Phonon-limited electron mobility in Si, GaAs, and GaP with exact treatment of dynamical quadrupoles

We describe a new approach to compute the electron-phonon self-energy and carrier mobilities in semiconductors. Our implementation does not require a localized basis set to interpolate the electron-phonon matrix elements, with the advantage that computations can be easily automated. Scattering potentials are interpolated on dense $\mathbf{q}$ meshes using Fourier transforms and ab initio models to describe the long-range potentials generated by dipoles and quadrupoles. To reduce significantly the computational cost, we take advantage of crystal symmetries and employ the linear tetrahedron method and double-grid integration schemes, in conjunction with filtering techniques in the Brillouin zone. We report results for the electron mobility in Si, GaAs, and GaP obtained with this new methodology.

Figure 1

Schematic representation of the filtering of (a) $\mathbf{k}$ and (b) $\mathbf{q}$ points for the computation of the SE in the case of a single parabolic conduction band at the $\mathrm{\Gamma }$ point. (a) States represented by green dots (red crosses) are included (excluded) in (from) the transport computation according to the input variable $\mathrm{\Delta }{\varepsilon }_c$. (b) For initial and final electron states ${\mathbf{k}}_{\mathbf{i}}$ and ${\mathbf{k}}_{\mathbf{f}}$, the wave vector $\mathbf{q}={\mathbf{k}}_{\mathbf{f}}-{\mathbf{k}}_{\mathbf{i}}$ requires a phonon mode with frequency ${\omega }_{\mathbf{q}\nu }={\varepsilon }_{{\mathbf{k}}_{\mathbf{f}}}-{\varepsilon }_{{\mathbf{k}}_{\mathbf{i}}}$. If this condition is not fulfilled, the $\mathbf{q}$ point is ignored.

Figure 2

Schematic representation of the DG technique. Black dots represent $\mathbf{q}$ points belonging to the coarse grid where the e-ph matrix elements $g_{mn\nu }(\mathbf{k},\mathbf{q})$ are explicitly computed using our interpolation procedure. Colored small dots are $\mathbf{q}$ points of the fine grid where the $g_{mn\nu }(\mathbf{k},\mathbf{q})$ are assumed to be constant in the region surrounding the black dot (the shaded green area).

Figure 3

Band structure of (a) Si and (b) GaAs. The Fermi level is located so that $n_e={10}^{18}$ and ${10}^{15}{\mathrm{cm}}^{-3}$ for Si and GaAs, respectively. The energy window used to filter the $\mathbf{k}$ points for transport computations is represented by horizontal dashed lines. Phonon dispersions of (c) Si and (d) GaAs.

Figure 4

Electron linewidths near the CBM of Si obtained with a $60\times 60\times 60 \mathbf{k}$-point grid at $T=300\mathrm{K}$. A very dense $180\times 180\times 180 \mathbf{q}$-point grid is used for all curves to achieve well-converged results. Results are obtained with Lorentzian (L) and tetrahedron (T) methods. The broadening parameter is given in parentheses in meV.

Figure 5

Average (solid line) and maximum (dashed line) error on the linewidths evaluated on a $9\times 9\times 9 \mathbf{k}$-point grid in (a) Si and (b) GaAs as a function of the $\mathbf{q}$ grid used for the integration. Only states within 1 eV from the CBM and VBM are considered. Results have been obtained with the tetrahedron method (T, blue) and a Lorentzian (L, red) broadening of 5 meV. The reference linewidths for each curve are the results obtained with the densest $\mathbf{q}$ grid and the corresponding integration technique. The insets show the linewidths obtained with the tetrahedron method and the densest $\mathbf{q}$ grid.

Figure 6

Average (solid line) and maximum (dashed line) error on the linewidths on a $9\times 9\times 9 \mathbf{k}$-point grid in (a) Si and (b) GaAs as a function of the fine $\mathbf{q}$-point grid used for the integration. The matrix elements are obtained on (a) coarse $9\times 9\times 9$ (blue), $18\times 18\times 18$ (red) and $27\times 27\times 27$ (green) grids for Si and (b) coarse $9\times 9\times 9$ (blue), $18\times 18\times 18$ (red) $36\times 36\times 36$ (green) and $72\times 72\times 72$ (black) grids for GaAs. KS energies are computed on the fine grids. Only states within 1 eV from the CBM and VBM are considered. The reference linewidths (insets) are the results obtained with a $90\times 90\times 90$ ($144\times 144\times 144$) $\mathbf{q}$-point grid for both matrix elements and energies for Si (GaAs).

Figure 7

(a) $xx$ component of the integrand in Eq. (28) convoluted with $\delta (\varepsilon -{\varepsilon }_{n\mathbf{k}})$ in the conduction band of Si, for $24\times 24\times 24$ (black), $48\times 48\times 48$ (red), and $72\times 72\times 72$ (green) $\mathbf{k}$-point and $144\times 144\times 144 \mathbf{q}$-point grids at $T=300\mathrm{K}$. (b) Electron mobility of Si as a function of the energy window used for the computation of the lifetimes in Eq. (28). $72\times 72\times 72 \mathbf{k}$-point and $144\times 144\times 144 \mathbf{q}$-point grids have been used. For these grids, the converged value is 1509 ${\mathrm{cm}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$. A 160-meV energy range is enough to reach a relative error in the mobility lower than 1%.

Figure 8

Electron mobility in Si ($T=300\mathrm{K}$) as a function of the $\mathbf{k}$- and $\mathbf{q}$-point grids. The density of $\mathbf{q}$ points is the same as the density of $\mathbf{k}$ points (black), or twice as dense (red). The results obtained with DG technique are also reported (green, dashed). In this case, the densities of the $\mathbf{k}$- and $\mathbf{q}$-point grids are the same for matrix elements and lifetimes, but a grid twice as dense is used for the energies in Eq. (26). The equivalent homogeneous $\mathbf{k}$ mesh is first reported, together with the corresponding number of points in the IBZ. The effective number of $\mathbf{k}$ points included in the computation of ${\tau }_{n\mathbf{k}}$ is also given.

Figure 9

Electron mobility in GaAs ($T=300\mathrm{K}$) as a function of the $\mathbf{k}$- and $\mathbf{q}$-point grids. The density of $\mathbf{q}$ points is the same as the density of $\mathbf{k}$ points (black), twice (red), or three times (blue) as dense in each direction. The DG technique results are also reported (green, dashed). In this case, the densities of the $\mathbf{k}$- and $\mathbf{q}$-point grids are the same for e-ph matrix elements and lifetimes, but a grid twice as dense in all directions is used for the energies in Eq. (26). The equivalent homogeneous $\mathbf{k}$ mesh is reported, together with the corresponding number of points in the IBZ. The effective number of $\mathbf{k}$ points included in the computation of ${\tau }_{n\mathbf{k}}$ is also given.

Figure 10

Electron mobility in GaP ($T=300\mathrm{K}$) as a function of the $\mathbf{k}$- and $\mathbf{q}$-point grids. The density of $\mathbf{q}$ points is the same as the density of $\mathbf{k}$ points (black), twice (red), or three times (blue) as dense in each direction. The DG technique results are also reported (green, dashed). In this case, the densities of the $\mathbf{k}$- and $\mathbf{q}$-point grids are the same for matrix elements and lifetimes, but a grid twice as dense in all directions is used for the energies in Eq. (26). The equivalent homogeneous $\mathbf{k}$ mesh is reported, together with the corresponding number of points in the IBZ. The effective number of $\mathbf{k}$ points included in the computation of ${\tau }_{n\mathbf{k}}$ is also given.