Exchange Interaction and Exchange Anisotropy

The origin of the interaction which lines up the spins in a magnetic system is the exchange interaction. When spin magnetic moments of adjacent atoms $i$ and $j$ make an angle $\phi_{ij}$, the exchange energy, $w_{ij}$, between the two moments can be expressed as[10]

$$w_{ij} = -2JS^{2}\cos\phi_{ij}$$ (5.4)

where $J$ is the exchange integral and $S$ is the total spin quantum number of each atom. For positive values of $J$ this gives a minimum when $\phi_{ij}=0$ ie the spins are aligned parallel to each other.

Although the exchange interaction produces strong interactions between neighbouring magnetic atoms it can also be mediated by various mechanisms, producing long range effects. As the exchange energy for neighbouring atoms is dependent only upon the angle between them it does not give rise to anisotropy.

In multilayers where magnetic layers are separated by a non-magnetic layer, there can be an exchange coupling between the two magnetic layers mediated by, for example, the RKKY interaction (see Section 5.4). The exchange coupling is composed of two terms: an isotropic exchange coupling and an anisotropic Dzialoshinski-Moriya exchange coupling.

The total exchange interaction energy is given by[21]

$$E_{EX} = -2J(z)\boldsymbol{M}_{1}\centerdot\boldsymbol{M}_{2}$$ (5.5)

where $J(z)$ is the exchange coupling constant, $\boldsymbol{M}_{1}$ and $\boldsymbol{M}_{2}$ are the magnetisation of the adjacent magnetic layers and $z$ is the non-magnetic layer thickness. This product is a maximum for ferromagnetically aligned layers.

The anisotropic exchange energy is given by[21]

$$E_{DM} = \boldsymbol{J}_{DM}(z)\centerdot\left(\boldsymbol{M}_{1}\times\boldsymbol{M}_{2}\right)$$ (5.6)

where $\boldsymbol{J}_{DM}$ is the Dzialoshinski-Moriya exchange constant. The cross product gives a resultant that is perpendicular to the direction of the layer magnetisation. This is at a maximum when the two layer magnetisations are at right angles to each other. This can favour inplane moment alignment for positive $\boldsymbol{J}_{DM}$ and out of plane alignment for negative $\boldsymbol{J}_{DM}$.