DOI: 10.1126/science.1242247, 1490 (2013);342 Science et al.H. Murakawa

Detection of Berry's Phase in a Bulk Rashba Semiconductor

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associated with some additional density in the EM maps(fig. S4). This density is partially discontinuous for themobile glycan components, providing a spiky appearance

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Acknowledgments: We thank Y. Cheng and X. Li for providingraw frame alignment scripts prior to publication, R. Hendersonfor making the makestack_HRnoise.exe program availablefor use to assess the overfitting of the EM data, J. Korzun fortechnical assistance, J.-C. Ducom (TSRI) for support withcomputational resources, and C. R. King and W. Koff forsupport and encouragement. Supported by NIH grants HIVRADP01 AI82362 ( J.P.M., I.A.W., and A.B.W.) and R01 AI36082(I.A.W.); the International AIDS Vaccine Initiative NeutralizingAntibody Consortium (D.R.B., J.P.M., I.A.W., A.B.W.); ScrippsCHAVI-ID (UM1 AI100663) (D.R.B., I.A.W., A.B.W.); a Vidigrant from the Netherlands Organization for ScientificResearch (R.W.S.); a Starting Investigator Grant from theEuropean Research Council (R.W.S.); and a Canadian Institutesof Health Research fellowship ( J.-P.J.). The EM work wasconducted at the National Resource for Automated MolecularMicroscopy at The Scripps Research Institute, which issupported by the Biomedical Technology Research Centerprogram (GM103310) of the National Institute of GeneralMedical Sciences (B.C., C.S.P.). 3D visualizations weregenerated using the UCSF Chimera package. The EMreconstructions have been deposited in the Electron MicroscopyData Bank under accession codes EMD-5779, EMD-5780,EMD-5781, and EMD-5782. The structure coordinates havebeen deposited in the Protein Data Bank under accession code3J5M. Sharing of other materials will be subject to standardmaterial transfer agreements. Raw EM data will be providedupon request. The content is the responsibility of the authorsand does not necessarily reflect the official views of NIGMS orNIH. IAVI has previously filed a patent relating to the BG505SOSIP.664 trimer: U.S. Prov. Appln. No. 61/772,739, titled“HIV-1 envelope glycoprotein,” with inventors M. Caulfield,A.C., H. Dean, S. Hoffenberg, C. R. King, P.-J.K., A. Marozsan,J.P.M., R.W.S., A.B.W., I.A.W., and J.-P.J. This is manuscript25060 from The Scripps Research Institute.

Supplementary Materialswww.sciencemag.org/342/6165/1484/suppl/DC1Materials and MethodsFigs. S1 to S20Movie S1References (55–74)6 September 2013; accepted 7 October 2013Published online 31 October 2013;10.1126/science.1245627

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Detection of Berry’s Phase in a BulkRashba SemiconductorH. Murakawa,1*† M. S. Bahramy,1,2 M. Tokunaga,3 Y. Kohama,3 C. Bell,4 Y. Kaneko,1N. Nagaosa,1,2 H. Y. Hwang,1,4 Y. Tokura1,2

The motion of electrons in a solid has a profound effect on its topological properties and may result in anonzero Berry’s phase, a geometric quantum phase encoded in the system’s electronic wave function.Despite its ubiquity, there are few experimental observations of Berry’s phase of bulk states. Here, wereport detection of a nontrivial p Berry’s phase in the bulk Rashba semiconductor BiTeI via analysis of theShubnikov–de Haas (SdH) effect. The extremely large Rashba splitting in this material enables theseparation of SdH oscillations, stemming from the spin-split inner and outer Fermi surfaces. For bothFermi surfaces, we observe a systematic p-phase shift in SdH oscillations, consistent with the theoreticallypredicted nontrivial p Berry’s phase in Rashba systems.

