Surface States of Topological Insulators

University of Pennsylvania ScholarlyCommons

Department of Physics Papers Department of Physics

8-10-2012

Fan Zhang University of Pennsylvania

Charles L. Kane University of Pennsylvania, [email protected]

Eugene J. Mele University of Pennsylvania, [email protected]

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Recommended Citation Zhang, F., Kane, C. L., & Mele, E. J. (2012). Surface States of Topological Insulators. Retrieved from https://repository.upenn.edu/physics_papers/256

This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/256 For more information, please contact [email protected]. Surface States of Topological Insulators

Abstract We introduce a topological boundary condition to study the surface states of topological insulators within a long-wavelength four-band model. We find that the Dirac point energy, the band curvature, and the spin texture of surface states are crystal-face dependent. For an arbitrary termination of a bulk crystal, the energy of the symmetry protected Dirac point is determined by the bulk physics that breaks particle-hole symmetry in the surface normal direction and is tunable by surface potentials that preserve time reversal symmetry. For a model appropriate to Bi2Se3 the constant energy contours are generically elliptical with spin textures that are helical on the cleavage surface, collapsed to one dimension on any side face, and tilted out of plane otherwise. Our findings identify a outer to engineering the Dirac point physics on the surfaces of real materials.

Disciplines Physical Sciences and Mathematics | Physics

Comments Zhang, F., Kane, C. L., & Mele, E. J. (2012). Surface states of topological insulators. Physical Review B 86(8), 081303. doi: 10.1103/PhysRevB.86.081303

This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/256 RAPID COMMUNICATIONS

PHYSICAL REVIEW B 86, 081303(R) (2012)

Surface states of topological insulators

Fan Zhang,* C. L. Kane, and E. J. Mele Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA (Received 28 March 2012; published 10 August 2012) We introduce a topological boundary condition to study the surface states of topological insulators within a long-wavelength four-band model. We ﬁnd that the Dirac point energy, the band curvature, and the spin texture of surface states are crystal-face dependent. For an arbitrary termination of a bulk crystal, the energy of the symmetry protected Dirac point is determined by the bulk physics that breaks particle-hole symmetry in the surface normal direction and is tunable by surface potentials that preserve time reversal symmetry. For a model

appropriate to Bi2Se3 the constant energy contours are generically elliptical with spin textures that are helical on the cleavage surface, collapsed to one dimension on any side face, and tilted out of plane otherwise. Our ﬁndings identify a route to engineering the Dirac point physics on the surfaces of real materials. DOI: 10.1103/PhysRevB.86.081303 PACS number(s): 73.20.−r, 71.70.Ej, 73.22.Dj

