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 es
pecially wish to thank Dr. Aldert van der Ziel for his contributions
to the PNP transistor research. I thank Dr. Gijs Bosman and Dr.
Peter H. Handel for sharing with me their insight and knowledge of
theoretical and experimental physics.
In addition, I wish to express my appreciation to Dr. Wolf and
Dr. Burhman of Cornell's NRFSS for producing the high quality Au
thin film arrays.
Furthermore, my sincere gratitude goes to Miss Mary Catesby
Halsey for her willing spirit and unstinted patience in typing this
manuscript.
This research was supported by the Air Force Office of Scien
tific Research, under Grant Number AFOSR 800050.
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS . . . . . . . . . .. .ii
LIST OF FIGURES . . . . . . . . ... . . v
ABSTRACT . . . . .... . . . . ... vii
INTRODUCTION . . . . . . . .
1.1 Fundamental Questions . . .
1.2 Competing Theories . . . .
1.2.1 A Thermally Activated Number
Fluctuation Theory . . .
1.2.2 Diffusion Theories . . .
1.2.3 The Quantum 1/f Noise Theory
1.2.4 The Temperature Fluctuation
Theory . . . . . .
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 1/N Dependence and the Hooge
Parameter . . . . . .
2.4 Temperature Dependence of 1/f
Mechanisms . . . . . .
CHAPTER III
THEORY OF HEAT TRANSFER AND OF TEMPERATURE
FLUCTUATION NOISE . . . . . ...
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
CHAPTER I
S 3
4
5
7
7
10
. . 4
. . 5
. . 7
. . 7
. . 10
. 11
. . 13
. . 16
S. 19
22
. 22
23
28
30
34
36
Page
CHAPTER IV
EXPERIMENTAL METHOD . . . . . .
41
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
CHAPTER V
CHAPTER VI
EXPERIMENTAL RESULTS . . . . . .
59
5.1 Thermal Transfer Function Experiment . 59
5.2 Thin Film Heating . . . . ... 61
5.3 Resistance Versus Temperature ... . 64
5.4 1/f Noise Versus Temperature . . .. 64
5.5 The Clearcut Evidence of 1/f
Mobility Fluctuations in Transistors 73
5.5.1 Discrimination Between Base and
Collector Noise Sources ...... .74
5.5.2 Interpretation of SHR andSLR . 80
s s
INTERPRETATION OF RESULTS . . . ... 87
The 0.5pm Devices . . . .
Noisy 1 and 2pm Devices . .
Quiet 1 and 2pm Devices . .
Conclusions . . . .
CHAPTER VII
RECOMMENDATIONS FOR FUTURE WORK . . . .
7.1 Continued Studies of the Low
Temperature MobilityFluctuation
Noise . . . . . . .
7.2 Continued Studies of the High
Temperature NumberFluctuation
Noise . . . . . .
7.3 Investigation of New Queries . . .
REFERENCES . . . . . . . . . . . .
BIOGRAPHICAL SKETCH . . . . . . . . . .
LIST OF FIGURES
Figure Number
21 Layout of Closely Spaced Gold Thin Film
Resistor Array . . . . . . . .
31 Layout of a Chip Containing Three Different
Groups of Thin Film Resistor Arrays . .
41 Calibrated Noise Measurement Scheme . .
42 Equivalent Circuit of the Calibrated Noise
Measurement Scheme . . . . . . .
43 Schematic of the Schmidt Low Noise Amplifier
44 Simplified Equivalent Circuit of the Low
Noise Amplifier . . . . . . .
51 Thermal Transfer Responses of Two Thin Film
Resistors . . . . . . . . .
52 Coherence Between Thermal Transfer Responses
53 Relative Thermal Noise Versus Device Bias
Current . . . . . . . . .
54 Device Resistance Versus Temperature . .
55 1/f Noise 12 Dependence . . . . .
56 Relative Noise Magnitude at 300K of all
Devices Measured . . . . . . .
57 Noise Magnitude and Slope Versus Ambient
Device Temperature (0.5pmdevice) . . .
58 Noise Magnitude and Slope Versus Ambient
Device Temperature (Noisy lpm device) . .
59 Noise Magnitude and Slope Versus Ambient
Device Temperature (Quiet 1pm device) . .
510 Typical Coherence Between the 1/f Noise of
Two Thin Films . . . . . . . .
Page
. 12
S. 37
. 44
. 45
48
. 55
S. 60
. 62
. 63
. 65
. 66
. 67
S. 69
. 70
. 71
S. 72
Figure Number
511 Equivalent Common Emitter Circuit . . . .
512 Measurements of High Source Impedance
Spectra (SHR ) and Low Source Impedance
s
Spectra (SLR ) . . . . . . . . .
S
513 Base 1/f Noise Magnitude Versus Base
Current .. . . . . . . .
514 Collector 1/f Noise Magnitude Versus C<
Current . . . . . . . .
61 Photograph of OpenCircuited Devices
62 Photograph of ShortCircuited Devices
63 SEM Photograph of Thin Film Resistors
64 true Versus Temperature . . . .
true
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
CoChairman: Gijs Bosman
Major Department: Electrical Engineering
In recent years, theorists have been trying to explain the pheno
menon of electrical 1/f noise. Presently some controversy exists over
the exact origin of the fluctuating physical quantities which gives
rise to these resistance fluctuations. The disputes have centered on
fluctuations of carrier mobility, carrier number, and temperature, any
of which could cause the observed resistance fluctuations. The pri
mary thrust of our research is to determine which of these possible
1/f noise producing mechanisms is responsible for the 1/f noise com
monly observed in two electronic devices (ie. this film resistors and
PNP transistors). First, we delve into the details fo two competing
1/f noise theories of metal thin 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
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
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,
AI
S (f) = (0.88 < y < 1.1) (11)
I fY
where the I indicates the current dependence. However, the theore
tical explanation of this noise has remained one of the oldest of the
unresolved problems of solidstate 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
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],
aHv2
S = (12)
fV N
where SV is spectral density of voltage fluctuations and aH is the
3
Hooge parameter which he assumed to be a constant aH = 2 X 10.
We see the magnitude of the voltage spectrum is proportional to the
,square ofthe DC voltage, V, across the device implying that the cur
rent does not drive the resistance fluctuations but merely, by Ohm's
law, reveals them as voltage fluctuations. Hooge's equation also
predicts the voltage spectrum to vary inversely with the number of
carriers, N, which implies 1/f noise is a bulk effect rather than a
surface effect since N ~ Volume. Finally the formula shows the em
pirical 1/fY dependence where y (an exponent close to unity) accounts
for the observed slope variations. Van Vliet et at. [7] observed
the relationship between the voltage spectrum and the current spec
trum, S measured from a device of differential resistance R,
SV = S R (13)
From this, we see we can write with Hooge [8],
S S ct
V I H
(14)
V2 2 fN
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
served for frequencies as low as 107Hz implying the need to postu
late implausibly long time scales!
1.1 Fundamental Questions
First, we begin our research with some fundamental questions.
Is the phenomenon a surface or bulk effect? Hooge found in gold that
SV 1/thickness but Calesco, et al. [9], suggested the noise in
films occurs at the interface of the film and substrate. Dutta and
Horn showed that the substrates can play a role if they are not good
thermal conductors; but, in general, the noise in metal films is of
a bulk origin. Recently, this has been verified by Fleetwood and
Giordano [10] who found S ~ 1/N over six decades of carriernumbers.
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!
4
Is the 1/f noise process stationary? If the power really does di
verge, it would be a nonstationary 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
nonlinearities.
From what fluctuating physical quantities does 1/f noise in elec
tronic conductors arise? Presently three physical quantities, the mo
bility of the carriers, the number of the carriers, and the temperature,
are in the theoretical arena. Fluctuations in any one of these quanti
ties can theoretically give rise to 1/f noise, las the next section will
show. It is the main purpose of our investigations to isolate which
fluctuating quantities account for the 1/f noise in some specific elec
tronic structures.
1.2 Competing Theories
A real "menagerie" of theories have been proposed and rejected
regarding 1/f noise. The McWhorter theory [13] is based on an inco
herent superposition of Lorentzians. Usually, such Lorentzians are
caused by number fluctuations. The overall spectrum then becomes
'2
S4AT
S(w) = 2 g(T)dT (15)
T1
5
where g(T) is the distribution of time constants for T < T < T In
the McWhorter model the distribution g(T) stems from a distribution of
tunneling widths (with a flat distribution of activation energies)
causing Equation 15 to yield a 1/f spectrum. In another model, the
distribution of time constants stems from a uniform distribution of
activation energies.
Dutta and Horn speculate that the flat distribution of activa
tion energies is not actually present in metal films. Instead, they
show how a smooth distribution of activation energies gives rise to
"generic"1/f noise which is more commonly observed (see next section).
