A Two-Parameter Nondiffusive Heat Conduction Model for Data. Analysis in Pump-Probe Experiments

Transcription

1 A Two-Parameter Nondiffusive Heat Conduction Model for Data Analysis in Pump-Probe Experiments Yanbao Ma School of Engineering, University of California at Merced, USA Nondiffusive heat transfer has attracted intensive research interests in last fifty years because of its importance in fundamental physics and engineering applications. It has unique features that cannot be described by the Fourier law. However, current studies of nondiffusive heat transfer still focus on studying the effective thermal conductivity within the framework of the Fourier law due to a lack of a well-accepted replacement. Here we show that nondiffusive heat conduction can be characterized by two inherent material properties: a diffusive thermal conductivity and a ballistic transport length. We also present a two-parameter heat conduction model and demonstrate its validity in different hotspot-size-dependent heat transfer experiments. This model not only offers new insights of nondiffusive heat conduction, but also opens up new avenues for the studies of nondiffusive heat transfer outside of the framework of the Fourier law. I. INTRODUCTION Heat conduction is ubiquitous in all substances under thermal nonequilibrium. Thermal conductivity is a material property that describes its ability to conduct heat and is defined based on the Fourier law for diffusive heat transfer. In last five decades, nondiffusive heat transfer has attracted intensive research interests because it not only incorporates a wide range of intriguing fundamental physics, but also is involved in numerous engineering applications. 1-3 Nondiffusive 1

2 heat transfer was first observed in heat pulse experiments in macroscale crystals at low temperatures in Later on, with relentless decreasing in the size of devices and structures, there is increasing interest in the study of micro-/nanoscale nondiffusive heat transfer. 5,6 From the microscopic point of view, heat conduction in dielectric materials and semiconductors is mediated by phonon transport. 1 When the characteristic timescale and length scales of heat transport are much larger than the typical phonon relaxation time and the phonon mean free path (MFP), the classical Fourier law can capture the diffusive transport behavior. However, when the heat transport time scale and length scale are comparable or even smaller than the phonon relaxation time and MFP, there is remarkable ballistic heat transport, which cannot be described by the Fourier law. 2,3,7,8 The state-of-the-art microscopic heat transfer models have been reviewed by Luo and Chen. 9 These models include the ab initio model, atomistic Green function, molecular dynamics (MD) simulations, and the phonon oltzmann transport equations (TE). Although these models can describe microscopic phonon transport, they are prohibitively expensive in terms of computation for length scales of even a few tens of nanometers. 9 In addition, when these models are applied to predict the effective thermal conductivity that is defined based on the Fourier law, a phenomenological ingredient is invariably reintroduced, which is unjustified on the basis of a purely microscopic theory from a mathematically rigorous point of view. In the past, non-fourier models derivable from the TE have also been developed to describe nondiffusive heat transfer, including the CV model, 10,11 a linear phonon hydrodynamic (or G-K) model, 12 a nonlinear G-K model, 13 an equation of phonon radiative transfer (EPRT), 14 a combined C- and F-processes model, 15 ballistic-diffusive equations (DE), 16 a dual-phase lag (DPL) model, 17 a thermomass model. 18 These models can capture 2

3 some features of nondiffusive heat transport, but they are not well accepted by the thermal science community due to a lack of experimental evidence for justification. 19 Recently, Wilson et al. presented a two-channel model for the time-domain thermoreflectance (TDTR) data analyses in nonequilibrium heat transfer, which can explain the frequency dependence of the ETC in Si 0.99 Ge 0.01, but it could not be applied to explain the grating size dependence of the ETC of Si in the transient thermal grating (TTG) experiments without effects from thermal interfaces. 20 Even though Fourier s law breaks down for nondiffusive heat transfer, it is still used in the state-of-the-art nanothermometry to extract the effective thermal conductivity (ETC) of nanosturctures due to a lack of high-fidelity non-fourier models. 21,22 Recently, puzzling results have been reported showing the dependence of measured ETC upon either modulation frequencies 23 or pump beam diameters. 24 These results indicate that the ETC is not an inherent material property and unsuitable to be used to characterize nondiffusive heat transfer. Therefore, inherent physical properties independent of probing methods are necessary for characterizing nondiffusive heat conduction. Inspired by recent experimental results on nondiffusive heat transfer at room temperature, 20,25 and building on our previous work, we present a two-parameter heat conduction (TPHC) model to replace the Fourier law for nondiffusive heat conduction in nonmetallic crystals. To validate the new model, we predict nondiffusive heat transfer using this model and compare against experiments with different methods, including transient thermal grating (TTG), frequency-domain thermoreflectance (FDTR), and time-domain thermoreflectance (TDTR). 3

