Bulk Fermi surface coexistence with Dirac surface state in Bi Se : a comparison of 2 3 photoemission and Shubnikov-de Haas measurements James G. Analytis,1,2 Jiun-Haw Chu,1,2 Yulin Chen,1,2 Felipe Corredor,1,2 Ross D. McDonald,3 Z. X. Shen,1,2 and Ian R. Fisher1,2 1Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA 2Geballe Laboratory for Advanced Materials and Department of Applied Physics, Stanford University, USA 0 3Los Alamos National Laboratory, Los Alamos, NM 87545, USA 1 ShubnikovdeHaas(SdH)oscillationsandAngleResolvedPhotoEmissionSpectroscopy(ARPES) 0 2 are used to probe the Fermi surface of single crystals of Bi2Se3. We find that SdH and ARPES probes quantitatively agree on measurements of the effective mass and bulk band dispersion. In n high carrier densitysamples, thetwo probes also agree in theexact position of theFermi level EF, a butforlowercarrierdensitysamplesdiscrepanciesemergeinthepositionofEF. Inparticular,SdH J revealsabulkthree-dimensionalFermisurfaceforsampleswithcarrierdensitiesaslowas1017cm−3. 2 We suggest a simple mechanism to explain these differences and discuss consequences for existing 2 and futuretransport studies of topological insulators. ] l l Recently, a new state of matter, known as a topolog- ject of careful and thorough research for much of the a h ical insulator, has been predicted to exist in a number 20th century. However, common to all the unambiguous - ofmaterials: Bi1−xSbx,Bi2Se3,Bi2Te3 andSb2Te3[1,2]. measurements of the Dirac cone is the use of surface- s e This state of matter is characterized by a full band gap sensitive probes. Only recently have transport measure- m in the bulk of the material, but with a gapless, dissipa- mentsemergedspecificallyinvestigatingthesurfacestate tionless surface state. The surface state is comprised of (Refs [9–11]), all of which note the dominance of the . t a counter-propagating spin states, which create a disper- bulk conductivity. It is thus of great interest to perform m sion of a single, massless Dirac cone that is protected by a coordinated study of these materials using both bulk - time-reversalsymmetry. The experimental realizationof transport experiments and surface sensitive ARPES ex- d this state could mean significant advances in spintronic periments. Herewereportresultsoftheseinvestigations. n devices, quantum computation and much more besides. The transport experiments reveal quantum oscillations o c As a resultthere has been greatexcitementin the last that indicate a bulk band structure and Fermi surface [ yearafter the discoveriesof variousARPES experiments volumethatmonotonicallychangewith doping. For car- [3–5] and more recently from scanning-tunneling mea- rier densities in the range ∼1019cm−3, the transport ex- 1 v surements [6–8] that such a state appears to exist in tractedbandstructure is in quantitative agreementwith 0 nature. Amidst this flurry of recent results, it is easy the bulk band structure determined by ARPES which 5 to forget that these same materials have been the sub- also observes the Dirac dispersion of the surface state. 0 The quantitative agreement between ARPES and SdH 4 provides additional support for the existence of novel . 1 band structure in these materials. For lower carrierden- 0 sity samples down to 1017cm−3 we observe SdH oscil- 0 lations which unambiguously pin the Fermi level in the 1 : bulk conduction band, with a high level of consistency v acrossallsamplesmeasuredfromthe samebatch. While i X ARPES places E near the SdH level for some samples, F r there are others from the same batch whose EF is found a to reside into the bulk gap. We discuss possible expla- nations for these discrepancies and the implications for transport studies of surface Dirac Fermions in samples near a metal-insulator transition. The material Bi Se can be grown without the intro- 2 3 duction of foreign dopants as either n or p type[12, 13] though is more commonly found as the former because FIG.1: (a)Temperaturedependenceoftwotypicalsamplesof the dominant defects tend to be Se vacancies. Quantum Bi2Se3 with carrier densities differing by two orders of mag- nitude. (b)Showsthecarrierdensityne,(c)theresistivityρ0 oscillatory phenomena, which provides evidence of bulk at T=2K and (d) the mobility for samples of different thick- metallicbehaviorhasbeenreportedbyKohleret al. [12] nesses. Each sample was a cleave from