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

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

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.

For Bi 2 Te 3, constant-energy contours of the band structure and the evolution of the height of E F 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)

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

The characteristic length scale:

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

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

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

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: LDOSFT-LDOS

Re Im

Numerical Results

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

Theoretical argument Time-reversal operator has the property

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.

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.

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

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

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

Impurity spin polarization along x-axis 2.2. Feature: 1. The model breaks x  -x mirror symmetry (m x ); 3. The strongest interference appears at wave vector ±q 13 & ±q 35. ≈ 0

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 σ z required, i.e. ? ?

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

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 exist allowed 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 k x reaches local minimum or maximum.

Thanks!

Discussion (i)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. (iii) our calculation shows behavior if the FS shape is dominated by the warping term, and if the warping term is negligible. (ii)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.