In the exploration of three-dimensional quaternary topological insulators, understanding surface states has become pivotal for unraveling the underlying physics and tapping into potential applications. Our study delves into the temperature and magnetic field-angle dependence of universal conductance fluctuations (UCF) and weak anti-localization (WAL) effects in a Bi1.9Sb0.1Te2Se topological insulator-based mesoscopic device. Conventionally, other low-temperature transport phenomena in probing surface states may inevitably face interference from three-dimensional bulk conductance. However, we experimentally demonstrate that, at low temperatures, UCF reflects the properties of two-dimensional topological surface states more accurately, thereby providing a more reliable and distinct way to confirm their existence. Moreover, we carefully analyze the temperature-dependent WAL using the Hikami–Larkin–Nagaoka model, proposing a crucial role for charge puddles associated with electrostatic fluctuations in the electron dephasing process. Our findings not only emphasize the key role of UCF in unveiling the underlying behavior of topological surface states but also offer a deeper understanding of phase-coherent transport in quaternary topological insulators.

Topological insulators (TIs) share similarities with conventional insulators as they are also gapped bulk insulators with a specific energy gap. However, TIs exhibit a unique feature with the existence of a new class of gapless conducting edge or surface states1,2 owing to the nontrivial topological character of their bulk band. These surface states are topologically protected by time-reversal symmetry3,4 and demonstrate remarkable resilience against weak disorder, making TI materials the core active candidates for the field of quantum computations.

Among the notable TI materials, Bi1−xSbx alloy5 in a particular range of x was the first theoretically predicted three-dimensional (3D) TI and later experimentally6 confirmed. Along with Bi1−xSbx, the materials of Bi2Se3,7 Bi2Te3,8,9 and Sb2Te38,10,11 gradually became the focus of research, motivated by the presence of a single simple Dirac cone8 on their surfaces. Although Bi2Se3, Bi2Te3, and Sb2Te3 are, in principle, excellent 3D topological materials for transport experiments, their Fermi levels are often unintentionally locked in the bandgap12 due to crystalline defects, leading to undesired doping effects. Consequently, this can result in a significant transport contribution from the bulk channel, bringing obstacles to the full exploration of the remarkable surface states. Hence, minimizing the bulk contribution and realizing surface-dominated transport are of significant importance in this field.

Technically, developing new ternary or quaternary compounds, such as Bi2Te2Se13 and Bi2−xSbxTe3−ySey,14 has been proven to be a practical strategy for engineering and manipulation of surface states. The quaternary compounds demonstrate even higher bulk-insulating properties than ternary materials due to the compensation between donor-type and acceptor-type defects achieved through alloying of pnictogens (Bi and Sb) and chalcogens (Se and Te). However, the stoichiometric inhomogeneity and the electrostatic fluctuations resulting from the compensation doping15,16 may substantially complicate the investigation of the surface states. Therefore, reliable identification of topological surface states and a comprehensive understanding of the quantum interference transport are highly needed, but they remain less fully explored in the prominent quaternary TI materials.

To address these challenges, we employ two popular experimental probes, namely, universal conductance fluctuations (UCF) and the weak anti-localization (WAL) effect, which have been widely utilized in exploring the low-temperature quantum transport properties in various systems, such as Bi2O2Se,17 Bi2Se3,18 and PdCoO2.19 UCF is characterized by aperiodic but reproducible conductance fluctuations with a universal value of e2/h,20 originating from the quantum interference of all possible electron paths traversed between two points in a sample.21 The WAL effect corresponds to the destructive interference of the electron wave function between the time-reversed paths.22 By analyzing UCF and the WAL effect together, a deeper understanding of the underlying physics of the quantum interference transport in quaternary TI-based mesoscopic devices should be available.

Thus, in this work, a weakly disordered Bi1.9Sb0.1Te2Se micro-flake was chosen, and its magneto-transport measurements have been investigated thoroughly. Through a comparative study of UCF and the WAL effect, we experimentally demonstrate that UCF is a more reliable method for identifying the existence of topological surface states. In addition, the dephasing mechanism has been discussed in detail, providing evidence for the presence of charge puddles in the Bi1.9Sb0.1Te2Se system.

Single crystals of Bi1.9Sb0.1Te2Se were grown by the modified Bridgman method. High-purity Bi (5N), Te (5N), Se (5N), and Sb (5N) powders were carefully mixed and sealed in an evacuated quartz tube. The tube was initially heated to 900 °C and held at this temperature for 2 days, after which it was slowly cooled down to 600 °C within 5 days and finally quenched in liquid nitrogen. The quality of the as-grown single crystals was probed by the x-ray diffraction (XRD) method.

