Spin-conserving and reversing photoemission from the surface states of Bi 2 Se 3 and Au (111)
SSpin-conserving and reversing photoemissionfrom the surface states of Bi Se and Au (111) Ji Hoon Ryoo and Cheol-Hwan Park ∗ Department of Physics, Seoul National University, Seoul 08826, Korea (Dated: October 16, 2018)We present a theory based on first-principles calculations explaining (i) why the tunability of spinpolarizations of photoelectrons from Bi Se (111) depends on the band index and Bloch wavevectorof the surface state and (ii) why such tunability is absent in the case of isosymmetric Au (111). Theresults provide not only an explanation for the recent, puzzling experimental observations but alsoa guide toward making highly-tunable spin-polarized electron sources from topological insulators.
Since the beginning of spintronics, constant effortshave been made to generate electrons with a high de-gree of spin polarization using transport [1], optical [2],and magnetic resonance methods [3]. In particular, op-tical methods, also known as optical spin orientation,use polarized-light irradiation. For example, electronsin the valence band of strained and surface-treated GaAscan be excited by circularly polarized light and emittedwith ∼
80 % spin polarization [4]. GaAs photocathodesare widely used as spin-polarized electron source in low-energy electron microscopy [5], in accelerators used inhigh-energy physics [6], etc.Recently, it has been proposed that topological insu-lators can serve as a spin-polarized electron source whenirradiated with polarized light [7]. By changing the po-larization of light and the direction toward which photo-electrons are collected, one can obtain an electron beamwhich is spin-polarized in an arbitrary direction, with a100 % degree of spin polarization [8] (the measured de-gree is over 80 % [9]). On the other hand, the direction ofspin polarization of electrons generated from a strained-GaAs photocathode is fixed by the surface-normal direc-tion perpendicular to which the strain is applied. More-over, unlike GaAs photocathodes, in which the photonenergy is fixed to ∼ . Se family of topologicalinsulators, whose space group is R¯3m, such as Bi Se ,Bi Te and Sb Te . Jozwiak et al. [7] studied photoelec-trons ejected from the Dirac-cone-like surface band ofBi Se and the Rashba-split surface band of Au (Fig. 1).When shone on a Bi Se (111) surface, p-polarized lightgenerates photoelectrons whose spin direction is paral-lel to that of the surface electrons, while s-polarized lightproduces photoelectrons with the opposite spin [7]. Sincethe Bi Se and Au (111) surfaces have the same symme- −0.4−0.2 0 Au s pol. p pol.p pol. Au s pol.s pol. e - e - e - e - h (cid:81) h (cid:81) h (cid:81) h (cid:81) k z k y k x k z k y k x k z k y k x k z k y k x Bi Se
Bi Se p r p t k x k y K (cid:42) M (cid:12)(cid:12) (cid:14)(cid:14) −0.4−0.2 0 (a)(b) (c) AuBi Se E n e r gy ( e V ) K ← Γ → M FIG. 1. (a) The Brillouin zone of Bi Se and Au surfaces.The p r and p t orbitals of a given Bloch state specified bythe Bloch wavevector k = k (cos φ k , sin φ k ) are defined as p r = cos φ k p x + sin φ k p y and p t = − sin φ k p x + cos φ k p y , re-spectively, where p x and p y are the valence atomic p orbitals.(b) The bandstructures (blue curves) of the (111) surfaces ofBi Se and Au along (0 . , − (0 , − (0 , .
