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a LIBRARY
, r ’., Michigan State
6" ‘ ' University
{ 7” ,1 i Z [I2

This is to certify that the
thesis entitled

Thermal conductivity measurements of bulk materials and thin
films by the 30) method

presented by
Romy Liske

has been accepted towards fulfillment
of the requirements for the

MS. degree in Material Science Enfleering

@Qmsgfieam

Major Professor's Signature

3 9,53

Date

 

 

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-———_——
2K5 WWhGO-DJS

THERMAL CONDUCTIVITY MEASUREMENTS OF BULK MATERIALS AND
THIN FILMS BY THE 30) METHOD
BY

Romy Liske

A THESIS
Submitted to
Michigan State University
In partial fulfillment of the requirements
for the degree of
MASTER OF SCIENCE
Department of Material Science and Chemical Engineering

2005

ABSTRACT

THERMAL CONDUCTIVITY MEASUREMENTS OF BULK MATERIALS AND
THIN FILMS BY THE 30) METHOD

By

Romy Liske

An experimental apparatus was assembled based on a four-wire 3m method to
determine the thermal conductivity of bulk and thin film materials. By driving a
current of angular frequency a) through a thin heating wire deposited on a
specimen surface, a localized thermal wave with a frequency 2w is created. Due
to the change of the metal line’s resistance with increasing temperature the
imposed temperature oscillation can be estimated by measuring the voltage drop
at a frequency 3m. The 300 measurements were performed in a frequency range
covering five orders of magnitude (1 Hz - 300 kHz).

Bulk materials with a low thermal conductivity (soda lime with 1.15 W/(m-K) and
Pyrex with 1.33 W/(moK)) as well as silicon that possesses a much higher thermal
conductivity (130 W/(m-K)) were measured by the 300 method. In addition, the
thermal conductivities of silica films of thicknesses between 0.3 pm to 3.2 pm
were measured. The determined thermal conductivity of those silica films (1.55
W/(moK)) was independent of film thickness. Finally, the possibility of measuring

the thermal conductivity of thin diamond films is discussed.

ACKNOWLEDGEMENTS

Especially, I would like to thank Professor Eldon D. Case for thesis direction, his
patience, thoughtful commentary and support. Without Professor Case this thesis
would not exist.

My appreciation goes to my other committee members as well: Professor
Subramanian and Professor Mueller. Thank you very much not just for being on
my committee but for all of your guidance and assistance.

I owe a great deal to staff and students of the MSU Material Science Department,
without whose cooperation and willingness to be inconvenienced this work would
not have been possible.

My gratitude to my co-workers in the Fraunhofer Center and in the Engineering
Research Center for helping to make this thesis a reality. In particular, I would
like to thank:

- Professor Timothy Hogan (Electrical and Computer Engineering
Department, Michigan State University) and his group for providing the
initial 300 background information, and for providing the probe station as
well as the lock-in amplifiers,

- Professor Timothy Grotjohn (Electrical and Computer Engineering
Department, Michigan State University) for his advice in all electrical
issues and his guidance in assembling the experimental apparatus,

- Thomas Wunderer for assembling the compensation circuit,

iii

- Syed Ahmed for always offering a helping hand and his support in the
day-to-day project work, and

- Fraunhofer CCL for establishing and funding the project.
A very special word of thanks goes to a great friend and colleague, Michael
Leonhardt, for going this journey with me, for supporting dR/dT-measurements
even on weekends, for carrying out laser acoustic measurements, and for all his
assistance.
I am grateful for the patience of my friends in overseas. Thank you for repeatedly
calling, when I missed my turns, for talking and for listening, and for making my
vacations back home unforgettable.
Most of all, I would like to thank my family, and especially my parents, for their
absolute confidence in me. The knowledge that they would always be there to

pick up the pieces is what allowed me to risk getting shattered.

iv

TABLE OF CONTENTS

 

 

 

LIST OF TABLES ................................................................................................. vi
LIST OF FIGURES ...................................... vii
1 INTRODUCTION __ ........................................................ 1
2 THEORY OF THERMAL CONDUCTIVITY .................. 4
3 MATHEMATICAL BACKGROUND. OF THE 30 METHOD ........................... 11
3.1 Measurements on bulk materials ........................................................... 19
3.2 Slope Method ......................................................................................... 20
3.3 Measurements on two-layer systems ..................................................... 22
3.4 Offset Method ......................................................................................... 24
3.5 High frequency measurements .............................................................. 27
3.6 Electrode design and measurement considerations ............................... 31

4 EXPERIMENTAL DETAILS ........................................................................... 42
4.1 Sample Preparation ................................................................................ 45
4.1.1 Sample cleaning ......................................................................... 46

4.1.2 Lithography ................................................................................ 47

4.1.3 Metal deposrtron ....................... 49

4.1.4 Lift-off ......................................................................................... 50

4.2 dR/dT measurements ............................................................................. 52
4.3 30) Measurements .................................................................................. 53

5 RESULTS AND DISCUSSION ....................................................................... 56
5.1 Measurements on bulk materials ........................................................... 57
5.2 Measurements of silica films on silicon .................................................. 70
5.3 Application of the 30) method for thin diamond films .............................. 79

6 SUMMARY AND CONCLUSION ................................................................... 81
7 OUTLOOK _ -- - - - - -- 83

 

APPENDIX A: Measuring the film thicknesses of silica films by laser acoustics 84

REFERENCES ................................................................................................... 87

LIST OF TABLES

Table 1. Literature values for the thin metal line parameters including the
material, thickness, width and length of the line as well as the
patterning method ................................................................................. 30

Table 2. Electrical and thermal parameters and their standard deviations
for the 300 measurements of Pyrex 7740, soda lime, and silicon .......... 32

Table 3. Comparison of the thermal conductivities K calculated by the slope
method (equation 34) and fit method (equation 39) for Pyrex 7740,
soda lime, and silicon for different heater line widths 2b ...................... 59

Table 4. Matrix of the process parameters used to prepare silicon dioxide
films on silicon by thermal oxidation ..................................................... 69

Table 5. Electrical and thermal parameters and their standard deviations
for the 300 measurements of silicon dioxide films on silicon .................. 71

Table 6. Comparison of the thermal conductivities K determined by the
slope method (equation 34) and fit method (equation 39) for silica
layers of different thicknesses t measured with a series of heater
line widths 2b ........................................................................................ 72

Table 7. Literature values for thermal conductivity K, thermal diffusivity D,

specific heat CF) and thermal expansion coefficient a at room
temperature .......................................................................................... 78

vi

Figure 1.

Figure 2.

Figure 3.

Figure 4.

Figure 5.

Figure 6.

Figure 7.

Figure 8.

Figure 9.

Figure 10.
Figure 1 1.

Figure 12.

Figure 13.

LIST OF FIGURES

Gas between two plates of different temperatures (adapted
from [11]) .................................................... g ........................................ 5

Schematics of the temperature oscillation as a function of
frequency assuming infinite sample thickness t ............................... 18

Schematic curvature of the temperature oscillation for a
coated sample .................................................................................. 23

Left: Schematic curvature of the temperature oscillation'for a
coated sample. Right: Cross section of the heater and
sample geometry for a coated substrate. ......................................... 25

Deposition through a mask: The specimen is covered with a
reusable hard mask. Then, the deposition of a thin metal film
is performed by evaporation or sputtering. ....................................... 28

Photolithography: Graphical scheme of the process steps
involved in the manufacturing of the electrode device on a
sample surface. ................................................................................ 34

Lift-off technique: Graphical scheme of the process steps to
manufacture the electrode device on a sample surface. .................. 36

Thermal penetration depth (given in pm) for diamond, silicon

and different glass types calculated from equation 24 over a
frequency range of 1 Hz to 1 MHz .................................................... 38
Design and dimensions of an electrode device consisting of

the thin metal line of a width 2b and a length I between the
lower two pads of 1mm. ................................................................... 41

Photograph of the measurement set-up for the 30) method .............. 44

Micrographs of a specimen supplied with 30) measurement

patterns (right: 25x; left: 125x magnification) .................................... 51
3a) measurement of soda lime glass: Probing of an electrode

device using a Janis Probe Station .................................................. 54
Electronic circuit of the 3m measurement set-up .............................. 54

vii

Figure 14.

Figure 15.

Figure 16.

Figure 17.

Figure 18.
Figure 19.

Figure 20.

Figure 21.

Figure 22.

Figure 23.

Figure 24.

Figure 25.

Silicon wafer supplied with an array of metal patterns ...................... 58

Plot of the measured 3w voltage for Pyrex 7740 at varying

metal line widths 2b .......................................................................... 61
Plot of the measured 30) voltage for soda lime at varying

metal line widths 2b .......................................................................... 62
Graphs of the 3m voltage for silicon measured between 200

Hz and 10 kHz .................................................................................. 63
Fit model applied to 3m measurements of Pyrex. ............................. 65
Fit model applied to 3m measurements of soda lime. ....................... 66

Graphs of the temperature oscillation for silicon determined
from the data shown in Figure 18 and equation 20 .......................... 67

Plot of the temperature oscillation (power normalized) for
silicon dioxide films of thickness t measured with a 5 pm wide
heater line ........................................................................................ 75

Plot of the temperature oscillation (power normalized) for
silicon dioxide films of thickness t measured with a 10 um
wide heater line. ............................................................................... 76

Plot of the temperature oscillation (power normalized) for
silicon dioxide films of thickness i measured with a 15 um
wide heater line. ............................................................................... 77

Dispersion curves measured by laser acoustics ............................... 86

Due to the linear behavior of the phase velocity versus

frequency in Figure 1, the slopes of the plots were used to

estimate the film thickness t of the silica films. A Young’s

Modulus of 69 GPa and a density of 2200 kg/m3 were

assumed for silica in these calculations. .......................................... 86

viii

1 Introduction

In the past years, the field of applications for thin films constantly increased. Thin
films function as protection layers, hard or decorative coatings or optical coatings
[1]. Either in form of single or multi layers, the term “thin” films covers thicknesses
from a few Angstroms to tens of microns. ln microelectronics electrical circuits or
sensors are entirely built by stacking and structuring thin films, as well as diode
and miniature solid-state are made of thin films [1]. Among other things, the
reliability of electronic devices or coated materials depends on heat transport and
conduction especially when high power or high temperature levels are involved
[2]-

The thermal conductivity has been investigated sufficiently for most materials on
the market. Although these literature values of thermal conductivity are a good
indication of how well a material conducts heat in general, those values should
be considered as “bulk values” assuming an ideal and infinite thick material.
Properties of thin films can differ from that of bulk materials with respect to their
microstructure and chemical constitution [3]. Furthermore, thin films are
characterized by their large surface to volume ratio. In addition boundary and
interface scattering become important issues for conducting heat [4, 5, 6].
Thermal conductivity measurements of thin films directly can provide helpful

information about the relationship between thermal properties and film

composition, thickness or structure. The measurements also provide a helpful

tool to enhance the search for suitable materials for thin film applications.

