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」
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
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
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 )
Calculation of electronic structure
M. Takumi et al.:Fukuoka University Science Reports
42 (1) 1 (2012).
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 points near the Fermi level in Te and Se.
LDA+SO
Fully relativistic two-component first-principles code
QMAS (Quantum MAterials Simulator) based on the
projector augmented wave (PAW) method
http://qmas.jp/
T. Kosugi et al.: J. Phys. Soc. Jpn. 80, 074713 (2011).
(k-point mesh: 6x6x6, Plane-wave energy cutoff : 40 Ry)
GW
Full-potential linear muffin-tin orbital (FP-LMTO) code
M. van Schilfgaarde et al.: Phys. Rev. B 74, 245125 (2006).
T. Miyake and F. Aryasetiawan: Phys. Rev. B 77, 085122 (2008).
(k-point mesh: 6x6x4)
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).
Various three-dimensional Dirac points exist near the Fermi level.
Without the SOI
Without the SOI, a Dirac point (3x2 fold
degenerate) emerges under pressure.
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))
5px’
5py’(Tei)+5pz’(Te(i+1))
One-dimensional system (non SO)
Unfolding
The states of
±π/3 (and
±2π/3) are
degenerate.
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
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 P3121).
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.