Sum-Rule Constraints on the Surface State Conductance of Topological InsulatorsK.W. Post, 1,* B.C. Chapler, 1 M.K. Liu, 1 J.S. Wu, 1 H.T. Stinson, 1 M.D. Goldflam, 1 A.R. Richardella, 2 J.S. Lee, 2A.A. Reijnders, 3 K.S. Burch, 4 M.M. Fogler, 1 N. Samarth, 2 and D.N. Basov 11 Physics Department, University of California-San Diego, La Jolla, California 92093, USA2 Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA3 Department of Physics & Institute for Optical Sciences, University of Toronto, Toronto, Ontario M5S 1A7, Canada4 Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA(Received 3 June 2014; revised manuscript received 5 July 2015; published 11 September 2015)We report the Drude oscillator strength D and the magnitude of the bulk band gap E g of the epitaxiallygrown, topological insulator ðBi;SbÞ 2 Te 3 . The magnitude of E g , in conjunction with the modelindependent f-sum rule, allows us to establish an upper bound for the magnitude of D expected in atypical Dirac-like system composed of linear bands. The experimentally observed D is found to be at orbelow this theoretical upper bound, demonstrating the effectiveness of alloying in eliminating bulk chargecarriers. Moreover, direct comparison of the measured D to magnetoresistance measurements of the samesample supports assignment of the observed low-energy conduction to topological surface states.DOI: 10.1103/PhysRevLett.115.116804 PACS numbers: 73.50.-h, 73.25.+i, 78.30.-jThe predictionanddiscoveryofDirac-likegapless surfacestates(SSs)(seeFig.1)attheinterfacebetweenatopologicalinsulator (TI) and a trivial insulator have vaulted TIs to thevanguard of condensed matter physics [1]. Surface probessuch as angle resolved photoemission spectroscopy andscanning tunneling spectroscopy have been extremely suc-cessful in verifying and discovering novel phenomenaassociated with the SSs (e.g., Refs. [2–5]). However,progress in the field has been plagued by native defects,resulting in significant concentrations of bulk charge car-riers. These bulk dopants inhibit isolation and utilization ofSS phenomena [6]. In order to realize many of the novelscientific and technological advances that could blossomfrom the unique electronic, spin, and optical properties ofSSs in TIs, it is paramount to eliminate the bulk dopants.Two archetypal strong TIs (topological invariant ν 0 ¼ 1[1]) that are known to suffer from materials issues related tobulk dopants are Bi 2 Te 3 and Sb 2 Te 3 . However, epitaxialfilmsofBi 2 Te 3 canben-typebulkconductors[6,7],whereasSb 2 Te 3 films are p-type bulk conductors [8]. Based on thisobservation, TI materials have been grown where the ratioBi∶Sb in ðBi;SbÞ 2 Te 3 (BST) is tuned to produce a com-pensated material with bulk insulating properties [9–11].Here, we prove, using optical spectroscopy, the acuteeffectiveness of alloying in reducing or eliminating bulkcharge carriers. The advantage of our optical experiments isthat they give direct access to the frequency dependentelectrodynamic response of free carriers in a metal via theDrude peak. In our ðBi;SbÞ 2 Te 3 film we find the Drudeoscillator strength D sufficiently low as to lie at or below theupper bound that is theoretically anticipated for an isolatedDirac SS Drude response (i.e., a Drude response with nobulk contribution) (see Table I). These infrared data arecomplemented bythecharge carrier density n and mobility μfrom Hall effect measurements, as well as the magneto-resistance, which are all consistent with low-energy con-duction arising from the topological SS.We begin by describing the optical response of Diracelectrons in a strong TI, neglecting potential interband bulk→ SS transitions (or vice versa) [15]. The simplest Diracelectron system is composed of linear bands (LB), withoutspin or valley degeneracy, where the dispersion is givenby EðkÞ ¼ kv F , as illustrated schematically in Figs. 1(a) and1(b). In this system, the total conductance G LBtotis the sum ofan intraband G LBintra ðωÞ and an interband GLBinter ðωÞ compo-nent. When the Fermi energy E F is at the Dirac point, theorypredicts that SS interband transitions give rise to a frequencyindependent conductance of G LBinter¼18 ðπe2 =hÞ[16].However, the area of the Fermi surface is zero, yieldingG LBintra¼ 0. When E F is shifted away from the Dirac point,empty states suppress interband transitions at energies below2jE F j, as illustrated in Fig. 1(b), resulting in G LBinter ðωÞ ¼ 0for ω < 2jE F j. Importantly, the f-sum rule [17] demandsthat the total spectral weight of the Dirac electronsR∞0G LBtot dω be conserved [18,19]. Therefore, the loss ofspectral weight in G LBinter ðωÞ must be compensated by a gainin G LBintra ðωÞ, leading toG LBinter 2jE F j ¼Z∞0G LBintra ?dω:ð1ÞIt is customary to express the relationship between intrabandspectral weight and Drude oscillator strength D sLBasZ∞0G intra;LB ðωÞ?dω ¼π30ΩD sLB ;ð2Þwhich implies the simple relationship between E F and D sLBPRL 115, 116804 (2015)PHYSICAL REVIEW LETTERSweek ending11 SEPTEMBER 20150031-9007=15=115(11)=116804(5) 116804-1 © 2015 American Physical Society