In 1987 Cahill and co-workers introduced a method (named 30) method) that
allows thermal conductivity measurements of dielectric materials [7]. The Cahill
technique involves depositing a thin rectangular metal line on a sample using
lithography and metal evaporation. Applying an alternating current of frequency a)
to the metallic line creates a periodic semi-cylindrical heat wave which diffuses
into the substrate. The heat wave is exponentially damped in the radial direction,
whereby the degree of damping is associated with the ability of the underlying
material to conduct thermal energy. A material of low thermal conductivity
exhibits a short damping distance, while a material with high thermal conductivity
transmits the heat wave over a longer distance. The metal line experiences a
temperature oscillation at 2m, twice of the exciting frequency. Since the
resistance of a metal is temperature dependent, the amplitude of the temperature
oscillation can be determined by measuring the change in resistance of the metal
line. It can be shown, that the temperature oscillation amplitude is proportional to
a voltage drop at 3a), which is three times the input frequency [7]. By measuring
the 3m voltage as a function of excitation frequency by a lock-in amplifier

technology, the thermal conductivity of the substrate can be determined.

Unlike the hot-wire technique [8, 9, 10], where a step function of power is applied

to a wire and the change of resistance is recorded as a function of time, the 3m

method has the advantage of moving the measurement into the frequency
domain. Hence, the long equilibration times required for the hot-wire technique

are reduced to a few periods of temperature oscillation for the Cahill technique.

Initially, the 30) method was used to measure the thermal conductivities of
materials exhibiting low values. In those materials the damping distance of the
heat wave is small resulting in a high temperature amplitude of the metal line.
Therefore, the resistance of the metal will change significantly, resulting in a
strong dependency of the 3m voltage on the excitation frequency. An improved
lock-in amplifier technology enables the detection of small changes in the 3m
voltage with high accuracy. This has the advantage of extending the
measurements to materials of high thermal conductivities.

The 300 method probes the sample to a depth q" referred to as the penetration
depth or more precisely the damping distance of the thermal wave. The
parameter q" is a function of the thermal conductivity and the frequency co. Low
frequencies allow the thermal wave to penetrate deeply into the substrate, while

high frequencies lead to a reduced penetration depth allowing the measurement

of thin films.

2 Theory of thermal conductivity

In a solid heat is transported by atomic lattice vibrations (called phonons) and by
charge carries (electrons and holes). The contribution cf charge carries to the
thermal conductivity is the reason, why materials that are good electrical
conductors, such as metals, are also good thermal conductors.

Compared to metals, electrical insulators do not exhibit free electrons due to
covalent (or ionic) bonding. Thus, heat conduction in dielectric materials is due to
lattice vibrations.

The bonds between atoms in a solid crystal are not rigid, but act like springs
creating a spring-mass system. When an atom or plane of atoms is displaced,
this displacement can travel as a wave through the crystal, transporting energy

as it propagates.

Considering phonons as a gas of particles, the thermal conductivity can be
derived from gas kinetics [11].
A gas between two plates of different temperatures T, and T2 has a temperature

gradient along the y-direction, dT/dy (Figure 1).

 

4111”le

 

 

 

1])
[1
m
fro
d

te

p

da

(a

5

re

tu

ra

pe

m

e

rent t

iffe

f d

o

s

te

la

P

o

tw

n

e

e

tw

be

5

Ga

1 .

re

u

'9

F

Let The a temperature between the two plates (demonstrated by a dotted line in

Figure 1) where the gas particles have a kinetic energy E,
w = gNkT (1)

where

f - Degree of freedom
N - Number of gas particles

k - Boltzmann’s constant.

The number of molecules per unit time crossing a unit area of the dotted line

from either side is int—I, where 17 denotes the mean velocity of the gas particles

[12]. It can be shown that, on the average, each molecule crossing the dotted line
makes its last collision at a height } equal to two-thirds of the mean free path of

the gas particles, xi.

VI

ll

|+
b.) l N

)9

(2)

At a height i above or below the dotted line in Figure 1 the temperature of the

gas is given by

Ti :Tizldl

3 dy . (3)

From equation 1, the energy carried across the dotted line by the gas particles
becomes
W: =inv-1k[T:-2—lt‘£]

3 dy (4)

The net rate of energy flow per unit area is the difference between the energy
that is carried by particles crossing the plane from above and the energy that is

carried by particles crossing the plane from below

g

H=W+ —W_ =lm7/kt

The energy flow H is similar to the heat flowing across an area per unit time. By
definition, the heat flow and the temperature gradient dT/dy are related to each

other by the thermal conductivity K.

H=K£ZZ

dy . A (6)

Thus, K is given by

K = Lnd/i
6 . (7)
Similar to the gas kinetics, where heat is transported by gas particles, in dielectric

solids, heat is conducted by phonons. By introducing the heat capacity of

phonons as C = '/2 f k n (Dulong Petit Law), v as the phonon velocity and A as the

phonon mean free path, the thermal conductivity for a dielectric material

becomes [13]

K=1Cwi
3 .

(8)

When discussing heat transport in solids by phonons, an important factor is the
degree to which these lattice waves are disrupted, or scattered. Phonons can be
scattered by defects or dislocations in the crystal, crystal boundaries, impurities
such as dopants or alloying species, or by interactions with other phonons.
These scattering mechanisms can be grouped into two categories: elastic
scattering and inelastic scattering. An important metric in the discussion of
phonon scattering mechanisms is the phonon mean free path 9», which is the

average distance a phonon travels between collision.

In an ideal, infinite crystal, lattice vibrations are purely harmonic and do not
interact with each other. Without collisions between phonons, the phonon mean
free path would be infinite resulting in an infinite thermal conductivity. However,
in non-ideal crystals with an-harmonic lattice interactions, the phonon mean free
path is limited due to phonon-phonon interactions [13].

Phonon-phonon interactions are divided into elastic collisions and inelastic

collisions, called “normal” and “umklapp” processes, respectively.

In a normal scattering process, two phonons with wave vectors k1 and k2

combine to form a third wave vector E.

kT+E=k—3'

(9)

In normal processes the phonon momentum is unchanged. Due to the
conservation of the phonon momentum, elastic collisions do not produce any
resistance to the heat flow [12].

The finite thermal conductivity of solids is due to the existence of umklapp
processes in which the phonon momentum is not conserved

The reciprocal lattice vector 5 reverses the direction of the phonon which slows
down the mass transport of heat [12]. Thus, the mean free path ,1 related to the

thermal conductivity in equation 8 is the mean free path between umklapp

collisions.

At low temperatures the umklapp process drastically decreases due to the
reduced number of phonons. Hence, the phonon mean free path increases until it
reaches the dimensions of crystal. However, the thermal conductivity may not
increase, because of the temperature dependent heat capacity C in equation 8.
The heat capacity C varies as T" at low temperatures. Therefore, the thermal
conductivity K will decrease towards lower temperatures whenever the phonon
mean free paths becomes comparable with the crystal size [13].

At high temperatures the number of phonons increases which results in a shorter

mean free path lengths. This in turn increases the probability for the occurrence

of umklapp processes. The thermal conductivity at high temperatures will be

inversely proportional to temperature [12].

Lattice imperfections may also reduce the phonon mean free path. Phonon

scattering can occur due to crystal boundaries, atomic” mass deviations due to

isotopic elements, chemical impurities, and vacancies [13, 14, 15].

10

3 Mathematical background of the 30) method

When an electrical current of 1(t) = 1,, cos(wt) is supplied by two contact pads,

the electrical power input, P(t), is given by

P(t) = 13 cos2 (at) - R = 113141 + cos(2at)]
2 . (11)

Equation 11 can be split into two terms, one frequency independent term and

one frequency dependent term [16].

P(t) = [113R] + [11&Rcos(2ax)]
2 DC 2 20). (12)
The dissipation of electrical energy generates a temperature increase in the

metal strip. The temperature change also consists of a frequency dependent as

well as frequency independent term.
T(t) = TDC + T2“, cos(2wt + (1)) . (13)

The Tao term describes a temperature increase caused by steady state or joule
heating while T2,, is the temperature oscillation at a frequency twice of the input
frequency.

An increase in temperature will cause a material specific change in resistance.

By knowing the dFi/dT behavior of the metal strip either from tabulated values or

11

from measurements, the strip can be used both as heater and also as

thermometer.
The constant a is defined as the relative change of resistance versus

temperature.

0” . (14)
Therefore the temperature dependent resistance of a metal can be written as

R = R0 (1 + aT(t))

. (15)
Combining equations 13 and 15 gives a relationship for the oscillation in
resistance with a frequency of 20).
R = R, (1 + aTDC + arm cos(2wt + ¢)) (16)

Both, the applied current and the resistance of the metal are connected by the

voltage drop across the strip.
V(t) = 1(t) - R_ (17)

Expressing the current as 1 (t) = 10 cos(wt) and using equation 16 the voltage

then can be written as

V(t) = 10 cos(a)t) ~R0(1+ aTDC + aTZw cos(2a)t + ¢)) (18)

12

Rearranging equation 18 gives a 3m term, after which the measuring method was

named [1 6].

W 008(3ax + ¢)

V(t) = (1 +aTDC)ROIO cos(ax)+L2“’21-eO—Igcos(at+¢)+ (19)

The first two terms of equation 19 describe the voltage drop across the metal
strip due to joule heating at an applied frequency 10). The last term in equation
19 adds a voltage at a frequency So), from which the temperature oscillation T2,,

can be estimated [16].

2V V dT
T2a1(w)= A = z'fi—RO
aRolo V10) dR . (20)
By measuring the first harmonic and third harmonic voltage as well as the relative

change of resistance of the metal versus temperature, the temperature oscillation

can be determined as a function of frequency.

The temperature oscillation of the metal strip can be related to the thermal
conductivity of the material underneath the heater/thermometer line using a
solution adapted from work by Cahill and co-workers.
We begin with Carslaw and Jéger’s solution for temperature change induced in
an infinite solid heated by a thin wire carrying electric current [17].
t iat'—r2
AT 2 _P_ I 8237—?) i

4an t—t'
. (21)

13

Equation 21 holds for a periodic line source heated with by an electrical power
per unit length P/I from time t: -oo to t: f’ with r being the radial distance from the

line given by

mix—2.72. . (22>

Laplace transformation of the integral term gives K0, a zeroth order Bessel

function of the second kind, which gives

AT

 

The reciprocal of the wave vector q defines the wavelength of the thermal heat

wave generated by electrical heating [18], such that

W . (24)

The wavelength q" is often referred as the penetration depth of the heat wave,
although damping distance might be more accurate. The magnitude of q" is an
important parameter for designing the 30) method to probe materials of different
thicknesses. As equation 24 shows the penetration depth of the thermal wave
into a material decreases with increasing frequency. In addition the penetration
depth is associated with the thermometric (or thermal) diffusivity D of a material.
Thermal diffusivity D and thermal conductivity K are related to each other by the

density p and heat capacity C of a material [18].

