Structure-Thermal Property Correlation of Aligned Silicon Dioxide. Nanorod Arrays

Transcription

1 Structure-Thermal Property Correlation of Aligned Silicon Dioxide Nanorod Arrays Jie Zhu 1, 3, Yu Zhu 2, Xuewang Wu 1, Helun Song 2, Yaohui Zhang 2, and Xiaojia Wang 1,* 1 Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA 2 Suzhou Institute of Nano-technology and Nano-bionics, Chinese Academy of Sciences, , Suzhou, China 3 Institute of Engineering Thermophysics, Chinese Academy of Sciences, , Beijing, China *Author to whom correspondence should be addressed: wang4940@umn.edu Abstract: Quantitative characterization of thermal properties of nanorod (NR) arrays appears to be challenging due to the complex combination of high volume of air voids, anisotropy, and structural non-uniformity. This work investigates the structure-thermal property correlations of arrays consisting of either vertically aligned or slanted silicon dioxide (SiO 2 ) NRs, fabricated by the dynamic shadowing growth technique. We apply the frequency-dependent time-domain thermoreflectance method to quantify the thermal properties of SiO 2 NR arrays that may possess inhomogeneity along the depth direction. The effective thermal conductivities of four SiO 2 NR array films and one reference capping layer for the SiO 2 NR array are obtained. The impact of the structure on the effective thermal conductivities of the SiO 2 NR array is discussed. The lowest effective thermal conductivity among all samples in this work is found to be 0.13 W m 1 K 1 for the slanted NR array. We attribute the reduction in the effective thermal conductivity of the NR array to the discontinuous nature of SiO 2 NRs, which reduces the density of the thermal transport channels and thus prevents heat flux from the top surface from propagating

2 downwards along the through-plane direction. The results from this work facilitate the potential applications of NR-array-based thermal insulators for micro-thermal devices. Thermal insulation is of vital importance for uncooled infrared (IR) detectors, which are widely used in security, motion detection, night vision, firefighting, and automotive brake assistance systems, 1,2 to achieve maximum signal amplitude and sensitivity. Currently, other than microelectromechanical systems (MEMS)-based micromachining techniques, 3,4 and aerogelbased mesoporous and macroporous membranes, 5-13 compatible thin films consisting of aligned SiO 2 nanorods (NRs) fabricated by dynamic shadowing growth (DSG) are more cost-effective with easier control of porosity and structural orientation. For this kind of NR-array structure, the thermal conductivity can also be reduced significantly with moderate porosities, while the durability of the structure is much better compared with the two other thermal insulating materials aforementioned. 14 Quantitative characterization of thermal conductivity and/or thermal diffusivity of SiO 2 NR array structures appears to be challenging due to the complex combination of high air void volumes, rough surface, discontinuity, inhomogeneity, as well as non-uniformity of the NRs. Though previous publications reported the thermal characteristics of semiconductor NR arrays, the thermal conductivity of SiO 2 NR arrays is still missing. In this work, we applied the frequency-dependent time domain thermoreflectance (TDTR) method to quantify the thermal properties of SiO 2 NR arrays that may possess inhomogeneity along the depth direction The DSG (a.k.a. glancing angle deposition, GLAD) method was applied to synthesize thermal insulating thin films with different thicknesses constructed by slanted or vertically aligned SiO 2 NR arrays. Ellipsometry and scanning electron microscope (SEM) measurements were performed to determine the samples structure. 2

3 A smooth and dense capping layer is applied on top of NR arrays to serve as a base to deposit metal transducers for TDTR measurements. This eliminates the requirement of filling the air gaps between nanorods with polymer matrix to form a continuous and smooth surface for transducer deposition, as typically adopted in thermal characterizations of array structures in the literature This significantly reduces the measurement uncertainty benefiting from avoiding additional thermal property calibrations of the matrix host materials. In the TDTR measurements, tuning the modulation frequency of the pump beam results in a varied penetration depth of thermal waves, allowing for probing the inhomogeneity of NR arrays along the through-plane (depth) direction. The modulation frequencies f in this work were 18 MHz, 9 MHz, and 1.6 MHz for each sample. The combination of these three modulation frequencies resulted in a thermal penetration depth ranging from 120 nm to 400 nm, predicted using the typical thermal properties of bulk SiO 19 2 according to ( Cf ). 27 Therefore, the depth-dependent thermal properties of the NR array with a thickness of hundreds of nanometers could be probed to extract information about array inhomogeneity. 3

