Transition from Population to Coherence-dominated Non-diffusive Thermal Transport

TL;DR

This work extends the phonon Wigner Transport Equation to include space-time dependent heat sources and phonon coherences arising from inter-branch tunneling, enabling the study of non-diffusive transport in low-κ crystals. By solving a linearized WTE with a relaxation-time collision term and a gradient-drive, the authors compute the dynamical thermal conductivity and introduce an energy-conservation closure for arbitrary sources. First-principles data for CsPbBr3 and La2Zr2O7 reveal large coherence contributions and pronounced size effects, with deviations from bulk diffusivity occurring at micron-to-submicron scales even at room temperature. The GreenWTE framework, offering both Green's-function and direct-solver options, provides a practical tool to connect ultrafast pump-probe experiments to non-diffusive phonon transport.

Abstract

Deviations from diffusive heat transport in high thermal conductivity crystalline insulators are generally understood within the framework of the phonon Boltzmann Transport Equation. However, for low thermal conductivity materials with large primitive cells or strong anharmonicity, the recently developed Wigner Transport Equation is more appropriate as it includes tunnelling between overlapping phonon bands. In this work, via solutions to the Wigner Transport Equation, we develop a scheme to obtain the dynamics of the phonon populations and coherences as a function of an arbitrary heat source. The approach is applied to predict size effects and dynamical thermal conductivities in CsPbBr3 and La2Zr2O7 using first-principles data as input. We predict significant deviations from the bulk thermal conductivity in these materials at length scales on the order of hundreds of nanometers to a few microns at room temperature, well within the reach of direct observation using current experimental techniques.

Figures (3)

• Figure 1: a Thermal conductivity of silicon (black), CsPbBr$_3$ (blue) and La$_2$Zr$_2$O$_7$ (red) computed with the source term $\tilde{\bm{\mathsf{Q}}}_{\nabla T}$. The contribution to the overall thermal conductivity (round markers) from populations and coherences is shown with triangles pointing up and down, respectively. The thermal conductivity of silicon is scaled by a factor of 0.01 for visual clarity. Experimental reference data glassbrenner1964thermal is shown with green crosses. b Thermal conductivity of La$_2$Zr$_2$O$_7$ as a function of grating frequency at 300 and 700 K, in blue and red respectively. c Dynamical thermal conductivity of La$_2$Zr$_2$O$_7$ at 500 K for two different grating periods. Large markers denote the real part and smaller markers connected with a dashed line denote the imaginary part of the thermal conductivity. Note that $\mathrm{Re}[\kappa_\mathrm{C}]$ for both grating sizes fully overlap and that $\mathrm{Im}[\kappa_\mathrm{C}]$ are negligible.
• Figure 2: a,b Two contour plots that sample the transport character $\chi(T,k,\omega)$ of La$_2$Zr$_2$O$_7$ at two different planes. The thick black contours indicate equal contributions from populations and coherences ($\chi=0$). a shows the plane for the static case. b shows the plane for a temperature of 500 K. The intersecting line of the planes is represented by a line that starts with an open circle and ends with a solid circle in a and b. c Phonon dispersion of La$_2$Zr$_2$O$_7$ along the $\overline{\mathrm{K}\Gamma\mathrm{L}}$ high-symmetry path. Encoded in the color of the phonon modes is the change in transport character $\Delta\chi$ between the static and the high-frequency case. The thickness of each phonon branch is proportional to the square root of its model contribution to the total thermal conductivity, i.e. line width $\propto\sqrt{\kappa_{\bm{q} s}}$. Note that the energy axis is truncated at 17 THz, because the high-energy optical phonons show no changes in $\chi$.
• Figure 3: Phonon dispersion of La$_2$Zr$_2$O$_7$ (CsPbBr$_3$) along the $\overline{\mathrm{K}\Gamma\mathrm{L}}$ ($\overline{\mathrm{S}\Gamma\mathrm{X}}$) high-symmetry path. Encoded in the color of the phonon modes is the transport character $\chi$ in the static case for a macroscopic grating period (a, b (d,e)) and at $\omega$=31(87) GHz (c(f)). The thickness of each phonon branch is proportional to the square root of its model contribution to the total thermal conductivity, i.e. line width $\propto\sqrt{\kappa_{\bm{q} s}}$. Panel a,d shows the full energy range, while panel b,c,e,f is a zoom into the low-energy part of the spectrum.