Effects of surface-bulk hybridization in 3D topological `metals'
Yi-Ting Hsu, Mark H. Fischer, Taylor L. Hughes, Kyungwha Park, Eun-Ah Kim
EEffects of surface-bulk hybridization in 3D topological ‘metals’
Yi-Ting Hsu, Mark H. Fischer,
1, 2
Taylor L. Hughes, Kyungwha Park, and Eun-Ah Kim Department of Physics, Cornell University, Ithaca, New York 14853, USA Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel Department of Physics, University of Illinois, 1110 West Green St, Urbana IL 61801, USA Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA (Dated: August 20, 2018)Identifying the effects of surface-bulk coupling is a key challenge in exploiting the topologicalnature of the surface states in many available three-dimensional topological ‘metals’. Here wecombine an effective-model calculation and an ab-initio slab calculation to study the effects of thelowest order surface-bulk interaction: hybridization. In the effective-model study, we discretize anestablished low-energy effective four-band model and introduce hybridization between surface bandsand bulk bands in the spirit of the Fano model. We find that hybridization enhances the energy gapbetween bulk and Dirac surface states and preserves the latter’s spin texture qualitatively albeitwith a reduced spin-polarization magnitude. Our ab-initio study finds the energy gap between thebulk and the surface states to grow upon an increase in the slab thickness, very much in qualitativeagreement with the effective model study. Comparing the results of our two approaches, we deducethat the experimentally observed low magnitude of the spin polarization can be attributed to ahybridization-type surface-bulk interaction. We discuss evidence for such hybridization in existingARPES data.
I. INTRODUCTION
Many discrepancies between experimental measure-ments and theoretical predictions of ideal topological in-sulators (TI’s) are attributed to the fact that the chem-ical potential lies in the conduction band and the bulkband interferes with measurements . That is, manyavailable TI materials are actually metallic. Recent de-velopments in thin-film experiments further call forstudies of surface-bulk electron interaction in films. How-ever, explicit first-principles calculations on TI films arelimited to very thin slabs with thickness less than 10nm due to the computational cost. Therefore, there is aneed for a simple microscopic model which incorporatessurface-bulk interactions that can be used to study howphysical properties depend on the film thickness.One important question such a model should address isthe effect of surface-bulk interaction on the spin-texture.The surface spin-texture is a key physical characteris-tic of topological surface states in ideal TIs. While low-energy effective theories guided by symmetries predictperfect spin-momentum locking for topological surfacestates within the bulk gap, spin- and angle-resolved pho-toemission spectroscopfy (SARPES) data show lower in-plane spin polarization and total spin magnitude .On the other hand, an ab-initio calculation on a few-quintuple-layer (QL) slab found both the spin polar-ization and the total spin magnitude to be much smaller.Although reduction in spin polarization and total spinmagnitude are to be anticipated at large surface Fermi-momenta where hexagonal warping manifests , lit-tle is understood about the reduction observed at smallFermi-momenta and how the surface-bulk interaction af-fects the spin texture. However, such understandingis crucial for pursuing technical applications of spin-momentum locking in thin films of topological insulators in the metallic regime .Our starting point is the observation made by Bergmanand Refael that the lowest order electron-electron in-teraction term between the surface state and the bulkstates can be viewed as a hybridization term in theFano model . The key effect of hybridization in thislow-energy effective theory is to spectroscopically sepa-rate surface states localized on the surface from the ex-tended metallic bands. Building on the principles under-lying this low-energy effective theory, we study the hy-bridization effects with a microscopic model of 3D time-reversal invariant strong topological insulators to addressthe thickness dependence of physical