Quantum Transport Behavior in Quasi-One-Dimensional Topological Matter Bi₄X₄ (X = Br, I)

Abstract

Quasi-one-dimensional (quasi-1D) topological matter Bi₄X₄ (X = Br, I) possesses versatile topological phases determined by its molar ratio of halide and the stacking mode. Establishing the intrinsic relationship between these topological orders and the quantum transport properties is extremely crucial for both of fundamental research and device applications. Here we review the recent work on the characteristic quantum transport behavior of the Bi₄X₄ system originating from various electronic states, including three-dimensional (3D) bulk states, two-dimensional (2D) surface states, and one-dimensional (1D) topological hinge states. Specifically, variable range hopping effect, Lifshitz transition, metal–insulator transition, and Shubnikov de Haas oscillations are evoked by the gapped bulk states with significant doping carriers. In 2D limits, the (100) surface states exhibit Dirac-type dispersion to produce weak antilocalization, which is a strong 1D nature due to quasi-1D crystal and electronic structure and evidenced by anomalous planar Hall effect. Last but not the least, coherent transport with Aharonov–Bohm oscillations is observed in thin-layer devices, implying the existence of 1D topological hinge states separated by the (100) surface. These unconventional quantum transport features verify the topological nature of Bi₄X₄ in different dimensions, signifying an ideal platform to design and utilize multiple topological orders in this quasi-one-dimensional material system.

1. Introduction

Searching for tunable topological quantum materials (TQMs) has been a critical challenge in the fields of condensed matter physics and materials science for decades, since the robust spin-polarized and high-mobility transport feature of topological non-trivial states are beneficial for low-energy information transmission [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]. TQMs are classified by the topology theory, which concentrates on the global property that remains unchanged under continuous deformation rather than the arrangement of local details. The unchanged quantities are defined as topological invariants. The topological invariants of the electronic energy bands essentially distinguish TQMs from ordinary insulators. The differentiation between topological single-particle and multi-particle effects constitutes a critical step toward a clear framework of topological states. The former is based on band theory and can be described by the Bloch wave-function defined in the Brillouin zone. Its topological properties are manifested by topological invariants (such as Chern number and Z₂ invariant) [16,17]. Most topological materials, e.g., topological insulators and Dirac semimetals, fall into this category. Their topological properties are essentially determined by the behavior of a single electron in a periodic lattice. And the latter depends on the strong electron–electron interaction. For instance, in the fractional quantum Hall effect, electron correlations give rise to emergent excitations with fractional charges, of which the topological properties are characterized by a topological cyclotron commensurability criterion [18,19]. Based on the mentioned comprehensive theorical foundation, TIs received much attention for their insulating bulk states and ballistic topological surface/edge/point states protected by time-reversal symmetry (TRS) [20,21,22,23]. However, the tunability of topological boundary states is rarely achieved due to hinderance for realizing topological phase transition, which requires bulk gaps to close and reopen accompanied by parity switching.

Recently, a group of TQMs with quasi-1D features, such as ZrTe₅, ZrTe₃, TaSe₃, and Bi₄X₄ (X = Br, I) [24,25,26,27,28,29,30,31], has been examined, whose anisotropic crystal structure provides extra freedom to engineer crystal fields by altering inter-chain interaction. Specifically, bismuth halide Bi₄X₄ has two van der Waals directions, which are sensitive to molar ratio of bromine and iodine, providing opportunities to control the overall crystal structure by controlling the halogen doping ratio. The crystal structure of Bi₄X₄ is composed of stacked quasi-1D bismuth halide chains. Through changing the stacking mode by regulating the halogen ratio, it can achieve multiple topological phase transitions including high-order topological insulator (HOTI), weak topological insulator (WTI), and dual topological phase with both strong topological insulator (STI) and HOTI orders [32,33,34,35,36,37,38,39,40,41]. Thus, the controllable multiple topological boundary states in one single material system can be achieved under different dimensions, laying the foundation for the future integration of topological quantum devices.

