[1] P. W. Anderson, More is different, Science 177, 393–396 (1972). 1
[2] F. Bloch, U¨ ber die quantenmechanik der elektronen in kristallgittern, Zeitschrift fu¨r physik 52, 555–600 (1929). 1
[3] J. Bardeen, L. N. Cooper, J. R. Schrieffer, Theory of superconductivity, Phys. Rev. 108, 1175 (1957). 1
[4] J. G. Bednorz, K. A. Mu¨ller, Possible high-Tc superconductivity in the Ba-La- Cu-O system, Zeitschrift fu¨r Physik B Condensed Matter 64, 189–193 (1986). 1
[5] E. Dagotto, Complexity in strongly correlated electronic systems, Science 309, 257 (2005). 1
[6] K. v. Klitzing, G. Dorda, M. Pepper, New method for high-accuracy determina- tion of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett. 45, 494 (1980). 1, 20
[7] R. B. Laughlin, Quantized Hall conductivity in two dimensions, Phys. Rev. B 23, 5632 (1981). 1, 20, 21
[8] D. J. Thouless, M. Kohmoto, M. P. Nightingale, M. den Nijs, Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49, 405 (1982). 1, 20, 21
[9] M. Z. Hasan, C. L. Kane, Colloquium: topological insulators, Rev. Mod. Phys. 82, 3045 (2010). 1, 3, 69, 71
[10] X.-L. Qi, S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011). 1, 3, 69, 71
[11] S. Tomonaga, Remarks on Bloch’s Method of Sound Waves applied to Many-Fermion Problems, Prog. Theor. Phys. 5, 544 (1950). 1
[12] R. E. Peierls, Quantum theory of solids, Clarendon Press, (1955). 1
[13] J. M. Luttinger, An Exactly Soluble Model of a Many‐Fermion System, J. Math. Phys. 4, 1154 (1963). 1
[14] K. S. Novoselov, A. K. Geim, S. Morozov, D. Jiang, M. Katsnelson, I. Grig- orieva, S. Dubonos, Firsov, AA, Two-dimensional gas of massless Dirac fermions in graphene, Nature 438, 197 (2005). 1
[15] Y. Zhang, Y.-W. Tan, H. L. Stormer, P. Kim, Experimental observation of the quantum Hall effect and Berry’s phase in graphene, Nature 438, 201 (2005). 1
[16] C.-C. Liu, W. Feng, Y. Yao, Quantum spin Hall effect in silicene and two- dimensional germanium, Phys. Rev. Lett. 107, 076802 (2011). 1
[17] Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman, M. S. Strano, Electronics and optoelectronics of two-dimensional transition metal dichalcogenides, Nat. Nanotech. 7, 699 (2012). 1, 69
[18] A. Zangwill, Physics at surfaces, Cambridge University Press, 1988. 2, 8, 39
[19] J. H. Davies, The physics of low-dimensional semiconductors: an introduction, Cambridge University Press, (1998). 2, 9, 14, 48, 49
[20] M.-C. Desjonqueres, D. Spanjaard, Concepts in surface physics, Springer Science & Business Media, (2012). 2
[21] P. Harrison, A. Valavanis, Quantum wells, wires and dots: theoretical and com- putational physics of semiconductor nanostructures, John Wiley & Sons, (2016). 2, 110
[22] J. Butterfield, Less is different: Emergence and reduction reconciled, Foundations of Physics 41, 1065 (2011). 2
[23] A. Y. Cho, J. Arthur, Molecular beam epitaxy, Progress in solid state chemistry 10, 157 (1975). 2, 48
[24] A. Roth, Vacuum technology, Elsevier, (2012). 2
[25] K. Oura, V. Lifshits, A. Saranin, A. Zotov, M. Katayama, Surface science: an introduction, Springer Science & Business Media, (2013). 2
[26] S. Hu¨fner, Photoelectron spectroscopy: principles and applications, Vol. 82, Springer Science & Business Media, (2013). 2, 3, 8, 9, 13, 35, 41, 71
[27] T. Chiang, Photoemission studies of quantum well states in thin films, Surf. Sci. Rep. 39,, 181 (2000). 2, 78, 83, 86
[28] R. M. Martin, Electronic structure: basic theory and practical methods, Cam- bridge University Press, (2004). 2, 50
[29] A. Brugmans, Magnetismus, Seu De Affinitatibus Magneticis Observationes Aca- demicae, Apud Luzac & Van Damme, (1778). 3, 27
[30] T. J. Seebeck, Ueber die magnetische Polarisation der Metalle und Erze durch Temperaturdifferenz, Ann. Phys. 82, 253 (1826). 3, 27
[31] A. v. Ettingshausen, W. Nernst, Ueber das Auftreten electromotorischer Kr¨afte in Metallplatten, welche von einem W¨armestrome durchflossen werden und sich im magnetischen Felde befinden, Ann. Phys. Chem. 265, 343 (1886). 3, 27
[32] L. Schubnikov, W. J. de Haas, Comm. Phys. Lab. Leiden 207d, 35 (1930). 3, 27
