Quantum transport in Dirac materials: signatures of tilted and anisotropic Dirac and Weyl cones

1.3 MB·English

by  Maximilian Trescher

QuantumtransportinDiracmaterials: SignaturesoftiltedandanisotropicDiracandWeylcones
Maximilian Trescher, Björn Sbierski, Piet W. Brouwer and Emil J. Bergholtz
Dahlem Center for Complex Quantum Systems and Institut für Theoretische Physik,
Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany
(Dated:April2,2015)
We calculate conductance and noise for quantum transport at the nodal point for arbitrarily tilted and
anisotropicDiracorWeylcones.Tiltedandanisotropicdispersionsaregenericintheabsenceofcertaindiscrete
symmetries,suchasparticle-holeandlatticepointgroupsymmetries. Whereasanisotropyaffectstheconduc-
tanceg,butleavestheFanofactorF (theratioofshotnoisepowerandcurrent)unchanged,atiltaffectsbothg
andF.SinceF isauniversalnumberinmanyothersituations,thisfindingisremarkable.Weapplyourgeneral
5
considerationstospecificlatticemodelsofstrainedgrapheneandapyrochloreWeylsemimetal.
1
0
PACSnumbers: 72.10.Bg,03.65.Vf,05.60.Gg
2

r
p
I. INTRODUCTION port, affecting both the conductance and the Fano factor in
A
absenceofdisorder. WefindthetiltdependenceoftheFano
1 Drivenbyacombinationofkeyadvancesinmaterialsfab- factorF remarkable,becauseinmanycasesofinterestF was
rication and profound conceptual progress, the past decade foundtobeanumberwithaconsiderabledegreeofuniversal-

] ity[22,24,26–28]. Ourresultsapply—withvariousdegrees
has witnessed an explosive increase in the study of elec-
l
l tronicsystemsdispersinglinearlyaroundisolatedbandtouch- ofnumericalrelevance—toanumberofexperimentallyrel-
a
evantsystemsforwhichtiltedandanisotropicconicaldisper-
h ing points [1]. Notably, this includes graphene [2, 3], vari-
- oustwo-dimensionalorganiccompounds[4],andthesurface sioneitheroccurgenerically,asinthecaseofWeylsemimet-
s als,orforwhichtheforbiddingsymmetriesareeasilybroken,
e states of three-dimensional topological insulators [5–8]. In
m three dimensions, a Dirac semimetal [9, 10], which has two suchasstrainedgraphene.
coinciding linear band touching points with opposite chiral-
.
at ity, was observed experimentally [11, 12], and the first Weyl
m semimetal[13,14],whichhasnon-degenerateband-touching II. TILTEDANDANISOTROPICCONES
points,wasobservedveryrecently[15–17]. Subsequentlythe
-
d first transport measurements on Weyl semimetals were per- In the vicinity of a nodal point, a generic Dirac or Weyl
n formed [18, 19]. A Weyl semimetal phase is also predicted Hamiltoniancanbewrittenas
o
tooccur,e.g.,inmultilayerstructures[20]andpyrochloreiri-
c (cid:88)
[ dates[13]. H = vijkiσj +(aiki u)σ0, (1)
Byvirtueofstochiometry,theFermilevelliesexactlyatthe −
i,j
2 nodalpointofthelow-energy“cones”inmanyofthesemate-
v
rials,andtheirelectronicbehaviorisneitherthatofinsulators where the sum is over i,j = x,y or i,j = x,y,z for di-
4
—thereisnogap—northatofconventionalmetals—there mensionalitiesd = 2andd = 3,respectively. Furtherσ
3 x,y,z
isavanishingdensityofstatesatthenodalpoint.Indeedithas are the Pauli matrices and σ is the 2 2 unit matrix. The
0 0
×
4 been shown experimentally [3, 21] and theoretically [22–24] dispersion is shown schematically in Fig. 1. The “tilt” term
0 thattheconductivityσreachesaminimalbutfinitevalueata proportionaltoa istypicallydiscarded, asitdoesnotaffect
i
. nodalpointintwodimensions,whereasanodalpointinthree the eigenspinors and, hence, the topology of the band struc-
1
0 dimensionsischaracterizedbyafiniteconductanceG,itscon- ture. As we show below, inclusion of this term does affect
5 ductivityσbeingzero[24,25]. TheFanofactorF,definedas transport at the nodal point, however. Tilts can occur only if
1 theratioofshotnoisepowerandcurrent,wasfoundtobean particle-holesymmetryisabsent,andtiltisadditionallycon-
: excellentindicatorofthequantumnatureofelectronictrans- strained by point group symmetries. With a suitable choice
v
i port at the nodal point, taking the universal sub-Poissonian ofthepseudospinquantizationaxis,theanisotropymatrixvij
X
valueF = 1/3ingraphene[22]. InWeylsemimetalsF was canbebroughttoupperdiagonalform,v =v =v =0.
