Robust Dual Topological Character with Spin-Valley Polarization in a Monolayer of the Dirac Semimetal Na$_3$Bi

2.2 MB·

by  Chengwang Niu

RobustDualTopologicalCharacterwithSpin-ValleyPolarizationinaMonolayeroftheDirac
SemimetalNa Bi
3
Chengwang Niu1,∗ Patrick M. Buhl1, Gustav Bihlmayer1, Daniel
Wortmann1, Ying Dai2, Stefan Blu¨gel1, and Yuriy Mokrousov1
1Peter Gru¨nberg Institut and Institute for Advanced Simulation,
Forschungszentrum Ju¨lich and JARA, 52425 Ju¨lich, Germany
2School of Physics, State Key Laboratory of Crystal Materials,
Shandong University, 250100 Jinan, People’s Republic of China
Topologicalmaterialswithbothinsulatingandsemimetalphasescanbeprotectedbycrystalline(e.g.mirror)
symmetry.Theinsulatingphase,calledtopologicalcrystallineinsulator(TCI),hasbeenintensivelyinvestigated
7
andobservedinthree-dimensionalmaterials.However,thepredictedtwo-dimensional(2D)materialswithTCI
1
0 phase are explored much less than 3D TCIs and 2D topological insulator, while so far considered 2D TCIs
2 almostexclusivelypossessasquarelatticestructurewiththemirrorChernnumberCM =−2.Here,wepredict
theoretically that hexagonal monolayer of Dirac semimetal Na3Bi is a 2D TCI with a mirror Chern number
n
C = −1. Thelargenontrivialgapof0.31eVistunableandcanbemademuchlargerviastrainengineering
M
a
whilethetopologicalphasesarerobustagainststrain,indicatingahighpossibilityforroom-temperatureobser-
J
vationofquantizedconductance. Inaddition,anonzerospinChernnumberCS = −1isobtained,indicating
5
the coexistence of 2D topological insulator and 2D TCI, i.e. the dual topological character. Remarkably, a
2
spin-valleypolarizationisrevealedinNa Bimonolayerduetothebreakingofcrystalinversionsymmetry.The
3
dualtopologicalcharacterisfurtherexplicitlyconfirmedviaunusualedgestates’behaviorundercorresponding
]
i symmetrybreaking.
c
s
-
l Thediscoveryoftopologicalinsulator(TI)1,2 hastriggered together with inversion symmetry breaking can lead to cou-
r
t anexplosionofnoveltopologicallynontrivialphases,suchas pled spin and valley physics, in which the new degree of
m
thetopologicalcrystallineinsulator(TCI),forwhichtherole freedom offers a promising route to the eventual realization
. ofthetime-reversalsymmetryisreplacedbythecrystal(mir- of valleytronic devices28,29. The spin-valley polarization has
t
a ror) symmetry3–5. The hallmark of a TCI, similar to a TI, is been observed experimentally in MoS monolayer30, which
2
m
thepresenceofgaplesssurface/edgestateswithDiracpoints isatopologicallytrivialinsulator. Therefore, anaturalques-
- insideoftheinsulatingbulkenergygap.Inpresenceofcrystal tion arises as to whether the spin-valley polarization in non-
d
mirror symmetry, the coexistence of TI and TCI phases has trivial insulators, such as 2D TIs and 2D TCIs, is possible.
n
been predicted in three dimensions (3D) for Bi Sb 6 and Recently, thin films of the Dirac semimetal Na Bi31,32 have
o 1−x x 3
c Bichalcogenides7–9, andthustheyexhibitadualtopological been fabricated by molecular beam epitaxy33 and, therefore,
[ character(DTC).Recently,unusualtopologicalsurfacestates in the present study, we take Na Bi as an example and pro-
3
for a 3D DTC system have been observed experimentally8,9. posetherealizationofthe2DDTCinamonolayerofNa Bi
1 3
In the 2D case, graphene maybe a prototypical example of with a band gap of 0.31 eV, which is well above the energy
v
1 DTC10,11.However,theextremelysmallbandgapofgraphene scale of room temperature. The calculated spin Chern num-
0 makes it very difficult to verify the DTC in this material ex- ber C = −1 and mirror Chern number C = −1 confirm
S M
3 perimentally12. Todate,the2DTIsareidentifiedexperimen- the 2D DTC phase directly. In addition, the spin-valley po-
7 tallyinHgTe/CdTe13 andInAs/GaSb14 quantumwellsatlow larizationduetothelackofthespatialinversionsymmetryis
0
temperatures, and a lot of 2D TIs with giant band gaps have investigated.
