![a) Schematic diagram of a typical pulsed power technique for thermal conductivity measurement where periodic square-wave current is applied. (b) The time-dependent temperature difference DT 5 T h 2T c across the sample, from Ref. [38] where the temperature of the heat sink is slowly drifted. The dots represent experimental data, and line represents calculation results.](https://www.researchgate.net/profile/Dongliang-Zhao-5/publication/303655808/figure/fig11/AS:668648817565703@1536429709324/a-Schematic-diagram-of-a-typical-pulsed-power-technique-for-thermal-conductivity_Q320.jpg)
![The measurement principle of the hot-wire method. (a) One-dimensional radial heat flow is assumed inside the test sample. (b) As an electric current of fixed intensity passes through the hot wire, the thermal conductivity of the test sample can be derived from the resulting temperature change at a known distance from the hot wire over a known time interval. This figure is reproduced with permission from Franco [50]. Copyright 2007 by Elsevier.](https://www.researchgate.net/profile/Dongliang-Zhao-5/publication/303655808/figure/fig12/AS:668648821772290@1536429710051/The-measurement-principle-of-the-hot-wire-method-a-One-dimensional-radial-heat-flow-is_Q320.jpg)
Content uploaded by Dongliang Zhao
Author content
All content in this area was uploaded by Dongliang Zhao on Oct 05, 2019
Content may be subject to copyright.
Dongliang Zhao
Department of Mechanical Engineering,
University of Colorado,
Boulder, CO 80309-0427
Xin Qian
Department of Mechanical Engineering,
University of Colorado,
Boulder, CO 80309-0427
Xiaokun Gu
Department of Mechanical Engineering,
University of Colorado,
Boulder, CO 80309-0427
Saad Ayub Jajja
Department of Mechanical Engineering,
University of Colorado,
Boulder, CO 80309-0427
Ronggui Yang
1
Department of Mechanical Engineering,
University of Colorado,
Boulder, CO 80309-0427
e-mail: ronggui.yang@colorado.edu
Measurement Techniques for
Thermal Conductivity and
Interfacial Thermal Conductance
of Bulk and Thin Film Materials
Thermal conductivity and interfacial thermal conductance play crucial roles in the design
of engineering systems where temperature and thermal stress are of concerns. To date, a
variety of measurement techniques are available for both bulk and thin film solid-state
materials with a broad temperature range. For thermal characterization of bulk material,
the steady-state method, transient hot-wire method, laser flash diffusivity method, and
transient plane source (TPS) method are most used. For thin film measurement, the 3x
method and the transient thermoreflectance technique including both time-domain and
frequency-domain analysis are widely employed. This work reviews several most com-
monly used measurement techniques. In general, it is a very challenging task to determine
thermal conductivity and interfacial thermal conductance with less than 5%error. Select-
ing a specific measurement technique to characterize thermal properties needs to be
based on: (1) knowledge on the sample whose thermophysical properties are to be deter-
mined, including the sample geometry and size, and the material preparation method; (2)
understanding of fundamentals and procedures of the testing technique, for example,
some techniques are limited to samples with specific geometries and some are limited to
a specific range of thermophysical properties; and (3) understanding of the potential
error sources which might affect the final results, for example, the convection and radia-
tion heat losses. [DOI: 10.1115/1.4034605]
Keywords: bulk solid materials, thin films, thermal conductivity, thermal contact resistance,
thermal boundary resistance, interfacial thermal conductance
1 Introduction
Thermal conductivity (denoted as k,j,ork) measures the heat
conducting capability of a material. As shown in Fig. 1(a), it can
be defined as the thermal energy (heat) Qtransmitted through a
length or thickness Lin the direction normal to a surface area A,
under a steady-state temperature difference ThTc. Thermal con-
ductivity of a solid-phase material can span for several orders of
magnitude, with a value of 0.015 W=m K for aerogels at the low
end to 2000 W=m K for diamond and 3000 W=m K for single-
layer graphene at the high-end, at room temperature. Thermal
conductivity of a material is also temperature-dependent and can
be directional-dependent (anisotropic). Interfacial thermal con-
ductance (denoted as Kor G) is defined as the ratio of heat flux to
temperature drop across the interface of two components. For
bulk materials, the temperature drop across an interface is primar-
ily due to the roughness of the surfaces because it is generally
impossible to have “atomically smooth contact” at the interface as
shown in Fig. 1(b). Interfacial thermal conductance of bulk mate-
rials is affected by several factors, such as surface roughness, sur-
face hardness, impurities and cleanness, the thermal conductivity
of the mating solids, and the contact pressure [1]. For thin films,
the temperature drop across an interface can be attributed to the
bonding strength and material difference. Note that thermal con-
tact resistance and thermal boundary resistance (or Kapitza resist-
ance [2]) are usually used to describe heat conduction capability
of an interface in bulk materials and thin films, respectively. Inter-
facial thermal conductance is simply the inverse of thermal
contact/boundary resistance. Knowledge of thermal conductivity
and interfacial thermal conductance and their variation with
temperature are critical for the design of thermal systems. In this
paper, we review measurement techniques for characterizing ther-
mal conductivity and interfacial thermal conductance of solid-
state materials in both bulk and thin film forms.
Extensive efforts have been made since the 1950s for the char-
acterization of thermal conductivity and thermal contact resist-
ance in bulk materials [3–8]. Table 1summarizes some of the
most commonly used measurement techniques, which in general
can be divided into two categories: steady-state methods and tran-
sient methods. The steady-state methods measure thermal proper-
ties by establishing a temperature difference that does not change
with time. Transient techniques usually measure time-dependent
energy dissipation process of a sample. Each of these techniques
has its own advantages and limitations and is suitable for only a
limited range of materials, depending on the thermal properties,
sample configuration, and measurement temperature. Section 2is
devoted to comparing some of these measurement techniques
when applied for bulk materials.
Thin film form of many solid materials with a thickness ranging
from several nanometers to hundreds of microns has been
extensively used in engineering systems to improve mechanical,
optical, electrical, and thermal functionality, including microelec-
tronics [9], photonics [10], optical coating [11], solar cells, and
thermoelectrics [12]. Thin film materials can be bonded on a sub-
strate (Fig. 1(c)), free-standing, or in a multilayer stack. When the
thickness of a thin film is smaller than the mean free path of its
heat carriers, which are electrons and phonons depending on
whether the material is electrically conducting or not, the thermal
conductivity of thin films is reduced compared to its bulk counter-
parts because of the geometric constraints. Thermal conductivity
of thin films is usually thickness-dependent and anisotropic, where
the heat conducting capability in the direction perpendicular to
the film plane (cross-plane) is very different from that parallel to
the film plane (in-plane), as shown in Fig. 1(c). The thermal con-
ductivity of thin films also depends strongly on the materials
1
Corresponding author.
Contributed by the Electronic and Photonic Packaging Division of ASME for
publication in the JOURNAL OF ELECTRONIC PACKAGING. Manuscript received May 26,
2016; final manuscript received August 30, 2016; published online October 6, 2016.
Assoc. Editor: Mehdi Asheghi.
Journal of Electronic Packaging DECEMBER 2016, Vol. 138 / 040802-1Copyright V
C2016 by ASME
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 10/19/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
preparation (processing) method, and the substrate that thin films
are sitting on. The conventional thermal conductivity measure-
ment techniques for bulk materials are usually too large in size to
measure the temperature drop and the heat flux across a length
scale ranging from a few nanometers to tens of microns. For
example, the smallest beads of commercial thermocouples have a
diameter of around 25 lm, which could be much larger than the
thicknesses of most electronic thin films.
Significant progresses have also been made for the characteriza-
tion of thermal conductivity and thermal boundary resistance of
thin films over the past 30 years due to the vibrant research in
micro- and nano-scale heat transfer [13–19]. Section 3reviews a
few measurement techniques for thin films including the steady-
state methods, the 3xmethod, and the transient thermoreflectance
technique in both time-domain (TDTR) and frequency-domain
(FDTR), as summarized in Table 1. We note that some techniques
(e.g., 3x, time-domain thermoreflectance (TDTR), and frequency-
domain thermoreflectance (FDTR)) are actually very versatile and
can be applied for the thermal characterization of both bulk and
thin film materials although the techniques reviewed here have
been divided into two categories for bulk materials and thin films
just for convenience.
2 Bulk Materials
2.1 Steady-State Methods. In the steady-state measurement,
the thermal conductivity and interfacial thermal conductance are
determined by measuring the temperature difference DTat a sepa-
ration (distance) under the steady-state heat flow Qthrough the
sample. Figure 2shows the schematic of four different steady-
state methods commonly adopted: absolute technique, compara-
tive cut bar technique, radial heat flow method, and the parallel
thermal conductance technique.
2.1.1 Absolute Technique. Absolute technique is usually used
for samples that have a rectangular or cylindrical shape. When
conducting this measurement, the testing block is placed between
a heat source and a heat sink as shown in Fig. 2(a). The sample is
heated by the heat source with known steady-state power input,
and the resulting temperature drop DTacross a given length (sepa-
ration) of the sample is measured by temperature sensors after a
steady-state temperature distribution is established. The tempera-
ture sensors employed can be thermocouples and thermistors.
Thermocouples are the most widely used sensors due to their wide
range of applicability and accuracy. The resulting measurement
error in DTdue to temperature sensor shall be less than 1% [20].
Thermal conductivity kof the sample can be calculated using
Fourier’s law of heat conduction
k¼QL
ADT(1)
Q¼pQloss (2)
where Qis the amount of heat flowing through the sample, Ais
the cross-sectional area of the sample, Land DTare the distance
and temperature difference between temperature sensors, pis the
applied heating power at heat source side, and Qloss is the parasitic
heat losses due to radiation, conduction, and convection to the
ambient.
The major challenge of the absolute technique is to determine
the heat flow rate Qthrough the sample at the presence of parasitic
heat losses Qloss and to measure temperature difference DTaccu-
rately. Parasitic heat losses include convection and radiation to
the surrounding and conduction through thermocouple wires. In
general, parasitic heat losses should be controlled to be less than
2% of the total heat flow through the sample. To minimize
Fig. 1 (a) Definition of thermal conductivity of a solid bulk
material using Fourier’s law of heat conduction. Temperatures
at the left and right sides are Thand Tc, respectively. Heat trans-
fer cross-sectional area is denoted by A. The heat flow is
assumed to be one-dimensional with no lateral heat loss. (b)
Definition of interfacial thermal conductance Kat the contact
between two bulk materials. Bulk materials are contacted
through limited contact points, which results in a temperature
drop DTacross the interface. (c) Schematic of a typical thin film
material on a substrate structure. The thermal conductivity of
thin films is, in general, anisotropic due to the geometric con-
straints, which means that the values are different in different
directions (i.e., in-plane thermal conductivity kkand cross-
plane thermal conductivity k?). Due to the difference in materi-
als, there usually exists a thermal boundary resistance (the
inverse of interfacial thermal conductance) at the bonded pla-
nar interface.
Table 1 Commonly used thermal characterization techniques
Bulk material Thin film
Steady-state Absolute technique Steady-state electrical heating methods
Comparative technique
Radial heat flow method
Parallel conductance method
Transient (frequency-domain) Pulsed power technique 3xmethod
FDTR technique
Transient (time-domain) Hot-wire method (needle-probe method) TDTR technique
Laser flash method
Transient plane source (TPS) method
040802-2 / Vol. 138, DECEMBER 2016 Transactions of the ASME
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 10/19/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
convection and radiation heat losses, most measurements are con-
ducted under vacuum with radiation shields [21]. Besides the con-
vection and radiation heat losses, another concern is that the heat
conduction through thermocouple wires. It is therefore preferable
to use thermocouples with small wire diameter (e.g., 0.001 in. [3])
and low thermal conductivity wires (e.g., chromel–constantan).
Also, in order to minimize the conduction heat loss, differential
thermocouple with only two wires coming off the sample can be
applied to directly obtain the temperature difference DT[3]. A
typical test apparatus of the absolute technique is the guarded-hot-
plate apparatus. ASTM C177 [20], European Standard EN 12667
[22], and International Standard ISO 8302 [23] have more details
about the apparatus and testing procedure. Major drawbacks of
the absolute technique include: (1) the testing sample should be
relatively large (in centimeter scale or even larger) and can form a
standard circular or rectangular shapes when thermocouples are
used for measuring temperatures. (2) The test usually suffers from
a long waiting time, up to a few hours, to reach steady-state.
Resistance thermometers (i.e., RTDs) [24,25] and infrared (IR)
thermography [26] are often employed for temperature sensing
when testing small samples (in micron scale or even smaller) by
using absolute technique.
The absolute technique has also been applied for measuring the
thermal contact resistance between two components, as shown in
Fig. 3(a). Testing samples are pressed together with controllable
contact pressures. Several thermocouples (usually four or six) are
placed inside the two mating samples at uniform intervals (separa-
tions) to measure local temperature in response to an applied heat-
ing power. Once the steady-state heat flow is achieved,
temperatures are recorded and plotted (denoted by solid dots) in
the temperature versus distance curves as depicted in Fig. 3(b).
Temperatures at both interfaces (i.e., Thand Tc, denoted by hollow
circles) can be deduced assuming the temperature distribution on
each side is linear. The thermal contact resistance is then calcu-
lated by the temperature drop (i.e., ThTc) divided by total heat
flow across the interface. In order to obtain an accurate thermal
contact resistance, the temperature drop across the interface
should be maintained relatively large (e.g., >2C[27]) through
the control of applied heating power.
In addition to bare contacts, thermal interface materials (TIMs)
are usually employed to reduce thermal contact resistance. Com-
monly used TIMs include pressurized gases (e.g., helium and
hydrogen), thermal greases (e.g., silicone oil and glycerin), ther-
mal adhesive, thermal conductive pad, and polymer matrix com-
posited with high thermal conductive particulates, such as silver
and ceramics as fillers, phase-change material, and solders [28].
The one-dimensional steady-state testing method is widely used
for characterizing the thermal conductivity of TIMs, as well as the
thermal contact resistance [29]. The testing instrument is similar
to Fig. 3(a), except that the two testing samples are replaced by
two metal blocks (usually copper) with known thermal conductiv-
ity, and the TIM to be investigated is inserted in between as
shown in Fig. 3(c).
