Density functional theory study of the structural and electronic properties of single and double acceptor dopants in $M{X}_2$ monolayers

Density functional theory calculations are used to systematically investigate the structural and electronic properties of $M{X}_2$ transition metal dichalcogenide monolayers with $M$ = Cr, Mo, W and $X$ = S, Se, Te that are doped with single (V, Nb, Ta) and double (Ti, Zr, Hf) acceptor dopants on the $M$ site with local $D_{3h}$ symmetry in the dilute limit. Three impurity levels that arise from intervalley scattering are found above the valence band maxima (VBM): an orbitally doubly degenerate $e^{\prime }$ level bound to the $K/K^{\prime }$ VBM and a singly degenerate $a_1^{\prime }$ level bound to the $\mathrm{\Gamma }$-point VBM. Replacing S with Se or Te lowers the $\mathrm{\Gamma }$ point VBM substantially with respect to the $K/K^{\prime }$ VBM bringing the $a_1^{\prime }$ level down with it. The relative positions of the impurity levels that determine the different structural and electronic properties of the impurities in $p$-doped $M{X}_2$ monolayers can thus be tuned by replacing S with Se or Te. Single acceptors introduce a magnetic moment of $1{\mu }_{\mathrm{B}}$ in all $M{X}_2$ monolayers. Out-of-plane magnetic anisotropy energies as large as 10 meV/dopant atom are found thereby satisfying an essential condition for long-range ferromagnetic ordering in two dimensions. For double acceptors in $M{\mathrm{S}}_2$ monolayers, both holes occupy the high-lying $a_1^{\prime }$ level with opposite spins so there is no magnetic moment; in $M{\mathrm{Se}}_2$ and $M{\mathrm{Te}}_2$ monolayers the holes occupy the $e^{\prime }$ level, a Jahn-Teller (JT) distortion wins the competition with exchange splitting resulting in the quenching of the magnetic moments. Even when the JT distortion is disallowed, magnetic double acceptors have a large in-plane magnetic anisotropy energy that is incompatible with long-range magnetic ordering in two dimensions. The magnetic moments of pairs of single acceptors exhibit long-range ferromagnetic coupling except for $M{\mathrm{S}}_2$ where the coupling is quenched for impurity pairs below a critical separation. For Se and Te compounds, the holes are accommodated in high-lying degenerate $e^{\prime }$ levels, which form triplets for all separations. However, for $X=\mathrm{Te}$, a JT distortion lifts the degeneracy of the $e^{\prime }$ levels leading to a reduction of the exchange interaction between impurity pairs. Deep, intrinsic, vacancy, and antisite defects that localize the holes might stabilize the magnetization of $p$-doped $M{X}_2$ monolayers. Our systematic study of the $p$-doped $M{X}_2$ monolayers identifies 1H ${\mathrm{CrTe}}_2$ and ${\mathrm{MoSe}}_2$ as the most promising candidates for room-temperature ferromagnetism. We combine the exchange interaction estimated from the energy difference calculated for ferromagnetically and antiferromagnetically coupled pairs with Monte Carlo calculations to estimate the Curie temperatures $T_{\mathrm{C}}$ for vanadium-doped ${\mathrm{CrTe}}_2$ and ${\mathrm{MoSe}}_2$ monolayers. Room-temperature values of $T_{\mathrm{C}}$ are predicted for V dopant concentrations of 5% and 9%, respectively. In view of the instability of ${\mathrm{CrTe}}_2$ in the 1H form, we suggest that the ${\mathrm{Cr}}_x{\mathrm{Mo}}_{1-x}{\left({\mathrm{Te}}_y{\mathrm{Se}}_{1-y}\right)}_2$ alloy system be studied. A single $d$ electron or hole is uncorrelated. However, in the single-impurity limit, the residual self-interaction of this carrier in the local spin density approximation (LSDA) can be corrected by introducing a Hubbard $U$. Doing so leads to a large increase of the ordering temperatures calculated in the LSDA (reducing the doping concentration needed to achieve room-temperature ordering) but at the expense of introducing an indeterminate parameter $U$.