1.2.1 A Thermally Activated Number Fluctuation Theory
To explain the observed temperature dependence of the magnitude
and the slope of 1/f noise, Dutta and Horn [141 postulated a modified
McWhorter theory where the total spectrum is again a superposition of
Lorentzians. However, now we assume the characteristic time, T, is
thermally activated and write T as a function of energy, E,
T =T eE (16)
o
where 8 = l/k T, and we let
SE
g(E)dE = Ke dE, E < E < E (17)
11 
Then,
(/)g(T1
g(T) = g(E) = (18)
d EdT SE T T
/ **
where K = 1/Eo. Hence,
S 2 4A2 (61/)1
4AT 1
E2 WT
o 1 +x o
The integral converges if 1 < (6/3) < 1. Thus,
(6/P) = 1 + 6, S(w) cc (1/OTO ) (110a)
o
(6/3) = 0, S(W) a 1/WT (1/f spectrum) (110b)
2E
(6/O ) = 1 E, S(M) 1/(6T ) 2 (110c)
Y
Thus, all spectra of the form C/ with 0 < y < 2, referred to as
"generic" 1/f noise [15], are possible. There is, however, scaling
in this theory, for there is a lowest al and an upper w2 for which the
spectrum 1/fY changes. In some cases these turnovers have been found
(see, e.g., Hanafi and van der Ziel's experiments on CdxHglxTe [16]).
It is also clear that only by exception the spectrum is exactly 1/f;
this requires a strictly uniform distribution of activation energies.
Without any reference to an exact type of thermallyactivated
random process, Dutta and Horn showthat:anyg(E) which has a sharp peak,
E ~ leV, and a width of a few tenths of an eV (ie. a "smooth" energy
distribution) will give the observed SV vs. T and y vs. T dependence.
They speculate that g(E), as a peaked function, is more "amenable to
physical justification" than a mere flat distribution, since E can
p
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 suggestedas a source of 1/f noise because they can give
the long time scales associated with 1/f noise. However, extensive
theoretical investigations by Van Vliet and Mehta [18] find a 1/f
spectrum can be produced by a diffusion theory only if a surface
source is assumed. Since we have seen 1/f noise in metals is a bulk
effect, the diffusion theory does not apply.
1.2.3 The Quantum 1/f Noise Theory
The only truly generalized theory of 1/f noise was suggested by
Handel in 1975 [19]. Until recently [20,21,22] experimental evidence
verifying the theory did not exist. Specifically Handel's Quantum 1/f
Noise theory was questioned as the source of 1/f noise in electronic
circuits because of the low value of the Hooge parameter, aH, calcu
lated from his theory. The theory states, the interference between
the part of the carrier's wave function which suffered energy losses
due to an inelastic or "bremsstrahlung" scattering and the part of
wave function which did not suffer losses produces a low "beatfrequency"
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,
A
i(k'rwt) 1 I (E td
ST(r,t) = a e ~ ~ (1 + bT( )ei dEe (111)
0
8
In Equation 111 the frequencyshifted components present in the in
tegral interfere with the elastic term, yielding beats of frequency
E/h. The particle density given by Equation 111 is
A
L1T 2 = a2 1 + 2 f b T()Icos e+ y dt
o
o o
the second term in large parentheses describes the particle beats.
If the particle fluctuation is defined by 6 i2 = 212 <2 I 2>, its
autocorrelation is found to be, if a term "noise of noise" is neglected,
A
<610S 26112 > = 21a 4 b(c) 2cos de
0
A/h
= 21a4f lb() 12cos2TfTdf (113)
f
0
From the WienerKhintchine theorem one sees that the integrand is the
spectral density:
Sl 12/ = 2fib(E) 2 (114)
f )
For the spectral density in the particle velocity v = ( V)
I ( 2im
one easily sees,
S /2 = Sl12/<1 2> = 2fltb()2 (115)
9
For the bremsstrahlung matrix element, under emission of photons, one
calculates easily, either classically or quantum mechanically,
Ib() 12 = CA/E = aA/hf ; (116)
here a = 1/137 is the fine structure constant and
2 Jv J
A 3 A 2 (117)
3Tr 2
c
Handel's theory predicts Umklapp process (Uprocess) scattering
to be the largest source of 1/f noise in metals since the 1/f noise
magnitude scales with the photon infraquanta coupling constant, aA,
ie. combining Equations 115, 116, and 117,
S
v 2aA
2 f
20A V
S2c (118)
3rf 2
c
Since Uprocess 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
AqIN
discredit Handel's theory explaining, "most electrons in a metal can
not emit lowenergy 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.
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, 0,
1 dR
R jd (119)
Using the Langevin diffusion equation,
T 2 V*F
S= DV T + (120)
t c
where D is thermal diffusivity, c is the specific heat and F is a
random driving term (uncorrelated in space and time). This equation
causes temperature fluctuations to be spatially correlated with the
mean square amplitude for temperature fluctuations divergent at long
wavelengths resulting in a 1/f spectrum over a limited frequency range.
The theory has some criticism in that the diffusion equation, as Voss
and Clarke have written it, cannot be rationalized on a microscopic
level since the source term, F, representing heat flow from the metal
film to the low thermal conductivity substrate [23] violates conserva
tion of energy [24]. The theory has had moderate successes in special
cases such as in Sn films near the superconducting transition, but it
is merely fortuitous that it works at room temperature for Au and Cu
films [4, p. 508]. Our research has been designed to distinguish
which theories are applicable to the 1/f noise observed in electronic
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 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 cryogenicambient
temperature range (10K to 300K).
Van Vliet has designed a configuration of three closelyspaced gold
thin film resistors (see Figure 21) 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
21) will be biased with an AC voltage source to act as a "heater" and
11
ha
Figure 21 Layout of Closely Spaced Gold Thin Film Resistor Array
L ,I
a
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
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
closedcycle 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 21 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
f I
and with R A low frequency flattening of the room tem
perature 1/f spectrum is observed with 5,000 A Al underlay,
TABLE 21
1/f Noise Experimental Results
from Metal Films
\
*The only substrates ever used in these measurements were glass or sapphire, therefore, substrate "divots"
or corregations were never a concern; except Fleetwood & Giordano who used oxygen glow discharge to "clean"
substrates and improve adhesion.
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 8 shows a very
small 1/f noise. Bi, which shows a comparable noise magni
tude as metals of the same geometry, but with a much smaller
.carrier density than the metals, suggests the noise should
be scaled by 1/Volume (or 1/Natoms) rather than 1/Ncarriers
4. Eberhard & Horn (1978) [31]. The temperature dependence of
the 1/f noise's magnitude and slope between 100K and 600K
is shown. Annealing decreases noise and increases the tem
perature dependence.
5. Dutta, Eberhard & Horn (1978) [32]. In Ag "Type B" noise
dominates at all temperatures. In Cu (which has a lower
room temperature noise) crossover of "Type A" noise can
be observed below room temperature. No exponent dependence
on substrate is indicated.
6. Dutta, Dimon & Horn (1979) [33]. T dependence of noise
changes slightly with film thickness. Variation of the
exponent vs. T is shown to be consistent with the noise mag
nitude changes vs. T. The sharper Guassian distribution of
activation energies in thicker films lends to the thought
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/Natoms 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 118). Since v = Ep, we can write Equa
tion 118 also as a normalized spectrum of mobility fluctuations of
a single carrier,
true (21)
2 f
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 21 (which obviously
excludes carrier number fluctuations since we are talking only about
a single carrier) has been postulated by Van Vliet and Zijlstra [36],
and van der Ziel and Jindal [37]. We now show how this formula is
modified for the case of a current spectrum (that which is actually
measured by the spectrum analyzer). A current is actually a macro
scopic quantity comprised of a cumulation of microscopic events; that
is, 
N
I = v (22)
L.=l i
where we assume there are N individual carriers of a charge q with
individual drift velocities, vd, and L is the device length. Now
we generalize Equation 22, for the purpose of noise investigations,
by assuming we have mobility fluctuations (ie. vd. v (t) = Ei (t))
d.
1 1
and also number fluctuations (ie. N + N(t)). Thus,
N(t)
1(t) = (t) (23)
i=l
where N(t) = N, p.(t) = p, and the individual carriers are incoherent
(ie. i.p. = Pi ( 1) ). We now take the autocorrelation of both
sides of Equation 23, applying Van Vliet's and van der Ziel's Exten
sion of Burgess' Variance Theorem [38], and find,
L2
Al(t)Al(t+s) =  p2 AN(t)AN(t+s) + N Ap(t)Ap(t+s) (24)
Lj
18
Now, applying the WienerKhintchine theorem we find,
2 2
S (f) = 2 S(f) + NS (f) (25)
2 j25 N SP M
L
2 2 22 2 2
Normalization of both sides with I = q E 2 N /L gives the generalized
expression for current noise when both mobility and number fluctuations
are present, first derived by Van Vliet and van der Ziel [38],
SI(f) SN(f) 1 S2
+ (26)
2 2 N 2
I N (1)
Ignoring for the moment the possibility of number fluctuations,
(SN/N2 0), and substituting the normalized mobility fluctuation
spectrum for a single carrier Equation 21 into Equation 26, we arrive
at Hooge's formula,
S a
 H (27)
12 fN
We see the N in the denominator accounts for the increase in the num
ber of degrees of freedom associated with incoherent scattering. 'When
ever the 1/N appears explicitly in the formula, a is defined as aH.
