Angular-dependent oscillations of the magnetoresistance in Bi_2Se_3 due to the three-dimensional bulk Fermi surface

1.3 MB·English

by  Kazuma Eto

Angular-dependent oscillations of the magnetoresistance in Bi Se due to the
2 3
three-dimensional bulk Fermi surface
Kazuma Eto, Zhi Ren, A. A. Taskin, Kouji Segawa, and Yoichi Ando
Institute of Scientific and Industrial Research, Osaka University, Ibaraki, Osaka 567-0047 Japan
Weobservedpronouncedangular-dependentmagnetoresistance(MR)oscillationsinahigh-quality
Bi2Se3 single crystal with the carrier density of 5×1018 cm−3, which is a topological insulator
with residual bulk carriers. We show that the observed angular-dependent oscillations can be well
simulatedbyusingtheparametersobtainedfromtheShubnikov-deHaasoscillations,whichclarifies
0
that the oscillations are solely due to the bulk Fermi surface. By completely elucidating the bulk
1
oscillations,thisresultpavesthewayfordistinguishingthetwo-dimensionalsurfacestateinangular-
0
2 dependentMRstudiesinBi2Se3withmuchlowercarrierdensity. Besides,thepresentresultprovides
a compelling demonstration of how the Landau quantization of an anisotropic three-dimensional

n Fermisurface can give rise to pronounced angular-dependent MR oscillations.
a
J PACSnumbers: 71.18.+y,73.25.+i,72.20.-i,72.80.Jc

9
2
I. INTRODUCTION gives rise to new types of oscillatory phenomena in the

] angular-dependence of the magnetoresistance (MR). It
i
c was demonstrated29 that the oscillations observed at
s The three-dimensional (3D) topological insulator is a
lower fields provide a way to distinguish the 2D Fermi
- rapidly growing field of research in the condensed mat-
l surface(FS)inBi Sb ,whileanothertypeofoscilla-
r ter physics.1–29 Because the bulk of a topologicalinsula- 0.91 0.09
t tions of unknown origin was found to become prominent
m tor belongs to a different Z2 topological class1–3 than athigherfields. Inthepresentwork,wehavetriedtoelu-
the vacuum, on the surface of a 3D topological insu-
. cidate whether this new tool, the angular-dependence of
t lator emerges an intrinsically metallic two-dimensional
a the MR, can distinguish the 2D surface state in Bi Se ,
m (2D) state which hosts helically spin-polarized Dirac 2 3
which is believed to be the most promising topological
fermions.1,3 Besides the profound implications of those
- insulator for investigating the novel topological effects
d spin-helical 2D Dirac fermions on future spintronics, the
because of its simple surface-state structure.11,14 We ob-
n topological insulator is expected to present various ex-
o otic quantum phenomena associated with its non-trivial served pronounced angular-dependent oscillations of the
[c topology.3–8 MdeRnsiitnyann n=-ty5p×e1B01i28Sec3m−si3n,glbeuctryosutraldwetitahiletdheancaarlyriseisr
Motivated by theoreticalpredictions,3,11 three materi- clarified theat the observed oscillations are solely due to
1 als have so far been experimentally confirmed12–18 to be
v the Landau quantization of the anisotropic bulk FS of
3 3Dtopologicalinsulators: Bi1−xSbx,Bi2Se3,andBi2Te3. this material. Nevertheless, to the best of our knowl-
The angle-resolvedphotoemission experiments12–18 have
5 edge,suchanangular-dependentMRoscillationsina3D
3 played decisive roles in indentifying the topologically material has never been explicitly demonstrated in the
5 nontrivial nature of their surface states. More recently,
literature, so the present results nicely supplement our
. scanning tunneling spectroscopy experiments19,20 have
1 general understanding of the angular-dependent MR os-
0 elucidated the protection of the surface state from spin- cillation phenomena. In addition, our analysis provides
0 nonconserving scattering. In addition to those surface- asolidgroundfordiscriminatingthecontributionsofthe
1 sensitiveprobes,transportexperimentsareobviouslyim-
2D and 3D FSs in the angular-dependence of the MR in
: portant for understanding the macroscopic properties of
v future studies of Bi Se single crystals with much lower
2 3
i topologicalinsulators and for exploiting the applications carrier density.
