Simultaneous measurement of in-plane and through-plane thermal conductivity using beam-offset frequency domain thermoreflectance

Simultaneous measurement of in-plane and through-plane thermal conductivity using beam-offset frequency domain thermoreflectance David Rodin 1 and Shannon K. Yee 1 1 George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA Transient thermoreflectance (TTR) techniques are ubiquitous methods for measuring thermal conductivity of bulk materials and thin-films. Both through-plane thermal conductivity k and inplane thermal conductivity k should be independently measured in transversely anisotropic materials. When these properties are measured using conventional TTR techniques, the accuracy of the k measurement is dependent on the accuracy of measuring k, and vice versa. This is especially problematic for thin-films measurements as uncertainty in k (~5%) can propagate and grow for uncertainty in k. In this paper, we present a method for the simultaneous measurement of k and k using beam-offset frequency domain thermoreflectance (FDTR) with robust uncertainty estimation. The conventional diffusive heat transfer solution is analyzed to show that offset and heating frequency can independently control the sensitivity to directional thermal conductivity and extract values for k and k. Numerical uncertainty analyses demonstrate that sweeping both heating frequency and beam offset results in a reduction of measurement uncertainty. This modified measurement technique is demonstrated on crystalline alumina (c- Al 2 O 3 ), amorphous alumina (a-al 2 O 3 ), quartz, fused silica, and highly-oriented pyrolytic graphite (HOPG). I. INTRODUCTION The continued use of thin-film technologies perpetuates interest in the thermal properties of nanostructured materials. 1-5 Nanostructuring and other size effects (e.g., nanoinclusions, block copolymer macrostructures, and grain boundaries) lead to significant differences in the thermal properties of thin-films compared to their bulk counterparts. 3 Additionally, some isotropic bulk materials show anisotropic behavior when processed into thin-films. 6 Materials whose thermal conductivity tensor can be fully defined by in-plane and through-plane values are referred to as transversely isotropic materials. Numerous techniques 7-11 have been developed to measure thin-film thermal properties with the 3-omega method and the transient thermoreflectance (TTR) methods, which includes time-domain thermoreflectance (TDTR) and frequencydomain thermoreflectance (FDTR), being most widely used. The ability of both 3-omega and TTR methods to measure the through-plane thermal conductivity of thin-films has been 1, 11-13 robustly demonstrated. Beam offset has been shown to improve the sensitivity of TDTR to in-plane thermal conductivity 10 and recent advancements in TDTR methodologies by Feser, Liu, and Cahill have demonstrated the ability to resolve full thermal conductivity tensors. 14 In-plane thermal conductivity measurements using FDTR with co-aligned beams have also been 1

demonstrated on transversely isotropic 15, 16 materials. Although FDTR measurement capabilities are lagging those of TDTR when it comes to anisotropic materials, FDTR still offers some important advantages. In particular, FDTR uses a continuous-wave (CW) laser as opposed to TDTR s femtosecond pulsed laser and does not require a delay stage. These two factors contribute to the lower cost of FDTR and the use of a CW laser allows for a single timescale. 17 Furthermore, the ability of FDTR to achieve high heating frequencies using a heterodyned signal may provide some important insights into nanoscale heat transport. 9, 17, 18 Reducing the experimental uncertainty of FDTR without significantly increasing the complexity or cost of the measurement is an important step towards improving measurement capabilities for anisotropic thin films. The general approach for anisotropic TTR measurements 10 is to perform two independent measurements: (i) to measure the through-plane thermal conductivity and then (ii) to measure the in-plane thermal conductivity where the previous throughplane value is used as a dependent parameter. One approach to improving the sensitivity to independent values in TTR measurements involves changing the spot size of the pump and probe beams. 