Anisotropic and strong negative magneto-resistance in the three-dimensional topological insulator Bi2Se3
S. Wiedmann, A. Jost, B. Fauque, J. van Dijk, M. J. Meijer, T. Khouri, S. Pezzini, S. Grauer, S. Schreyeck, C. Brune, H. Buhmann, L. W. Molenkamp, N. E. Hussey
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A ug Anisotropic and strong negative magneto-resistance in the three-dimensionaltopological insulator Bi Se S. Wiedmann, ∗ A. Jost, B. Fauqu´e, J. van Dijk, M. J. Meijer, T. Khouri, S. Pezzini, S. Grauer, S. Schreyeck, C. Br¨une, H. Buhmann, L. W. Molenkamp, and N. E. Hussey High Field Magnet Laboratory (HFML-EMFL) & Institute for Molecules and Materials,Radboud University, Toernooiveld 7, 6525 ED Nijmegen, The Netherlands LPEM (CNRS-UPMC), ESPCI, 75005 Paris, France Physikalisches Institut (EP3), Universit¨at W¨urzburg, Am Hubland, 97074 W¨urzburg,Germany (Dated: September 27, 2018)We report on high-field angle-dependent magneto-transport measurements on epitaxial thin filmsof Bi Se , a three-dimensional topological insulator. At low temperature, we observe quantumoscillations that demonstrate the simultaneous presence of bulk and surface carriers. The magneto-resistance of Bi Se is found to be highly anisotropic. In the presence of a parallel electric andmagnetic field, we observe a strong negative longitudinal magneto-resistance that has been consid-ered as a smoking-gun for the presence of chiral fermions in a certain class of semi-metals due tothe so-called axial anomaly. Its observation in a three-dimensional topological insulator implies thatthe axial anomaly may be in fact a far more generic phenomenon than originally thought. PACS numbers: 73.43.Qt, 73.25.+i, 71.70.Di, 71.18.+y
The role of topology in condensed matter systems, oncea rather esoteric pursuit, has undergone a revolution inthe last decade with the realization that a certain classof insulators and semi-metals play host to topologically-protected surface states. In 2009, band structure calcu-lations revealed that stoichiometric Bi Se , a well-knownthermoelectric material [1], bears all the hallmarks of athree-dimensional topological insulator (3D TI) [2] withan insulating bulk and conducting surface states providedthat the Fermi energy ǫ F is situated within the bulk bandgap [3]. These gapless surface states possess opposite spinand momentum, and are protected from backscatteringby time reversal symmetry. The existence of Dirac-likesurface states within the bulk band gap was confirmedin an angle-resolved photoemission spectroscopy studyperformed that same year [4].Though Bi Se is arguably the most simple represen-tative of the 3D TI family, accessing the topological sur-face states (TSS) in transport has been hindered by alarge residual carrier density in the bulk[5, 6]. WhileShubnikov-de Haas (SdH) oscillations are a powerfulmeans to distinguish between bulk and surface chargecarriers via their angle dependence, their analysis andinterpretation remain controversial. The literature is re-plete with results that have been attributed to single-bands of bulk carriers, TSS or multiple bands, empha-sizing the difficulty in distinguishing between bulk, TSSand a two-dimensional charge-accumulation layer [5–11].Apart from the TSS, the electronic bulk states in Bi Se are of particular interest since their spin splitting is foundto be twice the cyclotron energy observed in quantum os- ∗ Electronic address: [email protected] cillation [12, 13] and optical [14] experiments. Anotherpeculiar property of Bi Se and other 3D TIs is the ob-servation of a linear positive magneto-resistance (MR)that persists up to room temperature [15–20].The recent explosion of interest in 3D massless Diracfermions in ‘3D Dirac’ or ‘Weyl’ semi-metals [21] is basedprimarily on their unique topological properties thatcan be revealed in relatively straightforward magneto-transport experiments. Examples include the observa-tion of an extremely large positive MR [22], linear MR[23] and, more specifically, the negative longitudinal MR(NLMR) predicted to appear in Weyl semi-metals whenthe magnetic and electric field are co-aligned. ThisNLMR has been attributed to the axial anomaly, a quan-tum mechanical phenomenon that relies on a number im-balance of chiral fermions in the presence of an appliedelectric field [22, 24–26]. In a recent theoretical study,however, it was proposed that the NLMR phenomenonmay in fact be a generic property of metals and semicon-ductors [27], rather than something unique to topologicalsemi-metals.In this Rapid Communication, we present magneto-transport experiments on Bi Se epitaxial layers in mag-netic fields up to 30 T. At low-temperatures, we estab-lish the existence of both bulk and surface carriers viaangle-dependent SdH measurements. Moreover, we ob-serve a strong anisotropy in the MR which depends onthe orientation of the current I with respect to the ap-plied magnetic field B over a wide range of carrier con-centrations. When the magnetic field is applied parallelto I ( I k B x ), we observe a strong NLMR. This sur-prising finding confirms that the observation of NLMRis not unique to Weyl semi-metals and therefore cannotby itself be taken as conclusive evidence for the existenceof Weyl fermions in other systems. With this in mind,we consider possible alternative origins of this increas-ingly ubiquitous phenomenon, but argue finally that theaxial anomaly may indeed be generic to a host of three-dimensional materials [27].The present study has been performed on samples withdifferent layer thicknesses d =290, 190, 50 and 20 nm(referred to hereafter as samples L × width W - (30 × µ m ).The carrier concentration n = n Hall at 300 K (extractedfrom the linear part of the low-field Hall resistivity ρ xy )varies from 1.2 · -1.7 · cm − with decreasing thick-ness [29]. All magneto-transport measurements reportedhere were performed in a He flow cryostat in a resistive(Bitter) magnet up to 30 T using standard ac lock-in de-tection techniques with an excitation current of 1 µ A. Forthe SdH oscillation analysis, the magnetic field is appliedin a plane perpendicular to the current I .
