Introduction

In topological insulators (TIs), a finite band gap in the bulk is accompanied by metallic surface states with linear dispersion, resulting from band inversion1,2. These surface states have a variety of potential applications owing to their robustness resulting from topological protection, potentially high mobilities, and spin-momentum locking, making them ideal candidates for spin-transport3,4,5,6,7,8. In particular, devices for applications in advanced computing and logic beyond Moore’s law have been envisioned9,10,11,12,13.

Bi2Se3 and Bi2Te3 are prototypical TIs that have been widely studied in order to understand this vibrant class of materials14,15,16,17,18,19. This is because they are relatively simple to grow as thin films20,21,22, and both possess a single Dirac point at accessible doping levels23. Despite the great amount of research into their topological nature, little work has been done to understand how changing concentrations with either Se or Te affect the overall topological properties. Insight into the nature of the conducting surface states as well as their robustness to chemical substitution is vital to effectively utilizing these materials. Further rationale for investigating Bi2TexSe3−x alloys is that they offer robust Dirac dispersion for surface conduction combined with a path toward reduced bulk conductivity compared to their parent compounds at either x = 0 or 324. Specifically, the Bi2Te2Se alloy represents a special line compound whose bulk conductivity is sufficiently low that surface and bulk transport may be distinguished25. The interest, therefore, is to obtain high quality epitaxial thin films that facilitate transport mediated by topological surface states distinct from bulk electronic states. Moreover, in order to use these materials in the proposed applications, the material properties must be optimized.

In this work, we perform magnetoresistance (MR) measurements on four Bi-based topological material films: Bi2Se3, Bi2Te3, and two films with a mixture of Se and Te under constant growth conditions. We observe the 2D weak anti-localization (WAL) indicative of topological materials and we explore anisotropic magnetoresistance effects. We find a strong angle-independent WAL contribution in Bi2Se3 resulting from 3D bulk states. 2D states arise as Te replaces Se, creating a sinθ dependence to the WAL though also accompanied by an increasing classical MR background. However, these 2D states disappear in the pure Bi2Te3.

Thin Film Growth and Device Fabrication

MBE growth of Bi2TexSe3−x was performed on (001) semi-insulating GaAs. During MBE growth, the relative (Te + Se)/Bi beam-equivalent flux ratios for all epitaxial layers ranged from 15–20, and the nominal growth temperature was constant at 290 °C. For consistency, the thickness of all films was limited to 50 nm. The specific alloys that were obtained in this study include pure Bi2Se3 (x = 0), Bi2Te2Se1 (x = 2), Bi2Te2.5Se0.5 (x = 2.5) and pure Bi2Te3 (x = 3). X-ray, RHEED and TEM characterization confirmed high quality epitaxy throughout the alloy system with a sharp interface with the substrate. (see Supplemental Materials Figs. S5–7). Nonetheless, AFM analysis [Fig. 1(a–d)] of the highly specular films showed that the surface height varies between 15 nm for Bi2Se3 and 25 nm for Bi2Te3, similar to other MBE grown films in the literature22,26,27. This discrepancy is likely due to the highly polycrystalline nature of the films.

Figure 1
figure 1

(a–d) AFM images of the four alloys used in this study. The surface height variation for these same films was between 15 nm for Bi2Se3 and 25 nm for Bi2Te3. (e) Optical image of one 10 × 20 µm hall bar. (f) Schematic of hall bar and magnetic field orientation.

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Mesoscopic Hall bars (10 × 20 µm2, Fig. 1(e)) were defined lithographically with an argon plasma etch. While plasma is known to cause damage, the surface was protected by a 1.5 μm thick, spun-on layer of poly(methyl methacrylate) (PMMA) such that the edges of the mesa were sharp, and similar measurements using similar un-patterned films with pressed indium contacts showed similar carrier densities. Electron beam evaporated gold (150 nm) with a titanium adhesion layer (10 nm) provided Ohmic contact to the Hall bar. Gold wire bonds were affixed with a ball bonder using indium spheres without heating or ultrasonic agitation.

