Direct observation of spin-polarized bulk bands in an inversion-symmetric semiconductor

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1 J. M. Rley, 1 F. Mazzola, 2 M. Dendzk, 3 M. Mchard, 3 T. Takayama, 4, 5 L. Bawden, 1 C. Granerød, 2 M. Leandersson, 6 T. Balasubramanan, 6 M. Hoesch, 7 T. K. Km, 7 H. Takag, 4, 5 W. Meevasana, 8, 9 Ph. Hofmann, 3 M. S. Bahramy, 10, 11 J. W. Wells, 2 and P. D. C. Kng 1, 1 SUPA, School of Physcs and Astronomy, Unversty of St. Andrews, St. SUPPLEMENTARY INFORMATION Drect observaton of spn-polarzed bulk bands n an nverson-symmetrc semconductor Andrews, Ffe KY16 9SS, Unted Kngdom 2 Department of Physcs, Norwegan Unversty of Scence and Technology (NTNU), N-7491 Trondhem, Norway 3 Department of Physcs and Astronomy, Interdscplnary Nanoscence Center (NANO), Aarhus Unversty, 8000 Aarhus C, Denmark 4 Department of Physcs, Unversty of Tokyo, Hongo, Tokyo Planck Insttute for Sold State Research, Stuttgart, Germany 6 MAX IV Laboratory, Lund Unversty, P. O. Box 118, Lund, Sweden 7 Damond Lght Source, Harwell Campus, Ddcot, OX11 0DE, Unted Kngdom 8 School of Physcs, Suranaree Unversty of Technology, Nakhon Ratchasma, 30000, Thaland 9 NANOTEC-SUT Center of Excellence on Advanced Functonal Nanomaterals, Suranaree Unversty of Technology, Nakhon Ratchasma 30000, Thaland 10 Quantum-Phase Electroncs Center and Department of Appled Physcs, The Unversty of Tokyo, Tokyo , Japan 11 RIKEN center for Emergent Matter Scence (CEMS), Wako , Japan (Dated: July 22, 2014) NATURE PHYSICS Macmllan Publshers Lmted. All rghts reserved.

2 SUPPLEMENTARY INFORMATION a b c Mn Γ Mn K Mn M k z (Å -1 ) k z (Å -1 ) -8 Γ M K Γ FIG. S1: k z dependence of bulk electronc structure of WSe 2. (a) ARPES measurements of the k z dsperson of electronc states at the Γ, K, and M ponts, respectvely. The features whch appear to form sharp dagonal lnes are an expermental artefact arsng from hgher-order lght. (b) Our ab-nto calculatons are addtonally ncluded (red lnes), showng good agreement wth the measured dspersons. (c) The calculated electronc structure, shown projected onto the surface Brlloun zone for dfferent values of k z (coloured lnes), ndcates sgnfcant n-plane dsperson of all states, wth the presence (absence) of substantal k z dsperson for the dfferent bands, characterstc of ther three (two-) dmensonal character, respectvely. 2 NATURE PHYSICS Macmllan Publshers Lmted. All rghts reserved.

3 SUPPLEMENTARY INFORMATION a 0.4 b c K Γ K 0.2 g f c b VB e d VB2 Mn (Å -1 ) In Plane Out of Plane Intensty (arb. unts) Polarsaton (%) d e f g FIG. S2: Spn-resolved EDCs. (a) Dsperson measured by ARPES (hν = 125 ev, T = 30 K), along the K Γ K drecton. (b-g) Spn-resolved EDCs (hν = 25 ev, T = 300 K, measured along the red cuts n (a)), accountng for the fnte effcency of the spn detecton (Sherman functon of the Mott detectors). Here, I = Itot (1 + P )/2 and I = Itot (1 P )/2, wth I tot =(I + I + I ), I± as shown n the man text, = {, }, and the polarsaton P shown n the bottom row, calculated as defned n the methods secton. NATURE PHYSICS Macmllan Publshers Lmted. All rghts reserved.

