Phonon collisional broadening and heat transport beyond the Boltzmann equation

Abstract

In crystals, macroscopic technological properties such as thermal conductivity originate from the microscopic drift and scattering of phonons, commonly described by the Boltzmann Transport Equation (BTE). Despite its widespread use, the most general space-time nonlocal form of the BTE still lacks a rigorous derivation of its collisional part based on Fermi's Golden Rule (FGR), and becomes inadequate in several regimes, including when the energy-variation scale set by phonon dispersion approaches that of collisional broadening. A hallmark of this issue is the poor numerical convergence of conductivity with respect to the smearing used to evaluate FGR rates. This is often circumvented using adaptive schemes, which however violate detailed balance and allow unphysical negative eigenvalues in the collision operator. Here, we overcome these limitations by rigorously deriving the space-time-dependent BTE from the Kadanoff-Baym Equations (KBE), and introduce a linearized generalized BTE (LGBTE) that goes beyond the FGR framework, incorporating self-consistent, physically derived, fully anharmonic, and mode-resolved collisional broadening and energy-nonconserving scattering. More generally, we establish a hierarchy of ansätze on Green's functions, enabling controlled extensions of the semiclassical BTE and a roadmap toward quantum KBE accuracy. Finally, using first-principles simulations complemented by analytical arguments, we show that this approach addresses two long-standing problems of the FGR-based linearized BTE across crystal dimensionalities: (i) the lack of conductivity convergence, common to heat conductors like diamond; and (ii) its universal failure in all 2D systems, rooted in FGR predicting an unphysical overdamping for scattering channels involving flexural vibrations, as shown in the insulating α-GeSe monolayer.

Figures (12)

• Figure 1: Computational protocol for evaluating the self-consistent collisional broadening or phonon linewidths. $\Gamma_{\nu}$s are the computed linewidths, $\gamma_{\nu}$s are collisional broadening of Lorentzian spectral functions, while $\Delta$ represents convergence threshold for each phonon $\nu=\boldsymbol{q}s$.
• Figure 2: Crystal structure of orthorhombic monolayer $\alpha$-GeSe. (a,b) Side views highlighting the puckered (hinge-like) geometry characteristic of group IV–VI compounds. (c) Top view with the in-plane zigzag and armchair directions indicated; the lengths of the arrows are drawn proportional to the relative anisotropy of the lattice thermal conductivity along the two directions. (d) Perspective view of monolayer $\alpha$-GeSe. Ge atoms are shown in lavender and Se atoms in lime green.
• Figure 3: Room-temperature lattice thermal conductivity as a function of the smearing parameter for Gaussian (green) and Lorentzian (teal) numerical broadening schemes. (a) In-plane lattice thermal conductivity along the zigzag direction of $\alpha$-GeSe monolayer. Very small smearing values lead to strongly overdamped phonon lifetimes and, consequently, to unphysically low thermal conductivity. Even within the commonly used smearing range of $10^{-1}$--$10^{0}$ cm$^{-1}$, a variation of approximately 32% is observed. (b) Lattice thermal conductivity of bulk diamond. Both the overall trend and the zoomed inset, focusing on the smearing range $10^{0}$--$10^{1}$ cm$^{-1}$, demonstrate that convergence with respect to the smearing parameter is never achieved. The result obtained with the present Lorentzian self-consistent collisional broadening (SCF) approach is shown as a black dashed line and is independent of the smearing.
• Figure 4: Difference between consecutive iterations of the room-temperature Lorentzian phonon broadening (linewidths) for monolayer $\alpha$-GeSe (a) and bulk diamond (b). The results are obtained following the self-consistent procedure outlined in Fig. \ref{['fig:scf_broadening']}, starting from an initial Lorentzian numerical smearing of $10^{-8}$ cm$^{-1}$ for monolayer $\alpha$-GeSe and $8$ cm$^{-1}$ for bulk diamond. Owing to these markedly different initial smearings, the linewidths of bulk diamond clearly exhibit a faster convergence behavior. The relative percentage difference in linewidths between consecutive iterations, defined as $\Delta\Gamma_{\nu}^{i}$ = $\left|\Gamma_{\nu}^{i}-\Gamma_{\nu}^{i-1}\right|/\Gamma_{\nu}^{i-1}$, is shown for selected iteration pairs: 1--0 (first), 5--4, 11--10, 20--19, and 28--27 (last) for monolayer $\alpha$-GeSe, and 1--0 (first), 2--1, 3--2, 11--10, and 20--19 (last) for bulk diamond, clearly demonstrating the achievement of self-consistency. Importantly, the largest impact of collisional broadening (the largest value of $\Delta\Gamma_{\nu}^{i}$, highlighted by the red arrows) does not systematically occur in the acoustic phonon branches, despite their higher group velocities. This demonstrates that there is no strict correlation with the phonon energy scale $W_{\nu}$ used in the color bar. Instead, the self-consistent collisional broadening captures anharmonic effects beyond those described such adaptive smearing schemes. For visualization, $W_{\nu}$ is linearly normalized to the interval $[0,1]$ as $(W_{\nu}-W_{\nu}^{\mathrm{min}})/(W_{\nu}^{\mathrm{max}}-W_{\nu}^{\mathrm{min}})$. As self-consistency is approached, $\Delta\Gamma_{\nu}^{i}$$\to$ 0, reflecting the convergence of the phonon linewidths and, correspondingly, of the lattice thermal conductivity.
• Figure 5: Self-consistent iterations of the room-temperature lattice thermal conductivity of monolayer $\alpha$-GeSe (zigzag and armchair directions in panels a and b, respectively) and bulk diamond (c). For the zigzag lattice thermal conductivity of monolayer $\alpha$-GeSe, we report the self-consistent evolution starting from different initial smearing values, namely the same ones sampled in Fig. \ref{['fig:kappa_vs_smearing']}a, in order to explicitly demonstrate that the final self-consistent conductivity is independent of the initial smearing and therefore free from any computational smearing bias. The armchair direction of monolayer $\alpha$-GeSe exhibits an analogous behavior; therefore, we only show the representative case corresponding to $\gamma_{\rm{start}} = 10^{-8}$ cm$^{-1}$. For bulk diamond, we adopt $\gamma_{\rm{start}} = 8$ cm$^{-1}$, leading to a faster convergence because the initial value is physically more realistic. The convergence of the lattice thermal conductivity clearly mirrors the convergence behavior of the linewidths shown in Fig. \ref{['fig:3D_percent_gamma_diff']}.