Introduction. The discovery1–6 of topological insulators inversion near its bulk point. For a momentum linearized (TIs) and the synthesis7–9 of three-dimensional materials that theory of its cleavage surface our method reproduces its realize their physics has opened up a new field in solid state well-studied low-energy surface state dispersion and spin physics. Particular interest has focused on the TI surface states texture. Extending our method to include its symmetry allowed with point degeneracies that are topologically protected by quadratic terms reveals electronic physics that is unexpectedly time reversal symmetry (T ) when a trivial insulator with rich: The topological surface bands develop a pronounced 1,2 topological index Z2 = 1 is joined to a TI with Z2 =−1. On curvature and the Dirac point is displaced from the gap center, the cleavage surface of Bi2Se3, a single Dirac cone has been as seen experimentally. Notably, we find that the both Dirac identified9–11 by angle resolved photoemission (ARPES) and point energy and its associated spin texture are crystal-face scanning tunneling microscope (STM) experiments. The sur- dependent due to the R3¯m bulk symmetry of this material. face states form a spin-momentum-locked helical metal with We find that these properties can be compactly encoded in conduction and valence bands exhibiting opposite helicities the algebraic structure of a general surface state Hamiltonian near the Dirac point, and develop hexagonal warping9,11–13 of that couples the orbital and spin degrees of freedom for an its constant energy surfaces away from the Dirac point. arbitrary crystal face. We further identify the modification of Interest in the topological surface bands has led to the these features by surface-localized potentials that are absent derivation of various models to describe them using bulk con- from previous bulk-derived theories. Of the possible surface tinuum Hamiltonians terminated at the boundary to vacuum. potentials that are allowed by T symmetry we find that This approach is attractive since it provides a compact and the four-band model admits only two that affect the TI surface often analytic theory of the TI surface states. In practice this states. The combination of the TBC and these two surface approach has been problematic because its solution requires, terms provides an analytically tractable two-parameter family in addition, the specification of the surface boundary condition of Hamiltonians that dictate the surface electronic spectra for (SBC) that terminates the bulk-derived model. The nature an arbitrary crystal termination of a TI. of the SBC has remained unclear and different approaches We start from a description of the low-energy model of to it are discussed in the current literature. The surface has Bi2Se3, which applies generally to other TIs with the same been introduced either by an ad hoc fixed node boundary crystal structure with space group R3¯m. At the origin of Bi2Se3 condition7,13–17 which clamps the wave function to zero at the Brillouin zone , the effective Hilbert space near the bulk gap 18 =±1 boundary or by a “natural” boundary condition which posits is spanned by states with angular momentum mj 2 and an energy functional whose minimization yields the SBC. The parity P =±1. Because of the spin-orbit coupling (SOC), choice of SBC has profound consequences for the spatial form |pz ↑ and |pz ↓ states are mixed with |p+ ↓ and |p− ↑ states, of the surface state wave functions and their interactions with respectively, however, since the crystal-field splitting is much external fields and absorbed species. While these properties larger than the SOC, pz orbitals dominate and mj pseudospin should be derivable from a complete continuum theory of the is proportional to the electron spin. The P =±1 hybridized surface, a well-defined SBC for the termination of a TI has so states can be labeled approximately by the |+ state from Bi far been missing. atoms while |− from Se, due to the large energy difference In this Rapid Communication we point out that this diffi- between 4p (Se) and 6p (Bi) principal quantum levels. culty has a unique resolution due to the topological character Besides T and the parity inversion (P) symmetries, the of the interface between a TI and vacuum. We identify the Bi2Se3 crystal structure has threefold rotational (C3) symmetry physically relevant topological boundary condition (TBC) at along zˆ perpendicular to the quintuple layers, and twofold rota- the surface which correctly accounts for the band inversion at tional (C2) symmetry along the M direction. By convention the TI-vacuum interface responsible for its protected surface we choose the parity operator P = τz and the time reversal T = states. As an example we apply the method to Bi2Se3,a operator iKσy, where K is the complex conjugate well-studied topological insulator controlled by a single band operation. Therefore, to quadratic order in k,thek · p bulk

FAN ZHANG, C. L. KANE, AND E. J. MELE PHYSICAL REVIEW B 86, 081303(R) (2012)

