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新学術領域「コンピューティクスによる物質デザイン：複合相関と非平衡ダイナミクス」 計画研究「第一原理有効模型と相関科学のフロンティア」. 三方晶テルル・セレンにおける Dirac 分散とスピン軌道効果. Dirac Cone and Spin-Orbit Effects in Trigonal Tellurium and Selenium. 平山元昭 石橋章司 三宅隆. 産業技術総合研究所 ナノシステム研究部門. Introduction of Tellurium and Selenium 1. Group-VI element .
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新学術領域「コンピューティクスによる物質デザイン：複合相関と非平衡ダイナミクス」新学術領域「コンピューティクスによる物質デザイン：複合相関と非平衡ダイナミクス」 計画研究「第一原理有効模型と相関科学のフロンティア」 三方晶テルル・セレンにおけるDirac分散とスピン軌道効果 Dirac Cone and Spin-Orbit Effectsin Trigonal Tellurium and Selenium 平山元昭 石橋章司 三宅隆 産業技術総合研究所 ナノシステム研究部門
Introduction of Tellurium and Selenium 1 Group-VI element Structure : Trigonal P3121 or P3221 (D43 or D63) (ex. :α-quartz, HgS) Spiral symmetry S3, Rotational symmetry C2 3 atoms in unit cell Electron Configuration : Te (5p)4, Se (4p)4 Lattice Constant R. Keller et al.: Phys. Rev. B 16, 4404 (1977). (a, c) :Te(4.46, 5.92) Å Se (4.37, 4.60) Å (r, R) :Te (2.83, 3.49) Å Se (2.37, 3.44) Å Se has stronger one-dimensional character
Introduction of Tellurium and Selenium 2 Band Gap :Te 0.323, Se 2.0 (eV) V. B. Anzin et al.: Phys. Stat. Sol. (a) 42, 385 (1977). S. Tutihasi et al.: Phys. Rev. 158, 623 (1967). Insulator to metal transition (IMT) under pressure P. W. Bridgman:Proc. Am. Acad. Arts Sci. 74, 425 (1942). Te: ~4 GPa, Se: ~14-~22 GPa Structural transition (ST) also occurs near the IMT. (Relation between IMT and ST is not yet clarified ) M. Takumi et al.:Fukuoka University Science Reports 42(1) 1 (2012). Calculation of electronic structure k・p perturbation T. Doi et al.: J. Phys. Soc. Jpn. 28, 36 (1970). Pseudopotential technique J. D. Joannopoulos et al.: Phys. Rev. B 11, 6 (1975). Strong topological insulator under shear strain ? L. A. Agapito et al.: Phys. Rev. Lett. 110, 176401 (2013). We find that there are various three-dimensional Dirac pointsnear the Fermi level in Te and Se.
Method (GW+SO) LDA+SO Fully relativistic two-component first-principles code QMAS (Quantum MAterials Simulator) based on the projector augmented wave (PAW) method http://qmas.jp/ (k-point mesh: 6x6x6, Plane-wave energy cutoff :40 Ry) T. Kosugiet al.:J. Phys. Soc. Jpn. 80, 074713 (2011). GW Full-potential linear muffin-tin orbital (FP-LMTO) code M. van Schilfgaarde et al.:Phys. Rev. B 74, 245125 (2006). (k-point mesh: 6x6x4) T. Miyake and F. Aryasetiawan:Phys. Rev. B 77, 085122 (2008). Hamiltonian of the GW+SO method φ: maximally localized Wannier function N. Marzari and D. Vanderbilt:Phys. Rev. B 56, 12847 (1997). I. Souza et al.: Phys. Rev. B 65, 035109 (2001).
Electronic Band Structure Various three-dimensional Dirac pointsexist near the Fermi level.
3x2 fold degenerate Dirac cone Without the SOI Without the SOI, a Dirac point (3x2 fold degenerate) emerges under pressure.
Orbital Character Maximally localized Wannier function of Te Three types of the p bands J. D. Joannopoulos et al.: Phys. Rev. B 11, 6 (1975). originating mainly from 5py’(Tei)-5pz’(Te(i+1)) Anti-bonding 5px’ Lone-pair Bonding 5py’(Tei)+5pz’(Te(i+1))
Origin of the Degeneracy One-dimensional system (non SO) The states of ±π/3 (and ±2π/3) are degenerate. Unfolding Three-dimensional system (non SO) Degeneracy at H, K, A, and Γ is protected by the spiral symmetry S3 and the rotational symmetry C2. With the SOI (→ the space group becomes the double group.) H and K: spiral symmetry S3 and rotational symmetry C2 A, Γ, L, and M: Time-reversal symmetry
Origin of the 3x2 fold degenerate Dirac cone 2x2 fold degenerate Dirac cone in graphene 3x2 fold degenerate Dirac cone in Tellurium and Selenium
Fermi Surface and Spin Structure Spin on the ΓKK’-AHH’ line is directed parallel to the line (owing to the space group P3121). Direction of spin is changed by the hybridization.
Summary • Various three-dimensional Dirac points exist in Tellurium and Selenium. • Without the SOI, a 2x3 fold degenerate Dirac point emerges under pressure. • Spin on the ΓKK’-AHH’ line is directed parallel to the line.
Te Fermi surface 0 GPa (GW+SO) : lowest unoccuoied state at H
Te Fermi surface 0 GPa and 3.82 GPa : highest occuoied degenerate state at H