.
1 been predicted to exist as a result of substrate interaction ef- Thedensityfunctionalcalculationsareperformedusingthe
0 fect15,chemicalfunctionalization16–20,orglobalstructureop- generalizedgradientapproximation(GGA)ofPerdew-Burke-
7 timization21. In many cases the complex structures and the Ernzerhof (PBE)34 for the exchange correlation potential as
1
lack of mirror symmetry in such materials forbid the forma- implemented in the FLEUR code35 as well as in the Vienna
:
v tionofa2DTCIphase. Ontheotherhand,2DTCIsaresofar ab-initiosimulationpackage(VASP)36,37. A20A˚ thickvac-
Xi limitedtotheoreticalpredictionsthataremainlyrestrictedto uumlayerisusedtoavoidinteractionsbetweennearestslabs
SnTemultilayers10,(Sn/Pb)(Se/Te)monolayers22–24,Tl(S/Se) forVASPwhilethefilmcalculationsarecarriedoutwiththe
r
a monolayers25, and SnTe/NaCl quantum wells26 with mirror filmversionoftheFLEURcode38. SOCisincludedinthecal-
Chern number CM = −2. The even number of band inver- culationsself-consistently. ThemaximallylocalizedWannier
sions leads to a vanishing Z2 invariant. Therefore, 2D TCIs functions (MLWFs) are constructed using the wannier90
withCM =−2cannotbe2DTIsprotectedbythetimerever- codeinconjunctionwiththeFLEURpackage.39,40
salsymmetry. Thus,forfurtherinvestigationandapplications Bulk Na Bi in hexagonal P6 /mmc structure is a three-
3 3
ofDTCintwodimensions,itisessentialtoextendthedomain dimensional (3D) counterpart of graphene, which hosts 3D
ofcandidate2DTIsandTCIsbothwithrespecttotopological Dirac points in its electronic structure and is called a topo-
manifestations(i.e. differentCM27)andmaterialrealisation. logical Dirac semimetal31,32. The bulk crystal structure con-
For both 2D TIs and 2D TCIs, the spin-orbit coupling sistsofstackedtriplelayersalongthez-direction. Eachtriple
(SOC) is known to play a vital role. In addition, the SOC layerhasfouratomswhichareoneBiinWyckoff2cposition,
2
FIG.1. (a)Topand(b)sideviewofhoneycombNa Bimonolayer,
3
wheretheunitcellisindicatedbythedashedlines.(c)TheBrillouin
zoneof2DNa Bimonolayerandtheprojected1DBrillouinzones.
3
one Na(1) in 2b position, and two Na(2) in 4f position. In
Figs. 1(a) and (b) the side and top view of the Na Bi triple
3
layerarepresented,withBiandNaatomsformingthehoney-
comblattice. Unlikeinthebulkmaterial, theinversionsym-
metryisbrokeninaNa Bitriplelayerbutthemirrorsymme-
3
tryz → −z ispreserved, inexactanalogytoaMoS mono-
2
layer41. Hereafter,wecallsuchatriplelayeraNa Bimono-
3
layer. To check its energetic stability, the formation energy
FIG.2. Orbitally-resolvedbandstructuresforNa Bimonolayer(a)
3
iscalculatedbyEf = ENa3Bi −3µNa −µBi, whereENa3Bi withoutand(b)withSOC,weightedwiththecontributionofBi-sand
is the total energy of the Na3Bi monolayer, µNa and µBi are Bi-px,py states. TheFermilevelisindicatedwithadashedline. (c)
thechemicalpotentialsofNaandBiatoms,respectively. The Momentum-resolvedpolarizationofspinperpendiculartothemirror
calculated formation energy of −0.78 eV indicates that the planeforthehighestoccupiedband.(d)Berrycurvaturedistribution
Na Bimonolayerisenergeticallystable. ofthehighestoccupiedbandsintheK−Γ−K(cid:48)direction.
