Quantum anomalous Hall effect with field-tunable Chern number near Z2 topological critical point

0.96 MB·

by  Le Quy Duong

QuantumanomalousHalleffectwithfield-tunableChernnumbernearZ topologicalcriticalpoint
2
Le Quy Duong,1,2 Hsin Lin,1,2 Wei-Feng Tsai,3,∗ and Y. P. Feng1,2
1CentreforAdvanced2DMaterialsandGrapheneResearchCentre,NationalUniversityofSingapore,Singapore117546
2Department of Physics, National University of Singapore, Singapore 117542
3Department of Physics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan
(Dated:January6,2016)
We study the practicability of achieving quantum anomalous Hall (QAH) effect with field-tunable Chern
numberinamagneticallydoped,topologicallytrivialinsulatingthinfilm. Specificallyinacandidatematerial,
TlBi(S Se ) ,wedemonstratethattheQAHphaseswithdifferentChernnumberscanbeachievedbymeans
1−δ δ 2
oftuningtheexchangefieldstrengthorthesamplethicknessneartheZ topologicalcriticalpoint.Ourphysics
2
6
scenariosuccessfullyreducesthenecessaryexchangecouplingstrengthforatargetedChernnumber.ThisQAH
1
mechanismdiffersfromthetraditionalQAHpicturewithamagnetictopologicalinsulatingthinfilm,wherethe
0
“surface”statesmustinvolveandsometimescomplicatetherealizationissue.Furthermore,wefindthatagiven
2
Chernnumbercanalsobetunedbyaperpendicularelectricfield, whichnaturallyoccurswhenasubstrateis

n present.
a
J PACSnumbers:73.43-f,73.20-r,75.50.Pp,73.63.-b

5

] I. INTRODUCTION dereddopantsintheTIthinfilms22,30,31,referringtothesame
l
l mechanism: the spin-polarized band inversion from the in-
a
terplay between the FM ordering and the SOC in the TIs.
h Topologicalinsulators(TIs)belongtoanewphaseofmat-
- Recently, breakthrough experiments have realized the QAH
ter with their topological nature protected by time reversal
s phase with C = 1 in a magnetic TI thin film of Cr-doped
e symmetry(TRS)1–3.Usually,spin-orbitcoupling(SOC)plays
(Bi,Sb) Te 22,30,31 andwithalargeChernnumberinbosonic,
m akeyroleto“twist”theBlochwavefunctionsinthefirstBril- 2 3
photonic crystals32. Hence, these significant results are mo-
. louinzoneofthesolidsandsuchatwistcanbecharacterized
t tivation to disclose these following questions for fermionic
a byaZ number. Materialswithanon-trivialZ number,such
2 2 systems: 1) How to obtain a QAH phase with higher Chern
m asBi X (X=Se,Te)4–8,haveattractedstronginterestinre-
2 3 number? 2)IsitpossibletotunetheChernnumberbyother
- centyearsduetotheirunusualspin-momentumlocked,mass-
perturbations? 3) Is measuring the magnetic TI thin films a
d
less Dirac-like boundary states with potential for spintronic
n keycomponenttoobtainQAHeffect?
applications9–11 and realizing other exotic phenomena origi-
o In this paper, we provide an alternative route, compared
nallyappearedinhighenergyphysics12–17.
c to the previous studies33,34,37, to answer the aforementioned
[ Moreover,theTIshostvarietyofothertopologicalphases.
questions. We show the feasibility of exhibiting QAH ef-
1 OneofthesignificantexampleswithbrokenTRSisthequan- fectwithfield-tunableChernnumberinamagneticallydoped,
v tum anomalous Hall (QAH) effect18,19. This effect is first topologically trivial insulating thin film, in proximity to the
4 proposed by Haldane in a two-dimensional (2D) honeycomb Z topologicalcriticalpoint35. Specifically,wewillfocuson
6 lattice with locally non-vanishing magnetic flux but zero in 2
acandidatecompound,TlBi(S Se ) ,foritstunableprop-
7 average20. Similar to the integer quantum Hall effect, it can 1−δ δ 2
ertyfromatrivialinsulatortoannon-trivialonebychanging
0
0 exhibitthequantizedHallconductance,Ceh2,throughgapless thechemicalcomposition36. Althoughthemicroscopicmech-
. edgechannelswithanintegernumberC (theso-calledChern anismtorealizetheQAHphasesisstillthesameasthemag-
1
number21)indicatingitsnon-trivialbulktopologicalproperty neticallydopedTIs,ourscenariofurtheroffersafewattractive
0
6 without Landau levels. The existence of the edge channels featurestoachievehigherChernnumber: (1)Makingthesys-
1 makesitevenmoreattractiveduetoitspredictedapplications tem close to the Z2 critical point lowers the threshold value
: fordissipationlesselectronicsandintegratedcircuits19,22. So of the exchange field to induce the high order spin-polarized
v
far,withoutusingTIs,severalapproacheshavebeenproposed band inversion; (2) we start with a trivially insulating thin
i
X torealizeQAHstate,forinstance,viamainlybandstructures film, in sharp contrast to previous studies. As a result, all
r like in the mercury-based quantum wells with ferromagnetic 2D-like subbands have bulk nature and it has advantages to
a (FM) order23 and in graphene with Rashba interactions and avoid the treatment for the sample surfaces, usually exposed
the exchange field24; via electron-electron interactions like tocomplexenvironment;(3)wealsosuggestanimplementa-
in transition metal oxide heterostructures25, nearly flat band tionofthepotentialgradientorthedielectricsubstratetore-
systems26, and quadratic band-crossing systems27,28; via dis- ducethecriticalvalueoftheexchangefield. Thishasnotyet
ordereffectinAndersoninsulator29. However,theyhavenot beenexploredinpreviousrelatedstudies. Therefore,inbrief,
yetbeenmeasuredbyexperiments. ourworkstronglysuggeststhatthetrivialinsulatingthinfilm
Bycontrast,theQAHphasescanalsoberealizedbyusing withsufficientlylargeSOCcanprovideauniqueplatformto
the TIs as a platform. Theoretically, the QAH effect can be exhibitQAHeffectwithhighChernnumber.
