Metadynamics study of the temperature dependence of magnetic anisotropy and spin-reorientation transitions in ultrathin films

We employ metadynamics simulations to calculate the free-energy landscape of thin ferromagnetic films and perform a systematic study of the temperature dependence of magnetic anisotropy and of the spin-reorientation transitions. By using a simple spin model we recover the well-known power-law behavior of the magnetic anisotropy energy against magnetization and present a rather detailed analysis of the spin-reorientation transitions in ultrathin films. Based on tensorial exchange interactions and anisotropy parameters derived from first-principles calculations, we perform simulations for Fe double layers deposited on Au(001) and W(110). In the case of ${\mathrm{Fe}}_2\mathrm{W}\left(110\right)$ our simulations display an out-of-plane to in-plane spin-reorientation transition in agreement with experiments.

Figure 1

Temperature dependence of the magnetic anisotropy energy $K\left(T\right)$ of a monolayer with uniaxial on-site anisotropy $\lambda =0.01J$ and vanishing exchange anisotropy $d=0$. In the inset the free energy is shown at $T/T_{\mathrm{C}}=0.2$ as a function of the CV $\eta$. The simulations have been done on a $32\times 32$ lattice with the parameters $T_{\mathrm{m}}=10J, w_0=0.02J$, and $\sigma =0.03$ (see Sec. 2).

Figure 2

Log-log plot of the magnetic anisotropy energy of a $32\times 32$ square lattice with uniaxial anisotropy $\lambda =0.01J$ and vanishing exchange anisotropy $d=0$ as a function of the magnetization. Note that both the MAE and the magnetization are normalized to zero temperature.

Figure 3

Log-log plot of the magnetic anisotropy energy of a $32\times 32$ square lattice with anisotropic exchange coupling $d=0.01J$ and zero on-site anisotropy $\lambda =0$ as a function of the magnetization. Note that both the MAE and the magnetization are normalized to zero temperature.

Figure 4

Magnetic anisotropy energy as normalized to zero temperature of a model monolayer system as a function of the temperature. The metadynamics simulations have been done on a $64\times 64$ rectangular lattice with competing on-site and nearest-neighbor two-site anisotropy, $\lambda =0.05375J$ and $d=0.025J$, respectively. In the inset the free energy is shown for different temperatures as a function of the collective variable, $\eta =M_{\mathrm{z}}/M$.

Figure 5

Phase diagram of an fcc(001) ferromagnetic bilayer described by the model Hamiltonian Eq. (1) with nearest-neighbor exchange interactions, $J$ and $d$, and uniaxial anisotropies, ${\lambda }_1$ and ${\lambda }_2$. For the case of $d=0.005J$, the lower solid red line shows the boundary where the normal-to-plane and in-plane configuration have the same energy, while the region of canted ground states given by Eqs. (8) and (9) is comparable with the linewidth. The upper solid red line bounds the area where a normal-to-plane to in-plane spin reorientation occurs according to mean-field theory. This area becomes considerably narrower from metadynamics simulations, as shown by the colored area. The color bar to the right refers to $T_{\mathrm{r}}/T_{\mathrm{C}}$. The dashed and dotted lines are the boundaries of the region with a canted ground state for $d=0.05J$. For this case, points A (${\lambda }_1/d=8.66, {\lambda }_2/d=0$) and B (${\lambda }_1/d={\lambda }_2/d=4.33$) are chosen for further investigations (see the text).

Figure 6

Free-energy profiles from metadynamics simulations of a $2\times 64\times 64$ bilayer with nearest-neighbor exchange interactions, $d=0.05J$, and anisotropy constants ${\lambda }_1=8.66d, {\lambda }_2=0$ (point A in Fig. 5). The temperature is measured in units of $T_{\mathrm{C}}$. The low-temperature magnetic configuration is canted ($0<\eta <1$), while by increasing the temperature the system continuously turns into the phase with in-plane magnetization, showing the nature of a second-order phase transition.

Figure 7

Free-energy profiles from metadynamics simulations of a $2\times 64\times 64$ bilayer with nearest-neighbor exchange interactions, $d=0.05J$, and anisotropy constants ${\lambda }_1={\lambda }_2=4.33d$ (point B in Fig. 5). The temperature is measured in units of $T_{\mathrm{C}}$. At low temperature the magnetization points normal to the plane ($\eta =\pm 1$), while at the reorientation temperature, $T_{\mathrm{r}}\sim 0.366T_{\mathrm{C}}$, it suddenly jumps to in plane, as $\eta =0$ becomes the minimum position of the free energy, thus displaying a first-order phase transition.

Figure 8

Temperature dependence of the magnetic anisotropy energy of ${\mathrm{Fe}}_2\mathrm{Au}$(001). The simulations were performed on a $2\times 64\times 64$ lattice with the parameters $T_{\mathrm{m}}=10T_{\mathrm{C}}, w_0=0.16T_{\mathrm{C}}$, and $\sigma =0.04$.

Figure 9

Calculated magnetic anisotropy energy, $K\left(T\right)=F_{1\overline{1}0}\left(T\right)-F_{110}\left(T\right)$, of an Fe DL on top of W(110) as obtained by well-tempered metadynamics MC simulations on a $2\times 64\times 64\times$ lattice with the parameters $T_{\mathrm{m}}=2T_{\mathrm{C}}, w_0=0.02T_{\mathrm{C}}$, and $\sigma =0.04$.