Massive Dirac fermions and spin physics in an ultrathin film of topological insulator
Hai-Zhou Lu, Wen-Yu Shan, Wang Yao, Qian Niu, Shun-Qing Shen
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Massive Dirac fermions and spin physics in an ultrathin film of topological insulator
Hai-Zhou Lu, Wen-Yu Shan, Wang Yao, Qian Niu, and Shun-Qing Shen Department of Physics and Center for Theoretical and Computational Physics,The University of Hong Kong, Pokfulam Road, Hong Kong, China Department of Physics, The University of Texas, Austin, Texas 78712-0264, USA (Dated: February 11, 2010)We study transport and optical properties of the surface states which lie in the bulk energy gapof a thin-film topological insulator. When the film thickness is comparable with the surface statedecay length into the bulk, the tunneling between the top and bottom surfaces opens an energygap and form two degenerate massive Dirac hyperbolas. Spin dependent physics emerges in thesurface bands which are vastly different from the bulk behavior. These include the surface spin Halleffects, spin dependent orbital magnetic moment, and spin dependent optical transition selectionrule which allows optical spin injection. We show a topological quantum phase transition where theChern number of the surface bands changes when varying the thickness of the thin film.
PACS numbers: 72.25.-b, 85.75.-d, 78.67.-n
I. INTRODUCTION
A three-dimensional (3D) topological insulator is anovel quantum state of matter which possesses metal-lic surface states in the bulk energy gap.
The sur-face states consist of an odd number of massless Diraccones which are protected by Z topological invari-ants. The first 3D topological insulator is Bi x Sb − x ,an alloy with complex structure of surface states whichwas confirmed by using angle-resolved photoemissionspectroscopy (ARPES). Recently it was verified thatBi Se and Bi Te have only one Dirac cone near theΓ point by both experiments and the first principlescalculation, which attracts extensive attentions incondensed matter physics. The electrons or Diracfermions in the surface states of topological insulatorobey the 2+1 Dirac equations and reveal a lot of un-conventional properties such as the topological magne-toelectric effect. It was also proposed that the surfacestates interfaced with a superconductor can form Ma-jorana fermions for performing fault-tolerant quantumcomputation.
Since the surface states surround thesample, it is still a great challenge for both experimen-talists and theorists to explore the transport propertiesfor metallic surface states in the topological insulators.In this paper, we study an ultrathin film of topologicalinsulator where tunneling between the surface states onthe top and bottom surfaces opens a finite gap in theDirac cone centered at the Γ point ( k = 0). The low ly-ing physics of the ultrathin film can be described as twodegenerate massive Dirac hyperbolas which form timereversal copy of each other. Each massive band has a k -dependent spin configuration: one near the Γ point deter-mined by the energy gap, and the other at k large enoughdetermined by a spin-orbit coupling term quadratic in k .We show that the energy gap oscillates with the thinfilm thickness, and changes sign at critical thicknesses.Across the transition points, the k -dependent spin con-figuration near the Γ point is flipped while those at large k remains unchanged, leading to a topological quantum phase transition where the Chern numbers of the surfacebands change [Eq. (31)]. In the two Dirac hyperbolas ofopposite spin configurations, the k -dependent spin struc-ture results in a distribution of orbital magnetic momentand Berry curvature with opposite signs. In doped metal-lic regime, the Berry curvature drives the spin Hall effectof the extra carriers which leads to net spin accumula-tions on the thin film edges. We also discover a spin-dependent optical transition selection rule which allowsoptical injection of spins in the thin film.The paper is organized as follows. In Sec. II, wepresent the derivation of the effective Hamiltonian forthe thin film of topological insulator. In Sec. III, theoscillation of the gap and the k -dependent spin config-uration are discussed. In Sec. IV, the orbital magneticmoment and Berry curvature are addressed. In Sec. V,the spin Hall conductance is derived in detail. In Sec.VI, the spin-dependent optical transition selection rule isshown. Finally, a conclusion is given in Sec. VII. II. MODEL FOR AN ULTRATHIN FILM
