Opposite impact of thermal expansion and phonon anharmonicity on the phonon-limited resistivity of elemental metals from first principles

Understanding electrical resistivity in metals remains a central challenge in quantifying charge transport at finite temperature. Current first-principles calculations based on the Boltzmann transport equation often match experiments, yet they almost always neglect the effect of thermal expansion and phonon anharmonicity. We show that both effects exert an opposite impact on electron-phonon coupling and on electrical resistivity. Thermal expansion enhances the coupling and leads to overestimation of resistivity, whereas anharmonic effects reduce it. By explicitly incorporating both effects, we establish a more complete description of resistivity in elemental metals, demonstrated here for Pb, Nb, and Al.

💡 Research Summary

In this work the authors address a long‑standing gap in first‑principles predictions of the temperature‑dependent electrical resistivity of elemental metals. While modern Boltzmann‑transport calculations based on density‑functional theory (DFT) and electron‑phonon matrix elements have achieved impressive agreement with experiment, they almost universally neglect two temperature‑dependent phenomena that are intrinsic to real crystals: lattice thermal expansion (TE) and phonon anharmonicity. The paper demonstrates, through systematic calculations on lead (Pb), aluminum (Al) and niobium (Nb), that these two effects exert opposite influences on electron‑phonon coupling (EPC) and consequently on the resistivity ρ(T).

Methodologically the authors first compute the Helmholtz free energy F(V,T) as the sum of an electronic part (static DFT energy plus thermal electronic excitations) and a phononic part treated within the quasi‑harmonic approximation (QHA). By fitting the volume‑temperature dependence of F with a third‑order Birch‑Murnaghan equation they obtain temperature‑dependent lattice parameters, which encode the TE contribution. To capture anharmonicity they employ the temperature‑dependent effective potential (TDEP) method: stochastic sampling of atomic displacements generated from the canonical ensemble is used to extract temperature‑dependent second‑order force constants, which are then iterated until self‑consistency at each temperature. Both QHA and TDEP calculations are performed with the same DFT settings (Quantum ESPRESSO) as in earlier benchmark studies, and spin‑orbit coupling (SOC) is included for Pb.

With the temperature‑dependent crystal structures and anharmonic force constants in hand, the authors evaluate electron‑phonon matrix elements gₘₙᵥ(k,q) using density‑functional perturbation theory (DFPT) and Wannier interpolation (EPW, Wannier90). The linearized Boltzmann transport equation (BTE) is solved iteratively (IBTE) to obtain the electron scattering rates τ⁻¹(k) and the resistivity tensor ρ_αβ. Four computational scenarios are considered for each metal: (i) no TE, no anharmonicity (the conventional approach), (ii) TE only, (iii) anharmonicity only, and (iv) both TE and anharmonicity.

For Pb the results are striking. Including TE alone dramatically softens low‑frequency phonons, especially near the X point, leading to a 20 % increase in scattering rates at 310 K and a 92 % overestimation of ρ at 610 K relative to experiment. When anharmonicity is added, the softened modes harden, the scattering rates drop back toward the baseline, and the resistivity curve aligns almost perfectly with measured data across the whole 10–610 K range. Spectral decomposition of τ⁻¹ shows that TE concentrates scattering in the 2–4 meV phonon window, whereas anharmonicity shifts the dominant contribution to 3–5 meV, effectively canceling the TE‑induced excess.

Al exhibits the same antagonistic behavior: TE raises ρ, anharmonicity lowers it, and the combined calculation reproduces experimental values up to ~500 K. Above this temperature a modest underestimation remains, consistent with earlier variational‑approximation studies.

Nb, however, behaves differently because of its complex Fermi surface and a pronounced Kohn anomaly along Γ–H. While TE still softens phonons and tends to increase ρ, the anharmonic correction is larger than the TE effect, leading to a net reduction of resistivity. Consequently the two contributions do not cancel; the calculated ρ remains slightly below experiment at high temperature, a discrepancy the authors attribute to saturation effects and the highly dispersive nature of the phonon shifts in Nb. They argue that when TE‑induced volume changes and anharmonic‑induced temperature‑driven electronic screening affect different regions of the Brillouin zone, simple cancellation cannot be expected.

Overall, the study provides a clear microscopic picture: thermal expansion expands the lattice, reduces phonon frequencies, and enhances electron‑phonon scattering; phonon anharmonicity, by renormalizing the force constants at finite temperature, hardens the same modes and suppresses scattering. The balance between these mechanisms determines whether the resistivity is over‑ or under‑estimated. By explicitly incorporating both, the authors achieve quantitative agreement with experiment for all three metals, establishing a more complete first‑principles framework for transport in metals.

The implications are broad. Accurate high‑temperature resistivity predictions are essential for designing power electronics, thermoelectric devices, and for understanding failure mechanisms in metallic interconnects. The methodology can be extended to alloys, complex compounds, and to systems where strong electron‑phonon coupling coexists with other scattering channels (e.g., magnetic fluctuations). The work thus sets a new standard for predictive transport modeling in condensed‑matter physics.