Title Page
 Table of Contents
 List of Figures
 Scope of our investigations
 Theory of heat transfer and of...
 Experimental method
 Experimental results
 Interpretation of results
 Recommendations of future work
 Biographical sketch

Group Title: Identification of 1/f noise producing mechanisms in electronic devices
Title: Identification of 1f noise producing mechanisms in electronic devices
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00082444/00001
 Material Information
Title: Identification of 1f noise producing mechanisms in electronic devices
Physical Description: viii, 107 leaves : ill. ; 28 cm.
Language: English
Creator: Kilmer, Joyce Prentice, 1958-
Publication Date: 1984
Subject: Electronic noise   ( lcsh )
Current noise (Electricity)   ( lcsh )
Transistors   ( lcsh )
Thin films   ( lcsh )
Electrical Engineering thesis Ph. D
Dissertations, Academic -- Electrical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis (Ph. D.)--University of Florida, 1984.
Bibliography: Bibliography: leaves 103-106.
Statement of Responsibility: by Joyce Prentice Kilmer.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00082444
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 000473847
oclc - 11699479
notis - ACN9056

Table of Contents
    Title Page
        Page i
        Page ii
    Table of Contents
        Page iii
        Page iv
    List of Figures
        Page v
        Page vi
        Page vii
        Page viii
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
    Scope of our investigations
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
    Theory of heat transfer and of temperature fluctuation noise
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
    Experimental method
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
    Experimental results
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
    Interpretation of results
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
    Recommendations of future work
        Page 100
        Page 101
        Page 102
        Page 103
        Page 104
        Page 105
        Page 106
    Biographical sketch
        Page 107
        Page 108
        Page 109
Full Text








I am greatly indebted to Professor Carolyn Van Vliet for her

most generous and valuable assistance, guidance, and time. I es-

pecially wish to thank Dr. Aldert van der Ziel for his contributions

to the PNP transistor research. I thank Dr. Gijs Bosman and Dr.

Peter H. Handel for sharing with me their insight and knowledge of

theoretical and experimental physics.

In addition, I wish to express my appreciation to Dr. Wolf and

Dr. Burhman of Cornell's NRFSS for producing the high quality Au

thin film arrays.

Furthermore, my sincere gratitude goes to Miss Mary Catesby

Halsey for her willing spirit and unstinted patience in typing this


This research was supported by the Air Force Office of Scien-

tific Research, under Grant Number AFOSR 80-0050.



ACKNOWLEDGEMENTS . . . . . . . . . .. .ii

LIST OF FIGURES . . . . . . . . ... . . v

ABSTRACT . . . . .... . . . . ... vii

INTRODUCTION . . . . . . . .

1.1 Fundamental Questions . . .
1.2 Competing Theories . . . .
1.2.1 A Thermally Activated Number
Fluctuation Theory . . .
1.2.2 Diffusion Theories . . .
1.2.3 The Quantum 1/f Noise Theory
1.2.4 The Temperature Fluctuation
Theory . . . . . .



2.1 1/f Noise Correlation Experiment
and the Thermal Fluctuation Model
2.2 Previous Results from Experimental
Work on Metal Films . . . .
2.3 The 1/N Dependence and the Hooge
Parameter . . . . . .
2.4 Temperature Dependence of 1/f
Mechanisms . . . . . .




Introduction . . . . . ....
The Heat Transfer Function . . .
Heat Transfer Correlation . . .
Temperature Noise . . . . .
Temperature Noise Correlation . ..
The Green's Function for the Gold
Thin Film Array . . . . . .



S 3


. . 4

. . 5
. . 7
. . 7

. . 10

. 11

. . 13

. . 16

S. 19


. 22






4.1 Devices .... . . . . . . 41
4.2 Apparatus . . . . . . ... .42
4.2.1 The Closed Cycle Cryostat . . .. 42
4.2.2 The Flow Cryostat . . . ... 43
4.2.3 Calibrated Noise Measurements . .. 43
4.2.4 The Measurement System . . ... .47
4.3 Thermal Noise Measurements ...... .49
4.3.1 Thermal Noise Measurements Using
the Calibrated Noise Source ... . 50
4.3.2 Thermal Noise Measurements Without
Using the Calibrated Noise Source 51
4.3.3 Accuracy in Thermal Noise
Measurements . . . . ..... . 52
4.4 Thin Film Heating Effects ... . . 56





5.1 Thermal Transfer Function Experiment . 59
5.2 Thin Film Heating . . . . ... 61
5.3 Resistance Versus Temperature ... . 64
5.4 1/f Noise Versus Temperature . . .. 64
5.5 The Clearcut Evidence of 1/f
Mobility Fluctuations in Transistors 73
5.5.1 Discrimination Between Base and
Collector Noise Sources ...... .74
5.5.2 Interpretation of SHR andSLR . 80
s s


The 0.5pm Devices . . . .
Noisy 1 and 2pm Devices . .
Quiet 1 and 2pm Devices . .
Conclusions . . . .



7.1 Continued Studies of the Low
Temperature Mobility-Fluctuation
Noise . . . . . . .
7.2 Continued Studies of the High
Temperature Number-Fluctuation
Noise . . . . . .
7.3 Investigation of New Queries . . .

REFERENCES . . . . . . . . . . . .

BIOGRAPHICAL SKETCH . . . . . . . . . .


Figure Number

2-1 Layout of Closely Spaced Gold Thin Film
Resistor Array . . . . . . . .

3-1 Layout of a Chip Containing Three Different
Groups of Thin Film Resistor Arrays . .

4-1 Calibrated Noise Measurement Scheme . .

4-2 Equivalent Circuit of the Calibrated Noise
Measurement Scheme . . . . . . .

4-3 Schematic of the Schmidt Low Noise Amplifier

4-4 Simplified Equivalent Circuit of the Low
Noise Amplifier . . . . . . .

5-1 Thermal Transfer Responses of Two Thin Film
Resistors . . . . . . . . .

5-2 Coherence Between Thermal Transfer Responses

5-3 Relative Thermal Noise Versus Device Bias
Current . . . . . . . . .

5-4 Device Resistance Versus Temperature . .

5-5 1/f Noise 12 Dependence . . . . .

5-6 Relative Noise Magnitude at 300K of all
Devices Measured . . . . . . .

5-7 Noise Magnitude and Slope Versus Ambient
Device Temperature (0.5pmdevice) . . .

5-8 Noise Magnitude and Slope Versus Ambient
Device Temperature (Noisy lpm device) . .

5-9 Noise Magnitude and Slope Versus Ambient
Device Temperature (Quiet 1pm device) . .

5-10 Typical Coherence Between the 1/f Noise of
Two Thin Films . . . . . . . .


. 12

S. 37

. 44

. 45


. 55

S. 60

. 62

. 63

. 65

. 66

. 67

S. 69

. 70

. 71

S. 72

Figure Number

5-11 Equivalent Common Emitter Circuit . . . .

5-12 Measurements of High Source Impedance
Spectra (SHR ) and Low Source Impedance
Spectra (SLR ) . . . . . . . . .

5-13 Base 1/f Noise Magnitude Versus Base
Current .. . . . . . . .

5-14 Collector 1/f Noise Magnitude Versus C<
Current . . . . . . . .

6-1 Photograph of Open-Circuited Devices

6-2-- Photograph of Short-Circuited Devices

6-3 SEM Photograph of Thin Film Resistors

6-4 true Versus Temperature . . . .





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 Science



Joyce Prentice Kilmer

August 1984

Chairperson: Carolyn Van Vliet
Co-Chairman: Gijs Bosman
Major Department: Electrical Engineering

In recent years, theorists have been trying to explain the pheno-

menon of electrical 1/f noise. Presently some controversy exists over

the exact origin of the fluctuating physical quantities which gives

rise to these resistance fluctuations. The disputes have centered on

fluctuations of carrier mobility, carrier number, and temperature, any

of which could cause the observed resistance fluctuations. The pri-

mary thrust of our research is to determine which of these possible

1/f noise producing mechanisms is responsible for the 1/f noise com-

monly observed in two electronic devices (ie. this film resistors and

PNP transistors). First, we delve into the details fo two competing

1/f noise theories of metal thin films--Dutta & Horn's Number Fluctu-

ations theory and Handel's Quantum 1/f Noise theory (ie. mobility

fluctuations). Then, we review the results of previous investigators

in 1/f noise, in order to identify the areas needed to be researched.


Our central experimental effort has been to devise methods to distin-

guish between the three possible 1/f noise producing mechanisms. We

show a lack of correlation between the 1/f noises of two devices in

the same thermal environment and conclude that temperature fluctua-

tions do not produce the 1/f noise. We find two methods to discri-

minate between number and mobility fluctuations (ie. changing of

the source resistance for active devices and changing the ambient

temperature in passive devices). Results from both the thin films

and the transistors show for the observed current noise both number

fluctuations and mobility fluctuations may be present, but under

certain experimental conditions one mechanism will dominate. Finally,

the application of Handel's Quantum theory to the noise observed in

thin films at low temperatures leads us to believe that Quantum 1/f

noise does exist and sets a theoretical minimum to all 1/f noise.



The study of the spectral density of random fluctuations which

vary inversely with frequency (1/f noise) is an old subject. Speci-

fically, in 1937, Bernamont [1] observed a "current" noise in metal

films whose spectral density, S (f), followed an empirical formula,

S (f) = (0.88 < y < 1.1) (1-1)
I fY

where the I indicates the current dependence. However, the theore-

tical explanation of this noise has remained one of the oldest of the

unresolved problems of solid-state physics.

A 1/f spectrum has been observed from such a diversity of systems

(ie. from variation of traffic flow rates [2] to the biomedical noise

observed in axon membranes [3]!) that reviewers such as Dutta and

Horn [4] feel the physical origin of the phenomenon cannot be universal.

Still other theorists, such as Handel [5], fascinated by the univer-

sality of the phenomenon have concluded that any phenomenon governed

by nonlinear equations of motion with no contribution from-boundary

conditions and no characteristic times will necessarily generate a 1/f

spectrum. Clearly a divergence of opinions exists regarding 1/f noise,

hence, the recent interest within the scientific community.

Through the electrical properties of condensed matter is how we

find 1/f noise most commonly manifests itself. For this reason we will



be concerned mainly with the 1/f fluctuations in the conduction pro-

cesses of solid state devices. What is known about the phenomenon

can be summarized by Hooge's phenomenological equation [6],

S = (1-2)
fV N

where SV is spectral density of voltage fluctuations and aH is the
Hooge parameter which he assumed to be a constant aH = 2 X 10.

We see the magnitude of the voltage spectrum is proportional to the

,square ofthe DC voltage, V, across the device implying that the cur-

rent does not drive the resistance fluctuations but merely, by Ohm's

law, reveals them as voltage fluctuations. Hooge's equation also

predicts the voltage spectrum to vary inversely with the number of

carriers, N, which implies 1/f noise is a bulk effect rather than a

surface effect since N ~ Volume. Finally the formula shows the em-

pirical 1/fY dependence where y (an exponent close to unity) accounts

for the observed slope variations. Van Vliet et at. [7] observed

the relationship between the voltage spectrum and the current spec-

trum, S measured from a device of differential resistance R,

SV = S R (1-3)

From this, we see we can write with Hooge [8],

S S ct
V2 2 fN


where I is the DC current through the device. Clearly, the formula is

not complete since it gives no explicit temperature dependence of the

noise. More will be said about this in Chapter II.

