Purely rotational symmetry-protected topological crystalline insulator $\alpha$-Bi₄Br₄

Topological crystalline insulators (TCIs) are insulators in which the nontrivial band topology is protected by crystalline symmetries [1–4]. Although the availability of a large number of crystalline space groups suggests the existence of a rich variety of TCI states, to date, only mirror symmetry-protected TCIs have been experimentally identified [4–19]. Recently, a new class of TCIs, where the protection is through the presence of N-fold rotational symmetries, has been proposed [20]. The rotation symmetry protected TCIs are distinct from the common topological insulators (TIs). For example, the surface normal to the rotational axis in these TCIs hosts N Dirac cones where the Dirac points can appear at any general k point in the surface Brillouin zone (BZ) and are not restricted to lie on high symmetry points or lines in the BZ. Moreover, such a 3D TCI can support 1D topological edge states due to the higher order bulk-boundary correspondence involved in this case [21–27]. Additionally, in a rod with N-fold rotational symmetry that is finite sized along the two directions perpendicular to the rotational axis, the hinges of the rod host N helical 1D edge states.

Experimental realization of purely rotational symmetry-protected TCIs has, however, remained elusive in real materials. One challenge is that the rotational-symmetry related topological invariants are difficult to compute in first-principles calculations as they are defined in terms of Wannier center flows [28]. Recent progress in this connection relates crystalline symmetry-protected topological invariants with symmetry eigenvalues of the electronic states [29–36]. In particular, Song et al [32] and Khalaf et al [33], found that when a certain additional symmetry Y is present, the topological invariants of TCIs protected by symmetry X can be inferred from the Y-symmetry eigenvalues of energy bands. This proposal of symmetry indicators has facilitated first-principles studies of new TCIs [37–41].

Another challenge involves how to isolate the nontrivial topology protected by the rotational symmetry from that protected by the mirror symmetry. We illustrate this point by taking the example of a tetragonal crystal lattice with point group D_4h (space group P4/mmm). The nontrivial topology protected by the out-of-the-plane four-fold rotational symmetry $4_{[0\,0\,1]}$ would lead here to four Dirac fermions on the $(0\,0\,1)$ surface, see figure 1(a). (The conventional basis is used for $[h\,k\,l]$ and $(h\,k\,l)$, see supplementary materials (stacks.iop.org/TDM/6/031004/mmedia)). However, the band topology that gives rise to the nontrivial $4_{[0\,0\,1]}$ index is likely to harbor the mirror-symmetry-protected nontrivial topology in the mirror planes ($M_{[1\,0\,0]}$, $M_{[0\,1\,0]}$, $M_{[1\,1\,0]}$ and $M_{[1 - 1\,0]}$). Consequently, the four Dirac fermions must be located along the $[1\,0\,0]$, $[0\,1\,0]$, $[1\,1\,0]$ or $[1\bar{1}0]$ high-symmetry directions (figure 1(a)).

Zoom In Zoom Out Reset image size

We surmise that a particular class of monoclinic lattices with the point group C_2h are ideally suited to searching for purely rotational symmetry-protected TCIs. Our reasoning is as follows. As shown in figure 1(b), a C_2h lattice supports three symmetry operations, a two-fold rotational axis $2_{[0\,1\,0]}$, a mirror plane $M_{[0\,1\,0]}$, and the space inversion symmetry $\mathcal{I}$. Interestingly, the $(0\,1\,0)$ surface only preserves the two-fold rotational symmetry $2_{[0\,1\,0]}$ but breaks $M_{[0\,1\,0]}$. On the other hand, the presence of the inversion symmetry $\mathcal{I}$ in the bulk still allows for the application of the symmetry indicator scheme. These considerations suggest that C_2h crystals are ideal for searching for purely rotational symmetry-protected TCIs and that two unpinned Dirac fermions can appear at general k points on the $(0\,1\,0)$ surface due to the nontrivial topology protected by $2_{[0\,1\,0]}$.

