introduction to magnetic recording

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 \vec{M}=M_{s}\hat{m}:

\frac{d \hat{m}}{dt} = - \hat{m} \times \left[\gamma \vec{H}_{eff} - \alpha \frac{\partial \hat{m}}{dt}\right]

where \gamma is the gyromagnetic constant, \alpha is the Gilbert damping constant and \vec{H}_{eff} is the ‘effective field‘ found by taking the derivative of the thermodynamic free energy, G(\vec{M},\ldots), with respect to \vec{M} (i.e. \vec{H}_{eff} = - \nabla_{\vec{M}} \,G =- \partial G / \partial \vec{M}). One part of the free energy is the Zeeman energy, -\vec{M}\cdot\vec{H}_a, where \vec{H}_a is an externally applied magnetic field.

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

\frac{dG}{dt} = \int_{V} \left[ \frac{\delta G}{\delta \vec{M}} \cdot \frac{\partial \vec{M}}{\partial t} + \frac{\delta G}{\delta \vec{H}_{a}} \cdot \frac{\partial \vec{H}_{a}}{\partial t} \right] dV

=M_{s} \int_{V} \left[ -\vec{H}_{eff} \cdot \frac{\partial \hat{m}}{\partial t} - \hat{m} \cdot \frac{\partial \vec{H}_{a}}{\partial t} \right] dV

= M_s V \left[ -\frac{\alpha}{\gamma} \left| \frac{\partial \hat{m}}{\partial t}\right|^2 - \hat{m} \cdot \frac{\partial \vec{H}_{a}}{\partial t}\right]

where V is the volume of the magnetic grain and \frac{\partial \hat{m}}{dt} \cdot \left[\gamma \vec{H}_{eff} - \alpha \frac{\partial \hat{m}}{dt}\right] = 0 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 \hat{m}\cdot\vec{H}_{eff}=\pm H_{eff}. The second term is what we are mostly concerned about. If -\hat{m} \cdot \partial \vec{H}_a/ \partial t is less (greater) than zero then free energy is decreasing (increasing). That is, either the time variation of amplitude and/or direction of \vec{H}_a 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

G=\int_V \left[ - \vec{M}\cdot\vec{H}_a - \frac{1}{2}\vec{M} \cdot \vec{H}_m - \frac{H_k M_s}{2}  (\hat{k} \cdot \hat{m})^2 + A \left|\nabla \hat{m}\right|^2 \right] dV

where \vec{H}_m is magnetostatic field, H_k is the magnetocrystalline anisotropy (MCA) with ‘easy axis’ along \hat{k} and A is exchange energy. Correspondingly,

\vec{H}_{eff} =-\nabla_{\vec{M}} \, G =  \vec{H}_a + \vec{H}_m + H_k ( \hat{k} \cdot \hat{m})\, \hat{k} + \vec{H}_{exch}

Typically inside a grain, \hat{m} is assumed uniform and \vec{H}_{exch} can be ignored (\hat{m} \times \vec{H}_{exch}=0). If the isolated grain is in a shape of an ellipsoid, then \vec{H}_m is replaced by a modified MCA term: H_k \rightarrow H_k - (3 N /2 - 2\pi)M_{s} where N_ is the depolarization (or demagnetization) factor along \hat{k} ranging from 0 (a needle) to 4\pi (pancake) [see]. If not in isolation, other grains would contribute to \vec{H}_m, as a summation of dipole fields in the case of ellipsoidal grains.

Regardless, the MCA term determines stable \hat{m} orientations which \vec{H}_a must be able to switch between. Without loss of generality, we will write, or switch, a grain starting with \hat{m} ~ \hat{k} and going to -\hat{k} 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 (D~8nm) shows |\hat{m}\cdot H_a \partial \hat{H}_a/\partial t| \ll |\hat{m}\cdot\hat{H}_a \partial H_{a}/\partial t| ~100kOe/ns. Even at this rate, the magnetization is in quasi-equilibrium with \vec{H}_a(t). Without loss of generality, let \hat{z}(=\hat{k}) be normal to the medium plane and align \hat{y} along \vec{H}_a-(\vec{H}_a\cdot\hat{z})\,\hat{z} then equilibrium \hat{m}^0=(0,\sin\theta_0,\cos\theta_0) is given by the minimization of free energy, H_a \sin(\Theta-\theta_0)=H_k\cos\theta_0 \sin\theta_0 where \cos \Theta = \hat{z}\cdot\hat{H}_a. For H_a>H_k there is one energy minimum at \theta_0 (i.e. successful switch), H_a<H_k/2 has two energy minima separated by an energy maximum (incomplete switch), otherwise there can be one or two energy minima depending on the value of \Theta (e.g. \Theta=3\pi/4 has \theta_0>\pi/2 for H_a\ge H_k/2– the Stoner-Wohlfarth limit).