X
of their novel surface state. However, capturing a sig-
r nature of the surface state of a topological insulator in
a
its transport properties has proved difficult: The sur-
face state has been seen by quantum oscillations only in II. EXPERIMENTAL DETAILS
Bi1−xSbx at a particular Sb concentration of 9%;21 the
universal conductance fluctuations associated with the High-quality Bi Se single crystals were grown by
2 3
surface state were observed only in Ca-doped Bi2Se3 af- melting stoichiometric mixtures of 99.9999% purity Bi
tercarefultuning ofthecarrierdensity;22 the Aharonov- and 99.999% purity Se elements in sealed evacuated
Bohm oscillations through the surface state were ob- quartz tubes. After slow cooling from the melting point
servedonlyinverynarrownanoribbonsofBi2Se3.23Most downtoabout550◦Covertwodays,crystalswerekeptat
often,thetransportpropertiesaredominatedbythebulk this temperature forseveraldaysand then werefurnace-
conductivity due to residual carriers.22,28 cooled to room temperature. The obtained crystals are
Inthiscontext,wehaveveryrecentlyfound29 thatthe easily cleaved and reveal a flat shiny surface. The X-
2D surface state of the topological insulator Bi Sb ray diffraction measurements confirmed the rhombohe-
0.91 0.09
2

1.0 s)
m) 0.8 ensity (arb. unit 006 009 0012 0015 0018 0021
c 0.6 Int B (T)
m 20 40 60 -10 0 10
2 (deg)
(xx0.4 cm) 2 RH = -1.82 cm3/C
1
0.2 (myx-01
-2 n = 3.4. 1018cm-3
0.0
0 50 100 150 200 250 300
T (K)
FIG. 2: (Color online) SdH oscillations measured within the
FIG.1: (Coloronline)Temperaturedependenceofρxxin0T. C3−C2 plane. Thin solid lines are the result of our ρxx(B)
UpperinsetshowstheX-raydiffractionpatternoftheBi2Se3 simulation (see text). Inset shows the measurement configu-
single crystal used for transport measurements. Lower inset ration.
showsρyxforBkC3 measuredat1.5K.Theslopeofρyx(B),
shown by the thin solid line, suggests that the main carriers
are electrons whose density is 3.4×1018 cm−3.
Figure 2 shows ρ (B) measured at 1.5 K for several
xx
field directions in the transverse geometry (I k C and
1
B ⊥ I), after removing the antisymmetric components
dral crystal structure of Bi Se . Post-growth annealings
2 3
due to the leakage of ρ . Two features are evident:
at various temperatures under controlled selenium par- yx
First, pronounced Shubnikov-de Haas (SdH) oscillations
tial pressures were used to reduce the number of sele-
areseenforanyfielddirection,suggestingtheir3Dorigin.
nium vacancies that are responsible for creating electron
carriers.22,28 Second, the background of the SdH oscillations varies
significantly with the field direction, indicating that the
The resistivity ρ was measured by a standard four-
xx
transverse MR is very anisotropic. Both features are
probe method on rectangular samples, with the electric
taken into account in our simulation of the observed an-
current I directed along the C axis. Continuous rota-
1
gular dependence of the MR, as will be discussed below.
