Topological phase transition in quasi-one-dimensional bismuth iodide Bi4I4

Topological quantum phase transitions (TQPTs) represent a fascinating type of phase transitions characterized by global topological invariants, which diverge from the conventional symmetry-breaking phase transitions described by local order parameters. TQPTs not only hold significant scientific implications for elucidating topological properties under external perturbations^1,2,3,4,5 but also pave an avenue for the utilization of topological quantum materials with high tunability in future electronic and spintronic devices^6,7,8,9. Throughout a TQPT, the ground-state wavefunction evolves with a controlling parameter such as strain, chemical substitution, electric field, and temperature^{10,11,12,13,14,15,16,17}, typically associated with a change in the energy gap of the boundary states.

Recently, quasi-one-dimensional (quasi-1D) bismuth halides Bi₄X₄ (X = Br, I) have garnered great research attention owing to the abundant topological quantum phases in this family compounds, including strong/weak topological insulator^18,19,20, higher-order topological insulator^21,22,23, and Weyl semimetal²⁴. These intriguing topological quantum phases are sensitive to the stacking sequence and interlayer coupling of Bi₄X₄ layers constituting the bulk crystals^25,26,27. By manipulating external conditions, e.g., tuning temperature, applying strain or chemical substitution, TQPT can be established and feasibly controlled.

Taking Bi₄I₄ as an example, monolayer Bi₄I₄ consists of arrays of chains along the b axis, which has been verified as near a 2D quantum spin Hall (QSH) insulator/normal insulator phase boundary^28,29,30. Bulk α/β-Bi₄I₄ share the same chain structure except that the α-phase comprises two Bi₄I₄ chains relatively shifting along the a axis in a unit cell (Fig. 1a, b). Consequently, the interaction between the QSH edge states in each Bi₄I₄ monolayer is distinctive in the two phases. For β-Bi₄I₄, the monolayers stack along c with the metallicity of the side surface preserved by the symmetry, realizing a WTI phase^{4,12,31,32,33,34}. By contrast, the adjacent QSH edge states are in different chemical environments in α-Bi₄I₄. Therefore, the hybridization between adjacent QSH edge states induces gapped (100) surface states (Fig. 1b, c). Remarkably, for particular stacking sequences, gapless modes, namely topological hinge states, survive at the hinges of the crystal as enforced by the higher-order bulk-boundary correspondence (see the red dots in Fig. 1b)^27,35. Since the dimensional difference between the bulk and conducting boundary mode is larger than 1, this intriguing topological state is classified as a HOTI. Upon increasing temperature, α-Bi₄I₄ undergoes a structural transition to β-Bi₄I₄³⁶, establishing a temperature-induced TQPT.

a The crystal structure of α-Bi₄I₄. The black lines indicate the unit cell. b Schematic illustration of the boundary states on the (100) surfaces. Each dot represents the QSH edge state of a Bi₄I₄ monolayer. The yellow-dot pair in an ellipse and red dots represent the quantum-hybridized (100) surface states and the topological hinge states, respectively. c Illustration of the electronic structure of α-Bi₄I₄ on the (100) side surface. The gray, yellow, and red curves indicate the bulk, surface, and hinge states, respectively. d–f The same as (a–c) except for β-Bi₄I₄. The green dots in (e) and green curves in (f) represent the topological surface states on the (100) surface of β-Bi₄I₄. g Cross-section image of the (010) surface of α-Bi₄I₄ with atomic resolution measured with scanning transmission electron microscope (STEM). h X-ray diffraction measurements of α-Bi₄I₄. The STEM and XRD measurements are performed at 298 K.

