Mixed Wannier-Bloch Functions for Electrons and Phonons in Periodic Systems

We introduce mixed Wannier-Bloch functions for studying electronic and vibrational spectra of periodic systems. These functions carry both spatial localization and limited spectral broadening, thereby combining the advantages of descriptions based on energy eigenstates (Bloch states) and position eigenstates (Wannier states). For the analysis of vibrational modes, a lattice position operator is introduced, analogous to the electronic Berry-phase position operator. Application to vitreous ${\mathrm{SiO}}_2$ demonstrates that mixed Wannier-Bloch functions constitute a powerful tool for tracking fingerprints of short- and medium-range structural order in electronic and vibrational spectra.

Figure 1

Average spatial localization (blue) and average spectral broadening (red) vs the mixing parameter $\lambda$, for electrons and phonons in our model structure of $v\mathrm{\text{−}}{\mathrm{SiO}}_2$. The steplike behavior of the electronic spectral broadening around $\lambda \simeq 0.6$ reflects the large energy gap between O-$2s$ and O-$2p$ states (cf. Fig. 2). The vanishing spatial extent of optimally localized ($\lambda =1$) phonon WB functions indicates that, in this limit, only one atom is vibrating while all the other atoms remain fixed in their equilibrium positions.

Figure 2

From left to right: the adopted model structure of $v\mathrm{\text{−}}{\mathrm{SiO}}_2$ [yellow (red) balls indicate Si (O) atoms], together with the density isosurface ($0.03\mathrm{\text{electrons}}/{\AA }^3$) of an electron WB function; zoom on the WB function localized around the O atom; total WB projected density of states corresponding to the modes of a given type (shaded area), compared to the total density of states (solid line; a Gaussian broadening of 0.25 eV was used). From top to bottom: WB functions corresponding to $s$-like, and to stretching, bending and rocking $p$-like electronic states, respectively.

Figure 3

From left to right: model structure of $v\mathrm{\text{−}}{\mathrm{SiO}}_2$ ( gray), with a color code indicating the amplitude of the WB vibrational modes on every atom; zoom on the WB function indicating the displacement patterns by arrows; total WB projected vibrational density of states corresponding to the modes of a given type (shaded area), compared to the total vibrational density of states (solid line; Gaussian broadening of $20{\mathrm{cm}}^{-1}$). From top to bottom: WB functions corresponding to rocking, bending, and stretching vibrations.

Figure 4

Electron WB function associated to medium-range structures in $v\mathrm{\text{−}}{\mathrm{SiO}}_2$. Left: a three-membered ring with the charge density isosurface of a localized electronic state ($0.03\mathrm{\text{electrons}}/{\AA }^3$). Right: corresponding signature (shaded) in the electronic spectrum (Gaussian broadening of 0.25 eV). The peak width (0.5 eV) mainly results from the inhomogeneous broadening due to the disorder.

Figure 5

Phonon WB functions associated to medium-range structures in $v\mathrm{\text{−}}{\mathrm{SiO}}_2$. Left: vibrational patterns of three-membered rings. Right: corresponding signatures in the vibrational spectrum (Gaussian broadening of $20{\mathrm{cm}}^{-1}$). Top: breathing mode corresponding to the Raman line $D_2$, and spectral weights of the breathing mode of three- and four-membered rings (shaded line). The experimental Raman spectrum [28] is shown for comparison (red line), with the $D_1$ and $D_2$ lines highlighted. Bottom: out-of-phase in-plane rotations of Si and O atoms in a three-membered ring [29].