are
traveling in a direction opposite to the electric field of the
junction. Since electrons are negatively charged, they will be
accelerated as they cross the junction. Their transit time will be much
less than for electrons traveling from semiconductor to metal. It is
concluded that the spectral intensity of this component of current may
be considered to be white over the frequency range of interest.
The complete results of the theoretical calculations are presented
in Figure 5.6. The Schottky barrier height of 0.585 eV and series
resistance of 10 ohms were obtained from DC IV measurements. The diode
was assumed to have an anode diameter of 2 im and a doping density of
101 cm3. The substrate material is silicon.
The curves of Figure 5.6 indicate that for each frequency of opera
tion there is an optimum bias. It is at this bias that the lowest noise
temperature of the diode will be obtained. The lowest noise temperature
which can be achieved increases with the operating frequency. At high
frequencies the lowest noise temperature is not much reduced from the
ambient temperature.
At low frequencies, with no series resistance, the effective noise
temperature is 295 K at zero bias and drops sharply to 147.5 K. This is
the wellknown result for intermediate frequencies. By including the
series resistance of 10 ohms, it can be seen that for high biases the
series resistance dominates over the junction resistance and the noise
300
Tn [K]
200
100
0
V/VT
Figure 5.6.
Theoretical noise temperature
vs. bias voltage.
temperature again rises to the effective temperature of the series
resistance.
When operated at high frequencies, the noise temperature does not
begin to drop until a higher bias is reached. This is due to the fact
that at low bias the noise and conductance of the device are dominated
by the returning electrons. At high frequencies the noise temperature
of the diode remains nearly constant due to the returning electrons. At
97.5 GHz there is almost no decrease in the noise temperature.
The rolloff from the ambient temperature towards a noise
temperature of mT/2 occurs when the spectral density and conductance of
the passing electrons dominate. The bias at which the rolloff begins
is a function of frequency. If there were no series resistance, then
the noise temperature would always reach a value of mT/2. This is not
the case.
At 2.2 GHz the highbias portion of the curve is seen to be
dominated by the series resistance. It should be recalled that at low
bias the returning electrons cause both the junction conductance and
susceptance to be large compared to their lowfrequency values. This
allows the series resistance to play a role even at the lower biases.
The series resistance, in conjunction with the junction susceptance,
causes the effective conductance at the device terminals to be
increased. The circuit transformation which is made is indicated in
Figure 5.7. It is the transformed quantities which are measured.
Since the numerical analysis also gives the theoretical junction
admittance, the device reflection coefficient as a function of bias may
be calculated. The results of these calculations are presented in
Figure 5.8a. In this case the series resistance was 10 ohms.
Figure 5.7. Circuit transformation due to the combined
effect of the series resistance and
junction susceptance.
1.0
2.2 GHz
0.8
Ir12 0.6 12 GHz
0.4 
0.4 97.5 GHz
0.2
0.0
0 5 10 15
V/VT
(a)
1.0
0.8 2.2 GHz
Ir12 0.6 12 GHz
0.4
0.2 97.5 GHz
0 5 10 15
V/VT
(b)
Figure 5.8. Theoretical magnitude squared of
the reflection coefficient.
a) A series resistance of 10 Q
is assumed;
b) A series resistance of 15 a
is assumed.
Figure 5.8b presents the results of the same calculations, but for a
series resistance of 15 ohms. It will be seen that the results of
Figure 5.8b more closely approximate the measured results which will be
presented later.
The magnitude of the reflection coefficient depends more critically
on the value of the series resistance than does the noise temperature.
For this reason the noise temperature versus bias was presented for only
one case.
HighFrequency Measurements
The noise measurements were performed on a Hughes silicon diode
detector (model 47316H1111) using the circulator method of Gasquet et
al. [34]. Since four measurements are performed, there are also four
quantities which can be determined. It is possible to determine Ta, the
system background noise temperature; K*G*B, the gainbandwidth product
(K is a constant of proportionality); TDUT, the device under test noise
temperature; and Ir 2, the magnitude of the reflection coefficient
squared. Usually, only the latter two quantities are of interest.
Microwave measurements were performed at 2.2, 12 and 97.5 GHz. The
various front ends of the receivers are shown in Figures 5.9 through
5.11. The down conversion and detection were done using the setup shown
in Figure 5.12. This portion of the circuit was the same for all of the
measurements, although an HP desktop computer was used to record the
voltmeter readings during 97.5 GHz noise measurements. Use of the com
puter allows for long averaging times to obtain the required statistical
accuracy.
Measurements proceed as follows:
1) A short circuit is placed at the reference plane and the noise
source with a noise temperature of TN1 is applied. This gives a meter
HP
NOISE
SOURCE
AMPLICA
REF
PLANE
24 GHz
2.2 GHz
Figure 5.9. Measurement front end for 2.2 GHz.
bias
_T
NOISE SOURCE
STEP ATTEN
20 dB
812 GHz
f = 12 GHz
BW =1%
I
Noise measurement front end for 12 GHz.
REF
PLANE
I
CIRC
10 dB
Figure 5.10.
94 GHz
SOURCE
CIRC ISOLATOR
TN1 0 LN
REF
PLANE
TN2 DUT
Rbias
I
Figure 5.11. Noise measurement front end
for 97.5 GHz.
21.4
MHz
DET
PWR
METER
SPECTRUM
ANALYZER
LEVEL
SET
Figure 5.12. Detection system for noise measurements.
Desktop computer was needed only during
97.5 GHz measurements. See the appendix
for the required program.
reading of
M1 = K.G*B[Ta + TN]
(5.20)
2) The short circuit remains in place and a noise source of
equivalent temperature TN2 is applied. This gives a second meter
reading equal to
M = K.G.B[T + TN2]
(5.21)
3) The device under test is connected at the reference plane and
biased to the proper operating point. By applying the noise source of
temperature TN1 the meter reading becomes
M3 = KG.B[Ta + TDUT(1 Ir2) + TNlIr12] .
(5.22)
4) The device remains biased and in position, but the noise source
of TN2 is again applied. The meter reading for this case is
M = KGB[Ta + TDUT(1 Ir 2) + TN2 r2] .
(5.23)
These equations can be solved to yield all four of the unknown
quantities.
TN Y TN
T N2 N1
T =
a Y 1
M
H2
,where Y = ;
M1
Z T T M M1
T = Z 1 where Z= M3 M
DUT Z1 M M2
(5.24)
(5.25)
