Dynamical and anharmonic effects on the electron-phonon coupling and the zero-point renormalization of the electronic structure

G. Antonius, S. Poncé, E. Lantagne-Hurtubise, G. Auclair, X. Gonze, and M. Côté

Phys. Rev. B 92, 085137 – Published 21 August 2015

Abstract

The renormalization of the band structure at zero temperature due to electron-phonon coupling is explored in diamond, BN, LiF, and MgO crystals. We implement a dynamical scheme to compute the frequency-dependent self-energy and the resulting quasiparticle electronic structure. Our calculations reveal the presence of a satellite band below the Fermi level of LiF and MgO. We show that the renormalization factor $(Z)$ , which is neglected in the adiabatic approximation, can reduce the zero-point renormalization (ZPR) by as much as $40 %$ . Anharmonic effects in the renormalized eigenvalues at finite atomic displacements are explored with the frozen-phonon method. We use a nonperturbative expression for the ZPR, going beyond the Allen-Heine-Cardona theory. Our results indicate that high-order electron-phonon coupling terms contribute significantly to the zero-point renormalization for certain materials.

Received 7 May 2015

DOI:https://doi.org/10.1103/PhysRevB.92.085137

Authors & Affiliations

G. Antonius^1,*, S. Poncé², E. Lantagne-Hurtubise¹, G. Auclair¹, X. Gonze², and M. Côté¹

¹Département de physique, Université de Montréal, C.P. 6128, Succursale Centre-Ville, Montréal, Canada H3C 3J7
²Institute of Condensed Matter and Nanosciences, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium

^*gabriel.antonius@gmail.com

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Issue

Vol. 92, Iss. 8 — 15 August 2015

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Images

Figure 1
(Top) Real and imaginary parts of the self-energy for the top of the valence bands (VB) of LiF. The vertical line indicates the bare eigenvalue, the $x = y$ line gives the renormalized eigenvalue at the intersection of the real part of the self-energy, and the short-dashed line is the linearized self-energy, which approximates the renormalized eigenvalue at the intersection of the $x = y$ line. (Bottom) The corresponding spectral function. The position of the principal peak gives the quasiparticle energy. This narrow peak collects $\sim 60 %$ of the weight, and the rest of the charge forms a broad satellite peak below the bare eigenvalue. Note that the finite width of this peak is an artifact of the imaginary parameter used ( $0.1$ eV).
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Figure 2
Spectral functions summed over the bands at each $k$ point of the Brillouin zone (arbitrary units). The green lines are the DFT band structure, in eV. When a quasiparticle peak is visible in the spectral function, the renormalization is inferred from the difference of the position of that peak with the bare band structure. In the regions of flat bands, the band structure is being completely diffused. A satellite peak is seen below the last valence band of LiF and MgO.
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Figure 3
Some of the high-order electron-phonon coupling diagrams that contribute to the self-energy. Each vertex with $n$ phonon branches is associated with an $n th$ -order derivative of the one-particle Hamiltonian. The independent-phonon approximation presented in the text retains only the diagrams formed with multiple interactions involving the same phonon mode.
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Figure 4
Dependence of the CB eigenvalue (green, circles) and the total energy (blue, squares) on a phonon displacement for the mode $Ω_{4}$ of diamond. The circles and squares are the actual frozen-phonon calculations, the solid lines correspond to the harmonic approximation, and the filled curve is the phonon wave function.
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Figure 5
(Top) The band structure of diamond, in eV. The dashed lines show the energies of the VB and CB states. (Middle and bottom) Electron-phonon coupling energies (EPCE), in eV for the CB state (middle) and the VB state (bottom), computed with various methods. The blue line is the DFPT calculation, the yellow discs are the frozen-phonon method in the harmonic approximation, and the green triangles are the frozen-phonon method including anharmonic effects. A divergence is observed in the EPCEs of the CB state when a phonon wave vector couples this state to another one with close energy, while the EPCEs of both VB and CB states show a broad diverging peak at the center of the Brillouin zone.
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Figure 6
(Top) Convergence of the self-energy at the renormalized eigenvalue for the CB state of diamond, as a function of $q$ -point spacing, with various imaginary parameters. (Bottom) Frequency dependence of the self-energy for the CB state of diamond, with various imaginary parameters. The solid lines are obtained on a $32 \times 32 \times 32 q$ -point grid, and the dashed lines are obtained on a $24 \times 24 \times 24 q$ -point grid, corresponding to the two left-most points on the upper panel graph.
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