Topological Defects inside a Topological Band Insulator

Transcription

1 Topological Defects inside a Topological Band Insulator Ashvin Vishwanath UC Berkeley Refs: Ran, Zhang A.V., Nature Physics 5, 289 (2009). Hosur, Ryu, AV arxiv:

2 Part 1: Outline A toy model of a topological insulator. Node pairing picture of surface state. Hosur, Ryu, AV arxiv: Part 2 Topological defects in a topological insulator.

3 What are topological band insulators? Topology characterizes the identity of objects up to deformation, e.g. genus of surfaces Figure courtesy C. Kane

4 What are topological band insulators? Similarly, band insulator can be classified up to the deformation of band structure. Modify H smoothly preserving gap. Nontrivial topology in momentum space. For universal properties - can use a representative model in each topological class. Convenient in the Dirac limit.

5 Topological insulator in D=3 strong Topological insulator in D=3 : Characteristic feature Surface state: single Dirac node. Topological Metal cannot be localized without breaking time reversal symmetry. Proof: If localized, must have xy=(n+1/2) e 2 /h. Breaks T. Fu, Kane & Mele (2006), Moore & Balents (2006), Roy (2006). The edge is the window into the bulk X.G. Wen

6 Strong Topological Insulator in D=3 Single Dirac node impossible in a D=2 band structure with time reversal symmetry. Proof : 1 Node = 2 component Dirac theory Graphene 2 Dirac nodes (each with spin) Strong topological insulator ~¼ of graphene BUT psuedospin-> spin. Berry phase of must evolve to zero at Zone boundary. ky Time reversal invariance => Berry phase 0 or. No continuous evolution possible. kx Surface state can evade theorem

7 Constructing a Topological Insulator Inversion symmetric topological insulators (r-r) Fu and Kane, Phys. Rev. B 76, (2007) At Time Reversal Invariant Momenta (8 in 3D Brillouin Zone), energy eigenstates can be labeled by inversion eigenvalues=± i = -i +G

8 Constructing a Topological Insulator Inversion symmetric topological insulators (r-r) Fu and Kane, Phys. Rev. B 76, (2007) Simplest Model: 2 orbitals, s & p [ z =±1] per site

9 Constructing a Topological Insulator Cubic Lattice Model ( s & p on each site) d [ ] M m (cos cos cos ) H = τ k k k z x y z TRIMs Phase Diagram: Phase Diagram: Spin-orbit only opens gaps!

10 Constructing a Topological Insulator Orbitals: Work out overlaps: Opens a full gap. How do we see single surface Dirac node? Other toy models: diamond lattice, with s-orbitals (Fu- Kane-Mele)

11 Constructing a Topological Insulator Special point Quasi 2D:

12 Constructing a Topological Insulator Coupling Between Layers Nodes are paired between layers-leads to bulk gap BUT Top and Bottom nodes unpaired. At this special point Dirac states only on one layer At this special point Dirac states only on one layer Like AKLT. Surface Hamiltonian: H=px

13 Topological Indices in 3D Given a band structure, one can calculate the following topological indices: Strong index ν 0 (0,1): (with Inversion) 1 ) (- ν = Π δ Physical Interpretation: ν 0 =1=>Odd number of surface Dirac nodes i i Weak or Lattice indices: M ν time rev. invariant wave-vector (with Inversion) M ν = 1 M 2 δ = 1 i ν = G ν Γ i M ν Physical Interpretation: For Strong Insulator: Location of Dirac node on Surface B.Zone (project on surface) M ν

14 Topological Indices in 3D For Weak insulator: ν 0 =0 M ν 0 Physical Interpretation: Stacking of 2D quantum spin Hall layers along G = 2M ν ν OR Pair of Dirac Nodes on surface at Γand M // surface ν -surface disorder could localize states but phase is more robust.

15 3D Strong T-I in Experiments Bi 1-x Sb x Theory: L. Fu & C. L. Kane (07) Surface modes ARPES confirms strong T-I. Experiment: D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava and M. Z. Hasan, Nature (08) ν 0 = 1; Mν (111) 5-crossings Newer materials: Bi 2 Se 3, Bi 2 Te 3 Eg. Y. Chen et al. (2009). ν ν = 1; M = 0 0 = ν =

16 Topological Defects in a Topological Band Insulator PART II Ref: Nature Physics 5, 289 (2009). arxiv: Ying Ran (Berkeley Boston College) Yi Zhang

17 Topology in Condensed Matter Physics? Topological Insulators Nontrivial topology in momentum space. Other well known application of topology to physics: Topological defects in a broken symmetry phase. Nontrivial topology in real space. Real space topology + momentum space topology => Interesting things happen

18 Broken Symmetry + Exotic Band Topology Superconducting order parameter: Vortex defects: ϕ dr = 2π m Ψ 0 e iϕ `Superconducting Cosmic Strings (Vortex of Higgs Field) J. Kiskis. (1977), E. Witten (1985).

