measuring thermal conductivity of thin films

2015

Measuring Thermal Conductivity of Thin Films

MECHANICAL ENGINEERING FINAL YEAR PROJECT

KALE CROSBIE

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Table of Contents

Acknowledgements ....................................................................................................................................... 3

Abstract ......................................................................................................................................................... 4

Letter of Transmittal ..................................................................................................................................... 5

Introduction .................................................................................................................................................. 6

What is Thermal Conductivity? ............................................................................................................. 6

Project Aims .......................................................................................................................................... 6

Porous Silicon ............................................................................................................................................ 6

Manufacturing Porous Silicon Films ...................................................................................................... 7

Variable Properties ............................................................................................................................... 7

Thermal Conductivity Measurement Methods......................................................................................... 8

The Absolute Plate Method .................................................................................................................. 8

Time Domain Thermo-Reflectance ....................................................................................................... 9

The 3-Omega Technique ....................................................................................................................... 9

Comparison of Each Measurement Technique ................................................................................... 10

Methodology of the 3-Omega Technique ................................................................................................... 10

Application to Thin Films..................................................................................................................... 13

Samples Fabricated ..................................................................................................................................... 14

Sample 1 .............................................................................................................................................. 14

Sample 2 .............................................................................................................................................. 14

Resistance of the Metal Heater .................................................................................................................. 17

Theoretical Values- Gold/Chromium .................................................................................................. 17

Theoretical Value – Copper ................................................................................................................. 18

Measured Sample Resistances ............................................................................................................ 18

Temperature Coefficient of Resistance .............................................................................................. 19

Experimental Design ................................................................................................................................... 20

Lock-in Amplifier ................................................................................................................................. 21

Function Generator ............................................................................................................................. 21

Measurement Steps ............................................................................................................................ 22

Sample 1 Results ......................................................................................................................................... 22

Experiment 1 ....................................................................................................................................... 22

Experiment 2 ....................................................................................................................................... 23

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Results ................................................................................................................................................. 23

Improvements to be made on Experimental Design .......................................................................... 24

Noise from Resistors ........................................................................................................................... 25

Geometric Sample Considerations ..................................................................................................... 26

Sample 2 Results ......................................................................................................................................... 27

Sample 3 Results ......................................................................................................................................... 28

Conclusion ................................................................................................................................................... 29

Recommendations for Future Work ....................................................................................................... 29

Wheatstone Bridge ............................................................................................................................. 29

Low TCR Resistors ............................................................................................................................... 30

Use of a Vacuum Chamber.................................................................................................................. 30

References .................................................................................................................................................. 31

Appendices .................................................................................................................................................. 32

Sample 1 Data ......................................................................................................................................... 32

Sample 2 Data ......................................................................................................................................... 33

Sample 3 Data ......................................................................................................................................... 33

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Acknowledgements

It has been a great experience working together with my supervisors Professor Adrian Keating and

Professor Giacinta Parish this year on this project. Not only did they encourage me to push myself and

engage with the project in a friendly environment, but gave me new technical and analytical skills that

will help me throughout my life.

I’d like to also thank the other students who I’ve worked with this year, and in particular Xiao Sun who

has helped me in fabricating samples.

Last but not least, I’d like to thank my girlfriend Mayan and my friends and family for putting up with me

while undertaking this difficult project.

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Abstract

This study investigates modern methods of measuring thermal conductivity and its application to new

thin film materials such as porous silicon. As the thermal conductivity of a sample is found to be a

function of both characteristic length and material properties, traditional macroscopic scale

measurements cause large errors when applied to these thin films. New innovative techniques such as

the 3-Omega Technique or Time Domain Thermoreflectance are required to measure the thermal

properties of materials. Throughout the course of this year I have analysed current papers, journal

articles and reports on the subject of the 3-Omega Technique to design a measurement system within

the Optics Laboratory with the Microelectronics Research Group. I have manufactured several samples

using shadow masks which I designed where parts were subsequently produced by thermodeposition by

a PhD student (Xiao Sun). These samples were used for testing the validity of the 3-Omega experimental

setup I have created. New innovations are constantly happening within the field thermal

characterisation and there are many sources of potential experimental problems to be aware of when

designing such an experiment, so this project should serve as a guide to future students undertaking

thermal conductivity measurements. Values recorded of the thermal conductivity of FR4 fiberglass

ranged from 0.0083 to 0.007 W/mK (compared to the accepted value of 0.04 W/mK) and thermal

conductivity of glass was calculated to be 0.0037 to 0.00456 W/mK (compared to the accepted value of

0.8 W/mK). The cause of the difference is discussed within the work.

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Letter of Transmittal

Kale Vernon Crosbie

8 O’Hara Court

Greenwood, WA, 6024

23rd October, 2015

Winthrop Professor John Dell

Dean

Faculty of Engineering, Computing and Mathematics

University of Western Australia

35 Stirling Highway

Crawley, WA, 6009

Dear Professor Dell

I am pleased to submit this thesis, entitled “Measuring the Thermal Conductivity of Thin Films”, as part

of the requirement for the degree of Bachelor of Engineering.

Yours Sincerely,

Kale Vernon Crosbie

20929599

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Introduction

What is Thermal Conductivity? Thermal conductivity is a measure of how easily heat can flow through a material and is commonly

expressed in the units W/m.K. This material property is poorly understood in new materials and

particularly amongst thing films, which have thermal properties which differ from their bulk material

values. These thin films often require complex measurement techniques to correctly ascertain thermal

properties.