2.1.2 Comparative Technique. The biggest challenge in the
absolute technique is to accurately determine the heat flow
through sample. However, if one has a standard material whose
thermal conductivity is known, the comparative cut bar technique
can be applied and the direct measurement of heat flow is
unnecessary. Figure 2(b)shows the measurement configuration of
the comparative cut bar technique, which is similar to that of the
absolute method. At least two temperature sensors should be
employed on each bar. Extra sensors can be used for conforming
Fig. 3 (a) Measurement configuration and (b) an illustration of
data processing for thermal contact resistance of bulk materi-
als. Thermal contact resistance is calculated by dividing the
temperature drop (T
h
2T
c
) across the interface by the heat flow.
(c) Measurement configuration for thermal interface materials
(TIMs) according to ASTM D5470 standard (not to scale).
Fig. 2 Schematic of steady-state methods commonly used for
measuring the thermal conductivity of bulk materials. (a) Abso-
lute technique: The sample is placed between a heat source
and a heat sink, with known power output, which results in tem-
perature drop across a given length (separation) of the sample.
(b) Comparative cut bar technique: A standard material with
known thermal conductivity is mounted in series with the test
sample. (c) Radial heat flow method: A cylindrical sample is
heated from its axis and as heat flows radially outward, a
steady-state temperature gradient in the radial direction is
established. (d) Parallel thermal conductance technique for
measuring small needlelike samples that cannot support heat-
ers and thermocouples. A sample holder or stage is used
between the heat source and heat sink. A thermal conductance
measurement for the sample holder is carried out first to quan-
tify the heat loss associated with the sample holder. The testing
sample is then attached to the sample holder, and the thermal
conductance is measured again. Thermal conductivity of the
measured sample can then be deduced by taking the difference
of these two measurements.
Journal of Electronic Packaging DECEMBER 2016, Vol. 138 / 040802-3
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 10/19/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
linearity of temperature versus distance along the column. The
selection of temperature sensors depends on the system size, tem-
perature range, meter bars, specimen, and gas within the system
[30], while thermocouples are the most widely employed tempera-
ture sensors. Since the amount of heat flow through the standard
material equals to the heat flow through the measurement sample
target, the thermal conductivity of the measurement sample target
is given by
k1¼k2
A2DT2L1
A1DT1L2
(3)
where subscripts 1 and 2 are associated with the sample and the
standard material, respectively.
By implementing the standard material with known thermal
conductivity, the sample target thermal conductivity measured
can be extracted without heat flow measurement as shown in
Eq. (3), and the associated error due to heat flow measurement
can thus be eliminated. However, efforts are still needed to ensure
equal heat flow between the standard material and the testing
specimen. This technique achieves the best accuracy when the
thermal conductivity of the measurement target is comparable to
that of the standard material [3]. This is the most widely used
method for axial heat flow thermal conductivity testing. ASTM
E1225 [30] gives the experimental requirements and procedure
for the comparative cut bar technique.
Another type of comparative technique is the heat flow meter
method. A heat flux transducer is used, which essentially replaces
the standard material in the comparative cut bar method. By
appropriate calibration of the heat flux transducer using a speci-
men of known thermal conductivity, the thermal conductivity of
measurement sample can then be easily determined using Four-
ier’s law of heat conduction with measured heat flux. This method
is usually used to characterize low thermal conductivity materials,
such as building insulation materials. ASTM C518 [31], ASTM
E1530 [32], and European Standard EN 12667 [22] specify the
test apparatus and calibration method for heat flow meter method.
2.1.3 Radial Heat Flow Method. The two steady-state meth-
ods described above use a longitudinal arrangement of samples to
measure its thermal conductivity. This can be satisfactory at low
temperatures. However, for measurement at very high tempera-
tures (e.g., >1000 K), radiation heat loss from the heater and sam-
ple surfaces is not negligible and can cause large uncertainties
when quantifying the heat flow through the sample. In order to
overcome this, samples with cylindrical geometry are used in the
radial heat flow method. Flynn described the evolution of the
apparatus used for measurement of thermal conductivity by using
this technique [33]. The ASTM C335 [34] and the ISO 8497 [35]
cover the relevant measurement requirements and testing proce-
dure using this method. The sample is heated internally at the
axis, and the heat flows radially outward as depicted in Fig. 2(c).
A steady-state temperature gradient in the radial direction is estab-
lished. Thermocouples are used predominantly for temperature
sensing in radial heat flow method, with an accuracy within
60.1 C[34]. The thermal conductivity is derived from Fourier’s
law of heat conduction in cylindrical coordinate
k¼Qln r2=r1
ðÞ
2pHDT(4)
where r1and r2are the radius where the two temperature sensors
are positioned, His the sample height, and DTis the temperature
difference between the temperature sensors.
2.1.4 Parallel Thermal Conductance Technique. Characteri-
zation of small bulk materials with a size in the millimeter scale is
very challenging because temperature sensing by thermocouples
and the heat flux measurement is extremely difficult. The parallel
thermal conductance technique was introduced by Tritt and
coworkers [36] for small needlelike samples (e.g.,
2.0 0.05 0.1 mm
3
[36] and 10 11mm
3
[37]). Figure 2(d)
shows the typical experimental configuration, which is a variation
of the absolute technique for those samples that cannot support
heaters and thermocouples. A sample holder or stage is used
between the heat source and heat sink. Differential thermocouple
is positioned between the hot side and the post on the one junction
end, and between the cold side and the post on the other junction.
Before measuring the thermal conductivity of the specimen, a
thermal conductance measurement of the sample holder is per-
formed first to quantify the thermal losses associated with the
sample holder. The testing sample is then attached to the sample
holder, and the thermal conductance is measured again. Thermal
conductance of the sample can be deduced by taking the differ-
ence of these two measurements. Thermal conductivity is then
calculated from the thermal conductance by multiplying sample
length and dividing by the sample cross-sectional area. The major
drawback of this method is the requirement to measure cross-
sectional area of such small samples. Inaccuracies in cross-
sectional area measurement can lead to large uncertainties in the
calculated thermal conductivity.
2.2 Transient Technique. In order to overcome the draw-
backs associated with the steady-state methods described previ-
ously, such as parasitic heat losses, contact resistance of
temperature sensors, and long waiting time for establishing
steady-state temperature difference, a variety of transient techni-
ques have been developed. The heat sources used in transient
techniques are supplied either periodically or as a pulse, resulting
in periodic (phase signal output) or transient (amplitude signal
output) temperature changes in the sample, respectively. This sec-
tion focuses on the four commonly used transient techniques,
namely, pulsed power technique, hot-wire method, transient plane
source (TPS) method, and laser flash thermal diffusivity method.
2.2.1 Pulsed Power Technique. Pulsed power technique was
first introduced by Maldonado to measure both thermal conductiv-
ity and thermoelectric power [38]. This technique is a derivative
of the absolute technique in the steady-state methods, with the dif-
ference that a periodic electrical heating current IðtÞis used. This
technique is in principle very close to the Angstrom’s method
[39,40] in terms of the heating method. But the difference is that
the heat sink temperature of this technique is slowly varying dur-
ing the measurement. Figure 4(a)shows the schematic of a typical
setup for pulsed power technique. The sample (usually in cylindri-
cal or rectangular geometry) is held between a heat source and a
heat sink. The heating current used can be either a square wave of
constant-amplitude current or a sinusoid wave [41]. During the
experiment, a periodic electric current with a period of 2sis
applied to the heat source, while the temperature of the heat sink
bath Tcdrifts slowly. A small temperature gradient DT¼ThTc
(usually 0.3 K) is created between the heat source and the heat
sink, which can be measured by a calibrated Au–Fe chromel ther-
mocouple [38]. The heat balance equation between the heat dissi-
pated by the heater and conducted through the sample is given as
Q¼CT
h
ðÞ
dTh
dt ¼ReTh
ðÞI2t
ðÞKTcþTh
2
DTt
ðÞ (5)
where ReðThÞis the electrical resistance of the heater which
changes with temperature.
It is possible to obtain thermal conductance Kas a function of
temperature from the measured temperature Th. However, Eq. (5)
is nonlinear and difficult to be solved analytically. Therefore,
CðTcÞ,RðTcÞ, and KðTcÞare usually used in Eq. (5) to replace
CðThÞ,RðThÞ, and KððThþTcÞ=2Þto linearize the equation. This
assumption holds because temperature difference between Tcand
Th(i.e., DT) is very small (Fig. 4(b)). Also, Tcis considered to be
constant since it is drifted very slowly compared to the periodic
040802-4 / Vol. 138, DECEMBER 2016 Transactions of the ASME
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 10/19/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
current. The final solution has an oscillating sawtoothlike shape as
shown in Fig. 4(b). Smooth curves (i.e., the two dashed lines) are
drawn through the maxima and minima of the oscillations. The
difference between the two dashed smooth curves DTpp yields a
relation for the thermal conductance of the measured sample [38]
K¼RI2
0
DTpp
tanh Ks
2C
(6)
where sis the half period of the heating current, Cis the volumet-
ric heat capacity, Ris the thermal resistance, and I0is the ampli-
tude of electric current.
Numerical iteration can be applied to solve for the thermal con-
ductance term Kin Eq. (6) since all other parameters are known
as a function of temperature. This technique is capable of meas-
uring a wide temperature range from 1.9 to 390 K as reported in
literature [42–44], and an ultra-low thermal conductivity as low as
0.004 W=m K for ZrW
2
O
8
at temperature 2 K [43]. The measure-
ment uncertainty reported by Maldonado is less than 3% [38].
2.2.2 Hot-Wire Method. The hot-wire method is a transient
technique that measures temperature rise at a known distance
from a linear heat source (i.e., hot wire, usually platinum or tanta-
lum) embedded in the test sample. Stalhane and Pyk employed
this method in 1931 to measure the thermal conductivity of solids
and powders [45]. The method assumes an idealized “one-
dimensional radial heat flow” inside the isotropic and homogene-
ous test sample, which is based on the assumption that the linear
heat source has infinite length and infinitesimal diameter as shown
in Fig. 5. When there is an electric current of constant intensity
that passes through the hot wire, thermal conductivity of the test
sample can be derived from the resulting temperature change at a
known distance from the hot wire over a known time interval.
Hot-wire method is commonly used to measure low thermal con-
ductivity materials, such as soils [46], ice cores [47], and refracto-
ries (refractory brick, refractory fibers, plastic refractories, and
powdered materials [48]). This method has also been commonly
used for measuring the thermal conductivity of liquids. ASTM
C1113 and ISO 8894 specify more details on apparatus and test
procedure for the measuring of refractories using hot-wire method
[48,49].
As the hot wire produces a thermal pulse for a finite-time with
constant heating power, it generates isothermal lines in an infinite
homogeneous medium initially at thermal equilibrium. The tran-
sient temperature for sufficiently long time from the start of the
heating can be expressed with good approximation by [50]
Tr;t
ðÞ
¼p
4pkL ln 4at
r2
þr2
4at1
4
r2
4at
c
"#
(7)
For sufficiently large values of the time t, the terms r2=4at
inside the parenthesis are negligible because it is far less than one.
The above equation can then be simplified to
Tr;t
ðÞ
¼p
4pkL ln 4at
r2
c
(8)
The temperature rise at a point in the test sample from time t1
to t2is given by
DT¼Tt
2
ðÞ
Tt
1
ðÞ
¼p
4pkL ln t2
t1
(9)
Thermal conductivity is then obtained from the temperature
rise DTversus natural logarithm of the time lnðtÞexpressed below
Fig. 4 (a) Schematic diagram of a typical pulsed power tech-
nique for thermal conductivity measurement where periodic
square-wave current is applied. (b) The time-dependent temper-
ature difference DT5Th2Tcacross the sample, from Ref. [38]
where the temperature of the heat sink is slowly drifted. The
dots represent experimental data, and line represents calcula-
tion results.
Fig. 5 The measurement principle of the hot-wire method. (a)
One-dimensional radial heat flow is assumed inside the test
sample. (b) As an electric current of fixed intensity passes
through the hot wire, the thermal conductivity of the test sam-
ple can be derived from the resulting temperature change at a
known distance from the hot wire over a known time interval.
This figure is reproduced with permission from Franco [50].
Copyright 2007 by Elsevier.
Journal of Electronic Packaging DECEMBER 2016, Vol. 138 / 040802-5
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 10/19/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
k¼p
4pTt
2
ðÞ
Tt
1
ðÞ½
Lln t2
t1
(10)
It should be noted that when rin Eq. (8) equals to zero, the
wire will act both as a line source heater and a resistance ther-
mometer. Today’s hot-wire method instruments allow more than
1000 data readings of the transient temperature rise from times
less than 1 ms up to 1 s (or 10 s, in the case of solids) coupled with
finite-element methods to establish a very low uncertainty [51]. If
applied properly, it can achieve uncertainties below 1% for gases,
liquids, and solids, and below 2% for nanofluids [51]. Despite its
advantages, there are very few commercial hot-wire instruments
[52]. Possible reason is due to the delicacy of the very thin wire
which easily snaps.
The needle-probe method, also for testing isotropic and homo-
geneous materials, is a variation of the hot-wire method. Working
principle of the needle-probe method is same as the hot-wire
method. But the temperature sensing is based on a zero radius sys-
tem (i.e., r¼0 in Eq. (8)). The heating wire and the temperature
sensor (thermocouple) are encapsulated in a probe that electrically
insulates the heating wire and the temperature sensor from the test
sample. The probe helps protect the heating wire. This configura-
tion is particularly practical where thermal conductivity is deter-
mined by a probe inserted in the test sample. Therefore, the
method is conveniently applied to powderlike materials and soils.
A needle-probe device can be used to measure sample thermal
properties in situ, but most commonly a temperature-controlled
furnace is used to produce the base temperatures for the measure-
ments [53]. ASTM D5930 [54] and ASTM D5334 [55] standard-
ize the test procedure and data analysis method for the needle-
probe method.
2.2.3 Transient Plane Source (TPS) Method. The TPS method
(i.e., hot disk method) uses a thin metal strip or a disk as both a
continuous plane heat source and a temperature sensor as depicted
in Fig. 6(a). The metal disk is first sealed by electrical insulation
and then sandwiched between two identical thin slab-shaped test-
ing samples. All other surfaces of the testing samples are ther-
mally insulated. During the experiment, a small constant current
is applied to the metal disk to heat it up. Since the temperature
increase of the metal disk is highly dependent on the two testing
samples attached to it, thermal properties of the testing samples
can be determined by monitoring the temperature increase for a
short time period. This time period is generally only a few seconds
so that the metal disk can be considered in contact with infinite
size samples throughout the transient signal recording process.