14

PC . (25)

For the 30) method, temperature oscillation is detected at the metal/sample
interface by a metal line of the line width 2b. Therefore, only the temperature
gradient in the x-direction (along the sample surface) is important and for further
estimates of thermal conductivity thermal oscillations on the y-direction are
neglected. Ignoring the y-direction temperature oscillations implies the
assumption of a specimen that is isotropic in the x-y plane. The penetration depth
of the thermal wave in x and ydirection is assumed to be equal, which is in fact
only true for isotropic and homogenous materials. Furthermore, the thickness of

the metal line itself is neglected.

Equation 23 is transformed to Cartesian coordinates, and then the real space

coordinate x is replaced by the Fourier space k [19] to give

 

 

P Icos(qxt)dt= P COSUOC) dk

Inxo {f—z+1 17x0 (k—2+q2 . (26a)

Arm = 7;— K0 (qx) =

In general, eilor = cos(kx)+ isin (lcx). (26b)

The Fourier transform can be written as

15

AT(x) =J—i7? jAT(k)-e”°‘dk

i

1 °° 0° .
= Elam) - cos(lcx)dk + J5 LAND - sm(lor)dk

 

(27)

Since equation 26 is a symmetric function, the Fourier transformation can be

simplified to

AT(x) = E [An/c) - cos(kx)dk
° (28)
Replacing AT(x) in equation 16 with equation 28 and rearranging for AT(k)

yields [20]

1

P
211/2 2
H k”. (29)

AT(k) =

 

Assuming heat enters the sample uniformly across the line width 2b the Fourier

transformation of the finite heat source of line width 2b can be written as

 

1 x sin(kb)
—rect — I—>
2b 2b kb . (30)

The finite width of the metal line is included by taking the product of equation 28,
after AT(k) was substituted by equation 29, and equation 30. Then, the inverse

transformation is used to get

16

00

P J'sin(kb) cos(kx) dk

llzK / 2 2
0 [CD It +q . (31)

AT (x) =

 

 

Finally, equation 31 is integrated over the line width —b < x < b and averaged by

dividing by 2b [20].

- 2
8111 (kb) dk

P 00

AT:

 

Equation 32 gives a general solution for the total temperature oscillation on a
surface caused by a heat source of finite width 2b assuming one dimensional
heat flow or an isotropic material, and neglecting the thermal boundary
resistance between metal line and sample.

Figure 2 schematically shows the curvature of the temperature oscillation based
on equation 32 and the relationship between frequency and penetration depth of

the thermal wave.

17

Low freguency High freguency

High thermal penetration depth Low thermal penetration depth
" q'1 >> b q'1S 2b
linear curvature asymptotic curvature
__ heater
- b

line

\nmplc

Temperature oscillation

 

 

logarithmic frequency

Figure 2. Schematics of the temperature oscillation as a function of frequency
assuming infinite sample thickness t

3. 1 Measurements on bulk materials

The principal curvature of the temperature oscillation for an infinite thick sample
shown in Figure 2 can be divided into two regions. In the first region, the thermal
penetration depth is much larger than the heater width 2b. Therefore, the
temperature oscillation is nearly linear versus logarithmic frequency. The linear
curvature can be described by approximating equation 32. From the result of this
approximation, the thermal conductivity can be determined by the “slope
method”, which states that the thermal conductivity is inversely proportional to
the slope of the linear curvature.

In the second region, the penetration depth of the thermal wave is in the order of
the line width 2b or smaller than 2b. In this case, the temperature oscillation
shows an asymptotic curvature. Since an analytical solution for equation 32 in
the asymptotic region is unknown [21], extracting the thermal conductivity from
the temperature oscillation versus frequency plot is carried out by fitting equation

32 to measured data with the help of appropriate software.

19

5L2 snowmehhaflnmd

Cahill [22] explored an approximation for the case, that the thermal wavelength is
much larger than the line width, q" >> b. This approximation is appropriate for 3w
voltage measurements at low frequencies, where the thermal wave is allowed to
penetrate deep into the underlying material (left part of Figure 2).

Using a series expansion of the Bessel function and truncating the series in the

limit |qr| «1 equation 23 can be approximated by [7]:

. 2 .
AT=%[-;1.(2w)-m[%].mz-g-,Jzflgmmmw

it...

where 7 = 0.5 772... is Euler’s constant.

Substituting ATwith the temperature oscillation derived in equation 10 and

. . V . . .
consrdenng the slope 61—01% In order to eliminate the constant term, the thermal

conductivity can be calculated from [23]:

Va» .éfl

dT
4n1Rg[ég%9]

,1:

 

(34)

20

For a linear slope 3%"— measuring the third harmonic voltage at two different

frequencies should be sufficient to calculate the thermal conductivity K [7]. In
practice, data are taken over a wide range of frequencies and the slope of a
temperature oscillation versus frequency regression line is used rather than the
3m voltage at only two frequencies.

It should be pointed out that the linear approximation only holds when the
penetration depth of the thermal wave is much larger than the width of the metal
line. Otherwise the curvature of the temperature oscillation versus the logarithm

of the heater frequency no longer follows a straight line (see Figure 2).

21

3.3 Measurements on two-layer systems

The curvature of the temperature oscillation versus frequency for a sample
exhibiting a coating of lower thermal conductivity is schematically shown in
Figure 3. In general, the plot can be divided into two parts. For high frequencies,
the penetration depth of the thermal wave is low. As long as the thermal
penetration depth q‘1 is smaller than the coating thickness tthe temperature
oscillation contributes to the coating. The shape of the curvature depends on the
ratio of heater line width 2b and film thickness tas described in section 3.1. The
curvature is mainly asymptotic if the coating thickness is comparable with the line
width. If the coating thickness is higher than the line width 2b, the curvature will

consist of a linear region.

22

Low frguency High freguency

—— thermal wave penetrates
into the substrate

thermal wave stays
inside the coating

 
    

  

heater
line

U ulllllg

 

sample

Temperature oscillation

 

IJLLJLI A l Allllll 1 a 11.111]
lIlIIl l I'II

' v

logarithmic frequency

Figure 3. Schematic curvature of the temperature oscillation for a coated sample
showing the low frequency region where the offset model applies (Figure 2) and
the high frequency region that contributes to the coating.

23

34lommunmwnmr

Consider a sample that is coated with a low thermal conductivity film where the
film thickness tis small compared to the metal line width 2b. In this case, the
heat flow inside the coating can be assumed to be one dimensional. Since the
thermal wavelength is much larger than the film thickness for the low frequency
range, the slope of the temperature oscillation versus frequency will not be
affected. Instead, the ATcurve is shifted up by a constant value as the intercept
of our linear function changes (Figure 4) [22].

That offset can be used to calculate the thermal conductivity of the coating.

Combining equations 33 and 34 gives the solution for a normalized temperature

oscillation
2D
2 1n '——2-
8 (”V3 (”RD 1 71b
—?————=——————mf+
Via) dR/dT K substrate Kfilm (35)
Solving for ATexplicitly gives
2D
1 In F
ATtotal = —_lnf 'I'
K substrate K film . (36)

24

" “”31 Coated sample

meIOI II he \\ b

   
 
  

     

Temperature oscillation

 

 

.._ “:r ~~~~~~~ ATfiIm
A substrate .........
+ Unooated) “““ Wire's
sample
logarithmic frequency

Figure 4. Left: Schematic curvature of the temperature oscillation for a coated
sample. Right: Cross section of the heater and sample geometry for a coated
substrate.

25

Equations 35 and 36 are valid for the low frequency region, where the
temperature oscillation versus logarithmic frequency follows a straight line. The
slope of the total temperature oscillation A730,," is related to the thermal
conductivity of the substrate Ksubmm (as discussed in section 3.1). The slope is
not influenced by the thermal conductivity of the coating Km,n in contrary to the
intercept of the temperature oscillation curve. Related to the coating, there is an

offset in the temperature oscillation given by ATmm

ATtotal = ATsubstrate + ATfilm . (37)

The value of ATM“. is obtained by measuring or calculating the temperature
oscillation for the uncoated substrate, while the temperature oscillation A730,.) is
obtained from measuring the coated sample. Subtracting those temperature
oscillations gives ATM," which is proportional to the coating thickness tand

inverse proportional to the thermal conductivity of the coating Km," [24].

t

AT = —
film
2b] K mm . (38)

The dimensions of the heater line are given by its width 2b and length I.

26

3.5 High frequency measurements

The main disadvantage of the offset model is that it is restricted to coatings
exhibiting a lower thermal conductivity than the substrate. The reason for this
limitation is that heat accumulates at the substrate interface in case of high
thermal conductive coatings. That would in turn lead to heat wave reflection
which would degrade performance data.

For coatings with high thermal conductivity a frequency range has to be chosen
in such a way that the thermal penetration depth is smaller than the coating
thickness. In other words, the thermal wave should only stay inside the coating,
without touching the substrate underneath it as shown in the high frequency
region in Figure 3. According to equation 24, the thermal penetration depth
decreases with increasing frequency. In addition, the thermal penetration depth
depends on the thermal diffusivity D of a material. Figure 5 shows the frequency
dependence of the thermal penetration depths for three different materials: glass,
silicon and diamond. For example, a penetration depth of 5 pm is related to
frequencies of 5 kHz, 500 kHz and 5 MHz for glass, silicon and diamond,
respectively. The huge difference in the required excitation frequency is due to
the ability of the material to conduct heat. As higher the thermal conductivity of a

material as higher a frequency is required.

27

 

 

. fused silica . .3
-' soda lime "ea... 3
*%

pyrex ‘1

     

thermal penetration depth (um)

 

 

I. J LLIAL

105 106

 

10-1 4 J ......i 4 Hum: . .“um . .4.....| .
10° 101 1o2 103 1o4
frequency (Hz)

Figure 5. Thermal penetration depth (given in pm) for diamond, silicon and
different glass types calculated from equation 24 over a frequency range of 1 Hz
to 1 MHz

28

In case the coating possesses a thickness tcomparable to or smaller than the
heater line width 2b, the assumption made for low frequency measurements

(q‘1 >> b) is no longer valid. Instead of the linear behavior for low frequencies, the
measured temperature oscillation for the case q'1 5 2b asymptotically approaches
zero for high frequencies.