4 SUPPLEMENTARY INFORMATION 2-LAYER MODEL FOR PHOTOELECTRON SPIN POLARISATION To smulate the photon energy dependence of the measured spn polarsaton by spnresolved ARPES, shown n Fg. 2() of the man text, we start wth a mnmal model for the electronc structure of the upper valence band at K, as used n Ref. 1: VB1 = cos α [( ( ) ] d l x 2 2 y + dxy ) l + d u 2 x 2 y d u 2 xy + sn α [( ( ) ] d u x 2 2 y dxy ) u + d l 2 x 2 y + d l 2 xy (S1) =,σ c σ φ,σ, (S2) where = u, l s the layer ndex for the upper and lower layer n the unt cell, respectvely, and σ the spn ndex wth c u c u φ u, φ l, = sn α 2 = c l = cos α = c l 2 = d u x 2 y d u 2 xy = d l x 2 y 2 + d l xy. We take cos(2α) =0.9 such that each layer s 90% spn polarsed. We proceed to calculate the photoelectron spn polarsaton followng the method ntroduced n Refs. 2 and 3. We wrte the matrx element for photoemsson, assumng plane wave fnal states, as M VB1 = e k r A p VB1 = c σ m, where m = e k r A p φ,σ (S3) (S4) = e kzz e z /(2λ cos θ) e k r A p φ, (S5) explctly ncorporatng the effects of a layer-dependent phase and exponental attenuaton of photoelectrons emtted from deeper atomc layers n the crystal, takng 2me k z = (hν E B) k 2 2, (S6) θ as the photoelectron emsson angle for the K pont, λ the nelastc mean free path, and z the layer spacng of W planes along the c-axs. 4 NATURE PHYSICS Macmllan Publshers Lmted. All rghts reserved.

5 SUPPLEMENTARY INFORMATION From the Paul spn matrx σ z = 1 0, the z-component of the photoelectron spn polarsaton s gven by P z = I I I + I (S7) (S8) where Substtutng Eqn. S9 nto Eqn. S8, we get ( ) c c c c P z = ( c c + c I σ 2 = c σ m. (S9) c ) m 2 + ) (c c c c m m m 2 + ), (c c + c c m m (S10) where the latter terms gve rse to nterference between the layers. Takng only two layers, substtutng Eqn. S3 and assumng e k r A p φu = e k r A p φl, Eqn. S10 smplfes to ( sn 2 α cos 2 α )( c/2λ cos 1 e θ) P z = 1+e c/2λ cos θ + 4 sn α cos α cos ( k zc ). (S11) 2 e c/4λ cos θ An oscllatory dependence of the measured photoelectron spn polarsaton on photon energy s clearly apparent from the cos ( k zc ) 2 term, qualtatvely supportng our assgnment of the decrease of the measured photoelectron spn polarsaton at photon energes around 35 ev (Fg. 2() of the man text) to an nterference effect. To further valdate ths concluson, we numercally solve Eqn. S10 for a 20-layer bulk-lke model, takng nelastc mean free paths as calculated from the TPP-2M predctve formula [4], although we note that we fnd smlar results for fxed nelastc mean free paths on the order of 5 Å, and assumng a photon energy dependence of e k r A p φ gven by the W5d photoonsaton cross secton as calculated by Yeh and Lndau [5]. The calculated spn polarsaton s n good agreement wth our measured photon energy-dependent spn polarsatons (Fg. 2()). It valdates our measurement of hgh photoelectron spn polarsatons, confrmng that we can selectvely probe the top monolayer of the materal, and that we can tune these by changng photon energy. The photon energy dependence of the calculated total ntensty (Fg. S3) also well matches the varaton n spectral weght of these bands from spn-ntegrated ARPES over an extended photon energy range. Together, these model NATURE PHYSICS Macmllan Publshers Lmted. All rghts reserved.

6 SUPPLEMENTARY INFORMATION Intensty (arb. unts) 20-layer model Exp. sp. weght hv (ev) FIG. S3: Photon energy dependence of spectral weght. (a) Calculated (black lne) total ntensty predcted by the model descrbed here, compared to the measured spectral weght of the upper valence band at the K pont from spn-ntegrated ARPES (green ponts). calculatons and correspondng photon-energy dependent expermental measurements therefore strongly support our conclusons n the man text of an alternatng layer-dependent spn texture n WSe 2 for a gven valley ndex. To whom correspondence should be addressed: phlp.kng@st-andrews.ac.uk [1] Gong, Z., et al. Magnetoelectrc effects and valley-controlled spn quantum gates n transton metal dchalcogende blayers. Nature Commun. 4, 2053 (2013). [2] Zhu, Z.-H. et al. Layer-by-Layer Entangled Spn-Orbtal Texture of the Topologcal Surface State n B 2 Se 3. Phys. Rev. Lett. 110, (2013). [3] Zhu, Z.-H. et al. Photoelectron Spn-Polarzaton Control n the Topologcal Insulator B 2 Se 3. Phys. Rev. Lett. 112, (2014). [4] Tanuma, S. Powell, C. J. and Penn, D. R. Calculatons of electron nelastc mean free paths.. Surf. Interface Anal. 21, 165 (1994). [5] Yeh, J. J. and Lndau, I. Atomc subshell photoonzaton cross sectons and asymmetry parameters: 1 Z 103. Atomc Data and Nuclear Data Tables 32, 1 (1985). 6 NATURE PHYSICS Macmllan Publshers Lmted. All rghts reserved.