Towards predictive many-body calculations of phonon-limited carrier mobilities in semiconductors

We probe the accuracy limit of ab initio calculations of carrier mobilities in semiconductors, within the framework of the Boltzmann transport equation. By focusing on the paradigmatic case of silicon, we show that fully predictive calculations of electron and hole mobilities require many-body quasiparticle corrections to band structures and electron-phonon matrix elements, the inclusion of spin-orbit coupling, and an extremely fine sampling of inelastic scattering processes in momentum space. By considering all these factors we obtain excellent agreement with experiment, and we identify the band effective masses as the most critical parameters to achieve predictive accuracy. Our findings set a blueprint for future calculations of carrier mobilities, and pave the way to engineering transport properties in semiconductors by design.

Figure 1

Intrinsic electron and hole mobilities of silicon at 300 K, calculated using various levels of theory. The complexity of the theory increases as we move down the sequence of bars. The range of measured mobilities is indicated in light-gray vertical bars. Our most accurate theoretical predictions are ${\mu }_{\mathrm{e}}=1366{\mathrm{cm}}^2$/V s and ${\mu }_{\mathrm{h}}=658{\mathrm{cm}}^2$/V s; by replacing the GW hole effective mass with the experimental value we obtain ${\mu }_{\mathrm{e}}=502{\mathrm{cm}}^2$/V s, in much better agreement with experiment. Key: SOC, spin-orbit coupling; EXP, experimental lattice parameter; GW, calculations including quasiparticle corrections; SCR, electron-phonon coupling with corrected screening; IBTE, iterative Boltzmann transport equation; RE, change of effective mass due to electron-phonon renormalization; FIT, band structures calculated from the measured effective masses.

Figure 2

(a) Conduction bands of silicon calculated within scalar-relativistic PBE (gray), fully relativistic PBE (blue), the GW method (orange), and parabolic fit with measured effective masses (dashed). The zero of the energy axis is set to the conduction band minimum for clarity. (b) Valence bands of silicon, calculated within the same approximations as for (a), and shown using the same color code. The zero of the energy axis is set to the valence band top. In all panels the dots indicate explicit GW calculations carried out using uniform grids containing $12\times 12\times 12$ to $20\times 20\times 20$ points. The GW bands in orange are obtained via Wannier interpolation.

Figure 3

(a) Comparison between calculated and measured intrinsic (low carrier concentration $\le {10}^{15}{\mathrm{cm}}^{-3})$ electron and hole mobilities of silicon, as a function of temperature. The calculations are performed using our best computational setup. The blue lines are for holes and the orange line is for electrons. In the case of holes we show both our best ab initio calculations (solid blue), and the results obtained by setting the hole effective mass to the experimental value (dashed blue). The shading is a guide to the eye. Experiments are from [34] $\left(▵\right)$, [35] $(\diamond )$, [36] $\left(★\right)$, [37] $(\circ )$, and [19] $(\square )$. (b) Comparison between calculated and measured electron and hole mobilities of silicon at 300 K, as a function of carrier concentration, using the same color code as in (a). Experimental data are from [37] $(\circ )$. The impurity scattering is included via the model of Brooks and Herring with the Long-Norton correction [38, 39] as described in the Supplemental Material [27].