When the temperature oscillation versus logarithmic frequency leaves linearity
the integral formula (equation 32) has to be used to calculate the thermal

conductivity. Writing equation 32 by replacing q with equation 24 and 25 gives

sin 2 (kb)

AT- PT
MK '0) C
°(kb)2Jk2+[—' 2 P]

dk

 

 

 

(39)

Using MathCAD 2001 and its implemented “fit function” routine, the in-phase or
real part of equation 39 can be fitted to the measured temperature oscillation. For
the density p and the specific heat 0,, literature values according to Table 1 are

used.

29

Table 1. Literature values for thermal conductivity K, thermal diffusivity D,
specific heat 0,, and thermal expansion coefficient aat room temperature

 

 

 

 

 

 

 

 

 

K (w m"'K") D (10 2m2/s) c, (J kg'1K") 01(102 °C")
Diamond I 895 ‘
7.68 3 520 2 0.11-1.23 2
Diamond II a 2300 1
1
Silicon ‘48 2 0.84- 3 712 ‘ 2.5 2
141
1.5 I 1 1 2
$102 1 4 2 0.0086 740 0.4
1
Soda Lime 1; 2 0.0081 ‘ 840 2 9.0 2
1
Pyrex 112:2 0.0074 ‘ 850 ‘ 3.3 2
Silver 429 ‘ 1.74 ‘ 235 ‘ 18.9 2
Chromium 94 ‘ 0.291 ‘ 449 I 4.9 2

 

 

‘ Reference: Lide 2004 [29]
2 Reference: Callister 2003 [18]

3 Reference: Leung 1999 [41]

3O

3.6 Electrode design and measurement considerations

The key component of the 300 measurement system is a thin metal line deposited
on a dielectric material. As shown in chapter 3 the metal line functions both, as a
heater by carrying electrical current and as thermometer by detecting the change
in resistance (which in turn depends on the thermal properties of the coating or
substrate material). Four electrode pads are connected to the thin line: two pads
function as current leads l+ and l' and two pads are the voltage leads V+ and V.
The properties of the metal line and the electrode pads have to be optimized to
match the 30) measurement requirements. Also, the dimensions of the metal line
and electrode pad assembly should be feasible to be manufactured with
reasonable effort.

The metal of the wire and its contacts should have a high thermal conductivity
and a low electrical resistance in order to achieve a high electrical power input
without increasing the temperature of the metal line significantly. Furthermore,
the metal should be chemically inert to ensure reproducible measurements over
time. Both factors favor noble metals due to their resistance against reactions
with air and their good electrical and thermal properties. A drawback of noble
metals is their poor adhesion to most dielectric materials requiring an
intermediate adhesion layer.

Table 2 summarizes a literature survey regarding the metal line design and

device fabrication.

31

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32

There are three possibilities to fabricate the heater line and the electrode pads on

a specimen surface:

I Demsition through a mask

Probably the easiest way to form the metal line and the electrode pads is to
deposit an evaporated metal through a mask containing the desired features
(Figure 6). Deposition through a mask is preferred for devices having dimensions

not smaller than 50 pm.

33

 

Figure 6. Deposition through a mask: The specimen is covered with a reusable
hard mask. Then, the deposition of a thin metal film is performed by evaporation
or sputtering.

34

ll Photolithograghy

Fabricating structures in the nanometer range requires a photolithographic
process (Figure 7). First, the entire substrate surface is covered by a thin metal
layer that is deposited by metal evaporation or sputter deposition. Second,
photoresist patterns are applied using lithography. Those patterns serve as a
protection layer for the following etching step, where the metal layer is chemically
structured to achieve the final devices. A drawback of the photolithographic
process is the difficulty to pattern noble metals because of their high chemical

stability as well as the risk of damaging the underlying material.

35

 

 

Specimen coated with
thin metal film

 

 

Spinning a photoresist
layer on top of the
metal coating

 

 

Patterning the
photoresist

 

Sanfwi‘ 'i a.
M m

Metal film etching

 

Photoresist removal

 

Figure 7. Photolithography: Graphical scheme of the process steps involved in
the manufacturing of the electrode device on a sample surface. The device
structures are obtained by etching the deposited metal film.

36

Ill Lift-off technigue

The lift-off technique (Figure 8) overcomes the issues of the photolithographic
process by avoiding the etching step and is therefore suitable for almost all
substrate materials. Photoresist patterns are supplied to the specimen surface
prior to the deposition of the desired metal. When the resist layer is removed by
acetone in an ultrasonic bath, the metal is lifted off from those areas that were
covered by photoresist, but it remains in the areas that were not protected by the
photoresist.

Although this technique is suitable for sensitive substrate materials and for
patterning noble metals, the minimum structural dimensions that can be obtained

lie in the lower micrometer range.

37

 

 

Specimen surface
coated with photoresist

 

Patterning the
photoresist

 

Covering the entire
surface by a metal
layer

 

 

iv fl Lift-off the photoresist

 

Figure 8. Lift-off technique: Graphical scheme of the process steps to
manufacture the electrode device on a sample surface. This process bypasses
the metal etching shown in Figure 7.

38

The geometric aspect of the line length and width is considered by using
equation 39. The temperature oscillation, which is directly determined from the
30) voltage, is proportional to the power input per unit length of the metal line and
inversely proportional to the heat conductivity of the underlying material. Most
often the power input is limited by the capability of the pOwer supply to deliver a
harmonic-free, sine-wave power signal. Those signals are best realizable in the
lower power range, which favors shorter lines. On the other hand, the 3m voltage
drop is caused by an electrical resistance modulation at 200. The amplitude of the
modulation increases with increasing line length. According to the literature, line
lengths between 1 and 10 mm are typically used for 3w measurements. In this

study, our goal was to keep the length of the metal line at 1mm.

The role of the line width was discussed in chapter 3. It was shown, that the ratio
of the metal line half-width b and the thermal penetration depth q" defines the
curvature of the temperature oscillation plot. As long as the heater line width is
much larger than the penetration depth, the temperature oscillation follows a
linear line versus logarithmic frequency. When the penetration depth becomes
small and comparable to the width of the heater line, the curvature of the
temperature oscillation approaches an asymptotic behavior.

In general, as the thickness tof the coating to be analyzed decreases, the metal
line width should also decrease. In addition to the frequency dependency, the
thermal penetration depth q" also depends on the thermal diffusivity D

(equations 24 and 25).

39

In materials with a high thermal diffusivity the penetration depth will be high as
well. One requirement for measuring by the 30) method is that the thermal
penetration depth is smaller than the film thickness. Figure 5 shows that high
frequencies are required for analyzing materials by the 30) method exhibiting high
thermal diffusivities like diamond or silicon. To reach the same thermal
penetration depth, glass can be measured with frequencies that are 100 times

lower than for measuring silicon and 1000 times lower than for diamond.

To study the influence of the line width on the thermal conductivity measurement
of thin films, we decided to use four different line widths: 5 pm, 10 pm, 15 um and
20pm.

The final measurement device consisted of metal lines with a length of 1 mm and
line widths of 5 pm, 10 um, 15 um and 20 pm, which were connected to four
electrode pads with an edge length of 1 mm (Figure 9). To fabricate those metal

structures on specimens the lift-off technique is used (Figure 8).

40

1mm 1.2 mm

 

 

1mm

2!)

 

 

 

h

 

 

 

 

 

 

 

 

 

1mm

Figure 9. Design and dimensions of an electrode device consisting of the thin
metal line of a width 2b and a length Ibetween the lower two pads of 1mm. The
electrical connections are realized by four electrode pads each having an edge
length of 1 mm. Through the upper two pads an alternating current is supplied,
while the lower two pads are used for measuring the 3m voltage.

41

4 Experimental details

The 30) measurement can be applied to basically all dielectric materials and
films. Required is a sample surface of at least 3 x 4 mmz, which is the space
needed to supply one electrode device. Besides, a dense and very smooth
surface is essential to ensure good adhesion of the metal structure in order to
neglect the thermal boundary resistance between metal structures and sample.
The experimental of the 3m method can be divided into three parts:

- The sample preparation which includes deposition the electrode device
using a lithographic process for masking and a high vacuum electron
beam deposition system for metallization

- Evaluating the temperature coefficient of resistance (dFl/d'l) of the thin
metal line by measuring its resistance as a function of temperature

- Measuring the 3m voltage as a function of excitation frequency

The 300 method was tested on two different glass samples (Pyrex 7740 and soda
lime) and on a bare silicon wafer. Those three samples were supplied with an
array of electrode devices consisting of metal lines with different line widths 2b.
On each sample measurements were performed using 4 different line widths: 5,
10,15 and 20 pm.

To test the 30) method on thin films, four silicon wafers were thermally oxidized to
prepare SiOz layers of different thicknesses tranging from 0.3 pm to 3.1 pm.

Therefore, silicon wafers were placed into a furnace, which was heated to

42

typically 800 °C. Oxygen of purity 99.99 % was led through a water bath,
whereby the water temperature defined the amount of vaporized water that
contributed to the oxidation process in addition to the set amount of oxygen (60
sccm). Detailed information is found in Appendix A, which lists the process
parameters and describes the SiOz film thickness measurements for the 4

samples used for 30) measurements.

The 30) voltage of the silica films were measured using metal line widths of 5 pm,

10 pm and 15 pm for each of the 4 samples.

43

 

Figure 10. Photograph of the measurement set-up for the 30) method with the
probe station in the display foreground and the electronic devices in the
background

4. 1 Sample Preparation

The sample preparation including sample cleaning and supplying the electrodes
was carried out in the cleanroom facility at MSU Engineering Research Complex.
The manufacturing steps (schematically shown in Figure 8) are based on thin film
technologies including mainly masking by lithographic processes and
metallization by physical vapor deposition.

A mask consisting of 16 measuring device structures with varying widths of the
metal line between the electrodes was manufactured by PhotoSciences,
Torrance CA. One measuring device consists of a thin metal line with a length of
1 mm connected to four electrode pads, each having a square area of 1 mmz.
The electrode device is made of a 5 nm chromium layer followed by a 200 nm
silver layer, whereby the chromium is functioning as an adhesion layer between
the substrate and the silver layer. Silver was chosen as an electrode material
because of its high thermal and electrical conductivity. As the metal line works as
both heater and thermometer, those properties are essential for measuring
thermal conductivity of the underlying film or substrate.

To ensure an optimal contact between sample and measuring device all process

fabrication steps have to be carried out with high thoroughness.

