Local and global patterns in quasiparticle interference: A reduced response function approach

A physical system exposes to us in a real space, while its description often refers to its reciprocal momentum space. A connection between them can be established by exploring patterns of quasiparticles interference (QPI), which is experimentally accessible by Fourier transformation of the scanning tunneling spectroscopy. We here investigate how local and global features of QPI patterns are related to the geometry and topology of electronic structure in the considered physical system. A reduced response function (RRF) approach is developed that can analyze QPI patterns with clear physical pictures. The generalized joint density of states, which is the imaginary part of RRF, is justified for studying QPI. Moreover, we reveal that global patterns of QPI may be indicators of topological numbers for gapless systems, and we demonstrate the robustness of such indicators against distractive local features of QPI for topological materials with complicated band structures.

Figure 1

Behavior of ${\dot{E}}_{{\mathbf{k}}_{\omega }\left(t_0\right)+\mathbf{p}}$. The dashed red arrows stand for a small departure of scattered-in quasiparticle from ${\mathbf{k}}_{\omega }\left(t_0\right)$ but with the same $\mathbf{p}$. The left figure shows the general case, while the right demonstrates the case of backscattering.

Figure 2

CCE that changes sign of the curvature for a toy model. ${\mathbf{p}}_{1,2,3}$ are singularities of ${\mathcal{R}}_{\alpha \beta }(\mathbf{p},\omega )$ because they connect points with their group velocities (anti)parallel. According to Eq. (15), only the joint curvature of ${\mathbf{p}}_3$ is zero which leads to the fact that the order of ${\mathbf{p}}_3$ is higher than those of ${\mathbf{p}}_{1,2}$.

Figure 3

Stability of singularities for the real part of the reduced response function $\mathcal{S}$. (a) Amplitude of $A(\mathbf{k},\omega )$ and $B(\mathbf{k},\omega )$ with varying $\omega$ under positive finite $\eta$ that models a finite lifetime of quasiparticle. (b) Stable singularity. Contributions to $\mathcal{S}$ are all positive (dashed red arrows) under fixed $\mathbf{p}$ (solid green arrow). (c) Unstable singularity. Positive (thick dashed red arrow) and negative (thin dashed red arrow) contributions to $\mathcal{S}$ under fixed $\mathbf{p}$. (d) Stable singularity for near-nesting scattering. (e) Unstable singularity for near-nesting scattering.

Figure 4

Illustration of how spin direction (dashed red arrows) and group velocity (solid blue arrows) along the CCE jointly determine QPI pattern. For backscattering group velocities are reversed and the denominator becomes divergent. However, spin directions are reversed for $Q=1$ but are the same for $Q=2$. Thus, the divergence is kept only for $Q=2$.

Figure 5

Illustration of effective direction-selective prohibition of backscattering. The combination of scattering scattering channel $\alpha =x$ and probe channel $\beta =x$ effectively rotates the spin of quasiparticle around $x$ axis by $\pi$, leading to a twist of spin interference, and backscattering is prohibited along specified directions. For $Q=1$ and $Q=2$ QPI patterns have two and four hot arcs (brown color), respectively.

Figure 6

QPI patterns for surface state of 3D topological insulator ${\text{Bi}}_2{\text{Te}}_3$ for circlelike contour of constant of energy at low energy $\omega =0.25$. Backscattering is prohibited for charge impurity and charge probe, as evidenced ${\mathcal{S}}_{00}$, while is restored for ${\sigma }_z$ impurity and ${\sigma }_z$-resolved probe. The QPI pattern shows two bright arcs for ${\sigma }_x$ impurity and ${\sigma }_x$-resolved probe. Those QPI patterns under different channels reveal a topological number $Q=1$.

Figure 7

QPI patterns for nonconvex contour of constant energy for surface state of 3D topological insulator ${\text{Bi}}_2{\text{Te}}_3$. CCE at $\omega =1$ is given in the right-bottom [parameters in Eq. (21) are $v=1$ and $\lambda =2$]. Joint density of states $\mathcal{J}$ evidences high degrees of singularities with hotter spots, e.g., ${\mathbf{p}}_1,{\mathbf{p}}_4$ due to zero joint curvatures, and ${\mathbf{p}}_3$ due to near-nesting. QPI patterns are given by ${\mathcal{S}}_{\alpha \alpha }$ for $\alpha =0,x,z$ scattering and probe channels. As a comparison with ${\mathcal{S}}_{00}, {\mathcal{J}}_{00}$ gives extra false features around the center.

Figure 8

Quasiparticle interference for the surface state of BiTeI. In the middle is an inset of contours of constant energy at $\omega =-0.01$, which has two concentric CCEs. QPI for different combinations of scattering and probing channels are given by ${\mathcal{S}}_{00},{\mathcal{S}}_{zz},{\mathcal{S}}_{zx},{\mathcal{S}}_{xx}$, respectively. Inter-CCE (intra-CCE) scattering are favored (prohibited) in ${\mathcal{S}}_{00}$, while Intra-CCE (inter-CCE) scattering are favored (prohibited) in ${\mathcal{S}}_{zz}$. A global pattern of twofold symmetry appear in both ${\mathcal{S}}_{zx}$ and ${\mathcal{S}}_{xx}$. The four hot arcs in ${\mathcal{S}}_{xx}$ can be explained by direction selective spin-coherent factor due to an effective spin reflection around the $x$ direction ${\widehat{R}}_x$.

Figure 9

QPI pattern for graphene family evaluated by ${\mathcal{S}}_{00}(\mathbf{p},\omega )$ and ${\mathcal{S}}_{zz}(\mathbf{p},\omega )$, as shown in (a) and (b), respectively. Here we chose $\omega =0.3,0.15,0.1$ for $N=1,2,3$, respectively. Intervalley scattering leads to remarkable $2N$ hot arcs for $N=1,2,3$. CCEs and the joint density of states are shown in (c) for $N=1$.