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Identification of 1f noise producing mechanisms in electronic devices

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Title:
Identification of 1f noise producing mechanisms in electronic devices
Creator:
Kilmer, Joyce Prentice, 1958- ( Dissertant )
Vliet, C. Van ( Thesis advisor )
Bosman, Gijs ( Thesis advisor )
Chenette, E. R. ( Reviewer )
Ramaswamy, V. ( Reviewer )
Ballard, S. S. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1984
Language:
English
Physical Description:
viii, 107 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Low noise ( jstor )
Mechanical noise ( jstor )
Narrative devices ( jstor )
Noise measurement ( jstor )
Noise spectra ( jstor )
Noise temperature ( jstor )
Resistors ( jstor )
Thermal noise ( jstor )
Thin films ( jstor )
Trucks ( jstor )
Current noise (Electricity) ( lcsh )
Dissertations, Academic -- Electrical Engineering -- UF
Electrical Engineering thesis Ph. D
Electronic noise ( lcsh )
Thin films ( lcsh )
Transistors ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
In recent years, theorists have been trying to explain the phenomenom 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 primary thrust of our research is to determine which of these possible 1.f noise producing mechanisms is responsible for the 1.f noise commonly observed in two electronic devices (ie., thin film resistors and PNP transistors). first, we delve into the details of two competing 1.f noise theories 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 distinguish between the three possible 1/f noise producing mechanisms. We show a lack of correlation between 1/f noises of two devices in the same thermal environment and conclude that temperature fluctuations do not produce the 1/f noise. We find two methods to discriminate 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.
Thesis:
Thesis (Ph. D.)--University of Florida, 1984.
Bibliography:
Bibliography: leaves 103-106.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Joyce Prentice Kilmer.

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Copyright Joyce Prentice Kilmer. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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ACN9056 ( NOTIS )

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IDENTIFICATION OF 1/f NOISE PRODUCING MECHANISMS
IN ELECTRONIC DEVICES






BY

JOYCE PRENTICE KILMER


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


UNIVERSITY OF FLORIDA


1984

















ACKNOWLEDGEMENTS


I am greatly indebted to Professor Carolyn Van Vliet for her most generous and valuable assistance, guidance, and time. I especially 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 manuscript.

This research was supported by the Air Force office of Scientific Research, under Grant Number AFOSR 80-0050.

















TABLE OF CONTENTS


Page


ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . .

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

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


CHAPTER I


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 . . . . . . . . . . . .


3
4

5
7
7

10


CHAPTER II


SCOPE OF OUR INVESTIGATIONS . . . . . . . . .


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 11N Dependence and the Hooge
Parameter . . . . . . . . . . . .
2.4 Temperature Dependence of 1/f
Mechanisms . . . . . . . . . . . .


. . . 13

. . . 16

19


CHAPTER III


THEORY OF HEAT TRANSFER AND OF TEMPERATURE FLUCTUATION NOISE . . . . . . . . . . . .


22

22 23 28 30
34

36


3.1 3.2 .3.3
3.4 3.5 3.6


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


iii





Page


41


CHAPTER IV


EXPERIMENTAL METHOD . . . . . . . . . . . .


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


59


CHAPTER V


EXPERIMENTAL RESULTS . . . . . . . . . . .


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 S HR andS LR . . . 80
s s

INTERPRETATION OF RESULTS . . . . . . . . . . 87


CHAPTER VI


The 0.5pm Devices . . . . . . . Noisy 1 and 211M Devices . . . . Quiet 1 and 2pm Devices Conclusions . . . . . . . .


CHAPTER VII


RECOMMENDATIONS FOR FUTURE WORK . . . . . . .


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 . . . . . . . . . . . . . . . . . . . .





LIST OF FIGURES


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 . . . .
2
5-5 1/f Noise I Dependence . . . . . . . . . .

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

5-7 Noise Magnitude and Slope Versus Ambient
Device Temperature (0.5pm,.device) . . . . .

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

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

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


Page 12 37

44 45 48 55 60 62 63 65 66 67 69 70 71 72










Figure Number

5-11 Equivalent Common Emitter Circuit .

5-12 Measurements of High Source Impedance
Spectra (SHR) and Low Source Impedance
s
Spectra (S LR) . .
S


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 atrue Versus Temperature .


collector


Page 75 78

















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



IDENTIFICATION OF 1/f NOISE PRODUCING MECHANISMS IN ELECTRONIC DEVICES

By

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 phenomenon 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 primary thrust of our research is to determine which of these possible 1/f noise producing mechanisms is responsible f r the 1/f noise commonly observed in two electronic devices (ie(t 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 Fluctuations 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.

vii





our central experimental effort has been to devise methods to distinguish 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 fluctuations do not produce the 1/f noise. We find two methods to discriminate 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.


viii
















CHAPTER I
INTRODUCTION



The study of the spectral density of random fluctuations which vary inversely with frequency (1/f noise) is an old subject. Specifically, in 1937, Bernamont (1] observed a "current" noise in metal films whose spectral density, S I (f), followed an empirical formula,



S (f) = AI 2 (0.88 < y < 1.1) (1-1)
I fy



where the I indicates the current dependence. However, the theoretical 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 [310 that reviewers such as Dutta and Horn [4] feel the physical origin of the phenomenon cannot universal. Still other theorists, such as Handel [51, fascinated by the universality of the phenomenon have concluded that any phenomenon governed by nonlinear equations of motion with no contribution from-bound-ary 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







2


be concerned mainly with the 1/f fluctuations in the conduction processes of solid state devices. What is known about the phenomenon can be summarized by Hooge's phenomenological equation [6),




SV V (1-2)
fYN


where S is spectral density of voltage fluctuations and aH is the
VH

Hooge parameter which he assumed to be a constant acH 2 X 10-3 We see the magnitude of the voltage spectrum is proportional to the ,square of-the DC voltage, V, across the device implying that the current 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 empirical 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 spectrum, Sir measured from a device of differential resistance R,



SV =S IR 2 13




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



S S cO.
2 2 (1-4)
V 2 I fYN







3

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 complicated. Specifically, the total power or the integral of the spectrum diverges; yet we do not get a "shock" when we touch an unbiased device! 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 observed for frequencies as low as 10- Hz implying the need to postulate 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

1v /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 V-1/N over six decades of carrier-numbers.

Is 1/f noise present in thermal equilibrium? Is the current required 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!








4

Is the 1/f noise process stationary? If the power really does diverge, 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 question for which Voss [12] tried to rationalize a type of linearity by measuring a "conditional mean." The experiment gave proof of a macroscopic linearity, but did not exclude the possibility of microscopic non-linearities.

From-what fluctuating physical quantities does 1/f noise in electronic conductors arise? Presently three physical quantities, the mobility of the carriers, the number of the carriers, and the temperature, are in the theoretical arena. Fluctuations in any one of these quantities can theoretically give rise to 1/f noise, as 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 electronic 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 incoherent superposition of Lorentzians. Usually, such Lorentzians ,are caused by number fluctuations. The overall spectrum then becomes





SMw 4A2 g(TdT (1-5)

1 W2l







5

where g(T) is the distribution of time constants for T < T < T2. 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 activation 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 - E (1-6)
T=Te ,(-)
0,,


where S = l/kBT, and we let


g(E)dE Ke- 6EdE, E1 E (1-7)
1-1-7


Then,


( dE 1 o (1-8)
g(T) =g(E) -j = (18
dT SE 0T 0


where K = I/Eo. Hence,












ST 2 1 ) - 1




4AT X / 1+ T
o F dx(

LUl 2 LWTJ 19
o1 +x2



The integral converges if -1 < (0/3) < 1. Thus,



(O/M) = -1 + , S(M) cc (1/T 0 ) (1-10a)



(O/M) = 0, S(w) I/WT0 (1/f spectrum) (1-10b)



(O/M) = 1 - 6, S(M) 1 1/(0T ) 2-. (1-10c)
0



Thus, all spectra of the form C/w Y 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 i 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 CdxHg1x Te [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 show that-anyg(E) which has a sharp peak, E leV, and a width of a few tenths of an eV (ie. a "smooth" energy
P

distribution) will give the observed S 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
p







7

now be a material property and the width of the distribution may depend on parameters such as sample inhomogenity.


1.2.2 Diffusion Theories

Diffusion theories [17] (theories involving the diffusion equation) have been suggested as asource 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' calculated 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 Schr~dinger field version of Handel's theory, one considers the wave function upon scattering with small energy losses due to bremsstrahlung,


A
i~ (k-t 1 td
T(r,t) = a e ~(kr-. [1 + f bT()ed(E/) (1-i1)

0







8

In Equation 1-11 the frequency-shifted components present in the integral interfere with the elastic term, yielding beats of frequency C/h. The particle density given by Equation 1-11 is



PT = aI2 1 + 2 A cos + Y dt

0

A A
b bT (O)bT(C') e (06 t/fi d~dc' (1-12)


0 0


the second term in large parentheses describes the particle beats. If the particle fluctuation is defined by 6 IP2 = lI2- , its autocorrelation is found to be, if a term "noise of noise" is neglected,

A
2 2 14 2 s d
t tTf> = 21a lb(s)


0
A/h

= 21a4f j b(1)12 cs2ffTdf (1-13)

f
0

From the Wiener-Khintchine theorem one sees that the integrand is the spectral density:



sJ 12/ 2 fib(E)12 (1-14)



For the spectral density in the particle velocity hv = 2i(-m- V )

one easily sees,



2 2 2 2
S <>= S1J,12/ = 2,hib(C)l (1-15)







9

For the bremsstrahlung matrix element, under emission of photons, one calculates easily, either classically or quantum mechanically,



Ib(c) 12 = aA/ = cA/hf ; (1-16)



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



A 2 3 Av12 (1-17)
3Tr 2
c


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


S
v 2aA
2 f

- 2c I I (1-18)

3rrf 2
c


Since U-process give the largest L\, we expect them to be the largest contributor to i/f noise in metals. The changes in Ae of the scattered electrons give rise to mobility fluctuations which translate into resisZ2
tance fluctuation since R = - . Dutta and Horn in their review paper AqlIN

discredit Handel's theory explaining, "most electrons in a metal cannot 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 i/f noise.





10

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 (151. They assume temperature fluctuations give rise to resistance fluctuations since we measure a temperature coefficient of resistance,



1 dR
R jT (1-19)



Using the Langevin diffusion equation,



T 2 V-F
DV T + c (1-20)



where D is thermal diffusivity, c is the specific heat and Ris 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 conservation of energy [241. 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 devices.





CHAPTER II
SCOPE OF OUR INVESTIGATIONS



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 fluctuations, investigators have been trying to verify their theory. Scofield, et al. [251, 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 fluctuation model suggests. However, with the recent Japanese measurements of 1/f temperature fluctuations in a resistor [261, the theoretical possibility of 1/f noise stemming from surface sources [271, 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 transistors. 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 11





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





13

measurements of the responses in the other two "sensors" at an exciter frequency will exhibit thermal diffusion spectra and a correlation spectrum as a function of substrate temperature. A fully correlated coherence spectrum is expected if the devices are in the same thermal envi" . ronmient. 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 opportunity 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)- 1
2
and with R . A low frequency flattening of the room tem(dT)

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

















TABLE 2-1
1/f Noise Experimental Results


from Metal\Films


*The only substrates ver 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.







15

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 shows a very

small 1/f noise. Bi, which shows a comparable noise magnitude as metals of the same geometry, but with a much smaller

crirdensity than the metals, suggests the noise should

be scaled by 1/Volume (or 1/N atoms ) rather than l/N carriers*



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

the 1/f noise's magnitude and slope between l00K and 600K

is shown. Annealing decreases noise and increases the temperature dependence.


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

dominates at all temperatures. In Cu (which has a lower room temperature noise) crossover of "Type A"l 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 magnitude changes vs. T. The sharper Guassian distribution of

activation energies in thicker films lends to the thought







16

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 always 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]. 1I/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 individual carrier (see Equation 1-18). Since v = Ep, we can write Equation 1-18 also as a normalized spectrum of mobility fluctuations of a single carrier,


S P true (2-1)
-2 f










where a = 2UA (and not to be confused with the fine structure constant!).
true
The fact that the fundamental relationship for mobility fluctuations of an individual carrier is expressed by Equation 2-1 (which obviously exludes 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 macroscopic quantity comprised of a cumulation of microscopic events; that

-is,


N
I=q - , (2-2)




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)) d d.
1 1
and also number fluctuations (ie. N - N(t)). Thus,


N(t)
I(t) t) , (2-3)
L



where N(t) = N, i (t) = p, and the individual carriers are incoherent (ie. 'illj = ij P M2). We now take the autocorrelation of both sides of Equation 2-3, applying Van Vliet's and van der Ziel's Extension of Burgess' Variance Theorem [38], and find,



Ai(t)Ai(t+s) = q .E p 2 AN(t)AN(t+s) + N A(t)AP(t+s) (2-4) L 2







18

Now, applying the Wiener-Khintchine theorem we find,


2 2
S (f) -q E 2S()+ NSM(2-5)
Ij2f LS.2ifN



2 2 2-2 2 2
Normalization of both sides with I =q E 11 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 N 1__ (2-6)
2 - 2 N *2
I _ N (1f)


Ignoring for the moment the possibility of number fluctuations, (S IN 2 ->-0), and substituting the normalized mobility fluctuation
N
spectrum for a single carrier Equation 2-1 into Equation 2-6, we arrive at Hooge's formula,


S I q
Iii (2-7)




We see the N in the denominator accounts for the increase in the number of degrees of freedom associated with incoherent scattering. 'Whenever the 1/N appears explicitly in the formula, a is defined as a H However, we see a problem with using a H in that we need 1/fY (yF 1) in order to get a unique value for a

Looking only at number fluctuations as the source of the current noise (ie. S 5 2 + 0), we see Equation 2-6 has no explicit 1/N dependence. For number fluctuations to fit Hooge's phenomenological form, we must assume,

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










where a K is due to Klaassen [393. 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 a H1 for the total observed 1/f noise magnitude, which does not presume one mechanism is the sole source of the 1/f noise, is the dimensionless "noisiness" factor,



Noisiness 1 2 (2-9)



It is the normalized current noise spectrum evaluated at 1Hz (ie. f
0
l Hz). 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 a H and acK are avoided and trends associated with N can be readily discerned. 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 a true of the mobility fluctuations of a single carrier, S /P-2



2.4 Temperature Dependence of 1/f Noise mechanisms


In semiconductors, as we vary the ambient temperature, we effectively 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) - ___= - k Bt ,(2-10)



where C is the chemical potentialand m* is the effective mass [41].








20


1/2
Roughly, we have Jkj T. The incoming electron, it, is scattered

by a phonon, q, described by the elementary phonon emmission/absorbtion process,



k' k + q ,(2-11)



where k' is the electron wavevector after the scattering. The superposition of these elementary processes gives rise to macroscopic observables 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) q axThis defines the Debye temperature 0D= 'hqa/ where u is the

velocity of sound in the material. The e0 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 e 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 I at low temperatures will reveal which of the competing theories apply. If a peak in S I / is observed associated with a continuous slope variation, we may have another example of Dutta & Horn's number fluctuation theory. However, if we see







21

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- 11 have been calculated by G. Kousik using Handel's theory. However, for phonon scattering mechanisms with soft photon emission, a :5 10- 6 may be low enough for Handel's theory to apply (see Chapter VI).

