10 This requires a realignment of the optics for each measurement and prevents the two measurements from being performed in rapid succession. Hence, measurement time and accuracy could be improved if both inplane and through-plane thermal conductivity are measured simultaneously. Herein, we demonstrate the measurement realization of a simultaneous in-plane and through-plane thermal conductivity measurement using a beamoffset FDTR technique that works for both bulk and thin-film materials, can be applied across a wide range of thermal conductivities (~ 5 to ~2000 W/m-K), and results in lower in-plane measurement uncertainty when compared to alternative FDTR methodologies. 30 In Sec. II, the heat transfer model is analyzed to show how sensitivity to in-plane thermal conductivity, k, and through-plane thermal conductivity, k, can be determined without changing the laser spot size. In Sec. III, experimental data is presented alongside a robust method of uncertainty analysis to demonstrate the measurement capabilities. II. METHODOLOGY FDTR, which is well described in literature, 8, 9, 13 measures an unknown thermal property, most often thermal conductivity, using a pump-probe laser technique relying on a transducer s temperature dependent thermoreflectance, R -1 dr / dt. As an overview to FDTR, one experimentally measures the phase lag between a periodic heat source and its corresponding temperature response on a surface as a function of heating frequency. Then one matches the experimental frequency dependent phase lag data to an analytical model 19, 20 by adjusting the unknown thermal property until the best fit (minimizing the mean-squared error) is obtained. The value of the unknown parameter that yields the best fit is the measured value of the unknown thermal property. To absorb the periodic heat source delivered by a laser, FDTR requires the deposition of a thin (~50-100 nm) thermoreflectance transducer atop the sample; Au is the best transducer for the laser wavelengths used in many FDTR setups because the reflectance of Au exhibits a relatively strong temperature dependence at the wavelength of 532 nm, 2

FIG. 1. (a) Schematic of an offset FDTR set-up. The only difference between the experimental set-ups of offset FDTR and traditional FDTR is the addition of the picomotor mirror, which scans the probe beam across the sample to create precise offsets between the pump and probe beams. (b) Sample configuration for traditional FDTR. The pump beam (blue) is concentrically aligned with the probe beam (green). The interfacial thermal conductance between the transducer and film is labeled G 1 and the conductance between the film and substrate is labeled G 2. (b) Sample configuration for offset FDTR. The probe beam is offset from the pump beam and the film or substrate may have unique values for through-plane (k ) and in-plane (k ) thermal conductivity. R -1 dr / dt 10 4 K -1 and it absorbs well at phase-lag θ. The surface temperature oscillation induces a coherent oscillation in wavelengths near 488 nm. Thus, a pump the reflectance of the Au transducer. The laser (at 488 nm) creates a periodic heat incoming probe beam is initially at a source absorbed by the transducer and a constant intensity but, upon reflection, probe laser (at 532 nm) monitors the becomes modulated at frequency f and picks temperature response of the transducer. up the phase-lag θ. Upon reflection, the Therefore, the transducer serves two pump and the probe beams take the same purposes: it absorbs the pump beam, which path to a photodiode, where their intensity is enables isothermal surface heating (see converted into a voltage signal. The voltage Supplementary Materials 30 ), and it reflects signals from the pump and probe are the probe beam, which carries information measured individually (separated using about the temperature response of the inline band pass filters) as a function of surface to the periodic heat source encoded heating frequency. Some additional phase by the thermoreflectance of the transducer. 21 lag is probably acquired by the post-sample The pump laser s intensity is modulated optics. However, since the reflected paths at a frequency f and creates a temperature of the beams are identical, the only source of oscillation on the sample surface at that phase lag that is different between the pump same frequency f but with an additional and probe signals arises from the phase lag 3