[33] W. J. de Haas, P. M. van Alphen, Comm. Phys. Lab. Leiden 212a, 3 (1930). 3, 27
[34] L. Li, J. G. Checkelsky, Y. S. Hor, C. Uher, A. F. Hebard, R. J. Cava, N. P. Ong, Phase transitions of Dirac electrons in bismuth, Science 321, 547 (2008). 3, 27
[35] Z. Zhu, A. Collaudin, B. Fauqu´e, W. Kang, K. Behnia, Field-induced polarization of Dirac valleys in bismuth, Nat. Phys. 8, 89 (2012). 3, 27
[36] R. Ku¨chler, L. Steinke, R. Daou, M. Brando, K. Behnia, F. Steglich, Thermo- dynamic evidence for valley-dependent density of states in bulk bismuth, Nat. Mater. 13, 461 (2014). 3, 27
[37] B. E. Feldman, M. T. Randeria, A. Gyenis, F. Wu, H. Ji, R. J. Cava, A. H. MacDonald, A. Yazdani, Observation of a nematic quantum Hall liquid on the surface of bismuth, Science 354, 316 (2016). 3
[38] O. Prakash, A. Kumar, A. Thamizhavel, S. Ramakrishnan, Evidence for bulk superconductivity in pure bismuth single crystals at ambient pressure, Science 355, 52 (2017). 3
[39] V. Sandomirskii, Dependence of the forbidden-band width of semiconducting films on their thickness and temperature, Sov. Phys. JETP 16, 1630 (1963). 3
[40] V. Lutskii, Features of optical absorption of metallic films in the region where the metal turns into a dielectric, ZhETF Pis ma Redaktsiiu 2, 391 (1965). 3
[41] V. Sandomirskii, Quantum size effect in a semimetal film, Sov. Phys. JETP 25, 101 (1967). 3, 99
[42] L. Hicks, M. S. Dresselhaus, Effect of quantum-well structures on the thermo- electric figure of merit, Phys. Rev. B 47, 12727 (1993). 3
[43] C. Hoffman, J. Meyer, F. Bartoli, A. Di Venere, X. Yi, C. Hou, H. Wang, J. Ket- terson, G. Wong, Semimetal-to-semiconductor transition in bismuth thin films, Phys. Rev. B 48, 11431 (1993). 3, 99
[44] H. Chu, Comment on ”Semimetal-to-semiconductor transition in bismuth thin films”, Phys. Rev. B 51, 5532 (1995). 3, 99
[45] Z. Zhang, J. Y. Ying, M. S. Dresselhaus, Bismuth quantum-wire arrays fabricated by a vacuum melting and pressure injection process, J. Mater. Res. 13, 1745 (1998). 3, 99
[46] S. Lee, J. Ham, K. Jeon, J.-S. Noh, W. Lee, Direct observation of the semimetal- to-semiconductor transition of individual single-crystal bismuth nanowires grown by on-film formation of nanowires, Nanotechnology 21, 405701 (2010). 3, 99
[47] A. Nikolaeva, T. Huber, L. Konopko, A. Tsurkan, Observation of the semiconductor-semimetal and semimetal-semiconductor transitions in Bi quan- tum wires induced by anisotropic deformation and magnetic field, J. Low Temp. Phys. 158, 530 (2010). 3, 99
[48] L. Konopko, T. Huber, A. Nikolaeva, Magnetic Quantum Oscillations in Single Bi Nanowires, J. Low Temp. Phys. 159, 253 (2010). 3, 99
[49] G. Zhou, L. Li, G. Li, Semimetal to semiconductor transition and thermoelectric properties of bismuth nanotubes, J. Appl. Phys. 109, 114311 (2011). 3, 99
[50] S. Xiao, D. Wei, X. Jin, Bi(111) thin film with insulating interior but metallic surfaces, Phys. Rev. Lett. 109, 166805 (2012). 3, 98, 99, 112
[51] K. Zhu, L. Wu, X. Gong, S. Xiao, X. Jin, Quantum transport in the surface states of epitaxial Bi(111) thin films, Phys. Rev. B 94, 121401 (2016). 3, 98, 99, 112
[52] P. Kr¨oger, D. Abdelbarey, M. Siemens, D. Lu¨kermann, S. Sologub, H. Pfnu¨r, C. Tegenkamp, Controlling conductivity by quantum well states in ultrathin Bi(111) films, Phys. Rev. B 97, 045403 (2018). 3, 98, 99, 112
[53] P. Hofmann, The surfaces of bismuth: Structural and electronic properties, Prog. Surf. Sci. 81, 191 (2006). 3, 25, 26, 28, 69, 70
[54] C. Ast, H. H¨ochst, Electronic structure of a bismuth bilayer, Phys. Rev. B 67, 113102 (2003). 3, 70, 71
[55] Y. Ohtsubo, L. Perfetti, M. O. Goerbig, P. L. F`evre, F. Bertran, A. Taleb- Ibrahimi, Non-trivial surface-band dispersion on Bi(111), New J. Phys. 15, 033041 (2013). 3, 70, 71, 74, 76, 116, 123
[56] L. Perfetti, J. Faure, E. Papalazarou, J. Mauchain, M. Marsi, M. O. Goerbig, A. Taleb-Ibrahimi, Y. Ohtsubo, New aspects of electronic excitations at the bis- muth surface: Topology, thermalization and coupling to coherent phonons, J. Electron Spectrosc. Relat. Phenom. 201, 60 (2015). 3, 70, 71