4 II. CHARACTERIZING NONDIFFUSIVE HEAT CONDUCTION With the development of nanothermometry, nondiffusive heat conduction can be measured directly in experiments at room temperature. Here we would like to highlight two experimental results that inspire the development of our nondiffusive heat conduction model. The first work was reported by Hsiao and coworkers in the study of steady one-dimensional (1D) nondiffusive heat transfer in SiGe nanowires. 25 Figure 1a shows that the measured ETC versus nanowire length (L) (represented by solid circles). More than 20 SiGe nanowires with different alloy concentrations and diameters ranging from 50 nm to 185 nm were measured and the results were plotted together in Fig. 1a. For L < 8.3 μm, the ETC increases Figure 1. Size-dependent effective thermal conductivity from experiments at room temperature. (a) Effective thermal conductivity of SiGe nanowire versus nanowire length. 25 (b) Effective thermal conductivity of Si thin films versus thermal grating periods. 20 linearly with L (see solid line in Fig 1a), which was identified as a nondiffusive heat transfer region. For L > 8.3 μm, the ETC approaches a constant represented by a dashed line in Fig. 1a, which was identified as a diffusive heat transfer region. 25 Similar results were reported by Johnson and coworkers on the ETC in suspended Si thin films using a transient thermal gratings (TTG) method. 20 In their experiments, silicon thin films were heated up by two concurrent interfering laser pulses, leading to a spatially periodic thermal grating with period adjustable from 2 to 24 μm. Figure 1b shows the measured ETC versus thermal grating periods 4

5 (L). Two different symbols (empty circles and solid squares) represent measurements on two Si thin film samples with almost the same thickness (membrane 1: 392nm, membrane 2: 390nm). For L < 5 μm, the ETC almost increases linearly with L (see solid line in Fig. 1b), which indicates strong effects from ballistic heat transfer. For L > 15 μm, the ETC approaches a constant (dashed line in Fig. 1b), which belongs to the diffusive heat conduction regime. For 5 μm L 15 μm, there is transition from ballistic-diffusive to diffusive heat transfer. Comparing Fig. 1a with Fig. 1b, we notice that there is very similar trend for the ETC changing with nanowire length (Fig. 1a) or thermal grating periods (Fig. 1b). oth curves can be divided into two regions: a diffusive conduction region and a nondiffusive conduction region. In the diffusive conduction region, heat transfer process can be described by the Fourier law with a constant diffusive thermal conductivity. In the nondiffusive conduction region, both experiments show that the ETC will change almost linearly with a certain characteristic length. It should be noted that the linear relation in non-diffusive region shown in Fig. 1a and 1b is a specific other than a common case. However, the trend of increasing ETC from ballistic-diffusive region to diffusive region is common which has been found in other experiments on heat transfer in nanostructures. 5,6 ased on this observation, we assume nondiffusive heat transfer can be characterized by two parameters: (a) a constant diffusive thermal conductivity for characterizing heat conduction by diffusion; and (b) a ballistic heat transfer length Λ for characterizing the effect on thermal conduction from ballistic phonon transport. In the following, we will derive a two-parameter heat conduction (TPHC) model to describe nondifusive-diffusive heat transfer. Definitely, the diffusive thermal conductivity is an inherent material property, but we need to demonstrate the ballistic heat transfer length is also an inherent material property independent of probing methods. 5