14) in reciprocalspace in units of 2 π/a , where a is the lattice parameter. Pro-jected bulk bands are also shown in gray. (c) Schematics ofthe SARPES experimental setup and the results of Ref. [7].The horizontal arrows denote the direction of the spin polar-ization of the surface electrons and photoelectrons. try, the theoretical analysis predicts that the gold surfacewould also exhibit the same photo-induced spin modula-tion as Bi Se (111) [8]. The SARPES experiment for thegold surface [7] clearly resolves the two spin-split bands;however, both s- and p-polarized lights produce photo-electrons with the same spin direction as that of the ini-tial state in each surface band [Fig. 1(c)].Also, another experimental study on Bi Se (111) hasshown that photoemission from the upper branch of thesurface bands exhibits such photo-induced spin modula-tion, while that from the lower branch does not [10]. Inan effort to explain this observation, it was claimed thats-polarized light probes the spinor that couples to p r or-bital, because the electronic states in the lower branchhave more p r character than p t one [11] [for the definitionof p r and p t orbitals, see Fig. 1(a)]. However, the spinor a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b being measured is the one coupled to the orbital interact-ing with s-polarized light ( p t ), and not the one coupledto the dominant p orbital ( p r ). Therefore, the experi-mental observation cannot be understood from previoustheories [8, 11].In summary, we still do not have a good understand-ing of the photo-induced spin modulation phenomenoninvolving the Bi Se family of topological insulators. Inthis study, we perform first-principles calculations onthe spin polarization of photoelectrons ejected from theBi Se and Au (111) surfaces. First of all, our re-sults agree with the recent experimental observations inRefs. [7, 10] that were not understood before. We showthat the complicated, material-dependent coupling be-tween the spinor part and the orbital part of the wave-functions plays a central role in determining the spin po-larization from these surfaces. We also show that thisspinor-orbital coupling in the wavefunction of Bi Se , inparticular, depends heavily on both the direction andmagnitude of the Bloch wavevector; the pronounced de-viation of the spinor-orbital coupling from the one nearthe Dirac point is seen in the lower branch along ΓK,where the low-energy effective theories [8, 11] predict thatthe direction of the spin polarization of photoelectrons isthe opposite of the experimental observation [10]. Ourresults provide a theoretical background for developingnext-generation spin-polarized electron sources.To obtain the spin polarization of photoelectrons, wecalculate the matrix elements of A · p , where A is a vec-tor parallel to the polarization of light and p the momen-tum operator, between the initial surface state and thetwo (spin-up and spin-down) photoexcited states. Thismethod of using the dipole transition operator to accountfor light-matter interactions reproduces the measuredspin polarization of photoelectrons ejected from Bi Se quite successfully [12, 13]. For computational details, seeSupplemental Material [14]. To simulate low-energy pho-toemission experiments [7, 10], we set the photon energyto 6 eV.We denote the incoming direction of incidentphotons by ( − sin θ ph cos φ ph , − sin θ ph sin φ ph , − cos θ ph )and the outgoing direction of photoelectrons by(sin θ e cos φ e , sin θ e sin φ e , cos θ e ). We focus mainly ontwo cases: φ ph = ± ◦ and φ e = ± ◦ (i. e. , the in-planemomenta of light and photoelectrons are along ΓM) and φ ph = 0 ◦ or 180 ◦ and φ e = 0 ◦ or 180 ◦ (along ΓK).First, we compare the spin polarization of photoelec-trons emitted from the upper band of the surface states ofBi Se (111) [Fig. 1(c)] and of Au (111) [Fig. 1(d)] whenthe incident photons and photoelectrons both lie in themirror plane, which is perpendicular to ΓK ( φ ph = ± ◦ and φ e = 90 ◦ ). [For φ e = − ◦ , similar results are ob-tained provided the sign of φ ph is flipped in Fig. 2 (notshown)]. We define the spin polarization vector (with-out (cid:126) / P of a certain state as the expectation value ofthe Pauli spin operators taken for that state. Then, due −1−0.5 0 0.5 1 0 30 60 90 P x
0 30 60 90 s pol.( x pol.) p pol.( y pol.) k // Γ M (a) (b) φ ph = 90, Rotation angle of light polarization ( ° ) AuBi Se ± ° ° ° ° ° ± ° ± ° = ° θ ph φ e = ° = − ° θ ph α FIG. 2. The spin polarization along x , P x , of photoelectronsemitted from Bi Se (111) [(a)] and from Au (111) [(b)]. Notethat, for notational convenience, we have used negative θ ph to denote cases with φ ph = − ◦ . The initial surface state isin the upper band and has k = 0 .