[57] T. Hirahara, T. Shirai, T. Hajiri, M. Matsunami, K. Tanaka, S. Kimura, S. Hasegawa, K. Kobayashi, Role of quantum and surface-state effects in the bulk Fermi-level position of ultrathin Bi films, Phys. Rev. Lett. 115, 106803 (2015). 4, 98, 99, 101, 108, 112
[58] I. Tamm, U¨ ber eine m¨ogliche Art der Elektronenbindung an Kristalloberfl¨achen, Zeitschrift fu¨r Physik 76, 849 (1932). 8
[59] E. T. Goodwin, Electronic states at the surfaces of crystals: II. The approxima- tion of tight binding: finite linear chain of atoms, Math. Proc. Camb. Phil. Soc. 35, 221 (1939). 8
[60] W. Shockley, On the surface states associated with a periodic potential, Phys. Rev. 56, 317 (1939). 8
[61] H. Ibach, H. Lu¨th, Solid-state physics: An Introduction to Principles of Material Science, Springer Science & Business Media, 2009. 9
[62] E. V. Chulkov, V. M. Silkin, P. M. Echenique, Image potential states on metal surfaces: binding energies and wave functions, Surf. Sci. 437, 330 (1999). 10
[63] P. Echenique, J. Pendry, The existence and detection of Rydberg states at sur- faces, J. Phys. C 11, 2065 (1978). 12
[64] E. McRae, Electronic surface resonances of crystals, Rev. Mod. Phys. 51, 541 (1979). 12
[65] S. G. Davison, M. Steslicka, Basic Theory of Surface States, Oxford University Press, (1992). 13
[66] N. V. Smith, Phase analysis of image states and surface states associated with nearly-free-electron band gaps, Phys. Rev. B 32, 3549 (1985). 13, 125
[67] N. V. Smith, N. B. Brookes, Y. Chang, P. D. Johnson, Quantum-well and tight- binding analyses of spin-polarized photoemission from Ag/Fe(001) overlayers, Phys. Rev. B 49, 332 (1994). 13, 125
[68] J. Sakurai, Modern quantum mechanics, Addison Wesley, (1993). 16, 45
[69] 安藤陽一, トポロジカル絶縁体入門, 講談社, (2014). 17
[70] 斎藤英治, 村上修一, スピン流とトポロジカル絶縁体, 共立出版, (2014). 17
[71] Y. A. Bychkov, E. I. Rashba, Properties of a 2D electron gas with lifted spectral degeneracy, JETP Lett. 39, 78 (1984). 24
[72] 伏屋雄紀, ビスマス研究温故知新固体中ディラック電子とバンド間磁場効果, 物性研究 90, 537 (2008). 25, 27
[73] P. Cucka, C. Barrett, The crystal structure of Bi and of solid solutions of Pb, Sn, Sb and Te in Bi, Acta Crystallographica 15, 865 (1962). 25, 87
[74] Y. Liu, R. E. Allen, Electronic structure of the semimetals Bi and Sb, Phys. Rev. B 52, 1566 (1995). 25, 27, 28, 62, 63, 64, 67, 69, 71, 74, 82, 87, 100, 105
[75] H. M¨onig, J. Sun, Y. M. Koroteev, G. Bihlmayer, J. Wells, E. Chulkov, K. Pohl, P. Hofmann, Structure of the (111) surface of bismuth: LEED analysis and first- principles calculations, Phys. Rev. B 72, 085410 (2005). 26
[76] I. Aguilera, C. Friedrich, S. Blu¨gel, Electronic phase transitions of bismuth under strain from relativistic self-consistent GW calculations, Phys. Rev. B 91, 125129 (2015). 27, 69, 71, 82, 106, 116
[77] Y. Fuseya, M. Ogata, H. Fukuyama, Transport properties and diamagnetism of dirac electrons in bismuth, J. Phys. Soc. Japan 84, 012001 (2014). 27
[78] Y. Koroteev, G. Bihlmayer, J. Gayone, E. Chulkov, S. Blu¨gel, P. Echenique, P. Hofmann, Strong spin-orbit splitting on Bi surfaces, Phys. Rev. Lett. 93, 046403 (2004). 28, 69, 70, 76
[79] T. Hirahara, The Rashba and quantum size effects in ultrathin Bi films, J. Elec. Spectrosc. Relat. Phenom. 201, 98 (2015). 28
[80] R. Singh, Spin-orbit splitting in graphene, silicene and germanene: Dependence on buckling, Int. J. Mod. Phys. B 32, 1850055 (2018). 28
[81] F. Reis, G. Li, L. Dudy, M. Bauernfeind, S. Glass, W. Hanke, R. Thomale, J. Sch¨afer, R. Claessen, Bismuthene on a SiC substrate: A candidate for a high- temperature quantum spin Hall material, Science 357, 287 (2017). 28
[82] X. Gonze, J.-P. Michenaud, J.-P. Vigneron, First-principles study of As, Sb, and Bi electronic properties, Phys. Rev. B 41, 11827 (1990). 28
[83] A. Kimura, E. Krasovskii, R. Nishimura, K. Miyamoto, T. Kadono, K. Kanomaru, E. Chulkov, G. Bihlmayer, K. Shimada, H. Namatame, M. Taniguchi, Strong Rashba-type spin polarization of the photocurrent from bulk continuum states: Experiment and theory for Bi(111), Phys. Rev. Lett. 105, 076804 (2010). 29
