Abstract
The threedimensional topological insulator is a novel state of matter characterized by twodimensional metallic Dirac states on its surface. To verify the topological nature of the surface states, Bibased chalcogenides such as Bi_{2}Se_{3}, Bi_{2}Te_{3}, Sb_{2}Te_{3} and their combined/mixed compounds have been intensively studied. Here, we report the realization of the quantum Hall effect on the surface Dirac states in (Bi_{1−x}Sb_{x})_{2}Te_{3} films. With electrostatic gatetuning of the Fermi level in the bulk band gap under magnetic fields, the quantum Hall states with filling factor ±1 are resolved. Furthermore, the appearance of a quantum Hall plateau at filling factor zero reflects a pseudospin Hall insulator state when the Fermi level is tuned in between the energy levels of the nondegenerate top and bottom surface Dirac points. The observation of the quantum Hall effect in threedimensional topological insulator films may pave a way toward topological insulatorbased electronics.
Introduction
Quantum transport in Dirac electron systems has been attracting much attention for the halfinteger quantum Hall effect (QHE), as typically observed in graphene^{1,2}. A single Dirac fermion under a magnetic field is known to show the quantized Hall effect with the Hall conductance σ_{xy}=(n+1/2)e^{2}/h with n being an integer, e the elemental charge and h the Planck constant. This 1/2 is the characteristic of the Dirac fermion compared with the usual massive electrons. In graphene with such a Dirac electron, however, there is fourfold degeneracy due to the spin and valley degrees of freedom, and hence the quantized Hall conductance shows up experimentally as σ_{xy}=4(n+1/2)e^{2}/h. The recently discovered topological insulator (TI) possesses metallic Dirac states on the edge or surface of an insulating bulk^{3,4,5,6}. With the application of a magnetic field (B), the unique features of Dirac bands may be exemplified via the formation of Landau levels (LLs). The QHE is the hallmark of dissipationless topological quantum transport originating from onedimensional chiral edge modes driven by cyclotron motion of twodimensional (2D) electrons. Unlike the case of graphene, the degeneracy is completely lifted in the spinpolarized Dirac state of 2D and threedimensional (3D) TIs. The Hall conductance σ_{xy} of 3D TI is expected to be given by the sum of the two contributions from the top and bottom surfaces and hence σ_{xy}=(n+n’+1)e^{2}/h with both n and n’ being integers. When the two contributions are equivalent, that is, n=n’, only the odd integer QHE is expected. For such 3D TI films, the top and bottom surfaces support surface states with opposite spinmomentum locked modes when the top and bottom surfaces are regarded as two independent systems. Such a helicity degree of freedom in real space can be viewed as the pseudospin variable and is hence expected to yield a new quantum state via tuning of surface magnetism and/or Fermi level (E_{F}) that is applicable to quantum computation functions^{7,8,9}. Although intensive research has been carried out for bulk crystals, thin films and fieldeffect devices^{10,11,12,13,14,15,16,17}, parasitic bulk conduction and/or disorder in the devices continues to hamper efforts to resolve quantum transport characteristics of the Dirac states on chalcogenide 3D TIs surfaces. The most venerable example of the QHE with least bulk conduction has been achieved in a 70 nm strained HgTe film^{18}. Compared with the HgTe system, 3DTIs of Bichalcogenides such as Bi_{2}Se_{2}Te and (Bi_{1−x}Sb_{x})_{2}Te_{3} have a good potential for exploring the Dirac surface states with wide controllability of transport parameters (resistivity, carrier type and density) and band parameters (energy gap, position of Dirac point and Fermi velocity) by changing the compositions^{19,20}.
In the following, we report on the QHE in fieldeffect transistors based on 3D TI thin films of (Bi_{1−x}Sb_{x})_{2}Te_{3} (x=0.84 and 0.88). With electrostatic gatetuning of the Fermi level in the bulk band gap under magnetic fields, quantized Hall plateaus (σ_{xy}=±e^{2}/h) at the filling factor ν=±1 are resolved, pointing to the formation of chiral edge modes at the top/bottom surface Dirac states. In addition, the emergence of a σ_{xy}=0 state around the charge neutral point (CNP) reflects a pseudospin Hall insulator state when the location of Fermi level is between the nondegenerate top and bottom surface Dirac points.
