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, heat flow, 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 two-dimensional (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 ELPHBOLT, that capture mutual drag and ultrafast non-equilibrium in semiconductors; their results for Si, GaAs, and MoS 2 match the Time-Domain Thermo-Reflectance (TDTR) and isotope-controlled data within experimental error. For 2D Dirac crystals, mirror symmetry, carrier density, strain, and finite size rearrange the scattering hierarchy: flexural (ZA) modes dominate pristine graphene yet become the main resistive branch in nanoribbons once σ h symmetry is broken. At low Fermi energies where E F << k B T , the standard three-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 Fröhlich coupling, and show how doping, strain, or dielectric environment can tune e-ph damping. We conclude by outlining open challenges such as: developing femtosecond-resolved, coupled e-ph solvers, solving the full mode-to-mode Peierls-Boltzmann equation with four-particle terms, embedding correlated-electron methods (GW, dynamical mean-field theory, hybrid functionals) in e-ph workflows, implementing fully non-local, frequency-dependent screening for van-der-Waals stacks, and leveraging higher-order e-ph coupling and symmetry breaking to realise phononic thermal diodes and rectifiers. Solving these challenges will elevate electron-phonon 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.