We start with the low-lying effective model for bulkBi Se , in which surface states consist of a single Diraccone at the Γ point. We take the periodic bound-ary conditions in the x - y plane such that k x and k y are good quantum numbers, and denote the thicknessof the thin film along z direction as L . In the basisof {| p + z , ↑i , | p − z , ↑i , | p + z , ↓i , | p − z , ↓i} which are the hy-bridized states of Se p orbital and Bi p orbital, with even(+) and odd ( − ) parities, the model Hamiltonian is givenby H ( k ) = ( C − D ∂ z + D k ) + (cid:18) h ( A ) A k − σ x A k + σ x h ( − A ) (cid:19) (1)where h ( A ) = ( M + B ∂ z − B k ) σ z − iA ∂ z σ x , and σ α are the Pauli matrices, with k ± = k x ± ik y and k = k x + k y . This model is invariant under time re-versal symmetry and inversion symmetry. In this pa-per, the model parameters are adopted from Ref. 9: M = 0 . , A = 2 . A = 4 . B = 10eV˚A , B = 56 . , C = − . D = 1 . , D = 19 . . Though we adopt this concrete modelto study the properties of an ultrathin film of topologicalinsulator, the general conclusion in this paper should beapplicable to other topological insulators.To establish an effective model for an ultrathin film,we first find the four solutions to the surface states of themodel in Eq. (1) at the Γ point ( k x = k y = 0), H = (cid:20) h ( A ) 00 h ( − A ) (cid:21) , (2)where h ( A ) = C − D ∂ z + ( M + B ∂ z ) σ z − iA ∂ z σ x , The solution of the block-diagonal H can be found byputting a two-component trial solution into the eigenequation of the upper block h ( A ) (cid:18) ab (cid:19) e λz = E (cid:18) ab (cid:19) e λz , (3)with a , b , λ the trial coefficients defining the behaviorof the wavefunctions, and E the trial eigen energy. Notethat the trial coefficients may have multiple solutions, thefinal solution should be a linear superposition of thesesolutions, with the superposition coefficients determinedby boundary conditions. Then the problem becomes astraightforward calculation of the Schr¨odinger equation.Consider an open boundary condition that the wavefunc-tions vanish at the two surfaces located at z = ± L/ C − M − E − ( D + B ) λ ] λ [ C − M − E − ( D + B ) λ ] λ = tanh( λ α L/ λ ¯ α L/ , (4)note that where α =1 and 2, ¯ α = 2 if α = 1, vice versa,so there are two transcendental equations. In Eq. (4), λ α define the behavior of the wavefunctions along z -axis,and are functions of the energy Eλ α ( E ) = s − F + ( − α − √ R D − B ) , (5)where for convenience we have defined F = A +2 D ( E − C ) − B M and R = F − D − B )[( E − C ) − M ].The self-consistent solution to the two equations in (4)can be found numerically, and give two energies at the Γpoint, i.e., E + and E − , which define an energy gap∆ ≡ E + − E − . (6)Note that the bulk states with much higher or lower en-ergies due to the quantization of k z in the finite quantum well in principle can also be obtained in this way, but areignored here because we concentrate on the surface statesnear the gap. The eigen wavefunctions for E + and E − are, respectively, ϕ ↑ ( A ) = C + (cid:20) − ( D + B ) η +1 f + − iA f ++ (cid:21) ,χ ↑ ( A ) = C − (cid:20) − ( D + B ) η − f − + iA f −− (cid:21) , (7)where C ± are the normalization constants. The super-scripts of f ±± and η ± , stand for E ± , and the subscripts of f ±± for parity, respectively. The expressions for f ±± and η ± , are given by f ± + ( z ) = cosh( λ ± z )cosh( λ ± L/ − cosh( λ ± z )cosh( λ ± L/ ,f ±− ( z ) = sinh( λ ± z )sinh( λ ± L/ − sinh( λ ± z )sinh( λ ± L/ ,η ± = ( λ ± ) − ( λ ± ) λ ± coth( λ ± L/ − λ ± coth( λ ± L/ ,η ± = ( λ ± ) − ( λ ± ) λ tanh( λ ± L/ − λ ± tanh( λ ± L/ . (8)where λ ± α ≡ λ α ( E ± ) can be found by putting back E ± into Eq. (5). FIG. 1: (Color online) Schematic comparison between (a)the gapped K-K’ valleys in the staggered graphene, and (b)the two-fold degenerate hyperbolas described by our effectiveHamiltonian in Eq. (10).
