aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Tunable Surface Conductivity in Bi Se Revealed in Diffusive Electron Transport
J. Chen, X. Y. He, K. H. Wu, Z. Q. Ji, L. Lu, J. R. Shi, J. H. Smet, and Y. Q. Li Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China International Center for Quantum Materials, Peking University, Beijing 100871, China Max Planck Institute for Solid State Research, D-70569, Stuttgart, Germany (Dated: October 14, 2018)We demonstrate that the weak antilocalization effect can serve as a convenient method for detect-ing decoupled surface transport in topological insulator thin films. In the regime where a bulk Fermisurface coexists with the surface states, the low field magnetoconductivity is described well by theHikami-Larkin-Nagaoka equation for single component transport of non-interacting electrons. Whenthe electron density is lowered, the magnetotransport behavior deviates from the single componentdescription and strong evidence is found for independent conducting channels at the bottom andtop surfaces. Magnetic-field-dependent part of corrections to conductivity due to electron-electroninteractions is shown to be negligible for the fields relevant to weak antilocalization.
The surface of a 3D topological insulator (TI) [1, 2]hosts a 2D system of Dirac electrons with spins trans-versely locked to their translational momenta. Such spin-helical surface states [3] offer a new route for realizingexotic entities such as Majorana fermions and magneticmonopoles [4]. The unique surface spin structure also hasprofound impact on the transport properties of TI [1, 5].The Berry phase associated with the surface electronscauses suppression of backscattering [6] and hence im-munity to localization regardless of the strength of disor-der [5]. This weak antilocalization effect can be broughtout by applying a perpendicular magnetic field. It pro-duces a negative magnetoconductivity due to the break-ing of time-reversal symmetry. The negative magneto-conductivity has indeed been observed in various TI thinfilms by several groups [7–11]. However, most of thesemeasurements were carried out with samples in whichthe Fermi energy is not located in the band gap, sothat they are not in the so-called topological transportregime [12, 13]. Since topologically trivial 2D electronsystems (e.g. Au thin films) may also exhibit similar mag-netoconductivity behavior as long as the spin-orbit cou-pling (SOC) is sufficiently strong [14], concern has to beraised whether weak antilocalization can provide a reli-able method for identifying the surface state transport,which is a key starting point for future exploration ofvarious topological effects and novel devices [1, 15–19].Here we confirm unequivocally that the weak antilo-calization effect can be used to differentiate the surfacetransport from transport dominated by bulk carriers.This is demonstrated on Bi Se thin films with carrierdensities that can be tuned over a wide range with aback-gate. When the transport is not in the topolog-ical regime, the magnetoconductivity can be describedby a single component Hikami-Larkin-Nagaoka (HLN)equation [20]. This description is found to be validfor a remarkably wide range of electron densities (0.8-8.6 × cm − ) in samples with the Fermi energy locatedinside the conduction band, even if electron-electron in-teractions are taken into account. In contrast, in a regime where the electronic system is split up into an electronlayer at the top surface and a hole layer at the bot-tom, the magnetoconductivity deviates strongly from thesingle-component HLN equation. Our analysis provides aconvenient method for detecting decoupled surface trans-port. It complements existing techniques based on quan-tum oscillations that are limited to samples of high car-rier mobilities [21] or samples with a quasi-1D geome-try [22].The Bi Se thin films were