Introduction

Topological Insulators (TIs) are a new class of quantum matter where the bulk is insulating and the surface consists of an odd number of Dirac cones which along with a Berry phase of π on the Fermi surface enabling them to exhibit numerous exotic phenomena such as topological protection against backscattering from non-magnetic impurities and defects that do not break time reversal invariance1, 2. The Dirac like linear dispersion of these surface states (SS) along with helical spin texture arising out of spin-orbit coupling make them distinct from trivial surface states like 2D electron gas (2DEG) or other spin-orbit coupled (SOC) systems. These states are robust to any amount of non-magnetic disorder and defects There are a number of evidences of surface dominated magneto-transport experiments3,4,5,6,7,8 in thin films and nano-devices, arising out of two dimensional quantum interference phenomena like weak antilocalization (WAL), Shubnikov de – Hass (SdH) oscillations and Aharonov – Bohm (AB) Oscillations. These experiments provide direct evidence of topological protection of these robust surface Dirac states. WAL is a manifestation of quantum interference between electrons executing forward and time reversed paths, where the π Berry phase causes a destructive interference in absence of magnetic impurities or external magnetic fields and prevents any localization brought about by backscattering of forward moving electrons into time reversed paths. However, the application of external magnetic field destroys the time reversal symmetry resulting in localization of the electron. This phenomena is termed as weak antilocalization (WAL). WAL has been aptly described by Hikami, Larkin and Nagaoka (HLN) in two dimensional metals with spin-orbit interactions described later9.

Second generation TI materials such as Bi2Se3 (BS), Bi2Te3 (BT) and Sb2Te3 (ST) called the binary TIs suffered from parasitic bulk carriers which mask the contribution of surface states in conduction10. In the past few years there has been an intense search for better topological insulators than the earlier discovered binary TI materials, where, often there is a significant bulk conduction owing to carriers generated by defects and Se/Te vacancies, which shifts the Fermi level in to the bulk7, 11. Ternary and quaternary topological insulators materials, Bi2Se2Te (BST), Bi2Te2Se (BTS), Bi2−xSbxTe3, and BixSb1−xTeySe1−y (BSTS), on the other hand seem to be pushing the Fermi level inside the band gap4, 12,13,14,15. ARPES and magnetotransport experiments validate the above point and hence these ternary tetradymites fulfill the criteria of intrinsic topological insulators16,17,18. Thus these materials are perfect candidates for realizing surface dominated transport leading to fabrication of spintronic devices harnessing the surface Dirac electrons alone19. Currently the focus has shifted towards these materials in terms of compositional and band structure engineering owing to their superiority over their binary analogues in terms of yielding a stable bulk insulating ground state16, 20.

The transport signature of the surface Dirac states have been achieved unambiguously with very less parasitic bulk effects in such ternary and quaternary TI flakes/single crystals21, 22. Most of the transport studies on high quality BTS and BSTS has remained confined to single crystals or thin flakes exfoliated from these single crystals4, 5, 14, 18, 21, 23,24,25. Although BTS and related Sb doped crystals have been found to be consisting of highly insulating bulk and highly mobile surface states, in some cases these crystals have been found to be metallic in nature as well, for example see refs 11 and 26. Therefore, it is not certain that as grown crystals of BTS and BST will always show insulating character thus posing a serious uncertainty in growth parameters and compositions11, 25, 26. Both BST and BTS single crystals have been found to be insulating as well as metallic in nature with minor changes in composition ratio of Se/Te13, 14, 18, 27, 28. However, thin films of these TI’s lack thorough characterization and investigation of the surface electron states by means of transport studies. To the best of our knowledge there are no studies on thin films of these bulk insulating alloys except for few where thin films of BTS deposited by Molecular Beam Epitaxy have been found to be metallic in nature11, 29.

