2.4. Spin Wave Theory
2.4.2. Spin Wave Propagation
In analogy with conventional plane wave theory, a spin wave is characterized by an amplitude and a wave vector qv , defined as
n q=2 ⋅ˆ
λ
v π . (2-32)
Here, λ is the wavelength of the spin wave and nˆ is a unit vector that points in the direction of propagation. Figure 2-15 displays the wave vector associated with the spin wave shown there.
For the case of an extended isotropic magnetic thin film, various modes of spin wave propagation are analytically known. A distinction is usually made between the long wavelength, magnetostatic modes which are governed by long range dipolar interactions and the short wavelength, exchange dominated modes. The intermediate
yˆ
xˆ
qr zˆ
range is covered by spin waves that bear a combination of dipolar and exchange character. These are commonly referred to as the hybrid dipolar-exchange spin waves.
The term ‘magnetostatic’ originates from the fact that for a given wave number, the frequency of the spin wave mode is much smaller than the frequency of the associated electromagnetic mode, so that the magnetostatic approximation (see Appendix A of [35]) can be used for solving the Maxwell equations. Although the magnetostatic mode wavelengths are shorter than their electromagnetic counterparts, they remain sufficiently long so that exchange interactions can be neglected. Therefore, magnetostatic modes are effectively dominated by dipolar interactions.
The magnetostatic mode theory that was first established by Damon and Eshbach [37]
distinguishes between three main propagating modes, depending on the relative orientations of the magnon wave vector and the internal field in the thin film. These modes are known as the magnetostatic surface wave (MSSW), the magnetostatic backward volume wave (MSBVW) and the magnetostatic forward volume wave (MSFVW), whose field and wave vector orientations are depicted in Figure 2-16.
Note that this figure only shows the ‘pure’ modes for in-plane propagation at perpendicular (MSSW and MSFVW) and parallel (MSBVW) directions with the field.
Theoretically, spin wave propagation in an arbitrary direction relative to the field is possible, so that a mode can display a mixed character, although critical angles separate the modes between MSSW and MSBVW behavior [35].
Figure 2-16: MSSW, MSBVW and MSFVW magnetostatic mode geometries of in-plane wave-vector (qv||) and externally applied field (Br0
) for spin wave propagation in an extended magnetic thin film.
The modes introduced above are characterized by distinct dispersion relations that map the relation between mode frequency and wave vector. These dispersion relations can be calculated based on the energy contributions due to the different magnetic interactions at play. The relative energy contributions from the long range dipolar interaction and the short range exchange interaction (which is usually considered to act between nearest neighbors) determine whether a spin wave is dipolar or exchange dominated. Whereas the nature of the exchange interaction predicts a quadratic
MSSW MSBVW MSFVW
Br0
qv|| qv|| qv||
Br0
Br0
dependence of frequency on wave vector (see Section 2.3.1.4 or [36]), the dipolar interaction scales inversely with distance, leading to a linear wave vector dependence.
Only for sufficiently small wavelengths, the exchange contribution can no longer be neglected and should be included in the calculation of the modes. An excellent review of the magnetostatic mode theory for in-plane magnetized films, including a derivation of the governing dispersion relations is provided in [35]. These dispersion relations can also be found, in a slightly different form, in [38]. The following sections discuss the nature of the three main magnetostatic modes and introduce the dispersion relations that characterize them. These relations will be applied in the understanding of the optical Brillouin light scattering (BLS) experiments in Section 5.3 and are used to validate the results from the micro-magnetic simulations in Section 6.
2.4.2.1. Magnetostatic Backward Volume Wave
In the MSBVW dipolar mode geometry, the thin film is saturated in the in-plane xˆ -direction (Br0 = B0⋅xˆ
) and spin waves have their in-plane wave vector qv in the || xˆ -direction as well. These spin waves are characterized by deviations of the magnetic moments from the equilibrium xˆ-direction which can be decomposed in an out-of-plane ( zˆ ) and an in-out-of-plane ( yˆ ) part, as depicted in Figure 2-17.
Figure 2-17: In the MSBVW dipolar mode geometry, the deviations of the magnetic moments from their equilibrium x-direction can be decomposed in an out-of-plane and an in-plane phase. In the out-out-of-plane phase, the dipolar stray fields from neighboring volumes of magnetic moment lead to mode softening, whereas no stray fields are present in the in-plane phase, due to the infinitely extending thin film.
