# Weak Localization and Antilocalization in Topological Materials with Impurity Spin-Orbit Interactions

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## Abstract

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## 1. Introduction

## 2. General Model

## 3. Topological Insulators

## 4. Weyl Semimetal Thin Films

## 5. Other Topological Materials

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Phase diagram representing the evolution of the quantum correction to the conductivity ${\sigma}^{\mathrm{qi}}(0)$ at zero magnetic field for massless and massive Dirac fermions. The parameters a, b and $\lambda $ are defined in Section 2. The extrinsic spin-orbit scattering strength is quantified by $\lambda $, while the ratio $a/b$ represents the ratio of the spin-orbit energy to the quasiparticle mass, evaluated at the Fermi energy. The conductivity is expressed in units of ${e}^{2}/h$ with the color bar on the right. The blue dashed line separates the WAL and WL regimes. The numerical parameters here are the same as those used in Figure 6 below.

**Figure 2.**The diagrams for the weak (anti-)localization conductivity of Dirac fermions. (

**a**) definition of the Green’s functions as arrowed solid lines in which Greek letters are spin indices; (

**b**) definition of dashed lines: impurities lines expressed in both retarded and advanced cases; (

**c**) the retarded self-energy in the first-order Born approximation, where ${G}_{0}^{\mathrm{R}}$ is the bare retarded Green’s function. This contribution to the self energy represents the classical picture of electrons scattering off randomly distributed impurities; (

**d**,

**e**) are Keldysh self-energies $({\Sigma}_{\mathit{k},\gamma \delta}^{\mathrm{K},\mathrm{b}}$ and ${\Sigma}_{\mathit{k},\alpha \beta}^{\mathrm{K},\mathrm{R}})$ of maximally crossed diagrams in the bare and the retarded dressed cases, respectively, where $\Gamma $ is the Cooperon structure factor. Both of these represent contributions due to quantum interference in scattering processes: an electron travelling through a disordered region can backscatter and return to its starting point. The loop can be performed clockwise or anticlockwise, and, in quantum mechanics, the two trajectories interfere. Note that (

**e**) vanishes in the absence of spin-orbit coupling; (

**f**) the Bethe-Salpeter equation for the twisted Cooperon structure factor $\tilde{\Gamma}.$

**Figure 3.**Plot of the zero-field quantum-interference conductivity correction in terms of the spin-orbit scattering strength $\lambda $ for massless Dirac fermions (solid line), HLN formula for 2DEG (dashed line) and 3DEG (dotted line). $\delta {\sigma}_{0}=({e}^{2}/\pi h)ln({\tau}_{\varphi}/\tau )$. In order to emphasize that the WAL conductivity of massless Dirac fermions is also dependent upon the spin-orbit impurity strength, the inset shows WAL in units independent of $\lambda $, i.e., $({e}^{2}/\pi h)ln({\tau}_{\varphi}/{\tau}_{0}),$ where ${\tau}_{0}$ is the scattering time in the absence of spin-orbit impurities and the ratio ${\tau}_{\varphi}/{\tau}_{0}$ varies from 5 (purple), 10 (blue), 50 (green), 100 (orange) to 500 (red). Adapted from Figure 3 in [154].

**Figure 4.**The magnetic field dependence of the quantum correction to the conductivity for massless Dirac fermions (solid lines) and the 3D HLN formula (dashed lines) at different values of the spin-orbit impurities concentrations. Purple plots are for purely scalar disorder ($\lambda =0$), while blue (green) plots for $\lambda =0.2$ ($\lambda =1.25$). The magnetic field is given in the unit of ${\tilde{B}}_{0}=e/(2\pi \hslash {v}_{\mathrm{F}}^{2}{\tau}_{0}^{2})$, where ${\tau}_{0}$ is the elastic scattering time in the absence of spin-orbit impurities and ${v}_{\mathrm{F}}$ the Fermi velocity. Adapted from Figure 1 in [154], where $\delta \sigma (B)\equiv \Delta \sigma (B)$ as defined in Equation (16).

**Figure 5.**(

**a**) a minimal sketch of the energy dispersion of a Weyl semimetal. We have defined ${k}_{\parallel}^{2}={k}_{x}^{2}+{k}_{y}^{2}$ while $({k}_{x},{k}_{y},{k}_{z})$ represents the 3D wavevector. ${k}_{z}$ points along the preferred direction. The conductance and valence bands cross linearly at the Weyl nodes, i.e., the left and right cones, and the nodes appear in pairs with opposite chirality number. A Dirac node appears when two oppositely-chiral Weyl nodes emerge together; (

**b**) a schematic picture of the band structure in WSM thin films, where $\Delta $ is the band gap and ${\epsilon}_{\mathrm{F}}$ is the Fermi energy.

**Figure 6.**The magnetoconductivity $\Delta \sigma (B)$ plots at $\lambda =0.5$ for different masses M. The parameters used are $A=300$ meV·nm, ${n}_{e}=0.01\phantom{\rule{4.pt}{0ex}}{\mathrm{nm}}^{-2},$ ${n}_{\mathrm{i}}=0.0001\phantom{\rule{4.pt}{0ex}}{\mathrm{nm}}^{-2}$ and ${\ell}_{\varphi}=500$ nm, according to [131].

**Figure 7.**(

**a**–

**c**) the carrier density dependence of the quantum-interference conductivity ${\sigma}^{\mathrm{qi}}$ in the massless $(M=0$ meV), small mass $(M=10\phantom{\rule{4.pt}{0ex}}\mathrm{meV}),$ and large mass $(M=100\phantom{\rule{4.pt}{0ex}}\mathrm{meV})$ cases. The parameters are the same as in Figure 6 and $L={\ell}_{\varphi}$. Note that $\lambda ={\lambda}_{c}{n}_{e}$ and $A{k}_{\mathrm{F}}=106$ meV for ${n}_{e}=0.01\phantom{\rule{0.166667em}{0ex}}{\mathrm{nm}}^{-2}$.

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**MDPI and ACS Style**

Liu, W.E.; Hankiewicz, E.M.; Culcer, D. Weak Localization and Antilocalization in Topological Materials with Impurity Spin-Orbit Interactions. *Materials* **2017**, *10*, 807.
https://doi.org/10.3390/ma10070807

**AMA Style**

Liu WE, Hankiewicz EM, Culcer D. Weak Localization and Antilocalization in Topological Materials with Impurity Spin-Orbit Interactions. *Materials*. 2017; 10(7):807.
https://doi.org/10.3390/ma10070807

**Chicago/Turabian Style**

Liu, Weizhe Edward, Ewelina M. Hankiewicz, and Dimitrie Culcer. 2017. "Weak Localization and Antilocalization in Topological Materials with Impurity Spin-Orbit Interactions" *Materials* 10, no. 7: 807.
https://doi.org/10.3390/ma10070807