Project Aims The goal of this project is to create an experimental setup which can accurately measure the thermal

conductivity of thin films and bulk substrate materials. This is to be achieved by first investigating

different techniques available to measure thermal conductivity and the experimental design used by

others. If the technique is successful then it can be used to measure the thermal properties of porous

silicon films to further understanding of the material for the wider research community. Ongoing efforts

to manufacture micro-electromechanical systems (MEMS) from porous silicon have been hindered by

stresses within the structures caused by thermal expansion and oxidization amongst other sources. By

better understanding the thermal properties of porous silicon some of these stresses and failure modes

can be better understood and rectified. Thin films are used in a variety of fields from medical research,

semiconductor devices, optical coatings and laser technologies which will all benefit from the study of

thin films. The mechanical and thermal properties of thin films determine their suitability for each

application, for example the development of solar cell technology which is limited by thermal

conductivity in its ability to efficiently transform thermal energy into electrical energy.

Porous Silicon Silicon is the 2nd most common element in the Earth’s crust by mass and is used extensively in the

manufacturing of electrical systems as a semiconductor and in the materials sector to create iron and

aluminium alloys. Clearly silicon is a very important material in the modern world and understanding its

properties along with porous silicon will unlock new potential technologies.

While the application of this technique to porous silicon was not achieved, the 3-Omega technique is

suitable for measuring its thermal conductivity as the technique can be modified to measure thin films.

Future work will be able to take the lessons learned from this and other projects to apply to a

comprehensive measurement setup for porous silicon.

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Fig 1. Electrochemical Etching (Anrushin, 2005)

Manufacturing Porous Silicon Films Porous silicon is made via electrochemical etching of silicon in a hydrofluoric acid solution. This causes

the dissolution of Si particles as pores grow within the silicon. The level of porosity (defined as a

percentage of the removed mass from the initial mass) is controlled by current density throughout the

silicon and time spent in the solution. The porous silicon is then removed from the solution and several

processes are possible to stabilize the structure such as adding layers of oxidation or photoresist.

Variable Properties Many properties of porous silicon are altered as the porosity changes. If the mechanisms of these

changes are explored then porous silicon could be used to customize material properties based on

design requirements. Some of the interesting changes that porous silicon undergoes are:

Bioactivity- biological processes such as hydroxyapatite growth have been shown to occur on

porous silicon. (Canham, 1995)

Superhydrophobicity- pore morphology and geometry can control the wetting behaviour of

porous silicon. (Ressine, 2007)

Optical Properties- the refractive index is controlled by the refractive index of the medium

within the pores along with porosity.

Thermal Conductivity- As the porosity increases, conduction within the material is steeply

reduced.

Measuring the thermal conductivity of porous silicon under different porosity and geometrical

conditions is the motivation behind this project, which is primarily focused on the validation of the

3-Omega Technique. First of all, I will examine the traditional and innovative techniques for measuring

thermal conductivity and ascertain the problems in thin film thermal measurement.

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A = Cross sectional area (m2)

dT = Temperature difference (K)

L = Length (m)

Fig 2. Lattice Vibrations (Marquardt, 1996)

Thermal Conductivity Measurement Methods Traditional measurements using Fourier’s law of heat

conduction have been an ideal method of describing and

analysing heat flow on the macroscopic scale since the

1800s. However, the development of commercially viable

nano and microscale structures has precipitated the need for

more robust thermal characterization techniques. Non-

metals transfer heat throughout a material via the transport

of ‘phonons’, a discrete unit of vibrational mechanical

energy. These phonons are observed as lattice vibrations

which travel through a structure. These lattice vibration

interact with physical structures such as grain boundaries or

material surfaces which have an effect on the flow of heat

energy.

These interactions can cause unexpected results as the dimensions of a sample decrease. When the

thickness of a sample is approximately the same distance as the mean free path of a phonon (the

average distance between interactions) these interactions with boundaries will have a large impact on

the thermal conductivity (Zhang, 2007). The understanding of thermal properties in thin film materials

such as porous silicon requires new measurement techniques and models more complex than

traditional macro-scale conductivity measurements.

The Absolute Plate Method Typically, the thermal conductivity of thermally insulating solid specimens is measured using the

absolute plate method (Touloukian, 1973). In this method a heat source is applied to one side of a

material and waits for the system to reach steady state. At this point the thermal conductivity can be

found by using Fourier’s Law:

𝑑𝑄

𝑑𝑡=

𝑘𝐴𝛿𝑇

𝐿 𝐸𝑞𝑛 (1)

However if the thermal conductivity (k) is low (<5W/mK), then the time taken for the system to reach

steady state can increase up to several hours. This will cause heat to enter the system via radiation

which will cause inaccurate data. If the temperature difference (dT) is made to be very small to decrease

this equilibrium time, then the error bars from the temperature reading will result in a very low

precision result and the thermal conductivity will be poorly defined.

Q = Heat (J)

t = time (s)

k = Thermal conductivity (W/mK)

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Time Domain Thermo-Reflectance This method measures the thermal conductivity by analysing the change in reflectance of the surface,

which is a function of temperature. This method is typically used to measure thin films up to a few

hundred nanometers thick. The experimental setup consists of a pulsed laser beam which is focused

onto the surface to create localized heating in the material. This change in temperature induces a

thermal stress within the material, causing acoustic waves to be generated. These waves are analysed

by a second probe laser which uses the piezo-optic effect, a mechanism which causes a change in

refractive index of a material due to a change in pressure (Vedam, 1975). The experimental setup of this

technique is shown in figure 3.

This method provides a similar level of accuracy as other modern techniques, however due to the much

more complicated experimental setup was dismissed as a possible method to use for this project.

The 3-Omega Technique This method differs from the conventional absolute plate method as it is a transient measurement,

meaning that the system does not reach steady state and is in constant change. The benefits of this

technique are the short time required for measurement (less than one minute if the experiment is

computer controlled) and a higher precision than the absolute plate method in the measurement of

thermal conductivity due to lower exposure times to radiation. The experimental technique which was

employed is used to measure the thermal conductivity of bulk substrates (as opposed to a thin film of

porous silicon), but the principles are similar.