grown on SrTiO (111) sub-strates with molecular beam epitaxy [23]. The dielectricproperties of SrTiO are well suited for gating purposesand the carrier density in these devices can be varied byat least 2 × cm − [7]. All of the samples used inthis work were patterned into 50 µ m wide Hall bars withphotolithography, followed by Ar plasma etching (Fig. 1inset). This eliminates uncertainties in evaluating resis-tivities encountered in previous transport studies due tothe influence of electrical contacts or the irregular shapeof the sample. A set of more than ten samples with thick-nesses between 5 and 20 nm has been measured. Most ofthe samples have a back-gate deposited at the bottom ofthe substrate, and a few of them are equipped in additionwith a top-gate. The latter was deposited on an AlO x layer prepared with atomic layer deposition. Transportmeasurements were carried out in cryostats with temper-atures as low as 10 mK and magnetic fields up to 18 T.Fig. 1 displays typical magnetotransport data. All ofthe samples show a positive magneto-resistance with asharp cusp around zero magnetic field, consistent withprevious measurements of Bi Se thin films [7–11]. Asdemonstrated in Fig. 1(b), the low field magnetoconduc-tivity, defined as ∆ σ ( B ) = σ xx ( B ) − σ xx (0), can be fittedwell with the HLN equation in the strong SOC limit, i.e.when τ φ ≫ τ so , τ e :∆ σ ( B ) ≃ − α · e πh (cid:20) ψ (cid:18)
12 + B φ B (cid:19) − ln (cid:18) B φ B (cid:19)(cid:21) , (1)where τ so ( τ e ) is the spin-orbit (elastic) scattering time, ψ is the digamma function, B φ = ¯ h/ (4 Deτ φ ) is a char- R xy ( k W ) , M R B( T)
Rxy MR (a) a n e (10 cm -2 ) (c)
10 100 1000 a m (cm /V (cid:215) s) (d) Ds ( e / p h ) B (T) (b) B j FIG. 1. (color online) (a) Magnetoresistance (MR, defined as ρ xx ( B ) /ρ xx (0) −
1) and Hall resistance of a typical Bi Se thinfilm at T = 1 . ± B φ varies bynearly a factor of 30. Extracted values of α for 10 samplesare plotted as a function of electron density n e and mobility µ in (c) and (d), respectively. Both are evaluated based onthe low field transport measurements. acteristic field related to the dephasing time τ φ , D is thediffusion constant and h is the Planck constant. The coef-ficient α takes a value of 1/2 for a traditional 2D electronsystem with strong spin-orbit coupling. The same valueis expected for the electron transport on one surface of a3D TI with a single Dirac-cone [7].Fig. 1(c) shows that the extracted α values are dis-tributed in a narrow range near 1/2 for 10 sampleswith 2D electron densities n e spreading from 0.8 to8.6 × cm − [24]. No correlation is found between α and the electron mobility µ , which varies nearly two or-ders of magnitude (Fig. 1(d)). Based on angle-resolvedphotoemission measurements [12], the top and bottomsurfaces of a Bi Se thin film can only accommodate atotal electron density of ∼ . × cm − even if theFermi energy reaches the bottom of the conduction band.Thus we anticipate a significant number of bulk electrons(or quasi-2D electrons with parabolic dispersion) for theabove range of n e . The nonlinear Hall resistivity curves(see e.g. Fig. 1(a)) also suggest the coexistence of multi-ple charge carrier types. Even if so, the analysis of theweak antilocalization effect itself at small magnetic fieldsyields values of α close to 1/2. In this magnetic fieldregime where the antilocalization effect is observed, thesesamples do behave like 2D systems with a single type ofcharge carrier. This can only be understood when there isa strong mixing between the surface and the bulk electronstates or when the dephasing field of one of the conduct- -1 0 1-2-10 (cid:176) (cid:176) (cid:176) (cid:176) (cid:176) Ds ( e / p h ) B ^ (T) s xx ( e / p h ) T (K) k B (T) (a) (b)
