Room-Temperature Topological Phase Transition in Quasi-One-Dimensional Material Bi₄I₄

Topological phases and associated phase transitions in quantum materials have garnered tremendous interest in recent years since the discovery of the two-dimensional (2D) quantum spin Hall (QSH) effect [1, 2]. Extending this concept to three dimensions (3D) led to the idea of topological insulator (TI), which hosts gapless boundary states. Within the class of TIs, there are two schemes by which the topological class can be further distinguished. The first scheme is the distinction between strong and weak TI [3, 4]. While every surface of a strong TI possesses a gapless Dirac cone, such nontrivial surface states only appear on selected surfaces of a weak TI [4, 5, 6, 7]. The experimental identification between a strong and a weak TI therefore depends critically on the measurement of topological non-trivial surface states on multiple surfaces of a material, commonly probed via the technique of angle-resolved photoemission spectroscopy (ARPES). The challenge of such experimental efforts is that in natural materials, usually only a single preferred cleavage surface is experimentally accessible for ARPES. Most TIs that have been experimentally confirmed have been strong TIs, identified by a combination of a single-surface measurement and theoretical calculations, such as in the bismuth and antimony chalcogenides [8, 9, 10, 11]. Direct experimental verification of a weak TI, however, remains scarce due to the lack of materials with multiple cleavage surfaces where the existence of topologically nontrivial surface states could be probed [7, 12, 13, 14, 15, 16]. The second scheme to classify further a TI is by its topological order, which is distinguished by the dimension of its gapless boundary states compared to the dimension of its bulk state. This realization led to the identification of a novel phase of matter dubbed the higher-order TI [17, 18, 19, 20], which are materials that host protected gapless states on boundaries of two or three spatial dimensions lower than its bulk.

Interestingly, materials predicted to be candidate weak TIs are often in the vicinity of a topological critical point, where the exact topological classification depends sensitively upon the lattice parameters [7, 21]. On the one hand, this produces an uncertainty in theoretical predictions of their topological properties, which makes experimental determination a necessity [7, 21]. On the other hand, this renders an advantage that a topological phase transition between different topological phases can be easily induced by external perturbations such as strain and thermal effects [21, 7, 22, 23].

As a quasi-1D material, Bi₄I₄ provides a superior platform not only to realize but also to be experimentally examined for the aforementioned distinct TI classes [7]. Consisting of stacked quasi-1D chains, Bi₄I₄ naturally stabilizes in two crystal structures due to distinct stacking sequences, identified as $\alpha$-Bi₄I₄ and $\beta$-Bi₄I₄ [24]. The $\alpha$-phase is naturally stabilized at low temperatures and transitions into the $\beta$-phase via a first-order phase transition around 300 K. $\beta$-Bi₄I₄ was theoretically predicted to be topologically nontrivial, in the vicinity of a transition point between weak and strong TIs [7, 25]. Two subsequent ARPES studies reached conflicting conclusions [25, 26]. While one study reports strong TI characteristics for a low temperature phase identified as $\beta$-Bi₄I₄ [25], the other utilized a quenching method to stabilize the $\beta$-Bi₄I₄ phase at low temperatures and identified it as a weak TI [26] while categorized the $\alpha$-Bi₄I₄ phase as a trivial insulator with a distinct electronic structure [26]. Besides the controversy amongst the experimental reports, these results also differ from a recent theoretical work predicting the $\alpha$-phase to be a topologically nontrivial higher-order TI hosting helical hinge modes [20]. While beyond the scope of the recently established topological materials database [27, 28, 29], this predicted rare phase may be viewed as a QSH effect [1, 2] around the top or bottom surface of a 3D insulator, i.e., the time-reversal-invariant counterpart of the long-desired 3D quantum Hall effect [30, 31]. To resolve the aforementioned controversies, we systematically carried out electrical transport, X-ray diffraction, and ARPES measurements in combination with theoretical calculations to clarify the topological classification of $\alpha$-Bi₄I₄ and $\beta$-Bi₄I₄ as tuned by the natural structural transition via temperature.

Large Bi₄I₄ single crystals were synthesized with the quasi-1D chains oriented along the $b$ axis, with large flat (001) and (100) surfaces (Fig. 1) (see Supplementary Materials). The structural phase transition can be clearly identified from the temperature-dependent resistivity (Fig. 1a) and heat capacity (Supplemental Fig. S1) measurements, where a hysteresis indicating a first order phase transition appears around 300 K. Below 270 K, the resistivity data show a typical semiconductor behavior with a broad hump at 150 K and are consistent with previous reports [26].

