Laser-induced topological transitions in phosphorene with inversion symmetry

1 MB·

by  C. Dutreix

Laser-inducedtopologicaltransitionsinphosphorenewithinversionsymmetry
C. Dutreix, E. A. Stepanov, M. I. Katsnelson
Radboud University, Institute for Molecules and Materials,
Heyendaalseweg 135, 6525AJ Nijmegen, The Netherlands
Recentabinitiocalculationsandexperimentsreportedinsulating-semimetallicphasetransitionsinmultilayer
phosphoreneunderaperpendiculardcfield,pressureordoping,asapossibleroutetorealizetopologicalphases.
Inthiswork,weshowthatevenamonolayerphosphorenemayundergoLifshitztransitionstowardsemimetallic
andtopologicalinsulatingphases,provideditisrapidlydrivenbyin-planetime-periodiclaserfields.Basedona
four-orbitaltight-bindingdescription,wegiveaninversion-symmetry-basedprescriptioninordertoapprehend
6 the topology of the photon-renormalized band structure, up to the second order in the high-frequency limit.
1 Apartfromtheinitialbandinsulatingbehavior,twoadditionalphasesarethusidentified.Asemimetallicphase
0 withmasslessDiracelectronsmaybeinducedbylinearpolarizedfields, whereasellipticpolarizedfieldsare
2 likelytodrivethematerialintoananomalousquantumHallphase.

n
u
J Morethanacenturyago,Bridgmanreportedhisattemptto particle-hole-symmetry-protected topological transition to-
changephosphorusfromwhitetored,whichactuallyresulted wardananomalousquantumHallphase,inwhichanon-zero
6
1 inthe discoveryofthemost thermodynamicallystablephos- Chernnumbercharacterizestheexistenceofchiralboundary
phorus allotrope at room temperature, namely, black phos- modes. Therefore, these three distinct phases are predicted

] phorus (BP) [1]. This is, like graphite and transition metal in monolayer BP by simply varying the polarization and
l
l dichalcogenides,alayeredmaterialinwhichone-atomic-thick intensityofasingleexternalfield,whichismorecontrollable
a
h films are bound to each other via weak van der Waals inter- than pressure or doping, and is also suitable for realizations
- actions, enabling the isolation of a few layers by mechani- inultracoldatomicgasesandphotoniccrystals[22–26].
s
e calexfoliation[2–4]. Low-energyelectronsrevealastrongly
m anisotropicsemiconductorwithadirectenergygapandhigh Tight-binding description — Phosphorene consists of a
. chargecarriermobility,featuresthatmakethephosphorusal- rectangularBravaislatticeandanout-of-planeunitcellmade
t
a lotrope particularly suitable for electronic applications, such of four phosphorus atoms, each of which have three nearest
m as field-effect transistors [5, 6]. From a fundamental per- neighbors, as illustrated in Fig. 1. Electronic properties are
- spective,investigationsofpressure-inducedelectronicandop- reasonably described within a tight-binding description that
d
tical properties in bulk BP addressed, a few decades ago, involvesuptotenhoppingprocesses[13]. Thecorresponding
n
o the possibilities to realize structural phase transitions and to Blochbandstructureforspinlesselectronsisthengivenbya
c tunethesemiconductinggap[7–9]. Recentabinitiocalcula- 4×4matrixwhich,inthesublatticebasis{A1,B2,A2,B1},can
[ tionsandexperimentspointedoutthatgaptuningunderpres- genericallybewrittenas

sure, dc field, or doping, could even result in an insulating-
2
 
6559v saiceramColuiopmtnheetarttsoaaewrlsyliaictrnodLthtithfhisseehnriateobzanolt-virczaeoantwmsiiootpnirookounsfn,idnswymmemuafmtoletecirltuiarasyyl-eh[pr1erBr0oe–tPe1o,c6tnth]eum.dsotoonppoeolnaloiyngeg-r H(k)= FFFF4123∗∗∗((((kkkk)))) FFFF4321∗∗((((kkkk))))e−iϕ(k) FFFF1243∗((((kkkk))))eiϕ(k) FFFF3214((((kkkk)))) . (1)
0 BP, namely phosphorene, when it is rapidly driven by
. in-plane homogeneous time-periodic laser fields. Within a
1
0 four-orbital tight-binding approximation introduced recently
6 [13], we provide an inversion-symmetry-based prescription
1 that enables us to apprehend the band structure topology up
:
v tothesecondorderinthehigh-frequencylimit. Threedistinct
i phasesarethenidentified,dependingonthefieldpolarization
X
andmagnitude. Inparticular,linearpolarizedfieldsarelikely
r
a to drive phosphorene from a band insulating phase toward
a semimetallic one with two nonequivalent Dirac cones,
thus mimicking electronic properties of graphene massless
Diracquasiparticles[17,18]. Rightatthetransition,theuni-
versal semi-Dirac band structure, which describes massless
FIG.1. (Coloronline)Topviewofphosphorene. Theblackdashed
electrons only along one direction, is recovered [19–21]. lineoutlinestheunitcellmadeofatoms —A , —A , —B ,
1 2 2
Being the underlying mechanism of topological transitions — B1. Nearest-neighbor vectors are(cid:32)d1 = a0((cid:32)cosθ,sin(cid:32)θ), d2 =
[13–16],thisLifshitztransitionshowsthatnon-trivialphases (cid:32)a0(cosθ,−sinθ),andd3 = b0(1,0),withθ(cid:39)48.395◦,a0 (cid:39)2.216Å
may be induced in phosphorene too. Besides, we show andb0(cid:39)0.716Å[6].Numbersfrom1to10refertothetendifferent
hoppingsallowedfromthecentralbiggerA site.
