RKKY interaction and intervalley processes in $p$-doped transition-metal dichalcogenides

We study the Ruderman-Kittel-Kasuya-Yosida interaction in $p$-doped transition-metal dichalcogenides such as ${\mathrm{MoS}}_2$ and ${\mathrm{WS}}_2$. We consider magnetic impurities hybridized to the Mo $d$ orbitals characteristic of the valence bands. Using the Matsubara Green's-function formalism, we obtain the two-impurity interaction vs their separation and chemical potential of the system, accounting for the important angular dependence which reflects the underlying triangular lattice symmetry. The inclusion of the valence-band valley at the $\Gamma$ point results in a strong enhancement of the interaction. Electron-scattering processes transferring momentum between valleys at different symmetry points give rise to complex spatial oscillation patterns. Variable doping would allow the exploration of rather interesting behavior in the interaction of magnetic impurities on the surfaces of these materials, including the control of the interaction symmetry, which can be directly probed in scanning tunneling microscopy experiments.

Figure 1

Magnetic impurities (red circles) hybridized to Mo $d$ orbitals. Blue dashed arrows show two high-symmetry directions, i.e., zigzag and armchair, along which we compare the effective interaction between local moments. Black solid arrows indicate unit vectors.

Figure 2

ZZ and XX components of the RKKY interaction as a function of impurity separation $r$, along (a) zigzag and (b) armchair directions. (c) DM component in the zigzag direction. The latter vanishes in the armchair direction. In all cases, the interaction amplitude decays as $r^{-2}$. The Fermi level ${\epsilon }_F\simeq -0.067$ crosses the uppermost $K_{\tau }$ valleys, without intersecting the valley at the $\Gamma$ point, as indicated by the solid line in (d). (d) Schematic low-energy band structure for ${\mathrm{MoS}}_2$ and ${\mathrm{WS}}_2$, showing the spin inversion of the valence bands at $K_1$ and $K_{-1}$ valleys. The black (red) curve corresponds to up (down) spin projection. The blue valley at $\Gamma$ is quadratic and spin degenerate. Dashed and dotted lines indicate higher $p$-doping levels discussed in Fig. 3 and below.

Figure 3

ZZ, XX, and DM components of the RKKY interaction as a function of separation in the zigzag direction. ${\epsilon }_F=-0.174$, as indicated by dashed line in Fig. 2. Left panels show the full interactions, including contributions of the $\Gamma$ valley. The red horizontal and vertical lines indicate a fully isotropic interaction point. Right panels show the same quantities without including the $\Gamma$ valley. Notice the different vertical scales.

Figure 4

Comparison of the different components of the RKKY interaction for different Fermi energy regimes: (a) $-{\epsilon }_{\Gamma }<{\epsilon }_F<0$, (b) $-2\lambda <{\epsilon }_F<-{\epsilon }_{\Gamma }$, and (c) ${\epsilon }_F<-2\lambda$. The interimpurity distance is fixed along the zigzag direction at $r=50$. Notice different vertical scales.