Abstract
Helically spinpolarized Dirac fermions (HSDF) in protected topological surface states (TSS) are of high interest as a new state of quantum matter. In threedimensional (3D) materials with TSS, electronic bulk states often mask the transport properties of HSDF. Recently, the highfield Hall resistance and lowfield magnetoresistance indicate that the TSS may coexist with a layered twodimensional electronic system (2DES). Here, we demonstrate quantum oscillations of the Hall resistance at temperatures up to 50 K in nominally undoped bulk Bi_{2}Se_{3} with a high electron density n of about 2·10^{19} cm^{−3}. From the angular and temperature dependence of the Hall resistance and the Shubnikovde Haas oscillations we identify 3D and 2D contributions to transport. Angular resolved photoemission spectroscopy proves the existence of TSS. We present a model for Bi_{2}Se_{3} and suggest that the coexistence of TSS and 2D layered transport stabilizes the quantum oscillations of the Hall resistance.
Introduction
Among the new material class of topological insulators (TI), the chalcogenide semiconductor Bi_{2}Se_{3} has been long subject to intense investigations due to its potential integration in room temperature applications, such as dissipationless electronics and spintronics devices^{1,2,3,4}. Bi_{2}Se_{3} has a single Dirac cone at the Γpoint in the first surface Brillouin zone and a direct band gap of 0.3 eV between the valence and the conduction band^{5,6,7}. Due to the inversion symmetry in Bi_{2}Se_{3} the topological Z _{2} invariant ν = (1;000) is equal to the charge of parity of the valence band eigenvalues at the timereversalinvariant points of the first Brillouin zone caused by the band inversion^{8}. In the crystalline modification Bi_{2}Se_{3} has a tetradymite structure with R\(\overline{3}\)m symmetry. The unit cell consists of 15 atomic layers grouped in three quintuple layers with Se–Bi–Se–Bi–Se order stacked in an A–B–C–A–B–C manner. The quintuple layers are van der Waals bonded to each other by a double layer of Se atoms, the socalled van der Waals gap^{4}. The existence of TSS in Bi_{2}Se_{3} has been experimentally confirmed through angle resolved photoemission spectroscopy (ARPES)^{3,7,9} and scanning tunneling microscopy/scanning tunneling spectroscopy (STM/STS)^{10,11}. The asgrown crystals of Bi_{2}Se_{3} are typically ntype because of electron doping due to natural selenium vacancies^{12,13}. Therefore, the transport properties of Bi_{2}Se_{3} are generally dominated by bulk conduction. In particular, the temperature dependence of the electrical resistivity ρ is metalliclike^{3,14,15,16,17} and Shubnikovde Haas (SdH) oscillations in the longitudinal resistivity ρ _{xx} show the characteristic signatures for a 3D Fermi surface^{14,17}. For highly Sbdoped samples with lower carrier density \(n\sim {10}^{16}\) cm^{−3}, the TSS can be detected via additional SdH oscillations with a frequency B _{SdH} higher than that of the bulk and the Hall resistivity ρ _{xy} exhibits quantum oscillations for a carrier density n < 5 ⋅ 10^{18} cm^{−3} (ref.^{18}). Different from that, for n ≥ 2 ⋅ 10^{19} cm^{−3} a bulk quantum Hall effect (QHE) with 2Dlike transport behavior was reported^{3,16}. Its origin remains unidentified.
In this work we demonstrate that the quantum oscillations of the Hall resistance R _{xy} in highpurity, nominally undoped Bi_{2}Se_{3} single crystals with a carrier density of n ≈ 2 ⋅ 10^{19} cm^{−3} persists up to high temperatures. The quantum oscillations in R _{xy} scale with the sample thickness, strongly indicating 2D layered transport. These findings stand out because the Bi_{2}Se_{3} samples investigated here have a lower carrier mobility μ of about 600 cm^{2}/(Vs) than materials hosting a typical 2D Fermi gas^{19,20,21,22} or 3D Fermi gas^{23,24,25} showing QHE. We discuss the conditions of the QHE below in detail and present a model for the coexistence of 3D bulk, 2D layered and TSS transport.