Historically, mono-layer Bi₄Br₄ was theoretically predicted as a quantum spin Hall insulator with a large band gap in 2014, and its bulk energy gap is much larger than the thermal excitation energy at room temperature, which has been experimentally confirmed by scanning tunneling microscopy (STM) and angle-resolved photoemission spectroscopy (ARPES) [41,42,43,44]. Later, Yazyev’s team and Kondo’s team characterized the electronic structure of $\beta$-Bi₄I₄ single crystals using ARPES, and observed the Dirac cones corresponding to topological edge states [32,35]. Based on STM and ARPES, $\alpha$-Bi₄I₄ is indicated to form non-degenerate edge states and three-dimensional quantum spin Hall insulator characteristics through spontaneous symmetry breaking [45]. In 2023, with high-resolution laser ARPES, one-dimensional topological hinge states in ${\alpha }^{\prime }$-Bi₄Br₄ single crystals were determined, providing experimental implications that ${\alpha }^{\prime }$-Bi₄Br₄ is a HOTI [37]. At the same time, $\gamma$-Bi₄Br₂I₂ with a triple-layer structure is fabricated and demonstrated to be a non-degenerate WTI [38,46]. In 2025, the influence of the halogen ratio in Bi₄(Br_1−xI_x)₄ was thoroughly studied, where the dual topological phase crystal ${\alpha }^{\prime }$-Bi₄(Br_0.2I_0.8)₄ was realized with a promising topological phase diagram in this system [40]. In fact, numerous quantum transport phenomena have been reported, e.g., weak antilocalization (WAL), Shubnikov de Haas (SdH) oscillation, and Aharonov–Bohm oscillation, which prove the existence of the aforementioned topological features and corresponding electronic structure [47,48,49].

In this colloquium article, we will review the representative quantum transport behavior and classical transport behavior in these bismuth halides from the perspective of 3D, 2D, and 1D electronic structures, respectively. In the first part, we will discuss diverse stacking modes and the correlated electronic states in the Bi₄X₄ family. In the following three parts, we will introduce the typical quantum transport results in 3D, 2D, and 1D cases, respectively. The variable range hooping (VRH), Lifshitz transition, magnetic field-induced metal–insulator-transition, and Shubnikov de Haas oscillation are induced by 3D bulk states. The 2D spin-polarized surface states can evoke WAL and anomalous planar Hall effect (APHE), which is regarded as a low-cost alternative to detect topological features. Aharonov–Bohm oscillations are also identified in this system, of which the origin is attributed to 1D topological hinge states. In the last part, we will provide an outlook for future studies and the potential applications of this system.

2. Crystal Structure of Bi₄X₄

The basic building block of Bi₄X₄ is 1D atom chains extended along the crystal b axis, which contains four Bi atoms and four halide atoms in a unit cell. There are two kinds of Bi atoms, the inner Bi atoms connecting with each other to form infinite atoms chain, and the outer Bi atoms connecting two halide atoms, as depicted in the upper panel of Figure 1a. The atomic chains are assembled by van der Waals force along the a axis to form mono-layer Bi₄X₄, which is predicted as a quantum spin Hall insulator with large energy gaps at room temperature [42,43,44,50]. The different layers are attached together according to different stacking modes, resulting in 3D bulk Bi₄X₄ with different phases, namely ${\alpha }^{\prime }$ phase, $\gamma$ phase, $\beta$ phase, and $\alpha$ phase corresponding to HOTI, non-degenerate WTI, WTI, and normal insulator (NI), respectively (see the lower panel of Figure 1a). These stacking modes can be modulated by the ratio of Br atoms and I atoms, as atomic radius can regulate the inter-layer spacing and thereby affect the inter-layer interaction. For completely Br elemental configuration, the unit cell of bulk crystals contains double antiparallel-arranged mono-layers (${\alpha }^{\prime }$ phase), bringing about the topological hinge states at two edges of the (100) surface (i.e., HOTI) [37]. With an equal ratio of Br and I, the structure becomes triple-layer unit cells ($\gamma$ phase), where the three layers lose degeneracy. Consequently, the bulk states and topological boundary states all become non-degenerate within the triple-layer unit cell, giving triple Dirac cones at the (100) surface of $\gamma$-Bi₄Br₂I₂ (i.e., non-degenerate WTI) [38,46]. Differently, Bi₄X₄ crystals with pure I configuration engender two stacking modes. The unit cell of $\beta$-Bi₄I₄ comprises single identical mono-layers with inversion centers within layers and between layers (i.e., WTI), whereas that of $\alpha$-Bi₄I₄ consists of two mono-layers with inversion centers only located between layers [45]. The vanishing of in-plane inversion centers makes the two Dirac cones at opposite sides of a mono-layer $\alpha$-Bi₄I₄ become non-degenerate, leaving double Dirac cone features at the (100) surface [45]. The coupling between n neighboring layers in $\alpha$-Bi₄I₄ opens finite gap in the (100) surface, making it a topological trivial phase (i.e., normal insulator) [36]. By slowly lowering the temperature for $\beta$-Bi₄I₄, the $\beta$ phase structure can be changed into $\alpha$ phase at about 290 K [36]. It is noted that the ratio of halogen can be accurately tuned, where the global bulk gap is closed and reopened with topological phase transition at around 73% I dopant. After the critical point, ${\alpha }^{\prime }$-Bi₄(Br_0.2I_0.8)₄ exhibits a dual topological phase with both STI and HOTI orders, expanding the synthesizable material database with rare coexisting topological orders [40].