response of the periodic heating at the sample and the associated temperature response. Therefore, the phase difference between the pump and probe voltage signals is θ. An analytical solution to the heat diffusion equation can be used to predict θ. 19, 20 The inputs to the model are the thermal conductivity, volumetric heat capacity, thickness of each layer, as well as the 1/e 2 radius of the pump and probe beams. In a typical isotropic sample, the thermal conductivity, k, and the thermal interface conductance between the transducer and the sample, G, are unknown. In the case of a multilayered sample, as shown in Fig. 1(a) and 1(b), the unknown parameters may include: (i) the film conductivity, k film, (ii) the substrate conductivity, k sub, and (iii) the thermal interface conductances, G 1 and G 2. In many multilayer samples, the substrate conductivity is independently measured without the film and k sub can be treated as a known input parameter to the model. The transducer thickness and thermal conductivity, as well as the volumetric heat capacity of the sample, can be independently measured with the methodologies described in the Supplementary Materials. 30 The remaining unknown parameters are used as fitting parameters (primarily k film ) to match the model to the experimental data. The measurement results are the set of fitted values that minimize the mean-squared error between the analytical model and experimental data. A detailed configuration for traditional and broadband FDTR is presented by Regner et al. 9 and the experimental set-up for beam-offset FDTR is shown in Fig. 1(a). In our setup, both the pump (Coherent MX-488 SLM) and probe (Coherent Verdi G2) beams pass through optical isolators (ISO, Conoptics 711A and 711C-3) tuned to their specific wavelengths. The isolators act as optical diodes to prevent light from reentering and damaging the laser cavities. The signal generator (Rigol DG5252) produces a sinusoidal voltage at a controllable frequency, which is amplified (Conoptics Model 200) to drive the electrooptic modulator (EOM, Model 350-80-01 KD*P Series with reduced aperture). The frequency range of the modulation system is between 9 khz and 200 MHz. However, we found that the experimental data is most reliable between 400 khz and 5 MHz. The lower limit of 400 khz was established to minimize uncertainty due to beam spot size errors and the upper limit was established to avoid high-frequency ambient (RF) noise. The EOM is a polarization-dependent optics and must be rotated such that the incoming light is polarized at a 45 angle relative to the EOM s crystal. To more easily achieve this, a half-wave plate (HWP, ThorLabs WPH10M-488) rotates the polarization of the laser before it enters the EOM s aperture and eliminates the need for the user to rotate the EOM thus simplifying the alignment process. The pump and probe beams are coaligned as they pass through the beam splitter (BS, ThorLabs BS025) and are then directed towards the sample using a polarizing beam splitter (PBS, ThorLabs CM1-PBS251). The beam polarizations are tuned through a quarter-wave plate (QWP, ThorLabs WPQ10M-488) and focused through an objective (OBJ, Nikon MUE31100) onto the sample. The reflected beams are sent to the photodiode (PD, ThorLabs PDA36A) and their phase-lag relative to the voltage signal generated by the signal generator is measured using a lock-in amplifier (SRS SR844). The reason the phase is measured relative to the arbitrary reference signal is because the lock-in amplifier requires the magnitude of the reference signal to be greater than 200 mv. The voltage magnitudes associated with the pump and probe are 100 mv and 4

