Thin Films and Multilayers

The RFe$_{2}$ family of intermetallics have shown potential uses in a number of nano-scale applications such as magnetic media readheads and nanosensors and so their behaviour in thin films and other low-dimensional systems has become of great interest to fundamental and applied research. Going from bulk to a thin film increases the effect of otherwise negligible effects such as shape anisotropy, strain and interface effects.

Previous work[29] on expitaxial Laves Phase DyFe$_{2}$ thin film samples highlighted the importance of expitaxial strain in perturbing the magnetic behaviour seen in bulk samples. In pure rare earths the anisotropy energy is roughly an order of magnitude greater than the magnetoelastic energy. RFe$_{2}$ systems, however, show giant magnetostriction at room temperature. The magnetoelastic energies in these samples can be large enough to strongly influence the overall anisotropy energy and hence the magnetic easy axis.

In an unstressed sample the dominant influence on the direction of the easy axis is the magnetocrystalline anisotropy energy, $E_{mc}$, given by[30]

$$E_{mc} = K_{1}\left( \alpha^{2}_{x}\alpha^{2}_{y} + \alpha^{2}_{x... ...2}_{z} \right) + K_{2}\left( \alpha^{2}_{x}\alpha^{2}_{y}\alpha^{2}_{z} \right)$$ (6.3)

where $\alpha_{i}$ are the cosines of the magnetisation direction and $K_{1}$ and $K_{2}$ are the anisotropy constants. The direction of magnetisation is mainly related to the magnitude and sign of the anisotropy constants (eg in DyFe$_{2}$ $K_{1}$ is positive and $K_{2}$ is negative, and $\vert K_{2}\vert/K_{1} < 9$ leading to a magnetisation along the $\left[100\right]$ direction, see Section 5.2.1).

In the samples studied in this chapter there are strains caused by a difference in thermal contraction between film and substrate as the sample cools from the deposition temperature.[30] It is now necessary to take into account the magnetoelastic energy, $E_{me}$, given by

$$E_{me} = b_{0}\left( \varepsilon_{xx} + \varepsilon_{yy} + \varepsilon_{zz... ..._{xx} + \alpha^{2}_{y}\varepsilon_{yy} + \alpha^{2}_{z}\varepsilon_{zz} \right)$$

$$+ b_{2}\left( \alpha_{x}\alpha_{y}\varepsilon_{xy} + \alpha_{x}\alpha_{z}\varepsilon_{xz} + \alpha_{y}\alpha_{z}\varepsilon_{yz} \right)$$ (6.4)

where $\varepsilon_{ij}$ is the strain tensor and $b_{0}$, $b_{1}$ and $b_{2}$ are the magnetoelastic coefficients, and the elastic energy, $E_{el}$, given by

$$E_{el} = \frac{1}{2}C_{11}\left( \varepsilon^{2}_{xx} + \varepsilon^{2}_{y... ...} + \varepsilon_{yy}\varepsilon_{zz} + \varepsilon_{xx}\varepsilon_{zz} \right)$$

$$+\frac{1}{2}C_{44}\left( \varepsilon^{2}_{xy} + \varepsilon^{2}_{yz} + \varepsilon^{2}_{xz} \right)$$ (6.5)

where $C_{11}$, $C_{12}$ and $C_{44}$ are the cubic elastic constants.[30]

The main strain in the samples is a negative shear[30] of $\varepsilon_{xy} = -0.55\%$. This simplifies Equation 6.4 to $E_{me} = b_{2}\varepsilon_{xy}\alpha_{x}\alpha_{y}$. The total energy, $E_{tot}$, is thus

$$E_{tot} = E_{mc} + b_{2}\varepsilon_{xy}\alpha_{x}\alpha_{y} + \pi M^{2}\left( \alpha_{x} + \alpha_{y} \right)^{2}$$ (6.6)

where $\pi M^{2}\left( \alpha_{x} + \alpha_{y} \right)^{2}$ is the shape anisotropy as a function of the magnetisation, $M$.[30] The magnetic easy axis will orient so as to minimise the energies of the competing magnetocrystalline and magnetoelastic energies and the shape anisotropy. The shape anisotropy favours an in plane alignment. DyFe$_{2}$ in bulk has an easy axis along the $\left[001\right]$ direction, which lies in plane for thin film samples so we would expect these samples to show the same easy axis in the absence of significant magnetoelastic anisotropy. However, results from other studies do not show an in plane magnetisation[29,30].