Y
However, we see a problem with using aH in that we need 1/f (y 1)
in order to get a unique value for aH.
Looking only at number fluctuations as the source of the current
noise (ie. S /52 0), we see Equation 26 has no explicit 1/N depen
dence. For number fluctuations to fit Hooge's phenomenological form,
we must assume,
a N
S (f) = (28)
where aK is due to Klaassen [39]. However, this assumption implies a
surface controlled density fluctuation model [40], and we have already
seen we are looking for a bulk effect in metals.
A better gauge than aH for the total observed 1/f noise magni
tude, which does not presume one mechanism is the sole source of the
1/f noise, is the dimensionless "noisiness" factor,
S (fo)fo
Noisiness = (29)
2
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 aH
and aK are avoided and trends associated with N can be readily dis
cerned. This is the approach we will take in plotting our data. Only
for relative comparison purposes, at the end of our investigations, we
will consider N in our calculations, to derive an effective true of
2
the mobility fluctuations of a single carrier, S /2 .
2.4 Temperature Dependence of 1/f Noise Mechanisms
In semiconductors, as we vary the ambient temperature, we effec
tively vary the magnitude of the conduction electrons energy (or wave
vector), k, since the average energy transported by an electron is,
2 2
E(k) = 2m* a kBt (210)
where C is the chemical potential, and m* is the effective mass [41].
20
Roughly, we have Ik ~ T/2. The incoming electron, k, is scattered
by a phonon, q, described by the elementary phonon emmission/absorb
tion process,
+ + 
k' = k q (211)
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 electronphonon scattering
mechanisms both in semiconductors and metals. Conceivably at low
temperatures, we may see a different type of noise since we may reveal
a different scattering mechanism.
The Debye model is definitely required at low temperatures, since
it appropriately describes acoustic phonons which are the only phonons
available at low temperatures [42]. Specifically, the model assumes
an upper limit to the allowable phonon energy (or wave vector) qmax
This defines the Debye temperature 9D = Uqmax/kB where u is the
velocity of sound in the material. The 6D is merely an abstraction,
its significance being only a convenient way to express the maximum
phonon energy. Since no observable physical phenomenon is expected
to occur at D, it is not a measured quantity and consequently some
controversy exists concerning its exact value for a specific element.
Observation of dependence of S /I2 at low temperatures will reveal
which of the competing theories apply. If a peak in S /I2 is observed
associated with a continuous slope variation, we may have another ex
ample of Dutta & Horn's number fluctuation theory. However, if we see
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
6
photon emission, a 5 106 may be low enough for Handel's theory to
apply (see Chapter VI).
CHAPTER III
THEORY OF HEAT TRANSFER AND OF TEMPERATUREFLUCTUATION 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 21 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
iv = i R, where R is the resistance of the film. If i = I cosw t, the
o o
12
AC heater power delivered is AP = AP cos2w t, with Ph = R; the
h ho o ho 2 o t
exciting power frequency is thus twice that coming from the signal gen
erator, w = 2w We suppose the power is uniformly dissipated along
e o
the length of the film, thus, neglecting current crowding.
In the sensor resistor, the AC voltage across the film probes the
temperature variation of the substrate. Denoting the AC signal developed
across the load resistor, RL, by vs, and the AC temperature of the sensor
by AT we have v = V cos(w t + f), AT =AT cos(w t + 0), with V =
s s so e s so e so
mAT We have according to Kilmer et al. [28],
so
22
RL
m V R+ (31)
where V and R are the sensor film's voltage and resistance, and $ is
o x
the temperature coefficient of the film (defined by Equation 119).
Thus, in summary, the power transfer takes place in the silicon
substrate between the two thin film resistors. To the sensor is deli
vered a power AP APH hocos et, which causes a temperature rise AT =
AT cos(w t + A).
so e
3.2 The Heat Transfer Function
With no power introduced into the material, the heat conduction
equation and energy conservation theorem are,
K = OVT, (32)
aT
cd + V K = 0; (33)
Dt
here K is the heat current vector, indicating heat carried per second
through unit area, 0 is the heat conductivity, c is the specific heat
per gram and d. is the density. Substituting (32) into.(33) one obtains
the heat conductivity equation,
aV T = 0 (34)
7t
where a = a/cd is the heat diffusivity. For the silicon substrate [43],
one has,
0 = 1.45 Watt/cm C
c = 0.7 Joule/gram C
d = 2.328 gram/cm3
a = 0.89 cm2/sec. (35)
If we now introduce a heating power APh(t) in a volume V, and if
we denote by AE the AC energy contained in V, we have the conservation
law,
d AK dS + APh(t). (36)
dt h
Writing AE = jcdAT(r,t)d r and APh(t) = A (t)d r, we find by Gauss's
V h V h
V V
theorem,
dAT(rt) 33 13
cd dAr r = V AK d3r + APh(t)d3r. (37)
Dt V
V V V
Since this holds for an arbitrary volume, the integrands must be equal,
BAT 1
cd  + AK = APh(t). (38)
With K = cV(AT) this gives,
aT v2(AT) = 1 (r,t) (39)
t cdV Ph(
h
where APh(r,t) is a function equal to APh(t) for r Vh and zero outside
Vh, Vh being the heater layer volume. Equation (39) is the AC exten
sion of (34) under conditions of an external heat supply. The r.h.s.
25
of (39) will also be written as APh(t)E(r), where e(r) is the function
which is unity in Vh and zero outside. For APh = AP he (39) gives
if AT = AT ejt,
joAT (r) aV2AT (r) = AP (r) '(310)
W W cdV Wh
We define the Green's function of (310) as usual by the impulse
function response,
S jwG(r,r',jw) V2 G(r,r',jw) = 6(rr'), (311)
subject to boundary conditions which we discuss in Section 3.6. The
solution of (310) is then [44],
AP
wh
AT (r) = G(r,r',jW)E(r')d r'
ccdV
00
AP
= G(r,r',j)d r'. (312)
cdV
Vh
Denoting by ATs the averaged temperature increment in the sensor, we
cos
have,
S 3h
aTs cdV Vh f 3r d
V V
s h
Let AT s= AT sle. Then for a real input signal with amplitude Pho'
the real sensor output signal appearing in the sensor circuit is v =
s
26
V cos(w t + 0) where w = w and Vs = mIAT ,si
so e e so s
mP
vs = cdV h dr d r'G(r,r',jw)I cos(wet + 9). (314)
Vs Vh
V V
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 (314) by,
Vs (mPho/cd) IG(rsrhjy)I cos(wet + 4) (315)
where ( is the phase angle of the Green's function.
We must now consider the measurement of vs by a HewlettPackard fast
Fourier transform analyzer. This machine measures the power spectrums
of the signal at its entrance. Thus denoting the measured quantity by
d[v v* ],
sW sW
d[v vs] = 2Af v (t)v (t + T) e TdT (316)
sw sw s s
0
where v (t) = V cos(w t + () with V given by (314) or (315). Now,
s so e so
v (t)v (t + T) = V cos(w t + 4)cos(w t + w T + f)
s s so e e e
= V [cos2 ( t + f)cosW T sin(w t + ))cos(w t + f)sinw T]
so e e e e e
3jW T jWe T
S(e +e )so
4 so
Using further,
coj (C + (e)T
e e dT = 2Tr6(w e)
e
OO
we find,
d[v v~* = V2 [6(w W ) + 6(W + w )]Af.
s sw so e e
now realize
w, where Aw
centered on
that the analyzer integrates over a bandwidth Aw centered
= 2rAf. The output is therefore a signal over the range
we, of magnitude,
2
S V2 soE(We)
V* so e
sW sW 2
e e
(320)
where (W ) is the function which is unity in the interval AW centered
on w and zero outside. Hence,
e
22 2 f
mP
ho 3 3 2
v V* vh = I d rd r'G(rr', j) I
Se e 2c d V
sWe sWe 2c2d2V2V J
s Vh
for the case of Equation (314), or,
v v* W (m2P2 /2c2d2) IG(r ,rh ) 12
Sho s h
e e
(321)
(322)
(317)
(318)
(319)
for the case of Equation (315). The linear signal response is,
Iv 2= mPho d3r d3r'G(r,r',jW) (323)
e TcdV Vh
s h
Vs Vh
It should still be noted that, since the machine measures the output in
dBV, one can equally well read the logarithmic output power, by divid
ing by ten, or the logarithmic output amplitude, by dividing by twenty.
The latter is done for the figures of Chapter V, which give the wave
analyzer's input in rms volt. Notice from (320) that this is also
the sensor rms voltage signal.