CHAPTER III
THEORY OF HEAT TRANSFER AND OF TEMPERATURE-FLUCTUATION NOISE




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 microphotograph. The power delivered by a resistor biased as a heater is

.2
iv = i R, where R is the resistance of the film. If i = I cosw t, the
0 0
AC heater power delivered is AP =AP cos2w t, with P =12;th h ho o ho 2 oR h
exciting power frequency is thus twice that coming from the signal generator, We = 2w 0 We suppose the power is uniformly dissipated along the length of the film, thus,* neglecting current crowdin g.

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, RLI, by v s, and the AC temperature of the sensor by AT5, we have v s= V socos(w et + fl, AT= =AT socos(w et + 0), with V so mAT so. We have according to Kilmer et al. [28],












m V 0 LR+ R) + (3-1)



where V 0and R xare the sensor film's voltage and resistance, and $ is 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 delivered a power AP H= AP hcos et, which causes a temperature rise AT =

Asocos(wt t )+



3.2 The Heat Transfer Function


With no power introduced into the material, the heat conduction equation and energy conservation theorem are,



K = -GVT, (3-2)



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



here K is the heat current vector, indicating heat carried per second through unit area, G 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,



T 2 =0(3-4)




where a~ a/cd is the heat diffusivity. For the silicon substrate 143], one has,










0 = 1.45 Watt/cm 0C

c = 0.7 Joule/gram 0C

d = 2.328 gram/cm3
= 0.89 cm2/sec. (3-5)




If we now introduce a heating power AP h(t) in a volume V, and if we denote by AE the AC energy contained in V, we have the conservation law,



dAE= - AK - dS + AP (t) (3-6)
dt h



Writing AE = fcdAT(r,t)d 3r and APh(t) = 1 fAP (t)d 3r, we find by Gauss's
V V
theorem,




cd DAT(r,t) d3r = - AK d3r + 1 {APh(t)d3r. (3-7)

V V V


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


AT 1
cd -- + -V._ AK = APh(t). (3-8)




With K -oV(AT) this gives,



DT a?2(AT) = (r,t) (3-9)
Dt cdVh rt



where APh(r,t) is a function equal to APh (t) for r a 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.







25

of (3-9) will also be written as AP h(t)E(r), where �(r) is the function which is unity in Vh and zero outside. For APh =AP e jt, (3-9) gives if AT = AT ejWt,



jWAT (r) - av2AT (r) V- AP hE(r) '(3-10)
W cdV h



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



- jwG(r,r',jw) - aV2 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],

AP~h F

AT (r) = A- G(r,r',jw)E(r')d 3r'
dVhj
00
AP~h F

= Ad--h G(r,r',jw)d 3r'. (3-12)
cdVhJ
Vh



Denoting by AT the averaged temperature increment in the sensor, we
wls

have,



AT - CdVh d3r {d3r'G(r,r',j'w) (3-13)
V Vh
s h


Let AT = IAT sleJ. Then for a real input signal with amplitude Pho' the real sensor output signal appearing in the sensor circuit is v =
s







26

V soCOs(W et + ) where W = We and V = mIaT sI'




V= ds-----h Ifd3r 3r'G(r,r',jw)I cos(W et + *). (3-14)

V 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 Z (mPho/Cd) JG(rs'r hy cos(Wet + 4) (3-15)



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

We must now consider the measurement of vs by-aHewlett-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 vZSWI



d[v vs I 2Af Vs(t)v (t + T) e-JWTdT (3-16)
sw sw j 5

whrev t =V o (et+ )wihV ivnby(31) r-315 Nw

where v s(t) = V socos(We t + fl with V sogiven by (3-14) or (3-15). Now,
s so eso"



v (t)v (t + T) = V2 cos(W t + 4)cos(W t + W T + f) s s so e e e

= V2 [Cos2(W t + f)cosw T - sin(w t + )cos(w t + f)sinw T]
so e e e e e









1 2 jW e T -jW eT =IV2 (e +e e
4 so


Using further,


o -j (W + W )T e
� dT = 27B (w � w j e
--O



we find,



d[v s VsI = TrV2 [6(w - W ) + 6(W + W )]Af.
swsw so e e


now realize w, where Aw centered on


that the analyzer integrates over a bandwidth Aw centered = 2fAf. The output is therefore a signal over the range We" of magnitude,


__ _ _ V 2 C(W v * so e
se sw 2
e e


(3-20)


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

on W and zero outside. Hence,
e


v v* ho I d3r d3 r'G(r,rjw)12
Vswe sw c 2V 2 2
eW5 2c2d2VV J

Vs Vh


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



v v* (m2P2 /2c2d2) IG(r ,r 12
sW sW ho s h
e e


(3-21)


(3-22)


(3-17)


(3-18)


(3-19)









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




Ivd s ho I 3r d3r'G(rr',9w)l. (3-23)
v hh
V sV


It should still be noted that, since the machine measures the output in dBV, one can equally well read the logarithmic output power, by dividing 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 cos2w t are cross-correlated.

For each sensor the circuit output voltage is given by a result like (3-14). The analyzer then measures the spectral power,

CO

F - j WT
Re v vs*2w 2Re v sl(t)v s2(t + T)e dT (3-24)




where v (t) =V cos(wt + ) and vs(t) =V cos(w t+ 2).
sl slo e 1s2 s2o e 2
Writing,



Cos(Wet + WeT + ) = {cos(wet. + )cos(W e T + 42 - i)

-sin(wet + 1l)sin(weT + P2 - PI) (3-25)








29

we obtaining a similar way as before,



Re vslW s2W = 7Vslo Vs2oC�S( 2 - l)E( e
e e


(3-26)


The coherence factor is defined as,


y(we) =


Rev s1v s2W
e e
[Vsl v* N v* ]1/2
sw slw Vs2w s22w
e e e e


(3-27)


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


Y(We) = Cos (2 - i)


(3-28)


We notice that the machine measures lyl. Here ck(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 v refer to the sensors' noises. Then for y we obtain, noticing that signal and noise are uncorrelated,


y(W ) =
e


Rev + Rev V
slw s2w n n2
e e
{[v v' +v * v v'1}1/
svW Vsiw nlVl [Vs2w'v*2w :+ n2 n2'
e e e e


(3-30)







30

(for the connection with the notation in the next section, we have V V*l = S (W )AW, etc.). If the noise drops slower with frequency nini vs e


than the signal response and if VnlVn2 = 0, the above gives for sufficiently high frequency, in the case that (3-15) applies,



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

m2o2 Gr ,r2,J ) I IG(r s,rjw)1 cos(2 - I)
Ji ho sr51 h sh

cd [V nl n2 ] 1/2

(3-31)



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 q(r,t) is a source with spectrum [27],



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



where I is the unit tensor. We assume the steady state temperature T (r) = T is uniform, and 0(r) = a. For the conservation we have again,
o o
analogous to (3-3),









Cd~(AT)
cd 3--- _ + V � (AK) = 0. (3-34)



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


3AT 2 1 V , T (r,t) (3-35)
ct V2 (AT) =- c-(rt)
Dt cd



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


(r,r') =4k T 2 V'S(r - r)/c 2d2 (3-36)
Sorr)V k ro/
Bo0



where V' is the del-operator with respect to r'. We note that in contrast 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,w). From (3-35) we obtain the relation,



jWAT(r,w) - V AT(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)d 3r' (3-38)

Vtotal



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],







32

SAT(r,r',W) = lim 2T AT(r,w)AT*(r',W), (3-39)
T4


and similarly for S Thus from (3-38) and (3-36) we obtain the "response form" [45),



SAT (rrw) = d3rl {d3r2G(r'ri'jw)G(r'r 2,-jw)(4k T2a/c2d 2)V V 6(r -r

Vtotal Vtotal (3-40)



Providing that in Green's theorem for the V2 operator the bilinear con-comittant of G(r,rIjw) and G(r',r2,-jW) vanishesthe singular distribution V1 � V 2(r - r ) can be replaced by,



V V6(r r - 1 [V26(r r2) + 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 rule,



Jf(r)V 6(r - r)a. 7 f(r (3-42)
0 0we arrive at,



B jwV2G(r',r ,-jw)
SAT(r'r''w) 2 c--cd2 0dr[G(r'rO ,

Vtotal



+ G(rI,roL-J)V2G(r,rojw)]} (3-43)







33

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



- V2 G(r',r ,-jw) = jwG(r',r ,-jw) + 6(r' - r ) (3-44a)
cd o 0 0



and,

-o V2Grro
-V G(rr jw) = -jwG(rro'jw) + 6(r - r )' (3-44b)
d oo0



we find from (3-43),


2
2k T2

SAT (r,r, ) d [G(r,r',jw) + G(r',r,-jw)]. (3-45)



2
This is the van Vliet-Fassett form [17], since [kB T /cd]6(r - r') is the covariance AT(r,t)AT(r',t). For the spatially averaged tamperature fluctuations in a volume V we have,
s



T (t) = 1fAT(r,t)d3r, (3-46)
s V
V s


and for the autocorrelation function,



AT(t)AT (t + T) = 1 f d3rd3r' AT(r,t)AT(r',t +.T). (3-47)

VV
s s



Whence by the Wiener-Khintchine theorem,



SAT () = , d3rd3 rSAT (r,r',w). (3-48)
5 V V
s s









Employing (3-45) this yields,


2 r
SAT () = Re G(r,r',jw)d 3rd 3r'. (3-49)

s cdV2
s
s V V
ss



Since G has usually a singularity for r = r', the intervals of integration must be broken up accordingly. Also, it is not possible to approximate the integral by V2 times the integrand since G varies strongly in the neighborhood of r = r'.

For the noise in the sensor, we have,


2
5(to) mS (
Sv M= S ATM
s s

4kBT2
- Re G3(r,r',jw) 3rd3r' (3-50)
cdV2 V V
ss


Comparing the noise (3-50) with the linear response Iv 12 given in
e
(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 temperature cross-correlation, we now have, analogous to (3-47),



AT1 (t)AT 2(t + T) = 1 d3rd3r' AT(r,t)AT(r' ,t + T) (3-51)
s
V sl V s2







35

where we assumed that both sensors have an equal volume V . From the Wiener-Khintchine theorem,



SATsAT = v { {d3rd3r'SAT(r,r,,W). (3-52)
s Vsl Vs2



Using (3-45), this yields,


2k T 2I
SB2 o d 3rd 3r'[G(r,r',jw) + G(r',r,-jw)]. (3-53)
SATlATs2 cdV2 J sis
. VslVs2



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

2 2
2kB T2m 3
S - d rd r'[G(r,r',jw) + G(r',r,-jw)] (3-54)
Vsl,Vs2 cdV2

5 VV
sl s2


where we assumed for simplicity that both sensors have the same temperature coefficient m. For the coherence factor of the noise, we have,


Re S (W)

YV = sl s2 (3-55)
[S 1W) S (M))l
V sl V s2



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


2 Re (G(r,r',jw) + G(r',r,-jw)]d 3rd3 r


y(w) Vsl Vs2

[Re {G(r,r,jw)d3rd3rh]l/2[Re jG(r,r',jw)d3rd3ril/2

Vsl Vsl Vs2 Vs2 (3-56)







36
! 2
We note that because of the self-adjointness of V , G(r,r',jw) = G(r',r,jw), so the numerator is also Re f fGd 3rd 3r'; hence, VslV s2



Re f JG(r,r',jw)d3rd3r'


Y() = slV s2

[Re G(r,r,jw)d 3rd3r]l/2 [Re G(r,r',jw)) d3rd3r) ]/2

VslVsl Vs2 Vs2 (3-57)




If there is a frequency range for which the integration of G is insensitive to the change f f f f =f f, we find y = 1.
VslV s2 VslV sl Vs2 Vs2



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. lpm 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 satisfy,



DG(x,x',jw) =0 (3-58)
ax x = �C







t y-axis


TEeI


L]I-]I


LI


0-


LE-


m-


-DLI


E
--E]JE-


1i1


-LI


D1


I


-i
x-axis


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







38

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) = cothyC(sinhyx+tanhyCcoshyx)(sinhyx'-tanhyCcoshyx') (x ' =2ya

G(x,x',jw) = cothyC(sinhyx'+tanhyCcoshyx')(sinhyx-tanhyCcoshyx), ,(x 2ya

(3-59)

where,


-Y= \, /a.


We define a corner frequency,



2
Bound = a/c


(3-60)


(3-61)


Then yC = i/ For

and since IxI < ICi, sinhyx



G(x,x',jw)
W << Wboubd


<< bound" ycI -- 0. Thus tanhyC - - yC, yx, coshyx 1 1; for (3-59) we now find,


c


(3-62)


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

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


Note that there is an error in this formula. The W's should be in front of the cosh terms and not in front of the sinh terms.







39

G(x,x',jW) e -2V'5Y/e a I x - x (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 resistors symmetric to the y-axis on Figure 3-1, we have,



-L E+L

id3r id3rG(XX',iW) { dx' { dx G(x,x',jW)

Vh V -E-L L
h s


1 (e-YL _ e-y(E + L))2 (3-64)
2,,r aY


We introduce another corner frequency,

2

, = a/(L + 6) 2.(3-65)



For Wbound << W << Wl, we have jy(L + E)l << 1, and a fortiori IyLI << 1. Thus expanding the exponentials, we obtain,




E 2A2B2 VhV
=~ A s (3-66)
id3r'fd3rG(x'x'jI ) 2 a 2 j'-m

Vh Vs vh vs



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,




1s {d3r' d3rG(x,x.,jw)( = 1
VhVs (3-67)









(ie. we expect a square root frequency dependence). We also notice that for w << wi' = -f/4, independent of the coordinates. Thus for W << wil the signal correlation y(w e) x 1.

For w >> i' we find jy(L + C)f and IYLI are both greater than unity and (3-64) becomes,


d3hr' 3rG(x,x',jV) 2 (3-68)


Vh Vs



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



{ d3r' d3rG(xJx'i) 7 3/2 (3-69)


-3/2
(ie. we expect to see an w dependence above the corner frequency wl).
-1/2 -3/2
An W spectrum which becomes an w spectrum above a corner frequency

is typical of one-dimensional diffusion spectra.

















CHAPTER IV
EXPERIMENTAL METHODS



4.1 Devices


Gold thin film resistors, 2,000 A thick, deposited on a 200A 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 = 0Sp phm, and 2p.m 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 Hoppenbrouwers 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 T05 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 T05 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








42

Chemistry and Physics. The resistivities of metals deposited as films is expected to differ from that of bulk-- since the atomic stacking 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 co uld vary by-as much as a factor of 10 (351.



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 refrigerator 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 monitoring 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 copld head's compressor frequency) harmonics. The pickup was eliminated by electrically 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








43

(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 Continuous 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 temperature. 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 operate.