quantities and con-nect the results to ab-initio slab calculations. Specifi-cally, we study TI slabs with finite thickness in the pres-ence of surface-bulk interaction from two complementaryperspectives: a simple microscopic model including thelowest order surface-bulk interaction and an ab-initio cal-culation of a few-QL Bi Se . We focus on how the spec-troscopic properties and the spin texture evolve as a func-tion of film thickness.The paper is structured as follows. In section II weconstruct a lattice model for a slab with surface-bulk hy-bridization (S-B hybridization) and study the spectro-scopic properties of the model as well as the effects ofthe S-B hybridization on the spin texture. In section IIIwe present an ab-initio study on 4,5,6-QL Bi Se usingdensity functional theory (DFT) and discuss the insightthe simple hybridization model offers in understandingthe ab-initio results. We then conclude in section IVwith discussions of implications of our results and openquestions. a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l II. LATTICE MODEL FOR A SLAB WITH S-BHYBRIDIZATIONA. The Model
In order to introduce surface-bulk hybridization as aperturbation in the spirit of the Fano model and studyits effects on a slab with finite thickness, we first needa lattice model for a slab. For this, we discretize theeffective continuum model by Zhang et al. , which is afour band k · p Hamiltonian guided by symmetries andfirst-principle-calculation results. We then make the sys-tem size finite along the vertical axis, i.e., the film growthdirection.The low-energy effective four-band model in Ref. 21 de-scribes a strong 3D TI with rhombohedral crystal struc-ture such as Bi Se . In this effective model each QL istreated as a layer since the inter-QL coupling is weak,and the four lowest-lying spin-orbital bands come fromthe mixing of the two P z atomic orbitals from Bi and Sereferred to as P and P and the two spins ↑ , ↓ . We takethis effective model written in terms of 4 × k · p Hamiltonian inRef. 21 in the limit | k | →
0. The resulting lattice model inthe rotated spin-orbital basis {| P , ↑(cid:105) , − i | P , ↑(cid:105) , | P , ↓(cid:105) , i | P , ↓(cid:105)} is H = (cid:88) k (cid:107) ,k z h ( k (cid:107) , k z ) c † k (cid:107) ,k z c k (cid:107) ,k z , (1)where c † k (cid:107) ,k z is an operator creating a four spinor in therotated spin-orbital basis with an in-plane momentum k (cid:107) = ( k x , k y ) and perpendicular momentum k z , and thelattice Hamiltonian is h ( k (cid:107) , k z ) = (cid:15) ( k (cid:107) , k z ) I × + A a z sin( k z a z )Γ (2)+ A a x sin( k x a x )Γ + A a y sin( k y a y )Γ + M ( k (cid:107) , k z )Γ , with (cid:15) ( k (cid:107) , k z ) ≡ C + 2 D a z [1 − cos( k z a z )] + 2 D a x [1 − cos( k x a x )]+ 2 D a y [1 − cos( k y a y )] , (3) M ( k (cid:107) , k z ) ≡ m − B a z [1 − cos( k z a z )] − B a x [1 − cos( k x a x )] − B a y [1 − cos( k y a y )] . (4)The Gamma matrices are defined as Γ = σ z ⊗ τ x , Γ = − I × ⊗ τ y , Γ = σ x ⊗ τ x , Γ = σ y ⊗ τ x , and Γ = I × ⊗ τ z , where σ j and τ j are Pauli matrices acting on ( ↑ , ↓ ) and ( P , P ) spaces, respectively. a x , a y and a z arethe lattice constants in x, y, z directions respectively.We use the parameters for Bi Se obtained by fittingthe continuum model to ab-initio calculations : m =0 .
28 eV, A = 2 . A = 4 . B = 10 eV˚A , B = 56 . , C = − . D = 1 . and D = 19 . .We model a slab of Bi Se by imposing open bound-ary conditions at the top and the bottom surfaces whichbreak the translational symmetry along the z -axis. Since k z is no longer a good quantum number, while k (cid:107) stillis, we substitute c k (cid:107) ,k z = √ N (cid:80) j e ik z ja z c k (cid:107) ,j in Eq. (1)and label the four-spinor operators by the in-plane mo-mentum k (cid:107) = ( k x , k y ) and the index of layers j stackingin the z direction. Now the Hamiltonian for a slab with N layers is H ( N ) = (cid:88) k (cid:107) H ( k (cid:107) , N ) , (5)where H ( k (cid:107) , N ) = N (cid:88) j =1 M c † k (cid:107) ,j c k (cid:107) ,j + T c † k (cid:107) ,j +1 c k (cid:107) ,j + T † c † k (cid:107) ,j c k (cid:107) ,j +1 (6)and the 4 × T and M are defined as T ≡ − D a z I × + B a z Γ − iA a z Γ , (7)and M ≡ (cid:15) ( k (cid:107) ) I × + A a x