that, under elliptic polarizations, it may also undergo a 1
2
Matrix components Fi=1...4 are functions of hopping ampli- �
tudes and wavevector k such that H(k+G) = H(k), with G �
C
areciprocallatticevector[27]. Besidesϕ(k) = k −k fork +
x y x
andk giveninunitsofa−1anda−1,respectively. �� ��
y 1 2
+
Time-periodic laser fields — Suppose that an in-plane ho- �� �� ��
mogeneous time-periodic electric field of frequency Ω is
turned on, driving the system into a nonequilibrium steady
state.Howelectronsrelaxintosuchastabilizedregimeiscur-
rentlyunderinvestigations[28–30]. Thefieldyieldsavector
potential of the form A(t) = (A cosΩt,A sin[Ωt − φ],0), FIG. 2. (Color online) Parity products at the time-reversal invari-
x y
whereasthescalarpotentialiswipedout,forweconsiderthe antmomentaoftheBrillouinzone(left)andlowestenergybandsof
temporalgauge. Hereinafter,itisassumedthatc = (cid:126) = 1,so phosphorene(right).EnergyisgivenineV.
that A = eE Ω−1 with E the x(y)-componentofthe
x(y) x(y) x(y)
fieldmagnitude.Thefieldpolarizationisellipticforφ=0and
dynamics,whereasthesystemevolutionbetweentwostrobo-
linear for φ = π/2. The vector potential subsequently enters scopictimesisencodedin∆ (τ). Thisfinallyresultsin
n
matrix(1)viaPeierlsphases[27].
Thetimeevolutionofaquantumstateisruledbythetime- H˜ = H and H˜ =−(cid:88)[Hm,H−m] , (5)
dependent Schro¨dinger equation i∂τΨ(λ,τ) = λH(τ)Ψ(λ,τ), 1 0 2 m
where τ = Ωt and frequency is encoded in the dimension- m>0
less parameter λ = δE/Ω. Energy scale δE corresponds to whereH = (cid:82)+π dτ eimτH(τ)definestheFloquetmatrixform
m −π 2π
phosphorenebandwidth(14eV [13])whichinvolvesthefour of the Hamiltonian. The first order in λ refers to the time-
photon-renormalizedenergybands. Besides, H(τ)isgivenin averagedHamiltonian,sinceelectronscannotfollowthefield
unitsofδEandisdimensionlesstoo.Hereweareinterestedin dynamics.ItimpliesthatH˜ (k)onlydiffersfrommatrix(1)in
1
thehigh-frequencylimitλ(cid:28)1anduseanextensionofMag- thefield-renormalizedhoppingamplitudes. Thesecondorder
nusexpansion[31],inthespiritofRef.[32].Itreliesonauni- inλdescribesthesumofallvirtualemissionsandabsorptions
tarytransformationdefinedbyΨ(λ,τ) = exp{−i∆(τ)}ψ(λ,τ), of m photons. A nearest-neighbor approximation yields
where ∆(τ) = (cid:80)+n=∞1λn∆n(τ), and where operators ∆n(τ) re- H˜2(k) = 0andH˜2(k) = M(k)⊗σ3 forlinearandellipticpo-
move the τ-dependent terms of H(τ). It is also assumed that larizations,respectively[27]. Therefore,ellipticallypolarized
∆n(τ) has the same time periodicity as H(τ) and averages at fields yield a non-vanishing second-order contribution that
zero[33]. Thistransformationleadstoi∂τψ(λ,τ)= H˜ψ(λ,τ), breakstime-reversalsymmetry,asM∗(−k)=−M(k).