a parent sample, so onlowcarrierdensitysamplesandmorerecentlybyKul- that the surface area of each sample was kept constant. bachinskii [14] on high carrier density samples. Below a 2 which was linear in the low field limit was used to avoid mixingwithlongitudinalcomponents. Inadditiontothis precaution,signalfrompositiveandnegativefieldsweeps was subtracted to extract the odd (Hall) components of the signal, after which the carrier density is extracted in theusualway. ARPESmeasurementswereperformedat beam line 10.0.1 of the Advanced Light Source (ALS) at Lawrence Berkeley National Laboratory. Measurement pressurewaskept<3×1011Torr,anddatawererecorded by Scienta R4000 analyzers at 15K sample temperature. FIG. 2: (a) ARPES band dispersion on samples of Bi2Se3 The total convolved energy and angle resolutions were with carrier density 2.3×1019cm−3 (batch S4). (b) Due to 16meV and 0.2◦ (i.e. <0.007(˚A−1) or <0.012(1˚A−1) for the quantization of the energy spectrum into Landau levels photoelectronsgeneratedby48eVphotons),atwhichen- (LLs),oscillations appearinthemagnetoresistance knownas ergy the cross-section for both surface state and bulk SdH oscillations. The SdH oscillations here are for a sample bands is strong. taken from the same batch as in (a) at θ =0, corresponding to a oscillatory frequency of F=155T, consistent with EF ∼ In Figure 2 we show complimentary ARPES and SdH 160meV. ARPES and SdH are in good agreement for these data on samples from the same batch, with carrier den- high carrier density samples. sitydeterminedbytheHalleffectofn =2.3×1019cm−3. e The SdH reveals an anisotropic pocket of frequency 155 T,correspondingtoafillingofaround160meV(theband carrierdensityof7×1018cm−3,thebandstructureiswell structureisnotparabolicatthisfillingandsoweassume approximatedbyasingleparabolicband,makingthein- similar band structure parameters as Kohler et al.[12] terpretation of transport measurements transparent[14]. characterizingsimilarcarrierdensity samples ofBi Se ). 2 3 Two n-type samples with carrier densities differing by ARPES results on samples from the same batch, show two orders of magnitude are shown in Figure 1. For the theFermilevel150meVabovethebottomoftheconduc- low carrier density samples an upturn in the resistivity tionbandingoodquantitativeagreement. Similarly,the is seen, which levels off at sufficiently low temperature. effective mass (see below) extractedfromSdH is in good This behavior has been attributed to the presence of an quantitative agreement with that measured by ARPES. impuritybandwhosethermallyactivatedconductivity is In Figure 3 we illustrate angle dependent SdH data comparabletothebandconductivityuntilcarriersfreeze (a) taken at 1.8K, on a sample from batch S1 with a out at around 30K [13, 14]. This behavior is not ap- lower carrier density of 1017cm−3. The SdH signal re- parent in the higher carrier density materials, where the veals a pocket that is approximately an ellipsoid elon- band conductivity always dominates. Even though we gated about the c3 axis, consistent with measurements have reduced the carrier density by 2 orders of magni- by Kohler et al. from the 1970’s[12] on samples with tude,theresistivityincreasesbyone,suggestingthatthe similar carrier densities. For a two-dimensional pocket mobilityhasincreasedinthelowcarrierdensitysamples, expected from the surface state, quantum oscillations consistentwithpreviousmeasurements[14]. Thelowcar- should vary as 1/cosθ, where θ is the angle between the rier density samples are around an order of magnitude c3axisandthefielddirection,sothepresentobservations smaller than those of reported topological insulators in- must originate from a 3D Fermi surface existing in the cludingSndopedBi Te (n ∼1.7×1018cm−3)[4]orCa bulk. It has been shown by Kohler et al. and more re- 2 3 e dopedBi Se (n ∼5×1018cm−3)[5]andasa resultmay cently by Kulbachinskii et al. that the conduction band 2 3 e be better candidates in which to observe the transport structure for these low carrier densities is approximately properties dominated by the topological surface state. parabolic[12,14],andsothebandfillingcanbeestimated SinglecrystalsofBi Se havebeengrownbyslowcool- by E = ~2Ak,where A is the area of the Fermi surface 2 3 F 2πm∗ k ing a binary melt. Elemental Bi and Se were mixed in in Fourier space. We estimate the Fermi energy to be alumina crucibles in a molar