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Angle-resolved photoemission spectroscopy (ARPES), serving as a powerful tool to elucidate the electronic structure of single crystals, has been performed to unravel the band structure of bismuth halides^{26,37,38,39,40}. While high-resolution laser-ARPES experiments have shown evidence for the HOTI phase in α-Bi₄Br₄, the investigation of Bi₄I₄ is more intricate^35,41,42. Gapless (100) surface states have been observed in β-Bi₄I₄^35,41, confirming its non-trivial topological property. However, there are still debates on the existence of gapless surface states on the (001) surface, making it elusive whether β-Bi₄I₄ is a WTI or strong topological insulator (STI). As for α-Bi₄I₄, although a gapped (100) surface state has been detected at the ({bar{Gamma }}_{100}) point³⁵ in accordance with the theoretical prediction, further confirmation is still required to ascertain whether the side surface states are fully gapped over the whole surface Brillouin zone (especially at the (bar{Z}) point) and conclusively identify the HOTI phase. Moreover, only a subtle change of the width of ARPES spectra was observed across the structural transition³⁵, with the temperature-evolution of the fine electronic structure, especially the surface gap, yet to be investigated.

In this work, by performing laser-based micro-ARPES measurements with high energy and momentum resolution^37,43, we systematically investigate the temperature evolution of the electronic structure of Bi₄I₄. At low temperatures, we observe gaps of about 40 meV and 5 meV at the ({bar{Gamma }}_{100}) and (bar{Z}) points on the (100) surface, confirming the HOTI phase of α-Bi₄I₄ (Fig. 2c). With increasing temperature, we directly observe the evolution of the gapped surface states into a gapless Dirac fermion across the transition temperature T_S~300 K. The magnitude of the gap shows a thermal hysteretic behavior, confirming the first-order nature of the TQPT. Moreover, our measurements unveil that a fully gapped (001) surface of β-Bi₄I₄, which, together with the gapless topological surface states (TSSs) on the (100) surface, compellingly evidences the WTI phase of β-Bi₄I₄ (Fig. 2d). Our experimental observations are in excellent agreement with ab-initio calculations and provide more systematic insights into the TQPT of quasi-1D Bi₄I₄.

a, b Bulk and surface Brillouin zones of α/β-Bi₄I₄ with high-symmetry points indicated, respectively. c, d Schematics showing the surface band structure of HOTI phase of α-Bi₄I₄ and WTI phase of β-Bi₄I₄. The blue curves and the red line indicate the (100) surface states and in-gap hinge states, respectively. The red arrows indicate the location of the gapless Dirac fermions.

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Electronic structure and topological property of α-Bi₄I₄

Figure 3a shows the ARPES spectrum of α-Bi₄I₄ at the ({bar{Gamma }}_{100}) point on the (100) surface. We resolve valence and conduction bands of the (100) surface states, characterizing the splitting feature and surface gap caused by hybridization of the QSH edge states from adjacent Bi₄I₄ layers (Fig. 1c). According to the curvature spectra in Fig. 3c, the energy gap between the valence and conduction band is estimated to be about 31 meV (the peak-to-peak gap between the band crossing points at ({bar{Gamma }}_{100}) is about 40 meV). At the (bar{Z}) point, the surface states exhibit a Dirac-cone-like dispersion with a gap of about 7 meV (Fig. 3b, d). Our measurements agree with the calculation in Fig. 3e, f. The gapped dispersion at (bar{Z}) can be further evidenced by fitting the Dirac dispersion as shown in Fig. 3b. The band dispersion is firstly extracted by fitting the momentum distribution curves (MDCs) to the Lorentzians (represented by the green dots). Then the upper and lower branches of the extracted dispersion are fitted to linear dispersions. The energy interval between the crossing points of the upper and lower branches indicates an energy gap of about 5 meV at the Dirac point, in line with the curvature analysis (Fig. 3d) and the theoretical calculation (Fig. 3f).

a, b Band structure of the (100) surface states at the ({bar{Gamma }}_{100}) and (bar{Z}) points measured by laser-based micro-ARPES. The green dots in (b) are determined by fitting the momentum distribution curves (MDCs) to Lorentzians and the yellow dashed lines are the fit of the extracted dispersion to the massive Dirac-fermion dispersion. c, d Curvature plots of the ARPES spectra in (a) and (b). e, f (100) surface-projected band structures calculated at the ({bar{Gamma }}_{100}) and (bar{Z}) points, respectively. g, h ARPES spectra of α-Bi₄I₄ at the ({bar{Gamma }}_{001}) and (bar{M}) points on the (001) surface. i, j Curvature plots of the ARPES spectra in (g) and (h). (k, l) (001) surface-projected band structures calculated at the ({bar{Gamma }}_{001}) and (bar{M}) point, respectively. Data were measured along the chain direction at 80 K using LH-polarized 7-eV laser.