Guided by the preceding analysis, we have carefully searched for materials with a monoclinic C_2h lattice. From this, we identified the van der Waals material $\alpha$-Bi₄Br₄ as a TCI which is purely protected by rotational symmetry. Figure 1(c) illustrates the experimentally determined lattice structure of $\alpha$-Bi₄Br₄ with point group C_2h (space group C2/m ($\#12$)) [42, 43]. Consistent with figure 1(b), the two-fold rotational axis in $\alpha$-Bi₄Br₄ lies along the $[0\,1\,0]$ direction (figure 1(c)). Notably, recent experiments [44–46] have identified a topological insulator state in the sister compound $\beta$-Bi₄I₄, and a weak TI phase has been predicted in $\beta$-Bi₄Br₄ [47], see supplementary materials. On the other hand, the electronic and topological properties of $\alpha$-Bi₄Br₄ remain unexplored, although its monolayer has been proposed to host a 2D quantum spin Hall insulator state [48].

The first-principles calculations were based on the generalized gradient approximation (GGA) [57] using the full-potential projector augmented wave method [58, 59] as implemented in the VASP package [60, 61]. The experimental crystal structure of $\alpha$-Bi₄Br₄ was used [42]. The electronic structure of bulk $\alpha$-Bi₄Br₄ was calculated using a 15 $\times$ 15 $\times$ 5 Monkhorst–Pack k-mesh over the Brillouin zone (BZ) with the spin–orbit coupling (SOC) included self-consistently. We used Bi and Br p  orbitals to construct Wannier functions, without performing the procedure for maximizing localization [62]. The surface spectral weight was computed via a semi-infinite Green's function method [63] using our effective Wannier tight-binding Hamiltonian. We also constructed a C₂-preserved 1D tube with a size close to 4 $\times$ 1 $\times$ 4 supercell from our Wannier Hamiltonian to examine the 1D hinge states, see figures 2(e) or S3 of the supplementary material for details of the crystal structure.

Band structure calculations were carried out using the VASP package [49] and the experimental crystal structure of $\alpha$-Bi₄Br₄ [42]. The Wannier [50] based tight-binding model was used for investigating surface states. The bulk band structure with spin–orbit coupling (SOC) (figure 1(d)) shows a band gap around 0.4 eV. The bands at low energies are dominated by four bands, two conduction bands (CBs) and two valence bands (VBs), all of which mainly involve the Bi 6p -orbitals. By examining the evolution of the band structure as a function of SOC strength, we identified an even number of band inversions between bands of opposite parity eigenvalues at the L point. The first band inversion between VBs and CBs can be seen from figures 1(f)–(g), which show how a trivial insulator evolves into a strong TI (STI). Subsequently (figure 1(h)), the CBs and VBs experience internal band inversions without closing the band gap and maintain the STI phase. In the final stage, another band inversion between the VBs and CBs takes place as seen in figures 1(h)–(i). The presence of an even number of band inversions rules out the possibility that the $\alpha$-Bi₄Br₄ hosts a time-reversal symmetry-protected Z₂ STI state. Interestingly, a rotation symmetry-protected TCI state can be derived from bands hosting two strong TI representations [20] where two band inversions will be needed. In other words, the two band inversions between VBs and CBs in $\alpha$-Bi₄Br₄ can give rise to a rotation symmetry-protected TCI state.

We now turn to discuss the symmetry-based indicators (SIs). As shown in [32, 33], crystals in space group $\#12$ are characterized by four symmetry indicators, three ${\mathbb{Z}}_2$ indices and one ${\mathbb{Z}}_4$ (${\mathbb{Z}}_{2,2,2,4}$) index. By enumerating the symmetry eigenvalues of the electron states at high symmetry points, we obtain ${\mathbb{Z}}_{2,2,2,4} =\lbrace 0,0,0,2 \rbrace$ in $\alpha$-Bi₄Br₄. Table 1 shows the topological states corresponding to ${\mathbb{Z}}_{2,2,2,4} =\lbrace 0,0,0,2 \rbrace$, following [32], see the caption of table 1 for the definition of various invariants. Importantly, the symmetry indicators (${\mathbb{Z}}_{2,2,2,4} =\lbrace 0,0,0,2 \rbrace$) point to two possible topological states, one being a purely rotational symmetry-protected TCI with $n_{2_{[0\,1\,0]}}=1$, and the other a purely mirror symmetry-protected TCI with $n_{\mathcal{M}_{[0\,1\,0]}}=2$. Note that the $\mathcal{I}$-protected topology is referred to as a higher-order topological insulator (HOTI) [25], and is not considered here. In order to uniquely determine the topological state of $\alpha$-Bi₄Br₄, we have further calculated the mirror Chern number $n_{\mathcal{M}_{[0\,1\,0]}}$, and found $n_{\mathcal{M}_{[0\,1\,0]}}=0$ for $\alpha$-Bi₄Br₄. These results thus reveal that $\alpha$-Bi₄Br₄ is a purely rotational symmetry-protected TCI with $n_{2_{[0\,1\,0]}}=1$.