yx zx zy
r foundtodiscriminatebetweenapseudoballisticregime[24]at Anisotropies are generic if the cone is not located at a high
a
weakdisorderandadiffusiveregimeatstrongdisorder[25]. symmetrypointintheBrillouinzone.
UnliketheconductanceG,whichretainsadependenceonthe Considering graphene as an important example, we note
ratioW/LofsamplewidthW andsamplelengthL,theFano that the trigonal “warping” of Dirac cones respects the crys-
factorF isindependentofbothW andL. talline symmetries and leads to anisotropies, but only at
Anisotropy and tilt of the cones are often neglected, es- quadraticorderinthemomentumk. Theanisotropiestolin-
sentially for two distinct reasons: (i) they are forbidden by ear order (1), which amount to a “squeezing” of the cone
symmetry in important special cases, such as graphene, and along some direction, are, just as any tilt of the cone, for-
(ii) they do not alter the topology of the low-energy theory. bidden by the threefold point group symmetry of the honey-
Here, however, we demonstrate that tilts and, to a lesser ex- comblattice,combinedwiththelocationoftheDiracconesat
tent, anisotropies lead to clear signatures in quantum trans- high-symmetrypointsintheBrillouinzone.However,assoon
2
The Fano factor, the ratio of shot noise and current, is given
by[29]
(cid:82) dd−1k T(k )(1 T(k ))
⊥ ⊥ ⊥
F = − . (4)
(cid:82) dd−1k T(k )
⊥ ⊥
No anisotropy, no tilt.– For the isotropic cone without tilt
(a = 0,v = v δ ) the conductance and Fano factor are
i ij 0 ij
knownfromtheliterature[22,24,25],
1 1
g = , F = , (d=2), (5)
π 3
Figure1. (Coloronline)AtiltedDiracconeintwodimensions. The ln2 1+2ln2
g = , F = , (d=3). (6)
momentumcoordinatesarelabeledkx andky; thethirddimension 2π 6ln2
representsenergy.Transparentplanesindicatezero-energyplane(vi-
olet)andtilta=(−0.5,0)(greenplane),respectively. Anisotropy, no tilt.– For the general anisotropic case but
withouttilt,a =0,i=1,...,d,onefinds
i
asthethreefoldrotationsymmetryisrelaxedanisotropiesoc- g = 1vx2x+vx2y (d=2), (7)
cur. If, in addition, second-nearest-neighbor hopping is also π v v
xx yy
takenintoaccounttheparticle-holesymmetryislost,andthe
cones acquire finite tilts. This scenario applies to strained whiletheFanofactorisunaffectedbytheanisotropy; i.e., F
graphene and will be discussed in more detail below. For isgivenbyEq.(5). Forthediagonalcase(vxy =0)thisresult
three-dimensional Weyl semimetals, the band touching oc- can be understood as a simple scaling of the y coordinate,
curatlowersymmetrypoints;henceanisotropiesandtiltsare whichaffectstheconductanceg,butnottheFanofactorF. In
ubiquitous. threedimensionstheexactresultinthediagonalcase(vxy =
v =v =0)isgivenbythecorrespondingrescaling
xz yz
III. TRANSPORT:LOW-ENERGYTHEORY g = ln2 vx2x (d=3), (8)
2π v v
yy zz
WecalculatetheconductanceGandtheFanofactorF for
whilethereisnosimpleformulaforthegeneralcase.Still,the
a region of length L and width W, taking the limit W L
(cid:29) Fanofactorremainsunaffectedbyanyanisotropyandisgiven
inordertoeliminateaspuriousdependenceonthetransverse
byEq. (6).
boundaryconditions[22]. Wechoosethexaxisinthetrans-
No anisotropy, tilted cones.– Although a closed analytical
portdirection,sothatthenodalsemimetalcorrespondstothe
solutionforatiltedDiracconeispossibleintwodimensions,
region 0 < x < L, whereas the source and drain leads have
theexplicitexpressionsaretoolengthytobereproducedhere.
