Onsager reciprocity relation for ballistic phonon heat transport in anisotropic thin films of arbitrary orientation

A classic Onsager reciprocity relation for Fourier heat conduction in the absence of magnetic fields states that the thermal conductivity tensor in bulk anisotropic solids is symmetric. However, since Fourier's law fails in thin dielectric films due to ballistic phonon transport effects, it is natural to ask whether an analogous Onsager relation can be identified in the boundary scattering regime. To answer this question, we solve the Boltzmann transport equation (BTE) under the relaxation time approximation for in-plane and cross-plane heat transport for thin films with anisotropic phonon dispersion relations and scattering rates. We use these BTE solutions to show that the thermal conductivity tensor of thin films is symmetric from the diffusive regime through the boundary scattering regime. We illustrate this reciprocity by calculating thermal conductivity suppression functions for a model anisotropic material. We compare our BTE solution to previous atomistic simulations of arbitrarily aligned graphite thin films, and use published first-principles calculations to model anisotropic in-plane heat flow in aligned black phosphorus. Our derivation shows how Onsager reciprocity for anisotropic heat conduction extends into the boundary scattering regime, and reduces the number of independent measurements required to fully characterize heat transport in anisotropic thin films.

Figure 1

Schematic of anisotropic heat transfer in thin films due to (a) cross-plane and (b) in-plane temperature differences $\left(T_{\mathrm{h}}-T_{\mathrm{c}}\right)$. In arbitrarily aligned anisotropic materials, the heat flux $\mathbf{q}$ is not necessarily antiparallel to the temperature gradient $\nabla T$, as represented by nonorthogonal adiabats and isotherms in (a), (b) and mathematically described by off-diagonal components of the thermal conductivity tensor $\mathbf{\kappa }$. We use Boltzmann transport equation solutions to prove that $\mathbf{\kappa }$ remains symmetric from the bulk through the thin-film boundary scattering regime: for example, we show that the off-diagonal thermal conductivities ${\kappa }_{xy}$ and ${\kappa }_{yx}$ are equal. Here, ${\kappa }_{xy}$ is the ratio of the heat flow in $x$ per unit depth $Q_x^{\prime }$ to the imposed temperature difference in $y$ [as in (a)], and ${\kappa }_{yx}$ is the ratio of the heat flow in $y$ per unit depth $Q_y^{\prime }$ due to the temperature difference in $x$ [as in (b)].

Figure 2

A numerical demonstration of the reciprocity relation ${\kappa }_{xy}={\kappa }_{yx}$ (a) Reciprocal-space schematic of the isofrequency ellipsoid for a material with an anisotropic Debye dispersion relation [Eq. (18)). The $\widehat{c}$ axis of the material is tilted by an angle $\mathrm{\Psi }$ with respect to the $\widehat{y}$ direction of the film. For simplicity we consider a constant relaxation time $\tau$, high temperatures, and a spherical first Brillouin zone. (b) According to the reciprocity relation ${\kappa }_{xy}={\kappa }_{yx}$ [Eq. (17)], the off-diagonal thermal conductivity suppression functions $S_{ij}={\kappa }_{ij}/{\kappa }_{ij,\mathrm{bulk}}$ are supposed to be equal $(S_{xy}=S_{yx})$ for all values of the dimensionless mean free path ${\lambda }_c$ from the diffusive $({\lambda }_c\ll 1)$ through the boundary scattering $({\lambda }_c\gg 1)$ regimes. Here this $S_{xy}=S_{yx}$ equality is verified numerically using Eqs. (13) and (14) for the particular case of $\mathrm{\Psi }={30}^o$ and three values of the group velocity anisotropy ratio $v_{ab}/v_c$.

Figure 3

Comparing the analytical BTE solutions to non-equilibrium molecular dynamics (NEMD) simulations of arbitrarily aligned graphite [35]. The NEMD simulations were performed for 24 combinations of basal-plane alignment angles $\mathrm{\Psi }$ (see Fig. 2) and film thickness $t$. The BTE model for highly anisotropic layered materials agrees well with the numerical results using a gray mean free path $\mathrm{\Lambda }=103\mathrm{nm}$.

Figure 4

Thin-film boundary scattering reduces the off-diagonal in-plane thermal conductivity ${\kappa }_{zx}$ of black phosphorus. The BTE solutions show that ${\kappa }_{zx}$ can be determined for a given in-plane temperature gradient rotation angle $\theta$ using simple tensor transformation identities, even in the boundary scattering regime. We use recent first-principles calculations [6] of thickness-dependent $\kappa$ along the zigzag and armchair directions at room temperature to calculate ${\kappa }_{zx}(\theta ,t)$ for aligned black phosphorus thin films $\left(\mathrm{\Psi }=0\right)$.