Photo-Induced Topological Phase Transition and a Single Dirac-Cone State in Silicene

0.49 MB·English

by  Motohiko Ezawa

Photo-Induced Topological Phase Transition and Single Dirac-Cone State
in Silicene
Motohiko Ezawa
Department of Applied Physics, University of Tokyo, Hongo 7-3-1, 113-8656, Japan
Silicene(amonolayer ofsiliconatoms) isaquantum spinHallinsulator(QSHI),whichundergoes atopo-
logical phase transition to a band insulator under external electric field Ez. We investigate a photo-induced
topologicalphasetransitionbyirradiatingcircularpolarizedlightatfixedEz. Thebandstructureandalsothe
2 topologicalpropertyaremodifiedbyphotondressing. ByincreasingtheintensityoflightatEz = 0,aphoto-
1 induced quantum Hall insulator (P-QHI)isrealized, where thequantum Hall effect occurswithout magnetic
0 field. Itsedgemodesareanisotropicchiral,inwhichthevelocitiesofupanddownspinsaredifferent. There
2 appearsaspinpolarizedmetalcharactrizedbytopologicalflatbandsatthecriticalpointbetweentheQSHIand
P-QHI.AtEz > Ecr withacertaincriticalfieldEcr, aphoto-induced spin-polarizedquantum Hallinsulator
l
u emerge.Thisisanewstateofmatter,possessingoneChernnumberandonehalfspin-Chernnumbers.Theedge
J modesupportsaperfectlyspin-polarizedcurrent,whicharedifferentfromthechiralorhelicaledgemodes.We
newlydiscoveredasingleDirac-conestatealongaphaseboundary. Adistinctivehallmarkofthestateisthat
8
2 oneofthetwoDiracvalleysisclosedwithalineardispersionandtheotheropenwithaparabolicdispersion.

]
l Introduction: Topological insulator (TI) is a distinctive inducedquantum Hall insulator (P-QHI). Here, PS-QHI is a
l
a state of matter indexed by topological numbers, and char- newstate ofmatterindexedbyoneChernandonehalfspin-
h
acterized by an insulating gap in the bulk and topologically Chern numbers, where the edge state supports a perfectly
-
s protectedgaplessedges1,2. Thetopologicalclassificationhas spin-polarizedcurrenttransportingbothchargesandspins.
e beenappliedtostaticsystems3,butrecentlyextendedtotime- The phase diagram in the (E , 2/Ω) plane shows a rich
m periodic systems4–9. A powerful method to drive quantum structure. There are QSHI, BI,zPA-QHI and PS-QHI, as we
. systems periodicallyis to apply electromagneticradiation to have stated. There appear spin polarized metal (SPM) and
t
a them. It can change material properties. It has been argued spinvalley-polarizedmetal16 (SVPM) onthecrossingpoints
m
that TIs are obtained from a semimetal5 and from a band ofthetwophaseboundaries.Theyhavedifferentspinconfig-
- insulator8 (BI) in this way. Topologicalbandstructuresmay urations.Aparticularlyintriguingstateappearsalongaphase
d
well be engineeredby application of coherent laser beam in boundary,whichhasonlyoneclosedgapwithalineardisper-
n
grapheneandsemiconductors. sion.WecallitthesingleDirac-cone(SDC)state. Itisutterly
o
c anewstateasfarasweareawareof.Theelectricfieldbreaks
In this paper we propose a new type of topologicalphase
[ inversionsymmetry,whilethelightbreakstime-reversalsym-
transitioninsilicene,aphoto-inducedtransitionfromaTIto
metry.Whentheyarebothbroken,thegapcanbedifferentat
1 anotherTIbychangingitstopologicalclass. Siliceneconsists
KandK’points.