sin( k x a x )Γ + A a y sin( k y a y )Γ + M ( k (cid:107) )Γ (8)where (cid:15) ( k (cid:107) ) ≡ C + 2 D a z + 2 D a x [1 − cos( k x a x )]+ 2 D a y [1 − cos( k y a y )](9)and M ( k (cid:107) ) ≡ m − B a z − B a x [1 − cos( k x a x )] − B a y [1 − cos( k y a y )] . (10)For the sake of simplicity, the results presented inthe remainder of this section are calculated with a x = a y = a z = 1˚ A .We can diagonalize H ( k (cid:107) , N ) as H ( k (cid:107) , N ) = N − (cid:88) α =1 E B,α ( k (cid:107) ) b † α, k (cid:107) b α, k (cid:107) + (cid:88) β =1 E D,β ( k (cid:107) ) d † β, k (cid:107) d β, k (cid:107) , (11)where b α, k (cid:107) and d β, k (cid:107) are four-spinor annihilation op-erators for bulk and surface states (henceforth referredto as the “Dirac” states) respectively in the absence ofhybridization, E B,α and E D,β are their correspondingeigenenergies. Here, α and β label the unhybridizedbulk and Dirac states respectively. All energy eigenstatesare two-fold degenerate as required by inversion (P) andtime-reversal (T) symmetries. For each in-plane momen-tum k (cid:107) , four energy eigenstates with their energies closestto the Dirac point are labeled to be valence ( β = 1 , β = 3 ,
4) Dirac states. α label the re-maining 4 N − α = 1 , · · · , N − α = 2 N − , · · · , N − α . As the model is derived from a low-energyeffective model near the Dirac point, it will break downat large energies. However, we expect qualitatively cor-rect results when it comes to trends of physical propertiesover the hybridization strength and film thickness whichonly require knowledge of the low-energy physics near theinsulating gap.Now we introduce the S-B hybridization term that isallowed by symmetries in the spirit of Fano model .The Fano model is a generic model describing the mix-ing between extended states c k with energy (cid:15) k and alocalized state b with energy (cid:15) through Hamiltonian H F = (cid:15)b † b + (cid:80) k [ (cid:15) k c † k c k + A k ( c † k b + b † c k )], where A k repre-sents scattering strength. Bergman and Refael pointedout that H F can be used to describe the lowest order in-teraction between a helical surface state and a metallicbulk band, i.e. hybridization. They studied effects of hy-bridization in a field theoretic approach. Here we use asymmetry-preserving form of the surface-bulk hybridiza-tion term in the spirit of H F for the microscopic modelof Bi Se shown in Eq. (11) to study the effects of mixingbetween Dirac states and bulk states.For simplicity we consider the case where the hy-bridization strength preserves in-plane momenta k (cid:107) andis independent of energy and k (cid:107) , i.e. h (cid:48) ( k (cid:107) ) = (cid:88) α,β g b † α, k (cid:107) d β, k (cid:107) + H.c.. (12)We then impose T and P symmetries on the full hy-bridization perturbation H (cid:48) ( k (cid:107) ) by constructing H (cid:48) ( k (cid:107) )through H (cid:48) ( k (cid:107) ) = h (cid:48) ( k (cid:107) ) + P h (cid:48) ( k (cid:107) ) P † + T h (cid:48) ( k (cid:107) ) T † + P T h (cid:48) ( k (cid:107) )( P T ) † , (13)where the representations for T and P symmetry opera-tors with the spatial inversion center at the middle pointof the slab in the current 4 N tight-binding spin-orbitalbasis are T = K iσ y ⊗ I × ⊗ I N × N with k (cid:107) ↔ − k (cid:107) ,and P = I × ⊗ τ z with z : [0 , N/ ↔ [ N/ , N ] and k (cid:107) ↔ − k (cid:107) , respectively. Here, K is the usual complexconjugation operator. Finally, the full Hamiltonian in-cluding hybridization at a given in-plane momentum k (cid:107) reads H ( k (cid:107) ) = H ( k (cid:107) , N ) + H (cid:48) ( k (cid:107) ) . (14) After diagonalizing the full Hamiltonian in the tight-binding spin-orbital bases, we can write H ( k (cid:107) ) = N − (cid:88) α =1 E B,α ( k (cid:107) ) b † α, k (cid:107) b α, k (cid:107) + (cid:88) β =1 E D,β ( k (cid:107) ) d † β, k (cid:107) d β, k (cid:107) , (15)where b α, k (cid:107) , d β, k (cid:107) , E B,α and E D,β are defined similarlyto the corresponding symbols with a superscript 0 inEq. (11) but in the presence of hybridization. E B,α and E D,β are again two-fold degenerate as H ( k (cid:107) ) preservesparity and time-reversal symmetries by design. α and β label the bulk and Dirac states for H ( k (cid:107) ), where theterms bulk and Dirac states are defined in the same fash-ion as for H ( k (cid:107) , N ) in the absence of hybridization. B. Topological Metal Regime