It is this power law dependence or scale invariance of noise

with y = 1 which makes this 1/f noise problem fascinating yet compli-

cated. Specifically, the total power or the integral of the spectrum

diverges; yet we do not get a "shock" when we touch an unbiased de-

vice! Therefore, we assume, if the process is stationary, we should

see at high frequencies a regime with Y > 1, and at sufficiently low

frequencies a roll off where y < 1. However, this has not been ob-

served for frequencies as low as 10-7Hz implying the need to postu-

late implausibly long time scales!

1.1 Fundamental Questions

First, we begin our research with some fundamental questions.

Is the phenomenon a surface or bulk effect? Hooge found in gold that

SV 1/thickness but Calesco, et al. [9], suggested the noise in

films occurs at the interface of the film and substrate. Dutta and

Horn showed that the substrates can play a role if they are not good

thermal conductors; but, in general, the noise in metal films is of

a bulk origin. Recently, this has been verified by Fleetwood and

Giordano [10] who found S ~ 1/N over six decades of carrier-numbers.

Is 1/f noise present in thermal equilibrium? Is the current re-

quired to simply observe the resistance fluctuations, or does the

current actually induce the 1/f noise? This question was resolved by

Voss and Clarke [11] who found an ingenious method to show that 1/f

noise is an equilibrium process by measuring 1/f noise in thermal noise!


Is the 1/f noise process stationary? If the power really does di-

verge, it would be a non-stationary process; but since simple thermal

equilibrium noise (of a simple carbon resistor, for example) does not

give infinite power, we must assume that 1/f noise is stationary.

Is the 1/f noise mechanism linear? This is a more difficult ques-

tion for which Voss [12] tried to rationalize a type of linearity by

measuring a "conditional mean." The experiment gave proof of a macro-

scopic linearity, but did not exclude the possibility of microscopic


From what fluctuating physical quantities does 1/f noise in elec-

tronic conductors arise? Presently three physical quantities, the mo-

bility of the carriers, the number of the carriers, and the temperature,

are in the theoretical arena. Fluctuations in any one of these quanti-

ties can theoretically give rise to 1/f noise, las the next section will

show. It is the main purpose of our investigations to isolate which

fluctuating quantities account for the 1/f noise in some specific elec-

tronic structures.

1.2 Competing Theories

A real "menagerie" of theories have been proposed and rejected

regarding 1/f noise. The McWhorter theory [13] is based on an inco-

herent superposition of Lorentzians. Usually, such Lorentzians are

caused by number fluctuations. The overall spectrum then becomes

S(w) = 2 g(T)dT (1-5)



where g(T) is the distribution of time constants for T < T < T In

the McWhorter model the distribution g(T) stems from a distribution of

tunneling widths (with a flat distribution of activation energies)

causing Equation 1-5 to yield a 1/f spectrum. In another model, the

distribution of time constants stems from a uniform distribution of

activation energies.

Dutta and Horn speculate that the flat distribution of activa-

tion energies is not actually present in metal films. Instead, they

show how a smooth distribution of activation energies gives rise to

"generic"-1/f noise which is more commonly observed (see next section).

1.2.1 A Thermally Activated Number Fluctuation Theory

To explain the observed temperature dependence of the magnitude

and the slope of 1/f noise, Dutta and Horn [141 postulated a modified

McWhorter theory where the total spectrum is again a superposition of

Lorentzians. However, now we assume the characteristic time, T, is

thermally activated and write T as a function of energy, E,

T =T e-E (1-6)

where 8 = l/k T, and we let

g(E)dE = Ke dE, E < E < E (1-7)
11- --


g(T) = g(E) = (1-8)
d EdT SE T T
/ **

where K = 1/Eo. Hence,

S 2 4A2 (61/)-1

4AT 1

o 1 +x o

The integral converges if -1 < (6/3) < 1. Thus,

(6/P) = -1 + 6, S(w) cc (1/OTO ) (1-10a)

(6/3) = 0, S(W) a 1/WT (1/f spectrum) (1-10b)

(6/O ) = 1 E, S(M) 1/(6T ) 2 (1-10c)

Thus, all spectra of the form C/ with 0 < y < 2, referred to as

"generic" 1/f noise [15], are possible. There is, however, scaling

in this theory, for there is a lowest al and an upper w2 for which the

spectrum 1/fY changes. In some cases these turnovers have been found

(see, e.g., Hanafi and van der Ziel's experiments on CdxHglxTe [16]).

It is also clear that only by exception the spectrum is exactly 1/f;

this requires a strictly uniform distribution of activation energies.

Without any reference to an exact type of thermally-activated

random process, Dutta and Horn showthat:anyg(E) which has a sharp peak,

E ~ leV, and a width of a few tenths of an eV (ie. a "smooth" energy

distribution) will give the observed SV vs. T and y vs. T dependence.

They speculate that g(E), as a peaked function, is more "amenable to

physical justification" than a mere flat distribution, since E can


now be a material property and the width of the distribution may de-

pend on parameters such as sample inhomogenity.

1.2.2 Diffusion Theories

Diffusion theories [17] (theories involving the diffusion equa-

tion) have been suggestedas a source of 1/f noise because they can give

the long time scales associated with 1/f noise. However, extensive

theoretical investigations by Van Vliet and Mehta [18] find a 1/f

spectrum can be produced by a diffusion theory only if a surface

source is assumed. Since we have seen 1/f noise in metals is a bulk

effect, the diffusion theory does not apply.

1.2.3 The Quantum 1/f Noise Theory

The only truly generalized theory of 1/f noise was suggested by

Handel in 1975 [19]. Until recently [20,21,22] experimental evidence

verifying the theory did not exist. Specifically Handel's Quantum 1/f

Noise theory was questioned as the source of 1/f noise in electronic

circuits because of the low value of the Hooge parameter, aH, calcu-

lated from his theory. The theory states, the interference between

the part of the carrier's wave function which suffered energy losses

due to an inelastic or "bremsstrahlung" scattering and the part of

wave function which did not suffer losses produces a low "beat-frequency"

infraquanta,giving rise to 1/f noise. More exactly, in the simple

Schrodinger field version of Handel's theory, one considers the wave

function upon scattering with small energy losses due to bremsstrahlung,

i(k'r-wt) 1 I (E td
ST(r,t) = a e ~ ~ (1 + bT( )ei dEe (1-11)



In Equation 1-11 the frequency-shifted components present in the in-

tegral interfere with the elastic term, yielding beats of frequency

E/h. The particle density given by Equation 1-11 is

L1T 2 = a2 1 + 2 f b T()Icos e+ y dt


o o

the second term in large parentheses describes the particle beats.

If the particle fluctuation is defined by 6 i2 = 212 <2 I 2>, its

autocorrelation is found to be, if a term "noise of noise" is neglected,


<610S 26112 > = 21a 4 b(c) 2cos de


= 21a4f lb() 12cos2TfTdf (1-13)


From the Wiener-Khintchine theorem one sees that the integrand is the

spectral density:

Sl 12/ = 2fib(E) 2 (1-14)

f )
For the spectral density in the particle velocity v =- -( V)
I ( 2im
one easily sees,

S /2 = Sl12/<1 2> = 2fltb()2 (1-15)


For the bremsstrahlung matrix element, under emission of photons, one

calculates easily, either classically or quantum mechanically,

Ib() 12 = CA/E = aA/hf ; (1-16)

here a = 1/137 is the fine structure constant and

2 Jv J
A 3 A 2 (1-17)
3Tr 2

Handel's theory predicts Umklapp process (U-process) scattering

to be the largest source of 1/f noise in metals since the 1/f noise

magnitude scales with the photon infraquanta coupling constant, aA,

ie. combining Equations 1-15, 1-16, and 1-17,

v 2aA

2 f

20A V
S2c (1-18)
3rf 2

Since U-process give the largest Av, we expect them to be the largest

contributor to 1/f noise in metals. The changes in Av of the scattered

electrons give rise to mobility fluctuations which translate into resis-
tance fluctuation since R = Dutta and Horn in their review paper
discredit Handel's theory explaining, "most electrons in a metal can-

not emit low-energy photons because all the nearby states are occupied."

While Handel's theory has not been a popular one among the critics, we

believe the Quantum theory may set a fundamental lower limit to the

observed 1/f noise.


1.2.4 The Temperature Fluctuation Theory

One of the possible competitors of Handel's theory is the thermal

fluctuation model of Voss and Clarke [15]. They assume temperature

fluctuations give rise to resistance fluctuations since we measure a

temperature coefficient of resistance, 0,

1 dR
R -jd (1-19)

Using the Langevin diffusion equation,

T 2 V*F
S= DV T + (1-20)
t c

where D is thermal diffusivity, c is the specific heat and F is a

random driving term (uncorrelated in space and time). This equation

causes temperature fluctuations to be spatially correlated with the

mean square amplitude for temperature fluctuations divergent at long

wavelengths resulting in a 1/f spectrum over a limited frequency range.

The theory has some criticism in that the diffusion equation, as Voss

and Clarke have written it, cannot be rationalized on a microscopic

level since the source term, F, representing heat flow from the metal

film to the low thermal conductivity substrate [23] violates conserva-

tion of energy [24]. The theory has had moderate successes in special

cases such as in Sn films near the superconducting transition, but it

is merely fortuitous that it works at room temperature for Au and Cu

films [4, p. 508]. Our research has been designed to distinguish

which theories are applicable to the 1/f noise observed in electronic



2.1 1/f Noise Correlation Experiment and the Thermal Fluctuation Model

Ever since Voss and Clarke proposed the theoretical possibility of

1/f noise in metal films in terms of equilibrium temperature fluctua-

tions, investigators have been trying to verify their theory. Scofield,

et al. [25], have shown Voss and Clarke's theory is not valid at room

temperature since the coherence of 1/f noise spectra from two thermally

coupled Au films was orders of magnitude less than the temperature fluc-

tuation model suggests. However, with the recent Japanese measurements

of 1/f temperature fluctuations in a resistor [26], the theoretical

possibility of 1/f noise stemming from surface sources [27], and the

strong temperature dependence of the fundamental 1/f noise (Type B

noise according to Dutta and Horn) [4, p. 510], an investigation of 1/f

noise correlations is to be performed over a full cryogenic-ambient

temperature range (10K to 300K).

Van Vliet has designed a configuration of three closely-spaced gold

thin film resistors (see Figure 2-1) which enables us to perform the 1/f

noise correlation experiments. Plus, we wish to prove the devices are

in the same thermal environment, which is expected of microscopically

close films on a good thermal conducting substrate. The method used will

be the same method incorporated by Kilmer [28] for the case of transis-

tors. One device in the group of three thin film resistors (see Figure

2-1) will be biased with an AC voltage source to act as a "heater" and



Figure 2-1 Layout of Closely Spaced Gold Thin Film Resistor Array

L ,I


measurements of the responses in the other two "sensors" at an exciter

frequency will exhibit thermal diffusion spectra and a correlation spec-

trum as a function of substrate temperature. A fully correlated coher-

ence spectrum is expected if the devices are in the same thermal envi-

ronment. Observation of an uncorrelated 1/f noise coherence spectrum

from the two devices shown to be in the same thermal environment would

eliminate Voss and Clarke's theory as a possible explanation for 1/f

noise in metals.

While the experimental proof to exclude Voss and Clarke's theory

is the primary objective of our research, we also will have the oppor-

tunity to observe the nature of 1/f noise at cryogenic temperatures.