v ofa honeycomblatticeofsilicon atomswithbuckledsublat-
4 Tightbindingmodel: Thesilicenesystemisdescribedby
tices made of A sites and B sites. The states near the Fermi
9 thefour-bandsecond-nearest-neighbortightbindingmodel16,
energyareπorbitalsresidingneartheKandK’pointsatop-
6
posite cornersof thehexagonalBrillouinzone. Silicene, be-
6
07. ipnhgysrieccse1n3–t1ly8.sTynhteheloswiz-eedn1e0–r1g2y, disyngaifmteidcswiinththeenKormanodusKly’rviaclh- H =−t X c†iαcjα+i3λ√SO3 X νijc†iασαzβcjβ
leysisdescribedbytheDiractheoryasingraphene.However, i,j α i,j αβ
2 h i hh ii
:1 (DSiOra)cgealepctorfon1s.5a5remmeVassinivesidliuceentoe.arIetliastivreemlyalrakragbelesptihna-otrtbhiet +iλR1(Ez) X c†iα(σ×dˆij)zαβcjβ
v i,j αβ
masscanbecontrolled15byapplyingtheelectricfieldE per- h i
i z 2
X pendiculartothesilicenesheet. −i3λR2 X µic†iα(σ×dˆij)zαβcjβ
r Silicene is a particular type of TI, that is a quantum spin i,j αβ
a hh ii
Hall insulator14 (QSHI). It undergoes a topological phase +ℓXµiEzc†iαciα, (1)
transition15fromaQSHItoaBIas E increasesandcrosses
z iα
| |
the critical field E . In this paper, we investigate photo-
cr
induced topological phase transition at fixed Ez. The off- where c†iα creates an electron with spin polarization α at
resonance coherentlaser beam rearrangesthe band structure site i, and i,j / i,j runover all the nearest/next-nearest
h i hh ii
byphotondressing:Berrycurvaturesinthemomentumspace, neighbor hopping sites. We explain each term. (i) The first
originatingfromtheSOcoupling,aremodifiedinthephoton- term represents the usual nearest-neighborhopping with the
dressed bands so that the occupied electronic states change transfer energy t = 1.6eV. (ii) The second term19 repre-
topologicalproperties4. Weshowthat,byapplyingstrongcir- sents the effective SO coupling with λ = 3.9meV, where
SO
cularpolorizedlightwithfrequencyΩ,siliceneistransformed σ = (σ ,σ ,σ ) isthe Paulimatrixofspin, withν = +1
x y z ij
from a QSHI or a BI into a photo-induced spin-polarized if the next-nearest-neighboringhoppingis anticlockwiseand
quantumHallinsulator(PS-QHI)andeventuallyintoaphoto- ν = 1ifitisclockwisewithrespecttothepositivez axis.
ij
−
2
(iii)Thethirdterm16 representsthefirstRashbaSOcoupling
associated with the nearest neighbor hopping, which is in-
duced by external electric field. It satisfies λ (0) = 0 and
R1
becomes of the order of 10µeV at the critical electric field
E = λ /ℓ = 17meVÅ 1. (iv)Theforthterm20 represents
c SO −
the second Rashba SO coupling with λ = 0.7meV asso-
R2
ciated with the next-nearest neighbor hopping term, where
µ = 1 for the A (B) site, and dˆ = d / d with the
i ij ij ij
vector±dij connectingtwositesiandjinthesa|mes|ublattice. FIG.1: (Coloronline)Theband structureofasiliceneintheSDC
(v)Thefifthterm16 isthestaggeredsublatticepotentialterm. state. ThegapisopenattheKpointwithaparabolicdispersionbut
Duetothebuckledstructurethetwosublatticeplanesaresep- closedattheK’pointwithalineardispersion.
aratedbyadistance,whichwedenoteby2ℓwithℓ = 0.23Å.
Itgeneratesastaggeredsublatticepotential∝ 2ℓE between
z where is typically less than 1 for intensity of lasers and
siliconatomsatAsitesandBsitesinelectricfieldEz. pulsesaAvailableinthefrequencyregimeofourinterests. The
Low-energyDiractheory:Weanalyzethephysicsofelec-
gauge potential satisfies the time-periodicity, A(t + T) =
tronsneartheFermienergy,whichisdescribedbyDiracelec-
A(t), with T = 2π/Ω. The electromagnetic potential is
trons near the K and K’ points. We also call them the Kη introduced into the Ha|m|iltonian (2) by way of the minimal
points with η = . The effective Dirac Hamiltonian in the
momentumspace±reads16 substitution,thatis,replacingthemomentum~kiwiththeco-
variantmomentumP ~k +eA .
i i i
≡
H =~v (ηk τ +k τ )+λ σ ητ ℓE τ A question arises whether a topological classification is
η F x x y y SO z z z z
− possibleinnon-equilibriumsituation,thatis,whentheHamil-
+aητ λ (k σ k σ )
z R2 y x− x y tonian has an explicit time dependence. The answer is yes,
+λR1(Ez)(ητxσy τyσx)/2, (2) providedthepotentialistime-periodic. Aconvenientmethod
−
istousetheFloquettheory5–9. Thetopologicalclassification
whereσ andτ arethePaulimatricesofthespinandthesub-
a a ispossiblewiththeaidofthestaticeffectiveHamiltonianap-
latticepseudospin,respectively.Thefirsttermarisesfromthe
propriatelyconstructed.
