Comprehensive inelastic neutron scattering study of the multiferroic ${\mathrm{Mn}}_{1-x}{\mathrm{Co}}_x{\mathrm{WO}}_4$

Using high-resolution inelastic neutron scattering, we examine the spin dynamics of ${\mathrm{Mn}}_{1-x}{\mathrm{Co}}_x{\mathrm{WO}}_4$ in the collinear AF1, the $ac\text{−}b$ spiral AF2, and the $ac$ cycloidal AF5 phases. The spin wave excitations are well described by a Heisenberg model with competing long-range exchange interactions $(J_i$ up to 12th nearest neighbors) and the single-ion anisotropy $K$ induced by the spin-orbit interaction. While the exchange constants are relatively unchanged, the dominant effect of doping is to change the single-ion anisotropy from easy axis $\left(K>0\right)$ in the collinear AF1 phase to easy plane $\left(K<0\right)$ in the two multiferroic phases.

Figure 1

(a) The crystal structure of ${\mathrm{Mn}}_{1-x}{\mathrm{Co}}_x{\mathrm{WO}}_4$. The spin structures of (b) the collinear commensurate (CM) AF1 phase (magnetic point group $2/m{1}^{\prime })$ with spins in the $ac$ plane at an angle of $\sim {35}^{\circ }$ to the $a$ axis, (c) the noncollinear ICM AF2 phase $\left({21}^{\prime }\right)$ with one elliptical spiral axis in the $ac$ plane and the other along the $b$ axis, and (d) the noncollinear ICM AF5 phase $\left(m{1}^{\prime }\right)$ with the spin cycloid in the $ac$ plane. (The spin configuration plots are from previous publication of Ref. [29].) In (b), we label the magnetic interactions with increasing Mn-Mn bond distances.

Figure 2

SW dispersion spectra of pure ${\mathrm{MnWO}}_4$ along the (a) $[1,0,-2]$, (b) $[1,0,2]$, and (c) $[0,1,0]$ directions in the commensurate $\uparrow \uparrow \downarrow \downarrow$ AF1 phase. (d)–(f) The corresponding SW spectra calculated using the exchange parameters listed in Table 1.

Figure 3

SW dispersion spectra of pure ${\mathrm{MnWO}}_4$ along (a) the $[1,0,-2]$, (b) $[1,0,2]$, and (c) $[0,1,0]$ directions in the incommensurate AF2 phase with $ac\text{−}b$ spiral spin structure at $T=8.5$ K. (d)–(f) The corresponding SW spectra calculated using the exchange parameters listed in Table 1.

Figure 4

(a) SW spectra of ${\mathrm{Mn}}_{0.88}{\mathrm{Co}}_{0.12}{\mathrm{WO}}_4$ along (a) the $[1,0,-2]$, (b) $[1,0,2]$, and (c) $[0,1,0]$ directions in the incommensurate AF5 phase with an $ac$ cycloidal spin configuration. (d)–(f) The corresponding SW spectra calculated using the exchange parameters listed in Table 1.

Figure 5

The distance dependence of the isotropic exchange interactions in the AF1, AF2, and AF5 phases. The corresponding Mn-Mn distances are displayed in Fig. 1.