[84] K. Kuroda, K. Yaji, M. Nakayama, A. Harasawa, Y. Ishida, S. Watanabe, C.-T. Chen, T. Kondo, F. Komori, S. Shin, Coherent control over three-dimensional spin polarization for the spin-orbit coupled surface state of Bi2Se3, Phys. Rev. B 94, 165162 (2016). 29
[85] K. Yaji, K. Kuroda, S. Toyohisa, A. Harasawa, Y. Ishida, S. Watanabe, C. Chen, K. Kobayashi, F. Komori, S. Shin, Spin-dependent quantum interference in pho- toemission process from spin-orbit coupled states, Nat. Comm. 8, 14588 (2017). 29
[86] D. Hou, Z. Qiu, K. Harii, Y. Kajiwara, K. Uchida, Y. Fujikawa, H. Nakayama, T. Yoshino, T. An, K. Ando, X. Jin, E. Saitoh, Interface induced inverse spin Hall effect in bismuth/permalloy bilayer, Appl. Phys. Lett. 101, 042403 (2012). 29
[87] H. Emoto, Y. Ando, E. Shikoh, Y. Fuseya, T. Shinjo, M. Shiraishi, Conversion of pure spin current to charge current in amorphous bismuth, J. Appl. Phys. 115, 17C507 (2014). 29
[88] H. Emoto, Y. Ando, G. Eguchi, R. Ohshima, E. Shikoh, Y. Fuseya, T. Shinjo, M. Shiraishi, Transport and spin conversion of multicarriers in semimetal bis- muth, Phys. Rev. B 93, 174428 (2016). 29
[89] E. J. Tichovolsky, J. G. Mavroides, Magnetoreflection studies on the band struc- ture of bismuth-antimony alloys, Solid State Comm. 7, 927 (1969). 29
[90] L. S. Lerner, K. F. Cuff, L. R. Williams, Energy-Band Parameters and Rel- ative Band-Edge Motions in the Bi-Sb Alloy System near the Semimetal― Semiconductor Transition, Rev. Mod. Phys. 40, 770 (1968). 29
[91] N. B. Brandt, M. V. Semenov, L. A. Falkovsky, Experiment and theory on the magnetic susceptibility of Bi-Sb alloys, J. Low Temp. Phys. 27, 75 (1977). 29
[92] B. Lenoir, M. Cassart, J.-P. Michenaud, H. Scherrer, S. Scherrer, Transport prop- erties of Bi-rich Bi-Sb alloys, J. Phys. Chem. Solids 57, 89 (1996). 29
[93] T. Nagao, J. Sadowski, M. Saito, S. Yaginuma, Y. Fujikawa, T. Kogure, T. Ohno, Y. Hasegawa, S. Hasegawa, T. Sakurai, Nanofilm Allotrope and Phase Transfor- mation of Ultrathin Bi Film on Si(111)-7×7, Phys. Rev. Lett. 93, 105501 (2004). 30, 31
[94] S. Hatta, Y. Ohtsubo, S. Miyamoto, H. Okuyama, T. Aruga, Epitaxial growth of Bi thin films on Ge(111), Appl. Surf. Sci. 256, 1252 (2009). 30, 31, 72
[95] Y. Lu, W. Xu, M. Zeng, G. Yao, L. Shen, M. Yang, Z. Luo, F. Pan, K. Wu, T. Das, P. He, J. Jiang, J. Martin, Y. P. Feng, H. Lin, X.-s. Wang, Topological properties determined by atomic buckling in self-assembled ultrathin Bi(110), Nano Lett. 15, 80 (2014). 30, 31
[96] G. Bian, X. Wang, T. Miller, T.-C. Chiang, P. J. Kowalczyk, O. Mahapatra, S. Brown, First-principles and spectroscopic studies of Bi(110) films: Thickness- dependent Dirac modes and property oscillations, Phys. Rev. B 90, 195409 (2014). 30, 31
[97] T. Payer, C. Klein, M. Acet, V. Ney, M. Kammler, F.-J. M. zu Hering- dorf, M. Horn-von Hoegen, High-quality epitaxial Bi(111) films on Si(111) by isochronal annealing, Thin Solid Films 520, 6905 (2012). 30, 106
[98] 高橋隆, 光電子固体物性, 朝倉書店, (2011). 35, 41, 42, 44
[99] L. Hedin, J. Lee, Sudden approximation in photoemission and beyond, J. Elec. Spectrosc. Relat. Phenom. 124, 289 (2002). 36
[100] A. Altland, B. D. Simons, Condensed matter field theory, Cambridge University Press, (2010). 36, 37, 117
[101] J. Braun, K. Miyamoto, A. Kimura, T. Okuda, M. Donath, H. Ebert, J. Minar, Exceptional behavior of d-like surface resonances on W(110): the one-step model in its density matrix formulation, New J. Phys. 16, 015005 (2014). 37
[102] S. Karkare, W. Wan, J. Feng, T. C. Chiang, H. A. Padmore, One-step model of photoemission from single-crystal surfaces, Phys. Rev. B 95, 075439 (2017). 37
[103] G. A. Somorjai, Chemistry in two dimensions: Surfaces, Cornell University Press, (1981). 39
[104] SPECS Surface Nano Analysis GmbH, Excitation Sources, http://www.specs. de/cms/front_content.php?idart=685. 44
[105] 東北大学光電子固体物性研究室, https://arpes.phys.tohoku.ac.jp/contents/study.html. 44
[106] 広島大学放射光科学研究センター, 光源・ビームライン, http://www.hsrc. hiroshima-u.ac.jp/storagering_beamlines/sr_ring.html. 44
[107] 今井功, 流体力学(前編), 裳華房, (1997). 48
[108] J. C. Slater, Atomic radii in crystals, J. Chem. Phys. 41, 3199 (1964). 48