01 (2 π/a ) ˆ y . to the mirror symmetry, (i) P of any surface state withBloch wavevector k along ΓM is parallel or antiparallelto ΓK and (ii) the p- and s-polarized photons generatephotoelectrons characterized by P which is 100 % in mag-nitude and is, respectively, parallel to and antiparallel tothe P of the surface state [8]. This symmetry analysis isin agreement with the SARPES experimental results onBi Se (111) [7, 13, 15].Since Bi Se (111) and Au (111) have the same sym-metry, one would naturally expect that the same symme-try analysis holds for Au (111); however, it was observedthat photoelectrons from the gold surface have the samespin polarization independent of the direction of A [7].In order to understand these seemingly contradictoryresults for Au (111), we calculate P x of photoelectronsas a function of the rotation angle α of light polarization(Fig. 2). For s- and p-polarized light ( α being 0 ◦ and 90 ◦ ,respectively) the calculated P x for both Bi Se (111) andAu (111) is in accord with the symmetry-based theoreti-cal prediction. A first-principles study also reported thespin reversal of photoelectrons ejected from Au (111) bys-polarized light and the spin conservation by p-polarizedlight [16]. However, the Bi Se and Au surfaces exhibitdifferences in the manner P x changes in between (Fig. 2).We first consider the case θ ph = 45 ◦ (correspondingto the curves in Fig. 2 with θ ph = ± ◦ ). For Bi Se , P x -versus- α relations for φ ph = 90 ◦ and for φ ph = − ◦ (denoted by negative θ ph in Fig. 2) are qualitatively dif-ferent [Fig. 2 (a)]: (i) when φ ph = − ◦ , P x varies slowlywith α from − α = 45 ◦ ; (ii)when φ ph = 90 ◦ , P x remains negative as long as α < ◦ .On the other hand, for Au, the dependence of P x on α for φ ph = 90 ◦ and that for φ ph = − ◦ are essentiallythe same. In both cases, P x changes sharply from − α [ P x = 0 at α = 3 ◦ ; see Fig. 2(b)].This difference between the two materials on how P x changes with α originates from the difference in the Se lower Bi Se upper Aulower Auupper P r o j ec t e d p r ob a b ilit y ( % ) p r p t p z k // Γ M FIG. 3. The projected probability (i. e. , squared amplitude)to each p orbital of the surface state with k = 0 .
01 (2 π/a ) ˆ y . surface-state wavefunctions. Among the orbitals consti-tuting the (initial) surface states, we focus on p orbitalswhich play a dominant role in photoemission when thefinal states have s -like characters. This scheme success-fully describes the results from low-energy SARPES ex-periments on Bi Se [13, 17].Figure 3 shows squared projections of the surface statesnear Γ of Bi Se and Au to each valence p orbital,summed over atomic sites. In Bi Se case, the contri-bution of in-plane p -orbitals ( p r and p t ) to the surfacestates is 35 %, similar in magnitude to that of p z orbital(51 %). On the contrary, each in-plane orbital ( p r or p t )of Au contributes less than 2 % to the surface state ofAu (111). The results on Au (111) are consistent withprevious studies [18, 19].Although the symmetry analysis indicates that each p orbital comprising the gold surface states couples tospinors in the same way as in the case of Bi Se , sincethe surface states of Au has almost no in-plane p -orbitalcharacter, the spin degree of freedom is not entangledwith the orbital ones. Therefore, if A z is finite, even ifit is small, P of photoelectrons from the gold surface isalmost completely determined by the spinor coupled tothe p z orbital of the surface state. Thus, P x rises sharplyas α deviates from 0 ◦ [Fig. 2(b)].We compare P x ’s of photoelectrons associated with φ ph = 90 ◦ and that associated with φ ph = − ◦ . Thelight polarization vectors for these two cases are the sameexcept that the signs of their out-of-plane componentsare opposite. Because the spinors attached to in-planeand out-of-plane p orbitals interfere with each other dif-ferently in the two cases, the corresponding P ’s are inprinciple different. This effect is sizable for Bi Se (111)[Fig. 2(a)] but is negligible for Au (111) [Fig. 2(b)] be-cause, again, the contribution of in-plane p orbitals tothe surface states of Au (111) is small.The dependence of P x on θ ph (Fig. 2) further illus-trates the importance of the entanglement between thespin and orbital degrees of freedom in photoemission pro-cesses. When θ ph = 0 ◦ (i. e. , normal incidence), A z = 0, −1−0.5 0 0.5 1 −0.1 0 0.1 P y k x (1/Å) UpperBranch s pol.p pol.TSS −0.1 0 0.1 k x (1/Å) Lowerbranch (a) (b) φ e = φ ph = 0 ° k // Γ K θ ph = 45 ° FIG. 4. The spin polarization of photoelectrons along y , P y , emitted from Bi Se (111) surface states with k alongΓK. The dash-dotted or green curve shows P y of the initialtopological surface state (TSS). and only the in-plane p orbitals are probed. Therefore,in