Energy Spectrum and Broken spin-surface locking in Topological Insulator quantum dots

Transcription

1 Energy Spectrum and Broken spin-surface locking in Topological Insulator quantum dots A. Kundu 1 1 Heinrich-Heine Universität Düsseldorf, Germany The Capri Spring School on Transport in Nanostructures 2011 Arijit Kundu (Heinrich-Heine University) Topological Insulator quantum dot 1 / 10

2 Introduction Outline Outline Introduction Quantum dot spectrum and spin structure (surface) Dirac Fermion theory on the surface Tight binding approach Summary A. Kundu, A. Zazunov, A. Levy Yeyati, T. Martin & R. Egger, Phys Rev B 83, (2011) Arijit Kundu (Heinrich-Heine University) Topological Insulator quantum dot 2 / 10

3 Introduction Motivation Introduction Strong spin-orbit coupling and band inversion conspire to produce a time-reversal invariant topological insulator phase in meterials like Bi 2 Se 3 which has a bulk gap b 0.3 ev. Xia et all, Nature Physics (2009) 1 The measured spin texture of the surface is consistent with predictions obtained from 2D massless Dirac fermions. 2 The surface state is stable under weak TR invariant disorders. 3 The spin-momentum is locked on the surface. D. Hsieh et al, Nature 2009 Arijit Kundu (Heinrich-Heine University) Topological Insulator quantum dot 3 / 10

4 Introduction Motivation Introduction Strong spin-orbit coupling and band inversion conspire to produce a time-reversal invariant topological insulator phase in meterials like Bi 2 Se 3 which has a bulk gap b 0.3 ev. Xia et all, Nature Physics (2009) 1 The measured spin texture of the surface is consistent with predictions obtained from 2D massless Dirac fermions. 2 The surface state is stable under weak TR invariant disorders. 3 The spin-momentum is locked on the surface. D. Hsieh et al, Nature 2009 Arijit Kundu (Heinrich-Heine University) Topological Insulator quantum dot 3 / 10

5 Zhang et al, Nature Phys 5, 438 (2009) and Liu et al, PRB 82, (2010) H b = ɛ 0 σ 0 τ 0 + M k σ 0 τ z + [A 0 (k x σ x + k y σ y ) + B 0 kσ z ]τ x (1) This also defines the fermi velocities v 1 = B 0 and v 2 = A 0 in x y plane and along z axis. σ and τ acts on spin and orbital space to give rise spin-parity structure for the eigen state of the Hamiltonin. For cylindrically symmetric geometry: we have the conserved angular momenta: J = i φ + σ z /2 (2) Arijit Kundu (Heinrich-Heine University) Topological Insulator quantum dot 4 / 10

6 Zhang et al, Nature Phys 5, 438 (2009) and Liu et al, PRB 82, (2010) H b = ɛ 0 σ 0 τ 0 + M k σ 0 τ z + [A 0 (k x σ x + k y σ y ) + B 0 kσ z ]τ x (1) This also defines the fermi velocities v 1 = B 0 and v 2 = A 0 in x y plane and along z axis. σ and τ acts on spin and orbital space to give rise spin-parity structure for the eigen state of the Hamiltonin. For cylindrically symmetric geometry: we have the conserved angular momenta: J = i φ + σ z /2 (2) σ r,φ,z = ê r,φ,z. σ Arijit Kundu (Heinrich-Heine University) Topological Insulator quantum dot 4 / 10

7 Dispersion relation of surface states: E j,± (k) = ± (v 1 k) 2 + (jv 2 /R) 2 And a gap in the spectrum which is 1/R Spin is locked with the surface σ r = 0 Egger et al, PRL (2010) Finite semiconductor nanowire/ carbon nanotube experiments are successfull Nygard et all, Nature (2000), Postma et all, Science (2001) Nontrivial spin connection in spherical TI Parente et all, arxiv: New feature may arise due to sharp (non-differential) egdes, for eg. in cylindrical geometry Arijit Kundu (Heinrich-Heine University) Topological Insulator quantum dot 5 / 10

