Quasiparticle Scattering in 2-D Helical Liquid

1. Quasiparticle Scattering in 2-D Helical Liquid arXiv: 0910.0756 X. Zhou, C. Fang, W.-F. Tsai, J. P. Hu

2. Outline • Introduction • The Model and T-matrix Formalism • Numerical Results • Nonmagnetic point impurity • Classical magnetic point impurity • Nonmagnetic edge impurity

3. Introduction • 3D topological insulators property: • bulk insulating gaps, but gapless surface states protected by topological property of time-reversal symmetry; • Odd number of Dirac cones. • Spin helical Dirac fermions • Spin locked to the momentum, leading to the breakdown of spin rotation symmetry. • Why QPI? • QPI provides a direct evidence to justify the model.

4. For Bi2Te3, constant-energy contours of the band structure and the evolution of the height of EF referenced to the Dirac point for the doping 0.67%. Red lines are guides to the eye that indicate the shape of the constant-energy band contours and intersect at the Dirac point. X. L. Chen et al, Science 325,178 (2009)

5. Model σi here are real spin operators. isotropic 2D helical Dirac fermions Particle-hole symmetry holds Hexagonal distortion of the FS L. Fu, arXiv:0908.1418

6. The characteristic length scale:

7. Warping term effects: nonlinear Density of states based on the model:

8. Non-vanishing spin along z direction exist moments around the FS, due to σz in the warping term, except on the vertices. No out-of-plane spin polarization for 2D Dirac fermions Spin textures around the FS at ω=0.05eV in (a) and at ω=0.3eV in (b)

9. General N-impurity problem T-matrix method & Formalism Impurity-induced electronic Green’s function and T-matrix: Green’s function (in momentum space) of the pure system:

10. Consider the case of a single point impurity located at the origin, means the scattering potential is momentum independence, T-matrix can be simply written as: FT-LDOS LDOS

11. Re Im

12. Numerical Results

13. A. Nonmagnetic point impurity Important Feature: absence of backscattering between diagonal vertices, which is topologically prohibited, bytime reversal invariance.

14. Theoretical argument Time-reversal operator has the property

15. B. Classical magnetic point impurity Feature: Very little effect on the charge density, means Why? To the lowest order(), spin-up & spin-down electrons see scattering potentials with opposite signs.

16. Naturally we introduce the spin local density of states (SLDOS) to study the interference for magnetic impurity case, and focus on the FT of the z-component SLDOS. Similar to the LDOS, the real and imaginary parts of FT-SLDOS correspond to the symmetric and antisymmetric parts of respectively.

17. Impurity spin polarization along z-axis Important Feature: Presence of backscattering between diagonal vertices. Why? The z-component of impurity spin polarization flips in-plane spin moments.

18. Magnetic impurities with in-plane magnetic moments Feature: The antisymmetric part is larger than the symmetric part.

19. Impurity spin polarization along y-axis Features: 1.The model has y  -y mirror symmetry (my); 2. 3. The strongest interference appears at wave vector ±q51.

20. Impurity spin polarization along x-axis Feature: 1. The model breaks x  -x mirror symmetry (mx); ≈ 0 2. 3. The strongest interference appears at wave vector ±q13 & ±q35.

21. Experimental Suggestions ? Spin polarized along y-axis： ? Spin polarized along x-axis： both are held in-plane model, i.e. only one is held warping term with σz required, i.e.

22. C. Nonmagnetic edge impurity ky conserved • y • V Quantum state on the LHS: • x • 0 Boundary condition:

25. Important Features in LDOS with the presence of nonmagneticedge impurity: 1. Friedel oscillation exists (at fixed energy); 2. The major contribution comes from the opposite k-points on the constant energy contour, but will not hold if the scattering between the states at k & -k is forbidden; 3. The oscillation will decay as a form 1/√d, if there existallowed the opposite k-points on the constant energy contour; in other words, long distance decaying function depends on the Fermi energy. 4. When |x| is large enough, the stationary points approximation tells us that, if the edge impurity is along the y-axis, the interference pattern is dominated by the k-points where kx reaches local minimum or maximum.

26. Thanks!

27. Discussion We neglected the possibility of any ordering due to interaction-induced FS instability. This is valid as long as there is no significant FS nesting vector. Strong electron-electron interaction is not expected based on the following observation. In experiments on topological insulators, the Fermi level of the sample in general is closer to the bottom of the conduction band and is far away from the Dirac point. Such a system with finite density of states may provide enough screening effect to Coulomb interaction between surface electrons. (iii) our calculation shows behavior if the FS shape is dominated by the warping term, and if the warping term is negligible.