stochastic effects in magnetic recording

In previous posts, we discussed magnetic recording. Here, we extend this discussion by looking at the effect of a thermal bath that is coupled to magnetic moments during recording.

Once again, we start with the Landau-Lifshitz-Gilbert (LLG) equation

\frac{d\hat{m}}{dt} = -\gamma \hat{m}\times\vec{H} + \alpha \hat{m} \times \frac{d\hat{m}}{dt}

where \hat{m} is an unit vector describing the orientation of the magnetic moment, t is time, \gamma is the gyromagnetic ratio, \vec{H} is the effective field and \alpha \sim 0.02 is a damping constant. Landau added the second term in order that the magnetization moment would come into equilibrium (e.g. \hat{m}\cdot\hat{H} = 1 and therefore \frac{d\hat{m}}{dt}=0) eventually. The equation has been derived in multiple theories for particular cases. In particular, [1] recovered LLG from a quantum macrospin in the limit of constant coupling and uniform density of states of the thermal bath.

For small magnetic particles, Brown [2] derived the required stochastic field from the Fokker-Planck equation. The resulting stochastic field, \vec{H}_t, is a Gaussian with mean of zero and variance, \sigma^2 = 2\alpha kT/ \gamma M_s V \delta t in each dimension where T is temperature, M_s is the saturation magnetization and V is the volume of the magnetic grain.

The stochastic field will cause a diffusion of \hat{m} from its equilibrium. Then, since \hat{m}\cdot\hat{H} \neq 1 the precession term will introduce a rotational flux of \hat{m}. Ignoring this rotational flux, we can define a diffusion coefficient of the magnetization as

D_m = \frac{\langle \delta \hat{m} \cdot \delta \hat{m}\rangle}{2 \delta t} = 2\alpha\gamma kT / (1+\alpha^2) M_s V

where the last equality is calculated by replacing \vec{H} with Brown’s stochastic fields \vec{H}_t in the LLG and integrating over all \vec{H}_t. Reasonable values would give D_m \sim 10^8-10^9rad^2/s. Roughly speaking this would mean in 1ns \hat{m} would have diffused ~20^o60^o from equilibrium. This does not include the additional processional term which also depend on this angle.

Instead of diffusion of magnetization, one could look at diffusion of free energy, G, of the magnetic moment. In this case,

D_e = \frac{\langle (\delta G)^2 \rangle}{2\delta t}  = \frac{\delta t}{2}\langle \left( - M_s V \vec{H} \cdot \frac{d\hat{m}(\vec{H}_{t})}{dt}\right)^2 \rangle = \frac{\gamma \alpha M_s V kT}{1+\alpha^2} |\hat{m}\times\vec{H}|^2

and one can show Einstein’s relation holds, D_e/\mu = kT, where \mu is advective flux due to \alpha in the absence of stochastic fields, namely

\mu = M_s V \vec{H} \cdot \frac{d\hat{m}}{dt} = M_s V \vec{H} \cdot \left(\alpha \hat{m} \times \frac{d\hat{m}}{dt} \right)=\frac{\gamma \alpha M_s V}{1+\alpha^2} |\hat{m}\times\vec{H}|^2.

From above, any analysis should include \vec{H}_t since it’s a consequence of damping (\alpha > 0). We will show its effect during conventional magnetic recording.

For conventional writing, as discussed previously, the applied write field goes as \vec{H}_a = H_a(t) \hat{H}_a such that \hat{H}_a is a slowly varying function of time (i.e. medium motion). On the other hand, H_a(t) is quickly varying in ~ 0.1ns with a peak large enough to eliminate the free energy barrier (e.g H_a > \left[H_k - (3 N /2 - 2\pi)M_{s_i}\right] / 2). Therefore, one would expect the diffusive flux (e.g. D_e \partial f/\partial \epsilon) would lower required H_a.

We first confirm \vec{H}_t does, in fact, result in a Boltzmann distribution of the orientation of \hat{m}.

For micro-magnetic simulations, we have identical square prisms (extruded 10nm) grains of volume 503nm^3 packed in a periodic square lattice with H_k=28kOe, \hat{z}(=\hat{k}) is normal to media plane, M_s=700emu/cc and \alpha=0.02.

For simplicity, we take 65536 grains, T=318.15K and exclude applied and magnetostatic fields. We start with all the grains along \hat{z} and integrate in time until equilibrium is achieved.

65,536 identical grains subject to stochastic fields and H_k. Upper left, standard deviation of m_x, m_y and m_z as a function of time t (ns); lower left, histogram of \hat{m} as a function of energy/kT; right, isometric projection of all grain orientations on a unit sphere: each white circle represents one magnetic orientation with orange trail showing short time history of orientation.

We see not only is the standard deviation of \hat{m} is as expected at equilibrium but the time evolution to equilibrium is also correct. Further, the distribution of \hat{m} as a function of energy is, in fact, a Boltzmann distribution (thereby ensuring the diffusive and drift flux will exactly balance). Of course, without \vec{H}_t, the variance would be identically zero, the distribution function would be a delta function at zero energy and the phase space plot would show all 65,536 grain orientations at the north pole (i.e. \theta=0).

