Quantum transport in Weyl semimetal thin films in the presence of spin-orbit coupled impurities

Topological semimetals have been at the forefront of experimental and theoretical attention in condensed-matter physics. Among these, recently discovered Weyl semimetals have a dispersion described by a three-dimensional Dirac cone, which is at the root of exotic physics such as the chiral anomaly in magnetotransport. In a time-reversal symmetric (TRS) Weyl semimetal film, the confinement gap gives the quasiparticles a mass, while TRS is preserved by having an even number of valleys with opposite masses. The film can be tuned through a topological phase transition by a gate electric field. In this work, we present a theoretical study of the quantum corrections to the conductivity of a topological semimetal thin film, which is governed by the complex interplay of the chiral band structure, mass term, and scalar and spin-orbit scattering. We study scalar and spin-orbit scattering mechanisms on the same footing, demonstrating that they have a strong qualitative and quantitative impact on the conductivity correction. We show that, due to the spin structure of the matrix Green's functions, terms linear in the extrinsic spin-orbit scattering are present in the Bloch and momentum relaxation times, whereas previous works had identified corrections starting from the second order. In the limit of small quasiparticle mass, the terms linear in the impurity spin-orbit coupling lead to a potentially observable density dependence in the weak antilocalization correction. Moreover, when the mass term is around 30% of the linear Dirac terms, the system is in the unitary symmetry class with zero quantum correction, and switching the extrinsic spin-orbit scattering drives the system to the weak antilocalization. We discuss the crossover between the weak localization and weak antilocalization regimes in terms of the singlet and triplet Cooperon channels, and analyze this transition as a function of the mass and spin-orbit scattering strength. Experimental schemes to detect this transition are discussed.

Figure 1

(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 three-dimensional wave vector. $k_z$ 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 merge together. (b) A schematic picture of the band structure in WSM thin films, where $\mathrm{\Delta }$ is the band gap and ${\varepsilon }_{\text{F}}$ is the Fermi energy.

Figure 2

The diagrams for the weak (anti-)localization conductivity of Dirac fermions. (a) Define Green's function 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 $G_0^{\text{R}}$ is the bare retarded Green's function. (d, e) Keldysh self-energies $({\mathrm{\Sigma }}_{\mathit{k},\gamma \delta }^{\text{K},\text{b}}$ and ${\mathrm{\Sigma }}_{\mathit{k},\alpha \beta }^{\text{K},\text{R}})$ of maximally crossed diagrams in the bare and the retarded dressed cases, respectively, where $\mathrm{\Gamma }$ is the Cooperon structure factor. (f) The Bethe-Salpeter equation for the twisted Cooperon structure factor $\tilde{\mathrm{\Gamma }}.$

Figure 3

The magnetoconductivity plots obtained without considering the angular-dependent ($\propto \lambda$) part in the Green's functions. (a, b) Magnetoconductivity $\mathrm{\Delta }\sigma \left(B\right)$ for different masses $M$ at weak spin-orbit scattering $(\lambda =0)$ and strong spin-orbit scattering ($\lambda =0.5$). Comparison between the two panels confirms the suppression of WL due to spin-orbit scattering. For the effect of the linear-in-$\lambda$ terms, see Fig. 4. Parameters are $A=300$ meV nm, $n_e=0.01{\text{nm}}^{-2},n_{\mathrm{i}}=0.0001{\text{nm}}^{-2}$, and ${\ell }_{\varphi }=500$ nm. Our results by the Keldysh formalism are qualitatively consistent with those by the Kubo formula [89], verifying the validity of our approach.

Figure 4

The magnetoconductivity $\mathrm{\Delta }\sigma \left(B\right)$ plots at $\lambda =0.5$ for different masses $M$ with considering the angular-dependent part in the spin-orbit scattering. Compared with Fig. 3, it is shown that there is more suppression of the WL channel when the angular-dependent part of the spin-orbit scattering is taken into account.

Figure 5

Zero-field quantum conductivity ${\sigma }^{\text{qi}}\left(0\right)$ in terms of $a/b$ and $\lambda .$ The unit of the conductivity is $e^2/h$ with the color bar on the right. The blue dashed line separates the WAL and WL regimes. The parameters here are the same as in Fig. 3.

Figure 6

The carrier density dependence of the total conductivity ${\sigma }^{\text{tot}}={\sigma }^{\left(0\right)}$ at the massless $(M=0)$ (a) and massive $(M=100\text{meV})$ (b) limits. Parameters are the same as Fig. 3, and $L$ is set to be equal to ${\ell }_{\varphi }$ for simplicity. Note that $\lambda ={\lambda }_c{n}_e$ and $A{k}_{\text{F}}=106$ meV for $n_e=0.01{\text{nm}}^{-2}.$

Figure 7

The carrier density dependence of the quantum-interference conductivity ${\sigma }^{\text{qi}}$ at the massless $(M=0)$ (a), small mass $(M=10\text{meV})$ (b), and large mass $(M=100\text{meV})$ (c) cases. Parameters here are the same as Fig. 6.