1. Lei Hao (郝雷) and Ting-Kuo Lee (李定国)
Surface spectral function in the Superconducting state of a topological insulator & Two models of Bi2Se3
Lei Hao (郝雷) and Ting-Kuo Lee (李定国)
IOP, Academia Sinica
NTNU 05/12/2011

2. Topological insulator
Originates from the study of time-reversal symmetry protected edge modes in quantum spin Hall states. [PRL95, Kane&Mele]
Characterized by a Z2 index (0 or 1), instead of the first Chern number in time-reversal symmetry breaking systems (e.g., QHE).
Insulating in the bulk, metallic at the boundary with odd pairs of gapless helical conducting modes localized at the edge (2D) or surface (3D).

3. Topological insulator: Examples in 2D
Graphene with
spin-orbit coupling
CdTe/HgTe/CdTe quantum well
Science 314, 1757; ibid 318, 766.
The topological insulators mentioned here are all defined in time-reversal invariant systems, characterized by a Z2 integer. The famous integer quantum Hall state may be regarded as the most studied topological insulator that breaks TRS, characterized by a TKNN number or the first Chern number.
[PRL95, Kane & Mele]
spin-orbit coupling in real graphene is too small

4. Topological insulator: Examples in 3D
Nature Phys. 5,438
The topological insulators mentioned here are all defined in time-reversal invariant systems, characterized by a Z2 integer. The famous integer quantum Hall state may be regarded as the most studied topological insulator that breaks TRS, characterized by a TKNN number or the first Chern number.
TlBiSe2
EPL90,37002

5. Two conditions
Band inversion: Two bands of opposite parity (p like band and s like band) interchange their relative positions. Negative mass semiconductors.
[HgTe quantum well. Science 314, 1757; Phys. Today 63(1), 33]
Spin-orbit coupling: linear in k terms induce the coupling between opposite parity bands, which 1) opens a gap at the band inversion point in the bulk and 2) gives the linear dispersion of the gapless surface (edge) modes and make them helical.
There might be many possible reasons for the band inversion to occur! In some materials, the reason is just the same spin-orbit coupling. While in some others, like in the CdTe/HgTe/CdTe quantum well.
The existence and stability of the Dirac cone at special points (TRIM) of the BZ are ensured by the pi Berry’s phase around there. [M. Berry Nphys 6, 148]

6. Superconducting CuxBi2Se3
Superconducting only when copper are doped to the positions between consecutive Bi2Se3 quintuple units.
PRL104, (2010)
Nphys1762(2010)

7. Superconducting transition: Resistivity
The residual resistivity is attributed to sensitivity of the superconducting phase to processing and stoichiometry.
That is, some of the doped copper atoms may replace the Bismuth ions.
PRL104, (2010)

8. Superconducting transition: Meissner effect
The diamagnetic magnetization is ~20% of the full diamagnetism, indicating bulk superconductivity.
PRL104, (2010)

9. Well-defined topological surface states in superconducting CuxBi2Se3
Δ~0.6meV
Copper doping reduces the slope of the topological surface states.
BCS estimation
Nphys1762(2010)

10. Pressure induced superconductivity in Bi2Te3
arXiv: v1

11. Phase Diagram: Phases with different pressure dependence
Another experiment: arXiv
Phase Diagram: Phases with different pressure dependence
arXiv: v1

12. Odd-parity Topological Superconductors PRL105, 097001 Fu and Berg
TRI, with SOC (break SU(2) SRS), centro-symmetric
Class DIII [PRB78, (2008)]
Generalized parity operator
The symmetry of the model with respect to the generalized parity operator is called as Z2 symmetry.
TSC: a full pairing gap in the bulk and gapless surface ABS.
At TRI momenta

13. Z2 criterion for nontrivial odd-parity topological superconductors
v=1 is sufficient condition for the topological superconducting phase.
A TRI centrosymmetric superconductor is topological if
(1) Odd parity pairing symmetry with a full bulk superconducting gap;
(2) The Fermi surface encloses an odd number of TRI momenta.
The cases of even-parity pairing are Z2 trivial following a similar derivation.
PRL105, Fu & Berg

14. Mean-field phase diagram for short range pairing potential
PRL105, Fu & Berg
A Kramers pair of zero energy surface Andreev bound states exists for Δ2 .

15. Surface spectral functions in the superconducting state: A possible way to identify pairing symmetry
The spectral function is obtained from the retarded surface Green’s functions as
The retarded surface Green’s function is obtained iteratively in terms of the transfer matrix method (or, iterative GF method).
PRB81, (2010)

16. Normal state Bi2Se3 2D quintuple layers stacked along z direction.
Four relevant spin-orbit degrees of freedom for each quintuple unit.
n labels the quintuple units. 1 and 2 labels the pz orbit.
k=(kx, ky) is a 2D in-plane wave vector.
Nphys. 5, 438 (2009)

17. Model of bulk Bi2Se3
Two models for hxy and hz which give the same bulk and surface dispersions are used in the literature:
(I) Phys. Rev. B 81,035104(2010).
(II) Nature Physics 5,438 (2009).
With in-plane spin-orbit coupling. The SOC along the z direction is present or not for the two different models.
(II)
(I)

18. Normal state surface spectral function
Now, the superconducting pairing would be introduced to the material.
Concentrate on the region where the topological surface states coexist but well separated from the continuum.