45

4.1.1 Sample cleaning

To remove organic and inorganic residuals on the sample surface, the sample
was put in a beaker filled with acetone and cleaned in a commercial ultrasonic
bath for 20 min. After rinsing the sample with de-ionized water the sample was
placed into a beaker filled with methanol and agitated by ultrasonics for another
20 min. Meanwhile 150 ml isopropanol in a glass beaker was brought to a boiling
on a heating plate. After rinsing the sample thoroughly with de-ionized water it
was exposed to the isopropanol vapor. The sample was held by a pair of
tweezers face down towards the isopropanol. By slightly turning the sample
across its axes, it could be visually observed, how the isopropanol vapor carries
away all water drops leaving the sample surface clean and water-free. This
process was carried out for approximately 5 min. A dehydration bake at 150 °C
for 35 min in a furnace (BlueM Stable-Therm® Gravity Oven C-3991-O) under
laboratory air drove off any remaining moisture that might be present on the

sample surface.

46

4.1.2 Lithography

The cleaned substrate was placed on the chuck of the spinner (Laurell Techn.
Comp. USA, WS-400B-NPP-Lite). To enhance adhesion between photoresist
and substrate hexamethyldisilizane (HMDS 100%, Transene Comp. Inc.,
Danvers MA) was applied by a disposal pipette in such a way, that it covered the
entire sample surface. The sample was fixed on the chuck by vacuum and after
applying HMDS rotated with a speed of 3,000 rpm for 30 sec. Due to the spinning
the surface was dried and the remaining very thin HMDS layer could react with
the ambient air to form reactive silicon hydroxide bonds.

Immediately after rotation stopped, a photo-sensitive polymeric material, called
photoresist (Shipley 1813, MicroChem Corp., Newton MA), was applied by a
second disposal pipette on the entire sample surface. The same spinning
parameters (3,000 rpm and a time of 30 sec) were used to dispense the
photoresist across the substrate resulting in a uniform photoresist thickness of
1.2 pm. The photoresist coated sample was then placed into a BlueM furnace
and “soft baked” at 80 °C for 20 min under laboratory air to drive off excess
solvent before the sample was introduced into the exposure system.

A mask aligner (Karl Suss MJB 3 UV300) was used to expose the photoresist
through a chromium coated glass mask. UV light of wavelength between 380 —
500 nm exposes all parts not being shadowed by the chromium-coated mask.
The exposure time was set to 30 sec. When the image of the mask is projected

onto the sample, the photoresist undergoes wavelength-specific radiation-

47

sensitive chemical reactions, which cause the regions exposed to light to become
more acidic.

During the 30 sec exposure time a beaker large enough for the sample was filled
with a photoresist specific developing solution (MF319 developer, MicroChem
Corp., Newton MA) to a liquid level of 1 cm. The sample was placed into the
developer containing beaker immediately after the UV-light turned off. Exposed
acetic photoresist was now soluble in developer solution and was therefore
washed away. Photoresist that was shadowed by the mask and not exposed to
the UV light is resistant to developing solution and remained on the specimen
surface as photoresist patterns.

The developing process can be observed visually by a localized dark red color
that arises from the reaction between acetic photoresist and developer. Agitating
the beaker slightly by hand leads to a dilution of the red color, revealing if the
reaction is still ongoing. After the reaction was completed (approximately 1.5 min)
the patterns were absolutely color-free and the sample was taken out of the
solution, rinsed for 10 sec with de-ionized water and dried with nitrogen.

The sample is then “hardbaked” in a furnace under laboratory air at 130 °C for 35

min in order to solidify the remaining photoresist pattern.

48

4.1 .3 Metal deposition

Physical vapor deposition (PVD) was used to coat the sample with a thin metal
layer. An electron beam vaporized the desired metal in a highly evaporated
system (Kurt J. Lesker AXXIS). The sample was fixed using commercial razor
blades on the substrate table. The metals chromium (99.998 % pure, 1-6 mm
random size chromium pieces, Kurt J. Lesker, Clairton PA) and silver (99.99 °/o
pure, 1A” a x 1A”, Kurt J. Lesker, Clairton PA) were put into graphite crucibles
(Kurt J. Lesker, EVCFABEB-23, volume: 3.7 cm3). After loading the sample and
the crucibles, the system was pumped down to a pressure of less than 51045 torr,
which took at least 4 hours. Using vacuum level and ultra pure metals ensured a
deposited metal film of high quality. After reaching the desired vacuum level, a 5
nm thin chromium layer was deposited. Typical parameters for Or were a
chamber pressure of 110'7 torr and an electron beam current of 18 mA leading to
a deposition rate of 0.1 nm/s (measured by an implemented piezoelectric
transducer). When the 5 nm of Cr were deposited, the electron beam was turned
off and held for 20 min to let the melted Cr in the crucible cool down. Then, the
rotation stage housing the crucibles was turned to center the silver containing
crucible. A 200 nm silver layer was deposited with typical parameters of 1'10“
Torr for the chamber pressure and 50 mA for the electron beam current leading
to a deposition rate of again 0.1 nm/s. A temperature measurement device
controlling the substrate holder temperature indicated, that the temperature

during deposition remained nearly at room temperature. It never exceeded 27 °C.

49

4.1.4 Lift-off

After the metal deposition, the entire sample surface was metallized including the
surface of the photoresist pattern. By dissolving the photoresist all metal covering
the pattern got removed leaving behind only those metal parts, which touch the
sample surface directly. Photoresist can be stripped by acetone. Therefore, the
sample was placed into a beaker filled with acetone. The lift-off process was
enhanced by ultrasonic agitation and carried out until all metal structures were
clearly recognizable. Then, the acetone was changed against new acetone
solution for a 20 min final cleaning of the sample.

After rinsing thoroughly with de-ionized water and drying with nitrogen, the

samples were ready for the thermal conductivity measurements.

50

 

Figure 11. Micrographs of a specimen supplied with 300 measurement patterns
(right: 25x; left: 125x magnification)

51

4.2 dR/dT measurements

The temperature oscillation is a function of the resistance of the deposited metal
strip. In order to calculate the thermal conductivity the resistance versus
temperature behavior of the thin metal line has to be measured.

For measuring dR/dT, the sample consisting of the deposited metal pattern was
placed on a heatable substrate table, which is part of a Signatone Probe Station
(Signatone 8-1060 QuieTemp System). Two copper probes were put on the two
lower electrode pads. The probes were connected to a Fluke 1990 Scopemeter
functioning as a multimeter to read out the resistance between the two lower
pads. A stable resistance read-out ensures a sufficient contact between the
probes and the electrode pads.

The temperature was measured by a Fluke 80T-150U temperature probe placed
on the sample surface at a location approximately 5 mm from the electrodes. The
temperature was read by a Hewlett Packard Multimeter 34401A. The substrate
table was heated with a rate of 2.5 °C/min. Beginning at room temperature up to
150 °C the resistance was recorded every 5 °C. After reaching 150 °C the
substrate table heater was turned off. While the sample cooled down, the
resistance was again monitored every 5 °C until 50 °C were reached.

At least one more temperature rising procedure was carried out to get sufficient
data points for calculating the change of the resistance over temperature. This

slope gives the dR/dT value for the measured heater line.

52

4.3 3w - Measurements

The sample was placed on the chuck of a Janis ST-4LF-2MW-21-CX probe
station (Figure 12). Four probes were placed on the device electrodes. The two
upper pads were used to supply electrical current of angular frequency by a
source generator (Hewlett Packard 8116A Pulse/Function Generator 50MHz),
while with the two lower pads the voltage at 3m was read out. In serial to the
electrode device a 15 Ohm resistor was placed.

The signals of the resistor as well as of the device were sent to two differential
amplifiers (IC opamp 300 MHz high-speed MAX477EPA). The gain for the
amplifier connected to the serial resistor was adjusted in such a way that the 1m
voltage for the resistor and the electrode device were equal. That set-up
(schematically shown in Figure 13) was used in order to cancel out the 3m
voltage coming from the source generator.

Then, the signals of those two amplifiers were led to a third differential amplifier,
which took the difference of the two incoming signals. The third amplifier was
connected to a lock-in amplifier (EG&G Instruments 7265 DSP) giving the

voltage at the third harmonic of the thin heater line.

53

 

Figure 12. 3a) measurement of soda lime glass: Probing of an electrode device
using a Janis Probe Station

Source

R1

Sample

 

 

 

+ I Lock-In!
R4 R41 Spect. Analyser

Figure 13. Electronic circuit of the 3m measurement set-up. All numbered
resistors were of equal magnitude R = 320 ohm. The resistor R was adjusted to
have the same resistance as the heater line (sample).

54

Before starting a measurement, the probe chamber was evaporated and the
source generator preheated for 1 hour. Then, the wanted voltage was set at the
source generator. The value of the 1m voltage was zeroed by changing the gain
of the amplifier of the serial resistor. This adjustment‘was carried out at 1 kHz
and as no change in phase shift was observed up to 300 kHz it was valid for the
entire frequency range.

The readout of the lock-in amplifier was changed to give the voltage at 3w for the
metal line directly. Starting at low frequencies the set point of the frequency at
the source generator was logarithmically increased up to 80 kHz and the
corresponding 30) voltage was recorded.

As the lock-in amplifier was only capable for frequencies up to 250 kHz, which
would correspond to read out 300 values up to a third of 250 kHz, a spectrum
analyzer (T ectronic 2754P) was used for measuring 30) voltage from 50 kHz to
300 kHz. The overlapping range of the lock-in amplifier and the spectrum
analyzer between 50 kHz and 80 kHz could be used to control the accuracy and

reproducibility of our data.

55

5 Results and discussion

Initial tests of the experimental apparatus were performed on bulk materials that
model a one-layer system with infinite sample thickness. This assumption is
necessary for using one dimensional heat flow calculations (chapter 3.1), which
neglects heat wave reflections from the rear surface of the sample. Two glass
materials, soda lime and Pyrex, were chosen, because of their low thermal
conductivity values. In an extension of the 3m method to higher thermal
conductivities, silicon was measured, which is approximately two orders of
magnitude more-conductive than glass. A wide frequency range was used to
record the 3m voltage, which allowed the data to be analyzed by the slope model
for lower frequencies (chapter 3.2) as well as the fit method for the entire
frequency range (chapter 3.5).

In addition, two-layer systems consisting of thin silicon dioxide films on silicon
were investigated. As the thickness tof the silica films was smaller than the metal
line widths 2b, the slope model for determining thermal conductivity was no
longer valid. Instead, the offset-model for the low frequency range was compared

to the fit method for high frequencies.

56

5. 1 Measurements on bulk materials

Two glass wafers (soda lime and Pyrex) and one silicon wafer were equipped
with 30) measurement patterns using the lift-off technique (as described in
section 4.1. The patterns consisted of heater lines, where the width 2b was 5 pm,
10 um, 15 um and 20 um (Figure 14). One device of each line width was used on
a given sample resulting in four thermal conductivity measurements on one

sample.