3.3 Heat Transfer Correlation
We also describe the theory of experiments in which the signals,
vsl and vs2 of two sensors, having a temperature fluctuation due to a
common heater power APh = APh cos2o t are crosscorrelated.
h ho o
For each sensor the circuit output voltage is given by a result
like (314). The analyzer then measures the spectral power,
00
Re v v = 2Re v sl(t)vs2(t + T)e dT (324)
00
where vsl (t) = V scos(w et + 1) and vs2(t) = V s2cos(w t+ _).
sl slo e 1 s2 s2o e 2
Writing,
Cos(W t + W T + 2) = {cos(w t.+ (l)cos(W T + 42 cI)
sin(w t + ) )sin(e T + P2 1 ) (325)
29
we obtain in a similar way as before,
Rev l v2w = VsloV cos( )E(W). (326)
slw s2W slo s2o 2 1 e
e e
The coherence factor is defined as,
Re v v*
slw s2w
y(W ) = e (327)
[v v* \v v* ]
slwv slw s2w s2w
e e e e
From (326) and (320), we find,
Y(We) = cos(2 1). (328)
We notice that the machine measures lyl. Here pk(k = 1,2) is, for the
simplified case of (315), given by,
(k = phase angle of [G(rsk,rhj))]. (329)
Though y should be oscillatory, the higher mixima may not be
noticed due to the presence of noise. Let vnk refer to the sensors'
noises. Then for y we obtain, noticing that signal and noise are un
correlated,
Re v + Re vv*
slw s2w nl n2
( )e e (330)
{[v v* + v v* ][v v* + v v* ]}
slw sl w nl nl1 s2' s2w n2 n2
e e e e
30
(for the connection with the notation in the next section, we have
v v* = S (w )Aw, etc.). If the noise drops slower with frequency
nl n vs e
sl
than the signal response and if v nv* = 0, the above gives for suf
ficiently high frequency, in the case that (315) applies,
Re v v*
slw s2w
y(large w ) = e
e 1/2
[v v* v v*
nl nl n2 n2
Tm2 ho G(rs,rh,j) IG(rs2,r,jW) cos(2 4)
m ^ho rsi5 h" s^ 2 1
2 2 21/2
2d v* v v* l
d[V1 nlv n2V2
(331)
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 (32) we have,
AK = GV(AT) + n(r,t) (332)
where AK and AT are spontaneous fluctuations in K and T and where r(r,t)
is a source with spectrum [27],
S (r,r',W) = 4k [T (r)] 2(r)6(r r')I (333)
fl B o
where I is the unit tensor. We assume the steady state temperature
T (r) = T is uniform, and G(r) = 0. For the conservation we have again,
o o
analogous to (33),
cd + V (AK) = 0. (334)
9t
From (332) and (334) we obtain,
AT CaV (AT) = V T (r,t). (335)
t A cd
For the spectrum of 5, from (333) we have,
S (r,r') = 4k T2V V'6(r r')/c22 (336)
where V' is the deloperator with respect to r'. We note that in con
trast to the hypothesis by Voss and Clarke [15], the spectrum of the
source is not a delta function, but is the more singular function
V V'S(r r'). We now represent AT and by truncated Fourier series
on the interval (0,T), with amplitudes AT(r,w) and E(r,c). From (335)
we obtain the relation,
jWAT(r,w) aV2AT(r,w ) = (r,w). (337)
Using the Green's function (311) we find the solution,
AT(r,w) = G(r,r',jw)E(r',w)d3r' (338)
total
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
S T(r,r',W) = lim 2T AT(r,w)AT*(r',w), (339)
T4_
and similarly for S Thus from (338) and (336) we obtain the "re
sponse form" [45],
ST(r,r',w) = d3rl d3r2G(r,rljw)G(r',r2,jw)(4k T2/cd2)V V26(r r
V V
total total (340)
Providing that in Green's theorem for the V2 operator the bilinear con
comittantof G(r,rI,jw) and G(r',r2,jW) vanishes,the singular distribu
tion V1 V26(rl r2) can be replaced by,
V, V26(rl = [ 6(r r) + V26(r r2)]. (341)
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 wellknown
rule,
f(r)V26(r ro 3r V f(ro), (342)
we arrive at,
SA(r,r',) = c 2 dro[G(r,ro'j)2G(rlr ojw)
total
+ G(r',r jVG(r,ro,j )] (343)
33
Now substituting from the defining equation for G, see Equation (311),
V2G(r',r ,jw) = jwG(r',r ,j) + 6(r' r ), (344a)
cd o o o
and,
V (rr ,jw) = jWG(r,r ,jw) + 6(r r ) (344b)
cd o o o
we find from (343),
2
2k T
SAT (r,r',) = [G(r,r',jw) + G(r',r,jw)]. (345)
SAT(rrw) cd
2
This is the van VlietFassett form [17], since [kBT /cd]6(r r') is the
covariance AT(r,t)AT(r',t). For the spatially averaged temperature fluc
tuations in a volume V we have,
s
T (t) =  AT(r,t)d3r, (346)
s V
V
s
and for the autocorrelation function,
AT (t)AT (t + T) = d d3rd3r' AT(r,t)AT(r',t + .T). (347)
s s 2
s V V
s s
Whence by the WienerKhintchine theorem,
1 3 3
ST (W) = d rd rSA (r,r',w). (348)
AT VsAT
VV
s s
Employing (345) this yields,
2
4k T
SA () = 2 Re G(r,r',jw)d rd r'. (349)
s cdV
s
V V
ss
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 V times the integrand since G varies strongly
s
in the neighborhood of r = r'.
For the noise in the sensor, we have,
2
S (t) = mS (0)
v SAT
s s
4k T m2
= Re G(r,r',j)d 3 rd r'. (350)
cdV2
s
V V
ss
Comparing the noise (350) with the linear response vs 12 given in
e
(323), we note that both are double integrals over Green's functions,
though over different volumes.
3.5 Temperature Noise Correlation
We consider the spontaneous noise correlation of two devices (the
same as the previous two sensors). For the spatially averaged tempera
ture crosscorrelation, we now have, analogous to (347),
AT (t)AT (t + T) = d3rd3r' AT(r,t)AT(r',t + T) (351)
V V2
s
Vsl Vs2
35
where we assumed that both sensors have an equal volume V From the
s
WienerKhintchine theorem,
S = d3rdr'S (r,r', ). (352)
AT sIAT s2 = 2 AT
s S
SslVs2
Using (345), this yields,
2k T
S 2B d3rd3r'[G(r,r',ja) + G(r',r,jw)]. (353)
ATsl'ATs2 cdV2
Vs VV
sl s2
For the crosscorrelation of the noise voltages of the two sensors, we
have,
2 2
2k T2 m2 I
Bo 0 13 3
S d rd r'[G(r,r',jW) + G(r',r,jW)] (354)
l ,2 cdV2
V V
sl s2
where we assumed for simplicity that both sensors have the same tem
perature coefficient m. For the coherence factor of the noise, we have,
Re S (W)
Y(w) = sl s2 (355)
[S 1w) S (w)]
Vsl Vs2
Thus from (354) and (350),
1 3 3
Re [G(r,r',jw) + G(r',r,jw)]d rd r'
V V
ysl s2
[Re G(rr',ja)d3rd3r']/2[Re jG(r,r',j)3d3rd3ri']/2
VslVsl Vs2 s2 (356)
36
2
We note that because of the selfadjointness of V G(r,r',jw) = G(r',r,jw),
3 3
so the numerator is also Re f fGd rd r'; hence,
slVs2
Re G(r,r',jW)d3rd3r'
SVslVs2
y(w) = s2
[Re G(r,r',jw)d3rd3r']l/2[Re G(r,r',jl)d3rd3r'l /2
VsVs V V
VslVsl Vs2 s2 (357)
If there is a frequency range for which the integration of G is insensi
tive to the change f f f = f/ we find Y = 1.
VslVs2 VslV sl s2Vs2
3.6 The Green's Function for the Gold Thin Film Array
In the experimental arrangement for the heat transfer, we used one
thin film resistor as a heater and the other two resistors as sensors
(see Figure 31). The geometry of the thin film resistor (ie. 1pm wide
by 800pm long) allows us to regard the lengths of the devices as near
equal temperature fronts. If we assume the power is dissipated evenly
along the resistor length (ie. we neglect current crowding at the sharp
cornerssee Figure 31), the gap and the heat transfer is basically
onedimensional, being along the xaxis. 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,
8G(x,x',jw) = 0 (358)
3x x = C
t yaxis
ThIeI
O I
I
0
IE

DL
0 D
0
0
D
I
L L
xaxis
xaxis
Figure 31 Layout of a Chip Containing Three Different Groups of Thin Film Resistor Arrays
38
The solution for this onedimensional Green's function is given in the
article of Van Vliet and Fasset [17, eq. (343)*]. They find,
cothyC
G(x,x' ,j) = (c(sinhyx+tanhyCcoshyx)(sinhyx'tanhyCcoshyx'), (x
2ya
cothyC
G(x,x',ja) = cthy(sinhyx'+tanhyCcoshyx')(sinhyxtanhyCcoshyx), (x
2ya
(359)
where,
Y =\/jW/a ..