witheffectiveHamiltonian
Topologicaltransitionsfrombandparities—Thepurpose
H˜ =λei∆(t)H(t)e−i∆(t)−iei∆(t)∂e−i∆(t). (2)
t of the upcoming lines is to apprehend the topology of the
photon-renormalizedbandstructureofphosphorenegivenby
Snider’sidentity[34]implies
H˜(k) (cid:39) λH˜ (k)+λ2H˜ (k). To do so, we give a prescription
1 2
(cid:88)∞ (cid:8)(cid:0)−i∆(τ)(cid:1)n,−i∂ ∆(τ)(cid:9) in the spirit of Ref. [36] which relates the inversion eigen-
∂τe−i∆(τ) = (n+1)! τ e−i∆(τ), (3) valuesatthetime-reversalinvariantpointstotheBerryphase
n=0 oftheoccupiedBlochwavefunctionsalonghalftheBrillouin
zone. Note that, if the occupation distribution is a priori in-
where the repeated commutator is defined for two operators
X andY by{1,Y} = Y and{Xn,Y} = [X,{Xn−1,Y}][35]; the volvedintheobservationofnonequilibriumtopologicalprop-
erties, transport is well described by the effective static band
squarebracketsdenotetheusualcommutator.Usingtheseries
representationH˜ =(cid:80)∞ λnH˜ ,togetherwithEqs.(2)and(3), structureinthehigh-frequencylimit,becauseelectronscannot
onecanthendeterminne=1operantorsH˜ and∆ iteratively. Here, absorbphotonsoftheoff-resonantlight[37].
n n
First,phosphoreneisinvariantunderspatialinversion. This
werestrictthehigh-frequencyanalysistothesecondorderin
transformation consists of reversing all real-space coordi-
λ,whichleadsto
nates,andinterchangingelectronicorbitalsA withB andA
1 1 2
H˜1 = H(τ)−∂τ∆1(τ), (4) withB2.Inmomentumspace,thisimpliesI†H˜(k)I= H˜(−k),
wheretheinversionoperatoriswrittenintermsofPaulima-
i
H˜ =i[∆ (τ),H(τ)]− [∆ (τ),∂∆ (τ)]−∂ ∆ (τ). tricesasI=σ ⊗σ . Atthetime-reversalinvariantmomenta
2 1 2 1 t 1 τ 2 1 1
Γ=G/2,thisyieldsthecommutationrelation[H˜(Γ), I]=0.
Imposingthesetwooperatorstobestaticrequiresustoretain Consequently, there exists a basis of Bloch eigenstates that
only τ-independent terms in H˜ and H˜ , which is achieved arealsoeigenvectorsofIwitheigenvaluesπ(Γ) = ±1. This
1 2
by time averaging over a period because H(τ) and ∆ (τ) are eigenvalue, named parity, labels the energy bands at time-
n
time periodic. So this averaging method leads to effective reversal invariant momenta Γi=0,...3 of the Brillouin zone de-
time-independentHamiltoniansthatdescribethestroboscopic fined in Fig. 2. Since we are interested in crossing between
3
(cid:18)(cid:113) (cid:19)
φ=0 φ=π/2 hopping amplitudes u = t J α2+α2±2α α sinφ
10 10 1± 1 0 x y x y
TI SM and u = t J (β ), where α = |d |A , α = |d |A , and
BI BI 2 2 0 x x 1x x y 1y y
β = |d |A . Besides,thenatureofthedispersionrelationis
x 3x x
encodedinthedeterminantoftheHessianmatrix,namely
A A
y y
DetHess±(Γ0)=±u1+u1−u2sgn[∆±(Γ0)] . (9)
Intheabsenceoftime-periodicfield,hoppingamplitudesful-
fill u1+u1−u2 > 0 and ∆±(Γ0) > 0. Thus, DetHess±(Γ0) > 0
0 0
0 A 10 0 A 10 and the system lies in a band insulating phase, with elliptic
x x
parabolic dispersion relations. When tuning the field magni-
FIG.3.(Coloronline)Phasediagramsforelectricfieldswithelliptic tude, ∆+(Γ0) changes signs, and so does DetHess+(Γ0). It
(left)andlinear(right)polarizations.Lightpurpleareasrefertoband describes a hyperbolic parabolic dispersion relation, so that
insulating(BI)phasescharacterizedbyeiγC = +1. Darkpurplear- there are now two Dirac cones in the vicinity of Γ0. There-
eascorrespondtosemimetallic(SM)andtopologicalinsulating(TI) fore, thesystemhasundergoneaLifshitztransitiontowarda
phasesinwhicheiγC =−1.ComponentsAxandAyaregiveninÅ−1. semimetallic phase. From Eqs. (6) and (7), this is equiva-
lently characterized by eiγC = −1, where the non-vanishing
Berry phase provides a topological stability of every Dirac