Results
Experimental data
Highresolution ARPES dispersions measured at a temperature of 12 K for two representative photon energies of hν = 16 eV and 21 eV are shown in Fig. 1a and b, respectively. We clearly observe distinct intensity contributions from the bulk conduction band (BCB) and bulk valence band (BVB) coexisting with sharp and intense Dirac cone representing the TSS. The BCB crossing the Fermi level indicates that the crystals are intrinsically ntype, in agreement with our Hall measurements on the same samples. At binding energies higher than the Dirac node (\({E}_{{\rm{D}}}\sim 0.35\) eV), the lower half of the TSS overlaps with the BVB. By changing the photon energy we select the component of the electron wave vector perpendicular to the surface k _{ z }. Since the lattice constant of Bi_{2}Se_{3} is very large along the z direction (c = 28.64 Å), the size of the bulk Brillouin zone (BBZ) is very small (\(\sim 0.2\) Å^{−1}). With photon energies between 16 to 21 eV we cross practically the complete BBZ, enhancing the sensitivity to the outofplane dispersion of the bulk bands. We note that the ARPES intensity changes with the photon energy as well due to the k _{ z }dependence of the photoemission transitions. Differently from the BCB or BVB, the TSS exhibits no k _{ z }dependence due to its 2D character. Consistent with the direct nature of the gap, we find the BCB minimum (≈Γpoint of the BBZ) at a binding energy of \(\sim 0.154\) eV, while the BVB maximum is at \(\sim 0.452\) eV. In particular, from the ARPES measurements, we estimate a bulk carrier density of n _{3D,BCB} = 1.77 ⋅ 10^{19} cm^{−3} and a sheet carrier density of \({n}_{\mathrm{2D},\mathrm{TSS}}={k}_{F,\mathrm{TSS}}^{2}\mathrm{/(4}\pi )=1.18\cdot {10}^{13}\) cm^{−2}, with k _{F,3D} = 0.064 Å^{−1} and k _{F,TSS} = (0.086 ± 0.001) Å^{−1}, respectively.
In the following, we present data measured on one Bi_{2}Se_{3} macro flake. However, similar results were obtained for other samples from the same source bulk single crystal. The longitudinal resistance R _{xx} and the Hall resistance R _{xy} were measured simultaneously in a temperature range between 0.3 K and 72 K in tilted magnetic fields up to 33 T. R _{xx} as a function of perpendicular magnetic field B measured at T = 0.47 K is shown in Fig. 1c as symmetrized raw data \({R}_{{\rm{xx}}}^{{\rm{sym}}}(B)=[{R}_{{\rm{xx}}}^{{\rm{raw}}}(+B)+{R}_{{\rm{xx}}}^{{\rm{raw}}}(B)]\mathrm{/2}\). The temperaturedependent R _{xx} at zero magnetic field shows metalliclike behavior (see inset of Fig. 1c). A residual resistance ratio RRR = R _{xx}(288 K)/R _{xx}(4.3 K) = 1.63 indicates a high crystalline quality^{12} (see Supplementary Information Sec. 1).
The longitudinal resistivity ρ _{xx} and the Hall resistivity ρ _{xy} as a function of the perpendicular magnetic field B at a temperature of T = 0.47 K are shown in Fig. 1d (ρ _{xx}: blue curve, left axis; ρ _{xy}: red curve, right axis). The onset of quantum oscillations with plateauxlike features in ρ _{xy} and SdH oscillations in ρ _{xx} can be observed at fields B ≥ 10 T. The lowfield slope of ρ _{xy} yields a carrier density of n _{Hall} = 1.97 ⋅ 10^{19} cm^{−3} and a carrier mobility of μ _{Hall} = 594 cm^{2}/(Vs).
In order to analyze the plateauxlike features in R _{xy}, we use the highfield antisymmetrized Hall resistance \({R}_{{\rm{xy}}}^{{\rm{asy}}}(B)=[{R}_{{\rm{xy}}}^{{\rm{raw}}}(+B){R}_{{\rm{xy}}}^{{\rm{raw}}}(B)]\mathrm{/2}\) data for T = 0.47 K and an angle of θ = 0°. θ denotes the angle between the direction of \(\overrightarrow{B}\) and the surface normal \(\overrightarrow{N}\) of the Bi_{2}Se_{3} macro flake (i. e. θ = 0° means \(\overrightarrow{B}\)\(\overrightarrow{N}\)). \(\overrightarrow{N}\) is parallel to the caxis of the single crystal. The scaling behaviour of \({\rm{\Delta }}{R}_{{\rm{xy}}}^{{\rm{asy}}}={R}_{{\rm{xy}}}^{{\rm{asy}}}(N){R}_{{\rm{xy}}}^{{\rm{asy}}}(N+\mathrm{1)}\) with the thickness leads to \({Z}^{\ast }=[\mathrm{(1/}N\mathrm{1/(}N+\mathrm{1))/}{\rm{\Delta }}{R}_{{\rm{xy}}}^{{\rm{asy}}}]\cdot (h\mathrm{/(2}{e}^{2}))\) as the number of 2D spindegenerate layers contributing to the transport. Conclusively, an average number of 2D layers of Z ^{*} = 25250 is derived. The variation of Z ^{*} for different Landau level (LL) index N is given in the inset of Fig. 2a.