ratio of 35:65 for batch S1 18meV above the bottom of the conduction band. (n =5×1017),34:66forbatchS2(n =3×1017),34:66 In Figure 3 (b) we show the derivative of the lon- e e for batch S3 (n = 2.3×1017), and 40:60 for batch S4 gitudinal magnetoresistance of a sample from batch S3 e (n = 2.3×1019). The mixtures were sealed in quartz and a fit of the entire data set using the usual Lifshitz- e ◦ ampules and raised to 750 C and cooled slowly to 550 Kosevich formalism, to extract the effective mass and ◦ C, then annealed for an extended period. Crystals can Dingle temperature T , with fit shown in (c). Fitting D be cleaved very easily perpendicular to the (0 0 1) axis. the entire data set, which is often more accurate than MeasurementsoftheresistivityandHalleffectweremea- tracing the amplitude of the Fourier transform, our fit ∗ sured in a 14T PPMS using a standard 4-probe contact yields m = 0.15m and T = 3.5K, for this frequency e D configurationandHallmeasurementswereperformedus- (F=14T). Similar data for samples from batch S4 give ∗ ing a 6-probe configuration. For the latter, only data m = 0.125m , T = 4K and F=155T. The mean free e D 3 FIG. 4: ARPES data on samples of Bi2Se3 from batch S1. The horizontal lines show the crossing of the Fermi level (E−EF =0)andtheDiraccrossing. (a)Bandstructuremea- suredbyARPESresultsonsamplesfrombatchS1showedthe Fermi level near the SdH level of ∼15meV from the bottom oftheconductionband. (b)Measurementonanothersample frombatchS1showedtheFermilevelingap. Someothersam- FIG. 3: (a) Magnetotransport for samples from S1. As the ples from S2 and S3 also showed the Fermi level in the bulk angleissweptthefrequencyoftheoscillationvariesaccording gap. The variation might be due to the lower carrier density to the topology of the Fermi surface. For a two-dimensional ofthesesamples, and thesurface bandstructureismoresus- pockettheexpecteddependenceis 1/cosθ (shown ingreen in ceptible to small amounts of surface contamination. c) Band the inset). The observed angle dependence is clear evidence structureofasamplealsofrombatchS1whichwascleavedin for a closed ellipsoidal Fermi surface pocket, similar to that atmosphere and exposed for 10s, showing significant n-type observedbyKohleret al.[12]. SimilarSdHdatawasgathered dopingwith large bulk conduction band pocket. on batch S2 and S3 on a number of samples. Samples from batch S3 showing the temperature dependenceof the deriva- tive of the SdH signal in (b) and a fit to the data shown in effectivemassandotherbandparametersonthelowcar- (c) from which the effective mass, Dingle temperature and rier density samples, the discrepancy in the position of oscillatory frequency can be extracted. theFermilevelrequiresexplanation. Suchdifferencescan occurforanumberofreasons,forexampleduetosample path is calculated using the orbitally averaged velocity variation within a batch, or perhaps due to variation in andscatteringtimeextractedfromT yieldsl ∼60nm the exposure of cleaved surfaces before a photoemission D S3 andl ∼220nm. Thisdataiswhollyconsistentwiththe measurement. However,itshouldbe notedthatthe SdH S4 very complete SdH studies of Kohler et al.[12, 13] and frequency does not appear to vary significantly within a more recently by Kulbachinskii et al. [14]. In addition, batchforupto20samplesmeasuredinthepresentstudy thedatawasreproducedwithhighconsistencyonanum- and so the former seems an unlikely scenario. Another berofsamplesfromthesamebatch,andevenonsamples reason for the discrepancy may be that atmospheric ex- from different batches with similar growth parameters. posureoftransportsampleshascontaminatedthemwith ARPES data on samples from the same batches as an n-type dopant causing them to appear bulk n-type. those shown in Figure 3, determining the effective mass Figure 4 (c) illustrates photoemission data for a sam- as m∗ −0.13 in very good quantitative agreement with ple cleaved in air. The Dirac cone of the surface state SdH. However,the exact placement of the Fermi level in remains robust and the bulk conduction band appears the band structure reveals some disagreement. In Fig- partially occupied. Such doping may lead to a 3D Fermi ure 4 we illustrate photoemission data for two separate surface pocket appearing in SdH oscillations if the con- samplesfrombatchS1. In(a)the Fermilevelisnearthe taminationisdeepenoughandallowsforsufficientlylong bottom of the