Local currents in a two-dimensional topological insulator

Transcription

1 Local currents in a two-dimensional topological insulator Xiaoqian Dang, J. D. Burton and Evgeny Y. Tsymbal Department of Physics and Astronomy Nebraska Center for Materials and Nanoscience University of Nebraska, Lincoln, Nebraska,USA

2 Two-dimensional topological insulator Predicted (Bernevig, Hughes, and Zhang, 2006) and observed (König et al., 2007) in CdTe/HgTe/CdTe quantum well structures System exhibiting a quantum spin-hall effect 2D bulk insulator with topologically protected edge states Spin is locked to the wave vector of the electron Conductance of the edge state is insensitive to disorder which does not break time reversal symmetry 2

3 Bernevig-Hughes-Zhang (BHZ) model Inverted band structure BHZ, Science 314,1757(2006) Fu and Kane, PRB 76, (2007) Square lattice with four basis states α on each site i: s, s, p + ip, p ip, Hamiltonian: x y x y H = c c t c c ε α iασ iασ aσ, αβ i+ aασ iβσ i, σα, i, σα, ε ε < 4( t + t ) s p ss pp - band insulator - topological insulator t t t e iσθa ss sp aσ = iσθ a tspe t pp θ a - angle of bond a with x axis 3

4 Tight-binding Green s function technique Semi-infinite square lattice Dyson equation: 0 G = G Layer-dependent spectral density: 2 [ ] A( Ek, )= Im TrG ( Ek, ) n x nn x 3 4 Layer-dependent DOS: y ρn( E)= Im TrGnn ( E ) π [ ] x 4

5 Layer-dependent spectral density k x (π/a) Energy dependent oscillatory decay of the edge state into the bulk 5

6 Oscillatory decay of DOS E = LDOS E = E = y (a) Oscillatory decay of the edge state into the bulk 6

7 Complex band structure For a given energy, there are three types of solutions: iky y Bloch state : ϕ e - oscillation Evanescent state: ϕ e κ y - decay Complex solution: ϕ e iky y y e κ - oscillation and decay DOS is superposition of two waves with opposite Re(k y ): ( y φ) 2κ y 2 ρ( y) e cos ky+ 7

8 Oscillatory decay of DOS E = LDOS E = 0.25 E = y (a) Layer-dependent DOS using parameters extracted from the complex band structure ( y φ) 2κ y 2 ρ( y) e cos ky+ 8

9 Electronic transport: model and methods Finite width strip within the BHZ tight-binding model Green s function formalism GE ( ) = [ E H Σ Σ ] Landauer-Büttiker approach L R 1 9

10 Local conductance Local current: where J = e ij Hij i j H ji j i ih ALR, ( E) = ig( E) ΣLR, ( E) ΣLR, ( E) G ( E) Within linear response Local conductance (per spin): Total conductance (per spin): e 2 T Tr ( L L) G ( R R) G Γ= = Σ Σ Σ Σ h 10

11 Local conductance for an isolated edge Local conductance LDOS E F = 0.35 Oscillation in the local conductance Correlation between local conductance and local density of states Explained by the complex band structure T = 1, as expected 11

12 Effect of impurity i Perturbation due to impurity: Real space Green s function: V imp ε s 0 = 0 ε p ( 1 ) 1 G = G + G V GV G mn mn mi ii ii ii in 12

13 Effect of impurity Impurity E i = 0 Impurity Ei E F = 0.35 Strong effect on local current distribution No back-scattering: T = 1, as expected 13

14 Effect of impurity Impurity E i = 1.25 Impurity E = 0.45 i E F = 0.35 Intricate current distributions Current vortex due to impurity 14

15 Effect of vacancy Vacancy Hollow Current vortex due to internal edge Chirality is determined by propagating mode spin state Counterclockwise current 15

16 Narrow width strip L y x For a given spin, two propagating states on the two edges Coupling between the edge states Backscattering due to impurities 16