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Figure 3g, h shows the laser-ARPES spectra of α-Bi₄I₄ measured on the (001) surface with their curvature plots shown in Figs. 3i and 3j, respectively. Around the ({bar{Gamma }}_{001}) point, we observe a band gap over 395 meV without the conduction band observed. At the (bar{M}) point, we observe a direct band gap of about 170 meV. Both the experiments at ({bar{Gamma }}_{001}) and (bar{M}) are well reproduced by the theoretical calculation of the bulk states, suggesting the absence of topological surface states on the (001) surface of α-Bi₄I₄. Therefore, our results on the (001) and (100) surfaces confirm the HOTI phase of α-Bi₄I₄, resembling the results of α-Bi₄Br₄³⁷. We do not resolve explicit signals from the topological hinge states, possibly due to the weak photoemission intensity from the hinge states and relatively small (100) surface gap of α-Bi₄I₄.

Electronic structure and topological property of β-Bi₄I₄

Next, we examine the electronic structure of the (100) surface of β-Bi₄I₄ at 316 K. As shown in Fig. 4a, b, β-Bi₄I₄ shows a gapless Dirac fermion dispersion at both ({bar{Gamma }}_{100}) and (bar{Z}) point. As to (001) surface, β-Bi₄I₄ exhibits a band gap of 330 meV (between the valence band top and E_F) and 125 meV (between the valence band top and the conduction band bottom) at the ({bar{Gamma }}_{001}) and (bar{M}) points, respectively, similar to α-Bi₄I₄ (Fig. 4c, d). Compared to α-Bi₄I₄, the valence and conduction band of β-Bi₄I₄ shift up slightly with a weak residual spectral weight of the conduction band at E_F. We do not observe any signature of topological surface states on the (001) surface, which is in line with the WTI phase of β-Bi₄I₄.

a, b ARPES spectra of β-Bi₄I₄ at ({bar{Gamma }}_{100}) and (bar{Z}) point on the (100) surface, respectively. c, d ARPES spectra of β-Bi₄I₄ at ({bar{Gamma }}_{001}) and (bar{M}) point on the (001) surface, respectively. The red arrow indicates the energy gap at the (bar{M}) point. The spectra were measured along the chain direction at 316 K using LH-polarized 7-eV laser.

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Topological phase transition of Bi₄I₄

In order to study the topological phase transition of bismuth iodide, we perform systematic ARPES measurements at different temperatures across T_S. At 80 K, the data at the ({bar{Gamma }}_{100}) point (Fig. 5a) clearly shows a gap, consistent with the (100) surface states of α-Bi₄I₄ in Fig. 3a. With increasing temperature up to around 310 K, the energy interval between the surface valence and conduction band at the ({bar{Gamma }}_{100}) point slightly decreases. The spectra also get blurred at high temperatures, which may indicate a preclude of crystal transformation related to the structural transition instead of thermal broadening effect³⁵. Prominently, above T_S, the surface gap is suddenly suppressed and the dispersion shows a single gapless Dirac dispersion (Fig. 5g), which confirms the WTI phase of β-Bi₄I₄ together with the gapped spectra on the (001) surface (Fig. 4). Simultaneously, the spectrum becomes much sharper with the full width at the half maximum changed from about 0.01 ({{text{AA }}}^{-1}) (302 K) to 0.007 ({{text{AA }}}^{-1}) (316 K). This band sharpening of surface states is because of the bilayer splitting of α-Bi₄I₄ (Fig. 1c) and has been revealed by the previous experiments³⁵. As the temperature decreases, the spectrum recovers at low temperatures and the surface gap reemerges (Fig. 5h–k).

a–k ARPES spectra of the (100) surface states at the ({bar{Gamma }}_{100}) point of α-Bi₄I₄ (a–g) and (bar{Z}) point of β-Bi₄I₄ (h–k). The thick arrows indicate the heating or cooling process. Please note the ({bar{Gamma }}_{100}) point of α-Bi₄I₄ and the (bar{Z}) point of β-Bi₄I₄ correspond to almost the same photoelectron emission angle. l Temperature evolution of the (100) surface gap showing a hysteresis loop.

