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 G>0 K≠0 G≠0 G=0 K≠0 G=0 G<0 • Gauss-Bonnet theorem for 2-dim closed surface ∫ M Euler characteristic g=0 da G = 2πχ χ = 2(1 − g ) 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 Increasing B B=0 LLs • Each LL contributes one e2/h to Hall conductance • Thouless 1982 Hall conductance = topological number energy EF σH ⎡ 1 2 (n) ⎤ Ω d k ⎢ z ⎥ ∫ ⎣ 2π BZ ⎦ Robust against perturbation (n) Density of states e2 = h Edge state in quantum Hall system (Classical picture) skipping orbit (Chiral edge state) Robust against disorder (no back-scattering) (Semiclassical picture) Bulk-edge correspondence 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) Θψ k ↑ = ψ − k ↓ ε k↑ = ε −k↓ ↑ ↑ • Bloch state ε k Edge state: QH vs TI (2D case) Real space Robust chiral edge state Robust helical edge state backscattering by non-magnetic impurity forbidden Energy spectrum (for one edge) TRIM: Time Reversal Invariant Momenta (Fu and Kane, PRB 2007) BZ surface BZ Γ01 Γ00 Γ11 Γ10 Λb Λa TRIM: r r 1 r Γ n1n2 = n1b1 + n2b2 2 ( ) ε Γ↑ = ε −Γ↓ = ε 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) • odd pairs of surface states (robust) • even number of Dirac pts filled • odd number of Dirac pts filled Topological number ν=0 Topological number ν=1 Candidates of topological insulator • Graphene (Kane and Mele, PRLs 2005) 2D SO coupling only 10-3 meV • HgTe/CdTe QW (Bernevig, Hughes, and Zhang, Science 2006) • Bi bilayer (Murakami, PRL 2006) •… • Bi1-xSbx, α-Sn ... (Fu, Kane, Mele, PRL, PRB 2007) • Bi2Te3 (0.165 eV), Bi2Se3 (0.3 eV) … (Zhang, Nature Phys 2009) 3D • The half Heusler compounds (LuPtBi, YPtBi …) (Nature Material, July 2010) •… ARPES of Bi2Se3 DFT prediction: Helical Dirac cone Fermi energy is not located at Dirac pt. ～ Rashba model Dirac points in graphene E Dirac cone r r Ω(k ) = πδ (k ) → γ C = π B E (k ) = hν F k … Cyclotron orbits k ⎛ ⎝ 1 2 π k 2 = 2π ⎜ n + − γ C ⎞ eB 2π ⎟⎠ h ⇒ En =vF 2eBhn 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 1 e2 I= E 2 h m e2 M= = E A 2h Quantized magneto-electric coupling Effective Lagrangian term LME Θ 1 r r e2 1 r r “axion” coupling = E⋅B =α E⋅B 2 2h c 4π c e2 α = ; Θ =π hc For a system with TRS, Θ can only be 0 (usual insulator) or π (TI) Maxwell eqs with axion coupling Θ r⎞ ⎛r ∇ ⋅ ⎜ E + α B ⎟ = 4πρ π ⎠ ⎝ Θ r ⎞ 4π r ∂ ⎛ r Θ r⎞ ⎛r J+ ∇×⎜ B −α E ⎟ = ⎜ E +α B⎟ π ⎠ c c∂t ⎝ π ⎠ ⎝ r ∇⋅B = 0 r ∂ r B ∇× E = − c∂t Effective charge and effective current r ∇ ⋅ E = 4π ( ρ + ρΘ ) r α ρΘ = − 2 ∇ ⋅ ( ΘB ) 4π r r 4π r r 1 ∂E ∇× B = J + JΘ + c c ∂t r r r cα α ∂ ΘB J Θ = 2 ∇ × ΘE + 2 4π 4π ∂t ( ( ) ( ) r r cα JΘ = − δ ( z ) zˆ × E 4π α ρΘ = δ ( z ) Bz 4π E B Θ=π ) Θ=π → half–integer QHE 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 •… ⎛ E ''⊥ ⎞ ⎛ E⊥ ⎞ = R ⎜ ⎟ ⎜ ⎟ '' E ⎝ // ⎠ ⎝ E// ⎠ k B’’ γ B θ E n Θ y n’ Θ’ B’ E’ E’’ θ’ ⎛ E '⊥ ⎞ ⎛ E⊥ ⎞ T = ⎜ ⎟ ⎜ ⎟ ⎝ E '// ⎠ ⎝ E// ⎠ x z R is symmetric → two orthogonal eigenmodes Rotation of eigenmodes 2α neiδ tan ( 2γ ) = 2 n − n '2 − α 2 k B γ E α ≡α Θ '− Θ π n 2 − n '2 − α 2 e ≡ 2 n − n '2 − α 2 iδ γ→π/4 if n 2 = n '2 + α 2 One eigenmode allows Brewster angle, n ' 1 − ( n '/ n ) 2 tan θ B = n 1 − ( n '/ n) 2 (Reflected/refracted beam no longer perpendicular to each other) γ Effective refraction indices 1⎡ ( n + n ') 2 + α 2 + eiδ ( n − n ') 2 + α 2 ⎤ ⎦ 2⎣ 1 n ' = ⎡ ( n + n ') 2 + α 2 − eiδ ( n − n ') 2 + α 2 ⎤ ⎦ 2⎣ n= Use Brewster angle + eigenmode direction to determine γaccurately For (n,n’)=(10,9), γ～0.1 degree 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!