Temperature increase at the sensor surface DT(e.g., 1–3 C[57])
as a function of time can be measured. Measurement accuracy of
the temperature sensor (temperature resistance thermometer) is
usually 60.01 C[57]. Then, fitting the temperature curves by
Eqs. (11) and (12) to the measured DTrenders the inverse of ther-
mal conductivity 1=k[58]
DT/ðÞ¼ Q
p1:5rk D/ðÞ (11)
/¼ffiffiffiffi
ta
r2
r(12)
where ris the sensor radius, and Dð/Þis a dimensionless theoreti-
cal expression of the time-dependent increase that describes heat
conduction of the sensor (Fig. 6(b)).
The TPS method was reported to be capable for measuring
materials with thermal conductivity ranging from 0.005 to 500
W=m K in the temperature range from cryogenic temperatures to
500 K, including liquids, aerogels, and solids [59–62]. ASTM
D7984 [57] and ISO 22007-2 [63] specify the test apparatus
and procedure for the TPS method. One drawback of TPS
measurement is that each of the two sample pieces needs to have
one entirely planar side. This makes it difficult for some materials,
especially powders or granules [64]. The TPS measurement errors
come from several sources: (1) thermal contact resistance between
the sensor and the testing samples, (2) the thermal inertia of the
sensor, (3) the measured power input being influenced by the heat
capacity of the electrical isolation films, and (4) the electrical
resistance change of the metallic disk sensors. Model corrections
for these errors are necessary to improve measurement accuracy
when doing data analysis. Readers can refer to Refs. [59,60,62]
for more information. For example, when considering the influ-
ence of the sensor’s electrical resistance change with temperature,
Eq. (11) needs to be revised to [62]
DTs
ðÞ¼Q2V1V2
V1þV2
ðÞ
2Re;sCDTt
ðÞ
t
"#
D/
ðÞ
p1:5rk (13)
2.2.4 Laser Flash Method for Thermal Diffusivity. Contact
thermal resistance is a major source of error for temperature mea-
surement. The laser flash method employs noncontact, nondes-
tructive temperature sensing to achieve high accuracy [65]. The
method was first introduced by Parker et al. [66]. It uses optical
heating as an instantaneous heating source, along with a
Fig. 6 (a) Top and cross-sectional views of the TPS method
[56]. Vis a constant voltage source, V1and V2are the precision
voltmeters, Re;sis a standard resistor, and Reis the nonlinear
resistance of the sensor. The metal disk (i.e., sensor) is sealed
by electrical insulation and then sandwiched between two test
samples. A small constant current is applied to the metal disk
to heat it up. Since the temperature increase of the metal disk is
highly dependent on the two testing samples attached to it,
thermal properties of the testing samples can be determined by
monitoring this temperature increase for a short time period.
(b) A typical plot shows Dð/Þas a function of /./is defined in
Eq. (12).Dð/Þis a dimensionless theoretical expression of the
time-dependent increase that describes heat conduction of the
sensor. The curve is reproduced with permission from Bou-
guerra et al. [58]. Copyright 2001 by Elsevier.
040802-6 / Vol. 138, DECEMBER 2016 Transactions of the ASME
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 10/19/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
thermographic technique for quick noninvasive temperature sens-
ing. The testing sample is usually a solid planar-shaped material
when measuring thermal conductivity and is a multilayer structure
when characterizing thermal contact resistance. A typical mea-
surement configuration for laser flash method is depicted in Fig.
7(a). An instantaneous light source is used to uniformly heat up
the sample’s front side, and a detector measures the time-
dependent temperature rise at the rear side. Heat conduction is
assumed to be one-dimensional (i.e., no lateral heat loss). The
testing sample is usually prepared by spraying a layer of graphite
on both sides to act as an absorber on the front side and as an
emitter on the rear side for temperature sensing [68]. The rear-
side infrared radiation thermometer should be fast enough to
respond to the emitting signals, and the precision of temperature
calibration is usually 60.2 K [53]. Dynamic rear-side temperature
response curve (Fig. 7(b)) is used to fit the thermal diffusivity.
The higher the thermal diffusivity of the sample, the faster the
heat transfer and temperature rise on the rear side.
Theoretically, the temperature rise at the rear side as a function
of time can be written as [66]
Tt
ðÞ¼q
qcpd1þ2X
1
n¼11
ðÞ
nexp n2p2
d2at
"#
(14)
where dis the sample thickness, and ais the thermal diffusivity.
To simplify Eq. (14), two dimensionless parameters, Wand g, can
be defined
WðtÞ¼TðtÞ=Tmax (15)
g¼p2at=d2(16)
where Tmax denotes the maximum temperature at the rear side.
The combination of Eqs. (14)–(16) yields
W¼1þ2X
1
n¼1ð1Þnexp ðn2gÞ(17)
When Wis equal to 0.5, which means the rear-side temperature
reaches to one-half of its maximum temperature, gis equal to
1.38, and so the thermal diffusivity ais calculated by [66]
a¼1:38d2
p2t1=2
(18)
where t1=2is the time that takes for the sample to heat to one-half
maximum temperature on the rear surface.
ASTM E1461 [69] and ISO 22007-4 [70] specify the require-
ments of apparatus, test sample, and procedure for thermal diffu-
sivity measurement by the laser flash method. In addition to the
thermal diffusivity, a, measured using the laser flash method,
material density, q, and specific heat, cp, need to be measured
from separate experiments, to obtain the thermal conductivity
using the relationship k¼aqcp. The laser flash method is capable
of measuring thermal conductivity over a wide temperature range
(120 C to 2800 C[70]) with measurement uncertainty reported
to be less than 3% [71]. The advantages of the method are not
only its speed (usually 1–2 s for most solid) but also its capability
to use very small samples, e.g., 5–12 mm in diameter [53]. There
are, however, some considerations that should be kept in mind
before carrying out laser flash measurement. First of all, sample
heat capacity and density should be known or determined from
separate experiments, which may result in the “stack up” of uncer-
tainties and lead to larger errors. Another criticism of the laser
flash method is that the effect of sample holder heating could lead
to significant error in the measurements if not accounted for prop-
erly [72]. Though laser flash method can be used to measure thin
films, thickness of the measured sample is limited by the time-
scales associated with the heating pulse and the infrared detector.
Typical commercial laser flash instruments can measure samples
with a thickness of 100 lm and above depending on the thermal
diffusivity of the sample. For thin film sample with a thickness
less than 100 lm, one needs to resort to the 3xmethod or transient
thermoreflectance techniques developed over the past two
decades.
3 Thin Films
3.1 Steady-State Methods
3.1.1 Cross-Plane Thermal Conductivity Measurement. Tem-
perature drop across a thin film sample needs to be created and
measured to characterize the cross-plane thermal conductivity.
Creating and measuring the temperature drop are extremely chal-
lenging when the sample thickness is as small as a few nano-
meters to tens of microns. Figure 8shows the schematic of two
steady-state measurement configurations being frequently
employed. In both configurations, thin films with thickness dfare
grown or deposited onto a substrate with high thermal
Fig. 7 (a) Schematic of the laser flash method for thermal dif-
fusivity. A light source heats the front side of the planar-shaped
sample, and an infrared detector measures the time-dependent
temperature rise at the rear side. Thermal diffusivity of the sam-
ple is obtained by fitting the dynamic response of rear side tem-
perature rise of the sample. (b) An example showing the
measured rear side temperature rising curve. The curve is
reproduced with permission from Campbell et al. [67]. Copy-
right 1999 by IEEE.
Journal of Electronic Packaging DECEMBER 2016, Vol. 138 / 040802-7
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 10/19/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
conductivity and small surface roughness (e.g., polished silicon
wafer). A metallic strip with length Land width 2a(L2a)is
then deposited onto the thin film whose thermal conductivity is to
be determined. The metallic strip should have a high temperature
coefficient of resistance, such as Cr/Au film. During the experi-
ment, the metallic strip is heated by a direct current (DC) passing
through it. The metallic strip serves as both an electrical heater
and a sensor to measure its own temperature Th.
The temperature at the top of the film Tf;1is generally assumed
to be the same as the average heater temperature Th. The most
straightforward way would be using another sensor to directly
measure the temperature Tf;2at the bottom side of the film (Fig.
8(a)). But this approach complicates the sample preparation proc-
esses which usually involve cleanroom microfabrication. The
other approach is using another sensor situated at a known dis-
tance away from the heater/sensor to measure temperature rise of
the substrate right underneath it (Fig. 8(b)). A two-dimensional
heat conduction model is then used to infer the substrate tempera-
ture rise at the heater/sensor location from the measured substrate
temperature rise at the sensor location [15].
3.1.2 In-Plane Thermal Conductivity Measurement. The
major challenge of measuring in-plane thermal conductivity is the
evaluation of heat flow along the film in the presence of parasitic
heat loss through the substrate. In order to increase measurement
accuracy, Volklein et al. [73] concluded that it is desirable to have
the product of the thin film in-plane thermal conductivity kf;kand
film thickness dfequal to or greater than the corresponding prod-
uct of the substrate (i.e., kf;kdfkSdS). However, in order to com-
pletely remove parasitic heat loss through the substrate, a
suspended structure by removing the substrate as shown in Fig. 9
is desirable, which complicates microfabrication for sample prep-
aration [74].
Figure 9depicts the schematic of the two steady-state methods
for measuring in-plane thermal conductivity along suspended thin
films. Figure 9(a)shows the measurement configuration that was
first developed by Volklein and Starz [75,76], where a metallic
strip (Cr/Au film) is deposited on top of the thin film that serves
as both an electrical heater and temperature sensor. When a DC
passes through the heater/sensor, the temperature rise in the
heater/sensor is a function of the heating power, thin film thermal
conductivity, ambient temperature, thin film thickness df, and
width Lf. The in-plane thermal conductivity can then be deduced
from the difference in heater/sensor temperature rise of two meas-
urements using two different thin film widths with all other
parameters unchanged [76]. The other steady-state method is
shown in Fig. 9(b), another sensor is used to measure the heat sink
temperature where the thermal conductivity can then be straight-
forwardly written as
kf;k¼QLf
2dfTf;1Tf;2
ðÞ (19)
where Qis the power dissipated in the metallic heater per unit
length; Lf=2 is the distance from the heater to the heat sink; Tf;1is
the thin film temperature right underneath the heater/sensor,
which is assumed the same temperature as heater/sensor; and Tf;2
is the temperature of the thin film edge in contact with the
substrate.
For measuring the thermal conductivity of an electrically con-
ductive material or a semiconducting material, an additional elec-
trical insulation layer is needed between the electrical heater/
sensor and thin film for both methods, which significantly compli-
cated the data analysis. In order to ensure one-dimensional heat
conduction inside thin film, parasitic heat losses must be mini-
mized, which include heat conduction loss along the length direc-
tion of the heater/sensor and convection and radiation losses to the
ambient. Heat conduction along the heater/sensor can be mini-
mized through advanced microfabrication to minimize its cross-
sectional area. To minimize heat loss to the ambient, the experi-
mental measurement is usually carried out under vacuum. Usually
a small temperature difference is used to minimize the radiation
heat loss, while coating the surface with low-emissivity material
could be another option. Nevertheless, the most effective way to
deal with radiation heat loss would be using a transient heating
(e.g., using alternating current (AC)) and temperature sensing
technique, such as the 3xmethod.
3.2 Transient Methods
3.2.1 3xMethod. The 3xmethod is widely used to measure
thermal properties of both bulk materials and thin films after it
was first introduced in 1990 by Cahill et al. [77,78]. Figure 10
shows a typical schematic of the 3xmeasurement. The thin film
of interest is grown or deposited on a substrate (e.g., silicon and
sapphire [79]). A metallic strip (e.g., aluminum, gold, and plati-
num) is deposited on top of a substrate or the film-on-substrate
stack. Dimensions of the metallic strip are usually half-width a
¼10–50 lm and length L¼1000–10,000 lm, which is treated as
infinitely long in the mathematical model. The metallic strip
serves as both an electrical heater and a temperature sensor, as
shown in Fig. 10. An AC at frequency xpasses through the
heater/sensor, which is expressed as
IðtÞ¼I0cos ðxtÞ(20)
where I0is the current amplitude, which results in Joule heating
of the resistive heater/sensor at 2xfrequency because of its elec-
trical resistance. Such a 2xheating leads to temperature change
of the heater/sensor also at 2xfrequency
DTðtÞ¼DT0cos ð2xtþuÞ(21)
where DT0is the temperature change amplitude, and uis the
phase. The temperature change perturbs the heater/sensor’s elec-
trical resistance at 2x
Fig. 8 Steady-state methods for measuring the cross-plane
thermal conductivity of thin films. In both measurements, the
metallic strip serves as both an electrical heater and a tempera-
ture sensor measuring its own temperature rise. The tempera-
ture Tf;1on the top side of the thin film is generally assumed to
be the same as the heater/sensor average temperature Th.T‘is
the ambient temperature. In order to determine the temperature
Tf;2at the bottom of the thin film, either (a) a sensor is depos-
ited between thin film and substrate for direct measurement, or
(b) a sensor is deposited at a known distance away from the
heater. A two-dimensional heat conduction model is then used
to calculate the temperature at the bottom of the thin film.
Fig. 9 (a) In-plane thermal conductivity can be deduced from
the difference in heater/sensor temperature rise of two meas-
urements using two different thin film widths Lfwith all other
parameters unchanged [73]. (b) Steady-state thermal conductiv-
ity measurement with a heater/sensor and an additional temper-
ature sensor.