The negative differentiated Hall resistivity −dρ _{xy}/dB vs magnetic field B is shown for different angles θ at constant T= 1.47 K in Fig. 2c, and for different temperatures T at constant θ = 0° in Fig. 2d. In accordance with the angular and the temperature dependence of the SdH oscillations, as shown in Figs 3 a and 4a, respectively, a decreasing amplitude of the differentiated Hall resistivity with increasing θ and increasing T is detected. At a constant T = 1.47 K the typical signatures of quantum oscillations of the Hall resistance are observed up to θ = 61.8° and at θ = 0° the amplitude of dρ _{xy}/dB vanishes only for temperatures above 71.5 K.
The SdH oscillations in the longitudinal resistivity ρ _{xx} are periodic functions of the inverse magnetic field 1/B and the minima correspond to an integer number of filled LLs. Since the degeneracy of states in the LLs is proportional to B, the inverse magneticfield positions of the ρ _{xx} minima are linear functions of the LL index. The slopes of the linear functions depend on the extremal crosssectional area of the Fermi sphere (for 3D systems) or circle (for 2D systems), and the intercepts depend on the Berry phase of the charge carriers (see Supplementary Information Sec. 2). In the LL fan diagram shown in Fig. 2b the inverse magneticfield positions \(\mathrm{1/}{B}_{\bar{n}}\) are plotted vs the \(\bar{n}\)th minimum of the longitudinal resistivity ρ _{xx} for different values of θ. The straight dashed lines, which represent the best linear fits to the data, intersect jointly the \(\bar{n}\)axis at the point of origin (see Supplementary Information Sec. 2). Hence, we find for all angles θ and temperatures T investigated here a significant evidence for a trivial Berry phase of Φ_{B} = 0 (cf. inset of Fig. 2b) and conclude the dominance of nonrelativistic fermions. For an improved estimate of the Berry phase we have fitted the behavior of the relative longitudinal resistivity Δρ _{xx} vs magnetic field B assuming 2D and 3D transport (cf. Fig. 4c and d, respectively).
The relative longitudinal resistivity Δρ _{xx} vs magnetic field B measured at T = 4.26 K for different angles θ is shown in Fig. 3a. Δρ _{xx} was calculated from the measured ρ _{xx} by subtracting a suitable polynominal fit to the background to extract the oscillatory component. The amplitude of the SdH oscillations decreases with increasing angle θ, and is really marginal for θ > 70°. Furthermore, a change in the frequency of the SdH oscillations with increasing angle θ with respect to θ = 0° is observed. For all values of θ and T, we found one value of the SdH frequency B _{SdH}, deduced from the periodicity in the 1/B dependence. These values are in agreement with those determined from the slopes of the lines in the LL fan diagram (Fig. 2b) and from fast Fourier transforms of the same data. The absence of additional frequencies and beatings, as well as the angular dependence of B _{SdH} (see Fig. 3b), are significant evidence of a single 3D (nonspherical) Fermi surface (see Supplementary Information Sec. 3).
The temperature dependence of Δρ _{xx} is shown in Fig. 4a: the amplitude decreases with increasing temperature T, and oscillations are not observed for T > 71.5 K. From the fitting of the relative longitudinal resistivity ratio Δρ _{xx}(T)/Δρ _{xx}(T = 1.47 K), we deduce an effective mass of the charge carriers of m ^{*} ≅ 0.16 m _{e} (m _{e} = 9.10938356 ⋅ 10^{−31} kg denotes the electron rest mass) and a Fermi velocity of v _{F} = \(\bar{h}\) k _{F,3D}/m ^{*} = 0.46 ⋅ 10^{6} m/s, with k _{F,3D} = 0.064 Å^{−1}.