First, the heater line was calibrated. By recording and differentiating the
resistance of the metal line as a function of temperature, one determines the
resistance change per unit of temperature difference. This value, also referred to
as dFl/dT, is assumed to be constant over the temperature range of interest.
Dividing the dR/dT value by the resistance of the metal line at 20 °C gives a, the
relative change of resistance versus temperature.

In Table 3 the values of a determined for the bulk material measurements are
listed. Thee mean value of a is (0.0033 :1: 0.0004) K". In order to reduce the
measurement error, avalues were recorded for each metal line instead of using

the mean value.

57

 

Figure 14. Silicon wafer supplied with an array of metal patterns. For each row
values for the metal line widths 2b are indicated on the left side of this figure.

58

Table 3. Electrical and thermal parameters and their standard deviations for the
30) measurements of Pyrex 7740, soda lime, and silicon. These parameters were
subsequently used to determine the thermal conductivities at varying line widths
2b. From the voltage drop V1,... measured parallel to the voltage pads, and the
current across the line measured in series (not shown in the table), the power per
unit length P/I is calculated. To determine the temperature Tof the specimen, the
resistance of the metal line is measured twice: first with a small voltage to avoid
heating of the sample and the second time with a voltage V10, used for 30)
measurements. By knowing the temperature coefficient of the resistance a, the
temperature Tof the sample during measurement can be determined.

Line

 

 

 

 

 

 

 

 

 

 

 

 

width v1... (mV) P/I (W/m) (1 (K") T (°C)
5);? 5 pm 355 1 1 4.34 1 0.05 0.0038 1 0.0001 33 1 2
10 pm 217 1 1 3.13 1 0.04 0.0035 1 0.0001 27 1 1
15 pm 185 1 1 3.01 1 0.03 0.0034 1 0.0001 28 1 1
20 pm 255 1 1 2.85 1 0.04 0.0028 1 0.0001 31 1 2
33:: 5 pm 354 1 1 3.57 1 0.05 0.0037 1 0.0001 31 1 1
10 pm 278 1 1 3.89 1 0.04 0.0033 10.0001 42 1 2
15 pm 208 1 1 3.42 1 0.04 0.0034 1 0.0001 35 1 2
20 pm 159 1 1 2.94 1 0.03 0.0035 1 0.001 38 1 2
Silicon 5 pm 1159 :1: 1 49.6 1 0.3 0.0028 :1: 0.0001 33 :1: 1
10 pm 795 1 1 45.1 1 0.2 0.0032 1 0.0001 27 1 1
15 pm 550 1 1 38.9 1 0.2 0.0030 1 0.0001 28 1 1
20 pm 845 1 1 46.2 1 0.2 0.0023 1 0.0001 31 1 1

 

59

During the 30) measurements the input voltage was adjusted so that the resulting
3m voltage was between 1 and 3 mV at lower frequencies. This voltage range is
sufficient to achieve stable read-outs on the one hand and to avoid an
overheating of the thin metal line on the other hand. By measuring the 10) voltage
and current, the input power and the actual resistance of the metal line was
calculated. Using the resistance of the heater line and the input power, the
sample temperature was then determined (see Table 3).

Due to the difference between the thermal conductivities of glass and silicon,
silicon required much higher input powers to achieve similar 30) voltages. Since
silicon is a better thermal conductor than glass, those high input powers did not
increase the sample temperature. In general, sample temperatures were
determined to be slightly above room temperature at 27 — 43 °C.

Figure 15 and Figure 16 show the measured 30) voltage as a function of
logarithmic frequency for Pyrex and soda lime, respectively. Both glass samples
were 1 mm thick and were measured using input voltages with frequencies
between 1 Hz and 100 kHz. According to the damping distance of the thermal
wave (equation 24), this frequency range is equivalent to a probing depth of
approximately 350 pm to 1 pm. Since the maximum penetration depth of 350 pm
is smaller than the sample thickness of 1 mm, the assumption of neglecting heat
wave reflection from the sample backside is reasonable.

For the silicon sample, which was 350 pm thick, thermal conductivity
measurements were made at a frequency range of 200 Hz to 10 kHz (Figure 17),

which is equivalent to a probing depth of about 260 pm to 35 um.

60

‘v5m(nvv)

Penetration depth (pm)

 

 

 

 

 

 

 

 

343 100 10 1
. . ,.. - . . . . . , ..... . . .

2 0 LB Pyrex 0 2b = 5 pm 4
2531., o 2b==10|un
1&0}; ‘11. A 2b = 15 pm

A. ‘0. B.
1.5 51, “Ann ERR 9 2b = 20 pm -
W*1 3.19051 ---------- regression lines
l- “,x \A x ‘G. :— .1
*‘fit ‘A.G\O [3.11
1.0 P ‘*“* A‘A‘G‘O “E1~ -4
“‘1 ‘AA\Q~ E‘O
‘fi‘ “A ‘Q‘ ‘Q
r *‘Q “A ‘0‘ 01‘
“*N'Q‘A‘Z‘QQ QRQ‘
0.5 - ~11 {5 A19 ‘53.: D -
\‘x‘fi’ ‘~‘A“9 O “‘1. D
s“* h‘ AKA O x. C] C]
“a“ ”19.5% gg‘g‘ D D 4

0 0 1 1 1 1 11111 1 1 11111] 1 1 1 1 11111 “1“ l ‘I‘:‘;l‘1‘lb 12‘2811§1§

10° 101 1o2 103 104 105
Frequency (Hz)

Figure 15. Plot of the measured 30) voltage for Pyrex 7740 at varying metal line
widths 2b. A linear regression was performed for frequencies below 200 Hz to
determine the thermal conductivity by the slope method, which gave a mean
value of K = 1.3 W/(m'K). (For better clarity, only every third data point is included
inmeNM)

&

V3... (mV)

Penetration depth (um)

 

 

 

 

 

 

 

 

359 100 10 1
1' ,....1. Ln”... .
:51. Soda lime D 20 = 5 pm

2.0 ktoxo ‘8‘0‘ 0 2b = 10 pm -I
A . ‘
{“93930‘0 an: A 20 = 15 pm
*‘ 34:25 ”1,3 9 2b = 20 pm
1'5 ” ‘19::‘3 8‘: ---------- regression lines 7
.fi ‘ . .
h ‘fifol‘o [3313‘
‘k. ‘ ‘(3‘ E1
1 A 0
1 0 l- w‘u‘A “6)~ . _
. ‘g A. ‘Q 151
‘1‘9. ‘ ‘1.
l ‘15. , 01,,
‘3 00 “t!
0 5 no no ‘9‘ D
1 - “£5: A“? 0 “ND D -1
.x‘ ”a" A‘~AO 0 xx D
\:*a 9&8 8 8 D D 0“
00 1 . 1 ”“11 1 . . .....1 . 11L 1 .. ““1 - 1 . u...
100 1o1 102 103 104 105
Frequency (Hz)

Flgure16. Plot of the measured 30) voltage for soda lime at varying metal line
widths 2b. A linear regression was performed for frequencies below 200 Hz to
determine the thermal conductivity by the slope method, which gave a mean
value of K = 1.1 W/(m'K). (For better clarity, only every third data point is included
in this plot)

62

Penetration depth (pm)

 

 

 

 

 

366 100 30
l Silicon E] 2b = 5 pm
2 0 r 0 2b = 10 um _
' A 2b = 15 pm
* 2b=20um
1.5 - ~~~~~~ 3899388388 ---------- regression lines

v30) (mV)

----- A-AA
1.0 - AAAAAAAAAAAs

 

 

 

0 5 ~ *****‘*** ....... .1
00 1 L4 1 1 1 L14 1 1 1 11111 1
102 1o3 104

Frequency (Hz)

Flgure17. Graphs of the 30) voltage for silicon measured between 200 Hz and
10 kHz. The plotted graphs were taken from one silicon wafer that was equipped
with an array of electrode devices with varying line widths 2b. A logarithmic
regression of the form y = m‘ln(x)+n was performed (plotted by black solid lines)
to determine the thermal conductivity with equation 34. The thermal conductivity
values are included in Table 4.

63

From the 30) data (Figure 15, Figure 16, Figure 17) the lower frequency data was
included in a logarithmic regression to determine the thermal conductivity K by
the slope model (section 3.2). The thermal conductivity is inversely proportional
to the slope of the logarithmic regression line (equation 34). The higher slopes in
the 3m plots of the soda lime and Pyrex sample indicate a lower thermal
conductivity than the lower slope observed for the silicon sample (Figure 17).
For the silicon sample, the 3m voltage follows a straight line over the entire
measured frequency range (on a logarithmic scale), allowing one to use
frequencies up to 10 kHz for the slope model. For the glass samples, the
frequency at which the 3m voltage departs from linearity depends on the metal
line width 2b. For a line width of 5 pm the upper frequency limit is about 1 kHz,
while for a 20 pm wide line, this limit is reached at about 200 Hz. This confirms
that the linear range of the temperature oscillation is extended to higher
frequencies by using a thinner line width 2b.

In a second step, the application of the slope model was compared to the fit
model. Using equation 20, the 3m voltage was converted to a temperature
oscillation. The resulting temperature oscillation plots for Pyrex, soda lime and
silicon were fitted to equation 34 to determine the thermal conductivity values, K,
(Figure 18, Figure 19 and Figure 20). The measured and fitted data agree well
over the entire frequency range for both the thermally low conductive glass
materials possessing a non-linear behavior of the temperature oscillation as a
function of the logarithmic frequency (Figure 18, Figure 19), as well as the higher

thermal conductive silicon with its linear temperature oscillation (Figure 20).

64

1.25 T tTI’WTT' I I IIIIUI' I I U'IIII'

 

 

 

 

 

 

 

 

3 Pyrex [:1 2b: 5 pm 1

A KER 0 2b =10 pm
29 1.00;- :gsg A 2b=15pm -
g . a} mix 1: 2b .-. 20 pm .
E O 75 _ \‘ \A:§8\ """ fit ""35 .1
7'3 ' \*\ ”9&3
m T *\ A‘ojs‘
O fitk‘Ap‘n‘
a) 1. \ A\\O\S‘ -1
g 0.50 *\Q \A\O\F\S
E ' RNA; 6 8‘18
(D *\ ‘A\® \EL
0' O 25 - “K ‘ei‘e ‘51 -
g . *‘*\\A:&\Q\B“Q
l- &‘«Z%E§E§;g:~u;§

000 1111111 11! 1 11 11 1 1-1 15?”

10° 101 1o2 103 104 105
Frequency (Hz)

Figure 18. Fit model applied to 300 measurements of Pyrex. The temperature
oscillation, which is measured with different heater line widths 2b, is plotted as a
function of frequency for soda lime and fitted to equation 39. The fitting plots for
each line width are represented by dotted lines. In comparison to Figure 15 the
entire frequency range can be used for data fitting. (For better clarity, only every
fourth data point is included in this plot.)