040802-8 / Vol. 138, DECEMBER 2016 Transactions of the ASME
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 10/19/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
ReðtÞ¼Re;0ð1þaRDTÞ¼Re;0½1þaRDT0cos ð2xtþuÞ (22)
where aRis the temperature coefficient of resistance of the heater/
sensor, and Re;0is the heater/sensor’s electrical resistance at the
initial state. When multiplied by the 1xdriving current, a small
voltage signal across the heater/sensor at 3xfrequency can be
detected [80]
Vt
ðÞ¼It
ðÞ
Rt
ðÞ
¼Re;0I0cos xt
ðÞ
þ1
2Re;0I0aRDT0cos xtþu
ðÞ
þ1
2R0I0aRDT0cos 3xtþu
ðÞ (23)
This change in voltage at 3xfrequency (i.e., the third term in
Eq. (23)) has the information about thermal transport within the
sample [81]. However, since the 3xvoltage signal (amplitude
R0I0aRDT0=2) is very weak and usually about three orders of mag-
nitude smaller than the amplitude of the applied 1xvoltage, a
lock-in amplifier is usually employed for implementing such
measurement technique.
When measuring the thermal conductivity of an electrically
conductive material or a semiconducting material, an additional
electrical insulation layer is needed between the electrical heater/
sensor and thin film. Depending on the width of the heater, both
the cross-plane and the in-plane thermal conductivity of thin films
can be measured using the 3xmethod. Approximate analytical
expressions are usually employed to determine the cross-plane
and the in-plane thermal conductivity. Borca-Tasciuc et al. pre-
sented a general solution for heat conduction across a multilayer-
film-on-substrate system [13]. Dames also described the general
case of thermal and electrical transfer functions framework [81].
For the simplest case that a metallic heater/sensor is deposited on
an isotropic substrate without a thin film, the heater/sensor can be
approximated as a line source if the thermal penetration depth Lp¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi
aS=2x
pis much larger than the heater/sensor half-width a.By
choosing an appropriate heating frequency of the heating current,
thermal penetration can be localized within the substrate. The tem-
perature rise of the heater/sensor can then be approximated as [13]
DTS¼p
pLkS
0:5ln aS
a2
0:5ln x
ðÞ
þg
ip
4LkS
¼p
pLkS
flinear lnx
ðÞ (24)
where subscript Sis associated with substrate, gis a constant, iis
ffiffiffiffiffiffiffi
1
p,kis the thermal conductivity, p=Lis the peak electrical
power per unit length, and flinear is a linear function of lnx.Itis
clear that the isotropic thermal conductivity of substrate kScan be
determined from the slope of the real part of the temperature
amplitude as a linear function of the logarithm frequency lnðxÞ
(i.e., the “slope method”), according to Eq. (24).
With a film-on-substrate, one needs to estimate the temperature
drop across the thin film to find the cross-plane thermal conductiv-
ity kf;?(Fig. 11(a)). The temperature at the upper side of the film
is usually taken to be equal to the heater/sensor temperature
because the contact resistances are typically very small, 108
107m2K=W[82]. The most common method to determine the
temperature at the bottom side of the film is calculated from the
experimental heat flux with the substrate thermal conductivity kS,
which is usually known or can be measured using the 3xmethod
as shown in Eq. (24). Assuming one-dimensional heat conduction
across the thin film (Fig. 10(a)), the thermal conductivity of the
thin film can be easily determined from
DTSþf¼DTSþpdf
2aLkf;?
(25)
where subscript fdenotes thin film properties, and subscript Sþf
denotes thin film-on-substrate structure. kf;?is obtained by fitting
the experimentally measured temperature rise data under a variety
of heating frequency xto Eq. (25) (see Fig. 12).
The 3xmethod has also been extensively used to measure the
in-plane thermal conductivity of thin films. In comparison to the
cross-plane thermal conductivity measurement, a much narrower
heater is used so that the heat transfer process within the film is
sensitive to both in-plane and cross-plane thermal conductivity as
shown in Fig. 11(b). The half-width aof the heater should be nar-
row enough to satisfy [81]
a
df
kf;?
kf;k
!
1=2
0:1(26)
where kf;?and kf;kare the cross-plane and in-plane thermal con-
ductivities of the thin film, respectively, and dfis the film thick-
ness. Due to the lateral heat spreading which is sensitive to the in-
plane thermal conductivity, a two-dimensional heat transfer model
needs to be used for data reduction. The temperature drop across
the thin film is obtained as [13]
DTf¼p
pL
1
kf;?kf;k
1=2ð1
0
sin2k
k3tanh kdf
a
kf;k
kf;?
!
1=2
2
43
5dk
(27)
Equation (27) gives the temperature drop of thin film normal-
ized to the value for purely one-dimensional heat conduction
through the film, as a function of the in-plane and cross-plane
Fig. 10 (a) Top view and (b) cross-sectional view of a typical
3xmethod for thermal characterization of thin films. A metallic
strip (i.e., heater/sensor) with width 2ais deposited on top of
the thin film, which functions as both heater and temperature
sensor. An alternating current at angular frequency 1xheats
the heater/sensor at 2xfrequency. The change in temperature
of the heater/sensor causes a change in the resistance which in
turn produces a small change in the voltage at 3xfrequency.
An electrical insulation layer (not shown in the figure) is
required between the thin film and the heater/sensor when
measuring electrical conducting thin films.
Fig. 11 Schematic of the cross-plane and the in-plane thermal
conductivity measurement using the 3xmethod. For cross-
plane thermal conductivity measurement, the heater width
should be relatively large compared to thin film thickness in
order to satisfy the assumption that the heat conduction is one-
dimensional across the thin film. For the in-plane thermal con-
ductivity measurement, narrower-width heater is used so that
the in-plane thermal conductivity can be deduced through heat
spreading in the thin film.
Journal of Electronic Packaging DECEMBER 2016, Vol. 138 / 040802-9
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 10/19/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
thermal conductivities and the heater/sensor half-width a. In prac-
tice, kf;?is usually measured first with a heater/sensor with much
greater width that is only sensitive to cross-plane thermal conduc-
tivity. kf;kis then measured with a much smaller heater/sensor
width (see Fig. 12).
The previously-mentioned 3xmethod has been limited to sam-
ples with thermal conductivity tensors that are either isotropic or
have their principle axes aligned with the Cartesian coordinate
system defined by the heater line and the sample surface.
Recently, Mishra et al. introduced a 3xmethod that can measure
an arbitrarily aligned anisotropic thermal conductivity tensor [83].
An anisotropic thermal conductivity tensor with finite off-
diagonal terms is considered. An exact closed-form solution has
been found and verified numerically. The authors found that the
common slope method yields the determinant of the thermal con-
ductivity tensor, which is invariant upon rotation about the heat-
er’s axis. Following the analytic result, an experimental scheme is
proposed to isolate the thermal conductivity tensor elements. By
using four heater lines, this method can measure all the six
unknown tensor elements in the three dimensions.
A significant advantage of the 3xmethod over the conventional
steady-state methods is that the error due to radiation heat loss is
greatly reduced. Errors due to thermal radiation are shown to scale
with a characteristic length of the experimental geometry. The
calculated error of 3xmeasurement due to radiation is less than
2% even at a high temperature of 1000 K [84]. The 3xmethod
can be used for measuring dielectric, semiconducting, and electri-
cally conducting thin films. For electrically conducting and semi-
conducting materials, samples need to be electrically isolated
from the metallic heater/sensor with additional insulating layer
[79,85], which introduces extra thermal resistance and inevitably
reduces both sensitivity and measurement accuracy. Another chal-
lenge is that the 3xmethod involves microfabrication for the
metallic heater/sensor. Optical heating and sensing method (e.g.,
transient thermoreflectance technique), on the other hand, usually
requires minimal sample preparation.
3.2.2 Transient Thermoreflectance Technique. The transient
thermoreflectance technique is a noncontact optical heating and
sensing method to measure thermal properties (thermal conductiv-
ity, heat capacity, and interfacial thermal conductance) of both
bulk and thin film materials. Samples are usually coated with a
metal thin film (e.g., aluminum or tungsten), referred to as metal-
lic transducer layer, whose reflectance changes with the tempera-
ture rise at the laser wavelength. This allows to detect the thermal
response by monitoring the reflectance change. Figure 13 shows
the schematic diagrams of the sample configuration for a thin film
and a bulk material being measured using concentrically focused
pump and probe beams. The thermoreflectance technique was pri-
marily developed in the 1970s and 1980s when continuous wave
(CW) light sources are used to heat the sample [86,87]. With the
advancement of pico- and femtosecond pulsed laser after 1980, this
technique has been widely used for studying nonequilibrium
electron–phonon interaction [88,89], detecting coherent phonon
transport [90–92] and thermal transport across interfaces [93–95].
This technique has recently been further developed over the last few
years for measuring anisotropic thermal conductivity of thin films
[96–98] and probing spectral phonon transport [99–102].
The transient thermoreflectance technique can be implemented
as both time-domain thermoreflectance (TDTR) method [103,104]
and frequency-domain thermoreflectance (FDTR) method
[97,105]. The TDTR method measures the thermoreflectance
response as a function of the time delay between the arrival of the
probe and the pump pulses at the sample surface. Figure 14(a)
shows a typical experimental system. A Ti–sapphire oscillator is
used as the light source with wavelength centered at around
800 nm and a repetition rate of 80 MHz. The laser output is split
into pump beam for heating and probe beam for sensing. Before
focusing on the sample, the pump beam is modulated by an
acoustic-optic modulator (AOM) or electro-optic modulator (EOM)
at a frequency from a few kilohertz to a few megahertz. The probe
beam passes through a mechanical delay stage such that the temper-
ature responses are detected with a delay time (usually a few pico-
seconds to a few nanoseconds) after the sample is heated by the
pump pulse. The signal from thermoreflectance change is then
extracted by the lock-in amplifier. Spatially separating the pump
and the probe or spectrally screening the pump beam with a filter
(the two-color [106] and two-tint method [107]) can avoid the scat-
tered light from modulated pump beam entering the photodetector.
The other group of thermoreflectance technique is FDTR,
where the thermoreflectance change is measured as a function of
the modulation frequency of the pump beam. Therefore, FDTR
can be easily implemented using the same TDTR system
(Fig. 14(a)) by fixing the delay stage at a certain position and
varying the modulation frequency. FDTR system, however, can
Fig. 12 Experimental temperature rise (dots) measured by 30
and 2 lm width heater/sensor deposited on the Si/Ge quantum
dot superlattice and the reference samples without the super-
lattice film of interest. The experimental signal is compared to
the predictions for the temperature rise of the heater/sensor
calculated based on Eqs. (24),(25), and (27) for the fitted values
of in-plane and cross-plane thermal conductivity of the film.
The figure is reproduced with permission from Borca-Tasciuc
et al. [13]. Copyright 2001 by AIP Publishing.
Fig. 13 Typical sample configuration used for measuring ther-
mal properties of (a) thin film on a substrate and (b) bulk mate-
rial, using the transient thermoreflectance technique with
concentric pump and probe beams. The sample is heated by
the frequency-modulated pump laser pulse. The change in the
temperature-dependent reflectivity of the metal transducer is
measured by the probe laser pulse that is delayed in a picosec-
ond to nanosecond timescale. In the figure, G2is the interfacial
thermal conductance between transducer and the sample. G4is
the interfacial thermal conductance between thin film and
substrate.
040802-10 / Vol. 138, DECEMBER 2016 Transactions of the ASME
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 10/19/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
avoid the complexity of beam walk-off and divergence associated
with the mechanical delay stage because the probe delay is fixed.
FDTR system can also be implemented using the less expensive
continuous wave (CW) lasers as shown in Fig. 14(b), which
achieves similar accuracy to TDTR measurement for thermal con-
ductivity of many thin film materials [97,104,105,108]. Similar to
TDTR, the pump beam of CW-based FDTR (CW-FDTR) is
modulated by EOM and creates a time-dependent temperature
gradient. However, the heating process by pump incidence is con-
tinuous in CW-FDTR. The probe beam is directly focused on the
sample without passing through a mechanical delay stage. The
thermoreflectance change embedded in the reflected probe beam
is also extracted using a photodiode and a lock-in amplifier.
An illustration of the TDTR data acquisition is given in Fig. 15.
The pump pulses modulated at frequency x0(Fig. 15(a)) heat the
sample periodically. The oscillating temperature response of the
sample (Fig. 15(b)) is then detected by the probe beam arriving
with a delayed time s0. In this case, the thermoreflectance
response Zat modulation frequency x0is expressed as an accu-
mulation of the unit impulse responses hðtÞin time domain
Zx0;sd
ðÞ
¼Vin þiVout ¼b2p
xsX
1
n¼0
hn
2p
xsþsd
eix0nT0þsd
ðÞ
(28)
where Vin and Vout are the real and imaginary part of the response
usually referred to as in-phase and out-of-phase signal, respec-
tively, sdis the delay time, ð2p=xsÞis the time between two suc-
cessive pulses at laser repetition frequency xs, and bis a constant
coefficient determined by
b¼GdetP11R1
ðÞ
dR2
dT
P2R2(29)
where Gdet is the gain of photodetector; P1and P2are the power
of pump and probe beams, respectively; R1and R2are the reflec-
tivity at the wavelengths of pump and probe beams, respectively;
and ðdR2=dT Þis the thermoreflectance coefficient of the trans-
ducer at probe wavelength. Identically, the thermoreflectance
response Zcan be expressed in the frequency domain
Zðx0;sdÞ¼Vin þiVout ¼bX
1
l¼1
Hðx0þlxsÞeilxssd(30)
where Hðx0þlxsÞis the thermoreflectance response of the sam-
ple heated by a continuous Gaussian beam modulated at frequency
Fig. 14 (a) Schematic of a two-color TDTR system. After the
first polarized beam splitter (PBS), the probe beam passes
through a beam expander to minimize the spot size change
over length of the delay stage, and it is then directed onto the
sample through an objective lens at normal incidence. The
pump beam passes through a second harmonic generation
(SHG) crystal for frequency doubling and then the EOM for
modulation before being directed through the same objective
onto the sample, coaxial with the probe beam. This system can
also be used for FDTR measurement by fixing the delay stage
and varying the modulation frequency. (b) Schematic of an
FDTR system based on two CW lasers with different
wavelengths. The pump beam is modulated by EOM to create a
periodic heating, and the probe beam measures the thermore-
flectance change. This FDTR system is reproduced with per-
mission from Schmidt [106]. Copyright 2013 by Begell House
Digital Library.
Fig. 15 An illustration of the TDTR detection scheme: (a) the
pump pulse input to the sample is modulated by the electro-
optic modulator or acoustic-optic modulator, (b) sample sur-
face temperature response to the pump pulse input, (c) the
probe pulses arrive at the sample with a delayed time and are
reflected back to a detector with an intensity proportional to the
surface temperature, and (d) the fundamental harmonic compo-
nents of the reference wave and measured probe wave. The
amplitude and phase difference between these two waves are
picked up by a lock-in amplifier. (Reproduced with permission
from Schmidt et al. [98]. Copyright 2008 by AIP Publishing.)