Figure 1

Band structures without spin-orbit coupling of monolayers of undoped ${\mathrm{MoS}}_2$ (a), ${\mathrm{MoSe}}_2$ (b), and ${\mathrm{MoTe}}_2$ (c) and with a single substitutional vanadium dopant atom on a Mo site in a $6\times 6$ supercell [(d)–(f)]. The zero of energy is the Fermi level. In (a)–(c), the blue and red horizontal lines represent, respectively, the $e^{\prime }$ impurity levels bound to $\varepsilon (K/K^{\prime })$, the $K/K^{\prime }$ valence band maxima (VBM), and the $a_1^{\prime }$ impurity level bound to $\varepsilon \left(\mathrm{\Gamma }\right)$, the $\mathrm{\Gamma }$ VBM. The energy difference $\varepsilon (K/K^{\prime })-\varepsilon \left(\mathrm{\Gamma }\right)={\mathrm{\Delta }}_{K\mathrm{\Gamma }}$ increases in the sequence $\mathrm{S}\to \mathrm{Se}\to \mathrm{Te}$. In (d)–(f) the blue and red lines are the corresponding bands formed from these impurity levels because of the finite size of the unit cell and the periodic boundary conditions. Note that different energy scales are used in the top and bottom panels.

Figure 2

Side view and top view of the atomic structure of an $M{X}_2$ ($M$ = Cr, Mo, W; $X$ = S, Se, Te) monolayer. The unit cell is enclosed by the dotted lines.

Figure 3

Impurity levels for single (V, Nb, Ta) and double (Ti, Zr, Hf) acceptors in $M{X}_2$ monolayers including atomic relaxation but without spin polarization. All energies are shown in meV with respect to the top of the valence band. The short horizontal red and blue lines represent $a_1^{\prime }$ and degenerate $e^{\prime }$ levels, respectively. The dotted blue lines indicate the Jahn-Teller split $e^{\prime }$ levels. Holes represented by open circles occupy the highest levels in each system. Note that the smallest bandgap is large on the energy scale of the figure.

Figure 4

Schematic of the energy gain with and without Jahn-Teller distortion for (a) single and (b) double acceptors. The solid and dashed blue lines represent spin-degenerate and spin-resolved energy levels, respectively. $\delta$ denotes the energy splitting of the degenerate $e^{\prime }$ levels caused by JT distortion and $\mathrm{\Delta }$ is the exchange splitting. Solid and open circles denote electrons and holes, respectively. The completely filled (with electrons; empty of holes) $a_1^{\prime }$ level is not shown.

Figure 5

Top view of the dopant-induced Jahn-Teller distortion in an $M{X}_2$ monolayer. The purple ball represents the dopant atom and the red arrow indicates its displacement for single acceptors; for double acceptors, the displacement is in the opposite direction. Because of the threefold symmetry of the unrelaxed structure, the symmetry-lowering dopant displacement can be in any one of three equivalent directions.

Figure 6

Total energy on displacing a substitutional vanadium dopant ${\mathrm{V}}_{\mathrm{Mo}}$ in the central $xy$ metal plane away from the Mo lattice site in a $12\times 12$ supercell of monolayers of (a) ${\mathrm{MoSe}}_2$ and (b) ${\mathrm{MoTe}}_2$ without allowing the host atoms to relax. Displacing the V atom along the dashed black lines in (b) weakens the V-${\mathrm{Te}}_1$ bonds while strengthening the V-${\mathrm{Te}}_2$ and V-${\mathrm{Te}}_3$ bonds. The brown circles in (b) represent the projection of the Te positions onto the metal plane. (c) Band structure without spin-polarization of the V-doped ${\mathrm{MoTe}}_2$ monolayer for a $12\times 12$ supercell with JT distortion corresponding to one of the minima in (b). The contributions from the vanadium $d_{xy}, d_{x^2-y^2}$ and $d_{z^2}$ orbitals are shown as half-filled green, half-filled blue, and open red circles, respectively. Partial charge densities of the bands from (c) with $d_{xy}$ (d) and $d_{x^2-y^2}$ (e) character with an isovalue of 0.001 e ${\AA }^{-3}$. The red arrow indicates the displacement of the dopant atom (purple circle) when relaxed.

Figure 7

Schematic diagram of the energy gain with spin orientation in-plane ($\mathbf{M}\parallel a$) and out-of-plane ($\mathbf{M}\parallel c$) for single and double acceptors. The solid and dashed blue lines represent spin degenerate and spin resolved levels, respectively. Solid and open circles denote electrons and holes, respectively.