Hamiltonian that preserves the above four symmetries has a N (001) unique form: H = + 2 + 2 + − + 2 + 2 c0 czkz ck m0 mzkz mk τz + vzkzτy + v(kyσx − kxσy)τx, (1) where we assume mz,m,vz,v > 0 andh ¯ = 1 hereafter. The first parentheses is a scalar term that preserves T and P E(100) O E (100) symmetries but breaks the particle-hole (p-h) symmetry. The quadratic scalar terms intrinsically give rise to the curvature of Dirac surface bands and the rigid shift of Dirac point from k3 the middle of the bulk gap, as we will demonstrate in the kz following. Equation (1) has both T and P symmetries with a topological index Z2 = sgn(−m0). Topological boundary condition. The symmetry protected k1 S (001) kx physics at the TI surface occurs because of the sign reversal of the mass term in Eq. (1) across an interface with vacuum. FIG. 1. (Color online) The definition of an arbitrary face of = Accordingly we study a model in which the TI side has m0 m aBiSe -type topological insulator. The south (S) pole denotes = = 2 3 positive and half of the bulk gap at k 0, and c0 0 to define the (001)¯ surface parallel to the quintuple layers and the equator the middle of the k = 0 gap as energy zero. The vacuum side is represents the side faces perpendicular to the quintuple layers. For =− a trivial insulator with an infinite gap, i.e., m0 M, where any face (θ), kˆ3 ⊥ ,andk2 = ky. M →+∞.Bothcz and c vanish since the vacuum side is dominated by M and can be regarded as p-h symmetric around σ by S2. For an arbitrary surface (θ) defined in Fig. 1,the c0. algebraic structure is It is useful to consider the one-dimensional quantum ={ + − } ¯ S1 ατx βσyτy,ατy βσyτx,τz , mechanics of the (001) surface in which we turn off the (4) quadratic terms in Eq. (1). Note that the two spin flavors are S2 ={ασx − βσzτz,σy,ασz + βσxτz}, decoupled for this eigenvalue problem. The spin ↑ states in = 2 + 2 = {τz,σ } representation are determined by the following coupled where v3 (vz cos θ) (v sin θ) , α vz cos θ/v3, and = i j = Dirac equations: β v sin θ/v3. These pseudospins satisfy [Sa,Sb] 2iδ ijkSk. Rewritten in this pseudospin basis, Eq. (1) reads −m(z) − E −v ∂ ψ ab a z z 1 = 0. (2) v ∂ m(z) − E ψ H =−m Sz + (v k + v k )Sy z z 2 0 1 3 3 0 1 1 x y x + − Thewavefunctionψ↑ = (ψ1,ψ2) is continuous across the v kyS2 v1k1S2 S1 , (5) surface, integrating Eq. (2) over the vicinity of the surface. 2 2 where v = (v − v )sinθ cos θ/v and v = v v/v .By This continuity condition leads to a nontrivial solution which 0 z 3 1 z 3 matching the eigensystems of the TI and vacuum sides, we is isolated in the middle of the bulk gap and is localized on the obtain the (θ) surface states similar to Eq. (3) where τ surface. This midgap (E = 0) state has a simple and elegant and σ are replaced by S1 and S2, respectively. Note that exact solution, = + y κ m/v3 iv0k1/v3 in general. k1 coupling to S1 neither 1 i(k x+k y) 1 1 influences the energy spectrum nor the spin texture, only ψ ↑(x,y,z) = e x y φ(z) ⊗ , k, A 0 0 making the evanescent states oscillate and giving a phase σ τx (3) accumulation along kˆ1 away from the Dirac point. Therefore, e−κz,z>0(TI), φ(z) = κ z ignoring the quadratic corrections, we first derive the effective e 0 ,z<0(vac), surface state Hamiltonian for an arbitrary face (θ)tothe where κ = m/vz, κ0 = M/vz, and A is a normalization factor. linear order, φ(z) is evanescent on both sides of the surface where the mass H(1) = x − y (θ) vkyS2 v1k1S2 , (6) m0 changes sign, analogous to the Jackiw and Rebbi (JR) solution19 of a two-band Dirac model. The spin ↓ solution from which we can further explicitly demonstrate the surface can be obtained by ψ↓ = T ψ↑. This isolated midgap state at state spin texture on (θ): k = 0 is identified as the Dirac point and at finite k spreads vzvky cos θ into a perfect Dirac cone which forms the ideal topologically σxθ =± , 2 2 + 2 2 protected surface bands. Note that the evanescent solutions v3 v1k1 v ky of the JR model in the exterior region provide the appropriate −vzvk1 boundary condition for terminating of the bulk states of Eq. (1). σyθ =± , (7) 2 2 + 2 2 This TBC applies even when the quadratic terms are nonzero. v3 v1k1 v ky Surface state spin texture. On the (001)¯ surface, the midgap σzθ = 0, solution of the boundary problem at k = 0 is determined by the operators τ and is free under any rotation of the where + (−) denotes the conduction (valence) band. The operators σ .Thisτ ⊗ σ algebra of the boundary problem electron real spin s is proportional to but always smaller 20 can be generalized to any surface with τ replaced by S1 and than σ due to the SOC. In the local coordinates,

