Write head design for effective curvature reduction in heat-assisted magnetic recording by topology optimization

TL;DR

This paper uses topology optimization to design heat-assisted magnetic recording heads with large field gradients, aiming to reduce transition curvature and improve data density, but finds that conventional head optimization alone is insufficient for significant improvements.

Contribution

The study applies topology optimization to design HAMR heads with enhanced field gradients, revealing limitations of conventional designs for curvature reduction.

Findings

Optimized head design does not significantly improve transition curvature.

Higher field gradients alone are insufficient for effective curvature reduction.

New head concepts are necessary beyond simple optimization.

Abstract

The reduction of the transition curvature of written bits in heat-assisted magnetic recording (HAMR) is expected to play an important role for the future areal density increase of hard disk drives. Recently a write head design with flipped write and return poles was proposed. In this design a large spatial field gradient of the write head was the key to significantly reduce the transition curvature. In this work we optimized the write pole of a heat-assisted magnetic recording head in order to produce large field gradients as well as large fields in the region of the heat pulse. This is done by topology optimization. The simulations are performed with dolfin-adjoint. For the maximum field gradients of $8.1\,$mT/nm, $8.6\,$mT/nm and $11.8\,$mT/nm, locally resolved footprints of an FePt like hard magnetic recording medium are computed with a coarse-grained Landau-Lifshitz-Bloch (LLB) model and the resulting transition curvature is analysed. Additional simulations with a bilayer structure with $50\%$ hard and $50\%$ soft magnetic material are computed. The results show that for both recording media, the optimized head design does not lead to any significant improvement of the written track. Thus, we analyse the transition curvature for the optimized write heads theoretically with an effective recording time window (ERTW) model. Moreover, we check how higher field gradients influence the curvature reduction. The results show that a simple optimization of the conventional head design design is not sufficient for effective curvature reduction. Instead, new head concepts will be needed to reduce transition curvature.

Tables

Table 1. Table 1: Approximated dimensions of an initial flipped recording head design as marked in Figure   2 .

$x_{\mathrm{SUL}}$	$y_{\mathrm{total}}$	$z_{\mathrm{total}}$	$x_{\mathrm{magn}}$	$y_{\mathrm{magn}}$	$z_{\mathrm{magn}}$	$z_{\mathrm{opt}}$	$x_{\mathrm{write}}$	$x_{\mathrm{return}}$	distw,r	$x_{\mathrm{back}}$	distf,w	$x_{\mathrm{field}}$	$y_{\mathrm{field}}$
2 $\mu$m	5 $\mu$m	13 $\mu$m	1 $\mu$m	3 $\mu$m	3 $\mu$m	2 $\mu$m	varied	200 nm	1 $\mu$m	varied	50 nm	100 nm	100 nm

Table 2. Table 2: Resulting fields and field gradients produced by the write heads which are optimized with the help of topology optimization for a FeCo like head material. The field and its gradient of a conventional head are added for comparison.

Geometrie	$x_{\mathrm{write}}$ [nm]	$x_{\mathrm{back}}$ [nm]	${\mu }_{0}H$ [T]	Max. ${\mu }_{0}dH/dx$ [mT/nm]
Conventional	100	—	0.8	—
Initial flipped design	100	—	0.2	2.2
Basic100	100	—	0.4	4.1
Basic200	200	—	0.7	8.6
Backshield100	100	100	0.67	8.1
Backshield200	200	200	0.8	11.8

Table 3. Table 3: Material parameters of a FePt like hard magnetic granular recording medium. Note, that different material parameters for the recording medium were used compared to the original work of Vogler et al vogler_reduction .

Curie temp. $T_{\text{C}}$ (K)	Damping $\alpha$	Anisotropy const. $K_1$ (MJ/m3)	Anisotropy field ${\mu }_0{H}_k$ (T) at 300 K	$J_s$ (T)	height $h$ (nm)	diameter $d$ (nm)
693.5	0.02	$6.6$	10	1.35	8	5

Table 4. Table 4: Resulting transition curvature, P max subscript 𝑃 max P_{\mathrm{max}} and jitter parameters of the different flipped head designs in combination with a pure hard magnetic recording material compared to a conventional recording head design.