Fig 3. TDTR Experimental Setup (McLaren, 2009)

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Table 1. Comparison of Conductivity Measurements

Comparison of Each Measurement Technique After each technique was researched I compiled a table comparing their strengths and weakness to

identify the suitability of each to my final year project.

Technique Application to thin films

Application to bulk substrates

Complexity Level Equipment readily available

Absolute Plate No Yes Low Yes

TDTR Yes No Very High No

3-Omega Yes Yes High Yes

Comparison of the various factors in the table show that the 3-Omega Technique was the only suitable

option to further investigate due to its applicability to both thin films and bulk substrates, as well as the

relatively low requirement for additional materials to be obtained.

Methodology of the 3-Omega Technique

One important variable which makes this method possible is β – the Thermal Coefficient of Resistance

(TCR). This variable describes the rate at which electrical resistance is increased as temperature

increases. Pictured to the right is the resistance vs temperature graph for gold, which has a relatively

high gradient, and therefore high TCR value. Materials with high TCR values should be selected as the

measured voltage is proportional to this value.

Fig 4. Resistance vs Temperature of Gold (Hanninen 2013)

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r = distance from sample centre (m)

x = horizontal distance (m)

y = vertical distance (m)

The material to be measured has a thin metal line which is evaporated onto a substrate as shown in

figure 5. Once this line has been connected to circuit the metal line has an AC current applied to either

end at a fundamental frequency of ω. By applying this current, heat flows radially out from the line into

the substrate.

The analytical solution of the temperature oscillations at a distance of:

𝑟 = (𝑥2 + 𝑦2)12 𝐸𝑞𝑛 (2)

Has been shown to be (Jaeger, 1959):

∆𝑇(𝑟) = (𝑃

𝑙𝜋𝑘) ∗ 𝐾0(𝑞𝑟) 𝐸𝑞𝑛 (3)

The important factor in this equation is the P/l value, which is the magnitude of the power per unit

length generated at a frequency of 2ω in the metal heater. The frequency change from the initial omega

value results from the fact that an electrical signal at frequency ω causes joule heating at a frequency of

2ω (Cahill D. G., 1990).

Fig 5. Side view of heater and substrate geometry

P = Applied power (W)

l = Heater length (m)

k = Substrate thermal conductivity (W/mK)

K0 = The zeroth order modified Bessel Function

1/q = Thermal penetration depth (complex quantity)

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Fig 6. Temperature Oscillation Magnitude vs

Frequency (Cahill D. G., 1990)

Recalling the importance of the β value of TCR; as the metal heater oscillates in temperature at a

frequency of 2ω, so does its electrical resistance. Given that a current of constant amplitude I(ω) and

fundamental frequency ω is applied to the heater, the voltage due to this resistance oscillation can be

found through Ohm’s Law:

𝑉3ω = 𝐼(ω) × 𝑅(2ω) 𝐸𝑞𝑛 (4)

This relationship is results in the small voltage signal at the 3rd harmonic of the fundamental frequency,

but in reality the temperature oscillations of the heater are more complicated and have both a

component that is in-phase with the applied current and an out of phase component. It is however only

the in-phase component of temperature fluctuation that is of interest, as it is related to the thermal

conductivity.

Shown in figure 6 is the plot of both the in-phase and out-of-phase components of the temperature

oscillations. The slope of in-phase component against the logarithm of heater frequency gives the

thermal conductivity (k) of the substrate

(Cahill D. G., 1990).

Given that the magnitude of the in

phase temperature oscillations are a

function of the 3-Omega voltage:

∆𝑇 = 4𝑑𝑇

𝑑𝑅(

𝑅

𝑉) 𝑉3𝑤 𝐸𝑞𝑛 (5)

This relationship implies that while the

slope of the in-phase temperature is

required for calculating the thermal

conductivity, it can be inferred from a

series of 3rd harmonic voltage

measurements over the same

frequencies.

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The thermal conductivity can then calculated from any two points on the above graph or two 3rd

harmonic voltage measurements and is given by:

𝑘 =𝑉3 ln (

𝑓2𝑓1

) 𝑑𝑅/𝑑𝑇

4𝜋𝑙𝑅2(𝑉3𝑜𝑚𝑒𝑔𝑎,1 − 𝑉3𝑜𝑚𝑒𝑔𝑎,2) 𝐸𝑞𝑛 (6)

Note that all voltages are RMS values.

Application to Thin Films It is important to note that the above methods are used to measure the thermal conductivity of a

substrate on which the metal heater has been applied. This method is very applicable to measuring thin

films by the addition of a thermal resistance independent of driving frequency (Cahill D. , 1997).

In this case the thin film to be measured is laid onto a substrate of known thermal conductivity and

thermal diffusivity such that its own thermal properties can be inferred from the measured 3rd harmonic

voltage. In one paper, “Implementing the 3-OmegaTechnique for Thermal Conductivity Measurements”

(Hanninen, 2013) the effect of a thin film is added as a thermal resistance independent of the

fundamental frequency. This thermal resistance will affect the flow of heat from the metal heater, and

hence the oscillating resistance and 3rd harmonic voltage of the metal heater. By first understanding the

application of the technique to bulk substrates, experimental errors can be reduced once the technique

is ready to measure thin films without dealing with additional errors from thin film manufacturing.

V = Voltage applied over sample (V)

fn = Applied fundamental frequency (Hz)

dR/dT = Resistance vs Temperature Gradient (Ω/K)

l = Metal heater length (m)

R = Sample resistance (Ω)

V3omega = Measured 3rd harmonic voltage (V)

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Fig 7. Sample 1

Samples Fabricated

For this project I used both prefabricated samples and designed a shadow mask which a PhD student

Xiao Sun used to evaporate gold and chromium onto a glass slide. These samples have been used for

testing the 3-Omega technique on bulk substrates.