10 1000.00.10.20.30.40 10 20 30102030400 50 100 1500102030 0 10 20 30567
10 100141516 d=290 nm ( m / V s ) T (K) n ( c m - ) T=4.2 K xx ( m ) B (T)T=4.2 K 050100 n Hall =1.6 10 cm -3Hall =740 cm /Vs
290 nm190 nm50 nm20 nm d=290 nm, (d)(c) (b) xy ( m ) (a) d=20 nm, Hall =1.1 10 cm -3Hall =3990 cm /Vs xx ( m ) T (K) xx ( m ) B (T) 051015 xy ( m ) xx ( m ) T (K)
FIG. 1: (Color online) (a) Temperature dependence of theelectrical resistivity ρ xx for Bi Se MBE-grown films with dif-ferent thicknesses. (b) Mobility µ and carrier density n as afunction of T for sample ρ xx and ρ xy as a functionof B at T =4.2 K for The temperature dependence of the longitudinal resis-tivity ρ xx is shown in Fig. 1(a) for all four samples. InFig. 1(b), we plot the carrier mobility µ and concentra-tion n = 1 / ( R H e ) for ρ xx ( T ) sweep and the measured ρ xy at B =1 T, respec-tively. The overall temperature dependence is metallic( dR xx /dT >
0) though below 40 K, we observe a tinyupturn in ρ xx which is strongest for the sample with thelowest carrier density. This increase is accompanied by asmall decrease in µ and an apparent increase in n whichhas been interpreted to originate from the presence of an impurity band [7, 8, 30]. In Figs. 1(c) and (d), we plot ρ xx and ρ xy as a function of the magnetic field B up to30 T for samples ρ xy is found to be non-linear for B > ρ xy , we extract a carrier mobility of 3990 (740) cm / Vsfor sample ρ xx ( B ) at 4.2 K when subject to a out-of-planeand in-plane magnetic field. Quantum oscillations areclearly visible in the second derivative − d ρ xx /dB , re-spectively, as a function of the inverse field, plotted inFigs. 2(c,d) for both orientations. In the parallel fieldconfiguration, only one frequency is evident, whereas sev-eral frequencies are found in a perpendicular field. x xx ( m ) B || (T) (g)(f)(e) (d)(c) (b) f ( T ) ( ) bulksurface (a) || (1/T) - d xx / d B - d xx / d B I B z d=290 nm, T=4.2 Kd=290 nm, T=4.2 K xx ( m ) B (T)
44 T a m p li t ude ( a . u . ) f (T)
162 T100 T21.5 T a m p li t ude ( a . u . ) f (T) FIG. 2: (Color online) Longitudinal resistivity ρ xx as a func-tion of (a) perpendicular and (b) in-plane magnetic field at4.2 K for ρ xx ( B )as a function of 1 /B and 1 /B k to highlight the SdH oscilla-tions. (e) Extracted frequencies from the FFT analysis as afunction of angle showing contributions from surface and bulkcarriers. The straight (dashed) lines correspond to the 1 / cosΘdependence expected for a purely two-dimensional system.(f,g) FFTs for a perpendicular and parallel field sweep, respec-tively, with the primary oscillation frequencies highlighted. In order to identify the origin of the quantum os-cillations, we have performed Fast Fourier Transforms(FFTs) on a series of ρ xx curves measured at differenttilt angles Θ. The results are summarized in Fig. 2(e).For Θ=0 (perpendicular field configuration), we observethree frequencies at 21.5, 100 and 162 T (see Fig. 2(f)).All frequencies appear to follow a 1 / cosΘ up to tilt anglesof around 60 o characteristic of a two-dimensional elec-tronic state. Beyond 60 o , however, the lower frequencystarts to deviate from this behavior and saturates to-wards 90 o . We therefore attribute the observed frequen-cies to two surface states (top and bottom) and one bulkband. Taking the Onsager relation, i.e. the extremalcross section of the Fermi surface A ( E F ) ∝ f and as-suming an ellipsoid pocket with V = 4 / πa b , we obtain n bulk =8.1 · cm − for the bulk band corresponding tothe pocket with the lowest frequency. For the surfacestates, we obtain the carrier densities 2.4 · cm − and3.9 · cm − . From the quantum oscillation analysis,we thus obtain a total carrier concentration of n tot,SdH = 1.0 · cm − , in excellent agreement with n Hall . Incontrast, assuming that all three pockets were ellipsoidal(i.e. bulk), we would obtain a total carrier concentrationthat is one order of magnitude larger than n Hall .Let us now turn our attention to the peculiar MR weobserve in these samples. To avoid quantum oscillatoryand quantum interference phenomena [29], we first fo-cus here