6 III. ONE-DIMENSIONAL TPHC MODEL AND VALIDATION Derivation of the TPHC model in One Dimension (1D) The energy conservation equation in the one-dimensional (1D) regime is E q 0, t x (1) where E is internal energy and q is the heat flux. If heat transport processes were entirely diffusive, there would have existed a local thermal equilibrium and T could have been calculated directly from the internal energy as T = E/c v. The diffusive heat transport is described by the Fourier law, T q, (2) x where κ is thermal conductivity. ased on the kinetic theory, the diffusive phonon lifetime D is related to thermal conductivity κ through 2 cc v D / 3, (3) where c v is the volumetric specific heat, and c is the speed of sound. The diffusive phonon MFP is given by D c D. However, in nondiffusive heat transfer processes, including ballistic and ballistic-diffusive heat transfer, the system of interest is highly nonequlibrium. Following Cimmelli et al s approach, 13 a nonequilibrium absolute temperature β is defined as E/ c v. Substituting β into Eq. (1) gives 1 q t c x v. (4) 6

7 The nonequilibrium absolute temperature β can be lined to the equilibrium local temperature T with a relaxation time in the equilibration process as 13 T t, (5) To interpret the physical meaning of, we simply classify phonons into two different categories: diffusive phonons and ballistic phonons. In diffusive heat transport processes, is relatively small compared with typical heat diffusion time that is estimated as 2 Lcv where L is the length scale of the investigated spatial domain. Therefore, there exists local temperature equilibrium, i.e., β = T. On the contrary, in nondiffusive heat transport processes, the dynamic nonequilibrium temperature β is different from the equilibrium temperature T due to the effects from ballistic heat transport. Thus, represents characteristic time in ballistic heat transport. Accordingly, the characteristic ballistic transport length is calculated from c. From Eq.(5), T can be expressed as: T. t (6) After substituting Eq. (5) and (6) into Eq.(2), we obtain a constitutive equation for nonequilibrium nondiffusive heat transfer: 2 q q x c x v 2. (7) Using both diffusive transport length D and ballistic transport length, Eq. (7) can be rewritten as 7

8 q x 3 2 D q 2 x (8) The first term on the RHS of Eq. (8) stands for diffusive heat transfer by diffusive phonons, while the second term accounts for nonlocal effects resulting from ballistic phonon transport. Since D c D, and D can be calculated from thermal conductivity κ through Eq. (3), there are two independent parameters in Eq.(8): κ and. oth parameters can be determined from experiments. In addition, when the heat transport length scale is much larger than, the 1D- TPHC model in Eq. (8) reduces to Fourier law. Therefore, the 1D-TPHC model is a unified heat conduction model for both nondiffusive and diffusive heat transport. It should be noted that the intrinsic phonon relaxation times depend on temperature and phonon frequencies, 30,31 which can be determined from first-principle calculations for crystal lattices. 32 In diffusive heat transport processes, the occupation of all phonons excitations with a broad spectrum of MFPs and frequencies can be well described by a single temperature under a local equilibrium condition. However, for nondiffusive heat transfer processes, local equilibrium breaks down. Different types of thermal excitations can leads to drastically different occupations of phonons within different spectrum bands. The occupation of phonons at different frequencies is difficult, if not impossible, to be measured from experiments. This paper mainly focuses on developing a simplified two-parameter model to replace the Fourier law for data analyses in experiments on nondiffusive heat conduction in nonmetallic crystals. The two independent parameters (κ and ) in Eq.(8) will be determined from experiments. The frequency dependence of these two parameters and their derivation from the first principles will be left for future study. 8