3. Quantum Transport Behavior

3.1. Three-Dimensional Transport Behavior

Bulk electronic states are characterized by conventional bulk electronic transport methods with silver epoxy and gold wires, measuring the temperature-dependent resistance, magnetoresistance, and Hall resistance. Figure 2a demonstrates the curve of normalized resistance of ${\alpha }^{\prime }$-Bi₄Br₄ samples (S1~S5) with respect to temperatures. All the five samples maintain the same trend, where the semiconductor-like behavior at low temperatures can be observed, which also appears in other bismuth halides. The semiconductor-like behavior can be described by the VRH model, where carriers hop among different sites, controlled by both thermal excitation energy and energy barrier from disorder. From the STM, three types of defects are observed [51]. After scrutinization between the experimental STM data, simulated STM data, and the crystal structure, these three types of defects are ascribed to vacancies of the two non-equivalent Br atoms on the same side of the first layer and the Br atoms on the second layer, consistent with previous studies reporting that halogen vacancies are frequently observed in this material [29,46,52,53,54]. From the DFT calculation, after introducing Br vacancies, there are additional impurity states near the conduction band passing through the Fermi surface (FS), suggesting an n-type doping nature. Hence, Br vacancies provide charge carriers, which are regulated by the VRH model at low temperatures to form the semiconductor-like transport behavior. In Mott’s VRH theory, the charge carriers provided by Br vacancies achieve quantum tunneling through thermally activated traversal of localized potential barriers [55,56,57]; the resistivity is given by Equation (1), $\rho \left(T\right) = {\rho }_0\exp [{(T_m/T)}^{1/(d+1)}]$, where ${\rho }_0$ denotes the characteristic resistivity of the sample, $T_m$ represents the characteristic temperature scale for hopping transport, and $d$ corresponds to the effective dimensionality of the conduction pathway. The dimensionality parameter takes values $d$ = 0 for thermal excitation of the charge carriers, $d$ = 1 for the VRH model in a 1D system, i.e., with significant electron–electron correlations, while $d$ = 3 for the VRH model in a 3D system, which lacks significant electron–electron correlations. Figure 2b shows the resistance fitted by Equation (1) with $d$ = 3 for bulk resistance. Within the temperature range from 13 K to 25 K, the curve is well fitted, showing consistency with VRH theory. When the temperature drops below 10 K, the electron–electron interactions cause deviations from the VRH model. Crucially, as the pressure increases, Bi₄X₄ gradually exhibits superconductivity [58,59,60,61]. And the resistivity of ${\alpha }^{\prime }$-Bi₄Br₄ at different pressures can also be fitted by Equation (1), which evidences the accuracy of the VRH model [51].

As temperature shifts from the VRH temperature to the high temperature, another typical transport behavior in ${\alpha }^{\prime }$-Bi₄X₄ is shown in Figure 2c,d. Hall measurements are performed in ${\alpha }^{\prime }$-Bi₄(Br_0.2I_0.8)₄ to identify Lifshitz transition. Figure 2c shows the Hall resistivity at different temperatures. As temperature increases, the slope of the Hall resistivity changes signs from negative to positive and the critical temperature is around 100 K, which is corresponding with the temperature of the hump in the temperature-dependent longitudinal resistance of ${\alpha }^{\prime }$-Bi₄(Br_0.2I_0.8)₄. The mobility (μ) and density (n) of charge carriers can be acquired by Equation (2), $\rho = {\displaystyle \frac{1}{nq\mu }}$, where q, n, and μ represent the charge, density, and mobility of the carriers, respectively. The result is depicted in Figure 2d. As temperature rises, the dominate charge carriers change from electron to hole. Denoted by the black arrow, the transition locates at around 100 K, in accordance with Figure 2c. However, the carrier polarity switching happens at temperatures of about 230 K in ${\alpha }^{\prime }$-Bi₄Br₄ [43,62], which diverges from the critical temperature of 100 K for ${\alpha }^{\prime }$-Bi₄(Br_0.2I_0.8)₄, providing implication for iodine-induced band restructuring in this quantum material family.