By replacing A by − A in the above solutions, theenergies of the lower block h ( − A ) of H are found de-generate with those of h ( A ) and their wave functionsare denoted as ϕ ↓ ( − A ) and χ ↓ ( − A ), respectively. Nowwe have four states, namely, [ ϕ ↑ ( A ) , T , [ χ ↑ ( A ) , T ,[0 , ϕ ↓ ( − A )] T , and [0 , χ ↓ ( − A )] T , where ↑ and ↓ implythat the orbits with spin up and down are decoupled. Byusing these four solutions as basis states and rearrangingtheir sequence following (note that each basis state is afour component vector) (cid:18) ϕ ↑ ( A )0 (cid:19) , (cid:18) χ ↓ ( − A ) (cid:19) , (cid:18) χ ↑ ( A )0 (cid:19) , (cid:18) ϕ ↓ ( − A ) (cid:19) , (9)we can map the original Hamiltonian to the Hilbert spacespanned by these four states, and reach a new low-energyeffective Hamiltonian for the ultrathin film, H eff = (cid:20) h + ( k ) 00 h − ( k ) (cid:21) (10)in which h τ z ( k ) = E − Dk − ~ v F ( k x σ y − k y σ x )+ τ z ( ∆2 − Bk ) σ z . (11)Note that here the basis states of Pauli matrices stand forspin-up and spin-down states of real spin. In Eq. (11),we have introduced a hyperbola index τ z = ± ± ).As shown in Fig. 1, one can view the hyperbolas as the Kand K’ valleys in the staggered graphene [(a)], but beingrelocated to the Γ point[(b)]. Unlike the momentum cor-respondence in graphene, it is the σ z to − σ z correspon-dence in the present case. Therefore, the dispersions of h ± are actually doubly degenerate, which is secured bytime-reversal symmetry. Here, τ z = ± are used to distin-guish the two degenerate hyperbolas, h + ( k ) and h − ( k )describe two sets of Dirac fermions, each show a pair ofconduction and valence bands with the dispersions ε c/v ( k ) = E − Dk ± p (∆ / − Bk ) + ( ~ v F ) k , (12)where c and v correspond to the conduction and valencebands, respectively. The eigen states for ε c/v are u c/v ( k ) = 1 (cid:13)(cid:13) u c/v (cid:13)(cid:13) (cid:20) (∆ / − Bk ) τ z + ε c/v − i ~ v F k + (cid:21) (13)with (cid:13)(cid:13) u c/v (cid:13)(cid:13) = p [(∆ / − Bk ) τ z + ε c/v ] + ( ~ v F ) k . Besides the gap ∆ already defined in Eq. (6), the otherparameters in Hamiltonian (11) are given by v F = ( A / ~ ) h ϕ ( A ) | σ x | χ ( − A ) i ,D = ( B / h ϕ ↑ | σ z | ϕ ↑ i + h χ ↑ | σ z | χ ↑ i ) − D ,B = ( B / h ϕ ↑ | σ z | ϕ ↑ i − h χ ↑ | σ z | χ ↑ i ) ,E = ( E + + E − ) / , (14)and can be calculated numerically by using Eq. (7).The numerical results of ∆, v F , D , and B are pre-sented in Fig. 2. It is noted that | D | must be less than | B | , otherwise the energy gap will disappear, and all dis-cussions in the following will not be valid. The ∆ termsplay a role of mass term in 2+1 Dirac equations.In the large L limit, v F = ( A / ~ ) q − D /B . (15)The dispersion relation is given by ε c/v ( k ) = ± v F ~ k (16)for small k . As a result, the energy gap closes at k = 0.The two massless Dirac cones are located near the top and bottom surfaces, respectively, as expected in a 3Dtopological insulator.In a small L limit, v F = A / ~ , (17)and ∆ = 2 B π /L . (18)The ratio of the velocity between the two limits is η = 1 / q − D /B . (19)It is noted that the velocity and energy gap for an ultra-thin film are enhanced for a thinner film. -0.1 0.0 0.1-0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5-0.1 0.0 0.1-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0-0.1 0.0 0.1-1.5-1.0-0.50.00.51.01.5 -0.050.000.050.100.1520 30 40 50 60 70 806.166.176.186.196.206.21246820 30 40 50 60 70 80-60-50-40-30-20-10010 (a) (b) (c) L =32 Å L =25 Å L =20 Å L [Å] L [Å] Å Å k[ Å -1 ] Å (e)(d) E ne r g y [ e V ] B [ e V Å ] D [ e V Å ] (g)(f) ∆ [ e V ] v F [ m / s ] FIG. 2: (Color online) [(a)-(c)]: Two-fold degenerate ( τ z = ±