tionsofthesampleinconstantmagneticfieldBwereused
tomeasuretheangulardependenceofthetransverseMR Figure 3 presents the analysis of the observedSdH os-
withintheC3-C2 plane. Forsomeselectedmagnetic-field cillations. The oscillationsin dρxx/dB plotted asa func-
◦
directions, the field dependences of ρxx and the Hall re- tion of 1/B for B kC2 (θ =90 ) are shown in Fig. 3(a)
sistivity ρ were also measured by sweeping B between asanexample. TheverysimplepatternseeninFig. 3(a)
yx
±14 T. isaresultofthesinglefrequencyF =107T(seeinsetfor
the Fourier transform) governing the SdH oscillations.31
The sameanalysiswasappliedtothe data forotherfield
directions, and the obtained F as a function of θ are
III. RESULTS AND DISCUSSIONS
shown in Fig. 3(b). The same set of frequencies can be
extracted from the Landau-level “fan diagram” [inset of
A. Resistivity and SdH Oscillations
Fig. 3(b)], which is a plot of the positions of maxima
in ρ (B) as a function of the Landau level numbers.
xx
Figure 1 shows the temperature dependence of ρ of The slopes of the straight lines in the fan diagram give
xx
the Bi Se single crystal studied in this work. It shows exactly the same F(θ) as the Fourier transform result.
2 3
a metallic behavior dρ/dT > 0 down to ∼30 K, and Another piece ofinformationthat can be extractedfrom
saturate at lower temperature (there is actually a weak the fan diagramis the phase of the oscillations, γ, which
minimum near 30 K, as is usually observed28,30 in low- is determinedby ρ ∼cos[2π(F/B+γ)]. Inthe present
xx
carrier-density Bi Se ). The single-crystal nature of the case, all the straight lines in the inset of Fig. 3(b) in-
2 3
sample is evident from the X-ray diffraction data shown tersect the horizontalaxis at the same point, giving γ =
in the upper inset of Fig. 1. The lower inset of Fig. 1 0.4 that is independent of the field direction. The an-
shows the Hall resistivity ρ measured at 1.5 K for the gular dependence of the SdH frequency F(θ) points to a
yx
fielddirectionalongthe C axis,whichsuggeststhatthe singleellipsoidalFSlocatedattheΓpointwiththesemi-
3
main carries are electrons and the carrier density n is axes k = k = 4.5×106 cm−1 (⊥C ) and k = 7.3×106
e a b 3 c
3.4×1018 cm−3 (in a one band model). From the values cm−1 (kC ). The expectedF(θ) forthis FS is shownby
3
of the Hall coefficient R = 1.82 cm3/C and ρ = 0.28 the solid line in Fig. 3(b), which fits the data very well.
H xx
mΩcm at 1.5 K, the Hall mobility µ is estimated to be The carrier density corresponding to this FS is 5×1018
H
6500 cm2/Vs. cm−3,whichisabout50%higherthanthatobtainedfrom
3

0.36 (a)
)
m
c 14 T
m 12
( 0x.32 10
x
8
6
0.28
0
14 T
0.36 (b)
) 12
m
c
10
m
( 0x.32 8
x 6
0.28
0
0 30 60 90 120 150 180 210
(deg)
FIG. 4: (Color online) (a) Angular dependences of ρxx mea-
FIG. 3: (Color online) Analyses of the SdH oscillations. The sured within the C3−C2 plane in constant magnetic fields,
insetinthemiddleschematicallyshowstheobtained3DFermi
whose values were 0, 0.5, 1.5, 3, 4.5, 6, 8, 10, 12, and 14 T.
surfaceandthedefinitionofθ. (a)SdHoscillationsforBkC2 (b)Simulationofρxx basedonEq. (1)forthesamemagnetic
asafunctionof1/B. TheFouriertransform showninthein- fieldsas in (a).