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, 5,r 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,




R = 11.29K + (5.6KI 15.6)

2

5ca - 2a
al R 2Af
S































































Figure 4-1 Calibrated Noise Measurement Scheme















S.,
A

--,I
IGV SA 'ST TH -S Sxf A OUT





LOW NOISE AMP



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









and,

I2

R = RRA (4-1)
IRA + R XI I R LI jRS

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


2

1) DUT ON V-G 2(S + S +s)z + S (4-2)
Af x th a a

-2
2) CAL ON --G (S + S + S )Z2 + S (4-3)
V02 ca2 th i 2 a)(3

--2

3) DUT OFF 3 (S + Si)Z2 + Sa (4-4)



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

-2 -2 v Ol - o3

x -2 -2 cal
v02 -v03


In essence, four separate measurements must be performed in each frequency 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 without 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 coherence spectrum between the two 1/f spectra. This is the method

used by Scofield and by Xilmer 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 PNP transistor cascaded with a low noise Burr Brown op, amp (see Figure 4-3). The INA gives +90dB of power gain down to lHz below which excess cryostat noise becomes a problem at low temperatures. An HP9825A desktop calculator samples the spectrum analyzer and performs the calculations 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-= lOmA), since the LNA has 1/f noise itself. If the relative magnitude of the DUT ON measurement exceeds the relative magnitude of the DUT OFF measurement, we know we must be observing true device noise.

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
























487K


I IO0/4F INPUT - I


Figure .4-3 Schematic of the Schmidt Low Noise Amplifier


12K


+12V


0.5aF


OUTPUT


91K





49

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 frequencies will avoid the linear capacitive coupling since only the nonlinearity of thermal power transfer (Joule effect) can produce the responses 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 SiQ 2 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 conducting material (ie. alumina, Beo 21' 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,



S th 4kT/R (4-6)


accurately enough to calculate T.







5O

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 excess device noise term, Sx, which dominates the device noise spectrum 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 as,


-2 -2
V01 - V03
total -2 -2 Scal +4kTDUT/RDUT ' (4-7)
V02 - 03


where TDUT and RDUT are device under test's temperature and resistance. Equation 4-7 can be modified to calculate the absolute magnitude of the devices'thermal noise (and consequently the devices'actual temperature) by using a dummy load resistance, RDUMMY' at a known temperature 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
4kT /R - 01 03 S + 4kT /R " (4-8)
DUT DUT -2 -2 CAL DUMMY DUMMY
V02 - 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 - R DUT ). Then, we can theoretically solve for TDUT. We say theoretically because there are







51
-2 -2 -2 -2
many sources of error, especially in the (V 01 V 03)/(V 02- V 0 term. Since here, we divide two quantities which are close, the statistical accuracy is low. Measurements by this method gave results with close to an order of magnitude deviation! Therefore, we need an alternative method to measure thermal noise and to calculate T DUT.



4.3.2 Thermal Noise Measurements Without Using the Calibrated Noise
Source

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, S th' to the thermal noise of a dummy resistor at one fixed temperature (e.g. me lting ice T, = 273K), S I and to the thermal noise of a second dummy at a second fixed temperature (e.g. liquid Nitrogen T 2= 77.5K), S T. Thus, we have 22
again from Figure 4-2 with R - since we use no calibration source,
S


z R 2 ( R JR L]2 (4-9)




For the three difference measurement cases we have, since we measure at high enough frequencies that S -- 0,




1) DUT ON V01 =G2 SiZ V(4-10)
Af(Sth +S az Sa



2) DUMMY AT T1I A G [(S T1 + S a)z2 + S a (4-11)










3) DUMMY AT T2


52

-2
V 03 G2 + Si)2 IVi Af (ST 2 a)


(4-12)


Solving for Sth, we find,


-2 -2 4kT
V01 V02
th -2 -2 R (T2-TI) +1 RV03 V02


(4-13)


where we have assumed the necessary condition R = RDUT 2 RDUMMY. We see we can simplify (4-13) to express the actual device temperature, TDUT, as,


-2 -2 V01 - V0
T V 1 02 (T -T+T
DUT -2 -2 2 1 T I
V03 V02


(4-14)


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,


(4-15)


-2
where x represents the (V01

we see,


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


AT = AAx .


(4-16)


Now we normalize since we are interested in relative error,


AT AAx T T









A (Ax (4-17)


We see the statistical accuracy of the -2 -2 -2 -2 (V0 - V02)/(V0 - V02) term,
01 02 03 02
x , must be -1% in order to have at least a 10% accuracy in since 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
x

the measuring bandwidth, Af, and averaging time, T, of the detector will decrease the statistical accuracy since we have from Van der Ziel

[47],



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



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 Analyzer, 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" system was configured where we used the entire frequency range of the LNA. We used an active bandpass filter with a 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








54

predominates with the "super bandwidth" system giving rise to erroneous 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 measurements predominates. our devices have a typical resistance of about 100Q at temperatures below 100K,and we must calculate what percentage 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 transistor 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, 4kTR n and an equivalent current noise source, 4kTg n(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, R , 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
x x

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 R nand g nare 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 xI
seene" with a given amplifier (ie. R nand g nare kno wn), we must solve the equation,






















r - - - - - - - - -


LNA-I


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










4kT R = 1/2[4kTR + R2 4kTg] (4-19)



derived from Figure 4-4. This equation has the general form, T = A/R x+ BR x, which is a skewed hyperbola in the T x- R plane. The skewed hyperbola will have a minimum corresponding to the lowest resistance, (R x)min needed to see its thermal noise at the lowest temperature, (T ) .Setting the derivative equal to zero, we find,
x mn = Rn
CR). = - (4-20)
x min g



and,




Tx mi m= 1/ gn( xmn(-1



From Schmidt's PhD thesis [46, p. 33), we know for the PAl LNA; R n= 35Q and g n - 0 -4. This gives CR).mi = 5900 at CT) i l 7K. Bob Schmidt designed a PA2 LNA where the R nis decreased by employing n n



perimental results using the PA2 and a specially designed two transistors in parallel LNA were inconsistent indicating the difficulty of trying to measure the thermal noise of a small resistance at low temperatures 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








57

(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 O.211m only 2mA of current is required to generate current densities -10l6 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 determing 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 temperature 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 function 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, , is not large especially at low temperatures and can introduce some error.







58

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.431. This, combined with the fact that the closed cycle qryostat 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.
















CHAPTER V
EXPERIMENTAL RESULTS



In this chapter we describe what was observed from the experiments 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 thermal 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 spectrum changes from f- 1/2 to f- 3/2 (see Figure 5-1) predicted from the diffusion theory. We are able to see the corner frequency, f diff' since,



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



where D is the thermal diffusion constant and L is the distance between heater and sensors. In the present case, we were able to use a heater almost the length of a chip (ie. L - lmm) away from the sensors causing the corner frequency to be at an observably low frequency (ie. f diff = 30Hz). This was not possible in the case of transistors since the devices were directly adjacent to each other and had a corner 59

















I0




16
Volts RMS

67


H
0


10-0









heater Power [ k
408 jjW
\


U (U~v\


I S I I


10 fdirf 100


1000


fo (Hz) -b


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


a I I










frequency beyond the range of observation. Again the thermal diffusion appears to be one-dimensional in nature as was the case in the transistors.

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 measurements with low duty cycles gave the same results as the resistance measured directly with an HP3466A digital multimeter on the lmA 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 TDT but relative-changets 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



















j10/0

Coherence

10%




1%


#:=r- -8 - --o---e

Heater Power o
o 408 uW E 70 .,W


t0 100
fe (Hz)


1000


Figure 5-2 Coherence Between Thermal Transfer Responses
















3TF


T2T
Ref ive.
Temp.

T



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





64

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 multimeter 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 (491 and is referred to as residual resistance. At high temperature, the resistance. which has a linear dependence with temperature, exhibits Matthiessen's Rule [421.



5.4 1/f Noise Versus Temperature


The current spectra of the gold thin film resistors were measured at different currents to check the 1 2 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 1IN dependence predicted by Hooge and have noise magnitudes in the same "ballpark" 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, R ON . The calculated R ON is always a few ohms larger than the R OFF (measured in Section 5.3)
























250


150


10 50 100 150 200 250 T (K) 10


300


Figure 5-4 Device Resistance Versus Temperature


















I= 20.5 mA

xk
xk


1=10.8


m

m A x


f (Hz) -


Figure 5-5


1/f Noise 12 Dependence


10-18-


C,,



Cl)


10-2010

10-212 i b -2 2 -


I I i
10 16o IK


10- 19-'.






















10-14-


c'J
C\)



10-15















I0 -16


HOOGE


0 U


.4 I I


0.'8 1
112


1.6 3.2


N - 1


Figure 5-6


Relative Noise Magnitude at 300K of all Devices Measured. (N =
-12
0.8 X 10 - 0.5pm devices, N =
-12
1.6 X 10 - lpm devices, N =
-12
3.2 X 10 + 2pm 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 I f 1 2 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 representing the different devices. In general, the "quiet" devices (those that fall below the Hooge line) have larger error bars on both the S I f/I 2 and y plots because in those devices there is less difference 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 probably 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 eliminates the temperature-fluctuation model proposed by Voss and Clarke for the explanation of 1/f noise in metal films.



















-14
I0








- 15



C\I"










08
I I 1O 100 300
T (K)


Figure 5-7 Noise Magnitude and Slope Versus Ambient Device
Temperature. (0.5pm device)






70



















-15
10







-16

1.2




--0.8

10 100 300

T (K)



Figure 5-8 Noise Magnitude and Slope Versus Ambient Device
Temperature. (Noisy 1pm device)

















10- 15U-,

oa 10- 17



-17




I0




FO




Figure 5-9


1-.4

1.2



" 0.8

100 300
--T(K)



Noise Magnitude and Slope Versus Ambient Device Temperature. (Quiet lpm device)




























T 100
0 1
so
0

z
LU
,, , I



0 .






0.01




0.001 S10 160 IK

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 straightforward 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 recombination 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 Dp since by the Einstein relation,



q6Dp = kT6. . (5-2)



Thus we may expect the mobility fluctuations to modulate the emittercollector hole diffusion current and/or the base-emitter electron injection current.

Van der Ziel's appendixed derivation [53] of Kleinpenning's expression for the noise spectrum due to mobility fluctuations of emittercollector hole diffusion in P+PN transistors shows,



SI (f) = 2I 4fTP 9.n 5-]
Ep =q dp P() (53)








74

where a P is the Hooge parameter associated with hole current, T d w B2/2D Pis the diffusion time for holes through the base region, wB the base width, and P(O) and P(w B) are the hole concentrations for unit length at the emitter side and the collector side of the base, respectively. We see the magnitude of S IEp is inversely proportional to T dp,


which means that S is proportional to f Tsince
Ep


f 1 (5-4)
T2TrT p



Therefore, the hole mobility fluctuation 1/f noise source is larger in transistors with large f T(e.g. microwave transistors).

Also, for electron injection from base to emitter, we have, due to mobility fluctuations, 153, Eq. (4)],



S (f) = q In fn N(O)J (5-5)
1 n E n4 dn


where T =w 2/2D ,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(w E) 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, R5, and a load resistance, R L(see Figure 5-11). Here, S ifb 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


A= PRL
rTr


Figure 5-11


Equivalent Common Emitter Circuit







76

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 contributor of the collector 1/f noise current source, Sifc, and the collector shot noise current source, 2eI c, have been referred to the input equivalent circuit as noise voltage spectrum sources by multiplying by
2 2
(/gm) = (rT/ 3) (valid if r >> rb).

An HP3582A FFT spectrum analyzer measures the spectral density of the collector noise, M 2/Af. Calculations from Figure 5-11 reveal,



T-f A (S Rs+rb) R s + r b + r I
Rs b [s~~~j




r+R 2 (r+R22r
rb + s b Sif
+ 2kTr fR +b +r (R + +rb +) r 2
CR + rb + 7F



2kT Sifc (5-6)
gm gm2



If we use that r 7r >> rb and >> 1, then Equation 5-6 can be written so

that we obtain,



A2 = R + r + r 2kT(2rb + i/gm) + + Sifbifc
Af s +b +r gm .


4 ifcr +S
R kT + 2 2 Sif2 i b




R2 2 + Sifc (5-7)
s r7 + 2 SifbJ







77

We see that there are three regions to the magnitude of the measured noise versus R -an independent, a linear, and a quadratic regime.
s

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
s

have a high DC 3. So the experiment was performed on low-noise PNP transistors (GE 82 185) with a = 350 typically. A simply biasing scheme was used for the high R experiment, [54] and the noise was measured for
s

three different I B's. From Equation 5-7 and for the Ncase of high Rs, we see that we measure with the spectrum analyzer,


MHI2 2 R2 2I +, + fc] (5-8)
Af 82L2 2eB +Sifb + 82



kT
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) repres
2
sents the absolute magnitude of the physical noise sources (in amp sec) referred back to the (base) input,



S M HI2 121 = 2eIB + S + Si2 (5-9)
Sf i ifb



The high-frequency roll-off, which each of the plots indicates, is attributed to the Miller effect of the capacitance CT in the equivalent circuit (see Figure 5-11) where,


CT = Cbeo + Cbco (i + IAv )


(5-10)















































10 100 1K IK
f (Hz) -


Figure 5-12


Measurements of High Source Impedance Spectra (S HR) and Low Source Impedance Spectra (SLR s
s


-18


-19 -20 ? -21

(I)
\J -22


-23


-25


-26







79

Since IB is small, r is large, and the f = 1/CT r Miller cut-off frequency, is low -2 KHz. Shot noise, low-pass filtered across the parallel combination of r and CT, gives at sufficiently high frequencies,


2eIB.
Ss - B+2Tr (5-11)
HR 2 22
T Tr


the observed 1/f2 roll-off.

Biased with a low R configuration [541, we neglect the terms in
2
Equation 5-7, which are proportional with R and R . Using gm = /r s s

and neglecting R and rb with respect to r , we see that we can plot (again in amp sec).


M L 2 2
_ RL2 = 2eI + 4~
LR Af 1!C bkr~ ifc



+ Sifbrb2 gm 2 (5-12)



This was done in Figure 5-12 (Curves I, II, III) at the same three IB 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 froms

the "noisiest" device where we see S If - 10-18 with a crossover frequency 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 2 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 sconfiguration give S If - 0-1with 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 sensitivity.

It should be noted that Equation 5-12 is only valid for R � < r b.

In practice, however, R swas of the same order of magnitude as r b at low I (e.g. R 2 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 S LRbecomes,




S LR 2eI 4Tr b + , gm + Sifc + Sifb rb 2gm 2 (5-13)



We see that the low R measurement provides the means to measure r b [54]. Using the magnitude of the white noise levels of S LRin Figure 5-12, the calculated values of r b are indicated in Table 5-1.