4 FIG. 1. Scanning electron microscopic images of (A) top view and (B) side view of the SiO 2 NR array layers, and (C) corresponding sample schematic in layered structures. The structure of SiO 2 NR array sample consists of a silicon substrate, nanostructured SiO 2 arrays, and capping layer from bottom to top. The representative SEM images in Fig. 1 depict the sample structure features of five SiO 2 NR arrays for the TDTR thermal measurements, including one sample of capping layer only, two slanted (tilting angle ~45 o ) and two vertical SiO 2 NR arrays with capping layers on top. The capping layers deposited on top of the NR arrays are porous SiO 2 with smooth and continuous surfaces and much lower porosities (10~20%). Figure 1(A) shows the top view SEM images of the SiO 2 NR array surfaces. The top surface appears to be continuous but granular, resulting from the random aggregation of the columnar NRs as the deposition angle ( d ) decreases from 84 to 0 in DSG (Section S1 of the Supplementary Material 28 ). Broccoli-like structures on the top surfaces also can be seen in Fig. 1(A). The film density of the capping layer increases from bottom to top (with decreasing d ) due 4

5 to the less dominant shadowing effect at smaller d, 24,26 as shown in Fig. 1(B), which depicts the sample morphology along the sample cross-sections (side view). Structural schematics for three types of SiO 2 NR arrays are shown in Fig. 1(C). The thicknesses of all five samples obtained from SEM characterizations and root-mean-square surface roughness ( rms ) from atomic force microscopy (Section S2 of the Supplementary Material 28 ) characterizations are summarized in Table I. Table I. Structural parameters of the SiO 2 NR-array samples. Sample # Alignment h C (nm) h NR (nm) rms (nm) 1 Capping layer 267± Slanted 1 246±5 296± Vertical 1 258±5 254± Slanted 2 250±5 1500± Vertical 2 250±5 1200± Modeling of TDTR measurements has been previously discussed in detail. 19,20,29,30 To extract the sample thermal properties, the measurement ratio of the in-phase to out-of-phase signal ( Vin / Vout capacity ( C Al ) is analyzed. For thermal modeling, the thickness ( h Al ), volumetric heat ), and thermal conductivity ( Al ) of the Al transducer are input parameters. The value of h Al = 80 3 nm was determined from picosecond acoustics 31,32 and the values of CAl were taken from literature. 33 The thermal conductivity of Al transducer films were calibrated with the four-point probe method and the Weidemann Franz Law prior to TDTR measurements, which was in a range of 100 ~ 110 W m 1 K 1 for all the samples. The beam spot size was characterized using a beam-offset approach previously described. 34,35 5

6 The existence of air voids within the SiO 2 NR array makes it essentially a porous medium, which can be treated as an effectively homogeneous material with effective thermal properties. We consider this as a reasonable approximation owing to the fact that the wavelength of the thermal wave generated during the pump excitation is much longer than the characteristic dimensions of the SiO 2 NR array. The effective thermal properties, such as effective thermal conductivity ( eff ) and effective heat capacity (C eff ), of the SiO 2 NR array take into account the individual thermal properties of each component: 16 C xc (1 x) C, (1) eff NR air x (1 x), (2) eff NR air where x is the volumetric fraction of NR. Unlike semiconducting NRs previously studied in literature, 16,17,36-38 the suppression of thermal transport in NR structures due to size effects is not applicable to SiO 2 NRs with low thermal conductivity. 12 As a first-order of approximation, the phonon mean free path (MFP) in SiO 2 is calculated to be nearly 0.4 nm using the Debye theory for a gray medium. 39 Since this phonon MFP is much smaller than the average diameter (60 ± 10 nm) and length (> 200 nm) of SiO 2 NRs, the thermal properties of bulk SiO 2 ( NR = 1.3 W m 1 K 1 and C NR = J m 3 K 1 ) are used for SiO 2 NRs. 40 The thermal conductivity ( air = W m 1 K 1 ) and heat capacity (C air = J m 3 K 1 ) of air are extremely low compared with those of SiO 2, thus neglected in Equations (1) and (2). The value x is then left as the only free parameter in data fitting of the thermal modeling. 6