x < 0andx > L,respectively. Thepotentialuissettozero
Instead,wewillpresentanumericalevaluationofthesolution
for0<x<L,tomodeltransportatthenodalpoint. Wetake
forrepresentativevaluesofthetiltparametersa ,a ,anda
the limit u for x < 0 and x > L to model strongly x y z
dopedleads.→ ∞ forfixedvaluesofvij = δij. Withoutanisotropythedimen-
sionlessconductancegandtheFanofactordependonthetotal
The transverse momentum k = k (d = 2) or k =
⊥ y ⊥ magnitudea2 =a2+a2 (d=2)ora2 =a2+a2+a2ofthe
(k ,k )(d=3)isconserved,andforeachvalueofk wecal- x y x y z
y z ⊥ tiltandtheangleϕ=arccos(a /a)betweenthetiltaxisand
culate the transmission coefficient T(k⊥) by matching wave | x|
functions in the sample and the leads (see Appendix for de-
tails). TheconductanceGperconeisthengivenbytheLan-
g F
dauerformula 0.40 0.41
e2 (cid:18)W(cid:19)d−1(cid:90) 0.35 a0.0 00..3490
G= dd−1k T(k ). (2) 0.30 0.38
h 2π ⊥ ⊥ 0.25 0.1 0.37
0.20 0.3 0.36
The aspect-ratio dependence can be partially eliminated by 0.5 0.35
0.15 0.7 0.34
changingtothedimensionlessconductancereferredtoacube,
0.10 0.33
0.9
definedbytherelations 0 π8 π4 38π π2 ϕ 0 π8 π4 38π π2 ϕ
e2 (cid:18)W(cid:19)d−1
G= g. (3)
h L
Figure2.(Coloronline)DimensionlessconductancegandFanofac-
Intwodimensionsgisidenticaltotheconductivityσ.Inthree torF foratiltedtwo-dimensionalDiraccone, asafunctionofthe
dimensions, a finite value for g in the limit W, L im- angle ϕ between transport and tilt direction. The tilt strength a is
plies a vanishing conductivity σ = GL/W2 = (e→2/h∞)g/L. giveninthelegend.
3
g F
0.16 0.63 g
0.14 a 0.62
0.12 0.0 0.61 4 (1+(cid:15))a ϕ
0.10 0.1 ~s
0.08 0.3 0.60 3 0
00..0046 00..57 00..5589 2 a π/4
0.02 0.9 0.57 1 π/2
0 π8 π4 38π π2 ϕ 0 π8 π4 38π π2 ϕ 0
(cid:15)
0.2 0.0 0.2 0.4
Figure 3. (Color online) Same as in Fig. 2, but for a three- −
dimensionalWeylcone.
Figure4. (Coloronline)Dimensionlessconductancegofastrained
graphenesheetasafunctionofthestrain(cid:15)fordifferentorientationsϕ
thetransportdirectiononly. Thelimita=1correspondstoa (theanglebetweentransportdirectionand(cid:126)s)andsummedoverboth
maximallytiltedconewithaflatbandalongthetiltdirection. Dirac cones and spin. The inset shows a hexagon of the graphene
ResultsareshowninFigs.2and3ford = 2andd = 3and latticefor(cid:15)=0.2.
forrepresentativevaluesofthetiltstrengtha.
We note that the results are quantitatively different in two
andthreedimensions,butqualitativelyverysimilar. Thereis conesmustbetakenintoaccountsimultaneously. Below,we
animportantdifferencebetweenatiltparalleltothetransport provideexplicitresultsfortwospecifictight-bindingmodels.