Quantum mechanical systems undergoingadiabatic evolution on a closed path in pa-rameter space acquire a geometrical phase

known as Berry’s phase, fB (1, 2). A number ofemergent phenomena—including the anomalous

(3) and quantum (4) Hall effects, charge pumping(5), and topological insulating and superconduct-ing phases (6)—are driven by a nontrivial (that is,nonzero) fB. A nontrivial fB can, for example, berealized for charge carriers that have k-space cy-

clotron orbits enclosing a Dirac point (7–10). Ingeneral, any closed cyclotron orbit is quantizedunder an external magnetic field B, according tothe Lifshitz-Onsager quantization rule

AnℏeB

¼ 2p nþ 1

2−

fB2p

¼ 2pðnþ gÞ ð1Þ

Here An is the extremal cross-sectional area of theFermi surface (FS) related to the Landau level(LL) n; g is defined as g ¼ 1

2−fB2p and can take

values from 0 to 1, depending on the value of

1RIKENCenter for EmergentMatter Science (Center for EmergentMatter Science), Wako 351-0198, Japan. 2Department of Ap-plied Physics and Quantum Phase Electronics Center, Universityof Tokyo, Tokyo 113-8656, Japan. 3International MegaGaussScience Laboratory, Institute for Solid State Physics, TheUniversityof Tokyo, Kashiwa 277-8581, Japan. 4Stanford Institute for Ma-terials and Energy Sciences, SLAC National Accelerator Laboratory,Menlo Park, CA 94025, USA.

*Present address: Department of Physics, Graduate Schoolof Science, Osaka University, Osaka, Japan.†Corresponding author. E-mail: murakawa-h@riken.jp

20 DECEMBER 2013 VOL 342 SCIENCE www.sciencemag.org1490

fB (7); ħ is Planck’s constant h divided by 2p;and e is the elementary charge. The quantity gcan be experimentally accessed by analyzingthe LL fan diagram of Shubnikov–deHaas (SdH)oscillations. A nontrivial Berry’s phase has beenobserved in pseudo-spin Dirac systems such asgraphene (10), as well as elemental bismuth (11),bulk SrMnBi2 (12), and, potentially, graphite(13–15), although in that case the experimentalsituation remains unresolved. Clear detection inphysical-spin Dirac systems such as topologicalinsulators has been complicated by large Zeemanenergy effects and bulk conduction (16–19).

Systems described by the Rashba Hamiltonianalso possess a Dirac point and provide an alter-native path to realizing a nontrivial fB. The Diracpoint in this class of noncentrosymmetric systemsresults from the crossing between energy bandsspin-split by the Rashba spin-obit HamiltonianHR ¼ l

ℏ e ⋅ðs pÞ, where l is the Rashba param-eter, e is the unit vector along which the systembreaks inversion symmetry (20), s is the spin, andp is the momentum vector. Because of this inter-action, the energy bands are linearly dispersedaround p = 0, and the resulting FS consists of twopockets, an inner FS (IFS) and an outer FS (OFS).For each FS, s is locked to be normal to p, there-by forming a helical spin texture around the Diracpoint at p = 0. It has been predicted that in sys-tems where both Rashba and Dresselhaus spin-orbit interactions are present, both FSs obtain aBerry’s phase expressed as FB ¼ l2 − b2

jl2 − b2 j p, withb corresponding to the Dresselhaus parameter(21–23). Accordingly, in the case of pure Rashbaspin splitting (RSS) (b = 0), both inner and outerFSs are expected to obtain a nontrivial p Berry’sphase. Consequences of this Rashba coupling andphase have been studied in two dimensions in

semiconductor heterostructures using weak anti-localization (24), ensemble averaging in interfer-ometers (25), commensurability oscillations (26),and SdH oscillations (27, 28). Given the relativelysmall RSS in these systems, the IFS andOFS havesimilar areas, and the SdH oscillations show beat-ing patterns that obscure the underlying oscillationindex structure. In addition,when the spin is alignedto the external magnetic field by a larger Zeemanenergy, RSS disappears, and the Berry’s phasetakes on the trivial value.

Recently, a large RSS in a bulk material hasbeen found in the polar semiconductor BiTeI(29–32). This material is composed of alternatingBi, Te, and I atoms stacked along the hexagonal caxis and is typically electron-doped by native de-fects, as found in many chalcogenide semicon-ductors. Because of the absence of inversionsymmetry and the strong polarity of the system,accompanied by the strong spin-orbit interactionof Bi, an extremely large RSS occurs around thehexagonal face center of the Brillouin zone, re-ferred to as the A point (30). Both angle-resolvedphotoemission spectroscopy and optical spec-troscopy (29, 31–34) reveal that the RSS in BiTeIapproaches 400 meV, with a Dirac point located~110 meVabove the conduction band minimum,in good agreement with relativistic first-principlescalculations (30, 35).

