Theory of thermal expansion: Quasi-harmonic approximation and corrections from quasi-particle renormalization

Abstract

“Quasi-harmonic” (QH) theory should not be considered a low-order theory of anharmonic effects in crystals, but should be recognized as an important effect separate from “true” anharmonicity. The original and widely used meaning of QH theory is to put [Formula: see text] volume-dependent harmonic phonon energies [Formula: see text] into the non-interacting phonon free energy. This paper uses that meaning, but extends it to include the use of [Formula: see text] [Formula: see text]-dependent single-particle electron energies [Formula: see text]. It is demonstrated that the “bare” quasi-particle (QP) energies [Formula: see text] and [Formula: see text] correctly give the first-order term in the [Formula: see text]-dependence of the Helmholtz free energy [Formula: see text]. Therefore, they give the leading order result for thermal expansion [Formula: see text] and for the temperature-dependence of the bulk modulus [Formula: see text]. However, neglected interactions which shift and broaden [Formula: see text] with [Formula: see text], also shift the free energy. In metals, the low [Formula: see text] electron–phonon mass enhancement of states near the Fermi level causes a shift in free energy that is similar in size to the electronic QH term. Before [Formula: see text] reaches the Debye temperature [Formula: see text], the mass renormalization essentially disappears, and remaining electron–phonon shifts of free energy contribute only higher-order terms to thermal expansion. Similarly, anharmonic phonon–phonon interactions shift the free energy, but contribute to thermal expansion only in higher order. Explicit next order formulas are given for thermal expansion, which relate “true” anharmonic and similar free energy corrections to QP self-energy shifts.