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 Fröhlich model: a simple (nondegenerate) parabolic isotropic electronic band coupled to one dispersionless longitudinal optical phonon branch. The Fröhlich 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 Fröhlich 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.

Figure 1

Polaron effective mass breakdown limit in the uniaxial case. Note that when the two relevant effective masses $m_z^{*}$ and $m_{\perp }^{*}$, respectively, are equal we reach the isotropic limit breakdown of 6.

Figure 2

Polaron localization length in the uniaxial limit calculated at constant density of states expressed in terms of ${\mu }^{*}$ for the volumetric average case ($a_P$), the out-of-plane localization length ($a_{Pz}$), and the in-plane localization length ($a_{P\perp }$). In the low limit of ${\mu }^{*}$, the asymptotic analytical behavior is represented in blue. In the ${\mu }^{*}\to 1$ limit, the analytical expression is represented in green. Finally, in the large ${\mu }^{*}$ limit, the asymptotic behavior analytical limit is represented in red. The expressions for the analytical asymptotic behavior in the three cases are provided in the Supplemental Material [48]. We consider here ${\epsilon }^{*}=1$ and $m_{\mathrm{DoS}}^{*}=1$ (see definition in Supplemental Material).

Figure 3

Absolute value of the polaron formation energy in the uniaxial limit calculated at constant volumetric localization length (or spherical average $a_P$) as a function of ${\mu }^{*}$. In the low limit of ${\mu }^{*}$, the asymptotic analytical behavior is represented in blue. In the spherical limit, the analytical expression is represented in green. Finally, in the large ${\mu }^{*}$ limit, the asymptotic behavior analytical limit is represented in red. The expressions for the analytical asymptotic behavior in the three cases are provided in the Supplemental Material [48]. We consider ${\epsilon }^{*}=1$ and $a_P=1$ (see definition in Supplemental Material).

Figure 4

Electron polaron effective mass as a function of the ratio between in-plane, $m_{\perp }$, and out-of-plane, $m_z$, bare electron effective masses for the studied materials. For the full set of values refer to Table S12 in Supplemental Material [48].

Figure 5

Hole polaron effective mass as a function of the (minimum) ratio between the corresponding electronic effective masses for the studied materials. Negative values for the polaron effective masses indicate a crossing of the mass breakdown limit. For the full set of values refer to Table S6 in Supplemental Material [48].

Figure 6

Ratio between the electron polaron localization length and repetition distance $d$ between equivalent atoms as a function of the ratio between in-plane, $m_{\perp }$, and out-of-plane, $m_z$, effective masses for the studied materials. For the full set of values refer to Table S9 in Supplemental Material [48]. Note that when the two effective masses are equal, the isotropic case is reached and the characteristic localization length is isotropic (red dots), while for the other cases two localization lengths are necessary, for the in- and out-of-plane lengths (orange and blue dots).

Figure 7

Ratio between hole polaron localization lengths along the direction with lowest formation energy expressed and repetition distance between equivalent atoms as a function of the (lowest) ratio between the electronic effective masses for the studied materials. Note that in the case when the lowest polaron formation energy is along the (110) direction there is no degeneracy in the electronic effective masses and, consequently, the polaron has the shape of a triaxial ellipsoid. $a_1$ represents the polaron radius along the direction in which the polaron formation energy is the lowest, while $a_2$ (or $a_3$) represents that along the perpendicular direction.

Figure 8

Ratio between shortest electron and hole polaron localization lengths and repetition distance between equivalent atoms, as a function of the smallest ratio between electron (or hole) effective mass and polaron effective mass for the studied materials. When the latter drops below zero, perturbation theory breaks down, while when the former drops below 1 the continuum hypothesis breaks down.

Figure 9

In the nondegenerate electron polaron situation, relative difference between the ZPR determined using perturbation theory (fully taking into account the possible anisotropy) and Feynman variational path integral approach (with approximate treatment of the anisotropy). In all cases the ZPR determined using the perturbative method has a larger value than the ZPR determined using the variational approach. For the full set of values refer to Table S11 in Supplemental Material [48].

Figure 10

In the nondegenerate electron polaron situation, relative difference between the effective masses determined using perturbation theory (fully taking into account the possible anisotropy) and Feynman variational path integral approach (with approximate treatment of the anisotropy). $m_{P,\text{iso}}^{*}$ is the isotropic effective mass, while in the anisotropic case $m_{P,\perp }^{*}$ and $m_{P,z}^{*}$ are the in-plane and out-of-plane polaron effective masses.