We begin our numerical study with no hybridization.In the absence of hybridization, depending on the chemi-cal potential µ , we now define three regimes: topologicalinsulator(TI), metal(M), and topological metal(TM)(seeFig. 1(a)). The familiar topological insulator (TI) regimeis where the chemical potential lies within the bulk gapand the system is actually a bulk band insulator, i.e., E v ≤ µ ≤ E c with E c being the bottom of the conduc-tion band and E v the top of the valence band. Withinthe TI regime, the Dirac states feature Rashba-type spin-momentum locking and a spatial profile localized on thesurfaces. Of our particular interest is the distinction wewill draw between M and TM regimes based on whetherthe Dirac states retain the spin-momentum locking andthe surface localization when being away from the TIregime.In order to examine the above two properties ofDirac states, we define | ψ D, k (cid:107) (cid:105) ≡ d † , k (cid:107) | (cid:105) ( d † , k (cid:107) | (cid:105) ) and | ˜ ψ D, k (cid:107) (cid:105) ≡ d † , k (cid:107) | (cid:105) ( d † , k (cid:107) | (cid:105) ) in the presence(absence) ofhybridization to represent the pair of degenerate Diracstates above the Dirac point. Now the spatial pro-file of the conduction Dirac states is | Ψ D, k (cid:107) ( z ) | ≡| ψ D, k (cid:107) ( z ) | + | ˜ ψ D, k (cid:107) ( z ) | which is a function of z mea-sured from the bottom of the slab along the finite dimen-sion of the slab. This quantity will show whether theDirac states are localized on the surfaces or not. Let usidentify a particular k (cid:107) of interest for a given value ofchemical potential as the in-plane momentum at whichthe chemical potential µ intersects the Dirac branch; wedenote such in-plane momentum by k (cid:107) ,µ . To illustratethe features defining the three regimes, we will now showthe spatial profiles and the spin polarizations of the Diracstates in the three regimes in the absence of hybridiza-tion. In the next section we will add the hybridizationand examine its effects. (a) (b)(c) (d)(e) (f) FIG. 1. (a) Spectra of the model on a 300-layer-thick slab.The three chemical potentials µ TI , µ TM and µ M are takenas representative points for the three regimes TI( µ ≤ E c ), M( µ (cid:38) E M ), and TM( E c ≤ µ (cid:46) E M ) as defined in the text,respectively. (b)-(d)The corresponding unhybridized(dashed,blue) and hybridized(solid, green) spatial profiles of the pair ofdegenerate conduction Dirac states | Ψ D, k (cid:107) ( z ) | at k (cid:107) = k (cid:107) ,µ with µ = µ TI , µ TM , and µ M , respectively. (e) Effect of thehybridization on the spectra. (f)ARPES data on Bi Se . Inspecting | Ψ D, k (cid:107) ( z ) | at k (cid:107) = k (cid:107) ,µ at the represen-tative values of chemical potential for the three regimes µ T I , µ M , µ T M shown in Fig. 1, we find that the spatialprofile of the Dirac states | Ψ D, k (cid:107) ( z ) | indicates surfacelocalized states of the slab in the TI regime as expected(see Fig. 1(b)). On the other hand, in the M regime,where the chemical potential is well within the bulk con-duction band, | Ψ D, k (cid:107) ( z ) | is fully delocalized over theentire slab (see Fig. 1(c)). In this regime, the system can-not be distinguished from an ordinary metal. However,even with µ > E c there is an energy window between E c and a crossover energy scale E M , where the Dirac statesare still spatially localized on the surfaces in the sensethat | Ψ D, k (cid:107) ( z ) | is peaked on each surface of the slaband decays away from the surfaces (see Fig. 1(d)). Thecrossover energy scale E M is a threshold energy, where,within the regime µ (cid:38) E M (regime M), wavefunctionsfor all states at in-plane momentum k (cid:107) ,µ delocalize. We define the system to behave as a topological metal(TM)when the chemical potential lies within this window, i.e. E c < µ < E M , represented by µ T M .A detectable characteristic of TI and TM regimes is thespin-momentum locking. One measure to quantify spin-momentum locking at the surface is through the so-called“spin polarization” which is the expectation value of thespin component perpendicular to the in-plane momentumof a Dirac state, i.e., (cid:104) S ˆ n (cid:105) ( k (cid:107) ) ≡ (cid:104) ψ D, k (cid:107) | S ˆ n | ψ D, k (cid:107) (cid:105) (16)with ˆn · k (cid:107) = 0. Here the ˆ n component of the quantumspin operator is defined as S ˆ n ≡ (cid:126) (cid:126)σ · ˆn ⊗ I ⊗ I NxN ,where (cid:126)σ = ( σ x , σ y , σ z ) are Pauli matrices acting on spin, I acts on the orbital degree of freedom , and I NxN actson the layer index. (cid:104) S y (cid:105) ( k x ˆ x ) is evaluated at k x ˆ x = k x,µ ˆ x and shown for µ = µ T I , µ