5.5.2 Interpretation of S and S
HR LR
5 S

To calculate the magnitudes of S.f and S if'we look only at the 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,



S LR Sifb (r b 2gm 2 + Sifc' (5-14)
s


and,

















TABLE 5-1
Data Obtained From High and Low R Biased PNP Transistors
s





Low R Data High R Data
s s

Curve IE rb Curve I (n)MIN


I 2.25mA 340 4Q IV 6.711A 362 1.2 x 10

II 1.3mA 420 9Q V 31A 363 6.6 x 108










S S + Sifc (5-15)
HR ifb 2



Having two equations involving the two unknown Sifc and Sifb, we solve for Sifc and find,



SLRs 12 Ss

S - (5-16)
ifc (r1g]2 2





Now from inspection of Figure 5-12, we see S << S at 1 Hz, and
HR LR
s s
since


l_ r





we can neglect 1/ 2 and 5-16 simplifies to,



SLR = Sifc (5-17)
s


We see at low frequencies the low Rs bias configuration isolates Sifc* Solving Sifb' we find, SLR
s
Sif = SH~ - 2 - - (5-18)



2
Since even the SLR from the "noisiest" device, when diminished by 2
s

is a factor of ten less than SHR at 1 Hz and we see the high R confis
guration essentially isolates Sifb.










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 kn[P(0)/P(wb)],% n (N(O)/N(wE)], Tdp, and Tdn are weak functions of the current, so that we can expect,



Sifc Ic


and,


Sifb b,



if the mechanisms are mobility fluctuation noise. Measurements of S ifbf versus Ib and S ifcf versus Ic are shown in Figures 5-13 and 5-14. We see the base noise is roughly proportional Ib (a small slope deviation from unity is expected since Tdn and ),n (N(O)/N(wE)1 are weak functions of the current). This shows the base noise is definately 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

n
For the case of base 1/f noise, we have,



S H Sifb SI , (5-19)
s En


and using the base to emitter expression (5-5) we have, for cn'































1


7




3.

2


2





Figure 5-13


- B (A) 11-


Base 1/f Noise magnitude Versus Base Current










































I I


7 I 1--1c (A)-


Figure 5-14


Collector 1/f Noise Magnitude Current


Versus Collector


3+


1618


71


3.-


.lO-


2 *3


I I


2 3


� , !









SHR f 2Tdn
S s (5-20)
n kn N(0)]

B


since I Enz IB in a P+NP transistor, if we neglect recombination. We take kn [N(0)/N(w E)] < 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 an
n

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
n
mobility reduction factor. Kilmer et al. [54], using a ratio of imp/
-5
Platt -- 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-fluctuation mechanism. Unlike the collector noise, the base noise showed no device dependence, indicating the base noise is intrinsic to the material. 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 definately caused by mobility fluctuations.

















CHAPTER VI
INTERPRETATION OF RESULTS



In the transistor experiments, the results are fairly well understood. The theory for mobility-fluctuation noise is well defined and correctly describes the observed base 1/f noise. However, the interpretation 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 If/I 2vs. temperature is the relative noise minimum occuring around 150K in all the samples (except one O.51Jm 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 (6=165K).
D

To determine whether this is merely a coincidence or, perhaps, the first means to actually measure e0D would require noise measurements from thin films made from different metals'to exhibit noise minima at their respective e I's. Good metals to verify this are ment-ioneqdin
D

Section 7.1.

The difference between the trends observed from Figure 5-7 through 5-9, can be roughly classified into three groups. The 0.51Jm width devices all exhibit types of noise which appear less fundamental to the element gold and more related to their unusual geometry. Since these 87








88

devices'exhibit a more extrinsic noise, we will put them in their own classification. Between the 1 and 21Jm 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 analyze them separately from the quiet devices Cie. 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 section of this chapter.



6.1 The O.514m Devices


Only a few of the O.5pjm devices could be measured over the full cryogenic temperature range. This was due to the high number of defective 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.O.5lpm devices were very delicate and often "died", from the transient currents between DUT ON and DUT OFF, before measurements could be completed. The O.514m devices were prone to burst noise exhibiting a greater than I current dependence. A S If/I 2vs. T plot of a O.5lpm device which did not exhibit burst noise is given in Figure 5-7. It is the only S If/I plot which shows the noise increasing at the low temperatures. We believe the strange behavior of the O.511m 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 2pim devices, the width of the resistors 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 0.51dm device, the widthto-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 0.51im samples are less interesting, for the purpose of this study.



6.2 Noisy 1 and 21dm Devices


The devices which have noise magnitudes at 300K which are larger than the values predicted by Hooge, we have chosen to call "noisy devices." A good example, of this classification, is the noise from the 11dm sample in Figure 5-8. The noisy 1 -and 21dm 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 f 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-would--explain 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 activation energy between 0.1 and 1eV, to be the random process alluded to in Dutta and Horn's theory. By a simple argument,.he, shows a




Full Text
30
(for the connection with the notation in the next section, we have
v ^v*^ = S (0) )Aw, etc.). If the noise drops slower with frequency
si
than the signal response and if v v*£ = 0/ the above gives for suf
ficiently high frequency, in the case that (3-15) applies,
y(large w )
Re v v*
sloo s2w
e e
[v V* V V* ]
1 nl nl n2 n2J
1/2
TTm
2_2
Pho lG(rsl'VjJ)| lG(rs2vrh;j<)l co*<*2 ~ *1*
2 2 .1/2
c d [v v* v v* ]
nl nl n2 n2
(3-31)
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 = -aV (At) + ii(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,)) = 4k [T (r) ] 2 (r) 6 (r r)I (3-33)
T) BO
where I is the unit tensor. We assume the steady state temperature
T (r) = T is uniform, and cr(r) = a. For the conservation we have again,
o o
analogous to (3-3),


LIST OF FIGURES
Figure Number Page
2-1 Layout of Closely Spaced Gold Thin Film
Resistor Array 12
3-1 Layout of a Chip Containing Three Different
Groups of Thin Film Resistor Arrays 37
4-1 Calibrated Noise Measurement Scheme 44
4-2 Equivalent Circuit of the Calibrated Noise
Measurement Scheme 45
4-3 Schematic of the Schmidt Low Noise Amplifier . 48
4-4 Simplified Equivalent Circuit of the Low
Noise Amplifier 55
5-1 Thermal Transfer Responses of Two Thin Film
Resistors 60
5-2 Coherence Between Thermal Transfer Responses . 62
5-3 Relative Thermal Noise Versus Device Bias
Current 63
5-4 Device Resistance Versus Temperature 65
2
5-5 1/f Noise I Dependence 66
5-6 Relative Noise Magnitude at 300K of all
Devices Measured 67
5-7 Noise Magnitude and Slope Versus Ambient
Device Temperature (0.5ym device) . 69
5-8 Noise Magnitude and Slope Versus Ambient
Device Temperature (Noisy lym device) ..... 70
5-9 Noise Magnitude and Slope Versus Ambient
Device Temperature (Quiet lym device) 71
5-10 Typical Coherence Between the 1/f Noise of
Two Thin Films 72
v


99
2) by changing the ambient temperature of a passive device.
In either case, close analysis of the slope of the 1/f noise is a
key in distinguishing which mechanism is dominating. So called
"generic 1/f noise," with slope ranging 1.4 < y < 0.7, must be due
to number fluctuations; while pure 1/f noise, y = 1, is necessary for
mobility fluctuations. Another trend which was seen in both the tran
sistors and the resistors is that number-fluctuation noise is gener
ally larger than mobility fluctuation noise, ie.
is fl
f \
S f
I
>
I
2
2
I
N
We make the conjecture that mobility-fluctuation noise (such as Handel's
Quantum 1/f Noise) is always present and sets the theoretical minimum
of all 1/f noise.
At this point, we must congratulate NRFSS for producing such high
quality devices (especially the 1 and 2ym) that the same pattern of
noise versus temperature reproduced itself on almost every device;
and the impurity and defect concentrations were low enough that, for
the first time, Quantum 1/f noise could be observed. It is entirely
possible that even the high noise 0.5ym device, as well as the noisy
1 and 2ym devices, after annealing by age or other processes, may
loose the extrinsic number-fluctuation noise they presently-exhibit
and reveal the intrinsic Quantum 1/f noise which we believe is omni
present.


TABLE 2-1
1/f Noise Experimental Results from Metal(Films
\
T DEPENDENCE
OTHER DEPENDENCES
Near
Supercond.
Trans.
At
Room
Temp.
Extended
Temp.
Range
Substrate*
Dependence
Underlay
Dependence
Slope
Dependence
Film
Thickness
Dependence
Annealing
Dependence
Resistivity
Dependence
Sn
1,2
3,7,8
1,2,7
1,7
7
8
Pb
1
8
8
Cu
3,8
4,5 '
5
4
8
Ag
3,8
4,5,6
5
4,6
6
4
8
Au
3,8,10
4
4
8
Bi
3
6
Man-
ganin
3
8
Ni
4
4
Pt
8,9
8
In
1
8 1
8
*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.


85
Figure 5-14 Collector 1/f Noise Magnitude Versus Collector
Current


CHAPTER V
EXPERIMENTAL RESULTS
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 to f (see Figure 5-1) predicted from
the diffusion theory. We are able to see the corner frequency, f ,
since,
^diff
1 D
Tdiff L2
(5-1)
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 ~ 1mm) away from the sensors
causing the corner frequency to be at an observably low frequency (ie.
fdiff 30Hz). This was not possible in the case of transistors since
the devices were directly adjacent to each other and had a corner
59


31
(3-34)
From (3-32) and (3-34) we obtain,
- aV2(At) = r V n = 5(r,t).
(3-35)
For the spectrum of £, from (3-33) we have
Sr(r,r') = 4k T2aV V'<5(r r')/c2d2
c, BO
(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'6(r r'). We now represent At and § by truncated Fourier series
on the interval (0,T) with amplitudes AT(r,co) and £(r,w). From (3-35)
we obtain the relation,
ju)AT(r,c) aV2AT(r,w) = £(r,w). (3-37)
Using the Green's function (3-11) we find the solution,
/
(3-38)
V,
total
complete integrated circuit. For the spectra of AT(r,t)AT(r',t) we have
as usual [17]


38
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',jw)
G(x,x' j))
where,
cothYC
2ya(sinhyx+tanhyCcoshyx)(sinhyx'-tanhyCcoshyx'), (x cothYC
(sinhyx1+tanhyCcoshyx1)(sinhyx-tanhyCcoshyx), (x (3-59)
Y
= \/jw/a
(3-60)
We define a corner frequency,
a). = a/c .
bound
(3-61)
Then ye = For w Wkoun(j* |ycI r* 0* Thus tanhyc -* yC,
and since |x| < |c|, sinhyx yx, coshyx -> 1; for (3-59) we now find,
G(x,x' jw)
W 0).
c
2a
(3-62)
bound
the heat transfer function (3-22) thus becomes a constant at low fre
quencies .
For frequencies 0) ^ound' tan^YC 1. We then easily find,
k
Note that there is an error in this formula. The a's should be in
front of the cosh terms and not in front of the sinh terms.


21
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 < 10 ^ may be low enough for Handel's theory to
apply (see Chapter VI).


9
For the bremsstrahlung matrix element, under emission of photons, one
calculates easily, either classically or quantum mechanically,
|b(e)| = aA/e = aA/hf ?
(1-16)
here a = 1/137 is the fine structure constant and
1 Av 1 2
3tt 2
(1-17)
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, otA,
ie. combining Equations 1-15, 1-16, and 1-17,
S
v

2
2aA
f
2a |Av[2
3TTf 2
c
(1-18)
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-
%2
tance fluctuation since R = Dutta and Horn in their review paper
Aq)JN *
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.


93
reasonable defect concentration to yild the observed noise magnitude.
Since annealing changes the defect concentration and also the noise,
Robinson's argument appears reasonable. We believe, however, that
whatever the nature of these activated processes, the 1/f noise arising
from number fluctuations is still an "extrinsic noise." A more funda
mental source of 1/f noise will be discussed in the next section.
6.3 Quiet 1 and 2ym Devices
Fleetwood and Giordano [35] believe, "the best experimental mea
sure of the 1/f noise intrinsic to a metal should be the minimum
noise observed for a given metal." In attempting to apply Handel's
Quantum Noise theory, which we would assume to be responsible for the
intrinsic source of 1/f noise, we will concentrate only on the quiet
devices (ie. those that fall below the Hooge line in Figure 5-6).
From Equation 2-6, we see, discarding number fluctuations (since
Handel's theory is only for mobility fluctuations),
I %
N2
(6-1)
Thus, we see we need to have a value for N. Voss and Clark [11]
started using N as the number of atoms in the sample, and never consi
dered the possibility of N as a function of temperature. Butta__and
Horn, as well as Fleetwood and Giordano, continued to use the number
of atoms since it is an approximation to the number of carriers in
metal films. We believe for metal films; however, N = N^, the total
number of carriers available for scattering at room temperature. We
have used


97
true
Figure 6-4 a
Versus Temperature


25
!
of (3-9) will also be written as Ap, (t)e(r), where e(r) is the function
h
which is unity in and zero outside. For Ap^ = Ap^e3^/ (3-9) gives
if At = At e^wt,
00 '
jwATw(r)
- aV At (r)
co
1
cdV,
h
AP e(r)
coh
'(3-10)
We define the Green's function of (3-10) as usual by the impulse
function response,
j(0G(r,r', jco)
aV G(r,r', jco)
<5(r-r1),
(3-11)
subject to boundary conditions which we discuss in Section 3.6. The
solution of (3-10) is then [44],
G(r,r',jw)£(r')d3r'
00
G(r,r, jco)d3r'. (3-12)
Ap
coh
cdV,
Ap ,
At (r) =
co cdV,
h
Denoting by At the averaged temperature increment in the sensor, we
COS
have,
AT
'coh
cos cdV V,
s h
d3r
d r'G(r,r', jco)
v V.
s h
(3-13)
Let At^s = |ATwg|e^. Then for a real input signal with amplitude P^q,
the real sensor output signal appearing in the sensor circuit is v =
s


80
that for this "noisy" device we may be revealing a different noise
producing mechanism. Other transistors biased with the low R confi-
s
21
guration give S^f ~ 10 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
vity.
It should be noted that Equation 5-12 is only valid for Rg r^.
In practice, however, Rg was of the same order of magnitude as r^ at
low I_(e.g. R a 5ti). As a consequence, the thermal noise generated
£ S
by R cannot be neglected and has to be incorporated in Equation 5-12.
s
The expression for S becomes,
LR
S
3 = 2el + 4kT(r + R )g
LR c b s m
s
+ S.c
lfc
+ S. r
lfb b
m
(5-13)
We see that the low Rg measurement provides the means to measure r^ [54].
Using the magnitude of the white noise levels of S in Figure 5-12,
JjK
s
the calculated values of r. are indicated in Table 5-1.
b
5.5.2 Interpretation of S and S
HR LR
. s s
To calculate the magnitudes of S^c and we look only at the
1/f portion of our spectra (ie. at f < 100 Hz) where we areTabove the
shot-noise level and can write, at low f,
!LR Si£b(rbV>
s
+ S._ ,
lfc
(5-14)
and,


52
3) DUMMY AT T
-2 -
!03
Af
= G
i 2 i V
(S + S )Z +
l2 a a
(4-12)
Solving for S we find,
V
Sth ~
01
2
02
03
4k
T (T2
Tl)
4kT
R
(4-13)
where we have assumed the necessary condition R = R^m s R. We
1 DUT DUMMY
see we can simplify (4-13) to express the actual device temperature,
DUT'
as,
T,
DUT
-2 -2
V V
01 02 ,
3 (T T ) + T .
2 1 1
03 02
(4-14)
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,
(4-15)
2 2 2 2
where x represents the (VQ1 V02^/^V03 V02^ term Differentiating
we see,
At = aAx
(4-16)
Now we normalize since we are interested in relative error,
At = aAx
T T


57
(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 lym X 0.2ym only 2mA
6 2
of current is required to generate current densities ~10 A/cm .
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 determing 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 filmsthe resistance versus temperature coefficient, 0, is
not large especially at low temperatures and can introduce some
error.