achievedeitherbygappingoutthesurfacestatesofthethree Theremainderofthispaperisorganizedasfollows. Insec-
dimensional (3D) TIs19 or by inserting ferromagnetically or- tion II, we introduce our effective model for the thin films
2
and the methods for demonstrating the topological proper-
a) Z trivial thin film b) Z nontrivial TI film
ties of the system. In section III, in the absence of any ex- 2 2
change/electric field we discuss electronic properties of the 3 2
effective model with an emphasis on its tunable topological 2 Δ 1
phasetransitionbytheinter-layertunnelingparameter.Insec- Δ2 1 2 0
tion IV, we present our scenario to achieve the QAH effect Δ1 Δ1
viatuningeitheranexchangefieldoraperpendicularelectric E E
field. Lastly,wegiveabriefconclusioninsectionV. g g
1’
2’
0’
II. MODELANDMETHOD 3’
2’ 1’
Many of the usual 3D TI materials are layered materials. c) d) e)
For example, Bi Se can be viewed as consisting of Se-Bi- 0.1
Se-Bi-Sequintup2lela3yersstackedalong[111](z)direction.4,5 0.2 Trivial TI 0.3 band 0
E band 1
Thispictureisalsoapplicableforourfocused,Tl-basedcom- )V g
p(Soeu,nSd)-sBTi-lB(Sie(S,S1)−.3δ6S,3e8δW)2,hiennwthheiychatrheecloanyfienreudniatlobnecgozmedsirTecl-- ye( ΔΔ12 2Φ|0.2 band 2
g |0.1
tiontobethinfilms,theunderlyingmechanismfortheground re
n
statetobecomingtopologicallynon-trivialcouldbemorein- E
0 0 0
sightful if one considers each one-unit-cell layer as a build-
0.1 0.35 1 100 1 10 20
ingblockandthencouplesthemwithnearest-neighborinter-
t N l
layerhopping39,40. Asarepresentativesampletoachievethe z l
QAHeffect,westartwithaneffectiveone-unit-celllayerk·p
HamiltonianHl aroundΓpoint, FIG.1:(Coloronline)Theschematicenergyspectraofthethinfilm
 k2 vk D 0  modelaroundΓpointfromatrivialinsulator(a)toaZ2TI(b)with
2m + k twotypesofenergyscales: thesystembandgapEg andtheenergy
vk k2 0 D  differencebetweenneighboringbands∆ .Thepositiveintegernum-
H = − 2m k , (1) i
l D 0 k2 −vk  bers without (with) prime denote conduction (valence) bands. “0”
 0k D −2vmk k2+ and “0(cid:48)” denote bands with surface nature. (c) The calculated Eg
k − 2m and∆iasafunctionoftz. tzcisfoundtobe0.22eVhere. (d)The
where the basis vector for lth layer is taken to be φ† = bandgapEg ofthethinfilmattz =tzcasafunctionofthenumber
lk oflayersN. (e)Thewavefunctiondistribution|Φ|2atΓpointasa
[a†lk↑,a†lk↓,b†lk↑,b†lk↓]withaandboperatorscorrespondingto functionofllforthelowestthreeconductionbandsinanNl =20,Z2
localpz orbitalsofSeorSresidingonthetopandbottomunit TIthinfilmwithtz =0.23eV(listhelayerindex).Othernumerical
layerhybridizedwithneighboringatomicorbitalswithinit. k parametersaregiveninthemaintext.
(cid:113)
denotesa2Dmomentumwithmagnitudek = k2+k2and
x y
k = k ±ik ;mrepresentsthefermioneffectivemassand
± x y
Note that the tunneling between top and bottom positions
v isdeterminedbytheSOC.ThepresenceofD = d+ k2
inH isduetothenon-vanishingoverlapbetweenk thetopamndd of the two adjacent layers is tz. The real-space coordinates
l in z direction are positive integers with each number corre-
thebottomorbitalswithinthelayer. ThestructureofH can
l sponding to lth layer. This model is seen to correctly give
be clearly understood as two coupled Rashba systems (with
the spin texture for the surface Dirac cone states in Bi Se
2 3
oppositechiralities),whichpreservesbothTRSandinversion
and has been applied to transport simulations of a TI slab.41
symmetry(IS),formingakeycomponentinTIthinfilms.