We define a corner frequency,
2
bound = /C .
bound
(360)
(361)
Then yC = / / For
and since Ixl < ICI, sinhyx
bound
<< Wound. YCI 0. Thus tanhyC  yC,
yx, coshyx 1; for (359) we now find,
C
2a
(362)
the heat transfer function (322) thus becomes a constant at low fre
quencies.
For frequencies w >> Wbound' tanhyC  1. We then easily find,
Note that there is an error in this formula. The C's should be in
front of the cosh terms and not in front of the sinh terms.
39
G(x,x',jW) = _2 e xa (363)
which is the onedimensional infinite domain Green's function [17, eq.
(314)]. If we take specifically the heat flow between the two resis
tors symmetric to the yaxis on Figure 31, we have,
L +L
d3r' d3rG(x,x',j)) : dx' dx G(x,x',jW)
Vh V EL L
h s
1 (eYL e( L))2. (364)
2 'Y2
We introduce another corner frequency,
2
ti = a/(L + ) 2. (365)
For bound << W << WI, we have jy(L + E) << 1, and a fortiori IyL << 1.
Thus expanding the exponentials, we obtain,
2 2 2 VV
jG3 3 h, (366)
d3r' d3rG(x'x''JI ) 2 2 (366)
Vh V
h s
where A and B are the pertinent dimensions in the z and y directions.
The linear response corresponding to (323) now becomes proportional to,
d1 d3r' d3rG(x,x',jw) = (367)
VhVs
40
(ie. we expect a square root frequency dependence). We also notice that
for w << 0 c = TV/4, independent of the coordinates. Thus for 0 << wl'
the signal correlation y(w ) 1.
For w >> l, we find Iy(L + E) and IYL are both greater than unity
and (364) becomes,
r VhVs
d3r' drG(x,x',jw) VhVs .(368)
Vh V
h s
Writing this in the linear response form and using (360) we see,
1 1
Sd r' d 3rG(xx',jo)  (369)
VhVs v, 3/2/
3/2
(ie. we expect to see an w3/2 dependence above the corner frequency w ).
An 1/2 spectrum which becomes an 3/2 spectrum above a corner frequency
is typical of onedimensional diffusion spectra.
CHAPTER IV
EXPERIMENTAL METHODS
4.1 Devices
Gold thin film resistors, 2,000 A thick, deposited on a 200 A
Chromium layer adhering to a standard oxide coated silicon wafer
have been prepared for us by Dr. E. Wolf and Dr. R. A. Buhrman of
the National Research and Resource Facility for Submicron Structures
at Cornell University. The standard configuration of three resistors
closely spaced for high thermal conductivity (see Figure 21) is
repeated for varying widths and spacing of d = 0.5pm, lpm, and 2pm as
designed by Dr. Van Vliet (see Figure 31). The choice of Au films
on a Si substrate is interesting since the excess weakly temperature
dependent noise ("Type A", according to Dutta and Horn) is lowered
with a strong conducting Si substrate and the strongly temperature
dependent 1/f noise ("Type B") must predominate. Note that this is
in contrast to the original studies on Au films by Hooge and Hoppen
brouwers who had their films on glass and were supposedly observing
the "Type A" noise in Au.
Samples are diced and mounted with silver epoxy glue (for thermal
conductivity into a TO5 can). An ultrasonic bonder was used to bond
gold wires to the 100pm pads on the devices (dimension "a" indicated
on Figure 21) and to the TO5 can posts. From the measured resistance
and dimensions of the thin film resistors, we find the resistivity to
be slightly greater than 2.35pQcm as listed in the Handbook of
41
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 a CTI Cryogenics Model 21 liquid He closedcycle 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 feedthroughs are provided; however, the
leads are long (for precooling 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 precooled 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 precool
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 CT310 Cryotran Conti
nuous Flow cryostat requires an externally supplied dewar of liquid
nitrogen or liquid helium to cool the sample. A similar heater/
controller is used to stabalize the cold head to the desired temper
ature. A flow cryostat is preferable to a closedcycle 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
source (see Figure 41). Using this method, one can calculate the
absolute magnitude of the DUT's current noise spectrum, Sx, simply
by comparing the relative magnitude of the noise spectra of deviceon,
deviceoff, and calibration sourceon. This is illustrated by the
equivalent circuit shown in Figure 42. For our case, we define,
RS = 11.29K + (5.6KI 5.6)
2
Scal
cal R 2
R Af
Figure 41 Calibrated Noise Measurement Scheme
IA
GV
SCL S Is[ S. 4r @ R vS YVAOUT
LOW NOISE AMP
Figure 42 Equivalent Circuit of the Calibrated Noise Measurement Scheme
/
and,
S2
2 2 Rx RI l IRS
Z = R (41)
I RA + RxI IRLI RS
We can write three equations for the three different measurement cases,
2
v01 G2 s )z 2
1) DUT ON = G2 (S + + S)2 + S (42)
Af x th a a
2
V02 2 i 2 v (43)
2) CAL ON = G (S + S + S )Z2 + S (43)
2
af th 2 a v
3) DUT OFF = G (St + Si)Z + S (44)
Af th a a
where G is the amplifier's gain, S is the amp's equivalent current
a
noise source, and S is the amp's equivalent voltage noise source as
a
indicated in Figure 42. Solving for the excess device noise term, Sx,
we find,
2 2
v01 V03
x 2 2 cal
v
02 03
In essence, four separate measurements must be performed in each fre
quency range to reveal the device's current spectrum. However, the
tedious calibrated noise source method is worthwhile since it permits
us to find the absolute magnitude of the device's noise spectrum with
out using any amplifier parameters.
4.2.4 The Measurement System
We wish to measure simultaneously the current noise spectra
from two closely spaced thin film resistors in order to see a co
herence spectrum between the two 1/f spectra. This is the method
used by Scofield and by Kilmer for the case of transistors to refute
the temperature fluctuation model of 1/f noise. This is why, in
Figure 41, 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 41 consist of high P PNP
transistor cascaded with a low noise Burr Brown op amp (see Figure
43). The LNA gives +90dB of power gain down to 1Hz below which ex
cess cryostat noise becomes a problem at low temperatures. An HP9825A
desktop calculator samples the spectrum analyzer and performs the cal
culations indicated in Equation 45. 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
487 K
IN OOIF
1NPUTH
Figure 43 Schematic of the Schmidt
12K
+12V
0.5aF
OUTPUT
91K
Low Noise Amplifier
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 Sio2 substrate, Ag pasted
to the T05 can which is pressed to a gold plated copper mount isolated
from the cold finger by an electrically isolating but thermal conduc
ting material (ie. alumina, Be02, or sapphire). In addition to the
possibility of one of the aforementioned items causing a thermal
barrier (specifically the Cr adhesion layer), we have the direct heat
injection into the film by the nonprecooled leads mentioned in
Section 4.2.1. Consequently, we cannot assume the device is actually
at the cold finger temperature indicated by the digital readout.
Therefore, we have the need for thermal noise measurements. We must
measure the high frequency "white" current noise (thermal or Johnson
noise) which for a resistor is, 
Sth = 4kT/R (46)
accurately enough to calculate T.
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 45 shows the magnitude of the ex
cess device noise term, Sx, which dominates the device noise spec
trum at low frequencies. If we use our calibrated noise measurement
system at higher frequency where Sx is comparable to Sth, we must
consider the thermal noise term. The total device noise is expressed
as,
2 2
01 03
total 2 2 cal 4kTDUTDUT (47)
V V
02 03
where TDUT and RDUT are device under test's temperature and resis
tance. Equation 47 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 43) and DUT OFF (Equation 44)
measurements. Under these circumstances and at frequencies where the
DUT ON noise is white, Equation 47 is written,
2 2
V V
01 03
4kT /R 01= SAL + 4kT /RMY (48)
DUT DUT 2 2 CAL DUMMY DUMMY
02 03
This equation is only valid, if we have the dummy resistance at a
known temperature, such as room temperature (ie. TDUMMY = 300K) and
we match the dummy's resistance with the device's resistance at the
unknown temperature (ie. RDUMMY T=300K RDUT ). Then, we can
theoretically solve for T We say theoretically because there areDUT
theoretically solve for TDUT. We say theoretically because there are
51
2 ? 2 2 2
many sources of error, especially in the (V V )/(V V )
term. Since here, we divide two quantities which are close, the
statistical accuracy is low. Measurements by this method gave re
sults with close to an order of magnitude deviation! Therefore,
we need an alternative method to measure thermal noise and to calcu
late TDUT.
