Even-odd dependent optical transitions of zigzag monolayer black phosphorus nanoribbons
Pu Liu, Xianzhe Zhu, Xiaoying Zhou, Benliang Zhou, Wenhu Liao, Guanghui Zhou, Kai Chang
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Even-odd dependent optical transitions of zigzag monolayer black phosphorus nanoribbons
Pu Liu , Xianzhe Zhu , Xiaoying Zhou ∗ , Benliang Zhou , Wenhu Liao , Guanghui Zhou † , and Kai Chang ‡ Department of Physics, Key Laboratory for Low-Dimensional Structures and Quantum Manipulation (Ministry of Education) andSynergetic Innovation Center for Quantum E ff ects and Applications of Hunan, Hunan Normal University, Changsha 410081, China Department of Physics and Key Laboratory of Mineral Cleaner Production and Exploit ofGreen Functional Materials in Hunan Province, Jishou University, Jishou 416000, China and SKLSM, Institute of Semiconductors, Chinese Academy of Sciences, P.O. Box 912, Beijing 100083, China
We analytically study the electronic structures and optical properties of zigzag-edged black phosphorenenanoribbons (ZPNRs) utilizing the tight-binding (TB) Hamiltonian and Kubo formula. By solving the discreteSchordinger equation directly, we obtain the energy spectra and wavefunctions for a N -ZPNR with N number oftransverse zigzag atomic chains, and classify the eigenstates according to the lattice symmetry. We then obtainthe optical transition selection rule of ZPNRs based on the symmetry analysis and the analytical expressions ofthe optical transition matrix elements. Under an incident light linearly-polarized along the ribbon, importantly,we find that the optical transition selection rule for the N -ZPNR with even- or odd- N is qualitatively di ff erent.In specification, for even- N ZPNRs the inter- (intra-) band selection rule is ∆ n = odd (even), since the parity ofthe wavefunction corresponding to the n th subband in the conduction (valence) band is ( − n [( − ( n + ] due tothe presence of the C x symmetry. In contrast, all optical transitions are possible among all subbands due to theabsence of the C x symmetry. Our findings provide a further understanding on the electronic states and opticalproperties of ZPNRs, which are useful in the explanation of the optical experiment data on ZPNR samples. I. INTRODUCTION
Black phosphorus (BP) is a layered material similar tographite with the atomic layers coupled by van der Waals in-teractions [1–4]. Few-layer [1–5] and monolayer [2, 6–8] BP(termed as phosphorene) have been fabricated experimentally,attracted intensive attentions due to their unique electronicproperties and potential applications in nanoelectronics [9–11]. Unlike graphene, BP is a semiconductor possessing a di-rect band gap ranging from 0.3 eV to 1.8 eV depending on thethicknesses of BP samples [8, 11]. The field-e ff ect-transistor(FET) based on phosphorene is found to have an on / o ff ratio of10 and a carrier mobility of 800 cm / V · s [12]. Sizable bandgap and relatively high mobility in phosphorene bridge thegap between graphene and transition metal dichalcogenides(TMDs), which are important for electronics and optoelec-tronics [9–11]. Inside phosphorene, phosphorus atoms are co-valently bonded with three adjacent atoms to form a puckeredhoneycomb structure due to the sp hybridization [13]. Aris-ing from the low symmetric and high anisotropic structure,BP exhibits strongly anisotropic electrical [2, 14–17], optical[8, 11, 18, 19] and transport [20] properties.The band structure of 2D phosphorene can be well de-scribed by a four band tight-binding (TB) model [21, 22]. Tai-loring it into 1D nanoribbons o ff er us a way to tune its elec-tronic and optical properties due to the quantum confinementand unique edge e ff ects [19, 23–27]. The band structure ofphosphorene nanoribbons (PNRs) depends on the edge con-figurations [19, 23–27]. The armchair-edged PNRs (APNRs)are semiconductors with direct band gap sensitively depend-ing on the ribbon width with scaling law of 1 / N [19, 26, 27], ∗ [email protected] † [email protected] ‡ [email protected] while the bare zigzag-edged PNRs (ZPNRs) are metallic re-gardless of their ribbon width due to the quasi-flat edge states[23, 25]. In bare ZPNRs, the edge states are entirely de-tached from the bulk bands and localized at the boundaries.These edge states result in a relatively large density of statesnear the Fermi energy [25, 28]. Further, the band structureof ZPNRs can be e ff ectively modified by tensile strain [24]or electric field [24, 25, 29]. Very recently, few-layer ZPNRsare successfully synthesized in recent experiments [30, 31].Up to date, various interesting properties have been predictedfor ZPNRs, including those related to transverse electric fieldcontrolled FET [25], room temperature magnetism and halfmetal phase [32–34], strain induced topological phase transi-tion [35], and symmetry dependent response to perpendicularelectric fields [36], etc.On the other hand, although there are already many re-search works on 2D phosphorene and its 1D ribbons, the ana-lytical calculation on the band structure of ZPNR is still lack-ing. Most of the previous works on this issue are based on thefirst-principles calculation [19, 23, 24, 29] or numerical diag-onalization utilizing the TB model [25, 26]. As well, thereis also less attention has been paid to the optical property ofZPNR [27, 37], and particularly the optical transition selec-tion rule in relation to the lattice symmetry and wavefunc-tion parity are not fully understood. Optical spectrum mea-surements are fundamental approach to detect and understandthe crystal band structure, which have been successfully per-formed for 2D phosphorene [11]. To this end, in this workwe theoretically investigate the optical properties of ZPNRsbased on the TB model and the Kubo formula. By solvingthe discrete Schordinger equation analytically, we obtain theelectronic structures of ZPNRs and classify their eigenstatesaccording to the crystal symmetry. We then obtain the opti-cal transition selection rules of ZPNRs directly based on thesymmetry analysis and the analytical expressions of the op-tical transition matrix elements. When the incident light ispolarized along the ribbons (see Fig. 1), interestingly, we find z l+1l A l-1 l+1 l-1 a t t t t t C a a x ... N-1N
A BB AABAA BB AB o Y a FIG. 1: Top view of a even- N ZPNR, where the red (blue) spheresrepresent phosphorous atoms in the upper (lower) sub-layer withprimitive vectors | a | = .
32 Å and | a | = .