9 Analysis of experimental TTG data with the 1D-TPHC model To validate the 1D-TPHC model, we study the thermal decay processes in the TTG experiments by Johnson et al. 20 shown in Fig. 1b. The experimental results shown in Fig. 1a, will be studied in a separate paper. For quantitative comparison, we need to determine parameters in Eq. (8) based on experimental data. First, the diffusive thermal conductivity for the Si thin film is determined from experiments as κ = 92 W/mK in the diffusive heat transfer region (L 15 μm in Fig. 1b). It should be noted that the diffusive thermal conductivity κ = 92 W/mK for the Si thin film is dependent on film thickness 27, which is different from that for Si bulk materials (κ = 148 W/mK). Given volumetric specific heat c v = Jm -3 K -1 and speed of sound c = 6400 m/s, 33 the diffusive phonon MFP D is calculated from 3 / cc as 26 nm. We assume the decay of an initial sinusoidal temperature profile of thermal gratings satisfies Eq. (4) and (8), and the solutions of β and q take the form D v e, and q q e, where 2 / L, L is the grating period, and γ is the thermal ix t ixt 0 0 decay rate of gratings. Substituting the solutions of β and q into Eq. (4) and (8) gives 2 cv 2 1 D 3 (9) Given the ballistic transport length, Eq. (9) can be used to predict the thermal decay rate changing with grating wavenumber λ (or grating period L). After determining the thermal decay rates for different grating periods from Eq. (9), the ETC is given by 2 ef f cv (10) 2 1 D 3 9

10 The ballistic transport length can be determined from experimental thermal decay rate for nondiffusive heat transfer using 2 3( cv - 1) 2 D (11) From Eq. (9), there is decreasing effect from ballistic heat transport with increasing grating periods. When L > 15 μm, the ballistic effect is negligible (see Fig. 1b). The value of should be determined from experimental data with small grating periods where there are remarkable ballistic heat transfer. ased on experimental results shown in Fig. 1b, the ETC almost increases linearly with L for L < 5 μm, which indicates that there are strong effects from ballistic heat transfer in this region. For 2 μm L 4.9 μm, the extracted value of from the thermal decay rates is in the range of 5.8 μm 8.4 μm. Here, we take the average value 7.1 m for the TTG data analysis using the 1D-TPHC model. After determining the two parameters in the 1D-TPHC model, the thermal decay rates and the ETCs for different thermal grating periods are calculated from Eq. (9) and (10), respectively. The predictions based on the TPHC model are compared against experiments, which is shown in Fig. 2a and 2b. There is excellent agreement between theoretical predictions and experiments. The theoretical predictions based a two-fluid model by Johnson et al. 20 and TE results by Minnich 34 are also plotted together in Fig. 2b for comparison. Here, the TE results by Minnich were regenerated based on Fig. 3 in Ref. [34]. It shows that the TPHC model outperforms the two-fluid model and the TE results (see Fig. 1d). ased on thermal decay rates calculated from Eq. (9) and electronic decay rates provided by Johnson et al., 20 we can also numerically reconstruct the signal decay traces. The numerical 10

11 method is very similar to the bi-exponential fitting method used by Johnson et al. 20 The only difference is that we use theoretical thermal decay rates calculated from Eq. (9) instead of using experimental data. Again, there is very good agreement between the experimental data and numerical results for four representative grating periods (see Fig.2c, 2d). The good agreement shown in Fig. 2 demonstrates that the nondiffusive heat transfer in Si thin film can be described by the 1D-TPHC model. For TTG experiments with different thermal grating periods, the nonlocal effects from ballistic phonon transport can be characterized by a single parameter, the characteristic ballistic transport length. Further experimental studies are necessary to prove this parameter is an inherent material property independent of probing methods. Furthermore, it will be interesting to study how will change for different Figure 2. Comparison of predictions based on the TPHC model with TTG experiments 20 in silicon membrane at room temperature. (a) Thermal grating decay rates versus the grating wavenumber squared. (b) Comparison of the effective thermal conductivity versus transient grating period. (c) and (d): Comparison of predictions of thermal decay traces with experiments for four typical grating periods. 11