7 FIG. 2. Time-domain thermoreflectance data of (A) Sample 1-capping layer, (B) Sample 2- slanted 1 and (C) Sample 3-vertical 1 are measured at three modulation frequencies and compared to the 2D heat transfer model for the best-fitting curves. Representative TDTR ratio signals measured at three modulation frequencies (1.6 MHz, 9 MHz, and 18 MHz) on Samples 1, 2, and 3 are shown in Figs. 2(A)-2(C). The red solid lines are the best-fit curves of TDTR measurement data analyzed with a heat diffusion model for a multi-layer stack structure. 19,29 The same values of the thermal properties of the SiO 2 NR array are used for all three modulation frequencies. For Sample 1, the capping layer is the only layer with unknown thermal properties, while for Samples 2 5, both the volumetric fractions of capping layer (x CL,th) and SiO 2 NRs (x NR,th ) are obtained by fitting the TDTR data. The effective thermal conductivities of the NR array ( array ) and capping layer ( CL ) can be calculated based on Eq. (2) with individual volumetric fractions of x NR,th and x CL,th, respectively. The values are listed in Table II for all five samples. 7

8 Different from previous observations of the frequency-dependent thermal conductivity of semiconducting NR arrays, array does not show any frequency dependence from our thermal measurement. 15,16 There are two major reasons for the frequency-dependent array in literature: (1) from the material aspect, semiconducting materials such as Si and InAs, have long-mfp phonons contributing to thermal transport. In TDTR, lower modulation frequencies with deeper thermal penetration depths allow more long-mfp phonons to be captured, which leads to a higher thermal conductivity. 41 (2) From the sample structure aspect, the NRs studied in references 15,16 were fully filled by host media (PMMA or Siloxane) with low thermal conductivities. For such nanocomposite structures, when the modulation frequency is high ( 10 MHz), the thermal penetration depth in the host medium ( ~90 nm) is smaller than the average spacing distance between adjacent NRs (~100 nm in ref 16 and ~200 nm in ref 15 ). Thus, the temperature distribution is not uniform in the lateral direction. At this high-frequency limit, the average temperature response at the sample surface is indistinguishable from that of an effective medium with an effusivity calculated as: 16 ( C) x ( C) (1 x) ( C). (3) eff NR host However, if the modulation frequency is low, the temperature response from the NRs and the host medium is nearly isothermal along the in-plane direction, and the effective thermal conductivity and heat capacity can be averaged based on the volumetric fraction of the NRs separately, as expressed by Equations (1) and (2). In this study, the phonon MFPs of the amorphous SiO 2 is sufficiently short, resulting in a negligible phonon ballistic effect under high-modulation frequencies. In addition, the lateral thermal transport between the NRs is fully suppressed due to the large thermal resistance imposed by the air gaps between the NRs, regardless of the modulation frequency. This also 8

9 greatly reduces the complexity of calculating the effective thermal properties of the NR array in the multi-frequency regimes. The measured SiO 2 NR volumetric fraction from TDTR (x NR,th ) and ellipsometry (x NR ), as well as the effective thermal conductivity ( array ) of the NR-array layers without the capping layer are listed in Table II (Section S2 of the Supplementary Material 28 ). Table II. Time-domain thermoreflectance and ellipsometry data of the SiO 2 NR array samples Sample # Alignment CL (W m 1 K 1 ) array (W m 1 K 1 ) x NR (%) x NR,th (%) x NR,th / x NR 1 Capping layer ± ±8 90± Slanted ± ± ±3 20± Vertical ± ± ±3 22± Slanted ± ± ±4 10± Vertical ± ± ±5 24± Due to the tilting angle of the SiO 2 NRs, for Samples 2 and 4, the thermal transport along the NR channels has a lateral component. However, considering the thermal penetration depth in the SiO 2 NR matrix (~ m) is much smaller than the laser beam spot sizes (~12 m) used in TDTR measurements, the effective thermal transport is still approximately 1D along the through-plane direction. This allows the volumetric-average approach to be still validated for accurately describing the effective thermal properties. (Section S3 of the Supplementary Material 28 ). By varying the thermal penetration depth via tuning the modulation frequency, the sample inhomogeneity along the through-plane direction (capping layer and NR array) would be captured more easily with the full suppression of the lateral thermal transport. It should be noted here that although the sample systems in this study are likely to have anisotropic thermal properties due to the inhomogeneous structures, the in-plane thermal transport has negligible 9