conduction band in agreement with SdH, mean free paths. whilein(b)itisinthegap(about60meVbelowthecon- To investigate this possibility further we measure the duction band), crossing the Dirac cone with apparently thickness dependence of the transport by systematically no bulk contribution. A number of samples from similar thinning a single sample. Cleaving was achieved with batches,suchasbatchS2andS3,alsohaveshownsimilar tape,keepingthesurfaceareaoftheresultingsamplesrel- variationinE crossingthegaponsomesamples. While atively constant and allowing direct comparison of data F E determined by photoemission appears to show some sets of each cleave. Though the samples are vulnerable F variation, it is important to note that the measured E to deformations between cleaves, only data from mirror- F from ARPES is either near or below the SdH E . like flat samples is presented. In most cases these sam- F Despite the good agreement of SdH and ARPES on ples still exhibited quantum oscillatoryphenomena, con- high carrier density samples, and the agreement of the firming the high quality of the cleaved samples. Figure 4 study,suchpropertieshavenotbeenobserved. Recently, Aharanov-Bohm and universal conductance fluctuations havebeenobservedwhichmaybeduetothesurfacestate [10,11], buteveninthese casesthe conductanceappears bulk at the temperatures considered. Conventionally, suchintrinsicallydopedmaterialscanbecome‘insulators’ byeitheraMott-likeoranAndersontransition. Thefirst can occur when the Bohr radius a = κ~2/m∗e2 falls B below the Thomas-Fermi screening length λ , so that TF wavefunctionscannotoverlap. Thiscanbe estimatedus- ingλ2 =κǫ /(2πe2g(E )), whereg(E )is thenumber TF 0 F F of states per unit volume per unit energy, estimated by FIG.5: (a)Aschematicrepresentation ofthebandstructure Middendorff et al.[20]. In the present case, the large κ seenbyARPES(solid)redhorizontallinedenotingtheFermi and small m∗ tend to make a very large. For the low- B level as seen by SdH. (b) We infer band bending of about est carrier density samples investigated here a ∼ 3nm 60 meV at the surface from a comparison of ARPES and B and λ ∼ 4nm, which places is this material on the quantumoscillations. TF metal-insulator boundary. The carrier density can also be reduced by introducing foreign dopants which ‘drain’ 1 (b-d) shows a summary of the low-temperature car- the excess carriers and pin the chemical potential µ in rier density, resistivity and mobility. Within our error the gap. For hydrogenic like impurities this can be very bars, each quantity seems to vary weakly down to 3µm effective,but in the presentmaterialsimpurity bands of- inthickness. Althoughtheresidualresistivityandcarrier ten form instead. At high enough impurity densities the density varies slightly (possibly from disorder related to carriers may become Anderson localized. Such samples slight sample deformation, despite the precautions men- are characterizedby a high carrierdensity with very low tioned above), the mobility remains almost constant as mobility, leading to a negative gradient in the tempera- a function of thickness at µ ∼ 1m2/Vs. In summary, ture dependence of the resistivity. This may be the case the transport is insensitive to the thickness, suggesting for example in BixSbyPbzSe3 which has ρ ∼ 30mΩcm that the SdH oscillationsare nota consequence ofatmo- yeta carrierdensity ne ∼5×1018cm−3[21]. An Anderson spheric contamination, but originate from the intrinsic insulatorisgenerallybadnewsfortopologicalinsulators, band conductivity of the bulk. because even though at zero temperature the bulk con- A final scenario for the discrepancy is that the band ductivity σ =0,at finite temperature the transportmay structureisdistortednearthesurfaceduetospace-charge remain dominated by bulk hopping mechanisms. accumulation. This is known to occur in many semicon- In conclusion, the present study reveals substantial ductors, such as InSb or CdTe[15, 16], whereby the bulk agreementbetweentransportandARPESmeasurements bandstructure bends as the surfaceis approached. Typ- of the Fermiology of Bi2Se3, in particular for samples ically, such bending occurs overa surface depletion layer with large carrier densities. However, for samples with z , which canbe calculatedby solving the Poissonequa- carrier densities approaching 1017cm−3, discrepancies d tion to yield z2 = κǫ ∆V/en [17], where