Kinetic Monte Carlo approach to modeling thermal decay in perpendicular recording media

A procedure is developed to study the evolution of high anisotropy magnetic recording media due to thermally activated grain reversal. It is assumed that the system is composed of single domain grains that evolves by passing through a sequence of relatively long-lived metastable states punctuated by abrupt reversals of individual grains. Solutions to the rate equations describing the sequence of metastable states are calculated using kinetic Monte Carlo. Transition rates are formulated from the Arrhenius-Néel expression in terms of the material parameters, temperature, and applied field. Results obtained from this method are shown to be in good agreement with those calculated from finite-temperature micromagnetics. The method is applied to study the rate dependence of finite-temperature $MH$ loops and the thermal degradation of a recorded bit pattern in perpendicular recording media. A significant advantage of the procedure is its ability to extend simulations over time intervals many orders of magnitude greater than is feasible using standard finite-temperature micromagnetics with relatively modest computational effort.

Figure 1

Schematic diagram showing the relative alignment of the anisotropy axis, field, and magnetic moment.

Figure 2

Remanent coercivity $H_{\mathrm{cr}}$ as a function of the time interval for a system of $16\times 16$ grains of dimension $8\mathrm{nm}\times 8\mathrm{nm}\times 10\mathrm{nm}$ with $M_s=5\times {10}^5\mathrm{A}/\mathrm{m}$, $K=3.75\times {10}^5\mathrm{J}/{\mathrm{m}}^3$, $T=330\mathrm{K}$, and $f_0=1\mathrm{GHz}$. The anisotropy axis is aligned perpendicular to the plane of the film and results are presented for several values of ${\theta }_h$ as indicated. The solid line is calculated by the direct integration of the rate equations while the red circles are the results obtained using KMC.

Figure 3

Remanent coercivity $H_{\mathrm{cr}}$ as a function of the time interval for the model system as in Fig. 2. The anisotropy axis is normally distributed about perpendicular to the plane of the film with variance $\langle {\alpha }^2\rangle ={\sigma }_{\theta }^2$ and results are presented for several values of ${\sigma }_{\theta }$ as indicated. The solid line is calculated by the direct integration of the rate equations while the red circles are the results obtained using KMC.

Figure 4

Time evolution of the magnetization per spin for a system of $16\times 16$ grains with $M_s=5\times {10}^5\mathrm{A}/\mathrm{m}$, $A=5\times {10}^{-12}\mathrm{J}/\mathrm{m}$, $K=3.75\times {10}^5\mathrm{J}/{\mathrm{m}}^3$, ${\sigma }_{\alpha }=3^{\circ }$ for $T=330\mathrm{K}$. The LLG calculations were performed with a damping parameter $\alpha =0.01$. For the KMC calculations the attempt frequency was calculated from Eq. (18).

Figure 5

Remanent coercivity $H_{\mathrm{cr}}$ as a function of $t_0$ for $16\times 16$ rectangular array of cells with dimensions $8\mathrm{nm}\times 8\mathrm{nm}\times 10\mathrm{nm}$ with $M_s=5\times {10}^5\mathrm{A}/\mathrm{m}$, $K=3.75\times {10}^5\mathrm{J}/{\mathrm{m}}^3$, and ${\sigma }_{\alpha }=3^{\circ }$, with and without interactions, as indicated on the figure. In both cases $f_0=1\mathrm{Ghz}$ was assumed and, in the case of the exchange interaction, $A=5\times {10}^{-12}\mathrm{J}/\mathrm{m}$ was used.

Figure 6

Evolution of a synthetic bit pattern calculated using KMC, with $T=450\mathrm{K}$ and a constant attempt frequency $f_0=100\mathrm{GHz}$. Each pixel represents a single grain with dimensions of $6.2\mathrm{nm}\times 6.2\mathrm{nm}\times 20\mathrm{nm}$, a saturation magnetization of $M_s=5.5\times {10}^5$ A/m and an anisotropy constant of $K=3.5\times {10}^5$ J/m${}^3$. The unixial anisotropy orientation is normally distributed about the perpendicular axis of the film with a variance ${\sigma }_{\alpha }=3^{\circ }$.

Figure 7

Schematic showing the strip of empty cells used to calculate the average perpendicular stray field detected by the transducer as it passes over the bit pattern.

Figure 8

Bit pattern SNR calculated using Eq. (19) as a function of the product $f_{0}t$ for 12 different runs are plotted as a function of $f_{0}t$. The runs consisted of four different temperatures ($T=300$ K, $T=330$ K, $T=350$ K, and $T=450$ K) using three different attempt frequencies ($f_0=1$ GHz, $f_0=10$ GHz, and $f_0=100$ GHz).

Figure 9

SNR vs $f_{0}t$ for the parameters and temperatures given in Fig. 8 (solid lines) and the results for $T=450\mathrm{K}$ scaled according to the expressions given by Eqs. (20) and (21) (dotted lines).

Figure 10

Zero-temperature $MH$ loop for a system of $16\times 16$ grains of dimension $6.2\mathrm{nm}\times 6.2\mathrm{nm}\times 20\mathrm{nm}$ with $M_s=5.5\times {10}^5\mathrm{A}/\mathrm{m}$, $K=3.5\times {10}^5\mathrm{J}/{\mathrm{m}}^3$ and $A=5\times {10}^{-12}\mathrm{J}/\mathrm{m}$ with ${\sigma }_{\alpha }=3^{\circ }$. Data are shown for both LLG simulations and for the $T\to 0$ limit of the KMC algorithm.

Figure 11

$M\mathrm{vs}H$ loops at $T=330\mathrm{K}$ for an ensemble of $16\times 16$ grains of dimension $6.2\mathrm{nm}\times 6.2\mathrm{nm}\times 20\mathrm{nm}$ with $M_s=5.5\times {10}^5\mathrm{A}/\mathrm{m}$, $K=3.5\times {10}^5\mathrm{J}/{\mathrm{m}}^3$, and $A=5\times {10}^{-12}\mathrm{J}/\mathrm{m}$ with ${\sigma }_{\alpha }=3^{\circ }$. Data are shown for both LLG simulations, with $\alpha =0.1$ and KMC, with $f_0=50\mathrm{Ghz}$ for $\Delta H/\Delta t$ $=$ (a) $1\times {10}^{12}\mathrm{Oe}/\mathrm{s}$, (b) $1\times {10}^{11}\mathrm{Oe}/\mathrm{s}$, (c) $1\times {10}^{10}\mathrm{Oe}/\mathrm{s}$, and (d) $1\times {10}^9\mathrm{Oe}/\mathrm{s}$.

Figure 12

$M\mathrm{vs}H$ at $T=300$ for several sweep rates for a $16\times 16$ ensemble of grains of dimension $6.2\mathrm{nm}\times 6.2\mathrm{nm}\times 20\mathrm{nm}$ with $M_s=5.5\times {10}^5\mathrm{A}/\mathrm{m}$, $K=3.5\times {10}^5\mathrm{J}/{\mathrm{m}}^3$, and $A=5\times {10}^{-12}\mathrm{J}/\mathrm{m}$ with ${\sigma }_{\alpha }=3^{\circ }$, calculated using KMC with $f_0=50\mathrm{GHz}$. The $T=0$ $MH$ loop is included for comparison.

Figure 13

Coercive field as a function of sweep rate calculated from $MH$ loops shown in Fig. 12 together with corresponding results calculated using finite-temperature LLG simulations with $\alpha =0.1$. The discrepancy at the lowest sweep rates is attributed to approximations in the estimate of the attempt frequency.