What is a topological insulator? Ming-Che Chang Dept of Physics, NTNU
A mini course on topology extrinsic curvature K vs intrinsic (Gaussian) curvature G K 0 G 0 G>0 G=0 K 0 G=0 G<0
Gauss-Bonnet theorem for 2-dim closed surface M da G = 2 πχ Euler characteristic χ = 2(1 g) g = 0 g = 1 g = 2 Gauss-Bonnet theorem in higher dimensions Gauss-Bonnet theorem for fiber bundles ( 陳省身, 1940 s) Physical space Inner space (spin, gauge field )
Quantum Hall effect (von Klitzing, PRL 1980) classical quantum = 25.81202 k-ohms E F energy B=0 Increasing B Density of states LLs Each LL contributes one e 2 /h to Hall conductance Thouless 1982 Hall conductance = topological number σ e 1 = 2 ( n ) 2 ( n) H dk z h 2π Ω BZ Robust against perturbation
Edge state in quantum Hall system (Classical picture) skipping orbit (Chiral edge state) (Semiclassical picture) Bulk-edge correspondence Robust against disorder (no back-scattering) number of edge modes = Hall conductance
Different types of insulator Band insulator (due to lattice) Mott insulator (e-e interaction) Anderson insulator (disorder) Insulators with nontrivial topology Quantum Hall insulator (cyclotron-energy gap) Topological insulator (Spin-Orbit gap) TR symmetry is broken TR symmetry is NOT broken
Time reversal symmetry, a review Time reversal operator Kramer s degeneracy (for a system with j=half integer) Bloch state Θ ψ = ψ ε k k = ε k k ε k
Edge state: QH vs TI (2D case) Real space Energy spectrum (for one edge) Robust chiral edge state Robust helical edge state backscattering by non-magnetic impurity forbidden
TRIM: Time Reversal Invariant Momenta (Fu and Kane, PRB 2007) BZ surface BZ Γ 01 Γ 11 Λ b Γ 00 Γ 10 Λ a TRIM: r 1 r r Γ nn = nb + nb 2 ( ) 1 2 1 1 2 2 ε = ε = ε Γ Γ Γ Energy band is guaranteed to be 2-fold degenerate at TRIM.
Surface state: usual insulator vs TI usual insulator TI Dirac point (for 3D TI) (for one surface) TRIM even pairs of surface states (fragile) even number of Dirac pts filled odd pairs of surface states (robust) odd number of Dirac pts filled Topological number ν=0 Topological number ν=1
Candidates of topological insulator Graphene (Kane and Mele, PRLs 2005) SO coupling only 10-3 mev 2D HgTe/CdTe QW (Bernevig, Hughes, and Zhang, Science 2006) Bi bilayer (Murakami, PRL 2006) Bi 1-x Sb x, α-sn... (Fu, Kane, Mele, PRL, PRB 2007) 3D Bi 2 Te 3 (0.165 ev), Bi 2 Se 3 (0.3 ev) (Zhang, Nature Phys 2009) The half Heusler compounds (LuPtBi, YPtBi ) (Nature Material, July 2010)
ARPES of Bi 2 Se 3 DFT prediction: Helical Dirac cone Fermi energy is not located at Dirac pt. ~ Rashba model
Dirac points in graphene E Dirac cone B Cyclotron orbits k r r Ω ( k) = πδ( k) γc = π E( k) = hν Fk π k 2 1 E = v 2eBhn n γ C eb = 2π n+ 2 2π h F Similar shift exists in TI expecting half integer QHE
Difference of the Dirac points: Graphene vs TI 1. Even Dirac points vs odd Dirac points 2. Graphene s are locked at Fermi energy, TI s are not 3. TI s: spin locked with momentum (helical Dirac point) 4. Graphene s can be opened by substrate, TI s is robust 5. Note: Nielsen-Ninomiya s theo requires (massless) lattice Dirac fermons to appear in pairs A major obstacle to lattice QCD.
Topological transport regime half integer QHE gives Induced magnetization 2 1 e I = E 2 h 2 m e M = = A 2h E Quantized magneto-electric coupling Effective Lagrangian term 2 e 1 r r Θ 1 r r LME = E B= α E B 2 2hc 4π c 2 e α = ; Θ = π hc axion coupling For a system with TRS, Θ can only be 0 (usual insulator) or π (TI)
Maxwell eqs with axion coupling r Θ r E + α B = 4πρ π r Θ r 4π r r Θ r B α E = J + E+ α B π c c t π r B = 0 r r E = B ct Effective charge and effective current r r r 4π r r 1 E E = 4π ( ρ+ ρθ) B= ( J + JΘ) + c c t α r r α r α r ρθ = Θ = Θ + Θ 2 2 2 4π 4π 4π t c ( B) JΘ ( E) ( B) ρ Θ = α δ( zb ) z 4π r c J = α δ ( z) zˆ Θ E 4π r half integer QHE Θ=π B Θ=π E
A point charge induces an image monopole (Qi, Hughes, and Zhang, Science 2009) A point charge Circulating current An image charge and an image monopole
Optical signatures of TI (Chang and Yang, PRB 2009) axion effect on Snell s law Fresnel formulas Brewster angle Goos-Hänchen effect
γ n Θ n Θ E k B y θ θ B E z E B x E '' E = R E '' E // // E' E = T E' E // // R is symmetric two orthogonal eigenmodes Rotation of eigenmodes k tan 2 iδ 2α ne n' α ( γ ) = 2 2 2 n γ E B iδ n n' e n n' α 2 2 2 α 2 2 2 ' α α Θ Θ π γ π/4 if 2 2 2 n = n' + α
One eigenmode allows Brewster angle, tanθ B = n' 1 ( n'/ n) n 1 ( n'/ n) 2 2 (Reflected/refracted beam no longer perpendicular to each other) γ Effective refraction indices 1 2 2 iδ 2 2 n = ( n+ n') + α + e ( n n') + α 2 1 2 2 iδ 2 2 n' = ( n+ n') + α e ( n n') + α 2 For (n,n )=(10,9), γ~0.1 degree Use Brewster angle + eigenmode direction to determine γaccurately
Alternative optical probes: Rotation of polarization from reflected wave (Kerr effect) and transmitted wave (Faraday effect) for a TI thin film W.K. Tse and A.H. MacDonald, PRL July 2010 J. Maciejko, X.L. Qi, H.D. Drew, and S.C. Zhang, PRL, Oct 2010 Giant Kerr effect Universal Faraday effect Independent of material details!
An insulator with topological number ν=1 manifests itself in many ways: odd pair of robust, helical surface states odd number of helical Dirac points half integer QHE from the surface states quantized magneto-electric coupling novel optical effects from axion coupling Thank you!