3
Figures 2(a) and (b) present the orbitally resolved band
structuresofNa BimonolayerwithoutandwithSOC,respec-
3
tively,thatdeliverpreliminaryinsightintotopologicalproper- andΩ±i(k)istheBerrycurvatureofalloccupiedbandscon-
tiesofthesystem. Duetothepresenceoftime-reversalsym- structed from respective mirror projected states in the mirror
metry,thebandsatvalleysKandK(cid:48) areenergeticallydegen- plane,calculatedaccordingto
erate,andthusweonlyshowthedispersionaroundK.Inthe
absence of SOC, Bi-px and Bi-py orbitals contribute to the Ω(k)=−2Im (cid:88) (cid:104)ψnk|υx|ψmk(cid:105)(cid:104)ψmk|υy|ψnk(cid:105), (2)
valence band maximum (VBM) while the conduction band (εmk−εnk)2
m(cid:54)=n
minimum (CBM) is dominated by Bi-s orbitals with a direct
bandgapof0.16eV.SwitchingonSOCleadstoaninversion wherem,narebandindices,ψ andε arethecorre-
m/nk m/nk
oftheVBMandtheCBM,andans-pbandinversionoccursat spondingwavefunctionsandeigenenergiesofbandm/n, re-
Γpoint. Theinsulatingcharacterispreservedwithabandgap spectively, and υ arethevelocityoperators. TheMLWFs
x/y
of0.31eV,indicatingthefeasibilityofexperimentalobserva- are constructed to calculate the Berry curvature efficiently.
tionofthe2Dtopologicalpropertiesofthismaterialatroom With z → −z mirror symmetry, the calculated Chern num-
temperature.Tofurtherconfirmourresults,thebandstructure bers are respectively C±i = ∓1, leading to a mirror Chern
is checked by using the more sophisticated Heyd-Scuseria- numberCM = −1. ThisindicatesthattheNa3Bimonolayer
Ernzerhof hybrid functional method (HSE06)42. Similar to is a 2D TCI. Interestingly, the CM = −1 case we consider
1-T(cid:48) MoS 43, the nontrivial phase has a larger band gap (∼ hereistopologicallydistinctfromthepreviouslyreported2D
2
0.4eV)whenSOCisincluded. TCIs,suchas(Sn/Pb)Te10,23,24andTl(S/Se)25withCM =−2.
The existence of the mirror symmetry z → −z (see Here, in Na Bi monolayer, band inversion occurs at Γ
3
Figs. 1(a) and (b)) for Na Bi monolayer promises the pos- point, i.e., weacquireanoddnumberofbandinversions. To
3
sibility of realizing the TCI that is characterized by the so- identifytherelationshipbetweenthe2DTIandtheoddnum-
called mirror Chern number3,6, which is defined as C = berofband inversionsinNa Bimonolayer, wecalculate the
M 3
(C − C )/2, where C and C are the Chern numbers spinChernnumberC 45–47whichcanbedirectlyrelatedtothe
+i −i +i −i S
formirroreigenvalues+iand−i,44 Z topologicalinvariantofthesystem.C providesequivalent
2 S
characterization to Z number in that for time-reversal sym-
1 (cid:88) (cid:90) 2
C = Ω (k)d2k, (1) metric and inversion symmetric systems the even values of
±i 2π ±i
n<EF BZ CS correspondtoatopologicallytrivialinsulatorstate, while
3
mimic a magnetic environment, we compute the matrix el-
ementsofthePaulimatricesσ (α = x,y,z)inthebasisof
α
MLWFs,whichallowsustoconsidertheeffectofanexchange
fieldappliedalongdifferentdirections. Foranexchangefield
perpendiculartothemirrorplane,H =B ·σ ,thetime-
mag ⊥ z
reversal symmetry is broken while the mirror symmetry is
maintained. In this case, as shown in Fig. 3(c) for the Bi-
Na(1)termination,theDiracpointmovesslightlyawayfrom
theΓ¯ point,whileabandgapdoesnotopenasaconsequence
of2DTCIphase’ssurvival. Iftheexchangefield,ontheother
hand, is in the plane, H = B · σ , both time-reversal
mag (cid:107) x
andmirrorsymmetriesarebrokenandtheedgestatesbecome
gapped (Fig. 3(d)). This behavior is reminiscent of that in
Bi (Se/Te) 7,48, but with different directions of an exchange
2 3
field owing to a different sense of the mirror sysmmetry in
thesetwocompounds(B inthe3DTIcorrespondstoB in
z (cid:107)
the2DTI)7,48.
