Fröhlich polaron effective mass and localization length in cubic materials: Degenerate and anisotropic electronic bands

Polarons, that is, charge carriers correlated with lattice deformations, are ubiquitous quasiparticles in semiconductors, and play an important role in electrical conductivity. To date most theoretical studies of so-called large polarons, in which the lattice can be considered as a continuum, have focused on the original Fr\"ohlich model: a simple (nondegenerate) parabolic isotropic electronic band coupled to one dispersionless longitudinal optical phonon branch. The Fr\"ohlich model allows one to understand characteristics such as polaron formation energy, radius, effective mass, and mobility. Real cubic materials, instead, have electronic band extrema that are often degenerate (e.g., threefold degeneracy of the valence band), or anisotropic (e.g., conduction bands at $X$ or $L$), and present several phonon modes. In the present paper, we address such issues. We keep the continuum hypothesis inherent to the large polaron Fr\"ohlich model, but waive the isotropic and nondegeneracy hypotheses, and also include multiple phonon branches. For polaron effective masses, working at the lowest order of perturbation theory, we provide analytical results for the case of anisotropic electronic energy dispersion, with two distinct effective masses (uniaxial) and numerical simulations for the degenerate three-band case, typical of III-V and II-VI semiconductor valence bands. We also deal with the strong-coupling limit, using a variational treatment: we propose trial wave functions for the above-mentioned cases, providing polaron radii and energies. Then, we evaluate the polaron formation energies, effective masses, and localization lengths using parameters representative of a dozen II-VI, III-V, and oxide semiconductors, for both electron and hole polarons. We show that for some cases perturbation theory (the weak-coupling approach) breaks down. In some other cases, the strong-coupling approach reveals that the large polaron hypothesis is not valid, which is another distinct breakdown. In the nondegenerate case, we compare the perturbative approach with the Feynman path integral approach in characterizing polarons in the weak-coupling limit. Thus, based on theoretical results for cubic materials, the present paper characterizes the validity of the continuum hypothesis for a large set of 20 materials.

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