Ionization Potentials of Solids: The Importance of Vertex Corrections

The ionization potential is a fundamental key quantity with great relevance to diverse material properties. We find that state of the art methods based on density functional theory and simple diagrammatic approaches as commonly taken in the $GW$ approximation predict the ionization potentials of semiconductors and insulators unsatisfactorily. Good agreement between theory and experiment is obtained only when diagrams resulting from the antisymmetry of the many-electron wave function are taken into account via vertex corrections in the self-energy. The present approach describes both localized and delocalized states accurately, making it ideally suited for a wide class of materials and processes.

Figure 1

(a) Vertex correction in the screened interaction $W^{\mathrm{TC}\text{−}\mathrm{TC}}$, and (b) and (c) first order vertex correction in the self-energy yielding the second order exchange diagram. (a) The incoming photon (wiggly line) creates a particle-hole pair (forward and backward arrows). The vertex describes the electrostatic interaction (double-wiggly line) between the particles and holes. A static approximation is used for the screened interaction $W$ (double-wiggly lines), as commonly done in solid state Bethe-Salpeter calculations [5]. Panel (c) corresponds to (b) but shows explicitly that time ordering gives rise to two second order exchange diagrams for a static $W$ [compare two terms in Eq. (1)].

Figure 2

Ionization potentials of various semiconductors and insulators obtained for nonpolar surfaces using PBE, HSE, $G{W}_0@\mathrm{PBE}$, $G{W}^{\mathrm{TC}\text{−}\mathrm{TC}}@\mathrm{HSE}$, and $GW{\mathrm{\Gamma }}^1@\mathrm{HSE}$ [16]. Experimental (EXP) values for the corresponding nonpolar surfaces are shown with triangles, whereas those for different surface orientations are denoted by inverted triangles.

Figure 3

Band gaps for $G{W}^{\mathrm{TC}\text{−}\mathrm{TC}}$@HSE and $GW{\mathrm{\Gamma }}^1@\mathrm{HSE}$versus experimental band gaps for the semiconductors considered in this work [16].