nearest-neighborhopping,wherev = √3at=5.5 105m/s
F 2 × We summarize the result of the floquet theory. When
is the Fermi velocity with the lattice constant a = 3.86Å. the light frequency is off-resonant for any electron transi-
There is no recognizable effect from the term λR1(Ez) as tions, light does not directly excite electrons and instead ef-
far as we have numerically checked. Although we include fectively modifies the electron band structures through vir-
all terms in numerical calculations, in order to simplify the tual photonabsorptionprocesses. Such an off-resonantcon-
formulas and to make the physical picture clear, we set dition is satisfied for the frequency ~Ω t in our model
λR1(Ez)=0inallanalyticformulas. with π bands. The influence of suc|h |of≫f-resonant light is
There are four bands in the energy spectrum of Hη. The summarized7 in the static effective Hamiltonian defined by
band gap is located at the K and K’ points, and given by ∆H =(i~/T)logU,whereU = exp[ i/~ T H(t)dt]
2∆(E ),where16 eff T − R0
z isthetimeevolutionoperatorwith thetime-orderingoper-
| |
T
ator. Inthelimitof 1,∆H isparticularlysimplenear
∆(Ez)=−sztzλSO+ℓEz, (3) the Dirac points: ∆AH≪eff = (~Ωef)f−1[H 1,H+1] + O(cid:0) 4(cid:1),
wThitehythareesgpoinodszq=uan±tu1manndutmhebesursbalatttthiceeKpsaenuddoKs’pipnotiznt=s. T±h1e. wis,heHre 1H±=1 Ti1sRt0hTeHFo(Tur)ieer±cito|Ωm|dpto.neTnhte−ofmHodaimfiiclatotinoinano,Aftthhaet
spin s is an almost good quantum number even away from Hami±ltonian due to the time-periodic perturbation is under-
z
the K and K’ pointsbecauseλ is a smallquantity. On the stoodasthe sumoftwosecond-orderprocesses, whereelec-
R2
other hand, the pseudospin t is strongly broken away from tronsabsorbandthenemitaphoton,andviceversa.
z
theKandK’pointsforE =0. Byexplicitlycalculatingthecommutation,wehavetheef-
z
As E increases,thega6pdecreaseslinearly,andclosesat fectiveHamiltonian,Hη =H +∆H ,with
| z| eff η eff
thecriticalpoint E = E withE = λ /ℓ = 17meV/Å,
| z| cr cr SO 2
and then increases linearly. Silicene is a semimetal due to ∆H = A [(~v )2ητ (aλ )2σ
gapless modes at E = E , while it is an insulator for eff −~Ω F z − R2 z
z cr
|Ez|6=Ecr. | | −aλR2~vF(ητxσy −τyσx)]. (5)
Photo-inducedtopologicalinsulator:Weconsiderabeam
The modification of the band structure is remarkable, which
ofcircularlypolarizedlightirradiatedontothesilicenesheet.
wedemonstratebasedonanalyticformulasbyneglectingthe
Thecorrespondingelectromagneticpotentialisgivenby second Rashba terms (∝ λ ) since λ is a small constant.
R2 R2
A(t)=(AsinΩt,AcosΩt), (4) Thegapisgivenby2|mD|withtheDiracmass
m = s t λ +ℓE η~v2 2Ω 1. (6)
whereΩisthefrequencyoflightwithΩ>0fortherightcir- D − z z SO z− FA −
culationandΩ<0fortheleftcirculation.Thelightintensity HencewecancontroltheDiracmassbyapplyingelectricfield
is characterized by the dimensionless number = eAa/~, E and/orcoherentlaser beam∝ 2/Ω. Note thatthe band
z
A A
3
FIG.2: (Coloronline) Phasediagraminthe(Ez,A2/Ω)plane. A
circle shows a point where the energy spectrum is calculated and
showninFig.3. Heavylinesrepresentphaseboundariesindexedby
Kη and sz =↑↓. There appear a SDC state along the line, which
ischaracterizedbyasinglegaplessDiracconeattheK pointwith
η
spinsz.Thetopologicalcharges(C,Cs)arealsoindicated.
gapsat the K andK’ pointsare differentin general. We can
makeoneDiracconegaplessandtheotherDiracconegapped.