set reveals a single frequency F = 107 T. (b) F(θ) measured
withintheC3−C2plane;insetshowsthefandiagramforsev-
eral θ. (c) Temperature dependence of the SdH oscillations this gives an isotropic scattering time τ = 1.3×10−13 s,
for B k C3; inset shows the temperature dependences of the
implying the mean free path ℓ (= v τ) of ∼50 nm and
STd,Hyiealmdipnlgittuhdeescymcleoatrsounremdaaslsonogf0t.h1e4mCe3aanndd0C.224maxe,esreastpe1c2- themobilityµSdH ofabout1600cm2/FVs,whichisalmost
tively. (d) Dingle plots for B k C3 at several temperatures four times smaller than µH estimated from RH. Such a
give thesame TD = 9.5 K. discrepancybetweenµSdHandµH haslongbeennotedin
varioussystems,33 andtheessentialreasonliesinthedif-
ference in how the effective scattering time comes about
R , as was reported previously in the literature.32 This fromthemicroscopicscatteringprocess:34Thescattering
H
discrepancy is due, at least partly, to the fact that the rate 1/τtr determining transport properties acquires the
present R was measuredin the low-field limit and con- additionalfactor1−cosφuponspatialaveraging(φisthe
H
tains the so-called Hall factor, which is usually between scattering angle), while 1/τ to govern the dephasing in
1 – 2. the quantum oscillations is given by a simple spatial av-
The temperature dependence of the SdH oscillations eraging without such a factor. Hence, if the small-angle
was measured for two field directions, along the C and scattering becomes dominant (which is often the case at
3
C2 axes. Figure3(c)showstheρxx(B)datainB kC3 for low temperature), 1/τtr can be much smaller than 1/τ.
some selected temperatures, where one can see that the This means that the mean free path in our sample can
background MR is essentially temperature-independent be even larger than that estimated above.
and that oscillations are still visible even at 25 K. The
inset of Fig. 3 (c) shows the temperature dependence
of the SdH amplitude measured at 12 T for B along C B. Angular-Dependent MR Oscillations
3
◦ ◦
(θ = 0 ) and C (θ = 90 ). The fits with the standard
2
Lifshitz-Kosevichtheory33yieldthecyclotronmassmcof Thenew observationofthe presentworkisthe oscilla-
◦ ◦
0.14me and0.24me forθ =0 and90 ,respectively. The tory angular-dependence of the MR shown in Fig. 4(a),
energy dispersion near the conduction band minimum where, on top of the two-fold-symmetric backgroundan-
of Bi2Se3 is known to be parabolic,32 and the observed gular dependence, pronounced oscillations are evident
anisotropy in mc for the given FS is entirely consistent at higher fields. Importantly, the peak positions shift
with the parabolic dispersion to within 5%. Relying on with magnetic field, which is different from the ordinary
this dispersion, the Fermi energy EF is calculated to be angular-dependent MR oscillations (AMRO) in quasi-
56 meV and the Fermi velocity vF is 3.8×107 cm/s for low-dimensional systems.35 In the following, we show
any direction. that these oscillations are due to the Landau quantiza-
The Dingle plots [shown in Fig. 3(d) for B kC as an tion ofthe 3D FS and, hence, are essentially of the same
3
example]yieldtheDingletemperatureT =9.5Kwhich origin as the SdH oscillations. The SdH oscillations oc-
D
is almost isotropic and is constant at low temperature; cur as the Landau-quantized cylinders in the Brillouin
4

◦
0.295 2)cm/T0.6 ) cm0.2 f1o2r◦b,owthhicah2(iθs)parnodbaρbslayt(dθu)eistsohtifhteedrhfroommboθh=ed0ralbsyyambmouet-
m try of the crystal which leads to the appearance of cross
( 20.4 ( 0at.1 terms36 in the magnetic-field expansionof the resistivity
a s
m) 0.2 0.0 tensor.37
c 0.290 0 90 0 90 Once ρBG is known as functions of both θ and B, one
m (deg) (deg) cansuperposetheSdHoscillationstoobtaintheexpected
(xx ρ (θ,B). In the case of our Bi Se , thanks to the fact
xx 2 3
that the SdH oscillations are composed of only one fre-
0.285 quency, one can simulate experimental data as
ρ (θ,B) = ρ (θ,B)×
xx BG
-15 -10 -5 0 5 10 15
F(θ)
