Equilibrium and nonequilibrium properties of synthetic metamagnetic films: A Monte Carlo study

Synthetic antiferromagnets with strong perpendicular anisotropy can be modeled by layered Ising antiferromagnets. Accounting for the fact that in the experimental systems the ferromagnetic layers, coupled antiferromagnetically via spacers, are multilayers, we propose a description through Ising films where ferromagnetic stacks composed of multiple layers are coupled antiferromagnetically. We study the equilibrium and nonequilibrium properties of these systems where we vary the number of layers in each stack. Using numerical simulations, we construct equilibrium-temperature-magnetic-field phase diagrams for a variety of cases. We find the same dominant features (three stable phases, where one phase boundary ends in a critical end point, whereas the other phase boundary shows a tricritical point at which the transition changes from first to second order) for all studied cases. Using time-dependent quantities, we also study the ordering processes that take place after a temperature quench. The nature of long-lived metastable states are discussed for thin films, whereas for thick films we compute the surface autocorrelation exponent.

Figure 1

Sketch of a system where ferromagnetic stacks formed by $n=3$ layers are coupled antiferromagnetically. The black and green bonds indicate ferromagnetic and antiferromagnetic couplings, respectively.

Figure 2

The magnetization density as function of the magnetic field strength for a system with stacks containing four ferromagnetically coupled layers. The system is composed of $80\times 80\times 24$ spins. At low magnetic fields a first partial ordering takes place, followed by the alignment of all the layers with the field for a larger value of $h$. Data for different temperatures are shown. Here and in the following error bars are smaller than the symbol sizes.

Figure 3

The energy density as a function of the magnetic field strength for a system with three layers per stack. The size of the system is $80\times 80\times 18$. The two successive transitions show up as sudden increases followed by a decrease of the energy with increasing field strength.

Figure 4

The magnetic susceptibility vs the magnetic field strength for a system where every stack contains two layers. The system is composed of $20\times 20\times 8$ spins.

Figure 5

Temperature-magnetic-field phase diagrams for three different systems: $N=16$ and $n=2$ (black squares), $N=18$ and $n=3$ (red circles), $N=24$ and $n=4$ (blue triangles). The open symbols indicate the phase transition between the antiferromagnetically aligned stacks at low fields and the intermediate phase with a partial alignment to the magnetic field. For all studied cases this transition is discontinuous and ends in a critical end point. The lines with closed symbols show the phase transitions to the high-field paramagnetic phase. The large cyan circles indicate the tricritical points separating the discontinuous transition at low temperatures from the continuous transition at high temperatures. The phase diagrams have been obtained through a finite-size scaling analysis of the susceptibility and specific heat for systems with layer sizes ranging from $20\times 20$ to $320\times 320$ spins.

Figure 6

Temporal evolution of the layer magnetization densities for a metamagnetic film $\left(n=1\right)$ composed of eight layers. Whereas the left column presents runs where the system rapidly relaxes toward equilibrium, the middle and right columns show cases where the system gets trapped in a long-lived metastable state. The magnetic field strength is (a)–(c) $h=0$, (d)–(f) $h=0.6$, and (g)–(i) $h=1.1$, with the temperature being held constant at $T=2.5$. The system is prepared in a disordered initial state. Each layer contains $80\times 80$ spins.

Figure 7

Spin configurations in layers 1 to 4 at the end of some of the runs shown in Fig. 6 for which the system has not yet reached equilibrium after $t=10000$ MCSs. The left column shows Fig. 6, the middle column shows Fig. 6, and the right column shows Fig. 6. Spins with a value of $+1$ are shown as black squares.

Figure 8

Temporal evolution of the layer magnetization densities for a film of $N=8$ layers composed of four two-layer stacks $\left(n=2\right)$. Whereas the left column presents runs where the system rapidly relaxes toward equilibrium, the middle and right columns show cases where the system gets trapped in a long-lived metastable state. The magnetic field strength is (a)–(c) $h=0$, (d)–(f) $h=0.2$, and (g)–(i) $h=0.55$, with the temperature being held constant at $T=2.5$. The system is prepared in a disordered initial state. Each layer contains $80\times 80$ spins.

Figure 9

Spin configurations in layers 1 to 4 at the end of some of the runs shown in Fig. 8 for which the system has not yet reached equilibrium after $t=10000$ MCSs. The left column shows Fig. 8, the middle column shows Fig. 6, and the right column shows Fig. 6. Spins with a value of $+1$ are shown as black squares.

Figure 10

Time-dependent bulk and surface autocorrelation in the absence of a magnetic field for the metamagnet $\left(n=1\right)$ and the three-dimensional Ising model quenched inside the ordered phase. The slopes in the log-log plot yield the ratio between the bulk and surface autocorrelation exponent and the dynamic exponent $z$ (i.e., ${\lambda }_b/z$ and ${\lambda }_s/z)$. The solid line indicates a slope of 0.774, whereas the dashed line has a slope of 0.854. The size of the simulated systems is $100\times 100\times 80$.