Negative permeability due to exchange spin-wave resonances in thin magnetic films with surface pinning

We report a theory of the effective permeability of multilayered metamaterials containing thin ferromagnetic layers with magnetization pinned on either one or both surfaces. Because of the pinning and small film thickness, the lowest frequency magnetic resonances are due to nonuniform exchange spin waves with frequencies far above those expected for uniform ferromagnetic resonance in known magnetic materials. Yet, the coupling of the nonuniform spin-wave modes to the electromagnetic field is shown to be strong enough to lead, for magnetic parameters characteristic for conventional transition metal alloys, to negative values of the effective permeability at frequencies of several hundred gigahertzs. The permittivity of metals is already negative in this frequency range. Hence, this system represents a negative refractive index metamaterial at subterahertz frequencies. The ways by which to maximize the frequency and the strength of the negative magnetic response are analyzed.

Figure 1

(Color online) The sketch of the proposed metamaterial is schematically shown. Thin ferromagnetic layers are separated by layers of the host dielectric. The electromagnetic wave (light) propagates along $z$ direction. The light wavelength is much greater than the characteristic dimensions of the superlattice.

Figure 2

(Color online) The magnetic geometry of the ferromagnetic layers is shown for the (a) in-plane and (b) out-of-plane magnetizations.

Figure 3

(Color online) The calculated effective permeability is shown as a function of frequency for the cases of (1) in-plane and (2) out-of-plane magnetizations. The solid and dashed lines denote the real and imaginary parts of the effective permeability, respectively. The effective permeability is calculated for an array of CoFe films with thickness of 5 nm. The filling fraction of 0.25 is assumed. The spins are perfectly pinned at one surface of the film and are free at the other.

Figure 4

(Color online) The effective permeability ${\mu }_{\perp e}$ is plotted as a function of the frequency and dimensionless thickness $l$, for the same assumptions and parameter values as in Fig. 3.

Figure 5

(Color online) Effective permeability ${\mu }_{\perp e}$ is plotted as a function of the frequency and magnetization of saturation $M_0$, for the same assumptions and parameter values as in Fig. 3.

Figure 6

(Color online) The effective permeability ${\mu }_{\perp e}$ is plotted as a function of frequency and pinning parameter $\beta$. The calculations are for the same assumptions and parameter values as in Fig. 3, except the finite pinning strength assumed here.

Figure 7

(Color online) The effective permeability ${\mu }_{\perp e}$ is plotted as a function of the frequency and bias magnetic field $H_0$, for the same assumptions and parameter values as in Fig. 3 (spins are perfectly pinned at one side of the film).