Sample Heater Material Substrate Material Substrate Conductivity (W/mK) Heater TCR

1 Copper Fiberglass (FR-4) 0.04 0.003715

2 Gold/Chromium Glass 0.8 0.0039

3 Gold/Chromium Glass 0.8 0.0039

Sample 1 The sample used for initial measurements was a copper line on a

fiberglass substrate (FR4) shown below. This sample was chosen as it

was readily available from spare parts to use, so no additional

equipment had to be ordered to verify the experimental technique.

The sample (shown to the right) was connected via soldering two wires

to each end of one of the thin copper lines. Care was taken to avoid

shorting the circuit across one of the other lines, and a small segment

of the line was removed from the adjacent lines to ensure an open

circuit. A 47Ω resistor was connected in series with the sample to

ensure a steady current was applied to the sample, and also as the

function generator requires an output load of at least 50Ω.

Sample 2 This sample was several orders of magnitude smaller in size and manufactured much more precisely by

evaporating the heater metal onto ordinary glass used as a microscope slide. The sample is to be

created by depositing a 200nm layer of gold and chromium onto the glass slide by thermodeposition.

Gold is selected due to its high TCR (Temperature Coefficient of Resistance), and a 10nm Chrome

adhesion layer is necessary as gold does not adhere well to ceramic substrates.

Table 2. List of Samples

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Fig 9. Laser Cut Shadow Mask

The process of thermodeposition involves first creating a shadow mask from stainless steel, this mask

acts as a stencil to precisely determine where chrome and gold are deposited onto the porous silicon.

The heating element of the ‘3-Omega’ Method is the thin line in the figure below, while the large

squares are contact pads for administering the current and measuring the voltage drop. All dimensions

in the diagram below are in micrometres. The design contained 4 different masks of varying lengths and

line widths in case the sample geometry was not optimal.

The shadow mask was designed in AutoCAD

and a suitable manufacturer was soon

located. Two processes are most common

for fabrication of shadow masks; laser

cutting and chemical etching. Chemical

etching has the advantage of intricate

patterns or many samples being cut in the

same amount of time as simpler patterns,

however the minimum resolution of

geometry is larger than laser cutting. Laser

cutting was chosen as the method due to

its relatively low geometrical resolution of

approximately 25 microns and as there

were only four samples to be cut as seen in

figure 9, therefore a low cutting time and

cost. Once the shadow mask had arrived,

PhD student Xiao Sun conducted an optical

assessment of the masks to ensure

geometrical conformity.

Fig 8. Shadow Mask Design (Dimensions in μm)

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The shadow mask was then used to deposit the Chromium (10nm) and Gold (200nm) onto the glass

slide. Each of the patterns were inspected to ensure complete electrical connection, and 6 of the 12

were suitable to proceed with the experiment. The glass slide was glued to a copper wiring board using

a hot glue gun; this board is used to avoid unnecessary movement of the wires connecting the sample.

In order to connect to the small contact pads of the sample, I used thin wires of ~5cm in length which I

dipped into a 2-part epoxy solution and carefully placed onto each pad. This epoxy is conductive as not

to disturb the electrical properties of the sample. After 24 hours when the epoxy has dried, I soldered

the ends of the wires into an individual column of the copper board, and soldered another conductive

pin into this column to connect the circuit together.

Fig 10. Microscope images of Shadow Mask (Sun, 2015)

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Resistance of the Metal Heater

The initial resistance of the metal heater (R0) is required for the measurement of thermal conductivity.

Both the theoretical values for the metal heater as well as measured values are presented below.

Theoretical Values- Gold/Chromium Of the 12 samples which had been deposited onto glass, four of these samples were found to have

maintained their structure through optical analysis. Only two of these samples (B and C) maintained

electrical contact when tested later on, so samples B and C are later referred to as samples 2 and 3

respectively. Gold/Cr samples each had a different geometry, and given these values along with material

properties the resistance of the sample can be found by:

𝑅0 =𝜌𝐿

𝐴 𝐸𝑞𝑛 (7)

ρ = resistivity (Ohm*m) = 2.214*10-8 for Gold (Lide, 1997), and 1.3*10-7 Ohm.m for Chromium

(n.d., Resistivity of Common Materials, 2012)

L = Length (m)

A = Cross sectional area (m2)

The geometry of the samples and calculated resistances of Gold (first 4 entries) and their corresponding

Chromium layers (last 4) are shown below.

Sample Length (m) Thickness (m) Width (m) Resistivity (Ohm*m) Resistance (Ohms)

A 3.05*10-3 2.00*10-7 2.50*10-5 2.214*10-8 13.51

B (2) 6.05*10-3 2.00*10-7 5.00*10-5 2.214*10-8 13.40

C (3) 3.15*10-3 2.00*10-7 7.50*10-5 2.214*10-8 4.65

D 6.15*10-3 2.00*10-7 1.00*10-4 2.214*10-8 6.81

A 3.05*10-3 1.00*10-8 2.50*10-5 1.3*10-7 1586

B (2) 6.05*10-3 1.00*10-8 5.00*10-5 1.3*10-7 1573

C (3) 3.15*10-3 1.00*10-8 7.50*10-5 1.3*10-7 546

D 6.15*10-3 1.00*10-8 1.00*10-4 1.3*10-7 799

Table 3. Theoretical Resistance of Samples

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The chromium layer will have a relatively small effect on the sample resistance due to being only 5% of

the area of the gold layer. The total sample resistance can be found by modelling them as resistors in

parallel:

1

𝑅𝑡𝑜𝑡𝑎𝑙=

1

𝑅𝑐ℎ𝑟𝑜𝑚𝑖𝑢𝑚+

1

𝑅𝐺𝑜𝑙𝑑 𝐸𝑞𝑛 (8)

Sample Total Resistance (Ohms)

A 13.39

B (2) 13.28

C (3) 4.61

D 6.75

Theoretical Value – Copper By using the same formula above (equation 7), the theoretical resistance value of the copper resistor

used as the metal heater can be calculated. As the layer is only copper, there is no need for a parallel

resistance calculation.