Journal of Electronic Packaging DECEMBER 2016, Vol. 138 / 040802-11
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 10/19/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
ðx0þlxsÞ. The single frequency response HðxÞis determined by
thermal properties including thermal conductivity k, heat capacity
Cof each layer, and interfacial thermal conductance Gbetween
different layers.
Here, we show an outline for the derivation of the single fre-
quency response function HðxÞ, while the detailed derivation of
the heat transfer model can be found in Refs. [93,104,106]. The
heat conduction equation in cylindrical coordinates is written as
C@T
@t¼kr
r
@
@rr@T
@r
þkz
@2T
@z2(31)
where krand kzare the thermal conductivity in the in-plane and
cross-plane directions (see Fig. 16 for definition of in-plane and
cross-plane directions).
By applying the Fourier transformation to the time variable t
and Hankel transform to the radial coordinate r;a transfer matrix
can be obtained relating the temperature htand heat flux Fton the
top of the single slab and the temperature hband heat flux Fbon
the bottom side (Fig. 16(a))
hb
Fb
¼cosh qd 1
kzqsinh qd
kzqsinh qd cosh qd
0
@1
A
ht
Ft
(32)
where dis the thickness of the slab, and the complex thermal
wave number q¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ððkrx2þixCÞ=kzÞ
p, where xis the Hankel
transform variable. Multiple layers can be handled by multiplying
the matrices for individual layers together (Fig. 16(b))
hb
Fb
¼MnMn1…M1
ht
Ft
¼AB
CD
ht
Ft
(33)
For interfaces, the transfer matrix can be obtained by taking the
heat capacity as zero and choosing kzand dsuch that interfacial
thermal conductance Gequals to kz=d. The boundary condition
is approximately adiabatic in experimental condition, hence
Fb¼ChtþDFt¼0, the surface temperature can be obtained by
ht¼D
CFt(34)
In experiment, if a Gaussian laser with radius w0and power Q
is used as pump, the heat flux Ftafter Hankel transform is
Ft¼ðQ=2pÞexpððw2
0x2Þ=8Þ, then Eq. (34) becomes
ht¼D
C
Q
2pexp w2
0x2
8
(35)
The reflectivity response is then weighted by the Gaussian dis-
tribution of probe beam intensity with radius w1
Hx
ðÞ
¼Q
2pð1
0
xD
C
exp w2
0þw2
1
x2
8
dx (36)
We give an example of TDTR signal detected by the lock-in
amplifier in Fig. 17. Thermal properties including the thermal
conductivity, heat capacity, and interfacial thermal conductance
are encoded in the TDTR signal trace. The in-phase signal Vin rep-
resents the change of surface temperature. The peak in the Vin rep-
resents the surface temperature jump right after the pump pulse
incidence, and the decaying tail in Vin represents the cooling of sur-
face due to the heat dissipation in the sample. The out-of-phase sig-
nal can be viewed as the sinusoidal heating of the sample at
modulation frequency x0[109]. When processing the experimental
Fig. 16 Schematic of heat transfer model for deriving single
frequency response in cylindrical coordinate. The rdenotes in-
plane direction and zdenotes cross-plane direction. (a) The
transfer matrix of a single layer and (b) the transfer matrices of
multilayers. hand Fdenote temperature and heat flux, respec-
tively, after Fourier and Hankel transform, and the subscripts t
and bdenote top and bottom sides.
Fig. 17 Measured TDTR temperature responses in terms of (a)
in-phase signal, (b) out-of-phase signal, and (c) the ratio 2
Vin=Vout on amorphous SiO
2
sample coated with 100 nm Al as
metallic transducer. Here, a pump spot size of 9.1 lm, a probe
spot size of 7.5 lm, and a modulation frequency of 1.06 MHz
were used. The solid line in (c) is predicted by the heat transfer
model with thermal conductivity k51.4 W=m K and interfacial
thermal conductance G5120 MW=m2K.
040802-12 / Vol. 138, DECEMBER 2016 Transactions of the ASME
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 10/19/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
data, the ratio between in-phase and out-of-phase signalVin=Vout
is fitted with the heat transfer model to extract thermophysical prop-
erties. In this fitting process, numerical optimization algorithms
(e.g., quasi Newton [110] and simplex minimization [111]) are
used to minimize the squared difference between the experimental
data and the heat transfer model, until the value change in thermal
properties is smaller than the tolerance (e.g., 1%).
In the case of FDTR system based on pulsed laser (Fig. 14(a)), the
obtained signal can be fitted with Zðx0;sdÞin Eq. (30),wherethe
modulation frequency x0is varied as an independent variable and
the delay time sdis fixed. When an FDTR system is implemented
using CW lasers, the thermoreflectance signal is instead directly pro-
portional to single frequency response Hðx0Þ[96,112,113]
Zðx0Þ¼bHðx0Þ(37)
Figure 18 shows the calculated phase response /¼
arctanðVout=Vin Þof pulsed and CW-based FDTR measurement of
100 nm Al on sapphire with modulation frequency ranges from 50
kHz to 20 MHz. A clear phase difference is observed between the
pulsed FDTR and the CW-based FDTR, and attentions must be
paid to adopt the correct solution when processing signals from
different FDTR systems. Similar to the TDTR system, the least-
squared error method can be used to extract thermophysical
properties.
The challenge of thermoreflectance technique lies in its versa-
tile capability that multiple thermal properties including thermal
conductivity, interfacial thermal conductance, and heat capacity
can be determined depending on the measurement regime. The
interfacial thermal conductance can be determined at long delay
time (>2 ns) using TDTR [114,115]. A sensitivity parameter Spis
defined to characterize the dependence of signal on different
parameters p(interface thermal conductance, heat capacity, and
thermal conductivity, etc.)
Sp¼dln Vin=Vout
ðÞ
dlnp
ðÞ (38)
The sensitivity parameter describes the scaling law between
change in the signal Vin=Vout and the change in the parameter p.
For example, a sensitivity value of 0.4 means that there would be
0.4% change in the signal if the parameter pis changed by 1%. As
shown in Fig. 19, the TDTR signal is dominantly determined by
the interfacial thermal conductance when delay time is longer
than 2 ns. TDTR technique is therefore implemented extensively
to study the thermal transport mechanisms across interfaces,
including the effect of surface chemistry on interfacial thermal
conductance across functionalized liquid–solid boundary
[94,116,117] and solid–solid interfaces [118], interfacial thermal
conductance between metals and dielectrics [119–123], and inter-
face between low-dimensional materials and bulk substrates
[124–126].
With interfacial thermal conductance determined, heat capacity
and thermal conductivity can then be simultaneously obtained.
Liu et al. demonstrated that thermal conductivity and heat
capacity can be determined simultaneously for both bulk materials
and thin films by performing multiple measurements using differ-
ent combinations of modulation frequencies and spot sizes [115].
Let us consider bulk materials first to introduce the physical
picture we based on to determine heat capacity and thermal con-
ductivity simultaneously. Under periodic laser heating, the pene-
tration depth is defined as a characteristic length which describes
the depth the temperature gradient penetrates into the sample and
can be written as Lp;d¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2kd=x0C
p, where x0is the modulation
frequency, Cis the volumetric heat capacity, and the index of
directions dcorresponds to in-plane (d¼r) and cross-plane
(d¼z), respectively. If a very low modulation frequency and
Fig. 18 The calculated phase response for CW and pulsed
FDTR of 100 nm Al on sapphire substrate [96]. (Reproduced
with permission from Schmidt et al. [96]. Copyright 2009 by AIP
Publishing.)
Fig. 19 The sensitivity of 2Vin=Vout signal to the cross-plane
thermal conductivity kzand heat capacity Cof silicon, and the
interface thermal conductance Gbetween Al and Si at 6.8 MHz.
(Reproduced with permission from Liu et al. [115]. Copyright
2013 by AIP Publishing.)
Fig. 20 Schematic of temperature profile with different combi-
nations of modulation frequency x0and spot sizes. (a) Spheri-
cal thermal waves in the sample when heated by a small spot
size at low modulation frequency x0, and the radial penetration
depth Lp;ris much longer than ð1=4Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
w2
01w2
1
q, where w0and w1
are the pump and probe beam radius. (b) Plane thermal waves
in the sample when heated by a large beam spot at high modu-
lation frequency x0. The radial penetration depth is much
smaller than beam size ð1=4Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
w2
01w2
1
q.
Journal of Electronic Packaging DECEMBER 2016, Vol. 138 / 040802-13
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 10/19/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
small spot size are used to conduct TDTR measurement, the radial
heat transfer dominates the thermal dissipation (Fig. 20(a)), and
the signal is determined by the geometric average thermal conduc-
tivity ffiffiffiffiffiffiffiffi
krkz
p. This approximation holds true when the radial pene-
tration depth Lp;rð1=4Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
w2
0þw2
1
p, where w0and w1are 1=e
radius of pump and probe beam. On the other hand, if we use a
large spot size and high modulation frequency, the temperature
gradient only penetrates into a very thin layer into the sample and
cross-plane thermal transport is dominating (Fig. 20(b)). The ther-
mal response signal is determined by ffiffiffiffiffiffiffiffiffiffi
kz=C
p. If we assume the
material isotropic, we can first determine thermal conductivity
using a small beam spot at low frequency and then measure heat
capacity using a large beam spot at high frequency.
Measuring thermal properties of thin film is similar to the bulk
material with only slight differences. If a large beam spot modu-
lated at high frequency is used to measure thermal properties, the
cross-plane heat transfer dominates. Different from bulk samples
whose thickness is always much longer than the cross-plane pene-
tration depth, the thickness of a thin film might be comparable or
even smaller than the cross-plane penetration depth Lp;z.We
describe the sample “thermally thin” if the penetration depth Lp;z
is still much larger than film thickness d(Fig. 21(a)). In this case,
the thermal response is controlled by the thermal resistance d=kz
and heat capacity C. At high frequency limit, the penetration
depth Lp;zwould be so small that temperature gradient only pene-
trates into a limited depth of the layer (referred to as “thermally
thick,” see Fig. 21(b)), the thermal response is analogous to the
high frequency limit of the bulk material, only dominated by ther-
mal effusivity ffiffiffiffiffiffiffi
kzC
p. If the penetration depth is comparable to the
thin film thickness, both thermal effusivity ffiffiffiffiffiffiffi
kzC
pand diffusivity
kz=Caffect the temperature response of the thin film sample.
Based on the analysis above, both cross-plane thermal conductiv-
ity and heat capacity of thin films can be obtained by applying dif-
ferent modulation frequencies (Fig. 22).
If the beam is tightly focused to the thin film sample at low
modulation frequency, the heat transfer is dominated in the
in-plane radial direction (Fig. 21(c)). In this case, the in-plane
thermal conductivity krdominates the thermal response of the
material. A low thermal conductivity substrate can be used to fur-
ther improve the sensitivity to the in-plane thermal conductivity
of the thin film [96,98,115].
This method of measuring in-plane thermal conductivity using
a small spot size, however, assumes cylindrical symmetry of ther-
mal conductivity in the in-plane direction [93]. For anisotropic
Fig. 21 Schematic of temperature profile when (a) a thermally
thin film where the cross-plane penetration depth Lp;zis much
larger than thickness of the film d,(b) a thermally thick film
where the thermal excitation only penetrates into a limited
depth into the thin film, and (c) in-plane heat transfer dominates
with a small beam spot size at low modulation frequency
Fig. 22 (a) Measured TDTR signal for 110 nm SiO
2
film at multi-
ple modulation frequencies. (b) The thermal conductivity–heat
capacity (k,C) pairs yielding best fit to the experiment data are
not unique. Simultaneous determination of kand Ccan be
obtained using the k–C diagram. Data are obtained using a
110 nm SiO
2
, compared with data measured by Zhu et al. [104].
(Reproduced with permission from Liu et al. [115]. Copyright
2013 by AIP Publishing.)
040802-14 / Vol. 138, DECEMBER 2016 Transactions of the ASME
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 10/19/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
thin films lacking in-plane symmetry, the thermal conductivity
tensor can be extracted by offsetting the pump beam away from
the probe beam [127]. The schematic of implementing beam-
offset TDTR on measuring in-plane thermal conductivity tensor
of a-SiO
2
is shown in Fig. 23. Instead of measuring the ratio
between in-phase and out-of-phase signal Vin=Vout , beam-offset
TDTR measures the full-width half maximum (FWHM) of the
out-of-phase signal Vout, as the pump beam is detuned from the
probe beam (Fig. 23(b)). By sweeping the beam in different direc-
tions parallel to the thin film plane, beam-offset TDTR directly
detects the in-plane penetration length by monitoring the two-
dimensional temperature profile in the in-plane direction, thus
makes it possible to extract anisotropic in-plane thermal conduc-
tivity. For example, the thermal conductivity kcalong c-axis of
the SiO
2
can be measured by offsetting the pump beam parallel to
the c-axis (Fig. 23(a)), and the out-of-phase signal is then
recorded as a function of beam-offset distance y0, shown as the
open circles in Fig. 23(b). This step maps a Gaussian shaped out-
of-phase signal, and the experimental FWHM is extracted from
the Gaussian profile of Vout. Then, the FWHM can be calculated
as a function of thermal conductivity kashown as the curves in
Fig. 23(c). With the experimentally measured FWHM, the thermal
conductivity kccan be extracted from the FWHM– kccurve. The
FWHM perpendicular to the c-axis is also plotted in Fig. 23(d),
nearly independent of the thermal conductivity kcalong c-axis,
ensuring that the in-plane thermal conductivity along different
directions can be extracted independently by offsetting the pump
beam in different directions. Similarly, the thermal conductivity
kaperpendicular to the c-axis can also be extracted by offsetting
the pump beam along x-direction (Fig. 23(c)). The beam-offset
technique allows the TDTR technique to extract thermal conduc-
tivity tensor for materials lacking in-plane symmetry. This beam-
offset technique has also been extended to time-resolved
magneto-optic Kerr effect (TR-MOKE) based pump-probe mea-
surement [128] and has been applied to measure thermal conduc-
tivity tensor of two-dimensional materials like MoS
2
[128] and
black phosphorous [129,130].