direction,wheregdecreasesuponincreasingthetiltstrength, Strainedgraphene.–In“intrinsic,”unstrainedgraphenethe
andatiltperpendiculartothetransportdirection,whereg in- Dirac cones are located at high symmetry points in the Bril-
creaseswithincreasingtiltstrength. TheFanofactorF isun- louinzone. Theapplicationofstrainchangesthepositionsof
affectedbyatiltinthetransportdirection,andincreaseswith theDiracpointsandtheconesarenolongerprotectedbycrys-
increasingtiltifthereisafiniteanglebetweenthetiltdirection tallinesymmetries. Whereasthesimplesttight-bindingmodel
and the transport axis. Interestingly, upon averaging over all with nearest-neighbor hopping only is particle-hole symmet-
orientationsofthetiltaxis,wefindasystematicbutsmallde- ric, which rules out a tilt of the Dirac cones, realistic tight-
creaseinconductanceforbothtwoandthreedimensions.The binding models have longer-range hopping, which lifts the
mainconclusion,however,isthattheFanofactorisnolonger particle-hole symmetry [30]. As an example, we now apply
auniversalnumberoncethetiltofthedispersionistakeninto the above calculations to the model of quinoid-type strained
account, but depends on the magnitude and direction of the graphene,asdescribedbyGoerbigetal. inRef. 4. Transport
tilt. propertiesofstrainedgraphenehavebeenstudiedearlier[31],
The analytical solution for a two-dimensional Dirac cone butwithouttheinclusionofatiltoftheDiraccones.
takes a simple form if the tilt axis and the transport direc-
A schematic of the tight-binding model for quinoid-type
tion are collinear (ϕ = 0). In that case one finds g =
strained graphene is shown in the inset of Fig. 4. It con-
(1/π)√1 a2,F =1/3. Further,forsmalltiltstrengthsitis
sists of a honeycomb lattice which is extended/compressed
−
possibletoexpandtheanalyticalsolutionintwodimensions.
in the direction perpendicular to the lattice vector (cid:126)s, such
Wefind
that each hexagon has four “short” bonds of length a and
1 a2 (cid:18)4 (cid:19) two “long” bonds of length a(cid:48) for positive strain (cid:15) > 0.
g = + sin2ϕ 1 + (a4), (9)
π 2π 3 − O Strainismeasuredintermsofadimensionlessstrainparame-
ter(cid:15) = a(cid:48)/a 1. Thetight-bindingmodelofRef.4contains
1 2a2 −
F = + sin2ϕ+ (a4), (10) nearest-neighborhoppingamplitudesaswellasnext-nearest-
3 45 O neighborhopping,andwetakethemagnitudesofthehopping
which deviates less then 1% from the exact value up to a = amplitudes from Ref. 4. Figure 4 shows the conductance g
0.5. for strains 0 < (cid:15) < 0.3 and three representative angles ϕ.
| |
Anisotropy and tilted cones.– In the presence of both The strain is perpendicular to the(cid:126)s direction (as depicted in
anisotropy and tilt the dimensionless conductance and the Fig. 4). Theangleϕisdefinedastheanglebetweenthetrans-
Fano factor are qualitatively similar as in the absence of portdirectionand(cid:126)s. Thetiltisofordera/v 0.06(v being
∼
anisotropy. However,foratiltintransportdirectiontheFano thevelocityintiltdirection)forthe(alreadyquiteunrealistic)
factorchangesiftheanisotropiesarenotorientatedalongthe strain(cid:15)=0.3[4].Asaconsequenceofthisnumericallysmall
axis of the reference frames, i.e., if one of v ,v or v is value of the tilt strength, the relative change in Fano factor
xy xz yz
nonzero. remains small, (cid:46) 0.1% for (cid:15) < 0.3. While this variation is
probablyoutofreachofexperimentaldetection,itshowsthat
Fano factor F = 1/3 is not strictly “universal” in graphene
IV. APPLICATIONTOLATTICEMODELS butcanbechangedbythebreakingofsymmetries.
Weyl semimetal.– As an example in three dimensions we
Ingenericlatticemodels,theconesarebothanisotropicand consider a tight-binding model of a spin-orbit coupled py-
tilted, and moreover, contributions from an even number of rochloreslabwhichhostsaWeylsemimetalphasewithWeyl
4
V. DISCUSSION
g
F
0.9 0.62
We have investigated the effect of anisotropies and tilts of
0.8 0.61 DiracandWeylconesonquantumtransportpropertiesatthe
0.7 nodal point. Neither anisotropies nor tilts change the topol-
0.60
0.6 (cid:126)s ogy of the band structure and for this reason they are often
0.59
0.5 neglected. We showed that a tilt nevertheless affects the di-
0.4 0.58 mensionlessconductancegandFanofactorF. Thelatterob-
servation is remarkable, since the Fano factor is often found
0.3 0.57
0.4 0.2 0.0 0.2 to be a universal number, that does not depend on system-
− −
t2 specificdetails.
