Long-range quadrupole electron-phonon interaction from first principles

Lattice vibrations in materials induce perturbations on the electron dynamics in the form of long-range (dipole and quadrupole) and short-range (octopole and higher) potentials. The dipole Fröhlich term can be included in current first-principles electron-phonon (e-ph) calculations and is present only in polar materials. The quadrupole e-ph interaction is present in both polar and nonpolar materials, but currently it cannot be computed from first principles. Here we show an approach to compute the quadrupole e-ph interaction and include it in ab initio calculations of e-ph matrix elements. The accuracy of the approach is demonstrated by comparing with direct density functional perturbation theory calculations. We apply our method to silicon as a case of a nonpolar semiconductor and tetragonal PbTiO3 as a case of a polar piezoelectric material. In both materials we find that the quadrupole term strongly impacts the e-ph matrix elements. Analysis of e-ph interactions for different phonon modes reveals that the quadrupole term mainly affects optical modes in silicon and acoustic modes in PbTiO3, although the quadrupole term is needed for all modes to achieve quantitative accuracy. The effect of the quadrupole e-ph interaction on electron scattering processes and transport is shown to be important. Our approach enables accurate studies of e-ph interactions in broad classes of nonpolar, polar, and piezoelectric materials.