65

 

 

 

 

 

 

 

 

1 2 (‘3:0‘ Soda Lime 0 2b = 5 pm
9 “38:3 o 20 = 10 pm
: fi.*:1a‘ngp A 2b = 15 pm
0 *~,,“A P~ 1* 2b = 20 pm
:7 ‘xkns 6: .
m “A, B: ---------- fit lines
E 0 8 - ‘31, ‘A 8n -
0 \fi ‘A‘ ‘08“
8 “11' A1 “gm

\fi~AA “0:0.
2 ‘fifi ‘A‘ ~O‘U1n
a \fi‘A‘ Q‘OU
g 0 4 - fixifino PD ‘
Q *‘sfiA‘A‘Q‘O 0‘11
GE) 1 “£340,031; 1:1
.‘ ~ ‘0~ ‘U

r— 539 Q . 43-13..

0 0 1 1 1 11111L 1 1 1 111111 1 1 1 111111 1 1 1 1111ffi1§ 112891913

10° 101 102 103 104 105

Frequency (Hz)

Figure 19. Fit model applied to 30) measurements of soda lime. The temperature
oscillation, which is measured with different heater line widths 2b, is plotted as a
function of frequency for soda lime and fitted to equation 39. The fitting plots for
each line width are represented by dotted lines. In comparison to Figure 16 the
entire frequency range can be used for data fitting. (For better clarity, only every

fourth data point is included in this plot.)

66

1.5

 

 

 

 

 

Temperature oscillation (K)
O
01

s
D
E
D
E
E
f
D
f
E

 

Silicon 0 2b = 5 pm
1 o 2b=10pm,
A 2b = 15 urn
1: 2b = 20 m
1-0 ’ --------- 9988129118 l --------- fit p “‘
88899558813 a
........... BmGG-GBBGGBGBBHHN

 

 

Frequency (Hz)

Figure 20. Graphs of the temperature oscillation for silicon determined from the

data shown in Figure 17 and equation 20. The plotted data were fitted to
equation 39 to determine the thermal conductivity of silicon. The thermal

conductivity values are included in Table 4.

67

Table 4 summarizes the thermal conductivity values determined by the slope and
fit method are summarized showing the determined thermal conductivities for the
four different line widths investigated for Pyrex, soda lime and silicon. Both the
slope and the fit method show only minor deviations in the resulting thermal
conductivities K of the same sample. However, the thermal conductivities
calculated using a line width of 5 pm is slightly higher than the average value for
each of the specimens. In general, soda lime has the lowest thermal conductivity
value of 1.13 W/(m K) for the slope method and 1.17 W/(m K) for the fit method.
Both values agree well with reference values for soda lime of K = 1.1 W/(m K)
[29]. Pyrex possesses a slightly higher thermal conductivity (1.31 W(m K)") as
determined by the slope method compared to 1.36 W/(m K) determined from the
fit method. Reference values for the thermal conductivity of Pyrex range from
1.25 W/(m K) [29] to 1.4 W/(m K) [18], which agrees well with the thermal
conductivity values determined in this study. The observed thermal conductivities
for silicon of 131 W/(m K) for the slope method and 130 W/(m K) for the fit
method are only slightly below reference values of 141 W/(m K) [18] and 148
W/(m K) [29]. For silicon, the difference between the literature values and the
values determined in this study might be caused by the doping level of the silicon
specimen that was measured in our study. Doping can cause an increased
phonon scattering by impurity atoms resulting in a lower thermal conductivity

value [12].

68

Table 4. Comparison of the thermal conductivities K calculated by the slope
method (equation 34) and fit method (equation 39) for Pyrex 7740, soda lime,
and silicon for different heater line widths 2b. The given uncertainties are
calculated by the error extension law by using the values of Table 3 and an
additional 2% error for the line width 2b for the fit method. The coefficient of
determination R2 measures how well the data are described by the regression or
fitting curve. R2 makes no conclusions about the error of the thermal conductivity.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Slogg method Flt method
”"e ”at" K (w m'1K“) n2 K (w 111'1 K") n2

Pyrex 7740 5 pm 1.48 1 0.09 0.9997 1.64 1 0.08 0.9989
10 pm 1.31 1 0.07 0.9999 1.35 1 0.08 0.9994
15 pm 1.29 1 0.08 0.9994 1.31 1 0.09 0.9996
20 pm 1.33 1 0.09 0.9997 1.41 1 0.11 0.9990

average 1.31 1 0.02 1.36 1 0.05
Soda Lime 5 pm 1.37 1 0.07 0.9999 1.39 1 0.09 0.9989
10 pm 1.12 1 0.06 0.9999 1.15 1 0.08 0.9990
15 pm 1.13 1 0.06 0.9999 1.17 1 0.08 0.9995
20 pm 1.14 1 0.07 0.9998 1.18 1 0.08 0.9913

average 1.13 1 0.01 1.17 1 0.02
Silicon 5 pm 146 1 6 0.9997 139 1 8 0.9983
10 pm 11513 0.9999 11415 0.9997
15 pm 11914 0.9995 11616 0.9986
20 pm 144 1 4 0.9988 150 1 7 0.9994

average 131116 130118

 

 

 

 

69

 

5.2 Measurements of silica films on silicon

Four silicon wafers were oxidized to create thin SiOz films of different thicknesses
tvarying from 0.3 to 3.2 pm. Table 5 shows the parameters used for thermal
oxidation and the resulting film thicknesses measured by a laser acoustic wave
method (for further details please see Appendix A).

Each sample was supplied with metal patterns of line widths 5, 10 and 15 pm.
The 30) voltage was measured in a frequency range of 500 Hz to 300 kHz.
Values for the power supplied to the metal line, the resulting sample temperature
and the temperature coefficient of the metal line’s resistance (determined prior to
the 30.) measurements) are included in Table 6. The input power for measuring
thicker Slog films was comparable to the input power used to measure the glass
samples. For the thinnest film of 0.3 pm, a power input level similar to that used
for pure silicon was necessary. The mean temperature coefficient of the heater
line’s resistance is 0.0037 1 0.0003 K". Again, the sample temperature during 30)

measurements was slightly above room temperature at 25 — 35 °C.

70

Table 5. Matrix of the process parameters used to prepare silicon dioxide films
on silicon by thermal oxidation

 

 

 

 

 

Sample Furnace Water Oxidation SiOz Film
temperature temperature time thickness

1 800 °C 22 °C 4 hrs 0.3 pm

2 800 °C 76 °C 19 hrs 1.2 pm

3 800 °C 76 °C 52 hrs 2.3 pm

4 1000 °C 76 °C 77 hrs 3.2 pm

 

71

Table 6. Electrical and thermal parameters and their standard deviations for the
300 measurements of silicon dioxide films on silicon. These parameters were
subsequently used to determine the thermal conductivities at varying line widths
2b. From the voltage drop V1,», measured parallel to the voltage pads, and the
current across the line measured in series (not shown in the table), the power per
unit length P/l is calculated. To determine the temperature Tof the specimen, the
resistance of the metal line is measured twice: first with a small voltage to avoid
heating of the sample and the second time with a voltage V1,, used for 30)
measurements. By knowing the temperature coefficient of the resistance a, the
temperature Tof the sample during measurement can be determined.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

:3: 11115121215 v11. (mV) P/I (W/m) or (K") T (°C)
5 pm 0.3 pm 903 1 1 29.9 1 0.1 0.0034 1 0.0001 32 1 1
1.2 pm 601 11 12.9 1 0.1 0.0037 1 0.0001 26 1 1
2.3 pm 533 1 1 7.6 1 0.1 0.0042 1 0.0001 25 1 1
3.2 pm 524 1 1 8.6 1 0.1 0.0036 1 0.0002 25 1 2
10 um 0.3 pm 776 1 1 37.8 1 0.1 0.0033 1 0.0002 32 1 2
1.2 pm 721 1 1 27.0 1 0.1 0.0037 1 0.0001 34 1 1
2.3 pm 515 1 1 14.5 1 0.1 0.0041 10.0001 29 1 1
3.2 pm 550 1 1 16.5 1 0.1 0.0037 1 0.0001 34 1 1
15 pm 0.3 pm 733 1 1 43.1 1 0.1 0.0035 1 0.0001 32 1 1
1.2 pm 698 1 1 37.0 1 0.1 0.0037 1 0.0001 33 1 1
2.3 pm 586 1 1 21.0 1 0.1 0.0036 1 0.0001 32 1 1
3.2 pm 557 1 1 21.5 1 0.1 0.0036 1 0.0001 35 1 1

 

72

Figure 21, Figure 22 and Figure 23 show the resulting temperature oscillation for
films measured with the same heater line width 2b. All three plots show the
general curvature of a two-layer system, where the top layer exhibits a lower
thermal conductivity than the substrate. Starting at high frequencies the damping
distance of the thermal wave is short preventing the heat to diffuse into the
silicon substrate. The measured 30) voltage due to the silica layer is not
influenced by the thermal properties of the silicon because the silicon does not
“see” the heat wave supplied by the thin metal line. Therefore, the curvature of
the temperature oscillation plots for high frequencies appears to be similar to the
curvature of the plots for the glass samples. When the frequency gets low
enough for the thermal wave to penetrate the entire 8102 film into the silicon
substrate, the temperature oscillation follows the flat curvature of uncoated
silicon. However, it is shifted by an offset ATM”, that is a function of the film
thickness land the film’s thermal conductivity. By applying the offset model
(section 3.4) and equation 38 the thermal conductivities of the four thin film
samples were determined, along with the dependence on the line width 2b (Table
7). Thus, the thermal conductivity is independent of the film thickness f, but there
seems to be an influence of the line width, since the values of the thermal
conductivity decrease slightly with increasing line width. According to Lee [24] the
reason might be that the offset model assumes one-dimensional heat flow
requiring a heater width 2b much greater than the film thickness 1. In our case we
might have reached the limit of this assumption by using line widths 2b

comparable to the film thicknesses t. Despite these small deviations, the offset

73

model gives an average thermal conductivity of 1.55 W/(m K) for the silica films
on silicon, which agrees well with reference values of 1.4 - 1.5 W/(m K) [18, 29].
The thermal conductivity value obtained by the offset method could be confirmed
by fitting the upper frequency region contributing solely to the SiOz layers to
equation 39 yielding in a mean thermal conductivity of 1.6 W/(m K) (see Table 7).
The fit method can be applied only to the thickest silica layer of 3.2 pm. No
unambiguous solution was found by trying to fit films thinner than 3.2 pm.

However, the fit curvature appears to fit the thinner films visibly.

