Topological Dislocation Response in Elementary Semiconductors

Abstract

We study elementary semiconductors and insulators that are symmetric under spatial inversion: silicon, diamond, germanium, and black phosphorene. These materials are ideal candidates for realizing obstructed atomic insulators, which differ from trivial atomic insulators by a quantized spatial shift of their electronic Wannier centers with respect to the atomic lattice. We use symmetry indicator invariants that allow the prediction of non-trivial responses to crystal dislocations in these materials. We find that edge dislocations generically exhibit a non-trivial response, while screw dislocations always display a trivial response. With the aid of numerical simulations of realistic tight-binding models, we confirm the presence of mid-gap polarization bands localized along dislocations in silicon, diamond, and germanium.

Figures (2)

• Figure 1: Dislocation response of black phosphorene (BP). (a) Brillouin zone of BP, showing time-reversal invariant momentum (TRIM) Schindler_2022 points and their occupied inversion eigenvalues. (b) Band structure along the path shown in (a), with associated inversion eigenvalues at TRIMs. (c) Spectrum in presence of a pair of $\mathcal{I}$-related dislocations, plotted against the interpolation parameter $\lambda$ [see Eq. \ref{['eq:interpolating_hamiltonian']}]. Two polarization bands appear inside the gap. One of them is slightly shifted to aid visualization. We color-code their IPR, which is maximal at $\lambda = 0$ (90° model) and minimal at $\lambda = 1$ (realistic model) [see Eq. \ref{['eq:interpolating_hamiltonian']}]. Simulations are performed on a lattice of $30 \times 30$ unit cells spanned by $\boldsymbol{a}_1$ and $\boldsymbol{a}_2$. The dislocation is introduced by cutting the lattice (keeping unit cells intact), removing $7$ unit cells, and gluing back together. (d) Unit cell of BP, consisting of $4$ phosphorus atoms, each of which has $5$ valence electrons. The commonly used unit cell consists of the gray and blue atoms ($A$, $B$, $A'$ and $B'$). By contrast, the unit cell we use consists of the gray yellow atoms ($A$, $B$, $A'$ and $\hat{B}'$). This choice ensures that the edge dislocation preserves $\mathcal{I}$ symmetry. The positions of these atoms in terms of fractional lattice vectors are: $\tau_A = \left[ \left(1/2- u \right)a,\ b/2,\ c \right],\ \tau_B = (ua,\ 0,\ c),\ \tau_{A'} = -\tau_B,\ \tau_{B'} = -\tau_A + \boldsymbol{a}_2$ and $\tau_{\hat{B}'} = -\tau_A$, where $u = 0.08056$ and $c = 1.0654\,\text{\AA}$BP_structure. The pink sphere represents the Wannier center induced by the $p_z$ orbital; the red plane, which in our simulation connects the two $\mathcal{I}$-related edge dislocations, cuts through it. (e),(f) Real-space wavefunction density $|\psi|^2$ plots of polarization states for $\lambda = 0$ and $\lambda = 1$, respectively, with inversion center shown in red.
• Figure 2: Edge dislocation response of 3D silicon. (a) Brillouin zone of silicon, showing time-reversal invariant momentum (TRIM) Schindler_2022 points and their occupied inversion eigenvalues. (b) Band structure along the path shown in (a), with associated inversion eigenvalues at TRIMs. (c) Band structure with a pair of $\mathcal{I}$-related dislocations present, plotted against the only remaining momentum quantum number $\kappa_3=(\boldsymbol{k}\cdot{\boldsymbol{a}_3})/{2\pi}$. The band structure for positive $\kappa_3$ can be obtained by reflection along the $\kappa_3=0$ axis because of $\mathcal{T}$ symmetry. Simulations are performed on a lattice of $30 \times 30$ unit cells spanned by $\boldsymbol{a}_1$ and $\boldsymbol{a}_2$ with $100$ sample points for momentum $\kappa_3$. The dislocation is introduced by cutting the 3D material along a plane (keeping unit cells intact), removing $10$ unit cells, and then gluing the lattice back together Schindler_2022. We enforce periodic boundary conditions in all three directions. (d) Real-space unit cell geometry, common to all three 3D semiconductors that we study. There are two silicon atoms in a unit cell, with positions $\tau_A = (0,0,0)$ and $\tau_B = \frac{1}{4}(1, 1, 1)$ in fractional lattice vectors. (e),(f): Real-space wavefunction $|\psi|$ plots for polarization states at $\kappa_3=-1/2$ and $\kappa_3=0$, respectively, with inversion center shown in red. We plot the absolute value of the amplitude instead of the density here for a more detailed visualization.