this case, P x of photoelectrons from both Bi Se andAu surfaces changes slowly with α from − θ ph increases from 0 ◦ to 15 ◦ , A z becomes finite; therefore,the dependence of P x of photoelectrons from Au (111) on α significantly changes, becoming similar to that corre-sponding to θ ph = 45 ◦ . For Bi Se , however, this increasein θ ph does not have such a huge effect on P x .From the results of our calculations, we can under-stand the hitherto incomprehensible differences in theresults of SARPES experiments on Bi Se and Au sur-faces [7]. If the light with perfect s-polarization excitesa surface state, the measured P must be antiparallel tothe spin polarization of the surface state for both Bi Se and Au. In real experiments, however, the “s-polarized”light may contain a few percent of the p component dueto the imperfection of the polarizer, the inaccuracy inthe alignment, or the inhomogeneity of the surface. Ourcalculations [Fig. 2(b)] suggest that this small fraction ofp-polarized light may determine the spin polarization ofphotoelectrons from Au (111), which explains the exper-imental result [7] that s- and p-polarized lights producephotoelectrons with similar P ’s and that photo-inducedspin modulation is hard to achieve with Au (111).We now discuss the SARPES configuration φ ph = 0 ◦ and φ e = 0 ◦ , i. e. , photons and electrons have the in-plane momenta parallel to ΓK. In this case, no symmetryprinciple restricts the spin direction of surface electronsor photoelectrons. Nevertheless, when k of a surface stateis small, according to first-order k · p perturbation the-ory [11], the p r and p z orbitals in the surface states alwayscouple to the spinor |↓ t (cid:105) and the p t orbital to |↑ t (cid:105) in theupper branch, where |↑ t (cid:105) and |↓ t (cid:105) are the eigenspinors of σ t = σ · (ˆ z × ˆ k ) with eigenvalues 1 and −
1, respectively.(The three p orbitals couple to the opposite spinors in thelower branch.) Therefore, for a small k , P of the pho-toelectrons generated by p- and s- polarized lights areparallel to and antiparallel to the P of the surface state,respectively.
642 0 2 Se Bi Se Bi Se Se Bi Se Bi Se42 0 2 4 1 2 3 4 5 6 7 8 9 10Atomic layer index p y ↑ y 〉 p y ↓ y 〉 (a)(b) Lower branch (cid:12)(cid:14) p t p y = k // Γ K P r o j ec t e d p r ob a b ilit y ( % ) k x = 0.015 Å −1 k x = 0.106 Å −1 FIG. 5. Projected probability to the p t orbital in eachatomic layer of the surface states in the lower branch at k = 0 .
015 ˚A − ˆ x [(a)] and at k = 0 .
106 ˚A − ˆ x [(b)]. The spinis quantized along y . Atomic layer 1 is the topmost surfacelayer. However, first-order k · p theory is valid only at a small k : the couplings between orbitals and spinors that areforbidden near Γ (e. g. , p r and |↓ t (cid:105) or p t and |↑ t (cid:105) in thelower branch) are allowed if second or higher order effectsare considered. These couplings are anisotropic in thatif k is along ΓM they are strictly forbidden even at alarge k . For Au (111), these higher-order spin-orbitalentanglement effects are difficult to observe due to thedominance of p z character in the surface state; however,for Bi Se (111), in the lower branch along ΓK, theysignificantly affect the photoemission process if k is notsmall (Fig. 4).Figure 4 shows that, when probing the lower branchwith large k x , s-polarized light as well as p-polarized lightyields photoelectrons whose P y (the tangential compo-nent of P ) has the same sign as P y of the surface state,contrary to the small- k results. In the case of the upperbranch, this stark sign change of the spin polarization isnot observed in our calculation. The results at large k are confirmed by recent experiments [10].It was suggested that this lack of photo-induced spinmodulation associated with the surface state in the lowerbranch was due to the dominance of p r orbital in the cor-responding surface state which couples to |↑ t (cid:105) [10]. How-ever, since s-polarized light picks up the spinor coupledto p t orbital and not the spinor coupled to the dominant p orbital (i. e. , p r ), this explanation is not satisfactory.Instead, we show in the following that the origin of thisphenomenon is the complex spin-orbital coupling in theinitial surface state at large k , which is absent in the low-energy theory [8, 11]. P r o j ec t e d p r ob a b ilit y ( % ) k x (1/Å) p y y 〉 p y y 〉 ↑↓
0 0.05 0.1 k x (1/Å) (a) Topmost Se layerLower branch k // Γ KUpper branch (b)
FIG. 6. Projected probability to the p t orbital in the top-most atomic layer of the surface states with k along ΓK as afunction of k . Figure 5 shows the extent of contribution of the p t orbital (which is p y ) to the surface states in the lowerbranch with k along ΓK, resolved to each spinor. NearΓ ( k = 0 .