Table 1. The two possible topological states for the symmetry indicator ${\mathbb{Z}}_{2,2,2,4} =\lbrace 0,0,0,2 \rbrace$ [32], where $(\nu_0;\nu_1\nu_2\nu_3)$ are the $\mathbb{Z}_2$ invariants for a 3D $\mathbb{Z}_2$ TIs, and are all seen to be zero, indicating that the state is neither a strong nor a weak TI. $n_{2_{[0\,1\,0]}}$ is the topological invariant for the two-fold rotational symmetry $2_{[0\,1\,0]}$; it is also a $\mathbb{Z}_2$ invariant. $n_{2_{[0\,1\,0]}}=1$ corresponds to a rotational symmetry-protected TCI, which features two unpinned Dirac surface states on the $(0\,1\,0)$ surface. $n_{\mathcal{M}_{[0\,1\,0]}}$ is the topological invariant for the mirror plane $\mathcal{M}_{[0\,1\,0]}$ (the mirror Chern number), and is a ${\mathbb{Z}}_N$ invariant. $n_{\mathcal{M}_{[0\,1\,0]}}=N$ corresponds to a mirror symmetry-protected TCI, which features N Dirac surface states, see supplementary materials for a discussion of the symmetry indicators and the related topological invariants.

$(\nu_0;\nu_1\nu_2\nu_3)$	$n_{2_{[0\,1\,0]}}$	$n_{\mathcal{M}_{[0\,1\,0]}}$
(0;000)	1	0
(0;000)	0	2

We emphasize that according to [32], crystals in space group $\#12$ cannot host a weak TI and a rotational symmetry-protected TCI phase at the same time. Therefore, $\alpha$-Bi₄Br₄ cannot be a weak TI, and the predicted weak TI $\beta$-Bi₄Br₄ [47] cannot be a rotational symmetry-protected TCI.

In order to confirm the topological nature of $\alpha$-Bi₄Br₄, we investigated its novel topological boundary modes. We first focus on uncovering the unpinned Dirac surface states on the $(0\,1\,0)$ surface protected by the two-fold rotational symmetry $2_{[0\,1\,0]}$. By calculating the band structure throughout the $(0\,1\,0)$ surface BZ, we identified two Dirac cones. Their Dirac points (DPs) are located at $\pm(0.0434\;\frac{2\pi}{a},0.3550\;\frac{2\pi}{c})$. These are seen to be general k points, as expected, which are only related by the two-fold rotational symmetry $2_{[0\,1\,0]}$. The calculated surface states along a k path passing through the DP ($\bar{P}-\bar{{\rm DP}}-\bar{Z}$, figure 2(c)) directly shows the topological Dirac surface states. On the other hand, along the path $\bar{P}-\bar{Z}-\bar{\Gamma}$ that does not go through the DP, the surface states are gapped (figure 2(d)).