Results
Transport properties with electrostatic gatetuning
We grew 3D TI thin films of (Bi_{1−x}Sb_{x})_{2}Te_{3} (x=0.84 and 0.88; both 8 nm thick) using molecular beam epitaxy (MBE)^{19} and insulating InP (111) substrates. The E_{F} of asgrown film was tuned near to the bulk band edge by precisely controlling the Bi/Sb composition ratio^{19,20}. Films were then fabricated into photolithographydefined gated Hallbar devices to allow electrostatic tuning of E_{F}. A crosssectional schematic of the device structure and the topview image are shown in Fig. 1a,b, respectively. The device consists of a Hall bar defined by Ar ionmilling, and an atomiclayerdeposited AlO_{x} insulator isolated Ti/Au top gate with electronbeamevaporated Ti/Au electrodes (see Methods). The magnetotransport measurements were carried out in a dilution refrigerator by lowfrequency (3 Hz) lockin technique with a low excitation current of 1 nA to suppress heating (see Supplementary Fig. 1 and Supplementary Note 1).
First, device operation was examined at B=0 T. Figure 1c shows the electric field effect controlled conductivity σ_{xx} of the x=0.84 film as a function of top gate voltage V_{G} with changing temperature. The minimum conductivity of roughly 2e^{2}/h is observed at V_{G}=−1.7 V. On both sides of this minimum, σ_{xx} shows a linear but asymmetric increase with increasing or decreasing V_{G}. The weak temperature dependence of longitudinal conductivity σ_{xx} for a wide range of V_{G} is characteristic of the gapless nature of Dirac states under finite disorder^{21}. Thus, we ascribe the conductivity of this TI film below 1 K to the Dirac surface states with a small contribution from bulk conduction. To verify the ambipolar nature of the device, the longitudinal and transverse resistance R_{xx} and R_{yx} as a function of V_{G} at B=3 T were measured (Fig. 1d). As expected, this results in a sign change of R_{yx} at a certain V_{G}, which we hereafter define as the gate voltage corresponding to E_{F} being located at the charge neutral point (CNP), V_{CNP.} This point coincides closely with the V_{G} at which R_{xx} reaches a maximum. At the V_{CNP}, the Hall effects from the top and bottom surface states appear to cancel, resulting in R_{yx}∼0, although electronrich and holerich puddles are thought to still exist, as in the case of graphene^{22}. To capture the essence of the observed phenomena, we hereafter take the working hypothesis that E_{F} shifts equally on top and bottom states, which in turn retain their difference in energy position. The inverse of Hall coefficient 1/R_{H} is shown in Fig. 1e, which would be proportional to 2D charge carrier density n_{2D}=1/(eR_{H}), in the simplest case. Asymmetric behaviour of 1/R_{H} between positive and negative V_{G} regions with respect to V_{CNP} is in accord with the asymmetric σ_{xx} behaviour at zero magnetic field as shown in Fig. 1c. The more efficient increase in σ_{xx} with electron accumulation (positive V_{G}) is often observed in ambipolar TI transistors^{13,14,15,16,17} and may be related to the difference in v_{F} (Fermi velocity) above and below the Dirac point^{19}. In addition, the proximity of the Dirac point to the valence band edge needs to be taken into account for the present thin film system; in most of the negative region of V_{G}−V_{CNP}, E_{F} must go inside the valence band in which doped holes appears to be fully localized at low temperatures below 1 K, as argued later.
Observation of quantum Hall states
With applying higher magnetic field of B=14 T, clear signatures of QHE are revealed in the temperature dependence of R_{xx} and R_{yx}, as shown in Fig. 2a–d for the x=0.84 film and in Fig. 2e–h for the x=0.88 film. We first focus on the results of the x=0.84 film. R_{xx} increases steeply with lowering the temperature for V_{G} corresponding to the CNP, yet decreases rapidly toward zero on both sides at higher and lower V_{G}. Concomitantly to the decrease in R_{xx}, R_{yx} reaches the values of quantum resistance ±h/e^{2}=±25.8 kΩ and forms plateaus in V_{G} at T=40 mK. This correspondence between the rapidly declining R_{xx} value and the quantized R_{yx} plateaus at ±h/e^{2} are distinct evidence for QHE at Landau filling factor ν=±1, respectively, as schematically shown in Fig. 2i (magnetic field dependence is shown in Supplementary Fig. 2 and discussed in Supplementary Note 2). The LL splitting energy of Dirac dispersion (E_{n}) is given by , where n is the LL index.