12 materials with different thin film thickness and experimental temperatures. In the TTG data analysis, we use the diffusive thermal conductivity for the Si thin film determined from experiments as κ = 92 W/mK, which is worth of further discussion. It should be noted that the thermal conductivity κ in Si thin films is dependent on the film thickness, which is different from that for Si bulk materials (κ = 148 W/mK). In the TPHC model, it should be better to use diffusive thermal conductivity κ = 148 W/mK for Si bulk material that has been proven to be a material property. If we use diffusive thermal conductivity of κ = 148 W/mK for Si bulk materials to interpret the TTG experiments in Si thin films, we have to consider nonuniform heat flux across the thin film due to three-dimensional non-local effects from phonon scattering at free surfaces. Currently, there are no measurements or convincing theories on heat slip boundary condition to determine heat flux on free surfaces of the thin film. To circumvent this problem, we consider 1-D model with κ = 92 W/mK determined from experiments for TTG data analysis. We will use 2D-THPC model with appropriate heat slip boundary condition and diffusive thermal conductivity of κ = 148 W/mK to interpret the TTG experiments, which will be our future work. IV. THREE-DIMENSIONAL TPHC MODEL AND VALIDATION Following the phonon hydrodynamic equation derivable from linearized TE, 12 we extend the 1D-TPHC model to the three-dimensional (3D) regime to describe more general 3D nondiffusive heat transfer, cv q = Q t (12) q k q q 9 D 2 + ( 2 ( )) (13) 12

13 where Q stands for a heating source. While D can be calculated from κ, there are two independent parameters (κ and ) in the 3D-TPHC model. It should be noted that although the 3D-TPHC model takes similar form with the linear phonon hydrodynamic equation, there are different physical meanings for the parameters. In phonon hydrodynamics equations, 12 except for κ, there are other two parameters: resistive and normal phonon scattering times that are difficult to measure in experiments. Here the two independent parameters in the 3D-TPHC model can be determined directly from experiments. After taking divergence and substituting Eq. (13) into Eq. (12), we obtain 2 D 2 cv cv Q t 3 t (14) The validation of the TPHC model will be described below. Validation of the 2D-TPHC model in TTG Johnson et al. also performed reflection TTG measurements of thermal transport in a bulk GaAs sample 35, where there is two-dimensional (2D) heat transfer. To validate the TPHC model, we study the 2D thermal grating decay and compare against experiments. The thermal properties of bulk GaAs materials at room temperature are specified as κ = 45 Wm -1 K -1, c = 3,700 m/s, and c v = Jm -3 K -1, the diffusive phonon MFP D = 20.8 nm. 33 Following the analyses of the thermal grating decay described by Johnson et al., 36 we considered a finite absorption depth σ = 100 nm of a pulsed laser with duration of t p = 60 ps in GaAs 35. The source term Q can be expressed as: 2 p p Q Q0 cos( x)exp z / ( t 2 t ) / t (15) 13

14 where Q 0 is the laser power amplitude, λ is grating wavenumber, 2 / L, and L is the grating period. At z = 0, zero temperature gradient boundary condition is used, while temperature is set to be zero at z =. The thermal grating decay in term of temperature is simulated based on a second-order finite difference method. Unlike 1D TTG data analysis using TPHC where the value of can be determined directly from experimental results on thermal decay rates based on Eq. (11), there is no explicit expression to extract from the measurements of thermal decay rates for 2D TTG data analysis. To determine, we fit experimental data of thermal grating decay for a fixed grating period. Here we randomly choose a case for L = 4.9μm, and find good match when 10 m. Here, we demonstrate that the 2D-TPHC model with this rough estimation of ballistic transport length 10 m (based on L = 4.9 μm) can be used to predict thermal grating decay for other grating periods. The predictions are compared with experiments in Fig. 3. Further studies are required for more accurate measurements of ballistic transport length in bulk GaAs, which is out of the scope of this work. Figure 3. Comparison of 2D thermal grating decay traces between numerical predictions based on the 2D-TPHC model and experiments in GaAs at room temperature. 35 (a) Thermal decay traces for L = 2.4 μm and L = 3.7 μm. (b) Thermal decay traces for L = 2.8 μm and L = 4.9 μm. 14

