Abstract
Calculations of electronic and optical properties of solids at finite temperature including electron-phonon interactions and quantum zero-point renormalization have enjoyed considerable progress during the past few years. Among the emerging methodologies in this area, we recently proposed a new approach to compute optical spectra at finite temperature including phonon-assisted quantum processes via a single supercell calculation [Zacharias and Giustino, Phys. Rev. B 94, 075125 (2016)]. In the present work we considerably expand the scope of our previous theory starting from a compact reciprocal space formulation, and we demonstrate that this improved approach provides accurate temperature-dependent band structures in three-dimensional and two-dimensional materials, using a special set of atomic displacements in a single supercell calculation. We also demonstrate that our special displacement reproduces the thermal ellipsoids obtained from X-ray crystallography, and yields accurate thermal averages of the mean-square atomic displacements. At a more fundamental level, we show that the special displacement represents an exact single-point approximant of an imaginary-time Feynman's path integral for the lattice dynamics. This enhanced version of the special displacement method enables non-perturbative, robust, and straightforward ab initio calculations of the electronic and optical properties of solids at finite temperature, and can easily be used as a post-processing step to any electronic structure code. Given its simplicity and numerical stability, the present development is suited for high-throughput calculations of band structures, quasiparticle corrections, optical spectra, and transport coefficients at finite temperature.
Highlights
The calculation of the electronic and optical properties of materials at finite temperature is a long-standing challenge for ab initio electronic structure methods
The core of the special displacement method described in this paper is to identify one set of atomic displacements so that a single evaluation of
The agreement between our calculation and experiment is very good, except that we underestimate slightly the temperature slope. This effect is a well-known consequence of the fact that the strength of the electron-phonon interaction is underestimated by density functional theory (DFT)/local density approximation (LDA); the slope can be improved by using GW calculations in combination with the special displacement method (SDM), as demonstrated in Ref. [23]
Summary
The calculation of the electronic and optical properties of materials at finite temperature is a long-standing challenge for ab initio electronic structure methods. The matrix elements are employed to obtain temperature-dependent band structures in the Allen-Heine (AH) method [31,32], and to compute indirect optical absorption in the Hall, Bardeen, and Blatt theory [33,34] These approaches have enjoyed considerable success during the past decade across a broad range of materials [3,4,5,7,8,9,10,11,12,13,14,24,35,36,37,38]. Phonon wave vectors that coincide with their time-reversal partners are grouped in a finite set A, and their contribution to the atomic displacements vanishes in the limit of dense Brillouin-zone sampling.
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