Basics of topological insulator

Transcription

1 NTU Basics of topological insulator Ming-Che Chang Dept of Physics, NTNU

2 A brief history of insulators Band insulator (Wilson, Bloch) Mott insulator Anderson insulator Quantum Hall insulator Topological insulator Peierls transition Hubbard model Scaling theory of localization D TI is also called QSHI

3 Gauss-Bonnet theorem for a D surface with boundary M da G + ds k = πχ M g ( M, M ) χ = χ = 0 χ = 1 Quantum Hall effect (D lattice fermion in magnetic field) DBZ dk Ω Z = π C 1 σ H = C e 1 h

4 D Lattice fermion with time reversal symmetry (TRS) BZ Without B field, Chern number C 1 = 0 Bloch states at k, -k are not independent EBZ is a cylinder, not a closed torus. No obvious quantization. Moore and Balents PRB 07 C 1 of closed surface may depend on caps C 1 of the EBZ (mod ) is independent of caps (topological insulator, TI) types of insulator, the 0-type, and the 1-type

5 DTI characterized by a Z number (Fu and Kane 006 ) 1 ν = Ω dk π EBZ ( EBZ ) dka mod ~ Gauss-Bonnet theorem with edge How can one get a TI? : band inversion due to SO coupling 0-type 1-type

6 Bulk-edge correspondence Different topological classes Semiclassical (adiabatic) picture: energy levels must cross (otherwise topology won t change). E g gapless states bound to the interface, which are protected by topology. Topological Goldstone theorem?

7 Bulk-edge correspondence in TI Dirac point Bulk states Edge states Fermi level TRIM (-fold degeneracy at TRIM due to Kramer s degeneracy) helical edge states robust backscattering by non-magnetic impurity forbidden

8 Topological insulators in real life Graphene (Kane and Mele, PRLs 005) SO coupling only 10-3 mev D HgTe/CdTe QW (Bernevig, Hughes, and Zhang, Science 006) Bi bilayer (Murakami, PRL 006) Bi 1-x Sb x, α-sn... (Fu, Kane, Mele, PRL, PRB 007) Bi Te 3 (0.165 ev), Bi Se 3 (0.3 ev) (Zhang, Nature Phys 009) 3D The half Heusler compounds (LuPtBi, YPtBi ) (Lin, Nature Material 010) thallium-based III-V-VI chalcogenides (TlBiSe ) (Lin, PRL 010) Ge n Bi m Te 3m+n family (GeBi Te 4 ) strong spin-orbit coupling band inversion

9 A stack of D TI z D TI Helical edge state Helical fragiless y x 3 TI indices

10 3D TI: 3 weak TI indices: Eg., (x 0, y 0, z 0 ) z + z 0 x 0 x + y 0 y + 1 strong TI index: ν 0 ν 0 = z + -z 0 (= y + -y 0 = x + -x 0 ) difference between two D TI indices Fu, Kane, and Mele PRL 07 Moore and Balents PRB 07 Roy, PRB 09

11 Weak TI index Screw dislocation of TI A stack of D TI not localized by disorder half of a regular quantum wire Ran Y et al, Nature Phys 009 From Vishwanath s slides

12 Band inversion, parity change, spin-momentum locking (helical Dirac cone) S.Y. Xu et al Science 011

13 Dirac point: Graphene Even number located at Fermi energy half integer QHE ( 4) Spin is not locked with k vs. Topological insulator Odd number (on one side) not located at E F half integer QHE (if E F is located at DP) spin is locked with k can be opened by substrate cannot be opened k k σ H =+1 Top bottom σ H =+1 σ H =-1 σ H =-1

14 Electromagnetic response Axion electrodynamics

15 First, a heuristic argument: half-iqhe E TI J H E F Effective Lagrangian for EM wave L = L + L L EM 0 0 axion 1 E = B 8π c axion coupling e 1 Θ Laxion = E B= α E B hc 4π 4π e e 1 note: α = = c h c 137 Surface state ~ DEG e Hall current J H = E h e Induced magnetization M = E h magneto-electric coupling For systems with time-reversal symmetry, Θ can only be 0 (usual insulator) or π (TI) Cr O 3 : θ~π/4 (TRS is broken)

16 Maxwell eqs with axion coupling Θ E + α B = 4πρ π Θ 4π Θ B α E = J + E+ α B π c c t π B = 0 E = B ct Θ=π ρ Θ = α δ( zb ) z 4π B z Effective charge and effective current E = 4π ( ρ + ρθ ) α ρθ = ( ΘB) 4π 4π 1 E B= ( J + JΘ ) + c c t cα JΘ = ΘE B 4π 4π t α ( ) + ( Θ ) c 1 J = α Θ δ ( z)ˆ z E σ xy 4π = Θ=π E z e h

17 Static: Magnetic monopole in TI Dynamic: Optical signatures of TI? A point charge An image charge and an image monopole Circulating current axion effect on Snell s law Fresnel formulas Brewster angle Goos-Hänchen effect D Qi, Hughes, and Zhang, Science 009 Longitudinal shift of reflected beam (total reflection) Chang and Yang, PRB 009 (Magnetic overlayer not included)

18 Dimensional reduction Topological field theory

19 D quantum Hall effect 1D charge pump Laughlin s argument (1981): y B x x ψ compactification x ψ When Berry the curvature: flux is changed by Φ 0, integer fij charges = iaj are jai transported from edge Berry to connection: edge IQHE a = i u u k 1 = π C1 d kfxy k k ( ) A ( x, t) θ ( x, t), a parameter y polarization P ( θ ) 1 = π dk a x x C 4π δ S 1 μντ SCS = d xdtε Aμ νaτ j C μ CS 1 μντ = = ε ν δaμ π A τ S αβ = dxdtpε A j = ε α αβ β P α β Qi, Hughes, and Zhang PRB 008

20 4D quantum Hall effect 3D topological insulator w ψ compactification ψ (x,y,z) C = ε 4π μ C μνρστ j = ε A A ν ρ σ τ 8π nonlinear response. 4 μνρστ S d xdt A A A μ ν ρ σ τ 1 = ε 8π Θ α 1 αβγδ j = ε A βθ γ δ 4π 3 αβγδ S d xdt A A α β γ δ ( ij kl ) 1 3π fij = iaj jai i[ ai, aj ] mn a = i u u 4 ijkl C = d kε tr f f k m k n 1 i Θ ε 8π ijk = dk tr i jk i j, BZ af a a ak Qi, Hughes, and Zhang PRB 008

21 5 Without TRS NTU Phys bldg With TRS 4D QHE 4 TI 3 QHE Chern elevator QSHE Charge pump 1 Re-enactment from Qi s slide

22 Alternative derivations of Θ 1. Semiclassical approach (Xiao Di et al, PRL 009) Pi. B A. Essin et al, PRB 010 j M j 3. A. Malashevich et al, New J. Phys. 010 E i Pi B j M α α α δ j = = ij = ij + θ ij Ei Θ e αθ = π h Explicit proof of Θ=π for strong TI: Z. Wang et al, New J. Phys. 010

23 Physics related to the Z invariant graphene Quantum SHE Spin pump 4D QHE Topological insulator generalization of Gauss-Bonnett theorem Helical surface state Magneto-electric response Interplay with SC, magnetism Majorana fermion in TI-SC interface QC Time reversal inv, spin-orbit coupling, band inversion

24 Thank you!