RKKY interaction on the surface of three-dimensional Dirac semimetals

We study the RKKY interaction between two magnetic impurities located on the surface of a three-dimensional Dirac semimetal with two Dirac nodes in the band structure. By taking into account both bulk and surface contributions to the exchange interaction between the localized spins, we demonstrate that the surface contribution in general dominates the bulk one at distances larger than the inverse node separation due to a weaker power-law decay. We find a strong anisotropy of the surface term with respect to the spins being aligned along the node separation axis or perpendicular to it. In the many impurity dilute regime, this implies formation of quasi-one-dimensional magnetic stripes orthogonal to the node axis. We also discuss the effects of a surface spin-mixing term coupling electrons from spin-degenerate Fermi arcs.

Figure 1

(a) Spin-degenerate Dirac cones, and spin-up (spin-down) Fermi arcs at zero energy shown in red (blue). (b) Chiral dispersion of spin-up and spin-down Fermi arcs in $k_x$.

Figure 2

Surface (red curves) and bulk (blue curves) contributions to the RKKY interaction normalized to a characteristic energy scale $E_0\equiv J^2{k}_0^5/v$ plotted as a function of the dimensionless distances between the localized spins along the $x$ axis (left column) and $z$ axis (right column). We take $\mu =0$ and $\mu =0.1\mathrm{eV}$ for upper and lower panels correspondingly. We assume the distance between the impurities to be $z=10\AA$ (upper-left panel), $z=1\AA$ (lower-left panel), and $x=10\AA$ (upper-right panel), $x=1\AA$ (lower-right panel). We set: $S_{1i}=S_{2i}=1, J_{x,y}=J_{\perp }=J,J_z=\delta J_z=J$, where we set the exchange coupling constant is measured in $\mathrm{eV}{\AA }^3$, and we take realistic parameters for ${\mathrm{Cd}}_3{\mathrm{As}}_2$ given by Eq. (S45) from Ref. [38]: $v=0.889\mathrm{eV}\AA , k_0=0.033{\AA }^{-1}, m_1=18.77\mathrm{eV}{\AA }^2$, and $\gamma =0.513$.

Figure 3

Surface (red curves) and bulk (blue curves) contributions to the RKKY interaction normalized to a characteristic energy scale $E_0\equiv J^2{k}_0^5/v$ plotted as a function of the chemical potential $\mu$. We assume the distance between the impurities to be $z=10\AA$ along the $z$ axis and $x=100\AA$ along the $x$ axis. Other parameters are taken to be the same as in Fig. 2.