Magnetoresistance Measurements and Analysis

Magnetoresistance (MR) was measured for all four samples in a variable temperature cryostat set in a 1 T resistive magnet on a rotating platform. Sample measurement geometry was as shown in the schematic depicted in Fig. 1(f). For a majority of the measurements, the magnetic field is first applied in-plane with the film and perpendicular to the current direction, as indicated in the figure. The magnet is then incrementally rotated around the sample in the cryostat using the rotating platform such that the field is eventually out-of-plane with the sample. Other studies in the literature observed an anomalous negative magnetoresistance in Bi2Se3 when the magnetic field is both in-plane with the film and parallel to the applied current28. However, the negative component is expected to dominate at fields significantly higher than those used in our experiment. Nonetheless, we performed the measurement using this geometry on our samples as well and indeed observed similar results to the geometry shown in Fig. 1 (See Supplemental Materials, Fig. S1).

First, we measure mobility of all of the samples. Hall mobilities \({\mu }_{Hall}\) and carrier concentrations \(n\) were determined using the ordinary Hall effect \(d{R}_{xy}/dB\,=\,1/net\), where \(t\) is the sample thickness, e is the elementary electronic charge, and \({\mu }_{Hall}=1/ne\rho \). The temperature dependence of the carrier concentrations and Hall mobilities are shown in Fig. 2. In Bi2Se3, n is nearly independent of temperature due to the semimetallic nature of the Dirac state, while n in Bi2Te3 has a strong exponential dependence due to the stronger influence of the bulk bandgap in the transport. Conversely, the mobility in Bi2Te3 and Bi2Te2.5Se0.5 decreases inversely proportional to T, while the mobility in Bi2Se3 and Bi2Te2Se changes less with T. The strength of the decrease in the mobility with temperature relates to the amount of scattering within the material. For topologically protected states, the scattering should be reduced, thus maintaining a more constant mobility with temperature. The increase (decrease) in carrier concentration (mobility) is consistent with the Bi2Te3 and Bi2Te2.95Se0.05 samples having conventional conducting states, while the steadier mobility in Bi2Se3 and Bi2Te2Se is strong evidence of topological protection. The anomalous slight increase in mobility for Bi2Te2Se could result from ionized impurity scattering29,30. Our measured mobilities of 100–1000 cm2/Vs correspond to mean free path on the order of 10 nm, corresponding to diffusive transport within our samples6.

Figure 2
figure 2

Carrier concentration (a) and mobility (b) versus temperature for the various Te concentrations. The large carrier concentration of 1019 cm-3 indicates these samples are heavily doped into the conduction band. However, the reduced mobility for the two intermediate samples suggests a stronger proportion of the conduction occurs in the surface. (See Supplemental Materials Fig. S8 for resistance versus temperature).

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Figure 3 shows Magnetoconductance (\(\Delta \sigma \)) vs. Magnetic field for both in-plane and out-of-plane magnetic field for the samples that contained Se. Open circles in blue show the data for the in-plane applied magnetic field, while the dashed lines of the same color show the fits to the models, discussed below. Likewise, red “x” marks show the data for the out-of-plane magnetic field and the red dashed line shows the fits. The pure Bi2Te3 sample show a very weak WAL cusp, however the strength of the quadratic background drastically increases the uncertainty in the fitting. Further information can be found in the Supplemental Materials Figs. S3, S4. At 3 K, weak antilocalization (WAL) is observed with a magnetic field applied both in-plane (0°) and out-of-plane (90°), persisting up to 50 K (see Supplemental Materials Fig. S3). For an out-of-plane magnetic field B, the Hikami-Larkin-Nagaoka model for 2D WAL in the strong spin-orbit coupling limit gives the change in conductance as

$$\frac{\Delta {\sigma }_{xx,\perp }}{{\sigma }_{0}}=\frac{\alpha {e}^{2}}{2\pi h}\left[\psi \left(\frac{{B}_{\perp }}{B}+\frac{1}{2}\right)-\,\mathrm{ln}\left(\frac{{B}_{\perp }}{B}\right)\right]$$
(1a)
$${B}_{\perp }=\frac{\hslash }{2e{L}^{2}},$$
(1b)

where \(\psi \) is the Digamma function, \(L\) is the coherence length, and e and h are the electron charge and Plank constant, respectively31,32,33. For Dirac states, the coefficient α is expected to be 1/2 for each Dirac cone34. As is evident in Fig. 3, the model fits the data well. From the fits, we extract the localization length and α. The 3 samples give values of α = 0.43, 0.50, and 0.42 for Bi2Se3, Bi2Te2Se, and Bi2Te2,5Se0.5, respectively. This range is consistent with literature and indicates 2D topological surface transport14,34.

