Strong surface scattering in ultrahigh mobility Bi2Se3 topological insulator crystals
Nicholas P. Butch, Kevin Kirshenbaum, Paul Syers, Andrei B. Sushkov, Gregory S. Jenkins, H. Dennis Drew, Johnpierre Paglione
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Strong surface scattering in ultrahigh mobility Bi Se topological insulator crystals N. P. Butch, ∗ K. Kirshenbaum, P. Syers, A. B. Sushkov, G. S. Jenkins, H. D. Drew, and J. Paglione
Center for Nanophysics and Advanced Materials, Department of Physics,University of Maryland, College Park, MD 20742 (Dated: October 8, 2018)While evidence of a topologically nontrivial surface state has been identified in surface-sensitivemeasurements of Bi Se , a significant experimental concern is that no signatures have been observedin bulk transport. In a search for such states, nominally undoped single crystals of Bi Se withcarrier densities approaching 10 cm − and very high mobilities exceeding 2 m V − s − havebeen studied. A comprehensive analysis of Shubnikov de Haas oscillations, Hall effect, and opticalreflectivity indicates that the measured electrical transport can be attributed solely to bulk states,even at 50 mK at low Landau level filling factor, and in the quantum limit. The absence of asignificant surface contribution to bulk conduction demonstrates that even in very clean samples,the surface mobility is lower than that of the bulk, despite its topological protection. PACS numbers: 71.18.+y, 72.20.My, 78.30.-j, 78.20.Ci
Topological insulators are bulk insulators that featurechiral Dirac cones on their surface. This surface state,which is protected by time-reversal symmetry, is of fun-damental interest and may have potential for spintron-ics and quantum computation applications. A smallgroup of semimetallic stoichiometric chalcogenides havebeen theoretically shown to have the properties of three-dimensional topological insulators: Bi Se , Sb Te , andBi Te [1, 2]. The chiral surface state has been investi-gated by surface probes, notably angle-resolved photoe-mission spectroscopy (ARPES) [2–4] and scanning tun-neling microscopy (STM) [5, 6] and similar studies existof the alloy Bi − x Sb x [7, 8]. The observation of chiralDirac cones and forbidden backscattering in these exper-iments has generated a great deal of excitement.In principle, electrical conduction in topological insula-tors should occur only at the surface, but in practice thesestoichiometric materials have been known for decades tobe low carrier density metals. Bi Se is expected to havea 300 meV direct bandgap at zone center [1, 2], yet is al-most always n -type, long believed the result of charged Sevacancies. It remains an open question whether a bulk in-sulating state can be achieved in stoichiometric undopedBi Se . This problem has been recently sidestepped bycounter-doping with Ca [2, 9, 10], although this proce-dure introduces further defects. Over a range of carrierdensities, ARPES data show one conduction band at zonecenter [2, 4], and bulk Shubnikov de Haas measurementsindicate that the Fermi surface is ellipsoidal [11].There have been several recent studies of bulk trans-port in Bi Se . A thorough study of angle dependentShubnikov de Haas (SdH) oscillations on a sample witha moderate carrier density has mapped the anisotropyof the Fermi surface [12]. A comparative investigationdemonstrated that while ARPES may identify a Fermilevel in the band gap, SdH oscillations in similar samplesalways show metallic behavior, suggesting that the Fermilevel may vary between bulk and surface in Bi Se [13]. This raises questions about the correspondence betweensurface and bulk studies on the same material. While itis possible to tune the Fermi level of Bi Se into the gapvia Ca doping, the resulting samples show unusual mag-netoresistance fluctuations, but no SdH oscillations fromthe surface state [10]. In contrast, field-dependent oscilla-tions ascribed to the surface state have been observed intunneling spectra measured on