Geometry	$c$ ($\%$)	$P_{\max }$ ($\%$)	${\sigma }_{\mathrm{down}}$ ($\%$)
Basic200	$+1.4$	$-5.5$	$+118.7$
Backshield100	$+3.8$	$-7$	$+113.5$
Backshield200	$-1.35$	$-2.3$	$+28.4$

Table 5. Table 5: Material parameters of the soft magnetic composition of the bilayer structure.

Curie temp. $T_{\text{C}}$ (K)	Damping $\alpha$	Anisotropy const. $K_1$ (MJ/m3)	$J_s$ (T)
740.9	0.1	$0$	1.35

Table 6. Table 6: Resulting transition curvature, P max subscript 𝑃 max P_{\mathrm{max}} and jitter parameters of the different flipped head designs in combination with an exchange spring recording medium compared to a conventional recording head design with the same recording material.

Geometry	$c$ ($\%$)	$P_{\max }$ ($\%$)	${\sigma }_{\mathrm{down}}$ ($\%$)
Basic200	$+1.38$	$-2.5$	$-16.2$
Backshield100	$+1.9$	$-1.19$	$-4.7$
Backshield200	$-0.08$	$+/-0.0$	$-19$

Table 7. Table 7: Curvature reduction in % percent \% compared to a conventional design for pure hard magnetic recording material (HM) and an exchange spring bilayer structure (HM/SM) calculated with the ERTW model.

Field gradient [mT/nm]	$c$ ($\%$) HM	$c$ ($\%$) HM/SM
12	-1.0	-1.7
16	-1.6	-3.64
20	-2.09	-5.58
24	-2.59	-7.5
28	-3.38	-9.48
32	-4.6	-11.4
36	-5.2	-13.11
40	-5.8	-14.2

Equations

$${\text{B}}_{\text{s}}={\mu }_0({\text{H}}_{\text{s}}+{\text{M}}_{\text{s}})={\mu }_0{\mu }_r{\text{H}}_{\text{s}}$$

$${\text{B}}_{\text{p}}={\mu }_0{\mu }_m{\text{H}}_{\text{p}}+{\mu }_0{\text{M}}_{\text{0}}$$

$$\text{M}(\rho )={\rho }^k{\text{M}}_0$$

$$\chi (\rho )={\chi }_0\cdot {\rho }^k$$

$${\mu }_r(\rho )=({\mu }_{r0}-1){\rho }^k+1.$$

$$\mathrm{Find}:\underset{\rho }{\min }J(\rho )$$

$$\mathrm{subject}\mathrm{to}:{\int }_{{\Omega }_i}\rho (r)dr\le V;$$

$$0\le \rho (r)\le 1,r\in {\Omega }_i$$

$$J(\rho )={\int }_{{\Omega }_{\mathrm{field}}}\frac{1}{|\nabla H_z(\rho ){|}^2}dr+{\int }_{{\Omega }_{\mathrm{field}}}\frac{1}{|H_z(\rho ){|}^2}dr$$

$$+\underset{\mathrm{penalization}}{\underbrace{\lambda \cdot {\int }_{{\Omega }_{\mathrm{opt}}}\rho (1-\rho )dr}},$$

$$T(x,y,t)=(T_{\mathrm{write}}-T_{\min })e^{-\frac{(x-vt{)}^2+y^2}{2{\sigma }^2}}+T_{\min }$$

$$=T_{\mathrm{peak}}(y)\cdot e^{-\frac{(x-vt{)}^2}{2{\sigma }^2}}+T_{\min }$$

$$\sigma =\frac{\mathrm{FWHM}}{\sqrt{8\ln (2)}}.$$

$${\Phi }_{\mu ,{\sigma }^2}=\frac{1}{2}(1+\mathrm{erf}(\frac{x-\mu }{\sqrt{2{\sigma }^2}}))\cdot p$$

$$\mathrm{erf}(x)=\frac{2}{\sqrt{\pi }}{\int }_0^x{e}^{-{\tau }^2}d\tau ,$$

$$T_{\mathrm{peak}}(y)=(T_{\mathrm{write}}-T_{\min })e^{-\frac{y^2}{2{\sigma }^2}}+T_{\min }.$$

$$c=\frac{\Delta x}{\Delta y}\equiv cp\cdot \Delta y$$

$${\mathrm{ERTW}}_{\uparrow }=[t(T_{\mathrm{c}}),t(T_{\mathrm{freeze}})]\cap [t_{\uparrow ,\mathrm{start}},t_{\uparrow ,\mathrm{final}}]$$

$$p=\min \left(\frac{{\mathrm{ERTW}}_{\uparrow }}{{\Theta }_{\mathrm{ERTW}}},1\right)\left[1-\min \left(\frac{{\mathrm{ERTW}}_{\downarrow }}{{\Theta }_{\mathrm{ERTW}}},1\right)\right].$$