Sample# Length (m) Thickness (m) Width (m) Resistivity (Ohm*m) Resistance (Ohms)

1 2.50*10-2 1.0*10-5 5.00*10-4 1.68*10-8 0.084

Measured Sample Resistances The resistance of the sample can be measured by applying a small voltage of 100mV (peak to peak) to

the circuit (shown below).

Fig 11. Resistance Measurement Circuit

Table 4. Total Theoretical Resistance

Table 5. Theoretical Resistance of Sample 1

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The current running through the circuit is found using Ohms Law:

𝐼 =𝑉𝑠𝑜𝑢𝑟𝑐𝑒

𝑅𝑡𝑜𝑡𝑎𝑙=

100

2 ∗ √2 ∗ (50 + 47 + 𝑅𝑠) 𝐸𝑞𝑛 (9)

Next, the 1st harmonic voltage ( Vmeasured) is measured in rms volts over the sample to determine the

resistance:

𝑅𝑠 =𝑉𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑

𝐼=

𝑉𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 ∗ 2 ∗ √2 ∗ (50 + 47 + 𝑅𝑠)

100 𝐸𝑞𝑛 (10)

∴ 𝑅𝑠 =𝑉𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 ∗ 2 ∗ √2 ∗ (97)

100 ∗ (1 −2 ∗ √2 ∗ 𝑉𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑

100)

𝐸𝑞𝑛 (11)

In the table below, the results of testing the three working sample resistance have been compiled.

Sample# Vapplied (mV) Vmeasured (V) Resistance measured (Ω) Resistance Calculated (Ω)

1 100 3.68 * 10-5 0.101 0.084

2 100 5.98 * 10-3 19.75 13.28

3 100 2.32 * 10-3 6.80 4.61

There is some discrepancies between the measured and calculated values, and in fact the initial values

of resistance were a factor of 100 from the calculated value until the correction of the function

generator’s output load value was rectified. The difference in resistance could be caused by the

measured resistance taking into account not only the thin metal line’s resistance but also the short

connecting path to the contact pads, boundary resistances between interfaces and the small resistance

of connecting wires of the circuit.

Temperature Coefficient of Resistance The TCR, usually denoted as β can be found by testing the resistance of the sample over a wide range of

temperatures to obtain the connection between resistance and temperature. This method has the

benefit of taking into account any impurities or defects in the sample, but as it requires accurate

temperature data and can yield large error values if measured imprecisely, an approximate value was

used to derive the resistance vs temperature relationship.

Table 6. Comparison of Sample Resistances

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The resistance of a sample that has changed in temperature is given by:

𝑅 = 𝑅0(1 + 𝛽(∆𝑇)) 𝐸𝑞𝑛 (12)

This equation is then rearranged to find dR/dT, required by the conductivity formula (equation 6):

𝑑𝑅

𝑑𝑇=

𝑅0(𝛽∆𝑇)

∆𝑇= 𝑅0𝛽 𝐸𝑞𝑛 (13)

Where the β value for gold at 20oC is 0.003715 K-1, and the value for copper is 0.0039 K-1 (AAC, 2012).

Experimental Design

The circuit used in testing each of the samples was identical in each case. It consisted of a function

generator, in-series resistor, lock-in amplifier and the sample to be tested arranged into the system

shown below:

Fig 12. 3-Omega Measurement Circuit

Note that in the above diagram, everything to the left of the dotted line indicates that it is within the

function generator.

R0 = Initial sample resistance (Ω)

β = Temperature coefficient of

resistance (Ω/K)

ΔT = Change in temperature (K)

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Lock-in Amplifier The lock-in amplifier is a device used to isolate an electrical signal within a very specific frequency range,

as the 3-Omega voltage is approximately 1000x smaller than the applied voltage signal. The lock-in

amplifier available at UWA is the Stanford Research Systems SR830. Isolating the 3-Omega frequency in

the SR830 requires a reference frequency which in this case is the fundamental voltage frequency

omega which is applied to the sample. By comparing two sine waves, the average over time will only be

non-zero if the frequencies match. The amplifier is then set to find the 3rd harmonic of this frequency via

the ‘Harm #’ button.

The remaining signal which is output is in the form of a DC output (due to the multiplication of reference

and input signals) along with AC signal noise. To remove these unwanted AC signals a low-pass filter is

used, with two performance parameters:

Roll off Rate – The rate at which the signal decays above the cut off frequency. This function is

under the ‘Slope/Oct’ button.

Time constant / Cut off frequency – This value is the point at which the signal has been

attenuated by -3dB, and is given by:

𝑓𝑐 =1

2𝜋𝜏𝑓 𝐸𝑞𝑛 (14)

As the time constant increases, so does the stabilization time and accuracy of the output. The time

constant is varied from 1microsecond – 30ks, and while a lower time constant will allow faster initial

readings of results the value used for low frequency measurements should be at least triple the period

of the fundamental frequency.

Function Generator After it was found that the previous function generator was producing unacceptable levels of 3rd

harmonic noise, it was replaced for a better quality Agilent 33220A waveform generator. The

measurements of harmonic distortion can be found in table 9 and 10. The function generator is vital in

the measurement of the sample resistances, while the 3-Omega voltage does not use data from the

voltage source in the calculation of thermal conductivity. One important setting to input into this device

is the ‘output load’, a measure of the external resistance of the circuit. Data had been taken a few times

with the voltage incorrectly displayed before this error was rectified, resulting in incorrect sample

resistance measurements.

P a g e | 22

Fig 13. Sample 1 Experiment 1 Results

Measurement Steps To measure the 3rd harmonic voltage over a sample, I first ensured that all circuit elements were

connected properly and a reasonable first harmonic voltage is displayed over the lock-in amplifier.