74

5 pm metal line

 

 

    

 

 

 

 

 

 

 

 

 

 

..., . -...... . -f
U t=3.2 pm

A 0.20 mqpmmuumg-dééba ...................... 0 t=2.3 pm

£ 0:100 A t=1.2 pm

5 i qbo w t=0.3 pm

0 DO ............ '

.fi 0 15 _ COGOGW‘OWOUOOOQOOOO .. .......... fit 31.02

= '-'°fitSl|lCOl1

'6

8

m 010 _ MLLAAAI)--A‘A‘AA“

h

3

g

a 0.05 -

«E,

'— NTMIW *‘k W *mfi'fi'flllllld
0.00”“: n." """ . t. ----------- . -------

103 1o4 105
Frequency (Hz)

Figure 21. Plot of the temperature oscillation (power normalized) for silicon
dioxide films of thickness tmeasured with a 5 pm wide heater line.

The low frequency region is affected by the silicon substrate. Thus, the offset-
model can be applied by using the temperature oscillation offset A'I'mm (denoted
by arrows). The data for higher frequencies includes contributions only from the
Si02 films and is used to fit measured data to equation 39 (solid line).

75

10 um metal line

 

 

 

 

 

 

r 1 TI r iv?
o . . . D t: 3.2 pm
m ....................
A qUCmDDD CU DDDmD ................. O t: 23 pm
19010. A ”1'2”"
8 ° ‘ * 2:39: pm
2: Dooooooqgooooooooo I 2
g 00000qu . """""" — - - -fit silicon
._ F o
O 00
m ,
O
0
:3: 0.05 -
g.
E
G.)
.—
0.00

 

 

 

 

Frequency (Hz)

Figure 22. Plot of the temperature oscillation (power normalized) for silicon
dioxide films of thickness tmeasured with a 10 um wide heater line.

The low frequency region is affected by the silicon substrate. Thus, the
offset-model can be applied by using the temperature oscillation offset
ATmm (denoted by arrows). The data for higher frequencies includes
contributions only from the SiOz films and is used to fit measured data to
equation 39 (solid line).

76

15 um metal line

 

 

 

 

 

0,10 ...., . .. ...., . .f
m m: ............... D t: 32 um
WU DUDmDUDm DU .................... O t: 23 pm
”on A t=1.2 um
f ‘ w t= 0.3 pm

fit SiO2
mobmofiooooooooo-Ooowda) ............. - - -fit silicon:
0.05 - 000 a

  

 

 

Temperature oscillation (K)

 

 

 

000 ....n . . ......|

 

Frequency (Hz)

Figure 23. Plot of the temperature oscillation (power normalized) for silicon
dioxide films of thickness tmeasured with a 15 um wide heater line.

The low frequency region is affected by the silicon substrate. Thus, the offset-
model can be applied by using the temperature oscillation offset AT)”,n
(denoted by arrows). The data for higher frequencies includes contributions
only from the Si02 films and is used to fit measured data to equation 39 (solid
line).

77

Table 7. Comparison of the thermal conductivities K determined by the slope
method (equation 34) and fit method (equation 39) for silica layers of different
thicknesses tmeasured with a series of heater line widths 2b.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Line Film 89‘ "‘0'" ———F" meth°d
width thickness K (W m-tK-t) ‘K (W m-tK-t) R2
5pm 03pm L7102

12pm L5102

23pm L6102

320m L7104

average 1.63 :1: 0.15 1 .43 0.999
10wn 08pm L7t02

12pm 15:02

28pm L6102

32pm 15:02

mmmga 158:0ul La; can
15wn 03pm L6¢02

12pm L4¢02

23pm L5102

32pm t3102

average 1.45 :l: 0.18 1.78 0.990

 

 

 

 

 

 

78

5.3 Application of the 30 method for thin diamond films

Materials that are able to dissipate excess heat are of great interest especially for
the integrated circuit industry. Diamond, possessing an outstanding thermal
conductivity of up to 2300 W/(m K) at room temperature, has gained special
attention since scientists and engineers developed techniques to manufacture
artificial diamond [30, 31]. Depending on the method, diamond can be fabricated
in form of single crystals by a HPHT process (high pressure high temperature) or
diamond can be deposited in form of thin films by CVD techniques @hemical
yapor deposition), whereby films exhibit grain sizes ranging from hundreds of
microns down to a few nanometers.

The thermal conductivity of primary single diamond crystals and free standing
polycrystalline membranes has been measured by heated bar and laser flash
techniques [8, 10, 32, 33, 34, 35, 36, 37]. In addition, the 3m method has been
used to measure natural or isotropically enriched diamonds [27, 38, 39]. A broad
spread of room temperature thermal conductivity values can be found in the
literature, ranging from 2265 W/(m K) for natural diamond [34] to the probably
highest reported value of 3320 W/(m K) for isotropically enriched 12C diamond
[36]. For polycrystalline diamond films, the literature data ranges from 140 W/(m
K) for 100 um grain size [10], to 700 W/(m K) for a grain size of 2 pm [40] and
320 W/(m K) for 150 nm grains [32].

An undeniable dependency between thermal conductivity and grain size of

diamond films exists. Since grain boundaries are areas with a high concentration

79

of lattice distortions, phonons experience a resistance at grain boundaries, when
travelling through a crystalline solid. This thermal resistance increases with an
increasing number and area of grain boundaries (i.e. smaller grain sizes). That is
why, the thermal conductivity of diamond films is expected to decrease with
smaller grain size.

As a next step, it would be an interesting challenge to investigate the relationship

between grain size and thermal conductivity of diamond films by the 300 method.

80

6 Summary and Conclusion

A measurement method to determine the thermal conductivity based on the four-
wire 30) technique was successfully built up. An experimental apparatus was
assembled from a commercial probe station and differential amplifiers. Samples
were supplied with a thin metal structure using a lithographic process for masking
and a high vacuum electron beam deposition system for metallization. The final
patterns made of a chromium silver double-layer consisted of a thin metal line
functioning both as heater and thermometer and four electrode pads for
contacting.

By supplying a current of angular frequency to the heater line, a thermal wave is
created which diffuses into the underlying material. The damping distance (or
penetration depth q") of the thermal wave is a function of the excitation
frequency and the thermal conductivity of the sample. .

The parameter q" defines the probing depth, which should always be smaller
than the sample thickness t.

Related to the thermal penetration depth q", the sample thickness tand the
heater width 2b, different approaches to extract the thermal conductivity from
measurement data were tested.

The data analysis included the slope model for thick samples explored by Cahill
and co-workers. The slope model is valid as long as the sample thickness is

much larger than the heater line width. In this case, the temperature oscillation

81

versus logarithmic frequency can be approximated by a linear function. In
general, the slope model is used to measure the thermal conductivity of uncoated
bulk materials.

The thermal conductivity of very thin coatings can be calculated by the offset
model. To use the offset model, the film thickness has to be small compared to
the heater line width. In addition, the thermal conductivity of the coating must be
much smaller than the thermal conductivity of the substrate.

The main drawback of the slope model and offset model is their restrictions to
either very thick or very thin films, respectively. In the medium range, where
coatings exhibit thickness values comparable to the width of the heater line, the
thermal conductivity cannot be calculated with those models.

To fill that medium range, calculating thermal conductivities by the integral
formula was introduced. The integral formula can be applied to thin films
exhibiting a thickness of a few micrometers and is not limited by a maximum film
thickness. In contrast to the offset model, the integral formula is not restricted to

films of low thermal conductivity values.

82

7 Outlook

The 300 method was sucCessfully extended to high frequencies of up to 300 kHz
by assembling an experimental apparatus. Re-designing that apparatus by
decreasing the size of the probing station and drastically decreasing the length of
the cables that are directly connected to the electrode device can help to further
extend the frequency range. Especially for measuring materials with a high
thermal conductivity, measurements under vacuum and temperature stabilization
are suggested to enhance the stability and reproducibility of the recorded 3a)
voltages.

An interesting challenge would be the measurement of diamond films, because
diamond possesses an extremely high thermal conductivity of up to 2300
W/(m-K). Diamond films can have grain sizes from some hundred micrometers to
a few nanometers. Since smaller grain sizes result in a larger grain boundary
area, phonon scattering at grain boundaries is supposed to increase at
decreasing grain size. Measuring a series of diamond films and relating
measured thermal conductivity values to film properties like the grain size could

be an interesting and challenging task.

83

APPENDIX A

Measuring the film thicknesses of silica films by laser acoustics

Laser acoustics [42, 43] were used to determine the film thickness of the silicon
dioxide films. Although originally developed to determine the Young’s moduli of
thin and hard coatings, it can be extended to measure film thicknesses if film
parameters such as density and Young’s modulus are known. Laser acoustics
offer the advantage of being non-destructive and especially suited to characterize
coatings as thin as a few nanometers.

Laser-induced acoustic waves propagate along the surface with a velocity that
depends on the elastic properties of the material. In a homogeneous isotropic
material the surface acoustic wave (SAW) velocity is constant, while in coated
materials the velocity depends on frequency. This frequency dependence is
caused by the skin effect resulting in a reduced SAW penetration depth with
increasing frequencies. Hence, surface waves with higher frequencies are more
influenced by thin films, in a manner similar to the principle of the 300 method.
The dependence of the SAW velocity on the frequency can be plotted as a
dispersion curve. The shape of that curve depends on film parameters, i.e.
Young’s modulus, density, and thickness. The dispersion data is fit to a
theoretical curve which allows one to determine the values of the Young’s

modulus, film thickness and density.

84

For the SiOz films fabricated in this study the dispersion curves follow a straight
line for frequencies up to 200 MHz (Figure 24). Therefore, the film thickness of
the silica films can be determined directly from the slope of the phase velocity as

a function of frequency (Figure 25).

85

phase velocity (103 m/s)

9‘
o

:5
on

P
o:

 

I

I

 

(0)
\
N‘ a)

 

““-a>

‘*, ‘"‘(3)
‘700

 

 

160 200
frequency (MHz)

300

Figure 24. Dispersion curves measured by laser acoustics: Phase velocities for
the four SiOz films on silicon (Table 5). The plot marked (0) refers to an uncoated

silicon substrate.

SiO2 film thickness t (um)

 

l-

 

f r . , .
I measured slope
. . linear regression

‘4l.‘

(3i,~"”
(2)__

(1),-

..a'

l

 

 

kcis "fio A 135 gzio ‘ 2:5

absolute slope (10‘6m)

Figure 25. Due to the linear behavior of the phase velocity versus frequency in
Figure 24, the slopes of the plots were used to estimate the film thickness tof the
silica films. A Young’s Modulus of 69 GPa and a density of 2200 kg/m3 were
assumed for silica in these calculations.

86

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