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Low-temperature behavior of density-functional theory for metals based on density-functional perturbation theory and Sommerfeld expansion

Xavier Gonze, Christian Tantardini, Antoine Levitt

TL;DR

This work develops a first-principles framework to treat the electronic temperature response of metals within density-functional perturbation theory by viewing temperature as a perturbation around a reference state. The authors couple bare thermal density changes to self-consistent screening via the electronic dielectric operator, incorporating Adler–Wiser and Fermi-level shift contributions, and validate the approach with aluminum in DFTK. They extend the formalism to the $T\to 0$ limit through the Sommerfeld expansion, deriving explicit low-temperature formulas for density, energy, and entropy, and analyze how Van Hove singularities alter the leading temperature dependence. The resulting low-temperature DFT framework, valid for 1D–3D periodic metals, provides a basis for studying temperature-dependent electronic instabilities (e.g., charge-density waves) within DFT/DFPT and suggests pathways for broader investigations of purely electronic low-temperature phenomena in metals.

Abstract

The temperature dependence of most solid-state properties is dominated by lattice vibrations, but metals display notable purely electronic effects at low temperature, such as the linear specific heat and the linear entropy, that were derived by Sommerfeld for the non-interacting electron gas via the low-temperature expansion of Fermi-Dirac integrals. Here we treat temperature as a perturbation within density-functional perturbation theory (DFPT). For finite temperature, we show how self-consistency screens the bare, temperature-induced density change obtained in the non-interacting picture: the inverse transpose of the electronic dielectric operator, that includes Adler-Wiser and a term related to the shift in Fermi level, links the self-consistent density response to the bare thermal density change. This approach is implemented in DFTK, and demonstrated by the computation of the second-order derivative of the free energy, and the first-order derivative of entropy for aluminum. Then, we examine the $T\!\to\!0$ limit. The finite temperature formalism contains divergences, that we cure using the Sommerfeld expansion to analyze metallic systems at 0 K. The electronic free energy is quadratic in $T$ provided the Fermi level is not at a Van Hove singularity of the density of states. If the latter happens, another temperature behavior might appear, depending on the type of Van Hove singularity, that we analyze. Our formulation applies to systems periodic in one, two, or three dimensions, and provides a basis for studying temperature-dependent electronic instabilities (e.g., charge-density waves) within density-functional theory and DFPT.

Low-temperature behavior of density-functional theory for metals based on density-functional perturbation theory and Sommerfeld expansion

TL;DR

This work develops a first-principles framework to treat the electronic temperature response of metals within density-functional perturbation theory by viewing temperature as a perturbation around a reference state. The authors couple bare thermal density changes to self-consistent screening via the electronic dielectric operator, incorporating Adler–Wiser and Fermi-level shift contributions, and validate the approach with aluminum in DFTK. They extend the formalism to the limit through the Sommerfeld expansion, deriving explicit low-temperature formulas for density, energy, and entropy, and analyze how Van Hove singularities alter the leading temperature dependence. The resulting low-temperature DFT framework, valid for 1D–3D periodic metals, provides a basis for studying temperature-dependent electronic instabilities (e.g., charge-density waves) within DFT/DFPT and suggests pathways for broader investigations of purely electronic low-temperature phenomena in metals.

Abstract

The temperature dependence of most solid-state properties is dominated by lattice vibrations, but metals display notable purely electronic effects at low temperature, such as the linear specific heat and the linear entropy, that were derived by Sommerfeld for the non-interacting electron gas via the low-temperature expansion of Fermi-Dirac integrals. Here we treat temperature as a perturbation within density-functional perturbation theory (DFPT). For finite temperature, we show how self-consistency screens the bare, temperature-induced density change obtained in the non-interacting picture: the inverse transpose of the electronic dielectric operator, that includes Adler-Wiser and a term related to the shift in Fermi level, links the self-consistent density response to the bare thermal density change. This approach is implemented in DFTK, and demonstrated by the computation of the second-order derivative of the free energy, and the first-order derivative of entropy for aluminum. Then, we examine the limit. The finite temperature formalism contains divergences, that we cure using the Sommerfeld expansion to analyze metallic systems at 0 K. The electronic free energy is quadratic in provided the Fermi level is not at a Van Hove singularity of the density of states. If the latter happens, another temperature behavior might appear, depending on the type of Van Hove singularity, that we analyze. Our formulation applies to systems periodic in one, two, or three dimensions, and provides a basis for studying temperature-dependent electronic instabilities (e.g., charge-density waves) within density-functional theory and DFPT.

Paper Structure

This paper contains 11 sections, 1 theorem, 92 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Density-functional theory for metallic periodic solids
  3. DFPT : changing the temperature at finite temperature
  4. Validation of the DFPT treatment of a temperature change
  5. The Sommerfeld expansion
  6. Low-temperature free electron gas
  7. Convergence of the Sommerfeld expansion
  8. Beyond the free electron gas
  9. Van Hove singularities
  10. Low-temperature density-functional theory for metals based on the Sommerfeld expansion
  11. Conclusion

Key Result

Theorem 1

Let $f:\mathbb{R}\to \mathbb{C}$ be $C^{\infty}$ and $\mu \in \mathbb R$, and consider the formal Sommerfeld series where and $h^{(2n-1)}$ is the $2n-1$ derivative of the function $h$ evaluated at $\mu$. Then $Somm(T)$ has a nonzero radius of convergence as a power series in $T$ if and only if the odd part of $h$ about $\mu$, $h_{o}(\mu+x) = \tfrac{1}{2} (h(\mu+x)-h(\mu-x))$ extends to an entire

Figures (2)

  • Figure 1: The free energy per unit cell of aluminium as a function of Fermi-Dirac temperature, taking as reference the (extrapolated) $T=0$ result. The branches of parabolas are computed using free-energy values, and first- and second- derivative information obtained from the method described in the text. Quantities are given in atomic unit, with k$_B$ taken equal to 1. The conversion factor to Kelvin is 0.010Ha=3157.77K, corresponding to the highest temperature shown.
  • Figure 2: The entropy per unit cell of aluminium as a function of Fermi-Dirac temperature. The segments are computed using the first derivative information obtained from the method described in the text. Quantities are given in atomic unit, with k$_B$ taken equal to 1. The conversion factor to Kelvin is 0.010Ha=3157.77K, corresponding to the highest temperature shown.

Theorems & Definitions (1)

  • Theorem 1