Analytical analysis of the spin wave dispersion in the cycloidal spin structures under the influence of magneto-electric coupling

Abstract

Spin waves and coupling of the spin waves with electromagnetic waves are considered in the multiferroic materials with the electric dipole moment proportional to the scalar product of spins. Nature of this interaction is discussed within the spin current model. Dispersion dependence for the spin waves propagating as the perturbation of the equilibrium state described by spin cycloid is found analytically. Contribution of the wave vector of the equilibrium cycloid is traced and it is found that it decreases the contribution of the anisotropy constant, which can lead to the instability. Limit regime of the spin waves in systems of collinear spins is described to show the role of the magnetoelectric coupling for two regimes of the electric dipole moment: it proportional to the scalar product of spins or to the vector product of spins. The dielectric permeability as the response on the electromagnetic perturbations associated with the magneto-electric coupling for the same equilibrium state is calculated.

Figures (5)

• Figure 1: The figure illustrates comparison of the Jacobi elliptical function (at $\nu=0.95$) $F(x)=sn(x,\nu)$ (red thick curve) and the trigonometric sin(x) (we choose period to match sin(0.54x) and present it with black thin curve). At $\nu<0.9$ there is good agreement between sin of corresponding period and $sn(x,\nu)$.
• Figure 2: The figure shows the dispersion dependence of spin wave in cycloidal structure in the easy-axis regime for the equal signs of the cycloid amplitudes $S_c=S_b$ in accordance with equation (\ref{['MFMemf disp dep with YiX NOa k NOd dimless']}). Here $\xi=\omega/\omega_{0}$ is the dimensionless frequency, and $\nu=k/q$ is the dimensionless wave vector. Parameter $r=Aq^{2}/\kappa$ chosen to be equal to $r_1=2$ (the lower thick continuous red line), $r_2=1$ (the second from below thin continuous black line), $r_3=0.5$ (the second from above thick dashed blue line), $r_4=0.3$ (the upper thin dashed black line).
• Figure 3: The figure shows the dispersion dependence of spin wave in cycloidal structure in the easy-axis regime for the different signs of the cycloid amplitudes $S_{c}=-S_{b}$ in accordance with equation (\ref{['MFMemf disp dep with YiX NOa k NOd dimless']}). Parameter $r=Aq^{2}/\kappa$ chosen to be equal to $r_{1}=2$ (the upper thick continuous red line), $r_{2}=1$ (the second from above thin continuous black line), $r_{3}=0.5$ (the second from below thick dashed blue line), $r_{4}=0.3$ (the lower thin dashed black line).
• Figure 4: The figure shows the dimensionless form for the real part of the dielectric permeability $Re\Xi=\kappa^{yy}_{R}/(4q^2 S_{b}^{2}S_{c}^{2}c_{2}^{2}\delta^{2}k \mid\kappa\mid)$ (\ref{['MFMemf kappa yy Re dimless']}) as the function of the dimensionless frequency $\textrm{w}=\omega/(\mid\kappa\mid S_{b})$. The dimensionless wave vector of the spin cycloid is equal $\tilde{q}=\sqrt{A/\mid\kappa\mid}q=0.2$. The wave vector (dimensionless) of the perturbation is chosen as the parameter $\tilde{k}=\sqrt{A/\mid\kappa\mid}k$: $\tilde{k}=0.15$ for the red continuous line, $\tilde{k}=0.25$ for the black thin continuous line, and $\tilde{k}=0.4$ for the blue dashed line. The damping constant is chosen to be $aS_{b}=-0.1$. The Dzyaloshinskii-Moriya interaction contribution in $\mathcal{M}$ is dropped in this estimation as the correction to the Heisenberg exchange interaction.
• Figure 5: The figure shows the dimensionless form for the imaginary part of the dielectric permeability (\ref{['MFMemf kappa yy Im dimless']}) as the function of the dimensionless frequency $\textrm{w}=\omega/(\mid\kappa\mid S_{b})$. The dimensionless wave vector of the spin cycloid is equal $\tilde{q}=\sqrt{A/\mid\kappa\mid}q=0.2$. The wave vector (dimensionless) of the perturbation is chosen as the parameter $\tilde{k}=\sqrt{A/\mid\kappa\mid}k$: $\tilde{k}=0.15$ for the red continuous line, $\tilde{k}=0.2$ for the the black thin continuous line, $\tilde{k}=0.3$ for the blue dashed line, and $\tilde{k}=0.5$ for black thin dashed line. The damping constant is chosen to be $aS_{b}=-0.1$. The Dzyaloshinskii-Moriya interaction contribution in $\mathcal{M}$ is dropped in this estimation as the correction to the Heisenberg exchange interaction.