Figure 1 shows the calculated band dispersionaround the A point (Fig. 1A) and a typical FS fora Fermi level EF located slightly above the Diracpoint (Fig. 1B), corresponding to the samples mea-sured here. The large RSS causes the ratio of theextremal cross-sectional areas of the IFS and OFSto be extremely large, diverging as EF approachesthe Dirac point. In terms of SdH experiments forsamples with finite electron mobility, this conve-

niently decouples the two sets of oscillations byconfining the IFS and OFS to the low- and high-field regime, respectively. Moreover, the observedgiant RSS in BiTeI can dominate the Zeemaneffect, even in high magnetic fields. Therefore,BiTeI is an ideal system for investigating theBerry’sphase originating from the Rashba spin-split band.

Figure 2 shows the in-plane magnetoresistiv-ity (rxx) of a BiTeI sample (sample A) up to 14 Tapplied along the [001] c axis at 1.8 K. In thissample, the low-field Hall mobility is ~300 cm2/V·s at 1.8 K. The Hall density (~4.0 × 1019 cm−3)would placeEF slightly above the Dirac point. Ascan be seen, the SdH oscillations stemming fromthe IFS (<4 T) andOFS (>10 T) are well separatedfrom each other; thus, they can be separatelyanalyzed. Focusing first on the IFS, Fig. 3A showsthe SdH oscillations at various temperatures. Be-cause the IFS extremal cross-sectional area (AIFS)is so small, 3.4 T is sufficient to reach the quantumlimit, where all electron states in the IFS are con-densed into the lowest LL. This can be seen bytaking the negative second derivative of resistivity(–d2rxx/dB

2) (Fig. 3B). Here, the lowest-indexmaximum (corresponding to the peak in resistiv-ity) appears at 3.4 T (1/B = 0.294 T–1), and theoscillation disappears above 5 T. The period ofthe oscillation [∆(1/B) = 0.294 T–1] corresponds toAIFS = 3.1 × 10−4 Å−2.

The oscillatory component ∆rxx is plotted inFig. 3C, after subtraction of the nonoscillatingbackground deduced by fitting a fourth-orderpolynomial, based on the resistivity values at thenode positions of the SdH oscillations (Fig. 3A).From the temperature dependence of the oscillationamplitude at 3.4 T (Fig. 3D), the electron effectivemass m* for the IFS (m∗

IFS) is determined to be(0.023 T 0.001)m0 (where m0 is the free electronmass), following the Lifshitz-Kosevich formulafor a three-dimensional (3D) system (36–38)

AðB,TÞ ¼ Drr0

ºðℏwc=EFÞ12

expð−2p2kBTD=ℏwcÞ 2p2kBT=ℏwc

sinhð2p2kBT=ℏwcÞ ð2Þ

Here, r0 is the nonoscillatory component of theresistivity at B = 0, TD is the Dingle temperature,kB Boltzmann’s constant, and the cyclotron fre-quency wc = eB/m*. Considering AIFS = 3.1 ×10–4 Å–2, first-principles calculations indicateEF = 151 meVabove the conduction band mini-mum. At this EF, the calculated m∗

IFS is 0.021m0,in agreement with the observed value. We canthus confidently assign this set of SdH oscilla-tions to the IFS.

In the expression for rxx in 3D systems (36–40)

rxx ¼ r0½1þ AðB,TÞcos2pðBF=B − dþ gÞð3Þ

1/BF is the SdH frequency, and d is a phase shiftdetermined by the dimensionality, taking the valued = 0 (or d = T1/8) for the 2D (or 3D) case (41–43).In this formula, values of |g – d| = |1/2 – fB/2p – d|

EF

A Bkx(Å-1)ky(Å-1) kx(Å-1)

k z(Å

-1)

Ene

rgy

(eV

)

ky(Å-1)

Fig. 1. Electronic structure of BiTeI. (A) Energy band dispersion of the conduction band around thehexagonal face center A point of the BiTeI Brillouin zone. (B) Typical Fermi surface of BiTeI for EF located abovethe Dirac point. The Fermi surface consists of an inner Fermi surface (pink pocket) and an outer Fermi surface(purple-blue pocket). The extremal orbits normal to kz have helical spin textures (arrows), with opposite helicities.

www.sciencemag.org SCIENCE VOL 342 20 DECEMBER 2013 1491

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between 0 and 1/8 indicate a nontrivial p Berry’sphase, with the precise value determined by thedegree of two-dimensionality via d. To obtain theBerry’s phase, the relation between 1/B andLandau index number n is plotted in Fig. 3E. Here,peak and valley positions of the oscillation aredetermined using –d2rxx/dB