B (T) 1+AR R R cos 2π +γ ,
T D S
(cid:18) (cid:20) (cid:18) B (cid:19)(cid:21)(cid:19)
FIG.5: (Color online)FittingofthebackgroundMRforBk
C2. Open circles show the experimental data [symmetrical (1)
componentofρxx(B)];thedashedlinedemonstratesthelow-
field quadratic dependence of the resistivity, ρqd(B)=a2B2; whereRT,RD andRS aretemperature,Dingle,andspin
thedash-dottedlinerepresentsthehigh-fieldsaturationlimit, damping factors, respectively.33 As shown in Fig. 2 by
ρsat; thesolid line is theresulting reconstructed background, thinsolidlines,the experimentallyobservedSdH oscilla-
ρBG(B)(seetext). Theleftandrightinsetsshowtheobtained tions are reproduced reasonably well with Eq. (1) with
angulardependencesofthecoefficientsa2 andρsat withinthe a parameter A·R = 0.12 which is independent of θ.
S
C3−C2 plane. The same set of parameters can be used to calculate
the angular-dependent MR for fixed B, ρ (θ), with Eq.
xx
(1). The result is shown in Fig. 4(b), where one can
zone expand and cross the FS with increasing magnetic easilyseethatthe calculatedangulardependences ofthe
field, while the angular-dependent oscillations occur as MRfollowverycloselytherathercomplicatedpatternsof
the axis of those cylinders rotates in the Brillouin zone. theobservedρ (θ). Thisgivescompellingevidencethat
xx
Obviously, no oscillation is expected for rotating mag- the angular-dependent oscillations are essentially due to
netic field when the FS is spherical, but when the FS the Landau quantization of the anisotropic 3D FS.
is anisotropic, the number of cylinders residing within It was emphasized in Ref. 29 that a merit of mea-
the FSchangesas the cylinder axis is rotated,leading to suring the angular-dependence of the MR is that at suf-
resistivity oscillations. Therefore, an anisotropy in the ficiently high magnetic field, the 3D FS remains in the
FS is a necessary ingredient for the angular-dependent quantumlimit (where allthe electronscondense into the
oscillations. 1st Landau level) and do not contribute to the angular-
To show that the observed angular-dependent oscilla- dependent oscillations, while the 2D FS will always pro-
tionsareessentiallyduetotheLandauquantization,itis duce MR oscillations when the magnetic-field direction
most convincing to reconstruct the angular-dependence is nearly parallel to the 2D plane. Once the density of
of the MR based on the SdH oscillations data. For this the residual carriers in a topological insulator sample is
purpose, one needs to understand the exact magnetic- sufficiently reduced, the 3D FS should be easily brought
field dependence of the MR in the presence of the SdH intothequantumlimitandthe2DFSwouldbecomedis-
effect, and know how it evolves when the magnetic field tinguishable in the angular-dependent MR oscillations.
is rotated. Figure 5 shows the ρ (B) data for B k C Also, the SdH effect needs a certain range of magnetic
xx 2
(θ = 90◦), which we take as an example for presenting field for the oscillations to be recognized, whereas the
our procedure to extract the necessary information. At observation of the angular-dependent oscillations can be
low magnetic fields, the MR in Fig. 5 exhibits an al- made only at the highest field; this gives a certain prac-
most quadratic field dependence, which can be fitted by tical advantage to the latter method when the quanti-
ρ (B)=a B2. With increasing magnetic field, the MR zation condition is only barely achieved with the avail-
qd 2
tends to saturate, while pronouncedSdH oscillations de- able magnetic field. Therefore, one would expect that
velop at the same time. The overall background MR the angular-dependence of the MR will be a useful tool
ρ (B) can be described as 1/ρ = 1/ρ + 1/ρ , for distinguishing the 2D surface state of the topological
BG BG qd sat
which combines the low-fieldquadraticbehavior andthe insulator Bi2Se3 when a single crystal with much lower
high-field saturation of ρ (B). This background devel- carrier density becomes available.