T M , µ M in Fig. 2(a). We seehere that, in the absence of hybridization(g=0), the spinpolarization stays maximal in the TI and TM regimeswhile rapidly dropping upon entering the M regime. An-other quantity of experimental interest is the total spinmagnitude associated with the Dirac states with in-planemomentum k (cid:107) defined in terms of spin polarization as S ( k (cid:107) ) ≡ (cid:115) (cid:88) i = x,y,z (cid:2) (cid:104) S i (cid:105) ( k (cid:107) ) (cid:3) . (17)Fig. 2(a) and 2(b) show that the spin-momentum lockingquantified using these measures clearly distinguishes theTM regime from the ordinary metal regime (M) in theabsence of hybridization. (a) (b) FIG. 2. The effect of hybridization on the degree of spin-momentum locking in different regimes. The spin expectationvalues are calculated for a 300-layer-thick slab using a conduc-tion Dirac state | ψ D,k x ˆ x (cid:105) with k x = k x,µ at different represen-tative chemical potentials µ = µ TI , µ TM and µ M . (a)Spin po-larization (cid:104) S y (cid:105) ( k x ˆ x ). (b)Total spin magnitude S ( k x ˆ x )(definedin the text). C. Effects of S-B Hybridization
We now turn to the effects of hybridization. One ef-fect of hybridization that is manifest in the experimentaldetection of Dirac surface states in the TM regime is anincrease in the bulk-Dirac state energy gap. We quan-tify this energy gap, for a given chemical potential µ, using the energy difference between a Dirac state abovethe Dirac point and the energetically closest bulk statedefined by∆ DB ( µ ) ≡ E (0) B, N − ( k (cid:107) ,µ ) − E (0) D, ( k (cid:107) ,µ ) (18)in the presence(absence) of hybridization. ComparingFig. 1(a) to (e), we find that the key effect of hybridiza-tion that is spectroscopically detectable is the increase in∆ DB ( µ ) in both TM and M regimes compared to the TIregime. Otherwise the spectra in the absence or presenceof hybridization look similar. Note that most ARPESdata on 3D TIs exhibit a clear energy gap between theDirac branch and the bulk states at a chemical poten-tial well into the bulk band as shown in Fig. 1(f). Thisexperimental trend hints at the possibility that a sizablehybridization between Dirac states and the bulk states iscommon in 3D TI materials. In order to demonstrate theeffect of hybridization, we choose a value of g = 5meVthat is subdominant to all the hopping terms yet substan-tial in this paper. However, key effects of hybridizationdo not depend qualitatively on the value of g .Another effect of hybridization is to broaden the Diracstate wavefunctions in the TI and TM regimes. The de-gree of broadening depends on the chemical potential µ ,hybridization strength g , and the slab thickness N . How-ever, as long as g is the smallest energy scale in the totalHamiltonian as is the case for Figs. 1(b-d), the Diracstates in the TI and TM regimes remain localized on thesurfaces. A tangible consequence of the wavefunctionbroadening is the quantitative suppression of the spin-momentum locking. As mentioned earlier, in the absenceof hybridization the Dirac states of TI and TM exhibita maximal degree of spin-momentum locking. However,hybridization rotates the spin vectors of different atomicorbitals and layers away from the direction perpendicu-lar to the in-plane momentum. Hence, both measures ofspin-momentum locking shown in Fig. 2 show quantita-tive reduction upon hybridization. This is in qualitativeagreement with the low values of spin polarization andtotal spin magnitude found in a first-principle calculationof a thin slab in a previous work and our DFT resultsin the next section. Note that the hybridization still pre-serves the spin-texture winding despite the