[109] 高田康民, 多体問題特論 第一原理からの多電子問題, 朝倉書店, (2009). 50
[110] P. Hohenberg, W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136, B864 (1964). 50
[111] W. Kohn, L. J. Sham, Self-consistent equations including exchange and correla- tion effects, Phys. Rev. 140, A1133 (1965). 51
[112] J. C. Slater, Quantum theory of molecules and solids, McGraw-Hill New York, (1963). 51
[113] J. F. Janak, Proof that ∂E/∂ni= ϵ¯ in density-functional theory, Phys. Rev. B 18, 7165 (1978). 53
[114] H. J. Monkhorst, J. D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13, 5188 (1976). 56, 87
[115] J. C. Slater, G. F. Koster, Simplified LCAO method for the periodic potential problem, Phys. Rev. 94, 1498 (1954). 57
[116] 藤森淳, 強相関物質の基礎 原子、分子から固体へ, 内田老鶴圃, (2005). 57, 61
[117] C. Kittel, Introduction to solid state physics, John Wiley & Sons, (2005). 57
[118] N. Ashcroft, N. Mermin, Solid State Physics, (1976). 57
[119] G. H. Wannier, The structure of electronic excitation levels in insulating crystals, Phys. Rev. 52, 191 (1937). 57
[120] P.-O. L¨owdin, On the non-orthogonality problem connected with the use of atomic wave functions in the theory of molecules and crystals, J. Chem. Phys. 18, 365 (1950). 58
[121] Y. Ohtsubo, S.-i. Kimura, Topological phase transition of single-crystal Bi based on empirical tight-binding calculations, New J. Phys. 18, 123015 (2016). 64, 76, 100, 116
[122] K. Saito, H. Sawahata, T. Komine, T. Aono, Tight-binding theory of surface spin states on bismuth thin films, Phys. Rev. B 93, 041301 (2016). 64, 65, 66, 100
[123] J. C. Y. Teo, L. Fu, C. L. Kane, Surface states and topological invariants in three-dimensional topological insulators: Application to Bi1−xSbx, Phys. Rev. B 78, 045426 (2008). 65, 69
[124] L. Petersen, P. Hedeg˚ard, A simple tight-binding model of spin-orbit splitting of sp-derived surface states, Surf. Sci. 459, 49 (2000). 65
[125] C. R. Ast, I. Gierz, sp-band tight-binding model for the Bychkov-Rashba effect in a two-dimensional electron system including nearest-neighbor contributions from an electric field, Phys. Rev. B 86, 085105 (2012). 65
[126] G. Aut`es, A. Isaeva, L. Moreschini, J. C. Johannsen, A. Pisoni, R. Mori, W. Zhang, T. G. Filatova, A. N. Kuznetsov, L. Forr´o, W. V. d. Broek, Y. Kim, K. S. Kim, A. Lanzara, J. D. Denlinger, E. Rotenberg, A. Bostwick, M. Grioni, O. V. Yazyev, A novel quasi-one-dimensional topological insulator in bismuth iodide β-Bi4I4, Nat. Mater. 15, 154 (2015). 69
[127] C.-C. Liu, J.-J. Zhou, Y. Yao, F. Zhang, Weak Topological Insulators and Com- posite Weyl Semimetals: β-Bi4X4 (X = Br, I), Phys. Rev. Lett. 116, 066801 (2016). 69
[128] S. Golin, Band structure of bismuth: Pseudopotential approach, Phys. Rev. 166, 643 (1968). 69
[129] Y. M. Koroteev, G. Bihlmayer, E. V. Chulkov, S. Blu¨gel, First-principles inves- tigation of structural and electronic properties of ultrathin Bi films, Phys. Rev. B 77, 045428 (2008). 69, 74, 126
[130] T. Hirahara, N. Fukui, T. Shirasawa, M. Yamada, M. Aitani, H. Miyazaki, M. Matsunami, S. Kimura, T. Takahashi, S. Hasegawa, K. Kobayashi, Atomic and electronic structure of ultrathin Bi(111) films grown on Bi2Te3(111) sub- strates: evidence for a strain-induced topological phase transition, Phys. Rev. Lett. 109, 227401 (2012). 69
[131] N. A. Red’ko, N. A. Rodionov, Topological phase transitions in Bi1−xSbx alloys and composition dependence of the position of the heavy-hole band, Pis’ma Zh. Eksp. Teor. Fiz. 42, 246 (1985) [J. Exp. Theor. Phys. Lett. 42, 303 (1986)]. 69
[132] C. Ast, H. H¨ochst, Fermi surface of Bi(111) measured by photoemission spec- troscopy, Phys. Rev. Lett. 87, 177602 (2001). 70, 74
[133] W. Ning, F. Kong, C. Xi, D. Graf, H. Du, Y. Han, J. Yang, K. Yang, M. Tian, Y. Zhang, Evidence of topological two-dimensional metallic surface states in thin bismuth nanoribbons, ACS nano 8, 7506 (2014). 70, 83
[134] W. Ning, F. Kong, Y. Han, H. Du, J. Yang, M. Tian, Y. Zhang, Robust surface state transport in thin bismuth nanoribbons, Sci. Rep. 4, 7086 (2014). 70, 83
[135] K. Yaji, private communication (2015). 72