17 Oscillatory band gap Energy gap due to coupling between the edge states Oscillatory behavior as a function of strip width 17

18 Oscillatory band gap y L x Edge states: Energy gap: κ L α Eg ψ1ψ2 = e cos( kl) + sin( kl), L 18

19 Oscillatory band gap Excellent agreement 19

20 Complex band structure of Bi 2 Se 3 In collaboration with J. Velev (UPR) reals bands complex bands bulk gap Real component of the wave vector in the bulk gap region 20

21 Oscillatory band gap in Bi 2 Se Slab calculation CBS fit Band gap (ev) Thickness (nm) Nearly perfect agreement with the complex band structure parameters 21

22 Friedel oscillations: density of states E F, E imp 2 No Friedel oscillations for an isolated edge For a finite strip, LDOS oscillates away from the impurity with no decay Oscillation period depends on the Fermi energy 22

23 Friedel oscillations: model Model Hamiltonian: Green s function: ( ) H k vk = vk Dyson equation within Born approximation: where ( ) = λδ ( x) V x is a local perturbation due to impurity Resulting perturbation in LDOS: ( k x) sin 2 ρ ( x) = [ G G ] π = λ F ImTr π v kf

24 Friedel oscillations: local conductance Impurity Spin up Spin down Total Periodically repeated vortices in spin-resolved current distribution No Friedel oscillations in net local conductance 24

25 Resonant scattering Resonant channel: full back-scattering due to bound state created by impurity 25

26 Resonant scattering: antiresonance Energy dependent transmission 2 T (E) E Full suppression of net current Destructive interference of the incoming and reflected waves Antiresonances 26

27 Antiresonance E i = 0 2 Two bound states (electron-like and hole-like) Antiresonances in transmission at the bound state energies Decreasing antiresonance width with increasing strip width L 27

28 Antiresonance: model Scattering problem: φ ( ) = ikx k x e λ Impurity E, χ i ( x) G G ( ) ( ) ik x x i e θ x x 0 v 0 e θ x x = ( ) TI ik x x imp = χ ( x) χ ( x ) E E + iη i ( ) Transmission: T( E) = ( E Ei ) ( ) E E + γ i Antiresonance in transmission characterized by width γ = 2 λ χ v ( 0) 2 28

29 Antiresonance: local conductance Spin up contribution Perfect back scattering due to antiresonance No net local currents Total transmission is zero 29

30 Effect of magnetic impurity z x θ m φ 2D TI y Magnetic impurity Hamiltonian: Hex = mˆ σ 2 mˆ = (sinθcos φ, sinθsin φ, cos θ) Breaks time reversal symmetry Expected back-scattering due to mixing of spin channels Effect depends on the impurity magnetic moment angle Effect on transport spin polarization: SP = G G G + G 30

31 Magnetic impurity: local conductance Spin up Spin down Total SP = 0.33 Backscattering seen in spin-down channel 31

32 Magnetic impurity: local conductance Spin up Spin down Total SP = 0.93 Nearly perfect backscattering 32

33 Magnetic impurity: angular dependence θ = θ 0 Antiresonance in transmission due to magnetic impurity at the critical angle of the magnetic moment 33

34 Magnetic impurity: local conductance θ = θ 0 = 54 θ = 90 SP = 0 SP = 0.98 SP = 0.33 Current vertex at the antiresonance conditions 34

35 Summary The BHZ model for a 2D topological insulator implemented within the tight-binding Green s function technique reveals: Oscillatory decay of the local conductance away from the TI edge Intricate current distributions and formation of current vertices of different chirality around impurities Oscillatory behavior of the edge-state energy gap as a function of 2D TI width Impurity-driven Friedel oscillations in electron density and spindependent local conductance for sufficiently narrow TI strips Resonant back scattering and antiresonances in transmission for finite-size impurity system Back scattering produced by magnetic impurity and resonant-type transmission as a function of magnetic moment angle 35