FIG. 2. (color online) (a) Magnetoconductivity ∆ σ ( B ) at T =2 K and V G = plotted as a function of perpendicular magneticfield B ⊥ for several tilt angles. θ = 0 refers to B perpendicularto the thin film plane. (b) Temperature dependencies of σ xx (open symbols) recorded at B =0, 0.2, 1, and 5 T. The straightlines are linear fits of σ xx to ln T . In the upper inset, theslope, defined as κ = ( πh/e ) dσ xx ( B, T ) /d (ln T ), is plottedas a function B . The electron density is about 2 × cm − so that E F is located in the conduction band. ing components (i.e. the bulk or top/bottom surface) ismuch smaller than those of the others. We note that itwas demonstrated long ago that the two-valley 2D elec-tron system confined in a Si inversion layer displays α val-ues close to that for a single-valley system and not the oneexpected for two independent valleys [25]. Fukuyama at-tributed it to intervalley scattering [26]. Similar physicsmight take place here because of considerable scatteringbetween the surface and bulk states when the Fermi en-ergy is located in the conduction band.The robustness of α ≃ / σ ( B ) isdetermined by the ratio γ = E Z /E so = gµ B B/ (¯ hτ − ),where g is the electron g -factor. The Zeeman energyalso causes an extra change in ∆ σ ( B ) if the electron-electron interaction is not negligible in the diffusion chan-nel [28, 29]. The corresponding correction to the conduc-tivity is ∆ σ I ( B ) = e πh e F σ g (˜ h ) with ˜ h = E Z /k B T , where e F σ is a parameter reflecting the strength of dynamicallyscreened Coulomb interaction.Since the electron g -factor of bulk Bi Se is quitelarge [30], one would expect sizable Zeeman corrections to∆ σ ( B ). This appears to be in contradiction with the datarecorded in tilted magnetic fields and plotted in Fig. 2(a).The low field magnetoconductivity exhibits very little an-gular dependence for tilt angles less than 80 ◦ . Consid-ering that E Z nearly doubles (triples) for θ =60 ◦ (70 ◦ )with respect to the zero-tilt case, we conclude that theinfluence of the Zeeman energy can be neglected in caseof zero- or small tilts. In the non-interacting regime,this can be understood as a consequence of strong SOC,and hence small γ for the fields of interest. Also in theregime where e-e interactions are important, the strongSOC suppresses the Zeeman contribution. The Zeemanterm was derived under the assumption of weak SOC [28].Theories [28, 29, 31] and experiments [32] on other ma-terials have clearly shown that strong SOC can diminishand even entirely suppress the Zeeman-split term in thediffusion channel.The effects of strong SOC are further manifested inthe temperature dependence of σ xx displayed in Fig. 2(b).The slope of the ∆ σ ( B )-ln T plot, defined as κ =( πh/e ) dσ xx ( B, T ) /d (ln T ), is nearly constant for B =0.2-5 T. Both weak antilocalization and e-e interaction cancause the ln T dependence [14, 28]. The weak antilo-calization effect however only produces a pronounced T -dependence to σ xx at zero or low magnetic fields.The nearly constant slope at B > . T cor-rections proportional to e F σ [28, 31]. Hence, κ = 1 is ex-pected [5] for sufficiently large B . The observed κ ≃ . κ for B = 0 . σ ( B ) in lower fields,where the weak antilocalization is pronounced. As tothe zero-field conductivity, the combined effects of e-einteractions and weak antilocalization lead to a ln T de-pendence with κ = 1 − / Se film. The electron densityat V G =0, estimated from the low field Hall resistance,is about 2 . × cm − . The Hall resistance, R xy ( B ),increases as V G decreases. It reaches a maximum at V G = −
125 V, which would correspond to an electrondensity of n e ≈ . × cm − . Further decrease in V G leads to smaller R xy ( B ) and even reversal of its sign. For V G < −