Applying our results to the example of strained graphene,
we found that the inclusion of a tilt of the Dirac cone leads
Figure 5. (Color online) Dimensionless conductance g (solid line) toasizabledirectionaldependenceoftheconductanceg. For
andFanofactorF (dashedline)forapyrochloreslabasafunctionof realisticstrains,thetilteffectontheFanofactorF isnonzero,
thein-planenext-nearest-neighborhoppingamplitudet2(leftpanel). though numerically very small — underlining the symme-
Transportdirectionisparalleltothecrystalaxis(cid:126)s(asshowninthe
tryprotectednatureof“universal”quantumtransportintwo-
upper right panel). Hopping parameters are indicated in the lower
dimensional Dirac materials. The consequences of a tilted
rightpanel.
dispersion, including a shift of the Fano factor F, may be
moresignificantforothertwo-dimensionalmaterialspossess-
ingmorestronglytiltedDiraccones,suchastheorganiccom-
poundα–(BEDT-TTF) I [4,34].
2 3
conesthatmaybesignificantlytilted[32,33]. Inthiscasethe WhilethefirstobservationsofWeylsemimetalsareagreat
latticestructureislayered,seeFig.5,sothatitisintrinsically experimental success, they all observed time-reversal sym-
anisotropicandnoexternalstrainneedstobeappliedinorder metric Weyl semimetals [16? –19]. These Weyl semimet-
toliftanysymmetriesforbiddingatiltoftheWeylcone. The alshaveadditionalstatescrossingtheFermilevelopposedto
model consists of a tight-binding Hamiltonian that contains the still hypothetical time-reversal symmetry breaking Weyl
spin-orbit coupling, in-plane and interplane nearest-neighbor semimetals [13], where the only states crossing the Fermi
hoppingamplitudes,andin-planenext-nearest-neighborhop- level are Weyl nodes, as is the case in our example of a py-
ping amplitudes. It was found to have a Weyl-semimetal rochlore slab. Hence the full transport properties could be
phase for a certain range of parameter space, with a tilt of obtained from the combined contribution of the Weyl nodes,
the Weyl cone that depends on the magnitude of the next- whereasintheexperimentallyobservedmaterialsoneshould
nearest-neighbor hopping amplitude t . There are six Weyl accountalsofortheotherstatesattheFermilevel.
2
cones, located on the Γ–M lines [33] in the projected two-
dimensional Brilluoin zone of the slab geometry. The Weyl In a recent work three of us proposed that the Fano factor
pointsarerelatedtoeachotherbythesixfoldsymmetryofthe can be used as a universal quantity to discriminate different
underlyinglattice. Wehavenumericallydeterminedtheposi- transport regimes in a disordered Weyl semimetal [25]: In a
tionaswellastiltandanisotropyparametersasafunctionof calculationthatdidnotincludetiltoranisotropy,F wasfound
thenext-nearest-neighborhoppingt ,keepingtheothermodel totaketheballisticvalue(6)belowacriticaldisorderstrength,
2
parameters, defined in the lower right panel of Fig. 5, fixed whereasF approachesthesmallerdiffusivevalueF =1/3at
(t = 1,t =2,λ =0.3,λ =0.2),andcalculatedthedi- larger disorder. Our present results indicate that there is no
1 ⊥ 1 2
−
mensionlessconductancegandtheFanofactorF.Theresults universalvaluefortheFanofactorintheballisticlimit. How-
areshowninFig.5foranin-planetransportdirectionaligned ever,wealsofindthatatiltoftheWeylconecanonlyincrease
with one of the crystal axes as indicated in the inset. The F,sothattheFanofactorcontinuestobeapowerfulindicator
dependence on the orientation of the pyrochlore slab is very discriminatingthepseudoballisticanddiffusiveregimes.
weak, less than 1% for both g and F, which can be under-
stoodasaconsequenceoftherebeingsixdifferentcontribut- For tilted and anisotropic cones the conductance varies
ingcones: whenrotatingthesample, someconesarerotated strongly with transport direction and can be either higher or
“away” from the transport direction, while others are rotated lower than the conductance of the symmetric cone. In con-
“towards” the transport direction. The changes in transport trasttheFanofactorisonlysensitivetothetiltoftheconeand,
propertiesindifferentconesthenhaveoppositesigns(cf. Fig. whereasitstilldependsontheanglebetweentiltandtransport
3), leading to a very weak angular dependence of g and F. direction, the Fano factor always increases for tilted cones.
