2. P a g e | 1
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
3. P a g e | 2
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
4. P a g e | 3
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.
5. P a g e | 4
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.
6. P a g e | 5
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
7. P a g e | 6
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.
8. P a g e | 7
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.
9. P a g e | 8
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)
10. P a g e | 9
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)
11. P a g e | 10
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)
12. P a g e | 11
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)
1
2 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)
13. P a g e | 12
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 = 4
𝑑𝑇
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.
14. P a g e | 13
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 (
𝑓2
𝑓1
) 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)
15. P a g e | 14
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
16. P a g e | 15
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)
17. P a g e | 16
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)
18. P a g e | 17
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
19. P a g e | 18
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
𝑅 𝑐ℎ𝑟𝑜𝑚𝑖𝑢𝑚
+
1
𝑅 𝐺𝑜𝑙𝑑
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
20. P a g e | 19
The current running through the circuit is found using Ohms Law:
𝐼 =
𝑉𝑠𝑜𝑢𝑟𝑐𝑒
𝑅𝑡𝑜𝑡𝑎𝑙
=
100
2 ∗ √2 ∗ (50 + 47 + 𝑅 𝑠)
𝐸𝑞𝑛 (9)
Next, the 1st
harmonic voltage ( Vmeasured) is measured in rms volts over the sample to determine the
resistance:
𝑅 𝑠 =
𝑉 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑
𝐼
=
𝑉 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 ∗ 2 ∗ √2 ∗ (50 + 47 + 𝑅 𝑠)
100
𝐸𝑞𝑛 (10)
∴ 𝑅 𝑠 =
𝑉 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 ∗ 2 ∗ √2 ∗ (97)
100 ∗ (1 −
2 ∗ √2 ∗ 𝑉 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑
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
21. P a g e | 20
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
= 𝑅0 𝛽 𝐸𝑞𝑛 (13)
Where the β value for gold at 20o
C 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)
22. P a g e | 21
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
2𝜋𝜏 𝑓
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.
23. 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
3OmegaVoltage(μV)
Applied Frequency (Hz)
3 Omega Voltage vs Frequency
24. 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/o
C 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
25. P a g e | 24
Using the trend line obtained from the above trend line equation and other input parameters below:
Vapplied F2 F1 dR/dT L R V3w 1 V3w 2
3.695mV 1000Hz 1Hz 0.0734Ω/o
C 2.2cm 0.101Ω 20.263 μV 13.335 μV
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
3OmegaVoltage(μV)
Applied Frequency (Hz)
3 Omega Voltage vs Frequency
measured
Calculated
Log. (measured)
Fig 14. Sample 1 Experiment 2 Results
Table 8. Sample 1 Experiment 2 Data
26. 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.
27. 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
2𝑤2 𝜌𝐶
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)
28. 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:
Vapplied F2 F1 dR/dT L R V3w 1 V3w 2
5.98mV 10Hz 1Hz 0.0734Ω/o
C 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
3OmegaVoltage(μV)
Applied Frequency (Hz)
3 Omega Voltage vs Frequency
measured
Log. (measured)
Fig 15. Sample 2 Results
Table 12. Sample 2 Data
29. 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:
Vapplied F2 F1 dR/dT L R V3w 1 V3w 2
11.0mV 10Hz 1Hz 0.0265Ω/o
C 6.05mm 6.80Ω 33.28 μV 4.07μV
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
3OmegaVoltage(μV)
Applied Frequency (Hz)
3 Omega Voltage vs Frequency
measured
Log. (measured)
30. 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)
31. 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.
32. 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.
33. P a g e | 32
Appendices
Sample 1 Data
Vapplied F2 F1 dR/dT L R V3w 1 V3w 2
0.619mV 100Hz 1Hz 0.0734Ω/o
C 2.2cm 0.101Ω 2.365 μV 2.338 μV
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
Vapplied F2 F1 dR/dT L R V3w 1 V3w 2
3.695mV 1000Hz 1Hz 0.0734Ω/o
C 2.2cm 0.101Ω 20.263 μV 13.335 μV
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
34. P a g e | 33
Sample 2 Data
Vapplied F2 F1 dR/dT L R V3w 1 V3w 2
5.98mV 10Hz 1Hz 0.0734Ω/o
C 6.05mm 19.75Ω 2.46 μV 2.13μV
frequency (Hz) V3w microvolts Error
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
Vapplied F2 F1 dR/dT L R V3w 1 V3w 2
11.0mV 10Hz 1Hz 0.0265Ω/o
C 6.05mm 6.80Ω 33.28 μV 4.07μV
frequency (Hz) V3w microvolts Error
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