In a first step we assumed 2D transport in accordance with some other investigations^{14,15,26}. The Dingle plots (inset of Fig. 4b) at temperatures of T = 1.47 K, 4.22 K, 15.3 K, and 26 K yield the following Dingle scattering time (also known as singleparticle relaxation time) τ _{D} and the Dingle temperature T _{D} = h/(4π ^{2} k _{B} τ _{D}), assuming the fit function −πm ^{*}/(eτ _{D} B) with m ^{*} = 0.16 m _{e}: τ _{D} = 5.8 ⋅ 10^{−14} s (T _{D} = 20.8 K), 5.1 ⋅ 10^{−14} s (23.7 K), 3.9 ⋅ 10^{−14} s (30.9 K) and 2.7 ⋅ 10^{−14} s (45.5 K), respectively. For a more detailed analysis we have fitted the magneticfield dependence of Δρ _{xx} (see Supplementary Information Sec. 4) and have used as fit function the LifshitzKosevich formula^{4,27,28} for 2D transport. We found a reasonably good agreement between experimental data and the calculated behavior for Δρ _{xx}(B) (cf. Fig. 4c).
However, in a second step we also performed fits under the assumption of 3D transport^{29} (cf. Fig. 4d), because the angular dependence of the SdH frequency B _{SdH} in Fig. 3b clearly follows the function for 3D transport. In this case, we find for all curves a single value for the Dingle temperature T _{D} = 23.5 K and hence a single value for the Dingle scattering time τ _{D} = 5.2 ⋅ 10^{−14} s, consistent with a nearly constant R _{xx}(T) up to T = 30 K (see inset of Fig. 1c). From τ _{D} and the effective mass m ^{*} = 0.16 m _{e}, we determined a carrier mobility of μ _{D} = eτ _{D}/m ^{*} = 572 cm^{2}/(Vs).
Evaluation of experimental data
Most of the investigations of bulk Bi_{2}Se_{3} conclude that the Fermi surface is 3D^{3,12,14,17,30}, usually from the angular dependence of the SdH oscillations. However, in the search of TSS and QHE some works evaluated the Fermi surface as 2D^{15,16}. Our analysis of the SdH oscillations (see above) indicates that the Fermi surface is 3D. This is confirmed by our following analysis of the angular dependence of the SdH frequencies.
The angle dependence of the SdH oscillations determines that the Fermi surface has an ellipsoidal shape. For a plane 2D Fermi surface, the SdH oscillation frequency is equal to \({B}_{{\rm{SdH}}}^{{\rm{2D}}}(\theta )={B}_{\perp }/\,\cos \,\theta \), with \({B}_{{\rm{SdH}}}^{{\rm{2D}}}(\theta )\to \infty \) for θ → 90° (blue curve in Fig. 3b), and for an ellipsoidal 3D Fermi surface it is \({B}_{{\rm{SdH}}}^{{\rm{3D}}}(\theta ))={B}_{\perp }{B}_{}/\sqrt{{({B}_{}\cos \theta )}^{2}+{({B}_{\perp }\sin \theta )}^{2}}\) (red curve in Fig. 3b), with \({B}_{\perp }={B}_{{\rm{SdH}}}^{{\rm{3D}}}(\theta ={0}^{\circ })={B}_{{\rm{SdH}}}^{{\rm{2D}}}(\theta ={0}^{\circ })=166\) T and \({B}_{}={B}_{{\rm{SdH}}}^{{\rm{3D}}}(\theta ={90}^{\circ })=328\) T. Previous data^{15,16} may also be interpreted as 3D ellipsoidal Fermi surface (see Supplementary Information Sec. 3).
