Self-organized pseudo-graphene on grain boundaries in topological band insulators

Semimetals are characterized by nodal band structures that give rise to exotic electronic properties. The stability of Dirac semimetals, such as graphene in two spatial dimensions, requires the presence of lattice symmetries, while akin to the surface states of topological band insulators, Weyl semimetals in three spatial dimensions are protected by band topology. Here we show that in the bulk of topological band insulators, self-organized topologically protected semimetals can emerge along a grain boundary, a ubiquitous extended lattice defect in any crystalline material. In addition to experimentally accessible electronic transport measurements, these states exhibit a valley anomaly in two dimensions influencing edge spin transport, whereas in three dimensions they appear as graphenelike states that may exhibit an odd-integer quantum Hall effect. The general mechanism underlying these semimetals—the hybridization of spinon modes bound to the grain boundary—suggests that topological semimetals can emerge in any topological material where lattice dislocations bind localized topological modes.

Figure 1

Grain boundary semimetal on a 1D grain-boundary in a translationally active 2D TBI (the $M$ phase of the BHZ model on a square lattice; see Appendix pp1). (a) Schematic illustration of a GB. The coordination discrepancy due to the angular mismatch $\theta$ of lattice basis vectors results in an effective array of dislocations, each described by Burgers vector $\mathbf{b}$ and marked by a “T” symbol, of spacing $d=\left|\mathbf{b}\right|/tan\theta$. (b) Real-space numerical tight-binding calculation of the low-energy states in the presence of a GB, which are described by an effective tight-binding model (inset). The spinons bound to the dislocations hybridize with tunneling amplitude $t$, which gives rise to an extended 1D state along the GB. The radii of the circles indicate the amplitude of a wave function, associated with the node at $k_x=\pi /d$ of panel (c), while the colors indicate the phase. The system also includes slight disorder to separate clearly the dislocation modes from the bulk states. (c) The characteristic midgap bowtie dispersion of the hybridized spinon bands (orange) and bulk bands (grey), corresponding to GB opening angle $\theta =18.4^{\circ }$ for which isolated dislocations along the GB are well defined, as the function of the momentum $0\le k_x\le 2\pi /d$ along the GB. (d) The same plot for the maximal opening angle $\theta ={45}^{\circ }$. While the picture of isolated dislocations breaks down and the simple nearest-neighbor hybridization based sinusoidal dispersion is lost, the defining TRS semimetal dispersion with the nodes at the TRS momenta survives. The data are for a $30\times 60$ site or larger system with periodic boundary conditions.

Figure 2

Grain-boundary semimetal localized on a 2D grain boundary in a translationally active 3D topological band insulator (the $R$ phase of the BHZ model on a cubic lattice; see Appendix pp1 for details). (a) Schematic illustration of the 2D GB, which now realizes a sheet of parallel 1D edge dislocation lines that extend along the grain boundary. Each edge dislocation hosts a pair of propagating helical modes localized along the dislocation cores that cross at $k_z=\pi$. (b) On a GB of opening angle $18.4^{\circ }$, the hybridization of the helical modes results in the semimetal band structure with two anisotropic pseudo-relativistic fermions at $(k_x,k_z)=(0,\pi )$ and $(\pi /d,\pi )$. Along the GB we recover the same midgap bowtie dispersion as in the 2D case (shown top right for $k_z=\pi$), while along the edge dislocations the hybridized modes (orange) still flow into the bulk bands (gray) as a function of $k_z$ (bottom right for $k_x=\pi /d$). The data are for a $60\times 60\times 90$ system with periodic boundary conditions.