Bi Se thin films [14, 15]and Aharonov-Bohm interferometry of the surface stateis reported in Bi Se nanoribbons [16]. Because SdH os-cillations attributable to surface states were identified inintrinsically disordered Bi − x Sb x alloys [17], it is intrigu-ing that analogous bulk transport phenomena have notbeen seen in stoichiometric Bi Se .In order to look for signatures of the novel surfacestate in bulk nominally undoped Bi Se , single crystalswere synthesized over a wide range of carrier concentra-tions. Their longitudinal and transverse electrical resis-tivity were systematically characterized as a function offield and temperature, and infrared reflection and trans-mission were studied. This investigation confirms thatall measured transport properties can be ascribed to bulkelectron-like carriers, despite the samples having the low-est reported bulk carrier densities and highest mobili-ties. Even at low Landau level filling factor and in thequantum limit, no transport signature of surface statesis apparent, placing strong constraints on the mobility ofsurface carriers in these high quality samples.Single crystals of Bi Se were prepared by meltinghigh purity bismuth (6N) and selenium (5N) in sealedquartz ampoules. Multiple batches were prepared withvarying bismuth/selenium ratios and heating conditions,which were responsible for variations in the carrier con-centrations of the samples. Similar trends were reporteddecades ago [18] and more recently [13]. The resul-tant crystals cleaved easily perpendicular to the c trig-onal axis. Four-probe measurements of longitudinal andtransverse electrical transport between 1.8 K and 300 Kwere conducted using a Quantum Design 14 T PPMS.The current flowed in the plane perpendicular to c inboth longitudinal and transverse geometries, and themagnetic field H was always applied perpendicular toapplied current. Electrical transport measurements atlower temperatures were performed in a dilution refriger-ator with a rotating sample stage and 15/17 T magnet.Optical measurements were performed using a BomemDA3 FTIR spectrometer. Carrier density n was deter-mined by measurements of Hall effect, SdH oscillations,and fits to optical reflectivity data, which always yieldedidentical results within experimental uncertainty.The temperature dependence of the longitudinal elec-trical resistivity ρ ( T ) is summarized in Figure 1a. Thevalues of n in these samples were estimated via analy-sis of the SdH oscillations in ρ ( H ) and approximatingthe Fermi surface as spherical, which is sufficient for pur-poses of comparison at low n [11]. In sample vi , no SdHoscillations are observed up to 14 T, and the estimated n = 5 × cm − is based on Hall measurements on sam-ples from the same batch (Fig. 1b). For n > cm − ,the ρ ( T ) data reflect good metallic conductivity and gen-erally, as n decreases, the samples become more resistive.For n < cm − , a negative slope develops above250 K, which reflects a crossover to activated behaviorat T >
300 K [19]. For n < cm − , the ρ ( T ) dataalso develop a shallow local minimum at 30 K, which isassociated with a relatively small upturn that saturatesas T → T dependence of the Hall carrier den-sities n H = ( R H e ) − is shown for a range of samples.Values of the Hall coefficient R H were determined vialinear fits to symmetrized transverse magnetoresistancedata, which are linear to 5 T over the entire T range,except for at lowest n where linearity extends to 2 T atlow T . Generally, n H ( T ) has a gentle T -dependence, con-sistent with a gradual crossover from extrinsic to intrin-sic conduction ( E g >