In this article we explore the transport properties of BST thin films grown by pulsed laser deposition (PLD) technique. Two insulating films of thickness 500 nm (designated as B2) and 150 nm (B1) and one metallic film (B3) having the same thickness as B1(150 nm) have been prepared for this study. The main findings of this paper are; fitting the resistance vs. temperature (R-T) curve with a parallel conduction mechanism, signatures of WAL and surface to bulk coupling from the study of R-T curve itself and the consistent description of MR data with the simplified Hikami, Larkin and Nagaoka (HLN) equation in the metallic as well as in insulating thin films9. The HLN equation (Equation 4) was formulated to capture WAL in 2D metals. This description removes the discrepancies concerning the values of transport coefficient “α” (see Equation 4). “α” is a measure of the number of independent surface states contributing to the conductivity. Most papers report a single coherent channel and suggest that one of the surfaces has higher coherence length than the other. There have been efforts to control the number of channels by tuning the Gate voltage applied between the top and bottom surfaces30. Bulk carriers play a crucial role in TI coherent transport, and that channel coupling is controlled by a competition between the phase coherence time (τϕ) and the surface-to-bulk coupling time (τSB). The number of independent coherent channels can also be tuned by thickness, deposition conditions and temperature. PLD allows us to tune the Fermi level in the bulk band gap by varying these parameters, which is a cheaper analogue of the gating effect but sometimes more effective.

We fit the low field magneto-resistance (MR) data with the simplified HLN equation and the value of coefficient “α” remains around 0.5 for bulk metallic films, but changes from 0.5 to 1.0 in bulk insulating films corresponding to the two 2D conduction channels, the top and bottom surfaces, which are decoupled in the latter. With this finding we conclude that the surface states couple to the bulk in the films where the bulk is metallic, giving rise to one effective conduction channel, whereas, the bulk insulating films have no surface to bulk coupling and the two surfaces, top and bottom have separate contribution to the MR. Temperature dependence of α and phase coherence length \(({l}_{\phi })\) for both the insulating as well as metallic thin films are also explained in this work.

Results and Discussions

We report the transport properties on three thin films of thicknesses 150 nm (B1), 500 nm (B2) and 150 nm (B3). The films “B1” and “B2” show insulating character while “B3”, which has the same thickness as B1, displays metallic character as shown in Fig. 1(c). The film B3 was grown for the comparative study with the insulating films. The detailed film growth and characterization is described in the Supplementary section.

Figure 1
figure 1

(a), (b) and (c) Resistance variation with temperature. This figure depicts the insulating behavior of BST thin films B1, B2 and metallic behavior of B3 down to 2 K. B1 and B2 display an insulating character, whereas B3 is metallic in nature (c). From the Hall measurements the carriers are found to be ‘n’ type (see Supplementary section). The insets of Fig. 1(a) B1 and (b) B2 shows the logarithmic temperature dependent resistance upturn. (d) Application of perpendicular magnetic field changes the slope of the upturn in the R-T behavior for sample B1. The right panels of Fig. 1(a) and (b) display the band diagrams respectively of these two samples. The dashed lines show the impurity states in the bulk band gap. The chemical potential (not shown) lies between the Dirac point and the impurity states. The band diagram of sample B3 is shown in Fig. 1(c). Here it can be seen that the chemical potential (the black horizontal line) lies in the bulk conduction band.

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Temperature dependent quantum correction

Interesting features of the electron transport can be inferred from the temperature dependence of resistance (Fig. 1), especially in the low temperature regime. The existence of WAL depends on the competition between the phase coherence of the electron wave function and the dephasing factors present in the system. Temperature (electron- phonon scattering) and electron–electron scattering play a crucial role in this dephasing mechanism at low temperatures. With lowering temperature as the phonon contribution to dephasing reduces in the system, a pronounced WAL type behavior in B1 and B2 can be seen from the resistance vs. temperature (R-T) data in Fig. 1(a) and (b). This is captured from the portion below the dashed vertical line, which signifies the dominant role played by the surface states below this temperature. This can be considered as semiconductor to metal transition with decreasing temperature. Though traditionally, WAL is referred to in the context of MR data, it is also captured here in the R-T data below the dashed line as here the surface state dominated transport is protected from backscattering by the spin momentum locking and a π Berry phase.

In highly insulating TI samples WAL can be observed qualitatively by the nature of the R-T curve itself. This observation is somewhat analogous to the strong localization in two dimensional hetero- structures where metal – insulator transition (MIT) can be captured by R-T alone17, 31. One can draw an important conclusion from this observation that the fall in resistance in the R-T curve (below the indicated point TD on the graph) manifests robust delocalized nature of surface states against other interactions, disorders and grain boundaries at finite temperatures31. In our PLD grown BST thin films primary cause of disorder is from the granular structures (see SEM images in Supplementary section) which limit the mobility to a lower value of 25 cm2/V-sec. This is in contrast to the single crystals and MBE deposited thin films which have mobilities of the order of 2000–4000 cm2/V-s, where surface Dirac electrons face a nearly atomically flat surface. Therefore, PLD deposited thin films are real test for the topological protection of Dirac states against disorder.