1) energy spectra of surface states for thickness L = 20, 25,32 ˚A(solid lines), and L → ∞ (dash lines). The grey areacorresponds to the bulk states. The energy spectra are ob-tained by solving H ( k, − i∂ z )Ψ( z ) = E Ψ( z ) under the bound-ary conditions Ψ( z = ± L/
2) = 0. Please note that the scalesof energy axis in (a)-(c) are different. The model parametersare adopted from Ref. 9: M = 0 . , A = 2 . A =4 . B = 10eV˚A , B = 56 . , C = − . D = 1 . , D = 19 . . [(d)-(g)] The parameters forthe new effective model H eff : D , B , the energy gap ∆, andthe Fermi velocity v F vs L . III. ENERGY GAP AND K-DEPENDENT SPINCONFIGURATION
The opening of energy gap for the Dirac fermions isexpected as a result of quantum tunneling between thesurface states on the top and bottom surfaces. Whenthe thickness of the ultrathin film is comparable withthe decay length of the surface states into the bulk, thewavefunctions of the top and bottom surface states havea spatial overlap which leads to an energy gap at the Γpoint, analogous to the splitting of the bound and anti-bound orbitals in a double-well potential. The dispersionrelation of the surface states are plotted in Fig. 2(a), (b)and (c) for several thicknesses. A massless dispersion isobtained for a large L limit as expected. For a finitethickness, the energy gap at k = 0 is a function of thethickness L and decays quickly with L [see Fig. 2(f)]. Itis noticed that the gap ∆ even changes its sign at certainvalues of L . For instance, for the present case, at aboutinteger times of 25 ˚A. Correspondingly, the velocity ofthe Dirac fermions is also thickness dependent, which isenhanced for a small L . Strictly speaking, for an ultra-thin film, these so-called “surface states” emerge in theentire film. Nevertheless, they always lie in the bulk en-ergy gap and can thus be distinguished from those bulkoriginated quantum well states.From the solution to Eq. (10), it is obvious that thespin vectors in each band take a k -dependent spin config-uration, in the neighborhood of the Γ point determinedby the gap parameter ∆, and at large k determined bythe term Bσ z k , as schematically illustrated in Fig. 3.As the two Dirac hyperbolas are time reversal copy ofeach other, it is obvious that they have just the opposite k -dependent spin configurations, as can be seen from the τ z dependence of h τ z ( k ) in Eq. (11). (b) ∆ / B > 0(a) ∆ / B < 0 FIG. 3: (Color online) Schematic illustration of the k -dependent spin configurations in the conduction band (top)and valence band (bottom) in Dirac hyperbola τ z = +1 when(a) ∆ /B < /B >
0. The center and corners ofeach panel correspond to that k = 0 and k is large enough,respectively. IV. ORBITAL MAGNETIC MOMENT ANDBERRY CURVATURES
The opposite k -dependent spin configurations resultin opposite physical properties of the surface Bloch elec-trons in the two Dirac hyperbolas, including the orbitalmagnetic moments and Berry curvatures, as shown inthe context of graphene. These properties make pos-sible the manipulation of spin dynamics by electric andmagnetic fields in the thin film topological insulator.Orbital magnetic moment arises from a self-rotating motion of the surface Bloch electron and is defined as m ( k ) = − i e ~ h∇ k u ( k ) | × [ h τ z − ε ( k )] |∇ k u ( k ) i · ˆ z, (20)where ε ( k ) and u ( k ) are the dispersion and eigenstatesof h τ z . By putting Eqs. (12) and (13) into Eq. (20),we find the conduction and valence bands for each h τ z have the same orbital angular momentum m ( k ) = − τ z | e | ~ ~ v F (∆ / Bk )2[(∆ / − Bk ) + ~ v F k ] . (21)At the Γ point, the two degenerate Dirac hyperbolas ac-quire opposite orbital moments which add to the spinmagnetic moment.
