Unified Approach to Polarons and Phonon-Induced Band Structure Renormalization

Ab initio calculations of the phonon-induced band structure renormalization are currently based on the perturbative Allen-Heine theory and its many-body generalizations. These approaches are unsuitable to describe materials where electrons form localized polarons. Here, we develop a self-consistent, many-body Green’s function theory of band structure renormalization that incorporates localization and self-trapping. We show that the present approach reduces to the Allen-Heine theory in the weak-coupling limit, and to total energy calculations of self-trapped polarons in the strong-coupling limit. To demonstrate this methodology, we reproduce the path-integral results of Feynman and diagrammatic Monte Carlo calculations for the Fröhlich model at all couplings, and we calculate the zero point renormalization of the band gap of an ionic insulator including polaronic effects.

Figure 1

(a) Schematic illustration of the ground state of the $N$-electron system, with atoms vibrating around the equilibrium sites of the periodic crystal. (b) Schematic of phonon-induced band structure renormalization, as obtained by using the FM and DW self-energies. The dashed line is the noninteracting band, the brown lines are the renormalized band and its phonon sidebands. (c) Self-consistent set of equations for calculating electron-phonon renormalization of band structures including polaron localization effects, Eqs. (1)–(4). (d) Schematic illustration of the ground state of the $N+1$-electron system, where the excess electron forms a localized polaron. (e) In the scenario illustrated in (d), the energy of the conduction band bottom is lowered by the formation energy of the polaron.

Figure 2

(a) Ground-state energy of the Fröhlich polaron, $\mathrm{\Delta }E/\hslash \omega$, as a function of the coupling strength $\alpha$: present calculation (blue line), Feynman’s path integral results [42] (gray line), and diagrammatic Monte Carlo (DMC) data taken from Ref. [58] (black circles). Red lines indicate the asymptotic expansions at weak and strong coupling, respectively. The quasiparticle amplitudes $|A_k{|}^2$ in these limits are shown in the inset, superimposed to the free electron band. (b) Breakdown of the ground-state energy of the Fröhlich polaron into its self-energy contributions.

Figure 3

(a), (b) Conduction and valence bands of LiF, with energies referred to the conduction band minimum (CBM) and to the valence band maximum), respectively. The yellow disks indicate the square moduli of the quasiparticle amplitudes of the Dyson orbitals, as obtained by solving Eqs. (5), (8), and (9). In each panel we indicate the quasiparticle renormalization including polaronic effects, $\mathrm{\Delta }E$. (c), (d) Expectation values of ${\mathrm{\Sigma }}^{\mathrm{FM}}+{\mathrm{\Sigma }}^{\mathrm{DW}}$ along the bands. (e), (g) Calculated renormalization of the conduction and valence band extrema, respectively, using two methods: the standard perturbative approach that does not include polaron localization (gray), and the present approach including localization effects (blue). (f), (h) Calculated wave functions of the electron and hole polarons in LiF, respectively.