Electronic standing waves on the surface of the topological insulator Bi 2 Te 3

Transcription

1 Electronic standing waves on the surface of the topological insulator Bi 2 Te 3 Phys. Rev. B. 86, (212) P. Rakyta, A. Pályi, J. Cserti Eötvös Loránd University, Hungary Department of Physics of Complex Systems 1

2 Experimental background of electronic standing waves on a crystal surface a) Bi 2 Te 3 cleavage step (line defect) on the surface of the crystal: topography scans by current STM technique with constant b) studying electronic standing waves by measuring the differential I U ρ(x) conductivity. (ρ(x) is the local density of states) A. Varykhalov et al., Phys. Rev. Lett. 11, (28). Electronic standing waves on the surface of the topological insulator Bi 2 Te 3 slide 2.

3 Three-dimensional topological insulator: Bi 2 Te 3 Effective Hamiltonian of the surface states: (L. Fu, Phys. Rev. Lett. 13, (29).): Ĥ TI (k) = k2 2m +v k(k x σ y k y σ x )+ λ 2 (k3 + +k 3 )σ z k y [A 1 ] E TI ± (k) = k2 2m ± v 2 k k2 +λ 2 k 2 x(k 2 x 3k 2 y) ARPES data:.1.1 k x [A 1 ] A. Varykhalov et al., Phys. Rev. Lett. 11, (28). Electronic standing waves on the surface of the topological insulator Bi 2 Te 3 slide 3.

4 Asymptotic theory of electronic standing waves k sc δρ 1 v (B) k sc,2 k sc,1 y,ky = k x,k x = q line defect Ψ = Ψ in +rψ r Ψ in = χ in (q)e i(qx+k iny) Ψ r = χ r (q)e i(qx+k ry) k sc = k in k r dq Re( r(k) χ in (q) χ r (q) e ) ik scy v (B) (q) ; v (q) = E(k,q) q (q) δqα v (B),, r(k,q) r δq β, χ in (q) χ r (q) Ω δq γ, k sc + 1 δq η Stationary phase method around extremal pointsk sc (q). (R. R. Biswas and A. V. Balatsky, Phys. Rev. B 83, (211).) E Electronic standing waves on the surface of the topological insulator Bi 2 Te 3 slide 4.

5 Asymptotic theory of electronic standing waves δρ extremal points abs(r Ω ) v, sin( y +ϕ) y α+β+γ+1 η k sc k sc,2 k sc,1 y,ky = k x,k x = q line defect 1 v (q) δqα v (B),, r(k,q) r δq β, χ in (q) χ r (q) Ω δq γ, k sc + 1 δq η Isotropic dispersion:, 2DEG electrons:α =,β =,γ =,η = 2. δρ sin(k sc y +ϕ) y 1 2 Electronic standing waves on the surface of the topological insulator Bi 2 Te 3 slide 5.

6 Asymptotic theory of electronic standing waves δρ extremal points abs(r Ω ) v, sin( y +ϕ) y α+β+γ+1 η k sc k sc,2 k sc,1 y,ky = k x,k x = q line defect 1 v (q) δqα v (B),, r(k,q) r δq β, χ in (q) χ r (q) Ω δq γ, k sc + 1 δq η Isotropic dispersion, helical electrons:α =,β = 1,γ = 1,η = 2. δρ sin(k sc y +ϕ) y 3 2 Electronic standing waves on the surface of the topological insulator Bi 2 Te 3 slide 6.

7 Asymptotic theory of electronic standing waves δρ extremal points abs(r Ω ) v, sin( y +ϕ) y α+β+γ+1 η k sc k sc,2 k sc,1 y,ky = k x,k x = q line defect 1 v (q) δqα v (B),, r(k,q) r δq β, χ be (q) χ r (q) Ω δq γ, k sc + 1 δq η Hexagonally warped dispersion, helical electrons:α 2 =,β 2 =,γ 2 =,η 2 = 2. δρ A 1 sin(k sz,1 y +ϕ 1 ) y 3 2 +A2 sin(k sz,2 y +ϕ 2 ) y 1 2 A 1 A 2 Electronic standing waves on the surface of the topological insulator Bi 2 Te 3 slide 7.

8 Electronic standing waves on the surface of Bi 2 Te 3 crystal δρ sin(2k fit y +ϕ) y 1 Difference between experimental data and the asymptotic theory Energy contour fore = 33 mev..1 k nest 2k fit q [A 1 ] 2k Γ M A. Varykhalov et al., Phys. Rev. Lett. 11, (28). exact solution of the scattering problem, closer to the line defect. k nest : the reflected and transmitted states from the other side cancels each other. Electronic standing waves on the surface of the topological insulator Bi 2 Te 3 slide 8.