Using the data of R_{xx} and R_{yx}, σ_{xy} and σ_{xx} as functions of V_{G}−V_{CNP} for the x=0.84 film are plotted in Fig. 2c,d. Again, the Hall plateaus at σ_{xy}=±e^{2}/h as well as minima of σ_{xx} approaching zero (black triangles) are observed and are indicative of the QHE with ν=±1. In this plot, however, two additional features are to be noted. The first is an unexpectedly wide σ_{xy} plateau and thermally activated behaviour of σ_{xx} for the ν=+1 (σ_{xy}=e^{2}/h) state in the corresponding V_{G}−V_{CNP} (negative) region. As already noted in the V_{G} asymmetric change of σ_{xx} (Fig. 1c) in the negative region of V_{G}−V_{CNP}, that is, holedoping, E_{F} readily reaches the valence band top. The energy position of Dirac point of the x=0.84 film lies by at most 30 meV above the valence band top, while the LL splitting between n=0 and n=−1 levels amounts to 70 meV at 14 T, according to resonant tunneling spectroscopy on similarly grown thin films of (Bi_{1−x}Sb_{x})_{2}Te_{3} (ref. 19). From the consideration of the Fermi velocity of Dirac cone, the V_{G}−V_{CNP} value at which the E_{F} reaches the valence band top is estimated to be around −1.5 V or even a smaller absolute value. Therefore, in the V_{G}−V_{CNP} region where the quantum Hall plateau or its precursor is observed, the E_{F} locating between n=0 and n=−1 LLs of the surface state is close to or already buried in the valence band, as schematically shown in Fig. 2i. Although the doped but localized holes in the valence band may hardly contribute to transport, that is, σ_{xy}(bulk)≪σ_{xy}(surface), the relative E_{F} shift with negatively sweeping V_{G}−V_{CNP} becomes much slower as compared with the positive sweep case owing to the dominant density of states of the valence band. This explains a wider plateau region for ν=+1 in the holedoping side, contrary to the normal behaviour of electron accumulation side, ν=−1.
The second notable feature in Fig. 2c is the emergence of the ν=0 state around V_{G}=V_{CNP}, as seen in the step of σ_{xy} and (finite) minimum in σ_{xx} as functions of V_{G}−V_{CNP}. This state is more clearly resolved in the x=0.88 film, as shown in Fig. 2e–h, on which we focus hereafter. In a similar manner to the x=0.84 film, the x=0.88 film also shows with lowering temperature divergent behaviour of R_{xx} around V_{G}=V_{CNP}, while approaching zero around V_{G}−V_{CNP}=−1.5 V. The similar divergent (at V_{G}=V_{CNP}) and vanishing (at V_{G}−V_{CNP}=−1.5 V) behaviours of R_{xx} are observed also with increasing B at 40 mK (see Supplementary Fig. 3 and Supplementary Note 3). The R_{yx} reaches 25.8 kΩ around V_{G}−V_{CNP}=−1.5 V forming the ν=+1 quantum Hall state, whereas in the electrondoping regime, R_{yx} reaches −20 kΩ, short of the quantized value. The failure to form the fully quantized ν=−1 state is perhaps related to the disorder of the surface Dirac state, which is induced by the compositional/structural disorder of the asgrown film and cannot be overcome by gate tuning.
Nevertheless, the ν=0 feature is clearly resolved for the x=0.88 film, as shown in the V_{G} dependence of σ_{xy} (Fig. 2g) calculated from R_{xx} and R_{yx}. In addition to the σ_{xy}=e^{2}/h (ν=+1) plateau, a plateau at σ_{xy}=0 appears at around V_{G}=V_{CNP}. The plateau broadening occurs via centring at σ_{xy}=0.5 e^{2}/h as the isosbestic point with elevating temperature. In accordance with the plateaus in σ_{xy}, σ_{xx} takes a minima at ν=+1 and 0, as shown in Fig. 2h. Here, we can consider the contribution of both the top and bottom surface Dirac states to this quantization of σ_{xy}, as schematically shown in Fig. 2j. At the ν=+1 (ν=−1) state, both the top and bottom surfaces are accumulated by holes (electrons) with E_{F} being located between n=0 and n=−1 (n=+1) LLs, giving rise to the chiral edge channel. In contrast, we assign the ν=0 state to the gapping of the chiral edge channel as the cancellation of the contributions to σ_{xy} from the top and bottom surface states with ν=±1/2, when E_{F} locates in between the energy levels of the top and bottom surface Dirac points (n=0 levels), as shown in Fig. 2j. This ν=0 state can hence be viewed as a pseudospin Hall insulator, if we consider the top and bottom degree of freedom as the pseudospin variable. Such an observation of a zero conductance plateau has been reported also in disordered graphene under very high magnetic field^{23,24,25,26} and analysed theoretically^{27}, as well as in the 2D TIs, the quantum wells of HgTe^{28} and InAs/GaSb^{29}. From the analyses shown in the following, we propose here that the major origin for the presence of σ_{xy}=0 is more like the energy difference of the top/bottom Dirac points rather than other effects such as electronhole puddles due to composition inhomogeneity.