To minimize the bulk contribution to the transport properties, a thin flake of Bi1.9Sb0.1Te2Se, ∼67 nm in thickness, was mechanically exfoliated onto a Si substrate capped with a 300 nm thick silicon dioxide layer. Subsequently, standard ultraviolet lithography techniques were employed, followed by sputter-coating of 60 nm Au layer electrodes and contact leads. All the magneto-transport measurements were carried out in the Quantum Design Physical Property Measurement System (PPMS)-9.

Figure 1(a) shows the room temperature XRD pattern of the as-grown bulk Bi1.9Sb0.1Te2Se crystal, which exhibits well-resolved (00l)-oriented reflections, confirming the high-quality single crystalline nature of the prepared bulk samples. Obviously, a distinct shift to higher 2θ has been observed in Bi1.9Sb0.1Te2Se, compared to a single crystal of Bi2Te2Se, which is also shown in Fig. 1(a). This shift in the diffraction angle results from the alloying of Sb with a relatively smaller atomic radius into the Bi site within the lattice. The obtained Bi1.9Sb0.1Te2Se crystal exhibits easy cleavability, facilitating the fabrication of a four-terminal device based on mechanically exfoliated Bi1.9Sb0.1Te2Se micro-flakes. Figure 1(b) illustrates the schematic of the device geometry and the actual optical micrograph of the studied sample. It is hoped that the bulk regime of the fabricated Bi1.9Sb0.1Te2Se micro-flake behaves as an “insulator” due to compensation doping and its reduced thickness. The temperature dependence of resistance of the corresponding micro-flake is plotted in Fig. 1(c), unequivocally demonstrating its insulating behavior below 200 K. The fit of Rxx(T) by Arrhenius law, RxxeEa/kBT, yields an activation energy of about 1.65 meV, as shown in the inset of Fig. 1(c). Our extracted value is comparable with those of other reports on similar TIs.13,23 Generally, the activation energy is sensitive to the location of defect states in the energy band, and its absolute value can reflect the bulk-insulting behavior of the sample. To detect and characterize the surface state more easily, larger values of activation energy are desired normally.

FIG. 1.

(a) Room temperate x-ray diffraction of Bi1.9Sb0.1Te2Se and Bi2Te2Se single crystals. (b) The schematic diagram of device geometry and the optical micrograph of the studied device. (c) Temperature dependence of the resistance of the Bi1.9Sb0.1Te2Se micro-flake. The inset shows the Arrhenius fitting for the sample, and the fitting range is from 190 K to ∼60 K.

FIG. 1.

(a) Room temperate x-ray diffraction of Bi1.9Sb0.1Te2Se and Bi2Te2Se single crystals. (b) The schematic diagram of device geometry and the optical micrograph of the studied device. (c) Temperature dependence of the resistance of the Bi1.9Sb0.1Te2Se micro-flake. The inset shows the Arrhenius fitting for the sample, and the fitting range is from 190 K to ∼60 K.

Close modal

Temperature-dependent magnetoresistance measurements were conducted on the present Bi1.9Sb0.1Te2Se micro-flake in the range of −1 to 9 T, with the magnetic field perpendicular to the sample surface. As shown in Fig. 2(a), sharp cusps occur at zero magnetic fields, indicative of the onset of the WAL effect,24 which is expected in a strong spin-orbital coupled system. This effect will be discussed in detail later. Moreover, it is noted that there is some special “noise” in each curve. After subtracting a smooth background from each magneto-conductance curve, aperiodic but reproducible oscillations are observed, as depicted in Fig. 2(b). The amplitude of these fluctuations decreases with increasing temperature, mainly associated with the reduction in the phase-coherence length.25,26 The temperature-dependent conductance fluctuation matches with the typical characteristics of the UCF, a phenomenon commonly observed in other TI materials, such as Bi2Se3,27 Bi2Te2Se,28 quaternary Bi1Sb1Te1.5Se1.5,29 and Bi1.5Sb0.5Te1.8Se1.2.30 The currently observed fluctuation pattern is unique to the studied Bi1.9Sb0.1Te2Se micro-flake because UCF are sensitive to defect configurations.31 

FIG. 2.

(a) Magneto-conductance curve of the Bi1.9Sb0.1Te2Se micro-flake at different temperatures. (b) Temperature-dependent UCF curves. The magnetic field B is perpendicular to the surface of the sample.

FIG. 2.

(a) Magneto-conductance curve of the Bi1.9Sb0.1Te2Se micro-flake at different temperatures. (b) Temperature-dependent UCF curves. The magnetic field B is perpendicular to the surface of the sample.