10 impact on the through-plane thermal properties determined from TDTR measurements (Section S3 of the Supplementary Material 28 ). FIG. 3. SiO 2 volumetric fractions of both capping layer (x CL,th, open symbols) and NR layer (x NR,th, bars) extracted from frequency-dependent TDTR measurements, as well as x from ellipsometry (x NR or x CL, closed symbols) on the slanted (red symbols), vertical (blue symbols) and capping layer only (black symbols) samples. To better illustrate how the structure influences the effective thermal properties of the NR arrays, we summarize the volumetric fraction of SiO 2 NRs and capping layers determined from both thermal and ellipsometry characterizations for all five samples in Fig. 3. The measurement uncertainties of NR-array volumetric fraction ratios are estimated to be ~30% and 10% for thermal characterization and ellipsometry, respectively, while those of capping layers are 10% for both thermal characterization and ellipsometry (Section S4 of the Supplementary Material 28 ). For Sample 1 of the capping layer with a high packing density of the SiO 2 NRs, a good agreement is achieved between TDTR thermal measurements (x CL,th ) and the ellipsometry characterization (x CL ). In addition, the fitted x CL,th of the capping layers for Samples 2 5 obtained from TDTR also show consistent values (0.8 < x CL,th < 0.9). However, the volumetric fractions of the SiO 2 NRs beneath the capping layers (x NR,th ) for Samples 2 5 from TDTR are consistently 10

11 smaller than those from ellipsometry (x NR ). The ratio of x NR,th / x NR is sample dependent ranging from 0.23 to 0.63 (see Table II). After examination of the results, we present the following discussions comparing x NR,th and x NR from two characterization methods: (i) As can be seen from the side view of the SiO 2 NR arrays depicted in Fig. 1(B), shorter NRs exist that are not in contact with the capping layer, which do not contribute to thermal transport from the Al transducer to substrate; however, they do contribute to the ellipsometry signal by affecting the effective index of refraction. This leads to an underestimation of x NR,th compared with x NR. (ii) We also speculate that the kink structure illustrated in Fig. 1(C) would further hinder the thermal transport from the capping layer to the SiO 2 NRs beneath. The cross-sectional area of this kink structure gradually decreases with depth, where the crowding of the heat flux as it flows around a corner will increase the thermal resistance of the NR. 42 Due to the challenge of accurately separating this constrictive kink structure from the NR array, the thermal resistance of this layer is lumped into the x NR,th to form an effective homogenous layer in the model analysis, which subsequently leads to a further reduced value of x NR,th. (iii) By comparing the ratio of x NR,th / x NR, the underestimation of x NR,th in comparison with x NR looks more severe for Samples 4 and 5 with longer SiO 2 NRs, which is presumably due to a larger portion of the shorter SiO 2 NRs that are not in connection with the capping layer. NRs synthesized using the DSG method with the bottom-to-top approach tend to break more easily if the NR length is longer. We did notice that for longer SiO 2 NRs, x NR,th of the slanted NR array (Sample 4) is only nearly half of that of vertical NR array (Sample 5). For the vertical alignment, the thermal channels along SiO 2 NRs are in parallel with through-plane direction. While for the tilting 11

12 alignment, heat flux sees a series of NR-air-NR-air structural arrangement as it propagates downwards (Section S3 of the Supplementary Material 28 ). This indicates that the tilting alignment shall bring in additional interfacial thermal resistance at the NR-air interfaces for the through-plane thermal transport. Although this argument could explain the ~50% lower effective thermal conductivity of Sample 4 compared with that of Sample 5, it fails to elucidate the similar values of x NR,th for the case of NR arrays with shorter lengths (Samples 2 and 3). It is apparent that the correlations between structure parameters and thermal properties are rather complicated, and that some influential structural characteristics elude rapid measurement and standard thinfilm thermal modeling. Further systematic investigation in the future will be needed to reveal more details of how the structures of these materials would impact their thermal properties. In summary, this work experimentally studied the correlation of structure and thermal property of aligned SiO 2 NR arrays, as a kind of material with high volume of air voids, low thermal conductivity, anisotropy, and structural discontinuity. The volumetric fractions of the SiO 2 NRs (x NR,th ) determined from TDTR measurements were underestimated compared to those obtained from ellipsometry (x NR ). Analysis of the SiO 2 volumetric fraction from thermal measurement suggests a complicated heat transfer process inside the NR arrays, and provides guidance on further tuning down the thermal conductivity of NR arrays via structural engineering. The lowest effective thermal conductivity of the samples in this work is found to be 0.13±0.03 W m 1 K 1 for long slanted NR arrays, which demonstrates the promise of this class of materials for thermal insulation in micro-thermal device applications. 12