Transient thermoreflectance method has been widely applied to
explore thermal properties of novel materials ranging from low
thermal conductivity of 0.1 W=m K of hybrid materials
[131,132] and fullerene derivatives [133,134], to very high ther-
mal conductivity, such as graphene [135]. In addition, the capabil-
ity of determining multiple thermophysical properties
simultaneously makes transient thermoreflectance a versatile tech-
nique to characterize thermal properties of a wide range of materi-
als including diamond [136,137], pure and doped Si films [114],
disordered layered crystals [138], and superlattices [103,139].
4 Summary
This work reviews the measurement techniques for thermal
conductivity and interfacial contact resistance of both bulk materi-
als and thin films. For thermal characterization of bulk material,
the steady-state absolute method, comparative technique, laser
flash diffusivity method, and transient plane source (TPS) method
are most used. For thin film measurement, the 3xmethod and
transient thermoreflectance technique are employed widely.
Figure 24 gives a summary of sample size and measurement time
scale for different thermal conductivity measurement techniques.
The absolute technique and comparative technique measure large-
size sample but need longest time for data acquisition, while tran-
sient thermoreflectance technique is capable of measuring thinnest
film sample in a quickest way. In general, it is a very challenging
task to determine material thermal conductivity and interface con-
tact resistance with less than 5% error. Selecting a specific mea-
surement technique to characterize thermal properties needs to be
based on: (1) knowledge on the sample whose thermophysical
properties are to be determined, including the sample geometry
and size, surface roughness, and preparation method; (2) under-
standing of fundamentals and procedures of the testing technique
Fig. 23 (a) Schematic of beam-offset TDTR and the alignment
of the h110iSiO
2
principal thermal conductivity axes with the
experimental coordinate, (b) the experimental beam-offset data
of out-of-phase signal, (c) comparison of experimental and the-
oretical FWHM to extract thermal conductivity kaperpendicular
to c-axis, and (d) comparison of experimental and theoretical
FWHM to extract thermal conductivity kcparallel to c-axis.
(Reproduced with permission from Feser et al. [127]. Copyright
2014 by AIP Publishing, LLC.)
Journal of Electronic Packaging DECEMBER 2016, Vol. 138 / 040802-15
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 10/19/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
and equipment, for example, some techniques are limited to sam-
ples with specific geometries and some are limited to specific
range of thermophysical properties; and (3) understanding of the
potential error sources which might affect the final results, for
example, the convection and radiation heat losses.
Nomenclature
a¼width, m
A¼cross-sectional area, m2
C¼volumetric heat capacity, J m3K1
cp¼specific heat capacity, J kg1K1
d¼thickness, m
G¼interfacial thermal conductance, W m2K1
Gdet ¼gain of photodetector
H¼height, m
I¼electric current, A
I0¼electric current amplitude, A
k¼thermal conductivity, W m1K1
K¼thermal conductance, W K1
L¼length, m
Lp¼thermal penetration depth, m
p¼heating power, W
P1¼power of pump beam, W
P2¼power of probe beam, W
q¼complex thermal wave number, m1
q¼heat flux, W m2
Q¼heat transfer rate, W
r¼radius, m
R¼thermal resistance, W K1
Re¼electrical resistance, X
R1¼reflectivity at the wavelength of pump beam
R2¼reflectivity at the wavelength of probe beam
Sp¼sensitivity of signals to parameter p
t¼time, s
T¼temperature, CorK
Tc¼cold-side temperature, CorK
Th¼hot-side temperature, CorK
T1¼ambient temperature, CorK
V¼voltage, V
Vin ¼in-phase signal, mV
Vout ¼out-of-phase signal, mV
V1;V2¼voltmeter, V
W¼dimensionless number defined in Eq. (15)
w0¼pump beam radius, lm
w1¼probe beam radius, lm
Greek Symbols
a¼thermal diffusivity, m2s1
aR¼temperature coefficient of resistance, K1
b¼constant coefficient determined by Eq. (29)
d¼index of direction in transient thermoreflectance method
DT¼temperature difference, CorK
DTpp ¼peak-to-peak amplitude of the temperature signal
between two envelope curves in pulsed power tech-
nique, CorK
DT0¼amplitude of temperature change, CorK
g¼dimensionless number defined in Eq. (16)
h¼temperature after Fourier transform
j¼thermal conductivity, W m1K1
k¼thermal conductivity, W m1K1
s¼half period of the heating current in pulsed power tech-
nique, s
sd¼the delay time in TDTR method, ns
/¼defined as ffiffiffiffiffiffiffiffiffiffiffi
ta=r2
pin transient plane source method
x¼frequency, Hz
xs¼repetition frequency, Hz
x0¼modulation frequency, Hz
Subscripts
b¼bottom
f¼associated with thin film
loss ¼heat loss
p¼represent interface thermal conductance, heat capacity,
or thermal conductivity for sensitivity parameter
r¼coordinate
s¼standard
S¼associated with substrate
Sþf¼associated with thin film on substrate
t¼top
z¼coordinate
k¼in-plane
?¼cross-plane
References
[1] Savija, I., Culham, J. R., Yovanovich, M. M., and Marotta, E. E., 2003,
“Review of Thermal Conductance Models for Joints Incorporating Enhance-
ment Materials,” J. Thermophys. Heat Transfer.,17(1), pp. 43–52.
[2] Pollack, G. L., 1969, “Kapitza Resistance,” Rev. Mod. Phys.,41(1),
pp. 48–81.
[3] Tritt, T. M., and Wes ton, D., 2004, “Measurement Techniques and Considera-
tions for Determining Thermal Conductivity of Bulk Materials,” Thermal
Conductivity, T. M. Tritt, ed., Springer, New York, pp. 187–203.
[4] Hamilton, R. L., and Crosser, O. K., 1962, “Thermal Conductivity of Hetero-
geneous Two-Component Systems,” Ind. Eng. Chem. Fundam.,1(3),
pp. 187–191.
[5] Zeller, R. C., and Pohl, R. O., 1971, “Thermal Conductivity and Specific Heat
of Noncrystalline Solids,” Phys. Rev. B,4(6), pp. 2029–2041.
[6] Cooper, M. G., Mikic, B. B., and Yovanovich, M. M., 1969, “Thermal Contact
Conductance,” Int. J. Heat Mass Transfer,12(3), pp. 279–300.
[7] Madhusudana, C. V., and Fletch er, L. S., 1986, “Contact Heat Transfer: The
Last Decade,” AIAA J.,24(3), pp. 510–523.
[8] Prasher, R., 2006, “Thermal Inte rface Materials: Historical Perspective, Status,
and Future Directions,” Proc. IEEE,94(8), pp. 1571–1586.
[9] Machlin, E. S., 2006, Materials Science in Microelectronics: The Relation-
ships Between Thin Film Processing and Structure, Cambridge University
Press, New York,
[10] Peumans, P., Yakimov, A., and Forrest, S. R., 2003, “Small Molecular Weight
Organic Thin-Film Photodetectors and Solar Cells,” J. Appl. Phys.,93(7),
pp. 3693–3723.
[11] Xi, J.-Q., Schubert, M. F., Kim, J. K., Schubert, E. F., Chen, M., Lin, S.-Y.,
Liu, W., and Smart, J. A., 2007, “Optical Thin-Film Materials With Low
Refractive Index for Broadband Elimination of Fresnel Reflection,” Nat. Pho-
tonics,1(3), pp. 176–179.
[12] Dresselhaus, M. S., Chen, G., Tang, M. Y., Yang, R. G., Lee, H., Wang, D. Z.,
Ren, Z. F., Fleurial, J.-P., and Gogna, P., 2007, “New Directions for
Fig. 24 Sample size and measurement time for different ther-
mal conductivity measurement techniques. The sample size is
the length of temperature gradient inside the sample during the
measurement. The measurement time is the temperature varia-
tion time (continuous or pulsed heating source) or temperature
variation period (continuous periodic heating source) during
the test. The steady-state methods for bulk materials measure
large-size sample but need longest time for data acquisition,
while transient thermoreflectance technique (TDTR and FDTR)
is capable of measuring thinnest film sample in a quickest way.
Note that the steady-state methods for bulk materials include
absolute technique, comparative technique, and radial heat
flow method.
040802-16 / Vol. 138, DECEMBER 2016 Transactions of the ASME
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 10/19/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Low-Dimensional Thermoelectric Materials,” Adv. Mater.,19(8), pp.
1043–1053.
[13] Borca-Tasciuc, T., Kumar, A. R., and Chen, G., 2001, “Data Reduction in 3x
Method for Thin-Film Thermal Conductivity Determination,” Rev. Sci. Ins-
trum.,72(4), pp. 2139–2147.
[14] Cahill, D. G., Goodson, K., and Majumdar , A., 2002, “Thermometry and Ther-
mal Transport in Micro/Nanoscale Solid-State Devices and Structures,”
ASME J. Heat Transfer,124(2), pp. 223–241.
[15] Borca-Tasciuc, T., and Chen, G., 2004, “Experimental Techniques for Thin-
Film Thermal Conductivity Characterization,” Thermal Conductivity,T.M.
Tritt, ed., Springer, New York, pp. 205–237.
[16] Cahill, D. G., Braun, P. V., Chen, G., Clarke, D. R., Fan, S., Goodson, K. E.,
Keblinski, P., King, W. P., Mahan, G. D., Majumdar, A., Maris, H. J., Phillpot,
S. R., Pop, E., and Shi, L., 2014, “Nanoscale Thermal Transport—II:
2003–2012,” Appl. Phys. Rev.,1(1), p. 11305.
[17] Shi, L., Dames, C., Lukes, J. R., Reddy, P., Duda, J., Cahill, D. G., Lee, J.,
Marconnet, A., Goodson, K. E., Bahk, J.-H., Shakouri, A., Prasher, R. S.,
Felts, J., King, W. P., Han, B., and Bischof, J. C., 2015, “Evaluating Broader
Impacts of Nanoscale Thermal Transport Research,” Nanoscale Microscale
Thermophys. Eng.,19(2), pp. 127–165.
[18] Marco nnet, A. M., Panzer, M. A., and Goodson, K. E., 2013, “Thermal Con-
duction Phenomena in Carbon Nanotubes and Related Nanostructured Materi-
als,” Rev. Mod. Phys.,85(3), pp. 1295–1326.
[19] Tober er, E. S., Baranowski, L. L., and Dames, C., 2012, “Advances in Ther-
mal Conductivity,” Annu. Rev. Mater. Res.,42(1), pp. 179–209.
[20] ASTM, 2013, “Standard Test Method for Steady-State Heat Flux Measure-
ments and Thermal Transmission Properties by Means of the Guarded-Hot-
Plate Apparatus,” ASTM International, West Conshohocken, PA, Standard
No. ASTM C177-13.
[21] Pope, A. L., Zawilski, B., and Tritt, T. M., 2001, “Description of Removable
Sample Mount Apparatus for Rapid Thermal Conductivity Measurements,”
Cryogenics,41(10), pp. 725–731.
[22] EN, 2011, “Thermal Performance of Building Materials and Products: Deter-
mination of Thermal Resistance by Means of Guarded Hot Plate and Heat
Flow Meter Methods—Products of High and Medium Thermal Resistance,”
European Standard, London, Standard No. EN 12667:2001.
[23] ISO, 1991, “Thermal Insulation: Determination of Steady-State Thermal
Resistance and Related Properties—Guarded Hot Plate Apparatus,” Interna-
tional Organization for Standardization, New York, Standard No. ISO
8302:1991.
[24] Shi, L., Li, D., Yu, C., Jang, W., Kim, D., Yao, Z., Kim, P., and Majumdar,
A., 2003, “Measuring Thermal and Thermoelectric Properties of One-
Dimensional Nanostructures Using a Microfabricated Device,” ASME J. Heat
Transfer,125(5), pp. 881–888.
[25] Stojanovic, N., Berg, J. M., Maithripala, D. H. S., and Holtz, M., 2009, “Direct
Measurement of Thermal Conductivity of Aluminum Nanowires,” Appl. Phys.
Lett.,95(9), p. 91905.
[26] Marconnet, A. M., Yamamoto, N., Panzer, M. A., Wardle, B. L., and Goodson,
K. E., 2011, “Thermal Conduction in Aligned Carbon Nanotube–Polymer
Nanocomposites With High Packing Density,” ACS Nano,5(6), pp. 4818–4825.
[27] Zhang, X., Cong, P. Z., and Fujii, M., 2006, “A Study on Thermal Contact
Resistance at the Interface of Two Solids,” Int. J. Thermophys.,27(3),
pp. 880–895.
[28] Shahil, K. M., and Balandin, A. A., 2012, “Th ermal Properties of Graphene
and Multilayer Graphene: Applications in Thermal Interface Materials,” Solid
State Commun.,152(15), pp. 1331–1340.
[29] ASTM, 2012, “Standard Test Method for Thermal Transmission Properties of
Thermally Conductive Electrical Insulation Materials,” ASTM International,
West Conshohocken, PA, Standard No. ASTM D5470-12.
[30] ASTM, 2013, “Standard Test Method for Thermal Conductivity of Solids
Using the Guarded-Comparative-Longitudinal Heat Flow Technique,” ASTM
International, West Conshohocken, PA, Standard No. ASTM E1225-13.
[31] ASTM, 2015, “Standard Test Method for Steady-State Thermal Transmission
Properties by Means of the Heat Flow Meter Apparatus,” ASTM International,
West Conshohocken, PA, Standard No. ASTM C518-15.
[32] ASTM, 2011, “Standard Test Method for Evaluating the Resistance to Thermal
Transmission of Materials by the Guarded Heat Flow Meter Technique,” ASTM
International, West Conshohocken, PA, Standard No. ASTM E1530-11.
[33] Flynn, D. R., 1992, “Thermal Conductivity of Loose-Fill Materials by a
Radial-Heat-Flow Method,” Compendium of Thermophysical Property Mea-
surement Methods, K. D. Maglic´, A. Cezairliyan, and V. E. Peletsky, eds.,
Springer, New York, pp. 33–75.
[34] ASTM, 2010, “Standard Test Method for Steady-State Heat Transfer Proper-
ties of Pipe Insulation,” ASTM International, West Conshohocken, PA, Stand-
ard No. ASTM C335/C335M-10.