38 Å of 2D phospho-rene. The bond length between two adjacent atoms is a = θ = ◦ . The (black) dashed-line rectangles aresuppercells adopted here for TB diagonalizing calculation. that the optical selection rules change significantly for a N -ZPNR with even- or odd- N . In particular, for even- N ZP-NRs the electronic wavefunction parity of the n th subband inthe conduction (valence) band is ( − n [( − ( n + ] due to the C x symmetry, and therefore their inter- (intra-) band selectionrule is ∆ n = n − n ′ = odd (even). For odd- N ZPNRs without C x symmetry, in contrast, the optical transitions are all possibleamong subbands. Further, the edge states of both even- andodd- N ZPNRs play an important role in the optical absorp-tion. Moreover, impurities or external electric field can breakthe C x symmetry of even- N ZPNRs, which consequently en-hances the optical absorption.The paper is organized as follows. Sec. II mainly presentsthe analytical result. We first repeat the numerical diagonal-ization procedure to obtain the band structure for the system,and the detailed analytical calculations on the band structurewith particularity of approaching accurate edge bands are fol-lowed. Then the wavefunctions, the joint density of states andthe optical conductivity for ZPNRs are expressed. In Sec. III,we present some numerical examples and discussions on theband structure and optical absorptions of the ZPNRs. Finally,we summarize our results in Sec. IV.
II. ELECTRONIC STRUCTURE AND OPTICALPROPERTIESA. Numerical diagonalization on Hamiltonian
The puckered honeycomb structure of phosphorene isshown in Fig. 1, where the red and blue dots represent phos-phorous atoms in di ff erent sub-layers. There are four atomsin one unit cell with the primitive vectors | a | = .
32 Å and | a | = .
38 Å. The bond length between two adjacent atomsis a = θ = ◦ [21]. Tailoring phos- phorene into 1D nanoribbons along the zigzag direction lead-ing to ZPNRs. The length of the bond connecting di ff erentsub-layers is 2.207 Å and the layer spacing is l = .
14 Å. Theintegers 1, 2, · · · N describe the number of the zigzag atomicchains of a ZPNR along its transversal direction. In the TBframework [21, 22], the Hamiltonian of a phosphorene in thepresence of in-plane transverse and of-plane vertical electricfields as well as impurities can be generically written as H = X < i , j > t i j c † i c j + X i ( 12 eE v l µ i + eE t y i + U i ) c † i c i , (1)where the summation h i , j i runs over all neighboring atomicsites with hopping integrals t i j , and c † i ( c j ) is the creation(annihilation) operator for atom site i ( j ). It has been shownthat five hopping parameters (see Fig. 1) are enough to de-scribe the electronic band structure of phosphorene [21] withhopping energies t = − t = t = − t = − t = − E v will result in a staggered potential elE v between theupper ( µ i =
1) and lower ( µ i = −
1) sublayers due to the puck-ered structure [36, 38]. Applying a transverse electric field E t will shift the on-site energy to eE t y i with y i the atom coordi-nation in the y -direction, and U i is the impurity potential.For a N -ZPNR with the number of zigzag chains N acrossthe width, by applying the Bloch’s theorem the TB Hamilto-nian in the momentum space is [39] H = H + H e ik x a + H † e − ik x a , (2)where Hamiltonian H ( H ) describes the intra (inter)- su-percell [see the (black) dashed-line rectangles in Fig. 1] in-teractions, k x is the wavevector along the x -direction. Inour calculation, we accordingly choose the basis ordered as( | A i , | B i , | A i , | B i , · · ·| mA i , | mB i , · · · | NA i , | NB i ) T to writedone H and H in the form of (2 N × N ) matrix for the supercell adopted. Then, we can obtain the energy spectrum E n , k x and the corresponding wavefunction | n , k x i for the system bynumerical diagonalization. In real space, the wavefucntioncan be formly expressed as ψ n , k x ( r ) = N X m = X i = A , B e ik x x m , i √ L x c m , i √ N πα e − ( r − R m , i )2 α , (3)where r = ( x , y ) is the electron coordination, R m , i = ( x m , i , y m , i ) is the atomic position vector, { c m , i } T = [ c A , c B , c A , c B , · · · c NA , c NB ] T is the eigenvector of theHamiltonian matrix in Eq. (2) with the transpose operator T ,and α is a Guass broadening parameter. Up to now, the bandstructure of ZPNRs is well understood by the first-principlescalculations [19, 23, 24, 29] or numerical TB calculations[25, 26]. For comparison here, the band structure of ournumerical diagonalization for a 10-ZPNR is shown by the(black) solid lines in Fig. 2(a), which is in good agreementwith the existed results [25, 26]. We note that there is a littledi ff erence compared with that of the first-principles calcula-tion [19, 23, 24, 29] due to the relaxation of the edge atoms.Considering the limitation of the first-principles calculation,the TB model can be applied to study the ZPNRs with largewidths. More importantly, we give the analytical solutionsfor electronic states and optical transitions in the ZPNRs witharbitrary widths. In comparison with the previous numericalcalculations [19, 23–26, 29], the analytical results are moreconvenient to do further understand in the electronic propertyof ZPNRs, i.e., identifying the subband symmetry propertyand calculating the optical absorption. Hereafter, we willpresent the analytical calculations on the band structure ofZPNRs in the next subsection. B. Analytical calculation on electronic structure
In this subsection, we firstly outline a scheme to obtainthe analytical energy spectrum for ZPNRs by solving the TBmodel directly. According to the TB approximation, the dis-crete Schordinger equation for a N -ZPNR is E φ A ( m ) = t g k φ B ( m ) + t φ B ( m − + t g k φ B ( m − + t g k [ φ A ( m − + φ A ( m + + t φ B ( m + , E φ B ( m ) = t g k φ A ( m ) + t φ A ( m + + t g k φ A ( m + + t g k [ φ B ( m − + φ B ( m + + t φ A ( m − . (4)where g k = k x a / φ A / B ( m ) is the wavefuntion of the m thA / B atom, and the site index m = , , , · · · N +