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a b c E E E 0.4 0.4 0.4

0.2 0.2 0.2

k k k 0.1 0.1 0.1 0.1 0.1 0.1

0.2 0.2 0.2

0.4 0.4 0.4

Π Π a 0 Face b 4 Face c 2 Face FIG. 3. (Color online) Band curvature and Dirac point position of the (001) surface state (red) in the bulk (blue) gap. (a) plots the linear theory Eq. (6), (b) includes quadratic corrections in Eqs. (8) and (9), and (c) further considers a surface potential with E =−m/8in Eq. (10). E and k are in units of eV and A˚ −1. Parameters are adopted from Ref. 7. FIG. 2. (Color online) Dirac cones and spin textures of surface states on faces with (a) θ = 0, (b) θ = π/4, and (c) θ = π/2. The spectrum are changed. In the coupled Schrodinger-type¨ Eq. (1), equal-energy contours are from −80 to 160 meV with 10 meV the components of the wave functions and their slopes are increments relative to the Dirac point. Surface band curvatures are all continuous across the boundary. We find that the effective 21 taken into account. All three panels are in the same scale and the Hamiltonian for face (θ) inherit two important corrections: parameters are adopted from Ref. 7. The lower panels only show the H(2) = 2 + 2 + 2 2 (θ) cky (cz sin θ c cos θ)k1, (8) spin textures at 160 meV in the k1-ky planes, where ky is the vertical axis. 2 2 c cos θ + c sin θ HDP(θ) = z m. (9) 2 2 mz cos θ + m sin θ σ1θ = σxθ cos θ and σ3θ = σxθ sin θ. Equation (7) in- dicates that while the pseudospin (S2) texture has a universal The breaking p-h symmetry terms give the surface Dirac cone structure on the elliptic constant energy contour near the Dirac a parabolic curvature described by Eq. (8) and shifts the Dirac point, the surface state spin (σ) texture is rather different from point from the midgap to a nonzero energy given by Eq. (9). face to face. The absence of spherical symmetry in the bulk These two effects are responsible for the shape of the cleavage Hamiltonian requires that the spin and orbital structures are surface bands observed in ARPES experiments,8,9,22–24 as crystal-face dependent on the surface of a TI. The spin texture illustrated in Fig. 3. is helical on the south and north poles, though compressed to Although the surface state solution obtained by Eq. (2) with a single dimension along the equator, and tilted out of plane TBC remains topologically stable, it is essential to understand for a general crystal surface. The surface state anisotropy and how localized potentials that arise from surface reconstruction spin textures for different faces are compared in Fig. 2. or adsorbed species can influence the surface spectra and the The surface state spin textures on the poles and the equator associated wave functions as well. We focus on potentials of a TI sphere can be understood by symmetries. Note that that preserve T symmetry as well as the lattice translational the bulk Hamiltonian Eq. (1) has no coupling to σz in the symmetry in the plane of the surface. Among these six types, I, crystal frame due to the mirror symmetry with respect to τz, τx, and στ y listed in Table I, τx and στ y break P symmetry. M(xˆ). For the (100)¯ face, the C2 symmetry along the M It turns out that the same potential could play different roles direction forbids σx coupling to any in-plane momentum. In on different crystal faces, as shown in Table I. linear order C3 symmetry upgrades to continuous rotational On the south pole of a TI sphere, a potential 0δ(z)2vz/mI symmetry, consequently, the surface normal spin is zero along changes the wave function continuity condition obtained by the TI equator. On the two poles, the mirror symmetry with integrating Eq. (2), including surface potentials over the = respect to M and the C3 symmetry along the zˆ direction ensure vicinity of z 0. The amplitudes of ψ1 and ψ2 are either that spin and momentum lock into the form of kyσx-kxσy. weakened or enhanced across the surface, with changes always It is interesting to point out that the surface band is the opposite to each other. Consequently, to match the evanescent x positive eigenstate of S1 and the chiral counterpart is separated solutions on the vacuum and the matter sides, the surface band and localized on the opposite face. Thus the surface state energies are rigidly shifted by δEDP. τx potentials have similar Hilbert space is reduced by half and this pseudospin polarity effects to the I type although τx breaks P symmetry. For a τz (or chirality) blocks the backscattering on the surface as a potential, it simply modifies the mass term on the surface, and it result of the interplay between T symmetry and band inversion is not surprising that it does not affect the surface spectra. A τz physics with TBC. potential tunes the amplitudes of ψ1 and ψ2 in the same manner Dirac point energy and surface potentials. The absence of on the matter side, but this gives no observable effect since spherical symmetry in the bulk also requires that the Dirac ψ1(z) = ψ2(z) and ψ1,2(z<0) → 0 are still valid. Similarly, point has different energies on different crystal faces. When a potential such as σnτy (nσn ≡ · σ ) does nothing to the the quadratic mass terms and p-h symmetry breaking terms surface spectra and ψ1/ψ2; however, it couples the two spin in Eq. (1) are introduced, the surface state wave function and flavors and shifts their phase with opposite signs. Note that