Dynamic write

Let us consider the alternative \hat{m}\cdot \partial \hat{H}_a/\partial t \,\neq 0 (i.e. \partial H_a/\partial t = 0). Once again, let \hat{z}(=\hat{k}) be normal to the medium plane, we can solve LLG directly for the dynamic \hat{m}(t)=(\cos\phi(t)\sin\theta(t),\sin\phi(t)\sin\theta(t),\cos\theta(t)) when we apply an AC in-plane field such that \hat{m}\cdot\hat{H}_a=0, say \vec{H}_a=H_a(-\sin\phi(t),\cos\phi(t),0),

\frac{d\theta}{dt} =\gamma H_a -\alpha \sin\theta\, \frac{d\phi}{dt} = \frac{\gamma}{1+\alpha^2}\left(H_a-\alpha H_k \cos \theta \sin \theta\right)

\sin \theta \,\frac{d\phi}{dt} = \gamma H_k\sin\theta\cos\theta + \alpha \frac{d\theta}{dt} =\frac{\gamma}{1+\alpha^2} \left(H_k \cos \theta \sin \theta + \alpha H_a\right)

such that -\hat{m}\cdot\partial\vec{H}_a/\partial t = H_a \sin \theta\, \dot{\phi} (that is, sign of H_a \dot{\phi} determines if free energy increases or decreases). We see \dot{\phi}} switches sign when \theta passes through ~\pi/2 therefore the flow of free energy also switches sign.

The expression for \dot{\theta} gives the constraint |H_a| > \alpha H_k /2 and an analytic time for \theta=0 \rightarrow \pi/2 as (1+\alpha^2)/\gamma H_a \sqrt{1-(\alpha H_k/2H_a)^2}).

From expression for \dot{\phi}}, one see the angular frequency of \hat{H}_a goes as \omega \approx \omega_0 \cos \theta =  \frac{\gamma}{1+\alpha^2} H_k \cos \theta. Then average microwave frequency would be 2 \omega_0 /\pi~1.78GHz/kOe H_k. For H_k ~ 19.2kOe and we require ~0.1ns to switch then H_a \gtrsim 600 Oe or ~4 cycles to switch. Larger H_a 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^{3} packed in a periodic square lattice with H_k=28kOe, \hat{k}=\hat{z}, M_s=700emu/cc and \alpha =0.02. We include the full magnetostatic field H_m interactions. However, in the cases below, since the grains, their initial condition and applied field histories are identical then H_m=-4\pi M_s m_z. 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 \vec{H}_a = H_c(t) (0,\sin \pi/4,-\cos \pi/4) where H_c(t) is linearly increasing from zero to 10kOe (>H_k/2\rightarrow H_k/2 - 2 \pi M_s =9.6kOe) in 0.1ns:

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_c(0,\sin \pi/4,-\cos\pi/4) is ramped from 0 to 10kOe in 0.1ns.

As expected, all 1024 grains behave identically and are nearly in equilbrium with \vec{H}_a until switching. Damping will limit how quickly H_c can be reversed.

We next look at dynamic writing by \vec{H}_a(t) = H_d (-\langle m_y(t)\rangle, \langle m_x(t) \rangle, 0) where \langle \hat{m}(t)\rangle the average of \hat{m} over the 1024 grains at time t.

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_d(-\langle m_y(t)\rangle, \langle m_x(t)\rangle,0) is ramped from 0 to 1.5kOe in 0.1ns.

This is an interesting result, we can switch a grain with H_a < H_k/2 in less than 0.2ns. In fact, as long as H_a > \alpha H_k/2, the grain will switch, albeit with longer required time. Increasing H_a reduces the time to switch.

Finally, let us apply a conventional write field and then include an dynamic field to assist in writing. Here, \vec{H}_a=H_{c}(0,\sin \pi/4, -\cos \pi/4) + H_d (-\langle m_y(t) - m^0_x \rangle, \langle m_x(t) - m^0_y \rangle, 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_c(0,\sin \pi/4,-\cos\pi/4) is ramped from 0 to 6kOe in 0.1ns and held then \vec{H}_d=H_d(-\langle m_y(t) - m^0_x \rangle, \langle m_x(t) - m^0_y \rangle, 0) is ramped from 0 to 1.5kOe in 0.1ns and added to \vec{H}_a.

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

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_c(0,\sin \pi/4,-\cos\pi/4) is ramped from 0 to 6kOe in 0.1ns and held then \vec{H}_d=H_d(-\langle m_y(t) - m^0_x \rangle, \langle m_x(t) - m^0_y \rangle, 0) is ramped from 0 to 300 Oe in 0.1ns and added to \vec{H}_a.

This latter case is similar to microwave assisted magnetic recording (MAMR). Here, instead of having a time varying \omega it is a fixed frequency of about ~1.78GHz/kOe H_k and H_d large to reduce the number of required cycles.

While the simulation correctly follows the governing physics, we have not simulated an realizable physical device. While identical square prism grains are not realistic, they are not impossible.

No, we have made at least two mistakes. First, since the grains are identical including their initial orientations we have introduce a high degree of symmetry into the problem. In fact this is clearly shown in the results. Each simulation has 1024 grains, but only one phase space point appears. That is, all 1024 grains have identical trajectories. This happened due to our second mistake.

LLG includes damping. This damping is energy exchange between the magnetic system and the lattice it finds itself in. Since the magnetic grains are small, there are not insignificant thermal fluctuations in the magnetization orientation due to the lattice[1]. This effect needs to be included for realistic simulation of actual physical devices.

The result of thermal fluctuations is to introduce a variation of \hat{m} around the equilibrium orientation. In conventional recording this introduces a diffusive flux which allows some grains to switch even when H_a < H_k/2. In dynamic recording, \hat{m} has a distribution in \phi and \theta causing either a mismatched phase or distribution of resonant angular frequency \omega, respectively. And, of course, it breaks the high symmetry of the results reported here.

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

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