Nondiffusive thermal transport and prediction of the breakdown of Fourier's law in nanograting experiments

Zhengxian Qu, Dadong Wang, and Yanbao Ma

Citation: AIP Advances 7, 015108 (2017); doi: 10.1063/1.4973331 View online: http://dx.doi.org/10.1063Z1.4973331 View Table of Contents: http://aip.scitation.org/toc/adv/7/1 Published by the American Institute of Physics

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Nondiffusive thermal transport and prediction of the breakdown of Fourier's law in nanograting experiments

Zhengxian Qu, Dadong Wang, and Yanbao Maa

School of Engineering, University of California, Merced, California 95343, USA (Received 15 April 2016; accepted 2 December 2016; published online 20 January 2017)

An appropriate heat conduction model is indispensable for experimental data analysis in nanothermometry in order to extract parameters of interests and to achieve a fundamental understanding of phonon-mediated heat transfer in nanostructures and across interfaces. Recently, nanoscale periodic metallic gratings are used as a group of distributed heaters as well as transducers in nanothermometry. However, in this technique, there are coupled hotspot-size-dependent effective thermal conductivity (ETC) and hotspot-size-dependent thermal interface resistivity, which posts a challenge for experimental data analysis using Fourier's law that fails to extract both ETC and thermal interface resistivity simultaneously. To overcome this challenge, a novel two-parameter nondiffusive heat conduction (TPHC) model, which has been successfully applied to data analysis in different types of pump-probe experiments, is applied to analyze laser-induced nondiffusive heat transfer in nanoscale metallic grating experiments. Since the hotspot-size-dependent ETC is automatically captured by the TPHC model, the hotspot-size-dependent interface resistivity becomes the only parameter to be determined from experiments through data fitting. Thus, the hotspot-size-dependent thermal interface resistivity can be determined from experiments without the impact from the hotspot-size-dependent ETC. Currently, there is a lack of a criterion to predict when Fourier's law breaks down in nanoscale heat transfer. To fill this gap, a criterion based the TPHC model is identified to predict the valid range of Fourier's law, which is validated in both theoretical analyses and nanoscale metallic grating experiments. © 2017 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). (http://dx.doi.org/10.1063/L4973331]

I. INTRODUCTION

With the development of thermoelectronics and nanoelectronics cooling,1-3 nanoscale heat transfer has been attracting significant research interests in last two decades.4 The traditional macroscale heat transfer theory based on Fourier's law fails to capture the new features of heat conduction at the nanoscale, such as hotspot-size-dependent thermal conductivity.5 Microscopically, these deviations result from the contribution of nondiffusive phonon transport with the characteristic thermal transport length comparable to or even smaller than phonon mean free path (MFP) in nanosystems.6

To study the new features of nanoscale heat conduction, different nanothermometry techniques have been developed. Among them, transient thermoreflectance (TTR), including time-domain thermoreflectance (TDTR) and frequency-domain thermoreflectance (FDTR),7-9 is an established experimental technique contributing to the discovery of many nondiffusive heat transfer phenomena, such as frequency-dependent and hotspot-size-dependent effective thermal conductivity (ETC).10-12 However, the presence of interface between the metal transducer layer and the interrogated sample makes it very complicated for experimental data analysis. To exclude the interface effects on

aAddress all correspondence to this author; yma5@ucmerced.edu

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2158-3226/2017/7(1 )/015108/12 7,015108-1 ~ "nil if ) "fir I mt ■

nanothermometry, a transient thermal grating (TTG) method was developed to measure the decay of thermal gratings without a metallic transducer layer.5,13,14 In TTG experiments, it was found that there is hotspot-size-dependent ETC in silicon thin films at room temperature.5 For both TDTR and TTG, the size of heating source in the experiment is larger than 1 ^m due to the optical diffraction limit. To remove the optical diffraction limit on the size of heating source, Siemens et al. designed an experimental technique with a nanoscale heating source. In this technique, a laser beam is absorbed by metallic gratings (MG), and the thermal decay of metallic gratings is measured as the diffracted signal of soft X-ray.15 With this MG-X-ray experiment, Siemens et al. reported the first observation and quantitative measurement of transition from diffusive to quasi-ballistic thermal transport.15