Figure 8

Band structures for (a) a defect-free ${\mathrm{MoS}}_2$ monolayer, (b) a Mo vacancy ${Ø}_{\mathrm{Mo}}$, (c) a sulfur vacancy ${Ø}_{\mathrm{Mo}}$, (d) a ${\mathrm{Mo}}_{\mathrm{S}}$ antisite defect consisting of a Mo atom on a S site, (e) a vanadium atom on a Mo site ${\mathrm{V}}_{\mathrm{Mo}}$, (f) a Mo vacancy next to a substitutional vanadium atom ${Ø}_{\mathrm{Mo}}+{\mathrm{V}}_{\mathrm{Mo}}$, (g) a sulfur vacancy next to a substitutional vanadium atom ${Ø}_{\mathrm{S}}+{\mathrm{V}}_{\mathrm{Mo}}$, (h) a ${\mathrm{Mo}}_{\mathrm{S}}$ antisite defect next to a substitutional vanadium atom ${\mathrm{Mo}}_{\mathrm{S}}+{\mathrm{V}}_{\mathrm{Mo}}$ all modelled for ${\mathrm{MoS}}_2$ with a $6\times 6$ supercell. The zero of energy is the top of the valence band, which is not always visible in a finite supercell; its determination is discussed in [8]. The Fermi level is indicated by a dashed black line. The inset in (d) is a blow-up of the band structure near the Fermi level.

Figure 9

Orbital resolved band structure of an ${\mathrm{MoS}}_2$ monolayer with one Mo atom replaced with a Re atom in a (a) $6\times 6$ and (b) $12\times 12$ supercell. The open red circles and half-filled blue circles represent the contributions from Re $d_{3z^2-r^2}$ and $d_{x^2-y^2},d_{xy}$, respectively. The Fermi level is indicated by the black dashed line.

Figure 10

Difference between the total energies of antiferromagnetic and ferromagnetic states for pairs of substitutional V impurities in $12\times 12$ supercells plotted as a function of their separation for ${\mathrm{MoTe}}_2$ monolayers with (half-filled blue circles) and without (half-filled black circles) relaxation. The solid lines are a guide to the eye. The inset shows the spin resolved bonding antibonding levels of ferromagnetically coupled V pairs in which red and black lines represent different spin channels. $\mathrm{\Delta }$ represents the exchange splitting of the impurity states and $\delta$ the splitting of the degenerate $e^{\prime }$ levels as the V impurities get closer and interact. Open circles denote the holes.

Figure 11

Difference between the total energies of antiferromagnetic and ferromagnetic states for pairs of substitutional V impurities including relaxation in $12\times 12$ supercells plotted as a function of their separation for (a) $\mathrm{Cr}{X}_2$ and (b) $\mathrm{Mo}{X}_2$ ($X$= S, Se, Te) monolayers. The solid lines are a guide to the eye.

Figure 12

Curie temperature as a function of doping concentration for V-doped ${\mathrm{MoSe}}_2$ (red square) and ${\mathrm{CrTe}}_2$ (blue half-filled circle) monolayers.

Figure 13

Local magnetic moment per atom in ${\mu }_{\mathrm{B}}$ as a function of Coulomb $U$ (eV) on the $3d$ orbitals for hcp Ti, bcc V, bcc Fe, hcp Co, fcc Ni.

Figure 14

Radial distribution of the magnetization arising from a single substitutional V acceptor in ${\mathrm{WSe}}_2$ calculated for (a) the LSDA using a $12\times 12$ supercell and using a $9\times 9$ supercell (b) the $\mathrm{LSDA}+U$ with $U=3\mathrm{eV}$, (c) the $\mathrm{LSDA}+U$ with $U=3\mathrm{eV}$ with SOC included in the SCF cycle.

Figure 15

Difference between the total energies of antiferromagnetic and ferromagnetic states for pairs of substitutional V impurities including relaxation in $9\times 9$ supercells plotted as a function of their separation for ${\mathrm{WSe}}_2$ monolayers when $U=3\mathrm{eV}$ is included on V. The LSDA results calculated in a $12\times 12$ supercell are included for comparison ($U=0\mathrm{eV}$). The solid lines are a guide to the eye. The labeling of the W sites surrounding the central impurity atom used on the top $x$ axis is the same as that used in Fig. 3 of [8].