  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 non-polar 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 inclusion of on-shell iterated
2-phonon (2ph) scattering processes, indicating that scattering from
higher-order electron-phonon interactions is non-negligible even in non-polar
semiconductors. Further, a ~20% underestimate of the low-field mobility with
2ph scattering suggests that non-trivial cancellations may occur in the
perturbative expansion of the electron-phonon interaction.