An appropriate heat conduction model is indispensable for experimental data analysis in nan-othermometry in order to extract parameters of interests from detected signals and to achieve a fundamental understanding of phonon-mediated heat transfer in nanostructures and across interfaces. The measured hotspot-size-dependent ETC5,11,12 demonstrates that ETC is unsuitable to characterize nondiffusive heat transfer for at least three reasons: (1) ETC is defined within the framework of Fourier's law, but Fourier's law cannot describe nondiffusive heat transfer; (2) ETC is an artificial parameter rather than an inherent material thermal property because it depends on probing methods; and (3) The hotspot-size- and heating-frequency-dependent ETC makes the accurate measurement impossible. However, due to a lack of well-accepted nondiffusive model, Fourier's law is currently still used in experimental data analysis of nanothermometries to extract ETC. Similar to hotspot-size-dependent ETC and thermal interface resistivity found in TDTR experiments,11,12 the measured ETC and thermal interface resistivity in MG-X-ray experiments should also depend on the grating sizes. However, Fourier's law cannot determine the coupled ETC and thermal interface resistivity in MG-X-ray experiments simultaneously through data fitting to the thermal decay curves. Consequently, in data analysis of MG-X-ray experiments,15,16 the hotspot-size-dependent ETC of the substrate is not considered. Instead, thermal conductivity of the substrate was kept as a constant and only thermal interface resistivity was determined from thermal decay curves for different sizes of metallic gratings. Consequently, the measured hotspot-size-dependent thermal interface resistivity contains the effects of hotspot-size-dependent ETC of the substrate. To obtain pure size-dependent interface resistivity excluding the impact from ETC, it is necessary to apply high-fidelity nondiffusive heat transfer model in MG-X-ray data analysis in order to extract more physical results compared with current approaches based on Fourier's law. Unfortunately, so far to the best of our knowledge, the results in MG-X-ray experiments have not been repeated by any existing nondiffusive heat transfer model.

In the last six decades, lots of nondiffusive heat transfer models, such as Cattaneo-Vernotte model,17,18 Guyer-Krumhansl model,19 C and F process model,20 ballistic-diffusive model,21 two-channel model,22 were proposed. However, these models are not well-accepted by the heat transfer community due to a lack of experimental evidence for justification. Meanwhile, the phonon Boltzmann transport equation (BTE) can describe nondiffusive heat transfer, but it is prohibitively expensive to solve with many variables.23 This huge number of unknowns, and the corresponding needed computational resources and time, mark its unavailability for experimental analysis.

Recently, a unified diffusive-nondiffusive two-parameter heat conduction (TPHC) model has been proposed by Ma.24 This model has been successful in explaining multiple nanoscale heat transfer experiments, such as TTG experiments24 and TDTR experiments.25 In this study, the TPHC model is applied for experimental data analysis of MG-X-ray experiments conducted by Siemens et al.15 Based on the TPHC model, we define a nondimensional parameter as a criterion to predict when Fourier's law breaks down. In this paper, the TPHC model is briefly described first. Then it is applied for data analysis of MG-X-ray experiments in order to extract the interface thermal resistivity without contamination from the size-dependent ETC. This is then followed by the discussion on the size-dependence of interface resistivity. Finally, the main results are summarized in the conclusion section.

II. THE TPHC MODEL

Like Fourier's law that is aphenomenological model, the TPHC model is also a phenomenological model that cannot be derived from the first principles. Here, we briefly describe the formulation of the

TPHC model without rigorous derivation. For diffusive heat transfer, there exists a locally equilibrium temperature (T). Fourier's law describes the relationship between the heat flux q and temperature gradient,

q = (1)

where к is thermal conductivity. Microscopically, phonons are the dominant thermal energy carriers in insulators and intrinsic semiconductors. A locally equilibrium state means phonons follows Bose-Einstein distribution in diffusive heat transfer. However, for nondiffusive heat transfer, the system is in a highly nonequilibrium state without a locally equilibrium temperature. A nonequilibrium absolute temperature в is defined as в = Elcv, where E is energy density and cv is volumetric specific heat. With the nonequilibrium absolute temperature в, the energy conservation equation in one-dimensional regime can be expressed as

dl=-1 di (2)

dt cv 3x'

The dynamic nonequilibrium temperature в will relax to a locally equilibrium temperature T through a phonon equilibration process,

f = - ^. (3)

where tb represents the relaxation time of the phonon equilibration process in nondiffusive heat transfer. Accordingly, лв is defined as лв = ctb to characterize the nondiffusive heat transfer length, where c is the speed of sound. Substituting Eq. (2) into Eq. (3) gives

T = P - ^. (4)

Substituting this into Eq. (1) give a constitutive equation for one-dimensional (1D) nondiffusive heat transfer,

=-к— + клв d2q (5)

q K dx ccv dx2'

On the right-hand side of the equation, the first term represents the temperature gradient in Fourier's law, and the second term describes the nonlocal effects from nondiffusive heat transfer. Given volumetric specific heat cv and speed of sound c in Eq. (5), there are only two parameters: к and Лв. One parameter (к) is to characterize diffusive heat transfer, and the other (Лв) is to characterize nonlocal effects on the heat flux from nondiffusive heat transfer. Thus, this equation is called a two-parameter heat conduction (TPHC) model.