ThisistheSDCstate. Therealizationofdifferentbandgaps FIG.3: (Color online) The photon-dressed band structure of a sil-
icenenanoribbonatmarkedpointsinthephasediagram(Fig.2).The
atthetwovalleysisanentirelynewphenomenon.
verticalaxisistheenergyinuitoft,andthehorizontalaxisisthemo-
We are able to write down the tight-binding Hamiltonian
mentum.WecanclearlyseetheDiracconesrepresentingtheenergy
yielding ∆H as the low-energy theory near the K and K’
eff spectrumofthebulk.LinesconnectingthetwoDiracconesareedge
points.TheleadingtermisgivenbytheHaldanemodel24,
modes.Thespinszispracticallyagoodquantumnumber,whichwe
haveassignedtotheDiraccones.
∆Heff =iη~vF2A2/(3√3Ω) X νijc†iαcjβ+O(λR2). (7)
i,j αβ
hh ii
boundariesare given by m = 0 with (6). The topological
InFig.1wehaveillustratedthebandstructureoftheSDCstate D
numbersare (C,C ) = (0,0) in the BI, (0,1) in the QSHI,
calculatedbasedonthetight-bindingHamiltonian. s
( 2,0) in the P-QHI, and ( 1,1) in the PS-QHI for E >
Spin-Chern number: The topological quantum numbers − − 2
0 and Ω > 0 in Fig.2. In all these states the band gap is
aisreatghoeoCdhqeurnanntuummbneurmCbearn,dthteheZZ2indinedxexis. idIefntthiceaslptionthsze open,wheretheFermilevelispresent,andtheyareinsulators.
2 WehavederivedtheseresultswithouttheRashbainteractions.
spin-Chernnumber modulo2. Theyaredefinedwhenthe
stateisisthgeapspuemdm,Cati=oCnCso+f+thCe−BaenrrdyCcsur=vat12u(rCe+in−thCe−)m,owmheenre- ThTeyhereHmaallincotrnudeuwcthiveintythiesygiavreenswfoirtcehaecdhosnpiandciaobmaptiocnalelny.tby
tCTu±hmeyspaarceewoevlelrdaellfioncecdupevieednsitfattehseosfpeilnecitsronnostwaitghosozd=qu±an1-. uchsianrgget-hHeaTllKaNndNsfpoinrm-Hualall23c,oσnxsdzyuc=tivei2ti/e(s2aπr~e)σPxyη==±σCx↑ysηz+. Tσhx↓ye
tumnumber21,22. Inthepresentmodelthespinisnotagood andσs =σ σ . TheyareequaltotheChernandspin-
xy x↑y − x↓y
quantum number because of spin mixing due to the Rashba Chernnumbersuptothenormalizatione2/2π~.
couplingsλR1andλR2. Aconvenientwayistocalculatethem Photo-inducededgemodes:Wehavediscoveredaclassof
inthesystemwithouttheRashbacouplingsandthenadiabat- newphasesbyapplyingcircularpolorizedlighttosilicenein
icallyswitchingonthesecouplings21,22. thepresenceofelectricfieldE , assummarizedinthephase
z
WhenwesetλR1 =λR2 =0,theHamiltonian(2)becomes diagram(Fig.2). TheTIischaracterizedalsobygaplessedge
blockdiagonal. Foreachspinsz = 1andvalleyη = ,it modes. We have calculated the band structure of a silicene
describesa two-bandsystem in the f±orm,H = τ d, w±here nanoribbonwithzigzagedges,whichwegiveinFig.3fortyp-
dx = η~vFkx, dy = ~vFky, dz = mD. The sum·mation of ical points in the phase diagram (Fig.2). Let us summarize
theBerrycurvatureisreducedtothePontryaginindexinthe typicalfeaturesoftheP-QHI,PS-QHI,SDCandSPMstates,
two-bandsystem2. Weexplicitlyhave17 whichhaveneverbeendiscussedinliterature.
TheP-QHIexhibitsquantumHalleffectswithoutmagnet-
1
η = sgn(m ) (8) ics field. They have anisotropic chiral edge modes. It is re-
Csz 2 D
markable that the velocities of up and down spins in chiral
for the K valley. The Chern and spin-Chern numbers are edge states are different, as found by the different slopes of
η
givenby = ( η+ η)and = 1( η η), the edge modes in Fig.4. This is not the case in graphene5.
which arCe shoPwnη=i±n tCh+e phCa−se diagCrsam (PFiηg=.2±).2TCh+e−pCh−ase Similarly,thevelocitiesofupanddownspinsinhelicaledge
4
states are different in QSHI. The difference increases as the
intensityoflightincreases.
The edge mode of the PS-QHI is peculiar such that,
e.g., only down spins propagate. Namely, a perfectly spin-
polarized edge mode is realized. Note that helical or chiral
edge modes consist of two modes per edge, while the per-
fectly spin-polarized edge mode contain only one mode per
edge,asinFig.4. Similaredgemodesarerealizedinquantum
Hall effectsunderstrong magneticfield, where spin are per-
fectly polarized in each Landau level. However, the present
edgemodeis realizedwithoutmagneticfield. We also point
outthattopologicalstateswithbothnonzeroChernandspin-
Chernnumbershaveneverbeenknown.
