1. 三方晶テルル・セレンにおけるDirac分散とスピン軌道効果
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
Today, I will talk about the Dirac points and the spin-orbit effects in the single element materials, Tellurium and Selenium.
産業技術総合研究所　ナノシステム研究部門

2. 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
Trigonal Tellurium and Selenium are the group-VI elements. They have the characteristic helical structure.
The helical chains are arranged in a hexagonal array. The space group is P3121 or P3221 depending on the right-handed or left-handed screw axis. There are spiral symmetry S3, a rotation of 2pi/3 about the c axis, followed by the fractional translation c/3, and the rotational symmetry C2, a rotation of pi about the a axis. There is no inversion symmetry.
The unit cell contains three atoms. The electron configuration is (5p)4 in Te, and (4p)4 in Se.
The lattice constants are here. Se has stronger one-dimensional character than Te.
(r, R) :Te (2.83, 3.49) Å
Se (2.37, 3.44) Å
Se has stronger one-dimensional character

3. 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).
Both Te and Se are semiconducting, and especially, Te has a small band gap. Under pressure, the transition from insulator to metal takes place. There is also a structural phase transition. There are some calculations of the electronic structure of Te and Se. In this study, we find that there are various three-dimensional Dirac cones near the Fermi level in Te and Se.
We clarify the origin of the Dirac cone, effects of the spin-orbit interaction, and the spin structure in the k space.
Strong topological insulator under shear strain ?
L. A. Agapito et al.: Phys. Rev. Lett. 110, (2013).
We find that there are various three-dimensional Dirac points near the Fermi level in Te and Se.

4. Method (GW+SO) LDA+SO GW Hamiltonian of the GW+SO method
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)
T. Kosugi et al.: J. Phys. Soc. Jpn. 80, (2011). GW
Full-potential linear muffin-tin orbital (FP-LMTO) code
M. van Schilfgaarde et al.: Phys. Rev. B 74, (2006).
(k-point mesh: 6x6x4)
T. Miyake and F. Aryasetiawan: Phys. Rev. B 77, (2008).
Hamiltonian of the GW+SO method
The calculations are based on both the fully relativistic LDA and the GW approximation.
The Hamiltonian including both the SOI and the GW self-energy correction is expressed as this equation.
By diagonalizing this Hamiltonian, we obtain the electronic band structure.
φ: maximally localized Wannier function
N. Marzari and D. Vanderbilt: Phys. Rev. B 56, (1997).
I. Souza et al.: Phys. Rev. B 65, (2001).

5. Electronic Band Structure
First, we show the band structures of Te and Se at ambient pressure and under pressure. The correspondent Brillouin Zone is here. The band gap is in good agreement with that in the experiment. We find that there are several Dirac points near the Fermi level. For example, in Te at ambient pressure, the lowest unoccupied state at the H point is the Dirac point. As the Fermi level is shifted crossing the energy level of this state, the Fermi surface shrinks to the H point, then turns to grow larger. The accidental degenerate point at P1 also forms the Dirac point. Under pressure, the shape of the conduction band bottom becomes sharp as the insulator-to-metal transition is approached. As the pressure is increased further, the system turns to a metal. In 3.82 GPa, the highest occupied degenerate state at the H point becomes the Dirac point. In addition to P1, the accidental degenerate points P2 and P3 are also the Dirac points.
Various three-dimensional Dirac points exist near the Fermi level.

6. 3x2 fold degenerate Dirac cone
Without the SOI
For comparison, we show the electronic band structures of Te without the spin-orbit interaction. The system is insulating at ambient pressure. At the H point, the lowest unoccupied state is 1x2-fold degenerate, whereas the highest occupied state is 2x2-fold degenerate. The two energy levels approach each other under pressure, and eventually coincide at 5.3 GPa. Then, the 2x3 degenerate Dirac point emerges at the H point.
Without the SOI, a Dirac point (3x2 fold degenerate) emerges under pressure.

7. 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
Next, we analyze the orbital character by the 9x2 maximally localized Wannier functions originating from the p orbitals. The 18 eigenstates near the Fermi level are classified into three types. The deepest six states have bonding character between the nearest-neighobour atoms. The middle six states are the lone-pair states, and the unoccupied six states are anti-bonding states.
Lone-pair
5px’
Bonding
5py’(Tei)+5pz’(Te(i+1))

8. 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)
To understand the band dispersion, it is helpful to neglect inter-chain interactions and consider a one-dimensional helical chain. In such a case, we can unfold the Brillouin zone. The egde of the Brillouin zone corresponds to –pi/3, pi/3, or pi states in the unfolded Brillouin zone. Therefore, -pi/3 and pi/3 states are degenerate. Even in the three dimensional system, where the inter-chain interaction exists, the degeneracy is protected by the spiral symmetry S3 and the rotational symmetry C2. In the presence of the spin-orbit interaction, the degeneracy is also protected by S3 and C2 or time-reversal symmetry.
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

9. Origin of the 3x2 fold degenerate Dirac cone
2x2 fold degenerate Dirac cone in graphene
3x2 fold degenerate Dirac cone in Tellurium and Selenium
Now we discuss the origin of the 3x2 fold degenerate Dirac cone between the lone-pair and anti-binding bands at the H point without the spin-orbit interaction. The origin of the Dirac cone has similarities and differences with that in graphene. There are two atoms in the unit cell of graphene. Graphene has three kinds of transfer integrals, namely that between the atom A and B, between A and B', and between A and B". When these three kind of transfer integrals are equal, the Dirac cone is formed at the K point. On the other hand, there are three atoms in the unit cell of Te and Se.
The one intra-chain hopping shown by green triangles, and the two inter-chain hoppings shown by red and blue triangles, correspond to the three kind of the transfer integrals in graphene. The 3x2 fold degenerate Dirac cone is formed when the energy from the inter-chain transfer integral becomes the same as that from the intra-chain transfer integral.

10. Fermi Surface and Spin Structure
Spin on the ΓKK’-AHH’ line is directed parallel to the line (owing to the space group P3121).
Finally, we show the Fermi surface and the spin structure on the Fermi surface at ambient pressure and under pressure. The figures in left-side and center are the Fermi surface and the spin structure near the lowest unoccupied states at the H point at 0.31 eV. The figure in the right-side is at the Fermi level. The spin is parallel to the side on the sides of the triangular prism formed by the GammaKK'-AHH' points owing to the space symmetry P3121. The direction of the spin is changed by the hybridization Indeed, the spin rotates once around the H point at ambient pressure. On the other hand, the spin on the kxky-plane rotates twice at 3.82 GPa.
Direction of spin is changed by the hybridization.

11. Spin on the ΓKK’-AHH’ line is directed parallel to the line.
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.
In this study, we find that the various three-dimensional Dirac cones exist in Tellurium and Selenium. Without the SOI, 2x3 fold degenerate Dirac point emerges under pressure. Spin on ΓKK’-AHH’ line is directed parallel to the line.

13. Appendix

14. Te 3.82GPa and Se 14GPa (GW+SO)

15. Te Fermi surface 0 GPa (GW+SO)
: lowest unoccuoied state at H

16. Te Fermi surface 0 GPa and 3.82 GPa
: highest occuoied degenerate state at H

17. Te GW+SO 0GPa band structure (P1)

18. Te GW+SO 3.82GPa band structure (P3)

19. Band crossing (normal)

20. Te GW 0GPa band structure (H)

21. Te GW 5.3GPa band structure (H)

22. Te GW 5.3GPa band structure (P2)

23. Spin structure

24. title