Discussion
To further discuss the characteristics of these QH states, we investigate the B dependence of σ_{xy} (Fig. 3a,b). The analysis of the plateau width against V_{G} determines the phase diagram as shown in Fig 3c,d. The plateau edges are determined from the second derivative of σ_{xy} with respective to V_{G} (see Supplementary Fig. 4 and Supplementary Note 4), while the plateau transition points between ν=0 and ν=±1 are defined here by σ_{xy}=±e^{2}/2h. The plateau shrinks with decreasing B for the ν=−1 state of the x=0.84 film (Fig. 3a). However, the ν=+1 state for both films appears to be rather robust with reducing V_{G} (doping more holes), since E_{F} positions already below the top of the valence band, perhaps for V_{G}−V_{CNP}<−1.5 V. On the other hand, the ν=0 plateau is only weakly dependent on B, although the plateau width is wider for x=0.88 than for x=0.84. The observation of ν=0 requires the condition that the Fermi level is located in between the energy levels of the top and bottom surface Dirac points (Fig. 2j). From the Hall data in the relatively high positive V_{G}−V_{CNP} region (electrondoping) shown in Fig. 1e, we can know the relation between the sum of the top and bottom Dirac electron density versus V_{G}−V_{CNP}. Then, with the values of the ν=0 plateau width between the σ_{xy}=±0.5 e^{2}/h points (δV_{G}∼0.9 and 1.4 V; see Fig. 3c,d) and the Fermi velocity (v_{F}∼5 × 10^{5} ms^{−1}; ref. 19), we can estimate the energy difference (δE_{DP}) between the Dirac points at the top and bottom surface states to be ∼50 meV and ∼70 meV for the x=0.84 and x=0.88 film, respectively (see Supplementary Note 5). These values should be compared with a much larger band gap energy (∼250 meV). The energy difference δE_{DP} is, however, considerably larger than a Zeeman shift (∼9 meV at 14 T; ref. 19), which rationalizes the above analysis with ignoring the Zeeman shift of the n=0 LL. Although the reason why the two films (x=0.84 and 0.88) show such a difference in δE_{DP} is not clear at the moment, we speculate that the monolayer buffer layer of Sb_{2}Te_{3} (x=1.0) used for the growth of the x=0.88 film (see Methods) may cause the considerably higher energy position of the Dirac point at the bottom surface. Incidentally, for the region of V_{G}−V_{CNP}<δV_{G}/2, electron accumulation at the top surface and hole accumulation at the bottom surface should coexist. This may naturally explain the observed (Fig. 1e and see also Supplementary Fig. 5 and Supplementary Note 6) deviation from the linear relationship between 1/R_{H} and V_{G}−V_{CNP} as well as the extrema of 1/R_{H} observed at around ±δV_{G}/2.
Figure 4 summarizes the flow of conductivity tensor (σ_{xy}, σ_{xx}) plotted with the two experimental subparameters (T and V_{G}) at 14 T. With decreasing T, the flow in (σ_{xy}, σ_{xx}) tends to converge toward either of (σ_{xy}, σ_{xx})=(−e^{2}/h, 0), (0, 0) or (e^{2}/h, 0) at high magnetic field (for example, 14 T), which corresponds to ν=−1, 0 and +1 QH state, respectively. Incipient convergence to ν=0 is discerned for the x=0.84 film (Fig. 4a), while the ν=−1 state is not discernible for the x=0.88 film (Fig. 4b). Among these three QH states, the unstable fixed point appears to lie on the line of σ_{xy}=±0.5 e^{2}/h (approximately with the critical σ_{xx} value of ∼0.5 e^{2}/h) which corresponds to the crossing of E_{F} at the n=0 LL (or Dirac point) of the bottom and top surface state (see Fig. 2j), respectively^{30}.