on the angle-dependent MR response at roomtemperature. The longitudinal and Hall resistivities havebeen measured in two different configurations, as shownin Fig. 3(a). In configuration [i], the applied magneticfield is always perpendicular to I (field rotated in the or-thogonal plane) whereas in configuration [ii], the currentand field are parallel if φ = 90 o . The carrier concentra-tions (mobilities) extracted from ρ xy ( ρ xx ) at low fieldsare summarized in Table 1 in the Supplemental mate-rial [29].We first present our results and analysis for the highmobility sample ( ρ xx ( B ) − ρ ) /ρ at different angles Θand φ as indicated in each figure. The overall MR is sim-ilar to the one observed at low temperature, i.e. it firstincreases quadratically then tends towards saturation athigher field. In both configurations, the MR is stronglyanisotropic. Most surprisingly, we observe a large NLMR( ∼
15 %) when the magnetic field is applied parallel tothe current ( I k B x ). As the second bulk conductionband is far from the Fermi energy ǫ F at room temper-ature [30–32], we analyze the MR at Θ= φ =0 using astandard one-carrier Drude model (for completeness, atwo-carrier analysis is presented in the Supplemental Ma-terial [29]).The corresponding longitudinal and Hall conductivi-ties σ xx and σ xy in the transverse configuration are il-lustrated in Fig. 3(d). From ρ xy /ρ xx we extract µB andfinally the carrier mobility µ as a function of the appliedfield, as shown in Fig. 3(e), and find that µ (0 T)/ µ (30 T) ≃ ρ xy for both configurations and found =90 data B -fit (g) B (T) (d) = =0 n=1.2*10 cm -3 =1490 cm /Vs xx ( S / m ) xx ( S / m ) xy ( S / m ) o o o o o xy ( k ) B (T) (f) B B (T) ( m / V s ) T=300 K (e)[i] (c)(b) [ii][i] M R B (T)
T=300 K [ii] B z IB y I B x IB z y x M R B (T)z I (a) FIG. 3: (Color online) Anisotropic magneto-transport inBi Se at T =300 K: (a) Schematic diagram of the electricaltransport measurements for configurations [i] and [ii]. (b,c)Magneto-resistance of sample B for bothconfigurations indicating a strong negative MR if I k B x . (d)Longitudinal σ xx and Hall conductivities σ xy as a functionof the magnetic field and (e) extracted µB and µ using theDrude model. (f) Hall resistivity ρ xy as a function of B forboth configurations (solid line for [i], open symbols for [ii]).(g) The conductivity σ xx as a function of B (solid line) at φ = 90 o is found to be ∝ B (triangles). that the Hall resistivity follows a simple cosine depen-dence and does not depend on the orientation of B withrespect to I (see Fig. 3(f)). Finally, in Fig. 3(g), we plot σ xx ( B ) for the parallel field configuration ( I k B x - solidline) and observe a B -dependence up to 30 T (symbolsrepresent a quadratic fit σ xx ( B ) = σ + aB to the data).In Fig. 4(a), we present the temperature depen-dence of the normalized longitudinal magneto-resistivity ρ xx ( B ) /ρ xx (0) for several chosen temperatures for sam-ple ρ xx ( B ) /ρ xx (0) for sample φ ≃ o ( φ < o ) for4.2 K (300 K). At 4.2 K, the NLMR is superimposed by
500 1000 1500-3-2-10
300 K50 K29 K xx ( B ) / xx ( ) B || (T) =90 o
200 K
T=4.2 K (b) =90 o o o o xx ( B ) / xx ( ) (a) T=300 K (c) xx ( B ) / xx ( ) B (T) o o o =90 o (d) experimental data fit || (T) xx ( B ) / xx ( ) xx ( B ) / xx ( ) T (K) B || =25 T a ( * - ) (cm /Vs) FIG. 4: (Color online) (a) Normalized longitudinal magneto-resistivity ρ xx ( B ) /ρ xx (0) for sample ρ xx ( B ) /ρ xx (0) at B =25 T). ρ xx ( B ) /ρ xx (0) fordifferent angles φ at (b) T = 4.2 K (c) and T = 300 K. (d) Nor-malized magneto-conductivity σ xx ( B ) /σ xx (0) at T = 300 Kfor all samples φ = 90 o . All sam-ples follow a σ xx ( B ) /σ xx (0) ∝ B dependence (see fits). Theinset shows the fitting parameter a as a function of µ for allsamples. SdH oscillations which have previously been attributed toTSS from the sidewalls [33]. We have shown here, how-ever, that they originate from the lowest bulk conductionband.The anisotropy in the MR and the large NLMR ina parallel field are not unique to one particular waferor sample. Indeed, for all samples, we observe a posi-tive MR in a purely