Structure : Trigonal P or P (D 4 3 or D 6 3 ) (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 Lattice Constant 3 atoms in unit cell Se has stronger one-dimensional character (ex. :α-quartz, HgS) Electron Configuration : Te (5p) 4, Se (4p) 4 R. Keller et al.: Phys. Rev. B 16, 4404 (1977). Spiral symmetry S 3, Rotational symmetry C 2 Group-VI element
Band Gap :Te 0.323, Se 2.0 (eV) Insulator to metal transition (IMT) under pressure Te: ~4 GPa, Se: ~14-~22 GPa Structural transition (ST) also occurs near the IMT. Calculation of electronic structure k ・ p perturbation Pseudopotential technique Strong topological insulator under shear strain ? (Relation between IMT and ST is not yet clarified ) P. W. Bridgman: Proc. Am. Acad. Arts Sci. 74, 425 (1942). M. Takumi et al.:Fukuoka University Science Reports 42 (1) 1 (2012). J. D. Joannopoulos et al.: Phys. Rev. B 11, 6 (1975). T. Doi et al.: J. Phys. Soc. Jpn. 28, 36 (1970). L. A. Agapito et al.: Phys. Rev. Lett. 110, (2013). V. B. Anzin et al.: Phys. Stat. Sol. (a) 42, 385 (1977). S. Tutihasi et al.: Phys. Rev. 158, 623 (1967). We find that there are various three-dimensional Dirac points near the Fermi level in Te and Se.
Fully relativistic two-component first-principles code QMAS (Quantum MAterials Simulator) based on the projector augmented wave (PAW) method (k-point mesh: 6x6x6, Plane-wave energy cutoff : 40 Ry) LDA+SO GW Full-potential linear muffin-tin orbital (FP-LMTO) code M. van Schilfgaarde et al.: Phys. Rev. B 74, (2006). T. Miyake and F. Aryasetiawan: Phys. Rev. B 77, (2008). Hamiltonian of the GW+SO method φ: maximally localized Wannier function (k-point mesh: 6x6x4) T. Kosugi et al.: J. Phys. Soc. Jpn. 80, (2011). N. Marzari and D. Vanderbilt: Phys. Rev. B 56, (1997). I. Souza et al.: Phys. Rev. B 65, (2001).
Various three-dimensional Dirac points exist near the Fermi level.
Without the SOI, a Dirac point (3x2 fold degenerate) emerges under pressure. Without the SOI
Maximally localized Wannier function of Te J. D. Joannopoulos et al.: Phys. Rev. B 11, 6 (1975). Three types of the p bands 5p y’ (Te i )-5p z’ (Te (i+1) ) 5p x’ originating mainly from 5p y’ (Te i )+5p z’ (Te (i+1) )
One-dimensional system (non SO) Three-dimensional system (non SO) Degeneracy at H, K, A, and Γ is protected by the spiral symmetry S 3 and the rotational symmetry C 2. A, Γ, L, and M: Time-reversal symmetry With the SOI (→ the space group becomes the double group.) H and K: spiral symmetry S 3 and rotational symmetry C 2 The states of ±π/3 (and ±2π/3) are degenerate. Unfolding
2x2 fold degenerate Dirac cone in graphene 3x2 fold degenerate Dirac cone in Tellurium and Selenium
Spin on the ΓKK’-AHH’ line is directed parallel to the line (owing to the space group P3 1 21). Direction of spin is changed by the hybridization.
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.