Figure 3
figure 3

Magnetoconductance (\(\Delta \sigma \)) vs. Magnetic field for both in-plane and out-of-plane magnetic fields and the Hikami-Larkin-Nagaoka (HLN) and Tkachov-Hankiewicz (TH) fits with a linear background term (quadratic for x = 2.5). Fits at 3 K for the (a) Bi2Se3, (b) Bi2Te2Se, and (c) Bi2Te2.5Se0.5 samples for in-plane and out-of-plane fields. WAL was extremely weak compared to the much stronger quadratic background for Bi2Te3. Thus, it is not included in these plots. (d) Extracted coherence length from the HLN fit is used in the TH fit to calculate the skin depth.

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A linear background is subtracted for the Bi2Se3 and Bi2Te2Se samples, and a quadratic background for the Bi2Te2,5Se0.5. The character of the backgrounds is derived from the MR behavior up to 9 T (see Supplemental Materials Fig. S4). Similar linear magnetoresistance has previously been observed in these materials, likely arising from the polycrystalline nature of the films, observed here in the AFM images in Fig. 1 35,36,37,38,39. As evidenced by the magnetoresistance data, the quadratic contribution decreases as Te is replaced with Se. The normalization of using [R(B) − R(0)]/R(0) shows the relative contribution of the two component independent of the absolute resistance.

Unusually, TIs exhibit WAL with an applied in-plane field as well. This behavior can partially be explained by WAL originating from 3D states due to high spin-orbit coupling in the bulk, where the effect is independent of field angle40. Additionally an in-plane contribution for 2D states is hypothesized to arise due to a unique hexagonal warping of the surface state energy spectrum at all applied magnetic field angles. In a parallel magnetic field, the magnetic flux through the surface states decays exponentially up to a certain skin depth, defining an effective surface state depth. The magnetoconductance in a TI with an in-plane magnetic field is derived by Tkachov and Hankiewicz in ref. 41 to be

$$\frac{\Delta {\sigma }_{xx,\parallel }}{{\sigma }_{0}}=-\,\frac{{e}^{2}}{2\pi h}\,\mathrm{ln}(1+\frac{{B}^{2}}{{B}_{\parallel }^{2}})$$
(2a)
$${B}_{\parallel }=\frac{2L}{\lambda }{B}_{\perp }=\frac{\hslash }{\lambda eL},$$
(2b)

where, \(\lambda \) is the effective surface state skin depth. For a purely 2D surface state (\(\lambda =0)\), the argument of the log term reduces to 1, eliminating the change in magnetoconductance for an in-plane magnetic field. The existence of in-plane WAL requires a finite skin depth for the surface states, as well as a wide enough field range or small enough \(\lambda /L\) such that the effect is observably non-quadratic. For sufficiently large values of \(\lambda /L\), the in-plane magnetoconductance will be quadratic over a large magnetic field range, in which case the in-plane WAL behavior could be identified as trivial anisotropic magnetoresistance (AMR). The in-plane quadratic MR has previously been observed in Bi2Te340. However, to our knowledge, this is the first application of the Tkachov-Hankiewicz model to explain in-plane magnetoconductance data apart from the authors’ original culling of data available in the literature at the time of their derivation. As observed in Fig. 3, the data and model are in excellent agreement.

From the fits, values for the coherence length and effective surface state skin depth are extracted and shown in Fig. 3(d). The coherence length (skin depth) decreases (increases) with increasing Te concentration, with the WAL effect disappearing in Bi2Te3. The measured effective skin depth for each sample is on the order of the surface height roughness as seen in Fig. 1(a–d), which is much greater than the expected ~4 nm for the topological surface state skin depth from literature42,43. As this value is measure across a macroscopic sample, the effective surface depth is a convolution of the depth of the surface state and the roughness of the sample. Additionally, defects near the surface of the topological material can lead to an increase in the skin depth44. These samples are heavily electron doped (\(n\approx {10}^{19}\) cm−3) which, along with the relative complexity of the band structure for Bi2Te3 relative to Bi2Se3, provides an explanation for the disappearance of the surface states. Bi2Se3 has a bulk bandgap of 0.3 eV, compared to 0.1 eV for Bi2Te345,46. Additionally, the Dirac point in Bi2Te3 lies below the maximum of the bulk valence band while the Dirac point in Bi2Se3 lies above the bulk valence band. The larger band gap and less proximity of bulk states to the Dirac point allow Bi2Se3 to maintain its topological behavior over a wider range of dopings than Bi2Te3. In addition to the decrease in WAL signal, Te substitution is accompanied by an increase in the Hall mobility, owing to a Fermi energy firmly in the conduction band. Based on band structure calculations from the literature it is apparent that bulk bands are closer to the dirac point in Bi2Te3 than in Bi2Se3. Thus, the high carrier concentration in our samples has likely moved the Fermi level far enough into the bulk as to reduce the relative contribution of the surface states18,23,45. We expect the alloyed compositions to have band structures that are some linear combination of the Bi2Se3 and Bi2Te3 bands. From the measured effective penetration depth, we also expect some coupling of surface states on the top and bottom of the film.

