Zero-point renormalization of the band gap of semiconductors and insulators using the projector augmented wave method

We evaluate the zero-point renormalization (ZPR) due to electron-phonon interactions of 28 solids using the projector-augmented-wave (PAW) method. The calculations cover diamond, many zincblende semiconductors, rock-salt and wurtzite oxides, as well as silicate and titania. Particular care is taken to include long-range electrostatic interactions via a generalized Fröhlich model. The data are compared to recent calculations [Miglio et al., npj Comput. Mater. 6, 167 (2020)] and generally very good agreement is found. We discuss in detail the evaluation of the electron-phonon matrix elements within the PAW method. We show that two distinct versions can be obtained depending on when the atomic derivatives are taken. If the PAW transformation is applied before taking derivatives with respect to the ionic positions, then equations similar to the ones conventionally used in pseudopotential codes are obtained. If the PAW transformation is used after taking the derivatives, then the full-potential spirit is largely maintained. We show that both variants yield very similar ZPRs for selected materials when the rigid-ion approximation is employed. In practice, we find, however, that the pseudoversion converges more rapidly with respect to the number of included unoccupied states.

Figure 1

Feynman-diagrammatic representation of the lowest-order electron-phonon contributions to the electron self-energy in the context of KS DFT. Electronic states (solid lines) interact with phonon states (dashed lines) via the first and second-order electron-phonon matrix elements, $g_{mn\mathbf{k},\nu \mathbf{q}}$ and $g_{n\mathbf{k}}^{2,\text{DW}}$, respectively. (a) Fan-Migdal process; a phonon with wave vector $\mathbf{q}$ and branch index $\nu$ is emitted and later reabsorbed. (b) Debye-Waller process; a phonon is simultaneously emitted and reabsorbed. Note that there is no momentum transfer in this case.

Figure 2

Convergence behavior of the nonadiabatic ZPR with respect to the maximum number of included intermediate states for both the AE and PS method for (a) MgO, (b) AlAs, and (c) ZnS. The second $x$ axis scale above each panel displays the average KS energy with respect to the Fermi level that corresponds to the number of bands. The PS method converges much faster and smoother than the AE method, but both eventually converge to approximately the same value.

Figure 3

Same as Fig. 2 but for diamond and using three different PAW potentials: C (a), C_GW_new (b), and C_h_GW (c). AE and PS methods converge to approximately the same result only in panel (c). The AE results depend more strongly on the PAW potential.