MSBVW qv||
Br0 out-of-plane phase
in-plane phase
yˆ xˆ zˆ
It is illustrative to consider the pure out-of-plane and in-plane phases to gain insight into the MSBVW mode behavior. During the out-of-plane phase, the deviations in the magnetic moment of adjacent thin film volumes oppose each other. Apart from the restoring fields that are present due to demagnetization effects (i.e. thin film in-plane shape anisotropy), the dipole moments experience additional stray fields from neighboring volumes of magnetization that tend to anti-align adjacent units of magnetic moment. Note that, with increasing wavelength, the volumes that are to be considered as magnetic dipoles grow larger as well. However, the average magnetization deviation stays the same, while the dipole interaction strength decreases with distance (i.e. wavelength). Therefore, for larger wavelengths (decreasing qv ), the energy minimization due to the dipole interaction is less ||
effective. Accordingly, for smaller wavelengths (increasing qv ) the dipole interaction ||
is stronger and a unit of magnetization experiences a stronger stray field that pushes its magnetization further out-of-plane. This leads to a mode softening with associated frequency decrease, which is expressed through the characteristic negative dispersion that is typically observed for the MSBVW mode. During the in-plane phase, due to flux closure (actually, non-closure, since it is an infinitely extending film), adjacent volumes experience no in-plane stray fields and thus no qv -dependent restoring force. ||
The dispersion relation that results for the dipolar MSBVW mode is given by [39]
(note that both H and Ms are expressed in A/m here)
In this equation, the reduction of the thin film in-plane shape anisotropy (∝Ms) due to the out-of-plane phase softening is expressed through the qv -dependent negative ||
contribution to the in-plane anisotropy μ0Ms. Naturally, increasing the film thickness d also decreases the in-plane shape anisotropy due to the increased softening from stray fields associated with larger magnetic moments. In the limit situation of very large dq|| , the in-plane shape anisotropy is effectively lifted by the softening action of the spin waves and the spins behave as if they precess freely in a field Hrapp
with no restoring forces, with a frequency given by
app
For qv = 0, the MSBVW frequency coincides with that of the MSSW mode, which is ||
discussed in Section 2.4.2.3. Note that the negative dispersion continues until the exchange-dominated regime is entered. For large qv , the exchange interaction has to ||
be included in the dispersion relation, which then assumes an additional quadratic dependence in qv , following from the the Herring-Kittel equation [40] as ||
⎟⎟
2.4.2.2. Magnetostatic Forward Volume Wave
In the MSFVW dipolar mode geometry, the thin film is saturated in the out-of-plane zˆ -direction (Br0 =B0 ⋅zˆ
) and spin waves are characterized by an in-plane wave vector qv which is chosen in the positive || xˆ-direction. These spin waves are characterized by deviations of the magnetic moments from the equilibrium zˆ -direction (provided the thin film is fully saturated out-of-plane) which can be decomposed in an in-plane anti-aligned (xˆ) and an in-plane shear ( yˆ ) phase, as illustrated in Figure 2-18.
Figure 2-18: In the MSFVW magnetostatic mode geometry, the deviations of the magnetic moments from their equilibrium z-direction can be decomposed in an in-plane anti-aligned and an in-in-plane shear phase. In the in-in-plane anti-aligned phase, the dipolar stray fields from neighboring volumes of magnetic moment lead to mode hardening, whereas no stray fields are present in the in-plane shear phase, due to the infinitely extending thin film.
MSFW
The in-plane shear phase is analogous to the MSBVW in-plane phase discussed in the previous section, in that no resulting stray fields act on the yˆ -direction dipole deviations from the equilibrium zˆ -direction. During the in-plane anti-aligned phase, however, a qv -dependent stray field acts on adjacent dipoles, which in this case leads ||
to mode stiffening, with the dipoles being pushed back towards their equilibrium out-of-plane orientations. In the limit situation of very large dq|| , the only effective anisotropy the system experiences is that due to the out-of-plane bias field, and the MSFVW mode frequency is given by the Kittel formula [41]
(
app s)
Although the MSFVW mode will not be used explicitly in the remainder of this text, it was introduced here for the sake of completeness and to gain a full understanding of the behavior of the modes, based on the occurring dipolar stray fields.
2.4.2.3. Magnetostatic Surface Wave
In the MSSW dipolar mode geometry, the thin film is saturated in the inplane yˆ -direction (Br0 = B0 ⋅yˆ
) and spin waves have their in-plane wave vector qv in the || xˆ -direction. These spin waves are characterized by deviations of the magnetic moments from their equilibrium yˆ -direction which can be decomposed in an out-of-plane ( zˆ ) and an in-plane anti-aligned ( yˆ ) phase, as illustrated in Figure 2-19.
The MSSW mode can be regarded as a combination of the MSBVW and MSFVW modes considering the mode softening in the out-of-plane phase and hardening in the in-plane, anti-aligned phase. When only dipolar interactions are considered, the dispersion relation for the MSSW mode (also known as the Damon-Eshbach mode) reads (equation (11) from [20] or [42])
(
app S)
S(
qd)
The resulting dispersion of the MSSW modes is positive, due to the mode hardening being the most important contribution.
Combining the dipolar and exchange contributions in a single expression for the dipolar-exchange modes, which display a hybrid dipolar and exchange dominated character, results in
(
qd)
In conclusion, the dispersion relations for the MSBVW, MSFVW and MSSW magnetostatic modes have been introduced and intuitively explained. The resulting mathematical expressions for the dispersion relations will be applied in the sections that consider the micro-magnetic modeling of these modes and in the discussion of the results of the optical BLS experiments.
Figure 2-19: In the MSSW dipolar mode geometry, the deviations of the magnetic moments from their equilibrium y-direction can be decomposed in an out-of-plane and an in-plane anti-aligned phase. In the out-of-plane phase, the dipolar stray fields from neighboring volumes of magnetic moment lead to mode softening, whereas in the in-plane anti-aligned phase, the dipolar stray fields from neighboring volumes of magnetic moment induce mode hardening. The resulting dispersion of the MSSW modes is positive, due to the mode hardening being the most important contribution.