Structurally, the $\alpha$- and $\beta$-phases differ in the packing of the Bi₄I₄ chains. The $\alpha$-Bi₄I₄ has double-layered Bi₄I₄ chains per unit cell (Fig. 1c) whereas the $\beta$-Bi₄I₄ has only a single-layer of Bi₄I₄ chains per unit cell (Fig. 1e). This subtle difference in stacking causes a large change of the $c$ lattice and the associated doubling of the unit cell volume (Fig. 1d), as demonstrated by X-ray single crystal diffraction (Fig. 1f, h). Contrasting the single crystal diffraction patterns for the reciprocal lattice $a^{\ast}$-$c^{\ast}$ planes taken in the $\alpha$-phase at 293 K (Fig. 1f) and the $\beta$-phase at 313 K (Fig. 1h), a near-doubling of the $c^{\ast}$ in reciprocal space from $\alpha$-Bi₄I₄ to $\beta$-Bi₄I₄ is visible, as seen by the additional peaks in Fig. 1f marked by a circle. This structural change is well-captured in the temperature-dependent refined lattice parameters (Fig. 1g and Fig. S2). The lattice parameters of the $\alpha$-phase are refined at 300 K to be $a=14.251(2)\AA$, $b=4.4304(5)\AA$, $c=19.976(6)\AA$, and $\beta=92.93(2)^{\circ}$, while those of the $\beta$-phase are refined at 310 K to be $a=14.394(5)\AA$, $b=4.4288(13)\AA$, $c=10.494(5)\AA$, and $\beta=107.96(5)^{\circ}$.

The large facets on both the (001) ($>$ 0.5 mm) and (100) ($>$ 0.2 mm) surfaces are naturally cleavable, exposing flat surfaces that enable conventional ARPES studies on either surface using an incident beam spot size of 50 × 20 $\mu$m² (Fig. 1b). We note the $\beta$-angle difference between $\alpha$-Bi₄I₄ and $\beta$-Bi₄I₄ crystal structures, where the natural cleavable side (100) surface of $\beta$-Bi₄I₄ becomes the ($\bar{2}$01) surface in the $\alpha$-phase per definition in the true primitive unit cell (Fig. 1c, e), we label it as (100)’ in $\alpha$-Bi₄I₄ (Fig. 1i, j) to avoid confusion in the comparison with the (100) surface of the $\beta$-phase.

Taking advantage of the rare multi-surface experimental access to Bi₄I₄, we carried out ARPES measurements on both the (001) and (100) surfaces of the $\alpha$- and $\beta$-phases in their respective temperature regimes. We note that such two-surface ARPES measurement has also been desirable to identify unambiguously the topological characters of weak TIs and higher-order TIs. A comparison of the complete electronic structure of the two phases are summarized in Fig. 2. Two main observations can be made: i) there is a strong distinction between the electronic structures measured on the (100) and (001) surfaces for both phases, and ii) the change between the two phases across the structural transition is subtle.

We first examine the distinction between the (100) and (001) surfaces, which is most clearly seen in the measured Fermi surfaces (FS). On the (100) surface, both phases exhibit a quasi-1D FS along $k_{z}$ (Fig. 2a, c), revealing a rather weak interlayer coupling in the $c$-direction. The FS on the (001) surface, by contrast, consists of island-like bright spots for both phases (Fig. 2b, d and Fig. S9), suggesting a stronger interlayer coupling along the $a$ axis compared to that of the $c$ axis. Furthermore, the measured sample surface orientation can be unambiguously distinguished by the periodicity of the observed band dispersions, which corresponds to the clearly distinct lattice parameters associated with the (100) and (001) surfaces. The $\alpha$-phase, for example, has a period of 0.3 Å^-1 along the $k_{z}$ direction on the (100)’ surface (Fig. 2m) and 0.88 Å^-1 along the $k_{x}$ direction on the (001) surface (Fig. 2n). We note that we do not observe a drastic change of the fermiology across the phase transition, in contrast to the previous report that the quasi-1D FS only exists in the $\beta$-phase [26]. By clearly identifying single domains of each cleaved surface, we now clarify that the quasi-1D intensity and island-like intensity are due to the two different crystal surfaces measured rather than the two distinct structural phases.