quantitativereduction in the spin polarization.Finally, we study how the effects of hybridizationon the two experimentally accessible characteristics ofthe TM regime, namely how the bulk-Dirac energy gap∆ DB ( µ ) and the spin polarization (cid:104) S ˆ n (cid:105) ( k (cid:107) ) of a Diracstate, vary with the slab thickness. Since the quantizedenergy spacings due to finite size effects decreases with in-creasing slab thickness, we consider a dimensionless mea-sure that quantifies the bulk-Dirac energy gap: r ( µ ) ≡ ∆ DB ( µ ) / ∆ BB ( µ ) , (19)where ∆ BB ( µ ) ≡ E (0) B, N +1 ( k (cid:107) ,µ ) − E (0) B, N − ( k (cid:107) ,µ ) is theenergy spacing in the presence(absence) of hybridizationbetween the two lowest lying conduction bulk branches (a) (b) FIG. 3. Thickness dependence of the hybridization effects.(a)Dimensionless measure of bulk-Dirac energy gap r =∆ DB / ∆ BB (defined in the text) at different slab thickness for µ = µ TM . (b)The spin polarization of a conduction Diracstate (cid:104) S y (cid:105) ( k x ˆ x ) at µ = µ TM . measured at the same in-plane momentum k (cid:107) ,µ where∆ DB ( µ ) is calculated. This dimensionless quantity r ( µ )allows us to compensate for finite size effects though ∆ BB would be hard to measure experimentally for realisticbulk samples due to the lack of the required energy res-olution. Fig. 3(a) shows that the hybridization inducedenhancement in the bulk-Dirac energy gap becomes moreprominent with increasing slab thickness. Comparingthe existing ARPES data on bulk samples and on thinfilms , we find the Dirac branch to be better separatedfrom the bulk states in the bulk samples than in the thinfilms, which is consistent with the hybridization effectshown in Fig. 3(a). Finally Fig. 3(b) shows that the re-duction in spin-polarization magnitude |(cid:104) S y (cid:105) ( k x ˆ x ) | is alsointensified with increasing thickness. Such an enhance-ment in the impact of hybridization with the increasein slab thickness can be explained from the fact that athicker slab implies a larger number of bulk states thatmix with a fixed number of Dirac surface states for agiven strength of hybridization g. III. DFT CALCULATIONS OF THIN BI SE SLABS
Now we turn to an ab-initio study of thin slabs tocompare with the simple phenomenological model of hy-bridization we explored in the previous section. The ap-proach of the previous section is limited, in the sensethat it builds on a low-energy effective description of theband structure, and that there is no detailed knowledgeof the hybridization strength g which could in princi-ple be k (cid:107) -dependent. On the other hand, the DFT ap-proach on slabs, which does not require calculating sur-face and bulk separately as in the calculations of semi-infinite systems , is limited to very thin films of severalQLs due to computational limits. By combining the twoapproaches, we extract a more robust understanding ofthe effects of hybridization in the TM regime and impli-cations on their trends over film thickness.We calculate the electronic structure of Bi Se (111)slabs of 4-6 QLs using the VASP code withthe projector-augmented-wave method , within thegeneralized-gradient approximation (GGA) . Spin-orbitcoupling is included self-consistently. We use experimen-tal lattice constants and an energy cutoff of 420 eV witha 31 × × k -point grid. Our DFT calculations are lim-ited up to 6 QLs. For 5-6 QLs, the overlap between topand bottom surface states is already very small yieldingan energy gap of the order of meV at Γ. Expectationvalues of spin components (cid:104) S x (cid:105) , (cid:104) S y (cid:105) , (cid:104) S z (cid:105) are calculatedfrom the summation of the expectation values of eachatom. FIG. 4. (a) DFT-calculated band structure of a 6-QL slabof Bi Se . (b)DFT-calculated spin expectation values of theconduction Dirac state (cid:104) S i (cid:105) ( k x ˆ x ) for a 6-QL Bi Se slab.FIG. 5. (a) Ratio r = ∆ DB / ∆ BB within the TM regime,calculated at a fixed k (cid:107) . (b) (cid:104) S y (cid:105) of the conduction Diracstate calculated using DFT as a function of slab thickness N . Figure 4 shows the DFT-calculated band structure andspin expectation values (cid:104) S i (cid:105) ( k x ˆ x ) of a 6-QL slab. Thesurface states are doubly