Since no one has reported what 1/f noise does below 100K with our new

closed-cycle cryostat (more fully explained in Chapter IV), we will

be able to report these experimental findings.

2.2 Previous Results from Experimental Work on Metal Films

Briefly we have summarized what has been recently reported on 1/f

noise in metal films. Table 2-1 explicitly shows what temperature

ranges have been studied and what dependencies have been observed.

The numbers in the table indicate which researchers have studied which

metals. Below we list the key to the researcher numbers, and summarize

their results.

1. Clark and Hsiang (1976) [29]. Noise scales with (Volume)-
f I
and with R A low frequency flattening of the room tem-

perature 1/f spectrum is observed with 5,000 A Al underlay,

1/f Noise Experimental Results

from Metal Films

*The only substrates ever used in these measurements were glass or sapphire, therefore, substrate "divots"
or corregations were never a concern; except Fleetwood & Giordano who used oxygen glow discharge to "clean"
substrates and improve adhesion.


since underlay decreases thermal boundary resistance.

Measurements on both glass and sapphire substrates were made.

2. Ketchen & Clark (1978) [30]. Freely suspended films, (ie.

without substrate) show a flattening off of spectrum at

low frequencies and faster than 1/f above "knee."

3. Voss & Clark (1976) [15]. Manganin with low 8 shows a very

small 1/f noise. Bi, which shows a comparable noise magni-

tude as metals of the same geometry, but with a much smaller

.carrier density than the metals, suggests the noise should

be scaled by 1/Volume (or 1/Natoms) rather than 1/Ncarriers

4. Eberhard & Horn (1978) [31]. The temperature dependence of

the 1/f noise's magnitude and slope between 100K and 600K

is shown. Annealing decreases noise and increases the tem-

perature dependence.

5. Dutta, Eberhard & Horn (1978) [32]. In Ag "Type B" noise

dominates at all temperatures. In Cu (which has a lower

room temperature noise) crossover of "Type A" noise can

be observed below room temperature. No exponent dependence

on substrate is indicated.

6. Dutta, Dimon & Horn (1979) [33]. T dependence of noise

changes slightly with film thickness. Variation of the

exponent vs. T is shown to be consistent with the noise mag-

nitude changes vs. T. The sharper Guassian distribution of

activation energies in thicker films lends to the thought


that distribution width results from the number of sample

inhomogeneities. Bi shows peak just as Ag and Cu, however,

Au shows no peak--only flattening at 550K.

7. Fleetwood & Giordano (1982) [34]. Sapphire substrate al-

ways gave less noise than glass. They observed a trend

of better substrate adhesion (ie. by underlay) to lower

noise. Glass slides with underlay gave greater 1/f noise

slopes. Overnight aging removed burst noise.

8. -F-leetwood & Giordano (1983) [35]. They give a compilation of

room temperature resistivities, slopes, and noise magnitudes.

They suggest a 1/P modification to Hooge formula and use

N = number of atoms rather than carriers.

9. Fleetwood & Giordano (1983) [10]. 1/Natoms observed over
6 decades; no slope variation with sample volume are seen.

10. Scofield, Darling & Webb (1981) [25]. They show the

exclusion of Voss & Clarke's model at room temperature in

Au films.

2.3 The 1/N Dependence and the Hooge Parameter

Handel's theory is based on the velocity fluctuation of an indi-

vidual carrier (see Equation 1-18). Since v = Ep, we can write Equa-

tion 1-18 also as a normalized spectrum of mobility fluctuations of

a single carrier,

true (2-1)
-2 f


where a = 2aA (and not to be confused with the fine structure constant!).
The fact that the fundamental relationship for mobility fluctuations of

an individual carrier is expressed by Equation 2-1 (which obviously

excludes carrier number fluctuations since we are talking only about

a single carrier) has been postulated by Van Vliet and Zijlstra [36],

and van der Ziel and Jindal [37]. We now show how this formula is

modified for the case of a current spectrum (that which is actually

measured by the spectrum analyzer). A current is actually a macro-

scopic quantity comprised of a cumulation of microscopic events; that

-is, -

I = v (2-2)
L.=l i

where we assume there are N individual carriers of a charge q with

individual drift velocities, vd, and L is the device length. Now

we generalize Equation 2-2, for the purpose of noise investigations,

by assuming we have mobility fluctuations (ie. vd. v (t) = Ei (t))
1 1
and also number fluctuations (ie. N + N(t)). Thus,

1(t) =- (t) (2-3)

where N(t) = N, p.(t) = p, and the individual carriers are incoherent

(ie. i.p. = Pi ( 1) ). We now take the autocorrelation of both

sides of Equation 2-3, applying Van Vliet's and van der Ziel's Exten-

sion of Burgess' Variance Theorem [38], and find,

Al(t)Al(t+s) = -- p2 AN(t)AN(t+s) + N Ap(t)Ap(t+s) (2-4)


Now, applying the Wiener-Khintchine theorem we find,

2 2
S (f) = 2 S(f) + NS (f) (2-5)
2 j25 N SP M

2 2 2-2 2 2
Normalization of both sides with I = q E 2 N /L gives the generalized

expression for current noise when both mobility and number fluctuations

are present, first derived by Van Vliet and van der Ziel [38],

SI(f) SN(f) 1 S2
+-- (2-6)
2 2 N 2
I N (1)

Ignoring for the moment the possibility of number fluctuations,

(SN/N2 0), and substituting the normalized mobility fluctuation

spectrum for a single carrier Equation 2-1 into Equation 2-6, we arrive

at Hooge's formula,

S a
-- H (2-7)
12 fN

We see the N in the denominator accounts for the increase in the num-

ber of degrees of freedom associated with incoherent scattering. 'When-

ever the 1/N appears explicitly in the formula, a is defined as aH.
However, we see a problem with using aH in that we need 1/f (y 1)

in order to get a unique value for aH.

Looking only at number fluctuations as the source of the current

noise (ie. S /52 0), we see Equation 2-6 has no explicit 1/N depen-

dence. For number fluctuations to fit Hooge's phenomenological form,

we must assume,

a N
S (f) = (2-8)

where aK is due to Klaassen [39]. However, this assumption implies a

surface controlled density fluctuation model [40], and we have already

seen we are looking for a bulk effect in metals.

A better gauge than aH for the total observed 1/f noise magni-

tude, which does not presume one mechanism is the sole source of the

1/f noise, is the dimensionless "noisiness" factor,

S (fo)fo
Noisiness = (2-9)

It is the normalized current noise spectrum evaluated at 1Hz (ie. f
= 1Hz). Using this, we get a constant valve, regardless of slope,

which is especially beneficial when number fluctuation may give slopes

different from unity. This way the restrictions associated with aH

and aK are avoided and trends associated with N can be readily dis-

cerned. This is the approach we will take in plotting our data. Only

for relative comparison purposes, at the end of our investigations, we

will consider N in our calculations, to derive an effective true of
the mobility fluctuations of a single carrier, S /2 .

2.4 Temperature Dependence of 1/f Noise Mechanisms

In semiconductors, as we vary the ambient temperature, we effec-

tively vary the magnitude of the conduction electrons energy (or wave

vector), k, since the average energy transported by an electron is,

2 2
E(k) = 2m*- a kBt (2-10)

where C is the chemical potential, and m* is the effective mass [41].


Roughly, we have Ik ~ T/2. The incoming electron, k, is scattered

by a phonon, q, described by the elementary phonon emmission/absorb-

tion process,

+ -+ --
k' = k q (2-11)

where k' is the electron wavevector after the scattering. The super-

position of these elementary processes gives rise to macroscopic ob-

servables such as resistance and noise. Measuring the noise as a

function of temperature effectively probes electron-phonon scattering

mechanisms both in semiconductors and metals. Conceivably at low

temperatures, we may see a different type of noise since we may reveal

a different scattering mechanism.

The Debye model is definitely required at low temperatures, since

it appropriately describes acoustic phonons which are the only phonons

available at low temperatures [42]. Specifically, the model assumes

an upper limit to the allowable phonon energy (or wave vector) qmax

This defines the Debye temperature 9D = Uqmax/kB where u is the

velocity of sound in the material. The 6D is merely an abstraction,

its significance being only a convenient way to express the maximum

phonon energy. Since no observable physical phenomenon is expected

to occur at D, it is not a measured quantity and consequently some

controversy exists concerning its exact value for a specific element.

Observation of dependence of S /I2 at low temperatures will reveal

which of the competing theories apply. If a peak in S /I2 is observed

associated with a continuous slope variation, we may have another ex-

ample of Dutta & Horn's number fluctuation theory. However, if we see


the magnitude of the noise drastically drop with temperature, we may

find noise magnitudes which are on the order of the range predicted

by Handel's Quantum theory. For the case of impurity scattering in

metals, values of a 10- have been calculated by G. Kousik using

Handel's theory. However, for phonon scattering mechanisms with soft
photon emission, a 5 106 may be low enough for Handel's theory to

apply (see Chapter VI).


3.1 Introduction

In this chapter, we will give the theory underlying the thermal

fluctuation experiments to be described in Chapter IV and V. First,

we deal with the response function for the heat transfer. Next, we

consider the noise which would be observed if the noise were due to

spontaneous temperature fluctuations in the average temperature of the

film. Also, we compute the correlations between the noises of two

thermally coupled but electrically isolated resistors.

In Figure 2-1 we find the layout of the gold thin film resistor

array produced by Cornell's NRSSS facility as determined by a micro-

photograph. The power delivered by a resistor biased as a heater is
iv = i R, where R is the resistance of the film. If i = I cosw t, the
o o
AC heater power delivered is AP = AP cos2w t, with Ph =- R; the
h ho o ho 2 o t

exciting power frequency is thus twice that coming from the signal gen-

erator, w = 2w We suppose the power is uniformly dissipated along
e o

the length of the film, thus, neglecting current crowding.

In the sensor resistor, the AC voltage across the film probes the

temperature variation of the substrate. Denoting the AC signal developed

across the load resistor, RL, by vs, and the AC temperature of the sensor

by AT we have v = V cos(w t + f), AT =AT cos(w t + 0), with V =
s s so e s so e so
mAT We have according to Kilmer et al. [28],

m -V R+ (3-1)

where V and R are the sensor film's voltage and resistance, and $ is
o x
the temperature coefficient of the film (defined by Equation 1-19).