4.3.2 Thermal Noise Measurements Without Using the Calibrated Noise
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, Sth, to the thermal noise of a
dummy resistor at one fixed temperature (e.g. melting ice T1 = 273K),
S and to the thermal noise of a second dummy at a second fixed
T1
temperature (e.g. liquid Nitrogen T = 77.5K), S Thus, we have
2
again from Figure 42 with RS m since we use no calibration source,
2 2 Rx RL 2
Z = R. + (49)
A + Rx RL
For the three difference measurement cases we have, since we measure
at high enough frequencies that S + 0,
2
x
01 2 2 V
1) DUT ON = G2 + S )Z + S (410)
Af th a a
2
V02 2 i 2 V
2) DUMMY AT T = G (S + S )Z + S (411)
1 Af T a a
1 *
3) DUMMY AT T2
52
2,
03 2 i 2 Vi
v G 2 +S a)Z + S
Af (ST 2 +
(412)
Solving for Sth, we find,
2 2 4
V V 4kT
01 02 4
S (T T ) + 1
th 2 2 R 2 1 R
03 V02
(413)
where we have assumed the necessary condition R = RDUT = RDUMMY. We
see we can simplify (413) to express the actual device temperature,
DUT as,
DUT'
2 2
01 V02
T =  (T T ) + T
DUT 2 2 2 1 1
V 02
03 02
(414)
4.3.3 Accuracy in Thermal Noise Measurements
To get a grasp of the accuracy involved in calculating TDUT using
Equation 414, we see that this equation has the general form,
T = Ax + B,
(415)
2
where x represents the (V01
we see,
2 2 2
V )/(V V ) term. Differentiating
02 03 02
AT = AAx .
(416)
Now we normalize since we are interested in relative error,
AT AAx
T T
= ( (417)
T X
2 2 2 2
We see the statistical accuracy of the (V V )/V V 2 term,
01 02 03 02
X must be ~1% in order to have at least a 10% accuracy in since
x T
the factor Ax/T ~ 10 when T1 = 77K and T2 = 273K. By making T1 and T2
closer (ie. T1 at liquid nitrogen and T2 at liquid oxygen or liquid
argon) A can be reduced, but at low temperatures the factor Ax/T can
still cause problems.
Concerning the statistical accuracy of the ax term, increasing
x
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],
Ax 1/2
= (2AfT) (418)
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 418. Using the HP Wave Ana
lyzer, a maximum bandwidth of 3kHz can be used and the machine can
measure at frequencies far above the 25kHz limit of the FFT machine.
Theoretically the Wave Analyzer should have a 0.7% statistical accuracy.
In an attempt to improve even upon this, a "super bandwidth" sys
tem was configured where we used the entire frequency range of the LNA.
We used an active bandpass filter with high frequency cutoff of 100kHz
corresponding to the LNA's corner frequency. Once filtered the noise
power is measured using an HP digital true RMS meter. With this method
only a single temperature can be calculated, therefore, the statistical
error is low. However, experiments showed, a type of systematic error
predominates with the "super bandwidth" system giving rise to erro
neous temperatures. For the best tradeoff between systematic and
statistical error, the center frequency should be four times the
bandwidth. This means we would need to measure near 400kHz, and we
cannot do that with our LNA.
However, a more fundamental problem with our thermal noise mea
surements predominates. Our devices have a typical resistance of
about 100Q at temperatures below 100K,and we must calculate what per
centage of the total noise at the amplifier's front end is the device
noise we are interested in. In the case of a bipolar junction tran
sistor LNA, such as our PAl, two noise sources at the transistors base
(the amplifier's "front end") compete with the device noise. The noise
sources are characterized by an equivalent voltage noise source, 4kTRn,
and an equivalent current noise source, 4kTgn (see Figure 44). In
Figure 44, the current noise source is represented as a voltage noise
source by multiplying by the device resistance, Rx, squared, where we
assume the device resistance is small compared to the amplifiers input
resistance. In this way, both of the amplifier's noise sources can be
combined and directly compared with the device thermal noise 4kT R .
xx
If the device noise amounts to 50% of the total amplifier's noise,
we consider that the device noisecan be "seen". Herein lies the
fundamental problem; because, while Rn and gn are typically small in
a good LNA, the amp is at room temperature (ie. T = 300K) and we '..
wish to measure device noise near 10K. To realize what constraints
we have on T and R in order that the devices thermal noise can be
x x
"seen" with a given amplifier (ie. Rn and gn are known), we must solve
the equation,
r   
1 2
L LNA
L
Figure 44 Simplified Equivalent Circuit of the Low Noise Amplifier
4kT R = 1/2 4kTR + R 4kTg, (419)
x xn x n' (419)
derived from Figure 44. This equation has the general form, T
x
A/R + BR which is a skewed hyperbola in the T R plane. The
skewed hyperbola will have a minimum corresponding to the lowest re
sistance, (R ) min needed to see its thermal noise at the lowest
temperature, (T ) m. Setting the derivative equal to zero, we find,
x min
= Rn
(R)m=. ~ (420)
x min gn
and,
TR ]
(T ) = 1/2 n + Tg (R) (421)
xx min
From Schmidt's PhD thesis [46, p. 33], we know for the PAl LNA;
4
Rn = 352 and g ~ 10 4U. This gives (R ) = 5900 at (T ) e 17K.
n in x mn x mn
Bob Schmidt designed a PA2 LNA where the R is decreased by employing
five transistors in parallel at the front end. The tradeoff, of course,
is an increase in g and a decrease in the input resistance, RA. Ex
perimental results using the PA2 and a specially designed two transis
tors in parallel LNA were inconsistent indicating the difficulty of
trying to measure the thermal noise of a small resistance at low tem
peratures by these methods.
4.4 Thin Film Heating Effects
In the beginning of Section 4.3, we touched upon one reason why
we need to determine the true temperature of an unbiased thin film
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 0.21m only 2mA
of current is required to generate current densities ~106 A/cm2
With such a large current density, trying to measure the exact
temperature of the film is similar to trying to measure the exact
temperature of the filament of a heater! This is not a new problem,
however, and determine the actual temperature of the film has been
a major concern of all the researchers of 1/f noise at cryogenic
temperatures. Voss and Clark extensively talk of this problem and
show a nonlinear IV 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 VI 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 nonprecooled 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, 8, 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.43]. This, combined with the fact that the closed cycle
cryostat has a 2 watt cooling capacity above 20K, would seem to imply
that we should have no problem cooling one of our biased thin films
(typically dissipating only few milliwats of heat). The results of
our endeavors are given in the next chapter.
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/2 to f3/2 (see Figure 51) predicted from
the diffusion theory. We are able to see the corner frequency, fdiff'
since,
f = (51)
diff T 2f
diff L
where D is the thermal diffusion constant and L is the distance be
tween heater and sensors. In the present case, we were able to use a
heater almost the length of a chip (ie. L ~ Imm) away from the sensors
causing the corner frequency to be at an observably low frequency (ie.
f 30Hz). This was not possible in the case of transistors since
the devices were directly adjacent to each other and had a corner
59
'5
10
16
Volts
RMS
7
10
10
H
o
00
, a
\
\\
eater Power 0
408 pW
M\0
U (U ~v\
I S I I
10 fdr 100
1000
f. (Hz)
Figure 51 Thermal Transfer Responses of Two Thin Film Resistors
I I I
frequency beyond the range of observation. Again the thermal diffusion
appears to be onedimensional in nature as was the case in the transis
tors.
In Figure 52, we show the correlation between the two sensors'
thermal transfer responses at the exciter frequency. We see the two
resistors have full coherence at low frequencies where the responses
were well above the background noise. This proves the two devices are
indeed in the same thermal environment.
5.2 Thin Film Heating
According to the Dutta and Horn procedure mentioned in Section 4.4,
the severity of the sample, heating was determined. Pulsed V/I measure
ments with low duty cycles gave the same results as the resistance
measured directly with an HP3466A digital multimeter on the ImA test
current range (see Section 5.3). Essentially, we see no heating effect
when the device is biased with lmA of continuous current. This fact was
confirmed through relative thermal noise measurements versus device
bias current (see Figure 53). In Figure 53, 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 48 are shown for
increasing device current. As explained in Section 4.3..3, the accuracy
is not enough to determine an absolute TDUT, but relative changes in
thermal noise from the value of the unbiased thermal noise can be seen.
From Figure 53, 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
100%
Coherence
10%
1%
@@=@=BEEr FB=?fgo
Heater Power o
o 408 uW
E 70 .pW
10 100
fe (Hz)
1000
Figure 52 Coherence Between Thermal Transfer Responses
3TT
t2T
Relative
Temp.
I
T<
0 I 2 3 4 5 6 7 8 9 10 II 12 13 14
 Current (mA)
Figure 53 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 multi
meter on the lmA current range) is shown in Figure 54. We see at
low temperatures the resistance approaches a constant due to the limit
where scattering becomes impurity dominated. This is predicted by
Kittel [49] and is referred to as residual resistance. At high tem
perature, the resistance, which has a linear dependence with tempera
ture, exhibits Matthiessen's Rule [42].
5.4 1/f Noise Versus Temperature
The current spectra of the gold thin film resistors were measured
at different currents to check the I2 dependence (see Figure 55). We
have quantitatively compared the room temperature noise magnitudes of
all the devices we measured to the values expected by Hooge in Figure
56. From Figure 56, we see our devices roughly obey the 1/N depen
dence predicted by Hooge and have noise magnitudes in the same "ball
park" as observed by Hooge.