1. Since the0B and ( N + φ B (0) = φ A ( N + = . (5)According to the Bloch theorem, the generic solutions for φ A ( m ) and φ B ( m ) can be written as φ A ( m ) = Ae ipm + Be − ipm , φ B ( m ) = Ce ipm + De − ipm , (6)where A , B , C and D are arbitrary coe ffi cients and p thewavenumber in the transverse direction, which can be definedby the Schordinger equation combined with the boundary con-dition. Substituting Eq. (6) into (5), the wavefunction φ A / B ( m )can be simplified as φ A ( m ) = A ( e ipm − z e − ipm ) = A ϕ A ( p , m ) ,φ B ( m ) = C ( e ipm − e − ipm ) = C ϕ B ( p , m ) , (7)where z = e ip ( N + . Meanwhile, substituting Eq. (7) into (4),we obtain a matrix equation M AC ! = , (8)where M is a 2 × M = E ϕ A ( p , m ) − g k t [ ϕ A ( p , m − + ϕ A ( p , m + , M = − [ t g k ϕ B ( p , m ) + t ϕ B ( p , m − + t ϕ B ( p , m + + t g k ϕ B ( p , m − , M = − [ t g k ϕ A ( p , m ) + t ϕ A ( p , m + + t ϕ A ( p , m − + t g k ϕ A ( p , m + , M = E ϕ B ( p , m ) − g k t [ ϕ B ( p , m − + ϕ B ( p , m + . The condition for nontrivial solutions of A and C in Eq.(8), namely [ A , C ] T ,
0, is det( M ) =
0. However, it is worth tonote that the solutions of p = ± π should be excluded asunphysical results because these values of p yield φ A / B ( m ) = m . In other words, electrons are ab-sent in the system in these cases, which is unphysical. There-fore, we should find solutions that satisfy det( M ) = m except p = ± π . After some arithmetic, we find thatthe equation det( M ) = ve i pm + we − i pm + ξ = , (9)where v , w and ξ are functions of E , k x and p . Generally, Eq.(9) should be valid for arbitrary m . Thus, the two coe ffi cients( v and w ) of e ± i pm and the constant term ξ should be zero. Wethen obtain the energy spectrum for ZPNR as E = g k t cos( p ) ± (cid:12)(cid:12)(cid:12) t g k + t e ip + t e − ip + t g k e ip (cid:12)(cid:12)(cid:12) , (10)where ± represent the conduction and valence bands, respec-tively.On the other hand, from ξ =
0, we find a transcendental equa-tion for the transverse wavevector p which can be determinedby F ( p , N , k ) = t g k sin[ p ( N + + t sin( pN ) + t g k sin[ p ( N − + t sin[ p ( N + = . (11)This equation implies that the transverse wavenumber p = p ( k x , N ) depends not only on the ribbon width N but alsoon the longitudinal wavenumber k x . Obviously, we have F ( p , N , k ) = − F ( − p , N , k ), which means that Eq. (11) definesthe same subbands for p ∈ ( − π,
0) and p ∈ (0 , π ). Hence, wecan simply find the solutions of p from Eq. (11) in the laterinterval. If t = t and t = t =
0, Eq. (11) reduces to thetranscendental equation for a zigzag-edged graphene nanorib-bon (ZGNR) case [40, 41]. Similar to that in a N -ZGNR,there are only N -1 nonequivalent solutions of Eq. (11) for p ∈ (0 , π ), which defines 2 N -2 subbands, namely the bulkstates of a ZPNR.Notably, the two edge states are naturally missing in thescheme here since the transverse wavevector are purely imag-inary as is described by Eq. (6). But fortunately we canrestored them by setting p = i β and do the same procedureabove to obtain the eigenenergy and the corresponding tran-scendental equation. In this case, the eigenenergy in Eq. (10)can be rewritten as E = g k t ( e β + e − β ) ± p f ( β ) f ( − β ) , (12)where f ( β ) = t g k + t e β + t g k e β + t e − β , with the correspond-ing transcendental equation expressed by G ( β, N , k ) = t g k sinh[ β ( N + + t sinh( β N ) + t g k sinh[ β ( N − + t sinh[ β ( N + = , (13)where sinh( x ) is the hyperbolic sine function. Obviously, wehave G ( β, N , k ) = − G ( − β, N , k ), which means we only need tofind the solution of β for β > ff erent 10-, 15- and 30-ZPNR are also shown inFigs. 2(b-d), where the (black) solid and (blue) dash-dottedlines represent the numerical and analytical results, respec-tively. Unfortunately, we can see that the analytical results foredge states given by Eq. (12) are not in consistent with the nu-merical ones. This discrepancy was also revealed in a recentwork [42]. We think that the discrepancy mainly originatesfrom the hopping links t , t and t , with which their hoppingdistances are beyond a zigzag chain (see Fig. 1). This makesthe discrete Schordinger equation (4) is invalid for the edgeatoms, namely m equal to 1 or N . To resolve this problem,one solution is choosing four atoms to write down Eq. (4) anddouble the number of the boundary condition (5). But thismethod will enlarge the dimensions of matrix M unavoidablyand make the problem to be quite complicate and di ffi cult tosolve.Hereby, we propose an e ffi cient solution to eliminate thisdiscrepancy by simply adding a correction term. Generally,the two edge states can be described by a 2 × H edge = h h c h ∗ c h ! , (14)where h describes the two degenerate edge states when theribbon width N is large, and h c describes the coupling betweenthe edge states for small N . Based on this argument, accordingto Eq. (12), we have h = t g k sinh( β ) and h c = p f ( β ) f ( − β ).The band structure of the edge states in 10-, 15- and 30-ZPNRgiven by Eq. (12) are presented by the (blue) dash-dotted linesin Figs. 2(b-d), respectively. From these figures, we find that h c is finite for a narrow ribbon as shown in Fig. 2(b) for 10-ZPNR, but it vanished for the wider ones [e.g., Fig. 2(d) for30-ZPNR]. This means that h c is suitable to describe the cou-pling between the edge states. However, there is a observablediscrepancy between the analytical results [Eq. (12)] shownby the (blue) dash-dotted lines and the numerical ones [the(black) solid lines] in Figs. 2(b-d). This implies that h is un-suitable to describe the edge states and needs a correction soas to describe the edge bands accurately. Naturally, we can as-sume that the correction term is a superposition of the energyterm caused by hopping links ( t , t , t ) beyond one zigzagchain, which is expressed as h ′ = X s =+ , − ( b s t g k e s β + b s t g k e s β + b s t e s β ) , (15)where the coe ffi cients b + = . b − = . b ± = − . b + = − . b − = . E = h + h ′ ± h c . (16)The analytical edge states expressed by Eq. (16) are also de-picted by the (red) dashed lines in Figs. 2(b-d). Comparing -1.0 -0.5 0.0 0.5 1.0-6-4-202468 -1.0 -0.5 0.0 0.5 1.0-0.6-0.4-0.20.00.2-1.0 -0.5 0.0 0.5 1.0-0.6-0.4-0.20.00.2 -1.0 -0.5 0.0 0.5 1.0-0.4-0.20.00.2 E ne r g y ( e V ) E ne r g y ( e V ) (a) (b) (c) k x ( a ) k x ( a ) (d) FIG. 2: (a) The band structure of a bare 