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 ${\mathrm{PbTiO}}_3$ 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 ${\mathrm{PbTiO}}_3$, 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.

中文翻译：

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第一性原理的远程四极电子声子相互作用

材料中的晶格振动会以长距离（偶极和四极）和短距离（八极及更高）电势的形式对电子动力学产生扰动。偶极Fröhlich项可以包含在当前的第一性原理电子声子（e-ph）计算中，并且仅存在于极性材料中。极性和非极性材料中都存在四极e-ph相互作用，但是目前无法根据第一原理进行计算。在这里，我们展示了一种计算四极e-ph相互作用的方法，并将其包括在e-ph矩阵元素的从头算中。通过与直接密度泛函摄动理论计算进行比较，证明了该方法的准确性。我们将我们的方法应用于硅作为非极性半导体和四方晶的情况${\mathrm{钛酸铅}}_3$作为极性压电材料的情况。在这两种材料中，我们都发现四极子项强烈影响e-ph矩阵元素。分析不同声子模式的e-ph相互作用表明，四极子项主要影响硅中的光学模式和硅中的声学模式。${\mathrm{钛酸铅}}_3$，尽管所有模式都需要四极子项才能达到定量精度。已证明四极子e-ph相互作用对电子散射过程和传输的影响很重要。我们的方法可以在广泛的非极性，极性和压电材料中精确研究e-ph相互作用。