FIG. 2. Sensitivity analysis results for single-crystal quartz with input parameters listed in Table II. (a) Sensitivity to in-plane thermal conductivity as a function of beam offset and heating frequency. Regions of high (white), moderate (gray), and low (black) sensitivity are all found within the domain (see Supplemental Materials 30 ). (b) Sensitivity to through-plane thermal conductivity. (c) Sensitivity to thermal interface conductance demonstrates low sensitivity to G across the entire domain. (d) Total sensitivity shows regions where measurement is sensitive to both in-plane and through-plane thermal conductivity (white), only to through-plane conductivity (gray), and insensitivity to both (black). 100 μv, respectively, so neither signal can be used as a reference. Instead, the phase lag between the two signals is obtained by independently measuring their phase lag relative to the signal generator and taking the difference between the measured phase lag values. The individual measurement of the beams is enabled by bandpass filters for each laser respectively (BP, ThorLabs FL488-1 and ThorLabs FL532-10). In order to controllably offsetting the probe beam, a picomotor mirror mount (Newport Model 8807) with an open-loop driver (Newport Model 8742) is used. The mirror is controlled using high-precision piezo-actuators with each actuator step corresponding to a change in angle of 0.7 µrad. The probe beam passes through several optical elements between the picomotor mirror and the sample, so it is necessary to calibrate changes in mirror 5

angle to changes in spatial positioning of the probe beam on the sample. These calibrations were performed using a scanning slit beam profiler (DataRay Beam R2) with a spatial resolution of 100 nm, which allows for the full resolution of the laser spots even when fully focused. The calibration results showed that, although the probe s positioning is consistently linear with respect to picomotor steps, the slope of this relationship would vary each day by as much as 4% (Supplemental Materials 30 ). Therefore, we calibrate the Picomotor before and after each measurement to get an accurate measure of the beam offset distances and to ensure that the calibration does not drift over the course of the measurement. A. Analytical Heat Transfer Solution The heat transfer model for TTR experiments has been rigorously 19, 20 developed. The temperature fluctuation detected by the probe beam is given by 2 T G( f, h) P( h) S( h) hdh A, (1) s 0 where P(h) and S(h) are the Hankel transforms of the intensity profiles of the pump and probe beams, G(f,h) is the Hankel transform of the Green s function solution for the temperature response from a point heat source that oscillates sinusoidally with unit strength, and A s is the incident power of the probe beam. Feser et al. describe the formulations of G(f,h), P(h), and S(h) for the case of offset beams with transversely isotropic materials. 10 A full solution for Equation (1) can be found in the Supplemental Materials. 30 B. Offset Sensitivity Analysis A sensitivity analysis can be used to determine how a specific parameter influences the phase-lag θ. The sensitivity is TABLE I. Nominal parameters for quartz used in the sensitivity analysis shown in Fig. 2. Property Nominal Value Uncertainty C transducer 2.5 MJ/m 3 -K 5% k transducer 220 W/m-K 5% L transducer 110 nm 3% r probe 2.0 µm 3% r pump 3.2 µm 4% L sample 500 µm N/A C sample 1.65 MJ/m 3 -K 5% k 6.5 W/m-K 10% k 9.8 W/m-K 19% G transducer 51 MW/m 2 -K 22% defined as the logarithmic derivative of θ with respect to the system parameter of interest β. S d ln d ln (2) Equation (2) is evaluated numerically by changing the value of β by 1% about the nominal value and observing the corresponding change in the phase-lag of Equation (1). The results of a sensitivity analysis on single-crystal quartz are shown in Figure 2 with the nominal parameters listed in Table I. The sensitivity analysis allows one to determine which parameters most heavily influence the measurement response across a range of frequencies and beam offsets. The measurement should be more sensitive to the fitting parameters than the input parameters to prevent small uncertainties in the independent input measurements from propagating into the final result. This consideration is used to optimize the measurement accuracy by selecting proper film thickness, transducer thickness, heating frequency range, and beam offsets. If any 6

