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

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 $\text{textsc}elphbolt$, 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.