Electron-Phonon interaction and lattice thermal conductivity from metals to 2D Dirac crystals: a review

TL;DR

This review synthesizes how electron–phonon (e–ph) coupling governs lattice and electronic heat transport from bulk metals to 2D Dirac crystals. It highlights state‑of‑the‑art first‑principles workflows (DFT/DFPT, Wannier interpolation, and e–ph matrix elements $g_{mn}^{\nu}(\mathbf{k},\mathbf{q})$) and coupled electron–phonon Boltzmann transport solvers (e.g., elphbolt) that capture drag, ultrafast non‑equilibrium, and Onsager reciprocity without empirical parameters. In metals, e–ph damping can reduce lattice thermal conductivity by up to ~40% and significantly alter phonon lifetimes; in semiconductors, coupled BTE treatments reproduce experimental benchmarks and reveal how carrier density and dielectric environment tune heat transport. In 2D Dirac crystals like graphene, symmetry, strain, and finite size reorganize the scattering hierarchy, with ZA modes playing a major role only when symmetry is broken, and higher‑order four‑particle processes becoming necessary at low $E_{F}$ and high $T$. The work outlines open challenges—including fully coupled, beyond‑DFT, time‑dependent, and moiré–system frameworks—that could deliver predictive, parameter‑free control of heat and charge transport for next‑generation electronic and photonic devices.

Abstract

Electron--phonon (e--ph) coupling governs electrical resistivity, hot-carrier cooling, and critically, thermal transport in solids. Recent first-principles advances now predict e--ph limited thermal conductivity from d-band metals and wide-band-gap semiconductors to 2D Dirac crystals without empirical parameters. In bulk metals, ab-initio lifetimes show that phonons, though secondary, still carry up to 40\% of the heat once e--ph scattering is included. We next survey coupled Boltzmann frameworks, exemplified by \textsc{elphbolt}, that capture mutual drag and ultrafast non-equilibrium in semiconductors. For 2D Dirac crystals, mirror symmetry, carrier density, strain, and finite size rearrange the scattering hierarchy: ZA modes dominate pristine graphene yet become the main resistive branch in nanoribbons once symmetry is broken. At low Fermi energies and high temperatures, the standard 3-particle decay is partially cancelled, elevating 4-particle processes and necessitating dynamically screened, higher-order theory. Throughout, we identify the microscopic levers such as the electronic density of states, phonon frequency, deformation potential, and show how doping, strain, or dielectric environment can tune e--ph damping. We conclude by outlining open challenges such as: developing coupled e--ph solvers, solving the full mode-to-mode Peierls--Boltzmann equation with 4-particle terms, embedding correlated electron methods in e--ph workflows, and leveraging higher-order e--ph coupling and symmetry breaking to realise phononic thermal diodes and rectifiers. Solving these challenges will elevate e--ph theory from a diagnostic tool to a predictive, parameter-free platform that links symmetry, screening, and many-body effects to heat and charge transport in next-generation electronic, photonic, and thermoelectric devices.

Figures (9)

• Figure 1: (a) Phonon vs. electron share of $\kappa_{total}$ at 300 K: Phonons contribute $<10\%$ of $\kappa_{\mathrm{tot}}$ in noble/alkali/NIC metals but $10$–$40\%$ in transition/TIC metals (even though $\kappa_{\mathrm{ph}} \approx 3$–$15\ \mathrm{W\,m^{-1}\,K^{-1}}$). Higher $\sigma$ in nobles and stronger e-ph coupling in transitions explain the contrast. (b) Mean free paths at 300 K: Average mean free paths (MFPs) at $50\%$ accumulation of thermal conductivity show phonons $\lesssim 10\ \mathrm{nm}$ for all 18 metals, while electrons are $\sim 5$–$25\ \mathrm{nm}$; electron MFPs generally exceed phonon MFPs, implying a stronger size effect in $\kappa_{\mathrm e}$ for metal nanostructures. Figures taken from Tong et al.tong2019comprehensive
• Figure 2: Interpolated e-ph matrix elements for (a), (b) silicon and (c), (d) diamond, for different coarse grids. We observe the accuracy of e–ph matrix-element interpolation. Both AO and WF schemes reproduce direct-DFPT $|g_{mn}^{\nu}|$ along high-symmetry lines within numerical precision, validating the coarse$\to$dense interpolation. Figures taken from Agapito et al.Agapito2018_PRB
• Figure 3: (a) In monolayer MoS$_2$ at 300 K, phonon-limited electronic mean free paths cluster in the few–10 nm range (often $<\!10$ nm), illustrating the strong size/confinement sensitivity and providing context for the 2D transport lengths discussed in this subsection. Figure taken from Li et al.Li2015_PRB. (b) Lattice thermal conductivity of Si vs temperature at low and high doping from the fully coupled e–ph BTE agrees with experiment Inyushkin et al.inyushkin2004isotope and shows stronger low-$T$ suppression from e-ph scattering. At room temperature ph–ph scattering dominates and coupled/uncoupled predictions converge. Figure taken from Protik et al.Protik2022_npj
• Figure 4: (a) The dielectric response function $\chi$ vs $\Psi=q/2\mathrm{k}_{\mathrm{F}}\approx1$ for different 2D Dirac crystals with different $\mathrm{w_{F}} = c/\mathrm{v}_{\mathrm{F}}$ at T = 0 K. We observe that the dielectric response function diverges near the FSN at the point $\Psi\approx1$ even for small values of $\mathrm{w_{F}}\ne0$. The negligible difference in the dielectric response function of various 2D Dirac crystals with different values of $\mathrm{w_{F}}$ shows the generality of our solution for 2D Dirac crystals. (b) Comparing the dielectric response function for the static case $c=0$, designated with $\chi_{0}(\Psi)$, of a 2D Dirac crystal with that of a 1D and 2D Fermion gas at T = 0 K. We observe that $\chi_{0}(\Psi)$ of a 2D Dirac crystal does not follow a constant line like the 2D Fermi gas and it increases reaching a maximum at the FSN point, making the study of 2D Dirac crystals more intricate. Figure taken from Kazemian et al.kazemian2023dynamic.
• Figure 5: (a) Branch-resolved lattice thermal conductivity of suspended monolayer graphene from first-principles BTE. The ZA branch dominates over the full T-range ($\approx 76 \%$ at 300 K). Experimental data over 9.7-µm holes has been shown for comparison. Figure taken from Lindsay et al.Lindsay2014_PRB. (b) Phonon dispersion under uniaxial tensile strain showing linearization of the ZA branch at small $q$ (and corresponding DOS change), which underpins the strain-enhanced $\kappa_{\mathrm ph}$ in suspended graphene. Figure taken from Kuang et al.Kuang2016_IJHMT.