Optical signature of topological insulator 1/7/11 @ NCTU Optical signature of topological insulator Ming-Che Chang Dept of Physics, NTNU Min-Fong Yang Dept of Physics, Tunhai Univ.

Surface state in topological insulator ARPES of Bi2Se3 DFT prediction H. Zhang et al, Nature Phys 2009 Helical Dirac cone Dirac point at TRIM Robust against non-magnetic disorder Fermi energy is not located at Dirac pt. 2/14

… Landau levels of the Dirac cone LLs STM experiment E Dirac cone B Cyclotron orbits LLs k Berry phase Orbital area quantization Linear energy dispersion P. Cheng et al, PRL 2010 3/14

Dirac point: Graphene vs. Topological insulator (TI) Even number Odd number (on one side) located at Fermi energy not located at EF (so far) not locked spin is locked with k can be opened by substrate cannot half integer QHE (×4) in graphene Nielsen-Ninomiya’s theorem requires (massless) lattice Dirac fermons to appear in pairs A major obstacle in lattice QCD Tricky to separate the 2 in transport experiment half integer QHE in TI (if EF is located at DP) 4/14

An alternative probe of TI : EM wave To have the half-IQHE, Effective Lagrangian for EM wave EF “axion” coupling E JH TI For systems with time-reversal symmetry, Θ can only be 0 (usual insulator) or π (TI) Surface state ～2 DEG Hall current A. Essin et al, PRB 2010 A. Malashevich et al, New J. Phys. 2010 Z. Wang et al, New J. Phys. 2010 Induced magnetization “magneto-electric” coupling 5/14

Maxwell eqs with axion coupling Θ=π B + + + + + z Effective charge and effective current Θ=π E z 6/14

Magnetic monopole in TI Optical signatures of TI? Static: Dynamic: Magnetic monopole in TI Optical signatures of TI? axion effect on A point charge Snell’s law Fresnel formulas Brewster angle Goos-Hänchen effect … Circulating current An image charge and an image monopole D Qi, Hughes, and Zhang, Science 2009 Longitudinal shift of reflected beam (total reflection) Chang and Yang, PRB 2009 Magnetic overlayer not included 7/14

k γ γ reflection refraction Rotation of eigen-modes θ E’’ E n Θ E’ R and T are symmetric (non-diagonal) → two orthogonal eigen-modes (not the usual TE/TM mode) For (n,n’)=(10,9), γ～0.1 degree (2γ is the Kerr rotation angle) 8/14

Effective refraction indices (in the usual Fresnel formulas) Brewster angle (for one of the eigenmode) E k 9/14

Change of reflectance (due to the axion term) Change of Goos-Hanchen shift (due to the axion term) D～penetration depth θ=π θ=π θ=π/2 θ=π/2 θc θc and for 2 eigen-modes Multiple reflections amplify the GH shift (n,n’)=(10,9.5) θc=71.805 BiSb 10/14

Hall conductivity and Faraday effect in graphene 2DEG or graphene n2 Fig from I. Crassee et al, Nat Phys 2010 Normal incidence Determine from Faraday rotation K.W. Chiu et al, Surf Sci 1976 Faraday rotation For a free-standing graphene with QHE 11/14

TI thin film, normal incidence 1 2 3 TI substrate Faraday rotation Agree with calculations from axion electrodynamics gap opening (A.Khandker et al, PRB 1985) Kerr rotation Multiple reflections (λ>>d) 12/14

Kerr effect and Faraday effect for a TI thin film W.K. Tse and A.H. MacDonald, PRL 2010 J. Maciejko, X.L. Qi, H.D. Drew, and S.C. Zhang, PRL 2010 (Free-standing film) E E “Giant” Kerr effect 13/14

Thank you! These interesting optical effects in TI remain to be seen. 14/14