It should be noted that лв was called characteristic ballistic transport length in Reference 24. Here, we rename лв to nondiffusive heat transfer characteristic length because lots of experiments5,11,12 demonstrates the breakdown of Fourier's law due to nondiffusive heat transfer, but none of these experiments at room temperature presents direct evidence of ballistic heat transfer. Therefore, the nondiffusive heat transfer characteristic length is a more appropriate name for лв than the ballistic transport length. In addition, лв is used in TPHC, a phenomenological model, to characterize the nonlocal effects on the heat flux from nondiffusive heat transfer. The validation of the 1D-TPHC model is demonstrated by the successful interpretation of hotspot-size-dependent heat transfer in TTG experiments in silicon thin films,5 and лв is found to be лв = 7.1 ^m.24 The relationship between лв and phonon MFPs with broad-band spectrum still requires systematic theoretical and experimental studies.

After the validation, the 1D-TPHC model was extended to the three-dimensional (3D) regime without rigorous derivation by modifying Guyer-Krumhansl model, which is derived from linearized phonon BTE.19

cv ¥+v- q=q, (6)

q = -KVP + злв (V2q + 2V(V ■ q)), (7)

where Q is the internal heat generation. Using the traditional kinetic theory on diffusive heat conduction, a diffusive phonon MFP (AD) can be determined from thermal conductivity as follows

k = cv cad/3. (8)

Here, the same speed of sound is used for the calculation of AD and AB because an average speed of sound is used to link the relaxation time and characteristic length instead of using phonon group velocity. In addition, phonon polarizations with different group velocities are not considered here. Using the phonone MFP (AD), the 3D-TPHC model expressed in Eq. (7) can be rewritten as

q = -kVP + ADAB(V2q + 2V(V • q)). (9)

The 3D-TPHC model has been validated in 2D-TTG and TDTR experiments.24 In the following section, we will apply the TPHC model to study nondiffusive heat transfer in MG-X-ray experiments.

III. APPLICATION OF THE TPHC MODEL FOR MG-X-RAY DATA ANALYSIS A. Modeling of MG-X-ray experiments

Using nanoscale metallic gratings (MG) heaters and soft X-ray as thermal probe, Siemens et al. reported the first quantitative measurements of heat transfer from nanoscale hotspots covering both diffusive and nondiffusive regimes in 2010.15 Fig. 1(a) shows the schematic of MG-X-ray experiments where the nanoscale heaters are nickel strips with thickness of 20 nm, length of 120 ^m and various widths, installed periodically on the substrate. These nickel strips are attached on either sapphire or fused silica substrate. Here, the thickness of the substrate is several orders of magnitude larger than that of the nickel strips so that the substrate can be treated as a semi-infinite body. Considering that the length of MGs is at least two orders of magnitude larger than their width and thickness, heat transfer in MG-X-ray experiments can be simplified as two-dimensional (2D). Moreover, because the laser beam diameter (~ 100 ^m) is much larger than the spatial period of MGs, only one typical MG period is representative and selected as the computational domain shown in Fig.1(b). Here, horizontal (in-plane) direction along the width of the MG is chosen to be x direction and vertical (cross-plane) direction is chosen to be _y direction.

When the substrate and nickel gratings are exposed to an ultrashort (~ 25 fs) pulse of laser pump at a wavelength of 800 nm, only nickel gratings are heated up because the substrate (sapphire or fused silica) is transparent to the laser at this wavelength. After the ultrashort laser heating, electrons in nickel gratings are excited first. Then, thermal energy is transferred to phonons in nickel through electron-phonon interaction. In less than 10 ps, electrons and phonons can reach local thermal equilibrium. Compared to the whole thermal decay process (~ tens of nanoseconds) in MG-X-ray experiments, the duration of electron-phonon equilibration process in nickel gratings is negligibly short, so that Fourier's law is applicable to describe the thermal decay process of nickel gratings afterwards. For comparison, both TPHC model and Fourier's law will be used to calculate thermal transport in the substrate. We will discuss the boundary and initial conditions in the next subsection.