015 ˚A − ), the p y orbital in each layer couplesexclusively to |↓ y (cid:105) , as predicted by first-order k · p the-ory [11]. However, when k = 0 .
106 ˚A − , the coupling of p y to |↑ y (cid:105) is significant and, especially, in the case of thetopmost layer (which is the most important in photoe-mission processes), the projected probability to |↑ y (cid:105) ismore than twice as high as that to |↓ y (cid:105) .Figure 6 shows the projected probability of the tan-gential p orbital at the topmost surface layer. For thesurface state in the upper branch, the contribution fromthe term | p y ↓ y (cid:105) , albeit not forbidden at large k , is neg-ligible in the range of k considered. (In fact, this | p y ↓ y (cid:105) contribution is tiny up to the fourth atomic layers fromthe surface [14].) However, for the surface state in thelower branch at large k , the contribution of | p y ↑ y (cid:105) in thetopmost layer outweighs that of | p y ↓ y (cid:105) . (See Supplemen-tal Material [14] for the layer-resolved projection to thethree p orbitals.) This difference explains why, at large k , P y of the photoelectrons from the upper branch ex-cited by the s- and p-polarized lights have different signs[Fig. 4(a)], whereas P y of the photoelectrons from thelower branch have the same sign [Fig. 4(b)].In conclusion, we studied the possibility of modulatingthe electron spin through photoemission from the sur-faces of Bi Se and Au. We find that both (i) the in-tricate spin-orbital coupling and (ii) large- k effects arecrucial in understanding and predicting the possibilityof photo-induced spin modulation. Not only does ourstudy provide an explanation of the recent low-energy,spin-dependent photoemission experiments in a coherentmanner, it also establishes a designing principle for a newkind of spin-polarized electron sources using topologicalinsulators.We gratefully acknowledge fruitful discussions withX. J. Zhou on his experimental results in Ref. [10] andwith Chris Jozwiak and Choongyu Hwang on manyaspects of SARPES experiments on Bi Se and Au(111). This work was supported by Korean NRF-2013R1A1A1076141 funded by MSIP and computationalresources were provided by Aspiring Researcher Programthrough Seoul National University in 2014. ∗ [email protected][1] M. Johnson and R. H. Silsbee, Phys. Rev. Lett. , 1790(1985).[2] D. T. Pierce and F. Meier, Phys. Rev. B , 5484 (1976).[3] S. D. Sarma, J. Fabian, X. Hu, and I. ˇZuti´c, IEEE Trans.Magn. , 2821 (2000).[4] T. Nakanishi, H. Aoyagi, H. Horinaka, Y. Kamiya,T. Kato, S. Nakamura, T. Saka, and M. Tsubata, Phys.Lett. A , 345 (1991).[5] E. Bauer, T. Duden, and R. Zdyb, J. Phys. D: Appl. Phys. , 2327 (2002).[6] R. Alley, H. Aoyagi, J. Clendenin, J. Frisch, C. Garden,E. Hoyt, R. Kirby, L. Klaisner, A. Kulikov, R. Miller,G. Mulhollan, C. Prescott, P. Sez, D. Schultz, H. Tang,J. Turner, K. Witte, M. Woods, A. Yeremian, andM. Zolotorev, Nucl. Instr. Meth. Phys. Res. A , 1(1995).[7] C. Jozwiak, C.-H. Park, K. Gotlieb, C. Hwang, D.-H. Lee,S. G. Louie, J. D. Denlinger, C. R. Rotundu, R. J. Bir-geneau, Z. Hussain, and A. Lanzara, Nat. Phys. , 293(2013).[8] C.-H. Park and S. G. Louie, Phys. Rev. Lett. , 097601(2012).