Two-magnon scattering model

Two-magnon scattering, if applicable, can provide a simple model for the description of extrinsic damping in FMR. Inhomogeneous magnetic properties generate scattering of the resonant mode (uniform mode with q ∼ 0) into non-uniform modes (q = 0 magnons), as discussed in section 2.5. From Eq. 2.48 it follows that magnon scattering is confined to degenerate magnons following the path of lobes, q₀(ψ_q), around the direction of the magnetic moment (see Fig. 5.7).

Two-magnon scattering formally enters the in-plane rf susceptibility [12] as an additional termR in the denominator (cf. Eq. 2.53)

χ_||= M_sB

B_effH−(^ω_γ)²+i(H+B)α^ω_γ + [(R) +i(R)] , (5.1) whereB_eff =H+4πM_eff is the effective magnetic induction, and the in-plane anisotro-pies are neglected for simplicity. The real part, (R), leads to a shift in the FMR field and the imaginary part (R) provides additional damping. (R) and (R) have to satisfy the symmetry of the magnetic inhomogeneities. In a rectangular network of misfit dislocations one expects to get additional four-fold anisotropies,

M

[010]

[100]

M

Figure 5.7: The black lines show the lobes of degenerate magnons calculated for 73 GHz (big lobes) and 24 GHz (small lobes). The background is a contour plot of the in-plane Fourier components from Eq. 3.12. Note that Pukite’s type ofI(q), even if it were scaled to produce an anisotropic behavior for small q-vectors (large L), would result only in a weak anisotropy for ∆H.

affecting both the FMR field and magnetic damping. In addition both, (R) and (R) are dynamic effects and depend on the microwave frequency. This leads to an important conclusion: Magnetic anisotropies measured using FMR can have a frequency dependent contribution which is absent in dc measurements.

The strength of two-magnon scattering as a function of the angleϕM of the magne-tization with respect to the in-plane crystallographic axis can be tested by inspecting the expression (R). The imaginary part of the two-magnon contribution to the susceptibility is proportional to the Fourier components of the scattering potential.

Using this one can write [R(ϕ_M)]∼

I(q)δ(ω−ω_q)dq² = 2

ψ_max

−ψ_max

I(q₀, ϕ_q, ϕ_M) q0dψq

∂ω

∂q(q₀, ψ_q) , (5.2)

ϕ

Figure 5.8: The FMR fields for the 30Fe layer in 200Pd/30Fe/GaAs(001) as a function of the in-plane angle ϕ_H. Where ϕ_H = 0 corresponds to the [100]_Fe direction. Mea-surements were carried out at 73.0(✩), 36.4(◦), 24.9(), and 14.1( ) GHz. The solid lines were calculated using the following parameters: The effective demagnetizing field 4πMeff = 18.7 kG, in-plane uniaxial field 2K_U^||/MS=−360 Oe, four-fold in-plane anisotropy field 2K₁/M_S = 330 Oe, and g-factor, g= 2.11. The uniaxial in-plane anisotropy has the hard axis along the [1¯10]Fe direction. With these parameters the agreement between the measured and calculated FMR fields along the 110Fe direction is within 10 Oe for all frequencies. This is not true when the field applied along the 100Fe directions where the discrepancy is of the order of 100 Oe. It is not possible to get a perfect fit for all microwave frequencies by using a single set of parameters, indicating that the magnetic anisotropies are partly frequency dependent.

where I(q) represents the intensity of two-magnon scattering, and ϕ_q = ϕ_M +ψ_q is the angle of the q vector with respect to the [100]Fe (defect) axis. The magnon group velocity ^∂ω_∂q(q0, ψq) in Eq. 5.2 is proportional to the strength of the dipolar and exchange fields and represents the dipole-exchange narrowing of local inhomogeneities [49]. The term _∂ω ^q0

∂q(q₀,ψ_q) describes a weighting parameter along the path of the two-magnon scattering lobes q(ψ_q). It turns out that for a given microwave frequency this factor is nearly independent ofψ_q. This means that the whole two-magnon

scat-tering lobe contributes to (R) with an equal weight, independent of the angle ψ_q. The exception is the q-space close to the origin of the reciprocal space. It is important to realize that long wavelength (small q) variations in magnetic properties lead to a simple superposition of local FMR peaks. The extrinsic FMR linewidth in this case merely reflects large length scale sample inhomogeneities and should not be treated by two-magnon scattering. McMichael et al. [160] concluded that the FMR linewidth is given by a superposition of local resonances whenthe characteristic inhomogeneity field is larger than the interaction field [163]. In the range of long wavelength defects the important part of the interaction field between grains having different magnetic properties is the magneto-static contribution 2πM_sqt_F to the magnon energy dis-persion, where t_F is the film thickness. The FMR spectrum is given by a simple superposition of local FMR peaks when

H_pL≥3πM_st_F, (5.3)

where H_p is the root mean square value of random variations of a local anisotropy field satisfying a Gaussian distribution, andL is the average grain size (see Fig. 4 in [164]). The summation of local FMR signals can result in a genuine zero frequency offset ∆H(0), (see reference [164]).

The critical angle ψ_max appearing in Eq. 5.2 decreases with a decreasing angleθ_M of the magnetization with respect to the sample plane. For θ_M ≤π/4 no degenerate magnons are available [56]. The angle θ_H satisfying θ_M ≤ π/4 has to be calculated by minimizing the total magnetic energy. For the 200Pd/30Fe/GaAs sample this angle was θ_H = 12^◦ at 24 GHz, as can be seen in Fig. 5.10. This is an important criterion that allows one to test the applicability of two-magnon scattering to the interpretation of extrinsic damping. ∆H from the extrinsic damping has to disappear when the direction of the magnetization is in the vicinity of the film normal. It is interesting to note that the weighing parameter _∂ω ^q0

∂q(q₀,ψ_q) is nearly independent ofθ_M. It increases somewhat very close to the critical angleθ_M =π/4, where the two-magnon scattering is switched off.