Kondo effect and non-Fermi-liquid behavior in Dirac and Weyl semimetals

We study the Kondo effect in three-dimensional (3D) Dirac materials and Weyl semimetals. We find the scaling of the Kondo temperature with respect to the doping $n$ and the coupling $J$ between the moment of the magnetic impurity and the carriers of the semimetal. We consider the interplay of long-range scalar disorder and Kondo screening and find that it causes the Kondo effect to be characterized not by a Kondo temperature, but by a distribution of Kondo temperatures with features that cause the appearance of strong non-Fermi-liquid behavior. We then consider the effect of Kondo screening, and of the interplay of Kondo screening and long-range scalar disorder, on the transport properties of Weyl semimetals. Finally, we compare the properties of the Kondo effect in 3D and 2D Dirac materials such as graphene and topological insulators.

Figure 1

$T_{\mathrm{K}}$ as a function of $\widehat{J}=J\mathcal{N}\left(D\right)$ for different values of $\mu /D$ for a (a) 3D and (b) 2D Dirac SM.

Figure 2

$\overline{T_{\mathrm{K}}}P\left(\overline{T_{\mathrm{K}}}\right),\overline{T_{\mathrm{K}}}\equiv k_{\mathrm{B}}{T}_{\mathrm{K}}/D$, for a (a) 3D and (b) 2D SM with $v_{\mathrm{F}}={10}^8\mathrm{cm}/\mathrm{s},D=0.5$ eV, $J=0.6J_{\mathrm{cr}},N_{\mathrm{w}}=2$, and different values of $\overline{n}$ [in units of ${\mathrm{cm}}^{-3}$ in (a) and ${\mathrm{cm}}^{-2}$ in (b)]. In (a), ${\sigma }_n={10}^{16}{\mathrm{cm}}^{-3}$ and $g_s=1$, and in (b), ${\sigma }_n={10}^{11}{\mathrm{cm}}^{-2}$ and $g_s=2$.

Figure 3

(a), (b) ${\rho }_{\mathrm{K}}^{\left(\mathrm{EMT}\right)}\left(T\right)$ for the case in which $\overline{\mu }=60$ meV in 2D $\left(\widehat{J}=0.98\right)$ and 3D $\left(\widehat{J}=1.77\right)$, respectively. (c), (d) ${\rho }_{\mathrm{K}}^{\left(\mathrm{EMT}\right)}\left(T\right)$ for the case in which $\overline{n}=0$ $\left(J<J_{\mathrm{cr}}\right)$, in 2D and 3D, respectively. In this plot, ${\sigma }_n$ is in units of ${\mathrm{cm}}^{-3}$ in the 3D case, and of ${\mathrm{cm}}^{-2}$ in the 2D one.