Thus, in summary, the power transfer takes place in the silicon

substrate between the two thin film resistors. To the sensor is deli-

vered a power AP APH hocos et, which causes a temperature rise AT =

AT cos(w t + A).
so e

3.2 The Heat Transfer Function

With no power introduced into the material, the heat conduction

equation and energy conservation theorem are,

K = -OVT, (3-2)

cd + V K = 0; (3-3)

here K is the heat current vector, indicating heat carried per second

through unit area, 0 is the heat conductivity, c is the specific heat

per gram and d. is the density. Substituting (3-2) into.(3-3) one obtains

the heat conductivity equation,

aV T = 0 (3-4)

where a = a/cd is the heat diffusivity. For the silicon substrate [43],

one has,

0 = 1.45 Watt/cm C

c = 0.7 Joule/gram C

d = 2.328 gram/cm3

a = 0.89 cm2/sec. (3-5)

If we now introduce a heating power APh(t) in a volume V, and if

we denote by AE the AC energy contained in V, we have the conservation


d AK dS + APh(t). (3-6)
dt h

Writing AE = jcdAT(r,t)d r and APh(t) = A (t)d r, we find by Gauss's
V h V h

dAT(rt) 33 13
cd dAr r = V AK d3r + APh(t)d3r. (3-7)
Dt V

Since this holds for an arbitrary volume, the integrands must be equal,

cd -- + AK = APh(t). (3-8)

With K = -cV(AT) this gives,

aT v2(AT) = 1 (r,t) (3-9)
t cdV Ph(

where APh(r,t) is a function equal to APh(t) for r Vh and zero outside

Vh, Vh being the heater layer volume. Equation (3-9) is the AC exten-

sion of (3-4) under conditions of an external heat supply. The r.h.s.


of (3-9) will also be written as APh(t)E(r), where e(r) is the function

which is unity in Vh and zero outside. For APh = AP he (3-9) gives

if AT = AT ejt,

joAT (r) aV2AT (r) = AP (r) '(3-10)
W W cdV Wh

We define the Green's function of (3-10) as usual by the impulse

function response,

S jwG(r,r',jw) V2 G(r,r',jw) = 6(r-r'), (3-11)

subject to boundary conditions which we discuss in Section 3.6. The

solution of (3-10) is then [44],

AT (r) = G(r,r',jW)E(r')d r'

=- G(r,r',j)d r'. (3-12)

Denoting by ATs the averaged temperature increment in the sensor, we

S 3h
aTs cdV Vh f 3r d
s h

Let AT s= AT sle. Then for a real input signal with amplitude Pho'

the real sensor output signal appearing in the sensor circuit is v =


V cos(w t + 0) where w = w and Vs = mIAT ,si
so e e so s

vs = cdV h dr d r'G(r,r',jw)I cos(wet + 9). (3-14)
Vs Vh
s h

The sensor signal is, as expected, fully determined by the Green's

function. For two cases this can be simplified. Suppose that either

Vs and Vh are very small with respect to the area over which the Green's

function changes appreciably, or that there is a frequency range in

which G is independent of r and r'; then, we can approximate (3-14) by,

Vs (mPho/cd) IG(rsrhjy)I cos(wet + 4) (3-15)

where ( is the phase angle of the Green's function.

We must now consider the measurement of vs by a Hewlett-Packard fast

Fourier transform analyzer. This machine measures the power spectrums

of the signal at its entrance. Thus denoting the measured quantity by

d[v v* ],
sW sW

d[v vs] = 2Af v (t)v (t + T) e- TdT (3-16)
sw sw s s

where v (t) = V cos(w t + () with V given by (3-14) or (3-15). Now,
s so e so

v (t)v (t + T) = V cos(w t + 4)cos(w t + w T + f)
s s so e e e

= V [cos2 ( t + f)cosW T sin(w t + ))cos(w t + f)sinw T]
so e e e e e

3jW T -jWe T
S(e +e )so
4 so

Using further,

co-j (C + (e)T

e e dT = 2Tr6(w e)

we find,

d[v v~* = V2 [6(w W ) + 6(W + w )]Af.
s sw so e e

now realize

w, where Aw

centered on

that the analyzer integrates over a bandwidth Aw centered

= 2rAf. The output is therefore a signal over the range

we, of magnitude,

S V2 soE(We)
V* so e
sW sW 2
e e


where (W ) is the function which is unity in the interval AW centered

on w and zero outside. Hence,

22 2 f
ho 3 3 2
v V* vh = I d rd r'G(rr', j) I
Se e 2c d V
sWe sWe 2c2d2V2V J

s Vh

for the case of Equation (3-14), or,

v v* W (m2P2 /2c2d2) IG(r ,rh ) 12
Sho s h
e e






for the case of Equation (3-15). The linear signal response is,

Iv 2= mPho d3r d3r'G(r,r',jW) (3-23)
e TcdV Vh

s h
Vs Vh

It should still be noted that, since the machine measures the output in

dBV, one can equally well read the logarithmic output power, by divid-

ing by ten, or the logarithmic output amplitude, by dividing by twenty.

The latter is done for the figures of Chapter V, which give the wave

analyzer's input in rms volt. Notice from (3-20) that this is also

the sensor rms voltage signal.

3.3 Heat Transfer Correlation

We also describe the theory of experiments in which the signals,

vsl and vs2 of two sensors, having a temperature fluctuation due to a

common heater power APh = APh cos2o t are cross-correlated.
h ho o
For each sensor the circuit output voltage is given by a result

like (3-14). The analyzer then measures the spectral power,


Re v v = 2Re v sl(t)vs2(t + T)e dT (3-24)


where vsl (t) = V scos(w et + 1) and vs2(t) = V s2cos(w t-+ _).
sl slo e 1 s2 s2o e 2

Cos(W t + W T + 2) = {cos(w t.+ (l)cos(W T + 42 cI)

-sin(w t + ) )sin(e T + P2 1 ) (3-25)


we obtain in a similar way as before,

Rev l v2w = VsloV cos( )E(W). (3-26)
slw s2W slo s2o 2 1 e
e e

The coherence factor is defined as,

Re v v*
slw s2w
y(W ) = e (3-27)
[v v* \v v* ]
slwv slw s2w s2w
e e e e

From (3-26) and (3-20), we find,

Y(We) = cos(2 1). (3-28)

We notice that the machine measures lyl. Here pk(k = 1,2) is, for the

simplified case of (3-15), given by,

(k = phase angle of [G(rsk,rhj))]. (3-29)

Though y should be oscillatory, the higher mixima may not be

noticed due to the presence of noise. Let vnk refer to the sensors'

noises. Then for y we obtain, noticing that signal and noise are un-


Re v + Re vv*
slw s2w nl n2
( )--e e (3-30)
{[v v* + v v* ][v v* + v v* ]}
slw sl w nl nl1 s2' s2w n2 n2
e e e e


(for the connection with the notation in the next section, we have

v v* = S (w )Aw, etc.). If the noise drops slower with frequency
nl n vs e

than the signal response and if v nv* = 0, the above gives for suf-

ficiently high frequency, in the case that (3-15) applies,

Re v v*
slw s2w
y(large w ) = e
e 1/2
[v v* v v*
nl nl n2 n2

Tm2 ho G(rs,rh,j) IG(rs2,r,jW) cos(2 4)
m ^ho rsi5 h" s^ 2 1
2 2 21/2
2d v* v v* l
d[V1 nlv n2V2

3.4 Temperature Noise

When considering the noise, we can use the same approach, taking

into account the noise source of the Nyquist type in the heat current.

Thus instead of (3-2) we have,

AK = -GV(AT) + n(r,t) (3-32)

where AK and AT are spontaneous fluctuations in K and T and where r(r,t)

is a source with spectrum [27],

S (r,r',W) = 4k [T (r)] 2(r)6(r r')I (3-33)
fl B o

where I is the unit tensor. We assume the steady state temperature

T (r) = T is uniform, and G(r) = 0. For the conservation we have again,
o o
analogous to (3-3),

cd + V (AK) = 0. (3-34)

From (3-32) and (3-34) we obtain,

AT CaV (AT) = V T (r,t). (3-35)
t A cd

For the spectrum of 5, from (3-33) we have,

S (r,r') = 4k T2V V'6(r r')/c22 (3-36)

where V' is the del-operator with respect to r'. We note that in con-

trast to the hypothesis by Voss and Clarke [15], the spectrum of the

source is not a delta function, but is the more singular function

V V'S(r r'). We now represent AT and by truncated Fourier series

on the interval (0,T), with amplitudes AT(r,w) and E(r,c). From (3-35)

we obtain the relation,

jWAT(r,w) aV2AT(r,w ) = (r,w). (3-37)

Using the Green's function (3-11) we find the solution,

AT(r,w) = G(r,r',jw)E(r',w)d3r' (3-38)


where Vtotal is the entire volume subject to heat diffusion, ie. the

complete integrated circuit. For the spectra of AT(r,t)AT(r',t) we have

as usual [17],


S T(r,r',W) = lim 2T AT(r,w)AT*(r',w), (3-39)

and similarly for S Thus from (3-38) and (3-36) we obtain the "re-

sponse form" [45],

ST(r,r',w) = d3rl d3r2G(r,rljw)G(r',r2,-jw)(4k T2/cd2)V V26(r -r

total total (3-40)

Providing that in Green's theorem for the V2 operator the bilinear con-

-comittant-of G(r,rI,jw) and G(r',r2,-jW) vanishes,the singular distribu-

tion V1 V26(rl r2) can be replaced by,

V, V26(rl = [ 6(r r) + V26(r r2)]. (3-41)
1 2 1 2 2 1 2 2 1 2

(We note that this replacement amounts to a partial integration whence

the conditions on the bilinear concomittant). Employing the well-known


f(r)V26(r ro 3r V f(ro), (3-42)

we arrive at,

SA(r,r',) = c- 2 dro[G(r,ro'j)2G(rlr o-jw)


+ G(r',r -jVG(r,ro,j )] (3-43)


Now substituting from the defining equation for G, see Equation (3-11),

V2G(r',r ,-jw) = jwG(r',r ,-j) + 6(r' r ), (3-44a)
cd o o o


V (rr ,jw) = -jWG(r,r ,jw) + 6(r r ) (3-44b)
cd o o o

we find from (3-43),

2k T
SAT (r,r',) = [G(r,r',jw) + G(r',r,-jw)]. (3-45)
SAT(rrw) cd

This is the van Vliet-Fassett form [17], since [kBT /cd]6(r r') is the

covariance AT(r,t)AT(r',t). For the spatially averaged temperature fluc-

tuations in a volume V we have,

T (t) = -- AT(r,t)d3r, (3-46)
s V

and for the autocorrelation function,

AT (t)AT (t + T) = d d3rd3r' AT(r,t)AT(r',t + .T). (3-47)
s s 2
s V V
s s

Whence by the Wiener-Khintchine theorem,

1 3 3
ST (W) = d rd rSA (r,r',w). (3-48)
s s

Employing (3-45) this yields,

4k T
SA () = 2 Re G(r,r',jw)d rd r'. (3-49)
s cdV

Since G has usually a singularity for r = r', the intervals of inte-

gration must be broken up accordingly. Also, it is not possible to ap-
proximate the integral by V times the integrand since G varies strongly
in the neighborhood of r = r'.

For the noise in the sensor, we have,

S (t) = mS (0)
s s

4k T m2
=- Re G(r,r',j)d 3 rd r'. (3-50)

Comparing the noise (3-50) with the linear response vs 12 given in
(3-23), we note that both are double integrals over Green's functions,

though over different volumes.

