Reliability of Sharrocks equation for exchange spring bilayers

A Monte Carlo approach and a modified nudged elastic band method are used to study the dynamic coercivity of interacting particle arrays in particular perpendicular recording media and exchange spring bilayers. Monte Carlo simulations are performed to study the effect of the interactions on the dynamic coercivity of interacting particle arrays. It is shown that the interactions in magnetic recording media only slightly influence the dynamic coercivity. The reliability of energy barrier measurements based upon Sharrock’s equation for frequency-dependent coercivity data is investigated using a modified nudged elastic band method. It is shown that the extrapolated energy barrier at zero field may deviate from the correct one by up to 18% if the conventional exponent $n=1.5$ is assumed. Our micromagnetic simulations furthermore indicate that the accuracy of the extrapolated energy barrier can be improved by about a factor of 3 upon measuring the dynamic coercivity at an angle of 45° and using the exponent $n$ as an additional fit parameter.

Figure 1

(Color online) Energy barrier as a function of the external field and the angle $\theta$ between the easy axis and the external field. The grain has a rectangular basal plane with edge length of $1\mathrm{nm}$. The film thickness is $20\mathrm{nm}$. Left image: the energy barrier was calculated using the nudged elastic band method. The barrier is calculated for 20 different values of the external field and 14 values of the angle $\theta$. Right image: the Pfeiffer approximation is used to estimate the energy barrier.

Figure 2

Remanent hysteresis loops obtained by Monte Carlo simulations. The temperature $T=300\mathrm{K}$. No intergranular exchange field is assumed in the calculations. The waiting time is $t=1\mathrm{s}$, ${10}^{-1}\mathrm{s}$, ${10}^{-2}\mathrm{s},\dots ,{10}^{-9}\mathrm{s}$ for the curves $a,b,c,\dots ,h$, respectively. The angle between the easy axis and the external field is 15.9°. Dashed lines $a\text{−}h$: the stray field of neighboring grains is not taken into account. Solid lines $A\text{−}H$: same as $a\text{−}h$ but the demagnetizing field is taken into account.

Figure 3

(Color online) Compilation of the dynamic coercivity obtained from the waiting time experiments of Fig. 2. The temperature is $T=300\mathrm{K}$. Instead of the waiting time $t$ the logarithm $\ln \left(\kappa t\right)$ is used as the $y$ axis. The constant $\kappa =1∕\left[{\tau }_0\ln \left(2\right)\right]$. The effect of interaction fields (stray field and exchange field) on the dynamic coercivity is investigated.

Figure 4

Energy barrier of exchange spring media for different soft-layer thicknesses as a function of the external field strength. The grain diameter is $6\mathrm{nm}$; the hard-layer thickness is $18\mathrm{nm}$. The anisotropy constant in the hard layer is $K_{1,\text{hard}}=1\times {10}^6\mathrm{J}∕{\mathrm{m}}^3$. The numbers in the plot (0–36) denote the soft-layer thickness in nm. The solid lines are fitted to numerically calculated energy barriers using $E_0$, $H_0$, and $n$ as fit parameters. The angle between the easy axis and the external field is $\theta =0.5°$. The dotted line shows the results of the analytical formula that is valid for $\theta =0°$ and infinite thick layer thicknesses. Exchanging the $x$ axis and the $y$ axis in the above plot gives a curve which is usually called “time dependence of the remanent coercivity.”

Figure 5

Field dependence of the fitting parameter $n$. $\theta =0.5°$. The hard-layer thickness $t_h=18\mathrm{nm}$. The numbers next to the curves denote the soft-layer thickness.

Figure 6

Same as Fig. 4 except that the external field is applied at an angle of $\theta =45°$.

Figure 7

Field dependence of the fitting parameter $n$. $\theta =45°$. The hard-layer thickness $t_h=18\mathrm{nm}$. The numbers next to the curves denote the soft-layer thickness.

Figure 8

Same as Fig. 3. However, $t_h=10\mathrm{nm}$.

Figure 9

Energy barrier of exchange spring media for different soft-layer thicknesses and different angles $\theta$ between the external field and the easy axis. The anisotropy constant in the hard layer and in the soft layer is $K_{1,\text{hard}}=1\times {10}^6\mathrm{J}∕{\mathrm{m}}^3$ and $K_{1,\text{soft}}=2\times {10}^5\mathrm{J}∕{\mathrm{m}}^3$, respectively. The numbers in the figure denote the soft-layer thicknesses.

Figure 10

Hysteresis loops of a bilayer with an $18\text{−}\mathrm{nm}$-thick hard layer and an $11\text{−}\mathrm{nm}$-thick soft layer ($K_{1,\text{hard}}={10}^6\mathrm{J}∕{\mathrm{m}}^3$, $K_{1,\text{soft}}=2\times {10}^5\mathrm{J}∕{\mathrm{m}}^3$). The angle $\theta$ between the external field and the easy axis is varied. The inset shows the angular dependence of $H_0$ as a function $\theta$.

Figure 11

Same as Fig. 5 ($\theta =0.5°$, $t_h=18\mathrm{nm}$). However, the anisotropy in the soft layer is $K_1=2\times {10}^5\mathrm{J}∕{\mathrm{m}}^3$.

Figure 12

Same as Fig. 11 except that $\theta =45°$.