voafloacnccuepaineddceonnedrguyctbioanndbsaninddse,xweedinbtyrond:uδce=th(cid:81)epaπrit(yΓp)r.oTduhcist cone inside C. Right at the transition, when ∆+ = 0, the de-
i n n i terminant of the Hessian matrix is null, whereas its trace is
qcounandtuictytioonnlybacnhdasngoefsowpphoensittehepcarroitsiseisn,gwinhvicohlvdeesfivnaelesnacebaanndd not: TrHess+(Γ0) = u21+ +u21− > 0. Consequently, the dis-
persion relation is linear in one direction albeit parabolic in
inversion. Besides,nobandinversioncanoccuratΓi=1,3 and others. ThisspectrumprovidesauniversalsignatureofDirac
δi=1,3 =−1isfixed[27]. AtmomentaΓi=0,2,thisproductis conemergingintwodimensionsandissaidtobesemi-Dirac
δi =sgn[∆−(Γi)]sgn[∆+(Γi)] , (6) [19]. It happens for example for A ∼ 2.1Å−1 (see Fig. 4),
whichcanbeobtainedformagnitudesofthefieldlargerthan
where∆ (Γ) = F (Γ)±F (Γ)refertotheenergygapsof 30V/Å. Here, one canshow that, right atthe transition, the
± i 2 i 1 i
thetwooccupiedbandsatΓ. semi-Diracoccupiedbandsatisfies
i
Ontheotherhand,ellipticpolarizedfieldsbreakbothtime- (cid:113)
E (Γ +Q)(cid:39)− Q2c2+Q4/4m2, (10)
reversalandchiralsymmetriesindividually, meaningrespec- − 0 x x y
tively that H˜2(k) (cid:44) H˜2∗(−k) and SH˜2(k)S (cid:44) −H˜2(k) with (cid:113)
S = σ0 ⊗σ3. Nonetheless, theirproductalwaysleadstothe whichreferstoDiracelectronswithceleritycx = u21++u2−1
particle-hole symmetry S†H˜(k)S = −H˜∗(−k), regardless of along Q = u1√+Kx+u1−Ky, and Shro¨dinger electrons with effec-
x
the field polarization. Thus, eigenstates come in pairs with u21++u21−
opposite energies, and opposite parities because {I,S} = 0. tivemassm= u21++u2−1 alongQ = u1√+Ky−u1−Kx.
Thespectrumisthenparticle-holesymmetricandbandinver- (u1++u−1)u+1u−1 y u21++u21−
sions can only occur at zero energy. Based on inversion and
particle-hole symmetries, we can finally connect the Berry
phase γ picked up by the occupied Bloch wavefunctions
C
alongapathCwhichencloseshalftheBrillouinzone,tothe
parityproductsatΓi=0,2[27]. Thisresultsin
eiγC =δ δ . (7)
0 2
This quantity, which is illustrated in Fig. 3, changes signs
onlywhenphoton-inducedbandinversionsoccuratΓi=0,2. ky ky
0 0
Linear polarized fields — For linear polarizations, phos-
phoreneband-structureissimplydescribedbythetimeaver-
ageH˜ (k)uptothesecondorderinthehigh-frequencylimit.
1
AroundΓ ,occupiedenergybands“±”canbeexpandedas
0 2 3A 4 5 0 kx
1
E±(Γ0+K)(cid:39)−|∆±(Γ0)|− KT·Hess±(Γ0)·K. (8) FIG. 4. (Color online) Lowest-energy bands in the semimetallic
2
phase induced by linearly polarized fields (φ = π/2). Figures on
HereK = (k + k )/2,K = (k − k )/2,andHessianmatrix top describe semi-Dirac (left) and massless Dirac electrons (right).
x x y y x y
Hess describesthelocalcurvatureofenergybands. Energy EnergyisgivenineV.FiguresatthebottomshowtheDiraccones
gap ∆±±(Γ0) = u1+ + u1− ± u2 relies on field-renormalized trajectories(blacklines)fortheconfigurationAx =Ay=A.
4
φ=0 φ=0 rene is an anisotropic semiconductor in which engineering
1 1
topologically stable chiral Dirac charge carriers, either two-
or one-dimensional in the bulk, or one-dimensional along
0 0 a boundary, may be achieved by tuning the polarization
and intensity of a single external field. All these features
occurringinthesamenon-compoundthinfilmmakeitrather
1 1 appealingforelectronicnanodevicerealizations.
−−π ky π −−π ky π
FIG.5. (Coloronline)Spectraofaribbonwithbeardedtermination TheauthorswouldliketothankA.N.Rudenko,A.Itin,J.
along direction y, under circularly polarized fields A = 2.80Å−1 MentinkandS. Brenerforusefulcomments. Thisworkwas
x
(left)and Ax = 2.95Å−1 (right). Thelatteryieldschiraledgestates supportedbyNWOviaSpinozaPrizeandbyERCAdvanced
withinthebulkenergygap,inagreementwiththetopologicalphase Grant338957FEMTO/NANO.
predictedinFig.3.