input parameter shows greater sensitivity than the fitting parameters, an emphasis is placed on the accuracy of the independent measurement of that input parameter. Input parameters with low sensitivity (S β < 0.05) can exhibit relatively large uncertainty (U β ~ 25%) without significantly impacting the uncertainty of the final measurement so long as the sensitivity to the fitting parameters is relatively high (S β > 0.2) (see Supplemental Materials 30 ). In some unavoidable cases, the sensitivity to multiple input parameters may be higher than the sensitivity to the fitting parameters and measurement uncertainty is very high (U β > 50%). For these cases, a robust approach to uncertainty analysis plays a critical role in proper reporting of measurement results. The results of a sensitivity analysis on single-crystal quartz are shown in Figure 2 with the sample parameters listed in Table I. Since the sensitivity to the thermal interface conductance between the transducer and sample is low across the entire domain (see Fig. 2(c)), inaccuracies in the measured value of G are unlikely to propagate into the measurement uncertainty of k and k The total sensitivity plot in Figure 2(d) shows how using both heating frequency and beam offset creates regions that are sensitive to both k and k (white) as well as regions that are only sensitive to k (gray). The existence of both regions is important because it allows the measurement to produce unique results when fitting to multiple parameters simultaneously. C. Computational Uncertainty Analysis A Monte Carlo computational approach, similar to the methods of other TTR measurements, is used to calculate measurement uncertainty. 22-24 In this approach, the nominal input parameters are first collected through independent measurements (see Supplemental Materials 30 ) and the uncertainty associated with the independent measurement is recorded. The contribution of experimental noise to uncertainty is accounted for by making five independent measurements at each heating frequency and recording the measurement standard deviation. The errors in our input parameters and experimental phase data are normally distributed about their nominal and mean values, respectively. A new set of input parameters and a new set of experimental data are randomly sampled from these distributions. Next, using the input parameters, the generated data is fit to the heat transfer model to determine the fitting parameters that minimize the mean squared error. The resulting fitting parameters are treated as a single trial outcome and this process is repeated until the distribution of these outcomes converges to a normal distribution, typically after 2000 trials. 7

Table II. Literature and measured values for in-plane thermal conductivity (k ), through-plane thermal conductivity (k ), and volumetric heat capacity (C v ) and measured values for thermal interface conductance (G 1 and G 2 for a- Al 2 O 3 ). Literature Measured (this work) Sample C v (MJ/m 3 -K) k (W/m-K) k (W/m-K) k (W/m-K) k (W/m-K) G(W/m 2 -K) Fused Silica 1.6 ± 0.1 25 1.4 ± 0.1 25 1.4 ± 0.1 25 1.3 ± 0.5 4.6 ± 4.6 70±23 Quartz 2.0 ± 0.2 25 5.9 ± 0.9 25 11.0 ± 1.0 25 6.3 ± 1.2 10.3 ± 3.0 51±11 a-al 2 O 3 3.2 ± 0.3 26 0.8 ± 0.2 26 0.8 ± 0.2 26 0.9 ± 0.4 2.6 ± 2.6 71±16, 68±24 c-al 2 O 3 3.2 ± 0.3 27 25 ± 3 27 33 ± 4 27 25.5 ± 2.5 37.6 ± 7.0 390±56 HOPG 1.6 ± 0.1 10 8 ± 2 10 1800 ± 200 10, 28 9±2.7 1861±744 104±43 Once the outcome distribution is obtained, a 95% confidence interval is used to quantify the uncertainty in the fitting parameter. This approach is more conservative than reporting the uncertainty at one standard deviation, albeit at a perceived higher uncertainty. An example of the Monte Carlo histogram using this approach for the in-plane thermal conductivity of single crystal quartz is shown in Fig. 3. This approach is advantageous because it does not assume a hierarchy of dominant uncertainty sources and factors both parameter uncertainty propagating through the model and experimental fluctuations into the total reported measurement uncertainty. D. Sample Selection and Preparation The ability to simultaneously measure in-plane and through-plane thermal conductivity is demonstrated over a range of conductivities and degrees of anisotropy. The samples are listed with their nominal thermal conductivity and volumetric heat capacity values in Table II. With the exception of the highlyoriented pyrolytic graphite (HOPG), each sample is prepared by submersion in a sonication bath of deionized water, followed by isopropyl alcohol, and finally ethanol. Immediately before the deposition of the Au transducer, the samples are rinsed in a 3:1 piranha solution at 120 C for 30 minutes. This is to ensure that all organic matter is removed from the surface. Before the deposition of the gold, a thin (~ 5 nm) layer of chromium is deposited onto the cleaned sample surface to serve as an adhesion layer. Regner et al. found that a high transducer interface conductance improves sensitivity to the sample thermal conductivity. 