3.5 Temperature Noise Correlation

We consider the spontaneous noise correlation of two devices (the

same as the previous two sensors). For the spatially averaged tempera-

ture cross-correlation, we now have, analogous to (3-47),

AT (t)AT (t + T) = d3rd3r' AT(r,t)AT(r',t + T) (3-51)
V V2
Vsl Vs2


where we assumed that both sensors have an equal volume V From the

Wiener-Khintchine theorem,

S = d3rdr'S (r,r', ). (3-52)
AT sIAT s2 = 2 AT
s S

Using (3-45), this yields,

2k T
S 2B d3rd3r'[G(r,r',ja) + G(r',r,-jw)]. (3-53)
ATsl'ATs2 cdV2
sl s2

For the cross-correlation of the noise voltages of the two sensors, we


2 2
2k T2 m2 I
Bo 0 13 3
S d rd r'[G(r,r',jW) + G(r',r,-jW)] (3-54)
l ,2 cdV2
sl s2

where we assumed for simplicity that both sensors have the same tem-

perature coefficient m. For the coherence factor of the noise, we have,

Re S (W)
Y(w) = sl s2 (3-55)
[S 1w) S (w)]
Vsl Vs2

Thus from (3-54) and (3-50),

1 3 3
Re [G(r,r',jw) + G(r',r,-jw)]d rd r'

ysl s2

[Re G(rr',ja)d3rd3r']/2[Re jG(r,r',j)3d3rd3ri']/2

VslVsl Vs2 s2 (3-56)

We note that because of the self-adjointness of V G(r,r',jw) = G(r',r,jw),
3 3
so the numerator is also Re f fGd rd r'; hence,

Re G(r,r',jW)d3rd3r'

y(w) = s2

[Re G(r,r',jw)d3rd3r']l/2[Re G(r,r',jl)d3rd3r'l /2

VsVs V V
VslVsl Vs2 s2 (3-57)

If there is a frequency range for which the integration of G is insensi-

tive to the change f f f = f/ we find Y = 1.
VslVs2 VslV sl s2Vs2

3.6 The Green's Function for the Gold Thin Film Array

In the experimental arrangement for the heat transfer, we used one

thin film resistor as a heater and the other two resistors as sensors

(see Figure 3-1). The geometry of the thin film resistor (ie. 1pm wide

by 800pm long) allows us to regard the lengths of the devices as near

equal temperature fronts. If we assume the power is dissipated evenly

along the resistor length (ie. we neglect current crowding at the sharp

corners--see Figure 3-1), the gap and the heat transfer is basically

one-dimensional, being along the x-axis. Concentrating on the latter,

let us assume that there is a boundary at x = C, with boundary conditions

that the heat flow beyond C is zero. Thus the Green's function must


8G(x,x',jw) = 0 (3-58)
3x x = C

t y-axis


O- I






-0 D-




-L L


Figure 3-1 Layout of a Chip Containing Three Different Groups of Thin Film Resistor Arrays


The solution for this one-dimensional Green's function is given in the

article of Van Vliet and Fasset [17, eq. (343)*]. They find,

G(x,x' ,j) = (c(sinhyx+tanhyCcoshyx)(sinhyx'-tanhyCcoshyx'), (x 2ya

G(x,x',ja) = cthy(sinhyx'+tanhyCcoshyx')(sinhyx-tanhyCcoshyx), (x 2ya


Y =\/jW/a ..

We define a corner frequency,

bound = /C .



Then yC = / / For

and since Ixl < ICI, sinhyx


<< Wound. YCI 0. Thus tanhyC -- yC,

yx, coshyx 1; for (3-59) we now find,



the heat transfer function (3-22) thus becomes a constant at low fre-


For frequencies w >> Wbound' tanhyC -- 1. We then easily find,

Note that there is an error in this formula. The C's should be in

front of the cosh terms and not in front of the sinh terms.


G(x,x',jW) = _--2 e- |xa (3-63)

which is the one-dimensional infinite domain Green's function [17, eq.

(314)]. If we take specifically the heat flow between the two resis-

tors symmetric to the y-axis on Figure 3-1, we have,

-L +L

d3r' d3rG(x,x',j)) : dx' dx G(x,x',jW)

Vh V -E-L L
h s

1 (e-YL e-( L))2. (3-64)
2 '-Y2

We introduce another corner frequency,

ti = a/(L + ) 2. (3-65)

For bound << W << WI, we have jy(L + E) << 1, and a fortiori IyL << 1.

Thus expanding the exponentials, we obtain,

2 2 2 VV
jG3 3 h, (3-66)
d3r' d3rG(x'x''JI ) 2 2- (3-66)

Vh V
h s

where A and B are the pertinent dimensions in the z and y directions.

The linear response corresponding to (3-23) now becomes proportional to,

d1 d3r' d3rG(x,x',jw) = (3-67)


(ie. we expect a square root frequency dependence). We also notice that

for w << 0 c = -TV/4, independent of the coordinates. Thus for 0 << wl'

the signal correlation y(w ) 1.

For w >> l, we find Iy(L + E) and IYL| are both greater than unity

and (3-64) becomes,

r VhVs

d3r' drG(x,x',jw) VhVs .(3-68)

Vh V
h s

Writing this in the linear response form and using (3-60) we see,

1 1
Sd r' d 3rG(xx',jo) -- (3-69)
VhVs v, 3/2/

(ie. we expect to see an w-3/2 dependence above the corner frequency w ).

An -1/2 spectrum which becomes an -3/2 spectrum above a corner frequency

is typical of one-dimensional diffusion spectra.


4.1 Devices

Gold thin film resistors, 2,000 A thick, deposited on a 200 A

Chromium layer adhering to a standard oxide coated silicon wafer

-have been prepared for us by Dr. E. Wolf and Dr. R. A. Buhrman of

the National Research and Resource Facility for Submicron Structures

at Cornell University. The standard configuration of three resistors

closely spaced for high thermal conductivity (see Figure 2-1) is

repeated for varying widths and spacing of d = 0.5pm, lpm, and 2pm as

designed by Dr. Van Vliet (see Figure 3-1). The choice of Au films

on a Si substrate is interesting since the excess weakly temperature

dependent noise ("Type A", according to Dutta and Horn) is lowered

with a strong conducting Si substrate and the strongly temperature

dependent 1/f noise ("Type B") must predominate. Note that this is

in contrast to the original studies on Au films by Hooge and Hoppen-

brouwers who had their films on glass and were supposedly observing

the "Type A" noise in Au.

Samples are diced and mounted with silver epoxy glue (for thermal

conductivity into a TO5 can). An ultrasonic bonder was used to bond

gold wires to the 100pm pads on the devices (dimension "a" indicated

on Figure 2-1) and to the TO5 can posts. From the measured resistance

and dimensions of the thin film resistors, we find the resistivity to

be slightly greater than 2.35pQ-cm as listed in the Handbook of



Chemistry and Physics. The resistivities of metals deposited as

films is expected to differ from that of bulk--since the atomic stack-

ing in films is different from bulk, and dangling bonds and vacancies

may be present in thin film geometries. This is in agreement with

the observations of Fleetwood and Giordano who found that resistivities

for the same element deposited as thin films could vary by as much as

a factor of 10 [35].

4.2 Apparatus

4.2.1. The Closed Cycle Cryostat

Once in the T05 can, the device can be placed in the cryostat.

The cryostat is a CTI Cryogenics Model 21 liquid He closed-cycle re-

frigerator with a temperature controller that can be set to maintian a

stable temperature (ie. 0.1K over the duration of a low frequency

noise measurement) anywhere between 300K and 10K. The cold finger has

been designed to hold a T05 can, and the controller's temperature mon-

itoring diode can be mounted directly to the sample mount to get an

accurate reading of the T05 can's temperature. The T05 can mount and

cold finger are in a vacuum chamber to eliminate thermal conduction.

Six vacuum sealed coaxial feed-throughs are provided; however, the

leads are long (for pre-cooling purposes) and are spiralled down along

the cold finger causing magnetic pickup of 60Hz and 3Hz-(the cold head's

compressor frequency) harmonics. The pickup was eliminated by elec-

trically isolating the device from the cold finger and by using a short

lead entering through a vacuum chamber window. However, the short

leads are not pre-cooled and deliver heat directly to the metal film

causing its temperature to be higher than indicated by the diode sensor


(this is the subject of Section 4.3 and 4.4). Attempts to pre-cool

the short leads result in pickup; therefore, we trade off some low

temperature capabilities for a "clean" noise spectrum.

4.2.2 The Flow Cryostat

A flow cryostat was used to verify the trends of the noise

versus temperature observed from the device in the closed cycle

cryostat (see Section 5.4). The Cryosystems CT-310 Cryotran Conti-

nuous Flow cryostat requires an externally supplied dewar of liquid

-nitrogen or liquid helium to cool the sample. A similar heater/

controller is used to stabalize the cold head to the desired temper-

ature. A flow cryostat is preferable to a closed-cycle cryostat for

noise measurements since there are no mechanical vibrations from a

cold head compressor. However, the need for the continuous supply

of a liquid makes the instrument more complicated and costly to oper-


4.2.3 Calibrated Noise Measurements

The measurement scheme used, incorporated a calibrated noise

source (see Figure 4-1). Using this method, one can calculate the

absolute magnitude of the DUT's current noise spectrum, Sx, simply

by comparing the relative magnitude of the noise spectra of device-on,

device-off, and calibration source-on. This is illustrated by the

equivalent circuit shown in Figure 4-2. For our case, we define,

RS = 11.29K + (5.6KI 5.6)
cal R 2
R Af

Figure 4-1 Calibrated Noise Measurement Scheme


SCL S Is[ S. 4r @ R vS YVAOUT


Figure 4-2 Equivalent Circuit of the Calibrated Noise Measurement Scheme


2 2 Rx RI l IRS
Z = R (4-1)

We can write three equations for the three different measurement cases,

v01 G2 s )z 2
1) DUT ON = G2 (S + + S)2 + S (4-2)
Af x th a a

V02 2 i 2 v (43)
2) CAL ON = G (S + S + S )Z2 + S (4-3)

af th 2 a v
3) DUT OFF = G (St + Si)Z + S (4-4)
Af th a a

where G is the amplifier's gain, S is the amp's equivalent current
noise source, and S is the amp's equivalent voltage noise source as
indicated in Figure 4-2. Solving for the excess device noise term, Sx,

we find,

-2 -2
v01 V03
x -2 --2 cal
02 03

In essence, four separate measurements must be performed in each fre-

quency range to reveal the device's current spectrum. However, the

tedious calibrated noise source method is worthwhile since it permits

us to find the absolute magnitude of the device's noise spectrum with-

out using any amplifier parameters.

4.2.4 The Measurement System

We wish to measure simultaneously the current noise spectra

from two closely spaced thin film resistors in order to see a co-

herence spectrum between the two 1/f spectra. This is the method

used by Scofield and by Kilmer for the case of transistors to refute

the temperature fluctuation model of 1/f noise. This is why, in

Figure 4-1, one sees two parallel measurement schemes and amplifiers

feeding into a dual channel FFT spectrum analyzer (HP 3582A) which

can display the coherence between the two channels. The Schmidt PAl

low noise amplifier (LNA) [46] on Figure 4-1 consist of high P PNP

transistor cascaded with a low noise Burr Brown op amp (see Figure

4-3). The LNA gives +90dB of power gain down to 1Hz below which ex-

cess cryostat noise becomes a problem at low temperatures. An HP9825A

desktop calculator samples the spectrum analyzer and performs the cal-

culations indicated in Equation 4-5. From the Hooge formula we know

the magnitude of the 1/f noise is proportional to the square of the

current, and we expect we will not be able to observe any device

1/f noise unless we bias the device with an appreciable current (eg.

I = 10mA), since the LNA has 1/f noise itself. If the relative mag-

nitude of the DUT ON measurement exceeds the relative magnitude of

the DUT OFF measurement, we know we must be observing true device


For the case of the thermal transfer experiment (referred to

hereafter as the "heater experiment") the calibrated noise source

method is not needed since the induced response in the sensors is at

a specific frequency and is well above the LNA noise. To avoid the

problem of capacitive coupling in these closely spaced resistors, the

487 K


Figure 4-3 Schematic of the Schmidt






Low Noise Amplifier


response in the sensors will be measured at exciter frequencies which

are sum and difference frequencies of two close fundamentals driving

the heater. Measuring the response at these "mixed" heater frequen-

cies will avoid the linear capacitive coupling since only the non-

linearity of thermal power transfer (Joule effect) can produce the re-

sponses at the sum and difference frequencies.

