Spectrum of collective vibrations of vortex domain walls in a ferromagnetic nanostripe array

Abstract

A problem of the spectrum of the collective periodic motion of vortex domain walls in a system of parallel ferromagnetic nanostripe is theoretically solved. The magnetic subsystems of stripes are coupled by the magnetostatic interaction. The effect of the distribution of vortex core polarities and chiralities on the character of periodic motion and the spectrum of collective modes of the nanostripe magnetization is discussed. Analytical expressions for the dispersion law and damping parameter of the collective periodic motion of vortex domain walls are obtained.

Graphic Abstract

Appendix A

In the projections onto the coordinate axes for Eq. (6), we have

$$\begin{aligned} \begin{array}{ll} -Gv_y &{}+D v_x+\sum \limits _n\kappa _{x_n}(x-x_n)=F_{H_x}, \\ Gv_x &{}+D v_y+\sum \limits _n\kappa _{y_n}(y-y_n)+\chi y=F_{H_y}. \end{array} \end{aligned}$$

(A.1)

After simple transformations, system of Eq. (A.1) for the x and y components can be reduced to a single equation, e.g., for the x component. To do that, we multiple the upper equation by $\varrho$ and the lower equation, by G and summate them. Thus, we arrive at

$$\begin{aligned}&\left( G^2+D^2\right) v_x +\sum \limits _n\left[ G\kappa _{y_n}(y-y_n) +D\kappa _{x_n}(x-x_n)\right] \nonumber \\&\quad +G\chi y=GF_{H_y}+D F_{H_x}. \end{aligned}$$

(A.2)

Next, we take the time derivative of both sides of this equation and substitute $v_y$ expressed from upper Eq. (A.1) into it. As a result, we obtain the differential equation containing the core coordinate x as the only unknown function:

$$\begin{aligned}&\left( G^2+D^2\right) \frac{{\text {d}}v_x}{{\text {d}}t} +Dv_x\left( \chi +\sum \limits _n\left( \kappa _{y_n}+\kappa _{x_n}\right) \right) \nonumber \\&\quad -D\sum \limits _n\left( v_{x_n}\left( \kappa _{x_n}+\kappa _{y_n}\frac{G}{G_n}\right) \right) \nonumber \\&\quad +\left( \chi +\sum \limits _n\kappa _{y_n}\right) \sum \limits _n\left( \kappa _{x_n}\left( x-x_n\right) \right) \nonumber \\&\quad -\sum \limits _n\left[ \kappa _{y_n}\frac{G}{G_n}\sum \limits _{m\ne n }\kappa _{x_m}(x_n-x_m)\right] \nonumber \\&= -G\frac{dF_{H_y}}{{\text {d}}t}+D\frac{{\text {d}}F_{H_x}}{{\text {d}}t}\nonumber \\&\quad +F_{H_x}\left( \chi +\sum \limits _n\kappa _{y_n}\left( 1-\frac{G}{G_n}\right) \right) . \end{aligned}$$

(A.3)

An efficient technique for solving Eq. (A.3) is the complex variable method. Let the time dependence of the external magnetic field be described by a harmonic function varying at frequency $\omega$: $F_{H_x}=0$, $F_{H_y}=F_0e^{-i\left( \omega t+\varphi \right) }$). Then, as a trial solution for the steady regime, we choose

$$\begin{aligned} x_n(t)=x_0e^{i\left( \omega t-kn\right) }. \end{aligned}$$

(A.4)

Here, $x_0$ is the complex amplitude, k is the wavenumber, and i is the imaginary uni.

Substituting the solution of (A.4) into (A.3), we obtain the simple equation for unknown amplitude $x_0$

$$\begin{aligned} x_0\left( A\omega ^2-2i\omega B+C\right) =f. \end{aligned}$$

(A.5)

Here, for brevity, we introduced the designations

$$\begin{aligned} f= & {} -i\omega GF_0e^{-i\varphi }, \nonumber \\ A= & {} -\left( G^2+D^2\right) , \nonumber \\ B= & {} \frac{1}{2}D\left( \chi +4\sum \limits _{n>0}\kappa _{x_n}\sin ^2\left( \frac{kn}{2}\right) \right. \nonumber \\&\left. + 2\sum \limits _{n>0}\kappa _{y_n}\left( 1-\frac{G}{G_n}\cos \left( kn\right) \right) \right) , \nonumber \\ C= & {} 4\left( \sum \limits _{n>0}\kappa _{x_n}\sin ^2\left( \frac{kn}{2}\right) \right) \left( \chi \right. \nonumber \\&\left. +2\sum \limits _{n>0}\kappa _{y_n}\left( 1-\frac{G}{G_n}\cos \left( kn\right) \right) \right) , \end{aligned}$$

(A.6)

and used the fact that the quasi-elasticity coefficients $\kappa _x$ and $\kappa _y$ depend not on a specific number of an array element, but on the distance of this element from the isolated stripe for which the equation of motion is written. The same is valid for the ratio $G/G_n$.

For the absolute value of the amplitude, from Eq. (A.5) we obtain

$$\begin{aligned} |x_0|=\frac{\omega G}{\sqrt{\left( A\omega ^2+C\right) ^2+4B^2\omega ^2}}. \end{aligned}$$

(A.7)