10 20 30 40 50 60 -400-300-200-1000100200300400 -0.4-0.20.00.20.4-0.4-0.20.00.20.4 -30-1501530 -0.10 -0.05 0.00 0.05 0.10-200-150-100-50050100150200-0.10 -0.05 0.00 0.05 0.10-20-15-10-505101520 (e) k=0.01 Å -1 k=0.001 Å -1 k=0 m / µ B E ne r g y [ e V ] m / µ B k [ Å -1 ]k [ Å -1 ] L [ Å ] -100-50050100 (b)(a) L =32 Å L =20 Å (d) L =32 Å (c) Ω [ n m ] L =20 Å FIG. 4: (Color online) [(a)(b)]: Energy spectra of surfacestates (dash lines) and orbital angular moment m of the con-duction bands (solid lines) as functions of k for L = 20 and32 ˚A. [(c)(d)] Berry curvature Ω( k ) of the conduction band.(e) Orbital angular moment m of the conduction band as afunction of thickness L for k =0.01, 0.001, and 0 ˚ A − . Thesingularities occur at where ∆ changes sign. Only results forthe hyperbola τ z = +1 are shown. The results for τ z = − τ z = +1. The k -dependent spin configuration also results in agauge field in the crystal momentum space, known asthe Berry curvatureΩ( k ) = ˆ z · ∇ k × h u ( k ) | i ∇ k | u ( k ) i . (22)In an in-plane electric field, electrons acquire an anoma-lous transverse velocity proportional to the Berry cur-vature, giving rise to the Hall effect. In the conductionbands, the Berry curvature distribution near the Γ pointis Ω( k ) = − τ z ~ v F (∆ / Bk )2[(∆ / − Bk ) + ~ v F k ] / . (23)The Berry curvature in the valence bands is right op-posite to that of in the conduction bands. There areopposite distributions of the Berry curvature in the twoDirac hyperbolas. The in-plane electric field can there-fore drive the spin up and spin down electrons towardsthe opposite transverse edges of the thin film, which is asurface spin Hall effect. V. SPIN HALL EFFECT AND TOPOLOGICALQUANTUM PHASE TRANSITION
The ∆ term in our model plays a role as the magne-tization in the massive Dirac model exploited to studythe anomalous Hall effect. In principle, we can find theHall conductance for each h τ z . Note that h τ z in Eq. (11)can be explicitly written as h τ z = E − Dk + X i = x,y,z d i σ i , (24)where σ i are the Pauli matrices, and the d ( k ) vectors d x = ~ v F k y , d y = − ~ v F k x , d z = τ z ( ∆2 − Bk ) , (25)For the 2 × d ( k ) vec-tors and Pauli matrices, the Kubo formula for the Hallconductance can be generally expressed as σ xy = e ~ Z d k (2 π ) ( f k,c − f k,v ) d ǫ αβγ ∂d a ∂k x ∂d β ∂k y d γ , (26)where d is the norm of ( d x , d y , d z ), f k,c/v =1 / { exp[( ε c/v ( k ) − µ ) /k B T ] + 1 } the Fermi distributionfunction of the conduction ( c ) and valence ( v ) bands,with µ the chemical potential, k B the Boltzmann con-stant, and T the temperature.At zero temperature and when the chemical potential µ lies between ( − | ∆ | , | ∆ | ), the Fermi functions reduce to f k,c = 0 and f k,v = 1. By substituting Eq. (25) into (26)we arrive at σ xy (0 , τ z ) = − τ z e h Z ∞ d ( k ) ( ~ v F ) ( ∆2 + Bk )[( ~ v F ) k + ( ∆2 − Bk ) ] . (27)Note that by comparing above equation with Eq. (23),we can write the Hall conductance in the form of theBerry curvature of the conduction band, σ xy (0 , τ z ) = e ~ Z d k (2 π ) ( f k,v − f k,c )Ω( k ) . (28)By definingcos θ = ( ∆2 − Bk ) q ( ~ v F ) k + ( ∆2 − Bk ) , (29) Eq. (27) can be transformed into σ xy (0 , τ z ) = τ z e h Z ∞ d ( k ) ∂ (cos θ ) ∂ ( k ) (30)The values of cos θ at k = 0 and k → ∞ only dependon the signs of B and ∆, respectively. As a result, inthe insulating regime − | ∆ | ≤ µ ≤ | ∆ | , we find that theanomalous Hall conductance for each hyperbola has theform σ xy (0 , τ z ) = − τ z e h (sgn(∆) + sgn( B )) , (31)where τ z (sgn(∆)+sgn( B )) is the Chern number of the va-lence bands. It is a well known result in the field theorythat the Hall conductance of the massive Dirac fermionsis a half of quanta e /h , σ xy = − τ z e h sgn(∆) (if B = 0).