Violation of Fourier's law in homogeneous systems

TL;DR

This paper investigates the physical mechanisms behind the formation of hotspots and graded thermal conductivity in homogeneous nanoscale systems, revealing that limited phonon scattering leads to these phenomena regardless of material or size.

Contribution

It provides a detailed analysis of phonon scattering effects on thermal conductivity in homogeneous nanoscale systems using phonon Boltzmann transport equation.

Findings

Graded thermal conductivity occurs with insufficient phonon scattering.

The phenomenon is independent of material properties and system size.

Insights aid understanding of heat dissipation in microelectronics.

Abstract

Hotspot is a ubiquitous phenomenon in microdevices/chips. In homogeneous nanoscale graphene disk with a hotspot, a graded thermal conductivity is observed previously even when the system size is fixed. However, the underlying physical mechanism is not clear. In this work, the hotspots in homogeneous 2D disk/3D ball and graphene disk are studied based on phonon Boltzmann transport equation. The mechanisms of phonon scattering are analyzed. It is found that for a system with fixed size, the graded thermal conductivity is predictable as long as there is not sufficient phonon scattering, which is independent on material properties, dimensions or system size. This work may shed light on both theoretical and experimental studies on heat dissipation of microelectronics.

Equations

$$\kappa (r)=-\frac{q(r)}{dT/dr},l<r<L,$$

$$\mathbf{v}\cdot {\nabla }_{\mathbf{x}}e={\tau }_R^{-1}(e_R^{eq}-e)+{\tau }_N^{-1}(e_N^{eq}-e),$$

$$T(r)$$

$$T=\frac{l{T}_h+L{T}_c}{l+L}.$$

$$dT$$

$$\kappa (r)$$

Full text

$Introduction.$—Thermal conductivity, a fundamental physical property of materials, is a constant that independent of system size and geometry in bulk materials. It is an intrinsic property that depends only on the component of materials. Heat conduction in such materials generally follows the Fourier’s law, which implies that the heat carriers (phonons) undergo a diffusive process Kaviany (2008); Chen (2005). However, as the size or dimension of the system decreases, in particular when the size goes down to nanoscale and/or dimension is reduced to two dimension (2D) or quasi-one dimension (1D), there is still no rigorous mathematical proof that the Fourier’s law is still valid. In contract, many researchers discovered that the thermal conductivity is a function of size and geometry Zhang et al. (2020); Sverdrup et al. (2001); Li et al. (2012); Gu et al. (2018); Li et al. (2003); Hsiao et al. (2013); Lee et al. (2017); Chang et al. (2008); Xu et al. (2014); Mahan and Claro (1988); Chen (1996); Regner et al. (2014).

The underlying physical mechanisms of non-Fourier heat conduction mainly include: First, when the size of structures is comparable with the phonon mean free path, the phonon transport is largely affected by the boundary scattering Majumdar (1993); Hsiao et al. (2013), such that the thermal conductivity can be altered significantly by nanoengineering Volz and Chen (1999); Li et al. (2003); Hsiao et al. (2013); Lee et al. (2017); Chang et al. (2008); Xu et al. (2014). Second, a divergent thermal conductivity with system size was found in many low dimensional momentum conserved systems because of the existence of zero frequency and large wave length modes  Prosen and Campbell (2000); Narayan and Ramaswamy (2002); Lepri et al. (2003); Dhar (2008). Third, as the system size is close or comparable to the phonon wave length, the wave nature of phonons is non-negligible in thermal transport Yang and Chen (2003); Ma et al. (2019). Fourth, the possible existence of the second sound makes heat transfer like wave propagation Guyer and Krumhansl (1966a, b); Lee and Li (2020); Lee et al. (2015); Cepellotti et al. (2015); Guo and Wang (2015); Huberman et al. (2019). This regime is usually called phonon hydrodynamic regime.

