Weak localization properties of the doped $Z_2$ topological insulator

Localization properties of the doped $Z_2$ topological insulator are studied by weak localization theory. The disordered Kane-Mele model for graphene is taken as a prototype and analyzed with attention to effects of the topological mass term, intervalley scattering, and the Rashba spin-orbit interaction. The known tendency of graphene to antilocalize in the absence of intervalley scattering between $K$ and $K^{\prime }$ points is naturally placed as the massless limit of the Kane-Mele model. The latter is shown to have a unitary behavior even in the absence of magnetic field due to the topological mass term. When intervalley scattering is introduced, the topological mass term leaves the system in the unitary class, whereas the ordinary mass term, which appears if A and B sublattices are inequivalent, turns the system to weak localization. The Rashba spin-orbit interaction in the presence of $K\text{−}{K}^{\prime }$ scattering drives the system to weak antilocalization in sharp contrast to the ideal graphene case.

Figure 1

(Color online) Weak localization phase diagram of the doped Kane-Mele model in the presence of potential scatterers. Relation to other graphene-based models with different types of the mass or ripples is taken into account. WL, AL, and U refer to weak localization (orthogonal class), weak antilocalization (symplectic class) and absence of WL (unitary class), respectively.

Figure 2

(Color online) Weak localization properties for LRS without $K\text{−}{K}^{\prime }$ scattering. The ordinate shows crossover from WAL to WL tendency (${\lambda }_R=0$, as $E$ is decreased from the graphene limit toward the bottom of the band). Crossover from unitary to orthogonal symmetry class $\left({\lambda }_R\ne 0\right)$.

Figure 3

Particle-particle ladders. Bare and dressed Cooperons. Relevant diagrams in the $K{K}^{\prime }$ sector. cis and trans refers to specific configurations of the valleys: $K$ and $K^{\prime }$.

Figure 4

(Color online) Topological mass case. Weak localization properties in the presence of short-range scatterers: intervalley scattering is allowed ($K\text{−}{K}^{\prime }$ coupled).

Figure 5

Particle-particle ladders involving interbranch processes. In the presence of finite Rashba term ${\lambda }_R\ne 0$ (a) ${\gamma }_{-+}$ does contributes to $1/q^2$ singularity, whereas such diagrams as (b) and (c) are irrelevant to the singularity since $p_{\alpha }+p_{{\alpha }^{\prime }}$ cannot be smaller than the order of ${\lambda }_R$.