Influence of Fr\"ohlich polaron coupling on renormalized electron bands in polar semiconductors. Results for zincblende GaN

\ni We develop a simple method to study the zero-point and thermally renormalized electron energy $\varepsilon_{\mathbf{k}n}(T)$ for $\mathbf{k}n$ the conduction band minimum or valence maximum in polar semiconductors. We use the adiabatic approximation, including an imaginary broadening parameter $i\delta$ to supress noise in the density-functional integrations. Fr\"{o}hlich polaron methods provide analytic expressions for the contribution of the problematic optical phonon mode. We use this to correct the renormalization obtained from the adiabatic approximation. Test calculations are done for zincblende GaN for an 18x18x18 integration grid. The Fr\"ohlich correction is of order -0.02 eV for the zero-point energy shift of the conduction band minimum, and +0.03 eV for the valence band maximum; the correction to renormalization of the 3.28 eV gap is -0.05 eV, a significant fraction of the total zero point renormalization of -0.15 eV.