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First-principles calculations of $e$-ph interactions are becoming a pillar ofelectronic structure theory. However, the current approach is incomplete. Thepiezoelectric (PE) $e$-ph interaction, a long-range scattering mechanism due toacoustic phonons in non-centrosymmetric polar materials, is not accuratelydescribed at present. Current calculations include short-range $e$-phinteractions (obtained by interpolation) and the dipole-like Fr\"ohlichlong-range coupling in polar materials, but lack important quadrupole effectsfor acoustic modes and PE materials. Here we derive and compute the long-range$e$-ph interaction due to dynamical quadrupoles, and apply this framework toinvestigate $e$-ph interactions and the carrier mobility in the PE materialwurtzite GaN. We show that the quadrupole contribution is essential to obtainaccurate $e$-ph matrix elements for acoustic modes and to compute PEscattering. Our work resolves the outstanding problem of correctly computing$e$-ph interactions for acoustic modes from first principles, and enablesstudies of $e$-ph coupling and charge transport in PE materials.

Piezoelectric Electron-Phonon Interaction from Ab Initio Dynamical Quadrupoles: Impact on Charge Transport in Wurtzite GaN