The SPM appears at the critical point between the QSHI
andtheP-QHI.Itisinterestingtocomparethestate withthe
SVPMstateappearingatthecriticalpointbetweentheQSHI
andtheBI.DuetotheidenticalspinconfigurationattheKand
K’points, thereemergespin polarizedtopologicalflatbands
intheSPMstate(Fig.4).
Finally, a particularly intriguing state is the SDC state
emerging along the phase boundary, where, e.g., the gap is
open at the K point but closed at the K’ point. The spin is
uppolarizedatKpointanddownpolarizedatK’point. Thus
thenetspinispolarizedintheSDCstate. Hence,thisstateis
FIG.4: (Coloronline)Enlargededgestatesofsilicenenanoribbons
ferromagnetwithoutmagneticfieldorexchangeinteractions.
fortheQSHI,SPM,P-QHIandPS-QHI.Ananisotropichelical(chi-
ral)currentflowsinQSHI(P-QHI),wherethevelocitiesofupand I am very much grateful to N. Nagaosa and T. Oka for
down spins aredifferent ineach edge. A topological flat band ap- manyhelpfuldiscussionsonthesubject. Thisworkwassup-
pearsinSPM. portedin part by Grants-in-Aid for Scientific Research from
the Ministry of Education, Science, Sports and Culture No.
22740196.
1 M.ZHasanandC.Kane,Rev.Mod.Phys.82,3045(2010). Yamada-Takamura,Phys.Rev.Lett.108,245501(2012).
2 X.-L.QiandS.-C.Zhang,Rev.Mod.Phys.83,1057(2011). 13 S.Cahangirov,M.Topsakal,E.Aktürk,1H.Sahin,andS.Ciraci,
3 A.P.Schnyder,S.Ryu,A.FurusakiandA.W.W.Ludwig,Phys. Phys.Rev.Lett.102,236804(2009).
Rev.B78,195125(2008). 14 C.-C. Liu, W. Feng, and Y. Yao, Phys. Rev. Lett. 107, 076802
4 W.Yao,A.H.MacDonaldandQ.Niu,Phys.Rev.Lett99,047401 (2011).
(2007). 15 M.Ezawa,NewJ.Phys.14,033003(2012).
5 T.OkaandH.Aoki,Phys.Rev.B79,081406(R)(2009). 16 M. Ezawa, cond-mat/arXiv:1203.0705 (tobe published inPhys.
6 J.InoueandA.Tanaka,Phys.Rev.Lett.105,017401(2010). Rev.Lett.).
7 T.Kitagawa,T.Oka,A.Brataas,L.Fu,andE.Demler,Phys.Rev. 17 M.Ezawa,cond-mat/arXiv:1205.6541.
B84,235108(2011). 18 M.Ezawa,cond-mat/arXiv:1206.5378
8 N.Lindner,G.RefaelandV.Gaslitski,Nat.Phys.7,490(2011). 19 C.L.Kane andE.J. Mele, Phys. Rev. Lett.95, 226801 (2005);
9 B.Dóra,J.Cayssol,F.SimonandR.Moessner, Phys.Rev.Lett. ibid95,146802(2005).
108,056602(2012). 20 C.-C.Liu,H.Jiang,andY.Yao,Phys.Rev.B,84,195430(2011).
10 P.Vogt,,P.DePadova,C.Quaresima,J.A.,E.Frantzeskakis,M. 21 E.Prodan,Phys.Rev.B80,125327(2009).
C.Asensio,A.Resta,B.EaletandG.L.Lay,Phys.Rev.Lett.108, 22 D.N.Sheng,Z.Y.Weng,L.ShengandF.D.M.Haldane,Phys.
155501(2012). Rev.Lett.97036808(2006).
11 C.-L.Lin, R.Arafune, K.Kawahara, N.Tsukahara, E.Minami- 23 D.J.Thouless,M.Kohmoto,M.P.Nightingale,andM.denNijs,
tani, Y. Kim, N. Takagi, M. Kawai, Appl. Phys. Express 5, Art Phys.Rev.Lett.49,405(1982).
No.045802(2012). 24 F.D.M.Haldane,Phys.Rev.Lett.61,2015(1988).
12 A.Fleurence,R.Friedlein,T.Ozaki,H.Kawai,Y.Wang,andY.