We estimate the ellipsoidal crosssection of the 3D Fermi surface with the wave vectors \({k}_{F,\mathrm{SdH}}^{(a)}={k}_{F,\mathrm{SdH}}^{(b)}=\) \(\sqrt{2e{B}_{\perp }/\hslash }=0.071\) Å^{−1} and \({k}_{F,\mathrm{SdH}}^{(c)}=2e{B}_{}/(\hslash {k}_{F,\mathrm{SdH}}^{(a)})=0.14\) Å^{−1}. With these values we deduced an eccentricity for the 3D nonspherical Fermi surface of \({k}_{F,\mathrm{SdH}}^{(c)}/{k}_{F,\mathrm{SdH}}^{(a)}=1.98\). Köhler^{30} and Hyde et al.^{12} show, that the eccentricity of the Fermi surface decreases with decreasing carrier density n. In accordance with the present study, Eto et al.^{14} deduced for a Bi_{2}Se_{3} bulk single crystal with a lower carrier density of n = 3.4 ⋅ 10^{18} cm^{−3} an eccentricity of \({k}_{F,\mathrm{SdH}}^{(c)}/{k}_{F,\mathrm{SdH}}^{(a)}=1.62\), consistent with eccentricities obtained by Köhler^{30}. Assuming a parabolic dispersion and using the values of \({k}_{{\rm{F}}}^{(a)}\) and \({k}_{{\rm{F}}}^{(c)}\) from the SdH analysis and of E _{F} from the ARPES measurements, we estimate with E _{F} = (\(\bar{h}\) k _{F})^{2}/(2 m ^{*}) for the effective masses \({m}_{{\rm{a}}}^{\ast }={m}_{{\rm{b}}}^{\ast }=0.125\,{m}_{{\rm{e}}}\) and \({m}_{{\rm{c}}}^{\ast }=0.485\,{m}_{{\rm{e}}}\). An average value for the effective mass is then given by^{31} \(\mathrm{1/}{m}^{\ast }=\mathrm{(1/}{m}_{{\rm{c}}}^{\ast }+\mathrm{2/}{m}_{{\rm{a}}}^{\ast }\mathrm{)/3}\), which yields m ^{*} = 0.166 m _{e}. This value is consistent with the value obtained from the temperature dependence of the SdH oscillations: \({m}_{{\rm{SdH}}}^{\ast }=0.16{m}_{{\rm{e}}}\).
Discussion
Generally, a bulk or 3D QHE is attributed to parallel 2D conduction channels, each made from one or a few stacking layers. A bulk QHE, where quantized values of the Hall resistance R _{xy} inversely scale with the sample thickness, has been observed in a number of anisotropic, layered electronic bulk materials, e.g., GaAs/AlGaAs multiquantum wells^{20}, Bechgaard salts^{24,32} and also in Fedoped Bi_{2}Se_{3} bulk samples^{33}, where transport by TSS was excluded. However, the observation of the quantum oscillations of the Hall resistance in Bi_{2}Se_{3} at elevated temperatures calls for a special condition considering the usual requirement of \(\mu B\gg 1\). In the present case B _{max} = 33 T and the carrier mobility μ ≈ 600 cm^{2}/(Vs) yields only μB _{max} ≈ 2. Furthermore, the deduced effective mass m ^{*} = 0.16 m _{e} yields for a magnetic field of B = 10 T, where we observe the onset of the quantum oscillations, a value for the LL energy splitting of \(\bar{h}\) ω _{c} = \(\bar{h}\) eB/m ^{*} ≈ 7 meV. However, the thermal energy amounts to k _{B} T ≈ 4 meV at T = 50 K, while \(\hslash {\omega }_{{\rm{c}}}\gg {k}_{{\rm{B}}}T\) is usually required for a QHE. Nevertheless, we observe unambiguous quantum oscillations in −dρ _{xy}/dB (Fig. 2c and d) as signature of a QHE.
In order to explain the experimental observations, we propose the following model. The Bi_{2}Se_{3} bulk sample investigated here may consist of three different conducting regions: a semiconductinglike core region, surrounded by a metalliclike shell region and the topological surface (see Fig. S1 in Supplementary Information Sec. 5). The semiconductinglike core was proven by the preparation of semiconducting micro flakes^{3}. The metalliclike shell region due to Se depletion dominates the transport mechanism observed here as metallic and 2D layered effects. From our experiments, we assume the shell to form a stacked system of 2D layers with a periodic potential^{23} either due to the van der Waalsgaps or the unit cell along the caxis because of the carrier density modulation due to Se vacancies. In magnetic fields B ≠ 0 the thickness scaling of the plateauxlike features in the Hall resistance yields an effective thickness for the shell of stacked 2D layers. For the charge carrier density, we estimate three different values for the core (from ref.^{3}), the shell (from the Hall measurements) and the topological surface (from the ARPES measurements): n _{core} ≈ 1.2 ⋅ 10^{17} cm^{−3}, n _{shell} ≈ 2 ⋅ 10^{19} cm^{−3} and n _{TSS} = 1.2 ⋅ 10^{13} cm^{−2}, respectively. In the semiconductinglike core region, the Fermi level (chemical potential) is in the gap close to the bottom of the conduction band, whereas in the metalliclike shell region the Fermi level is in the conduction band (see Supplementary Information Sec. 5). Because of a finite scattering rate between the 2D layers in the shell region ρ _{xx}(B) shows considerable 3D character in the SdH oscillations for higher angles θ, and the quantization in ρ _{xy}(B) even at the lowest temperature T is not exact and the plateaux have a finite slope.
