Magnetic relaxation and formation of magnetic domains in ultrathin films with perpendicular anisotropy

We have simulated the magnetic relaxation and the formation of magnetic domains in very thin films with strong anisotropy perpendicular to the film plane. We have used Monte Carlo simulation based on a two-dimensional classical Ising model on a square lattice including the long-range dipole-dipole interaction and an external magnetic field. Starting with all magnetic moments aligned, the magnetic relaxation exhibits two distinct behaviors, depending upon the relation $\alpha$ between the dipole-dipole and the exchange interactions. For $\alpha <{\alpha }_c\cong 0.37$, the magnetization follows an exponential decay, $\frac{M\left(t\right)\mathrm{}}{M\left(0\right)\mathrm{}}=\exp [-\frac{t}{\tau \left(T^{\prime }\right)}]$ ($\tau$ is the relaxation time and $T^{\prime }$ the reduced temperature) and the domain pattern at the beginning of the relaxation process is characterized by the nucleation of a few domains, followed by a rapid growth of the magnetic domain size. For $\alpha <{\alpha }_c$, the magnetization follows a power-law time decay, $\frac{M\left(t\right)\mathrm{}}{M\left(0\right)\mathrm{}}=a(\alpha , T^{\prime })t^{-\gamma \left(T^{\prime }\right)}$, with a demagnetization process associated with the nucleation of many domains at random positions in the film. In both cases, the system relaxes towards the ground state which, depending on the value of $\alpha$, exhibits a striped structure. With the present model we were able to obtain the energy of domain nucleation, $E_N$, for $\alpha <{\alpha }_c$, despite the complexity associated with long-range dipole-dipole interactions; we have also obtained the $\alpha$ dependence of the energy of domain nucleation, $E_N\left(\alpha \right)$.