[9] C. Jozwiak, Y. L. Chen, A. V. Fedorov, J. G. Analytis,C. R. Rotundu, A. K. Schmid, J. D. Denlinger, Y.-D.Chuang, D.-H. Lee, I. R. Fisher, R. J. Birgeneau, Z.-X.Shen, Z. Hussain, and A. Lanzara, Phys. Rev. B ,165113 (2011).[10] Z. Xie, S. He, C. Chen, Y. Feng, H. Yi, A. Liang, L. Zhao,D. Mou, J. He, Y. Peng, X. Liu, Y. Liu, G. Liu, X. Dong,L. Yu, J. Zhang, S. Zhang, Z. Wang, F. Zhang, F. Yang,Q. Peng, X. Wang, C. Chen, Z. Xu, and X. J. Zhou, Nat.Commun. , 3382 (2014).[11] H. Zhang, C.-X. Liu, and S.-C. Zhang, Phys. Rev. Lett. , 066801 (2013).[12] Z.-H. Zhu, C. Veenstra, G. Levy, A. Ubaldini, P. Syers,N. Butch, J. Paglione, M. Haverkort, I. Elfimov, andA. Damascelli, Phys. Rev. Lett. , 216401 (2013).[13] Z.-H. Zhu, C. N. Veenstra, S. Zhdanovich, M. P. Schnei-der, T. Okuda, K. Miyamoto, S.-Y. Zhu, H. Namatame,M. Taniguchi, M. W. Haverkort, I. S. Elfimov, andA. Damascelli, Phys. Rev. Lett. , 076802 (2014).[14] See Supplemental Material at http://link.aps.org/xxx forcomputational details and for a layer-resolved projectionof the surface-state wavefunctions of Bi2Se3 to each p or-bital.[15] Y. Cao, J. A. Waugh, N. C. Plumb, T. J. Reber,S. Parham, G. Landolt, Z. Xu, A. Yang, J. Schneeloch,G. Gu, J. H. Dil, and D. S. Dessau, arXiv:1211.5998v1.[16] J. Henk, A. Ernst, and P. Bruno, Phys. Rev. B ,165416 (2003).[17] J. S´anchez-Barriga, A. Varykhalov, J. Braun, S.-Y.Xu, N. Alidoust, O. Kornilov, J. Min´ar, K. Hummer,G. Springholz, G. Bauer, R. Schumann, L. V. Yashina,H. Ebert, M. Z. Hasan, and O. Rader, Phys. Rev. X ,011046 (2014). [18] H. Lee and H. J. Choi, Phys. Rev. B , 045437 (2012).[19] H. Ishida, Phys. Rev. B , 235422 (2014). Supplemental Material
CALCULATION DETAILS
We obtained the wavefunctions of surface states us-ing Quantum Espresso package [S1]. We have modeledBi Se (111) and Au (111) surfaces by 30- and 24-atomic-layer slabs, respectively. We set the inter-slab distanceto 20 ˚A for Bi Se and 30 ˚A for Au. We fully relaxed thelattice parameters and the atomic positions taking intoaccount, for Bi Se , van der Waals interactions. (Wehave checked that the relaxed structure of bulk Bi Se is in very good agreement - less than 1 % differences inthe lattice parameters and in the inter-quintuple-layerdistance - with the measurement [S2].)To describe ion-electron interactions, we used fully-relativistic, norm-conserving pseudopotentials. To ac-count for exchange-correlation interactions, we used themethod of Ref. [S3] for Bi Se and that of Ref. [S4] forAu. The k-point meshes that we used for Bi Se (111)and for Au (111) are 13 × × × ×
1, respec-tively. The kinetic energy cutoff is set to 60 Ry.Photoexcited states are described, following Refs. [S5,S6], as the Bloch sum of s -like orbitals inside thecrystal. A photoexcited state | f (cid:105) can be approxi-mated as, taking into account the phase of an elec-tron emitted from each atom and inelastic collisionsinside the crystal, | f (cid:105) ≈ | ψ (cid:105) ⊗ | χ (cid:105) with (cid:104) r | ψ (cid:105) ∝ (cid:80) R , α e z α / λ e i k f · ( R + τ α ) φ α ( r − R − τ α ) being the orbitalpart of the wavefunction and | χ (cid:105) the spin part (eitherspin-up or spin-down). Here, R denotes the (in-plane)lattice vector, τ α the position of an atom α within eachunit cell of the slab, λ the inelastic mean free path ofelectrons, set to 7 ˚A [S5], and φ α ( r − R − τ α ) the orbitallocalized at each atomic site (see below for details).The wavevector of a photoelectron k f = ( k f, (cid:107) , k f, z )and the wavevector of the initial surface elec-tron k i are related by k f, (cid:107) = k i and k f, z = (cid:112) m ( hν − E B ) / (cid:126) − ( k i, x ) − ( k i, y ) , where E B is thebinding energy of the surface state, m the mass of an electron, and the photon energy, hν , is set to 6 eV [S5].The calculations based on this scheme are proven to re-produce the measured spin polarization of photoelectronsfrom Bi Se quite well [S6, S7].We set φ α ( r ) = c α e − r /R , where the parameter R isset to be 0 . c α is an atomic-type-dependent constant. We checked that using the atomicvalence s orbitals as φ ( r )’s yields essentially the sameresults. For the final states of Bi Se , we set c α for Seto be twice as large as that for Bi in order to match theatomic cross sections of the two elements [S5]. ∗ [email protected][S1] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car,C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ-cioni, I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris,G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis,A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari,F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello,L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P.Seitsonen, A. Smogunov, P. Umari, and R. M. Wentz-covitch, J. Phys.: Condens. Matter , 395502 (2009).[S2] S. Nakajima, J. Phys. Chem. Solids , 479 (1963).[S3] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.Lett. , 3865 (1996).[S4] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov,G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke,Phys. Rev. Lett. , 136406 (2008).[S5] Z.-H. Zhu, C. Veenstra, G. Levy, A. Ubaldini, P. Syers,N. Butch, J. Paglione, M. Haverkort, I. Elfimov, andA. Damascelli, Phys. Rev. Lett. , 216401 (2013).[S6] Z.-H. Zhu, C. N. Veenstra, S. Zhdanovich, M. P. Schnei-der, T. Okuda, K. Miyamoto, S.-Y. Zhu, H. Namatame,M. Taniguchi, M. W. Haverkort, I. S. Elfimov, andA. Damascelli, Phys. Rev. Lett. , 076802 (2014).[S7] J. S´anchez-Barriga, A. Varykhalov, J. Braun, S.-Y.Xu, N. Alidoust, O. Kornilov, J. Min´ar, K. Hummer,G. Springholz, G. Bauer, R. Schumann, L. V. Yashina,H. Ebert, M. Z. Hasan, and O. Rader, Phys. Rev. X ,011046 (2014). P r o j ec t e d p r ob a b ilit y ( % ) ↑ y 〉 ↓ y 〉 k x = 0.015 Å −1 ↓ 23.2% Se Bi Se Bi Se Se Bi Se Bi Se k x = 0.106 Å −1 θ S ( ° ) Atomic layer index φ S = 90 ° φ S = − 90 ° Atomic layer index1 2 3 4 5 6 7 8 9 10 (a)(c) (d)(b) p r p t p z Upper branch k // Γ K
840 4 8 12 16 20 P r o j ec t e d p r ob a b ilit y ( % ) Se Bi Se Bi Se Se Bi Se Bi Se k x = 0.015 Å −1 ↑ y 〉 ↓ y 〉 Se Bi Se Bi Se Se Bi Se Bi Se k x = 0.106 Å −1 θ S ( ° ) Atomic layer index φ S = 90 ° φ S = − 90 ° (e)(g) (h)(f) Lower branch k // Γ K p r p t p z FIG. S1. (a) and (b) Projected probability of the surface states in the upper branch with k along ΓK and at k = 0 .
015 ˚A − [(a)]and at k = 0 .