The magnitude of the dimensionless conductance g and the Theseinsightsshouldbeusefulfortheexperimentalidentifi-
Fano factor F can however differ substantially from the val- cationandcharacterizationofarangeofWeylandDiracma-
uescalculatedintheabsenceofatilt. terialsbymeansoftransportmeasurements.
5
ACKNOWLEDGMENTS AppendixA
We calculate ballistic transport in a scattering region of
length L and width W as described in the main text. The
Hamiltonianisgivenby
(cid:88)
H = v k σ +(a k u)σ . (A1)
ij i j i i 0
−
ij
We acknowledge related discussions with Teemu Ojanen Thea k termscanbeinterpretedasatiltoftheconeandv
i i ij
and Masafumi Udagawa. E.J.B. and M.T. are supported by asad dmatrixdescribingtheanisotropyofthedispersion,
×
DFG’s Emmy Noether program (BE 5233/1-1). This work whereitissufficienttohavenonzeroentriesontheuppertri-
was in part supported by the Helmholtz VI “New States of angular to describe all possible anisotropies of a cone. The
MatterandTheirExcitations.” dispersionisgivenby
(cid:15) (k)= u+a k +a k +a k
1,2 x x y y z z
−
(cid:113)
k2(v2 +v2 +v2 )+2k (k (v v +v v )+k v v )+k2(v2 +v2 )+2k k v v +k2v2 . (A2)
± x xx xy xz x y xy yy xz yz z xz zz y yy yz y z yz zz z zz
Whenever the above square-root expression occurs we will calculate the transmission and reflection amplitude by wave
abbreviateitas√ . Thenthespinorsare functionmatchingatthebeginningandtheendofthescatter-
···
ingregion(x=0,L):
 
k v i(k v +k v )
x xx x xy y yy
χ1,2 =−kxvxz+−kyvyz1+kzvzz∓√··· (A3) √1vinχin+ √rvrχr =αχ1+βχ2, (A5)
t
and the velocities used to normalize incoming and outgoing √vtχt =αχ1expik˜1L+βχ2expik˜2L . (A6)
plane waves are v(k) = ∂ (cid:15)(k). We consider the limit of
k
highly doped leads (u ) in u = 0 in the scattering
→ ∞
region. The transverse momentum k = k (d = 2) and The total transmission probability can be obtained by sum-
⊥ y
k =(k ,k )(d=3)isquantizedduetothefinitewidthW, mingoverallmodes:
⊥ y z
2πn 2πm
k = k = . (A4) (cid:88)
y W z W T = t(k⊥)2. (A7)
| |
Foranymodeofgivenmomentumk ,k wedeterminethex k⊥
y z
componentofthewavevectorsk ,k ,k oftheincoming,re-
in r t
flected,andtransmittedwaveandthexcomponentinthescat- In the limit L 0 one may replace the sum by an integral
teringregionk˜ bysolvingEq. (A2)for(cid:15)(k)=0. Thenwe whichgivesWEq.→(2)ofthemaintext.
1,2
[1] T.Wehling,A.Black-Schaffer, andA.Balatsky,Adv.Phys.63, [6] J.E.MooreandL.Balents,Phys.Rev.B75,121306(2007).
1(2014). [7] R.Roy,Phys.Rev.B79,195322(2009).
[2] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, [8] Y. Zhang, K. He, C.-Z. Chang, C.-L. Song, L.-L. Wang,
Y.Zhang, S.V.Dubonos, I.V.Grigorieva, andA.A.Firsov, X. Chen, J.-F. Jia, Z. Fang, X. Dai, W.-Y. Shan, S.-Q. Shen,
Science306,666(2004). Q.Niu,X.-L.Qi,S.-C.Zhang,X.-C.Ma, andQ.-K.Xue,Nat.
[3] K.S.Novoselov, A.K.Geim, S.V.Morozov, D.Jiang, M.I. Phys.6,584(2010).
Katsnelson,I.V.Grigorieva,S.V.Dubonos, andA.A.Firsov, [9] Z.Fang,N.Nagaosa,K.S.Takahashi,A.Asamitsu,R.Math-
Nature438,197(2005). ieu,T.Ogasawara,H.Yamada,M.Kawasaki,Y.Tokura, and
[4] M.O.Goerbig,J.-N.Fuchs,G.Montambaux, andF.Piechon, K.Terakura,Science302,92(2003).
