Topological Insulators in 2D and 3D

Transcription

1 Topologicl Insultors in D nd 3D I. Introduction - Grphene - Time reversl symmetry nd Krmers theorem II. D quntum spin Hll insultor - Z topologicl invrint - Edge sttes - HgCdTe quntum wells, expts III. Topologicl Insultors in 3D - Wek vs strong - Topologicl invrints from bnd structure IV. The surfce of topologicl insultor - Dirc Fermions - Absence of bckscttering nd locliztion - Quntum Hll effect - θ term nd topologicl mgnetoelectric effect

2 Energy gps in grphene: σ ~ sublttice z τ z ~ vlley ~ spin s z H = σ p+ V v F E( p) =± v F p +. Stggered Sublttice Potentil (e.g. BN) V = CDW z σ Broken Inversion Symmetry. Periodic Mgnetic Field with no net flux (Hldne PRL 88) B V = Hldne z στ z Broken Time Reversl Symmetry e Quntized Hll Effect σ xy = sgn h 3. Intrinsic Spin Orbit Potentil V = SO στ z z z s Respects ALL symmetries Quntum Spin-Hll Effect

3 Quntum Spin Hll Effect in Grphene The intrinsic spin orbit interction leds to smll (~0mK-K) energy gp Simplest model: Hldne (conserves S z ) H H 0 H Hldne 0 = = * 0 H 0 H Hldne Bulk energy gp, but gpless edge sttes Spin Filtered or helicl edge sttes vcuum QSH Insultor J Edge bnd structure J E 0 π/ k Edge sttes form unique D electronic conductor HALF n ordinry D electron gs Protected by Time Reversl Symmetry

4 Time Reversl Symmetry : Anti Unitry time reversl opertor : Spin ½ : ψ ψ * Θ = ψ ψ * Θ ψ = Θ = i S e π y / [ H, Θ ] = 0 ψ * Krmers Theorem: for spin ½ ll eigensttes re t lest fold degenerte Θ χ = c χ Proof : for non degenerte eigenstte Consequences for edge sttes : Sttes t time reversl invrint moment k*=0 nd k*=π/ (=-π/) re degenerte. The crossing of the edge sttes is protected, even if spin conservtion is volted. Θ χ = c χ k* Θ = c D Dirc point Absence of bckscttering, even for strong disorder. No Anderson locliztion ψ in r=0 T invrint disorder t =

5 Time Reversl Invrint Ζ Topologicl Insultor D Bloch Hmiltonins subject to the T constrint ( k) ΘH Θ = H( k) with Θ = re clssified by Ζ topologicl invrint (ν = 0,) Understnd vi Bulk-Boundry correspondence : Edge Sttes for 0<k<π/ ν=0 : Conventionl Insultor ν= : Topologicl Insultor Krmers degenerte t time reversl invrint moment k* = k* + G k*=0 k*=π/ k*=0 k*=π/ Even number of bnds crossing Fermi energy Odd number of bnds crossing Fermi energy

6 Physicl Mening of Ζ Invrint Sensitivity to boundry conditions in multiply connected geometry ν=n IQHE on cylinder: Lughlin Argument Φ = φ 0 = h/e Q = N e Flux φ 0 Quntized chnge in Electron Number t the end. Quntum Spin Hll Effect on cylinder Φ = φ 0 / Flux φ 0 / Chnge in Electron Number Prity t the end, signling chnge in Krmers degenercy. Krmers Degenercy No Krmers Degenercy

7 Formul for the Ζ invrint Bloch wvefunctions : T - Reversl Mtrix : Antisymmetry property : u ( ) n k (N occupied bnds) w ( k) = u ( k) Θ u ( k ) U( N) mn m n Θ = w k = w T ( ) ( k) T T - invrint moment : k = Λ = Λ w( Λ ) = w ( Λ ) k y Λ 4 Λ 3 kx Λ Λ Bulk D Brillouin Zone Pfffin : det[ w( Λ )] ( Pf[ ( )]) = w Λ Fixed point prity : Z invrint : ν ( ) = δ ( Λ ) =± = Guge invrint, but requires continuous guge 0 z e.g. det = -z 0 Pf[ w( Λ )] δ ( Λ ) = =± det[ w( Λ )] Guge dependent product : δ( Λ ) δ( Λ ) b e time reversl polriztion nlogous to ( ) π A k dk 4 z

8 ν is esier to determine if there is extr symmetry:. S z conserved : independent spin Chern integers : n = n (due to time reversl) Quntum spin Hll Effect : J J E ν = n, mod. Inversion (P) Symmetry : determined by Prity of occupied D Bloch sttes P ψ ( Λ ) = ξ ( Λ ) ψ ( Λ ) n n n ξ ( Λ ) =± n In specil guge: δ( Λ ) = ξ ( Λ ) n n 4 υ ( ) = ξ ( Λ ) = n n Allows strightforwrd determintion of ν from bnd structure clcultions.

