Introduction to topological insulator

Transcription

1 NTHU Introduction to 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 A brief review on topology extrinsic curvature K vs intrinsic (Gaussian) curvature G Positive and negative curvature K 0 G=0 G>0 G=0 G<0 K 0 G 0

4 Gauss-Bonnet theorem (for a -dim closed surface) M da G = 4 π (1 g) g = 0 g = 1 g = Gauss-Bonnet theorem for a surface with boundary M da G + ds k = πχ M g ( M, M ) Marder, Phys Today, Feb 007

5 Topology in condensed matter D lattice electron in magnetic field B ω C Cyclotron motion D Brillouin zone Berry connection A( k) i u u k k k Berry curvature Ω ( k) = A( k) k Cell-periodic Bloch state = topological number (1st Chern number) 1 C1 = d kωz Z π BZ ~ Gauss-Bonnet theorem

6 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

7 D TI 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

8 Kramer s degeneracy (for fermions with TRS) Time reversal operator Θ = 1 for fermions If H commutes with Θ, then ψ, Θψ are degenerate in energy (Kramer pair) Bloch state Θ ψ = ψ ε k k = ε k k w/ inv symm w/o inv symm ε ε k k Time Reversal Invariant Momenta (TRIM) Γ 01 Γ 11 Dirac point ε Γ 00 Γ 10 k TRIM

9 Bulk-edge correspondence Different topological classes Semiclassical (adiabatic) picture: energy levels must cross. E g gapless states bound to the interface, which are topologically protected. Topological Goldstone theorem?

10 Example 1: QHE (Classical picture) skipping orbit (Chiral edge state) (Semiclassical picture) Gapless excitations at the edge LLs Robust against disorder (no back-scattering) number of edge modes = Hall conductance

11 Example : Topological insulator Usual insulator Topological insulator Dirac point TRIM 0 or even pairs of surface states odd pairs of surface states fragile robust backscattering by non-magnetic impurity forbidden

12 Edge state: Quantum Hall vs topological insulator Real space Energy spectrum (for one edge) Robust chiral edge state Robust helical edge state

13 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

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

15 3 weak TI indices: (ν 1, ν, ν 3 ) Eg., (x 0, y 0, z 0 ) z + z 0 x 0 D TIs stacked along x + y 0 y + Mν = ν G + ν G + ν G ( ) / 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

16 Weak TI indices

17 Screw dislocation of TI A stack of D TI Ran Y et al, Nature Phys 009 1D helical metal exists iff not localized by disorder half of a regular quantum wire From Vishwanath s slides

18 Strong TI index

19 D If there is inversion symm then Bloch state at TRIM Γ i has a definite parity Parity eigenvalue P ψ ( Γ ) = ξ ( Γ ) ψ ( Γ ) ; δ n i n i n i ξ ( Γ ) i n i n filled ξ ( Γ ) =± 1 n i Γ 01 Γ 11 Z class ( 1) ν δδδδ = (normal phase) Γ 00 Γ 10 ( 1) ν δδδδ = (TI phase) If there is NO inversion symm T-reverse matrix (antisymm) w ( k) u ( k) Θ u ( k) i mn m n δ [ w Γi ] [ w Γ ] Pf ( ) det ( ) i Or, count zeros of the Pfaffian Pf[w] (mod ) Fu and Kane PRB 006, 007

20 Similar analysis applies to 3D TRIM: strong weak 1 Γ nn n = nb + nb + nb ( 1) ν 0 = ( 1) ν = 1,,3 n = 0,1 i = ( ) i δ δ δ 1 3 n = 0; n = 0,1 i n n n j i δ δ δ n n n 1 3 For example,

21 Reality check: Surface state in topological insulator ARPES of Bi Se 3 H. Zhang et al, Nature Phys 009 Helical Dirac cone Σ 1 W. Zhang et al, NJP 010

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

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24 Dirac point, Landau levels

25 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

26 Landau levels of the Dirac cone STM experiment E Dirac cone B LLs Cyclotron orbits k Berry phase γ C = π Orbital area quantization π k γ C eb = π n+ π 1 Linear energy dispersion Ek ( ) = ν Fk E = v eb n n F P. Cheng et al, PRL 010

27 Magnetoelectric response Axion electrodynamics

28 To have the half-iqhe, Effective Lagrangian for EM wave E E F L = L + L EM 0 axion 1 E L0 = B 8π c axion coupling e 1 Θ Laxion = E B= α E B hc 4π TI J H 4π e e 1 note: α = = c h c 137 Θ -Θ for TRS 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)

29 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

30 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)