For the SdH frequency B _{SdH} we estimate at θ = 0° for the three regions the following values: B _{SdH,core} = 4.82 T, B _{SdH,shell} = 166 T and B _{SdH,TSS} = 248 T. The small value B _{SdH,core} corresponds to a slowchanging background which is out of the measurement range of our experimental setup. The larger value of the TSS is caused only by the small number of surface electrons with respect to the large number of bulk electrons (N _{bulk} ≈ N _{shell} ≈ 2 ⋅ 10^{15} and N _{TSS} ≈ 3 ⋅ 10^{11} yield a ratio N _{TSS}/N _{bulk} ≈ 10^{−4}). Therefore, from the experimental data we deduce only the B _{SdH} value for the shell (see Fig. 3a) and find the dominant contribution of the bulk (core + shell) in the transport behavior. A periodic modulation of the charge carrier density along the cdirection would result in a miniband structure for the LLs and, as long as the Fermi level is in a gap between these minibands, the Hall resistivity ρ _{xy} will be quantized and scale with the periodicity of the potential^{23}.
According to our estimate of the width of the LLs (see above), the persistence of the quantum oscillations in the Hall resistance up to high temperatures requires a special condition: We propose a Fermi level pinning in the miniband gap, which could be the result of an interaction with the existing TSS. Theoretically, due to the interlayer coupling, it is expected that in the quantum Hall state the edge states of the stacked 2D layers form a sheath at the surface^{34}. Due to the finite width of the wave functions at the surface, this sheath can interact with the TSS. This opens the possibility that the TSS act as electron reservoirs to pin the Fermi level in a miniband gap as the magnetic field is varied over a finite range. Therefore, we conclude that the observation of the quantum oscillations of the Hall resistance at higher temperatures in Bi_{2}Se_{3} (n ≈ 2 ⋅ 10^{19} cm^{−3}) with a majority of nonDirac fermions is related to the existence of the TSS. Based on our results, we propose that other 3D materials with TSS and a periodic potential modulation may show quantization effects in the Hall resistance at elevated temperatures.
Methods
Highquality single crystalline Bi_{2}Se_{3} was prepared from melt with the Bridgman technique. The growth time, including cooling was about 2 weeks for a ∼50 g crystal. The whole crystal was easily cleaved along the [00.1] growth direction, indicating crystal perfection. The macro flake was prepared by cleaving the bulk single crystal with a thickness of around 110 μm to investigate bulk properties.
We explored the structural properties of the bulk single crystal^{3} with atomic force microscopy (AFM), scanning transmission electron microscopy (STEM) and highresolution transmission electron microscopy (HRTEM). The composition and surface stability were investigated using energydispersive xray spectroscopy (EDX) and spatially resolved corelevel Xray PEEM. Structural analysis using HRTEM and STEM was carried out at a JEOL JEM2200FS microscope operated at 200 kV. The sample preparation for HRTEM characterization consisted of ultrasonic separation of the flakes from the substrate, followed by their transfer onto a carboncoated copper grid. Using adhesive tape, the surface was prepared by cleavage of the crystal along its trigonal axis in the direction perpendicular to the vanderWaalstype (0001) planes. The ARPES measurements were performed at a temperature of 12 K in an ultrahigh vacuum (UHV) chamber at a pressure of \(\sim 5\cdot {10}^{10}\) mbar with a VG Scienta R8000 electron analyzer at the UE112PGM2a beamline of BESSY II using ppolarized undulator radiation.