Zoom In Zoom Out Reset image size

We next discuss the 1D topological hinge states protected by $2_{[0\,1\,0]}$. For this purpose, we calculated the band structure of a 1D $\alpha$-Bi₄Br₄ rod that is finite-sized along $[\bar{1}\,1\,0]$ and $[0\,0\,1]$ but periodic along $[0\,1\,0]$ (see the supplementary material for details of the crystal structure). Figure 2(f) shows the existence of 1D helical edge states inside the bulk band gap. We also explored the real-space distribution of the electron wave functions of the 1D helical edge states. As shown in figure 2(e), these states are localized on the edges shared by adjacent side surfaces, confirming that they are indeed topological hinge states. Experimentally, the unpinned Dirac surface states could be directly imaged via photoemission spectroscopy, whereas the helical hinge states could be probed via scanning-tunneling spectroscopy. Although it was challenging, the band structure of the hinge states (figure 2(f)) indicates that a quantized longitudinal charge conductivity ($\sigma_{yy}$) could be measured on the 1D structure proposed here.

We further highlight the novel electronic properties enabled by the unpinned topological Dirac fermions on the $(0\,1\,0)$ surface. We first discuss the existence of van Hove singularities (VHSs) on the $(0\,1\,0)$ surface. At an energy close to the Dirac point (E  =  2.9 meV), the surface state exhibits two separated electron-like contours (figure 3(c)). As one goes to higher energies above the Dirac point, the contours expand and eventually merge. At E  =  11.9 meV, two concentric contours emerge (figure 3(d)). Such a Lifshitz transition signals the presence of a saddle point in the surface electronic band structure, and an associated VHS with diverging density of states (DOS). In order to gain insight into the nature of Dirac surface states and their relationship to the VHS, we have constructed an effective $k\cdot p$ model. Note that the $(0\,1\,0)$ surface supports two symmetries, the two fold rotation $2_{[0\,1\,0]}$ and time-reversal $\mathcal{T}$. Based on our first-principles calculations, we have chosen the following symmetry adapted bases for the $k\cdot p$ model. $|\Psi_{\mathbf{k}} \rangle = {(\psi_{+\uparrow}, \psi_{+\downarrow} , \psi_{-\uparrow}, \psi_{-\downarrow})}^{T}$ as,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:bases} |\psi_{\pm,s} \rangle &= \sum_{\alpha = x,z} \lambda_{\alpha}s (| p^A_{\alpha}, s \rangle \pm | p^B_{\alpha}, s \rangle) \nonumber \\ & \quad + \;\lambda_{y}s (| p^A_{y}, s \rangle \mp | p^B_{y}, s \rangle), \nonumber \end{align} \tag{ 1 }$$

where $\pm$ is the eigenvalue of $2_{[0\,1\,0]}\times\mathcal{T}$, $s\;(=\uparrow\downarrow)$ is the spin index, and $\lambda_{i}$ is the orbital weight for the p _i orbital. The A and B sub-lattices are indicated by the blue and green triangles in figure 1(c) (the (0 1 0) surface here is spanned by the x-z plane). The simplest $4 \times 4$ dimensional model including all the allowed constant terms $\delta_{ij}$ takes the form,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:Hamiltonian} H(\mathbf{k}) &=v_R\tau_{0} (k_{x} \sigma_{1} - k_{z} \sigma_{3}) + \delta_{30} \tau_3 \sigma_0 + \delta_{21} \tau_2 \sigma_1 \nonumber \\ & \quad + \delta_{23} \tau_2 \sigma_3, \nonumber \end{align} \tag{ 2 }$$

where the Pauli matrices $\sigma_i$ and $\tau_i$ denote the spin and sub-lattice degrees of freedom, respectively. As shown in figures 3(e) and (f), the Lifshitz transition is reproduced by our $k\cdot p$ model. The DOS as a function of energy based on this $k\cdot p$ model, shows the expected divergent behavior at the energy position of the VHS (figure 3(f)). The VHS and the related diverging DOS are favorable conditions for the appearance of correlated quantum phenomena [51–53].