Substrate (Semi-infinite)

FIG. 1. (a) Schematic of metallic gratings for the measurement of nondiffusive heat transfer; (b) Computational domain for one period of metallic grating.

B. Boundary and initial conditions

In 2D Cartesian coordinate shown in Fig. 1(b), the computational domain is composed of two subdomains: nickel heater and dielectric substrate. In heater subdomain, all boundaries except for the interface are insulated. In the substrate subdomain, two boundary conditions are required for the TPHC model. This is because Eq. (9) in the TPHC model is a second-order PDE in space, unlike the equation describing heat flux q in Fourier's law is a first-order PDE in space. As shown in Fig. 1(b), periodic boundary condition is applied in x direction for both qx and qy in the substrate. In y direction, = 0 is applied at boundary y = 0 and y = ymax ^ œ. On the top surface of the substrate, the boundary condition for the normal flux is specified as

where Theater and fi represent the temperature of nickel strip heater and the nonequilibrium temperature of the substrate at the interface, respectively, r is the thermal interface resistivity.15 The bottom boundary of the substrate is insulated.

It should be noted that there are two singularity points located at the two bottom corners of the nickel heater contacting the substrate (see Fig. 1(b)). To eliminate discontinuity which may cause numerical issues, interface heat flux is smoothed to be an error function shape profile during simulation process with the total energy conserved. The effects from this treatment are found to be negligible based on our numerical experiments.

For the initial conditions, the normalized temperature in nickel heater and substrate are specified as Theater = 1, and fi = 0, respectively.

C. Numerical methods

As mentioned in the previous subsection, Fourier's law and the TPHC model will be adopted to describe the thermal decay process of nickel heater and substrate, respectively. In Fourier's law, the heat flux q can be calculated directly from temperature gradient. Therefore, only one dependent variable Theater is adequate in the simulation, since the dependent heat flux can be extracted out from the temperature field. However, in Eq. (9) of the TPHC model, the heat flux q is unable to be explicitly expressed as a function of the nonequilibrium temperature fi. Consequently, both nonequilibrium temperature fi and heat flux q have to be solved simultaneously.

An implicit second-order finite difference method is applied to discretize Eqs. (6) and (9) and an iterative method with successive over-relaxation is implemented to solve for Theater, fi, qx and qy simultaneously. To eliminate numerical oscillation, staggered structured grids26 are implemented. An in-house code is developed to solve the nonequilibrium temperature and heat flux field in the heater-substrate system. In addition, grid-independence has been checked for every case to assure the convergence of the simulation results.

D. Signal model

In MG-X-ray experiment, the dynamic soft X-ray diffraction signal is the direct measurement. In References 15 and 27, this directly measured signal is linked to the surface deformation of the system through a Fresnel optical propagation model. Physically, the surface deformation of the system is a superposition of two components: the thermal expansion that decays monotonically with time, and the periodic surface acoustic waves (SAWs) induced by the impulsive expansion of the metallic grating. The SAWs lead to the oscillatory behavior of the signal, while the thermal expansion defines the decay around which the SAWs oscillate. For this study, we only focus on the thermal expansion component of the signal, so the SAWs are simplified as a sinusoidal oscillating term with exponentially decayed amplitude, and superimposed on top of the thermal expansion. Because the thermal expansion is related to the temperature field, we are able to start from the calculated temperature field to calculate the thermal expansion, and finally simulated the diffracted signal with the SAWs superimposed. The model for calculating the thermal expansion based on the temperature field is briefly described below.

on heater-substrate interface, else.

Considering the relatively small thickness and volume of the nickel heater, the averaged thermal expansion of the nickel heater is described by15

Ahheater (x) = ah ATheater (x) , (11)

where ah is the coefficient of linear thermal expansion of nickel heater, and Theater is the average temperature of nickel heater at corresponding x position. Meanwhile, the thermal expansion of substrate surface is given by

AhSUbstmte (x) = f f p (Xl,y) ydx1dy 2, (12)

3n JxxJy (x - x1)2 + y2

where as is the coefficient of linear thermal expansion of the substrate, vs is Poisson's ratio. Although this formula is applicable for an infinite body,28 it is also applicable for this case since the total deformation at one certain location can be decomposed into individual contribution of each grating heater. Hence, this integral should be carried out for multiple neighboring periods of gratings to account for the thermal expansion contributed by neighboring periods (i.e., Ah (x) = £°=-TC Ah (x + iP)).