The 3rd harmonic voltage value should be approximately 1000x smaller than the measured fundamental

voltage. If necessary, the source voltage is increased until the 3rd harmonic voltage approaches an ideal

magnitude of measurement above ambient noise. Next, frequencies are swept starting from the highest

frequency down to 1Hz, while changing frequency a low time constant is used to quickly reset the

integration time and at lower frequencies this constant must be set to a suitably large value.

Sample 1 Results

Experiment 1 After I had measured the samples resistance and the first harmonic applied voltage over the sample, the

measure 3rd harmonic voltage over the sample was measured from frequencies 1-10,000Hz. Given that

it is to be plotted on a log scale, values were taken at the 1, 7 and 10 multiples of each log value. Error

bars were recorded by noting the range of values that the lock-in amplifier displayed at each frequency.

The 3-Omega Voltage data for the copper line on FR4 is shown below:

y = -0.004ln(x) + 2.3652

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

1 10 100 1000 10000

3 O

meg

a V

olt

age

(μV

)

Applied Frequency (Hz)

3 Omega Voltage vs Frequency

P a g e | 23

This data was compressed into the above trend line using the logarithmic regression function in excel,

however from the error bars of 20-50% across the data this line is a relatively poor prediction of the

slope with a R2 value of only 0.0006. The calculation of thermal conductivity using equation 6 and the

data below was found:

Vapplied F2 F1 dR/dT L R V3w 1 V3w 2

0.619mV 100Hz 1Hz 0.0734Ω/oC 2.2cm 0.101Ω 2.365 μV 2.338 μV

Giving a k-value of 0.0083 W/mK, a factor of 5 from the accepted FR4 conductivity value of 0.04W/mK.

While this value is somewhere within the ballpark, there does not appear to be a general downward

trend of the voltage values, and the increase in voltage at the higher frequencies skews the rest of the

data.

It is expected that the 3-Omega voltage should be approximately a factor of 1000 smaller than the first

harmonic voltage applied over the sample. Because the applied voltage was low, it appears that the 3-

Omega voltage of the sample may have been drowned out by ambient electrical noise within the optics

lab as well as from the in-series resistor and inbuilt resistor in the function generator used. This can be

seen by the relatively flat profile of the trend line, indicating a low frequency dependence of the

measured voltage.

Experiment 2 Due to the noise recorded in the previous trial, a method of reducing possible electromagnetic

interference was introduced. Fluctuations within the electromagnetic field in a space can be reduced be

surrounding it with a barrier made of conductive materials such as metal. A metal box was used to

contain the sample and shield it from ambient electrical noise. This box was grounded via a cable from

the lock in amplifier. The same sample and circuit was used in this trial, but using an increased voltage in

an attempt to boost the 3-Omega signal of the sample above the noise.

Results The recorded 3-Omega data is displayed in figure 14, along with a line displaying the correct slope of the

data. This calculated line was created by assuming that the voltage data point for 1Hz was correct, and

solving the conductivity equation for the second voltage. I had overlaid this line over the data to see

what data I should be expecting and whether the error bars were too big in the measured values to

ascertain an accurate slope of the data.

Table 7. Sample 1 Experiment 1 Data

P a g e | 24

Using the trend line obtained from the above trend line equation and other input parameters below:



The thermal conductivity was found using equation 6 to be k = 0.0070 W/mK, still too far from the

accepted value of k = 0.04 W/mK for the experimental design to be reliable.

Improvements to be made on Experimental Design From the calculated slope it is clear that this technique requires a sensitive voltage measurement due to

the very low gradient. The only changes I made during this trial were the applied voltage and addition of

the electromagnetic shield, so given that the magnitude of the error bars had increased since the

previous trial, I had come to the conclusion that the electrical noise in each of the measurements was

due components within the circuit.

y = -1.003ln(x) + 20.263

0.00

5.00

10.00

15.00

20.00

25.00

1 10 100 1000 10000

3 O

meg

a V

olt

age

(μV

)



measured

Calculated

Log. (measured)

Fig 14. Sample 1 Experiment 2 Results

Table 8. Sample 1 Experiment 2 Data

P a g e | 25

Table 9. Harmonic Distortion from Voltage Source

Table 10. Harmonic distortion of new function generator

Noise from Resistors While researching other papers that had conducted the 3-Omega Technique experiment, I came across

one which had encountered similar problems from system noise, “Implementing the 3-Omega

Technique for Thermal Conductivity Measurements” (Hanninen, 2013). In the conclusion of this paper

Tuomas Hanninen found that “The goal of validation of the 3ω method for thermal conductivity

measurements was not achieved. Obtained 3ω signal did not contain information about the thermal

properties of the measured samples. At first this was thought to be due to low power level in the metal

line heater, but further measurements with the Wheatstone bridge setup suggest that the cause is

spurious 3ω signals from the components. The actual source of this signal remains unknown.”

After reading this I decided to stop and investigate the sources of noise within the circuit, starting with

the function generator used.

In order to measure the 3rd harmonic noise generated from the function generator, I connected the

leads directly into the lock-in amplifier and recorded the first and third harmonic voltages generated:

V (1st Harmonic) V (3rd Harmonic) Ratio (V 3rd / V 1st )

0.7499 V 1.65 mV 2.2 *10-3

0.0906 V 0.29 mV 3.2 *10-3

These values show that the 3rd harmonic of the function generator decreases with the voltage of the

first harmonic, but peak voltages used for my measurements are higher than either of these values

(voltage used is limited by the 1V overload of the lock in amplifier). A 3rd harmonic of ~1mV in the

function generator is too large for any measurements on the scale of microvolts to take place in series.