Apart from downturn in sheet resistance (Rs) from T = 70 K and 50 K (denoted as TD) in B1 and B2 thin films respectively, an upturn in the Rs can also be observed at around 10 K. This upturn in Rs could be due to freezing of bulk carriers in the impurity band or e-e correlation among surface states on the surface of TI in presence of disorder or combined effect of these two mechanisms. In order to find out the possible mechanism behind this upturn one needs to separate the positive quantum correction due to WAL from the sheet conductance GS. The WAL correction can be destroyed by application of perpendicular magnetic field as shown in the Fig. 1(d) for sample B1.The R-T curves in applied magnetic fields B = 0.5 and 1.0 Tesla are distinct from the zero field R-T resulting in an upward shift, which is affected by the destruction of 2D WAL in the topologically protected surface states by the external field. The combined form of temperature dependent quantum corrections in the conductance due to WAL and e-e interaction can be written as refs 32 and 33;

$${\rm{\Delta }}{G}_{S}={\rm{\Delta }}{G}_{S(WAL)}+{\rm{\Delta }}{G}_{S(e-e)}$$
(1)
$${\rm{\Delta }}{G}_{S}=\frac{{\rm{\Delta }}{R}_{s}(T)}{{{R}_{s}}^{2}(T)}=c\frac{{e}^{2}}{2{\pi }^{2}\hslash }\,\mathrm{ln}(T)-\beta \frac{{e}^{2}}{2{\pi }^{2}\hslash }\,\mathrm{ln}(T)$$
(2)

\({\rm{\Delta }}{G}_{S(WAL)}=c\frac{{e}^{2}}{2{\pi }^{2}\hslash }\,\mathrm{ln}(T)\) is a positive quantum correction due to 2D WAL effect33. Whereas e-e interaction driven negative correction in the conductance is given as; \({\rm{\Delta }}{G}_{S(e-e)}=-\beta \frac{{e}^{2}}{2{\pi }^{2}\hslash }\,\mathrm{ln}(T).\) On fitting the temperature dependent correction in the conductance due to e-e interaction when WAL is destroyed (Fig. 1(d)), we get the value of the fitted interaction parameter β (see Supplementary section for details). This value is not consistent with the theoretical value of the e-e interaction correction in the conductance in 2D34. It is therefore concluded that the upturn in these thin films is not entirely due to the e-e interaction but other mechanisms like localization or freezing out of the bulk electronic states35.

The deviation of the coefficient β from the expected theoretically calculated values indicate that even in the low temperature regime in the two insulating thin films (B1 and B2) do not show fully 2D transport and the total conduction is a combination of conduction of bulk and surface connected in parallel, henceforth referred to as the parallel channel conduction mechanism. The value of the parameter β could be used to estimate the truly 2D character of surface states of the bulk insulating TIs. Earlier electron-electron interactions have been observed in doped metallic thin films and ultrathin metallic films33, 36. In these cases it could not be conclusively inferred whether the interaction is among surface states or the bulk states.

Parallel channel conductance model has been used to fit the conductance vs T plot as shown in Fig. 2. The two parallel channels being bulk conductance (G b ) and surface conductance (G S ). The combination of bulk and surface conductance, the total conductance; G t is represented by:

$$\begin{array}{rcl}{G}_{t} & = & {G}_{b}+{G}_{S}\\ & = & \,1/({R}_{S}{e}^{{\rm{\Delta }}/T})+1/(A+BT)\end{array}$$
(3)

\({G}_{b}(T)=1/({R}_{S}{e}^{{\rm{\Delta }}/T})\,{\rm{and}}\,{G}_{S}(T)=1/(A+BT)\). Here Δ represents the activation energy for impurity states in the bulk. The fitted temperature dependent conductance using Equation (3) yields the value of ∆ = 25.85 meV and Δ = 23.59 meV for samples B2 and B1 respectively. This value of activation gap can be explained by compensation of donor and acceptor levels and formation of Coulomb disorder37. Coefficient A stands for disorder scattering which is independent of temperature and B represents electron phonon scattering17. At high temperatures, conductance is dominated by the thermally activated carriers form the impurity states residing below the conduction band edge. As the temperature is reduced a freezing out of bulk carriers takes place and conductance is dominated by the surface states in thin films.