2 (Fig. 3B).We assigninteger indices to the rxx peak positions in 1/B andhalf integer indices to the rxx valley positions. Theinterpolation line of n versus 1/B in sample A hasan intercept between –1/8 and 0, indicating a non-trivial Berry’s phase for the IFS. A similar trend isreproduced for other samples B, C, andDwithEF ~244, 194, and 154meV, respectively (Fig. 3E). Ina pure Rashba system (g = 0), these results in-dicate that d deviates from the 3D limit |d| = 1/8,

likely because of the quasi-2D nature of the sys-tem. In sample B, the SdH oscillation originatingfrom the IFS can be seen up to 39 T with the re-spective ∆(1/B) = 0.0259 T–1, corresponding to amuch larger AIFS = 3.65 × 10–3 Å–2. The strictlinearity of this index plot up to the quantum limitis a consequence of negligible Zeeman splittingand the fact that theOFS dominates the variation ofthe chemical potential at this magnetic field. Thisallows us to avoid index shifting near the quantumlimit, as observed, for example, in graphite (14), inmuch the same manner as has been observed forspecific field orientations of bismuth (11).

Next, we turn to SdH oscillations origi-nating from the OFS. These oscillations areclearly observed in the higher–magnetic field

region. Figure 4A shows rxx of sample A up to56 T at various temperatures. In this case, clearSdH oscillations are observed above 10 T at 1.5 Kand can be discerned even at 100 K above 40 T. Theoscillatory component is deduced by subtracting afourth-order polynomial, as discussed previously,and is plotted as a function of 1/B in Fig. 4B. Theperiod of the oscillation [∆(1/B) = 0.00288 T–1]corresponds to anOFSextremal cross-sectional areaof AOFS = 3.4 × 10–2 Å–2, which is very consistentwith the calculated AOFS = 3.43 × 10–2 Å–2 at EF =151meV.This provides an important self-consistencycheck, given that this EF was derived from the IFSSdH results. From the temperature dependence ofthe peak amplitude at 53.6 T,m∗

OFS is determinedto be (0.183 T 0.003)m0 (Fig. 4C), also in goodagreementwith the calculated valuem*= 0.182m0.To obtain the Berry’s phase for OFS, the fan dia-gram is plotted in Fig. 4D. Because a longerextrapolation is required in the case of the OFS,we measured five samples with varying EF forthe determination of the intercept value. Here theinteger index n, corresponding to the rxxmaximum,is assigned such that a linear extrapolation of theindex plots yields an intercept closest to zero index(n = 0). As can be seen in Fig. 4D, the index plotsuniformly exhibit a linear dependence on n withthe lowest integer index n = 6, This linearity againsuggests that the Zeeman effect is negligible andthat the observed oscillations are far from the OFS

Fig.2. SdHoscillationsfor the inner and outerFermisurfaces inBiTeI.Transverse magnetoresis-tivity (rxx) of BiTeI withB || [001] at 1.8 K. Twotypes of SdH oscillations(indicated by arrows)originate from the innerand outer Fermi surfaces,as illustrated in Fig. 1B. 0.66

0.67

0.68

0.69

ρ xx (

mΩ

cm)

B // [001]

IFS

B (T)

OFS

1.8 K

0 2 4 6 8 10 12 14

0 20 600.00

0.05

0.10

0.15

∆ρxx

/ ρxx

(0)

(%

)

T (K)

B = 3.4 T

0 1 2 6

0. 66

0. 67

0. 68

0. 69

23 K17 K

8 K

ρ xx (

mΩ

cm)

B (T)

1.8 K

B // [001]A

B DC

E

00. 0

0. 5

1. 0

1/B

(T

-1)

D

D

0

C

n

B

A

B

1/ 8

n

A

0

1

2

C

1/B

(10

-2 T

-1)

-1/8

0.0 0 .2 0.4 0 .6-0.5

0.0

0.5

1.0

∆ρxx

( µΩ

cm)

1/B (T-1)

1.8 K8 K

15 K23 K

33 K45 K

0.0 0 .5 1.0

n=3

n=2.5n=1.5

n=2

- d

2 ρ xx/d

B2 (

a.u.