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6
SUPPLEMENTALMATERIAL:LASER–INDUCEDTOPOLOGICALTRANSITIONSINPHOSPHORENEWITHINVERSION
SYMMETRY
Firstorderinthehigh-frequencylimit
Effectivetight-bindingdescription
The first order in the high-frequency limit shows that the system can be described by the time-independent Hamiltonian
H˜ = H , where H = (cid:82)+π dtH(t) is the time-average of the Hamiltonian over a period. In momentum space, its derivation
1 0 0 −π 2π
involvesthecalculationoftermslike
(cid:90) +π dt (cid:18)(cid:113) (cid:19)
2πeidxAxcosteidyAysin(t−φ) = J0 (dxAx)2+(dyAy)2+2dxdyAxAysinφ ,
−π
whichessentiallyresultsintherenormalizationofhoppingamplitudes. Itisthenstraightforwardtoshowthat
 
F (k) F (k) F (k) F (k)
H˜1(k)= FF341∗∗((kk)) FF432∗((kk))e−iϕ(k) FF123((kk))eiϕ(k) FF214((kk)) ,
F∗(k) F∗(k) F∗(k) F (k)
2 1 4 3
whereF1(k)= f1+ f5+ f8,F2(k)= f2+ f6+ f9,F3(k)= f3+ f7+ f10andF4(k)= f4,andwherefunctions fi=1...10aregivenby
(cid:18)(cid:113) (cid:19) (cid:18)(cid:113) (cid:19)
f1 =t1J0 α2x+α2y −2αxαysinφ +t1J0 α2x+α2y +2αxαysinφ eiky ,
f =t J (β )e−iϕ(k),
2 2 0 x
f =2t J (2α )cosk ,
3 3 0 y y
(cid:18)(cid:113) (cid:19) (cid:18)(cid:113) (cid:19)
f4 =2t4J0 (αx+βx)2+α2y +2(αx+βx)αysinφ e−iϕ(2k) cosKx+2t4J0 (αx+βx)2+α2y −2(αx+βx)αysinφ e−iϕ(2k) cosKy,
(cid:18)(cid:113) (cid:19) (cid:18)(cid:113) (cid:19)
f5 =t5J0 (αx+2βx)2+α2y +2(αx+2βx)αysinφ e−ikx +t5J0 (αx+2βx)2+α2y −2(αx+2βx)αysinφ e−iϕ(k),
f6 =t6J0(2αx+βx)eiky ,
f =2t J (2α +2β )cosk ,
7 7 0 x x x
(cid:18)(cid:113) (cid:19) (cid:18)(cid:113) (cid:19)
f8 =t8J0 α2x+9α2y +6αxαysinφ e2iky +t8J0 α2x+9α2y −6αxαysinφ e−iky ,
(cid:18)(cid:113) (cid:19) (cid:18)(cid:113) (cid:19)
f9 =t9J0 (2αx+βx)2+4α2y +4(2αx+βx)αysinφ e2iky +t9J0 (2αx+βx)2+4α2y −4(2αx+βx)αysinφ ,
(cid:18)(cid:113) (cid:19) (cid:18)(cid:113) (cid:19)
f =2t J 4(α +β )2+4α2+8(α +β )α sinφ cos2K +2t J 4(α +β )2+4α2−8(α +β )α sinφ cos2K ,
10 10 0 x x y x x y x 10 0 x x y x x y y
with α = |d |A , α = |d |A , β = |d |A and K = (k + k )/2, K = (k −k )/2. The expressions above rely on the
x 1x x y 1y y x 3x x x x y y x y
tight-bindingmodelintroducedin[13],wherehoppingparameters,givenineV,havebeenestimatedast =−1.486,t =3.729,
1 2
t =−0.252,t −0.071=,t =−0.019,t =0.186,t =−0.063,t =0.101,t =−0.042andt =0.073. Itisworthmentioning
3 4 5 6 7 8 9 10
herethatthenearest-neighborhoppingamplitudesareatleastoneorderofmagnitudelargerthanfurtherhoppingamplitudes.
7
Parityproductsofoccupiedenergybands
As long as the phosphorene crystal has inversion symmetry, the Bloch Hamiltonian matrix commutes with the inversion
(cid:104) (cid:105)
operatorI = σ ⊗σ atthetime-reversalinvariantmomenta,i.e. I,H˜(Γ) = 0. Thus,theeigenvaluesofI,namelyπ = ±1,
1 1 i n
becomegoodquantumnumbersthatlabeltheenergybands. AtmomentaΓ andΓ ,thefourenergybandssatisfy
0 2
E (Γ )= F +F −F −F with π (Γ )=+1,
1 0,2 3 2 1 4 1 0,2
E (Γ )= F +F +F +F with π (Γ )=+1,
2 0,2 3 2 1 4 2 0,2
E (Γ )= F −F −F +F with π (Γ )=−1,
3 0,2 3 2 1 4 3 0,2
E (Γ )= F −F +F −F with π (Γ )=−1.