Apart from Lifshtiz transition demonstrated by Hall resistance in Figure 2c, as the magnetic field increases, the longitudinal resistance also exhibits unique quantum properties, as shown in Figure 2e. All the longitudinal resistance curves under different temperatures grow with the magnetic field and converge into one point at the critical magnetic field B_c = 4.22 T. Below 4.22 T, the resistance exhibits a positive temperature dependence (i.e., resistance increases with rising temperature), corresponding to metallic behavior. The trend reverses when above 4.22 T (i.e., resistance decreases with rising temperature), indicating a magnetic field-induced MIT. The characteristic signature of magnetic field-induced MIT is evidenced by a universal scaling plot of the normalized longitudinal resistivity ${\displaystyle \frac{{\rho }_{xx} (B)}{{\rho }_{xx} (B_c)}}$ versus the scaled magnetic field $\left|B-B_c\right| T^{ -1/\xi }$, where $\xi$ denotes the critical exponent [63]. This scaling analysis is performed in ${\alpha }^{\prime }$-Bi₄(Br_0.2I_0.8)₄ with the fitting parameter $\xi = 7.14$, as displayed in Figure 2f. The transition also resides in $\beta$-Bi₄I₄ with the critical magnetic field B_c = 15.9 T and critical exponent $\xi = 6.54$ [63]. Large $\xi$ implies strong electron–electron interaction according to previous work, which may provoke FS instabilities that manifest as CDW-induced band gap and electronic phase transitions to insulating states after the amplification from quantum confinement of quasi-1D systems. Longitudinal resistivity in Figure 2e exhibits oscillation at low temperatures, corresponding to SdH oscillation. The resistance of the oscillation part decreases rapidly with the increase in temperature, which can be described by Lifshitz–Kosevich (LK) theory [64] using Equation (3), $R_T = {\displaystyle \frac{2{\mathsf{\pi }}^2{k}_{\mathrm{B}}T/\hslash {\omega }_{\mathrm{c}}}{\sin \mathrm{h}(2{\mathsf{\pi }}^2{k}_{\mathrm{B}}T/\hslash {\omega }_{\mathrm{c}})}}$, where k_B denotes Boltzmann’s constant, and ${\omega }_{\mathrm{c}}$ represents the cyclotron frequency defined as ${\omega }_{\mathrm{c}} = eB/m^{\ast }$ with $m^{\ast }$ signifying the quasiparticle effective mass. $m^{\ast }$ can be derived from fitting results according to Equation (3). The values of $m^{\ast }$ are around 0.15 m_e and 0.30 m_e for $\alpha$-Bi₄I₄ and ${\alpha }^{\prime }$-Bi₄Br₄, respectively [62,65]. Subtracting the background noise as $\Delta {\rho }_{xx} = {\rho }_{xx} - \overline{{\rho }_{xx}}$, where $\overline{{\rho }_{xx}}$ represents the background resistance stemming from quadratic fitting, is shown in Figure 2g [65]. Together with the first-order derivative of the magnetic field in Hall resistance (${\mathrm{d}\rho }_{xy}/\mathrm{d}B$), the oscillation part of the resistance can be obtained, and the oscillating frequency is about 3.5 T. Increasing magnetic field B causes persistent crossings of Landau bands through the Fermi energy E_F, driving the periodic oscillations in both longitudinal (ρ_xx) and Hall resistivities. Furthermore, Landau indices N extracted from Δρ_xx and dρ_xy/dB extrema are demonstrated, where the linear relationship between N and 1/B confirms the SdH oscillations [66,67]. The phase shift in the oscillations $\mathsf{\gamma } = {\displaystyle \frac{{\mathsf{\varphi }}_{\mathrm{B}}}{2\mathsf{\pi }}}-{\displaystyle \frac{1}{2}}$ (where ${\mathsf{\varphi }}_{\mathrm{B}}$ stands for the Berry phase) is determined by a vertical intercept of −0.04, which resides within the critical ±1/8 regime as evidence of the non-zero Berry phase [68]. It should be noted that SdH oscillation also resides in ${\alpha }^{\prime }$-Bi₄Br₄ ($F_1 = 7.69 T$ and $F_2 = 106 T$) and $\beta$-Bi₄I₄ ($F = 9.2 T$) [63,65,69]. The oscillation frequencies of both materials are higher, indicating larger electron pockets than ${\alpha }^{\prime }$-Bi₄(Br_0.2I_0.8)₄. Diminished quantum oscillation frequency lowers the magnetic threshold for entering the extreme quantum limit (EQL) in ${\alpha }^{\prime }$-Bi₄(Br_0.2I_0.8)₄, enabling access of EQL at a comparatively small magnetic field.