When taking into account the Zeeman effect introduced by
To obtain the full Hamiltonian H for the thin films, we
theferromagneticorder(withpolarizationalongz direction),
thenstackeachquintuplelayeralongz directionbycoupling
asweshallconsiderlateron,wefurtheraddinanextraterm,
nearest-neighbor layers with a finite, spin-non-flip tunneling
H =M σ forH withσ actingonthespinspace.
M z z l z
termT asfollows
The parameter specific model for TlBi(S Se ) in the
  1−δ δ 2
H1 T 0 ··· trivial side is close to the Z2 phase transition critical points.
H =T0† HT†2 HT3 ······, (2) Wd e=h−av0e.2m2e=V,v0.0=6225.5eVeV−·1A˚A˚;−tz2;=m0d.2=eV−.402.0B4yeiVnc−re1aA˚s−in2g;
 ... ... ... ... tz,wecanobtainanontrivialZ2 topologicalinsulatorphase.
This model has been used to describe the topological phase
where transitioninTlBi(S Se ) anddeliveredthespin-polarized
1−δ δ 2
0 0 0 0 surfacerelatedstatesbothinthetrivialandthenontrivialre-
0 0 0 0 gioningoodagreementwithrecentspin-andangle-resolved
T =t 0 0 0. (3) photoemissionmeasurements.42TheHallconductivityinunit
z
0 t 0 0 of e2 iscalculatedbasedonKuboformalism21,43 whichisan
z h
3
integration of the Berry phase of Bloch wave function. The a) M b) C=1 c) C=2
z
Hallconductivityisthus: T
d
lE 2
e z
σ =e2(cid:126) (cid:88) Im[(cid:104)nk|Vx|mk(cid:105)(cid:104)mk|Vy|nk(cid:105)]. c fi 1’ 1’
xy m,n,k (Enk−Emk)2 ectri T EF 22 ’
(nf(Enk)−nf(Emk)), (4) El 1 1
2’
Substrate
where m,n are band indices, V are velocity operators,
x,y
and n (E ) stands for Fermi-Dirac distribution function at
f nk
eigen-energyE . FIG. 2: (Color online) The left panel (a) schematically shows our
nk
proposedscenariotoachievethefield-tunableQAHeffect: Letlay-
eredTlBi(S Se ) thinfilmwithferromagneticorderbegrownon
1−δ δ 2
III. TOPOLOGICALPHASETRANSITION:FROM a substrate, which provides an electric potential along z direction.
TRIVIALTONON-TRIVIALINSULATORS Therightpanel,(b)and(c),showsschematicbandstructures,spin-
texture(“skyrmion”-like),andband-inversionwithsuitablebandla-
belsforC =1andC =2QAHphases,respectively,att <t .
z zc
It would be helpful to begin with a warm-up example
to demonstrate how the effective model can be driven into
non-trivial Z insulators by tuning the parameter t which
2 z
is related to the lattice constants and the size of atomic or- zerobyincreasingthenumberoflayersN toat -dependent
l z
bitals in general. For concreteness, we adapt the parame- thresholdvalueN .Forinstance,ast approachestot from
lc z zc
ters mentioned above which are fitted to band structures of above,theenergygapisclosedonlywhenN goestoinfinity
lc
BiTl(S Se ) 38,42.However,itisworthmentioningthatthis inthetrue3Dlimit[seeFig. 1(d)]. Finally,itisquiteimpor-
1−δ δ 2
modelisalsoapplicablefortheotherBi X compounds.41 tant to notice that typically E < ∆ < ∆ for TIs, while
2 3 g 2 1
ThetypicalenergyspectraaroundΓpointforZ trivialand neart ∼ t evenonthetrivialinsulatingside,theseenergy
2 z zc
non-trivialthinfilmsareshowninFigs. 1(a)and1(b),respec- scalesareallcomparablewitheachother. Thisisthekeyob-
tively. Without breaking both TRS and IS, each band in the servation of our proposed scenario for realizing QAH effect
spectraisobviouslyspindegeneratedandthushasazeroex- withfield-tunableChernnumber,aswewillexplainbelow.
pectationvalueforthenetspinpolarization. BydefiningE
g
astheenergygapbetweenthelowestconductionbandandthe
highestvalencebandatΓpointinthethinfilm,and∆ asthe
i
gap between ith lowest conduction band and i+1th lowest
IV. QAHEFFECTFROMTRIVIALINSULATINGTHIN
one, on the trivial insulator side, we find that E is typically
g FILMS
larger than ∆ , while among ∆ they are comparable, as a
i i
resultofallquantumwellstates.