Magnetotransport experiments were performed using standard lownoise lockin techniques (Stanford Research Systems SR830 with a Keithley 6221 as current source), with low excitation to prevent heating of the sample. The Bi_{2}Se_{3} macro flake was mounted in a flow cryostat (1.3 K to 300 K), as well as in a ^{3}He insert (down to 0.3 K), in a Bitter magnet with a bore diameter of 32 mm and magnetic fields up to 33 T at the High Field Magnet Laboratory of the Radboud University Nijmegen. In both setups, a Cernox thermometer in the vicinity of the sample was used to monitor the temperature in situ. In the ^{3}He system, the temperature between 0.3 K and 1.3 K was stabilized by the ^{3}He vapour pressure prior to the magnetic field sweep to assure a constant temperature. However, the temperature between 1.3 K and 4.2 K was stabilized by the ^{4}He pressure. Above a temperature of 4.2 K, we have used the flow cryostat and stabilized the temperature using a capacitance.
References
 1.
Xue, Q.K. Nanoelectronics: A topological twist for transistors. Nature Nanotechnol. 6, 197 (2011).
 2.
Zhang, H. et al. Topological insulators in Bi_{2}Se_{3}, Bi_{2}Te_{3} and Sb_{2}Te_{3} with a single Dirac cone on the surface. Nature Phys. 5, 438 (2009).
 3.
Chiatti, O. et al. 2D layered transport properties from topological insulator Bi_{2}Se_{3} single crystals and micro flakes. Sci. Rep. 6, 27483 (2016).
 4.
Ando, Y. Topological InsulatorMaterials. J. Phys. Soc. J. 82, 102001 (2013).
 5.
Checkelsky, J. G., Hor, Y. S., Cava, R. J. & Ong, N. P. Bulk band gap and surface state conduction observed in voltagetuned crystals of the topological insulator Bi_{2}Se_{3}. Phys. Rev. Lett. 106, 196801 (2011).
 6.
Betancourt, J. et al. Complex band structure of topological insulator Bi_{2}Se_{3}. J. Phys.: Condens. Matter 28, 395501 (2016).
 7.
Xia, Y. et al. Observation of a largegap topologicalinsulator class with a single Dirac cone on the surface. Nature Phys. 5, 398 (2009).
 8.
Fu, L. & Kane, C. L. Topological insulators with inversion symmetry. Phys. Rev. B 76, 045302 (2007).
 9.
Bianchi, M. et al. The electronic structure of clean and adsorbatecovered Bi_{2}Se_{3}: an angleresolved photoemission study. Semicond. Sci. Technol. 27, 124001 (2012).
 10.
Alpichshev, Z. et al. STM Imaging of Impurity Resonances on Bi_{2}Se_{3}. Phys. Rev. Lett. 108, 206402 (2012).
 11.
Liu, Y. et al. Tuning Dirac states by strain in the topological insulator Bi_{2}Se_{3}. Nature Phys. 10, 294 (2014).
 12.
Hyde, G. R., Beale, H. A., Spain, I. L. & Woollam, J. A. Electronic properties of Bi_{2}Se_{3} crystals. J. Phys. Chem. Solids 35, 1719 (1974).
 13.
Yan, B., Zhang, D. & Felser, C. Topological surface states of Bi_{2}Se_{3} coexisting with Se vacancies. Phys. Status Solidi RRL 7, 148 (2013).
 14.
Eto, K., Ren, Z., Taskin, A. A., Segawa, K. & Ando, Y. Angulardependent oscillations of the magnetoresistance in Bi_{2}Se_{3} due to the threedimensional bulk Fermi surface. Phys. Rev. B 81, 195309 (2010).
 15.
Petrushevsky, M. et al. Probing the surface states in Bi_{2}Se_{3} using the Shubnikovde Haas effect. Phys. Rev. B 86, 045131 (2012).
 16.
Cao, H. et al. Quantized Hall Effect and Shubnikovde Haas Oscillations in Highly Doped Bi_{2}Se_{3}: Evidence for Layered Transport of Bulk Carriers. Phys. Rev. Lett. 108, 216803 (2012).
 17.
Analytis, J. G. et al. Bulk Fermi surface coexistence with Dirac surface state in Bi_{2}Se_{3}: A comparison of photoemission and Shubnikovde Haas measurements. Phys. Rev. B 81, 205407 (2010).
 18.
Analytis, J. G. et al. Twodimensional surface state in the quantum limit of a topological insulator. Nature Phys. 6, 960 (2010).
 19.
Beenakker, C. W. J. & van Houten, H. Quantum transport in semiconductor nanostructures. Sol. St. Phys. 44, 1 (1991).
 20.
Störmer, H. L., Eisenstein, J. P., Gossard, A. C., Wiegmann, W. & Baldwin, K. Quantization of the Hall effect in an anisotropic threedimensional electronic system. Phys. Rev. Lett. 56, 85 (1986).