In Fig. 1(b), the deformation at point A (peak deformation) is a superposition of two parts: deformation of the heater and deformation of the substrate, while deformation at point B (valley deformation) is just the local deformation of the substrate. Therefore, the difference between calculated deformation of point A and point B represents the overall relative surface deformation Ah, i.e.,

Ah = AhA,heater + AhA,substrate — AhB,substrate. (13)

The deformation Ah is normalized before it is compared to the diffracted signals in MG-X-ray experiments.

E. Uncertainty of data analysis in MG-X-ray experiments using Fourier's law

Currently, modified Fourier's law with an effective thermal conductivity (ETC) is still widely used in data analysis of pump-probe nanothermometry, including TDTR, TTG, and MG-X-ray. If the modified Fourier's law is applied here for data analysis of MG-X-ray experiments, size-dependent ETC should be used. The size-dependent ETC is unknown for different heater sizes in MG-X-ray experiments. Therefore, it needs to be determined from MG-X-ray experiments. Except for ETC, there is another unknown parameter, the size-dependent thermal interface resistivity r, that also needs to be determined from MG-X-ray experiments.

However, because there are similar response of thermal decay to the change of the ETC and r, different combinations of the ETC and r can fit the same thermal decay curve. Therefore, it is almost impossible to extract both ETC and r simultaneously from MG-X-ray experiments. For example, Fig. 2 shows the comparison of experimental signal with two sets of predictions using Fourier's law for a MG-X-ray experiment with nickel strip width L = 80 nm on a sapphire substrate.15 The solid line is the experimental signal. The dashed line represents the predictions with the thermal conductivity remaining to be bulk value, Kbulk = 41.13 Wm-1K-1, and r = 4.4 x 10-9 Km2/W. The dashed-dot line

FIG. 2. Comparison of experiments with two predictions using Fourier's law with different combinations of Keff and r.

shows the predictions with ETC Kf = 0.53Kbulk, and r = 2.0 x 10-9Km2/W. Both predictions are able to fit the experimental signals. Here, the thermal response in MG-X-ray experiments is different from that in TDTR measurements. In TDTR experiments, the initial thermal decay is more sensitive to thermal interface resistance, while the long-time decay is more sensitive to the substrate thermal conductivity. Therefore, both ETC and r can be fitted simutaneously from the same thermal decay curve. However, in the MG-X-ray experiments, the ETC and r cannot be decoupled from data fitting. The main difference between the TDTR experiments and MG-X-ray experiments is in the heater geometry. While there is a single micron-scale axisymmetric heater on the uniform metal thin film in the TDTR setup, there are distributed heaters consisting of nanoscale metallic gratings in the MG-X-ray setup. The axisymmetric heating in TDTR experiments induces three-dimensional heat transfer, but laser heating on nanoscale metallic gratings lead to two-dimensional heat transfer in MG-X-ray experiments. Especially, the thermal decay in MG-X-ray experiments is sensitive to the width of the metallic gratings as distributed heaters. Therefore, the signals in two types of experiments responses differently to substrate thermal conductivity and interfacial conductance. In fact, for the same MG-X-ray experiment, there are innumerous combinations of the ETC and r that can fit the experimental data. Consequently, Fourier's law fails to provide an unequivocal interpretation to MG-X-ray experiments. In order to reveal the mechanisms of nondiffusive heat transfer in MG-X-ray experiments,15 we apply the TPHC model to reinterpret the experimental data.

F. Determination of Ag in the TPHC model

There are two parameters related to sapphire thermal properties in the TPHC model: thermal conductivity k and nondiffusive thermal transport characteristic length AB. The thermal conductivity of a bulk sapphire15 is given as k = 41.13 Wm-1K-1. Given specific heat cv = 2.63 x 106 Jm-3K-1, and speed of sound c = 11061.73 m/s, the diffusive phonon MFP (AD) is calculated from the kinetic theory as Ad = 4.24 nm. Other material properties are: substrate Poisson ratio vs = 0.25, linear coefficient of thermal expansion for nickel heater ah = 12.77 x 10-6 K-1 and substrate as = 5.31 x 10-6 K-1. The methodology to determine AB is described next.

In the experiments by Siemens et al.,15 the width of nickel strip (L) ranges from 80 to 810 nm. It is found that when L = 80 nm, there is most significant nondiffusive heat transfer compared with other cases.15 Therefore, this case is used to determine the value of AB. Although there is another unknown parameter, size-dependent thermal interface resistivity r, a least square method provided in MATLAB curve fitting toolbox is still able to extract both parameters (AB and r) by fitting the experimental thermal decay curve.