The aforementioned feature can be changed via chemical Inthissection,wedemonstratehowtheQAHEwithfield-
(Se)doping,oreffectively,increasethetunnellingenergyt . tunable Chern number can be achieved theoretically through
z
Close to a “critical” value of t (called t , at which the 3D our scenario. As shown schematically in Fig. 2(a), the
z zc
bulkbandgapcloses,independentofthethicknessofthethin TlBi(S Se ) thinfilm,growingonasubstrate,onthetriv-
1−δ δ 2
films),E becomescomparableto∆ andturnsintoasmaller ialinsulatingsidewitht ∼ t (doping-tunable)istakento
g 1 z zc
value rapidly after passing t , as can be seen in Figs. 1(b) be our prototype sample. During the process toward QAH
zc
and 1(c). This crossover phenomenon between the values of effect, we also assume that certain magnetic dopant can be
E and∆ marksatopologicalphasetransition(TPT):From distributed homogeneously over the whole sample and are
g 1
athinfilmoftrivialinsulatortoathinfilmofaZ topological ferromagneticallyorderedtoprovidethenecessaryexchange
2
insulator.Thelatterphaseissharplyidentifiedbythepresence field. Inthepresenceoftheexchangefield, thecoreconcept
ofthesurfacestates,aconsequenceofthebandinversion,with toexhibitQAHeffectissimplyfromthebandinversionphe-
Dirac-likeenergydispersion[seethebandlabeledby“0”and nomenon occurred between pairs of the conduction and va-
“0(cid:48)”inFig.1(b)].Notethatthesurfacestatesaredistinctfrom lence bands with different spin polarizations. Figs. 2(b) and
theusualquantumwell(bulk)states,becausetheirwavefunc- 2(c) schematically show one pair of the bands inverted (thus
tionswouldbealmostlocalizedateitherthetoporthebottom withChernnumberC =1)andtwopairsinverted(withChern
layer,ascomparedinFig. 1(e)byshowingwavefunctiondis- numberC =2),respectively.Afterbandinversion,eachband
tribution|Φ|2asafunctionofthelayerindexl. formsaskyrmion-likespintexturearoundΓpointinmomen-
There are a few things in the TI regime worth mentioning tum space, leading to non-vanishing Chern number. In ad-
here. Firstly,whent >t ,itisunderstoodthatthemassive- dition, in order to achieve the QAH phase with even higher
z zc
nessoftheDiracspectrumforthesurfacestatesisduetothe Chernnumber,morepairsofthebandinversionsareneeded.
tunnelingbarrier(orinevitablewavefunctionoverlappingbe- Accordingtothecomparisonamongvariousenergyscalesin
tweenboundaries),thatis,witharelativelysmallt (butstill the previous section, with a given strength of the exchange
z
> t ), in thin films. Secondly, similar to the usual case for field M , our focus on the trivial side near t would more
zc z zc
growingBi Se thinfilms44,theDiracmasscanbereducedto likelyarriveataphasewithhighC.
2 3
4
A. ExchangefieldtunableQAHeffect a) M =0 b) M <M
z z z1
0.3
WenowshowthebandstructureevolutionaroundΓpoint 2 1 2
2
asafunctionofM foraTlBi(S Se ) thinfilmwithN = Δ
z 1−δ δ 2 l 1
6inFig. 3.45 FermilevelE isalwayssetatzeroenergyand 0 ΔE 1
F g 1’
thelabelsi=1,2(i=1(cid:48),2(cid:48))denotetheithlowest(highest) 1’
1’ 2’
conduction(valence)bands,intheabsenceofMz,withspin- 2’
upandspin-downshownindifferentcolors. 2’
-0.3
StartingwithM =0,eachbandisspin-degenerateandthe
z c) M =M d) M <M
systemisintheC = 0(trivial)phase. IncreasingM causes z c1 z x1
z 0.3
spin splitting, which shifts the bands with spin-up and spin-
downpolarizationsinoppositedirectionswithrespecttoE ,
F 2
2
and hence reduces E [Fig. 3(b)]. As M reaches M , the
energy gap is closinggwith vanishing out-zof-plane spicn1mo- 0 1=1’ 1 1’
2’
mentatthetouchingpoint[Fig. 3(c)]. FurtherincreasingM 2’
z
reopensthegapagain, formsskyrmion-likespintexture, and
)
indicatesatopologicalphasetransitionduetobandinversion V-0.3
e
from C = 0 trivial phase to C = 1 QAH effect [Fig. 3(d)]. ( e) M =M f) M <M
y z x1 z c2
Wenotethatnowthebandlabeled“1”and“1(cid:48)”areswitched. rg 0.3
e
AsM =M ,theband“2”(“2(cid:48)”)meetswith“1(cid:48)”(“1”)[Fig. n
z x1 E
3(e)]andtheband“2”withspin-downand“2(cid:48)”withspin-up
become prominent near E if M continues increasing until 0 2=1’ 2
F z 2’=1 2’
M = M [Fig. 3(f)]. At M = M , the gap is closing
z c2 z c2
[Fig. 3(g)] andimplies asecondTPT, whichadds theChern
number by one and hence C = 2 after the gap-reopening as
-0.3
M >M [Fig.3(h)].Thefashionshownhereisinfactquite
z c2 g) M =M h) M >M
similartosomepreviousproposalsforgettingtheQAHeffect z c2 z c2
0.3
withhighChernnumberintheTIthinfilmregime33,34,37.