Full text

I Introduction

In heat-assisted magnetic recording (HAMR) ersteshamr ; fan ; hamr1 ; hamr a heat pulse is included to the writing process to overcome the so-called recording trilemma evans and make high-anisotropy grains writable with the available head fields. However, in granular media, the curved thermal profile of the heat pulse in combination with a spatially relatively homogeneous head field gives rise to a significant curvature at transitions between bits nft1 ; nft2 . This transition curvature is expected to be a serious problem for the read-back process in HAMR since the signal-to-noise ratio (SNR) is reduced zhu_correcting . Different methods to efficiently reduce transition curvature in HAMR have been proposed, for example a write head field design by Zhu et al zhu_field ; zhu_correcting ; zhuli . The present work is based on the publication by Vogler et al vogler_reduction , where a recording head design with flipped write and return pole to efficiently reduce transition curvature is suggested. For the flipped head design writing of the bit happens between the near field transducer (NFT) and the returnpole, whereas for the conventional design writing happens near the write pole. The position where the bit is written for the different design is indicated by the grey arrows in Figure 1(a) and (b). The different writing position leads to a different behavior of the write field during the cooling process of the heat pulse. For the conventional design, the applied field is spatially relatively homogeneous during the cooling process (see Figure 1(c)). In contrast to this, shown in Figure 1(d), the field decreases during cooling for the flipped design which leads to a field gradient in down-track direction. This field gradient turned out to be the key for the curvature reduction. For this reason, the head field gradient in down-track direction is an important parameter in the simulations.

In this work, we compute the realistic field gradient for a flipped design, that follows the model of a state-of-the-art recording head design suh_head ; lee_head , at the position where the heat pulse is cooling down. To optimize the head, we design a write pole that maximizes both the field and the field gradient at the required position. This is done by topology optimization bendsoe_topology_2013 , an application of the inverse magnetostatic problem bruckner_solving . With the resulting fields and field gradients we compute locally resolved switching probability phase diagrams with a coarse-grained LLB model LLB for both pure hard magnetic recording material and an exchange spring structure with $50\%$ hard and $50\%$ soft magnetic material. From this locally resolved phase diagrams, we can then analyse the transition curvature.

Additionally, we interpret the results in the context of the effective recording time window (ERTW) model by Vogler et al basic ; vogler_reduction . This paper is structured as follows: In Section II, a theoretical framework of the simulation methods is presented and the simulation and material parameters are given. The results are presented in Section III and discussed in Section IV.

II Theoretical framework

II.1 Topology optimization

To solve the topology-optimization problem, the density method, which is also known as solid isotropic microstructure with penalization (SIMP), is used huber_topology ; abert2017fast . First, the geometry is meshed with a tetrahedral mesh. At each element of the mesh the density $\rho$ of the material is considered and a value between 0 (void) and 1 (material) is assigned to it. This leads to one optimization variable per element. The final design should only have density values of 0 or 1. This is achieved by using a penalization parameter $k$ to penalize those densities that are intermediate.

The materials in the simulations are approximately described by linear material laws. For the soft magnetic material, the material law reads as

[TABLE]

where $\mu_{r}$ is the relative permeability. $\mu_{0}$ denotes the vacuum permeability. For the hard magnetic material, there holds

[TABLE]

with the recoil permeability $\mu_{m}$. For a permanent magnetic region $\Omega_{\mathrm{p}}\subset\mathbb{R}^{3}$, the magnetization as a function of the density $\rho$ can be written as

[TABLE]

with the density value $\rho\in[0,1]$ of one element and the penalization parameter $k$. For hard magnetic materials, $k=1$ is a good choice choi . For a soft magnetic design region $\Omega_{\mathrm{s}}\subset\mathbb{R}^{3}$, the magnetic susceptibiliy $\chi$ can be reformulated as a function of the density to

[TABLE]

and thus for the relative permeability, there holds

[TABLE]

In this way it can be used for topology optimization. In the case of soft magnetic material, $k=4$ leads to good results huberthesis .