10-ZPNR, where the(red / black) dashed / solid lines represent the analytical / numerical re-sults. The scale-enlarged edge bands for (b) 10-ZPNR, (c) 15-ZPNRand (d) 30-ZPNR with the comparison of analytical and numericalresults, where the [red (blue)] dashed (dash-dotted) lines representthe analytical result for the edge bands with (without) the correctionterm h ′ . them with the numerical data [the (black) solid lines], we findthat they are in excellent agreement with each other regardlessof the ribbon width, which indicates that our method is validand reliable.On the other hand, for the wavefunction (3), owing to thetranslational invariance along the x -direction we can rewrite itin another generic form ψ n , k x ( r ) = N X m = C A ϕ A ( p , m ) e ik x x m , A C B ϕ B ( p , m ) e ik x x m , B ! , (17)where C A and C B are the normalization coe ffi cient, N thenumber of A and B atoms in a super-cell of a ZPNR, and x A / B , m the x -coordination of the m th A / B atoms. Notably, Eq.(17) is the wavefunction for the bulk states of ZPNRs. Foredge states, we should replace the transversal wavevector p by i β .As for the wavefunction of even- N ZPNRs, we can obtainthe relation C = ∓ Az in Eq. (7) from the parity of a ZPNR,namely φ A ( N + − m ) = ± φ B ( m ) [40]. The reason is that thewavefunction of even- N ZPNRs is either symmetric or anti-symmetric which is similar to that in ZGNRs [43]. Specif-ically, combined with the translational invariance along the x -direction, the wavefuntion of even- N ZPNRs is specified as ψ n , k x ( r ) = C √ L x N X m = − sz − ϕ A ( p , m ) e ik x x m , A ϕ B ( p , m ) e ik x x m , B ! = C ′ √ L x N X m = − s sin[ p ( N + − m )] e ik x x m , A sin( pm ) e ik x x m , B ! , (18)where C ′ = [ P Nm = sin ( pm )] − / √ ffi -cient, s = ± N ZPNRs.For edge states, the wavefunction ( p = i β ) is ψ n , k x ( r ) = C e √ L x N X m = − s sinh[ β ( N + − m )] e ik x x m , A sinh( β m ) e ik x x m , B ! , (19)where C e = [ P Nm = sinh ( pm )] − / √ ffi cient. On the contrary, owing to the absence of the C x symmetry, there is no such a simple expression of wavefunc-tion for the odd- N ZPNRs.
C. Optical property and transition selection rules
In order to detect the above calculated band structure ofZPNRs, we study its optical response in this subsection. Oneuseful physical quantity to understand the optical property isthe joint density of states (JDOS) representing all possible op-tical transitions among the subbands, which is generally givenby D J ( ω ) = g s L x X n , n ′ , k x [ f ( E n , k x ) − f ( E n ′ , k x )] δ ( E n , k x − E n ′ , k x + ~ ω ) , (20)where the sum runs over all states | n , k x i and | n ′ , k x i , g s is2 for spin degree, L x the ribbon length, ~ ω the photon en-ergy, and f ( E ) = / [exp( E − E F ) / k B T +
1] the Fermi-Diracdistribution function with Boltzman constant k B and tempera-ture T . Here, we take a Guass broadening Γ √ π exp[ − ( E n , k x − E n ′ , k x + ~ ω ) / Γ ] to approximate the δ -function, where Γ isa phenomenological constant accounting for the energy levelbroadening factor. Meanwhile, assuming the incident lightis polarized along the longitudinal ( x -) direction, the opticalconductance based on the Kubo formula is given by [44, 45] σ ( ω ) = g s ~ e iL x X n , n ′ , k x [ f ( E n , k x ) − f ( E n ′ , k x )] |h n , k x | v x | n ′ , k x i| ( E n , k x − E n ′ , k x )( E n , k x − E n ′ , k x + ~ ω + i Γ ) , (21)where v x = i ~ ∂ H ∂ k x is the velocity operator, which is valid and in-dependent of the band structure model, and | n , k x i = φ ( r ) ϕ ( K )[46] is the total electron wavefunction in a ZPNR. Here φ ( r )is the envelop function which describes the slowly vary-ing electron sharing movement in the crystal, while ϕ ( K ) isthe band edge wavefunction (BEW) connecting to the atomorbits directly describing the fast movement in the crystal.In a ZPNR, ϕ ( K ) is composed by | s i , | p x i , | p y i , and | p z i atomic-orbits with di ff erent weights [47, 48]. For a lin-ear polarized light, the optical transition matrix elementssatisfy h n , k x | v x | n ′ , k x i = h ψ n , k x | v x | ψ n ′ , k x ih ϕ n ( K ) | ϕ n ′ ( K ) i . Obvi-ously, v n , n ′ ( k x ) = h ψ n , k x | v x | ψ n ′ , k x i determines the optical transi-tion selection rules. A zero matrix element v n , n ′ ( k x ) meansa forbidden transition. The inner product between the twoBEWs is subband dependent, with which only a ff ects the am-plitude of the optical conductance but does not change the op-tical selection rules. We take the inner production around the Γ -point ( h ϕ n ( Γ ) | ϕ n ′ ( Γ ) i ) as an approximation and treat it as aconstant. This approximation has also been used in the pre-vious work [18] for 2D phosphorene. We have omitted this -1.0 -0.5 0.0 0.5 1.0-6-4-202468 -1.0 -0.5 0.0 0.5 1.0-6-4-202468 nv=1nc=1 (b) E ne r g y ( e V ) k x ( /a ) (a)
13 2
WFWF -2.140-1.640-1.140-0.6400-0.14000.36000.86001.3601.8602.140 k x ( /a ) (d) (c) -2.080-1.780-1.480-1.180-0.8800-0.5800-0.28000.000 WFWF -2.120-1.470-0.8200-0.17000.48001.1301.7802.120
FIG. 3: The band structures of bare (a)10-ZPNR and (c)11-ZPNR, where the (red / blue) dashed / solid lines represent the sym-metric / antisymmetric states. (b) and (d) show the spatial distributionof the wave function of the subband n v = n c = k x = constant in our calculations because the specific expression ofthe BEWs in ZPNRs are currently unknown. This approxima-tion would not change the essential physics, i.e., the even-odddependent optical selection rule, reported here. Note that insome topological none-trivial system, the dipole optical ma-trix result in the winding number [49, 50]. But there is no suchan e ff ect in phosphorene because it is a topologically trivialsystem. The real part of σ ( ω ) indicates the optical absorptionwhen an laser beam incidents on the sample. Moreover, wecan obtain the dielectric function ε ( ω ) from optical conduc-tance by using ε ( ω ) = + π i ω σ ( ω ) [51].In optical transition process, selection rules determinedby the matrix elements v n , n ′ ( k x ) are the most important in-formation. The integral of the velocity matrix elements V n , n ′ = R dk x | v n , n ′ ( k x ) | is proportional to the optical transitionprobability