Fig. 1: Crystal structures of Bi₄(Br_1-xI_x)₄ and calculated band structures of Bi₄Br₄.

a Crystal structure with the unit cell labeled by red lines of α’ phase. The Bi and Br/I atoms are marked by blue and gray balls, respectively. The two sites of Bi atoms are denoted as Bi_in and Bi_out, respectively. The chain direction is along the crystal b axis direction and highlighted in red. b Three-dimensional (3D) Brillouin zone (BZ) with (001) and (100) projected surface BZs. Schematics represent two nondegenerate bands near the Fermi level at \(M\) and \(L\). c Calculated bulk bands of α’-Bi₄Br₄ without (W/O) spin-orbit coupling (SOC) (left panel) and with SOC (right panel), respectively. Parity eigenvalues of each band are labeled. The red and blue dots represent the contributions of the p-orbitals of Bi_in and Bi_out atoms, respectively. d, e Left: 3D topography of the (001) surface and the (100) surface of α’-Bi₄Br₄ with step height of 0.96 nm and 0.66 nm, respectively. Scale bar, 5 nm. Right: Atomic-resolved topography of the red squares in the left panels. Scale bar, 1 nm. The blue rectangle and yellow rectangle denote the (001) plane and (100) plane of the unit cell with corresponding atoms, respectively. Lattice constants of a, b, and c are ~ 1.33 nm, ~ 0.44 nm, and ~ 2.01 nm, respectively. f Atomic-resolved images of the (010) planes of the samples with different I content. Scale bar, 1 nm. The blue, yellow, and pink colors present different layers in each stacking mode. The red, yellow, and blue rhombuses stand for the (010) planes of the unit cells for α’ phase, γ phase, and α phase, respectively. g Crystal parameters summarized from our data as well as the previous references^{21,25,29,30,31,32,33}. Error bars are determined by average variation of X-ray diffraction (XRD), transmission electron microscopy (TEM), and scanning tunneling microscopy (STM) results, respectively. All the lattice constants increase with I content.