Figure 2. Three-dimensional transport behaviors in Bi₄X₄. (a) Temperature-dependent normalized resistance R/R_{300 K} for ${\alpha }^{\prime }$-Bi₄Br₄ samples S1–S5. Inset: Derivative dR/dT versus T, revealing a sharp peak near 10 K and. The shaded orange region marks the VRH regime. (b) The relation between LnR and $T^{ -1/4}$ according to Equation (1) (fitted by solid lines). (c) Temperature-varied Hall resistivity of ${\alpha }^{\prime }$-Bi₄(Br_0.2I_0.8)₄ single crystal. Inset: A measurement schematic of Hall resistance. (d) Temperature evolution of carrier mobility μ (left scale) and carrier density n (right scale). The black arrow indicates the sign-reversal temperature. (e) Longitudinal resistance R_xx versus magnetic field at selected temperatures; the black arrow pinpoints the metamagnetic transition field. (f) Collapse of normalized longitudinal resistance obtained via scaling analysis. (g) The oscillation part of longitudinal and Hall resistance, together with Landau index N versus 1/B. Dashed red and green lines indicate peak and valley positions extracted from both longitudinal and Hall traces, respectively. Reprinted with permission from Refs. [51,65].

Figure 2. Three-dimensional transport behaviors in Bi₄X₄. (a) Temperature-dependent normalized resistance R/R_{300 K} for ${\alpha }^{\prime }$-Bi₄Br₄ samples S1–S5. Inset: Derivative dR/dT versus T, revealing a sharp peak near 10 K and. The shaded orange region marks the VRH regime. (b) The relation between LnR and $T^{ -1/4}$ according to Equation (1) (fitted by solid lines). (c) Temperature-varied Hall resistivity of ${\alpha }^{\prime }$-Bi₄(Br_0.2I_0.8)₄ single crystal. Inset: A measurement schematic of Hall resistance. (d) Temperature evolution of carrier mobility μ (left scale) and carrier density n (right scale). The black arrow indicates the sign-reversal temperature. (e) Longitudinal resistance R_xx versus magnetic field at selected temperatures; the black arrow pinpoints the metamagnetic transition field. (f) Collapse of normalized longitudinal resistance obtained via scaling analysis. (g) The oscillation part of longitudinal and Hall resistance, together with Landau index N versus 1/B. Dashed red and green lines indicate peak and valley positions extracted from both longitudinal and Hall traces, respectively. Reprinted with permission from Refs. [51,65].

3.2. Two-Dimensional Transport Behavior

In ${\alpha }^{\prime }$-Bi₄Br₄ single crystals, the bulk state, together with two cleavage planes (001) and (100), offers advantages for characterizing 2D electronic structures with the transport method. With angular scaling approaches, the magnetoconductivity (MC) exhibits clear two-dimensional quantum diffusive transport at low temperatures, namely WAL [70]. In order to extract the contribution of surface states, bulk conductivity is subtracted by $∆\mathsf{\sigma } (\alpha , B) = \mathsf{\sigma } (\alpha , B)-\mathsf{\sigma } (90°, B)$, where MC $\mathsf{\sigma } (B) = 1/\mathrm{R} (B)-1/\mathrm{R} \left(0\right)$, and $\alpha$ is the angle between magnetic field (perpendicular to crystal a axis) and normal direction of (001) surface, as shown in the inset of Figure 3a. All the MC curves exhibit the same trend, i.e., a sharp negative cusp appears in the low field region, corresponding to WAL [71,72,73,74], as shown in Figure 3a. All the curves at the low field are scaled into one single line with out-of-plane magnetic field component $B\cos \alpha$, showing that WAL originates from 2D surface states. With increasing temperature, the cusp at the low field appears and the MC curves become conventional parabolic, denoting the shortening dephasing length caused by rising temperature [75]. The dephasing length $l_{\varphi }$ can be derived through the Hikami–Larkin–Nagaoka (HLN) theory as Equation (4) [76]: $∆\mathsf{\sigma } = \epsilon \left({\displaystyle \frac{e^2}{\mathsf{\pi }h}}\right) \left[\psi ({\displaystyle \frac{1}{2}}+{\displaystyle \frac{h}{8\mathsf{\pi }\mathrm{e}B{l_{\varphi }}^2}}) - \ln ({\displaystyle \frac{1}{2}}+{\displaystyle \frac{h}{8\mathsf{\pi }\mathrm{e}B{l_{\varphi }}^2}})\right]$, where $l_{\varphi }$ defines the maximum coherent length for electronic wavefunction, $\mathsf{\epsilon }$ parameterizes quantum channel multiplicity, ψ(x) is the digamma function, and e, h are fundamental constants. It should be noted that the pre-factor $\epsilon$ can reflect the quantity of conductive channels, which requires further investigations in the future [77]. Figure 3b shows the temperature-dependent dephasing length, with the fitted exponent $T^{ -0.56}$. The dimensionally sensitive nature of WAL manifests through the temperature scaling law $l_{\varphi } \propto T^{ -p/2}$, where the exponent p encodes the dominant dephasing mechanism [78,79,80]. Specifically, 2D systems exhibit p = 1 for electron–electron scattering and p = 2 for electron–phonon scattering, whereas 3D systems follow p = 3/2 (electron–electron) and p = 3 (electron–phonon) [78,79,80]. Crucially, the experimental fitting result (i.e., $l_{\varphi } \propto T^{ -0.56}$) aligns with the 2D electron–electron scattering exponent, thereby confirming the surface state origin of the WAL effect and excluding bulk contributions. Since the normal direction of the (100) plane can also be projected to the normal direction of the (001) plane, further distinguishment of the two surfaces is needed. Therefore, for separating the contribution of the (100) and (001) planes, one introduces a second rotation angle β, which denotes the angle between the magnetic field (perpendicular to crystal b axis) and the normal direction of the (001) surface, as shown in the inset of Figure 3c. Towards the normal direction of the (100) surface, MC curves are angular scaled by $B\cos (\beta -73°)$, showing consistent collapse behavior, as shown in the enlarged plot in Figure 3d. On the other hand, the failed scaling towards $B\cos \beta$, which is shown in Figure 3e and corresponding enlarged view in Figure 3f, reconfirms that the WAL is introduced by 2D surface states with special 73° configuration, which is the (100) surface of ${\alpha }^{\prime }$-Bi₄Br₄. The 2D WAL that originated from the (100) surface is related to Dirac-cone type dispersion, which provides non-trivial Berry phases to generate WAL rather than WL [70,77].