4.3 Thermal Noise Measurements

The cryostat has a well controlled temperature and a digital

readout indicating the cold finger's (thermal reservoir's) temperature.

The gold film is on a Cr layer adhering to a Sio2 substrate, Ag pasted

to the T05 can which is pressed to a gold plated copper mount isolated

from the cold finger by an electrically isolating but thermal conduc-

ting material (ie. alumina, Be02, or sapphire). In addition to the

possibility of one of the aforementioned items causing a thermal

barrier (specifically the Cr adhesion layer), we have the direct heat

injection into the film by the non-precooled leads mentioned in

Section 4.2.1. Consequently, we cannot assume the device is actually

at the cold finger temperature indicated by the digital readout.

Therefore, we have the need for thermal noise measurements. We must

measure the high frequency "white" current noise (thermal or Johnson

noise) which for a resistor is, --

Sth = 4kT/R (4-6)

accurately enough to calculate T.


4.3.1 Thermal Noise Measurements Using the Calibrated Noise Source

In a fashion similar to that described in Section 4.2.3, thermal

noise can be measured. Equation 4-5 shows the magnitude of the ex-

cess device noise term, Sx, which dominates the device noise spec-

trum at low frequencies. If we use our calibrated noise measurement

system at higher frequency where Sx is comparable to Sth, we must

consider the thermal noise term. The total device noise is expressed


-2 -2
01 03
total -2 -2 cal 4kTDUTDUT (4-7)
V -V
02 03

where TDUT and RDUT are device under test's temperature and resis-

tance. Equation 4-7 can be modified to calculate the absolute magni-

tude of the devices'thermal noise (and consequently the devices'actual

temperature) by using a dummy load resistance, RDUMMY, at a known tem-

perature for the CAL ON (Equation 4-3) and DUT OFF (Equation 4-4)

measurements. Under these circumstances and at frequencies where the

DUT ON noise is white, Equation 4-7 is written,

-2 -2
01 -03
4kT /R 01= SAL + 4kT /RMY (4-8)
02 03

This equation is only valid, if we have the dummy resistance at a

known temperature, such as room temperature (ie. TDUMMY = 300K) and

we match the dummy's resistance with the device's resistance at the

unknown temperature (ie. RDUMMY T=300K RDUT ). Then, we can

theoretically solve for T We say theoretically because there areDUT
theoretically solve for TDUT. We say theoretically because there are

-2 ? -2 -2 -2
many sources of error, especially in the (V V )/(V V )

term. Since here, we divide two quantities which are close, the

statistical accuracy is low. Measurements by this method gave re-

sults with close to an order of magnitude deviation! Therefore,

we need an alternative method to measure thermal noise and to calcu-

late TDUT.

4.3.2 Thermal Noise Measurements Without Using the Calibrated Noise

The calibrated noise source is not a good guage for the thermal

noise of the devices we have. We need to gauge our thermal noise to

known thermal noise of a comparable magnitude. With this in mind,

Dr. Bosman devised a thermal noise measurement scheme which involves

comparing the noise of our device, Sth, to the thermal noise of a

dummy resistor at one fixed temperature (e.g. melting ice T1 = 273K),

S and to the thermal noise of a second dummy at a second fixed

temperature (e.g. liquid Nitrogen T = 77.5K), S Thus, we have
again from Figure 4-2 with RS m since we use no calibration source,

2 2 Rx RL 2
Z = R. + (4-9)
A + Rx RL

For the three difference measurement cases we have, since we measure

at high enough frequencies that S -+ 0,


01 2 2 V
1) DUT ON = G2 + S )Z + S (4-10)
Af th a a

V02 2 i 2 V
2) DUMMY AT T = G (S + S )Z + S (4-11)
1 Af T a a
1 *



03 2 i 2 Vi
v G 2 +S a)Z + S
Af (ST 2 +


Solving for Sth, we find,

-2 -2 4
V V 4kT
01 02 4
S (T T ) + 1
th -2 -2 R 2 1 R
03 V02


where we have assumed the necessary condition R = RDUT = RDUMMY. We

see we can simplify (4-13) to express the actual device temperature,

DUT as,

-2 -2
01 -V02
T = -- (T T ) + T
DUT -2 -2 2 1 1
V 02
03 02


4.3.3 Accuracy in Thermal Noise Measurements

To get a grasp of the accuracy involved in calculating TDUT using

Equation 4-14, we see that this equation has the general form,

T = Ax + B,


where x represents the (V01

we see,

-2 -2 -2
V )/(V V ) term. Differentiating
02 03 02

AT = AAx .


Now we normalize since we are interested in relative error,


= ( (4-17)

2 -2 -2 -2
We see the statistical accuracy of the (V V )/V V 2 term,
01 02 03 02
X must be ~1% in order to have at least a 10% accuracy in since
x T
the factor Ax/T ~ 10 when T1 = 77K and T2 = 273K. By making T1 and T2

closer (ie. T1 at liquid nitrogen and T2 at liquid oxygen or liquid

argon) A can be reduced, but at low temperatures the factor Ax/T can

still cause problems.

Concerning the statistical accuracy of the ax term, increasing
Concerning the statistical accuracy of the term, increasing

the measuring bandwidth, Af, and averaging time, T, of the detector

will decrease the statistical accuracy since we have from Van der Ziel


Ax -1/2
= (2AfT) (4-18)

The maximum bandwidth available with the HP 3582A FFT spectrum analyzer

is 726Hz and with the maximum number of averages, 256, we obtain a

2.1% statistical accuracy from Equation 4-18. Using the HP Wave Ana-

lyzer, a maximum bandwidth of 3kHz can be used and the machine can

measure at frequencies far above the 25kHz limit of the FFT machine.

Theoretically the Wave Analyzer should have a 0.7% statistical accuracy.

In an attempt to improve even upon this, a "super bandwidth" sys-

tem was configured where we used the entire frequency range of the LNA.

We used an active bandpass filter with high frequency cutoff of 100kHz

corresponding to the LNA's corner frequency. Once filtered the noise

power is measured using an HP digital true RMS meter. With this method

only a single temperature can be calculated, therefore, the statistical

error is low. However, experiments showed, a type of systematic error

predominates with the "super bandwidth" system giving rise to erro-

neous temperatures. For the best tradeoff between systematic and

statistical error, the center frequency should be four times the

bandwidth. This means we would need to measure near 400kHz, and we

cannot do that with our LNA.

However, a more fundamental problem with our thermal noise mea-

surements predominates. Our devices have a typical resistance of

about 100Q at temperatures below 100K,and we must calculate what per-

centage of the total noise at the amplifier's front end is the device

noise we are interested in. In the case of a bipolar junction tran-

sistor LNA, such as our PAl, two noise sources at the transistors base

(the amplifier's "front end") compete with the device noise. The noise

sources are characterized by an equivalent voltage noise source, 4kTRn,

and an equivalent current noise source, 4kTgn (see Figure 4-4). In

Figure 4-4, the current noise source is represented as a voltage noise

source by multiplying by the device resistance, Rx, squared, where we

assume the device resistance is small compared to the amplifiers input

resistance. In this way, both of the amplifier's noise sources can be

combined and directly compared with the device thermal noise 4kT R .

If the device noise amounts to 50% of the total amplifier's noise,

we consider that the device noise-can be "seen". Herein lies the

fundamental problem; because, while Rn and gn are typically small in

a good LNA, the amp is at room temperature (ie. T = 300K) and we '..

wish to measure device noise near 10K. To realize what constraints

we have on T and R in order that the devices thermal noise can be
x x
"seen" with a given amplifier (ie. Rn and gn are known), we must solve

the equation,

r - - -
1 -2


Figure 4-4 Simplified Equivalent Circuit of the Low Noise Amplifier

4kT R = 1/2 4kTR + R 4kTg, (4-19)
x xn x n' (4-19)

derived from Figure 4-4. This equation has the general form, T
A/R + BR which is a skewed hyperbola in the T R plane. The

skewed hyperbola will have a minimum corresponding to the lowest re-

sistance, (R ) min needed to see its thermal noise at the lowest

temperature, (T ) m. Setting the derivative equal to zero, we find,
x min

= Rn
(R)m=. ~ (4-20)
x min gn


TR ]
(T ) = 1/2 n + Tg (R) (4-21)
xx min

From Schmidt's PhD thesis [46, p. 33], we know for the PAl LNA;
Rn = 352 and g ~ 10 4U. This gives (R ) = 5900 at (T ) e 17K.
n in x mn x mn
Bob Schmidt designed a PA2 LNA where the R is decreased by employing

five transistors in parallel at the front end. The tradeoff, of course,

is an increase in g and a decrease in the input resistance, RA. Ex-

perimental results using the PA2 and a specially designed two transis-

tors in parallel LNA were inconsistent indicating the difficulty of

trying to measure the thermal noise of a small resistance at low tem-

peratures by these methods.

4.4 Thin Film Heating Effects

In the beginning of Section 4.3, we touched upon one reason why

we need to determine the true temperature of an unbiased thin film


(ie. to see the effect of thermal barriers and lead heat injection).

A still more important reason is: In order to observe the device's

1/f noise, a sizable current must bias the device (typically a few

milliamps). For a film with a cross section of lpm X 0.21m only 2mA

of current is required to generate current densities ~106 A/cm2

With such a large current density, trying to measure the exact

temperature of the film is similar to trying to measure the exact

temperature of the filament of a heater! This is not a new problem,

however, and determine the actual temperature of the film has been

a major concern of all the researchers of 1/f noise at cryogenic

temperatures. Voss and Clark extensively talk of this problem and

show a nonlinear I-V curve due to local sample heating causing a

resistance increase with larger currents [11, Figure 3].

Eberhard and Horn [31, p. 6634] give a solution to the temper-

ature measurement problem which we have adopted. First, an accurate

plot of the sample's resistance versus temperature is made. For

resistance measurements the "pickup" introduced from the precooled

leads is not critical and pulsed V-I measurements with short pulses

and a low repetition rate give resistance values corresponding to an

unheated device (refer to Chapter V). Then the device is mounted in

the "clean" spectrum configuration (with the non-precooled leads),

and the noise is measured (with the large bias current) as a func-

tion of the device's resistance. In this fashion the device serves

"as its own thermometer." The only drawback of this method is in

metal films--the resistance versus temperature coefficient, 8, is

not large especially at low temperatures and can introduce some



According to Scofield et al. [25], local heating was not a

problem with their high thermal conducting sapphire substrate,

however, their measurements were only at room temperature. Most

substances which are considered good thermal conductors (eg. sapphire,

Si, and Cu) have a peak in their thermal conductivities below 50K

[48, 43, p.43]. This, combined with the fact that the closed cycle

cryostat has a 2 watt cooling capacity above 20K, would seem to imply

that we should have no problem cooling one of our biased thin films

(typically dissipating only few milliwats of heat). The results of

our endeavors are given in the next chapter.


In this chapter we describe what was observed from the experi-

ments set forth in Chapter IV.