100 meV) [19]. However, metallic ρ ( T ) (Fig. 1a) is always observed, indicating that phononscattering dominates any increase in conductivity dueto increasing carrier number for most of the T range.A detailed comparison of ρ ( T ) (sample iii ) and n H ( T )between two samples from the same batch is shown inFig. 2a. These data highlight two regimes: below 30 K,where n H is constant and ρ ( T ) exhibits an upturn, andbetween 150 and 250 K, where there is a change in cur-vature in n H ( T ) and ρ ( T ). The magnitude of these lat-ter features is greatest in samples with n < cm − .The two T ranges are accentuated in a plot of the Hallmobility µ ( T ) = R H ( T ) /ρ ( T ) (Fig. 2b), highlighting thesubstantial low- T mobility µ > V − s − in thesesamples, which is the highest value reported for Bi Se and corresponds to a mean free path longer than 300 nm.A comparison to µ ( T ) of batches with n ∼ cm − and n ∼ cm − indicates that µ does not simply in-crease with decreasing n , although at low n it still exceeds r ( m W c m ) T (K) a) iiiiiiivvvi n H ( c m - ) T (K) b) Bi Se FIG. 1. (Color online) a) Comparison of the electrical re-sistivity ρ ( T ) between samples of Bi Se . Carrier densities n (cm − ) are estimated from SdH oscillations: i × , ii . × , iii . × , iv . × , v . × , vi ∼ .In samples with n < cm − , a shallow minimum developsat 30 K. b) Comparison of Hall carrier density n H betweenvarious samples. Low temperature values span 3 orders ofmagnitude. V − s − .In Fig. 2c, ρ ( H ) of sample iii is shown. In additionto the dramatic anisotropy, oscillations are readily ap-parent. The non-oscillatory part is sufficiently well de-scribed by quadratic polynomial or weak power law func-tions. In contrast, sample vi exhibits no oscillations.The T -dependence of SdH oscillations is shown in Fig. 3in panels a) and b) for two different field orientations(c.f. Fig. 2c). The SdH oscillations have a frequency f SdH = 20 T when H k c and f SdH = 25 T when H ⊥ c , which correspond to an ellipsoidal Fermi sur-face with n = 6 . × cm − . A Lifshitz-Kosevichfit to the T -dependence of oscillation amplitude yieldsmasses m k = 0 . m e and m ⊥ = 0 . m e , where m e is the bare electron mass, consistent with otherreports [12, 13]. In order to check whether loweringthe temperature would uncover a second frequency ofSdH oscillations arising from the surface state, measure-ments were performed at 50 mK on two samples with n H = 4 − × cm − from the same batch as sample iv . The field angle dependence of both longitudinal andtransverse resistance are shown in Fig. 3 panels c) andd). Figure 3c shows oscillations in the transverse magne-toresistance R xy ( H ), whose frequencies range from 20 Twith H k c to 27 T with H ⊥ c ( n = 7 × cm − ). Bi Se r ( m W c m ) n H ( c m - ) T (K)a) 00.511.522.50 100 200 300 m ( m V - s - ) T (K) b) r ( m W c m ) H (T)
H || c H | c c) T = 1.8 K ( r / 10)10 FIG. 2. (Color online) a) A comparison of electrical resistivity ρ ( T ) ( iii from Fig. 1) and Hall carrier density n H ( T ) in twosamples from the same batch. The minimum in ρ ( T ) coincideswith the saturation of n H ( T ) below 30 K. Despite increasing n H , ρ increases with T up to 300 K. b) The calculated mobilityexceeds 2 m V − s − at low T (labeled 10 ). The mobilitiesof samples with lower and higher n are compared. c) Fieldorientation dependence of ρ at 1.8 K. Compare to ρ ( H ) ofsample vi (rescaled by factor of 10), which exhibits no SdHoscillations. As expected, the magnitude of the effective Hall signaldiminishes as H rotates into the plane and the evolutionof the oscillation frequency reflects the expected Fermisurface anisotropy. The longitudinal SdH oscillations areshown in Fig. 3d. The most prominent oscillations, with f SdH = 15 T ( n = 3 × cm − ) occur with H ⊥ c .In this field orientation, the electrons are almost all inthe first Landau level. Weak oscillations with H k c have f SdH = 11 T, yielding an anisotropy close to thatof the other two samples. The lack of bulk SdH oscilla-tions in sample vi (Fig. 2c) having even lower n is dueto its low estimated f SdH = 4 T, meaning that most ofthe accessible field range is in the bulk quantum limit.Samples were also characterized by infrared reflectionand transmission, which confirm the estimates of n basedon Hall and SdH data. Figure 4a shows an experimentalreflectivity spectrum of a sample from the same batch assample iv . Data were taken at 6 K and are presentedwith a fit to a sum of Lorentz oscillators to model thecomplex dielectric function. The fit shown in Fig. 4a in-cludes only bulk carriers as a Drude-type term togetherwith two low frequency phonons. A strong low frequencyphonon is centered at 67 cm − with a plasma frequencyof 640 cm − and relaxation rate of 5 cm − and a sec- -0.500.5 Dr xx ( m W c m ) a) H || c -0.100.1 D R xy ( W ) c) T = 50 mK
