Edge magnetism in transition metal dichalcogenide nanoribbons: Mean field theory and determinant quantum Monte Carlo

Edge magnetism in zigzag transition metal dichalcogenide nanoribbons is studied using a three-band tight-binding model with local electron-electron interactions. Both mean field theory and the unbiased, numerically exact determinant quantum Monte Carlo method are applied. Depending on the edge filling, mean field theory predicts different phases: gapped spin dimer and antiferromagnetic phases appear for two specific fillings, with a tendency towards metallic edge-ferromagnetism away from those fillings. Determinant quantum Monte Carlo simulations confirm the stability of the antiferromagnetic gapped phase at the same edge filling as mean field theory, despite being sign-problematic for other fillings. The obtained results point to edge filling as yet another key ingredient to understand the observed magnetism in nanosheets. Moreover, the filling dependent edge magnetism gives rise to spin-polarized edge currents in zigzag nanoribbons which could be tuned through a back gate voltage, with possible applications to spintronics.

Figure 1

(Left) Schematic for the structure coordination in the trigonal prismatic (2H) phase of a TMD. (Right) Projection of the 2H structure onto the $xy$ plane, yielding a honeycomb lattice. Hopping terms directly involving the $X$ atoms are neglected in our model. Thus we identify only the relevant nearest $M$ neighbors by the vectors ${\mathit{R}}_{i=1,2,\cdots ,6}$. This lattice represents part of the nanoribbon with a width of 5 $M$ atoms we shall consider later. Each row of the ribbon is defined as a set of $M$ atoms for which $y$ is constant.

Figure 2

Band structures of the three-band tight-binding model, i.e., Eq. (1) with $U=0$, for various infinitely long zTMDNRs with a width of 64 $M$ atoms.

Figure 3

Folded band structure of the three-band tight-binding model, i.e., Eq. (1) with $U=0$, for a $\text{Mo}{\text{S}}_2$ nanoribbon with a width of 20 $M$ atoms and a doubled unit cell in the longitudinal direction. The bands corresponding to bulk states are faded. The dashed horizontal lines indicate band fillings where two types of antiferromagnetic order develop.

Figure 4

Mean field magnetic ordering for $U=2.94\text{eV}$ at zero temperature for varying edge filling. The size of the circles indicates the magnitude of the local spin density. Red corresponds to spin-up and grey to spin-down. Left to right: ferromagnetic phase on both edges (${\text{F}}_{MX}$); dimer phase (AF2); ferromagnetic phase on the $M$ edge (${\text{F}}_M$); antiferromagnetic phase (AF1).

Figure 5

$U\text{−}T$ mean field phase diagrams at: half edge filling (left); three-quarter edge filling (right). As indicated, blueish regions correspond to the paramagnetic phase. A ribbon of width $N_y=5 M$ atoms was used.

Figure 6

Mean field band structure for edge fillings of ${\nu }_{\mathrm{edge}}=0.23$ (top left - ${\text{F}}_{MX}$), ${\nu }_{\mathrm{edge}}=0.5$ (AF2), ${\nu }_{\mathrm{edge}}=0.60$ (${\text{F}}_M$) and ${\nu }_{\mathrm{edge}}=0.75$ (AF1). We used $U=2.94\text{eV}$ and $T=0$, and a ribbon width of 10 $M$ atoms. Spin-up bands are in dashed-red and spin-down in dotted-blue. Unoccupied bands are faded and the horizontal green line marks the Fermi energy. Ferromagnetic phases (left panels) are characterized by spin-splitting of edge bands and present edge metallicity (absence of a gap). In the ${\text{F}}_{MX}$ phase, both edges are polarized while in the ${\text{F}}_M$ phase only the $M$ edge is magnetized. The antiferromagnetic phases AF1 and AF2 (right panels) are insulating.

Figure 7

Comparison between ribbons with widths $N_y=5,10,20 M$ atoms at $T=0$. (Top) Staggered magnetization $m_{\mathrm{st}}$ and electron density $\langle n\rangle$ profiles as the row position $y$ is changed for $U=2.94\text{eV}$. Results for $N_y=10$ and $N_y=20$ are indistinguishable. (Bottom) $m_{\mathrm{st}}$ vs $U$ at the $X$ edge (blue) and $M$ edge (orange).