5.1 Thermal Transfer Function Experiment

We have measured the thermal transfer function between a resistor

biased as a "heater" and two resistors biased as "sensors". The ther-

mal diffusion spectra obtained are similar to the ones observed by

Kilmer in transistors. However, the results are even more pleasing

because we are able to observe the characteristic frequency where the
-1/2 -3/2
spectrum changes from f/2 to f3/2 (see Figure 5-1) predicted from

the diffusion theory. We are able to see the corner frequency, fdiff'


f =-- (5-1)
diff T 2f
diff L

where D is the thermal diffusion constant and L is the distance be-

tween heater and sensors. In the present case, we were able to use a

heater almost the length of a chip (ie. L ~ Imm) away from the sensors

causing the corner frequency to be at an observably low frequency (ie.

f 30Hz). This was not possible in the case of transistors since

the devices were directly adjacent to each other and had a corner







, a


eater Power 0
408 pW

U (U ~v\


10 fdr 100


f. (Hz)-

Figure 5-1 Thermal Transfer Responses of Two Thin Film Resistors


frequency beyond the range of observation. Again the thermal diffusion

appears to be one-dimensional in nature as was the case in the transis-


In Figure 5-2, we show the correlation between the two sensors'

thermal transfer responses at the exciter frequency. We see the two

resistors have full coherence at low frequencies where the responses

were well above the background noise. This proves the two devices are

indeed in the same thermal environment.

5.2 Thin Film Heating

According to the Dutta and Horn procedure mentioned in Section 4.4,

the severity of the sample, heating was determined. Pulsed V/I measure-

ments with low duty cycles gave the same results as the resistance

measured directly with an HP3466A digital multimeter on the ImA test

current range (see Section 5.3). Essentially, we see no heating effect

when the device is biased with lmA of continuous current. This fact was

confirmed through relative thermal noise measurements versus device

bias current (see Figure 5-3). In Figure 5-3, the reservoir was set at

10K (where the sample heating is expected to be most pronounced) and

values of thermal noise as determined from Equation 4-8 are shown for

increasing device current. As explained in Section 4.3..3, the accuracy

is not enough to determine an absolute TDUT, but relative changes in

thermal noise from the value of the unbiased thermal noise can be seen.

From Figure 5-3, we can grasp the severity of the sample heating. We

conclude that the silicon substrate is a good enough thermal conductor

that the thin film heating effect is only pronounced at low temperatures

and under high bias currents. In general, the effect simply offsets





@@=@=BEEr- FB=?f-g---o

Heater Power o
o 408 uW
E 70 .pW

10 100
fe (Hz)--


Figure 5-2 Coherence Between Thermal Transfer Responses



0 I 2 3 4 5 6 7 8 9 10 II 12 13 14
-- Current (mA)-

Figure 5-3 Relative Thermal Noise Versus Device Bias Current


the displayed cold finger temperature by a few degrees over most of

the cryogenic temperature range.

5.3 Resistance Versus Temperature

For each device measured, an accurate R versus T plot is required

to correct for the thin film heating (refer to Sections 4.4 and 5.2).

A typical resistance versus T plot (as measured by the HP3466A multi-

meter on the lmA current range) is shown in Figure 5-4. We see at

low temperatures the resistance approaches a constant due to the limit

where scattering becomes impurity dominated. This is predicted by

Kittel [49] and is referred to as residual resistance. At high tem-

perature, the resistance, which has a linear dependence with tempera-

ture, exhibits Matthiessen's Rule [42].

5.4 1/f Noise Versus Temperature

The current spectra of the gold thin film resistors were measured

at different currents to check the I2 dependence (see Figure 5-5). We

have quantitatively compared the room temperature noise magnitudes of

all the devices we measured to the values expected by Hooge in Figure

5-6. From Figure 5-6, we see our devices roughly obey the 1/N depen-

dence predicted by Hooge and have noise magnitudes in the same "ball-

park" as observed by Hooge.

Upon reducing the ambient temperature, separate current through

the device and voltage across the device measurements were made to

calculate the resistance of the biased device, RON. The calculated

RON is always a few ohms larger than the ROFF (measured in Section 5.3)


Figure 5-4 Device Resistance Versus Temperature



10 50 100 150 200 250
T (K)---

= 20.5 mA


=x \


mA^ ^\

Sf (Hz)--

Figure 5-5

1/f Noise 12 Dependence







10 100 IK

10-19 t



I -1 5




.4 I I

o.8 I

1.6 3.2

-N --

Figure 5-6

Relative Noise Magnitude at 300K
of all Devices Measured. (N =
0.8 X 10 -1 0.5pm devices, N =
1.6 X 10 -12 lpm devices, N =
3.2 X 10 -+ 2um devices)

for each temperature. Using the R versus T plot of Section 5.3, the

true device temperature is determined. The typical trends of S f/I2

and the slope, y, versus the true device temperature are indicated

for a few of the devices in Figures 5-7 through 5-9. The symbols used

in Figure 5-7 through 5-9 correspond to the symbols in Figure 5-6 re-

presenting the different devices. In general, the "quiet" devices

(those that fall below the Hooge line) have larger error bars on both

the S f/I2 and y plots because in those devices there is less differ-

ence between the device noise and the systems background noise (ie.

between the DUT ON and DUT OFF measurements mentioned in Section 4.2.3).

The program used to calculate the mean of the slope and y-intercept

and the standard deviation of the slope and y-intercept from noisy

data is based on the least-squares approximation algorithm derived by

Legendre in 1806 [50].

The definitive results of our 1/f noise measurements between the

same two films shown to be in the same thermal environment (and of

all the samples measured) show no coherence between the film's 1/f

noises at any temperature. A typical coherence spectra (see Figure

5-10) shows the maximum coherence is 2.4% (and this value would pro-

bably go even lower with longer averaging) and most coherence values

are less than 1% over the same frequency range which we observed 100%

coherence in the "heater" experiment. With these results, the same

at all the temperatures, we feel safe to say that our research elimi-

nates the temperature-fluctuation model proposed by Voss and Clarke for

the explanation of 1/f noise in metal films.





0 100 300

(K) *0

Figure 5-7

Noise Magnitude and Slope Versus Ambient Device
Temperature. (0.5pm device)





S 15-





10 100 300

-T (K)--

Figure 5-8 Noise Magnitude and Slope Versus Ambient Device
Temperature. (Noisy lpm device)






Figure 5-9

Figure 5-9





100 300

Noise Magnitude and Slope Versus Ambient
Device Temperature. (Quiet lpm device)

T 100.

0 10-
I I0


o *



0.001 *
I 10 100 1K

f (Hz)--

Figure 5-10 Typical Coherence Between the 1/f Noise
of Two Thin Films-

5.5 The Clearcut Evidence of 1/f Mobility Fluctuations in Transistors

With the exclusion of the temperature fluctuation model of 1/f

noise, the controversy between the mobility fluctuation model and the

number fluctuation model is heightened. Toward the reconciliation of

the two competing theories Van der Ziel proposed a rather straight-

forward experiment to verify mobility fluctuation in transistors.

In older transistors the predominant 1/f noise source was the

recombination current because those devices had large surface recom-

bination velocities. The purpose of our present investigation is to

determine whether 1/f noise due to mobility fluctuations, as presented

first by Hooge [51] and recently by Kleinpenning [52], is present in

contemporary devices with small surface recombination velocities.

Mobility fluctuations imply fluctuations in the diffusion constant D ,

since by the Einstein relation,

qDD = kT6p (5-2)

Thus we may expect the mobility fluctuations to modulate the emitter-

collector hole diffusion current and/or the base-emitter electron in-

jection current.

Van der Ziel's appendixed derivation [53] of Kleinpenning's ex-

pression for the noise spectrum due to mobility fluctuations of emitter-

collector hole diffusion in P PN transistors shows,

SI (f) = 2qP in (5-3)
dp [P(O)
s (f) = 2I 4f n B
Ep dp PB


where a is the Hooge parameter associated with hole current, Tdp
w /2D is the diffusion time for holes through the base region, w the
B p B
base width, and P(O) and P(w ) are the hole concentrations for unit

length at the emitter side and the collector side of the base, respec-

tively. We see the magnitude of SIEp is inversely proportional to Tdp

which means that S is proportional to fT since

f = T (5-4)
T 2TrT

Therefore, the hole mobility fluctuation 1/f noise source is larger in

transistors with large f (e.g. microwave transistors).

Also, for electron injection from base to emitter, we have, due to

mobility fluctuations, [53, Eq.(4)],

an N(O))
S (f) = 2qIE In f N(O)J (5-5)
I E En 4fT Nwn
En dn E

where T = w /2D w the width of the emitter region, D the electron
dn E n E n
diffusion constant in the emitter region, whereas N(O) and N(wE) are

the electron concentrations for unit length at the base side of the

emitter and at the emitter contact, respectively.

5.5.1 Discrimination Between Base and Collector Noise Sources

We now draw an equivalent common-emitter noise circuit of the PNP

transistor biased with a source resistance, RS, and a load resistance,

RL (see Figure 5-11). Here, Sifb represents the 1/f contribution to

the total spectrum arising from the base 1/f noise current sources.

r b 4kTrb

2kT/gm Sifc/gm .1


Figure 5-11

Equivalent Common Emitter Circuit


The base 1/f noise sources are comprised of the electron injection term

of Equation 5-5 and a possible emitter-base recombination current term

which we assume to be small in this modern device. The spectral con-

tributor of the collector 1/f noise current source, Sif, and the collec-

tor shot noise current source, 2eIc, have been referred to the input

equivalent circuit as noise voltage spectrum sources by multiplying by

(/gm)2 = (r / )2 (valid if r >> rb).

An HP3582A FFT spectrum analyzer measures the spectral density of

the collector noise, M2/Af. Calculations from Figure 5-11 reveal,

M rs
rf +R r R 2 ( + r
s b s +b +T

2 2 2
r 2kTr R + R+ ( rb + Rs) r 2
+ 2kTr + S.
SC R++ +r ifb r)2
s b F

S2kT ifc (5-6)
g 2
gm gm

If we use that r >> rb and >> 1, then Equation 5-6 can be written so

that we obtain,

f r Ir2
M A l--- = Ai 2kT(2rb + 1/g) + + S r
Af R + r + r b + m 2 ifb b

4S fcS r T
+ Rs 4kT + 2 2 + 2Sifbrb

2 2kT Sife
+ 2 + S. (5-7)
s r 7 2 ifb


We see that there are three regions to the magnitude of the measured

noise versus R -an independent, a linear, and a quadratic regime.
Ideally, the mobility-fluctuation 1/f noise measurements should

be made on microwave transistors biased with low currents for both

high and low R Unfortunately, microwave transistors usually do not
have a high DC 8. So the experiment was performed on low-noise PNP

transistors (GE 82 185) with 3 = 350 typically. A simply biasing scheme

was used for the high R experiment, [54] and the noise was measured for

three different I 's. From Equation 5-7 and for the case of high R ,
we see that we measure with the spectrum analyzer,

MHI2 2 if (5-8)
= 2R2 2eI + S. + (5-8)
AfL B ifb

using r = where we have neglected the small rb and r compared to
a high R and the terms independent of and proportional with R The
s s
measured high R noise plotted in Figure 5-12 (curves IV, V, VI) repre-
sents the absolute magnitude of the physical noise sources (in amp

sec) referred back to the (base) input,

HI 1 Sifc
SR I 121= 2el + S. + Si (5-9)
HR af B ifb 2

The high-frequency roll-off, which each of the plots indicates, is at-

tributed to the Miller effect of the capacitance CT in the equivalent

circuit (see Figure 5-11) where,

CT = Cbeo + Cbco ( + A v1)
T beo bco v


10 100 1K 10
--f (Hz)----

Figure 5-12

Measurements of High Source Impedance Spectra (SHR)
and Low Source Impedance Spectra (S )




? -21

\J -22

o -23





Since IB is small, r is large, and the f = 1/C r Miller cut-off

frequency, is low ~2 KHz. Shot noise, low-pass filtered across the

parallel combination of r and C gives at sufficiently high frequen-


s = (5-11)
HR 2 2 2
s 1 + w C r
T Tr

the observed 1/f2 roll-off.