Close modal
To quantitatively verify whether the fluctuations originate from the surface states or not, we continue to investigate the temperature-dependence of the root mean square of the conductance fluctuation (δGrms), as displayed in Fig. 3(a). Here, δGrms is defined as follows:13,
δGrms=[δG(B)δG(B)]2,
(1)
where 〈⋯〉 expresses the ensemble average. In a two-dimensional system, δGrms usually follows the power-law temperature dependence, given by δGrmsT−0.5.32 As to the Bi1.9Sb0.1Te2Se micro-flake in this work, the data fitting at temperatures below 10 K reveals that δGrms scales as T−0.58, in good agreement with the anticipated 2D surface transport. To further support the 2D nature of UCF, we performed the magneto-conductance measurements at 2 K while varying the direction of the magnetic field (θ). Here, θ is defined as the angle between the direction of the magnet field and the normal of the sample surface. As shown in Fig. 3(b), by scaling the magnetic field with the vertical component B cos θ, the reproducibility of UCF has been obtained, a characteristic often observed in ideal 2D electronic systems.31,33 More interestingly, when the direction of the magnetic field is varied, δGrms is shown to be angle-dependent. Specifically, δGrms dampens from 0.018 to 0.006 e2/h as θ increases from 0° to 90°. For a given 3D electronic system, the anisotropy of UCF is typically suppressed compared to a 2D system. This is because δGrms is proportional to the phase coherence length Lφ31,34 and Lφ is hardly dependent on magnetic field directions in a 3D system. Therefore, the observed anisotropy in UCF here provides compelling evidence of surface-dominant conduction in the Bi1.9Sb0.1Te2Se micro-flake under investigation.
FIG. 3.

(a) Temperature dependence of the root mean square of the conductance fluctuation. The fitting curve using power law is also shown by the red solid line. To observe the deviation above 10 K more clearly, we intentionally extend the fitting line from 10 to 30 K. (b) The conductance fluctuations (δG) vs B cos θ for different field directions (θ = 0°, 22.5°, 45°, and 67.5°). As there is no perpendicular component when θ = 90°, the corresponding data are not shown here. Inset: δGrms vs different field directions.

FIG. 3.

(a) Temperature dependence of the root mean square of the conductance fluctuation. The fitting curve using power law is also shown by the red solid line. To observe the deviation above 10 K more clearly, we intentionally extend the fitting line from 10 to 30 K. (b) The conductance fluctuations (δG) vs B cos θ for different field directions (θ = 0°, 22.5°, 45°, and 67.5°). As there is no perpendicular component when θ = 90°, the corresponding data are not shown here. Inset: δGrms vs different field directions.

Close modal
As mentioned earlier, the WAL effect has been observed, which can help us get more insights into the quantum transport properties of Bi1.9Sb0.1Te2Se. The temperature dependence of conductance ΔG = G(B) − G(0) at small magnetic fields is displayed in Fig. 4(a). With increasing temperature, the cusp-like maxima around zero field vanish gradually due to enhanced channel decoherence.35 To analyze the experimental magneto-conductance, we focus on the low-field regime ranging from −0.5 to 0.5 T and employ the Hikami–Larkin–Nagaoka (HLN)36 model in the limit of strong spin–orbit interaction for the 2D system. In this model, ΔG(B) is described by
ΔGB=αe2πhψ12+BφBlnBφB,
(2)
where α is a prefactor and α = 0.537,38 means that there is only one TI surface state providing a channel for electron transport. ψ(x) is the digamma function, and Bφ=4eLφ2 is the phase coherence field related to Lφ.
FIG. 4.

(a) Low field dependence of magnetoconductance ΔG and the best fits by the HLN model. (b) The parameters of α and Lφ extracted from the fitting by the HLN model as a function of temperature. (c) The dephasing length Lφ vs temperature, using log scale for both the x-axis and y-axis. The dashed lines show LφT−0.22 and LφT−0.54 for 2–7 K and 10–30 K, respectively. (d) The angle-dependent magneto-conductance at T = 2 K.

FIG. 4.

(a) Low field dependence of magnetoconductance ΔG and the best fits by the HLN model. (b) The parameters of α and Lφ extracted from the fitting by the HLN model as a function of temperature. (c) The dephasing length Lφ vs temperature, using log scale for both the x-axis and y-axis. The dashed lines show LφT−0.22 and LφT−0.54 for 2–7 K and 10–30 K, respectively. (d) The angle-dependent magneto-conductance at T = 2 K.