between the n th and n ′ th subbands. Generally, theselection rule is always constrained by the symmetry of thesystem. Hence, in order to obtain a general optical selectionrule for ZPNRs, we firstly check their lattice symmetry. Ac-cording to Fig. 1, we find the lattice symmetry of a N -ZPNRis even-odd- N dependent. In particular, the even- N ones havea C x ( x , y , z ) → ( x , − y , − z ) operator with respect to the ribboncentral axis (see the dotted horizontal line in Fig. 1). Thisis equivalent to the symmetry operator σ zx σ xy , where σ zx and σ xy are the mirror symmetry operators corresponding to the xoz and the xoy planes, respectively. However, the odd- N onesdo not have this symmetry. In even- N ZPNRs, the constrainton the eigenstates h x , y , z | n , k x i imposed by C x symmetry is C x h x , y , z | n , k x i = h x , − y , − z | n , k x i . Assuming λ is the eigen-value of C x operator, we have h x , y , z | n , k x i = ( C x ) h x , y , z | n , k x i = λ h x , y , z | n , k x i . (22)Then we obtain λ =
1, i.e., λ = ±
1, where + / − means the n v =2,n c =2 V Ribbon Width (N)
Ribbon Width (N) n v =1,n c =1 (b) V (a) V n , n ’ FIG. 4: The integral of the optical transition matrix elements V n , n ′ = R dk x | v n , n ′ ( k x ) | for (a) the n v = n c = n v = n c = N , where the insets show the amplified picture of V n , n ′ on rib-bon width with large N . even / odd parity provided by the C x operator. This indicatesthat h x , y , z | n , k x i is either symmetric or antisymmetric alongthe y and z directions, namely h− y , − z | n , k x i = ±h y , z | n , k x i .Thus, we can classify the eigenstates for even- N ZPNRs aseven or odd parity according to the eigenvalues of the C x op-erator.In order to confirm the above argument on the symmetryand parity for the systems, we present the band structure andthe wavefunction in real space of the first subband in the con-duction and valence bands for 10- and 11-ZPNR in Figs. 3(a-b) and 3(c-d), respectively. The wavefunction correspondingto states indicated by the red (blue) dots in Figs. 3(a) and3(c) are shown in the left (right) panels in Figs. 3(b) and3(d), respectively. According to the left (right) panel in Fig.3(b), we find that the first subband in the conduction (valence)band is even (odd) under C x transformation. By checkingthe eigenstates in other subbands, we observe that the parityof wavefunctions varies alternatively from odd [(blue) solidlines] to even [(red) dashed lines] with the increase of sub-band index n . This is consistent with the previous results ob-tained by the first-principles calculation [19, 27]. Hence, theparity of the subband in the conduction (valence) band is re-lated to its subband index via ( − n [( − ( n + ]. Further, un-der the C x operation, the velocity operator v x is even, i.e., C x : v x → v x . Hence, we obtain the condition for none zeromatrix element v n , n ′ ( k x ) is that the parity of the initial ( | n , k x i )and final ( | n ′ , k x i ) states are the same. In other words, only thetransitions among the states with identical parity are allowed.This can also be verified by calculating the optical transitionmatrix element. For example, using the relation v = i ~ [ r , H ]combined with the wavefunction Eq. (18), the inter-band op-tical transition matrix element between the bulk states is [51] h ψ v | v x | ψ c i = i ~ h ψ v | Hx − xH | ψ c i , (23) ψ c / v is the wavefunction in Eq. (18) or (19). After some alge-bra, we have h ψ vn , kx | v x | ψ cn ′ , kx i = ( i ~ C ′ L x ( A t + A t + A t ) , s ′ = s , s ′ , s , (24) where A t = − it b sin( bk x ) N X m = sin( pm ) sin[ p ′ ( N + − m )] , A t = − it b sin( bk x ) N − X m = sin( pm ) sin[ p ′ ( N − − m )] , A t = it b sin( bk x ) cos( p ′ ) N X m = sin( pm ) sin( p ′ m ) . Here b = a / s = s ′ for the transitionbetween the edge bands as well as that between the bulk bandsand edge bands. Hence, we conclude that only the transitionsbetween the subbands with same parity are allowed. Conse-quently, in even- N ZPNRs, the inter (intra)-band selection ruleis ∆ n = n − n ′ = odd (even). This is in good agreement with theabove analysis based on the lattice symmetry. It is importantthat although the band structure of odd- N ZPNRs is similar tothat of even- N ones as shown in Fig. 3(c), the optical selec-tion rule is qualitatively di ff erent from that for even- N ZPNRs.According to the left (right) panel in Fig. 3(d), by checkingthe eigenstates within the whole band, we know that there isno subband owning definite parity in 11-ZPNR due to the ab-sence of C x symmetry. Thus, the optical transitions in odd- N ZPNRs between two arbitrary subbands are all possible. In or-der to illustrate the even-odd dependent optical selection rulemore clearly, in Fig. 4 we show the integral of the optical tran-sition matrix elements V n , n ′ as a function of the ribbon width N , where (a) for V n v = , n c = and (b) for V n v = , n c = , respectively,and the insets show the amplified picture of V n , n ′ on ribbonwidth with large N . Physically, V n , n ′ is proportional to the op-tical transition probability between the n th and n ′ th subbands.According to the figure, we find that the transition probabilityoscillates with the ribbon width N and shows an even-odd N dependent feature. The transitions between the subband n v = n c = N -ZPNRs due to thepresence of the C x symmetry. In contrast, the transitions be-tween the subband n v = n c = N -ZPNRs due to the absence of the C x symmetry. This even-odd dependent selection rule is also reflected in the opticalabsorption spectrum which will be discussed in the nest sec-tion. III. NUMERICAL RESULTS AND DISCUSSIONS
In this section, we present some numerical examples for theoptical absorption spectrum of ZPNRs and discuss the corre-sponding results. We take N =
10 and 11 to represent the evenand odd cases, respectively, which would not qualitatively in-fluence the results here. The temperature is 4 K and the levelbroadening Γ is 4 meV throughout the calculation unless spec- R e () (b) J D O S R e () R e () J D O S (a) (eV) (c) FIG. 5: The inter-band JDOS [(red) dash-dotted lines] and the opticalabsorption [(blue) solid lines] as a function of the incident photonenergy with σ = e / h for (a) 10-ZPNR and (b) 11-ZPNR. TheFermi level E F is chosen as − · · ·