[35] ISO, 1994, “Thermal Insulation: Determination of Steady-State Thermal
Transmission Properties of Thermal Insulation of Circular Pipes,” Interna-
tional Organization for Standardization, New York, Standard No. ISO
8497:1994.
[36] Zawilski, B. M., Iv, R. T. L., and Tritt, T. M., 2001, “Description of the Parallel
Thermal Conductance Technique for the Measurement of the Thermal Conduc-
tivity of Small Diameter Samples,” Rev. Sci. Instrum.,72(3), pp. 177 0–1774.
[37] Dasgupta, T., and Umarji, A. M., 2007, “Thermal Properties of MoSi
2
With
Minor Aluminum Substitutions,” Intermetallics,15(2), pp. 128–132.
[38] Maldonado, O., 1992, “Pulse Method for Simultaneous Measurement of Elec-
tric Thermopower and Heat Conductivity at Low Temperatures,” Cryogenics,
32(10), pp. 908–912.
[39] Sundqvist, B., 1991, “Thermal Diffusivity Measurements by Angstrom’s
Method in a Fluid Environment,” Int. J. Thermophys.,12(1), pp. 191–206.
[40] Angstrom, A. J., 1863, “New Method of Determining the Conductibility of
Bodies,” Philos. Mag.,25, p. 130.
[41] Romao, C. P., Miller, K. J., Johnson, M. B., Zwanziger, J. W., Marinkovic, B.
A., and White, M. A., 2014, “Thermal, Vibrational, and Thermoelastic Proper-
ties of Y
2
Mo
3
O
12
and Their Relations to Negative Thermal Expansion,” Phys.
Rev. B,90(2), p. 24305.
[42] Miller, K. J., Johnson, M. B., White, M. A., and Marinkovic, B. A., 2012,
“Low-Temperature Investigations of the Open-Framework Material
HfMgMo
3
O
12
,” Solid State Commun.,152(18), pp. 1748–1752.
[43] Kenne dy, C. A., and White, M. A., 2005, “Unusual Thermal Conductivity of
the Negative Thermal Expansion Material, ZrW
2
O
8
,” Solid State Commun.,
134(4), pp. 271–276.
[44] Whitman, C. A., Johnson, M. B., and White, M. A., 2012, “Characterization of
Thermal Performance of a Solid–Solid Phase Change Material, Di-n-
Hexylammonium Bromide, for Potential Integration in Building Materials,”
Thermochim. Acta,531, pp. 54–59.
[45] Stalhane, B., and Pyk, S., 1931, “New Method for Determining the Coeffi-
cients of Thermal Conductivity,” Tek. Tidskr.,61(28), pp. 389–393.
[46] Abu-Hamdeh, N. H., Khdair, A. I., and Reeder, R. C., 2001, “A Comparison
of Two Methods Used to Evaluate Thermal Conductivity for Some Soils,” Int.
J. Heat Mass Transfer,44(5), pp. 1073–1078.
[47] Festa, C., and Rossi, A., 1999, “Apparatus for Routine Measurements of the
Thermal Conductivity of Ice Cores,” Ann. Glaciol.,29(1), pp. 151–154.
[48] ASTM, 2009, “Standard Test Method for Thermal Conductivity of Refractories
by Hot Wire (Platinum Resistance Thermometer Technique),” ASTM Interna-
tional, West Conshohocken, PA, Standard No. ASTM C1113/C1113M-09.
[49] ISO, 2010, “Refractory materials—Determination of Thermal Conductivity—
Part 1: Hot-Wire Methods (Cross-Array and Resistance Thermometer),” Inter-
national Organization for Standardization, New York, Standard No. ISO 8894-
1:2010.
[50] Franco, A., 2007, “An Apparatus for the Routine Measurement of Thermal
Conductivity of Materials for Building Application Based on a Transient Hot-
Wire Method,” Appl. Therm. Eng.,27(14–15), pp. 2495–2504.
[51] Assael, M. J., Antoniadis , K. D., and Wakeham, W. A., 2010, “Historical Evo-
lution of the Transient Hot-Wire Technique,” Int. J. Thermophys.,31(6), pp.
1051–1072.
[52] Assael, M. J., Antoniadis, K. D., Metaxa, I. N., Mylona, S. K., Assael, J.-A.
M., Wu, J., and Hu, M., 2015, “A Novel Portable Absolute Transient Hot-
Wire Instrument for the Measurement of the Thermal Conductivity of Solids,”
Int. J. Thermophys.,36(10–11), pp. 3083–3105.
[53] Tong, X. C., 2011, “Character ization Methodologies of Thermal Management
Materials,” Advanced Materials for Thermal Management of Electronic Pack-
aging, Springer, New York, pp. 59–129.
[54] ASTM, 2009, “Standard Test Method for Thermal Conductivity of Plastics by
Means of a Transient Line-Source Technique,” ASTM International, West
Conshohocken, PA, Standard No. ASTM D5930-09.
[55] ASTM, 2014, “Standard Test Method for Determination of Thermal Conduc-
tivity of Soil and Soft Rock by Thermal Needle Probe Procedure,” ASTM
International, West Conshohocken, PA, Standard No. ASTM D5334-14.
[56] Gong, L., Wang, Y., Cheng, X., Zhang, R., and Zhang, H., 2013, “Porous
Mullite Ceramics With Low Thermal Conductivity Prepared by Foaming and
Starch Consolidation,” J. Porous Mater.,21(1), pp. 15–21.
[57] ASTM, 2016, “Standard Test Method for Measurement of Thermal Effusivity
of Fabrics Using a Modified Transient Plane Source (MTPS) Instrument,”
ASTM International, West Conshohocken, PA, Standard No. ASTM D7984-16.
[58] Bouguerra, A., A
€
ıt-Mokhtar, A., Amiri, O., and Diop, M. B., 2001,“Measurement
of Thermal Conductivity, Thermal Diffusivity and Heat Capacity of Highly
Porous Building Materials Using Transient Plane Source Technique,” Int. Com-
mun. Heat Mass Transfer,28(8), pp. 10 65–1078.
[59] Gustafsson, S. E., 1991, “Trans ient Plane Source Techniques for Thermal
Conductivity and Thermal Diffusivity Measurements of Solid Materials,” Rev.
Sci. Instrum.,62(3), pp. 797–804.
[60] Gustavsson, M., Karawac ki, E., and Gustafsson, S. E., 1994, “Thermal Con-
ductivity, Thermal Diffusivity, and Specific Heat of Thin Samples From Tran-
sient Measurements With Hot Disk Sensors,” Rev. Sci. Instrum.,65(12),
pp. 3856–3859.
[61] He, Y., 2005, “Rapid Thermal Conductivity Measurement With a Hot Disk Sensor—
Part 1: Theoretical Considerations,” Thermochim. Acta,436(1–2), pp. 122–129.
[62] Li, Y., Shi, C., Liu, J., Liu, E., Shao, J., Chen, Z., Dorantes-Gonzalez, D. J.,
and Hu, X., 2014, “Improving the Accuracy of the Transient Plane Source
Method by Correcting Probe Heat Capacity and Resistance Influences,” Meas.
Sci. Technol.,25(1), p. 15006.
[63] ISO, 2015, “Plastics—Determination of Thermal Conductivity and Thermal
Diffusivity—Part 2: Transient Plane Heat Source (Hot Disc) Method,” Inter-
national Organization for Standardization, New York, Standard No. ISO
22007-2:2015.
[64] Afriyie, E. T., Karami, P., Norberg, P., and Gudmundsson, K., 2014, “Textural
and Thermal Conductivity Properties of a Low Density Mesoporous Silica
Material,” Energy Build.,75, pp. 210–215.
[65] Min, S., Blumm, J., and Lindemann, A., 2007, “A New Laser Flash System
for Measurement of the Thermophysical Properties,” Thermochim. Acta,
455(1–2), pp. 46–49.
[66] Parker, W. J., Jenkins, R. J., Butler, C. P., and Abbott, G. L., 1961, “Flash
Method of Determining Thermal Diffusivity, Heat Capacity, and Thermal
Conductivity,” J. Appl. Phys.,32(9), pp. 1679–1684.
Journal of Electronic Packaging DECEMBER 2016, Vol. 138 / 040802-17
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 10/19/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
[67] Campbell, R. C., Smith, S. E., and Dietz, R. L., 1999, “Measurements of
Adhesive Bondline Effective Thermal Conductivity and Thermal Resistance
Using the Laser Flash Method,” Fifteenth Annual IEEE Semiconductor Ther-
mal Measurement and Management Symposium, Mar. 9–11, pp. 83–97.
[68] Ruoho, M., Valset, K., Finstad, T., and Tittonen, I., 2015, “Measurement of
Thin Film Thermal Conductivity Using the Laser Flash Method,” Nanotec h-
nology,26(19), p. 195706.
[69] ASTM, 2013, “Standard Test Method for Thermal Diffusivity by the Flash
Method,” ASTM International, West Conshohocken, PA, Standard No. ASTM
E1461-13.
[70] ISO, 2008, “Plastics—Determination of Thermal Conductivity and Thermal
Diffusivity—Part 4: Laser Flash Method,” International Organization for
Standardization, New York, Standard No. ISO 22007-4:2008.
[71] Abdulagatov, I. M., Abdu lagatova, Z. Z., Kallaev, S. N., Bakmaev, A. G., and
Ranjith, P. G., 2015, “Thermal-Diffusivity and Heat-Capacity Measurements
of Sandstone at High Temperatures Using Laser Flash and DSC Methods,”
Int. J. Thermophys.,36(4), pp. 658–691.
[72] Khuu, V., Osterman, M., Bar -Cohen, A., and Pecht, M., 2011, “Considerations
in the Use of the Laser Flash Method for Thermal Measurements of Thermal
Interface Materials,” IEEE Trans. Compon. Packag. Manuf. Technol.,1(7),
pp. 1015–1028.
[73] V€
olklein, F., Reith, H., and Meier, A., 2013, “Measuring Methods for the
Investigation of In-Plane and Cross-Plane Thermal Conductivity of Thin
Films,” Phys. Status Solidi A,210(1), pp. 106–118.
[74] Lee, J., Li, Z., Reifenberg, J. P., Lee, S., Sinclair, R., Asheghi, M., and Good-
son, K. E., 2011, “Thermal Conductivity Anisotropy and Grain Structure in
Ge
2
Sb
2
Te
5
Films,” J. Appl. Phys.,109(8), p. 84902.
[75] V€
olklein, F., 1990, “Thermal Conductivity and Diffusivity of a Thin Film
SiO
2
-Si
3
N
4
Sandwich System,” Thin Solid Films,188(1), pp. 27–33.
[76] Volklein, F., and Starz, T., 1997, “Thermal Conductivity of Thin Films:
Experimental Methods and Theoretical Interpretation,” XVI International
Conference on Thermoelectrics (ICT’97), Aug. 26–29, pp. 711–718.
[77] Cahill, D. G., Fischer, H. E., Klitsner, T., Swartz, E. T., and Pohl, R. O., 1989,
“Thermal Conductivity of Thin Films: Measurements and Understanding,”
J. Vac. Sci. Technol. A,7(3), pp. 1259–1266.
[78] Cahill, D. G., 1990 , “Thermal Conductivity Measurement From 30 to 750 K:
The 3xMethod,” Rev. Sci. Instrum.,61(2), pp. 802–808.
[79] Feser, J. P., Chan, E. M., Majumdar, A., Segalman, R. A., and Urban, J. J.,
2013, “Ultralow Thermal Conductivity in Polycrystalline CdSe Thin Films
With Controlled Grain Size,” Nano Lett.,13(5), pp. 2122–2127.
[80] Bourlon, A. B., and Van der Tempel, L., 2006, “Thermal Conductivity Mea-
surement by the 3 Omega Method,” Philips Electronics, Koninklijke, Amster-
dam, The Netherlands, Technical Note PRTN 2005/01035.
[81] Dames, C., 2013, “Measuring the Thermal Conductiv ity of Thin Films: 3
Omega and Related Electrothermal Methods,” Annu. Rev. Heat Transfer,
16(16), pp. 7–49.
[82] Cahill, D. G., Ford, W. K., Goodson, K. E., Mahan, G. D., Majumdar, A.,
Maris, H. J., Merlin, R., and Phillpot, S. R., 2003, “Nanoscale Thermal Trans-
port,” J. Appl. Phys.,93(2), pp. 793–818.
[83] Mishra, V., Hardin, C. L., Garay, J. E., and Dames, C., 2015, “A 3 Omega
Method to Measure an Arbitrary Anisotropic Thermal Conductivity Tensor,”
Rev. Sci. Instrum.,86(5), p. 54902.
[84] Cahill, D. G., and Pohl, R. O., 1987, “Thermal Conductivity of Amorphous
Solids Above the Plateau,” Phys. Rev. B,35(8), pp. 4067–4073.
[85] Roy-Panzer, S., Kodama, T., Lingamneni, S., Panzer, M. A., Asheghi, M., and
Goodson, K. E., 2015, “Thermal Characterization and Analysis of Microliter Liquid
Volumes Using the Three-Omega Method,” Rev. Sci. Instrum.,86(2), p. 24901.
[86] Rosencwaig, A., 1982, “Thermal-Wave Imaging,” Science,218(4569),
pp. 223–228.
[87] Rosencwaig, A., and Gersho, A., 1976, “Theory of the Photoacoustic Effect
With Solids,” J. Appl. Phys.,47(1), pp. 64–69.
[88] Eesley, G. L., 1983, “Observation of Nonequilib rium Electron Heating in
Copper,” Phys. Rev. Lett.,51(23), pp. 2140–2143.
[89] Wilson, R. B., Feser, J. P., Hohensee, G. T., and Cahill, D. G., 2013, “Two-
Channel Model for Nonequilibrium Thermal Transport in Pump-Probe
Experiments,” Phys. Rev. B,88(14), p. 144305.
[90] Wright, O. B., and Kawashima, K., 1992, “Coherent Phonon Detection From
Ultrafast Surface Vibrations,” Phys. Rev. Lett.,69(11), pp. 1668–1671.
[91] Luckyanova, M. N., Garg, J., Esfarjani, K., Jandl, A., Bulsara, M. T., Schmidt,
A. J., Minnich, A. J., Chen, S., Dresselhaus, M. S., Ren, Z., Fitzgerald, E. A.,
and Chen, G., 2012, “Coherent Phonon Heat Conduction in Superlattices,”
Science,338(6109), pp. 936–939.
