Phase-field-crystal model for magnetocrystalline interactions in isotropic ferromagnetic solids

An isotropic magnetoelastic phase-field-crystal model to study the relation between morphological structure and magnetic properties of pure ferromagnetic solids is introduced. Analytic calculations in two dimensions were used to determine the phase diagram and obtain the relationship between elastic strains and magnetization. Time-dependent numerical simulations in two dimensions were used to demonstrate the effect of grain boundaries on the formation of magnetic domains. It was shown that the grain boundaries act as nucleating sites for domains of reverse magnetization. Finally, we derive a relation for coercivity versus grain misorientation in the isotropic limit.

Figure 1

Two-dimensional phase diagram of the magnetic-PFC model, calculated using the single-mode approximation. $\Delta B$ is the temperature and $n_0$ is the mean density of the system. The dots denote the liquid-solid coexistence lines and solid line denotes the Curie line of ferromagnetic phase transition.

Figure 2

The structure factor of the system of size $256\times 256$ at coexistence. We first allow the system to equilibrate in the absence of the magnetic field and then apply a magnetic field of $\left|\mathcal{B}\right|=9.5$ at the angle of $\theta$ with respect to the $x$ axis. The red hexagons (solid line) show the structure factor of the initial configuration and the green deformed hexagons (dashed line) in (a)–(d) show the magnetically induced deformation of the hexagonal lattice at different magnetic field directions.

Figure 3

The vertical axis represents the magnetization $M=m/m_s$ and the dotted lines represent the two roots of the magnetic free energy $f_{\varphi }$ given in Eq. (7), corresponding to positive and negative magnetization values that minimize the free energy. When an external magnetic field is applied, the spins tend to flip to align with the magnetic field. If the magnetic field is large enough $\left(\right|\mathcal{B}|>H_c=4.7\times {10}^{-3})$ one of the minima disappears. The value of ${\mathcal{B}}_a$ gives an estimation for the coercivity of the model with parameters $(B_s,t,v,\alpha ,W_0,\beta ,r_c,\omega ,\gamma )=(0.98,0.5,1/3,{10}^{-3},0.5,4\times {10}^{-2},{10}^{-2},1,1)$.

Figure 4

Hysteresis curves for single-crystal and bicrystalline systems. The two-grain system is produced by placing two hexagonal crystal lattices that are rotated symmetrically with respect to each other. To produce the perfect fit inside the boxes, the lattice spacing was chosen to be $dx=28\pi /\left(64\sqrt{3}\right)$ and the time step $dt={10}^{-3}$. It can be seen that the coercivity of a single crystal is larger than that of a bicrystal.

Figure 5

The functional form ${[a_1-a_2\sin \left(a_3\theta \right)]}^{3/2}$ fitted to the simulation data of the coercivity, $H_c$, vs. the grain boundary angle, $\theta$ (degrees), for a system of $L=256$. The fit parameters are $a_1=0.0054$, $a_2=0.0013$, and $a_3=0.086$. The coercivity decreases as we increase the grain boundary angle as predicted by the mean-field estimation.

Figure 6

Hysteresis curves for two-grain systems for different grain sizes.

Figure 7

Simulation coercivity data, $H_c$, as a function of system size, $D$, compared with the experimental data [19, 51]. To make the comparison more clear, the coercivity values from the simulation data are scaled by a factor of $5.5\times {10}^3$ to better compare the peak positions of both data sets.