The Scaling of Signal-to-Noise in Heat-Assisted Magnetic Recording

This article presents a systematic micromagnetic modeling study on recording characteristics in heat-assisted magnetic recording (HAMR). Utilizing a novel micromagnetic model developed based on the <inline-formula> <tex-math notation="LaTeX">$\text{L}1_{0}$ </tex-math></inline-formula> ordering atomic structure of the FePt grains, the study shows that the medium noise caused by grain-to-grain Curie temperature variations can be effectively suppressed with sufficiently high thermal recording gradient along with sufficient head field amplitude. The rest of the article focuses on the thermal noise and its correlations with various recording parameters.

It has been well recognized that thermal gradient (TG) in recording is critically important to suppress medium noise [25].However, in practice, the bit error rate (BER) benefit seems to saturate at relatively low TG [25], [26], limiting the expected areal density gain for the recording.In searching for the answer to this puzzle, systematic micromagnetic modeling and simulation work is conducted and presented here.The study focuses on the understanding of the correlation between TG and grain-to-grain Curie temperature variation.At the same time, I will try to establish correlations between the thermal noise and other recording parameters, such as head field amplitude and disk speed during writing.

II. MODEL AND SIMULATIONS
The micromagnetic model employed here has been developed to incorporate the physical nature of the distinctive atomic structure in L1 0 FePt grains of the recording media [21].In a perfectly ordered L1 0 FePt single crystal grain, monolayers of pure Fe and pure Pt alternate in the ordering direction [27], [28].Because of this ordered atomic structure, near but below Curie temperature, the ferromagnetic exchange coupling within each Fe monolayers would dominate whereas the exchange coupling between adjacent Fe monolayers, mediated by the spin-polarized Pt monolayer in between, is relatively weaker [29].Considering this important fact, each FePt grain in the HAMR granular media studied here is modeled by a stack of 30 macro-spins [21] with each macro-spin representing the magnetization of a single Fe atomic monolayer, as illustrated in Fig. 1.This approach effectively assumes that each Fe monolayer is always uniformly magnetized.The 30 Fe monolayer stack corresponds to a single L1 0 FePt grain of a height 11.52 nm, using the lattice spacing of c = 0.384 nm in the ordering direction [28].While the magnetic moment of Pt monolayers is ignored in this model, the Pt-mediated exchange coupling between adjacent Fe monolayers is incorporated.Each macro-spin is assumed to follow the temperature dependence of the magnetization: and anisotropy field [26] H with easy axis oriented along the ordering direction, which is perpendicular to the film plane for this article.The model, thus, enables the modeling of nonuniform temperature as well as different T c through the depth of a grain.However, in this study, the Curie temperature T c is assumed to be the same for all the monolayers within the grain while T c can vary from grain to grain according to a Gaussian distribution.As shown in Fig. 1, a practical HAMR media is represented by a Voronoi assembly of the modeled L1 0 FePt grains.For all the cases presented in this article, the mean T c is assumed to be 675 K.The created grain assembly has a 15% grain size distribution with mean grain diameter at D = 7 nm.No correlation between the grain Curie temperature and the grain size is assumed.The anisotropy field of every grain is set to be the same with H k (RT) = 8 Tesla at room temperature without any variation from grain to grain.
Three-dimensional recording temperature profiles, with one shown in Fig. 1, are calculated using COMSOL software.The peak temperature is 800 K for all the cases in this article.In mapping the thermal profile to the actual grains in the medium at any moment, the average temperature over the lateral area of a grain is used.The dynamic orientation of each macro-spin follows the Landau-Lifshitz-Gilbert gyromagnetic equation of motion: where ⃗ m is the unit vector of the magnetic moment for the macro-spin, γ is the gyromagnetic ratio, α is the Gilbert damping constant, and ⃗ H is the effective field on the macrospin, which includes where the right-hand side are magnetic anisotropy field, the exchange coupling field between adjacent Fe monolayers within a grain, magnetostatic fields, head field, and thermal magnetic Langevin field, respectively.In particular, the interlayer exchange field is where ⃗ m i+1 and ⃗ m i−1 are the unit vector of the macro-spins for the adjacent Fe monolayer above and below, respectively, c is the distance between adjacent Fe monolayers, and A * is the effective exchange stiffness constant between the adjacent Fe monolayers within a gran.Since this exchange coupling is mediated by the spin-polarized Pt atoms in between, it is reasonable to assume that the exchange coupling would also be proportional to the magnetization level of the two adjacent Fe monolayers This is because the Pt spin polarization should be proportional to the magnetic moment of Fe monolayers.For all the calculation results presented here, A * 0 = 0.45 × 10 −6 erg/cm is used and the reason for choosing this particular value will be discussed later in the article.The head field is assumed to be spatially uniform and its direction is assumed to be tilted toward the down track direction at an angle of 20 • with respect to the perpendicular direction.The thermal Langevin magnetic field is used to model the thermal effect which has been described in detail in [7], [31], [32], and [33].For all the calculations presented here, a Gilbert damping constant α = 0.2 is used which gives reasonable match between experimental measurements and modeling results.Existing theoretical and experimental investigations have indicated significant rise of the Gilbert damping constant near Curie temperature [34], [35], [36].