Zoom In Zoom Out Reset image size

We now discuss the spin texture of the unpinned Dirac fermions and their implications for spin transport. At energies close to the surface Dirac points (see, for example, figure 3(c)), the spin texture is quite similar to that of existing topological insulators in that the spin polarization winds tangentially around the closed constant energy contours. On the other hand, as we move to energies further away from the Dirac point (e.g. E  =  33 meV in figure 4(b)), the constant energy contours evolve into separated lines that traverse the whole surface BZ. Interestingly, in this regime, the spin texture of two left movers are dominantly up spin-z whereas the two right movers are down spin-z (the spin orientation is indicated by the red arrow in figure 4(b)), realizing a 3D quantum spin Hall effect [9]. Importantly, in $\alpha$-Bi₄Br₄, this spin texture is realized in a large energy window (0–300 meV) driven by the large bulk energy band gap. This unique spin texture could lead to a large spin transport effect. In this connection, we have calculated the current-induced spin accumulation ($\delta S$) due to the Rashba–Edelstein effect [54], i.e. passing an electrical current tilts the Fermi surface out of its equilibrium distribution, yielding a net spin polarization. We thus expect a sizeable accumulation of the spin polarization along z (S^z) when passing an electrical current along the x direction. Note that the spin accumulation can be calculated within the linear response theory [55] as follows.

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \delta\mathbf{S}=\frac{e\hbar}{\Gamma V}\sum_n \delta(\varepsilon_{{\bf F}}-\varepsilon_{n}) \langle n |\mathbf{S}|n \rangle \langle n |\mathbf{v}\cdot\mathbf{\Xi}|n \rangle, \nonumber \end{align} \tag{ 3 }$$

where e, $\hbar$, $V$ and $\Gamma$ are the unit charge, the Planck constant, volume of the unit cell, and broadening due to impurities, respectively, $\varepsilon_{{\bf F}}$ is the Fermi energy, $\varepsilon_{n}$ is the energy eigenvalue of the state $|n \rangle$, and $\mathbf{S}$, $\mathbf{v}=\frac{\partial H(\mathbf{k})}{\hbar\partial \mathbf{k}}$ and $\mathbf{\Xi}$ denote the spin, velocity and the external electric field. The calculated results are shown in figure 4(c), where $\delta S^{\alpha}_{\beta}$ gives the spin polarization along the $\alpha$ direction by following an external current along the $\beta$ direction. The largest response is indeed $\delta S_{x}^{z}$. In regime (I), the surface states (SSs) and the spin texture resemble two copies of TI centered at generic momenta (referred to as type-A behavior, figure 4(a)). $\delta S_{x}^{z}$ increases with increasing energy since the two Fermi surface contours enlarge. For regime (III), the four parallel Fermi contours with high s_z polarization lead to the largest $\delta S_{x}^{z}$, which is almost constant in this regime due to its quasi-1D characteristic (referred to as type-B behavior, Fig 4(a)). Regime II is a crossover region that connects regimes I and III. We expect that this unique spin accumulation can be directly probed by Kerr rotation microscopy [56].

Zoom In Zoom Out Reset image size

In conclusion, our analysis based on first-principles computations shows that $\alpha$-Bi₄Br₄ is a candidate material for the newly proposed C₂-protected TCIs. Existence of this unique TCI phase is confirmed by the presence of unpinned Dirac surface states on the $(0\,1\,0)$ surface, and of 1D topological hinge states on a C₂ preserving $\alpha$-Bi₄Br₄ nanorod. Moreover, we have delineated the evolution of (0 1 0) surface states as a function of energy above the Dirac points to show that $\alpha$-Bi₄Br₄ hosts a 3D quantum-spin-Hall-effect, and can thus be expected to support large spin transport.

T-RC and XZ were supported by the Young Scholar Fellowship Program by Ministry of Science and Technology (MOST) in Taiwan, under MOST Grant for the Columbus Program MOST108-2636-M-006-002, National Cheng Kung University, Taiwan, and National Center for Theoretical Sciences (NCTS), Taiwan. This work is supported partially by the MOST, Taiwan, Grants No. MOST 107-2627-E-006-001. The work at Northeastern University was supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences Grant No. DE-FG02-07ER46352, and benefited from Northeastern University's Advanced Scientific Computation Center (ASCC) and the NERSC supercomputing center through DOE grant number DE-AC02-05CH11231. LF was supported by DOE Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-SC0018945. NG and SYX acknowledge support by the STC Center for Integrated Quantum Materials, NSF Grant No. DMR-1231319.

Note added. After we had submitted our manuscript, [38] came to our attention, which also proposes $\alpha$-Bi₄Br₄ as a rotation-symmetry protected TCI.