2’
0 2=2’
B. TowardhigherChernnumber 2
+
The model simulation and discussion in the previous sub- -
-0.3
section give us a clear physical picture of our mechanism to -0.1 0 0.1 -0.1 0 0.1
obtainhighChernnumber. Givensufficientlylargeexchange
field, the band gap can close and reopen multiple times due kx(Å-1) kx(Å-1)
to the presence of relatively intensive 2D subbands (quan-
tumwellstates),whichisaconsequenceofatrivialinsulating FIG.3: (Coloronline)Evolutionofthesub-bandstructureatt =
z
samplewith(nearly)criticalSe-doping. 0.2eVandN = 6aroundΓpointuponincreasingexchangefield
l
This mechanism is completely based on the “twist” of the Mz. The band labels are used as usual. Additionally, the color
spin-polarized bulk states in the quasi-2D system. Thus, it dressingineachbandrepresentsthemomentum-dependent,out-of-
is distinctive from the original proposal, where the QAH ef- the-plane spin polarization with arrows indicating the main spin-
polarized component. (a) M = 0; (b) 0 < M < M before
fectisachievedbygappingoutthesurfaceDiracconesin3D z z c1
firsttopologicalphasetransition(TPT);(c)M = M ,wherethe
TIs18,46. Toseethis,wefirstpresentenergyspectrumaround z c1
gap closes at the first time; (d) M < M < M in the QAH
Γ point and the corresponding spin texture for C = 2 QAH c1 z x1
phasewithC = 1; (e)M = M , wheretwolowestconduction
state with N = 20, i.e., in the thick film limit [see Fig. z x1
l bands meet; (f) M < M < M and the system approaches
x1 z c2
4(a)]. The initially lowest, spin-down polarized conduction secondbandinversion;(g)M = M atanothercriticalpoint;(f)
z c2
bandnowbecomesthethirdhighestvalenceband,indicating M > M , where the second TPT occurs, and the system enters
z c2
that the spin-polarized bands inverted twice. To explain the QAHstatewithC =2.
underlying physics, from the wave function distribution as a
function of l in Fig. 4(b), the first three lowest conduction
bandsallhavebulknature. Similarpropertiesarealsofound suggestsatleasttwoways: 1)Applylargeexchangefieldon
fortheotherbands. Suchafeatureresultsinasubtlebutim- the sample, and 2) increase the thickness of the sample. In
portant difference from previous studies33,34, because in our Figs. 4(c)and4(d), weexplicitlycalculatethebandgapand
case no surface bands are involved in the whole process. As the Hall conductivity as a function of M , respectively. The
z
we will consider later, this might affect the real experiments gap closes and reopens multiple times with the presence of
inwhicheachsampleisusuallygrownoncertainsubstrate. thecorrespondingquantizedplateausinσ ,indicatingarich
xy
To invert more spin-polarized bulk bands, our mechanism phase diagram of the QAH system. The Chern number in-
5
a) M =0.06 eV; C=2 b) t =0.2 eV a) b)
0.06 z z 0.1
1 CB M=0 M<M
) 0.06 2 CB z z c
V 2 CB 3 CB
e 3 CB
( ygr 0 1 CB 2Φ||0.04 )Ve 0
e (
nE yg -0.1
r
-0.06 0.02 e
n
-0.03 0 0.03 1 10 20 E
- + k (Å-1) l c) 0.1 d)
c) x d) Mz=Mc Mz>Mc
5
0.16 N=4
)V 4 l + 0
e 0.12 )h Nl=5
( pag 0.08 2e(/y 23 Nl=6 - -0.1
d ϭx -0.1 0 0.1 -0.1 0 0.1
n
aB 0.04 1 kx (Å-1) kx (Å-1)
0 0 e) f)
0 0.1 0.2 0.3 0 0.1 0.2 0.3 0.07
Exchange energy M (eV) )V Ez=0.0 eV 0.01
z e E =-0.03 eV
( z
FIG. 4: (Color online) (a) Band structure of the QAH phase with pagd EEzz==--00..0045 eeVV M2555z mmeeVV
C = 2inthethickfilmlimit, N = 20(M = 0.06eV)andthe n
l z a 0
colordressingisusedforspinpolarizations. (b)Thecorresponding B 0
wavefunctiondistributionsofthethreelowestconductionbandsat 0 0.1 0 0.04 0.07
theΓpointasafunctionoflayerindexl. (c)and(d)showtheband M (eV) E (eV)
gapevolutionandthequantizedHallconductivity,respectively,asa z z
functionofM forthegivenN = 4,5,6cases. Allplotshereuse
z l
FIG.5: (Coloronline)Theevolutionofthebandstructureuponin-
t =0.2eV.
z
creasingM withnon-vanishingE = −0.05eV:(a)M = 0;(b)
z z z
0 < M < M ; (c)M = M ; (d)M > M ,wherethesystem
z c z c z c
turnstotheC =1QAHphase. Thedegreeofout-of-the-planespin
polarizationisdenotedbycolor. (e)Thenon-linearbandgapevolu-
tionasafunctionofM withvariouselectricfieldstrength. Chern
z
creasesinoneintegerstepwhenM increases,asimilartrend numberC (startingfrom0)isaddedbyoneeachtimewhenthegap
z
compared with the usual quantum Hall system. In addition, closes.Thearrowsshownherearethechosenexchangefieldstrength
in the same figures by using different colors we also present tobecomparedin(f). (f)ElectricfieldtunableQAHphasesfortwo
given exchange field strength. The gap closing points separate the
bothquantitieswithdifferentnumberoflayers. Clearly,fora
C = 0andC = 1phases(bluecurve),andtheC = 1andC = 2
thickerfilmwithagivenM ,itismorelikelytoendupwith
z phases(greencurve),respectively.
