10.3 Interacting particles

In order to study the interaction between particles, pairs of two small ferromagnetic spheres have been simulated in different arrangements with respect to their easy axes. The same material parameters as in section  with a magnetocrystalline anisotropy constant of $K_1=2\times 10^{5}~\frac{\mathrm{J}}{\mathrm{m}^3}$ have been used and a temperature of 500 K was assumed.

The metastable lifetime has been measured as the time, until the average magnetization over both particles has decreased to zero, because this quantity can also be measured in experiments (cf. section ).

If the two particles are placed in the normal plane to their easy axes (figure ) and far apart (the centre to centre distance is four times their diameter, plot ``2f'' in figure ), then we find a probability of not switching, which is almost identical to that of a single particle (plot ``1'' in figure ). Thus, the influence of the other particle is negligible. However, if the centre to centre distance between the two particles is reduced to $1.2$ times the diameter (plot ``2c'' in figure ), the probability of not switching changes dramatically. The slope increases and the relaxation time $\tau$ decreases. For a single sphere we find $\tau=0.114 \mathrm{~ns}$, for the two spheres far apart $\tau=0.116 \mathrm{~ns}$, whereas the two close spheres give $\tau=0.0987 \mathrm{~ns}$. We can interpret this as a stray field effect. The stray field of one particle (for a homogeneously magnetized sphere it is the field of a dipole) is antiparallel to the initial magnetization in the other. Thus, it ``strengthens'' the external field and leads to an earlier magnetization reversal process.

Figure: Horizontal arrangement of two particles with a centre to centre distance of $1.2$ times the diameter

Figure: Probability of not switching for a single (1) and two interacting particles at a centre to centre distance of 4 (2f) and 1.2 (2c) times the diameter

Another interesting configuration is the vertical alignment of the particles parallel to their easy axes. In this configuration the stray field is much stronger and tries to align the magnetization of both particles parallel. Hence, this arrangement leads to a kind of ``shape anisotropy'', which stabilizes the magnetization. The average metastable lifetime for the vertically aligned spheres (plot ``2v'' in figure ) $\tau=0.113 \mathrm{~ns}$ is considerably higher than for the horizontally aligned spheres (plot ``2h'' in figure ), where we find $\tau=0.0987 \mathrm{~ns}$. It is also interesting to note, that the slope for the interacting particles is much larger than that for a single particle (plot ``1'' in figure ), which was identical to that of two horizontally arranged particles far apart.

A similar behaviour is observed for vertically aligned particles at a centre to centre distance of four times the diameter (figure ). In this case the mean metastable lifetime is again reduced as compare to the vertically aligned spheres at a distance of only $1.2$ times the diameter.

Figure: Vertical arrangement of two particles with a centre to centre distance of $1.2$ times the diameter

Figure: Probability of not switching for a single (1) and two horizontally (2h) and vertically (2v) aligned particles with a centre to centre distance of 1.2 times the diameter

Figure: Probability of not switching for a single (1) and two vertically aligned interacting particles at a centre to centre distance of 1.2 (2vc) and 4 (2vf) times the diameter