Upon reducing the ambient temperature, separate current through
the device and voltage across the device measurements were made to
calculate the resistance of the biased device, RON. The calculated
RON is always a few ohms larger than the ROFF (measured in Section 5.3)
300
Figure 54 Device Resistance Versus Temperature
250
150
10 50 100 150 200 250
T (K)
= 20.5 mA
x
x\
=x \
1=10.8
mA
mA^ ^\
Sf (Hz)
Figure 55
1/f Noise 12 Dependence
1018
I0
C,,:
Cl)
1020
10
1021
i022
10
I I K
10 100 IK
1019 t
1014
10
cJ
CM)
I 1 5
1015
1016
HOOGE
*0
.4 I I
o.8 I
1012
1.6 3.2
N 
Figure 56
Relative Noise Magnitude at 300K
of all Devices Measured. (N =
12
0.8 X 10 1 0.5pm devices, N =
12
1.6 X 10 12 lpm devices, N =
12
3.2 X 10 + 2um devices)
for each temperature. Using the R versus T plot of Section 5.3, the
true device temperature is determined. The typical trends of S f/I2
and the slope, y, versus the true device temperature are indicated
for a few of the devices in Figures 57 through 59. The symbols used
in Figure 57 through 59 correspond to the symbols in Figure 56 re
presenting the different devices. In general, the "quiet" devices
(those that fall below the Hooge line) have larger error bars on both
the S f/I2 and y plots because in those devices there is less differ
ence between the device noise and the systems background noise (ie.
between the DUT ON and DUT OFF measurements mentioned in Section 4.2.3).
The program used to calculate the mean of the slope and yintercept
and the standard deviation of the slope and yintercept from noisy
data is based on the leastsquares 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
510) shows the maximum coherence is 2.4% (and this value would pro
bably go even lower with longer averaging) and most coherence values
are less than 1% over the same frequency range which we observed 100%
coherence in the "heater" experiment. With these results, the same
at all the temperatures, we feel safe to say that our research elimi
nates the temperaturefluctuation model proposed by Voss and Clarke for
the explanation of 1/f noise in metal films.
14
10
c\J
'I
U)
1015
16
10
S1
0 100 300
(K) *0
Figure 57
Noise Magnitude and Slope Versus Ambient Device
Temperature. (0.5pm device)
1.2
0.8
0.8
T
70
S 15
10
16
10
1.2
0.8
10 100 300
T (K)
Figure 58 Noise Magnitude and Slope Versus Ambient Device
Temperature. (Noisy lpm device)
1015
16
1016
U,
17
10
Figure 59
Figure 59
1.4
1.2
SY
0.8
100 300
T'(K)
Noise Magnitude and Slope Versus Ambient
Device Temperature. (Quiet lpm device)
T 100.
0 10
I I0
z
LU I
o *
0
O O
0.01
0.001 *
I 10 100 1K
f (Hz)
Figure 510 Typical Coherence Between the 1/f Noise
of Two Thin Films
5.5 The Clearcut Evidence of 1/f Mobility Fluctuations in Transistors
With the exclusion of the temperature fluctuation model of 1/f
noise, the controversy between the mobility fluctuation model and the
number fluctuation model is heightened. Toward the reconciliation of
the two competing theories Van der Ziel proposed a rather straight
forward experiment to verify mobility fluctuation in transistors.
In older transistors the predominant 1/f noise source was the
recombination current because those devices had large surface recom
bination velocities. The purpose of our present investigation is to
determine whether 1/f noise due to mobility fluctuations, as presented
first by Hooge [51] and recently by Kleinpenning [52], is present in
contemporary devices with small surface recombination velocities.
Mobility fluctuations imply fluctuations in the diffusion constant D ,
since by the Einstein relation,
qDD = kT6p (52)
P P
Thus we may expect the mobility fluctuations to modulate the emitter
collector hole diffusion current and/or the baseemitter electron in
jection current.
Van der Ziel's appendixed derivation [53] of Kleinpenning's ex
pression for the noise spectrum due to mobility fluctuations of emitter
collector hole diffusion in P PN transistors shows,
SI (f) = 2qP in (53)
dp [P(O)
s (f) = 2I 4f n B
Ep dp PB
74
where a is the Hooge parameter associated with hole current, Tdp
2
w /2D is the diffusion time for holes through the base region, w the
B p B
base width, and P(O) and P(w ) are the hole concentrations for unit
length at the emitter side and the collector side of the base, respec
tively. We see the magnitude of SIEp is inversely proportional to Tdp
which means that S is proportional to fT since
Ep
f = T (54)
T 2TrT
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)],
an N(O))
S (f) = 2qIE In f N(O)J (55)
I E En 4fT Nwn
En dn E
2
where T = w /2D w the width of the emitter region, D the electron
dn E n E n
diffusion constant in the emitter region, whereas N(O) and N(wE) are
the electron concentrations for unit length at the base side of the
emitter and at the emitter contact, respectively.
5.5.1 Discrimination Between Base and Collector Noise Sources
We now draw an equivalent commonemitter noise circuit of the PNP
transistor biased with a source resistance, RS, and a load resistance,
RL (see Figure 511). Here, Sifb represents the 1/f contribution to
the total spectrum arising from the base 1/f noise current sources.
r b 4kTrb
2
2kT/gm Sifc/gm .1
A= PRL
rTT
Figure 511
Equivalent Common Emitter Circuit
76
The base 1/f noise sources are comprised of the electron injection term
of Equation 55 and a possible emitterbase recombination current term
which we assume to be small in this modern device. The spectral con
tributor of the collector 1/f noise current source, Sif, and the collec
tor shot noise current source, 2eIc, have been referred to the input
equivalent circuit as noise voltage spectrum sources by multiplying by
(/gm)2 = (r / )2 (valid if r >> rb).
An HP3582A FFT spectrum analyzer measures the spectral density of
the collector noise, M2/Af. Calculations from Figure 511 reveal,
M rs
rf +R r R 2 ( + r
s b s +b +T
2 2 2
r 2kTr R + R+ ( rb + Rs) r 2
+ 2kTr + S.
SC R++ +r ifb r)2
s b F
S2kT ifc (56)
g 2
gm gm
If we use that r >> rb and >> 1, then Equation 56 can be written so
that we obtain,
f r Ir2
M A l = Ai 2kT(2rb + 1/g) + + S r
Af R + r + r b + m 2 ifb b
4S fcS r T
fifbb
+ Rs 4kT + 2 2 + 2Sifbrb
2 2kT Sife
+ 2 + S. (57)
s r 7 2 ifb
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 mobilityfluctuation 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 8. So the experiment was performed on lownoise PNP
transistors (GE 82 185) with 3 = 350 typically. A simply biasing scheme
was used for the high R experiment, [54] and the noise was measured for
three different I 's. From Equation 57 and for the case of high R ,
B S
we see that we measure with the spectrum analyzer,
MHI2 2 if (58)
= 2R2 2eI + S. + (58)
AfL B ifb
kT
using r = where we have neglected the small rb and r compared to
eb
a high R and the terms independent of and proportional with R The
s s
measured high R noise plotted in Figure 512 (curves IV, V, VI) repre
2
sents the absolute magnitude of the physical noise sources (in amp
sec) referred back to the (base) input,
HI 1 Sifc
SR I 121= 2el + S. + Si (59)
HR af B ifb 2
The highfrequency rolloff, which each of the plots indicates, is at
tributed to the Miller effect of the capacitance CT in the equivalent
circuit (see Figure 511) where,
CT = Cbeo + Cbco ( + A v1)
T beo bco v
(510)
10 100 1K 10
f (Hz)
Figure 512
Measurements of High Source Impedance Spectra (SHR)
and Low Source Impedance Spectra (S )
s
S
18
19
20
? 21
(I)
\J 22
o 23
24
25
26
79
Since IB is small, r is large, and the f = 1/C r Miller cutoff
frequency, is low ~2 KHz. Shot noise, lowpass filtered across the
parallel combination of r and C gives at sufficiently high frequen
cies,
2eI
s = (511)
HR 2 2 2
s 1 + w C r
T Tr
the observed 1/f2 rolloff.
Biased with a low R configuration [54], we neglect the terms in
s
2
Equation 57, which are proportional with R and R Using gm = /r
and neglecting R and rb with respect to r we see that we can plot
(again in amp sec).
MLO2 1 2+
SLR = f 2= 2eIC + 4kTrg m2 ifc
s R
2 2
+ Sifbrb g (512)
This was done in Figure 512 (Curves I, II, III) at the same three I 's
used in the high R experiment in order that the high and low R spectra
s s
can be quantitatively compared.
There are a few interesting points about the SLR spectra. First,
S ..
it was found that the magnitude of the 1/f portion of SLR was quite
as~.
device dependent. The noise plotted for SLR in Figure 512 was from
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 512 shows the
slope of SLR to be y: 1.18 while we see SHR has y 1. This implies
LR HR
s s
that for this "noisy" device we may be revealing a different noise
producing mechanism. Other transistors biased with the low R confi
s
guration give S f 1021 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 512 is only valid for R << rb.
