Itinerant ferromagnetism in $p$-doped monolayers of ${\mathrm{MoS}}_2$

Density functional theory is used to explore the possibility of inducing impurity band ferromagnetism in monolayers of semiconducting ${\mathrm{MoS}}_2$ by introducing holes into the narrow Mo $4d$ band that forms the top of the valence band. A large out-of-plane anisotropy is found for unpaired spins bound to the substitutional acceptor impurities V, Nb, and Ta that couple ferromagnetically for all but the shortest separations. Using the separation-dependent exchange interactions as the input to Monte Carlo calculations, we estimate ordering temperatures as a function of the impurity concentration. For about 9% of V impurities, Curie temperatures in excess of 160 K are predicted. The singlet formation at short separations that limits the ordering temperature is explained and we suggest how it can be circumvented.

Figure 1

(a) The band structure of an ${\mathrm{MoS}}_2$ monolayer and (b) the corresponding total, and $\mathrm{Mo}-d$ and $\mathrm{S}-p$ projected densities of states. The spin-degenerate (point) defect levels that result from substituting Mo with various transition metals are indicated: group IV (pink), group V (black), group VII (green), and group VIII (blue).

Figure 2

The difference between the total energies of antiferromagnetic and ferromagnetic states (black) and the pair binding energy (red) for pairs of V substitutional impurities plotted as a function of their separation for unrelaxed (open symbols) and fully relaxed (solid symbols) configurations in $12\times 12 {\mathrm{MoS}}_2$ supercells. The dashed black curve extrapolates the relaxed results to separations where quenching occurs.

Figure 3

Variation of the ferromagnetic Curie temperature as a function of the doping concentration calculated using Binder's cumulant method and the exchange interactions shown in Fig. 2 for an ${\mathrm{MoS}}_2$ monolayer doped with V. The dashed curve was calculated by using extrapolated relaxed exchange interactions at separations where quenching occurs. Inset: Fourth-order cumulant calculated as a function of the temperature for three different lattice sizes.

Figure 4

Unpolarized electronic structure for a $12\times 12 {\mathrm{MoS}}_2$ supercell with two V atoms on nearest-neighbor Mo sites. Band structure and DOS for the system (a) without and (b) with atomic relaxation. Defect bands are highlighted in red and green.The insets show the coupling scheme of two adjacent ${\mathrm{VS}}_6$ units with unrelaxed (left) and relaxed atomic positions (right).