Fig. 2 shows simulated recording process (left) with color spectrum representing the perpendicular component of the grain magnetization as yellow for RT M s ↑ and deep blue for RT M s ↓.The thermal heating spot moves from up to down in a constant speed relative to the granular medium while recording head field reverses its direction over time.Over a rectangular area of 30 nm width (crosstrack) and 2 nm length (downtrack resolution) is performed over a simulated written track (∼320 nm in length).For each calculation case, 60 tracks, each with different granular pattern, are simulated.Mean magnetization profile and corresponding variances are calculated.The signal power Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.where B is the bit length, and noise power The signal-to-noise ratio (SNR) is defined as III. RESULTS AND DISCUSSION Fig. 3 shows calculated SNR as a function of recording TG for a range of head field amplitudes.A standard deviation of σ T C = 2% is assumed for grain Curie temperature distribution.At each field amplitude, SNR rises rapidly with the initial increase of TG followed by slowing down of the increase.The SNR saturation begins around TG = 7 K/nm.The TG dependence is essentially the same for all the cases with different field amplitudes, however, the exact SNR level increases with increasing head field amplitude for the field amplitude range shown in the figure.
The dependence of SNR on recording TG shown in Fig. 3 has been widely observed in practice [25], [26].The agreement with the experimental measurements not only provides certain degree of validation for the modeling excise presented here, but also enables us to examine the saturation effect.To understand the SNR saturation at relatively high recording TGs, recording simulations on media with different values of σ T C are performed.Fig. 4 shows the calculated SNR as a function of recording TG for three media of different σ T c values: σ T c = 1%, σ T c = 3%, and σ T c = 4%.For the three corresponding SNR calculations, the head field amplitude is adjusted such that they have the same SNR value at TG = 15 K/nm.The head field values are H head = 0.64 (Tesla), 0.88, and 0.96 for the three cases, respectively.First, we see that the three media with different σ T c values can reach the same SNR level with sufficient high TG and head field amplitudes.In other words, the SNR degradation caused by grain-to-grain T C distribution can be completely recovered if recording TG and head field amplitude are sufficient.The broader the Curie temperature distribution, the higher the TG is required along with higher head field amplitude.Conversely, the measurement of the TG at onset of SNR saturation can help to determine the σ T c value of a medium.Fig. 5 shows the SNR versus recording TG for the three media at the same field amplitude.Without raise the field amplitude, increase TG alone will not recover the SNR degradation due to grain Tc distribution.Raising field amplitude is equally important to eliminate the SNR impact of grain-tograin Curie temperature variations.
Sufficiently high head field amplitude, especially when recording TG is high, is important for achieving optimum recording performance.Fig. 6 shows three simulated recording tracks for three different field amplitudes.The recorded bit patterns at the lower field amplitudes appear to be incomplete: some of the grains magnetized in the wrong direction.The situation evidently improves with the higher field amplitudes  as the percentage of wrongly magnetized grains becomes less.Fig. 7 shows the calculated SNR as a function of head field amplitude.For the calculated results, the recording TG is set at TG = 15 K/nm.As shown in the figure, initial increase of head field amplitudes yields significant increase of the recording SNR.As the head field approaches H head = 1.2 Tesla, SNR starts to level off, continue to increase head field amplitude, the SNR starts to decrease because erasureafter-write starts to become more significant [7].
In HAMR, magnetic bits, or transitions, are formed at temperatures only slightly below the Curie temperature, which is significantly higher than the ambient.At such recording temperatures, the grain magnetization level is significantly below that of the room temperature.The ratio of the magnetic potential energy (due to the recording head field) and thermal The difference between the two energy minima is due to the head field and the energy barrier due to the grain magnetocrystalline anisotropy.Upper: The case with a relatively smaller head field and the limited magnetic potential energy was not strong enough such that a number of grains have their magnetization opposite to the field direction after the energy barrier becomes too high.Lower: A stronger head field leads to much reduced the number of wrongly magnetized grans due to higher magnetic potential energy as well as extended RTW.
energy is defined below [37] where k B is Boltzmann's constant, T is the recording temperature, V eff is the effective grain volume, and M is the magnetization level at the recording temperature.For HAMR, the value of η is nearly one order of magnitude smaller than that for conventional perpendicular magnetic recording, assuming the same medium grain volume.The impact of thermally excited magnetization fluctuation during recording is, thus, substantially greater in HAMR.The upper row of Fig. 8 graphically illustrates this thermal impact.The red dots in the figure indicate the magnetization state of the medium grains in the recording zone.As the medium grains cool down from Curie temperature (left to right), the energy valley for the grains whose magnetization is in the head field direction grows deeper as the grain magnetization level increases.At the same time, the barrier between the two energy valleys grows higher due to the rise of grain crystalline anisotropy energy.If time is sufficient, few grains will end in the "up" state (against downward head field) since the equilibrium probability has an exponential dependence on the difference between the two energy valleys.However, the rise of energy barrier in between the two states could block the state transition if the cooling time is too faster: grains will be magnetized, or frozen, in the wrong states, causing the "incomplete" magnetization pattern recorded, as shown in the lower field amplitude cases of Fig. 5.