The 3-Omega signal arises from the temperature coefficient of resistance (TCR) of circuit components,

for this reason all components except for the sample should have very low TCR values. While the in-

series resistor is another source of error, the old function generator was clearly a large source of

electrical noise so this was replaced with a newer one, the Agilent 33220A Waveform Generator. This

function generator boasts a much smaller harmonic distortion of -70dBc for frequencies up to 20Hz and

peak to peak voltages less than 1V (Keysight Technologies, 2011). To compare these two function

generators the same voltage was applied directly into the function generator and values recorded:

V (1st Harmonic) V (3rd Harmonic) Ratio (V 3rd / V 1st )

0.7500 V 0.9 μV 1.2 *10-6

0.0905 V 0.4 μV 4.4 *10-6

By changing the function generator to the newer low distortion Agilent model, this source of harmonic

noise had been reduced by approximately a factor of 1000.

P a g e | 26

Table 11. Sample 1 frequency restriction

Geometric Sample Considerations One of the biggest discrepancies that I noticed between other papers samples and my own was the

geometrical values of the original sample I used. Looking further into why the samples other papers

have used were so small, I encountered an important relationship governing the solution of the heat

transfer equations. This equation states that the wavelength of the diffusive thermal wave through the

substrate or ‘thermal penetration depth’ 1/q must be much greater than the sample width w (Cahill D.

G., 1990).

1

𝑞= √

𝑘

2𝜌𝐶𝜔≫ 𝑤 𝐸𝑞𝑛 (15)

Rearranging this equation to find the restrictions imposed on the applied first harmonic ω yields:

𝜔 ≪𝑘

2𝑤2𝜌𝐶 𝐸𝑞𝑛 (16)

Substituting in the relevant data below:

k (W/mK) w (m) ρ (kg/m3) C (J/kg K)

0.04 5*10-4 1,850 800

Yields the required maximum first harmonic frequency for this sample:

𝜔 ≪ 0.054 𝐻𝑧

This value is clearly far too small for the experimental design which has been created, and it is for this

reason that the width of the metal heater line on substrates must be very small, ideally less than 100μm

for this relationship to permit a measurement on the order of 1Hz.

The relationship of maximum fundamental frequency to the geometry of the sample has been found to

be a limiting factor in these trials, and care must be taken in selecting a geometry which is not too

difficult to manufacture (laser cutting a shadow mask has a maximum allowable resolution of

approximately 25 microns) while still maintaining a reasonable edge smoothness of the sample. For the

measurements of following samples, only the frequencies from 1-10Hz have been analysed to comply

with this limitation.

k = Substrate conductivity (W/mK)

ρ = Substrate density (kg/m3)

C = Substrate specific heat capacity (J/kgK)

1/q = Thermal penetration depth (complex quantity)

ω = Applied voltage frequency (Hz)

w = Heater width (m)

P a g e | 27

Sample 2 Results

The frequencies from 1-10Hz were tested and recorded below. Full results are available in the appendix.

By using the logarithmic trend line regression in excel, the equation:

𝑦 = −0.142 ln(𝑥) + 2.4598 𝐸𝑞𝑛 (17)

Is used to approximate the downward trend in voltage. Using frequencies of 1Hz and 10Hz from the

trend line and plugging these values into Eqn 6:


5.98mV 10Hz 1Hz 0.0734Ω/oC 6.05mm 19.75Ω 2.46 μV 2.13μV

Returns a value of: 𝑘 = 0.0037𝑊/𝑚𝐾

This value is about a factor of 200 lower than the accepted thermal conductivity value for glass of k =

0.8W/mK.

y = -0.142ln(x) + 2.4598

0

0.5

1

1.5

2

2.5

3

1 10

3 O

meg

a V

olt

age

(μV

)



measured

Log. (measured)

Fig 15. Sample 2 Results

Table 12. Sample 2 Data

P a g e | 28

Fig 16. Sample 3 Results

Table 13. Sample 3 Data

Sample 3 Results

The frequencies 1-10Hz were used to examine the 3ω Voltage response of sample 3 with a larger source

voltage applied. In addition, the in-series resistor has been removed from the circuit which has a large

effect on the measured voltage range within the data, however as the function generator is designed for

an output load of at least 50Ω it is possible that additional inaccuracies have been introduced to the

experiment.

Once again, the trend line obtained above is used to generate two voltages to be used in the thermal

conductivity equation.

By using the following data:



The calculated thermal conductivity of glass is found to be k = 0.00456 W/mK as compared to the

accepted value for glass of 0.8W/mK. This measurement is a factor of 175 from the accepted value,

however it appears in this trial that the data follows the trend line very well, with an R2 value of 0.9939.

y = -4.229ln(x) + 33.278

0

5

10

15

20

25

30

35

40

1 10

3 O

meg

a V

olt

age

(μV

)



measured

Log. (measured)

P a g e | 29

It is often difficult to ascertain the reliability of the error bars for this lock-in measurement, as while

there was very little movement in the recorded values above, the lack of in series resistor along with a

larger time constant used in low frequencies to obtain a stable measurement value reduce the error

range.

Conclusion

The 3-Omega Technique has not been validated in this experiment, returning thermal conductivity data

for FR-4 fiberglass as 0.007 and 0.0083 W/mK (compared to the accepted value of 0.04W/mK) and data

for glass as 0.0037 and 0.00456W/mK (compared to the accepted value of 0.8W/mK). The causes of

these experimental errors are thought to be due to experimental design errors causing excessive 3rd

harmonic noise to be generated within the circuit. The development of microscale material

measurement systems is vital for the ongoing technological contributions that UWA makes to the wider

community. By better understanding thermal properties of thin films the development of porous silicon

and other thin film technologies will be made easier. For these reasons I have compiled a list of

recommendations to the experimental design to return more accurate thermal data on thin films and

substrates.