degenerate and have a Diracdispersion and we show five confined states in the bulkconduction band region [Fig. 4(a)]. For small | k x | values, (cid:104) S y (cid:105) of a Dirac conduction state is clearly dominant overother components and exhibits spin-momentum locking[Fig. 4(b)]. As | k x | increases, a small z component ofspin expectation value develops. However, over the en-tire range of k x , (cid:104) S y (cid:105) is much less than the maximal value,in agreement with previous DFT study . A comparisonbetween Fig. 2 and Fig. 4(b) indicates that our hybridiza-tion model is an effective way to capture the broadeningof the Dirac surface state wavefunction and the result- ing reduction in the spin polarization and the total spinmagnitude .Now we discuss the thickness dependence in the bulk-Dirac energy gap measure and the spin polarization. Wecalculate the dimensionless measure of bulk-Dirac en-ergy gap r = ∆ DB / ∆ BB in the TM regime at the k (cid:107) point where the Dirac surface state branch has slightlyhigher energy than the bottom of the conduction band E c , as indicated in Fig. 4(a). We find that the ra-tio ∆ BB ( N ) / ∆ BB ( N ) is close to ( N /N ) at the k (cid:107) point of interest as expected of finite-size-effect originof the scale ∆ BB ( N ). Surprisingly, despite the smallrange of thickness accessible to the slab DFT calcula-tion, the dimensionless measure of bulk-Dirac energy gap r = ∆ DB / ∆ BB in Fig. 5(a) shows a significant increaseupon an increase in the slab thickness. This is quali-tatively consistent with observations from the effectivemodel and hybridization effects in Sec. II. On the otherhand, the range of thickness in the present calculationappears to be too small to show any change in the (cid:104) S y (cid:105) as a function of slab thickness [Fig. 5(b)]. IV. CONCLUSION
We combined a Fano-type hybridization model calcula-tion with an ab-initio slab calculation to study the lowestorder effects of surface-bulk interaction in topological in-sulators with a particular focus in the TM regime. Wedefined the TM regime of a topological insulator to bewhere the Dirac surface states and bulk states coexist andinteract, yet the spin-winding is preserved albeit with areduced spin-polarization magnitude. The hybridizationmodel presented in Sec. II captures the spin-polarizationreduction of the Dirac states originating from the hy-bridization with bulk states. Given the metallic behav-ior of most TIs, and the experimental evidence of re-duced spin polarization, our simple model offers a usefulstarting point for applications of TIs which need to takereal materials in the TM regime into account. Moreover,the hybridization-driven bulk-Dirac energy gap explainswhy the Dirac branch shows up so well separated frombulk states in ARPES experiments. Note that this en-ergy gap and the suppression of total spin magnitudesare both experimentally observed phenomena that can-not be accessed by the typical approach of coupling asingle “surface layer” to a bulk electronic structure to in-clude surface states in semi-infinite systems as in Ref. 21.We propose SARPES experiments for films of varyingthickness to test our predictions for hybridization-drivensuppression of spin polarization for further vindication ofthe model.Promising future directions include DFT tools to studyslightly thicker systems. This might reveal thickness de-pendence in spin polarization and compared to the re-sults of the simple model. Also this would reveal moredetailed knowledge of the magnitude and k (cid:107) -dependenceof the hybridization strength g . Preliminary DFT resultsshow that g ( k (cid:107) ) has a significant k (cid:107) -dependence. An-other interesting direction will be to study consequencesof the hybridization effect on transport properties. Manypuzzling aspects of transport experiments have beenattributed to the presence of bulk states or surface-bulkinteraction. There is growing theoretical interest on thetransport properties of topological edge states in the pres-ence of metallic bulk states as well. Our microscopicmodel offers a simple starting point to theoretically ad-dress effects of surface-bulk interaction on transport in 3D TIs. Acknowledgements.
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