(see Ref. 24.) Our result demonstrates that a non-zeroquadratic term in k ( B = 0) will give a reasonable resultsince a half quantized Hall conductance is not possiblefor a non-interacting system. For small L , the parameter B is always negative, and ∆ changes its sign accompa-nying a gap close-and-reopen while the thickness of thethin film increases [Fig. 2(f)]. The sign change of ∆flips the k -dependent spin configuration near the Γ pointand results in a jump τ z in the Chern number, i.e. ,a topological quantum phase transition as discussed inquantum spin Hall effect in HgTe/CdTe quantum well. In the ultrathin limit, ∆ > < σ xy (0 , τ z ) = τ z ( e /h ), as shown in Fig. 5.By the edge-bulk correspondence, gapless helical edge-states shall appear accompanying such a transition. Thisresult is indeed supported by the solutions of the differ-ential equation of h ± ( k ) in a geometry of a semi-infiniteplane, in which there exists a gapless edge state onlywhen ∆ /B >
0, and the sign of the Hall conductancecan be justified from the chirality of the edge states. -0.2 -0.1 0.0 0.1 0.2-1.0-0.50.00.51.0-0.2 -0.1 0.0 0.1 0.2-0.2-0.10.00.10.2 τ z = −1 τ z = +1 τ z = +1 (a) µ [ eV ] σ xy [ e / h ] L =20 Å ( ∆/Β <0) τ z = −1 (b) L =32Å ( ∆/Β >0) FIG. 5: (Color online) The hyperbola-dependent Hall con-ductance vs Fermi level µ for L = 20˚A (a) and 32˚A (b),respectively. When the Fermi level µ lies in the electron band, ac-cording to Eq. (26) σ xy ( µ > | ∆ | , τ z ) = σ xy (0 , τ z ) + τ z σ + xy where at zero temperature, σ + xy = e h Z k F d ( k ) A ( ∆2 + Bk )[ A k + ( ∆2 − Bk ) ] / . Near the Γ point, the B term in the electron band dis-persion can be neglected, so that ε c ≈ ( ~ v F ) k + ( ∆2 ) ,and k F = [ µ − ( ∆2 ) ] / ( ~ v F ) . After a straightforward calculation and Taylor expansionof the result up to the linear term of µ , the Hall con-ductance is obtained when the Fermi surface is in theconduction bands, σ xy ( µ > | ∆ | , τ z ) = − τ z e h [sgn( B )+sgn(∆) − π ( ~ v F ) δn ∆ | ∆ | ] . where the Fermi level µ is replaced by δn , the carrierdensity near the band bottom, by using the relation | ∆ | ( µ − | ∆ | ) / (4 π ~ v F ) ≈ δn , which is found by using thedispersion ε c and the same Taylor expansion approach.One can check that when the Fermi level lies in the va-lence bands, σ xy is the same as the above result due tothe particle-hole symmetry.As shown in Fig. 2(e), B is negative for small L . For∆ > σ xy ( µ > ∆2 , τ z ) = τ z e h π ∆ ( ~ v F ) δn, (32)where δn is the carrier density. In the two Dirac hyper-bolas, the Hall conductance are opposite (see Fig. 5) andspin vectors near the band bottom point in the + z and − z directions respectively. Thus net spin accumulationswith out-of-plane polarization are expected on the twoedges of the thin film. For ∆ <
0, doping reduces thequantized Hall conductance and we find σ xy ( µ > | ∆ | , τ z ) = τ z e h [1 − π ∆ ( ~ v F ) δn ] . (33)The Hall conductance as a function of the Fermi level µ are plotted in Fig. 5. Note that because the Hallconductances for the two hyperbolas are always equal inmagnitude and opposite in sign, and the two hyperbolashave right opposite spin orientations, here the Hall con-ductances are referred to as the ordinary and quantumspin Hall effects. VI. SPIN OPTICAL SELECTION RULE