In practice, however, R was of the same order of magnitude as rb at
low IE(e.g. R s 50). As a consequence, the thermal noise generated
E s
by R cannot be neglected and has to be incorporated in Equation 512.
s
The expression for SLR becomes,
s
2 2 2 2
SLR = 2e + 4kT(rb + R)g + Sif + S.fbr g (513)
LR c b s m ifc ifb b
s
We see that the low R measurement provides the means to measure rb [54].
Using the magnitude of the white noise levels of SLR in Figure 512,
s
the calculated values of rb are indicated in Table 51.
5.5.2 Interpretation of SH and SL
HR LR
s s
To calculate the magnitudes of S. and Sif, we look only at the
ifc ifb
1/f portion of our spectra (ie. at f < 100 Hz) where we are above the
shotnoise level and can write, at low f,
2 2
SLR Sifb(rb gm ) + Sifc' (514)
s
and,
TABLE 51
Data Obtained From High and Low R Biased PNP Transistors
s
Low R Data High R Data
s s
Curve IE r rb Curve IB (n)MIN
I 2.25mA 340 4Q IV 6.71A 362 1.2 x 107
II 1.3mA 420 90 V 3A 363 6.6 x 108
II 1.3mA 420 9Q V 3&A 363 6.6 x 10 o
H
ifc
SHR = Sifb + S2 (515)
HR ifb 2
s B
Having two equations involving the two unknown S. and Sifb we
ifc ifb
solve for Sc and find,
ifc
SLR r1 SHR
S r H (516)
ifc r g]2 2
rbgm 82
Now from inspection of Figure 512, we see SH << SL at 1 Hz, and
HR LR
s s
since
1 T
rb m (Brb
we can neglect 1/82 and 516 simplifies to,
SLR = Sifc (517)
s
We see at low frequencies the low R bias configuration isolates Sifc.
Solving Sifb, we find,
SLR
Sif = SHR (518)
ifb HR 2
2
Since even the SLR from the "noisiest" device, when diminished by 8 ,
LR
s
is a factor of ten less than SHR at 1 Hz and we see the high Rs confi
s
guration essentially isolates S. fb.
Now that the 1/f noise sources have been identified, we must apply
the results of the mobility fluctuations theory (Equations 53 and 55).
Before we attempt to calculate a Hooge parameter from Equations 53 and
55, we want to check the current dependence of the base and collector
noise sources. According to van der Ziel, the terms An P(0)/P(wb )
An N(O)/N(wE) Tdp, and Tdn are weak functions of the current, so that
we can expect,
S. ~ I ,
ifc c
and,
Sifb Ib '
if the mechanisms are mobility fluctuation noise. Measurements of
Sifbf versus Ib and Sifc f versus I are shown in Figures 513 and 514.
ifb b ifc c
We see the base noise is roughly proportional Ib (a small slope deviation
from unity is expected since Tdn and in (N(O)/N(wE) are weak functions
of the current). This shows the base noise is definitely a candidate for
mobility fluctuation noise. However, the collector noise's slope with
current is much greater than unity'and van der Ziel has found slopes
~ 1.5 imply a number fluctuation noise mechanisms. Thus, we will only
apply Kleinpenning's formula to the base noise source and calculate an
n
For the case of base 1/f noise, we have,
SHR s Sifb SI (519)
HR i fb I (
s En
and using the base to emitter expression (55) we have, for an',
21
7
3.
2
22
10 5
Figure 513
 B (A)
Base 1/f Noise Magnitude Versus Base Current
I I
7 163
 Ic (A)
Figure 514
Collector 1/f Noise Magnitude
Current
Versus Collector
3+
Id0'
71
31
lO4
2 3
I I
2 3
 !
SHR f 2Tdn
a = s (520)
n
n I N(0)]
q IBn IN(wE)
since IEn IB in aP NP transistor, if we neglect recombination. We
take An [N(0)/N(w )] < 5, since we expect the ratio of electrons in the
emitter to be a few orders of magnitude greater than the ratio of holes
in the base due to the high recombination of electrons in the heavily
doped emitter [54]. Using this and the approximation that Tdn Tdp
suggested by van der Ziel [53], we calculated the minimum values of a
n
which are tabulated in Table 51 for SHR evaluated at 1 Hz. The values
s+
seem a bit low; however, we realize that we have a P NP device where the
emitter is heavily doped and our observed an is diminished by an impurity
mobility reduction factor. Kilmer et al. [54], using a ratio of imp /
5
platt 2: 1/10, obtain a minimum value z 2 X 10 for (a ) true. Bosman
et al. [55, Figure 5) report a values ranging between 105 and 103 for
electrons in ntype silicon. Hence, we conclude that the 1/f noise in
the base of transistors can also be attributed to a mobilityfluctua
tion mechanism. Unlike the collector noise, the base noise showed no
device dependence, indicating the base noise is intrinsic to the mater
ial. This combined with the facts of unity 1/f noise slope and near
unity current dependence, gives clear cut evidence that the 1/f noise
in the base of modern transistors (which dominates the low frequency noise
spectrum in the grounded emitter configuration) is definitely caused
by mobility fluctuations.
CHAPTER VI
INTERPRETATION OF RESULTS
In the transistor experiments, the results are fairly well under
stood. The theory for mobilityfluctuation noise is well defined and
correctly describes the observed base 1/f noise. However, the inter
pretation of the thin film results are not as distinctly defined.
While the temperaturefluctuation theory does not apply, we must see
which of the possible theories (see Section 1.2), or combination of
theories, can explain the observed trends of the thin film resistors
1/f noise as a function of temperature.
The most obvious trend which is discerned from the plots of
S f/I2 vs. temperature is the relative noise minimum occurring around
150K in all the samples (except one 0.5pm device which exhibited burst
noise). The dip in the noise around 150K is most interesting since it
is surprisingly close to the Debye temperature of Au (8 = 165K).
To determine whether this is merely a coincidence or, perhaps, the
first means to actually measure 8D, would require noise measurements
from thin films made from different metals'to exhibit noise minima at
their respective D's. Good metals to verify this are mentioned in
Section 7.1.
The difference between the trends observed from Figure 57 through
59, can be roughly classified into three groups. The 0.5pm width de
vices all exhibit types of noise which appear less fundamental to the
element gold and more related to their unusual geometry. Since these
87
88
devices exhibit a more extrinsic noise, we will put them in their own
classification. Between the 1 and 2pm width devices a distinction can
be made between noisy and quiet devices. Those devices that fall
above the Hooge line in Figure 56 we will call noisy devices and ana
lyze them separately from the quiet devices (ie. those that fall below
the Hooge line). A section of this chapter will be dedicated to each
of these three classes before we make the conclusions in the last sec
tion of this chapter.
6.1 The 0.5pm Devices
Only a few of the 0.5pm devices could be measured over the full
cryogenic temperature range. This was due to the high number of de
fective devices which is typical of the submicron geometry. Many
could not be measured because they formed open circuits (see Figure
61) or shorted together (see Figure 62). The.0.5pm devices were
very delicate and often "died", from the transient currents between
DUT ON and DUT OFF, before measurements could be completed. The 0.5pm
devices were prone to burst noise exhibiting a greater than 12 current
dependence. A S f/I2 vs. T plot of a 0.5pm device which did not exhi
bit burst noise is given in Figure 57. It is the only S f/I2 plot
which shows the noise increasing at the low temperatures. We believe
the strange behavior of the 0.5pm devices is due to the narrowcross
section of the device. The SEM photograph (Figure 63) shows the
top surface of the resistors to be relatively smooth while the edges
are noticeably rough. In the 1 and 2pm devices, the width of the re
sistors cross section is five times, or ten times that of the height,
respectively. Therefore, surface noise contribution from the rough
Figure 61 Photograph of OpenCircuited Devices
Figure 62 Photograph of ShortCircuited Devices
Figure 63
SEM Photograph of Thin Film Resistors
edges is not so pronounced. However, for the 0.5pm device, the width
toheight ratio of the cross section is only 5:2 permitting the rough
edges to dominate the observed noise. Since excess noise from surface
or edge effects is not the fundamental bulk noise we seek to identify,
the 0.5pm samples are less interesting, for the purpose of this study.
6.2 Noisy 1 and 2pm Devices
The devices which have noise magnitudes at 300K which are larger
than the values predicted by Hooge, we have chosen to call "noisy de
vices." A good example, of this classification, is the noise from
the lpm sample in Figure 58. The noisy 1 and 2pm devices are good
candidates for Dutta and Horn's thermally activated numberfluctuation
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 110, slopes ranging from 0 <
y < 2 are theoretically plausible, and in Figure 58, 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, thiswouldexplain
the large spread in the data as we see in Figure 56 and Fleetwood and
Giordano's Figure 1 [35]. F.N.H. Robinson [56] has suggested that the
random motion of frozenin 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