4 0,2 3 2 1 4 4 0,2
Althoughwevoluntarilyomittomentionitintheexpressionsaboveformoreclarity,themomentumdependentfunctionsFj=1,...4
areevaluatedattime-reversalinvariantpoints. Besides,theorderofmagnitudesofhoppingamplitudes(t1,2 (cid:29) ti=3...10)implies
thatinversionsbetweenvalenceandconductionbandsarelikelytooccuronlybetweenbandsofoppositeparities,eitherwhen
E (Γ )=E (Γ )orwhenE (Γ )=E (Γ ). BothconditionsareindependentofF andF ,andcanberewrittenas
2 0,2 3 0,2 1 0,2 4 0,2 3 4
F ±F =0.
2 1
Theparityproductofthe(two)occupiedenergybandsatΓi=0,2isthengivenby
(cid:89)
δ = π (Γ)
i n i
occupiedn
=sgn[F (Γ)−F (Γ)]sgn[F (Γ)+F (Γ)].
2 i 1 i 2 i 1 i
Althoughthiscriterionisvalidwithinthetenhoppingmodel,itsuggeststhatbandinversionsarealreadywelldescribedwithina
nearest-neighbortight-bindingapproximationas,first,distantprocessesareatleastoneorderofmagnitudesmallerthannearest-
neighboronesand,second,theirassociatedzeroth-orderBesselfunctionsdecayfasterwiththefieldthanfornearestneighbors.
This is illustrated in Fig. 6 which depicts the four energy levels at Γ for elliptically and linearly polarized fields within both
0
frameworksthetenhoppingmodelandthenearest-neighborapproximation.
AtmomentaΓ andΓ ,onesimilarlyobtains
1 3
(cid:113)
E (Γ )= F + F2+(F +F )2 with π (Γ )=+1,
1 1,3 3 2 1 4 1 1,3
(cid:113)
E (Γ )= F + F2+(F −F )2 with π (Γ )=−1,
2 1,3 3 2 1 4 2 1,3
(cid:113)
E (Γ )= F − F2+(F −F )2 with π (Γ )=−1,
3 1,3 3 2 1 4 3 1,3
(cid:113)
E (Γ )= F − F2+(F +F )2 with π (Γ )=+1.
4 1,3 3 2 1 4 4 1,3
φ=0 φ=π/2
8 8
+1 +1
1 1
− −
0 0
8 8
− 0 10 − 0 10
A A
FIG. 6. (Color online) Illustration of the four energy levels at Γ for elliptic (φ = 0) and linear (φ = π/2) polarizations as a function of
0
A = A = A,giveninÅ−1. Inbothcases, F hasbeentakenastheenergyoriginandenergyisgivenineV. Thedashedlinesrefertothe
x y 3
ten-hoppingmodelandareingoodagreementwiththefullonesthatrefertothenearest-neighborapproximation. Lightanddarkbluesrefer
topositive(+1)andnegative(−1)parities,respectively
8
Importantly,whenE (Γ )=E (Γ ),orwhenE (Γ )=E (Γ ),nobandinversioncanoccuratΓ andΓ ,wheretheparity
2 1,3 3 1,3 1 1,3 4 1,3 1 3
productoftheoccupiedenergybandsisfixed: δ =−1. Thisresultsin
1,3
(cid:89)3
δ =δ δ .
i 0 2
i=0
ThisquantityonlydependsonbandinversionslikelyoccuringattimereversalinvariantmomentaΓ orΓ , amechanismthat
0 2
plays a central role in topological features of phosphorene band-structure and will be the purpose of the upcoming sections.
Notefinallythat,inthenearest-neighborapproximation, F ±F correspondstotheenergygap,hencethenotation∆ (Γ ) =
2 1 ± 0,2
F2(Γ0,2)−F1(Γ0,2). TheparityproductoftheoccupiedenergybandsΓi=0,2canthenberewrittenas
δi =sgn[∆+(Γi)]sgn[∆−(Γi)] . (11)
Bandinversionsfromparityproducts
In the high-frequency regime, the Bloch band structure is described by the time-average of the Hamiltonian matrix which,
limitedtonearest-neighborprocesses,reads
 
0 0 F (k) F (k)
H˜1(k)=0F1∗(k) 0F2∗(k)e−iϕ(k) 0F21(k)eiϕ(k) 0F12(k).