xx
ops on top of the zero-field resistivity ρ (0) of about
xx
0.28 mΩcm. The same fitting of the background of MR
can be made for the whole range of the magnetic-field IV. CONCLUSIONS
direction. The evolutions of the parameters a and ρ
2 sat
withθintheC −C planeareshownintheleftandright In conclusion, we observed pronounced angular-
3 2
insets of Fig. 5. Note that the center of the symmetry dependent oscillations of the MR in high-quality single
5
crystals of n-type Bi Se with n = 5×1018 cm−3. We dependentMR,bycompletelyelucidatingtheoscillations
2 3 e
show, by simulating the angular-dependent oscillations due to the bulk FS.
based on the information obtained from the SdH analy-
sis,thattheoscillationsareessentiallyduetotheLandau
quantizationofthe3DbulkFermisurface. Thisprovides
acompellingdemonstrationofhowtheLandauquantiza-
Acknowledgments
tion ofananisotropic3D FS cangiverise to pronounced
angular-dependent MR oscillations. Furthermore, the
present results pave the way for distinguishing the 2D This work was supported by JSPS (KAKENHI
surface state in Bi Se in future studies of the angular- 19340078and 2003004)and AFOSR (AOARD-08-4099).
2 3
1 L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, 20 T. Zhang, Peng Cheng, X.Chen, J.-F. Jia, X.Ma, K.He,
106803 (2007). L. Wang, H. Zhang, X. Dai, Z. Fang, X. Xie, and Q.-K.
2 J. E. Moore and L. Balents, Phys. Rev. B 75, 121306(R) Xue,Phys. Rev.Lett. 103, 266803 (2009).
(2007). 21 A.A.TaskinandY.Ando,Phys.Rev.B80,085303(2009).
3 L. Fu and C. L. Kane, Phys.Rev.B 76, 045302 (2007). 22 J. G. Checkelsky, Y. S. Hor, M.-H. Liu, D.-X. Qu, R. J.
4 X-L.Qi,T.L.Hughes,andS.-C.Zhang,Phys.Rev.B78, Cava,andN.P.Ong,Phys.Rev.Lett.103,246601(2009).
195424 (2008). 23 H. Peng, K. Lai, D. Kong, S. Meister, Y. Chen, X.-L. Qi,
5 L.FuandC.L.Kane,Phys.Rev.Lett.100,096407(2008). S.-C. Zhang, Z.-X. Shen, and Y. Cui, Nature Materials,
6 X-L. Qi, R. Li, J. Zang, and S.-C. Zhang, Science 323, doi:10.1038/nmat2609.
1184 (2009). 24 R. Shindou and S. Murakami, Phys. Rev. B 79, 045321
7 B. Seradjeh, J. E. Moore, and M. Franz, Phys. Rev. Lett. (2009).
103, 066402 (2009). 25 A. P. Schnyder,S. Ryu, A. Furusaki, and A. W. W. Lud-
8 Y. Tanaka, T. Yokoyama, and N. Nagaosa, Phys. Rev. wig, Phys. Rev.B 78, 195125 (2008).
Lett. 103, 107002 (2009). 26 Y. Ran, Y. Zhang, and A. Vishwanath, Nat. Phys. 5, 298
9 J. C. Y. Teo, L. Fu, and C. L. Kane, Phys. Rev. B. 78, (2009).
045426 (2008). 27 D.-H.Lee, Phys. Rev.Lett. 103, 196804 (2009).