Most studies so far have focused on the length-dependent thermal conductivity Zhang et al. (2020); Gu et al. (2018); Lepri et al. (2003); Dhar (2008); Hsiao et al. (2013); Lee et al. (2017); Chang et al. (2008); Xu et al. (2014). The quasi-ballistic thermal transport effects Hua and Minnich (2018); Chen et al. (2018) are also measured with a nanoscale heat source comparable to the phonon mean free path Sverdrup et al. (2001); Minnich et al. (2011); Hu et al. (2015); Siemens et al. (2010). The difference from Fourier’s law is just the value of thermal conductivity depends on the size of heat source. Thermal conductivity, defined through the Fourier’s law, is homogeneous in nanostructures Hu et al. (2015); Siemens et al. (2010).

Recently, in a system with fixed size, an abnormal phenomenon - graded thermal conductivity - the thermal conductivity in the radial direction increases with the distance from the disk center, has been observed in homogeneous nanoscale graphene disk and carbon nanocone by molecular dynamics simulations Yang et al. (2015); Ma et al. (2017).

Due to the limitation of computational resources, the diameter of system in previous molecular dynamics simulations Yang et al. (2015); Ma et al. (2017) is below $25$ nanometers. Does the graded thermal conductivity exist in a macro-system? What is the physical understanding on the mechanisms of graded thermal conductivity in homogeneous system?

In this Letter, we shall answer above mentioned questions by studying the graded thermal conductivity in homogeneous 2D disk/3D sphere (Figs. 1 and 2) with a fixed macroscopic size from ultra-low temperature to high temperature, without limiting to any specific material. The underlying physical mechanisms are to be analyzed by ballistic phonon transport, normal (N) scattering and resistive (R) scattering, respectively. The general conclusion will be exemplified by the graphene disk (Fig. 3).

The schematics of the 2D disk and 3D sphere are shown in Fig. 1(a) and Fig. 2(a), respectively, where the radii of the inner and outer heat baths are $l$ and $L$, respectively. The temperatures of the inner and outer heat baths are fixed at $T_{h}=T_{0}+\Delta T/2$ and $T_{c}=T_{0}-\Delta T/2$, where $\Delta T/T_{0}\rightarrow 0^{+}$. The local radial thermal conductivity $\kappa$ is calculated by

[TABLE]

where $q(r)$ is the local heat flux, namely the heat energy flow along the radial direction per unit area in a unit time. $T(r)$ is the local temperature, $r$ is the distance from the center,

$Model~{}equation.$—We start with the steady-state phonon Boltzmann transport equation (BTE) under the Callaway model and Matthiessen’s rule Chen (2005); Callaway (1959); Guo and Wang (2017); Allen (2018); Luo et al. (2019), in which both the normal (N) scattering and resistive (R) scattering are included.

[TABLE]

where $e$ is the phonon distribution function of energy density, $\bm{v}$ is the group velocity, $\bm{x}$ is the spatial position. The heat flux and temperature in Eq. (1) are obtained by taking the moment of the distribution function. $e^{eq}_{R}$ and $e^{eq}_{N}$ are the associated phonon equilibrium distribution functions of energy density for R and N scattering, respectively. $\tau_{R}$ and $\tau_{N}$ are the relaxation times for R and N scattering, respectively. In the BTE simulations Li and Cao (2019); Chen (2005); Guo and Wang (2017); Luo et al. (2019), the wave nature of phonons is not taken into account Yang and Chen (2003); Ma et al. (2019). The distribution functions of all phonons emitting from the inner (or outer) heat bath are $e_{R}^{eq}(T_{h})$ (or $e_{R}^{eq}(T_{c})$) Guo and Wang (2017); Luo et al. (2019). More details of phonon BTE and boundary conditions is shown in Supplemental Material(SM) I.

The phonon transport will be simulated by solving phonon BTE numerically by the implicit discrete ordinate method Guo and Wang (2017); Zhang et al. (2019). In simulation of 2D disk/3D sphere with a fixed macroscopic size, the Debye approximation and gray model Chen (2005) are used, where no phonon dispersion and polarization are considered. Note that the heat conduction in 2D disk/3D sphere is not limited by specific materials properties so that all physical variables are dimensionless. The radii of inner and outer heat baths are fixed at $l=0.1$ and $L=0.5$, respectively (Figs. 1 and 2). The group velocity is $|\bm{v}|=1$ and the specific heat is $C=1$. The thermal effects of N (R) scattering on graded thermal conductivity will be investigated by adjusting the values of $\tau_{N}^{-1}$ or $\tau_{R}^{-1}$.