a higher Chern number insulator due to the shrinking of ∆ ,
i
whichisinverselyproportionaltoN .
l
C. ElectricfieldtunableQAHE
From recent experiments in magnetic topological insula-
tors such as Cr or Fe doped (Bi, Sb) Te , people observed
2 3
thatthesemagneticdopantscanbeferromagneticallyordered One of the practical issues, based on the mechanism we
attemperatureoforder100K.47–50 Thecorrespondingeffec- have mentioned above, is the presence of certain substrates
tive exchange field strength M can be estimated as large whenpreparingthethinfilmsepitaxiallyintheexperiments.It
z
as 0.2 eV with 10% doping19,34 and thus strongly indicates isequivalenttothepresenceofaneffectiveelectricfield‘Ez’
the feasibility of our scenario. To roughly estimate what alongzdirectionandthisleadstoabrokenz →−zreflection
the largest Chern number could be achieved, one can sim- symmetryforthethinfilms. Hence, westudysystematically
plycounthowmanyquantumwellstates(subbands),labeled theeffectoftheelectricfieldbyaddingagivenlinearpotential
x, are able to be inverted by applying Mz. By noticing term along z direction, with potentials ∓Ez on the top and
that E ∼ ∆ ∼ 0.035 eV in Fig. 1(c) with N = 20, bottom surfaces of the sample, respectively, in Eq.(2) under
g i l
the number can be estimated through the following formula, variousappliedMz. Notethatthedispersionrelationdoesnot
x=[(Mz −0.0175)/0.035+1],where[···]denotesafloor changeifEz changessign.
function. Inserting the value of M ≈ 0.2 into the formula Wefirstconsidertheevolutionoftheenergyspectrumnear
z
yieldsx = 6,i.e.,thelargestC = 6inthiscase. Inaddition, ΓpointasafunctionofM withE = −0.05eV,asshown
z z
theenergyrangeforthisQAHphasetobestablecouldbeas in Figs. 5(a)-(d). In the absence of any exchange field, the
largeas0.035eV,aboveroomtemperature. electricfieldsimplyintroducesRashbatypeinteractionsinto
6
thesystemandconsequentlyeachspin-degeneratebandnow Fig. 5(f), it is feasible to apply an external electric field to
splitswiththeoriginallybandminimumshiftedawayfromΓ drive our focused system, namely, TlBi(S Se ) thin film
1−δ δ 2
point, while the spin degeneracy still keeps intact at Γ point from an originally Chern number C QAH phase to another
[see Fig. 5(a)]44. Assuming N = 2N, it is worth noting QAHphasewithChernnumberC +1. However, wewould
l
that the wave function distribution of these lowest (highest) liketopointoutthatthistuningapproachisefficienttoobtain
conduction(valence)bandsaroundΓpointismainlyfromthe QAHphaseuptoC = 2. ForgettinghigherChernnumbers,
contributionsofNthandN+1thlayersinthemiddle.Thisis it might become unstable because several sub-bands would
insharpcontrastwiththeusualTIthinfilms(i.e.,t > t ), come into play around E and the system may not maintain
z zc F
inwhichthelowestconduction(highestvalence)bandhasthe itsinsulatingnatureduringtheprocess.
largestweightfromthetopandthebottomlayers.
Uponturningontheexchangefield,asonecanseeinFig.
5(b),allthespin-degeneratepointsofRashba-likebandsatΓ
V. CONCLUSION
pointopenupgapsandthebandgapofthesystemreducesto
zeroasM reachestoacriticalvalueM [seeFig. 5(c)]. Fur-
z c
therincreasingM resultsinaspin-polarizedbandinversion In summary, we have presented our scenario to achieve
z
and reopens the band gap, leading again to the QAH phase QAH effect with field-tunable Chern number via a model
[seeFig. 5(d)]. Itisinterestingtonoticeafewsubtlediffer- study. Remarkably,themodelcandescribetopologicalphase
encesfromthecasewithout E : 1)WhenM = 0theband transitionfromaZ trivialtoannon-trivialinsulatingthinfilm
z z 2
gap is smaller due to the shift of the conduction band mini- for realistic materials such as TlBi(S Se ) . By showing
1−δ δ 2
mum;2)atM =M ,theenergydispersionisnon-linearand theband-structureevolution,spin-texture,andhencethespin-
z c
the spin texture around Γ point is relatively simple. Impor- polarizedbandinversion,weclearlydemonstratethefeasibil-
tantly,theaboveobservationsshowthatthecriticalexchange ity of our approach to tune the Chern numbers of the QAH
fieldstrengthisloweredandonecanpossiblyachieveQAHE phases through changing either the exchange field strength
with high Chern number by tuning the electric field, as we ortheelectricfieldstrengthintopologicallytrivialthinfilms
explainnext. neartheZ criticalpoint(toTIphase). Inparticular,westress
2
Fig. 5(e)illustratesthebandgapevolutionasafunctionof that the necessary exchange field strength to exhibit high-
M withagivenelectricfieldE inanN =20thinfilm.The ChernnumberQAHeffectcanbereducedfurtherwhenpush-
z z l
band gap repeatedly closes and reopens, indicating that the ingthesystemclosertothecriticalpointandcombiningwith
system undergoes topological phase transitions several times the benefit from the substrate. Therefore, we hope that this
uptotheQAHphasewithhighChernnumber(C = 3inour papercouldstimulateexperimentalworksalongthisdirection
plot). Significantly, afterconsideringseveraldifferentvalues inthenearfuture.