9 Exact solution of the scattering problem characteristics of the standing waves, that are independent of the exact shape of the potential. (decay, oscillation wavenumber) We model the cleavage step with a step potential:v(y) = V θ(y) (which is translational invariant in x) Determine the 6 eigenstates of H TI with given energy E and parallel to line defect momentum componentq x. (E TI ± (k r,q) = E,r = 1...6) Matching eigenstates at the potential step ψ (L/R) E,q stand for the incident plane-wave coming from the left/right.) scattering wave function. (L/R The electronic standing waves correspond to the oscillations in the local DOS: ρ(e,y) = 1 (2π) 2 d=l,r Γ (d) E dκ ψ(d) k,q (y) 2 v(k, q) Γ (L/R) E stand for the energy contour segments on the left/right side of energy E. Electronic standing waves on the surface of the topological insulator Bi 2 Te 3 slide 9.

10 Pre-asymtotic contribution to the oscillations (E = 33 mev) P. Rakyta et al., Phys. Rev. B. 86, (212). (a) δρ [1 4 ρ ] (b) FFT [A 1 mev 1 ] y [A] 5 1 k nest 2k fit f 1 f 2 (d) (c) q [A 1 ].1.1 k nest 2k fit 2k Γ M k.24 fit 2k k Γ M.18 nest E [mev] f 1 (x,n) = A 1 sin(2k fit x+ϕ 1 )x 3/2 f 2 (x,l) = A 2 sin(2k fit x+ϕ 2 )e x/l Electronic standing waves on the surface of the topological insulator Bi 2 Te 3 slide 1.

11 Pre-asymtotic contribution to the oscillations (a) (b) 1 r χ + χ (c) 1/v (d) Szorzat k fit δρ(e,y) dk q [A 1 ].1.1 k nest 2k fit 2k Γ M The characteristick fit wavenumber is related to the non-monotonic behaviour of the prallel to line defect component of the group velocity. ( ) r k,q χ k,q χ k,q e i2ky +c.c. ; F 2k Re v (E,k) ( rk,q χ k,q χ k,q v (E,k) ) Electronic standing waves on the surface of the topological insulator Bi 2 Te 3 slide 11.

12 Exponential decay of the oscilaltions Direct data f 2 (x,l) = A 2 sin(2k fit x+ϕ)e x/l a) b) FT [A 1 mev 1 ] k nest 2k fit c) d) FT [A 1 mev 1 ].1.2 2k fit.5 1 FT [A 1 mev 1 ] FT [A 1 mev 1 ] k fit k fit f 1 (x) = A 2 sin(2k fit x+ϕ)x 3/2 f 1 (x) = A 2 sin(2k fit x+ϕ)x 1/2 Electronic standing waves on the surface of the topological insulator Bi 2 Te 3 slide 12.

13 Conclusions Pre-asymptotic standing waves: The characteristic wavenumber in the oscillations (closer to the line defect) is related to the non-monotonic behavior of the parallel to line defect component of the group velocity. The decay of the pre-asymptotic contribution to the oscillations is exponential when a dominant nesting vector is missing. Asymptotic standing waves: The characteristic wavenumbers in the oscillations are given by nesting vectors on the constant energy contour. The decay of the asymptotic oscillations is power-like (x n ) for the dominant nesting vectors. Electronic standing waves on the surface of the topological insulator Bi 2 Te 3 slide 13.

14 Three-dimensional topological insulator: Bi 2 Te 3 The exact shape of the constant energy contours is important to describe standing waves around line defects. (v k = v +αk 2 ) E TI ± (k) = k2 2m ± v 2 k k2 +λ 2 k 2 x(k 2 x 3k 2 y) 2.15 ARPES data: k y [A 1 ] k x [A 1 ] A. Varykhalov et al., Phys. Rev. Lett. 11, (28). Electronic standing waves on the surface of the topological insulator Bi 2 Te 3 slide 14.

15 Bi 2 Te 3 : band-structure parameters from ARPES data (+,., DOS) BCB: bulk conduction band, BVB: bulk valance band a) b) 3 BCB 2 BVB 1 E [mev] c) d) E [mev] BCB BVB theory ARPES solid E(ΓM) dashed E(ΓK) ρ [mev 1 µm 2 ] BVB E [mev] BCB ρ [mev 1 µm 2 ] v = 2.55 evå,λ = 25 evå 3, 1 2m =,α = 6 BVB E [mev] BCB Density of states (only theory shown) v = 3.5eVÅ,λ = 15eVÅ 3, α = 21Å 2,γ = 19.5eVÅ 2 Electronic standing waves on the surface of the topological insulator Bi 2 Te 3 slide 15.

16 Fourier analysis of oscillating functions 1.5 x min x max 2x max f 2 (x) x The Fourier transform of the symmetrized function are real. Electronic standing waves on the surface of the topological insulator Bi 2 Te 3 slide 16.

17 Electronic standing waves around line defect ΓK (E = 33 mev) a) 1 b).2 δρ [1 4 ρ ] FT Γ K k nest y [A] k q [A 1 ].1.1 Γ K k nest 2k Γ K f 1 (x) = A 2 sin(2k fit x+ϕ)x 1/2 Power-like decay recovered with nesting vectork ΓK nest Electronic standing waves on the surface of the topological insulator Bi 2 Te 3 slide 17.