perpendicular magnetic field and aNLMR in the longitudinal configuration [29]. Moreover,for the samples ( I ⊥ B y .The negative MR for I ⊥ B y can be explained using theclassical Drude model provided the bulk carriers have alow mobility [29]. In Fig. 4(a), we plot σ xx ( B ) /σ xx (0)as a function of B k and find that the NLMR gets pro-gressively weaker with decreasing d (increasing carrierconcentration and decreasing carrier mobility) at roomtemperature. Significantly, the longitudinal conductiv-ity follows the B -behavior for all samples (The fittingparameter a as a function of µ is shown in the inset ofFig. 4(d) for all samples). Standard Boltzmann theory does not predict any lon-gitudinal magneto-resistance in the presence of a mag-netic field that is parallel to the applied electric field. ANLMR has been observed previously in both 1D [34] and2D [35] charge ordered systems for currents applied par-allel to the conducting chains (planes). In both cases, theNLMR exhibited ( B/T ) scaling attributed to a closingof the charge gap due to Zeeman splitting. In Bi Se , bycontrast, there is no strong T -dependence in the NLMR.A classical origin, found in inhomogeneous conductorsand attributed to macroscopic inhomogeneities and thusdistorted current paths [36] can be excluded since theanisotropic MR does not depend on the lateral samplesize [37]. The origin of the anisotropy of the MR and inparticular, the large NLMR in Bi Se is likely to arisefrom the underlying scattering mechanism, as inferredfrom our simple Drude analysis. In 1956, Argyres andAdam predicted a NLMR for a 3D electron gas in the caseof non-degenerate semiconductors where ionized impu-rity scattering is present [38] as observed, for example, inindium antimonide in the extreme quantum limit [39]. Incontrast, recently triggered by the discovery of new Diracmaterials [22–26], it has been proposed that a quantummechanical phenomenon called the axial anomaly cangive rise to a NLMR [27]. In a magnetic field, chargecarriers are subject to Landau quantization with a one-dimensional (1D) dispersion along B . If in addition anelectric field is applied parallel to B , a uniform accelera-tion of the center of mass in this field-induced 1D systemproduces the same axial anomaly effect as charge pump-ing between Weyl points in a Weyl semi-metal and thesubsequent charge imbalance leeds to a NLMR [22, 27].This effective reduction in the dimensionality of the elec-tronic dispersion is also the proposed origin for the re-cent observation of NLMR in the interplanar resistivityof 2D correlated metals [40]. Remarkably, the appear-ance of the NLMR is not tied to the band structure ofa particular material, but rather related to the type ofscattering mechanism present in the system and as inthe classical model [37], ionized impurity scattering isproposed to give rise to a positive magneto-conductivity σ ∝ B [27]. Depending on the dominant contributionof the underlying scattering mechanisms, the magneto-conductivity may be temperature-dependent as observedin indium antimonide [39]. For Bi Se , we estimate thatthe quantum limit is reached at a field strength B ≃
43 T for the sample with the lowest carrier concentrationof 1.2 · cm − (sample Se epilayers. The low-temperature angle-dependent SdH data suggests a coexistence of bulk andsurface charge carriers. At room temperature, we finda strong positive MR with a field dependence that canbe explained by a field-dependent carrier mobility. Themagnetoresistance itself is strongly anisotropic and de-pends on the orientation of the current I with respectto the parallel component of the magnetic field B . Wehave demonstrated that the observation of a NLMR akinto the axial anomaly is not specific to Dirac or Weylsemi-metals, but may in fact occur in generic three-dimensional materials.Part of this work has been supported by EuroMagNETII under the EU contract number 228043 and by theStichting Fundamenteel Onderzoek der Materie (FOM)with financial support from the Nederlandse Organisatievoor Wetenschappelijk Onderzoek (NWO). 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