Exposinghoneycomblatticestoinversionsymmetrybreak-
ing provides a new, so-called valley, tunable degree of free-
dom in addition to spin and charge. The valley degree of
freedom is receiving considerable attention these days due
FIG. 3. Localization-resolved edge states of Na Bi monolayer for
3
differentconfigurations.(a)Bi-Na(1)terminationwithoutamagnetic to potential application in valleytronics28,30. To demonstrate
field, (b) Bi-Na(1)-Na(2) termination without a magnetic field, (c) theeffectoftheinversionsymmetrybreaking,wefocusnow
Bi-Na(1) termination a magnetic field perpendicular to the mirror on the spin-valley coupling of the Na Bi monolayer by con-
3
plane,and(d)Bi-Na(1)terminationwithamagneticfieldwithinthe sideringthespin-polarizationofoccupiedstatesinreciprocal
mirrorplane. Insetsshowthecorrespondingzoom-inattheΓ¯point. space. Since the in-plane components of the spin polariza-
Colorfromlightgreentoredrepresentstheweightofatomslocated tionarevanishingduetothepresenceofmirrorsymmetry,in
frommiddletooneedgeoftheribbonstructures.
Fig. 2(c) we plot the momentum-resolved out-of-plane spin
polarizationofthehighestoccupiedband. Itisclearlyvisible
that the highest occupied state exhibits the spin polarization
odd values of CS indicate the emergence of a TI phase45–47. whichisofoppositesignatvalleysK andK(cid:48). Wefurtherin-
The CS is given by the difference of the Chern numbers for specttheBerrycurvatureofthehighestoccupiedbandalong
the spin-up (C+) and spin-down (C−) projected manifolds, the K −Γ−K(cid:48) path, plotted in Fig. 2(d). An odd behav-
CS = (C+−C−)/245. Theσz matrix,(cid:104)φmk|σz|φnk(cid:105),iscon- ior of the Berry curvature with respect to the valley agrees
structedanddiagonalizedtodistinguishthespin-upandspin- with the symmetry analysis in terms of time- and structural-
downmanifolds47. TheChernnumberforeachspinmanifold inversionsymmetry,similarlytothewell-knowncaseofspin-
is C+ = −1 and C− = 1, yielding the spin Chern number valleycouplinginMoS2monolayer49.Valleypolarizationthat
CS = −1. Thisclearlydemonstratesthe2DTInatureofthe iscoupledwithspinwillsuppressesspinandvalleyrelaxation
Na3Bi monolayer. Therefore, Na3Bi monolayer exhibits the and is promising to prepare the information carriers for the
DTCwithrespecttothe2DTIand2DTCIphases. next-generationelectronicandoptoelectronicdevices30.
TofurtherconfirmtheDTC,weinvestigatetheedgestates Havingestablishedourmaterial’sDTC,accompaniedbya
of 1D nanoribbons of Na3Bi monolayer. The nonzero CM bandgapof0.31eV(whichislargeenoughforpracticalap-
and/or CS should support the gapless edge states bridging plications at room temperature), we finally test its stability.
