Spin response and collective modes in simple metal dichalcogenides

Transition-metal dichalcogenide monolayers are interesting materials in part because of their strong spin-orbit coupling. This leads to intrinsic spin splitting of opposite signs in opposite valleys, so the valleys are intrinsically spin polarized when hole doped. We study spin response in a simple model of these materials, with an eye to identifying sharp collective modes (i.e., spin waves) that are more commonly characteristic of ferromagnets. We demonstrate that such modes exist for arbitrarily weak repulsive interactions, even when they are too weak to induce spontaneous ferromagnetism. The behavior of the spin response is explored for a range of hole dopings and interaction strengths.

Figure 1

Absorptive part of the spin response function Im ${\chi }_{\tau }(\mathbf{q},\omega )$ for $\mathbf{q}=0$, chemical potential ${\mu }_0=-0.49\mathrm{\Delta }$, and $U_0=0.2\mathrm{eV}$, with $\tau =+1$. Model parameters for band structure are given in Table 1. A sharp collective mode near $\omega \approx -0.0845\mathrm{\Delta }$ is prominent above a particle-hole continuum in the interval $-0.092\lesssim \omega /\mathrm{\Delta }\lesssim -0.087$, where $\mathrm{\Delta }=1.66$ eV.

Figure 2

The top panel, for ${\mu }_0=-0.49\mathrm{\Delta }$, in which there is only a single Fermi surface in the valley (demonstrated in Fig. 3), has a continuum of particle-hole excitations (shown in green) below some minimum frequency. The bottom panel has ${\mu }_0=-0.57\mathrm{\Delta }$, for which there are two Fermi surfaces in the valley, giving rise to the continuum modes with vanishingly small energies for $q_x>0.4k_0$, with $k_0=\mathrm{\Delta }/2ta$. For both panels, $U_0=0.2\mathrm{eV}$ and $\tau =+1$. Other parameters are listed in Table 1. Blue lines illustrate the collective spin-wave mode dispersion.

Figure 3

The band dispersion of Hamiltonian (1) showing a direct band gap $E_g$ between the valence and conduction bands and the separation of spin-polarized bands in the conduction band. Positions for two ${\mu }_0$ are marked in the right margin. $k_0=\mathrm{\Delta }/2ta$ is the scale of momentum. The parameters used are listed in Table 1, and $\tau =+1$.

Figure 4

Plot of a typical $\chi (\mathbf{q},\omega )$, Eq. (22), showing the particle-hole excitations of the spin-split valence bands below an energy ${\omega }_c$. At ${\omega }_1$, there is a single collective mode visible for which the real part of the denominator of Eq. (22) is zero. Here we have used $\mathbf{q}=\mathbf{0},{\mu }_0=-0.49\mathrm{\Delta },\tau =+1$, and $U_0=0.2\mathrm{eV}$.

Figure 5

Schematic representation of the left- and right-hand sides of Eq. (32) as functions of $\omega$, shown in red and blue, respectively. For low enough $k_F$, an isolated spin-wave mode is always present.

Figure 6

Spin-wave excitations and the particle-hole continuum as a function of the chemical potential ${\mu }_0$ shown for three different values of the interaction strength $U_0$ when $\mathbf{q}=\mathbf{0}$. The green band corresponds to the particle-hole continuum, as shown in Fig. 4. The blue dashed line corresponds to the isolated mode at frequency ${\omega }_1$ described in Figs. 4 and 5. The mode corresponding to Eq. (35) is barely visible as a red line. The vertical lines indicate the boundary beyond which the stability condition is violated (see main text for details).

Figure 7

The blue line depicts the dispersion of the isolated spin-wave excitation, i.e., the $\omega ,q_x$ points for which the real part of the denominator of the spin susceptibility given by Eq. (22) vanishes. The green continuum represents the particle-hole excitations for which the denominator of Eq. (22) has a nonvanishing imaginary component, as shown in Fig. 4. Here we have taken $U_0=0.2\mathrm{eV}$.

Figure 8

Spin susceptibility for $U_0=0.5\mathrm{eV}$, $\tau =+1$, and ${\mu }_0=-0.57\mathrm{\Delta }$. Two discrete spin-wave modes (indicated by arrows) are visible near $\omega =-0.055\mathrm{\Delta }$ and $\omega =-0.092\mathrm{\Delta }$, with the second mode very close to the continuum. However, the positivity $\omega (-\text{Im}\chi )>0$ does not hold for all $\omega$, implying that our assumed Hartree-Fock state is not the true ground state.