The topology-optimization problem that needs to be solved is given by

[TABLE]

where $J$ is the objective function, $V$ is the maximum volume of the design as a constraint and $i\in\{s,p\}$ defines the soft and permanent magnetic region, respectively.

The objective function used to maximize the magnetic field and the $z-$field gradient is given by minimizing

[TABLE]

where $\Omega_{\mathrm{opt}}$ is the region of the material that is optimized, $H_{z}(\rho)$ is the $z-$field and $\nabla H_{z}(\rho)$ is the gradient of the $z-$field. $\lambda$ is a penalization parameter. The topology-optimization problem is solved by a finite element method (FEM) which is based on the open-source library FEniCS fenics1 ; fenics2 for solving partial differential equations (PDEs) and the library dolfin-adjointdolfin1 ; dolfin2 . Dolfin-adjoint automatically determines and solves adjoint linear equations using PDEs which are discretized with finite elements. The minimization problem is solved using the L-BFGS-B method lbfgsb1 ; lbfgsb2 , a limited-memory quasi-Newton solver for bound-constrained optimization.

II.2 Head design parameters

In the topology-optimization simulations, a write head recording_head ; brug_magnetic_1996 is optimized, which consists of a write and a return pole, a permanent magnet to simulate the core magnetized by a coil and a 50 nm thick soft magnetic underlayer (SUL). Additionally, in some simulations a backshield is considered. All components and the dimensions of the write head are marked in Figure 2 and summarized in Table 1. The tip of the write pole and the backshield, if considered, are the parts of the write head that are optimized by topology optimization. Except the permanent magnet, all other parts of the write head consist of soft magnetic material.

The material of the recording head is assumed to be FeCo kief_materials . Hence, a relative permeability $\mu_{\mathrm{r}}=18000$ and a saturation polarization of 2.4 T are assumed feco . It is considered that the coil magnetizes the core with 0.8 T in $x-$direction which is modeled by the means of a permanent magnet that is magnetized in $x-$direction with 0.8 T. This field is chosen because then the maximum magnetization of the initial design is slightly below the saturation polarization of the material.

The field and the field gradient are computed and optimized at the position where the recording medium hitted by the heat spot produced by the NFT is already cooling down. This position is assumed to be 50 nm away from the pole tip in $x-$direction.

Since small head to media spacings (HMS) are needed, to get high areal storage density hmsmedia , the HMS is assumed to be 5 nm.

Four different starting geometries are considered. The starting geometries are basic geometries of dimensions $x_{\mathrm{write}}\times y_{\mathrm{total}}\times z_{\mathrm{opt}}$, where the density initially is 1 for each element. During the optimization process, the density of each element is adjusted. The difference between the geometries is the dimension in $x-$direction and the fact if a backshield is considered or not. The dimensions in $y-$ and $z-$direction are the same for all starting geometries and equal to $y_{\mathrm{total}}$ and $z_{\mathrm{opt}}$, respectively. The first geometry to be optimized, is one with dimension $x_{\mathrm{write}}=100$ nm and no backshield. This one is referred to as Basic100. The second geometry also has dimension $x_{\mathrm{write}}=100\,$nm. However, for this geometry a backshield with $x_{\mathrm{back}}=100\,$nm is additionally considered and optimized. Afterwards, this geometry is called Backshield100. The last geometries are similar to the former ones but with dimensions $x_{\mathrm{write}}=200$ nm and $x_{\mathrm{back}}=200$ nm. They are labeled Basic200 and Backshield200. The $x_{\mathrm{write}}$ and $x_{\mathrm{back}}$ values of the different geometries are summarized in Table 2.

After the optimal head designs are determined via topology-optimization, additional simulations with magnum.fe abert_magnum.fe:_2013 that include a coil instead of a permanent magnet are performed. These coil-simulations are performed in order to determine realistic fields with the optimized head designs. Here, all parts of the write head are considered to be soft magnetic with the above material parameters. The current density inside the coil is assumed to be $2.5\times 10^{10}$ A/m2, which is below the current density limit of $10^{12}\,$A/m2, hamann2007heating .