150 V, the Hall curves become strongly nonlinear.The high field Hall coefficient is plotted in Fig. 3(b) anddepends non-monotonously on gate voltage. Also shownis the longitudinal resistivity at B = 0, denoted as ρ xx (0)throughout this paper. It also exhibits a non-monotonicdependence on V G . This, together with the fact that theHall coefficient R H does not reach a minimum, points R xy ( k W ) B (T) -125V -90V 0V -170V -180V (a) -200 -100 0 -1012 r xx ( ) ( k W ) V G (V) (b) R H / ( W(cid:215) T - ) -200 -100 00.51.0 048 a V G (V) B f ( m T ) (c) -200 -100 0 0.1110 V G (V) B f , B f ( m T ) (d) FIG. 3. (color online) (a) Hall resistance curves for V G = − , − , , − , −
180 V (from top to bottom). (b) Gate-voltage dependence of ρ xx at B = 0 and high field Hall coeffi-cient, defined as dR xy /dB and fitted from the data in B = 16-18 T. (c) Gate-voltage dependence of α and B φ obtained fromfits to Eq. (1). Shown in the left and right insets are the banddiagrams for large and small negative gate voltages, respec-tively. The top (bottom) surface is depicted on the left(right).(d) V G dependence of B φ (hexagons) and B φ (triangles) ob-tained from fits to Eq. (2). B φ and B φ can be assigned tothe bottom and the top surfaces, respectively. This 10 nmthick sample only has a back-gate. to the coexistence of electrons and holes for large nega-tive gate voltages. It is noteworthy that the maximumin ρ xx (0) appears at a V G smaller than that of the R H maximum. Therefore, the crossover from the pure elec-tron system to the electron-hole system must take placebefore the appearance of the ρ xx (0) maximum.For the gate voltages smaller than that at the R xy max-imum, the Fermi energy on the bottom and top surfacesare expected to lie below and above the Dirac point,respectively, even though the precise position of E F isnot known. As a consequence, the Fermi energy in thebulk (or at least part of the bulk) must be located in theband gap. The nearly one order of magnitude increase in ρ xx (0) as V G is lowered from 0 to −
150 V is much largerthan what has been reported for cleaved Bi Se flakescleaved on SiO /Si substrates[8]. The significantly en-hanced ρ xx (0) is an encouraging signature that much ofthe bulk conductivity can be suppressed. It can reachvalues as high as ∼ h/e [7].The magnetoconductivity also exhibits a strong gate-voltage dependence, especially for V G < −
50 V. Best fitsto Eq. (1) yield the data plotted in Fig. 3(c). The moststriking feature is that α is close to 1/2 for V G > −
70 V,and it increases to values close to 1 for V G < −
140 V.In the crossover region ( −
70 to −
140 V), the Bi Se thinfilm undergoes a transition from a low density electronsystem to a separated electron-hole system. Hence, forlarge negative gate voltages, a fit of the magnetoconduc-tivity data to a two-component HLN equation is moreappropriate:∆ σ ( B ) ≃ − e πh X i =1 (cid:20) ψ (cid:18)
12 + B φi B (cid:19) − ln (cid:18) B φi B (cid:19)(cid:21) (2)Here B φ and B φ are dephasing fields for conductingcomponents 1 and 2, respectively. As shown in Fig. 3(d),they have opposite dependencies on V G . B φ ( B φ ) de-creases(increases) as V G is lowered. They approach ap-proximately the same value for large negative gate volt-ages. The dephasing field is proportional to ( Dτ φ ) − ∝ ( v F τ e τ φ ) − , so B φ is expected to increase for the electroncomponent on the top surface, while it should decreasefor the holes at the bottom surface (interface) with de-creasing V G . Therefore, the two curves with larger andsmaller values of B φ in Fig. 3(d) could be assigned to thebottom and top surfaces, respectively.The observation that α increases toward 1 based on fitsof the magnetoconductivity data to the single-componentHLN equation (Eq. 1) implies that the top and bottomsurfaces of the film make separate contributions to theconductivity [34]. Obtaining α values close to 1, how-ever, not only requires two decoupled conduction chan-nels, but also demands that both conduction channelshave nearly identical dephasing fields. This is in generalhard to achieve, in particular for samples where the gatetunability is not sufficient or the substrate surface is toorough [7]. Caution should also be taken to ensure thatthe transport is in the diffusive and weakly disordered( k F l ≫