Additionally, anisotropic WAL is a universal approach to identify spin texture for relativistic electrons in topological materials without access to sophisticated techniques [35]. One can use low-temperature WAL-corrected PHE to identify the anisotropic electronic structures with antiparallel spin texture in topological materials such as the (100) surface states in ${\alpha }^{\prime }$-Bi₄Br₄. On the one hand, PHE arises from the anisotropic magnetoresistance (AMC) when the magnetic field and electric field are oriented within the same plane. The corresponding expressions in terms of AMC are given by Equations (5) and (6) [81,82,83,84,85,86,87,88], $G_{xx} = G_{\perp } + (G_{\parallel } - G_{\perp }){\cos }^2\phi$ and $G_{yx}^{PHE} = (G_{\parallel } -{ G}_{\perp })\sin \phi \cos \phi$, where $G_{xx}$ is the longitudinal magnetoconductance with $G_{yx}^{PHE}$ being the transverse magnetoconductance. $G_{\parallel }$ and $G_{\perp }$ denote the conductance aligned parallel and perpendicular to the magnetic field B, respectively. And $\phi$ signifies the angle between the electric field and the magnetic field. At low temperatures, low-dimensional electronic systems in topological materials exhibit a phase coherence length sufficiently extended to support WAL. Equations (5) and (6) will be modified to Equations (7) and (8), $G_{xx} = G_{\perp } + G_{WAL }+ (G_{\parallel } -{ G}_{\perp } +{ G}_{WAL}){ \cos }^2\phi$ and $G_{yx}^{PHE} = (G_{\parallel } -{ G}_{\perp } +{ G}_{WAL}) \sin \phi \cos \phi$, where $G_{WAL}$ denotes the WAL-induced corrections of conductance dependent on the dimension, specific Fermi surface geometry, and spin texture. In the 2D isotropic Dirac-type electronic system, the $G_{WAL}$ is independent from $\phi$ [89]. Consequently, $G_{yx}^{PHE}$ exhibits no angular modulation and yields the intrinsic sine or cosine curves depicted in the top panel of Figure 4a. By contrast, highly anisotropic 2D Dirac systems (e.g., ${\alpha }^{\prime }$-Bi₄Br₄, comprising weakly coupled 1D chains) exhibit strong confinement of energy–momentum dispersion to the b axis due to their quasi-1D lattice. This is accompanied by spin–momentum locking in topological surface states, which rigidly orients spins perpendicular to momentum. Hence, the spin is locked perpendicular to the chain direction (left panel, Figure 4b). Owing to the significantly higher conductivity along the b axis compared to transverse directions, an in-plane magnetic field restricts carrier quantum diffusion loops to the 1D chains (right panel, Figure 4b). In this context, a 1D field-dependent localization equation can be used to adjust the WAL correction in Equations (7) and (8) to Equations (9) [90,91], $G_{WAL}={\displaystyle \frac{2e^2}{\hslash L}}{[{\displaystyle \frac{1}{{l_{\mathsf{\varphi }}}^2}}+{\displaystyle \frac{e^2{\left(B\sin \phi \right)}^2{w}^2}{3{\hslash }^2}}]}^{-0.5}$, where L represents the length of 1D transport channel with w being the width or diameter of the channel. And $e$ and $\hslash$ denote the elementary charge and reduced Planck constant, respectively. In this case, the curves of $G_{xx}$ and $G_{yx}^{PHE}$ deviate from intrinsic trigonometric function and adopt the distorted curves as shown in the blue lines in the bottom panel of Figure 4a. This phenomenon is defined as the APHE. Since the APHE arises from the anisotropy of the system, its line shape encodes the WAL anisotropy and, by extension, the spin texture. Hence, the APHE can be viewed as an effective tool for testing the spin texture or spin–orbit interaction of anisotropic 2D electron states. In Figure 4c, the isothermals ranging from 2 K to 50 K of angle-dependent $G_{xx}$ (the left panel) and $G_{yx}^{PHE}$ (the right panel) collected at $B = 9 T$ are presented. A pronounced transformation from APHE to typical PHE was observed above 30 K, indicating the progressive suppression of WAL correction. Equation (9) was applied to reproduces all experimental traces across the entire temperature range with excellent fidelity. An enlarged view of the 2 K data is shown in Figure 4d (top: AMC; bottom: PHE) for further clarity. In addition, Figure 4e demonstrates the fitted dephasing length $l_{\mathsf{\varphi }}$ from T = 2 to 30 K, indicating obvious temperature dependence. The $l_{\mathsf{\varphi }}~T^{ -0.42}$ tendency extracted from Figure 4e is in accordance with previous reports and also coincides with the quasi-1D feature of ${\alpha }^{\prime }$-Bi₄Br₄ [47]. Collectively, quantitative links between APHE and 1D WAL correction were built by angular-dependent electric transport measurement in the (100) surface of ${\alpha }^{\prime }$-Bi₄Br₄ in this work, indicating that the information about anisotropic features can be encoded into unique transport behavior such as APHE. The valuable result of this work provides an alternative methodology for detecting the spin texture of charge carriers in topological materials. It is noticeable that WAL and VRH coexist in the low temperature region, of which the interplay is beneficial to understanding the quantum behavior in future investigations [51,92].