9 The chromium layer serves to increase the thermal interface conductance by acting as an adhesion layer. HOPG was prepared by exfoliating the top layers using Scotch tape several times prior to deposition. FIG. 3. Uncertainty distribution for in-plane thermal conductivity of single crystal quartz. 95% of the outcomes fall between 7.3 W/m-K and 13.3 W/m-K, yielding a measured in-plane conductivity of W/m-K. III. EXPERIMENTAL RESULTS A. Experimental Data and Analysis We present the raw experimental data and uncertainty analysis results for single crystal quartz, and compare the measured 8

thermal conductivity values for all samples against their nominal values from Table II. For each sample, the in-plane and throughplane conductivities are treated as fitting parameters fitting the analytical solution to the experimental data. The model and experimental data for quartz are shown in Fig. 4, alongside the results from the computational uncertainty analysis. Raw data of the other samples can be found in the Supplemental Materials. 30 The measured phase-lag is in good agreement with the analytical model using the nominal values. Residuals never exceed 0.35 and are typically less than 0.2. The experimental phase-lag was observed to fluctuate by about 0.3 from ambient noise, so residuals below 0.2 are considered insignificant. Furthermore, the residuals between the model and experiment do not exhibit a dependence on any of the independent variables (beam offset and heating frequency), indicating that the technique is free of systematic errors. For quartz, the measured values for k and k are each within 7% of their nominal values. The thermal interface conductance between the transducer and substrate, G, is used as an additional fitting parameter, however this technique is not very sensitive to the value of G because the sensitivity to thermal conductivity is much higher than the sensitivity to G across the range of heating frequencies. The sensitivity to G tends to increase at higher heating frequencies and lower thermal conductivities, so a sensitivity analysis must be performed on each sample before making assumptions about the relevance of thermal interface conductance. FIG. 4. (a) Experimental phase data (dots) with fitted analytical solution (solid lines) for single crystal quartz. The beam-offset values were swept from 0 (blue) to 0.75 µm (red). Monte Carlo uncertainty estimations for both throughplane (top) and in-plane (bottom) thermal conductivity. 9

The measurement uncertainty was calculated to be 29% for k and 19% for k, which is comparable to the uncertainty in the reported nominal literature values 29. The same analysis shown for quartz was applied to all samples in Table II. Figure 5 compares the measured values of both k and k to their respective nominal values. Overall, the through-plane measurement results show good agreement with nominal literature values across the range of conductivities (~1-25 W/m-K). However, the uncertainty for the in-plane thermal conductivity for low thermal conductivity materials is large, particularly in fused silica, amorphous alumina, and HOPG. The surface roughness of HOPG created uncertainty in the transducer thickness which propagated into the final measurement uncertainty. For amorphous alumina and fused silica, we attribute the high uncertainty to large in-plane heat spreading in the Au transducer; the majority of in-plane heat spreading occurs in the transducer (not the sample), which causes the measurement to be less sensitive to the in-plane thermal conductivity of the sample. Two potential avenues to improve this sensitivity, both discussed in the Supplemental Materials, 30 include using a less conductive transducer or patterning the transducer to force heat to travel through the sample. IV. CONCLUSION We have developed a modified version of FDTR for simultaneous measurement of in-plane and through-plane thermal conductivity using beam-offset control. The technique sweeps over a range of both heating frequencies and beam offsets to independently achieve sensitivity in both k and k. Measurement capabilities have been demonstrated over a range of thermal conductivities (~1-2000 W/m-K) and materials ranging from isotropic (k /k =1) to FIG. 5.