In conclusion, we have successfully observed the QHE at ν=±1 and 0 in 3D TI thin films of (Bi_{1−x}Sb_{x})_{2}Te_{3} (x=0.84 and 0.88). Due to a considerable difference of the Dirac point (or n=0 LL) energies of the top and bottom surfaces of the thin film, the ν=0 state observed at σ_{xy}=0 is interpreted as a pseudospin Hall insulator with the top/bottom degree of freedom as the pseudospin. Further studies on nonlocal transport in mesoscopic structures will open the door to dissipationless topologicaledge electronics on the basis of the 3D topological insulators.
Methods
MBE film growth
Thin films of (Bi_{1−x}Sb_{x})_{2}Te_{3} (x=0.84 and 0.88) were fabricated by MBE on semiinsulating InP (111) substrate. The Bi/Sb composition ratio was calibrated by the beam equivalent pressure of Bi and Sb, namely 8 × 10^{−7} Pa and 4.2 × 10^{−6} Pa for x=0.84 and 6 × 10^{−7} Pa and 4.4 × 10^{−6} Pa for x=0.88. The Te flux was oversupplied with the Te/(Bi+Sb) ratio kept at ∼20. The substrate temperature was 200 °C and the growth rate was ∼0.2 nm per minute. Fabrication procedures for the x=0.84 and 0.88 films are slightly different at the initial growth on InP surfaces. We grew the 0.84 film with supplying Te and (Bi+Sb) from the initial stage. For the x=0.88 film, we started with supplying Te and Sb for a monolayer growth of Sb_{2}Te_{3} buffer layer followed by Bi shutter opening. This difference may be an origin of the larger energy gap δE_{DP} of the Dirac points between the top and bottom surfaces (Fig. 2j) for the x=0.88 film. After the epitaxial growth of 8 nmthick thin films, in situ annealing was performed at 380 °C to improve surface morphology.
Device fabrication
The AlO_{x} capping layer was deposited at room temperature with an atomic layerdeposition system immediately after the discharge of the samples from MBE. This process turned out to be effective to protect the surface from degradation. The device structure was defined by subsequent photolithography and Ar ionmilling processes. Ohmiccontact electrodes and top gate electrode were Ti/Au and deposited with an ebeam evaporator. Here, ionmilling was performed under 45degree tilt condition on a rotating stage, resulting in the ramped side edge as schematically shown in Fig. 1a. This ensured electrical contact to the top and bottom of the film.
Additional information
How to cite this article: Yoshimi, R. et al, Quantum Hall effect on top and bottom surface states of topological insulator (Bi_{1−x}Sb_{x})_{2}Te_{3} films. Nat. Commun, 6:6627 doi: 10.1038/ncomms7627 (2015).
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Acknowledgements
R.Y. is supported by the Japan Society for the Promotion of Science (JSPS) through a research fellowship for young scientists. This research was supported by the Japan Society for the Promotion of Science through the Funding Program for WorldLeading Innovative R & D on Science and Technology (FIRST Program) on ‘Quantum Science on Strong Correlation’ initiated by the Council for Science and Technology Policy and by JSPS GrantinAid for Scientific Research(S) No. 24224009 and 24226002. This work was carried out by joint research of the Cryogenic Research Center, the University of Tokyo.
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R.Y. performed thin films growth and device fabrication. R.Y., Y.K. and J.F. performed the low temperature transport measurements. R.Y. analysed the data and wrote the manuscript with contributions from all the authors. A.T., K.S.T., J.G.C., N.N., M.K. and Y.T. jointly discussed the results and guided the project. Y.T. conceived and coordinated the project.
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Supplementary Figures 15, Supplementary Notes 16 and Supplementary References (PDF 590 kb)
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Yoshimi, R., Tsukazaki, A., Kozuka, Y. et al. Quantum Hall effect on top and bottom surface states of topological insulator (Bi_{1−x}Sb_{x})_{2}Te_{3} films. Nat Commun 6, 6627 (2015). https://doi.org/10.1038/ncomms7627
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DOI: https://doi.org/10.1038/ncomms7627
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