Abstract |
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The renormalization of electronic eigenenergies due to electron-phonon interactions (temperature dependence and zero-point motion effect) is important in many materials. We study the Allen-Heine-Cardona (AHC) theory of the renormalization in a first-principle context. The AHC theory relies on the rigid-ion approximation, and naturally leads to a self-energy (Fan) contribution and a Debye-Waller contribution. The adiabatic AHC theory allows for an easy computation of the effect of electron-phonon interactions but cannot be used in the case of polar materials. Indeed, the Born effective charge present in these materials leads to a 1/q diverging behavior of the electron-phonon coupling matrix elements, for small q, where q is the norm of the phonon wave vector. However, such divergence is also observed in non-polar materials when the Born effective charge neutrality sum rule is broken. As it turns out, accurate evaluation of the Born effective charge requires a dense sampling of the electronic wavevectors in the Brillouin Zone and is a bottleneck is AHC computations. We here present a scheme to renormalize the perturbed Hartree potential and restore the charge neutrality. Such renormalization allows for a significant speed-up with respect to the electronic wavevector sampling (k-points). For example, in diamond, 216 k-points in the BZ are only required with the renormalization instead of 1000 k-points without. To avoid numerical instabilities, an ad-hoc small imaginary component is often added in the denominator of the GKKs. To have a parameter free and well defined theory, we also propose a fitting scheme to systematically extrapolate the zero-point motion renormalization and temperature-dependent corrections of the electronic structure to an infinite sampling of the phonon wavevector Brillouin Zone, and zero imaginary parameters. Finally, using this fitting scheme, we will present the temperature-dependence of the eigenenergies for Diamond and Silicon (non-polar materials) as well as $alpha$-Aluminum Nitride, $eta$-Aluminium Nitride and Boron Nitride (polar materials) within the rigid-ion approximation. We show that only the non-adiabatic extension to AHC can be used for polar materials. References [1] S. Poncé et al., Comp. Mater. Sci. 83, 341 (2014). [2] G. Antonius et al., Phys. Rev. Lett. 112, 215501 (2014). [3] S. Poncé et al., Phys. Rev. B 90, 214304 (2014). [4] A. Marini et al., Phys. Rev. B (submitted) (2015) [arXiv:1503.00567]. [5] S. Poncé et al., J. Chem. Phys. (submitted) (2015) [arXiv:1504.05992]. |