Weak Antilocalization and Anisotropic Magnetoresistance as a Probe of Surface States in Topological Bi2TexSe3−x Thin Films

Topological materials, such as the quintessential topological insulators in the Bi2X3 family (X = O, S, Se, Te), are extremely promising for beyond Moore’s Law computing applications where alternative state variables and energy efficiency are prized. It is essential to understand how the topological nature of these materials changes with growth conditions and, more specifically, chalcogen content. In this study, we investigate the evolution of the magnetoresistance of Bi2TexSe3−x for varying chalcogen ratios and constant growth conditions as a function of both temperature and angle of applied field. The contribution of 2D and 3D weak antilocalization are investigated by utilizing the Tkachov-Hankiewicz model and Hakami-Larkin-Nagaoka models of magnetoconductance.


Thin Film Growth and Device Fabrication
MBE growth of Bi 2 Te x Se 3−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 Bi 2 Se 3 (x = 0), Bi 2 Te 2 Se 1 (x = 2), Bi 2 Te 2.5 Se 0.5 (x = 2.5) and pure Bi 2 Te 3 (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 Bi 2 Se 3 and 25 nm for Bi 2 Te 3 , similar to other MBE grown films in the literature 22,26,27 . This discrepancy is likely due to the highly polycrystalline nature of the films.
Mesoscopic Hall bars (10 × 20 µm 2 , 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 Bi 2 Se 3 when the magnetic field is both in-plane with the film and parallel to the applied current 28 . 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  . The temperature dependence of the carrier concentrations and Hall mobilities are shown in Fig. 2. In Bi 2 Se 3 , n is nearly independent of temperature due to the semimetallic nature of the Dirac state, while n in Bi 2 Te 3 has a strong exponential dependence due to the stronger influence of the bulk bandgap in the transport. Conversely, the mobility in Bi 2 Te 3 and Bi 2 Te 2.5 Se 0.5 decreases inversely proportional to T, while the mobility in Bi 2 Se 3 and Bi 2 Te 2 Se 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 www.nature.com/scientificreports www.nature.com/scientificreports/ concentration (mobility) is consistent with the Bi 2 Te 3 and Bi 2 Te 2.95 Se 0.05 samples having conventional conducting states, while the steadier mobility in Bi 2 Se 3 and Bi 2 Te 2 Se is strong evidence of topological protection. The anomalous slight increase in mobility for Bi 2 Te 2 Se could result from ionized impurity scattering 29,30 . Our measured mobilities of 100-1000 cm 2 /Vs correspond to mean free path on the order of 10 nm, corresponding to diffusive transport within our samples 6 . Figure 3 shows Magnetoconductance ( σ ∆ ) 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 Bi 2 Te 3 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 2 where ψ is the Digamma function, L is the coherence length, and e and h are the electron charge and Plank constant, respectively [31][32][33] . For Dirac states, the coefficient α is expected to be 1/2 for each Dirac cone 34 . 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 Bi 2 Se 3 , Bi 2 Te 2 Se, and Bi 2 Te 2,5 Se 0.5 , respectively. This range is consistent with literature and indicates 2D topological surface transport 14,34 . A linear background is subtracted for the Bi 2 Se 3 and Bi 2 Te 2 Se samples, and a quadratic background for the Bi 2 Te 2,5 Se 0.5 . 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 angle 40 . 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 where, λ is the effective surface state skin depth. For a purely 2D surface state (λ = 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 λ L / such that the effect is observably non-quadratic. For sufficiently large values of λ 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 Bi 2 Te 3 40 . 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. www.nature.com/scientificreports www.nature.com/scientificreports/ 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 Bi 2 Te 3 . 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 literature 42,43 . 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 depth 44 . These samples are heavily electron doped ( ≈ n 10 19 cm −3 ) which, along with the relative complexity of the band structure for Bi 2 Te 3 relative to Bi 2 Se 3 , provides an explanation for the disappearance of the surface states. Bi 2 Se 3 has a bulk bandgap of 0.3 eV, compared to 0.1 eV for Bi 2 Te 3 45,46 . Additionally, the Dirac point in Bi 2 Te 3 lies below the maximum of the bulk valence band while the Dirac point in Bi 2 Se 3 lies above the bulk valence band. The larger band gap and less proximity of bulk states to the Dirac point allow Bi 2 Se 3 to maintain its topological behavior over a wider range of dopings than Bi 2 Te 3 . 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 Bi 2 Te 3 than in Bi 2 Se 3 . 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 states 18,23,45 . We expect the alloyed compositions to have band structures that are some linear combination of the Bi 2 Se 3 and Bi 2 Te 3 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 λ 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 Bi 2 Te 3 compared to the other samples, the relative size of the WAL cusp decreases leading to the near disappearance of WAL in Bi 2 Te 3 . 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 Bi 2 Se 3 indicates a 3D origin, while the stronger theta-dependence in the alloyed samples indicates some 2D surface contribution.
In various other studies on Bi 2 Se 3 , a negative longitudinal magnetoresistance is observed when the field is applied along the current, often attributed to the chiral anomaly [47][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 θ sin [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 Bi 2 Te 2.5 Se 0.5 than in Bi 2 Se 3 .  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 Bi 2 Se 3 and Bi 2 Te 2 Se samples is due to the disappearance of WAL, while Bi 2 Te 3 decreases steadily from the steadily decreasing mobility. The WAL decrease follows nearly the same divergent dependence in T as does σ ∆ B ( ) due to the temperature dependence of the coherence length (L T 1 − ) and the divergence of ψ x ( ). Bi 2 Te 2.5 Se 0.5 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 Bi 2 Se 3 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 Bi 2 Te 3 . This results in distinctive weak antilocalization and non-trivial anisotropic magnetoresistance in Se-containing samples which does not appear in Bi 2 Te 3 . 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.