19. In the superconducting state, introduce the Nambu basis
The intra-quintuple-layer part
The inter-quintuple-layer part
The difference between the two models enter through the inter-quintuple-layer hopping part.
From the 8X8 retarded surface Green’s function

20. Even-parity Intra-orbital s wave pairing
Two models give identical results.
Note: Since the material is inversion symmetric, parity is well defined to label states.
PRB83, Hao & Lee

21. Even-parity Inter-orbital s wave pairing
Two models give identical results.
PRB83, Hao & Lee

22. Even-parity Inter-orbital s wave pairing
Two models give identical results.
PRB83, Hao & Lee

23. Odd-parity inter-orbital triplet
Model (I)
Model (II)
PRB83, Hao & Lee

24. Odd-parity inter-orbital triplet
Model (I)
Model (II)
PRB83, Hao & Lee

25. Odd-parity intra-orbital s wave
Model (I)
Model (II)
PRB83, Hao & Lee

26. Odd-parity intra-orbital s wave
Model (I)
Model (II)
PRB83, Hao & Lee

27. Odd-parity inter-orbital equal-spin triplet
Two models give identical results, which are anisotropic.
This pairing is two fold degenerate. For another pairing channel, the results along kx and ky exchange with each other.
A circle of zero energy points on the ky-kz plane.

28. Thin film of topological superconductor
The surface Andreev bound states.
N=50
Linear close to the
point.
PRB83, Hao & Lee

29. Decay of the surface mode into the bulk
Normal phase

30. Summary
For odd-parity pairing, when a full gap opens in the bulk conduction band while no gap opens in the topological surface states (if well defined at EF), gapless Andreev bound states appear on the surface.
When the topological surface states are well separated from the bulk conduction band at EF, the surface Andreev bound states connect continuously to the topological surface states which are both nondegenerate.
Two models for Bi2Se3 give rise to identical normal state surface and bulk spectrum are in fact qualitatively different.
Explain the first more clearly, if time is enough.

31. Two Models of Bi2Se3
Two models for hxy and hz which give the same bulk and surface dispersions are used in the literature:
(I) Phys. Rev. B 81,035104(2010).
(II) Nature Physics 5,438 (2009).
With in-plane spin-orbit coupling. The SOC along the z direction is present or not for the two different models.
(I)
(II) 31

32. Topological surface states
Parameterizations of Wang et al, close to
point.
For a sample occupying z<=0, the zero energy surface mode (for the PHS model considered) is obtained from
The two zero energy modes
The zero energy surface mode could be searched since the model is explicitly particle-hole symmetric!
decay length of the two zero energy modes. 32

33. Topological surface states
basis
For model I:
The effective model of the surface states has the same form as in the bulk model
For model II:
Two differences between the two models: The orbits involved in the surface states; The mathematical form of the effective model. The dispersion is the same for the two models.
The effective model 33

34. Gap opening in the topological surface states Model I
Eigenvector for the surface states
Pairing in terms of the original basis

35. Gap opening in the topological surface states Model II
Eigenvector for the surface states
Pairing in terms of the original basis

36. Proposed experiment to identify the right model: Dynamical responses
For the surface states of an ideal infinite xy surface, charge current is directly related to surface spin:
For model I:
For model II:
Charge responses are thus simply related to spin responses.
PRL104,116401(2010).

37. Ribbon or Slab of Bi2Se3
For a real sample in the form of a ribbon or slab, presence of lateral surfaces makes the relationship between charge current and surface spin complicated.
However, one thing for sure is that the surface states reflect clearly properties of the bulk band structures and thus should reveal this connection in some way.
The dynamical spin susceptibility appears as an ideal tool to tell the correct model.

38. A ribbon running along x

39. Optical conductivity
Two models give identical results.

40. Dynamical spin susceptibility
Imaginary parts of the dynamical spin susceptibilities, divided by the frequency.

41. Summary
Surface states from two models of Bi2Se3 have different orbital characters.
Gap opening of the topological surface states upon the introduction of superconducting pairing is different.
Dynamical spin susceptibilities could be used to tell the correct model.
Explain the first more clearly, if time is enough.

42. Thank you for your attention !

43. Thin film of topological superconductorΔ4
N=50
Linear close to the
point.

44. PRL 105,097001

45. Surface states on the xz surface of Bi2Se3: for y<=0
Model I
Model II
Up and down spin mix together for the two bases.