Size Effect on the Thermal Conductivity of Ultrathin Polystyrene Films
Jun Liu1, Shenghong Ju1,2, Yifu Ding1, and Ronggui Yang1,*
1
Department of Mechanical Engineering, University of Colorado, Boulder, CO, 80309, USA
2
Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China
A short summary on the time-domain thermoreflectance (TDTR) measurement system,
heat transfer modeling, sensitivity analysis, and some measurement data are shown in this
Supporting Information, while the full details of the measurement technique can be found
in Ref. [1]. Since the detailed measurement procedure and fitting method vary with
different film thicknesses, the frequency-dependent TDTR measurements on thermal
properties of the 18.5 nm, 60 nm, and 191 nm-thick set-3 PS thin films are shown as
examples.
Figure S1 shows (a) the photo and (b) the schematics of the ultrafast laser-based TDTR
setup used for thermal property measurement in this work. The pulsed laser output from the
Spectra-Physics Tsunami femtosecond Ti-sapphire laser which emits a train of 150 fs pulses at a
repetition rate of 80 MHz is divided into pump and probe beams. The pump beam thermally
excites a sample and the probe beam measures the changes of the reflectivity in metal transducer
due to temperature rise (thermoreflectance). Since the thermoreflectance is generally small, on
the order of 10-5 - 10-4 K-1, the pump beam is modulated by an electro-optic modulator (EOM) at
a modulation frequency f between 0.1 and 20 MHz and the thermoreflectance signals are
detected by a lock-in amplifier at the same frequency. By changing the optical path length
*
Tel: +1-303-735-1003, Fax: +1-303-492-3498, E-mail address: Ronggui.Yang@Colorado.Edu
1
through a mechanical moving stage, the probe beam arrives at the sample surface at a different
time interval after the pump beam. The temporal decay of the optical signals is measured and
used to deduce the thermal properties with a heat transfer model through a multi-parameter
fitting process. The details of experiment setup and the data reduction scheme have been
presented in literature.2-4
(a)
(b)
Figure S1. (a) Photo of the two-color ultrafast laser-based TDTR experiment setup used in this
work. (b) Schematics of the TDTR setup used for thermal property measurement in this work.
Figure S2 shows the sample configuration for the frequency-dependent TDTR
measurement on PS thin films. Changing the modulation frequency f of the pump beam in TDTR
results in different thermal penetration depths L into the polymer thin film, which are defined as2
L
3
C3 f
,
(S1)
where κ3 and C3 is thermal conductivity and volumetric heat capacity of the thin film,
respectively. For instance, thermal waves only penetrate into a limited depth of the PS thin film
using a relatively higher modulation frequency f1 in Figure S2a compared to a much deeper
penetration throughout the thickness of the film using modulation frequency f2 (f1 > f2) in Figure
2
S2b. Reference [4] summarized the dominant thermophysical properties in the frequencydependent TDTR signal when the thermal penetration depth L is much smaller than, comparable
to, or much larger than the thickness of the thin film. We have then demonstrated that frequencydependent TDTR can be used to measure multiple thermophysical properties,1,
3, 5
such as
intrinsic thermal conductivity, volumetric heat capacity and interfacial thermal conductance.
Figure S2. Sample configuration for the frequency-dependent TDTR measurement on PS thin
films. PS thin films with thickness d3 were spin-coated on Si substrate. Al thin film was then
thermally-evaporated onto the PS thin film, which serves as both heater and sensor. By
modulating the pump beam with different modulation frequencies ((a) f1 or (b) f2), the thermal
waves penetrates into the PS thin film with different depths, which allows the measurement of a
variety of thermal properties, such as interfacial thermal conductance G2 between Al and PS film,
intrinsic thermal conductivity  3 of PS film, and interfacial thermal conductance G4 between PS
film and Si substrate.
3
When the thermal penetration depth L is much larger than the thickness d of the PS thin film,
L d 3  4.24 ,
(S2)
the thermal resistance of the PS thin film dominates the TDTR signal.1 The critical thickness of
PS thin film, below which only the total thermal resistance can be extracted from the TDTR
signal, is estimated using Eqs. (S1) and (S2). The critical thickness of PS film is 32 nm when the
TDTR measurement is conducted at a modulation frequency of 2.08 MHz, which is calculated
using a thermal conductivity value of 0.15 W/mK6 and a heat capacity value of 1.24 J/cm3K.6
Figure S3 shows the TDTR measurement signal –X/Y, which is the ratio of the in-phase and outof-phase signal from the lock-in amplifier, of the 18.5 nm-thick set-3 PS film at a modulation
frequency of 2.08 MHz, which satisfies the measurement condition given by Eq. S2, where the
thermal resistance or effective thermal conductivity of the PS film is fitted directly.