6) are associatedwith the subband transitions illustrated in Fig. 2(a). (c) The opticalabsorption spectra for 20-ZPNR [(orange) solid line] and 21-ZPNR[(purple) dashed line]. ificated. In all following figures, the green solid line (if avail-able) indicates the Fermi level.As discussed in Sec. IIC, the inter- (intra-) band opticaltransition selection rule in even- N ZPNRs satisfies ∆ n = n − n ′ = odd (even) due to the C x symmetry. On the contrary,the optical transitions in odd- N ZPNRs between two arbitrarysubbands are all possible resulting from the C x symmetrybreaking. Keeping this in mind is important to understandthe optical properties of even- N ZPNRs. Fig. 5 shows theinter-band JDOS and the optical absorption spectrum for (a)10-ZPNR and (b) 11-ZPNR with Fermi energy E F = − ff erent photon energy known as van Hovesingularities. The JDOS peaks range from the mid-infrared(155-413 meV) to the visible region due to the edge states andthe quantum confinement, which is di ff erent from that of 2Dphosphorene case [11]. However, there is no optical absorp-tion around zero frequency, which is contradict to the fact thatZPNRs are metallic. The reason is that the transition betweenthe edge states is forbidden by the C x symmetry in even- N ZPNRs since their parities are di ff erent from each other. Com-pared with the JDOS, we find that more peaks are missing inthe optical absorption spectra Re σ ( ω ) [the (blue) solid line]due to the optical selection rule ∆ n = odd arising from the C x symmetry, which is similar to that in ZGNRs [41, 52, 53].The remained optical absorption peaks (labeled by 1, 2, · · · (a) R e () R e () J D O S J D O S E ne r g y ( e V ) E ne r g y ( e V ) (c) E F (b) (eV) E F k x ( /a ) (d) FIG. 6: The intra-band JDOS [(red) dash-dotted lines] and the opti-cal absorption [(blue) solid lines] as a function of the incident pho-ton energy for (a) 10-ZPNR and (b) 11-ZPNR. The band structureand the corresponding optical transitions are shown in (c) and (d)for 10-ZPNR ( E F = − E F = − transitions among subbands are all possible for 11-ZPNR ow-ing to the C x symmetry breaking. All the optical absorptionpeaks appear one to one correspondence to the JDOS. Owingto the edge states, the absorption peaks range from the mid-infrared to the visible frequency, etc. However, the absorptionpeak in the mid-infrared frequency (the first peak) disappearsfor wider ribbons as shown in Fig. 5(c), which is di ff erentfrom that in ZGNRs [41, 52, 53]. The reason comes fromtwo sides: i) the edge states become degenerate for wider ZP-NRs and ii) unlike that in ZGNRs, the edge states of ZPNRsare slightly dispersed [see Fig. 3(a)] due to the electron-holeasymmetry. This fact means that only two k x states contributeto the optical absorption for a certain Fermi level, leading tozero optical conductance. Again, from Fig. 5(c), we find thatthere are more absorption peaks for 21-ZPNR than that of the20-ZPNR arising from the C x symmetry breaking. Moreover,it should be noted that there is a little discrepancy between theJDOS peaks and the optical absorption peaks because that theoptical transition matrix element v n , n ′ ( k x ) depends on the sub-bands’ derivatives ∂ E /∂ k x .Figure 6 shows the intra-band JDOS [(red) dash-dottedline] and optical absorption spectrum [(blue) solid line] for(a) 10-ZPNR and (b) 11-ZPNR with the corresponding bandstructures and Fermi levels shown in (c) and (d), respectively.As depicted in Fig. 6(a), the first JDOS peak for 10-ZPNRlocated at ~ ω = n v = n v = ff erent [see Fig. 6(c)] and the tran-sitions are forbidden by the C x symmetry. In other words,this transition violates the intra-band optical transition selec-tion rule ( ∆ n = even) for even- N ZPNRs. By the same to-ken, the second and third JDOS peaks are contributed bythe transitions between the subbands with the same parities[see Fig. 6(c)], hence the corresponding absorption peaks R e () (a) J D O S E F (c) E ne r g y ( e V ) R e () (eV) (b) J D O S E F E ne r g y ( e V ) k x ( /a )(d) FIG. 7: The inter-band JDOS [(red) dash-dotted line] and optical ab-sorption spectrum [(blue) solid line] for (a) 10-ZPNR ( E F = -0.31 eV)and (b) 11-ZPNR ( E F = − E v = / Å, respectively, where the (black) dash line is theabsorption spectrum for bare ZPNRs. The band structures of 10- and11-ZPNRs under electric field are shown in (c) and (d), respectively,where the Fermi levels for both cases are lying between the two edgestates. appear [see the (blue) solid line]. On the contrary, we findthat the optical absorption peaks are almost presented for 11-ZPNR [see Fig. 6(b)] arising from the C x symmetry break-ing which means that the all optical transitions are principallypossible among all subbands. The corresponding transitionsare shown in Fig. 6(d). On the other hand, some of thematrix elements h n , k x | v x | n ′ , k x i may be tiny (weak), i.e, the h , k x | v x | , k x i , and the corresponding absorption peaksare missing in this case [see Fig. 6(b)].Next, we turn to the e ff ect of externally applied electric fieldon the optical property of ZPNRs. Fig. 7 depicts the inter-band JDOS [(red) dash-dotted line] and optical absorptionspectra [(blue) solid line] for (a) 10-ZPNR and (b) 11-ZPNRunder a uniform vertical electric field (VEF) with strength E v = / Å, where the corresponding band structures with theoptical transition indications are shown in (c) and (d), respec-tively. The Fermi level for both cases are lying between theedge states. In real experiment, the VEF corresponding to thetop gate or substrate e ff ect. It maybe be generated by using thepolar semiconductors interface [54]. Owing to the puckeredlattice structure of ZPNRs, the band structure of the ZPNRunder a VEF is even-odd dependent [36] since the edge statesof even (odd)- N ribbons located on the di ff erent (same) sub-layers. The VEF opens a gap between the two edge bandsfor even ribbons [see Fig. 7(c)], but for odd ones the twoedge bands are always (nearly) degenerated [see Fig. 7(d)].Further, the VEF also breaks the C x symmetry in even- N ZPNRs. These features are also reflected in the optical ab-sorption spectrum. From