Phase diagram of an Ising model for ultrathin magnetic films: Comparing mean field and Monte Carlo predictions

We study the critical properties of a two-dimensional Ising model with competing ferromagnetic exchange and dipolar interactions, which models an ultrathin magnetic film with high out-of-plane anisotropy in the monolayer limit. In this work, we present a detailed calculation of the $(\delta ,T)$ phase diagram, $\delta$ being the ratio between exchange and dipolar interaction intensities. We compare the results of both mean field approximation and Monte Carlo numerical simulations in the region of low values of $\delta$, identifying the presence of a recently detected phase with nematic order in different parts of the phase diagram, besides the well-known striped and tetragonal liquid phases. We also found that, in the regions of the phase diagram where Monte Carlo simulations display nematic order, the mean field approximation predicts hybrid solutions composed by stripes of different widths. Another remarkable qualitative difference between both calculations is the absence, in this region of the Monte Carlo phase diagram, of the temperature dependency of the equilibrium stripe width predicted by the mean field approximation.

Figure 1

Equilibrium stripe width as a function of $\delta$ at $T=0$. Filled circles indicate the values of delta at which two adjacent striped configurations take the same energy.

Figure 2

Stability lines $T_s(\delta ,h)$ for the MF striped solutions at finite temperatures (dashed). Full line: critical temperature $T_c\left(\delta \right)$.

Figure 3

Mean field phase diagram. The numbers in the low-temperature region indicate the equilibrium stripe width $h$ of the phases. Dashed lines correspond to first-order phase transition between ordered phases; the full line corresponds to the critical temperature $T_c\left(\delta \right)$. Inset: Zoom of the hybrid region near the boundary between the striped phases $h=1$ and $h=2$.

Figure 4

(Color online) Order-disorder transition temperature $T_c^{\left(2\right)}\left(\delta \right)$ and striped stability lines $T_s\left(\delta \right)$ for different values of $h$ (see text for details).

Figure 5

Staggered magnetization for different $\delta$ and $h$ values: $h=2$, $\delta =3.4$, and $L=60$ (×); $h=3$, $\delta =4.1$, and $L=48$ (◇); and $h=5$, $\delta =2.9$, and $L=40$ (엯).

Figure 6

(Color online) Structure factor for a $L=72$ system with $\delta =2.1$, $T=0.78$ (엯) and $\delta =2.7$, $T=1.12$ (◇). Continuous lines correspond to fittings using Eq. (19): for $\delta =2.1$, $T=0.78$, we obtained $\lambda =0.033$, $k_0=1.39$; for $\delta =2.7$, $T=1.12$, $\lambda =0.0149$, $k_0=0.947$. $S(0,k_y)=0\forall k_y$ in both cases.

Figure 7

Typical nematic spin configurations for $L=72$. (A) $\delta =2.10$, $T=0.78$; (B) $\delta =2.70$, $T=1.12$.

Figure 8

(Color online) Monte Carlo phase diagram. The numbers indicate the equilibrium width of the low-temperature striped phases. The order of the different phase-transition lines is indicated in the inset (see text for details). The continuous lines are a guide to the eyes.

Figure 9

Specific heat and fourth-order cumulant as a function of $T$ for $\delta =2.25$ and $L=48$.