the conduction and valence bands, which exhibit a band gap The phonon spectrum calculation shows imaginary frequen-
when both the time-reversal and mirror symmetries are bro- ciesaroundtheMpoint,butnotattheΓpoint,indicatingthat
ken. Based on a description in terms of MLWFs of the thebandinversionatthispointisrobust. Wedemonstratethis
Na3Bimonolayer,thetight-bindingHamiltoniansofnanorib- by investigating the band inversion under various strains as
bons along two different directions with a width of 60 unit wellassubstrates50. Themagnitudeofstrainisdescribedby
cells are constructed. The calculated band structures on one the ratio a/a , where ”a ” and ”a” denote the lattice parame-
0 0
sideoftheribbonsarepresentedinFigs.3(a)and(b),respec- ters of the unstrained (5.31 A˚) and strained systems, respec-
tively. Onecanclearlyseeapairofgaplessedgestatesinthe tively.TheresultsofthecalculationsshowninFig.4,indicate
2Dgap. TheDiracpointaroundΓ¯ isaclearconsequenceof that the band gap of Na Bi monolayer can be significantly
3
thenon-trivialtopologicalcharacterofthesystem. enhanceduponstraining,similarlytotheresultsreportedpre-
Generally, time-reversal symmetry breaking generates a viously18,25. Thereisnoband-gapclosing-reopeningprocess
gap in the surface/edge states of TIs1,2 while mirror symme- between the valence and conduction bands (∆E and E )
Γ gap
trybreakingisindispensablefortheformationofabandgap inalargerangeofstrainfrom−12%to20%. Thepersistent
inthesurface/edgestatesofTCIs3. Onewaytodestroythese band inversion indicates that the topological character of the
symmetries is to introduce the magnetism in the system. To systemisrobustagainstthesubstrate-imposedstrain,whichis
4
FIG.4. (a)VariationoftheenergygapsforNa Bimonolayerasafunctionofstrain. Thegaps(∆E andE ininset)betweenthevalence
3 Γ gap
andconductionbandsremainwhileastrain-inducedbandinversionbetweenthevalencebands√(∆EV√ininset)occurs. (b)Orbitally-resolved
bandstructuresforNa Bimonolayersandwichedbetweenasubstrateof(b)BNwith2×2and 7× 7supercellsand(c)Na Sb,weighted
3 3
withthecontributionofBi-sandBi-p ,p states.TheDTCremainsintactwithcorrespondingsubstrates.
x y
important for further experimental investigations and device In summary, based on first-principles calculations, we re-
applicationsduetothefactthatafreestandingfilmisusually vealedthatthe2DTIand2DTCIphasescancoexistinNa Bi
3
hard to grow. Figure 4(b) shows the calculated band struc- monolayer with a large band gap of 0.31 eV. Nonzero spin
tures of quantum well structures, that retain the mirror sym- ChernnumberandmirrorChernnumber,aswellasnontrivial
m√etry,√withNa3Bimonolayersandwichedbetween2×2and topologicaledgestatesconfirmthedualtopologicalcharacter
7× 7BNmonolayers. Aswecansee,theenergygapat clearly. As the mirror symmetry is perserved for an out-of-
theMpointchangesmuchstrongerthanatΓwiththeinverted planeexchangefield,thegaplessedgestatessurvivebutmove
band gap at the Γ point surviving in both cases (although a away from the time-reversal invariant momenta, while a gap
f√urther√band inversion occurs between valence bands for the opensforin-planeexchangefields. Atlast,strainengineering
7× 7case,seealso∆E ofFig.4(a)),whichprovesthat showsthatthedualtopologicalcharacterinNa Bimonolayer
V 3
DTC is preserved with respective substrate-induced strain of isrobustinalargerangeofstraininwhichthenontrivialband
−6.2%and19.7%. Aweakinterlayerinteractionisexpected gapcanbetunedefficiently.
betweenBNandNa Bibecauseofthelargerelaxedinterlayer This work was supported by the Priority Program 1666 of
3
distances∼4.0A˚.Wethenconsiderastronginterlayerinter- theGermanResearchFoundation(DFG)andtheVirtualInsti-
action via the topologically trivial substrate Na Sb that has tuteforTopologicalInsulators(VITI).Weacknowledgecom-
3
thesamecrystalstructureasNa Bibutcreates−2%epitaxial putingtimeonthesupercomputersJUQUEENandJURECA
3
strain. As shown in Fig. 4(c), the band inversion remains at at Ju¨lich Supercomputing Centre and JARA-HPC of RWTH
theΓpoint. AachenUniversity.
∗ [email protected] arXiv:1604.08886.
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