Close modal

Through fitting the data with the HLN model, the temperature dependence of prefactor α and coherence length Lφ can be derived, both shown in Fig. 4(b). At 2 K, the prefactor α turns out to be about 0.6, indicating that the main contributions to the WAL are from the surface states. When the temperature increases, α slightly decreases, suggesting the possibility of non-negligible conduction from the bulk, which may explain the lower value of α at higher temperatures where the WAL is suppressed.39 The observation of the decrease in Lφ with increasing temperature is consistent with other TI systems.13,29 Nevertheless, a quantitative evaluation of Lφ as a function of temperature reveals some distinct behaviors. Figure 4(c) shows that the curve exhibits a T−0.22 dependence for the temperature range 2–7 K and a T−0.54 dependence for the temperature range 10–30 K. Typically, in a 2D system, Lφ decays with Tβ, where a different exponent β corresponds to different scattering mechanisms, such as β = 1/216 for electron-electron scattering and β = 113 for electron–phonon scattering. Here, the slower dephasing rate at low temperatures (T−0.22 dependence) is unusual and requires further discussion, indicating the presence of other possible dephasing channels. Comparable exponents have also been reported in other quaternary topological materials, such as the Bi1.5Sb0.5Te1.8Se1.2 nanoflake device (LφT−0.19)40 and in the Bi1Sb1Te1.5Se1.5 flake (LφT−0.15).29 In this regard, it is assumed that the slow dephasing process in our sample may be attributed to the same reason as in these two similar quaternary TI materials, which is related to the presence of electron–hole charge puddles.30 Indeed, the existence of charge puddles on the surface of the TI material has been proved in both theoretical15 and experimental studies.41,42 Our results on the temperature dependence of Lφ suggest the presence of electron–hole charge puddles associated with the electrostatic fluctuations from the compensation and highlight its crucial role in the intrinsic electron dephasing at low temperatures. Since the studied Bi1.9Sb0.1Te2Se has some similarities in structure to Bi2Te2Se, as confirmed by XRD, here, we would like to have some more discussion between our sample and that in the publication by Li et al.13 In both cases, the temperature dependent of α is comparable, while the behavior of phase coherence length Lφ differs, and thus, the electron dephasing mechanism varies. The lack of consistency indicates that the stoichiometric inhomogeneity can affect the quantum transport substantially in the TI material.

As we have previously discussed based on the experimental results of UCF, the quantum transport in the studied Bi1.9Sb0.1Te2Se micro-flake originates from the 2D surface states. It is interesting to further confirm this argument from the perspective of the tilted magnetic field dependence of the WAL effect.24 As a 2D system, the surface states should be solely B cos θ dependent. In Fig. 4(d), we present the ΔG vs B cos θ (the normal component of B) plots. At low magnetic fields, all traces follow the same curve, consistent with the UCF analysis. However, as the magnetic fields increase to higher values, the curves start to deviate from each other, indicating the non-negligible influence of bulk conductance. This behavior is in stark contrast to the UCF results, where the influence from bulk can be safely neglected, as indicated by the lower values of δGrms for the in-plane field (δGrms = 0.006e2/h) than the out-of-plane field (δGrms = 0.018e2/h). Taking all the discussions into consideration, the quantum transport in the Bi1.9Sb0.1Te2Se micro-flake is predominantly governed by the surface states. Moreover, the unique advantage of UCF in reliably identifying the existence of topological surface state has been demonstrated, while other low-temperature transport phenomena inevitably face interference from bulk conductance.

In summary, we have carried out low-temperature magneto-transport property measurements on a bulk-insulating three-dimensional Bi1.9Sb0.1Te2Se topological insulator micro-flake, revealing the simultaneous observation of two characteristic phase coherence transport phenomena: UCF and WAL. Through temperature and magnetic field angle-dependent UCF, we have confirmed that the conduction electrons are predominantly governed by the surface states. The intriguing temperature dependence of coherence length, derived from the WAL effect, leads us to suggest the presence of electron–hole charge puddles. Furthermore, we argue that UCF has distinct advantages over WAL in the reliable identification of topological surface states, a vital aspect previously overshadowed. Our findings offer a richer and deeper understanding of phase-coherent transport in quaternary topological insulators, shedding new light on their complex behaviors and setting a valuable foundation for future explorations.

This work was financially supported by the Natural Science Foundation of Nanjing University of Posts and Telecommunications (Nos. NY221063 and NY220186) and the Postgraduate Research and Practice Innovation Program of Jiangsu Province (No. SJCX22_0256).

The authors have no conflicts to disclose.

Wei Wang: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Project administration (equal); Writing – original draft (equal); Writing – review & editing (equal). Shengjing Hu: Funding acquisition (equal); Investigation (equal); Validation (equal). Qiyun Xie: Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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