F∗(k) F∗(k) 0 0
2 1
Since it satisfies the relation SH˜ (k)S = −H˜ (k), where S = σ ⊗σ , H˜ is said to have chiral symmetry. As a result, the
1 1 0 3 1
eigenstates come in pairs with opposite energies and the spectrum is particle-hole symmetric. One can then square the above
matrixtoshowthatthedispersionrelationsoftheoccupiedenergybandslabelledbytheindex±satisfy
E2±(K)=[u1+cosKx+u1−cosKy±u2]2+[u1+sinKx+u1−sinKy]2,
(cid:18)(cid:113) (cid:19)
whereu =t J α2+α2±2α α sinφ andu =t J (β ). Bandinversionsbetweenvalenceandconductionbandsoccurat
1± 1 0 x y x y 2 2 0 x
zeroenergyandextremaofthedispersionrelation. Thoseextremafulfil∇ E2 =0andE (cid:44)0,orequivalently
k ± ±
0=u2(cid:2)u1+sinKx+u1−sinKy(cid:3)
0=(cid:2)2u1−cosKy±u2(cid:3)u1+sinKx−(cid:2)2u1+ cosKx±u2(cid:3)u1−sinKy.
Ifu (cid:44)0andu (cid:44)0,then
1± 2
0=(cid:2)u1+sinKx+u1−sinKy(cid:3)
0=(cid:2)u1+ cosKx+u1−cosKy±u2(cid:3)u1+sinKx,
whereE±(K)=[u1+cosKx+u1−cosKy±u2](cid:44)0. Thus,theextremaofthedispersionrelationarenecessarilylocatedatΓ0and
Γ . Theyaretheonlymomentawherebandinversionsareallowedtotakeplace. Consequently,theparityproductsδ andδ are
2 0 2
sufficienttokeeptrackofbandinversions.
Notethatothercasesforwhichatleastoneoftheparametersu oru vanishesturnouttobeunstableas,forexample,they
1± 2
mayberemovedbydistanthoppingprocesses.
Secondorderinthehigh-frequencylimit
Effectivetight-bindingdescription
Asdetailedinthemaintext,thesecondorderinthehigh-frequencylimitisdescribedbythetime-independentmatrix:
(cid:88)[H ,H ]
H˜ =− m −m .
2 m
m>0
9
Sincethesystemisqualitativelywellcharacterizedwithinthenearest-neighbortight-bindingapproximation,onefindsthat
 
0 0 F (k) F (k)
Hm(k)=0FF1∗∗,−m((kk)) 0FF2∗∗,−m((kk))e−iϕ(k) 00F12,,mm(k)eiϕ(k) 00F21,,mm(k) ,
2,−m 1,−m
where
(cid:90) +π dt (cid:90) +π dt
F1,m(k)=t1 2π eimt eiaxcost eiaysin(t−φ)+t1eiky 2π eimt eiaxcost e−iaysin(t−φ)
−π −π
(cid:20) (cid:18)(cid:113) (cid:19) (cid:18)(cid:113) (cid:19) (cid:21)
=t1e−imπ/2 eimθ1 Jm a2x+a2y +2axaycos(φ+π/2) +eimθ2 Jm a2x+a2y −2axaycos(φ+π/2) eiky ,
and
F (k)=t e−imπ/2J (c)e−iϕ(k).
2,m 2 m
Intheexpressionsabove,itisimpliedthatθ = −arctan aysin(φ+π/2) ,θ = +arctan aysin(φ+π/2) ,a = d A anda = d A .
1 ax+aycos(φ+π/2) 2 ax−aycos(φ+π/2) x 1x x y 1y y
Besides,onecanshowthat
H (k)H (k)=(cid:32) Km(k) 0 (cid:33) , with K (k)=(cid:32) F1,mF1∗,m+F2,−mF2∗,−m F1,mF2∗,me−iϕ+F2,−mF1∗,−m (cid:33) .
m −m 0 K (k) m F F∗ eiϕ+F F∗ F F∗ +F F∗
−m 2,m 1,m 1,−m 2,−m 1,m 1,m 2,−m 2,−m
Thecommutationrelationcanthenberewrittenas[H ,H ]=(K −K )⊗σ . Twocaseshavetobedistinguished.
m −m m −m 3
• Forlinearpolarizations(φ=π/2andθ =θ =0),
1 2
H˜ =0.
2
As a result, the system is still described by its time-average. A fortiori chiral and time-reversal symmetries, as well as
invarianceunderspatialinversionarepreserved.
• Forellipticpolarizations(φ=0andθ=θ =θ ),
1 2
(cid:32) (cid:33) (cid:32) (cid:33)(cid:34) (cid:32) (cid:33) (cid:32) (cid:33) (cid:35)
(cid:16) (cid:17) k k k −k k −k
H˜ (k)=τ sin k σ ⊗σ +τ cos x sin y cos x y σ ⊗σ +sin x y σ ⊗σ ,
2 3 y 0 3 4 1 3 2 3
2 2 2 2
where

τ3 =4t12(cid:88)m>0 Jm2 (a)smin(2mθ)
τ4 =8t1t2(cid:88) Jm(a)Jmm(c)sin(mθ) .
m>0
Thoughthesystemisstillinvariantunderinversionsymmetry,i.e. I†H˜ (k)I= H˜ (−k),time-reversalandchiralsymme-
2 2
triesarebothbrokenindividually,sincerespectivelyH˜∗(−k)=−H˜ (k)andS†H˜ (k)S= H˜ (k).