H || c -0.100.10.05 0.1 0.15 0.2 0.25 Dr xx ( m W c m ) b) H | c -1010.05 0.1 0.15 0.2 0.25 D R xx ( W ) d) T = 50 mK Bi Se H | c H -1 (T -1 ) FIG. 3. (Color online) Temperature dependence of SdH os-cillations with a) H k c trigonal axis and b) H ⊥ c . Os-cillation frequencies f SdH = 20 T and 25 T, respectively, andthe mass m SdH ≈ . m e . SdH oscillations from two differ-ent samples at 50 mK in the c) transverse and d) longitudinalMR. The largest amplitude and smallest frequency in c) isobserved with H k c and the frequency ranges from 20 Tto 27 T. In d) the largest amplitude is observed with H ⊥ c and f SdH = 15 T. Only one SdH frequency is observed in eachmeasurement. ond weaker phonon is centered second at 124 cm − witha plasma frequency of 76 cm − and a width of 2 cm − .The calculated optical conductivity is shown in Fig. 4btogether with two dc values (sample iv ) for rough com-parison. This yields a bare plasma frequency of free carri-ers to be 382 cm − , corresponding to n = 2 . × cm − for 0 . m e . The effective scattering rate γ = 8 cm − ( τ = 0 . γ = 100 cm − ) is detected.For small n , the competition between negative ε of thefree carriers and the positive ε from the phonons givesrise to an additional low frequency plasma edge ( ε = 0)that leads to a transmission window near 30 cm − . Thestrong bismuth-dominated low frequency phonon pro-duces a large static dielectric constant ε (0) ≈
100 forBi Se . This implies a large reduction in the Coulombpotential and thus the scattering rate from ionized impu-rities, which can account for the small scattering rate ofthe bulk carriers and the good metallic behavior of this FIG. 4. (Color online) Spectra of a) reflectivity, b) opticalconductivity, and c) dielectric constant. Curves b) and c)were calculated from the fit to the reflectivity data in a). Forcomparison, the blue circle and red square in b) are dc valuesfrom sample iv for 6 K and room temperature, respectively. doped semiconducting material, and protecting againstlocalization at low n . The Coulomb mediated electron-electron interaction is also reduced [20] and, along withthe presence of low frequency optical phonons, impliesa dominance of the electron-phonon interaction both forthe carrier mobility and for the electron-electron interac-tion.ARPES measurements show that surface states inBi Se always have a larger Fermi surface than the bulkand that the areal difference increases as the chemical po-tential decreases [4]. Quantum oscillations arising fromthe surface should have an easily identifiable higher fre-quency of about 300 T ( k F = 0 . − ) when the chemi-cal potential is near the bottom of the conduction band.However, no higher frequency oscillation is ever observedin our measurements, even in the quantum limit, whereno bulk SdH oscillations exist to obscure surface signal.As the existence of the surface states is not in doubt, theabsence of their transport signature can only arise fromvery high scattering rates. A conservative assumptionthat the chemical potential is the same at the surfaceand bulk, consistent with ARPES surface aging data [4],yields n surf ≈ cm − . In our samples, the bulk arealcarrier density is about 100 times that of the surface,while the peak-to-noise ratio in the Fourier transform ofthe SdH oscillations is greater than 500, so the surfacemobility has to be at least 5 times worse than the bulkmobility for SdH oscillations to be below our detectionthreshold. We calculate scattering times τ bulk = 2 ps and τ surf < . τ surf < .