Figure 2
figure 2

(a) Conductance data fitted to a non-linear curve given in Equation (3) for sample B1which includes both surface and bulk contribution to the conductance. (b) Conductance as a function of temperature data fitted with the same expression for B2. The deviation in the low temperature regime in Fig. 2(a) is due to the onset of WAL which yields a logarithmic correction to conductance (Equation 2) which was not considered in this fit.

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Observation of various quantum interference phenomena in TI has been one of the important tools to characterize the backscattering immuned topological surface states. In systems with large spin orbit interaction, having an odd number of Dirac cones the electron wave function gathers a π Berry phase as it circulates along a closed path on the iso-energic Fermi circles in a Dirac cone. This results in destructive interference between the two time-reversed coherent paths of electron waves in a diffusive medium, effectively increasing the probability amplitude for forward scattering. These quantum interference phenomena manifest itself in lowering the resistance or making positive quantum correction to the Drude conductivity at low temperatures in zero field. This is observed in the downturn in R-T plot below 70 K and 50 K in B1 and B2 respectively. This phenomenon is known as weak antilocalization as opposed to its counterpart weak localization.

Magnetoresistance Data

From the temperature dependence of MR in Fig. 3 it is clear that quantum coherent transport of surface electrons persists even at higher temperatures (20 K), though reduced, showing a remarkable robustness of the surface states against phonon induced scattering. The sharp cusp like feature in MR at low temperature in all three samples is the manifestation of WAL and as the temperature increases this feature nearly vanishes above 40 K resulting in parabolic MR which is classical in nature as opposed to WAL.

Figure 3
figure 3

Magnetoresistance (MR) measured at different temperatures for the film B1, B2 and B3 thin films. A sharp rise in MR around zero field (WAL) at lower temperatures is clearly seen in the above plots. B3 owing to metallic nature has the WAL cusp relatively less pronounced in comparison to B1 and B2.

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We first consider the low field (0–1 T) correction to the conductance. In the presence of strong spin-orbit interaction, quantum correction to the conductivity in a two dimensional system can be described by HLN formula9. This expression is applicable in the diffusive quantum transport regime in samples with weak disorder, such as in thin films deposited by PLD. The HLN equationis valid for strong spin-orbit coupling limit and can be written in the simplified form for the following condition: τ so  < τ e  < τ φ , where τ so , τ e and τ φ are the characteristic time scales corresponding to the spin-orbit scattering, elastic scattering and inelastic scattering of charge carriers. Conductance correction due to WAL (HLN equation) consists of two components, the logarithmic and digamma components and is given as:

$${\rm{\Delta }}{G}_{S}=-\frac{\alpha {e}^{2}}{2{\pi }^{2}\hslash }[{\rm{\Psi }}(\frac{B\varphi }{B}+\frac{1}{2})-\,\mathrm{ln}(\frac{B\varphi }{B})]$$
(4)

where, \(\Psi \) is a digamma function and \({B}_{\varphi }=\frac{\hslash }{4e{l}_{\phi }^{2}}\) is the characteristic magnetic field corresponding to the coherence length \({l}_{\phi }\).The value of the coefficient “α” signifies effective number of independent 2D conducting channels and is related to the dimensionality of the conduction process in the system. For TI thin films the value of transport coefficient “α” should be equal to 1, a contribution of 0.5 corresponding to each of the two surfaces (the top and bottom surface) when they are decoupled. But experimentally this value has been found to vary from 0.3 to 1.5 and in some cases as large as 3.022. The value of α = 0.5 found in some earlier reports is similar to that found in thin films of Bismuth which is also a strongly spin - orbit coupled semimetal but not a TI38.

