Polarons from First Principles, without Supercells

We develop a formalism and a computational method to study polarons in insulators and semiconductors from first principles. Unlike in standard calculations requiring large supercells, we solve a secular equation involving phonons and electron-phonon matrix elements from density-functional perturbation theory, in a spirit similar to the Bethe-Salpeter equation for excitons. We show that our approach describes seamlessly large and small polarons, and we illustrate its capability by calculating wave functions, formation energies, and spectral decomposition of polarons in LiF and ${\mathrm{Li}}_2{\mathrm{O}}_2$.

Figure 1

Large electron polaron in LiF. (a) Isosurface of the polaron density $|\psi {|}^2$ and ball-stick model of LiF, with Li in green and F in silver. (b) Cross section of the polaron density $|\psi {|}^2$ along a [010] line cutting through the center. (c) Modulus of the atomic displacements projected along the [010] direction and taken on a -Li-F- chain of atoms nearest to the polaron center. The horizontal axes in (a)–(c) are aligned. (d),(e) Band structures and phonon dispersions of LiF, respectively. The Fourier amplitudes $A_{n\mathbf{k}}$ and $B_{\mathbf{q}\nu }$ are superimposed to the bands, with the radius of the circle proportional to their square modulus. In (d), the zero of the energy is aligned with the valence-band top. (f) Polaron formation energy $\mathrm{\Delta }{E}_f$ and eigenvalue $ϵ$ as a function of supercell size. The dashed lines are the Makov-Payne extrapolations. The shading indicates that no localized solution was found, and MIT stands for metal-to-insulator transition. The numbers next to the circles indicate the unit cells in each supercell, e.g., 12 means $12\times 12\times 12$ supercell. (g) Polaron energies (triangles) and eigenvalues (circles) obtained with our method using the model Fröhlich electron-phonon coupling compared to the solution of the Pekar polaron model.

Figure 2

Small electron polaron in ${\mathrm{Li}}_2{\mathrm{O}}_2$. (a) Isosurface of the polaron density $|\psi {|}^2$ and model of ${\mathrm{Li}}_2{\mathrm{O}}_2$, with Li and O atoms in green and red, respectively. (b) Cross section of $|\psi {|}^2$ along a [100] line cutting through the center. (c) Modulus of the atomic displacements along a [001] line passing through the O atom at the center. (d), (e) Band structures and phonon dispersion relations of ${\mathrm{Li}}_2{\mathrm{O}}_2$, respectively. The amplitudes $A_{n\mathbf{k}}$ and $B_{\mathbf{q}\nu }$ are superimposed as circles. The energy zero in (d) is aligned with the valence-band top. (f) Polaron formation energy and eigenvalue vs supercell size. The dashed gray lines represent the Makov-Payne extrapolations. The green triangle is the result of an explicit DFT calculation from Ref. [48]. (g) Comparison between the polaron wave function obtained from our method (left), and an explicit DFT calculation (right, Ref. [48]), and the corresponding formation energies.