Ferromagnetic Resonance Modes in the Exchange-Dominated Limit in Cylinders of Finite Length

We analyze the magnetic mode structure of axially magnetized finite-length nanoscopic cylinders in a regime where the exchange interaction dominates, along with simulations of the mode frequencies of the ferrimagnet yttrium iron garnet. For the bulk modes, we find that the frequencies can be represented by an expression given by Herring and Kittel by using wavevector components obtained by fitting the mode patterns emerging from these simulations. In addition to the axial, radial, and azimuthal modes that are present in an infinite cylinder, we find localized “cap modes” that are “trapped” at the top and bottom cylinder faces by the inhomogeneous dipole field emerging from the ends. Semiquantitative explanations are given for some of the modes, in terms of a one-dimensional Schrodinger equation, which is valid in the exchange-dominant case. The assignment of the azimuthal-mode number is carefully discussed, and the frequency splitting of a few pairs of nearly degenerate modes is determined through the beat pattern emerging from them.

Figure 1

Transformation of a vector field.

Figure 2

Dashed lines show the analytical demagnetization field calculated from Eq. (36) using $4\pi M_0=1750\mathrm{Oe}$ for YIG cylinders with a diameter of 75 nm and five different lengths of 75, 150, 300, 600, and 1200 nm in a field of 2000 Oe. For comparison, the solid lines show the field computed by oommf along the line r = 0. Note how for long cylinders, the field profile near one cap is insensitive to the presence of the other cap.

Figure 3

Dashed lines show the behavior of mode function Z(z) versus z obtained from integrating Eq. (34) for the (00g) cap mode for cylinders with a diameter of 75 nm and heights of 75, 150, 300, 600, and 1200 nm. Solid lines show the oommf simulation results for the same parameters.

Figure 4

Dashed lines show mode function Z(z) versus z obtained from integrating Eq. (37) for the (00g), (000), (002), and (004) modes. Solid lines show the corresponding forms arising from oommf. Note how the behavior at the cylinder faces closely conforms to the boundary condition, Eq. (26).

Figure 5

Example of a FFT spectrum. Left and right y axes show ${\overline{S}}_x(\omega )$ and ${\overline{S}}_z(\omega )$. Power spectra are given in arbitrary units, but with relative scales for left and right axes as shown. See Eq. (40). A broad sinc pulse, as described in Eq. (38), is used to excite this spectrum.

Figure 6

Beat patterns emerging from the time evolution arising from splitting of (a) the $(\pm 105)$ modes with an average frequency 17.29 GHz and a splitting of 30.60 MHz, (b) $p=0,n_r=0$, g and u cap modes with an average frequency of 6.54 GHz and a splitting of 14.4 MHz.

Figure 7

(a) Typical fit of the radial oommf amplitude to the function $J_1(k_{\mathrm{\perp }}r)$ for the (−105) mode. (b) Typical fit of the axial oommf amplitude to the function $\cos k_z(z-h/2)$ for the (006) mode.

Figure 8

Frequencies of $(00n_z)$ modes versus $k_z$, as obtained from oommf simulations (triangles), Herring-Kittel formula (solid line), and Schrodinger equation (squares), as discussed in Sec. 2b. For Schrodinger equation data, $k_{z}h/\pi$ is a proxy for $n_z+2$.

Figure 9

Symbols show oommf simulations for mode frequencies versus dimensionless wavevector $k_z$ for p = 0 and $n_r=0,1\text{,}\mathrm{and}2$; solid lines show predictions of the HK expression.

Figure 10

oommf mode frequencies versus $k_z$ for $n_r=0$ and $p=0,-1,\pm 2,\pm 3$. Curves show predictions of the HK equation.

Figure 11

Mode pattern for lowest (Kittel-like) bulk (000) mode, with a frequency of 7.813 GHz. (a),(b) x-y Cross section of the imaginary and real parts of the Fourier transform amplitude through the cylinder midpoint; (c) real y-z cross section containing the cylinder axis. Arrows show the direction of spins. Spin orientations for the real part correspond to a time 1/4 cycle later than that for the imaginary part. Lines show contours of constant $|\mathbf{m}(r,\omega )|$ in (a),(b) and of constant $m_x(r,\omega )$ in (c); these values are also color coded according to the scale given to the right.

Figure 12

p = 0, n_r = 0, cap modes. (a) Real x-y cross section at z = 0 of the $(00g)$ symmetric cap mode; this mode has the globally lowest frequency of 6.543 GHz. (b),(c) Real and imaginary y-z cross sections of the g and u modes, respectively, containing the cylinder axis from which surface confinement is apparent.

Figure 13

(a),(b) Imaginary and real x-y cross sections at z = 0 of (−10u) antisymmetric cap mode with a frequency of 10.06 GHz. (c) Real y-z cross section containing the cylinder axis from which surface confinement is apparent.

Figure 14

(a),(b) Imaginary and real x-y cross sections at z = 0 of the (10u) antisymmetric cap mode with a frequency of 10.35 GHz. Important to note here is the retrograde character of spin winding.

Figure 15

(a),(b) Imaginary and real x-y cross sections at z = 39 nm of $(\pm 105)$ modes with a frequency of 17.29 GHz. Note the standing-wave behavior of these cross sections arising from superposition of azimuthally counterpropagating modes. However, at a fixed point in space, spins still precess counterclockwise. (c) Real y-z cross section containing the cylinder axis.

Figure 16

The projection technique described in Appendix pp2 is used to separate modes with p = +1 and p = −1 from the same data as that used to construct Fig. 16. Notably, azimuthal intensity is now constant, as appropriate for running waves.

Figure 17

Real x-y cross section of the $(\pm 305)$ bulk mode with a frequency of 31.25 GHz, showing multiplicity of azimuthal nodes. Note also the deep central node due to $r^3$ behavior of $J_3(k_{\mathrm{\perp }}r)$.

Figure 18

Real x-y and y-z cross sections of the (020) mode with a frequency of 56.64 GHz exhibiting multiple radial nodes.

Figure 19

Dependence of frequency on h for the lowest-$k_z$ (000) bulk mode, symmetric (00g) cap mode, and antisymmetric (00u) cap mode of a YIG cylinder with d = 75 nm. Also shown is the frequency predicted by the Kittel expression. The inset shows the region where the symmetric cap mode crosses over the antisymmetric cap mode for small h. In the region below about 100 nm, the (00g) mode replaces the (000) mode as the quasi-uniform mode we associate with the Kittel formula.

Figure 20

Contours of constant intensity for three modes of a YIG cylinder with d = 75 nm and h = 37.5 nm. (a) 4.59-GHz symmetric (00g) cap mode, corresponding to the Kittel mode in the thin-disk limit; notably, it has no nodes. (b) 13.57-GHz antisymmetric (00u) cap mode. (c) 42.97-GHz (000) bulk mode.