Since an energy gap is opened in the Dirac hyperbo-las, interband transitions between the conduction andvalence surface bands can be excited by optical field. In the two gapped Dirac hyperbolas being time-reversal ofeach other, interband transition couples preferentially toright-handed ( σ +) or left-handed ( σ − ) circular polarizedlight, as was first discovered in the context of graphene. In the thin film topological insulator where the two Dirachyperbolas are associated with opposite spin configura-tions (Fig. 3), such an optical transition selection ruleis of significance since it allows the spin dynamics to beaddressed by optical means. The interband couplings inthe two hyperbolas to normally incident circular polar-ized lights can be studied by calculating the degree ofcircular polarization, defined as η ( k ) ≡ | π + cv ( k ) | − | π − cv ( k ) | | π + cv ( k ) | + | π − cv ( k ) | , where ± corresponding to σ ± lights, respectively, andthe interband matrix element of the velocity operator isdefined by π ± cv ( k ) ≡ h u c ( k ) | ∂h τ z ∂k x ± i ∂h τ z ∂k y | u v ( k ) i , where u c ( k ) and u v ( k ) are the eigen states for the con-duction and valence bands of h τ z , and have already beengiven in Eq. (13). By ignoring Bk terms near the Γpoint, one finds | π ± cv ( k ) | ≃ ( ~ v F ) (1 ± τ z cos θ ) , where cos θ = ∆ / ( ε c ( k ) − ε v ( k )), with ε c ( ε v ) the dis-persion of the conduction (valence) band of h τ z given inEq. (12). Then the degree of polarization is found outfor each τ z η ( k ) = τ z θ θ . Near the Dirac hyperbola center where ε c ( k ) − ε v ( k ) ≃| ∆ | , σ + ( σ − ) light couples only to the Dirac hyperbola τ z = sgn(∆) ( τ z = − sgn(∆)), causing transition betweenthe spin down (up) valence state and the spin up (down)conduction state. This is in sharp contrast to the spinoptical transition selection rule in conventional semicon-ductor which leaves the spin part of the wavefunction un-changed. This spin optical transition selection rule makespossible optical injection of spins in the thin film. Forexample, with band-edge optical excitation by σ + circu-larly polarized light, spin up electrons in the conductionband and spin up holes (by convention, a spin up holerefers to an empty spin down valence state) in the valenceband are created in the Dirac hyperbola τ z = sgn(∆),which can be separated by an in-plane electric field toprevent the radiative recombination. VII. CONCLUSIONS
In this paper, we derive an effective model for an ul-trathin film of topological insulator, in which tunnelingbetween the surface states on the top and bottom sur-faces opens a finite gap in the Dirac cone centered at theΓ point ( k = 0). The low lying physics of the ultrathinfilm can be described as two degenerate massive Dirachyperbolas which form time reversal copy of each other.Each massive band has a k -dependent spin configuration:one near the Γ point determined by the energy gap, andthe other at k large enough determined by a spin-orbitcoupling term quadratic in k . It is found that the energygap oscillates with the thin film thickness, and changessign at critical thicknesses. Across the transition points,the k -dependent spin configuration near the Γ point isflipped while those at large k remains unchanged, lead-ing to a topological quantum phase transition where theChern numbers of the surface bands change [Eq. (31)].In the two Dirac hyperbolas of opposite spin configura- tions, the k -dependent spin structure results in a distri-bution of orbital magnetic moment