Topological Insulators in 2D and 3D
I. Introduction - Graphene - Time reversal symmetry and Kramers‟ theorem II. 2D quantum spin Hall insulator
- Z2 topological invariant - Edge states - HgCdTe quantum wells, expts III. Topological Insulators in 3D - Weak vs strong - Topological invariants from band structure IV. The surface of a topological insulator - Dirac Fermions - Absence of backscattering and localization - Quantum Hall effect - q term and topological magnetoelectric effect Energy gaps in graphene:
~ sublattice z HvF p + V 2 z ~ valley 2 2 2 sz ~ spin E( p ) vF p + 1. Staggered Sublattice Potential (e.g. BN)
z Broken Inversion Symmetry V CDW
2. Periodic Magnetic Field with no net flux (Haldane PRL ‟88)
+- +- +- + zzBroken Time Reversal Symmetry B +- +- + e2 +- +- +- + V Haldane Quantized Hall Effect sgn xy h 3. Intrinsic Spin Orbit Potential z z z Respects ALL symmetries VsSO Quantum Spin-Hall Effect Quantum Spin Hall Effect in Graphene
The intrinsic spin orbit interaction leads to a small (~10mK-1K) energy gap
Simplest model: H 0 H 0 J↑ J↓ |Haldane|2 H Haldane (conserves S ) * z 0 H 0 HHaldane E Bulk energy gap, but gapless edge states “Spin Filtered” or “helical” edge states Edge band structure vacuum ↓ ↑ ↓ ↑ QSH Insulator
0 p/a k Edge states form a unique 1D electronic conductor • HALF an ordinary 1D electron gas • Protected by Time Reversal Symmetry Time Reversal Symmetry : [H , ] 0
y Anti Unitary time reversal operator : eiSp / *
* 2 Spin ½ : -1 - *
Kramers‟ Theorem: for spin ½ all eigenstates are at least 2 fold degenerate
c Proof : for a non degenerate eigenstate 22 |c | - 1 22||c Consequences for edge states :
States at “time reversal invariant momenta” k*=0 and k*=p/a (=-p/a) are degenerate. 1D “Dirac point” The crossing of the edge states is protected, k* even if spin conservation is volated. Absence of backscattering, even for strong disorder. No Anderson localization in r=0 T invariant disorder |t|=1 Time Reversal Invariant 2 Topological Insulator
2D Bloch Hamiltonians subject to the T constraint HHkk -1 () - 2 with -1 are classified by a 2 topological invariant (n = 0,1)
Understand via Bulk-Boundary correspondence : Edge States for 0 n=0 : Conventional Insulator n=1 : Topological Insulator Kramers degenerate at time reversal invariant momenta k* = -k* + G k*=0 k*=p/a k*=0 k*=p/a Even number of bands Odd number of bands crossing Fermi energy crossing Fermi energy Physical Meaning of 2 Invariant Sensitivity to boundary conditions in a multiply connected geometry n=N IQHE on cylinder: Laughlin Argument F f0 = h/e Q = N e Flux f0 Quantized change in Electron Number at the end. Quantum Spin Hall Effect on cylinder F f0 / 2 Kramers Degeneracy Flux f0 /2 Change in Electron Number Parity No Kramers at the end, signaling change Degeneracy in Kramers degeneracy. Formula for the 2 invariant (N occupied bands) • Bloch wavefunctions : un ()k • T - Reversal Matrix : wmn()()()()k u m k u n - k U N • Antisymmetry property : 2 -1 ww (kk ) -T ( - ) T • T - invariant momenta : k a - a ww ( a ) - ( a ) 2 0 z ky 2 • Pfaffian : det[ww (aa )] Pf[ ( )] e.g. det z -0z 4 3 kx Pf[w (a )] • Fixed point parity : (a ) 1 1 2 det[w (a )] • Gauge dependent product : ()()ab e “time reversal polarization” analogous to A() k dk Bulk 2D Brillouin Zone 2p 4 • Z invariant : n 2 (- 1) ( a ) 1 a1 Gauge invariant, but requires continuous gauge n is easier to determine if there is extra symmetry: 1. Sz conserved : independent spin Chern integers : (due to time reversal) nn- J↑ J↓ Quantum spin Hall Effect : n n mod 2 E , 2. Inversion (P) Symmetry : determined by Parity of occupied 2D Bloch states P ( ) ( ) ( ) n a n a n a 4 ( ) 1 na (- 1) 2na ( ) an1 In a special gauge: ()()a n a n Allows a straightforward determination of n from band structure calculations. Quantum Spin Hall Effect in HgTe quantum wells Theory: Bernevig, Hughes and Zhang, Science „06 HgTe HgxCd1-xTe d HgxCd1-xTe d < 6.3 nm : Normal band order d > 6.3 nm : Inverted band order E E G6 ~ s k G8 ~ p k G8 ~ p G6 ~ s Egap~10 meV Band inversion transition: Switch parity at k=0 Quantum spin Hall Insulator Conventional Insulator with topological edge states ( ) + 