Comparison between first-principles supercell calculations of polarons and the ab initio polaron equations

Abstract

Polarons are composite quasiparticles formed by excess charges and the accompanying lattice distortions in solids, and play a critical role in transport, optical, and catalytic properties of semiconductors and insulators. The standard approach for calculating polarons from first principles relies on density functional theory and periodic supercells. An alternative approach consists of recasting the calculation of polaron wavefunction, lattice distortion, and energy as a coupled nonlinear eigenvalue problem, using the band structure, phonon dispersions, and the electron-phonon matrix elements as obtained from density functional perturbation theory. Here, we revisit the formal connection between these two approaches, with an emphasis on the handling of self-interaction correction, and we establish a compact formal link between them. We perform a quantitative comparison of these methods for the case of small polarons in the prototypical insulators TiO2, MgO, and LiF. We find that the polaron wavefunctions and lattice distortions obtained from these methods are nearly indistinguishable in all cases, and the formation energies are in good (TiO2) to fair (MgO) agreement. We show that the residual deviations can be ascribed to the neglect of higher-order electron-phonon couplings in the density functional perturbation theory approach.

Figures (3)

• Figure 1: Supercell calculations of the formation energy of the small one-center hole polaron in LiF. The pink symbols are from PBE0 hybrid functional calculations of a charged supercell, as a function of the fraction of exact exchange $\alpha$. The green symbols are from the pSIC method of Ref. Sadigh_Aberg_2015 with the PBE0 functional, and correspond to charge-neutral supercells; also in this case we perform calculations for varying $\alpha$. For both methods and for all datapoints, the structure of the hole polaron is the same and is fixed to that obtained from the ab initio polaron equations Sio_Giustino_2019a, using the PBE functional and a 3$\times$3$\times$3 supercell.
• Figure 2: Comparison between the electron charge density and atomic displacements corresponding to the small hole polarons in anatase TiO2, MgO, and LiF, as obtained from the supercell method of Ref. Sadigh_Aberg_2015 and from the ab initio polaron equations of Ref. Sio_Giustino_2019a. (a) Ball-and-stick models of anatase TiO2 with Ti in light blue and O in red. The isosurface is the polaron charge density obtained from the supercell method, and the arrows indicate the corresponding atomic displacements with respect to the undistorted crystal. For ease of visulization, the atomic displacements are magnified so that the maximum displacements computed from two methods coincide. The differences of the actual atomic displacements can be found in Table \ref{['Tab:eform']}. (d) Same as in (a), but obtained from the polaron equations. (b) and (e): Same as in (a) and (d), but for the hole polaron in MgO. Mg is shown in orange. (c) and (f): Same as in (a) and (d), but for the one-center hole polaron in LiF. Li is in gray and F is in green.
• Figure 3: Decomposition of the potential energy surface of the small hole polarons in anatase TiO2 [panels (a) and (d)], MgO [panels (b) and (e)], and LiF [panels (c) and (f)] in terms of elastic energy (top row) and electronic excitation energy (bottom row), as defined in the main text. The leftmost horizontal coordinate in each panel corresponds to the undistorted crystal, and the rightmost coordinate corresponds to the polaron structure, as shown by the arrows in (a). The horizontal axis indicates the maximum atomic displacement along the potential energy surface. Orange symbols are from the supercell method [Ref. Sadigh_Aberg_2015], green symbols are from the polaron equations [Ref. Sio_Giustino_2019a].