κ is the DC emerge as to the exact position of the Fermi level. We d 0 e dielectric permittivity (estimated from these samples as have confirmed the bulk nature of the transport by the ∼ 113 [18])and ∆V is the difference in energy between thickness dependence of the Hall effect, resistivity and the surface and bulk state. We estimate zS1 ∼40nm for mobility. Furthermore SdH data is highly consistent be- d the low carrier density samples and zS4 ∼ 2nm for the tween different samples from the same batch. Interest- d highcarrierdensitysamples. Aschematicrepresentation ingly, the carrier densities measured here are an order of the band bending is shown in Figure 5. The present of magnitude smaller than those of the topological in- argument suggests that discrepancies between ARPES sulators recently reported in the literature[3–5, 10, 11]. and SdH can be explained, even expected for low carrier ARPES and STM have been invaluable tools in reveal- density samples. In addition, due to the small value of ing the physics of topological insulators, providing com- n, these samples are likely more susceptible to a small pellingevidenceforthepresenceofthetopologicallypro- amounts of surface contamination, especially if the un- tected Dirac surface state. The present results should contaminated surface E is in the gap (as illustrated in stimulatefurthertheoreticalworkastotheconsequences F by ARPES on atmosphere exposed samples). This may of the coexistence of bulk and surface states in a single help explain why there is some variability in the Fermi sample as well as innovation in novel ways to fabricate level of ARPES data but not in the SdH data. these materials so the bulk state can be cleanly elimi- Much theoretical work has emerged on the dra- nated. matic consequences of the surface state on transport We would like to thank D. Goldhaber-Gordon, J. R. properties[19]. Yet over several decades of experimental Williams,X.Qi,S.-CZhang,K.Lai,J.Koralek,J.Oren- 5 stein and T. Geballe for useful discussions. Work was [9] A. Taskin and Y. Ando,Physical Review B 80 (2009). supported by the U.S. DOE, Office of Basic Energy Sci- [10] H. Peng, K. Lai, D. Kong, S. Meister, Y. Chen, ences under contract DE-AC02-76SF00515. X. Qi, S. Zhang, Z. Shen, and Y. Cui, Nat Mater10.1038/nmat2609 advance online publication (2009). [11] J. G. Checkelsky, Y. S. Hor, M. H. Liu, D. X. Qu, R.J.Cava,andN.P.Ong,PhysicalReviewLetters103, 246601 (2009). [1] J. C. Y. Teo, L. Fu, and C. L. Kane, Physical Review [12] H. Kohler, Physica Status Solidi (b) 58, 91 (1973). B(CondensedMatterandMaterialsPhysics)78,045426 [13] H.KohlerandA.Fabbicius,physicastatussolidi(b)71, (2008). 487 (1975). [2] H.Zhang, C. Liu,X.Qi,X.Dai, Z.Fang,and S.Zhang, [14] V. A. Kulbachinskii, N. Miura, H. Nakagawa, H. Ari- Nat Phys5, 438 (2009), ISSN 1745-2473. moto, T. Ikaida, P. Lostak, and C. Drasar, Physical Re- [3] D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, view B 59, 15733 (1999). J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, [15] P. D. C. King, T. D. Veal, M. J. Lowe, and C. F. Mc- et al., Science 323, 919 (2009). Conville,JournalofAppliedPhysics104,083709(2008). [4] Y. L. Chen, J. G. Analytis, J. Chu, Z. K. Liu, S. Mo, [16] R. K. Swank,Physical Review 153, 844 (1967). X.L. Qi, H. J. Zhang, D.H. Lu, X.Dai, Z. Fang, et al., [17] W. Monch, Semiconductor surfaces and interfaces Science 325, 178 (2009). (Springer, 2001), ISBN 3540679022, 9783540679028. [5] D. Hsieh, Y. Xia, D. Qian, L. Wray,J. H. Dil, F. Meier, [18] U. R.O. Madelung and M. Schulz, in Non-Tetrahedrally J.Osterwalder,L.Patthey,J.G.Checkelsky,N.P.Ong, Bonded Elements and Binary Compounds I (1998), pp. et al., Nature 460, 1101 (2009), ISSN0028-0836. 1–12. [6] Z. Alpichshev, J. G. Analytis, J. H. Chu, I. R. Fisher, [19] D. Lee, Physical ReviewLetters 103, 196804 (2009). Y. L. Chen, Z. X. Shen, A. Fang, and A. Kapitulnik, [20] A. Middendorff, H. Kohler, and G. Landwehr, Physica 0908.0371 (2009). Status Solidi (b) 57, 203 (1973). [7] P. Roushan, J. Seo, C. V. Parker, Y. S. Hor, D. Hsieh, [21] J. Kasparova, C. Drasar, A. Krejcova, L. Benes, D. Qian, A. Richardella, M. Z. Hasan, R. J. Cava, and P. Lost’ak, W. Chen, Z. Zhou, and C. Uher, Journal of A.Yazdani, Nature460, 1106 (2009), ISSN0028-0836. Applied Physics 97, 103720 (2005). [8] K. K. Gomes, W. Ko, W. Mar, Y. Chen, Z. Shen, and H.C. Manoharan, 0909.0921 (2009).