 21.
Novoselov, K. S. et al. RoomTemperature Quantum Hall Effect in Graphene. Science 315, 1379 (2007).
 22.
Khouri, T. et al. Hightemperature quantum Hall effect in finite gapped HgTe quantum wells. Phys. Rev. B 93, 125308 (2016).
 23.
Halperin, B. I. Possible States for a ThreeDimensional Electron Gas in a Strong Magnetic Field. Jap. J. Appl. Phys. 26, 1913 (1987).
 24.
Hannahs, S. T., Brooks, J. S., Kang, W., Chiang, L. Y. & Chaikin, P. M. Quantum Hall effect in a bulk crystal. Phys. Rev. Lett. 63, 1988 (1989).
 25.
Hill, S. et al. Bulk quantum Hall effect in ηMo_{4}O_{11}. Phys. Rev. B 58, 10778 (1998).
 26.
Yan, Y. et al. HighMobility Bi_{2}Se_{3} Nanoplates Manifesting Quantum Oscillations of Surface States in the Sidewalls. Sci. Rep. 4, 3817 (2014).
 27.
Lifshitz, E. M. & Pitaevskii, L. P. Statistical Physics. (Pergamon Press, Oxford, 1986).
 28.
Taskin, A. A. & Ando, Y. Berry phase of nonideal Dirac fermions in topological insulators. Phys. Rev. B 84, 035301 (2011).
 29.
Ridley, B. K. Quantum Processes in Semiconductors. (Oxford University Press, Oxford, 2013).
 30.
Köhler, H. Conduction Band Parameters of Bi_{2}Se_{3} from Shubnikovde Haas Investigations. Phys. Stat. Sol. (b) 58, 91 (1973).
 31.
Ziman, J. M. Electrons and Phonons–The Theory of Transport Phenomena in Solids. (Clarendon Press, Oxford, 2001).
 32.
Balicas, L., Kriza, G. & Williams, F. I. B. Sign Reversal of the Quantum Hall Number in (TMTSF)_{2}PF_{6}. Phys. Rev. Lett. 75, 2000 (1995).
 33.
Ge, J. et al. Evidence of layered transport of bulk carriers in Fedoped Bi_{2}Se_{3} topological insulators. Sol. State Commun. 211, 29 (2015).
 34.
Balents, L. & Fisher, M. P. Chiral Surface States in the Bulk Quantum Hall Effect. Phys. Rev. Lett. 76, 2782 (1996).
Acknowledgements
Financial support from the Deutsche Forschungsgemeinschaft within the priority program SPP1666 (Grant No. FI932/71, FI932/72 and RA1041/71) and the Bundesministerium für Bildung und Forschung (Grant No. 05K10WMA) is gratefully acknowledged.
Author information
Affiliations
Contributions
M.B., O.C., S.P., S.W. and S.F.F. contributed to the transport experiments, analyzed the data and wrote the manuscript, J.S.B. and O.R. conducted the ARPES experiments and L.V.Y. conducted the bulk crystal growth. All authors contributed to the discussion and reviewed the manuscript.
Corresponding author
Ethics declarations
Competing Interests
The authors declare that they have no competing interests.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Busch, M., Chiatti, O., Pezzini, S. et al. Hightemperature quantum oscillations of the Hall resistance in bulk Bi_{2}Se_{3} . Sci Rep 8, 485 (2018). https://doi.org/10.1038/s41598017189600
Received:
Accepted:
Published:
Further reading

Transport evidence of linear Dirac dispersion of nontrivial surface states in Fesubstituted PbBi2Te4 3D Topological insulator
Physica E: Lowdimensional Systems and Nanostructures (2021)

Electrical Transport Properties of Vanadium‐Doped Bi 2 Te 2.4 Se 0.6
physica status solidi (b) (2020)

SurfaceInduced Enhanced Band Gap in the (0001) Surface of Bi2Se3 Nanocrystals: Impacts on the Topological Effect
ACS Applied Nano Materials (2020)

Signatures of Quantum Transport Steps in Bi2Se3 Single Crystal
Journal of Superconductivity and Novel Magnetism (2019)

Flux free single crystal growth and detailed physical property characterization of Bi1−xSbx (x = 0.05, 0.1 and 0.15) topological insulator
Materials Research Express (2019)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.