3.3. One-Dimensional Transport Behavior

As mentioned above, the unique features of quantum transport behavior in Bi₄X₄ from 3D and 2D perspectives were introduced in a comprehensive manner, respectively. Moreover, 1D distinctive topological hinge states in ${\alpha }^{\prime }$-Bi₄Br₄ have also been investigated systematically [93,94,95]. Recently, the existence of the quantum transport response of 1D topological hinge modes in four-layer ${\alpha }^{\prime }$-Bi₄Br₄ was verified comprehensively by Hossain et al. in 2023 [49]. For ${\alpha }^{\prime }$-Bi₄Br₄, gapless topological hinge states can only appear at the two outermost hinges of an ideal crystal, since other (100) in-plane topological boundary states couple with each other to open gaps [44]. In four-layer flake devices, the pair of hinge states reside on the same side surface, either the (100) or (−100) plane, as the total number of layers is even [43,44]. Restricted by the finite chain length, the surface states in (010) and its opposite plane degenerate to two gapless hinge states, which connect another two hinge states extended along the b axis to form a conductive loop in the bc plane, as shown in Figure 5a,b [49].

Thereby, the transport evidence can be discussed after establishing the presence of a hinge-mode loop within the (100) surface. A defining characteristic of quantum coherent transport is the manifestation of electron Aharonov–Bohm interference, observed as oscillations in resistance with the magnetic field. The period is defined as $∆B = {\varphi }_0/S$, where ${\varphi }_0 = h/e$ represents the flux quantum, S denotes the area enclosed by the coherent electron trajectory, h and e are fundamental physical constants, as mentioned before.

