Electronic transport in topological insulators

Transcription

1 Electronic transport in topological insulators Reinhold Egger Institut für Theoretische Physik, Düsseldorf Alex Zazunov, Alfredo Levy Yeyati Trieste, November 011

2 To the memory of my dear friend Please don t ignore these signs

3 Overview Brief intro to Topological Insulators (TI) M.Z. Hasan & C.L. Kane, Rev. Mod. Phys. 8, 3045 (010) Helical Luttinger liquid in TI nanowires RE, A. Zazunov & A. Levy Yeyati, PRL 105, (010) Coulomb blockade of Majorana fermion induced transport A. Zazunov, A. Levy Yeyati & RE, PRB 84, (011)

4 Topological Insulator Recently discovered new state of matter Bulk band gap like ordinary insulator, but band inversion due to strong spin orbit coupling Time reversal symmetry: topological protection of gapless modes at surface D quantum spin Hall TI: helical edge states Close cousin of integer quantum Hall state Initially predicted for graphene Kane & Mele, PRL 005 observed in HgTe quantum wells König et al., Science 007 Here: 3D strong topological insulator Predicted independently by three groups Moore & Balents, PRB 007; Fu, Kane & Mele, PRL 007; Roy, PRB 009 First observed via ARPES, present reference material is Bi Se 3 Hasan & Kane, RMP 010

5 Helical edge state of D TI up and down spins propagate in different direction: a peculiar new spin-filtered 1D liquid Hasan & Kane, RMP 010

6 D massless Dirac fermions as surface states of 3D topological insulators Map from Brillouin zone to Hilbert space has nontrivial topology Avoid fermion doubling theorem Spin momentum locking Mathematically characterized by Z invariant Observed by ARPES in Bi Se 3 Hasan & Kane, RMP 010

7 Bi Se 3 nanowires: fabrication Rhombohedral phase, space group 5 D 3d Au catalysed VLS growth Peng et al. (Stanford), Nature Materials 010

8 Surface state in transport experiments Clear evidence for the surface state: Aharonov-Bohm oscillations of conductance Upper bound for surface state width: 6 nm Peng et al., Nat. Mat. 010

9 TI nanowires: theory Consider strong TI material (e.g. Bi Se 3 ): cylindrical nanowire of radius R Bandstructure from kp approach Zhang et al., Nature Phys. 009 anisotropy axis z: conserved momentum k Rotational xy symmetry: total angular momentum conservation, J i / half-integer j z Egger, Zazunov & Levy Yeyati, PRL 010 Wave function vanishes at boundary r=r: expand in orthonormal set of radial functions J j1/ n, j1/ u r z nj RJ j1/ n, j1/ r R

10 Bandstructure of TI nanowire R=15 nm Density Spin density s ϕ Spin density s z Bulk gap 0.3 ev Conduction and valence bands: indexed by j surface states gapped

11 Properties of surface states Kramers degeneracy: Inversion symmetry: k E jk k E k Qualitatively same results from tight-binding calculations Spin-momentum locking: Local spin is always oriented tangential to surface and perpendicular to momentum j s keˆ z eˆ eˆ r R Large k : R-mover counterclockwise spin-polarized L-mover clockwise spin-polarized E E j j j

12 Surface gap TB kp Dirac fermion description yields s v R Magnetic flux can close the surface gap!

13 Analytical description Surface states well described by D massless Dirac fermions on cylinder surface Zhang & Vishwanath PRL 010 spin-momentum locking, spin direction tangential to surface Dirac Hamiltonian in curved space: 1 iv Covariant derivative: H F D D Spin connection defined via relation involving Christoffel symbols j j j, 1 D i i i k i i i, z ik ê i

14 Surface Dirac fermion theory Take into account anisotropy & dim.less flux yields surface Hamiltonian H surf e v k v R Now perform unitary transformation i z / i e i z / y z Then: antiperiodic boundary conditions around circumference, i.e., half-integer j Dispersion relation: waveguide modes k v k j E, j v R U e i z /

15 Surface state properties Spin-momentum locking changes dispersion compared to nanotubes! Scattering of Kramers pair j, k j, forbidden (zero overlap of eigenstates!) Gap vanishes for half-integer flux and special band j spin-conserving single-particle back-scattering forbidden, protected against weak disorder chemical potential near zero: precisely one gapless helical mode, effectively time reversal invariant from now on we focus on this case! k

16 Bosonization Chiral fermions with opposite spin surf 1 ik F z ikf z z, e Rz e Lz Bosonize: 4 ( R / L) 1 ie i 1 1 i ie i z e surface layer width Spin momentum locking: Particle density: s J 1 z 1 z z dual boson fields