We can furthermore examine the corresponding low-energy band dispersions. At the high symmetry points of both surfaces in both phases, an intense hole-like bulk band is observed below $E_{F}$, revealing a gap in the bulk state in all cases. The bulk nature of this band is confirmed by photon energy dependence as it disperses along $k_{z}$ (Fig. S6). However, two observations can be noted. Firstly, as the electronic structure is quasi-1D along $k_{z}$ on the (100) surfaces, the bands near the $\bar{{\Gamma}}$ and $\bar{Z}$ points are quite similar in both phases (Fig. 2e,f,i,j) while those near the $\bar{{\Gamma}}$ and $\bar{M}$ points on the (001) surface are notably distinct. Secondly, while we observe a hole-like bulk valence band on all the cuts shown, within the bulk band gap on the (100) surface, a sharp Dirac surface state is observed for both the $\alpha$- and $\beta$-phases (Fig. 2e,f,i,j). On the contrary, no Dirac surface state is observed inside the bulk band gap on the (001) surface of either phase (Fig. 2g,h,k,l). At the $\bar{\Gamma}$-point, the bulk valence band tops at -0.3 eV. No surface state is observed. At the $\bar{M}$-point, the bulk valence band approaches $E_{F}$, with a visible gap near $E_{F}$ between the bulk valence and conduction bands.

Having distinguished the two surfaces, we now focus on the transition between the two structural phases. Overall, no dramatic change of the bulk band dispersions is observed across the phase transition. There is, however, a doubling in the number of bands from the $\beta$-phase to the $\alpha$-phase [20] due to the doubling of the stacking layers in the unit cell. This band doubling is well-demonstrated by our theoretical calculations of the bulk bands projected onto the (001) surface for both the $\alpha$ and $\beta$ phases (Fig. 2p, r and Supplementary Materials). While the band splitting is small due to the weak interlayer coupling, it is clearly resolved in our corresponding dispersions measured at low temperatures in the $\alpha$-phase (see pairs of arrows in Fig. 2o,q as well as in a larger energy window shown in Fig. S3), consistent with the doubled stacking in the low temperature phase.

Next, we demonstrate that this band doubling is indeed induced via the structural transition by systematically measuring across the structural transition near room temperature. We note that while it is challenging to resolved the small band splitting near the valence band top at elevated temperatures, this band doubling manifests as a broadening in the momentum distribution curves (MDCs) when a single band splits into two from the higher temperature $\beta$-phase to the lower temperature $\alpha$-phase. Clearly, this effect is in contrast to thermal broadening effects, which would have the opposite trend with respective to temperature. In Fig. 3a-c, we plot the dispersions across the $\bar{{\Gamma}}$ point measured on (100) at selected temperatures. A clear abrupt broadening is indeed observed from the high temperature $\beta$-phase to the lower temperature $\alpha$-phase as exemplified by the MDCs at -1.2 eV (Fig. 3i,j). Importantly, this abrupt change is accompanied by a hysteresis behavior revealed in the warm up (Fig. 3i) and cool down (Fig. 3j) processes. To quantify this behavior, we fit the MDCs and plot both their full-width-at-half-max (Fig. 3l) and peak positions (Fig. 3m), both of which show the hysteresis behavior reminiscent of that in the resistivity measurement, confirming the electronic structure’s response to the first order structural transition.

Having confirmed the phase transition as observed from the electronic structure, we examine the topological properties of the two phases. In Fig. 4 we present high resolution measurements near $E_{F}$ for both phases. We first examine the $\beta$-phase. On the (001) surface (Fig. 2k, l and Fig. 4b), we observe a clear gap in all the bulk gaps and no additional surface Dirac crossings. This can be confirmed from the energy distribution curve (EDC) stacks of the bands across $\bar{M}$ (Fig. 4b), where the green dots track the bulk band dispersion and reveal a gap of $\sim$100 meV. Similar suppressed intensity can also be clearly identified in the MDC stacks (Fig. S7). On the (100) surface, a quasi-1D Dirac surface state exists inside the bulk band gap (Fig. 2i, j and Fig. 4d, f). Particularly at the $\bar{\Gamma}$ and $\bar{Z}$ points, the surface states are gapless within the resolution of measurements at 350 K. These combined measurements on both the (001) and (100) surfaces demonstrate that the high temperature $\beta$-Bi₄I₄ is a weak TI where only selected surfaces possess an even number (two here) of gapless Dirac cones [3, 7] (Fig. 4h). In the $\alpha$-phase, all the observed bands on the (001) surface are also gapped (Fig. 2g, h and Fig. 4a). A gap of $\sim$85 meV can be further seen from the bands tracked by the EDC stacks in Fig. 4a. On the (100)’ surface, the quasi-1D Dirac-like surface state still persists (Fig. 2e, f). From a high-resolution measurement taken with 30 eV photons at 10 K (Fig. 4c), we observe a gap of $\sim$35 meV clearly in the EDC stacks, indicating a gapped surface state within a bulk band gap. This is consistent with our theoretical prediction and calculation of a higher-order TI scenario of $\alpha$-Bi₄I₄ [20], illustrated in Fig. 4g, where both the bulk bands and surfaces bands are required to be gapped, hosting gapless hinge states inside the surface gap. A comparison of our calculated bulk and surface states with our measured bands show strong agreement (Fig. S11). We summarize our experimental findings and classification of the two phases in Table 1.