[136] T. Hirahara, T. Nagao, I. Matsuda, G. Bihlmayer, E. Chulkov, Y. Koroteev, S. Hasegawa, Quantum well states in ultrathin Bi films: Angle-resolved pho- toemission spectroscopy and first-principles calculations study, Phys Rev. B 75, 035422 (2007). 73, 74, 80, 83, 86
[137] A. Takayama, T. Sato, S. Souma, T. Takahashi, Giant out-of-plane spin compo- nent and the asymmetry of spin polarization in surface Rashba states of bismuth thin film, Phys. Rev. Lett. 106, 166401 (2011). 73, 80
[138] G. Kresse, J. Furthmu¨ller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci. 6, 15 (1996). 74
[139] J. P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996). 74, 87
[140] P. E. Bl¨ochl, Projector augmented-wave method, Phys. Rev. B 50, 17953 (1994). 74
[141] T. Hirahara, T. Nagao, I. Matsuda, G. Bihlmayer, E. Chulkov, Y. Koroteev, P. Echenique, M. Saito, S. Hasegawa, Role of spin-orbit coupling and hybridiza- tion effects in the electronic structure of ultrathin Bi films, Phys Rev. Lett. 97, 146803 (2006). 74, 76, 80, 106
[142] A. Takayama, T. Sato, S. Souma, T. Oguchi, T. Takahashi, Tunable spin polar- ization in bismuth ultrathin film on Si(111), Nano Lett. 12, 1776 (2012). 74, 76, 86, 106
[143] G. Bian, T. Miller, T. Chiang, Electronic structure and surface-mediated metasta- bility of Bi films on Si(111)-7×7 studied by angle-resolved photoemission spec- troscopy, Phys. Rev. B 80, 245407 (2009). 74, 86
[144] Y. Ohtsubo, J. Mauchain, J. Faure, E. Papalazarou, M. Marsi, P. L. F`evre, F. Bertran, A. Taleb-Ibrahimi, L. Perfetti, Giant anisotropy of spin-orbit splitting at the bismuth surface, Phys. Rev. Lett. 109, 226404 (2012). 74
[145] Y. Fuseya, H. Fukuyama, Analytical Solutions for the Surface States of Bi1−xSbx (0 ≤ x ≤ 0.1), J. Phys. Soc. Japan 87, 044710 (2018). 76
[146] P. Zhang, P. Richard, T. Qian, Y.-M. Xu, X. Dai, H. Ding, A precise method for visualizing dispersive features in image plots, Rev. Sci. Instrum. 82, 043712 (2011). 77
[147] H. Hayasaka, Y. Fuseya, Crystalline spin-orbit interaction and the Zeeman split- ting in Pb1−xSnxTe, J. Phys.: Cond. Matter 28, 31LT01 (2016). 83
[148] H. Du, X. Sun, X. Liu, X. Wu, J. Wang, M. Tian, A. Zhao, Y. Luo, J. Yang, B. Wang, J. G. Hou, Surface Landau levels and spin states in bismuth (111) ultrathin films, Nat. Comm. 7, 10814 (2016). 83
[149] Z. K. Liu, B. Zhou, Y. Zhang, Z. J. Wang, H. M. Weng, D. Prabhakaran, S. K. Mo, Z. X. Shen, Z. Fang, X. Dai, Z. Hussan, Y. L. Chen, Discovery of a three- dimensional topological Dirac semimetal, Na3Bi, Science 343, 864–867 (2014). 83, 117
[150] M. Hirayama, R. Okugawa, S. Ishibashi, S. Murakami, T. Miyake, Weyl node and spin texture in trigonal tellurium and selenium, Phys. Rev. Lett. 114, 206401 (2015). 83
[151] S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, M. Z. Hasan, Discovery of a Weyl fermion semimetal and topological Fermi arcs, Science 349, 613 (2015). 83, 117
[152] B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, H. Ding, Experimental discovery of Weyl semimetal TaAs, Phys. Rev. X 5, 031013 (2015). 83, 117
[153] J. Ruan, S.-K. Jian, H. Yao, H. Zhang, S.-C. Zhang, D. Xing, Symmetry-protected ideal Weyl semimetal in HgTe-class materials, Nat. Comm. 7, 11136 (2016). 83
[154] A. Kiejna, K. Wojciechowski, Work function of metals: Relation between theory and experiment, Prog. Surf. Sci. 11, 293 (1981). 85, 94
[155] T. Aruga, Y. Murata, Alkali-metal adsorption on metals, Prog. Surf. Sci. 31, 61 (1989). 85, 94
[156] C. Stampfl, M. Scheffler, Theoretical identification of a (2 × 2) composite double layer ordered surface alloy of Na on Al(111), Surf. Sci. 319, L23 (1994). 85, 86, 94
[157] R. Diehl, R. McGrath, Current progress in understanding alkali metal adsorption on metal surfaces, J. Phys. Condens. Matter 9, 951 (1997). 85, 86, 94
[158] Z. H. Zhu, G. Levy, B. Ludbrook, C. N. Veenstra, J. A. Rosen, R. Comin, D. Wong, P. Dosanjh, A. Ubaldini, P. Syers, N. P. Butch, J. Paglione, I. S. Elfimov, A. Damascelli, Rashba spin-splitting control at the surface of the topo- logical insulator Bi2Se3, Phys. Rev. Lett. 107, 186405 (2011). 85