17 Interacting helical Luttinger liquid Include Coulomb interaction Helical liquid protected against weak disorder Long-ranged Coulomb tail dominates Hard-core interaction subdominant (marginal) Single-mode helical Luttinger liquid H u dz K 1 K z Interaction parameter K smaller than in HgTe: K e 1 v z L 0.51 ln R Almost same expression in SWNT Egger & Gogolin, PRL / v u K

18 Physical properties of helical LL Anisotropic spin correlations / RKKY interaction: Slow z -K power law decay with k F oscillations in z direction Formally: anisotropic SDW phase Linear conductance G(T) very sensitive to presence of magnetic impurities Maciejko et al., PRL 009 Proximity induced superconductivity: Majorana fermion states are induced at ends of TI nanowire Cook & Franz, arxiv:

19 Majorana fermions: end states of topological superconductors BdG Hamiltonian has particle-hole symmetry Quasiparticle operator for zero mode (E=0) state bound to the end of the TSC must be Majorana fermion Then: Quasiparticle is its own antiparticle Never clearly observed so far in any system Hasan & Kane, RMP 010

20 Parity issues Fermion parity (-) n of closed dot is conserved Parity determines dot occupation i L R n 1 ( ) n1 Imposes constraint on Hilbert space Equivalent but technically simpler: number operator N for Cooper pairs, with conjugate phase ϕ Instantaneous charge state: (N,n d ) d Fu, PRL 010 Parity only determined by n d, no constraint n d d d 0,1

21 Coulomb blockade of Majorana fermion induced transport Zazunov, Levy Yeyati & Egger, PRB 84, (011) Low-energy transport through topological superconductor wire with decoupled Majorana end states & source/drain metallic contacts Assume all energy scales small against proximityinduced gap, i.e., no quasiparticles on grain Zero mode Majorana fermions: L / R L/ R Build auxiliary complex d fermion d 1 i, j ij i Assume floating island (not grounded): current must be conserved L R

22 Schematic setup

23 Model Wire has electrostatic charging energy H Offset gate charge is continuous H H H H Full Hamiltonian: c E C N n n d gate leads Tunnel Hamiltonian: source contact couples to & drain only to H t k R Fermi liquid leads: Wide band approximation & Fermi functions with chemical potential μ L/R Hybridizations c t L i i c d e d i c d e d h.c. L Lk L/ R ~ 0 L/ R R Rk

24 Noninteracting solution: resonant Andreev reflection Solution for E c =0: (0) jl / R Fermi functions: lead distribution Majorana spectral function Linear T=0: resonant Andreev reflection I j d F F A tanh Bolech & Demler, PRL 007 Law, Lee & Ng, PRL 009 j j T ret j Im A j G j G jl / R ei (0) j j e h j

25 Strong Coulomb blockade: Electron teleportation Resonant case of half-integer n gate Charging energy allows only two configurations For n gate =1/,5/, : fixed Cooper pair number N, states n d =0,1 degenerate For n gate =3/,7/, : (N-1,1) and (N,0) degenerate In both cases, model can be mapped to spinless resonant tunneling Hamiltonian Linear conductance (T=0): G R L / e / h Interpretation: Electron teleportation Fu, PRL 010

26 Crossover from resonant Andreev reflection to electron teleportation Arbitrary charging energy: Keldysh approach with (Majorana-generalized) AES action General expression: Zazunov, Levy Yeyati & Egger, PRB 011 Here study weak Coulomb blockade regime: interaction corrections to noninteracting result Full crossover from resonant Andreev reflection (E c =0) to teleportation (large E c ) can be captured as well (work in progress)

27 Weak Coulomb blockade regime Phase fluctuations are small & allow for semiclassical expansion Keldysh-AES functional is then equivalent to Langevin equation for classical phase Inverse RC time of effective circuit: Dimensionless damping strength (higher energy scales: damping retardation!) Gaussian random force c c t E C j j j t t 4ECKt t j

28 How to obtain the current K has lengthy expression in equilibrium satisfies fluctuation dissipation theorem Current: 1 J ( t t) solution c t Some algebra: I j j d G t ret j for given noise realization Interactions always decrease current! K eq coth eq T J sin F e t 0 c J c 1 0 j d K 1 cos 1 /

29 Nonlinear conductance Symmetric system & T=0 Observable: I( V ) g( V ) e V / h Noninteracting case (resonant Andreev reflection): g ev V tan 1 1 ( 0) Analytical result for E C : universal power law suppression of linear conductance with increasing charging energy 1/8 g 0 ev L L 0.96 EC R R ev / /

30 Linear conductance: numerics interaction induced suppression numerics analytical result

31 Nonlinear conductance

32 Conclusions Topological insulators provide interesting new playground for transport through interacting nanostructures TI nanowire as realization of single-mode helical Luttinger liquid with strong correlations Egger, Zazunov & Levy Yeyati, PRL 105, (010) Coulomb blockade in Majorana fermion induced transport: from resonant Andreev reflection to electron teleportation Zazunov, Levy Yeyati & Egger, PRB 84, (011)