III Results

III.1 Field gradients

Forward simulations show that the field gradient of the initial flipped design, which follows a state-of-the-art head design suh_head ; lee_head , at 50 nm distance from the write pole is approximately 2.2 mT/nm. In Figure 3 the resulting fields from the coil simulations with the optimized geometries are plotted. The results show that the best outcome can be achieved for a geometry with a pole tip with $x_{\mathrm{write}}=200$ nm and an additional backshield with $x_{\mathrm{back}}=200$ nm. Here, the maximum field gradient is 11.8 mT/nm. The optimized Backshield200 geometry is shown in Figure 4. Note that all optimized geometries are similar and show a tapered shape. Recapitulating, the resulting write fields and field gradients for the different geometries are summarized in Table 2.

III.2 Curvature reduction

To analyse the curvature reduction, switching probability phase diagrams are computed with the help of a coarse-grained model based on the stochastic Landau-Lifshitz-Bloch equation (LLB) LLB for a FePt like granular recording medium. The material parameters of the medium can be seen in Table 3. In the simulations a continuous laser pulse with Gaussian shape and a full width at half maximum (FWHM) of 60 nm is considered. The temperature profile of the heat pulse is given by

[TABLE]

with

[TABLE]

The speed $v$ of the write head is assumed to be 15 m/s. $x$ and $y$ are the down-track and the off-track position of the grain, respectively. In the simulations both the down-track position $x$ and the off-track position $y$ are variable. The initial and final temperature of all simulations is $T_{\mathrm{min}}=300$ K. The write temperature $T_{\mathrm{write}}$ is chosen to be $800\,$K. The applied field is modeled as a trapezoidal field with a field duration of 0.57 ns and a field rise and fall time of 0.1 ns. The angle of the applied field with respect to the normal is assumed to be 22 deg. In each phase point 128 HAMR simulation simulations of a recording grain are performed.

In the phase diagrams the switching probability of a recording grain is shown as a function of the down-track position $x$ and the off-track position $y$. First, the phase diagram is computed for a spatially homogeneous field which approximates a conventional recording head design. Additionally, footprints for optimized geometries with field gradients of 8.6 mT/nm (Basic200), 8.1 mT/nm (Backshield100) and 11.8 mT/nm (Backshield200) are computed. The geometries with field gradients 2.2 mT/nm (Initial flipped) and 4.1 mT/nm (Basic100) are not further considered since the phase diagrams show too much noise to get reliable results. Note, that different write fields (see Table 2) are used for the simulations with different field gradients as seen in Table 2. In Figure 5, the resulting switching probability phase diagrams can be seen. There are some visible differences between the footprints of the different field gradients.

One can see that for the flipped designs the recording performance is worse than for the conventional design where a homogeneous field is assumed. This can be seen by the increase of jitter in down-track direction and the reduction of the maximum switching probability $P_{\mathrm{max}}$. $P_{\mathrm{max}}$ and the jitter can be determined by fitting the $P(x)-$curve at one temperature with a Gaussian cumulative distribution function

[TABLE]

with

[TABLE]

where the fitting parameters are the mean value $\mu$, the standard deviation $\sigma$ and the mean maximum switching probability $p\in[0,1]$. The standard deviation $\sigma$ determines the steepness of the transition function and is a measure for the transition jitter. The temperature at which the down-track jitter is determined is $T_{\mathrm{peak}}=760$ K. The resulting jitter and $P_{\mathrm{max}}$ values are summarized in Table 4. Additionally, it can be seen that the grains are written at larger off-track positions for increasing field gradients and the resulting lower write fields. The various write fields lead to writing of the grains at different peak temperatures. For smaller write fields the grains are written at higher temperatures only whereas they are written at lower temperatures for higher fields. The convention

[TABLE]

shows that with higher write fields, the off-track edge of a bit shifts to larger off-track positions, because the minimum temperature necessary to write a grain is smaller.