Phys.Rev.B78,045415(2008). [10] S.Murakami,NewJ.Phys.9,356(2007).
[5] L.Fu,C.L.Kane, andE.J.Mele,Phys.Rev.Lett.98,106803 [11] Z.K.Liu,B.Zhou,Y.Zhang,Z.J.Wang,H.M.Weng,D.Prab-
(2007). hakaran, S.-K. Mo, Z. X. Shen, Z. Fang, X. Dai, Z. Hussain,
6
andY.L.Chen,Science343,864(2014). [20] A. A. Burkov and L. Balents, Phys. Rev. Lett. 107, 127205
[12] Z. K. Liu, J. Jiang, B. Zhou, Z. J. Wang, Y. Zhang, H. M. (2011).
Weng,D.Prabhakaran,S.-K.Mo,H.Peng,P.Dudin,T.Kim, [21] Y.Zhang,Y.-W.Tan,H.L.Stormer, andP.Kim,Nature438,
M.Hoesch,Z.Fang,X.Dai,Z.X.Shen,D.L.Feng,Z.Hus- 201(2005).
sain, andY.L.Chen,NatureMater.13,677(2014). [22] J.Tworzydło,B.Trauzettel,M.Titov,A.Rycerz, andC.W.J.
[13] X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Beenakker,Phys.Rev.Lett.96,246802(2006).
Phys.Rev.B83,205101(2011). [23] M.I.Katsnelson,Eur.Phys.J.B51,157(2006).
[14] P.HosurandX.Qi,C.R.Phys.14,857(2013). [24] P.Baireuther,J.M.Edge,I.C.Fulga,C.W.J.Beenakker, and
[15] S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, C. Zhang, J.Tworzydło,Phys.Rev.B89,035410(2014).
R. Sankar, S.-M. Huang, C.-C. Lee, G. Chang, B. Wang, [25] B.Sbierski,G.Pohl,E.J.Bergholtz, andP.W.Brouwer,Phys.
G.Bian,H.Zheng,D.S.Sanchez,F.Chou,H.Lin,S.Jia,M.Z. Rev.Lett.113,026602(2014).
Hasan,arxiv:1502.03807v1. [26] C. W. J. Beenakker and M. Büttiker, Phys. Rev. B 46, 1889
[16] L.Lu,Z.Wang,D.Ye,L.Ran,L.Fu,J.D.Joannopoulos, and (1992).
M.Soljacˇic´,arxiv:1502.03438v1 (2015). [27] R.A.Jalabert,J.-L.Pichard, andC.W.J.Beenakker,Europhys.
[17] B.Q.Lv,H.M.Weng,B.B.Fu,X.P.Wang,H.Miao,J.Ma, Lett.27,255(1994).
P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, [28] H.U.BarangerandP.A.Mello,Phys.Rev.Lett.73,142(1994).
X.Dai,T.Qian, andH.Ding,arxiv:1502.04684v1 (2015). [29] M.Büttiker,Phys.Rev.Lett.65,2901(1990).
[18] X. Huang, L. Zhao, Y. Long, P. Wang, D. Chen, Z. Yang, [30] J.-C.Charlier,J.-P.Michenaud,X.Gonze, andJ.-P.Vigneron,
H. Liang, M. Xue, H. Weng, Z. Fang, X. Dai, and G. Chen, Phys.Rev.B44,13237(1991).
arxiv:1503.01304v1 (2015). [31] F. M. D. Pellegrino, G. G. N. Angilella, and R. Pucci, Phys.
[19] C. Zhang, S.-Y. Xu, I. Belopolski, Z. Yuan, Z. Lin, B. Tong, Rev.B84,195404(2011).
N. Alidoust, C.-C. Lee, S.-M. Huang, H. Lin, M. Neupane, [32] M. Trescher and E. J. Bergholtz, Phys. Rev. B 86, 241111
D.S.Sanchez,H.Zheng,G.Bian,J.Wang,C.Zhang,T.Neu- (2012).
pert,M.Z.Hasan, andS.Jia,arxiv:1503.02630v1 (2015). [33] E.J.Bergholtz,Z.Liu,M.Trescher,R.Moessner, andM.Uda-
gawa,Phys.Rev.Lett.114,016806(2015).
[34] S. Katayama, A. Kobayashi, and Y. Suzumura, J. Phys. Soc.
Jpn.75,054705(2006).