2 2 2 2
Bandinversionsfromparityproducts
Thesystemisdescribedbythetime-independenteffectiveHamiltonianH˜ (cid:39) λH˜ +λ2H˜ . Inthecaseofalinearlypolarized
1 2
field,H˜ =0. Consequently,thecriterionalreadygiveninEq. (11)forthetimeaverageinordertokeeptrackofbandinversions
2
stillholds. Inthecaseofellipticallypolarizedfields,however,H˜ (cid:44) 0andwehavetofindthemomentawherebandinversions
2
are likely to occur. To do so, one can first remark that, although chiral and time-reversal symmetries are individually broken,
theirproductleadstothefollowingparticle-holesymmetryrelation:
S†H˜(k)S=−H˜∗(−k).
10
Thus,eigenstatescomeinpairs|ψ (k)(cid:105)andS|ψ∗(−k)(cid:105)withoppositeenergies,respectivelyE(k)and−E(−k). Thespectrumis
n n
particle-holesymmetricandbandinversionscanonlyoccuratzeroenergy. Inordertoevaluateatwhatmomentatheyarelikely
totakeplace,wecansimplysolveDetH˜(k)=0,whichcanequivalentlyberewrittenas
 (cid:32)k (cid:33)
u1cos 2y =±u2e−ik2x
τ3cos(cid:32)ky(cid:33)sin(cid:32)ky(cid:33)=±τ4cos(cid:32)kx(cid:33)sin(cid:32)ky(cid:33) .
2 2 2 2
When u1τ4 (cid:44) u2τ3, where u1 = u1+ = u1− in the case of elliptic polarization, the determinant only vanishes at k = Γ0 for
∆ (Γ ) = 0. Asaresult,ellipticallypolarizedfieldsyieldanenergygapwhichisonlyallowedtocloseatΓ . Thisimpliesthat
± 0 0
δ =+1isfixed,andfinally
2
(cid:89)3
δ =δ .
i 0
i=0
Topologicaltransitionsfrombandinversions
In this section, we first define some properties of the Berry connexion and the Berry curvature, before establishing their
relationtotopologicalfeaturesoftheBlochbandstructure,namelyaπ-quantizedBerryphaseandafirstChernnumber.
BerryconnectionandBerrycurvature
Startingfromitsdefinition,theBerryconnectionassociatedtoallbands(occupiedandempty)satisfies
(cid:88)
A(k)=i (cid:104)ψ (k)|∇ |ψ (k)(cid:105)
n k n
n
(cid:88)
=i (cid:104)ψ∗(−k)|S†|∇ |S|ψ∗(−k)(cid:105)
n k n
n
(cid:88)
=i (cid:104)ψ∗(−k)|S†I†|∇ |IS|ψ∗(−k)(cid:105)
n k n
n
(cid:88)
=i (cid:104)ψ∗(−k)|S†I†|∇ |u (−k)(cid:105)(cid:104)u (−k)|IS|ψ∗(−k)(cid:105)
n k m m n
mn
(cid:88)
=i (cid:104)u (−k)|IS|ψ∗(−k)(cid:105)(cid:104)ψ∗(−k)|S†I†|∇ |u (−k)(cid:105)
m n n k m
mn
(cid:88)
+i (cid:104)ψ∗(−k)|S†I†|u (−k)(cid:105)∇ (cid:104)u (−k)|IS|ψ∗(−k)(cid:105)
n m k m n
mn
(cid:104) (cid:105)
=−A(−k)+iTr M†(k)∇ M(k)
k
A(k)=−A(−k)+i∇ lnDetM(k)
k
whereM (k)=(cid:104)ψ (−k)|IS|ψ∗(−k)(cid:105)isaunitarymatrix.
mn m n
Moreover,theBerryconnectionsforemptyandoccupiedbandsarerelatedinthefollowingway:
(cid:88)
A+(k)=i (cid:104)ψ (k)|∇ |ψ (k)(cid:105)
n k n
En>0
(cid:88)
=i (cid:104)ψ∗(−k)|S†∇ S|ψ∗(−k)(cid:105)
n k n
En<0
(cid:88)
=i (cid:104)ψ∗(−k)|∇ |ψ∗(−k)(cid:105)
n k n
En<0
(cid:88)
=i (cid:104)ψ (−k)|∇ |ψ (−k)(cid:105)
n −k n
En<0
=A−(−k) ,