In detail, first, the thermal decay curve is extracted from experimental data by filtering out the oscillatory SAW component in the signal. Second, the least square method and parametric studies are performed to find the best match of thermal decay curves from numerical simulation to experiments. Fig. 3 shows the best fitting results with AB = 1.6 ^m and r = 2.9 x 10-9 Km2/W, and the original experimental signal is plotted as reference.

FIG. 3. Best fitted thermal decay (red solid line) by curve fitting is plotted with the original signal (black solid line) as reference. Sensitivity of the thermal decay curve for L = 80 nm to the change of thermal interface resistivity (r) and nondiffusive phonon MFP (Ab) is also shown as dashed and dashed-dot lines, respectively.

To study the sensitivity of thermal decay curve to the changes of two parameters, we either fix AB = 1.6 ^m while change r by ±20% around r = 2.9 x 10-9 Km2/W, or fix r = 2.9 x 10-9 Km2/W while change AB by ±50% around AB = 1.6 ^m. The results are also plotted in Fig. 3. It shows that within 300 ps, the numerical calculated thermal decay curve is sensitive to the change of r, but insensitive to the change of AB. Thus, r can be fitted using the signal within 300 ps, while AB is fitted with the rest of the signal. Figure. 3 shows that the whole thermal decay curve is not sensitive to the change of AB. This can be explained by two reasons. Firstly, as shown in the second term on the right-hand side of Eq. (9), the coefficient of the nonlocal effects from nondiffusive heat transfer is proportional to the product of AB and AD. The relatively small value of AD (4.24 nm) in sapphire significantly suppresses the impacts of AB to the signal. Secondly, the nonlocal effects in two-dimensional heat transfer in MG-X-ray experiments are not as significant as that in 1D situations, because the thermal decay signal becomes a weighted average of thermal expansion in the cross-plane direction calculated from Eq. (12).

Although the value of AB in the sapphire substrate is determined to be AB = 1.6 ±0.8 ^m with high uncertainty (about 50%) due to the insensitive response of the thermal decay curve to the change of AB in the MG-X-ray experiments, the TPHC model can automatically take the hotspot-size-dependent effective thermal conductivity (ETC) into account. Thus, we expect that the TPHC model will provide more reliable results on the thermal interface resistivity (r) than Fourier's law in the data analysis of MG-X-ray experiments.15

G. Reinterpretation of MG-X-ray experiments using the TPHC model

Here we use the TPHC model with AB = 1.6 ^m to reinterpret MG-X-ray experiments on sapphire.15 In total, there are four different cases with the nickel heater width L = 810,350,190 and 80 nm, respectively. For different heat widths, the ratio between grating period and heater width is fixed at 4, i.e., P = 4L. In the data analysis by the TPHC model, r is the only unknown parameter to be extracted from MG-X-ray experiments for different cases since AB has been fitted. To extract the value of

FIG. 4. Comparison of the fitting results from TPHC model and Fourier's law with the experimental data.15 The unit of the thermal interface resistivity r is 10-9Km2/W.

TABLE I. Summary of fitted interface resistivity r with different nickel heater width L. The unit of r is 10-9 Km2/W.

L 810 nm 350 nm 190 nm 80 nm

r eff ,Fourier 2.1 2.9 3.1 4.4

reff ,TPHC 2.1 3.3 3.6 2.9

Heater width L (nm)

FIG. 5. Comparison of size-dependent thermal interface resistivity changing with nickel heater width L between the TPHC model and Fourier's law.

r from experimental data, the same procedure as the one fitting for AB is performed, except that r is the only output now. After r is fitted, thermal decay curve is reconstructed and the component of surface acoustic waves is superimposed as a sinusoidal oscillating term with exponentially decayed amplitude. The final fitting signals with oscillatory SAW component using the TPHC model are compared with experiments in Fig. 4. The signal calculated using Fourier's law with Kbulk = 41.13 Wm-1K-1 are also plotted together for comparison. For four different cases, simulation results from both TPHC model and Fourier's law can fit the experimental signals very well. Through data fitting, we extract the value of thermal interface resistivity r for four different cases that are summarized in Table I. The unit of r is 10-9 Km2/W. The results of r vs. L are also plotted in Fig.5. It is found that there is monotonic increase of thermal interface resistivity with decreasing width of the nickel heater (L) based on the results of Fourier's law. Compared with the results of Fourier's law, there is faster increase of the thermal interface resistivity as the width of the nickel heater (L) decreases before L reaches 190 nm. When the width of the nickel heater (L) further decreases to 80 nm, the thermal interface resistivity from the TPHC model decreases. We will discuss this phenomenon in the following section.

