Shear-resistant topology in quasi one-dimensional van der Waals material Bi$_4$Br$_4$

TL;DR

This study uncovers a new $b/3$ in-plane shift of Bi$_4$Br$_4$ chains on the (001) surface, maintaining AB stacking and revealing a shear-strain–driven origin for the structure with a residual in-plane strain of $\gamma \approx 7.5\%$. Using low-temperature STM/STS, the authors observe a bulk insulating gap of $E_g \approx 240$ meV and metallic hinge-like edge states at monolayer steps, indicating higher-order topology consistent with a HOTI. Complementary DFT (HSE06/VASP and QE) shows that both the $b/2$ and $b/3$ configurations are quantum spin Hall insulators due to SOC-induced parity exchange at the Y point, though the $b/3$ geometry sits closer to a trivial transition with a smaller inverted gap. The results demonstrate the robustness of topological edge features under in-plane chain shifts and highlight strain engineering as a route to access or tune topological phases in quasi-1D van der Waals materials.

Abstract

Bi$_4$Br$_4$ is a prototypical quasi one-dimensional (1D) material in which covalently bonded bismuth bromide chains are arranged in parallel, side-by-side and layer-by-layer, with van der Waals (vdW) gaps in between. So far, two different structures have been reported for this compound, $α$-Bi$_4$Br$_4$ and $β$-Bi$_4$Br$_4$ , in both of which neighboring chains are shifted by $\mathbf{b}/2$, i.e., half a unit cell vector in the plane, but which differ in their vertical stacking. While the different layer arrangements are known to result in distinct electronic properties, the effect of possible in-plane shifts between the atomic chains remains an open question. Here, using scanning tunneling microscopy and spectroscopy (STM/STS), we report a new Bi$_4$Br$_4$(001) structure, with a shift of $\mathbf{b}/3$ between neighboring chains in the plane and AB layer stacking. We determine shear strain to be the origin of this new structure, which can readily result in shifts of neighboring atomic chains because of the weak inter-chain bonding. For the observed $b/3$ structure, the (residual) atomic chain shift corresponds to an in-plane shear strain of $γ\approx7.5\%$. STS reveals a bulk insulating gap and metallic edge states at surface steps, indicating that the new structure is also a higher-order topological insulator, just like $α$-Bi$_4$Br$_4$, in agreement with density functional theory (DFT) calculations.

Figures (6)

• Figure 1: (a) Atomic model of $\alpha$-Bi4Br4, with AB layer stacking. The black line shows a projection of the monoclinic bulk unit cell. Each monolayer is a quantum spin Hall (QSH) insulator with edge states (gray circles). On the right the edge states hybridize, while on the left, they form two hinge states. See main text for more details. (b) Scanning tunneling spectrum of the $b/3$ surface with band gap $E_\text{g}$, measured at the setpoint $V_\text{tip} = 0.6V$ and $I_\text{t} = 50p A$. (c) Schematic band structure with topological gap and edge states. (d) Top-view of the $\alpha$-Bi4Br4(001)-A surface. The lower Br atoms are indicated by smaller size. Neighboring chains are shifted by $\mathbf{b}/2$ with respect to each other. The rectangle shows the projected monoclinic bulk unit cell (which equals the non-primitive centred surface unit cell), the parallelogram is the primitive surface unit cell. (e) The experimentally observed surface structure in which neighboring chains are shifted by $\mathbf{b}/3$ against each other. The primitive surface unit cell is shown. (f) STM topography recorded at $I_\text{t} = 0.4n A$ and $V_\text{tip} = -0.4V$. The atomic model from (e) is superimposed on the chains, showing good agreement. The image was upscaled using a linear interpolation. The spectrum displayed in (b) was smoothed using a moving average with a 14m eV window.
• Figure 2: (a) STM topography of a step edge on the Bi4Br4(001) surface, acquired at $I_\text{t} = 0.1n A$ and $V_\text{tip} = -0.3V$. The image shows a one-monolayer step edge with atomic resolution on both the upper and the lower terraces. (b) Line profile indicated by the orange box in (a).
• Figure 3: STM topography of the upper terrace (a) and the lower terrrace (b) in Fig. \ref{['fig:Figure2_new']}. Both images were recorded at $I_\text{t} = +0.1n A$ and $V_\text{tip} = +0.4V$. They were upscaled using a linear interpolation. Comparing the location of the dark depression in the unit cell to the location of the lower Br atom in the atomic models, we conclude that the upper terrace exposes an A surface, the lower terrace a B surface. (c) Total energy per unit cell relative to the $b/2$ structure as a function of the shift $s=-d\cot\varphi$ (plotted in units of $b$) between neighboring chains, where $d$ is the inter-chain distance. $\varphi$ is the angle between the constant unit cell vector $\mathbf{b}_\text{s}$ and the changing $\mathbf{a}_\text{s}$ associated with $s$. The inset defines $\varphi$, $d$, $s$, $a'_\text{s}$, and $b'_\text{s}$.
• Figure 4: (a) STM topography of a one monolayer deep deep trench on the surface, bordered by two Bi4Br4 monolayer steps. Scanning tunneling spectra in (b) and (d) were recorded on terraces right and left of the trench and directly at the two step edges. The filled circles indicate the approximate positions were the spectra were measured. All spectra are normalized to 1 at 0.3eV. Spectra at the step edges (orange and blue) show metallic edge states instead of a band gap. Note that the band gaps in the spectra from the terraces (green, yellow, and red) vary in width and alignment with the Fermi level. We explain this with sample degradation, see main text for more details. (c) An atomic model of the trench with the edge states indicated by the filled circles. The spectra displayed in (b) and (d) were smoothed using a moving average with a 14m eV window.
• Figure 5: (a) and (b) show the Bi$_\text{in}$-$p_x$ and Bi$_\text{ex}$-$p_x$ projected orbital character of the conduction and the valence bands around the Y point without (a) and with (b) SOC for the $b/2$ structure, see zhouLargeGap2014. (c) and (d) show the same projected orbital character of the bands around the Y point for the $b/3$ structure. (c) and (d) were calculated using the lattice constants indicated in the main text. For both structures, SOC exchanges the two orbitals leading to an exchange of parity, which makes both the $b/2$ (b) and the $b/3$ (d) structure a QSH insulator. (e) - (g): evolution of the inverted band gap $E_\text{g}$ as a function of the inter-chain distance $d$ and the lattice constant $b_\text{g}$. The dashed lines are a guide for the eyes. (e) Dependence of $E_\text{g}$ on the inter-chain distance $d$, when fixing $b = 0.4338n m$. The inverted band gap decreases, until it closes at $d = 0.66n m$ and then transitions to a trivial gap. (f) Fixing $d = 0.6532n m$ and increasing the lattice vector $b_\text{s}$ enhances the the inverted band gap. (g) The same applies if the ratio $b_\text{s}/d = 0.664$ is kept constant.