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

2. 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.
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3. … 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
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4. 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)
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5. 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
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6. Maxwell eqs with axion coupling
Θ=π B z
Effective charge and effective current
Θ=π E z
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7. 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
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8. 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)
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9. Effective refraction indices (in the usual Fresnel formulas)
Brewster angle (for one of the eigenmode) E k
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10. 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
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11. 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
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12. TI thin film, normal incidence1 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)
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13. 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
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14. Thank you! These interesting optical effects in TI remain to be seen.
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