1) regime for which the HLN equation is valid.For highly resistive samples, e.g. ρ xx ∼ h/e as shown inRef. [7], the condition k F l ≫ k dielectric such as SrTiO for back-gating enablesus to tune the transport properties of both the top andthe bottom surfaces. This device geometry is particu-larly useful for the future exploration of hybrid devicesin which a topological insulator is interfaced with a su-perconductor [15, 16] or a ferromagnet [15–17]).We are grateful to L. Fu, D. Goldhaber-Gordon, I. V.Gornyi, P. M. Ostrovsky, V. Sacksteder, K. von Klitz-ing, X. C. Xie, P. Xiong, and in particular A. D. Mir-lin for valuable discussions. The work was supported byMOST-China, NSF-China, and Chinese Academy of Sci-ences and German Ministry of Science & Education. [1] X. L. Qi and S.-C. Zhang, Phys. Today (1), 33 (2010),and arXiv:1008.2026; M. Z. Hasan and C. L. Kane,Rev. Mod. Phys. , 3045 (2010); M. Z. Hasan and J. E.Moore, arXiv:1011.5462.[2] D. Hsieh et al., Nature , 970 (2008); L. Fu andC. L. Kane, Phys. Rev. B , 045302 (2007).[3] D. Hsieh et al., Science , 919 (2009); Y. Xia et al.,Nature Phys. , 398 (2009); H. J. Zhang et al., NaturePhys. , 438 (2009).[4] L. Fu and C. L. Kane, Phys. Rev. Lett. , 096407(2008); X. L. Qi et al., Science ,1184 (2009).[5] P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin, Phys.Rev. Lett. , 036803 (2010).[6] P. Roushan et al., Nature , 1106 (2009); T. Zhang etal., Phys. Rev. Lett. , 266803 (2009).[7] J. Chen et al., Phys. Rev. Lett. , 176602 (2010).[8] J. G. Checkelsky et al., arXiv:1003.3883.[9] H. T. He et al., Phys. Rev. Lett. , 166805 (2011).[10] M. Liu et al., arXiv:1011.1055 (2010).[11] J. Wang et al., arXiv:1012.0271 (2010).[12] D. Hsieh et al., Nature , 1101 (2009).[13] Y. L. Chen et al., Science , 178 (2009).[14] G. Bergmann, Phys. Rep. , 1 (1984); S. Kobayashiand F. Komori, Prog. Theor. Phys. Suppl. , 224 (1985).[15] L. Fu, C. L. Kane, Phys. Rev. Lett. , 216403 (2009).[16] A. R. Akhmerov, J. Nilsson, and C. W. J. Beenakker,Phys. Rev. Lett. , 216404(2009).[17] I. Garate and M. Franz, Phys. Rev. Lett. , 146802(2010).[18] B. Seradjeh, J. E. Moore, and M. Franz, Phys. Rev. Lett. , 066402 (2009).[19] R. Yu et al., Science , 61 (2010).[20] S. Hikami, A. I. Larkin, and Y. Nagaoka, Prog. Theor.Phys. , 707 (1980).[21] A. A. Taskin and Y. Ando, Phys. Rev. B , 085303(2009); D. X. Qu et al., Science , 821 (2010); J. G.Analytis et al., Nature Phys. , 960 (2010); Z. Ren et al.,Phys. Rev. B , 241306 (2010).[22] H. Peng et al., Nature Mater. , 225 (2010).[23] G. H. Zhang et al., Adv. Func. Mater. (in press).[24] α ≃ / Se thinfilms on Si by F. Yang et al. (unpublished).[25] Y. Kawaguchi and S. Kawaji, Surf. Sci. , 505 (1982).[26] H. Fukuyama, in Electron-Electron Interactions in Dis-ordered Systems , edited by A. L. Efros and M. Pollak,(North-Holland, Amsterdam, 1985).[27] S. Maekawa and H. Fukuyama, J. Phys. Soc. Jpn. ,2516 (1981).[28] P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. ,287 (1985).[29] B. L. Altshuler, A. G. Aronov and A. Y. Zuzin, SolidState Commun. , 137 (1982).[30] H. Kohler and E. Wuchner, Phys. Stat. Sol. B , 665(1975).[31] B. L. Altshuler and A. G. Aronov, in Electron-ElectronInteractions in Disordered Systems , edited by A. L. Efrosand M. Pollak (North-Holland, Amsterdam, 1985).[32] A. Sahnoune, J. O. Strom-Olsen, and H. E. FischerPhys. Rev. B , 10035 (1992).[33] A. D. Mirlin, private communications.[34] Qualitatively similar variation of α with V G was also re- ported in Ref. [8], in which the weak antilocalization sig-nal is however superimposed on fluctuating background of unknown origin. The non-Hall-bar device-geometryalso precluded accurate determination of αα