From the fits, we see a coherent picture of the development of the topological states as Se is replaced by Te, with \(\lambda \) being on the order of the sample roughness. This observation demonstrates the importance of film surface roughness to obtaining surface-pinned topological states that mix minimally with the bulk. At low temperature, there is a competition between the WAL from the surface states and a low field quadratic background. As the mobility at 3 K increases significantly in pure Bi2Te3 compared to the other samples, the relative size of the WAL cusp decreases leading to the near disappearance of WAL in Bi2Te3. A small cusp is visible, though overwhelmed by a quadratic background which precludes fitting to the Hikami-Larkin-Nagaoka model. The lack of angular dependence in the WAL signal from Bi2Se3 indicates a 3D origin, while the stronger theta-dependence in the alloyed samples indicates some 2D surface contribution.

In various other studies on Bi2Se3, a negative longitudinal magnetoresistance is observed when the field is applied along the current, often attributed to the chiral anomaly47,48,49. We do not observe this effect over the measured field range as the negative magnetoresistance effect is overwhelmed by the WAL at low fields. The in-plane WAL of the Tkachov-Hankiewicz model still occurs as it depends on the relative direction of the field to the surface, not the current direction. Thus, we consider the geometry that we used to be representative of the total angle-dependent field behavior at small fields.

MR was also measured as the angle between field and sample is varied from 0° (in-plane) to 90° (out-of-plane). The change at B = 1 T as a function of angle and normalized to the value at 90° is plotted in Fig. 4(a) (see Supplemental Materials Fig. S2 for additional data). In conventional materials, anisotropic magnetoresistance arises from coupling to the out-of-plane magnetic field, leading to a dependence on B \(\sin \,\theta \)50,51,52,53. However, the increased θ-dependence of WAL for increasing Te contribution, up to x = 2.5, suggests a stronger surface contribution in Bi2Te2.5Se0.5 than in Bi2Se3.

Figure 4
figure 4

(a) MR vs. field angle, normalized to the value at θ = 90°. Bi2Te3 nearly follows a \(\sin \,\theta \) dependence while Bi2Se3 is nearly constant with angle, with the Se, Te mixed samples decreasing between the extremes. (b) MR vs. Temperature. The Se containing samples decrease more quickly than the pure Bi2Te3 due to the rapid disappearance of WAL.

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Figure 4(b) shows the temperature dependence of the MR. As expected, the change in MR decreases with increasing temperature. The samples that show WAL have a cusp that broadens with T and develops into fully quadratic MR by 100 K. The MR at 1 T is plotted vs. temperature in Fig. 4(b). The rapid decrease in MR in the Bi2Se3 and Bi2Te2Se samples is due to the disappearance of WAL, while Bi2Te3 decreases steadily from the steadily decreasing mobility. The WAL decrease follows nearly the same divergent dependence in T as does \(\Delta \sigma (B)\) due to the temperature dependence of the coherence length (\(L\, \sim \,{T}^{-1}\)) and the divergence of \(\psi (x)\). Bi2Te2.5Se0.5 decreases slower than the other Se-containing samples due to a relatively stronger background MR.

Conclusion

In conclusion, we have demonstrated non-trivial AMR in Bi2Se3 arising from a non-zero skin depth of the topological surface states. As the Se is substituted for Te, the surface states are overshadowed by bulk conduction, leading to more conventional behavior in Bi2Te3. This results in distinctive weak antilocalization and non-trivial anisotropic magnetoresistance in Se-containing samples which does not appear in Bi2Te3. The Tkachov-Hankiewicz coupled with the Hikami-Larkin-Nagaoka model provides a straightforward means of measuring both the coherence length and characterizing the quality of the surface states through the effective skin depth. We have also demonstrated that careful investigation of the low field in-plane MR in topological materials can provide valuable insight into surface states and sample quality, information that will prove vital to best utilizing topological surface states and optimizing film growths.