05 ps determined from ourtransmission measurements. These estimates indeed in-dicate substantial surface scattering, yet are entirely con- sistent with ARPES data, which show significant surfaceband broadening, with τ = 0 .
02 ps [2] or 0.04 ps [21] evenfor samples cleaved in high vacuum.It is clear that topological protection does not guaran-tee high surface mobility, so increasing the surface con-duction will require considerable effort. Reducing samplethickness is one possibility, but bulk transport proper-ties appear to be insensitive to sample thickness downto 5 µ m [13]. Even in 50 nm thick nanoribbons, the ob-served f SdH = 70 T and 110 T [16] are well below 300 Tand are not likely due to surface states. Relative sur-face conduction may alternatively be improved by fur-ther reducing bulk n . Attempts to do so via chemicaldoping yield no surface SdH oscillations [10], possiblydue to even further decreased mobility. On the otherhand, doping may improve the surface signal if it pref-erentially decreases the bulk mobility without reducingsurface mobility, which may be possible considering thealready high surface scattering rate. A cleaner approachis to instead preferentially improve the surface mobilityvia controlled surface preparation. Such a task will bechallenging, as it is difficult to improve upon the vac-uum cleaving employed in ARPES studies, which yieldsbroad surface state linewidths. Nonetheless, high surfacemobility is a prerequisite for precision experiments likeoptical Kerr rotation and future applications potential oftopological insulators.We note a recent report of the observation of surfaceSdH oscillations in pulsed fields [22]. The cleaved sur-faces of the Sb-doped Bi Se were minimally exposed, yetthe requirement of high magnetic fields points to strongsurface scattering even under controlled conditions.We thank M. S. Fuhrer, S. K. Goh, R. M. Lutchyn,and M. L. Sutherland for helpful discussions. This workwas supported in part by the National Science Founda-tion MRSEC under Grant No. DMR-0520471. NPB issupported by CNAM. ∗ [email protected][1] H. Zhang, et al ., Nat. Phys. , 438 (2009).[2] Y. Xia, et al ., Nat. Phys. , 398 (2009).[3] Y. L. Chen, et al ., Science , 178 (2009).[4] D. Hsieh, et al ., Nature (London) , 1101 (2009).[5] T. Zhang, et al ., Phys. Rev. Lett. , 266803 (2009).[6] Z. Alpichshev, et al ., arXiv:0908.0371v2 (unpublished).[7] D. Hsieh, et al ., Science , 919 (2009).[8] P. Roushan, et al ., Nature (London) , 1106 (2009).[9] Y. S. Hor, et al ., Phys. Rev. B , 195208 (2009).[10] J. G. Checkelsky, et al ., Phys. Rev. Lett. , 246601(2009).[11] V. A. Kulbachinskii, et al ., Phys. Rev. B , 15733(1999).[12] K. Eto, Z. Ren, A. A. Taskin, K. Segawa, and Y. Ando,Phys. Rev. B , 195309 (2010).[13] J. G. Analytis, et al ., Phys. Rev. B , 205407 (2010). [14] P. Cheng, et al ., arXiv:1001.3220 (unpublished).[15] T. Hanaguri, K. Igarashi, M. Kawamura, H. Takagi, andT. Sasagawa, arXiv:1003.0100v1 (unpublished).[16] H. Peng, et al ., Nat. Mat. , 225 (2010).[17] A. A. Taskin, K. Segawa, and Yoichi Ando,arXiv:1001.1607v1 (unpublished).[18] G. R. Hyde, H. A. Beale, I. L. Spain, and J. A. Woollam, J. Phys. Chem. Solids , 1719 (1974).[19] H. K¨ohler and A. Fabricius, Phys. Stat. Sol. b71 , 487(1975).[20] S. Baldwin and H. D. Drew, Phys. Rev. Lett. , 2063(1980).[21] S. R. Park, et al ., Phys. Rev. B , 041405(R) (2010).[22] J. G. Analytis, et alet al