IV. DISCUSSION

Due to the existence of similar and coupled effect from both hotspot size-dependent ETC and thermal interface resistivity, MG-X-ray experiments have not been numerically reproduced by non-diffusive heat transfer models prior to this study. Since the TPHC model can automatically capture the hotspot-size-dependent nondiffusive heat transfer in the substrate, the hotspot size-dependent thermal interface resistivity can be extracted from experimental data without contamination from the hotspot-size-dependent ETC. In this section, we discuss the characterization of nondiffusive heat transfer in the sapphire substrate and hotspot-size-dependent thermal interface resistivity separately.

A. A criterion for the breakdown of Fourier's law

It is well accepted that Fourier's law becomes invalid in nanosystems due to significant nondif-fusive heat transport. However, there is a lack of a criterion to predict when Fourier's breaks down. Here, we define this criterion based on the TPHC model.

In Eq. (9) of the TPHC model, the nondiffusive effects are described by the last two terms. Therefore, the contribution of these two terms determines the significance of nondiffusive effects in the TPHC model. To quantify the nondiffusive effects, we define a nondimensional parameter Z for

spatially periodic heating with a period P based on the coefficient of those two terms:

Z = D B 2. (14)

3(P/2n)2

Later on in this section, we will demonstrate that a criterion based on the value of Z can be used to to predict when Fourier's law breaks down.

To study nondiffusive heat transfer without effects from the thermal interface, we consider nondiffusive heat transfer induced by artificial 2D-TTG in the sapphire substrate. In these artificial 2D-TTG cases, the grating periods are chosen to be the same as these in metallic grating cases. In total, four different cases are considered with the grating periods P = 3.24, 1.40, 0.76 and 0.32 ^m, respectively. The heater width to period ratio is fixed at 0.25, i.e., L = 0.25P. Similar to the 2D-TTG experiments in GaAs and Si conducted by Johnson et al. ,29 the sapphire substrate is heated up by two concurrent interfering pulsed-lasers, leading to a spatially periodic micron-sized thermal grating with sinusoidal shaped temperature distribution in the in-plane direction. In the cross-plane direction, there is exponential decay of the temperature change, which is detected by the diffracted signal from a probe laser. Since the heating process and the interaction between electron and phonon happen so fast, the laser heating process can be modeled as a source term with Dirac delta function given as

Q(t,x,y) = exp |iyxj exp(p |y|)5(t), (15)

where p1 is the reciprocal of optical depth corresponding to the pump beam in the solid, and P represents for the period of thermal grating. x and y direction are the same as chosen in metallic grating case.

Take divergence on both sides of Eq. (9) and substitute it into Eq. (6), then perform Fourier transform on both sides to obtain the solution

x Q (n, k, €y)

A/? n, €x, €y) = (---^TT, (16)

v ' (icvn + Kef (£) £2)

where Q (n, , £y) is the Fourier transform of Q (t, x, y) in Eq. (15), Kef (£) = Kbulk/ (1 + AbAd£2)

is effective thermal conductivity (ETC), £ = ^J^ + is the magnitude of spatial frequency, and n accounts for temporal frequency.

Similar to Minnich's approach,30 the optical signal H (t, x) is given by the following correlation

4p1 f32e^xx r exp [-k ) £2t/cv]

H (t, x) = - ,L . , --Vd£y, (17)

(P + $ (p2 +

where = P, >P2 is the reciprocal of optical depth corresponding to the probe laser.

Given J61, j62, and material thermal properties, the optical signal can be predicted from Eq. (17). All the needed material properties of sapphire have been specified in the previous section, and AB = 1.6 ^m in the sapphire is determined from the MG-X-ray experiments through data fitting. Our numerical results shows that the normalized optical signal based on Eq. (17) is not sensitive to the values of p1 or p2 if both values are greater than 0.25 ^m. The values of p1 and p2 are set as 0.5 ^m. The normalized optical signals predicted from Fourier's law and the TPHC model for four different grating periods are compared in Fig. 6. Solid lines represent the TPHC results and dashed lines represent the results of Fourier's law. Both the TPHC model and Fourier's law predict that the decay rate of signals increases with decreasing thermal grating period. For P = 3.24 ^m, the corresponding non-dimensional parameter Z can be calculated from Eq. (14) as Z = 8.5 x 10-3. Figure 6 shows that there is negligible difference between the predictions from the two models. For P = 1.4 ^m (Z = 0.0455), there is noticeable difference between the two models in Fig. 6. As P decreases, there is increasing difference between the results from the two models. For P = 0.32 ^m (Z ~ 0.872), there is significant difference between the two models.