10 H.-J.Zhang,C.-X.Liu,X.-L.Qi,X.-Y.Deng,X.Dai,S.-C. 28 J. G. Analytis, J.-H. Chu, Y. Chen, F. Corredor, R. D.
Zhang, and Z.Fang, Phys.Rev.B. 80, 085307 (2009). McDonald, Z.X. Shen,and I. R.Fisher, arXiv:1001.4050.
11 H.-J. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and 29 A. A.Taskin, K. Segawa, and Y. Ando,arXiv:1001.1607.
S.-C. Zhang, Nat. Phys. 5, 438 (2009). 30 H. K¨ohler and A. Fabricius, Phys. Status Solidi (b) 71,
12 D.Hsieh,D.Qian,L.Wray,Y.Xia,Y.S.Hor,R.J.Cava, 487 (1975).
and M. Z. Hasan, Nature452, 970 (2008). 31 Strictlyspeaking,thereisasubtleshoulderstructurenear
13 D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, 80 T in the Fourier transform shown in the inset of Fig.
J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. 3(a), which suggests that there may be a weak contribu-
S. Hor, R. J. Cava, and M. Z. Hasan, Science 323, 919 tion from a second frequency. In fact, it is known32 for
(2009). Bi2Se3 that the second conduction-band minima begin to
14 D.Hsieh,Y.Xia,D.Qian,L.Wray,J.H.Dil,F.Meier, J. bepopulatedwhenne becomeshigh,andinoursamplene
Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong, A. is actually close to this onset. Nevertheless, the possible
V.Fedorov,H.Lin,A.Bansil, D.Grauer,Y.S.Hor,R.J. contribution of this second electron pocket in our sample
Cava, and M. Z. Hasan, Nature 460, 1101 (2009). is so weak that it has no consequence in the transport
15 D.Hsieh,Y.Xia,D.Qian,L.Wray,F.Meier,J.H.Dil,J. properties being analyzed.
Osterwalder,L.Patthey,A.V.Fedorov,H.Lin,A.Bansil, 32 H. K¨ohler and H. Fisher, Phys. Status Solidi (b) 69, 349
D.Grauer,Y.S.Hor,R.J.Cava,andM. Z.Hasan,Phys. (1975).
Rev.Lett. 103, 146401 (2009). 33 D.Shoenberg,MagneticOscillationsinMetals(Cambridge
16 Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. University Press, 1984).
Bansil,D.Grauer,Y.S.Hor,R.J.Cava,andM.Z.Hasan, 34 S.DasSarmaandF.Stern,Phys.Rev.B32,8442(1985).
Nat. Phys. 5, 398 (2009). 35 B. K. Cooper and V. M. Yakovenko,Phys. Rev. Lett. 96,
17 Y.L.Chen,J.G.Analytis,J.-H.Chu,Z.K.Liu,S.-K.Mo, 037001 (2006).
X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. 36 R.N. Zitter, Phys. Rev.127, 1471 (1962).
Zhang, I. R. Fisher, Z. Hussain, and Z.-X. Shen, Science 37 If we take C1, C2, and C3 as the principal axes and write
325, 178 (2009). H=(H1,H2,H3),theresistivitycomponentρ11 inamag-
18 A.Nishide,A.A.Taskin, Y.Takeichi,T. Okuda,A.Kak- netic field of arbitrary direction may be expanded to the
izaki,T.Hirahara,K.Nakatsuji,F.Komori,Y.Ando,and lowestorderinHi asρ11 =ρ11(0)+ρ11,11H12+ρ11,22H22+
I. Matsuda, Phys. Rev.B. 81, 041309(R) (2010). ρ11,33H32+2ρ11,23H2H3(seeRef.36).Thecrosstermρ11,23
19 P. Roushan, J. Seo, C. V. Parker, Y. S. Hor, D. Hsieh, is probably the source of the offset of the center of sym-
D. Qian, A. Richardella, M. Z. Hasan, R. J. Cava, and A metry seen in theinsets of Fig. 5.
Yazdani, Nature460, 1106 (2009).