1 2na 2na( ) - 1 Experiments on HgCdTe quantum wells Expt: Konig, Wiedmann, Brune, Roth, Buhmann, Molenkamp, Qi, Zhang Science 2007 d< 6.3 nm Landauer Conductance G=2e2/h normal band order conventional insulator ↑ ↓ V I 0 ↓ ↑ d> 6.3nm G=2e2/h inverted band order QSH insulator 2 Measured conductance 2e /h independent of W for short samples (L kx Related to layered 2D QSHI ; (n1n2n3) ~ Miller indices Fermi surface encloses even number of Dirac points n0 = 1 : Strong Topological Insulator ky EF Fermi circle encloses odd number of Dirac points Topological Metal : kx 1/4 graphene Berry‟s phase p Robust to disorder: impossible to localize Topological Invariants in 3D 1. 2D → 3D : Time reversal invariant planes The 2D invariant kz p/a a 4 Pf[w ( )] n () a (- 1) ( a ) a a1 det[w (a )] ky Each of the time reversal invariant planes in the 3D Brillouin zone is characterized by a 2D invariant. p/a p/a k Weak Topological Invariants (vector): x Gn 4 2p (- 1)ni ( ) G n,, n n a ki=0 n 1 2 3 a1 plane a “mod 2” reciprocal lattice vector indexes lattice planes for layered 2D QSHI Strong Topological Invariant (scalar) 8 no (- 1) ( a ) a1 Topological Invariants in 3D 2. 4D → 3D : Dimensional Reduction Add an extra parameter, k4, that smoothly connects the topological insulator to a trivial insulator (while breaking time reversal symmetry) H(k,k ) is characterized by its second Chern number 4 HH(k ,1) 0 (Trivial insulator) 1 4 k HH(kk ,0) ( ) n d kTr[]FF 4 2 8p -1 H(,)(,)kk- k44 H k n depends on how H(k) is connected to H0, but due to time reversal, the difference must be even. n 0 n mod 2 Express in terms of Chern Simons 3-form : Tr[]FFdQ3 1 2 n d3kQ (k ) mod 2 Qd()[]kTr A A + A A A 034p 2 3 3 Gauge invariant up to an even integer. Bi1-xSbx Pure Bismuth Alloy : .09 EF L EF Ls a La Egap La Ls Ls E EF T L T L T L k 8 0 Inversion symmetry (- 1) 2ni ( G ) in1 Predict Bi1-xSbx is a strong topological insulator: (1 ; 111). Theory: Predict Bi1-xSbx is a topological insulator by exploiting Bi1-xSbx inversion symmetry of pure Bi, Sb (Fu,Kane PRL‟07) Experiment: ARPES (Hsieh et al. Nature ‟08) • Bi1-x Sbx is a Strong Topological Insulator n0;(n1,n2,n3) = 1;(111) • 5 surface state bands cross EF between G and M ARPES Experiment : Y. Xia et al., Nature Phys. (2009). Bi2 Se3 Band Theory : H. Zhang et. al, Nature Phys. (2009). • n0;(n1,n2,n3) = 1;(000) : Band inversion at G • Energy gap: ~ .3 eV : A room temperature topological insulator EF • Simple surface state structure : Similar to graphene, except Control EF on surface by only a single Dirac point exposing to NO2 Unique Properties of Topological Insulator Surface States “Half” an ordinary 2DEG ; ¼ Graphene EF Spin polarized Fermi surface • Charge Current ~ Spin Density • Spin Current ~ Charge Density p Berry‟s phase • Robust to disorder • Weak Antilocalization • Impossible to localize, Klein paradox Exotic States when broken symmetry leads to surface energy gap: • Quantum Hall state, topological magnetoelectric effect Fu, Kane ‟07; Qi, Hughes, Zhang ‟08, Essin, Moore, Vanderbilt „09 • Superconducting state Fu, Kane „08 Surface Quantum Hall Effect Orbital QHE : E=0 Landau Level for Dirac fermions. “Fractional” IQHE 2 2 e xy 1 2h B 2 0 e 1 xy +n -1 h 2 -2 e2 n=1 chiral edge state xy 2h Anomalous QHE : Induce a surface gap by depositing magnetic material † 2 2 Hi0 ( - v - + M z ) e e + - Mass due to Exchange field 2h 2h M↑ M↓ e2 sgn( ) xy M TI EF 2h Egap = 2|M| Chiral Edge State at Domain Wall : M ↔ -M Topological Magnetoelectric Effect Qi, Hughes, Zhang ‟08; Essin, Moore, Vanderbilt „09 Consider a solid cylinder of TI with a magnetically gapped surface M 2 e 1 J xy E n + E M h 2 J Magnetoelectric Polarizability topological “q term” 2 L EB E e 1 ME +n e2 h 2 q 2p h TR sym. : q 02 or p mod p The fractional part of the magnetoelectric polarizability is determined by the bulk, and independent of the surface (provided there is a gap) Analogous to the electric polarization, P, in 1D. L formula “uncertainty quantum” e d=1 : Polarization P PE Tr[]A e (extra end electron) 2p BZ 2 d=3 : Magnetoelectric e 2 eh2 / (extra surface EB 2 Tr[]AAAAAd + poliarizability 43p h BZ quantum Hall layer)