8 Dispersion relation of surface states: E j,± (k) = ± (v 1 k) 2 + (jv 2 /R) 2 And a gap in the spectrum which is 1/R Spin is locked with the surface σ r = 0 Egger et al, PRL (2010) Finite semiconductor nanowire/ carbon nanotube experiments are successfull Nygard et all, Nature (2000), Postma et all, Science (2001) Nontrivial spin connection in spherical TI Parente et all, arxiv: New feature may arise due to sharp (non-differential) egdes, for eg. in cylindrical geometry Arijit Kundu (Heinrich-Heine University) Topological Insulator quantum dot 5 / 10

9 For finite length cylinder : m = j σ/2 ψ j,ν,n,σ,τ = 2/V sin[nπ(z/l 1/2)]e imφ J m(γ m/r ) J m+1 (γ mν ) η τ χ σ (3) Kramers degeneracy j j k n nπ/l additional n = 0 states emerges one for valence band and one for conduction band for j > 1/2, a pair of subgap states inside the gap s. σ φ = 0 and σ r 0 Arijit Kundu (Heinrich-Heine University) Topological Insulator quantum dot 6 / 10

10 For finite length cylinder : m = j σ/2 ψ j,ν,n,σ,τ = 2/V sin[nπ(z/l 1/2)]e imφ J m(γ m/r ) J m+1 (γ mν ) η τ χ σ (3) Kramers degeneracy j j k n nπ/l additional n = 0 states emerges one for valence band and one for conduction band for j > 1/2, a pair of subgap states inside the gap s. σ φ = 0 and σ r 0 Arijit Kundu (Heinrich-Heine University) Topological Insulator quantum dot 6 / 10

11 For finite length cylinder : m = j σ/2 ψ j,ν,n,σ,τ = 2/V sin[nπ(z/l 1/2)]e imφ J m(γ m/r ) J m+1 (γ mν ) η τ χ σ (3) Kramers degeneracy j j k n nπ/l additional n = 0 states emerges one for valence band and one for conduction band for j > 1/2, a pair of subgap states inside the gap s. σ φ = 0 and σ r 0 Arijit Kundu (Heinrich-Heine University) Topological Insulator quantum dot 6 / 10

12 For finite length cylinder : m = j σ/2 ψ j,ν,n,σ,τ = 2/V sin[nπ(z/l 1/2)]e imφ J m(γ m/r ) J m+1 (γ mν ) η τ χ σ (3) Kramers degeneracy j j k n nπ/l additional n = 0 states emerges one for valence band and one for conduction band for j > 1/2, a pair of subgap states inside the gap s. σ φ = 0 and σ r 0 Arijit Kundu (Heinrich-Heine University) Topological Insulator quantum dot 6 / 10

13 For finite length cylinder : m = j σ/2 ψ j,ν,n,σ,τ = 2/V sin[nπ(z/l 1/2)]e imφ J m(γ m/r ) J m+1 (γ mν ) η τ χ σ (3) Kramers degeneracy j j k n nπ/l additional n = 0 states emerges one for valence band and one for conduction band for j > 1/2, a pair of subgap states inside the gap s. σ φ = 0 and σ r 0 Arijit Kundu (Heinrich-Heine University) Topological Insulator quantum dot 6 / 10

14 For finite length cylinder : m = j σ/2 ψ j,ν,n,σ,τ = 2/V sin[nπ(z/l 1/2)]e imφ J m(γ m/r ) J m+1 (γ mν ) η τ χ σ (3) Kramers degeneracy j j k n nπ/l additional n = 0 states emerges one for valence band and one for conduction band for j > 1/2, a pair of subgap states inside the gap s. σ φ = 0 and σ r 0 Arijit Kundu (Heinrich-Heine University) Topological Insulator quantum dot 6 / 10