31 TI thin film, normal incidence 1 3 TI substrate ni tij, ± = 4π ni + nj + σ ± c 4π ni nj σ Agree with ± r c ij, ± = calculations from 4π axion electrodynamics ni + nj + σ ± c (A.Khandker et al, PRB 1985) Multiple reflections (λ>>d) 1 t = t t r3r1 1 r = r + t r t r3r1 σ xy σ xy Faraday rotation 1 θf = θ+ θ 1+ ξ = α n + n ( ) 1 3 Kerr rotation ( ξ ) assume σ = 0, σ = ξσ = ± 1 θ i t t e θ ± ± = ± i r r e θ ± ± = ± K = n xx xy xy 3 1 ( + ξ ) n1 ( 1 ξ ) 1 n + + α = α α 4π σ c xy

32 Kerr effect and Faraday effect for a TI thin film W.K. Tse and A.H. MacDonald, PRL 010 J. Maciejko, X.L. Qi, H.D. Drew, and S.C. Zhang, PRL 010 σ xy = σ xy σ xy = σ xy E E tanθ F θ F = α α n + n 1 3 (Free-standing film) θ F = 0 tanθ K 4n α 1 = n n + 4α α Giant Kerr effect θ K = 0

33 Electric polarization Thouless charge pump Z spin pump

34 Electric polarization of a periodic solid 1 = V r ρ( ) 3 P d r r NOT well defined for an infinite solid P Unit cell Choice Choice - + P nevertheless ΔP Start from j = dp/dt, which is well-defined, then dp Δ P = dλ = P P dλ λ= 1 λ= 0 λ: atomic displacement Resta, Ferroelectrics 199

35 Polarization and Berry phase (1D example) King-Smith and Vanderbilt, PRB 1993 q Pλ = dkak (, ) π λ BZ Berry potential: ak (, λ) = i u λ k k u λ k P P+ nq under a "large" GT: iφ ( k) uk e uk with φ( π ) φ(- π)=πn 1 λ ΔP has no such ambiguity q Δ Pλ = dk a( k) π -π 0 π k

36 Thouless charge pump (Thouless PRB 1983) Periodic variation: H(k,t+T)=H(k,t) q Δ Pλ = dk a( k ) nq π = k = ( k, t) n is the first Chern number -π T 0 t π k Quantized pumping at Temp=0, for adiabatic variation. Pumping charges without using a DC bias Example 1: a sliding potential t=0 t=t Trivial

37 Example : SSH model with staggered field (Rice and Mele PRL 1983) t () t i H = c c + δ ( 1 ) c c + h ( t) ( 1 ) c c + hc.. i i i+ 1 i i+ 1 st i i i i i t=0 t=t/4 t=t/ t=3t/4 t=t t=5t/4 π t π t ( δ( t), hst ( t)) = δ0cos, h0sin T T adiabatic variation After a period of T, one electron is pumped to the right

38 Spin pump: Fu-Kane model (PRB 006) Spin-dep staggered field t δ () t i i z H = ciαci+ 1α + ( 1 ) ci αci+ 1α + hst( t) ( 1 ) ci σσαβciβ + hc.. i i i t=0 t=t/4 t=t/ t=3t/4 t=t S z is conserved After t=t/, one unpaired up (down) spin appears on the R(L) edge. (Effectively one up-spin is pumped to the right.) S z is not conserved After t=t/, again unpaired spins (not necessarily up/down) at each end. (some spin is pumped to the right)

39 Fu-Kane model t δ () t i H = c c + ( 1 ) c c + h ( t) ( 1 ) c c + hc.. i z iα i+ 1α iα i+ 1α st iσσ αβ iβ i i i H is time-reversal invariant at t=0, T/, T Kramer degeneracy at t=0, T/, T ( δ ( t), h ( t)) st π t π t δ cos, h sin T T = 0 0 Fu and Kane, PRB 006 Mid-gap edge states 3T/ T Different ground states t=0, T. Z symmetry no crossing, no pumping

40 Dimensional reduction Topological field theory

41 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

42 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

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

44 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

45 Not in this talk, Effective Hamiltonian for the SS [Shen] Disorder, non-crystalline TI [Shen] Electron-electron interaction [Xie] Hybrid materials (TI+M, TI+SC) [Fang] TI-SC interface, Majorana fermion [Law, Wan] Periodic table of TI/SC

46 Physics related to the Z invariant graphene Quantum SHE Spin pump 4D QHE Topological insulator Gauss-Bonnett theorem parity eigenvalue zero of Pfaffian 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

47 Thank you!