In simulation of graphene disk (Fig. 3), the phonon dispersion and polarization of graphene are calculated using Vienna Ab initio Simulation Package (VASP) combined with phonopy. And the effects of both frequency-dependent N and R scattering will be considered. More details on phonon properties of graphene and numerical solutions can be found in SM II-III.

$Results.$—The phonon transport in a homogeneous 2D disk with a fixed macroscopic system size is studied first. In addition to numerical results, the analytical solutions in the ballistic Olfe (1968); Li and Cao (2019) ($\tau_{R}^{-1}=0,~{}\tau_{N}^{-1}=0$), diffusive ($\tau_{R}^{-1}\rightarrow\infty,~{}\tau_{N}^{-1}=0$) and hydrodynamic Guyer and Krumhansl (1966a, b); Yang et al. (2019); Lee et al. (2015); Cepellotti et al. (2015) ($\tau_{N}^{-1}\rightarrow\infty,~{}\tau_{R}^{-1}=0$) limits are also plotted in Fig. 1 to show the separate thermal effects of N or R scattering (Derivations of three limits are shown in SM IV).

At ultra-low temperature, phonon-phonon interaction/scattering can be totally neglected and ballistic phonon transport dominates heat conduction Majumdar (1993); Li and Cao (2019) (e.g., $\tau_{R}^{-1}=0,~{}\tau_{N}^{-1}=0.1$). As shown in Fig. 1(c)(d), the temperature profile is nonlinear and the radial thermal conductivity is not a constant anymore, instead it depends on $r$. The results are consistent with the analytical solutions in the ballistic limit Olfe (1968); Li and Cao (2019) (see Fig. 1(b) or SM IV), i.e.,

[TABLE]

This suggests the graded thermal conductivity, similar to what was observed in nanodisks Yang et al. (2015) and nanocones Ma et al. (2017) by molecular dynamics simulations.

At low temperature, R scattering is weak and N scattering dominates the heat conduction so that phonon transports in the phonon hydrodynamic regime Li and Lee (2019); Guyer and Krumhansl (1966a, b); Yang et al. (2019); Guo and Wang (2015); Lee et al. (2015); Cepellotti et al. (2015). It can be observed that with the increase of $\tau_{N}^{-1}$, the slopes of the numerical profiles of graded thermal conductivity in Fig. 1(d) increase first and then decrease gradually. As N scattering is much stronger than R scattering, the radial temperature goes to a constant and recovers the phonon hydrodynamic limit Guyer and Krumhansl (1966a, b); Yang et al. (2019); Lee et al. (2015); Cepellotti et al. (2015) (see SM IV), i.e.,

[TABLE]

At high temperature, R scattering starts to dominate the heat conduction so that phonon transport goes to the diffusive regime. It can be observed that with the increase of $\tau_{R}^{-1}$, the temperature profile comes to linear and the graded thermal conductivity phenomenon disappears. The results agree well with the analytical solutions in the diffusive limit (see SM IV), i.e.,

[TABLE]

$Physical~{}mechanisms.$—As shown in Fig. 1, in homogeneous 2D disk with a fixed macroscopic size, the non-Fourier’s thermal transport phenomenon depends on scattering, i.e., $\tau_{N}^{-1}$ and $\tau_{R}^{-1}$. In the following, the underlying physical mechanisms of phonon scattering are discussed in details.

$Ballistic.$—In the ballistic regime, corresponding to ultra-low temperature, phonon-phonon interaction/scattering rarely exists. Phonon advection dominates heat conduction Li and Lee (2019); Li and Cao (2019). For any point in the interior domain, phonons reach this point from the inner and outer thermal baths with different directions Chen (1996); Li and Cao (2019). Both analytical (Eq. (3)) and numerical results predict that the temperature profile along radial direction has a non-linear dependence on the distance $r$ in 2D disk (Fig. 1(c)). It is different from ballistic phonon transport in a symmetric system, in which the temperature is a constant Majumdar (1993). In the symmetric system, all phonons emitting from one heat bath will be totally received by the other (see FIG. S2). So that the temperature gradient inside the system vanishes Majumdar (1993).