(a) Comparison between measured and nominal values for k. Vertical error bars indicate measurement uncertainty and horizontal bars indicate uncertainty in nominal value. (b) Comparison between measured and nominal values for k. The bottom half of the error bars are omitted for a-al 2 O 3 and fused silica. highly anisotropic (k /k >300) with a reduced in-plane measurement uncertainty compared to alternative FDTR approaches 30. Measurement uncertainty for through-plane thermal conductivity ranged between 10-45% with uncertainty increasing as thermal conductivity decreased. Measurement 10

uncertainty in the in-plane thermal conductivity varied between 10-20% for materials with conductivities greater than 5 W/m-K. However, uncertainties are larger for in-plane thermal conductivity below 5 W/m-K because of lateral heat spreading in the Au transducer. Despite its high in-plane conductivity, measurement uncertainty for in-plane thermal conductivity of HOPG approached 40% due to uncertainty in the transducer thickness. SUPPLEMENTARY MATERIALS See supplementary materials for the (a) analytical solutions, (b) beam diameter measurement protocol, (c) picomotor calibration, (d) full data sets, (e) sensitivity threshold analysis, (f) additional independent measurements, (g) low thermal conductivity threshold measurements, (h) uncertainty measurement bounds, (i) comparison of measurement uncertainty to alternative FDTR techniques. ACKNOWLEDGEMENTS This work was supported by the AFOSR under Award No. FA9550-15-1-0145. D. Rodin would like to acknowledge support from the NSF-IGERT fellowship. The authors would like to thank Professor Jonathan Malen and Dr. Keith Regner at the Carnegie Melon University for insightful conversations. REFERENCES 1. J. Liu, J. Zhu, M. Tian, X. Gu, A. Schmidt and R. Yang, Review of Scientific Instruments 84 (3), 034902 (2013). 2. D. G. Cahill, W. K. Ford, K. E. Goodson, G. D. Mahan, A. Majumdar, H. J. Maris, R. Merlin and S. R. Phillpot, Journal of Applied Physics 93 (2), 793 818 (2003). 3. A. Jain, S. Rogojevic, S. Ponoth, W. N. Gill, J. L. Plawsky, E. Simonyi, S. T. Chen and P. S. Ho, Journal of Applied Physics 91 (5), 3275 3281 (2002). 4. D. G. Cahill, K. Goodson and A. Majumdar, Journal of Heat Transfer 124 (2), 223 241 (2001). 5. S. Vineis, Majumdar, Kanatzidis, Adv. Mater. 22 (36), 10 (2010). 6. T. Bing Yue, Y. Chen Chi and F. Kuo Lung, presented at the VLSI Technology, Systems, and Applications, 2003 International Symposium on, 2003 (unpublished). 7. T. Tong and A. Majumdar, Review of Scientific Instruments 77 (10), 104902 (2006). 8. A. J. Schmidt, R. Cheaito and M. Chiesa, Review of Scientific Instruments 80 (9), 094901 (2009). 9. K. T. Regner, S. Majumdar and J. A. Malen, Review of Scientific Instruments 84 (6), 064901 (2013). 10. J. P. Feser and D. G. Cahill, Review of Scientific Instruments 83 (10), 104901 (2012). 11. D. G. Cahill, Review of Scientific Instruments 61 (2), 802 808 (1990). 12. C. Dames and G. Chen, Review of Scientific Instruments 76 (12), 124902 (2005). 13. J. A. Malen, K. Baheti, T. Tong, Y. Zhao, J. A. Hudgings and A. Majumdar, Journal of Heat Transfer 133 (8), 081601 081601 (2011). 14. J. P. Feser, J. Liu and D. G. Cahill, Review of Scientific Instruments 85 (10), 104903 (2014). 15. J. Yang, C. Maragliano and A. J. Schmidt, Review of Scientific Instruments 84 (10), 104904 (2013). 16. V. V. Medvedev, J. Yang, A. J. Schmidt, A. E. Yakshin, R. W. E. van de Kruijs, E. Zoethout and F. Bijkerk, Journal of Applied Physics 118 (8), 085101 (2015). 17. K. T. Regner, J. P. Freedman and J. A. Malen, Nanoscale and Microscale Thermophysical Engineering 19 (3), 183 205 (2015). 18. K. T. Regner, D. P. Sellan, Z. Su, C. H. Amon, A. J. H. McGaughey and J. A. Malen, Nature Communications 4, 1640 (2013). 19. D. G. Cahill, Review of Scientific Instruments 75 (12), 5119 5122 (2004). 11

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