and Berry curvaturewith opposite signs. In doped metallic regime, the Berrycurvature drives the spin Hall effect of the extra carrierswhich leads to net spin accumulations on the thin filmedges. We also discover a spin-dependent optical transi-tion selection rule which allows optical injection of spinsby circular polarized lights into the thin film.This work was supported by the Research Grant Coun-cil of Hong Kong under Grant No.: HKU 7037/08P, andHKU 10/CRF/08. Note added : After posting this paper on arXiv, welearnt about the works by Liu et al and Linder et al ,in which the similar finite size effect of the surface stateswas studied. L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. ,106803 (2007). J. E. Moore and L. Balents, Phys. Rev. B , 121306(R)(2007). S. Murakami, New. J. Phys. , 356 (2007). J. C. Y. Teo, L. Fu, and C. L. Kane, Phys. Rev. B ,045426 (2008). D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava,and M. Z. Hasan, Nature (London) , 970 (2008). D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J.Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S.Hor, R. J. Cava, M. Z. Hasan, Science , 919 (2009). A. Nishide, A. A. Taskin, Y. Takeichi, T. Okuda, A. Kak-izaki, T. Hirahara, K. Nakatsuji, F. Komori, Y. Ando, I.Matsuda, arxiv: condmat/0902.2251 (2009). Y. Xia, D. Qian, D. Hsieh, L.Wray, A. Pal, H. Lin, A.Bansil, D. Grauer, Y. S. Hor, R. J. Cava and M. Z. Hasan,Nat. Phys. , 398 (2009). H. J. Zhang, C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S.C. Zhang, Nat. Phys. , 438 (2009). D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J.Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong, A.V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J.Cava, and M. Z. Hasan, Nature , 1101 (2009). Y. L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K. Mo,X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C.Zhang, I. R. Fisher, Z. Hussain, Z.-X. Shen, Science 325,178 (2009). X. L. Qi, T. L. Hughes, and S. C. Zhang, Phys. Rev. B ,195424 (2008). L. Fu and C. L. Kane, Phys. Rev. Lett. , 096407 (2008). J. Nilsson, A. R. Akhmerov, and C. W. J. Beenakker, ,120403 (2008). L. Fu and C. L. Kane, Phys. Rev. Lett. 102, 216403 (2009). A. R. Akhmerov, J. Nilsson, and C. W. J. Beenakker, Phys. Rev. Lett. 102, 216404 (2009). Y. Tanaka, T. Yokoyama, and N. Nagaosa, Phys. Rev.Lett. 103, 107002 (2009). K. T. Law, P. A. Lee, and T. K. Ng, Phys. Rev. Lett. 103,237001 (2009). B. Zhou, H. Z. Lu, R. L. Chu, S. Q. Shen, and Q. Niu,Phys. Rev. Lett. , 246807 (2008). D. Xiao, W. Yao, and Q. Niu, Phys. Rev. Lett. , 236809(2007). M. C. Chang and Q. Niu, J. Phys.: Cond. Mat. , 193202(2008). D. Xiao, M. C. Chang and Q. Niu, arXiv: 0907.2021. D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M.den Nijs Phys. Rev. Lett. , 405 (1982). A. N. Redlich, Phys. Rev. D 29, 2366 (1984). S. Q. Shen, Phys. Rev. B 70, 081311(R) (2004). B. Zhou, C. X. Liu, and S. Q. Shen, EPL 79, 47010 (2007). B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Science314, 1757 (2006). Y. Hatsugai, Phys. Rev. Lett. , 3697 (1993). W. Yao, D. Xiao, and Q. Niu, Phys. Rev. B , 235406(2008). N. A. Sinitsyn, A. H. MacDonald, T. Jungwirth, V. K.Dugaev, and J. Sinova, Phys. Rev. B , 045315 (2007). X. L. Qi, Y. S. Wu, and S. C. Zhang, Phys. Rev. B ,085308 (2006). B. Zhou, L. Ren, and S. Q. Shen, Phys. Rev. B , 165303(2006). C. X. Liu, H. J. Zhang, B. H. Yan, X. L. Qi, T. Frauen-heim, X. Dai, Z. Fang and S. C. Zhang, arxiv: condmat/0908.3654 (2009). J. Linder, T. Yokoyama, and A. Sudb, Phys. Rev. B80