Fitting of magneto-conductance (MC) by HLN Equation (4) in the low field regime at different temperatures is shown in the Fig. 4(a), (b) and (c) for samples B1, B2 and B3 respectively and the corresponding extracted parameters “α” and phase coherence length“l Φ ” are plotted as a function of temperature in Figs 5(a), (b), (c) and 6. In the bulk insulating and decoupled regime we have observed strong robustness of the conduction channel parameter “α” which hovers around “1.0”, consistent with the two surface transport. This point towards a two independent channel coherent transport. Whereas, in the metallic film “α” remains around “0.7”. This value is consistent with ultrathin TI films where the top and bottom surfaces are directly coupled and in thick TI films with metallic bulk (mainly BS and BT), where the coupling is mediated by the bulk electrons39. This is in contrast with the earlier case, where the top and bottom surfaces are decoupled with α = 1. It should be noted here, WAL characteristics in these films can only be seen at low temperatures as evident from the sharp dip in MC at low field. At higher temperatures it takes a parabolic nature (a broaden cusp) where surface and bulk both contribute to MC. Therefore, fitting the MCcurves with simple HLN model at higher temperature is not appropriate and the equation needs to be modified. Thus, in order to capture the WAL from surface states at relatively higher temperatures we fit MC data with the modified HLN equation where a parabolic field dependent term was added (see Supplementary).

Figure 4
figure 4

HLN fit of conductance data at different temperatures of samples B1, B2 and B3.

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Figure 5
figure 5

Variation of α with temperature for the three films B2, B1 and B3. For the insulating samples the behavior of α is opposite to that for the metallic film B3. The direction of the arrow signifies increasing or decreasing trend of α for metallic or insulating films.

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Earlier studies based on the two channel conduction in TI thin films and gated devices in ultrathin and thick limit have shown that the value of the coefficient “α” depends on various parameters such as gate voltage, compensation doping, and thickness of the film30, 32, 40, 41. At sufficiently low temperature, i.e. 2 K, we can expect that the bulk electrons will not contribute to the conductivity (in B1 and B2) as evident from the thermal activation energy of the order of ∆ = 25.85 meV, where, there are hardly any excitations from the impurity states to the bulk conduction band. This energy scale is much higher for any excitation at the lowest temperatures and there is no bulk electronic conduction at these temperatures.

Variation of the coefficient α with temperature is shown in Fig. 5. For the bulk insulating films, B1 and B2, we see that α decreases from 1 to 0.5, whereas, of the bulk metallic film B3 it increases. In the bulk insulating case, at very low temperatures the number of bulk carriers is low (see Hall data in Supplementary) and the top and bottom surface channels are nicely decoupled from the bulk, yielding a value of α to be ‘1’. From the perspective of WAL, the surface and bulk states can be regarded as independent phase coherent channels as long as the carriers in one channel lose coherence before being scattered into the other. Here, the majority of closed loops responsible for WAL will involve states from a single channel, and each channel will exhibit its own WAL correction. This condition can be formulated as τSB > τφ, where τSB is the effective surface-to-bulk scattering time and τφ the coherent dephasing time. In the opposite limit τSB < τφ, charge carriers scatter between the bulk and surface states while maintaining phase coherence, effectively becoming a single phase-coherent channel. Thus the temperature dependence of α is related to the decrease in τSB, brought about by the increase in number of carriers in the bulk with temperature owing to thermal excitations. This results in increase in surface to bulk coupling and the separation between the top and bottom surface is gradually diminished with an increase in the temperature effectively increasing the effective conduction channels. Once we are at a fairly high enough temperature, the top and bottom surfaces and the bulk behave as a single coherently coupled conduction channel and hence the effective α is ‘0.5’. The coefficient α can be used phenomenologically as a measure for channel separation. α decreases with temperature, suggesting a decreasing channel separation with temperature. The effective number of phase coherent channels reflected in the magnitude of α depends on the ratio τSB to τφ, which was controlled through changing the bulk carrier density (as reflected in our Hall data reported in the Supplementary), which plays an important role in the surface to bulk coupling. An intuitive picture of surface to bulk coupling is given by in Fig. 3(d) of ref. 39. Steinberg et al., have varied the gate voltage to tune the τφ/τsbratio capturing a somewhat similar variation in “α”, whereas we were able to achieve a similar tuning by varying the temperature and deposition parameters30.