13
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-r
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)
2
-1
and with
f \
dR
. dT.
V /
. A low frequency flattening of the room tem
perature 1/f spectrum is observed with 5,000 Al underlay,


90
!
Figure 6-2 photograph of Short-Circuited Devices


2
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],
2
(1-2)
where is spectral density of voltage fluctuations and is the
-3
Hooge parameter which he assumed to be a constant a ~ 2 X 10
rl
We see the magnitude of the voltage spectrum is proportional to the
^square qf_the 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/f^ 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,
2
(1-3)
From this, we see we can write with Hooge [8]
(1-4)


82
S = S + lfC
HR Sifb + q2 *
s 3
(5-15)
Having two equations involving the two unknown and we
solve for S., and find,
ife
LR
S. _
ife
-rb^nv
- S.
HR
(5-16)
^rbV
Now from inspection of Figure 5-12, we see S S at 1 Hz, and
HR LR
S S
since

1
2
f \
r
IT
r,g
o in
K
~ 1,
we can neglect 1/3 and 5-16 simplifies to,
S = S
LR ife
s
(5-17)
We see at low frequencies the low R bias configuration isolates S._ .
.S 1IC
Solving we find,
LR
_ s
Sifb HR q2
s 3
(5-18)
Since even the S from the "noisiest" device, when diminished by 3 ,
LR
S
is a factor of ten less than S at 1 Hz and we see the high R confi-
HR S
s
guration essentially isolates S^^-


28
for the case of Equation (3-15). The linear signal response is,
mP.
v
ho
SO)
i/2cdV V,
s h
d3r
d r'G(r,r', jto)
v V.
s h
(3-23)
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,
vg^ and vg2 of two sensors, having a temperature fluctuation due to a
common heater power AP, = P. cos2to 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
sito s2(0
v (t)v (t + T)e ^dx
si
s2
(3-24)
where v (t) = V cos (to t + c|> ) and v (t) = V cos (to tH- ) .
S JL S-LC) 6 J. Su ShO 6 m
Writing,
Cos (to t + (0 T + (¡)) = {cos (to t.+ cj) ) COS (to T + (J> C¡) )
Q 4 61 J.
-sin(to t + tb,) sin (to T + <)>,) (3-25)
e 1 e 2 1


101
noise of the semimetal to be proportionally larger than the metal.
If the proportional increase in the noise agrees with the proportional
decrease in Nscat/ our hypothesis used in Chapter VI would be verified.
7.2 Continued Studies of the High Temperature
Number Fluctuation Noise
Continued investigations of the number fluctuation noise is sug
gested since this noise, while less important to the fundamentals of
solid-state physics, is from a practical standpoint the 1/f noise most
often encountered in devices.
Van der Ziel will use the technique of noise analysis versus tem
perature to study electro-migration. Electro-migration is a phenomenon
which accounts for the deterioration of VLSI metal film buses carrying
large current densities. Ion diffusion causes lattice breakdown and
scattering centers bunch together thereby increasing the resistance.
In these experiments, the slopes of the low frequency noise will be
carefully studied since y ~ 1.5 implies ion-diffusion while y ~ 1.2
suggests Dutta & Horn's narrow band distribution of time constants
theory. Observing the noise magnitude versus temperature permits cal
culation of the noise processes activation energies.
Another approach to study the extrinsic device noise is to analyze
the "aging effect" of a resistor's 1/f noise. Handel models the 1/f
noise source of the resistor as two noise sources in series. One noise
source is the stationary fundamental 1/f noise source and the second is
a time dependent "aging" noise source. Studies of the 1/f noise over
long periods of time would permit the aging effect, A(t), to be deter
mined and the characteristics of the fundamental 1/f noise source could


94
N = nV
c
(6-2)
where n is the density of carriers in gold and V is the sample volume
3 3
(800 X 1 X 0.2ym ). For n we calculate the number of atoms per cm
using Avagadro's number, the density of gold, and gold's atomic weight.
Then multiplying the chemical valence of gold by the atomic density,
we arrive at the order of magnitude approximation of the carrier (elec
tron) density, ie.,
22 -3
n s 10 cm
(6-3)
This is in agreement with the carrier density of a highly degenerate
semiconductor which is a close approximation to a metal.
As was pointed out by van Vliet and Zijlstra [36], the basic for
mula for the mobility fluctuations for the scattering of a single car
rier is (see Equation 2-1),
S.(f) a
yi true
2 f
(6-4)
where we subscripted the a-value as a. To obtain the fluctuations
in the band mobility, or for that matter of the current I, we must sum
over the scattering fluctuations of all carriers in the band [57,58].
For a nondegenerate semiconductor this provides a factor 1/N in the -
denominator (see Equation 6-1). In metals, however, most of the car
riers are "frozen" in the Fermi sea, a fact also noted by Dutta and
Horn in their review paper [4]. However, the concentration of carriers
in a metal is so large that we cannot assume (as we did in Section 2.3)


106
[53] A. van der Ziel, Solid State Electronics 25, p. 141 (1982).
[54] J. Kilmer, A. van der Ziel, G. Bosman, Solid State Electronics
26, p. 71 (1983).
[55] G. Bosman, R. J. J. Zijlstra, and A. van Rheenen, Physica 112B,
p. 193 (1982).
[56] F. N. H. Robinson, Physics Letters 97A, p. 162 (1983).
[57] F. N. Hooge, Physica (Utrecht) 114B, p. 391 (1982).
[58] A. van der Ziel, C. M. van Vliet, R. J. J. Zijlstra, and R.
Jindal, Physica (Utrecht) 12IB, p. 420 (1983).
[59] K. M. van Vliet and A. van der Ziel, Solid State Electronics 20,
p. 931 (1977).
[60] L._Brillouin, Helv. phys. Acta 7, Suppl. 2, p. 47 (1934).
[61] J. Bernamont, C. R. Academy of Science (France) 198, p. 1755
and p. 2144 (1934).
[62] M. B. Weissman, R. D. Black, P. J. Restle, Noise in Physical
Systems and 1/f Noise, M. Savelli et al. (Eds.) North Holland,
Amsterdam, p. 197 (1983).
[63] D. M. Fleetwood and N. Giordano, Noise in Physical Systems and
1/f Noise, M. Savelli et al. (Eds.) North Holland, Amsterdam,
p. 201 (1983).


26
V cos(O) t + 4>) where 0) = 0) and V = ml At I ,
so e e so 1 cos1 '
f
(3-14)
The sensor signal is, as expected, fully determined by the Green's
function. For two cases this can be simplified. Suppose that either
Vg and 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,
v (mP, /cd) IG (r ,r., j(0) [ cos (to t + )
s ho 1 sh e
(3-15)
where 4 is the phase angle of the Green's function.
We must now consider the measurement of vg 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
00
f
d[v v* ] = 2Af v (t)v (t + T) e ^UTdT
seo sco s s
(3-16)
where v (t) = V cos((0 t + ) with V given by (3-14) or (3-15). Now,
s so 0 so
2
v (t)v (t + x) = V cos (co t + )cos() t + co T + d>)
ss soe ee
so e e e e e


71
Figure 5-9 Noise Magnitude and Slope Versus Ambient
Device Temperature. (Quiet lym device)


72
Figure 5-10 Typical Coherence Between the 1/f Noise
of Two Thin Films.


23
ffi
V
o
R
(s* + V
(3-1)
where V and R are the sensor film's voltage and resistance, and 3 is
ox
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^ = Apeos ^t, which causes a temperature rise ATg =
At cos (to t + d>).
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
3t
3t
+ V
K
0;
(3-3)
here K is the heat current vector, indicating heat carried per second
through unit area, O 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,
3t n2
3t aV T
= 0
(3-4)
where a = a/cd is the heat diffusivity. For the silicon substrate [43],
one has,


49
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 SiO^ 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, BeC^, or sapphire). In addition to the
possibility of one of the aforementioned items pausing 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, --
Sfch = 4kT/R (4-6)
accurately enough to calculate T.


7
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 suggested as 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, a calcu-
H
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,
^T(,t}
A
a e
i(k*r-wt)
1 +
bT(e)ei(E/,i)tde
e
o
(1-11)



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PAGE 116

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PAGE 117

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TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ..... ii
LIST OF FIGURES v
ABSTRACT vii
CHAPTER I INTRODUCTION 1
1.1 Fundamental Questions 3
1.2 Competing Theories 4
1.2.1 A Thermally Activated Number
Fluctuation Theory 5
1.2.2 Diffusion Theories 7
1.2.3 The Quantum 1/f Noise Theory 7
1.2.4 The Temperature Fluctuation
Theory 10
CHAPTER II SCOPE OF OUR INVESTIGATIONS 11
2.1 1/f Noise Correlation Experiment
and the Thermal Fluctuation Model ... 11
2.2 Previous Results from Experimental
Work on Metal Films 13
2.3 The 1/N Dependence and the Hooge
Parameter 16
2.4 Temperature Dependence of 1/f
Mechanisms 19
CHAPTER III THEORY OF HEAT TRANSFER AND OF TEMPERATURE
FLUCTUATION NOISE 22
3.1 Introduction 22
3.2 The Heat Transfer Function 23
.3.3 Heat Transfer Correlation ....... 28
3.4 Temperature Noise 30
3.5 Temperature Noise Correlation 34
3.6 The Green's Function for the Gold
Thin Film Array 36
iii


66
1
OJ
co
CO
10
-18-r-
10
-19'
10
-2 0-
10
-21-
sx.
\
\
\
1 = 20.5 m A
\
V \
\ ",
x \
X
X\
X\X
\*
\ V
Njr
I=l0.8mA^ N
\ Jr
* >.
Mr
\>r
-2 2
10 j
10
100
IK
f (Hz)'
Figure 5-5 1/f Noise I Dependence


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the degree
of Doctor of Science.
C. Van Vliet, Chairperson
Professor of Electrical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the degree
of Doctor of Science.
G. Bosman, Co-Chairman
Assistant Professor of Electrical
Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the degree
of Doctor of Science.
E. R. Chenette
Professor of Electrical Engineering
:s I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the degree
of Doctor of Science.
V. Ramaswamy
Professor of Electrical Engineering


10
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, 3,
3 =
1 dR
R dT
(1-19)
Using the Langevin diffusion equation,
9t
at DV T
V*F
c
(1-20)
where D is thermal diffusivity, c is the specific heat and F is .
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
devices.


69
C\J
*i
CO
10 100 300
T (K)
Figure 5-7 Noise Magnitude and Slope Versus Ambient Device
Temperature. (0.5ym device)


IDENTIFICATION OF 1/f NOISE PRODUCING MECHANISMS
IN ELECTRONIC DEVICES
BY
JOYCE PRENTICE KILMER
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF SCIENCE
UNIVERSITY OF FLORIDA
1984


102
be isolated by subtracting from the total observed noise the Fourier
transform of the aging effect, A(t). Such procedures would signifi
cantly affect the value of the slope of the fundamental 1/f noise
source, possibly causing it to approach y ~ 1. Such a result would
open the possibility of a mobility-fluctuation theory (and even Handel's
Quantum Theory) to apply even at the high temperatures. These inves
tigations would be interesting to try to correlate with the changes in
noise found from annealing, and further investigations are suggested
along these lines.
7.3 Investigation of New Queries
The experiments, such as the one reported by Weissman at the 3rd
International Conference on 1/f Noise [62], to determine if 1/f noise
is scalar or anisotropic should be continued to resolve this question.
Finally, the role of stress application to metal films and how this
affects the 1/f noise, as proposed by Fleetwood and Giordano [63],
should be further investigated since it may give some insight into the
low temperature noise where induced stress due to thermal contraction
is a possibility. In conclusion, further measurements of the 1/f
noise of different materials, under varied experimental conditions,
should be performed at varied ambient temperatures. Thus, the area of
1/f electrical noise in solid-state devices is ripe for further inves
tigation.


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
IDENTIFICATION OF 1/f NOISE PRODUCING MECHANISMS
IN ELECTRONIC DEVICES
By
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 f 'he l/.f noise com
monly observed in two electronic devices (ie^ thisj) film resistors and
PNP transistors). First, we delve into the details fo two competing
1/f noise theories of metal thin filmsDutta & 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.
vii


74
where is the Hooge parameter associated with hole current, x^ =
2 .
w^ /2Dp 1S the diffusion time for holes through the base region, wfi the
base width, and P(0) and P(w ) are the hole concentrations for unit
B
length at the emitter side and the collector side of the base, respec
tively. We see the magnitude of is inversely proportional to x^ ,
JEp P
which means that S is proportional to f since
IEp
fT 2TTT
(5-4)
dp
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)],
S (f)
En
a
2ql
n
En 4fx
£n
dn
' N (0) *
N(we)_
(5-5)
2
where x, = w/2D w the width of the emitter region, D the electron
dn E n E 3 n
diffusion constant in the emitter region, whereas N(0) and N(w ) 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, R and a load resistance,
(see Figure 5-11). Here, represents the 1/f contribution to
the total spectrum arising from the base 1/f noise current sources.


24
O = 1.45 Watt/cm C
c = 0.7 Joule/gram C
3
d = 2.328 gram/cm
2
a = 0.89 cm /sec. (3-5)
If we now introduce a heating power AP^(t) in a volume V, and if
we denote by Ae the AC energy contained in V, we have the conservation
law,
AK dS + AP, (t). (3-6)
h
3 13
Writing Ae = /cdAT(r,t)d r and AP, (t) = /Ap, (t)d r, we find by Gauss's
V h V v h
theorem,
dAE
dt
, 3AT(r,t) ,3
cd 9t^ d r =
AK d3r
V
AP, (t)d r.
h
V
V
(3-7)
Since this holds for an arbitrary volume, the integrands must be equal,
cd ^ + V AK = i APh(t).
(3-8)
With K = aV(AT) this gives,
3AT
3t
- aV2(AT)
1
cdv,
h
(r,t)
(3-9)
where AP^(r,t) is a function equal to AP^(t) for r& and zero outside
Vh' Vh kein9 t^ie 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.