13 Acknowledgments This work was supported partially by the National Science Foundation (NSF) through the University of Minnesota MRSEC under Award Number DMR J.Z. would like to thank the support from the National Natural Science Foundation of China (Grant No and No ). Y.Z. acknowledges the support from Suzhou International Sci. & Tech. Cooperation Program (SH201111); H.S. and Y.Z. were supported by the Strategic Leading Science and Technology Special of the Chinese Academy of Sciences (No. XDA ). 13

14 References 1 V. R. Mehta, S. Shet, N. M. Ravindra, A. T. Fiory, and M. P. Lepselter, J. Electron. Mater. 34 (5), 484 (2005). 2 T. Kvisterøy, H. Jakobsen, C. Vieider, S. Wissmar, P. Ericsson, F. Niklaus, F. Forsberg, G. Stemme, J. E. Källhammer, H. Pettersson, D. Eriksson, J. Franks, J. VanNylen, H. Vercammen, and A. VanHulsel, in Advanced Microsystems for Automotive Applications, edited by J. Valldorf and W. Gessner (Springer, Berlin Heidelberg New York, 2007), pp F. Niklaus, C. Vieider, and H. Jakobsen, in Conference on Mems/Moems Technologies and Applications III: Proceedings of SPIE Vol. 6836, MEMS/MOEMS Technologies and Applications III, San Jose, CA, USA, 4 January 2007, edited by J. C. Chiao, X. Chen, Z. Zhou et al. (SPIE, Bellingham, WA, 2008), p D. 4 P. Muralt, in Electroceramic-Based MEMS, edited by N. Setter (Springer, New York, 2005), pp A. Katti, N. Shimpi, S. Roy, H. B. Lu, E. F. Fabrizio, A. Dass, L. A. Capadona, and N. Leventis, Chem. Mat. 18 (2), 285 (2006). 6 A. C. Peirre and A. Rigacci, in Aerogels Handbook, Advances in Sol-Gel Derived Materials and Technologyies, edited by M. A. Aegerter, N. Leventis, and M. M. Koebel (Springer, New York Dordrecht Heidelberg London, 2011), pp R. M. Costescu, A. J. Bullen, G. Matamis, K. E. O'Hara, and D. G. Cahill, Phys. Rev. B 65 (9), (2002). 8 L. Qiu, Y. M. Li, X. H. Zheng, J. Zhu, D. W. Tang, J. Q. Wu, and C. H. Xu, Int. J. Thermophys. 35 (1), 76 (2014). 9 D. Lee, P. C. Stevens, S. Q. Zeng, and A. J. Hunt, J. Non-Cryst. Solids 186, 285 (1995). 14

15 10 11 M. C. Garcia-Payo and M. A. Izquierdo-Gil, J. Phys. D-Appl. Phys. 37 (21), 3008 (2004). P. E. Hopkins, B. Kaehr, L. M. Phinney, T. P. Koehler, A. M. Grillet, D. Dunphy, F. Garcia, and C. J. Brinker, J. Heat Transf.-Trans. ASME 133 (6), (2011). 12 P. E. Hopkins, B. Kaehr, E. S. Piekos, D. Dunphy, and C. Jeffrey Brinker, J. Appl. Phys. 111 (11), (2012). 13 L. Qiu, X. H. Zheng, P. Yue, J. Zhu, D. W. Tang, Y. J. Dong, and Y. L. Peng, Int. J. Therm. Sci. 89, 185 (2015) M. W. Seto, B. Dick, and M. J. Brett, J. Micromech. Microeng 11, 582 (2001). A. I. Persson, Y. K. Koh, D. G. Cahill, L. Samuelson, and H. Linke, Nano Lett. 9 (12), 4484 (2009). 16 J. P. Feser, J. S. Sadhu, B. P. Azeredo, K. H. Hsu, J. Ma, J. Kim, M. Seong, N. X. Fang, X. L. Li, P. M. Ferreira, S. Sinha, and D. G. Cahill, J. Appl. Phys. 112 (11), (2012). 17 J. M. Weisse, A. M. Marconnet, D. R. Kim, P. M. Rao, M. A. Panzer, K. E. Goodson, and X. L. Zheng, Nanoscale Res. Lett. 7, 5 (2012) T. Zhang, S. L. Wu, R. T. Zheng, and G. A. Cheng, Nanotechnology 24 (50), 8 (2013). J. Zhu, D. W. Tang, W. Wang, J. Liu, K. W. Holub, and R. G. Yang, J. Appl. Phys. 108 (9), (2010). 20 J. Liu, J. Zhu, M. Tian, X. K. Gu, A. Schmidt, and R. G. Yang, Rev. Sci. Instrum. 84 (3), (2013). 21 X. J. Wang, C. D. Liman, N. D. Treat, M. L. Chabinyc, and D. G. Cahill, Phys. Rev. B 88 (7), (2013). 22 K. Zheng, F. Y. Sun, X. Tian, J. Zhu, Y. M. Ma, D. W. Tang, and F. S. Wang, ACS Appl. Mater. Interfaces 7 (42), (2015). 15

