Spin-orbit spillage as a measure of band inversion in insulators

We propose a straightforward and effective approach for quantifying the band inversion induced by spin-orbit coupling in band insulators. In this approach we define a quantity as a function of wave vector in the Brillouin zone (BZ) measuring the mismatch, or “spillage,” between the occupied states of a system with and without SOC. Plots of the spillage throughout the BZ provide a ready indication of the number and location of band inversions driven by SOC. To illustrate the method, we apply this approach to the two-band Dirac model, the 2D Kane-Mele model, a 2D Bi bilayer with applied Zeeman field, and to first-principles calculations of some 3D materials including both trivial and ${\mathbb{Z}}_2$ topological insulators. We argue that the distribution of spillage in the BZ is closely related to the topological indices in these systems. Our approach provides a fresh perspective for understanding topological character in band theory, and should be helpful in searching for new materials with nontrivial band topology.

Figure 1

The spillage of the half-filled Dirac Hamiltonian as $\lambda$ increases from 0.4 to 1.9.

Figure 2

Spin-orbit spillage of the Kane-Mele model in the QSH phase, with $t=1$, ${\lambda }_{\mathrm{so}}=0.1t$, and $\epsilon =0.1t$. (a) Spin-resolved spillage without Rashba coupling; solid (green) and dashed (red) lines denote spin-up and spin-down spillage. Inset shows $\Gamma -M-K-M-K^{\prime }-\Gamma$ path used here (blue) and $K\text{−}\Gamma \text{−}M\text{−}K$ path used in Fig. 3 (magenta). (b) Total spillage without (solid line) and with (dashed line) Rashba coupling.

Figure 3

(a) Spin-orbit spillage of the Bi bilayer for $C=1$ (dashed blue) and $C=-2$ (solid red) phases, plotted along the $K\text{−}\Gamma \text{−}M\text{−}K$ path [magenta path in inset of Fig. 2]. (b) Spillage for $C=1$ phase plotted in the 2D BZ ($k_x$ and $k_y$ in units of Å${}^{-1}$). (c) Same for $C=-2$ phase.

Figure 4

(a) Lattice structure of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. (b) The BZ of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$; the spillage and band structures shown in Figs. 5 and 6 are plotted along the black path.

Figure 5

(a) Spin-orbit spillage of rhombohedral ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, ${\mathrm{Sb}}_2{\mathrm{Se}}_3$, and ${\mathrm{In}}_2{\mathrm{Se}}_3$ as indicated by blue, green and red lines respectively. Solid (dashed) lines show the spillage computed from direct plane-wave (Wannier-based) calculations. (b) Spillage of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ in the $(k_x,k_y)$ plane at $k_z=0$ (units of Å${}^{-1}$).

Figure 6

(a) Wannier-interpolated band structure of ${\mathrm{Sb}}_2{\mathrm{Se}}_3$. (b) Same for ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. Color coding indicates weight of Sb $5p$ or Bi $6p$ orbitals.