Precursor nanoscale modulations in ferromagnets: Modeling and thermodynamic characterization

Using a Ginzburg-Landau model for the magnetic degrees of freedom with coupling to disorder, we demonstrate through simulations the existence of stripelike magnetic precursors recently observed in $\mathrm{Co}-\mathrm{Ni}-\mathrm{Al}$ alloys above the Curie temperature. We characterize these magnetic modulations by means of the temperature dependence of local magnetization distribution, magnetized volume fraction, and magnetic susceptibility. We also obtain a temperature-disorder strength phase diagram in which a magnetic tweed phase exists in a small region between the paramagnetic and dipolar phases.

Figure 1

Snapshots of stable magnetic structures at different temperatures. White (black) regions are up (down) magnetized regions and gray represents nonmagnetized areas. The system size is $L_x=L_y=512$ and $n_x=4$, $n_y=32$.

Figure 2

Local magnetization distribution at different temperatures for a system size defined by $L_x=L_y=256$, $n_x=8$, and $n_y=64$.

Figure 3

Inverse magnetic susceptibility vs temperature for three individual disorder realizations. The inset shows the detailed structure of one of the peaks at the onset of magnetic tweed.

Figure 4

Averaged inverse magnetic susceptibility vs temperature. Dashed lines indicate the expected behavior for the dipolar and paramagnetic phases. Arrows delimit the region of tweed.

Figure 5

Temperature behavior of the magnetized volume fraction. The pseudo-order parameter for different values of the strength of disorder. $L_x=L_y=256$, $n_x=8$, and $n_y=64$. The inset shows the scaling of different curves.

Figure 6

Phase diagram: disorder strength vs temperature. Note that $T_c>1$ at zero disorder strength due to the effect of the dipolar interaction (see Ref. 30).

Figure 7

Size distribution of magnetic domains in the $\widehat{x}$ direction at different temperatures.

Figure 8

Scaled function $S_{y^2-x^2}(0,\mathbf{k})$ for different degrees of shape anisotropy.