Finally, we comment on the inconsistencies in the previous two ARPES studies on Bi₄I₄. In the first report [25], since no quenching was performed, the phase measured at low temperatures was most likely the $\alpha$-phase instead of the claimed $\beta$-phase. The conclusion that this phase is a strong TI was based on the observation of a gapless Dirac crossing on the (001) surface. Here we show that this Dirac band is rather a bulk band, where high resolution measurements reveal a gap (Fig. 4a). In the second report with nanoARPES [26], the $\beta$-phase was claimed to be stabilized at low temperature by quenching directly from high temperature. The authors ascribed point-like FS to $\alpha$-Bi₄I₄ and quasi-1D FS to $\beta$-Bi₄I₄ based on a superposition of signals from the (001) and (100) surfaces. Based on this assumption, they concluded that the $\beta$-phase is a weak TI and the $\alpha$-phase is a trivial insulator. Benefiting from the relatively large cleavable (100) and (001) surfaces free of domains, here we unambiguously show that both phases have similar quasi-1D FS on the (100) side surface, and that the distinction between the two phases is the subtle band doubling in the $\alpha$-phase, which was not resolved in previous studies. Reminiscent of the bulk Peierls distortion in the Su-Schrieffer-Heeger model [32], this yields a gap opening in the Dirac surface state on the (100)’ surface, which is the prerequisite for the appearance of mid-gap hinge states in $\alpha$-Bi₄I₄ as a higher-order TI [20].

Topological properties of a variety of quantum materials have been verified by a combination of ARPES and first-principles calculations during the last decades [10, 11, 33, 34, 35, 36, 37, 38, 39, 40]. For the quasi-1D material Bi₄I₄, not only does the uncertainty of the theoretical prediction for the topological classification of both $\alpha$-Bi₄I₄ and $\beta$-Bi₄I₄ call for experimental exploration, but also the change of the electronic structures between $\beta$-Bi₄I₄ and $\alpha$-Bi₄I₄ is subtle such that careful ARPES measurements with considerable energy resolution is necessary. Unambiguously, the $\beta$-phase Bi₄I₄ is shown to be the first weak TI by our ARPES measurements on large natural cleavable planes of the top and side surfaces, which resolves the controversy of the previous reports [25, 26]. For $\alpha$-Bi₄I₄, two distinctions are clearly observed as compared with $\beta$-Bi₄I₄: band doubling and gap opening in the side surface state. Together with the identification of the $\beta$-phase as a weak TI, the existence of the Dirac-like surface state with a small gap indicates that the $\alpha$-phase is consistent with the identification as a rare higher-order TI [20] instead of the trivial insulator based on symmetry indicators alone [27, 28, 29]. Since the $\beta$-phase has been shown as a weak TI here, each (001) monolayer is a 2D Z₂ TI [20]. As the $\alpha$-phase has been shown to have two monolayers per unit cell here, i.e., a 2D trivial insulator, the 3D bulk must be Z₂ trivial given the weak interlayer couplings [3, 7]; in other words, the surface states are gapped. Moreover, for an $\alpha$-phase thin film with a large odd number of monolayers, it can be viewed as a 2D Z₂ TI with a helical edge state. Apparently, this edge state can only appear at hinges and form a loop, demonstrating that the α-phase is a higher-order TI [20]. However, we do note that further direct evidence of the hinge states is desirable.

Fundamentally, our result implies that even without strong correlation and magnetism, there are still many topological materials that are beyond the scope of the already efficacious topological quantum chemistry or symmetry indicators [27, 28, 29]. Significantly, our work reveals the strong tunability of the quasi-1D TI Bi₄I₄ as a novel material platform where the abundance of topological phases could be explored. In the future, it would be exciting to investigate in this platform, for example, the strain-induced topological phases and their transitions, the spatial locations and Luttinger parameters of helical hinge states, the possible intrinsic and extrinsic topological superconductivity, as well as applications of our identified room-temperature topological phase transition as a topological switch that controls the dimensions of the boundary conduction channels.