Figure 3

The experimental setups for identifying the signatures of the graphenelike semimetals. (a) The one-dimensional bowtie dispersion of the spinon semimetal on a 1D GB implies a parity anomaly per spin component when an electric field $\vec{E}$ is applied along the grain boundary. The arrows indicate the shift in the spectrum from one valley to the other: When $S_z$ is conserved, valley $v_1$ accumulates an excess of spin down (red), while valley $v_2$ accumulates excess spin up (blue). These valleys can be associated with two coexisting channels that connect the helical edge states from the opposing surfaces. When $\vec{E}$ is applied along the GB, a current for both spin orientations is driven parallel to it. At the GB termination points this current is predicted to flow into the edge states that propagate to opposite directions resulting in a doubled spin Hall effectlike helical imbalance of the edge states on the two grains. Measuring the current imbalance $I$ is the hallmark signature of the valley anomaly exhibited by the spinon semimetal. (b) The spinon semimetal on a 2D GB may feature an odd-integer Hall effect with transverse conductivity ${\sigma }_{xz}=(2n+1)e^2/h$ in the presence of the perpendicular magnetic field $\vec{B}$. In the absence of external fields, another signature is provided by the diagonal ballistic optical conductivity of ${\sigma }_{zz}=(\pi /4)e^2/h$, which is half that of graphene.

Figure 4

Illustration of the grain-boundary lattice used in the simulations in the particular instance of spacing $d=8$.

Figure 5

(a) The hybridization energy $\epsilon$ as the function of the separation $d$ between two dislocations in units of the bulk band gap $\mathrm{\Delta }$. For different values of $M/B$ the hybridization energy always exhibits exponential decay and characteristic oscillations that exhibit clear sinusoidal oscillations at larger separations (inset). The kinks as functions of $d$ are artefacts of simulating smooth dislocation transport by adiabatically switching on and off the relevant couplings. (b) The two dislocation hybridization energies $\epsilon$ in the GB geometry as the function of the GB angle $\theta$ in units of the bulk band gap $\mathrm{\Delta }$. (c) The midgap spectra of the 2D BHZ model across the reduced Brillouin zone $0<k_x<2\pi /d$ for different GB angles $\theta =\arctan \left(\right|b|/d)$ and the approximation of each by the unperturbed minimal effective model $H_{2\mathrm{D}}=tsinq$ over $0<q<2\pi$ with the hybridization energy $\epsilon \left(d\right)=t/2$ as the only input. As the dislocation separation $d\sim {\theta }^{-1}$ increases, the nearest-neighbor approximation becomes more valid and we find excellent agreement. All data have been produced using a $90\times 60$ system with $M/B=6$.

Figure 6

Stability of the graphenelike spinon semimetal. (a) Inclusion of a Rashba term of magnitude $R$ breaks $S_z$ conservation, which leads to suppression of the bandwidth and different velocities at the cones, but TRS guarantees that the cones are stable. (b) An $S_z$ conservation preserving Zeeman term of magnitude $h_z$ gives rise to a mass for the spinons. In the hybridized spectrum it shifts the spinon bands to opposite directions in energy and thereby moves the cones towards each other. The cones annihilate and gap out at a quadratic band touching point when $h_z=2\epsilon$. (c) When local chemical potential disorder of magnitude $w$ is introduced, i.e., $\mu$ is a random variable picked from box distribution $\left[\right(1-w\left)\right(M-4B),(1+w\left)\right(M-4B\left)\right]$, the hybridization energies $\epsilon$ become also disordered with the fluctuations, as quantified by the standard deviation $\mathrm{Std}\left(\epsilon \right)$ around the mean $\langle \epsilon \rangle$ increasing monotonously. The rate of increase depends on the microscopic details around the nearest-neighbor two-dislocation hybridization energy corresponding to different GB angles $\theta$ as show in Fig. 5. (d) In the full 2D TBI, we find that the gapless spinon semimetal persists at least up to $50\%$ chemical potential disorder ($n$ enumerates the states), which is insufficient to close the bulk gap (inset). We have also simulated randomized dislocation positions (dislocations along a GB of spacing $d$ are randomly displaced by up to $d$ lattice constants) and find similar stability. All data have been produced using a $90\times 60$ system in two dimensions with $M/B=6$. The disorder data have been produced for system size $42\times 30$ with periodic boundary conditions and averaged over 100 disorder realizations.