Stationary phase approximation approach to the quasiparticle interference on the surface of a strong topological insulator

Topological insulators have surface states with unique spin-orbit coupling. With impurities on the surface, the quasiparticle interference pattern is an effective way to reveal the topological nature of the surface states, which can be probed by scanning tunneling microscopy. In this paper, we present a general analytic formulation of the local density of states using the stationary phase approximation. The power laws of Friedel oscillations are discussed for a constant energy contour with a generic shape. In particular, we predict unique signature of magnetic impurities in comparison with nonmagnetic impurities for a surface state trapped in a “magnetic wall.”

Figure 1

Illustration of charge and spin interference patterns between two counterpropagating helical waves. The gray block is a 3D TI with a magnetic edge impurity (green stripe with thick arrows pointing up) lying along the $x$ axis on the surface. An incident helical wave along the $y$ direction with spin polarized in the $x$ direction (the blue dashed line) is backscattered by the magnetic edge and the spin is flipped (the red solid line). The interference of the two orthogonal helical waves leads to a constant LDOS in the charge channel, but a spiral LDOS in the spin channel (purple arrows between the solid and dashed lines) in the $yz$ plane.

Figure 2

Standing wave of spin interference between two helical waves inside a closed “magnetic wall” on top of a 3D TI surface. The magnetic wall is surrounded by a magnetic layer deposited on top of the 3D TI surface, which opens a gap in the helical surface states and plays the role of a barrier. The out-of-plane spin LDOS is exhibited by the colored (dark and bright) rings, and the in-plane spin LDOS is indicated by the black arrows.

Figure 3

Schematic picture of CEC and stationary points for point and edge impurities. (a) Convex CEC where there is only one pair of stationary points connected by wavevector $q_1$ (the red solid arrow) along any given direction for both point and line impurities. (b) Concave CEC for point impurity where there are multiple pairs of stationary points. Examples of nonstationary points are shown by $q_2$ and $q_3$ (blue thick solid arrows). (c) The concave CEC for edge impurities (brown shaded line) where the slopes (green dashed lines) at the pair of stationary points are the same.

Figure 4

Fourier transformation of the LDOS with $R^{-1}$ and $R^{-2}$ power laws.

Figure 5

Schematic CEC of (a) quadratic, (b) Dirac, and (c) Rashba dispersions. The spin orientations for each degenerate band are indicated respectively by the green (solid) and purple (dotted) arrows. The stationary points are represented by red dots and blue triangles respectively, which are connected by the scattering vector $\mathbf{q}$ shown as dashed arrows. The intraband scattering occurs between the stationary points with the same color (shape), while the interband scattering occurs between those with different colors (shapes).