For ballistic transports in 2D disk, all phonons emitting from the inner bath will be received by the outer bath. However, phonons emitting from the outer heat bath will be received by both the inner and outer heat baths (see Fig. 1(b)). That means a portion of phonons are not received by the inner bath, which do not contribute to heat flux, but contribute to local energy or temperature. The temperature gradient is built by the asymmetric phonon advection, instead of phonon-phonon scattering. Because the heat flux from inner to outer is conserved, graded thermal conductivity can be observed in 2D disk in the ballistic regime (Fig. 1(d)).

$Scattering.$—With the increase of temperature, phonon-phonon scattering becomes strong and dominates heat transfer Li and Lee (2019). In this case, the thermal effects of N scattering ($\tau_{N}^{-1}$, momentum conserved) and R scattering ($\tau_{R}^{-1}$, momentum not conserved) are discussed as follows.

$N~{}scattering.$—At low temperature, R scattering is weak and N scattering dominates the phonon transport. N scattering does not cause thermal resistance Guyer and Krumhansl (1966a, b); Yang et al. (2019); Guo and Wang (2015); Lee et al. (2015); Cepellotti et al. (2015), but affects energy distribution and temperature profile. When N scattering is weak $(\tau_{N}^{-1}\leq 10)$, the graded thermal conductivity is attributed to asymmetric scattering. This means that N scattering is frequent far from the center. But near the center, the N scattering is less, which limits the exchange of thermal energy. As $\tau_{N}^{-1}\gg 10$, the N scattering is very strong inside the whole domain and goes to a constant temperature profile Yang et al. (2019); Guyer and Krumhansl (1966a, b) (Eq. (4)), namely, graded thermal conductivity disappears. In a word, it can be observed that as N scattering increases, the graded thermal conductivity phenomenon increases first, and then fades away (Fig. 1(d)).

$R~{}scattering.$—At high temperature, R scattering starts to play the leading role on heat conduction. Different from N scattering, R scattering does not conserve momentum, and causes thermal resistance Kaviany (2008); Li and Lee (2019). With the increase of R scattering, the frequent energy exchange and heat dissipations decrease the temperature jump near the heat baths Li and Cao (2019) (Fig. 1(c)). As $\tau_{R}^{-1}\gg 10$, the heat conduction follows Fourier’s law and there are a linear temperature profile and a constant thermal conductivity (Fig. 1(d)). In other words, in a structure with frequent R scattering, no graded thermal conductivity appears.

$Graded$ $thermal$ $conductivity.$—Motivated by previous studies Yang et al. (2015); Ma et al. (2017), an experimental formula of graded thermal conductivity is used to fit the numerical data approximately (Fig. 1(d)), i.e.,

[TABLE]

where $\kappa_{0}$ is a constant, $R_{2D}^{*}$ is the normalized coordination in 2D disk and $\alpha$ is the graded rate Yang et al. (2015); Ma et al. (2017). The detailed fitting parameters can be found in SM IV and TABLE. S1. So that for a fixed disk size, there is no more homogenous value of thermal conductivity, instead a graded increasing thermal conductivity from the disk center to the outer.