We next use 1024 identical square prism grains with full magnetostatic fields and apply an uniform field of \vec{H}_a = H_a(t)(0,\sin \pi/4,-\cos\pi/4) linearly increasing H_a(t) from zero to H_a in 0.1ns. For H_a=9.45 kOe<9.6 kOe (=(H_k-4 \pi M_s)/2) no grains should flip. We confirm this by running with \vec{H}_t =0:

Upper left: applied field (Oe) as a function of time (ns), white, red, yellow, green are magnitude, x, y, z components, respectively; bottom left: \langle \hat{m} \rangle as a function of time (ns), white, red, yellow, green are magnitude, x, y, z components, respectively; right: isometric projection of unit sphere, dot represent the orientation of one of the 1024 grains updated every 1ps with line connecting to previous orientation. \vec{H}_a=H_a(t)(0,\sin \pi/4,-\cos\pi/4) and H_a(t) is ramped from 0 to 9.45 kOe in 0.1ns, full magnetostatic field and H_t=0.

We use the same identical set up, but now \vec{H}_t \neq 0:

Upper left: applied field (Oe) as a function of time (ns), white, red, yellow, green are magnitude, x, y, z components, respectively; bottom left: \langle \hat{m} \rangle as a function of time (ns), white, red, yellow, green are magnitude, x, y, z components, respectively; right: isometric projection of unit sphere, dot represent the orientation of one of the 1024 grains updated every 1ps with line connecting to previous orientation, with line reset every 50ps. \vec{H}_a=H_a(t)(0,\sin \pi/4,-\cos\pi/4) and H_a(t) is ramped from 0 to 9.45 kOe in 0.1ns, full magnetostatic field and H_t \neq 0. At 1ns, 62% of the grains have switched (not shown).

We also note that in the absence of \vec{H}_t, H_a=9.6kOe was sufficient to flip all 1024 grains. However, since \vec{H}_t breaks the symmetry, one requires H_a ~ 17.1kOe to switch all grains in 1ns. Therefore we see H_t does lower H_a for both first grain flip (~ 7kOe) and all grains flip.

For completeness, we repeat the previous calculations but we ramp H_a in 0.1ns (instead of 5ns) and include H_t:

microstructuremagnetostatic fieldinitial orientationH_a to flip first (kOe)
[with H_t]
H_a to flip all (kOe)
[with H_t]
\Delta H_a (kOe)
[with H_t]
1024 grains, square prism, h=10nm, any packing fractionnoneany13.9
[12.2]
13.9
[13.8]
0
[1.6]
1024 grains, square prism, h=10nm, any packing fractionself demagnetizationany14.4
[11.8]
14.4
[14.3]
0
[2.5]
1024 grains, square prism, h=10nm, 100% packingfullup9.5
[6.6]
9.5
[17.3]
0
[10.7]
1024 grains, square prism, h=10nm, 100% packingfullrandom11.4
[9.7]
17.2
[17.3]
5.8
[7.6]
849 grains, square prism, h=10nm, 83% packingfullup9.5
[7.3]
16.4
[16.8]
6.9
[9.5]
849 grains, square prism, h=10nm, 83% packingfullrandom11.6
[10.6]
16.6
[16.8]
5.0
[6.2]
481 grains, \langle D \rangle = 7nm, \sigma_D = 0.4nm, 83% packingnoneany13.9
[12.4]
13.9
[13.8]
0
[1.4]
481 grains, \langle D \rangle = 7nm, \sigma_D = 0.4nm, 83% packingself demagnetizationany14.1
[11.9]
14.7
[14.3]
0.6
[2.4]
481 grains, \langle D \rangle = 7nm, \sigma_D = 0.4nm, 83% packingfullup9.8
[7.6]
16.8
[16.4]
7.0
[8.8]
481 grains, \langle D \rangle = 7nm, \sigma_D = 0.4nm, 83% packingfullrandom11.7
[10.7]
16.8
[16.6]
5.1
[5.9]
Required \vec{H}_a = H_a(t) (0, \sin \pi/4, -\cos \pi/4) when H_a(t) is ramped from 0 to H_a in 0.1ns and kept there for an additional 0.5ns. Values in ‘[]‘ are with H_t \neq 0 and are dependent on the time H_a is held constant.

The one clear conclusion we see is that the ‘switching field distribution’ (the difference between H_a for switching all grains to switching one grain) increases when H_t \neq 0. This is usually due to the reduction of H_a needed to switch the first grain. When H_t \neq 0, the results above depend on the time that H_a is applied. For longer times (0.5ns used here), H_a would decrease.

This, once again, shows the importance of including the full magnetostatic and the stochastic fields during conventional writing. Be aware that many simulations reported in the literature don’t included either one or both (i.e. “effective write field” for switching)- they will lead you to wrong conclusions if you’re trying to design a new storage system.

[1] A. Rebei and G.J. Parker, Fluctuations and dissipation of coherent magnetizationPhys. Rev. B67, 104434, (2003).

[2] W.F. Brown, Thermal fluctuations of a single-domain particle, Phys. Rev., 130, 1677 (1963).

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