Because the non-dimensional parameter Z can characterize the nonlocal effects in nondiffusive heat transfer that is not considered in Fourier's law, a criterion based on the value of zeta can be

0 5 10 15 20 25

Time (ns)

0.32 /j.m

FIG. 6. Comparison of temperature decay curves in 2D-TTG in sapphire for four different grating periods (P, P = 4L) between analytical solutions of the TPHC model (solid lines) and Fourier's law results (dashed lines).

identified to predict the validity of Fourier's law. Based on the results shown in Fig. 6, we find that Z < 0.01 can be used as the criterion. For Z < 0.01, the nondiffusive effects become negligible, indicating that Fourier's law is valid. Thus, we demonstrate the validity of this criterion for nondiffusive heat transfer in the sapphire substrate without effects from the thermal interface. The validity of the criterion for the application of Fourier's law in other materials will be discussed in a separate paper.

B. Hotspot-size-dependent thermal interface resistivity

In the previous subsection, nondiffusive heat transfer in the sapphire substrate without the effects from the thermal interface is discussed. The hotspot size dependent nondiffusive heat transfer can be described by the TPHC model. In this subsection, we will continue to discuss the size-dependent thermal interface resistivity in the MG-X-ray experiments. Theoretically, the size-dependence of interface resistivity can be due to different scattering mechanisms for diffusive and nondiffusive heat transfer, which can be described by diffuse mismatch model (DMM) and acoustic mismatch model (AMM),31 respectively. Different weighted contribution by nondiffusive heat transfer in overall heat transfer may lead to different values of thermal interface resistivity. For the nickel-sapphire interface, there is higher thermal interface resistivity in nondiffusive heat transfer than diffusive heat transfer. This may explain the increase of the TPHC results on thermal interface resistivity as L decreases from 810 nm to 190 nm as shown in Fig. 5. The similar hotspot size-dependent behavior of thermal interface resistivity is found in our previous study of nondiffusive heat transfer in TDTR experiments.25 However, as L decreases further from 190 nm to 80 nm, there is significant decrease in the TPHC results of the thermal interface resistivity (see Fig. 5). Similar results are also reported by Hoogeboom et al. in their MG-X-ray experiments using Fourier's law for data analysis.27 The abnormal decrease in the thermal interface resistivity for smaller heater size is called "collective diffusive"effect.27 The cause of this phenomenon requires further studies and will be our future work.

V. CONCLUSION

MG-X-ray provides an important technique to investigate hotspot size-dependent thermal transport with spatial resolution about tens of nanometers. The coupled grating size-dependent effective thermal conductivity and size-dependent thermal interface resistance in the MG-X-ray experiments cannot be determined by using Fourier's law. In this paper, a unified diffusive-nondiffusive two-parameter nondiffusive heat conduction (TPHC) model is implemented to reinterpret MG-X-ray experiments on a sapphire substrate at room temperature. Because the size-dependence of the effective thermal conductivity (ETC)can be automatically captured by the two parameters in the TPHC model, namely the diffusive thermal conductivity k and the nondiffusive heat transfer characteristic length KB, the size-dependent thermal interface resistance can be determined through data fitting without contamination from the size-dependent ETC. Using this TPHC model with fitted KB = 1.6 ± 0.8 ^m, for the first time, MG-X-ray experiments are numerically reproduced outside the

framework of Fourier's law. It is found that the extracted thermal interface resistivity increases as the heater width L decreases from 810 nm to 190 nm, while abnormally decreases as L decreases further from 190 nm to 80 nm. This interesting "collective diffusive" phenomenon requires further studies. Finally, based on the TPHC model, we define a non-dimensional parameter Z to characterize the nonlocal effects on diffusive heat conduction. A criterion of Z > 0.01 is identified to predict the breakdown of Fourier's law. The validity of this criterion is demonstrated by the application for MG-X-ray experiments in sapphire substrate. The identification of the criterion fills the knowledge gap on the prediction when Fourier's law breaks down due to the effects from nondiffusive heat transfer.

ACKNOWLEDGMENTS

We would like to thank Prof. Murnane and Prof. Kapteyn et al for sharing their experimental data. The support for our research by the National Science Foundation (Award Number:1637370) is greatly appreciated.

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