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The temperature evolution of the (100) surface gap extracted by double-Lorentzian fitting the EDC at ({bar{Gamma }}_{100}) is summarized in Fig. 5l, which exhibits a thermal hysteresis loop near the structural transition. Specifically, an α-phase characterization is maintained up to about 310 K in the heating process, which can be confirmed by features such as surface gap and band sharpness mentioned above (Fig. 5a–f). While in the cooling process, a β-phase spectrum with a sharp single Dirac cone is evident even at 295 K in Fig. 5i, in significant contrast to Fig. 5e detected at the same temperature. The hysteresis in the (100) surface gap indicates that Bi₄I₄ undergoes a first-order structural phase transition accompanied by the topological phase transition from a HOTI to a WTI, in accordance with our ab-initio calculation.

For a long time, 2D QSH insulators, hosting symmetry-protected 1D edge states immune to backscattering, are expected to promote the development of dissipationless electronics and spintronics devices^4,5,44. However, the complicated fabrication of 2D-material devices limits the exploration and application of the 1D edge states. By contrast, 1D topological quantum materials such as Bi₄X₄ can be relatively easily synthesized and naturally harbor highly directional topological boundary states with spin-momentum locking property, which are beneficial for the preservation of coherent quantum states. The quasi-1D nature of Bi₄X₄ further allows for the integration into nanowire configurations that are natural and ideal for device engineering. In particular, the novel HOTI phase with hinge states is promising for dissipationless transport, efficient charge-to-spin conversion, topological quantum computation based on the Majorana zero modes, and the realization of spin-triplet superconductivity^{45,46,47,48,49,50,51,52,53}. Furthermore, the TQPT between HOTI and WTI phases in Bi₄I₄ bridged by a first-order structural transition provides a unique opportunity for the controllable manipulation of the topological boundary states near room temperature.

To conclude, we systematically investigate the electronic structure and its temperature evolution of Bi₄I₄. We observe a fully gapped (100) surface state of α-Bi₄I₄, supporting the identification of its HOTI phase. Our measurements at high temperatures confirm the WTI phase of β-Bi₄I₄ by detecting its gapped (001) band structure and gapless (100) TSS. More importantly, we detect the thermal hysteresis loop of the gap in the (100) surface states. Our study unravels the fine band structure of Bi₄I₄ with unprecedented precision and provides solid evidence on the TQPT between HOTI and WTI phases bridged by the structural transition. Our results provide deeper scientific insight and pave the way to promising applications on the novel topological quantum properties of quasi-1D Bi₄I₄.

Single crystal synthesis and characterization

High-quality Bi₄I₄ single crystals were synthesized using a solid-state reaction and chemical vapor transport method⁵⁴. Figure 1g presents the cross-section image by atomic-resolved scanning transmission electron microscope, which confirms the crystal structure and the high quality of our samples.

Laser-based μ-ARPES measurements

Laser-based μ–ARPES measurements with a sub-micron spatial resolution were conducted at Tsinghua University with the 7-eV laser generated by a KBe₂BO₃F₂ (KBBF) crystal and focused by an optics lens⁴³. The samples were cleaved in-situ under UHV better than 1 × 10⁻¹⁰mbar and the data were collected by Scienta DA30L electron analyzer. The total energy and angular resolutions were set to 3 meV and 0.2°, respectively.

Ab-initio calculations

First-principles calculations were conducted to investigate the electronic properties and topological band characteristics of bulk Bi₄I₄ using the Vienna ab-initio simulation package⁵⁵. The Heyd-Scuseria-Ernzerhof hybrid functional (HSE06) was employed to describe the exchange-correlation potential⁵⁶. Experimental lattice parameters were used, with the energy cutoff for the plane-wave basis set to 300 eV. Maximally localized Wannier functions (MLWFs) for the p-orbitals of Bi and I atoms were constructed using the WANNIER90 code on a 6 × 6 × 3 k-point mesh⁵⁷. These MLWFs were subsequently applied to build ab initio tight-binding models. Surface electronic structures were calculated by combining the ab-initio tight-binding models with the surface Green function method using our in-house code⁵⁸.