)

1/B (T-1)

T = 1.8 K

0

n=1

Sample A

m*=(0.023+_0.001)m0 -1/8 1/8

0

A: γ-δ = 0.01+_0.03B: γ-δ = 0.00+_0.05C: γ-δ = -0.04+_0.05D: γ-δ = -0.05+_0.03

1 2 3 4 5

3 4 5

40

Fig. 3. Berry’s phase of the inner Fermi surface. (A) rxx of sample A withB || [001] in the lower–magnetic field region, at various temperatures. Thenonoscillatory background component is deduced at each temperature basedon the SdH node positions (crosses), as shown in the case at 1.8 K. (B)Negative second derivative of rxx (–d

2rxx /dB2) as a function of 1/B. a.u.,

arbitrary units. (C) Oscillatory component at various temperatures. (D) Tem-

perature dependence of the SdH oscillation amplitude at 3.4 T [vertical dashedline in (A)]. (E) Landau index plot of the IFS (error bars are smaller than thesymbol size). Closed circles denote the integer index (rxx peak), and opencircles indicate the half integer index (rxx valley). The intercept is between–1/8 and 1/8 for samples A and D, as well as for samples with a higher EF(samples B and C). (Inset) Magnified view around the intercept.

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quantum limit. Similar to the case of the IFS, hereagain the intercept |g – d| is in the range 0 to 1/8.Thus, we observe a systematic p-phase shift in SdHoscillations, consistent with the theoretical predic-tion that a pure Rashba system will exhibit a non-trivial p Berry’s phase for both the IFS and OFS.

Observation of this effect is made possible bythe extremely large Rashba energy in BiTeI. It isinteresting to note that, unlike the other candidatespin Berry’s phase systems such as semiconduc-tor heterostructures and surface state of topologi-cal insulators, BiTeI is a 3D electronic system(albeit, with a quasi-2D electronic structure). Ac-cordingly, this system may provide the opportunityto explore the dependence of Berry’s phase on thetrajectory-dependent spin evolution. Moreover,the prediction that pressure can drive BiTeI througha quantum phase transition to a topological phaseprovides a test to compare Berry’s phase stemmingfrom different physical origins (35).

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Acknowledgments: We thank J. G. Checkelsky for fruitfuldiscussions. This work was supported by the Funding Programfor World-Leading Innovative Research and Development onScience and Technology (FIRST Program), Japan, and the U.S.Department of Energy, Office of Basic Energy Sciences,Materials Sciences and Engineering Division, under contractDE-AC02-76SF00515 (C.B. and H.Y.H). This research waspartly supported by Grants-in-Aid for Scientific Research (B)(no. 23340096) and (S) (no. 24224009) from the Ministry ofEducation, Culture, Sports, Science and Technology, Japan.

Supplementary Materialswww.sciencemag.org/content/342/6165/1490/suppl/DC1Materials and MethodsFigs. S1 to S3Reference (44)31 July 2012; accepted 18 November 201310.1126/science.1242247

A

B

D

C

106 8420 12 140.00

0.02

0.04

0.06

0 1/8-1/8

-1/8

0

1/8

FF

E

C

B

A

E

C

B

A1/B

(m

T-1)

1/B

(T

-1)

n

n

0

1

0 10

0.8

1.0

1.2

B // [001]

ρ xx (

mΩ

cm

)

B (T)

30 K

100 K80 K

50 K

1.5 Sample AK

0.02 0.03 0.04

-50

0

50

∆ρxx

(µ

Ω cm

)

1/B (T-1

)

80 K100 K

1.5 K30 K50 K

0 50 100 1500

2

4

6

8

∆ρxx

/ ρxx

(0)

(%

) B = 53.6 T m*=(0.183+_0.003)m0

T (K)

20 30 40 50 60

A: γ-δ = 0.01+_0.03B: γ-δ = -0.02+_0.06C: γ-δ = -0.09+_0.04E: γ-δ = 0.01+_0.04F: γ-δ = -0.03+_0.04

Fig. 4. Berry’s phase of the outer Fermi surface. (A) rxx of sample Awith B || [001] up to 56 T at various temperatures. (B) The oscillatorycomponent as a function of 1/B. (C) Temperature dependence of the oscillationamplitude at 53.6 T. (D) Landau index plots of the OFS with various EFsamples A, B, C, E, and F. Closed circles denote the integer index (rxx

peak), and open circles indicate the half integer index (rxx valley). Theinteger index n is assigned such that a linear extrapolation of the indexplots yields an intercept closest to zero index (n = 0). (Inset) Magnifiedview around the intercept, showing an intercept between –1/8 and 1/8 forall samples.

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