[159] P. Zhang, P. Richard, N. Xu, Y. Xu, T. Ma, J .and Qian, A. Fedorov, J. Den- linger, G. Gu, H. Ding, Observation of an electron band above the Fermi level in FeTe0.55Se0.45 from in-situ surface doping, Appl. Phys. Lett. 105, 172601 (2014). 85
[160] F.-F. Zhu, W.-J. Chen, Y. Xu, C.-L. Gao, D.-D. Guan, C.-h. Liu, D. Qian, S.-C. Zhang, J.-F. Jia, Epitaxial growth of two-dimensional stanene, Nat. Mater. 14, 1020 (2015). 85, 86
[161] Y. Feng, D. Liu, B. Feng, X. Liu, L. Zhao, Z. Xie, Y. Liu, A. Liang, C. Hu, Y. Hu, S. He, G. Liu, J. Zhang, C. Chen, Z. Xu, L. Chen, K. Wua, Y.-T. Liu, H. Lin, Z.-Q. Huang, C.-H. Hsu, F.-C. Chuang, A. Bansil, X. J. Zhou, Direct evidence of interaction-induced Dirac cones in a monolayer silicene/Ag(111) system, Proc. Natl. Acad. Sci. 113, 14656 (2016). 85, 86
[162] T. Ohta, A. Bostwick, T. Seyller, K. Horn, E. Rotenberg, Controlling the elec- tronic structure of bilayer graphene, Science 313, 951 (2006). 85
[163] J. Kim, S. Baik, S. Ryu, Y. Sohn, S. Park, B. Park, J. Denlinger, Y. Yi, H. Choi, K. Kim, Observation of tunable band gap and anisotropic Dirac semimetal state in black phosphorus, Science 349, 723 (2015). 85
[164] Y. Miyata, K. Nakayama, K. Sugawara, T. Sato, T. Takahashi, High-temperature superconductivity in potassium-coated multilayer FeSe thin films, Nat. Mater. 14, 775 (2015). 85
[165] M. Ren, Y. Yan, X. Niu, R. Tao, D. Hu, R. Peng, B. Xie, J. Zhao, T. Zhang, D.- L. Feng, Superconductivity across Lifshitz transition and anomalous insulating state in surface K-dosed (Li0.8Fe0.2OH)FeSe, Sci. Adv. 3, e1603238 (2017). 85
[166] S.-W. Kim, H. Jung, H.-J. Kim, J.-H. Choi, S.-H. Wei, J.-H. Cho, Microscopic mechanism of the tunable band gap in potassium-doped few-layer black phospho- rus, Phys. Rev. B 96, 075416 (2017). 86
[167] I. Belopolski, S.-Y. Xu, Y. Ishida, X. Pan, P. Yu, D. S. Sanchez, H. Zheng, M. Neupane, N. Alidoust, G. Chang, T.-R. Chang, Y. Wu, G. Bian, S.-M. Huang, C.-C. Lee, D. Mou, L. Huang, Y. Song, B. Wang, G. Wang, Y.-W. Yeh, N. Yao, J. E. Rault, P. Le F`evre, F. Bertran, H.-T. Jeng, T. Kondo, A. Kaminski, H. Lin, Z. Liu, F. Song, S. Shin, M. Z. Hasan, Fermi arc electronic structure and Chern numbers in the type-II Weyl semimetal candidate MoxW1−xTe2, Phys. Rev. B 94, 085127 (2016). 86
[168] A. V. Matetskiy, L. V. Bondarenko, A. Y. Tupchaya, D. V. Gruznev, S. V. Ere- meev, A. V. Zotov, A. A. Saranin, Adsorbate-induced modification of electronic band structure of epitaxial Bi(111) films, Appl. Surf. Sci. 406, 122 (2017). 86, 89
[169] S. Ito, B. Feng, M. Arita, A. Takayama, R.-Y. Liu, T. Someya, W.-C. Chen, T. Iimori, H. Namatame, M. Taniguchi, C.-M. Cheng, S.-J. Tang, F. Komori, K. Kobayashi, T.-C. Chiang, I. Matsuda, Proving nontrivial topology of pure bismuth by quantum confinement, Phys. Rev. Lett. 117, 236402 (2016). 86
[170] L. Davis, N. C. MacDonald, P. W. Palmberg, G. E. Riach, R. E. Weber, Handbook of Auger electron spectroscopy, Physical Electronics Industries, (1976). 87
[171] X. Gonze, J.-M. Beuken, R. Caracas, F. Detraux, M. Fuchs, G.-M. Rignanese, L. Sindic, M. Verstraete, G. Zerah, F. Jollet, M. Torrent, A. Roy, M. Mikami, P. Ghosez, J.-Y. Raty, D. C. Allan, First-principles computation of material prop- erties: the ABINIT software project, Comput. Mater. Sci. 25, 478 (2002). 87
[172] C. Hartwigsen, S. Goedecker, J. Hutter, Relativistic separable dual-space Gaus- sian pseudopotentials from H to Rn, Phys. Rev. B 58, 3641 (1998). 87
[173] J. N. Crain, M. C. Gallagher, J. L. McChesney, M. Bissen, F. J. Himpsel, Doping of a surface band on Si(111) √3 × √3-Ag, Phys. Rev. B 72, 045312 (2005). 87
[174] W. Halperin, Quantum size effects in metal particles, Rev. Mod. Phys. 58, 533 (1986). 97
[175] Y. Guo, Y. F. Zhang, X. Y. Bao, T. Z. Han, Z. Tang, L.-X. Zhang, W. G. Zhu, E. G. Wang, Q. Niu, Z. Q. Qiu, J. F. Jia, Z. X. Zhao, Q. K. Xue, Superconduc- tivity modulated by quantum size effects, Science 306, 1915 (2004). 97
[176] P. S. Kirchmann, L. Rettig, X. Zubizarreta, V. M. Silkin, E. V. Chulkov, U. Bovensiepen, Quasiparticle lifetimes in metallic quantum-well nanostructures, Nat. Phys. 6, 782 (2010). 99