15 For the case of L = 4.9μm, numerical results based on the Fourier law with κ = 45 Wm - 1 K -1 is also plotted in Fig. 3b for comparison. It shows that the Fourier law remarkably overpredicts the thermal grating decay. Overall, there is very good agreement between predictions based on the 2D-TPHC model and experiments, which demonstrates that the 2D-TPHC model can capture nondiffusive heat transfer in 2D TTG experiments. Analytical solutions of the 2D-TPHC model To validate the 2D-TPHC model, we apply this model to analyze the transient thermoreflectance (TTR) data, including frequency-domain thermoreflectance (FDTR), and time-domain thermoreflectance (TDTR). Currently, analytical solutions based on Fourier law are used in data analyses of transient thermoreflectance (TTR) experiments. 37,38 Here we replace Fourier law with the 2D-TPHC model and try to maintain the mathematical simplicity of analytical solutions in order to facilitate data fitting in TTR data analyses with the new model. Without distributed a heating source in the volume, Eq. (14) in cylindrical coordinate sytem with axixymmetric heat transfer can be rewritten as: D 1 c v r c 2 v r 2 t z r r r 3 t z r r r (16) The zeroth-order Hankel transform of Eq. (16) gives 2 2 ˆ ˆ ˆ 2 ˆ D 2 c ˆ v s c 2 v s 2 t z 3 t z (17) where ˆ is the temperature in Hankel transform and s is the transform variable. After taking Fourier transform, Eq. (17) is rewritten in the frequency domain as 2 2 u ˆ 2 z (18) 15

16 where u i / 3 / c 2 s (19) i D v The solutions provided by Cahill 37 and Schmidt et al. 38 are still valid except that we replace u for Fourier law with the new one in Eq. (19) for the TPHC model. Application of the 2D-TPHC model in FDTR data analysis The FDTR experiments were conducted in Si bulk materials at room temperature by Regner et al. 39 In their experiments, the sample temperature drifted to K due to laser heating. We redraw their experimental data and fitting results on phase angle versus modulation frequencies in Fig. 4. Symbol circles stands for experimental data. The dashed-dotted-dotted line represents their fitting results based on the Fourier law with κ = 99 W/mK, which is much lower than bulk Figure4. Comparison of phase angle (degree) versus modulation frequency between theoretical results and FDTR experiments. 39 = 143 W/mK measured by the steadstate heat flow technique. 40 In other words, the FDTR data could not be fitted using the Fourier law due to nondiffusive heat transfer. Using bulk four different cases with = 143 W/mK, G = 300 W/m 2 K, we calculate = 0, 7.5, 12.5, and 17.5 μm, respectively. The predictions based on the TPHC model are compared against experiments in Fig. 4. It shows that there is very good agreement with experimental data in phase angle for 12.5 m when modulation frequency is less than 30 MHz, even though there is poor match for higher modulation frequencies. Therefore, we estimate the ballistic transport length in Si bulk materials to be 12.5 m at room 16

17 temperature based on the analyses of FDTR data with the 2D-TPHC model. For TPHC model is reduced to the Fourier law. Comparing the phase angle curves between = 0, the = 0 and 12.5 m, we find that there is negligible difference for modulation frequency less than 2 MHz, but the difference increases for higher modulation frequencies due to more effects from ballistic heat transfer. There is better agreement between the TPHC predictions and experiments compared with that from the Fourier law. Therefore, we demonstrate that the 2D-TPHC model can be used in the analyses of FDTR data to achieve better interpretation of the measurements than the Fourier law. It should be noted that we could not determine the ballistic transport length from the FDTR experiments because we could not fit two parameters: and thermal interface conductance G at the same time. The disagreement between the 2D-TPHC prediction and the FDTR data at high frequencies (f > 30 MHz) is also due to the uncertainty of thermal interface conductance. Currently, almost all theoretical or experimental studies on thermal transport across interfaces focus on predicting or measuring G under the assumption that the G of an interrogated interface is a constant around a fixed experimental temperature. For diffusive heat transfer, there is similar heat diffusion across the interface for different laser beam diameters. However, for nondiffusive heat transfer with contribution from ballistic phonon transport, there is wave-like ballistic phonon reflection and transmission at the interface, which is different from diffusive phonon transport at the interface. For different modulation frequencies, the thermal penetration depth in the substrate immediately underneath the metal transducer is reversely proportional to the square root of modulation frequency 39. When phonon transport length is comparable or even smaller than the thermal penetration depth, there is remarkable ballistic phonon transport which will affect both the measured ETC and thermal interface conductance G. In addition, there is increasing effect on both ETC and G for higher modulation 17