Biased with a low R configuration [54], we neglect the terms in
Equation 5-7, which are proportional with R and R Using gm = /r

and neglecting R and rb with respect to r we see that we can plot

(again in amp sec).

MLO2 1 2+
SLR = f 2= 2eIC + 4kTrg m2 ifc
s R

2 2
+ Sifbrb g (5-12)

This was done in Figure 5-12 (Curves I, II, III) at the same three I 's

used in the high R experiment in order that the high and low R spectra
s s

can be quantitatively compared.

There are a few interesting points about the SLR spectra. First,
S ..
it was found that the magnitude of the 1/f portion of SLR was quite
device dependent. The noise plotted for SLR in Figure 5-12 was from
the "noisiest" device where we see S f ~ 10 with a crossover fre-

quency above 100Hz. With this device it was possible to get an accurate

picture of the slope of its noise. Inspection of Figure 5-12 shows the

slope of SLR to be y: 1.18 while we see SHR has y 1. This implies
s s

that for this "noisy" device we may be revealing a different noise

producing mechanism. Other transistors biased with the low R confi-

guration give S f 10-21 with crossover frequencies on the order of

a few Hz. With these devices it was not possible to determine the

slope of their 1/f noise, since accurate spectra could not be measured

as we were measuring at the limits of the spectrum analyzer's sensiti-


It should be noted that Equation 5-12 is only valid for R << rb.

In practice, however, R was of the same order of magnitude as rb at

low IE(e.g. R s 50). As a consequence, the thermal noise generated
E s

by R cannot be neglected and has to be incorporated in Equation 5-12.

The expression for SLR becomes,

2 2 2 2
SLR = 2e + 4kT(rb + R)g + Sif + S.fbr g (5-13)
LR c b s m ifc ifb b

We see that the low R measurement provides the means to measure rb [54].

Using the magnitude of the white noise levels of SLR in Figure 5-12,

the calculated values of rb are indicated in Table 5-1.

5.5.2 Interpretation of SH and SL
s s

To calculate the magnitudes of S. and Sif, we look only at the
ifc ifb

1/f portion of our spectra (ie. at f < 100 Hz) where we are above the

shot-noise level and can write, at low f,

2 2
SLR Sifb(rb gm ) + Sifc' (5-14)


Data Obtained From High and Low R Biased PNP Transistors

Low R Data High R Data
s s

Curve IE r rb Curve IB (n)MIN

I 2.25mA 340 4Q IV 6.71A 362 1.2 x 10-7

II 1.3mA 420 90 V 3A 363 6.6 x 10-8
II 1.3mA 420 9Q V 3&A 363 6.6 x 10 o

SHR = Sifb + S2 (5-15)
HR ifb 2
s B

Having two equations involving the two unknown S. and Sifb we
ifc ifb
solve for Sc and find,

S r H (5-16)
ifc r g]2 2

rbgm 82

Now from inspection of Figure 5-12, we see SH << SL at 1 Hz, and
s s

1 T
rb m (Brb

we can neglect 1/82 and 5-16 simplifies to,

SLR = Sifc (5-17)

We see at low frequencies the low R bias configuration isolates Sifc.

Solving Sifb, we find,

Sif = SHR (5-18)
ifb HR 2

Since even the SLR from the "noisiest" device, when diminished by 8 ,

is a factor of ten less than SHR at 1 Hz and we see the high Rs confi-
guration essentially isolates S. fb.

Now that the 1/f noise sources have been identified, we must apply

the results of the mobility fluctuations theory (Equations 5-3 and 5-5).

Before we attempt to calculate a Hooge parameter from Equations 5-3 and

5-5, we want to check the current dependence of the base and collector

noise sources. According to van der Ziel, the terms An P(0)/P(wb )

An N(O)/N(wE) Tdp, and Tdn are weak functions of the current, so that

we can expect,

S. ~ I ,
ifc c


Sifb Ib '

if the mechanisms are mobility fluctuation noise. Measurements of

Sifbf versus Ib and Sifc f versus I are shown in Figures 5-13 and 5-14.
ifb b ifc c
We see the base noise is roughly proportional Ib (a small slope deviation

from unity is expected since Tdn and in (N(O)/N(wE) are weak functions

of the current). This shows the base noise is definitely a candidate for

mobility fluctuation noise. However, the collector noise's slope with

current is much greater than unity'and van der Ziel has found slopes

~ 1.5 imply a number fluctuation noise mechanisms. Thus, we will only

apply Kleinpenning's formula to the base noise source and calculate an

For the case of base 1/f noise, we have,

SHR s Sifb SI (5-19)
HR i fb I (
s En

and using the base to emitter expression (5-5) we have, for an',





10 5-

Figure 5-13

- B (A)--

Base 1/f Noise Magnitude Versus Base Current


7 16-3

- Ic (A)-

Figure 5-14

Collector 1/f Noise Magnitude

Versus Collector






2 3


2 3

- !

SHR f 2Tdn
a = s (5-20)

n I N(0)]
q IBn IN(w-E)

since IEn IB in aP NP transistor, if we neglect recombination. We

take An [N(0)/N(w )] < 5, since we expect the ratio of electrons in the

emitter to be a few orders of magnitude greater than the ratio of holes

in the base due to the high recombination of electrons in the heavily

doped emitter [54]. Using this and the approximation that Tdn Tdp

suggested by van der Ziel [53], we calculated the minimum values of a

which are tabulated in Table 5-1 for SHR evaluated at 1 Hz. The values
seem a bit low; however, we realize that we have a P NP device where the

emitter is heavily doped and our observed an is diminished by an impurity

mobility reduction factor. Kilmer et al. [54], using a ratio of imp /
platt 2: 1/10, obtain a minimum value z 2 X 10 for (a ) true. Bosman

et al. [55, Figure 5) report a values ranging between 10-5 and 10-3 for

electrons in n-type silicon. Hence, we conclude that the 1/f noise in

the base of transistors can also be attributed to a mobility-fluctua-

tion mechanism. Unlike the collector noise, the base noise showed no

device dependence, indicating the base noise is intrinsic to the mater-

ial. This combined with the facts of unity 1/f noise slope and near

unity current dependence, gives clear cut evidence that the 1/f noise

in the base of modern transistors (which dominates the low frequency noise

spectrum in the grounded emitter configuration) is definitely caused

by mobility fluctuations.


In the transistor experiments, the results are fairly well under-

stood. The theory for mobility-fluctuation noise is well defined and

correctly describes the observed base 1/f noise. However, the inter-

pretation of the thin film results are not as distinctly defined.

While the temperature-fluctuation theory does not apply, we must see

which of the possible theories (see Section 1.2), or combination of

theories, can explain the observed trends of the thin film resistors

1/f noise as a function of temperature.

The most obvious trend which is discerned from the plots of

S f/I2 vs. temperature is the relative noise minimum occurring around

150K in all the samples (except one 0.5pm device which exhibited burst

noise). The dip in the noise around 150K is most interesting since it

is surprisingly close to the Debye temperature of Au (8 = 165K).

To determine whether this is merely a coincidence or, perhaps, the

first means to actually measure 8D, would require noise measurements

from thin films made from different metals'to exhibit noise minima at

their respective D's. Good metals to verify this are mentioned in

Section 7.1.

The difference between the trends observed from Figure 5-7 through

5-9, can be roughly classified into three groups. The 0.5pm width de-

vices all exhibit types of noise which appear less fundamental to the

element gold and more related to their unusual geometry. Since these



devices exhibit a more extrinsic noise, we will put them in their own

classification. Between the 1 and 2pm width devices a distinction can

be made between noisy and quiet devices. Those devices that fall

above the Hooge line in Figure 5-6 we will call noisy devices and ana-

lyze them separately from the quiet devices (ie. those that fall below

the Hooge line). A section of this chapter will be dedicated to each

of these three classes before we make the conclusions in the last sec-

tion of this chapter.

6.1 The 0.5pm Devices

Only a few of the 0.5pm devices could be measured over the full

cryogenic temperature range. This was due to the high number of de-

fective devices which is typical of the submicron geometry. Many

could not be measured because they formed open circuits (see Figure

6-1) or shorted together (see Figure 6-2). The.0.5pm devices were

very delicate and often "died", from the transient currents between

DUT ON and DUT OFF, before measurements could be completed. The 0.5pm

devices were prone to burst noise exhibiting a greater than 12 current

dependence. A S f/I2 vs. T plot of a 0.5pm device which did not exhi-

bit burst noise is given in Figure 5-7. It is the only S f/I2 plot

which shows the noise increasing at the low temperatures. We believe

the strange behavior of the 0.5pm devices is due to the narrow-cross

section of the device. The SEM photograph (Figure 6-3) shows the

top surface of the resistors to be relatively smooth while the edges

are noticeably rough. In the 1 and 2pm devices, the width of the re-

sistors cross section is five times, or ten times that of the height,

respectively. Therefore, surface noise contribution from the rough

Figure 6-1 Photograph of Open-Circuited Devices

Figure 6-2 Photograph of Short-Circuited Devices

Figure 6-3

SEM Photograph of Thin Film Resistors

edges is not so pronounced. However, for the 0.5pm device, the width-

to-height ratio of the cross section is only 5:2 permitting the rough

edges to dominate the observed noise. Since excess noise from surface

or edge effects is not the fundamental bulk noise we seek to identify,

the 0.5pm samples are less interesting, for the purpose of this study.

6.2 Noisy 1 and 2pm Devices

The devices which have noise magnitudes at 300K which are larger

than the values predicted by Hooge, we have chosen to call "noisy de-

vices." A good example, of this classification, is the noise from

the lpm sample in Figure 5-8. The noisy 1 and 2pm devices are good

candidates for Dutta and Horn's thermally activated number-fluctuation

theory. Specifically, the rapid increase in slope coinciding with the

rapid increase in noise magnitude is predicted from their theory [4,

Eqn. 20]. Also, we see from Equations 1-10, slopes ranging from 0 <

y < 2 are theoretically plausible, and in Figure 5-8, we see a slope

change from 0.75 < y < 1.2. Only a number fluctuation theory, such

as Dutta and Horn's, could give slopes as low as 0.75.

As to the exact origin of the random processes which give this

form of noise, we can say bulk and surface dislocations, specifically

in the form of dangling bonds, are a good possibility. Since methods

of thin film production cannot control such defects, this-wouldexplain

the large spread in the data as we see in Figure 5-6 and Fleetwood and

Giordano's Figure 1 [35]. F.N.H. Robinson [56] has suggested that the

random motion of frozen-in lattice defects, which diffuse with an acti-

vation energy between 0.1 and leV, to be the random process alluded

to in Dutta and Horn's theory. By a simple argument, he shows a

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