Spin susceptibility of two-dimensional transition-metal dichalcogenides

We have obtained analytical expressions for the $q$-dependent static spin susceptibility of monolayer transition-metal dichalcogenides, considering both the electron-doped and hole-doped cases. Our results are applied to calculate spin-related physical observables of monolayer ${\mathrm{MoS}}_2$, focusing especially on in-plane/out-of-plane anisotropies. We find that the hole-mediated Ruderman-Kittel-Kasuya-Yosida exchange interaction for in-plane impurity-spin components decays with the power law $R^{-5/2}$ as a function of distance $R$, which deviates from the $R^{-2}$ power law normally exhibited by a two-dimensional Fermi liquid. In contrast, the out-of-plane spin response shows the familiar $R^{-2}$ long-range behavior. We also use the spin susceptibility to define a collective $g$ factor for hole-doped ${\mathrm{MoS}}_2$ systems and discuss its density-dependent anisotropy.

Figure 1

Spin-resolved band dispersions for monolayer ${\mathrm{MoS}}_2$ at the $\mathrm{K}$ point. Spin-$\uparrow$ $(\downarrow )$ bands are shown as the blue solid (red dashed) curves. The unit scales for energy $E$ and wave vector $\mathrm{k}$ (measured from $\mathrm{K})$ are given in terms of band-structure parameters as $E_0\equiv \Delta /2$ and $k_0\equiv \Delta /\left(2at\right)$. Reversal of all spin labels yields the corresponding band dispersions at the ${\mathrm{K}}^{\prime }(\equiv -\mathrm{K})$ point.

Figure 2

In-plane component ${\chi }_{xx}\left(q\right)\equiv {\chi }_{xx}^{\left(\text{ext}\right)}\left(q\right)+{\chi }_{xx}^{\left(\text{int}\right)}\left(q\right)$ of the wave-vector-dependent spin-susceptibility tensor for electron-doped monolayer ${\mathrm{MoS}}_2$ (blue solid curves). We also plot the extrinsic (intrinsic) contributions ${\chi }_{xx}^{\left(\text{ext}\right)}$$\left({\chi }_{xx}^{\left(\text{int}\right)}\right)$ separately as the blue (red) dashed curves. Panel (a) [(b)] is for electron density $n=1\times {10}^{12}$ ${\mathrm{cm}}^{-2}$ $(5\times {10}^{13}$ ${\mathrm{cm}}^{-2})$. Notice the sharp feature exhibited by ${\chi }_{xx}$ and ${\chi }_{xx}^{\left(\text{ext}\right)}$ for $q\approx 2k_{\text{F}}$ and the different ordinate scale for ${\chi }_{xx}^{\left(\text{int}\right)}$.

Figure 3

Spin-susceptibility component ${\chi }_{zz}\left(q\right)\equiv {\chi }_{zz}^{\left(\text{ext}\right)}\left(q\right)+{\chi }_{zz}^{\left(\text{int}\right)}\left(q\right)$ for electron-doped monolayer ${\mathrm{MoS}}_2$ (blue solid curves). The extrinsic (intrinsic) contributions ${\chi }_{zz}^{\left(\text{ext}\right)}$ $\left({\chi }_{zz}^{\left(\text{int}\right)}\right)$ are also plotted as the blue (red) dashed curves. Panel (a) [(b)] shows results for electron density $n=1\times {10}^{12}$ ${\mathrm{cm}}^{-2}$ [$5\times {10}^{13}$ ${\mathrm{cm}}^{-2}$]. Sharp features are exhibited by ${\chi }_{zz}$ and ${\chi }_{zz}^{\left(\text{ext}\right)}$ for $q=k_{\text{F}}^{(+)},k_{\text{F}}^{(-)}$. Note the different scale for ${\chi }_{zz}^{\left(\text{int}\right)}$.

Figure 4

Spin response of hole-doped monolayer ${\mathrm{MoS}}_2$. Panel (a) [(b)] shows ${\chi }_{zz}\left(q\right)$ [${\chi }_{xx}\left(q\right)$] obtained for a hole sheet density $n=1\times {10}^{12}$ ${\mathrm{cm}}^{-2}$ (blue solid curve), $n=1\times {10}^{13}$ ${\mathrm{cm}}^{-2}$ (red dashed curve), and $n=3\times {10}^{13}$ ${\mathrm{cm}}^{-2}$ (black dot-dashed curve). Noteworthy features are the strong in-plane/out-of-plane anisotropy of the spin response, the deviations from ideal 2D-electron-gas behavior even for ${\chi }_{zz}\left(q\right)$, which are getting more pronounced as the density increases, and the nonanalyticity exhibited by ${\chi }_{zz}\left(q\right)$ [${\chi }_{xx}\left(q\right)$] for $q=2k_{\text{F}}\equiv 2k_{\text{F}}^{(+)}$ $\left(q=k_1>2k_{\text{F}}^{(+)}\right)$.

Figure 5

Spin response functions in real space. To highlight deviations from the ordinary 2D-electron-gas behavior [42], we plot the quantities $A_j{\left(k_{\text{F}}R\right)}^2{\chi }_{jj}\left(R\right)$ in panel (a) [(b)] obtained for a hole density $n=1\times {10}^{12}$ ${\mathrm{cm}}^{-2}$ $(n=3\times {10}^{13}$ ${\mathrm{cm}}^{-2})$. The red dashed (blue solid) curve shows ${\chi }_{xx}$ $\left({\chi }_{zz}\right)$. The decrease of the oscillation amplitude for ${\left(k_{\text{F}}R\right)}^2{\chi }_{xx}\left(R\right)$ shows that the in-plane spin response decays faster than the $R^{-2}$ power law expected for ordinary 2D electron systems. The scale factors are $A_x=4\times {10}^4$ (4) and $A_z={10}^3$ (1) in panel (a) [(b)], and ${\tilde{\chi }}_0={\chi }_0{k}_{\text{F}}^2/\left(2\pi \right)\equiv 2n{k}_0^2/\left(\pi E_0\right)$.

Figure 6

Density dependence of the Curie temperature for hole-mediated easy-axis Ising ferromagnetism of impurity spins in monolayer ${\mathrm{MoS}}_2$.

Figure 7

Density dependence of the effective in-plane $g$ factor for hole-doped ${\mathrm{MoS}}_2$.