I certify that I have read this study and! that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the degree
of Doctor of Science.
S. S. Ballard
Professor of Electrical Engineering
This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate School, and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Engineering.
August 1984
Dean, College of Engineering
Dean, Graduate:School


78
!
Figure 5-12 Measurements of High Source Impedance Spectra (S )
HR
g
and Low Source Impedance Spectra (S )
LR
S




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


CHAPTER III
THEORY OF HEAT TRANSFER AND OF TEMPERATURE-FLUCTUATION NOISE
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
. 2
xv = x R, where R is the resistance of the film. If x = I cosa) t, the
o o
1 2
AC heater power delivered is AP, = AP, cos2cd t, with P, = I R; the
e h ^ ho o ho 2 o >
exciting power frequency is thus twice that coming from the signal gen
erator, = 2o)q. We suppose the power is uniformly dissipated along
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, R by v and the AC temperature of the sensor
L S
by AT we have v = V cos (to t + 6) AT = AT cos (a) t + ) with V =
2 s' s so e y s so e y so
mATgQ. We have according to Kilmer et al. [28],
22


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.
viii


51
2 2 2 2
many sources of error, especially in the vq3^^V02 ~ V03^
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-
4.3.2 Thermal Noise Measurements Without Using the Calibrated Noise
Source
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, S^, to the thermal noise of a
th
dummy resistor at one fixed temperature (e.g. melting ice T^ = 273K),
S and to the thermal noise of a second dummy at a second fixed
1
temperature (e.g. liquid Nitrogen T = 77.5K), S Thus, we have
; 2 T2
again from Figure 4-2 with R - 00 since we use no calibration source,
O
z
2
(4-9)
For the three difference measurement cases we have, since we measure
at high enough frequencies that -* 0,
1) DUT ON
2) DUMMY AT T
v2
01
Af
= G
(s., + s1)z2 + sv
th a a
v2
02
Af
= G
(s_ + S1)Z2 + sv
T^ a a
(4-10)
(4-11)


86
a =
n
SHR £ 2Td
s
(5-20)
qlgi-n
N(0)
n(we)
since I cz. I inaP NP transistor, if we neglect recombination. We
En 5
take £n [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
suggested by van der Ziel [53], we calculated the minimum values of
which are tabulated in Table 5-1 for S evaluated at 1 Hz. The values
HR
s +
seem a bit low; however, we realize that we have a P NP device where the
emitter is heavily doped and our observed a is diminished by an impurity
mobility reduction factor. Kilmer et al. [54], using a ratio of V^mp^
-5
y 2: 1/10, obtain a minimum value 2 X 10 for (a ) true. Bosman
XcltZu XI
-5 -3
et al. [55, Figure 5) report a values ranging between 10 and 10 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 definately caused
by mobility fluctuations.


89
Figure 6-1 Photograph of Open-Circuited Devices


34
Employing (3-45) this yields,
G(r,r', j(jO)d3rd3r'. (3-49)
s
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-
2
proximate the integral by Vg times the integrand since G varies strongly
in the neighborhood of r = r'.
For the noise in the sensor, we have,
4k T
sat s cdV
sv M = m SAT (w)
s s
2 2
4k T m
B o
cdV2
Re
G(r,r' j(jo)d3rd3r'
(3-50)
V V
s s
Comparing the noise (3-50) with the linear response^ lvsa3 I given in
e
(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 twodevices (the
same as the previous two sensors). For the spatially averaged tempera
ture cross-correlation, we now have, analogous to (3-47),
V
d3rd3r' At(r,t)At(r',t + T) (3-51)
V V
si s2
At 1(t)At (t + t)
si s2


16
that distribution width results from the number of sample
inhomogeneities. Bi shows peak just as Ag and Cu, however,
Au shows no peakonly 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. .Fleetwood & 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/N observed over
atoms
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 = Ey, we can write Equa
tion 1-18 also as a normalized spectrum of mobility fluctuations of
a single carrier,
S a
y true
-2 f
y
(2-1)


c
rb 4kTr¡r) 2kT/gm S¡fc/gm
vw (^)
r0-
n
>rTr
1 s
%b
rCr s,
4kTRs
3 2ftTr
T
A =
/3R|_
,rr
E
Figure 5-11 Equivalent Common Emitter Circuit


ACKNOWLEDGEMENTS
' 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.
I
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
manuscript.
This research was supported by the Air Force Office of Scien
tific Research, under Grant Number AFOSR 80-0050.
ii


95
that the electrons are independent. Van Vliet shows [59, Section (3)]
that, in this case, the result of the summation of Equation 6-4 must
2
be multiplied by 00/00, where the subscript 00 refers to the grand
canonical ensenble. From statistical mechanics the above factor is
kT 9(lnN)/9e_ where £ is the Fermi energy. Explicitly, we have,
F F
kT9(In N) = F1/2[(£c ~ V
9£F = F-1/2[(£c £F)/kT] '
(6-5)
where e is the bottom of the conduction band, and F, the Fermi inte-
c k
k+1
gral of order k. For total degeneracy, Fk(r|) = r) /r (k + 2). Thus
the ratio (Equation 6-5) becomes 2Ae /3kT with Ae = £ £ Conse-
quently, Equation 6-4 followed by the proper statistical summation
leads to,
sI(f)
Cl.
true
fN
2Ae
F
3kT
01.
true
fN
ef f
(6-6)
indicating that the number of carriers available for scattering is
N ff = Nc(3kT/2AeF). This is also intuitively obvious: The Fermi
function differs only appreciably from 1 or 0 in a slice of order kT.
That such a reduction in noise must occur in metals was perhaps
first pointed out in a classic paper by Brillouin [60] on the first
noise observations in metals, by Bernamont [61]. Comparing (6-1),
(6-6), and (6-4), we see,
y i i
3kT
or
a
true
S f S f
2 2 eff
V I
3kT
2Ae aH
(6-7)


G(x,x', jw) =
1
X'
(3-63)
39
-\/jw/a|x -
2/jaw
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'
3
d rG(x,x' jw)
dx'
J 4
4
dx G(x,x',jw)
V, v
h s
-e-L L
-I (e^L e-^)2.
2/jawY^
We introduce another corner frequency,
(3-64)
w = a/(L + e) .
(3-65)
For ^bound a) w1, we have |y(L + e) | 1, and a fortiori |yL| 1
Thus expanding the exponentials, we obtain,
d3r'
d rG(x,x', jCO)
2, 2 2
GAB
V.V
h s
2/jaw 2/jaw
V. Vo
h s
(3-66)
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,
V, V
h s
*

d3r'
J
3
d rG(x,x',jw)
1
/2aw
(3-67)


77
We see that there are three regions to the magnitude of the measured
noise versus R an independent, a linear, and a quadratic regime,
s
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
s
have a high DC B. So the experiment was performed on low-noise PNP
transistors (GE 82 185) with $ 350 typically. A simply biasing scheme
was used for the high Rg experiment, [54] and the noise was measured for
three different I 's. From Equation 5-7 and for the base of high R ,
we see that we measure with the spectrum analyzer,
M,
HI
Af
e\2
2el.
B
+ S. +
ifb
ifc
(5-8)
using r^ = where we have neglected the small r^ and r^ compared to
b
a high R and the terms independent of and proportional with R The
s s
measured high Rg noise plotted in Figure 5-12 (curves IV, V, VI) repre-
2
sents the absolute' magnitude of the physical noise sources (in amp
sec) referred back to the (base) input,
M.
HI
HR
Af
82r 2
JL
2el + S. +
B ifb
S. _
xfc
(5-9)
The high-frequency roll-off, which each of the plots indicates, is at
tributed to the Miller effect of the capacitance in the equivalent
circuit (see Figure 5-11) where,
CT =
"beo
+ C, (1 +
bco
A )
v
(5-10)


79
Since I is small, r is large, and the f = 1/C r Miller cut-off
B '77 3 m T 77
frequency, is low ~2 KHz. Shot noise, low-pass filtered across the
parallel combination of r^ and C^, gives at sufficiently high frequen
cies ,
2el,
B
HR 2_ 2 2
s 1 + u) C r
T 77
(5-11)
the observed 1/f roll-off.
Biased with a low Rg configuration [54], we neglect the terms in
2 ,
Equation 5-7, which are proportional with Rg and Rg Using g^ = p/r^.
and neglecting Rg and r^ with respect to r^, we see that we can plot
(again m amp sec).
M.
f \
LO
LR Af
s
= 2el_ + 4kTr, g + S>jr
C b^m lfc
2 2
+ S, r, g
lfb b rm
(5-12)
This was done in Figure 5-12 (Curves I, II, III) at the same three I_'s
B
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 S spectra. First,
JLK
S
it was found that the magnitude of the 1/f portion of S was quite
IjK
s-___
device dependent. The noise plotted for S in Figure 5-12 was from
LK
S
18
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 S to be y 1.18 while we see SUT5 has 1. This implies
LR HR
S S


84
Figure 5-13 Base 1/f Noise Magnitude Versus Base Current


91
!
Figure 6-3 SEM Photograph of Thin Film Resistors


96
With Aep = 5.5eV [49, p. 154], the values of atrue were computed from
Figure 5-9 to yield the data of Figure 6-4. We note we now obtain a-
values low enough to become in the ballpark expected from the quantum
theory of 1/f noise. In the latter theory, the a^.rue of Equation 6-4
is just twice the infrared exponent, (see Equation 1-18 and 2-1),
ot = 2aA
true
(6-8)
With |AvI = 2v sin Tp/2, where v is the Fermi velocity (1.39 X 108
F F
cm/s) and \¡J the scattering angle (# 150 for U-processes) we obtain
-7
from Equation 1-17, 2aA f 2.4 X 10 This value is approximate since
Nc is not exactly known and since more correctly we must take into
account the detailed geometry of the Fermi surface, being a sphere
with eight "necks" (see Ziman [42]). However, this value comes close
-7
to the observed values of (a. ) in Figure 6-4 being 4.9 X 10
true max
Qualitatively, we believe that the observed data of Figure 6-4
can be well understood. Above the Debye temperature 0^, region C,
some non-fundamental 1/f noise occurs, similar to the "type B" noise
observed by Dutta and Horn [4]. Below 9^, where y 1, we have for
the first time a clear indication of the occurrence of quantum 1/f
noise. U-processes dominate in the region B. In the region A, U-
processes freeze out and normal phonon processes (N-processes) take
over. Finally, in a region D (not yet observed), ionized impurity
scattering may give rise to a plateau at very low temperatures.
A quantitative theory has not yet been fully developed. However,
with U-processes dominating the noise and N-processes and ionized im
purity scattering dominating the resistance, one expects the tempera
ture dependence to be of the form,


REFERENCES
[1 ] J. Bernamont, Annals de Physics, 1_, p. 71 (1937).
[ 2 ] T. Musha and H. Higuchi, Proceedings of the Symposium of 1/f
Fluctuations, Tokyo, Japan, p. 187 (1977).
[ 3 ] H. M. Fishman, Proceedings of the Second International Symposium
on 1/f Noise, Orlando, Florida, p. 132 (1980).
-[ 4] P. JDutta and P. M. Horn, Reviews of Modern Physics, 53, No. 3,
p. 497 (1981).
[ 5 ] P. H. Handel, T. Sherif, A. van der Ziel, K. M. van Vliet, and
E. R. Chenette, "Towards a More General Understanding of 1/f
Noise," Physics Letters, Submitted.
[6] F. N. Hooge, Physics Letters A 29, p. 139 (1969).
[7 ] K. M. van Vliet, C. J. van Leeuwen, J. Blok, and C. Ris, Physica
20, p. 486 (1954).
[8 ] F. N. Hooge, T. G. M. Kleinpenning, and L. K. J. Vandamme, Reports
on Progress in Physics, 44, p. 483 (1981).
[9 ] M. Celasco, F. Fiorello, and A. Mosoero, Physics Review B 19,
p. 1304 (1979).
[10] D. M. Fleetwood, J. T. Masden, and N. Giordano, Physics Review
Letters 5£, p. 450 (1983)
[11] R. F. Voss and J. Clarke, Physics Review Letters _36, p. 42-4 0
(1976).
[12] R. F. Voss, Physics Review Letters 4£, p. 913 (1978).
[13] A. van der Ziel, Advances in Electronics and Electron Physics,
L. Marson (Ed.) 49, p. 225 (1979).
[14] 'P. Dutta, P. Dimon, and P. M. Horn, Physical Review Letters 43,
No. 9, p. 646 (1979).
[15] R. F. Voss and J. Clarke, Physics Review B 13, p. 556 (1976).
[16] H. I. Hanafi and A. van der Ziel, Physica 94B, p. 798 (1978).
103


CHAPTER VI
INTERPRETATION OF RESULTS
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 interr
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
2
S^f/l vs. temperature is the relative noise minimum occuring around
150K in all the samples (except one 0.5ym 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 (@D = 165K).
To determine whether this is merely a coincidence or, perhaps, the
first means to actually measure 0D, would require noise measurements
from thin films made from different metals'to exhibit noise minima at
their respective 0D'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 0.5ym 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
87


4
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
non-linearities.
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, tas 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
T
r
2
4AT
1 +
2 2
0) T
S(W)
g(T)dx
(1-5)


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92
edges is not so pronounced. However, for the 0.5ym 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 0.5ym samples are less interesting, for the purpose of this study.
6.2 Noisy 1 and 2ym 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 lym sample in Figure 5-8. The noisy 1 and 2ym 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 would^explain
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


32
S. (r,r',co) = lim 2T AT(r,w)AT* (r' ,w) ,
aT T-*
(3-39)
and similarly for S^. Thus from (3-38) and (3-36) we obtain the "re
sponse form" [45],
SAT(r,r',U)) =
dV
2_ 2 ,2.
V V
total total
d r^ir,^,jw)G(r' ,r ,-jto) (4kBT0a/c d V^^-^)
(3-40)
Providing that in Green's theorem for the V operator the bilinear con-
comittantof G(r,r^,jw) and G(r',r2,-jw) vanishes,the singular distribu
tion r2^ can rePlacec^ by,
V1 V26(rl r2) =
- IV6(rl r2> +'#(r1-r2)l. (3-41)
(We note that this replacement amounts to a partial integration whence
the conditions on the bilinear concomittant). Employing the well-known
rule,
f(r)V2(r r )d3r = V2f(r ),
o o
(3-42)
we arrive at,
SAT(r,r',w)
-2k T a
B o
2 2
c d
3 2
d r [G(r,r ,jw)V G(r',r ,-jw)
o o o
V
total
+ G(r',r ,-jw)V G(r,rQ,jw)]
(3-43)


20
i 1 1/2 ^ ->
Roughly, we have |k| ~ T 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) q^ .
This defines the Debye temperature 0^ = iuq^^/kg w^ere u t*ie
velocity of sound in the material. The 0Q 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 0^, it is not a measured quantity and consequently some
controversy exists concerning its exact value for a specific element.-
, 2
Observation of dependence of S^/I at low temperatures will reveal
2
which of the competing theories apply. If a peak in S^/I 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