In this study, the researchers confirmed that the observed Aharonov–Bohm oscillations in the sample originate from hinge state carriers. Figure 5c illustrates a series of AMR curves with θ denoting the angle between magnetic field B (always perpendicular to a axis) and normal direction of the ab plane, as shown in Figure 5a and the inset of Figure 5d. The oscillation period $∆B$ increased monotonically with θ. Figure 5e demonstrates a series of magnetoresistance curves of another sample with $\delta$ denoting the angle between magnetic field B (always perpendicular to b axis) and the a axis, as shown in Figure 5b and the inset of Figure 5f. The oscillation period $∆B$ also demonstrates an interesting evolution trend with the magnetic field.

To further elucidate these trends and provide more robust evidence, the oscillation frequency is extracted by the fast Fourier transform (FFT) method, as the function of θ, which is depicted in Figure 5d. The dominant peak in FFT exhibits a distinct dependence with θ, and reaches its maximum when perpendicular to the ab plane. The variation in FFT frequency coincides with Figure 5a: as the θ increases, the component of the bc plane (filled with the purple color in Figure 5a) perpendicular to the magnetic field gradually decreases and reaches zero when B is parallel to the b axis. Meanwhile, FFT analysis is also performed in another sample, depicted with $\delta$ as demonstrated in Figure 5f. By fitting the data in Figure 5f to a cosine function, the variation mode of the frequency is obtained as $1.99 \cos (\delta - (16.4 \pm 0.3)°)$. The frequency processes a trend of initial increase followed by a decline, and reaches its maximum when $\delta$ is around 16.4$°$, i.e., perpendicular to the bc plane. The special 16.4$°$ angle is consistent with the identical monoclinic crystal structure of ${\alpha }^{\prime }$-Bi₄Br₄, indicating that an Aharonov–Bohm effect clearly results from coherent charge carriers moving at the perimeter of the bc plane. Figure 5g displays the FFT results of the first derivative $dR/dB$, where $R$ denotes the MR with the magnetic field $B$ parallel with the normal vector of the ab plane at low temperatures ($T \approx 30 \mathrm{mK}$). In addition to the dominant Aharonov–Bohm peak, a secondary feature at twice the fundamental frequency, characteristic of Altshuler–Aronov–Spivak (AAS) oscillations, is distinctly resolved.

Another experimental investigation on the hinge states is based on the Josephson junction of $\alpha$-Bi₄I₄ flake contacting with Ti/Nb. Figure 6a shows the differential resistance dV/dI versus bias current I and gate voltage (V_g), which implies two sharp peaks at ∼±15 and ∼±40 μA (Figure 6b), corresponding to the superconducting critical current [28]. It should be noted that these critical currents are considerably larger than those observed in graphene Josephson junctions with higher mobilities and similar device parameters [96,97,98]. This phenomenon is attributed to the gate-tunable additional channels, such as edge states in quantum spin Hall insulator, the gapped side surface states, and/or gapless hinge states in HOTI.

4. Conclusions and Outlook

In this review, we summarized several typical quantum transport phenomena in the quasi-1D TQM family Bi₄X₄, which are related to bulk states or stacking-modulated topological electronic states, leading to effects such as VRH, WAL, APHE, Aharonov–Bohm oscillations, and so on [47,48,49,51]. The electronic transport behavior is dependent on the crystal structure, where there are four types of stacking modes (${\mathsf{\alpha }}^{\prime }$ phase, $\mathsf{\gamma }$ phase, $\mathsf{\beta }$ phase, and $\mathsf{\alpha }$ phase, respectively), resulting in four different topological phases (HOTI, non-degenerate WTI, WTI, and NI, respectively). One efficient way to control the stacking mode with topological phases is to alter the molar ratio of Br and I atoms, leading to tunable topological phase transition with multiple topological orders. Therefore, with proper regulation of the halogen ratio, one may achieve the tunable conducting of selective topological states, which may expand from 1D to 3D space. Promisingly, the high tunability and double-side cleavable feature promises high-resolution in situ characterization under different perspectives. The existing studies mainly focus on STM, ARPES, and bulk crystal electric transport characterization [37,43,51]. The quantized conductance in thin-layer Bi₄X₄ devices has not been verified experimentally, which is the most solid evidence to confirm the quantum spin Hall insulating nature of the mono-layer sample. Additionally, the predicted room temperature quantum spin Hall effect, which is also supported by the ARPES and STM results, still needs to be confirmed by quantum transport measurements. In order to achieve this goal, the method with the fine protection of samples during exfoliation and lithography processes should be adopted to stabilize the outer halogen in atomic chains [49] 2. With state-of-the-art device fabrication approaches, in the future, the Bi₄X₄ family is expected to become the ideal platform for next-generation low-power spintronics and fault-tolerant quantum computing, initiating innovation from the discovery of quasi-1D topological materials to their integration and application.