Recommendations for Future Work One of the problems that is common throughout data acquired with the 3-Omega technique is the noise

generated within circuit components. Only by eliminating these noise sources can accurate thermal

conductivity data be obtained, so the following noise reduction techniques are recommended:

Wheatstone Bridge A Wheatstone bridge can be used as a common mode cancellation

technique, whereby certain frequencies of a signal can be filtered

out by comparing it to a reference signal which contains the same

signal components. This means that the signal of the fundamental

frequency can be filtered out from the final voltage measurement

of the sample without affecting the measured 3rd harmonic

voltage. From figure 17 on the right, this is achieved by sending

the differential voltage W3w to the lock in amplifier. By selecting

the values of R1 to be 100 times smaller than R2 the current sent

through the sample can be maximized as the V3ω is proportional to

I2, resulting in a larger 3rd harmonic signal.

Fig 17. Wheatstone bridge (Hanninen, 2013)

P a g e | 30

Low TCR Resistors One of the most important aspects of noise reduction within the circuit is the selection of resistors with

a low thermal coefficient of resistance, ideally less than 1% of the heater’s TCR value (Koninck, 2008). In

my experiment only the harmonic distortion of the function generator was reduced, while a standard

resistor was used in series causing additional noise. Care must also be taken in the connections between

circuit components, by using wires both shorter in length and larger in cross sectional area the effective

resistance of these connections can be reduced resulting in smaller voltage fluctuations.

Use of a Vacuum Chamber Vacuum chambers can eliminate the experimental error caused by convection of the metal heater, as

well as providing a more controlled temperature environment. There are drawbacks to using a vacuum

chamber however, notably the increased difficulty of experimental setup. Sample holders are required

to keep the sample in place and an electrical feedthrough to the chamber requires careful calibration.

Not being able to see within the chamber can mean that open circuits are only found once the chamber

has been decompressed and opened. This technique has been used by all of the papers that I have

looked at previously so this important experimental design technique should not be overlooked in

future.

Word count of main body: 6423.

P a g e | 31

References AAC. (2012). Temperature Coefficient of Resistance. Retrieved from All About Circuits Textbook Volume

1: http://www.allaboutcircuits.com/textbook/direct-current/chpt-12/temperature-coefficient-

resistance/

Cahill, D. (1997). Journal of Applied Physics, 2590.

Cahill, D. G. (1990). Thermal Conductivity measurement from 30 to 750K: The 3w method. Review of

Scientific Instruments, 802.

Canham, L. T. (1995). Bioactive silicon structure fabrication through nanoetching techniques. Advanced

Materials 7.

Hanninen, T. (2013). Implementing the 3-Omega Technique for Thermal Conductivity Measurements.

Finland: University of Jyvaskyla.

Jaeger, H. C. (1959). Condcution of Heat in Solids. Oxford: Oxford University Press.

Keysight Technologies. (2011). 20 MHz Function/Arbitrary Waveform Generator Data Sheet. Retrieved

from http://literature.cdn.keysight.com/litweb/pdf/5988-8544EN.pdf?id=187648

Koninck, D. d. (2008). Thermal Conductivity Measurements Using the 3-Omega Technique. Montreal,

Canada: McGill University.

Lide, D. (1997). Resistivity of Gold. In Handbook of Chemistry and Physics, 75th edition (pp. 11-41). New

York: CRC Press.

McLaren, R. C. (2009). Thermal Conductivity Anisotropy in Molybdenum Disulfide Thin Films. Illinois:

University of Illinois.

n.d. (2011, 09 25). American Heritage® Dictionary of the English Language, Fifth Edition. . Retrieved from

Thermal Conductivity: http://www.thefreedictionary.com/thermal+conductivity

n.d. (2012). Resistivity of Common Materials. Retrieved from Engineering Toolbox:

http://www.engineeringtoolbox.com/resistivity-conductivity-d_418.html

Ressine, M.-V. L. (2007). Porous silicon protein microarray technology and ultra/superhydrophobic

states for improved bioanalytical readout. Biotechnology Annual Review 13, 149-200.

Sun, X. (2015). Microscope Images of Shadow Mask. Perth: UWA.

Touloukian, Y. (1973). Thermal Conductivity: Nonmetallic Solids. In Thermophysical Properties of Matter.

New York: IFI/Plenum.

Vedam, K. a. (1975). Piezo-optic Behavior of Water and Carbon Tetrachloride under High Pressure.

Physics Review , 1014.

Zhang, Z. (2007). Nano/Microscale Heat Transfer. New York: McGraw-Hill.

P a g e | 32

Appendices

Sample 1 Data



frequency (Hz) V3w microvolts

Error

1 2.9 0.6

3 2.5 0.5

7 2.3 0.6

10 2.5 0.6

30 2 0.6

70 2.5 0.5

100 2.2 0.8

300 1.7 0.7

700 1.8 0.4

1000 2.1 0.9

3000 1.9 0.4

7000 2.6 1

10000 3.5 0.8



frequency (Hz) V3w microvolts Error

1 18.95 4.5

3 18.76 5

7 18.61 4.3

10 18.55 5.1

30 18.35 4.2

70 18.2 4.8

100 18.14 4.5

300 17.95 4.6

700 17.8 3.5

1000 17.73 4.2

3000 17.54 3.1

7000 17.39 5.1

10000 17.33 4.8

P a g e | 33

Sample 2 Data




1 2.4 0.2

2 2.35 0.1

3 2.4 0.2

4 2.28 0.1

5 2.23 0.2

6 2.32 0.2

7 2.12 0.2

8 2.15 0.2

9 2.11 0.2

10 2.1 0.1

Sample 3 Data




1 32.9 1.2

2 30.5 1.3

3 28.8 0.9

4 27.7 0.8

5 26.7 1.3

6 25.7 1.4

7 25.1 1.2

8 24.4 1.3

9 24 1

10 23.1 1.2

measuring thermal conductivity of thin films

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