15 For finite length cylinder : m = j σ/2 ψ j,ν,n,σ,τ = 2/V sin[nπ(z/l 1/2)]e imφ J m(γ m/r ) J m+1 (γ mν ) η τ χ σ (3) Kramers degeneracy j j k n nπ/l additional n = 0 states emerges one for valence band and one for conduction band for j > 1/2, a pair of subgap states inside the gap s. σ φ = 0 and σ r 0 Spin surface locking breaking reported previously Fu, PRL (2009) & Yazyev, Moore, Louie, PRL (2010) Arijit Kundu (Heinrich-Heine University) Topological Insulator quantum dot 6 / 10

16 For finite length cylinder : m = j σ/2 ψ j,ν,n,σ,τ = 2/V sin[nπ(z/l 1/2)]e imφ J m(γ m/r ) J m+1 (γ mν ) η τ χ σ (3) Kramers degeneracy j j k n nπ/l additional n = 0 states emerges one for valence band and one for conduction band for j > 1/2, a pair of subgap states inside the gap s. σ φ = 0 and σ r 0 Spin surface locking breaking reported previously Fu, PRL (2009) & Yazyev, Moore, Louie, PRL (2010) Arijit Kundu (Heinrich-Heine University) Topological Insulator quantum dot 6 / 10

17 Surface Dirac fermion Theory Surface Dirac fermion Theory [ v 1 σ φ ( i z ) v 2 ] τ z H D = R σ zj ( H c = v 2 [ i r + 1 ) σ r + J ] 2r R σ φ τ x H D = [ v 1 σ φ ( i z ) v2 R σ zj ] ˆτ the operator ˆτ can be determined from symmetries of the Hamiltonian, namely azimuthal symmetry, time reversal symmetry & inversion symmetry. H = H D (φ, z)δ(r R) + H cap (r, φ) s=± δ(z sl/2) (4) Arijit Kundu (Heinrich-Heine University) Topological Insulator quantum dot 7 / 10

18 Surface Dirac fermion Theory Surface Dirac fermion Theory [ v 1 σ φ ( i z ) v 2 ] τ z H D = R σ zj ( H c = v 2 [ i r + 1 ) σ r + J ] 2r R σ φ τ x H D = [ v 1 σ φ ( i z ) v2 R σ zj ] ˆτ the operator ˆτ can be determined from symmetries of the Hamiltonian, namely azimuthal symmetry, time reversal symmetry & inversion symmetry. H = H D (φ, z)δ(r R) + H cap (r, φ) s=± δ(z sl/2) (4) Arijit Kundu (Heinrich-Heine University) Topological Insulator quantum dot 7 / 10

19 Surface Dirac fermion Theory which gives the energy quantisation: On the surface of the cap, we obtain E n,j,± = ± (πnv 1 /L) 2 + (jv 2 /R) 2 (5) σ φ = 0 σ r 0 For each total angular momentum j, there are two zero-momentum states corresponding to conduction and valence band, respectively. For the physically allowed k = 0 we find spatially uniform densities. Arijit Kundu (Heinrich-Heine University) Topological Insulator quantum dot 8 / 10

20 Tight Binding Approach Tight Binding Model A simple microscopic model for strong TI was previously proposed by Fu, Kane, Mele PRL (2007) H tb = <i,j> t ij c i c j + 4iλ so a 2 We observe sub gap states as from the low energy model As similar to both the previous predictions, we observe < σ φ >= 0, < σ r > 0 (7) Also, note that, for infinite nanowire, TB calculations give < σ r >= 0. <<i,j>> c i ( σ).[ d 1 ij d 2 ij ]c j (6) Arijit Kundu (Heinrich-Heine University) Topological Insulator quantum dot 9 / 10

21 Summary Summary All three approaches shows that the spin surface locking is broken due to the presence of such edges. A nontrivial eigenstate with k = n = 0. In such a zero-momentum state the charge and spin densities along the trunk are basically homogeneous. The finite-length nanowire dot has subgap states when electron-hole symmetry is broken. The wavefunction of such a subgap state is localized on both caps simultaneously. It may be interesting to observe the effect of coulomb interactions and applied magnetic field Arijit Kundu (Heinrich-Heine University) Topological Insulator quantum dot 10 / 10