The ab initio theory of charge transport in semiconductors typically employs the lowest-order perturbation theory in which electrons interact with one phonon (1ph). This theory is accepted to be adequate to explain the low-field mobility of nonpolar semiconductors but has not been tested extensively beyond the low-field regime. Here, we report first-principles calculations of the electric field dependence of the electron mobility of Si as described by the warm electron coefficient $\beta$. Although the 1ph theory overestimates the low-field mobility by only around 20%, it overestimates $\beta$ by over a factor of two over a range of temperatures and crystallographic axes. We show that the discrepancy in $\beta$ is reconciled by the inclusion of on-shell iterated two-phonon (2ph) scattering processes, indicating that scattering from higher-order electron-phonon interactions is non-negligible even in nonpolar semiconductors. Further, a $\sim 20\%$ underestimate of the low-field mobility with 2ph scattering suggests that nontrivial cancellations may occur in the perturbative expansion of the electron-phonon interaction.

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• Electrical conductivity
• Electrical properties
• Electron-phonon coupling
• Electronic structure
• Electronic structure of atoms & molecules
• First-principles calculations
• Elemental semiconductors