For a detailed analysis of the transition curvature, the full down-track range $\Delta x$ in which the bit is written with $P_{\mathrm{max}}\geq 50\%$ is computed. $\Delta x$ is marked in Figure 5 as the distance between the black dashed lines. Because of the different off-track widths $\Delta y$ in the phase diagrams for the different geometries, it is necessary to scale the curvature parameter with $\Delta y$. This is done, since the track width is usually kept constant in magnetic recording. In reality the track-width can for example be steered by controlling the peak temperature $T_{\mathrm{peak}}$ or the full width at half maximum huang2014heat . With the curvature parameter $c=\Delta x/\Delta y$, the curvature can be reliably analyzed. Note that $c$ is equivalent to the curvature parameter used by Vogler et al vogler_reduction when multiplied by $\Delta y$ for scaling reasons. There holds

[TABLE]

where $cp$ is the curvature parameter defined by Vogler et al vogler_reduction . The curvature analysis shows that the curvature is increased for both the Basic200 and the Backshield100 design compared to the conventional design. The curvature reduction for the Backshield200 flipped head design with a field gradient of $11.8$ mT/nm is about $1.3\%$. Detailed information about the curvature parameters can be seen in Table 4.

III.3 Exchange Spring Recording Medium

Since the curvature reduction is negligible for pure hard magnetic recording media in combination with the flipped head design, a bilayer structure with $50\%$ hard and $50\%$ soft magnetic material is tested as recording material. In the original paper by Vogler et al vogler_reduction , a bilayer structure showed significantly higher curvature reduction than the pure hard magnetic recording medium. The total height of the grains is again $h=8\,$nm. The material parameters of the soft magnetic composition can be seen in Table 5. Phase diagrams for the different head designs are calculated and the off-track width is again normalized for comparability reasons. They are shown in Figure 6. Again, the jitter $\sigma_{\mathrm{down}}$ and $P_{\mathrm{max}}$ are calculated for the footprints and compared in Table 6. Due to the higher Curie temperature of the exchange spring recording medium, the down-track jitter is determined at $800\,$K. For the exchange spring recording material the behavior of the curvature is similar to that for pure hard magnetic recording media and seems to be even worse. For the Backshield200 head design the decrease of the curvature is only 0.08$\%$ compared to the conventional design. However, the maximum switching probability stays 100$\%$ and the down-track jitter decreases.

III.4 Comparison with ERTW model

To understand why both the pure hard magnetic and the exchange spring media show almost no curvature reduction, the results are compared to theoretical considerations with the effective recording time window (ERTW) model described by Vogler et al basic ; vogler_reduction . The effective recording time window (ERTW) is defined by zhu2013understanding ; basic

[TABLE]

The first term on the right hand side gives the time window in which the grains can be written in HAMR, namely in the temperature range between the Curie temperature $T_{\mathrm{c}}$ and the freezing temperature $T_{\mathrm{freeze}}$. At the freezing temperature the coercivity decreases below the given field strength such that the grain cannot be written any longer. To calculate the ERTW, the freezing temperature has to be estimated. For this reason, hysteresis loops are simulated for various temperatures with VAMPIRE evans . At each fixed temperature, 128 hysteresis loops are computed and the temperature dependence of the coercive field d$H_{\mathrm{c}}/$d$T$ is determined. If the coercive field at a fixed temperature is lower than the write field, the temperature is a possible write temperature. Since the magnitude of the write field is not constant for the flipped head designs, a correction factor is included in the model. With the correction factor the magnitude of the field is updated according to the field gradient and the resulting write field is used for the determination of the freezing temperature.

The second term gives the time window in which the field points in the desired write direction, which in the following is regarded as pointing upwards without loss of generality. With this ERTW definition a switching probability phase diagram without noise can be computed for different materials and various fields and field gradients. The switching probability of a recording grain can be computed via basic

[TABLE]

The first term gives the probability that a bit is written in write direction. Since the probability cannot exceed 1, the minimum between 1 and $\Theta_{\mathrm{ERTW}}$ is taken. $\Theta_{\mathrm{ERTW}}$ is a threshold of the ERTW that gives the time that must be exceeded for successful switching to occur. It can be determined as a fit parameter to reproduce the results by the LLB model. The second term is the probability that a grain is overwritten after it was already aligned in write direction. Together, $p$ describes the joint probability to switch a bit in write direction and not reverting it afterwards. From the phase diagrams, the transition curvature is calculated and it is determined how much the theoretical curvature reduction should be.