19 Line Defects in a Crystal Dislocations: Defined by location R() and strength B (Burgers vector). B Burgers Vector, must stay constant along the length and is quantized B Burgers Vector, must stay constant along the length and is quantized to lattice vectors. (Like vorticity)

20 Volterra Process: Visualizing Dislocations Cut with an imaginary plane, that ends on the dislocation line R() Move all atoms on one side of the plane by the Burgers vector B Add/remove atoms if required. SCREW DISLOCATION: t // B EDGE DISLOCATION: t B B B

21 Always present Dislocations in Solids n d to 10 Control mechanical properties eg. Plastic Flow 12 m -2

22 Dislocation in a Topological Insulator 1D Helical Metal occurs in a dislocation {R(), B} embedded in a topological insulator { ν 0 = 0,1; M ν } iff: E B M = π ( mod π ) ν 2 B

23 Illustration Diamond Lattice Top. Ins. ν = 1; M ν = 0 π (1,1,1) 2 Introduce a screw dislocation: B=(1,1,0). Easily introduced in tight binding. Momentum dependent phase factor for cut bonds.

24 Results: Screw Dislocation in Diamond Lattice Top. Ins. Insert a pair of screw dislocations (36x36x18 periodic BC). Momentum along the dislocations is a good quantum number. Two propagating modes per dislocation. `Helical metal.

25 Absence of localization Stable to disorder: (TR symmetry no backscattering) An atomically thin one dimensional wire that does not localize. (non-magnetic) Similar to 2D quantum spin Hall edge states

26 Proof for Weak Top. Ins. Weak Top.Ins. Adiabatically connected to a stack of decoupled 2D Top.Ins., stacking along M ν Different proof for Strong TI Cut Surface only one of the helical mode pairs is shown. Glued Surface Dislocation must carry helical modes

27 Proof For General Top. Ins. 1 Screw dislocation if surface Dirac node is at momentum m Dirac B = π then (-1) phase acquired on crossing the dislocation. In the weak surface connection limit => Dirac equation that changes mass term sign. ( p 1 σ 1 + p 2 σ 2 ) µ z + m( x 2 ) x H = µ m( x m( x 2 2 > 0) = m < 0) = + m

28 Proof For General Top. Ins. 2 = ( p 1 σ 1 + p 2 σ 2 ) µ z + m ( x 2 ) x H µ m( x > 0) = m 2 ( x 2 2 m ( x ') dx ' m ( x < 0) = + m Pair of zero modes at p 1 =0. ψ ( x Propagating 1D helical modes for general p 1. 2 ) = ψ e Location of Surface Dirac Node controlled by M m Dirac 0 0 B = π Mν B = π (mod 2 π ) M ν

29 Experimental Signatures Resistivity: dislocation contribution could dominate over surface conduction. h 1 ρ = e l n 2 d 2 Ωm n d 10 l 1µmµ m 12 m 2 (in current samples impurity band conduction dominates)

30 Experimental Signatures Generate and grow dislocations with stress. Stress dependent conductivity (strongly direction dependent - connected to lattice indices ) M ν (111)

31 Experimental Signatures - STM Can determine atomic defect structure and enhanced Local Density of States (LDOS). Demonstration Edge dislocation on (-1,-1,1) Surface. B=(1,1,0) [with modes] OR B=(-1,0,1). Integrated LDOS in [-0.1,0.1]. B M ν = π B Mν = 0

32 Interaction Effects Luttinger Liquid Physics With interactions dislocation mode is a Luttinger liquid. e 2 If K = >> 1 ε v significant deviations from free F electron physics (in 1D). Eg. carbon nano-tubes Here v F typically smaller controlled by spin orbit interactions. However, dielectric constants may be larger. A Liu-Allen type tight binding model for (111) screw dislocation gives: v F =1.3x10 5 m/sec (nanotube v F =8x10 5 m/sec). However, may be large too.

33 Future Directions Topological quantum computing? Surface of TI Majorana (real) fermion (Kitaev, ) Dislocation mode or 2D edge Cold atom realizations? (Dislocations in optical Cold atom realizations? (Dislocations in optical lattices)

34 Conclusions 3D topological Band Insulator has protected helical mode in those dislocations that satisfy B M = π ( mod π ) ν 2 M (111) M = 0 Should occur in Bi 1-x Sb x but not in Bi 2 Se 3, Bi 2 Te 3 ν M ν Weak topological insulator stable to moderate disorder. More stable than surface modes