of E , we find that the critical exchange energies to achieve
z
C = 1 and C = 2 phases, respectively, are less than the
cases in the absence of the electric potential. To examine it
Acknowledgments
morecarefully,wetaketworepresentativeinitialphasesinour
system at E = 0, as indicated by the arrows shown in Fig.
z
5(e): 1) a trivial C = 0 phase with fixed M =0.025 eV and ThisresearchissupportedbytheNationalResearchFoun-
z
2)aC = 1QAHphasewithfixedM =0.055eV.Purposely, dation,PrimeMinister’sOffice,SingaporeunderitsNRFfel-
z
they are chosen just prior to QAH phases with C = 1 and lowship (NRF Award No. NRF-NRFF2013-03). W.F.T. ac-
C = 2separately. Wethencomputethecorrespondingband knowledges the support from MOST in Taiwan under Grant
gap as a function of E for each of them. As one can see in No.103-2112-M-110-008-MY3.
z
∗ Electronicaddress:[email protected] S.Hor,R.J.Cava,andM.Z.Hasan,Phys.Rev.Lett.103,146401
1 M.Z.HasanandC.L.Kane,Rev.Mod.Phys.82,3045(2010). (2009).
2 X.-L.QiandS.-C.Zhang,Rev.Mod.Phys.83,1057(2011). 8 Jeffrey C. Y. Teo, Liang Fu, and C. L. Kane, Phys. Rev. B 78,
3 Y.Ando,J.Phys.Soc.Jpn.82,102001(2013). 045426(2008).
4 Haijun Zhang, Chao-Xing Liu, Xiao-Liang Qi, Xi Dai, Zhong 9 Forinstance,seeD.A.PesinandA.H.MacDonald,NatureMa-
FangandShou-ChengZhang,NaturePhysics5,438(2009). terials11,409(2012),andreferencestherein.
5 Y.Xia,D.Qian,D.Hsieh,L.Wray,A.Pal,H.Lin,A.Bansil,D. 10 F.Mahfouzi,N.Nagaosa,andB.K.Nikolic,Phys.Rev.Lett.109,
Grauer,Y.S.Hor,R.J.CavaandM.Z.Hasan,NaturePhysics5, 166602(2012).
398-402(2009). 11 M.Gotte,T.Paananen,G.Reiss,andT.Dahm,Phys.Rev.Applied
6 Y.L.Chen,J.G.Analytis,J.-H.Chu,Z.K.Liu,S.-K.Mo,X.L. 2,054010(2014).
Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. 12 L.FuandC.L.Kane,Phys.Rev.Lett.100,096407(2008).
Fisher,Z.HussainandZ.-X.Shen,Science325,5937(2009). 13 Xiao-LiangQi,TaylorL.Hughes,andShou-ChengZhang,Phys.
7 D.Hsieh,Y.Xia,D.Qian,L.Wray,F.Meier,J.H.Dil,J.Oster- Rev.B78,195424(2008).
walder,L.Patthey,A.V.Fedorov,H.Lin,A.Bansil,D.Grauer,Y. 14 Xiao-LiangQi, RundongLi, JiadongZang, Shou-ChengZhang,
7
Science323,1184(2009). 35 Pallab Goswami and Sudip Chakravarty, Phys. Rev. Lett. 107,
15 RundongLi,JingWang,Xiao-LiangQiandShou-ChengZhang, 196803(2011).
NaturePhysics6,284(2010). 36 Su-Yang Xu, Y. Xia, L. A. Wray, S. Jia, F. Meier, J. H. Dil, J.
16 G.RosenbergandM.Franz,Phys.Rev.B82,035105(2010). Osterwalder, B. Slomski, A. Bansil, H. Lin, R. J. Cava, M. Z.
17 H.OoguriandM.Oshikawa,Phys.Rev.Lett.108,161803(2012). Hasan,Science332,560(2011).
18 LiangFuandC.L.Kane,Phys.RevB.76,045302(2007). 37 HuaJiang,ZhenhuaQiao,HaiwenLiu,andQianNiu,Phys.Rev.
19 R.Yu,W.Zhang,H.-J.Zhang,S.-C.Zhang,X.Dai,andZ.Fang, B85,045445(2012)
Science329,61(2010). 38 HsinLin,R.S.Markiewicz,L.A.Wray,L.Fu,M.Z.Hasan,and
20 F.D.M.Haldane,Phys.Rev.Lett.61,18(1988). A.Bansil,Phys.Rev.Lett.105,036404(2010)
21 MahitoKohmoto,Ann.ofPhys.160,343(1985). 39 T.Das,andA.V.Balatsky,Nat.Commun.4,1972(2013).
22 Cui-ZuChang,JinsongZhang,XiaoFeng,JieShen2,Zuocheng 40 L.FuandE.Berg,Phys.Rev.Lett.105,097001(2010).
Zhang, Minghua Guo, Kang Li, Yunbo Ou, Pang Wei, Li-Li 41 G.Gupta,H.Lin,A.Bansil,M.B.A.Jalil,andG.Liang,Phys.