[92] Ravichandran, J., Yadav, A. K., Cheaito, R., Rossen, P. B., Soukiassian, A., Suresha,
S. J., Duda, J. C., Foley, B. M., Lee, C.-H., Zhu, Y., Lichtenberger, A. W., Moore, J.
E.,Muller,D.A.,Schlom,D.G.,Hopkins,P.E.,Majumdar,A.,Ramesh,R.,and
Zurbuchen, M. A., 2014, “Crossover From Incoherent to Coherent Phonon Scatter-
ing in Epitaxial Oxide Superlattices,” Nat. Mater.,13(2), pp. 168–172.
[93] Cahill, D. G., 2004, “Analysis of Heat Flow in Layered Structures for Time-
Domain Thermoreflectance,” Rev. Sci. Instrum.,75(12), pp. 5119–5122.
[94] Schmidt, A. J., Alper, J. D., Chiesa, M., Chen, G., Das, S. K., and Hamad-Schifferli,
K., 2008, “Probing the Gold NanorodLigandSolvent Interface by Plasmonic
Absorption and Thermal Decay,” J. Phys. Chem. C,112(35), pp. 13320–13323.
[95] Ge, Z., Cahill, D. G., and Braun, P. V., 2006, “Thermal Conductance of
Hydrophilic and Hydrophobic Interfaces,” Phys. Rev. Lett.,96(18), p. 186101.
[96] Schmidt, A. J., Cheaito, R., and Chiesa, M., 2009, “A Frequency-Domain
Thermoreflectance Method for the Characterization of Thermal Properties,”
Rev. Sci. Instrum.,80(9), p. 94901.
[97] Schmidt, A. J., Cheaito, R., and Chiesa, M., 2010, “Characterization of Thin
Metal Films Via Frequency-Domain Thermoreflectance,” J. Appl. Phys.,
107(2), p. 24908.
[98] Schmidt, A. J., Chen, X., and Chen, G., 2008, “Pulse Accumulation, Radial
Heat Conduction, and Anisotropic Thermal Conductivity in Pump-Probe Tran-
sient Thermoreflectance,” Rev. Sci. Instrum.,79(11), p. 114902.
[99] Siemens, M. E., Li, Q., Yang, R., Nelson, K. A., Anderson, E. H., Murnane,
M. M., and Kapteyn, H. C., 2010, “Quasi-Ballistic Thermal Transport From
Nanoscale Interfaces Observed Using Ultrafast Coherent Soft X-Ray Beams,”
Nat. Mater.,9(1), pp. 26–30.
[100] Hoogeboom-Pot, K. M., Hernandez-Charpak, J. N., Gu, X., Frazer, T. D.,
Anderson, E. H., Chao, W., Falcone, R. W., Yang, R., Murnane, M. M., Kap-
teyn, H. C., and Nardi, D., 2015, “A New Regime of Nanoscale Thermal
Transport: Collective Diffusion Increases Dissipation Efficiency,” Proc. Natl.
Acad. Sci.,112(16), pp. 4846–4851.
[101] Wilson, R. B., and Cahill, D. G., 2014, “Anisotropic Failure of Fourier Theory in
Time-Domain Thermoreflectance Experiments,” Nat. Commun.,5, p. 5075.
[102] Hu, Y., Zeng, L., Minnich, A. J., Dresselhaus, M. S., and Chen, G., 2015,
“Spectral Mapping of Thermal Conductivity Through Nanoscale Ballistic
Transport,” Nat. Nanotechnol.,10(8), pp. 701–706.
[103] Capinski, W. S., Mar is, H. J., Ruf, T., Cardona, M., Ploog, K., and Katzer, D.
S., 1999, “Thermal-Conductivity Measurements of GaAs/AlAs Superlattices
Using a Picosecond Optical Pump-and-Probe Technique,” Phys. Rev. B,
59(12), pp. 8105–8113.
[104] Zhu, J., Tang, D., Wang, W., Liu, J., Holub, K. W., and Yang, R., 2010,
“Ultrafast Thermoreflectance Techniques for Measuring Thermal Conductiv-
ity and Interface Thermal Conductance of Thin Films,” J. Appl. Phys.,108(9),
p. 94315.
[105] Malen, J. A., Baheti, K., Tong, T., Zhao, Y., Hudgings, J. A., and Majumdar,
A., 2011, “Optical Measurement of Thermal Conductivity Using Fiber
Aligned Frequency Domain Thermoreflectance,” ASME J. Heat Transfer,
133(8), p. 081601.
[106] Schmidt, A. J., 2013, “Pump-Probe Thermoreflectance ,” Annu. Rev. Heat
Transfer,16(1), pp. 159–181.
[107] Kang, K., Koh, Y. K., Chiritescu, C., Zheng, X., and Cahill, D. G., 2008,
“Two-Tint Pump-Probe Measurements Using a Femtosecond Laser Oscillator
and Sharp-Edged Optical Filters,” Rev. Sci. Instrum.,79(11), p. 114901.
[108] Schmidt, A., Chiesa, M., Chen, X., and Chen, G., 2008, “An Optical Pump-
Probe Technique for Measuring the Thermal Conductivity of Liquids,” Rev.
Sci. Instrum.,79(6), p. 64902.
[109] Koh, Y. K., Cahill, D. G., and Sun, B., 2014, “Nonlocal Theory for Heat
Transport at High Frequencies,” Phys. Rev. B,90(20), p. 205412.
[110] Davidon, W., 1991, “Variable Metric Method for Minimization,” SIAM J.
Optim.,1(1), pp. 1–17.
[111] Nelder, J. A., and Mead, R., 1965, “A Simplex Method for Function Mini-
mization,” Comput. J.,7(4), pp. 308–313.
[112] Yang, J., Maragliano, C., and Schmidt, A. J., 2013, “Thermal Property Microscopy
With Frequency Domain Thermoreflectance,” Rev. Sci. Instrum.,84(10), p. 104904.
[113] Regner, K. T., Majumdar, S., and Malen, J. A., 2013, “Instrumentation of
Broadband Frequency Domain Thermoreflectance for Measuring Thermal
Conductivity Accumulation Functions,” Rev. Sci. Instrum.,84(6), p. 64901.
[114] Cahill, D. G., and Watanabe, F., 2004, “Thermal Conductivity of Isotopically
Pure and Ge-Doped Si Epitaxial Layers From 300 to 550 K,” Phys. Rev. B,
70(23), p. 235322.
[115] Liu, J., Zhu, J., Tian, M., Gu, X., Schmidt, A., and Yang, R., 2013,
“Simultaneous Measurement of Thermal Conductivity and Heat Capacity of
Bulk and Thin Film Materials Using Frequency-Dependent Transient Ther-
moreflectance Method,” Rev. Sci. Instrum.,84(3), p. 34902.
[116] Huang, J., Park, J., Wang, W., Murphy, C. J., and Cahill, D. G., 2012,
“Ultrafast Thermal Analysis of Surface Functionalized Gold Nanorods in
Aqueous Solution,” ACS Nano,7(1), pp. 589–597.
[117] Harikrishna, H., Ducker, W. A., and Huxtable, S. T., 2013, “The Influence of
Interface Bonding on Thermal Transport Through Solid–Liquid Interfaces,”
Appl. Phys. Lett.,102(25), p. 251606.
[118] Losego, M. D., Grady, M. E., Sottos, N. R., Cahill, D. G., and Braun, P. V.,
2012, “Effects of Chemical Bonding on Heat Transport Across Interfaces,”
Nat. Mater.,11(6), pp. 502–506.
[119] Lyeo, H.-K., and Cahill, D. G., 2006, “Thermal Conductance of Interfaces
Between Highly Dissimilar Materials,” Phys. Rev. B,73(14), p. 144301.
[120] Costescu, R. M., Wall, M. A., and Cahill, D. G., 2003, “Thermal Conductance
of Epitaxial Interfaces,” Phys. Rev. B,67(5), p. 54302.
[121] Ziade, E., Yang, J., Brummer, G., Nothern, D., Moustakas, T., and Schmidt,
A. J., 2015, “Thermal Transport Through GaN–SiC Interfaces From 300 to
600 K,” Appl. Phys. Lett.,107(9), p. 91605.
[122] Donovan, B. F., Szwejkowski, C. J., Duda, J. C., Cheaito, R., Gaskins, J. T.,
Yang, C.-Y. P., Constantin, C., Jones, R. E., and Hopkins, P. E., 2014,
“Thermal Boundary Conductance Across Metal-Gallium Nitride Interfaces
From 80 to 450 K,” Appl. Phys. Lett.,105(20), p. 203502.
[123] Freedman, J. P., Yu, X., Davis, R. F., Gellman, A. J., and Malen, J. A., 2016,
“Thermal Interface Conductance Across Metal Alloy–Dielectric Interfaces,”
Phys. Rev. B,93(3), p. 35309.
[124] Huxtable, S. T., Cahill, D. G., Shenogin, S., Xue, L., Ozisik, R., Barone, P., Usrey,
M.,Strano,M.S.,Siddons,G.,Shim,M.,andKeblinski,P.,2003,“Interfacial
Heat Flow in Carbon Nanotube Suspensions,” Nat. Mater.,2(11), pp. 731–734.
[125] Schmidt , A. J., Collins, K. C., Minnich, A. J., and Chen, G., 2010, “Thermal
Conductance and Phonon Transmissivity of Metal–Graphite Interfaces,”
J. Appl. Phys.,107(10), p. 104907.
040802-18 / Vol. 138, DECEMBER 2016 Transactions of the ASME
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 10/19/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
[126] Gao, Y., Marconnet, A. M., Xiang, R., Maruyama, S., and Goodson, K. E., 2013,
“Heat Capacity, Thermal Conductivity, and Interface Resistance Extraction for Single-
Walled Carbon Nanotube Films Using Frequency-Domain Thermoreflectance,” IEEE
Trans. Compon. Packag. Manuf. Technol.,3(9), pp. 1524–1532.
[127] Feser, J. P., Liu, J., and Cahill, D. G., 2014, “Pump-Probe Measurements of
the Thermal Conductivity Tensor for Materials Lacking In-Plane Symmetry,”
Rev. Sci. Instrum.,85(10), p. 104903.
[128] Liu, J., Choi, G.-M., and Cahill, D. G., 2014, “Measurement of the Anisotropic
Thermal Conductivity of Molybdenum Disulfide by the Time-Resolved
Magneto-Optic Kerr Effect,” J. Appl. Phys.,116(23), p. 233107.
[129] Jang, H., Wood, J. D., Ryder, C. R., Hersam, M. C., and Cahill, D. G., 2015,
“Anisotropic Thermal Conductivity of Exfoliated Black Phosphorus,” Adv.
Mater.,27(48), pp. 8017–8022.
[130] Zhu, J., Park, H., Chen, J.-Y., Gu, X., Zhang, H., Karthikeyan, S., Wendel, N.,
Campbell, S. A., Dawber, M., Du, X., Li, M., Wang, J.-P., Yang, R., and
Wang, X., 2016, “Revealing the Origins of 3D Anisotropic Thermal Conduc-
tivities of Black Phosphorus,” Adv. Electron. Mater.,2(5), p. 1600040.
[131] Liu, J., Yoon, B., Kuhl mann, E., Tian, M., Zhu, J., George, S. M., Lee, Y.-C.,
and Yang, R., 2013, “Ultralow Thermal Conductivity of Atomic/Molecular
Layer-Deposited Hybrid Organic–Inorganic Zincone Thin Films,” Nano Lett.,
13(11), pp. 5594–5599.
[132] Ong, W.-L., Rupich, S. M., Talapin, D. V., McGaughey, A. J. H., and Malen,
J. A., 2013, “Surface Chemistry Mediates Thermal Transport in Thr ee-
Dimensional Nanocrystal Arrays,” Nat. Mater.,12(5), pp. 410–415.
[133] Wang, X., Liman, C. D., Treat, N. D., Chabinyc, M. L., and Cahill, D. G.,
2013, “Ultralow Thermal Conductivity of Fullerene Derivatives,” Phys. Rev.
B,88(7), p. 75310.
[134] Duda, J. C., Hopkins, P. E., Shen, Y., and Gupta, M. C., 2013, “Exceptionally
Low Thermal Conductivities of Films of the Fullerene Derivative PCBM,”
Phys. Rev. Lett.,110(1), p. 15902.
[135] Yang, J., Ziade, E., Maragliano, C., Crowder, R., Wang, X., Stefancich, M.,
Chiesa, M., Swan, A. K., and Schmidt, A. J., 2014, “Thermal Conductance
Imaging of Graphene Contacts,” J. Appl. Phys.,116(2), p. 23515.
[136] Bozorg -Grayeli, E., Sood, A., Asheghi, M., Gambin, V., Sandhu, R., Feygel-
son, T. I., Pate, B. B., Hobart, K., and Goodson, K. E., 2013, “Thermal Con-
duction Inhomogeneity of Nanocrystalline Diamond Films by Dual-Side
Thermoreflectance,” Appl. Phys. Lett.,102(11), p. 111907.
[137] Cho, J., Li, Z., Bozorg-Grayeli, E., Kodama, T., Francis, D., Ejeckam, F.,
Faili, F., Asheghi, M., and Goodson, K. E., 2013, “Improved Thermal Interfa-
ces of GaN-Diamond Composite Substrates for HEMT Applications,” IEEE
Trans. Compon. Packag. Manuf. Technol.,3(1), pp. 79–85.
[138] Chiritescu, C., Cahill, D. G., Nguyen, N., Johnson, D., Bodapati, A., Keblin-
ski, P., and Zschack, P., 2007, “Ultralow Thermal Conductivity in Disordered,
Layered WSe
2
Crystals,” Science,315(5810), pp. 351–353.
[139] Ezzahri, Y., Grauby, S., Dilhaire, S., Rampnoux, J. M., and Claeys, W.,
2007, “Cross-Plan Si /SiGe Superlattice Acoustic and Thermal
Properties Measurement by Picosecond Ultrasonics,” J. Appl. Phys.,101(1),
p. 013705.
Journal of Electronic Packaging DECEMBER 2016, Vol. 138 / 040802-19
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 10/19/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