9 Quntum Spin Hll Effect in HgTe quntum wells Theory: Bernevig, Hughes nd Zhng, Science 06 d Hg x Cd -x Te Hg x Cd -x Te HgTe d < 6.3 nm : Norml bnd order E d > 6.3 nm : Inverted bnd order E Γ 6 ~ s k Γ 8 ~ p k Γ 8 ~ p Bnd inversion trnsition: Switch prity t k=0 Γ 6 ~ s E gp ~0 mev Conventionl Insultor ξ ( ) n Λ =+ Quntum spin Hll Insultor with topologicl edge sttes ξ ( ) n Λ =

10 Experiments on HgCdTe quntum wells Expt: Konig, Wiedmnn, Brune, Roth, Buhmnn, Molenkmp, Qi, Zhng Science 007 d< 6.3 nm norml bnd order conventionl insultor d> 6.3nm inverted bnd order QSH insultor G=e /h Lnduer Conductnce G=e /h V 0 I Mesured conductnce e /h independent of W for short smples (L<L in )

11 3D Topologicl Insultors There re 4 surfce Dirc Points due to Krmers degenercy k y Λ 4 Λ 3 k x Λ Λ D Dirc Point Surfce Brillouin Zone ν 0 = 0 : Wek Topologicl Insultor E E k=λ k=λ b OR k=λ k=λ b How do the Dirc points connect? Determined by 4 bulk Z topologicl invrints ν 0 ; (ν ν ν 3 ) k y Relted to lyered D QSHI ; (ν ν ν 3 ) ~ Miller indices Fermi surfce encloses even number of Dirc points k x ν 0 = : Strong Topologicl Insultor Fermi circle encloses odd number of Dirc points Topologicl Metl : /4 grphene Berry s phse π Robust to disorder: impossible to loclize k y k x E F

12 Topologicl Invrints in 3D. D 3D : Time reversl invrint plnes The D invrint ν ( ) = δ ( Λ ) 4 = δ ( Λ ) = Pf[ w( Λ )] det[ w( Λ )] π/ k z Λ Ech of the time reversl invrint plnes in the 3D Brillouin zone is chrcterized by D invrint. π/ π/ k y Wek Topologicl Invrints (vector): k x 4 δ k i =0 = plne νi ( ) = ( Λ ) ν π = ν ν ν 3 (,, ) mod reciprocl lttice vector indexes lttice plnes for lyered D QSHI G G ν Strong Topologicl Invrint (sclr) ν o ( ) = δ ( Λ ) 8 =

13 Topologicl Invrints in 3D. 4D 3D : Dimensionl Reduction Add n extr prmeter, k 4, tht smoothly connects the topologicl insultor to trivil insultor (while breking time reversl symmetry) H(k,k 4 ) is chrcterized by its second Chern number = [ F F] 8π 4 n dktr n depends on how H(k) is connected to H 0, but due to time reversl, the difference must be even. ν 0 = n mod ν Express in terms of Chern Simons 3-form : H ( k,) = H (Trivil insultor) k 4 H ( k,0) = H( k) Tr[ F F] = dq 3 0 H ( k, k ) =ΘH ( k, k ) Θ 4 4 = d kq ( ) mod 4π k Q3 ( k) = Tr[ A da+ A A A] 3 Guge invrint up to n even integer.

14 Pure Bismuth semimetl Bi -x Sb x Alloy :.09<x<.8 semiconductor E gp ~ 30 mev Pure Antimony semimetl L s E F L E F L E L E F L s E gp L s T L T L k 8 T L Inversion symmetry υ0 ( ) = ξ ( Γ ) i= n n i Predict Bi -x Sb x is strong topologicl insultor: ( ; ).

15 Bi -x Sb x Theory: Predict Bi -x Sb x is topologicl insultor by exploiting inversion symmetry of pure Bi, Sb (Fu,Kne PRL 07) Experiment: ARPES (Hsieh et l. Nture 08) Bi -x Sb x is Strong Topologicl Insultor ν 0 ;(ν,ν,ν 3 ) = ;() 5 surfce stte bnds cross E F between Γ nd M Bi Se 3 ARPES Experiment : Y. Xi et l., Nture Phys. (009). Bnd Theory : H. Zhng et. l, Nture Phys. (009). ν0;(ν,ν,ν3) = ;(000) : Bnd inversion t Γ Energy gp: ~.3 ev : A room temperture topologicl insultor Control E F on surfce by exposing to NO Simple surfce stte structure : Similr to grphene, except only single Dirc point E F

16 Unique Properties of Topologicl Insultor Surfce Sttes Hlf n ordinry DEG ; ¼ Grphene Spin polrized Fermi surfce Chrge Current ~ Spin Density Spin Current ~ Chrge Density E F π Berry s phse Robust to disorder Wek Antilocliztion Impossible to loclize, Klein prdox Exotic Sttes when broken symmetry leds to surfce energy gp: Quntum Hll stte, topologicl mgnetoelectric effect Fu, Kne 07; Qi, Hughes, Zhng 08, Essin, Moore, Vnderbilt 09 Superconducting stte Fu, Kne 08

17 Orbitl QHE : Surfce Quntum Hll Effect E=0 Lndu Level for Dirc fermions. Frctionl IQHE σ xy e = n + h e σ xy = h e σ xy = h B ν= chirl edge stte Anomlous QHE : Induce surfce gp by depositing mgnetic mteril H0 = ψ ( iv σ µ + Mσz ) ψ e e E F Mss due to Exchnge field e σ xy = sgn( M ) h + h M M TI h E gp = M Chirl Edge Stte t Domin Wll : M M

18 Topologicl Mgnetoelectric Effect Consider solid cylinder of TI with mgneticlly gpped surfce E M J e J = σ xye = n+ E = M h Mgnetoelectric Polrizbility M Qi, Hughes, Zhng 08; Essin, Moore, Vnderbilt 09 = α E e α = n + h topologicl θ term L = αeb α e = θ π h The frctionl prt of the mgnetoelectric polrizbility is determined by the bulk, nd independent of the surfce (provided there is gp) Anlogous to the electric polriztion, P, in D. TR sym. : θ = 0 or π mod π d= : Polriztion P d=3 : Mgnetoelectric polirizbility α L P E αeb e π formul e π BZ Tr[ A] Tr[ A da+ A A A] 4 h BZ 3 uncertinty quntum e e / h (extr end electron) (extr surfce quntum Hll lyer)