Wang,Zhong-QingJi,YangFeng,ShuaihuaJi,XiChen,Jinfeng Rev.B89,245419(2014).
Jia,XiDai,ZhongFang,Shou-ChengZhang,KeH,YayuWan, 42 Su-Yang Xu, Madhab Neupane, Ilya Belopolski, Chang Liu,
LiLu,Xu-CunMa,Qi-KunXue,Science340,6129(2013). Nasser Alidoust, Guang Bian, Shuang Jia, Gabriel Landolt,
23 Chao-XingLiu,Xiao-LiangQi,XiDai,ZhongFang,andShou- BatoszSlomski,J.HugoDil,PavelP.Shibayev,SusmitaBasak,
ChengZhang,Phys.Rev.Lett.101,146802(2008). Tay-Rong Chang, Horng-Tay Jeng, Robert J. Cava, Hsin Lin,
24 ZhenhuaQiao,ShengyuanA.Yang,WanxiangFeng,Wang-Kong ArunBansilandM.ZahidHasan,Nat.Commun.6,6870(2015).
Tse,JunDing,YuguiYao,JianWang,andQianNiu,Phys.Rev. 43 Titus Neupert, Luiz Santos, Claudio Chamon, and Christopher
B82,161414(2010). Mudry,Phys.Rev.B86,165133(2012).
25 DiXiao,WenguangZhu,YingRan,NaotoNagaosaandSatoshi 44 YiZhang, KeHe, Cui-ZuChang, Can-LiSong, Li-LiWang, Xi
Okamoto,Nat.Commun.2,596(2011). Chen, Jin-Feng Jia, Zhong Fang, Xi Dai, Wen-Yu Shan, Shun-
26 S.-M.Huang,S.-T.Lee,andC.-Y.Mou,Phys.Rev.B89,195444 Qing Shen, Qian Niu, Xiao-Liang Qi, Shou-Cheng Zhang, Xu-
(2010). CunMaandQi-KunXue.NaturePhysics6,584(2010).
27 K.Sun,H.Yao,E.FradkinandS.A.Kivelson,Phys.Rev.Lett. 45 When the warping effect is relatively small, such as in
103,046811(2009). TlBi(S Se ) ,applyingM istheonlyeffectivewaytomake
1−δ δ 2 z
28 W.-F.Tsai,C.Fang,H.Yao,andJ.P.Hu,NewJournalofPhysics bands magnetically inverted; when the warping effect is large,
17,055016(2015). such as in Bi Te , one may also apply in-plane magnetization
2 3
29 M.OnodaandN.Nagaosa,Phys.Rev.Lett.90,206601(2003). alongcertaindirectiontoinduceQAHphase.
30 J.G.Checkelsky,R.Yoshimi,A.Tsukazaki,K.S.Takahashi,Y. 46 Dung-HaiLee,Phys.Rev.Lett.103,196804(2009).
Kozuka, J. Falson, M. Kawasaki and Y. Tokura, Nature Physics 47 V.Kul?bachinskii,A.Kaminskii,K.Kindo,Y.Narumi,K.Suga,
10,731(2014). P. Lostak, and P. Svanda, Pis?ma Zh. Eksp. Teor. Fiz. 73, 396
31 XufengKou,Shih-TingGuo,YabinFan,LeiPan,MurongLang, (2001)[JETPLett.73,352(2001)].
YingJiang, QimingShao, TianxiaoNie, KoichiMurata, Jianshi 48 Z. Zhou, Y.-J. Chien, and C. Uher, Phys. Rev. B 74, 224418
Tang, Yong Wang, Liang He, Ting-Kuo Lee, Wei-Li Lee, and (2006).
KangL.Wang,Phys.Rev.Lett.113,199901(2014). 49 C.-Z.Chang, J.Zhang, M.Liu, Z.Zhang, X.Feng, K.Li, L.-L.
32 Scott A. Skirlo, Ling Lu, Yuichi Igarashi, John Joannopoulos, Wang,X.Chen,X.Dai,Z.Fang,X.-L.Qi,S.-C.Zhang,Y.Wang,
MarinSoljacic(2015),preprintavailableatarXiv:1504.04399 K.He,X.-C.Ma,andQ.-K.Xue,Adv.Mater.25,1065(2013).
33 Zhenhua Qiao, Hua Jiang, Xiao Li, Yugui Yao, and Qian Niu, 50 XufengKou,YabinFan,MurongLang,PrameyUpadhyaya,and
Phys.Rev.B85,115439(2012). KangL.Wang,SolidStateCommunications215-216,34(2015).
34 JingWang,BiaoLian,HaijunZhang,YongXu,andShou-Cheng
Zhang,Phys.Rev.Lett.111,136801(2013).