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TABLE I. Summary of the influence of T symmetry allowed Our model is quite different from the fixed boundary momentum-independent surface potentials δ(r3)2v3/m on the condition (FBC) which arbitrarily clamps the surface state inversion symmetry, and the wave-function (Ref. 25) continuity and wave function to zero at the boundary. The FBC solution is the Dirac point energy of surface states. This represents different actually insensitive to the mass inversion at the surface which surface potentials and their corresponding energy scales: 0I, mτz, topologically protects the surface bands, and consequently x n y ES1 ,andnS2 S1 . For the σnτy column, only the results for the it cannot be used to calculate the energy of the symmetry Sn = 1 state are shown and their complex conjugates represent the 2 protected degeneracy relative to the bulk bands or their in- results for the Sn =−1 state. 2 teraction with surface-localized potentials. Indeed, the energy of the Dirac degeneracy is used as an input to these theories. (0) Iτz τx σnτy Furthermore, FBC admits an infinite number of (physically (θ) IτSx SnSy z 1 2 1 spurious) solutions with nonzero energies for k1 = k2 = 0 that P ++−−satisfy an (incorrect) surface boundary condition. By contrast 4m2 (m2−2) 4m2 (m2+2 ) TBC guarantees that there is only one isolated solution, i.e., δE 0 0 0 E E 0 DP (m2−2)2+4m22 (m2+2 )2+4m22 0 0 E E the Dirac point of surface bands protected by the change in + 2− − 2 2 ψ1(0 ) m 2m0 0 m−E bulk topology. Moreover, TBC demonstrates that the energy + 1 1 ψ (0 ) m2+2m −2 m+ 2 0 0 E position of the Dirac point is tunable in the bulk gap via the + m2−2m −2 − ψ1(0 ) 0 0 m+m m E m−in symmetry allowed scalar terms and surface potentials. ψ (0−) 2+ 2 m− m+ m+i 1 m 0 m E n + 2+ − 2 On a noncleavage surface, dangling bonds and their recon- ψ2(0 ) m 2m0 0 m+m m+E m−in ψ (0−) 2+ 2 m− m− m+i struction may add complexity to the surface state spectrum 2 m 0 m E n which can be modeled as surface potentials encoded in the parameters and . Our formulation for the surface ψ = ψ (Ref. 25) is always true on the vacuum side since 0 E 1 2 potentials is applicable within the four-band model whenever M →∞. The above results are summarized in Table I,pro- these perturbations are spatially localized to the surface region viding sufficient information for constructing the surface state (much smaller than the decay length for the TI evanescent wave function and to engineer the Dirac point energy position. states) and are energy independent. Since the bulk gap is On an arbitrary face (θ), the types of surface potentials are small in these materials, this will generally be the case for the same but their combinations and the corresponding roles most surface perturbations. However, for situations where are rearranged. This can be fully understood by the fact that the perturbing potential penetrates the bulk, either by the the spin-orbital structure τ ⊗ σ on the south pole is replaced propagation of electrostatic or strain fields, one needs to by a pseudospin-pseudospin structure S (θ) ⊗ S (θ)on (θ). 1 2 replace the surface matching condition by a scheme in which Note that the combination only occurs for potentials with the the bulk and vacuum evanescent states are separately matched same parity. to electronic states calculated within a finite depth near the Although the surface state solutions are stable in the surface. In principle, these states can be obtained numerically, presence of localized surface potentials that preserve T in which case Table I gives the symmetry allowed spatially symmetry, Table I shows that the two terms proportional to varying potentials that can enter the theory. Note that if I and Sx play an essential role in determining the energy 1 the penetration depth is sufficiently large, this can introduce position of the Dirac point, a strongly energy-dependent matching condition, reflecting 2 2 2 2 4m (0 + E) m − + the presence of bound and resonant states in the interme- E = HDP + 0 E , DP 2 (10) diate region. This physics is responsible for the quantum 4m2( + )2 + m2 − 2 + 2 0 E 0 E well states that arise from strong band bending near a TI x 27 which implies that I and S1 potentials are able to tune the Dirac surface. point from the midgap to the band edges ±m. This effect is The novel spin texture that we predict near the Dirac point illustrated in Fig. 3. on noncleavage surfaces is determined by the bulk symmetries Discussions. The interactions of the topologically protected along with the topological stability of the surface spectrum bands with surface potentials provide a robust route to and may be accessible in ARPES and STM experiments. On engineering and manipulating the topological surface states. an equatorial face, Zeeman coupling or magnetic disorder In particular, an external surface potential can raise or lower coupled to the spin degree of freedom do not generally open the Dirac point to the middle of the bulk gap, providing a gap, and the diamagnetic susceptibility is anticipated to be experimental access to the topologically protected band. unusually anisotropic. More spectacularly, there is intrinsic This goal could be achieved by surface oxidation,26 or charge redistribution in the surface bands near the corners by other possible chemical processes,27 perturbations,28 and of a TI that connect different crystal faces with intrinsically interactions29 on the surface. We also point out that the different Dirac point energies. 22,27,30,31 electrostatic gating Eeff r3sf can act to Stark shift the Finally, we point out that our theory can be applied to TI surface state into the gap, with a field strength determined many other topological states of matter that are related to band by the penetration length of the surface states. Typically, the inversion (or mass reversal), besides T symmetry protected screened field ∼100 meV nm−1 is required for the surface strong TIs. A recent example is SnTe, a topological crystalline states that decay in a couple of quintuple layers. Our present insulator,32 where band inversion occurs at four bulk L points results provide a framework to study the self-consistent band and its surface states are protected by mirror symmetry. bending physics of real TI materials8,9,22–24 and mean-field Acknowledgment. This work has been supported by DARPA models of surface state many-body interactions. under Grant No. SPAWAR N66001-11-1-4110.

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