**Theory**

In this blog post, we introduce magnetic recording by focusing on the dynamics of writing granular magnetic medium. We start with the Landau-Lifshitz-Gilbert (LLG) equation which is sufficient for describing the dynamics of a grain’s magnetic moment :

where is the gyromagnetic constant, is the Gilbert damping constant and is the ‘effective field‘ found by taking the derivative of the thermodynamic free energy, , with respect to (i.e. ). One part of the free energy is the Zeeman energy, , where is an externally applied magnetic field.

It is easy to show the time rate of change of the thermodynamic free energy to be

where is the volume of the magnetic grain and was used.

This equation describes the ** only** way that the free energy of the a magnetic moment can change in time according to LLG. The first term on the rhs is simply free energy loss due to Gilbert damping, vanishing when . The second term is what we are mostly concerned about. If is less (greater) than zero then free energy is decreasing (increasing). That is, either the time variation of amplitude and/or direction of can affect the free energy. The former tends to change the energy surface (e.g. eliminating one or more of the free energy minima) while the latter tends to change the free energy of the magnetic moment while not perturbing the energy surface itself.

**Magnetic recording**

For a magnetic grain used in magnetic recording, the free energy is

where is magnetostatic field, is the magnetocrystalline anisotropy (MCA) with ‘easy axis’ along and is exchange energy. Correspondingly,

Typically inside a grain, is assumed uniform and can be ignored (). If the isolated grain is in a shape of an ellipsoid, then is replaced by a modified MCA term: where is the depolarization (or demagnetization) factor along ranging from (a needle) to (pancake). If not in isolation, other grains would contribute to , as a summation of dipole fields in the case of ellipsoidal grains.

Regardless, the MCA term determines stable orientations which must be able to switch between. Without loss of generality, we will write, or switch, a grain starting with ~ and going to orientation, both being minima of the MCA energy.

**Conventional write**

In typical HDD write, the linear velocity (~20m/s), field rise time (~0.2ns peak-to-peak), write field gradient (~1kOe/nm) and grain size (~8nm) shows ~100kOe/ns. Even at this rate, the magnetization is in quasi-equilibrium with . Without loss of generality, let be normal to the medium plane and align along then equilibrium is given by the minimization of free energy, where . For there is one minimum (i.e. successful switch), has two minima separated by a maximum (incomplete switch), otherwise there can be one or two minima depending on the value of (e.g. has for – the Stoner-Wohlfarth limit).

**Dynamic write**

Let us consider the alternative (i.e. ). Once again, let be normal to the medium plane, we can solve LLG directly for the dynamic when we apply an AC in-plane field such that , say ,

such that (that is, sign of determines if free energy increases or decreases). We see switches sign when passes through ~ therefore the flow of free energy also switches sign.

The expression for gives the constraint and an analytic time for as .

From expression for , one see the angular frequency of goes as . Then average microwave frequency would be ~GHz/kOe . For ~ 19.2kOe and we require ~0.1ns to switch then 600 Oe or ~4 cycles to switch. Larger could achieve switching with fewer cycles.

**Micromagnetic simulations**

Detailed micro-magnetic simulations coincide with the analysis for both conventional and dynamic writing. Here we have 1024 identical extruded (10nm) square grains of volume 503nm packed in a periodic square lattice with kOe, , emu/cc and . We include the full magnetostatic field interactions. However, in the cases below, since the grains, their initial condition and applied field histories are identical then . Consider it a test of the underlying numerical routines (all 1024 grains should have identical trajectories even though the numerical procedure does not enforce this unique condition).

We first do *conventional writing* by where is linearly increasing from zero to 10kOe (9.6kOe) in 0.1ns:

As expected, all 1024 grains behave identically and are nearly in equilbrium with until switching. Damping will limit how quickly can be reversed.

We next look at dynamic writing by where the average of over the 1024 grains at time t.

This is an interesting result, we can switch a grain with in less than 0.2ns. In fact, as long as , the grain will switch, albeit with longer required time. Increasing reduces the time to switch.

Finally, let us apply a conventional write field and then include an dynamic field to assist in writing. Here, .

In this case, even of 300 Oe can still switch the grain, albeit in ~0.4ns:

This latter case is similar to microwave assisted magnetic recording (MAMR). Here, instead of having a time varying it is a fixed frequency of about ~GHz/kOe and large to reduce the number of required cycles.

However, this is not a real physical device. Gilbert damping arises from coupling of a thermal bath to the magnetization . In order to comply with thermodynamics, we require stochastic fields acting on the magnetization [1]. The result of this stochastic field is to introduce a variation of around the equilibrium orientation. In conventional recording this introduces a diffusive flux which allows some grains to switch even when . In dynamic recording, has a distribution in and causing either a mismatched phase or distribution of resonant angular frequency , respectively.

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

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