In the bulk metallic case (B3) on the other hand, there are enough bulk carriers even at the lowest temperature and the system behaves effectively as a single coherently coupled conducting channel reflected in an α of ‘0.5’. In this case the τSB < τφ. One can see that the carrier density obtained from Hall data (supplementary section) is independent of temperature. Hence one can surmise that the τSB is temperature independent. However, as l φ goes down with temperature (Fig. 6), so does \({{\rm{\tau }}}_{{\rm{\phi }}}({l}_{\phi }=\sqrt{D{\tau }_{\phi }})\), the τφ/τSB ratio increases with temperature and the magnitude of α goes up to 1.

Temperature dependent variation of phase coherence length (l φ ) is shown in Fig. 6 for all the three samples. While the temperature variation of “α” in B1/B2 and B3 is different, \({l}_{\phi }\) decreases monotonically with increasing temperature (Fig. 6) in all the films. This is a manifestation of dephasing by phonons on the surface of the film. From the SEM images of these films it is seen that, the average size of the grains on the thin films is larger than the coherence length \({l}_{\phi }\), which implies that granular structures of the films play a minimal role in dephasing surface electrons. The variation of \({l}_{\phi }\) with temperature signifies 2D surface dominant transport. Phase coherence length decreases with temperature with a power law variation \({l}_{\phi }\) ~T−p, where p depends on the dimensionality of the system and the agents responsible for the dephasing mechanism42, 43. For the three films under study in our case i.e., B1, B2 and B3 the value of p = 0.61, 0.31and 0.64 respectively35. Detailed investigation of this coefficient ‘p’ is being carried out for a variety of films. This will be reported elsewhere.

Figure 6
figure 6

Phase coherence length vs. temperature for the three thin films B3, B2 and B1. The variation with temperature follows the expected power law behavior with \({l}_{\phi }\) ~T−p.

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We have also fitted the magneto-conductance (MC) curves up to high magnetic fields (8 T) with the modified HLN equation where we added the classical Kohler term (MC \(\propto \) B2) to it. This along with fitting to higher temperature MC curves (40 K and 100 K) are shown in the Supplementary section. In the low temperature MC, at high fields the mixture of the classical Kohler term and the quantum MC gives the MC a linear appearance. At intermediate temperatures (40 K) the classical nature kicks in at lower fields in comparison to the low temperature curves and hence the classical MC dominates giving it a somewhat parabolic appearance (Fig. 3).The exact reason for the linear MC in the high field regime is still debated44 and will be addressed by us in details elsewhere. Though some reports attribute linear nature of MC to the surface states but at such high temperatures (say 100 K), there can be sizable bulk carrier contribution and electron-phonon scattering which may override the surface Dirac state contribution to MC. Therefore, we can attribute it to the mixed phase of MR arising from classical as well as quantum coherence effects.

Conclusion

It is remarkable that PLD grown films, studied in this work are insulating in the bulk and exhibit topological transport behavior, clearly manifested in the bulk insulating nature of R vs. T curves. PLD is a better deposition technique in comparison to other synthesis techniques as it is cost effective and offers ready integration with other materials for device applications. Our films were prepared to be rather thick (500 nm–150 nm) to facilitate a direct decoupling of the two surface states with the bulk such that the condition τφ/τsb < 1 is satisfied at low temperatures, where the value of the transport channel coefficient “α” was found to be 1 for these decoupled samples. We were able to tune the τφ/τsb ratio by varying the temperature, thereby tuning “α”. This was studied for both bulk insulating and bulk metallic films yielding contrasting results. A coherent and comprehensive understanding has been given for the different values of the transport coefficient “α” for different experimental conditions (temperature variation) and sample properties, i.e., for both bulk insulating and metallic films. This gives a better understanding of the underlying physics of topological insulator which coupled with tunability of PLD can lead to synthesis of better TI materials and devices.

Methods

Thin films of BST were grown by Pulse Laser Deposition method, where BST targets were ablated using high power Excimer Laser and deposited on Si (100) substrate. The thickness of the samples were controlled by monitoring the number of pulse shots on the target and the thickness was verified by profilometer. The devices were fabricated using optical lithography technique and a standard Hall bar pattern was inscribed on top of the samples. Indium contacts were put on the sample and then the samples were subjected to transport measurements down to low temperatures (300 K–1.6 K) and high magnetic fields (upto8 T).