For the Backshield200 geometry, the theoretically expected curvature reduction is about 1$\%$ for the pure hard magnetic recording material compared to the conventional design. This agrees very good with the curvature reduction resulting from the LLB simulations. For the exchange spring bilayer structure, the ERTW model predicts a curvature reduction of about $1.7\%$. The very small value achieved by the LLB simulation ($-0.08\%$) results from the switch of the transition around zero. This switch does not happen in the analytical ERTW model and comes from the stochastic nature of the LLB model.

Additionally, with the ERTW model, it can be analyzed with reasonable computational effort how the transition curvature reduction depends on the field gradient. The resulting curvature reduction values can be seen in Table 7. They display that the curvature reduces linearly with the field gradient. Moreover, a higher potential for the curvature reduction can be seen for the exchange spring bilayer structure. However, even with field gradients up to 40 mT/nm, the bilayer structure did not show as high curvature reduction as the exchange spring media used by Vogler et al. This can most likely be explained by the different damping constants used in the hard magnetic material. In the work by Vogler et al, the damping constant is $\alpha_{\mathrm{HM}}=0.1$ whereas it is $\alpha_{\mathrm{HM}}=0.02$ in this work. This leads to different $dH_{\mathrm{c}}/dT$ gradients and thus to the different curvature behavior. This shows that damping plays a key role for the curvature reduction.

IV Discussion

To conclude, in this work we tried to optimize the design of the write pole of a recording head for heat-assisted magnetic recording in order to reduce transition curvature. The write pole of the head was optimized in a way to maximize both the $z-$field and the $z-$field gradient at the position where the applied heat pulse is cooling down. This was done with the help of topology optimization which is an application of the inverse magnetostatic problem. Different starting geometries were considered. The comparison of the different geometries to a conventional recording head design showed the best results for a write pole and an additional backshield which both have dimension $200\,$nm in down-track direction. The resulting field gradient is then 11.8 mT/nm. The optimized geometries are all similar. They all have smooth edges, a tapered shape with a tip in the middle and the peak is skewed in $x-$direction such that the distance of the pole tip to the recording medium is close to the fieldbox and larger on the side away from it (see Figure 4). Since all optimized geometries are similar, the field and the field gradient could be maximized further by optimizing even larger write poles with an additional backshield. Another option to further increase the field gradient produced by the write pole is a design where the NFT is closer to the write pole.

We calculated switching probability phase diagrams of FePt like head magnetic recording media for the different optimized field gradients. It is noteworthy that the switching performance, in terms of down-track jitter and maximum switching probability, for all flipped head designs is worse than that of the conventional head design. Both, the AC and the DC noise increase for the flipped head designs. The performance loss results most likely from the smaller write fields that are produced by the flipped head designs. Analyzing the transition curvature showed that the curvature reduces only marginally even for the Backshield200 design where the highest write field and field gradient can be achieved. An idea how to improve the curvature reduction is to use an exchange spring medium suessexchange ; wangexchange ; victora ; suess ; coffey ; suess1 . The soft magnetic layer acts as a write assist and thus smaller write fields are required to write the grains. For the conventional design the curvature of the exchange spring recording medium is larger than that of the pure hard magnetic medium. Thus, the curvature reduction potential was expected to be higher for the exchange spring media. We computed switching probability phase diagrams for the different head designs in combination with the exchange spring recording media. Surprisingly, the curvature reduction for the Backshield200 design was even smaller than for the pure hard magnetic material. To understand why the curvature reduces only so little, we analyzed the results in the context of the effective recording time window (ERTW) model that was used in the paper by Vogler et al vogler_reduction . The ERTW model confirmed the results of the LLB model. Hence, we can conclude that the marginal reduction of the curvature is not a consequence of noise. The analysis also pointed out that the exchange spring recording medium shows a larger potential for curvature reduction if higher field gradients are used but even here the curvature reduction is only marginally.

In conclusion, we did not see any improvement of the transition curvature for the resulting field gradients of the optimized flipped head design in combination with pure hard magnetic and exchange spring recording media. The results showed that a simple optimization of the conventional head design design is not sufficient for effective curvature reduction but that new head concepts have to be introduced to reduce transition curvature.

V Acknowledgements

The authors would like to thank the Vienna Science and Technology Fund (WWTF) under grant No. MA14-044, the Advanced Storage Technology Consortium (ASTC), and the Austrian Science Fund (FWF) under grant No. I2214-N20 for financial support. The computational results presented have been achieved using the Vienna Scientific Cluster (VSC).