16 23 X. J. Wang, T. Mori, I. Kuzmych-Ianchuk, Y. Michiue, K. Yubuta, T. Shishido, Y. Grin, S. Okada, and D. G. Cahill, APL Mater. 2 (4), (2013) K. Robbie and M. J. Brett, J. Vac. Sci. Technol. A 15 (3), 1460 (1997). J. Q. Xi, J. K. Kim, E. F. Schubert, D. X. Ye, T. M. Lu, S. Y. Lin, and J. S. Juneja, Opt. Lett. 31 (5), 601 (2006). 26 Y. P. Zhao, D. X. Yeb, G. C. Wangb, and T. M. Lub, in SPIE 2003: Proceedings of SPIE Vol Nanotubes and Nanowires, San Diego, CA, USA, 27 October 2003, edited by A. Lakhtakia and S. Maksimenko (SPIE, Bellingham, WA, 2003), p Y. K. Koh, S. L. Singer, W. Kim, J. M. O. Zide, H. Lu, D. G. Cahill, A. Majumdar, and A. C. Gossard, J. Appl. Phys. 105 (5), (2009). 28 See supplemental material at [URL will be inserted by AIP] for 1: DSG technique, 2: sample structure characterization, 3: analysis on the anisotropic effect, 4: uncertainty analysis of the TDTR measurements D. G. Cahill, Rev. Sci. Instrum. 75 (12), 5119 (2004). A. J. Schmidt, X. Chen, and G. Chen, Rev. Sci. Instrum. 79 (11), (2008). G. T. Hohensee, W.-P. Hsieh, M. D. Losego, and D. G. Cahill, Rev. Sci. Instru. 83 (11), (2012) G. L. Eesley, B. M. Clemens, and C. A. Paddock, Appl. Phys. Lett. 50 (12), 717 (1987). D. A. Ditmars, C. A. Plint, and R. C. Shukla, Int.J. Thermophys. 6 (5), 499 (1985). J. P. Feser and D. G. Cahill, Rev. Sci. Instrum. 83 (10), (2012). 35 J. Zhu, H. Park, J.-Y. Chen, X. K. Gu, H. Zhang, S. Karthikeyan, N. Wendel, S. A. Campbell, M. Dawber, X. Du, M. Li, J.-P. Wang, R. G. Yang, and X. J. Wang, Adv. Electron. Mater. 2 (5), (2016). 16

17 36 A. I. Hochbaum, R. Chen, R. D. Delgado, W. Liang, E. C. Garnett, M. Najarian, A. Majumdar, and P. Yang, Nature 451 (7175), 163 (2008). 37 D. Li, Y. Wu, P. Kim, L. Shi, P. Yang, and A. Majumdar, Appl. Phys. Lett. 83 (14), 2934 (2003). 38 M. G. Ghossoub, K. V. Valavala, M. Seong, B. Azeredo, K. Hsu, J. S. Sadhu, P. K. Singh, and S. Sinha, Nano Lett. 13 (4), 1564 (2013) R. Berman, Thermal Conduction in Solids. ( Oxford University Press, Oxford, 1976). D. G. Cahill, Rev. Sci. Instru. 61 (2), 802 (1990). Y. K. Koh, D. G. Cahill, and B. Sun, Phys. Rev. B 90, (2014). T. Smy, D. Walkey, K. D. Harris, and M. J. Brett, Thin Solid Films 391, 88 (2001). 17

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