Figure 8

$U^{\prime }\text{−}U$ phase diagrams at $T=0$ for the multiorbital case, obtained for a ribbon width $N_y=16$ at the edge fillings ${\nu }_{\mathrm{edge}}=75\%$ [(a) and (b)], ${\nu }_{\mathrm{edge}}=62.5\%$ (c), and ${\nu }_{\mathrm{edge}}=65\%$ (d). On (a), we consider $J=J^{\prime }=0$ and assume that both edges become polarized simultaneously. On the remaining panels, we have $J=J^{\prime }=(U-U^{\prime })/2$. Dark-blue regions are paramagnetic, and the labels in the remaining phases stand for ferromagnetic on the $X$ edge (Ferro-$X$), on the $M$ edge (Ferro-$M$), or on both edges (Ferro-$MX$), and antiferromagnetic on the $X$ edge (AF-$X$), on the $M$ edge (AF-$M$), or on both edges (AF-$MX$).

Figure 9

(Top) Average sign obtained in the Monte Carlo sampling for varying $N_x$ and $U$ at fixed temperature $T=267\text{K}$, for ${\mathrm{MoS}}_2$ nanoribbons of 5 $M$ atoms of width for electron densities corresponding to ${\nu }_{\mathrm{edge}}\approx 0.75$ (error bars are negligibly small). (Bottom) Chemical potential used in the DetQMC algorithm to obtain the required edge filling for each system size (other parameters are kept the same as on the top panel).

Figure 10

Evidence for the AF1 phase obtained with DetQMC for ${\mathrm{MoS}}_2$ nanoribbons, with $U=2.76\text{eV}$ and $T=267\text{K}$. (Top) Spin-spin correlations for a $10\times 5$ ribbon for edge filling ${\nu }_{\mathrm{edge}}=0.75\pm 0.01$, measured with respect to the leftmost site of the $X\left(M\right)$ edge. The size of the circles indicates the magnitude of the correlations (stronger on the $X$ edge). Red corresponds to a positive correlation and blue to a negative correlation. (Bottom) Magnetic structure factor per row, normalized to its maximum value ($q=\pi$, at the $X$ edge). Here, we consider a $22\times 5 {\mathrm{MoS}}_2$ ribbon with ${\nu }_{\mathrm{edge}}=0.745\pm 0.008$. The error bars are negligibly small.

Figure 11

Spin-spin correlators between even and odd sites, along the $X$ edge of ${\mathrm{MoS}}_2$ nanoribbons measured with DetQMC for ${\nu }_{\mathrm{edge}}\approx 0.75$. On the top panels, we vary the Hubbard interaction $U$ and fix the temperature $T=267\text{K}$ for a $20\times 5$ ribbon. On the bottom panels, we vary $T$ and fix $U=2.94\text{eV}$ for a $12\times 5$ ribbon.

Figure 12

(Top) Magnetic structure factor $S_{\mathrm{row}}(q,y_X)$ normalized to the peak $S_{\mathrm{row}}(\pi ,y_X)$ for nanoribbons with a width $N_y=5 M$ atoms and $U=2.76\text{eV}$ at $T=267\text{K}$ for ${\nu }_{\mathrm{edge}}\approx 0.75$. The $q=\pi$ peak sharpens as $N_x$ increases. (Bottom) Spin-spin correlations corresponding to the curves on the top panel with additional data shown. We use $N_x=10,\cdots ,26$ in steps of 2. The error bars are smaller than the symbols. The green curves are fits to the DetQMC data using functions that decrease as a power law.

Figure 13

(Top) DetQMC results for $S_{\mathrm{row}}(\pi ,y)/N_x$, with $y=y_{\mathrm{edge}}$, as a function of $1/N_x$, for nanoribbons with a width of $N_y=5 M$ atoms at $T=267\text{K}$ and ${\nu }_{\mathrm{edge}}\approx 3/4$. Red circles are for $X$ edge and blue triangles for $M$ edge. The lines are fits to the data using Eq. (12). Dashed lines are extrapolations to the thermodynamic limit (we repeat the extrapolation illustrated on this panel for varying Hubbard interaction $U$ to produce the bottom panel). (Bottom) Comparison between the staggered magnetizations on the $X$ edge obtained with MFT (circles) and DetQMC (triangles) as a function of $U$. MFT results are for $N_y=5$ and $T=0$.

Figure 14

(Top) Variation of the mean field staggered magnetizations at the $M$ edge (orange) and the $X$ edge (blue) for the AF1 phase for different TMD nanoribbons, all of them with a width of 5 $M$ atoms and with $U=2.94\mathrm{eV}$. (Bottom) Example of the gapped AF2 (left) and AF1 (right) phases obtained with mean field theory at ${\nu }_{\text{edge}}=0.5$ for a ${\text{WSe}}_2$ nanoribbon with a width of 5 $M$ atoms and considering $U=2.94\mathrm{eV}$. We also recover the edge ferromagnetic phases for ${\nu }_{\text{edge}}\ne 0.5\text{and}0.75$ shown in Fig. 4 for all TMDs in the family, but we omit them here for the sake of brevity.