76
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, S._ and the collec-
ltc
tor shot noise current source, 2elc, have been referred to the input
equivalent circuit as noise voltage spectrum sources by multiplying by
(l/gm)2 = (r^/3)2 (valid if r^ rfa) .
An HP3582A FFT spectrum analyzer measures the spectral density of
2
the collector noise, M /Af. Calculations from Figure 5-11 reveal,
(SR +V
s b
"TT
R + r, + r
s b IT
+ 2kTr
r, + R '2
b s
7T
R + r, + r
S b TT
+ S._
ifb
(R + r, + r )'
s b ir
, 2kT Sifc
+ + 7T~
g 2
m g
rm
(5-6)
If we use that r^ rfa and 3 1, then Equation 5-6 can be written so
that we obtain,
M2 2
f = A
R + r, + r
s b TT
2
ifc 2
2W(2rb + l/gm) + + si£brb
gm
+ R
S, r
4kT + 2 + 2S. r
02 lfb b
P
+ R
2kT ifc
+ 2 + Sifb
TT 6
(5-7)


60
Figure 5-1 Thermal Transfer Responses of Two Thin Film Resistors


40
(ie. we expect a square root frequency dependence). We also notice that
for w << 0)^, = -tt/4, independent of the coordinates. Thus for 00
the signal correlation Y(Wg) 1.
For (0 0) we find |y(L + e) | and | yL | are both greater than unity
and (3-64) becomes,
d3r'
3 VhV
d rG(x,x' joo) S
2/jaooy^
v.
V
(3-68)
Writing this in the linear response form and using (3-60) we see,
V.V
h s
d3r'
d rG(x,x',jw)
/2/au)
3/2
(3-69)
-3/2
(ie. we expect to see an U) dependence above the corner frequency oo^) .
-1/2 -3/2
An OJ spectrum which becomes an oo spectrum above a corner frequency
is typical of one-dimensional diffusion spectra.


f y-axis
Hf
>
x-axis
-C
L L
C
U>
Figure 3-1 Layout of a Chip Containing Three Different Groups of Thin Film Resistor Arrays


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29
we obtain in a similar way as before,
Re v
v*
sito s2to
- msloVB2oCOsW2 Ve The coherence factor is defined as,
(3-26)
ttae) -
Re v v*
slU) s2to
e e
[v V* V V* 1
sito sito s2to s2to
e e e e
1/2
From (3-26) and (3-20), we find,
(3-27)
Y(to ) = cos($2 4^)'* (3-28)
We notice that the machine measures jY| Here (k = 1,2) is, for the
K
simplified case of (3-15), given by,
= phase angle of [G(rs^,r^, jto) ] (3-29)
Though y should be oscillatory, the higher mixima may not be
noticed due to the presence of noise. Let v ^ refer to the sensors'
noises. Then for y we obtain, noticing that signal and' noise are un
correlated,
Re v
v*
sito s2to
e e
+ Re v ,v*
nl n2
{[v
sito slu
e e
+ v
nl nl
] [v
s2to Vs2to
e ; e
+ v v* ]}
n2 n2
1/2
(3-30)


88
devices exhibit a more extrinsic noise, we will put them in their own
classification. Between the 1 and 2ym 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 0.5ym Devices
Only a few of the 0.5ym 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.0.5ym devices were
very delicate and often "died", from the transient currents between
DUT ON and DUT OFF, before measurements could be completed. The 0.5ym
2
devices were prone to burst noise exhibiting a greater than I current
2
dependence. A S f/l vs. T plot of a 0.5ym device which did not exhi-
, 2
bit burst noise is given in Figure 5-7. It is the only S f/I plot
which shows the noise increasing at the low temperatures.. We believe
the strange behavior of the 0.5ym 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 2ym 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


5
where g(T) is the distribution of time constants for < X < In
the McWhorter model the distribution g(x) 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 [14]: postulated a modified
McWhorter theory where the total spectrum is again a superposition of
assume the characteristic time, X, is
X as a function of energy, E,
(1-6)
< E2 (1-7)
(6/3)-l
' *
x
X
l J
Lorentzians. However, now we
thermally activated and write
X = X e
o
-Be
where 3 = 1/k T, and we let
B
_
g(E)dE = Ke dE, < E
Then,
g(x) = g(E)
dE
dx
V J
Be x
o o
where K = 1/E Hence,
(1-8)


73
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 mall surface recombination velocities.
Mobility fluctuations imply fluctuations in the diffusion constant Dp,
since by the Einstein relation,
q<$D = kT6y .
P P
(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-
f*
collector hole diffusion in P PN transistors shows,
S
I.
Ep
(5-3)


53
Ax Ax'
T x
V. J
(4-17)
_2 2 2 2
We see the statistical accuracy of the (VQ1 vq2^/^V03 V02^ term'
Ax At
must be ~1% in order to have at least a 10% accuracy in since
the factor Ax/T ~ 10 when T^ = 77K and = 273K. By making T^ and
closer (ie. T 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.
Ax
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
[47],
i,;
Ax
x
(2Afx)-1/2
(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 aJiigh frequency cutoff of 100kHz
corresponding to the LNA1s 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


Figure Number
Page
5-11
5-12
5-13
5-14
6-1
6-2
6-3
6-4
Equivalent Common Emitter Circuit 75
Measurements of High Source Impedance
Spectra (S ) and Low Source Impedance
HR
S
Spectra (S ) 78
LR
S
Base 1/f Noise Magnitude Versus Base
Current 84
Collector 1/f Noise Magnitude Versus Collector
Current 85
Photograph of Open-Circuited Devices 89
Photograph of Short-Circuited Devices 90
SEM Photograph of Thin Film Resistors 91
a. Versus Temperature 97
true r
vi


6
S(u =
2
* *
4AT 1
T
3E T , 2 2
T.
o o 1 + (1) I
J
(6/3)1
dT
4AT
~w
6/3
(6/3)+l
dx
O 1 + X
WT
(1-9)
The integral converges if -1 < (6/3) < 1. Thus,
(6/3) = -1 + e, S(t) (1/tox )
o
(l-10a)
(6/3) = 0, S(w) 1/WT0 (1/f spectrum) (l-10b)
(6/3) = 1 e, S(w) 1/(6to)2-£ (l-10c)
Y
Thus, all spectra of the form C/w 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 and an upper a) for which the
spectrum 1/f changes. In some cases these turnovers have been found
(see, e.g., Hanafi and van der Ziel's experiments on Cd Hg Te [16]).
X X X
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 show that any g(E) which has a sharp peak,
E ~ lev, and a width of a few tenths of an eV (ie. a "smooth" energy
P
distribution) will give the observed 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


15
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 3 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/N ) rather than 1/N
atoms carriers
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


18
Now, applying the Wiener-Khintchine theorem we find,
2 2
si(f) v
L
y sN(f) + ns -(f)
(2-5)
2 2 2-2 2 2
Normalization of both sides with I = q E y N /L gives the generalized
xpression for current noise when both mobility and number fluctuations
aire present, first derived by Van Vliet and van der Ziel [38] ,
sl(f)
Vf)
N
1
N
(2-6)
Ignoring for the moment the possibility of number fluctuations,
2
(Sn/N -> 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,
_I
2
a
H
fN *
(2-7)
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 a
H
pY
However, we see a problem with using a in that we need 1/f (y = 1)
H
in order to get a unique value for a .
H -
Looking only at number fluctuations as the source of the current
-2
noise (ie. S /y 0) we see Equation 2-6 has no explicit 1/N depen-
y
dence. For number fluctuations to fit Hooge's phenomenological form,
we must assume,
a N
SN(f) =
(2-8)


36
We note that because of the self-adjointness of V G(r,r',joj) = G(r',r,jo)),
3 3
so the numerator is also Re / /Gd rd r'; hence,
V V
si s2
Re
3 3
G(r,r', jw)d rd r'
Y(w)
V V
si s2
[Re
-
-
3 3 1/2
G(r,r',ju)d rd r'] [Re
.
.
*
J J .,1/2
G(r,r',3W)d rd r']
V V
si si
V V
s2 s2
(3-57)
If there is a frequency range for which the integration of G is insensi
tive to the change / /->/ f f /, we find y = 1.
V V V V V V
si s2 si si s2 s2
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
(se.e Figure 3-1). The geometry of the thin film resistor (ie. lym wide
by 800ym 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
cornerssee 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 = iC, with boundary conditions
that the heat flow beyond C is zero. Thus the Green's function must
satisfy,
3G(x,x jco)
3x
x = C
= 0.
(3-58)


98
a
true
C + e
0u/T
C1 + C2 T
(6-9)
where 0y is the umklapp temperature, 0^ = hq u /kg, in which uQ is
the transverse velocity of sound in gold, k Boltzmann's constant,
B
and qQ a phonon vector associated with the length of the necks be
tween adjacent Fermi surfaces in the extended zone scheme. We com-
7-1 5
puted qQ = 6.3 X 10 cm while u = 1.2 X 10 cm/s. This yields
0y = 57.6 K. The observed maximum in the noise occurs at about 65 K.
Though many details need fuller consideration, we believe that the ob
served noise can be reasonably well explained by the proposed proces
ses. Development of a full theory and detailed 1/f noise measurements
in metals may give much insight into the nature of the phonon proces
ses undergone by the Fermi surface electrons.
6.4 Conclusions
The definitive conclusion of our research is that temperature
fluctuations are not causing the 1/f noise we observe in metal film
resistors. We have seen both number fluctuations and mobility fluc
tuations present in the devices we have studied. The point we wish
to make is that neither mobility nor number fluctuations can be con
sidered the sole source of 1/f noise. Rather, the two compete, and
under certain experimental conditions one or the other mechanism
dominates the noise spectrum. We have demonstrated two methods which
can isolate either of the two mechanisms, ie.,
1) by changing the source resistance of an active device, or


19
where a 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 a 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_(f )f
.. . loo .
Noisiness = (2-9)
It is the normalized current noise spectrum evaluated at 1Hz (ie. f
o
= 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 a
H
and a are avoided and trends associated with N ^can be readily dis-
is.
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 ot of
true
_2
the mobility fluctuations of a single carrier, S /y .
y
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) ? = ? kfit (2-10)
where £ is the chemical potential, and m* is the effective mass [41].


17
where a = 2aA (and not to be confused with the fine structure constant!),
true
The fact that the fundamental relationship for mobility fluctuations of
an individual carrier is expressed by Equation 2-1 (which obviously
exludes 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)
1 = 1 1
where we assume there are N individual carriers of a charge q with
individual drift velocities, v^, 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. v^ v^ (t) = Ey^(t))
i 1
and also number fluctuations (ie. N -* N(t)). Thus,
N(t)
I(t) = ^ l y.(t) ,
(2-3)
i=l
where N(t) = N, Vt (t) = y, and the individual carriers are_jlncoherent
(ie. y .y. = y.y. = (y) ). We now take the autocorrelation of both
13 i :
sides of Equation 2-3, applying Van Vliet1s and van der Ziel's Exten
sion of Burgess' Variance Theorem [38], and find,
2_2
Al(t)Al(t+s)
= SUL
y2 An(t)An(t+s) + n Ay(t)Ay(t+s)
(2-4)


42
Chemistry and Physics. The resistivities of metals deposited as
films is expected to differ from that of bulksince 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 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


58
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 2OK, 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.


CHAPTER VII
RECOMMENDATIONS FOR FUTURE WORK
7.1 Continued Studies of the Low Temperature Mobility-Fluctuation Noise
An exact caluclation of the expected Hooge parameter, predicted by
Handel's theory, due to soft-photon emission scattering mechanisms such
as Umklapp-scattering, N-acoustical phonon scattering, and optical pho
non scattering should be computed for different temperatures. For metals
the mobility relaxation time approximation cannot be used, thus the
Boltzmann equation must be solved using the variational method. These
results then must be compared to our measured values of ^ Agree
ment between theory and experiments involving many different metals
would give the first definitive proof of Handel's theory.
Experimentally, the main recommendation is for an extensive survey
of the temperature dependence of the 1/f noise of many different metals
with different Fermi surfaces and Debye temperatures. Good candidates
ares Pd (0Q = 274K), Ag;(0D = 225K), Pt (6 = 240K), and In (0Q = 108K).
Other candidates are Cu"(0 = 343) and Al (@D = 428); however, these
metals oxidize and care must be taken to assure one is measuring the
intrinsic bulk noise rather than a surface trap noise.
Van der Ziel has proposed a simple experiment to verify that the
1/f noise scales with 1/N (the number of carriers available for
SCclu
scattering). Comparison of the noise between a metal, with more N ,
S C ci"C
and a semimetal (e.g. Sn), with less N should show the intrinsic
SC3X
100


54
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 haye a typical resistance of
about lOOfi 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, 4kTR^,
and an equivalent current noise source, 4kTg^ (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, R^, 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
xx
"seen" with a given amplifier (ie. Rn and g^ are known), we must solve
the equation,


I
CHAPTER II
SCOPE OF OUR INVESTIGATIONS
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
11


50
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, S which dominates the device noise spec-
X
trum at low frequencies. If we use our calibrated noise measurement
system at higher frequency where is comparable to S we must
consider the thermal noise term. The total device noise is expressed
as,
V
01
*03
total
v2
02
VL
Scal + 4kTDUT/RDUT '
(4-7)
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,
4kTDUT//RDUT -2
01
- V
03
02
- V
S + 4kT /R
CAL DUMMY' DUMMY
(4-8)
03
This equation is only valid, if we have the dummy resistance at a
known temperature, such as room temperature (ie. T, = 300K) and
DUMMY
we match the dummy's resistance with the device's resistance at the
unknown temperature (ie. R,
DUMMY
V R
T=300K
DUT
). Then, we can
T=T,
DUT
theoretically solve for T.
DUT
We say theoretically because there are


Figure 5-4 Device Resistance Versus Temperature


47
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 3 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-
-v
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
noise.
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


83
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 in
in
we can expect,
N(0)/N (wE)
P(0)/P (wb)
, T, and are weak functions of the current, so that
dp dn
and,
S._ ~ I ,
lfc c
Sifb *b '
if the mechanisms are mobility fluctuation noise. Measurements of
S,f versus I, and S.^ f versus I are shown in Figures 5-13 and 5-14.
ifb b ife c
We see the base noise is roughly proportional 1^ (a small slope deviation
from unity is expected since and £n
N(0)/N(we)
are weak functions
of the current). This shows the base noise is definately 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
a
n
For the case of base 1/f noise, we have,
HR ~ ifb I '
s En
(5-19)
and using the base to emitter expression (5-5) we have, for a ,


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


68
for each temperature. Using the R versus T plot of Section 5.3, the
2
true device temperature is determined. The typical trends of S^f/I
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
2
the S^f/I 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 sed 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 th "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.