Fig. 7(a), we find that several extraabsorption peaks [the (blue) solid line] appear due to the C x symmetry breaking by the VEF compared with the bare 10-ZPNR [see the (black) dashed line]. Especially, the first ab-sorption peak in mid-infrared frequency is greatly enhanced R e () (a) J D O S (c) E F E ne r g y ( e V ) R e () (b) (eV) J D O S (d)k x ( /a ) E F E ne r g y ( e V ) FIG. 8: The inter-band JDOS [(red) dash-dotted line] and opticalabsorption spectrum [(blue) solid line] for 10-ZPNR with impuritieslocalized at (a) one edge (the 1st atomic row) with E F = -0.1954 eVand (b) the center (the 10th row) with E F = − U i is 0.5 eV and 1.5 eV corresponding to(a) and (b), and the (black) dashed line indicates the optical spectrumof a pristine 10-ZPNR. While (c) and (d) respectively to (a) and (b)but for band structure. due to the degeneracy lifting of the edge states. In compar-ison, as shown in Fig. 7(b), the absorption spectrum of 11-ZPNR is slightly changed compared to the bare case [also seethe (black) dashed line] since the band structure is nearly un-a ff ected by the VEF. These features o ff er a useful approach toidentify the even-odd property of ZPNR samples by experi-mentally detecting the optical absorption under VEF.Experimentally, it is di ffi cult to avoid impurities and defectsin samples. This may consequently a ff ect the optical proper-ties of ZPNRs by changing the band structure or breaking the C x symmetry. Figs. 8 (c) and 8(d) show the band structureof 10-ZPNR with impurities distributing on its one edge (the1st atomic row) and the center (the 10th row), respectively,where we have defined a N -ZPNR with 2 N atomic rows. For azigzag chain of ZPNR there are two phosphorus atomic rows,hence a N -ZPNR has 2 N rows. We model the impurity ef-fect by adding a impurity potential U i to the on-site energyof the corresponding impurity atoms, which is widely used inprevious works [55, 56]. As shown in Fig. 8(c) for impuri-ties located on the edge, we find that the nearly degeneratededge states are separated [see the (orange) solid line] due tothe variation of the on-site energies, but the other subbandsremain unchanged. This is consistent with the result obtainedby the first-principles calculation [57, 58]. On the contrary,comparing Fig. 8(d) with Fig. 3(a), as impurities localizedon the center the subband contributed by the impurities areshifted [see the (orange) solid line] but the other subbands re-main unchanged. This means that the impurities only have alocal e ff ect on the electronic structure of a ZPNR. But, theywill play an important role in the optical absorption spectrumbecause of the lattice symmetry breaking. Figs. 8(a) and 8(b)show the inter-band JDOS [(red) dash-dotted line] and opticalabsorption spectrum [(blue) solid line]for 10-ZPNR with im- R e () R e () (a) J D O S E ne r g y ( e V ) (c) E ne r g y ( e V ) (b) (eV) J D O S (d)k x ( /a ) FIG. 9: The inter-band JDOS [(red) dash-dotted line] and optical ab-sorption spectrum [(blue) solid line] for (a) 10-ZPNR ( E F = − E F = − E t = / Å, respectively, where the Fermi level forboth cases are lying between the two edge states and the (black) dashline represents the absorption spectrum for bare ZPNRs. The bandstructures of 10- and 11-ZPNR under TEF are shown in (c) and (d),respectively. purities located at its one edge (the 1st atomic row) with E F = -0.1954 eV and the center (the 10th row) with E F = − C x symmetry since the wavefunctions corresponding to mostof the subbands are partially distributed on the edge. Hence,we observe the first and some extra optical absorption peaksreappeared [see the (blue) solid line] compared with the pris-tine 10-ZPNR shown in Fig. 8(a) [(black) dash line]. This issimilar to that of the VEF e ff ect discussed above. Similarly,from Fig. 8(b), we also find some extra peaks when the im-purities localized at the center. However, the absorption peakat the mid-infrared frequency (the first peak) is still missingalthough the C x symmetry is broken in this case. The rea-son is that the edge states are mainly localized on the edgeatoms, in consequence the band structure is nearly una ff ectedby the impurities localized on the center atoms. For 11-ZPNR,the optical absorption spectrum is just slightly changed by theimpurities, hence we do not present the result here for savingspace.Finally, a transverse electric field (TEF) can induce a Starke ff ect (potential di ff erence) arising from the finite width ofZPNRs [25], which can make a significant change of the bandstructure, especially the edge bands. And this e ff ect has beenexperimentally observed for few-layer BPs [59]. As a re-sult, this can also change the optical properties of ZPNRsby breaking the C x symmetry. Therefore, Fig. 9 displaysthe inter-band JDOS [(red) dash-dotted line] and optical ab-sorption spectrum [(blue) solid line] for (a) 10-ZPNR and (b)11-ZPNR under a uniform TEF with strength E t = / Å.And the corresponding band structures are shown in Figs. 9(c)and 9(d), respectively. As shown in the figure, we find thatin the presence of the TEF the optical absorption peaks areshifted compared to the bare ribbons [see the (black) dashed line]. In Figs. 9(c) and 9(d), unlike the VEF case, we cansee the degeneracy of the edge states for 10- and 11-ZPNRare both lifted by the Stark e ff ect. Owing to the C x symme-try breaking, all possible absorption peaks in 10-ZPNR cor-responding to the JDOS appear [see Fig. 9(a)], which meansthat the optical absorption of 10-ZPNR is greatly enhanced,especially the absorption peak in the mid-infrared frequency.Further, the first absorption peak for both 10- and 11-ZPNRis greatly enhanced due to the degeneracy lifting as show inFigs. 9(a) and 9(b). Hence we conclude that the e ff ect of TEFon the optical absorption in ZPNRs is di ff erent from that ofthe VEF or impurity. A TEF can induce di ff erent potentialson all atoms within a super-cell, which leads to a global C x symmetry breaking. IV. SUMMARY