For a given 2D disk with a fixed macroscopic size, the above results (Fig. 1) and analysis show that the graded thermal conductivity depends on the amount of phonon scattering. When phonon scattering is not sufficient and $\tau_{N}^{-1}$ and $\tau_{R}^{-1}$ are small, the temperature profiles are nonlinear and graded thermal conductivity appears. In the ballistic regime, the graded thermal conductivity is caused by the asymmetric phonon advection Olfe (1968); Li and Cao (2019) due to the spatial asymmetry of 2D disk as mentioned in Fig. 1(b) (or FIG. S2) and preceding paragraph (see $ballistic$). As the phonon-phonon scattering increases, the energy and momentum exchange among phonons break the asymmetric phonon advection gradually. However, the effects of N and R scattering on graded thermal conductivity are quite different (Fig. 1(d)). With the increase of $\tau_{N}^{-1}$ (see $N~{}Scattering$), the graded phenomenon is enhanced first and then fades away due to diverging thermal conductivity Yang et al. (2019); Guyer and Krumhansl (1966a, b). As $\tau_{R}^{-1}$ increases (see $R~{}Scattering$), the graded phenomenon fades away gradually.

$3D~{}ball.$—Does the graded thermal conductivity exist in 3D structures with a fixed macroscopic size? In order to look for the answer, the radial thermal conduction in a 3D sphere (Fig. 2(a)) is also investigated. The numerical results in different regimes are shown in Fig. 2(b)(c). It is found that the temperature profiles and thermal conductivities in 3D sphere are similar to those in the 2D disk. In addition, an exponential function of graded thermal conductivity is also used to fit the numerical data approximately (Fig. 2(c)), i.e.,

[TABLE]

where $R_{3D}^{*}$ is the normalized coordination in 3D sphere and $\gamma$ is a coefficient. The detailed fitting parameters can be found in SM IV and TABLE. S2. Therefore, the graded thermal conductivity can appear in both 2D and 3D radial homogeneous systems with fixed sizes.

Besides, the dimensional analysis Barenblatt (1987) and more results of 2D disk/3D sphere are shown in SM VI.

$Graphene~{}disk.$—Graphene, a very excellent thermal conductor that has been studied extensively Balandin et al. (2008); Gu et al. (2018); Yang et al. (2015), is used to illustrate our analysis.

Firstly, the size of graphene disk is fixed at $L=20~{}\mu$ m and $l=4~{}\mu$ m. Then, the temperature $T_{0}$ is decreased gradually, as shown in Fig. 3(a)(b). As the temperature is changed from $300~{}\text{K}$ to $3~{}\text{K}$, it can be observed that graded thermal conductivities and non-Fourier’s phenomenon happen, which can be explained that the R scattering becomes weak, and the N scattering starts to dominate the heat transfer Cepellotti et al. (2015); Lee et al. (2015); Guo and Wang (2017) as the temperature decreases. At $T_{0}=30~{}\text{K}$, the normalized temperature near the inner heat bath is even smaller than that in the ballistic limit, which is impossible if N scattering is weak. At ultra-low temperature $(3~{}\text{K})$, ballistic phonon transport dominates heat conduction so that the temperature profile recover the analytical solutions in the ballistic limit.

Secondly, the temperature of graphene disk is fixed at $300~{}\text{K}$. Then, the system size $L$ is decreased, as shown in Fig. 3(c)(d), where $L=5l$. It can be observed that as system size decreases, the thermal conductivity along the radial direction is not a constant. Because as system size decreases, the ballistic phonon transport starts to play an important role on heat conduction Sverdrup et al. (2001); Li and Lee (2019); Bae et al. (2013). It is noted that as the size of graphene disk is tens of nanometers, the graded thermal conductivity has been predicted by molecular dynamics Yang et al. (2015); Ma et al. (2017), which is beyond the applications of phonon BTE Chen (2005). According to the results of graphene disk (Fig. 3), it can be concluded that graded thermal conductivity occurs at low temperature or for a small sized system, which are consistent with the results in 2D disk/3D sphere.

$Summary~{}and~{}Conclusion.$—The thermal conductivity in homogeneous 2D disk/3D sphere and graphene disk with a spot heat source at the center is studied from the phonon Boltzmann transport equation. The results show that, for a homogenous system with fixed size, as long as phonon scattering is not sufficient, the thermal conductivity becomes inhomogeneous, namely, it increases from the center to the outer. This study may inspire a better understanding thermal transport in structures with hotspots.

Supported by National Natural Science Foundation of China (51836003, 11872024), National Key Research and Development Project of China No. 2018YFE0127800.