8
In Equation 1-11 the frequency-shifted components present in the in-
*
tegral interfere with the elastic term, yielding beats of frequency
£/h. The particle density given by Equation 1-11 is
k
A
l 12
k ,
f \
et
= 1 a 1
1 + 2
.
|bT(e)|cos
-rr- + Y
ft e
dt
A A
(e)bT(e')
i(e'-e)t/f) ,
e deds'
(1-12)
the second term in large parentheses describes the particle beats.
2 2 2
If the particle fluctuation is defined by 61xp| = |^| <|^| >, its
autocorrelation is found to be, if a term "noise of noise" is neglected,
t+T
2
cos
ST
ft
de
A/h
|b(e)
2
cos2TTfTdf
J
f
o
(1-13)
From the Wiener-Khintchine theorem one sees that the integrand is the
spectral density:
s^|2/ = 2fi|b(e)
(1-14)
For the spectral density in the particle velocity
one easily sees,
v
ft
2im
* *
(W V\p) ,
S / =
v
>1
2/ 2ft b(e)
(1-15)


64
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 1mA 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 [491 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
2
at different currents to check the I 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, RQN. The calculated
R^ is always a few ohms larger than the R (measured in Section 5.3)
ON OFF


48
-12 v
Figure 4-3 Schematic of the Schmidt Low Noise Amplifier


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


CHAPTER I
INTRODUCTION
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,
AI2
S (f) = (0.88 < y < 1.1) (1-1)
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 fromboundary
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
1


46
and,
(4-1)
We can write three equations for the three different measurement cases,
1) DUT ON
2) CAL ON
3) DUT OFF
01 = G2
Af
(S + s + s1)z2 + sv
x th a a
v
02 G2
Af
(S + s.. + sx)z2 + sv
cal th a a
03 G2
Af
(S + s^z2 +
tin a. Si
(4-2)
(4-3)
(4-4)
where G is the amplifier's gain, S1 is the amp's equivalent current
cl
V
noise source, and is the amp's equivalent voltage noise source as
indicated in Figure 4-2. Solving for the excess device noise term, S ,
X
we find,
S
x
-2
V01
- v
03
-2
v
02
- v
cal
03
(4-5)
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.


104
[17] K. M. van Vliet and J. R. Fassett, Fluctuation Phenomenon in
Solids, R. E. Burgess (Ed.), Adademic: New York, p. 267 (1964).
[18] H. Mehta, "Transport Noise Arising from Diffusion and Bulk of
Surface Generation-Recombination in Semiconductors," Doctoral
Dissertation, University of Florida, (1981).
[19] P. H. Handel, Physical Review Letters 3j4, p. 1492-1495 (1975).
[20] F. N. Hooge and L. K. J. Vandamme, Physics Letters 66A, p. 315
(1978).
[21] R. R. Schmidt, G. Bosman, C. M. Van Vliet, L. Eastman, and M.
Hollis, "Noise in Near-Balistic n+nn+ and n+pn+ Gallium Arsenide
Submicron Diodes," Solid State Electronics (revised version),
in press.
[22] J. Gong, C. M. Van Vliet, G. Bosman, W. Ellis, Jr., and P. H.
Handel, "Observations of a Flicker Noise Floor in a-Partical
241
Counting Statistics from cAm ." to be submitted to Physical
Review.
[23] S. H. Liu, Physics Review B JL6, p. 4218
[24] M. B. Weissman, Applied Physics Letters _32, p. 198
[25] J. H. Scofield, D. H. Darling, and W. W. Webb, "Exclusion of
Temperature Fluctuations as the Source of 1/f Noise in Metal
Films," Physics Review B, 24^, p. 7450 (1982).
[26] S. Hashiguchi, Japanese Journal of Applied Physics 22_, No. 5,
p. L284 (1983).
[27] C. M. Van Vliet, A. van der Ziel, and R. R. Schmidt, "Tempera
ture Fluctuation Noise of Thin Films Supported by a Substrate,"
Journal of Applied Physics, 51, p. 2947 (1980).
[28] J. Kilmer, E. R. Chenette, C. M. Van Vliet, and P. H. Handel,
"Absence of Temperature Fluctuations in 1/f Noise Correlation
Experiments in Silicon," Physica Status Solidi (a), 70, p. 287
(1982).
[29] J. Clarke and T. Y. Hsiang, Physics Review B 12_, p^_4790 (1976).
[30] M. B. Ketchen and J. Clarke, Physics Review B 17, p. 114 (1978).
[31] J. W. Eberhard and P. M. Horn, Physics Review B 18, P. 6681 (1978).
[32] P. P. Dutta, J. W. Eberhard, and P. M. Horn, Solid State Commu
nications 27, p. 1389 (1978).
[33] P. P. Dutta, P. Dimon, and P. M. Horn, Physical Review Letters
43, p. 646 (1979).


3
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-
-7
served for frequencies as low as 10 Hz implying the need to postu
late implausibly long time scales!
l
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 Sv ~ 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


56
4kT R = 1/2
x x
4kTR + R 4kTg
n x n
(4-19)
derived from Figure 4-4. This equation has the general form, T =
X
A/R + BR which is a skewed hyperbola in the T R plane. The
XX XX
skewed hyperbola will have a minimum corresponding to the lowest re
sistance, (R ) needed to see its thermal noise at the lowest
x mm
temperature, (T ) . Setting the derivative equal to zero, we find,
x min
(R ) .
x mm
(4-20)
and,
(T ) = 1/2
x mm
TR
7r + Tg (R ) .
(R ) n x mm
x mm
(4-21)
From Schmidt's PhD thesis [46, p. 33], we know for the PAl LNA;
R = 352 and g ~ 10 ^15. This gives (R ) = 5902 at (T ) ~ 17K.
n n x mm x mm
Bob Schmidt designed a PA2 LNA where the Rn 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, R,. Ex-
. n c A
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


BIOGRAPHICAL SKETCH
Joyce Prentice Kilmer was born in Pittsburgh, Pennsylvania
on June 28, 1958. When he was six, his family moved to Palm Beach,
Florida, and presently resides there.
As a freshman, he attended Carnegie-Mellon University in Pitts
burgh. His remining undergraduate years were completed at the
University of Florida, where he received his Bachelor of Science in
Electrical Engineering.
Joyce received the degree of Master of Science in Electrical
Engineering in 1982. Since then, Joyce has been pursuing a Ph.D.
in Electrical Engineering at the University of Florida
He is a member of Eta Kappa Nu, Electrical Engineering Honor
Society, for which he was Secretary in 1980.
107


Figure 5-2 Coherence Between Thermal Transfer Responses


TABLE 5-1
Data Obtained From High and Low Rg Biased PNP Transistors
Bias
Low R Data
s
Curve I 3 r,
E b
High Rg Data
Curve I 3 (Jmtki
B n MIN
IB ~ 6yA
IB it 3yA
IB Cz. lyA
I 2.25mA 340 40
II 1.3mA 420
, III 505yA 413 20
IV 6.7yA 362 1.2 x 10"7
-8
V 3yA 363 6.6 x 10
VI 1.2yA 307 9.2 x 10_8


27
1 2 j(V ~j(V i
=4Vso(e +e > (3-17>
Using further,
00
-j (CO (0 )T
e e dT = 27T<5(tO w ) (3-18)
we find,
d [v
v* ]
SCO SCO
7TV2 [6
so
(to to ) + 6 (to + to )]Af.
e e
(3-19)
We now realize that the analyzer integrates over a bandwidth Ato centered
on to, where Ato = 2irAf. The output is therefore a signal over the range
Ato centered on to of magnitude, :
v v*
SCO SCO
e e
v e(to )
so e
2
(3-20)
where £(tOe) is the function which is unity in the interval Ato centered
on to and zero outside. Hence,
e '
v v*
SCO sto
e e
2 2
m P.
ho
2 2 2 2
2c d V vf
s h
d3r
d3r'G(r,r' jto) | 2
(3-21)
v
s h
for the case of Equation (3-14), or,
v v* s;
sto sto
e e
2 2 2 2
(mV /2c d )
G(rs'rh,3W)
(3-22)


105
[34] D. M. Fleetwood and N. Giordano, Physics Review J5 25_, p. 1427
(1982).
[35] D. M. Fleetwood and N. Giordano, Physics Review jB 25, p. 667
(1983).
[36] K. M. van Vliet and R. J. J. Zijlstra, Physica 111B, p. 321
(1981).
[37] R. P. Jindal and A. van der Ziel, Journal of Applied Physics 52,
p. 2884 (1981).
[38] A. van der Ziel and K. M. van Vliet, Physica 113B, p. 15 (1982).
[39] F. M. Klaassen, IEEE Trans. Electron Devices ed-18, p. 887
(1971).
[40] A. van der Ziel, Advances in Electronics and Electron Physics
49, p. 260 (1979) ,
[41] C. M. Van Vliet, Lecture Notes EEL 6935.
[42] J. M. Ziman, Electrons and Phonons, Oxford University Press,
London (1967).
[43] S. M. Sze, Physics of Semiconductor Devices, Wiley-Interscience,
New York, p. 57-58 (1969). :
[44] P. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-
Hill, New York, Vol. I, Chapter 7 (1953) .
[45] C. M. van Vliet and H. Mehta, Physics Status Solidi (b) 106,
p. 11 (1981).
[46] R. R. Schmidt, "Noise and Current-Voltage Characteristics of
Near-Ballistic GaAs Devices," Doctoral Dissertation, University
of Florida (1983).
[47] A. van der Ziel, Noise: Sources, Characterization, Measurement,
Prentice Hall, Englewood Cliffs, New Jersey, p. 174 (1970).
[48] A. C. Rose-Innes, Low Temperature Techniques, D. van Nostrand
Co., Inc.; Princeton, New Jersey, Figure 7.6 (1964).
[49] C. Kittel, Introduction to Solid State Physics, McGraw-Hill,
New York (1976).
[50] G. L. Squires, Fysisch Experimenteren, McGraw-Hill, London
(in Dutch), p. 52-53 (1968).
[51] F. N. Hooge, Physica 60, p. 130 (1972); Physica 83b, p. 14 (1976).
[52]. T. G. M. Kleinpenning, Physica 98B, p. 289 (1980).


61
frequency beyond the range of observation. Again the thermal diffusion
appears to be one-dimensional in nature as was the case in the transis
tors.
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 1mA test
current range (see Section 5.3). Essentially, we see no heating effect
when the device is biased with 1mA 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 TDUTf 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


CHAPTER IV
EXPERIMENTAL METHODS
4.1 Devices
Gold thin film resistors, 2,000 & thick, deposited on a 200 it
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 = :0.5ym, lym, and 2ym 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 T05 can). An ultrasonic bonder was used to bond
gold wires to the lOOym pads on the devices (dimension "a" indicated
on Figure 2-1) and to the T05 can posts. From the measured resistance
and dimensions of the thin film resistors, we find the resistivity to
be slightly greater than 2.35yfi-cm as listed in the Handbook of
41


35
where we ass limed that both sensors have an equal volume V From the
Wiener-Khintchine theorem,
ATS1'ATS2 V2
s
d3rd3r'S^T(r,r',u).
(3-52)
V V
si s2
Using (3-45), this yields,
2k T2 (
B o
AT ,At ,2
si s2 cdV
s
.3 .3
d rd r'[G(r,r', jo)) + G(r* ,r,-jw) ] (3-53)
V V
si s2
For the cross-correlation of the noise voltages of the two sensors, we
have,
2 2
2k T m
B o
Vsl'Vs2 cdV3
,3 ,3
d rd r[G(r,r,jw) + G(r',r,-jw)] (3-54)
V V
si 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,
Y (w) =
Re S (0))
Vsl,Vs2
[S Xu) S (0))]
Vsl Vs2
1/2
(3-55)
Thus from (3-54) and (3-50),
1
Re
3 3
[G(r,r',jo)) + G(r',r,-ju)) ]d rd r'
Y (u) =
[Re
V V
si s2
*

G(r,r' j(jJ)d3rd3r' ] [Re
*
3 3 1/2
G(r,r',jw)d rd r']^
J
.
V V
si si
Vs2Vs2
(3-56)


44
i
Figure 4-1
Calibrated Noise Measurement Scheme


33
Now substituting from the defining equation for G, see Equation (3-11),
-a
cd
V2G(r',ro,-jw)
j0JG(r',rQ,-ja)) + 6(r' rQ) ,
(3-44a)
and,
V2G(r,rQ,jo)) = -jwG(r,ro,jw) + 6(r rQ),
(3-44b)
we find from (3-43),
2k T2
SAT(r,r',J) = [G(r,r,jw) + G(r' ,r,-jo)) ]
(3-45)
This is the van Vliet-Fassett form [17], since [k T /cd]6(r r') is the
B o
covariance AT(r,t)AT(r',t). For the spatially averaged tamperature fluc
tuations in a volume V we have,
s
Ts(t) -V
' 3
AT(r,t)d r,
V
and for the autocorrelation function,
(3-46)
At (t)At (t + t) =
s s v2
\ s
d3rd3r' AT(r,t)AT(r',t + T). (3-47)
V V
s s
Whence by the Wiener-Khintchine theorem,
SAT (J)
s
v2
s '
d3rd3r'SAT(r,r',w)
V V
s s
(3-48)


43
(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 <3ewar 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
ate.
4.2.3 Calibrated Noise Measurements
The measurement scheme used, incorporated a calibrated noise m
source (see Figure 4-1). Using this method, one can calculate the
absolute magnitude of the DUT's current noise spectrum, S simply
X
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,
11.29K + (5.6K|15.6)
vcal
EsAf


67
Relative Noise Magnitude at 300K
of all Devices Measured. (N =
-12
0.8 X 10 0.5ym devices, N =
-12
1.6 X 10 - lym devices, N =
-12
3.2 X 10 + 2ym devices)
Figure 5-6


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


Page
CHAPTER IV
CHAPTER V
CHAPTER VI
CHAPTER VII
REFERENCES
BIOGRAPHICAL
EXPERIMENTAL METHOD .
4.1 Devices ....... . .
4.2 Apparatus
4.2.1 The Closed Cycle Cryostat
4.2.2 The Flow Cryostat
4.2.3 Calibrated Noise Measurements . .
4.2.4 The Measurement System .
4.3 Thermal Noise Measurements
4.3.1 Thermal Noise Measurements Using
the Calibrated Noise Source
4.3.2 Thermal Noise Measurements Without
Using the Calibrated Noise Source .
4.3.3 Accuracy in Thermal Noise
Measurements .
4.4 Thin Film Heating Effects .......
EXPERIMENTAL RESULTS
5.1 Thermal Transfer Function Experiment .
5.2 Thin Film Heating ....
5.3 Resistance Versus Temperature
5.4 1/f Noise Versus Temperature
5.5 The Clearcut Evidence of 1/f
Mobility Fluctuations in Transistors .
5.5.1 Discrimination Between Bas and
Collector Noise Sources
5.5.2 Interpretation of S and.S . .
HR LR
S S
INTERPRETATION OF RESULTS
6.1 The 0.5ym Devices
6.2 Noisy 1 and 2ym Devices
6.3 Quiet 1 and 2ym Devices .
6.4 Conclusions .
RECOMMENDATIONS FOR FUTURE WORK
7.1 Continued Studies of the Low
Temperature Mobility-Fluctuation
Noise ~r~v
7.2 Continued Studies of the High
Temperature Number-Fluctuation
Noise
7.3 Investigation of New Queries
SKETCH
41
41
42
42
43
43
47
49
50
51
52
56
59
59
61
64
64
73
74
80
87
88
92
93
98
100
100
101
102
103
107
iv