In summary, we have theoretically studied the electronicand optical properties of ZPNRs under a linearly polarizedlight along the longitudinal direction based on the TB Hamil-tonian and Kubo formula. We have obtained analytically theenergy spectra of ZPNRs and the optical transition selectionrules based on the lattice symmetry analysis. Owing to the C x symmetry, the eigenstates of even- N ZPNRs are trans-versely either symmetric or antisymmetric, which makes theiroptical response qualitatively di ff erent from that of the odd- N ones. In particular the inter (intra) -band selection rule foreven- N ZPNRs is ∆ n = odd (even) since the parity factor ofthe wavefunction corresponding to the conduction (valence)band is ( − n [( − ( n + ] (with the subband index n ) providedby the C x symmetry. For odd- N ZPNRs, however, the alloptical transitions are possible among all subbands. Further,the edge states play an important role in the optical absorp-tion and are involved in many of the absorption peaks. Theoptical absorption of even- N ZPNRs can be enhanced by thesubstrate and impurity e ff ect as well as the transverse electricfield via breaking the C x symmetry. While the optical absorp-tion of odd- N ones can be e ff ectively tuned by lattice defectsor external electric fields. Our findings provide a further un-derstanding on the electronic states and optical properties ofZPNRs, which are essential for the explanation of the opticalexperiment data on ZPNR samples. V. ACKNOWLEDGMENTS
This work was supported by the National Natural Sci-ence Foundation of China (Grant Nos. 11804092, 11774085,61674145, 11704118, 11664010), and China PostdoctoralScience Foundation funded project Grant No. BX20180097,and Hunan Provincial Natural Science Foundation of China(Grant No. 2017JJ3210).0
Appendix A
In this appendix, we calculate the optical transition matrixelements in Eq. (24). Utilizing the relation v = i ~ [ r , H ] com-bined the wavefunction in Eq. (18), the optical matrix elementcan be written as [51] h ψ v | v x | ψ c i = i ~ h ψ c | Hx − xH | ψ v i , (A1) ψ c / v is the wavefunction in Eq. (18) or (19). According to Eq.(18), the transition matrix element between the bulk states is h ψ vn , k x | v x | ψ cn ′ , k x i = C ′ L x i ~ N X m = N X n = ( ( x A , n − x A , m ) e − ik x x A , m e ik x x A , n ss ′ sin[ p ( N + − m )] sin[ p ′ ( N + − n )] h A m | H | A n i + ( x B , n − x B , m ) e − ik x x B , m e ik x x B , n sin( pm ) sin( p ′ n ) h B m | H | B n i− s ( x B , n − x A , m ) e − ik x x A , m e ik x x B , n sin[ p ( N + − m )] sin( p ′ n ) h A m | H | B n i− s ′ ( x A , n − x B , m ) e − ik x x B , m e ik x x A , n sin( pm ) sin[ p ′ ( N + − n )] h B m | H | A n i ) , (A2)where m ( n ) is the atom site index, and s ( s ′ ) indicates the parity of the subbands. There are five hoppings, including h A m | H | B n i = h B m | H | A n i = t for n = m , h A m | H | B n i = h B m | H | A n i = t for n = m ± h A m | H | B n i = h B m | H | A n i = t for n = m ± h A m | H | B n i = h B m | H | A n i = t for n = m ±
1, and h A m | H | B n i = h B m | H | A n i = t for n = m ±
1. Then, Eq. (A2) can be written as h ψ vn , k x | v x | ψ cn ′ , k x i = i ~ C ′ L x ( A t + A t + A t ) , (A3)where A t , A t and A t represent the term of transition matrix related to the hopping t , t and t , and the corresponding term canbe written as A t = − N X m = s { t be ik x b sin[ p ( N + − m )] sin( p ′ m ) − t be − ik x b sin[ p ( N + − m )] sin( p ′ m ) } + s ′ { t be ik x b sin( pm ) sin[ p ′ ( N + − m )] − t be − ik x b sin( pm ) sin[ p ′ ( N + − m )] } = − it b sin( bk x ) N X m = { s sin[ p ( N + − m )] sin( p ′ m ) + s ′ sin( pm ) sin[ p ′ ( N + − m )] } , (A4)where the sum of m runs from 1 to N . Defining, n = N + − m , we find n also runs from 1 to N when m ǫ [1 , N ]. Applying thesummation transform n = N + − m , A t can be rewritten as A t = − it b sin( bk x ) { N X n = s sin( pn ) sin[ p ′ ( N + − n )] + N X m = s ′ sin( pm ) sin[ p ′ ( N + − m )] } = ( − it b sin( bk x ) P Nm = sin( pm ) sin[ p ′ ( N + − m )] , s = s ′ , s , s ′ . (A5)Meanwhile, the term A t is A t = − N X m = s { be ik x b sin[ p ( N + − m )] sin[ p ′ ( m − t − be − ik x b sin[ p ( N + − m )] sin[ p ′ ( m − t }− N − X m = s ′ { be ik x b sin( pm ) sin[ p ′ ( N + − m − t − be − ik x b sin( pm ) sin[ p ′ ( N + − m − t } = − ib sin( bk x ) t { N X m = s sin[ p ( N + − m )] sin[ p ′ ( m − + N − X m = s ′ sin( pm ) sin[ p ′ ( N + − m − } , (A6)1in this case, the atoms at the edges should be excluded because the hopping links of t is beyond one zigzag chain. Applyingsimilar summation transform in A t to A t , we have A t = − ib sin( bk x ) t { N − X n = s sin( pn ) sin[ p ′ ( N + − n − + N − X m = s ′ sin( pm ) sin[ p ′ ( N + − m − } = ( − it b sin( bk x ) P N − m = sin( pm ) sin[ p ′ ( N − − m )] , s = s ′ , s , s ′ . (A7)Finally, the A t term is A t = N X m = { ss ′ be ik x b sin[ p ( N + − m )] sin[ p ′ ( N + − m + t + ss ′ be ik x b sin[ p ( N + − m )] sin[ p ′ ( N + − m − t − ss ′ be − ik x b sin[ p ( N + − m )] sin[ p ′ ( N + − m + t − ss ′ be − ik x b sin[ p ( N + − m )] sin[ p ′ ( N + − m − t + be ik x b sin( pm ) sin[ p ′ ( m − t + be ik x b sin( pm ) sin[ p ′ ( m + t − be − ik x b sin( pm ) sin[ p ′ ( m − t − be − ik x b sin( pm ) sin[ p ′ ( m + t = N X m = it b sin( bk x ) cos p ′ { ss ′ sin[ p ( N + − m )] sin[ p ′ ( N + − m )] + sin( pm ) sin( p ′ m ) } . (A8)Here, we have used the relation sin( x ) + sin( y ) = x + y ) /
2] cos[( x − y ) /
2] to simplify it. Similarly, replacing all thesummation index N + − m with n , we obtain A t = N X n = iss ′ t b sin( bk x ) sin( pn ) sin( p ′ n ) cos p ′ + N X m = it b sin( bk x ) sin( pm ) sin( p ′ m ) cos p ′ = ( it b sin( bk x ) cos p ′ P Nm = sin( pm ) sin( p ′ m ) , s = s ′ , s , s ′ . (A9)Therefore, the transition matrix elements is h ψ vn , k x | v x | ψ cn ′ , k x i = ( C ′ L x i ~ ( A t + A t + A t ) , s ′ = s , s ′ , s , (A10)where A t = − it b sin( bk x ) N X m = sin( pm ) sin[ p ′ ( N + − m )] , A t = − it b sin( bk x ) N − X m = sin( pm ) sin[ p ′ ( N − − m )] , A t = it b sin( bk x ) cos( p ′ ) N X m = sin( pm ) sin( p ′ m ) . From Eq. (A10), we can explicitly find that only the inter band transition between the bulks states with the same symmetry areallowed. Using the wavefunction in Eq. (19), we can obtain the same selection rules s = s ′ for the transition between the edgebands as well as the bulk bands to the edge bands. 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