Figure S3. The TDTR measurement signal and the best-fit of the set-3 18.5 nm-thick PS film at
2.08 MHz.
Thermal effusivity or thermal diffusivity dominates the TDTR signal when
4
L d 3  4.24 ,
(S3).
Under this condition, intrinsic thermal conductivity and volumetric heat capacity can be
simultaneously measured by using multiple modulation frequencies as described in Ref. [1] if the
interfacial thermal conductance G2 and G4 is not sensitive in the TDTR signal, which
corresponds to a critical film thickness. Assuming κ3 as 0.15 W/mK, G2 as 20 MW/m2K, and G4
as 10 MW/m2K, which are roughly the thermal properties of the PS thin films measured, this
critical film thickness is calculated to be ~180 nm, where the thermal resistance of PS film is
much larger than the interfacial thermal resistance and approaches the total thermal resistance.
Figure S4a shows the TDTR measurement signal –X/Y and the best-fit of the 191 nm-thick set-3
PS film at modulation frequencies of 0.98 MHz, 2.08 MHz, and 6.8 MHz. Figure S4b shows the
κ-C diagram, similarly as shown in Ref [1], which is used to extract the thermal conductivity
and volumetric heat capacity of the 191 nm-thick set-3 PS film, where all the possible fitting
pairs of κ3 and C3 are shown under each modulation frequency and the cross-point or region
indicates the true values of κ3 and C3. The fitting results are shown in Table II in the main text.
The effective thermal conductivity can be approximated by the intrinsic thermal conductivity.
When the PS film thickness is ~60 nm, both the interfacial thermal conductance and
intrinsic thermal conductivity of the PS film is sensitive in the TDTR signal and could be fitted.
Figure S5 shows the sensitivity analysis and measurement procedure for the thermal properties
of the 60 nm-thick set-3 PS thin films. Similar to Ref. 7, the sensitivity S of the TDTR signal
 X / Y to the thermal property p is defined as
S
d ( X / Y ) ( X / Y )
,
dp p
(S5)
5
where d ( X / Y ) ( X / Y ) is the fluctuation of TDTR signal –X/Y and p is the property that are
of measurement interest, which could be thermal conductivity, heat capacity, or interface thermal
conductance in the multilayer structure presented in Figure S2. Figure S5a show the sensitivity
analysis of measuring the 60 nm-thick set-3 PS film under modulation frequencies of 2.08 MHz
and 6.8 MHz. The thermal conductivity κ3 of PS thin film is sensitive under both modulation
frequencies. The total resistance of the two interfaces G2 and G4 is sensitive under a modulation
frequency of 2.08 MHz while the interfacial thermal conductance between polymer film and Si
G4 is sensitive under a modulation frequency of 6.8 MHz. Thus, all the three unknown thermal
properties κ3, G2, and G4 can be extracted. Figure S5b shows the TDTR measurement signal and
the best-fit of the 60 nm-thick set-3 PS film at 2.08 MHz, 3.4 MHz, and 6.8 MHz. The fitting
results are shown in Table III in the main text. The effective thermal conductivity is then
calculated using Eq. (1) in the main text.
(a)
(b)
Figure S4. The measurement procedure for the thermal properties of 191 nm-thick set-3 PS thin
films. (a) The TDTR measurement signal of the 191 nm-thick set-3 PS film at 0.98 MHz, 2.08
6
MHz, and 6.8 MHz. (b) The κ-C diagram that used to extract the thermal conductivity and
volumetric heat capacity of the 191 nm-thick set-3 PS film.
(a)
(b)
Figure S5. (a) The sensitivity analysis of measuring the 60 nm-thick set-3 PS film under
modulation frequencies of 2.08 MHz and 6.8 MHz. (b) The TDTR measurement signal and the
best-fit of the set-3 60 nm-thick PS film at 2.08 MHz, 3.4 MHz, and 6.8 MHz.
The power of the pump beam was carefully selected during the measurement by limiting the
steady-state heating on the sample surface less than 20 K. The uncertainty of the measurement
were estimated by taking into account the individual uncertainties and measurement sensitivities
of the parameters in the heat transfer model, similarly as in Ref [8].
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7
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