Landau-Lifshitz-Bloch equation for exchange-coupled grains

Heat-assisted recording is a promising technique to further increase the storage density in hard disks. Multilayer recording grains with graded Curie temperature is discussed to further assist the write process. Describing the correct magnetization dynamics of these grains, from room temperature to far above the Curie point, during a write process is required for the calculation of bit error rates. We present a coarse-grained approach based on the Landau-Lifshitz-Bloch (LLB) equation to model exchange-coupled grains with low computational effort. The required temperature-dependent material properties such as the zero-field equilibrium magnetization as well as the parallel and normal susceptibilities are obtained by atomistic Landau-Lifshitz-Gilbert simulations. Each grain is described with one magnetization vector. In order to mimic the atomistic exchange interaction between the grains a special treatment of the exchange field in the coarse-grained approach is presented. With the coarse-grained LLB model the switching probability of a recording grain consisting of two layers with graded Curie temperature is investigated in detail by calculating phase diagrams for different applied heat pulses and external magnetic fields.

Figure 1

Zero-field equilibrium magnetization $m_{\mathrm{e}}$ versus temperature, calculated from an atomistic model of the HM material (see Table 1). The red solid line shows a fit, representing an infinite system.

Figure 2

Transverse and parallel susceptibilities versus temperature of a HM material (see Table 1), obtained by the fluctuations of the magnetization components in an atomistic model. The lines show fit functions extrapolating to the critical behavior of an infinite system. The dashed line indicates $T_{\mathrm{C}}$.

Figure 3

Susceptibilities versus temperature of the SM material (see Table 1), obtained by the fluctuations of the magnetization components in an atomistic LLG model. The dashed line indicates $T_{\mathrm{C}}$. The data belong to a cylindrical particle consisting of the SM material (see Table 1) and are simulated with vampire.

Figure 4

Variance of the magnetization length fitted to the average of the Cartesian components of the susceptibility, above $T_{\mathrm{C}}$. The resulting function of ${\tilde{\chi }}_m$ serves as parallel susceptibility. The dashed line indicates $T_{\mathrm{C}}$. The data belong to a cylindrical particle consisting of the SM material (see Table 1) and are simulated with vampire.

Figure 5

Fit of the parallel susceptibility of the SM material according to Eq. (16). The dashed line indicates $T_{\mathrm{C}}$.

Figure 6

Identical fluctuations of the $z$ component of the magnetization $\left({\tilde{\chi }}_{\parallel }\right)$ and its length $\left({\tilde{\chi }}_m\right)$, after scaling the latter. The dashed line indicates $T_{\mathrm{C}}$. The data belong to a cylindrical particle consisting of the HM material (see Table 1) and are simulated with vampire.

Figure 7

Grain model consisting of a stack of two layers with high and low Curie temperatures, coupled via an intergrain exchange interaction on the boundary surface. Each layer is represented as a single magnetization vector $({\mathbf{m}}_1$ and ${\mathbf{m}}_2)$.

Figure 8

Comparison of atomistic switching probability curves (green lines with circles) with the results of the coarse-grained LLB model (red solid lines) for different intergrain exchange constants $[A_{\mathrm{iex},n}\left(0\right)=2.575\times {10}^{-11}/2^n$ J/m, $0\le n\le 5]$. The investigated high/low $T_{\mathrm{C}}$ grain is subject to a Gaussian heat pulse with $t_{\mathrm{pulse}}=100$ ps and an external field with 0.5 T strength.

Figure 9

Linear fit of the inverse correction factors $1/k_{\mathrm{cor}}$ of the intergrain exchange field in the LLB model consisting of two HM layers (see Table 1). The bulk exchange interaction within the layers is assumed to be $A_{\mathrm{ex}}=2.158\times {10}^{-11}/2^n$ J/m.

Figure 10

Switching probabilities as in Fig. 8, with a corrected intergrain exchange field at different $A_{\mathrm{iex}}\left(0\right)$. The correction function is constructed as described in Sec. 4a.

Figure 11

Same switching probabilities as presented in Fig. 8 for a Gaussian heat pulse with $t_{\mathrm{pulse}}=10$ ps. The exchange field in the LLB model is corrected as described in Sec. 4a.

Figure 12

Same switching probabilities as presented in Fig. 8 for a Gaussian heat pulse with $t_{\mathrm{pulse}}=100$ ps and an external field with 0.8 T strength. The exchange field in the LLB model is corrected as described in Sec. 4a.

Figure 13

Switching probabilities of a high/low $T_{\mathrm{C}}$ recording grain subject to a Gaussian heat pulse with different lengths $t_{\mathrm{pulse}}$ and peak temperatures. The material parameters of the layers are given in Table 1 and the intergrain exchange constant at zero temperature is $A_{\mathrm{iex}}=2.575\times {10}^{-11}$ J/m. Additionally, an external magnetic field of 0.5 T is applied to the grain.

Figure 14

Same as Fig. 13 with an external magnetic field of 0.8 T.