18 frequencies. Therefore, it is impossible to use a fixed G to fit experimental data shown in Fig. 4. How modulation frequencies affect the ETC and G needs further study in the future. Application of the 2D-TPHC model in TDTR data analysis We also apply the 2D-TPHC model to analyze data from the TDTR measurements in Si bulk materials at room temperature conducted by Minnich and coworkers. 24 It was shown that the measured ETC almost reduces 30% (from 148 W/mK to 113 W/mK) when the laser pump diameter changes from 60 μm to 3.65 μm. In our data analyses, we fix κ = 148 W/mK, and =10 μm in the 2D-TPHC model, and fit the TDTR data with thermal interface conductance as the only free parameter. While can be determined from TTG experiments without the effects from thermal interface boundaries, it cannot be determined from the TDTR experiments. Here we use one example to show why cannot be determined from the TDTR experiments. We use one case with D = 3.65 μm, and f = 2.01 MHz to show the effects from on the fitting results. When we choose different in the TPHC model, i.e., = 7.5, 10, 12.5, and 20 μm, to extract G, the extracted G based on phase fitting is 174.1, 163.7, 152.1, and MW/m2K for = 7.5, 10, 12.5, and 20 μm, respectively. For different pairs of and G, we obtained very similar results in data fitting (results are not shown here). Therefore, we cannot extract both and G from data fitting. Here we use =10 μm to demonstrate that the TPHC model can offer better interpretation of the TDTR data for nondiffusive heat transfer. The real needs to be measured using TTG methods. Figure 5 shows the comparison of analytical solutions based on the 2D-TPHC model with TDTR data for two cases with pump laser diameter D = 60 μm, and D = 5.44 μm, respectively. The analytical solutions based on the Fourier law are also plotted together for comparison. In 18

19 data fitting, we can either fit amplitude curve (A-fitting) or phase angle curve (P-fitting) versus pump-probe time delay. For D = 60 μm, the analytical solutions of the Fourier law can fit the experimental data very well for both amplitude and phase angle curves no matter whether we consider A-fitting or P-fitting (see Fig 5a and 5b). Here, only the A-fitting results based on the Fourier law are shown because the P-fitting results are almost the same. This confirms that there is dominant diffusive heat transfer which can be described by the Fourier law. Figure 5a show that both the A-fitting and P-fitting of the TPHC model can give good predictions on amplitude curve, while there is noticeable mismatch in phase angle curves between the TPHC results and experiments shown in Fig. 5b. For D = 5.44 μm, the fitting results of the TPHC model and the Fourier law are compared with experiments in Fig. 5c and 5d. It shows that the P-fitting results of the Fourier law can barely fit phase angle (see Fig. 5d), and there is also remarkable mismatch with amplitude curve (see Fig. 5c). Similarly, while the A-fitting results of the Fourier law can fit amplitude curve in Fig. 5c, they could not match the phase angle curve in Fig. 5d. This indicates that there is remarkable nondiffusive heat transfer that could not be described by the Fourier law. Different from the Fourier law, the P-fitting results of the TPHC model can fit both the amplitude curve in Fig. 5c and the phase angle curve in Fig. 5d, even though there is noticeable mismatch for A-fitting results of the TPHC model in phase angle curve in Fig. 5d. We have also applied the TPHC model to analyze the TTR data at other different modulation frequencies (f = 2.01, 4.77, 6.65, 9.03, 9.13 and MHz) and different pump laser diameters (D = 3.65, 5.44, 8.9, 12, 15, 30, 45 and 60 μm). We find that the TPHC model always give better fitting results than the Fourier law when there is strong nondiffusive heat transfer for D 5.44 μm. 19

