Wannier interpolation of the electron-phonon matrix elements in polar semiconductors: Polar-optical coupling in GaAs

We generalize the Wannier interpolation of the electron-phonon matrix elements to the case of polar-optical coupling in polar semiconductors. We verify our methodological developments against experiments, by calculating the widths of the electronic bands due to electron-phonon scattering in GaAs, the prototype polar semiconductor. The calculated widths are then used to estimate the broadenings of excitons at critical points in GaAs and the electron-phonon relaxation times of hot electrons. Our findings are in good agreement with available experimental data. Finally, we demonstrate that while the Fröhlich interaction is the dominant scattering process for electrons/holes close to the valley minima, in agreement with low-field transport results, at higher energies, the intervalley scattering dominates the relaxation dynamics of hot electrons or holes. The capability of interpolating the polar-optical coupling opens new perspectives in the calculation of optical absorption and transport properties in semiconductors and thermoelectrics.

Figure 1

GaAs. Deformation potentials for the total contribution of the three highest valence bands [see Eq. (4)]. The initial electronic state is at $\mathbf{k}=\mathrm{\Gamma }$, and the phonon wave vector $\mathbf{q}$ is varying along high symmetry lines in the BZ. Black line: Direct DFPT calculation. Gray squares: Wannier interpolation extended to polar-optical coupling. Circles: Standard Wannier interpolation for metals and nonpolar semiconductors.

Figure 2

GaAs. LO deformation potentials for the initial electronic state with $\mathbf{k}=X$ and the phonon wave vector changing along the (100) direction: $\mathbf{q}=(q_x,0,0)2\pi /a$, for $q_x\ge$ 0.01. Panel (a): Deformation potentials calculated within DFPT for the lowest conduction band (dashed line) and two lowest valence bands [dot-dashed line, visible in panel (b)], compared with the deformation potentials calculated from the model of Eq. (9) (solid line). Panel (b): The difference between the deformation potentials calculated with DFPT, and the model of Eq. (9).

Figure 3

GaAs. Deformation potentials for the total contribution of the three highest valence bands [see Eq. (4)] multiplied by the modulus of the phonon wave vector $\left|\mathbf{q}\right|$. The initial electronic state is at $\mathbf{k}=\mathrm{\Gamma }$, and the phonon wave vector $\mathbf{q}$ is varying along high symmetry lines in the BZ. Black line: Direct DFPT calculation. Gray squares: Our results obtained with the Wannier interpolation extended to the polar-optical coupling.

Figure 4

GaAs. Panel (a): Highest valence and lowest conduction bands broadened by the electron-phonon interaction. The broadening is due to all electron-phonon interaction processes. For the sake of visibility, the calculated broadening ${\mathrm{\Gamma }}_{\mathbf{k}}$ of the electronic bands has been multiplied by 5 in this panel. For panels (b)–(d), the black solid line describes the behavior of the initial electronic state in the highest valence band and the gray solid line is that of the initial state in the lowest conduction band. Panel (b): Broadening ${\mathrm{\Gamma }}_{\mathbf{k}}$ due to electron-phonon interaction, as a function of the $\mathbf{k}$ vector of the initial electronic state. Panel (c): Electronic density of the final states allowed by conservation laws, as a function of the $\mathbf{k}$ vector of the initial electronic state. Panel (d): Relative contribution of the Fröhlich interaction only to the total broadening. The calculations were done at $T=300$ K.

Figure 5

GaAs. Broadening (meV) of the critical point $E_1$ as a function of temperature. Points: Experimental data of Ref. [7] obtained by fit with the Fano-type excitonic line [8]. Squares: Experimental data of Ref. [7], fit with the two-dimensional critical point model [8]. Solid black line: This work, Wannier interpolation method extended to polar-optical coupling. Dashed-dotted line: Theoretical calculation from Ref. [7] with empirical pseudopotential method. Both theoretical broadenings are due to electron-phonon coupling only and are calculated as the sum of the broadenings of the lowest conduction and highest valence bands, averaged over the four points along the $\mathrm{\Lambda }$ direction as explained in the text. The inset represents the same experimental and theoretical results of Ref. [7] as the main figure, with the same notations. Gray solid line on the inset figure: This work, Wannier interpolation method without extension to polar-optical coupling.

Figure 6

GaAs. Broadening (meV) of the critical point $E_2$ as a function of temperature. Squares: Experimental data of Ref. [7], fit with the two-dimensional critical point model [8]. Dashed-dotted line: Theoretical calculation from Ref. [7] with empirical pseudopotential method. Solid black line: This work, Wannier interpolation method extended to polar-optical coupling, sum of the broadenings of the lowest conduction and highest valence bands, averaged over three points (see text).