[177] O. Lopez-Acevedo, K. A. Kacprzak, J. Akola, H. H¨akkinen, Quantum size effects in ambient CO oxidation catalysed by ligand-protected gold clusters, Nat. Chem. 2, 329 (2010). 99
[178] A. Campos, N. Troc, E. Cottancin, M. Pellarin, H.-C. Weissker, J. Lerm´e, M. Ko- ciak, M. Hillenkamp, Plasmonic quantum size effects in silver nanoparticles are dominated by interfaces and local environments, Nat. Phys. 1 (2018). 99
[179] D. Mahendra, R. Grassi, J. Y. Chen, M. Jamali, D. R. Hickey, D. Zhang, Z. Zhao, H. Li, P. Quarterman, Y. Lv, M. Li, A. Manchon, A. Mkhoyan, K, T. Low, J. P. Wang, Room-temperature high spin–orbit torque due to quantum confinement in sputtered BixSe1−x films, Nat. Mater. 17, 800 (2018). 99
[180] R. Yukawa, K. Ozawa, S. Yamamoto, R.-Y. Liu, I. Matsuda, Anisotropic effective mass approximation model to calculate multiple subband structures at wide-gap semiconductor surfaces: Application to accumulation layers of SrTiO3 and ZnO, Surf. Sci. 641, 224 (2015). 101, 127
[181] F. Schindler, Z. Wang, M. G. Vergniory, A. M. Cook, A. Murani, S. Sengupta, A. Y. Kasumov, R. Deblock, S. Jeon, I. Drozdov, H. Bouchiat, S. Gu´eron, A. Yaz- dani, A. Bernevig, B, T. Neupert, Higher-order topology in bismuth, Nat. Phys. 14, 918 (2018). 116, 121
[182] G. Bian, T.-R. Chang, R. Sankar, S.-Y. Xu, H. Zheng, T. Neupert, C.-K. Chiu, S.-M. Huang, G. Chang, I. Belopolski, D. S. Sanchez, M. Neupane, N. Alidoust, C. Liu, B. Wang, C.-C. Lee, H.-T. Jeng, C. Zhang, Z. Yuan, S. Jia, A. Bansil, F. Chou, H. Lin, M. Z. Hasan, Topological nodal-line fermions in spin-orbit metal PbTaSe2, Nat. Comm. 7, 10556 (2016). 117
[183] M. Dzero, K. Sun, V. Galitski, P. Coleman, Topological kondo insulators, Phys. Rev. Lett. 104, 106408 (2010). 117
[184] M. Dzero, K. Sun, P. Coleman, V. Galitski, Theory of topological Kondo insula- tors, Phys. Rev. B 85, 045130 (2012). 117
[185] F. Lu, J. Zhao, H. Weng, Z. Fang, X. Dai, Correlated topological insulators with mixed valence, Phys. Rev. Lett. 110, 096401 (2013). 117
[186] M. Neupane, N. Alidoust, S. Xu, T. Kondo, Y. Ishida, D.-J. Kim, C. Liu, I. Be- lopolski, Y. Jo, T.-R. Chang, H. T. Jeng, T. Durakiewicz, L. Balicas, H. Lin, A. Bansil, S. Shin, Z. Fisk, M. Z. Hasan, Surface electronic structure of the topo- logical Kondo-insulator candidate correlated electron system SmB6, Nat. Comm. 4, 2991 (2013). 117
[187] H. Weng, J. Zhao, Z. Wang, Z. Fang, X. Dai, Topological crystalline Kondo insu- lator in mixed valence ytterbium borides, Phys. Rev. Lett. 112, 016403 (2014). 117
[188] K. Hagiwara, Y. Ohtsubo, M. Matsunami, S.-i. Ideta, K. Tanaka, H. Miyazaki, J. E. Rault, P. Le F`evre, F. Bertran, A. Taleb-Ibrahimi, R. Yukawa, M. Kobayashi, K. Horiba, H. Kumigashira, K. Sumida, T. Okuda, F. Iga, S.-i. Kumira, Surface Kondo effect and non-trivial metallic state of the Kondo insula- tor YbB12, Nat. Comm. 7, 12690 (2016). 117
[189] M. Kitzler, S. Gr¨afe, Ultrafast Dynamics Driven by Intense Light Pulse, Vol. 86, Springer, (2015). 117
[190] T. Frigge, B. Hafke, T. Witte, B. Krenzer, C. Streubu¨hr, A. S. Syed, V. M. Trontl, I. Avigo, P. Zhou, M. Ligges, D. v. d. Linde, U. Bovensiepen, M. Horn- von Hoegen, S. Wippermann, A. Lu¨cke, S. Sanna, U. Gerstmann, W. G. Schmidt, Optically excited structural transition in atomic wires on surfaces at the quantum limit, Nature 544, 207 (2017). 118
[191] C. W. Nicholson, A. Lu¨cke, W. G. Schmidt, M. Puppin, L. Rettig, R. Ernstorfer, M. Wolf, Beyond the molecular movie: Dynamics of bands and bonds during a photoinduced phase transition, Science 362, 821 (2018). 118
[192] M. R. Norman, H. Ding, M. Randeria, J. C. Campuzano, T. Yokoya, T. Takeuchi, T. Takahashi, T. Mochiku, K. Kadowaki, P. Guptasarma, D. Hinks, Destruction of the Fermi surface in underdoped high-Tc superconductors, Nature 392, 157 (1998). 119
[193] D. R. Lide, CRC handbook of chemistry and physics, CRC press, (2004). 126
[194] I. Matsuda, T. Ohta, H. W. Yeom, In-plane dispersion of the quantum-well states of the epitaxial silver films on silicon, Phys. Rev. B 65, 085327 (2002). 126