20 The TPHC model is developed to replace Fourier s law to study nondiffusive heat transfer outside the frame work of Fourier s law. Here, we demonstrate that the TPHC model can be used in the analyses of TDTR data to achieve better interpretation of the experimental data than the Fourier law for nondiffusive heat transfer. Using Fourier s law for TDTR data analysis, Ding et al. found that there is significant reduction (30%) in the measured ETC of Si at room temperature when the laser pump diameter changes from 60 μm to 3.65 μm. In TDTR data analysis using the TPHC model, we found that there is decreasing thermal interface conductance across the Al-Si interface with decreasing laser beam diameter. This is an important finding from the TPHC model. The reduction of the ETC and thermal interface conductance is due to nonlocal effects from the ballistic phonon transport. For TDTR experiments with large laser beam diameter (D 60 μm), heat transfer length is larger than the phonon MFP. Consequently, diffusive heat transfer is dominant. At the interface between metal transducer and dielectric substrate, there is similar heat diffusion across the interface for different laser beam diameters. Therefore, the measured thermal interface conductance will not change with the laser beam diameter. However, when the size of the laser beam becomes comparable or even smaller than the phonon mean free path, there is increasing effects from ballistic phonon transport with decreasing laser beam diameter. The wave-like ballistic phonon reflection and transmission at the interface highly depends the incident angle, which is totally different from memoryless diffusion process. The ballistic phonon transport will not only reduce the ETC in the substrate but also reduce the thermal interface conductance. The results of hotspot-size-dependent thermal interface conductance will be discussed in more detail in a separate paper

21 Figure 5. Comparison of data fitting results of the 2D-TPHC model and the Fourier law in the TDTR data analyses. 24 A-fitting stands for amplitude fitting, and P-fitting means phase angle fitting. (a) and (b): Amplitude and phase angle versus time delay for pump laser diameter D = 60 μm, modulation frequency ω = 6.85 MHz. (c) and (d): Amplitude and phase angle versus time delay for pump laser diameter D = 5.44 μm, modulation frequency ω = 2.01 MHz. V. DISCUSSION AND OUTLOOK Thermal conductivity is defined based on the Fourier law for diffusive heat transfer. Due to a lack of a well-accepted replacement, the Fourier law with an effective thermal conductivity (ETC) is still used in current studies of nondiffusive heat transfer. The predictions and measurements of the ETC within the framework of the Fourier law will provide little insights for nondiffusive heat transfer because the ETC is not an inherent material property and unsuitable for characterizing nondiffusive heat transfer. Here we showed that nondiffusive heat conduction can be characterized by two independent parameters: a diffusive thermal conductivity and a ballistic transport length. We have presented a two-parameter heat conduction (TPHC) model to 21

22 describe nondiffusive heat transfer. We have demonstrated the validity of the TPHC model in different experiments with different methods, including transient thermal grating, frequencydomain thermoreflectance, and time-domain thermoreflectance. Further experiments are necessary to prove that the ballistic transport length is a material property independent of probing methods. We conclude with an outlook on exciting future work. Firstly, the TPHC model provides a practical theoretical and numerical tool for both scientific research and engineering design involving nondiffusive heat transfer. Secondly, the TPHC model may open up new avenues for theoretical, numerical, and experimental studies of nondiffusive-diffusive heat transfer outside the framework of the Fourier law by predicting and measuring two independent inherent material properties: the diffusive thermal conductivity and the ballistic transport length. Acknowledgements We would like to thank Ashok Ramu for proofreading this paper, and thank Jeremy Johnson, Alexei Maznev, Keith Regner, Jonathan Malen, Xiangwen Chen, and Austin Minnich for sharing their experimental data. We also thank Alexei Maznev for helpful discussion. This work is supported in part by the United States Department of Energy under the ARPA-E program (DE- AR ). References 1 J. M. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids (Oxford University Press, 1960). 2 C. L. Tien, A. Majumdar, and F. M. Gerner, Microscale energy transport (Taylor & Francis, 1997). 22

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