The noise associated with this mechanism is referred to as thermal noise.In the case with a larger field amplitude, shown in the lower row of Fig. 8, the level difference of the two energy valleys is greater and the time to the "frozen" point becomes longer, leaving fewer grains wrongly magnetized, thereby lowering thermal noise.The wrongly magnetized grains not only cause magnetization level to be lower than the saturation remanence for Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.10.Simulated recording tracks at three different disk speed over the same medium.Same head field amplitude H head = 0.64 T is used for all recording simulations shown here.At higher speeds, the number of wrongly magnetized grains is evidently more than that at the lower recording speed.
the regions in between adjacent transitions, but also cause transition position to shift randomly, i.e., transition jitter noise, if the grains happen to be in the transitions.
In the case of insufficient head field amplitude, slowing down the recording process can help to suppress the percentage of those "wrongly" magnetized grains since the system will have more time to relax to the state with lower magnetic potential energy before the anisotropy of the grain becomes too high.Fig. 9 shows the calculated SNR as a function of recording TG for a range of disk speed at the same head field amplitude: H head = 0.64 Tesla.The reduction of disk speed from 30 to 10 m/s yields a significant increase of SNR.With high recording TG, slowing down disk speed can significantly raise the SNR.In practice, lower head field amplitude should cause greater difference in SNR between recording at the inner diameter and the outer diameter of a disk.Fig. 10 shows simulated recording tracks for three different writing disk speeds.The recorded magnetization pattern is evidently more "regular" at the lower speeds whereas more "random" at the higher ones.Similar to the lower field amplitude cases shown in Fig. 6, the higher speed cases in this figure have more grain with their magnetization "frozen" in wrong directions while at the lower speed cases, the wrongly magnetized grains are apparently less.Since the head field amplitude in this case is relatively low, even at v = 5 m/s, we still can find oppositely magnetized grains in the middle of a bit region due to the recording time window (RTW) is still insufficient.In this case, further reduction of disk speed could still yield a further reduction of thermal noise.
From the above discussion, one can see that the SNR dependence on disk speed should be a function of head field amplitude.In Fig. 11, SNR as a function of disk speed is plotted for a range of head field amplitudes.Note the recording TG is fixed at TG = 15 K/nm.The speed dependence of the SNR is stronger for lower field amplitudes and is weaker for higher field amplitudes.Increasing recording field amplitude decreases the possibility for grains' magnetization to be frozen in wrong directions, thereby alleviating the need for longer RTW.However, if the field amplitude is too high, erasure-after-write starts to become significant.Reducing disk speed will further aid this degradation.In principle, if the erasure-after-write can be avoided with sufficiently high TG, thermal noise in HAMR media can completely be eliminated with sufficient long RTW and sufficient high field amplitude.
With finite recording TG, even at TG = 15 K/nm, erasureafter-write and thermal noise caused by insufficient RTW co-exist at relatively high field amplitude, as Fig. 12 illustrates.As shown in the figure, the SNR leveling off at higher field amplitudes indicates the increased presence of erasure-afterwrite while thermal noise still not yet vanished as the SNR can still be enhanced with decreasing disk speed.To completely eliminate thermal noise with either high head field amplitude or/and slow disk writing speed, sufficiently high TG is critical to avoid erasure-after-write.
To further illustrate the interdependence of the SNR on both the linear density and recording speed, Fig. 13 plots the calculated recording SNR contours (numbers are in dB) as a function of recording linear density and writing data rate (note no electronic noise is included here.).Three cases with different head field amplitudes, H head = 0.64 Tesla (upper), 0.80 Tesla (middle), and 1.2 Tesla (lower) are shown here.Recording TG is fixed at TG = 15 K/nm.When the field amplitude is relatively low, lowering the write data rate enables significantly higher linear density while maintaining SNR.However, at relative high head field amplitude, the SNR gain by reducing the write data rate becomes much limited and the dependence on the write data rate is significantly less.These results are very much consistent with the ones shown in Fig. 11.
The SNR dependence on both the head field amplitude and the recording linear speed can be explained by the RTW which was first proposed in [7] with further in depth studies in [8].Substituting recording temperature T Recording with head field amplitude H head , one obtains the following expression: where T c is Curie temperature of media in Kelvin, TG is recording TG in K/nm, v is disk linear velocity (speed), H k,RT is anisotropy field at room temperature (300 K), and β H head is the maximum anisotropy field of the grain can be switched by the head field, H head , with the value of β in the between [1.0, 2.0], depending on the field angle at the location where transition is written.Insufficient value of RTW can lead to low SNR whereas prolonged RTW yields erasure-afterwrite [7], causing SNR to degrade.Relatively low head field amplitude combining with higher recording speed always leads to insufficient RTW.Equation ( 11) can be used to estimate the RTW for both modeling and spin stand measurements.
To put things in perspective, Fig. 14 shows the comparison of recording SNR as a function of recording TG for two cases: media with and without grain-to-grain Curie temperature variation.Here the recording head field amplitude is purposely chosen to be relatively high.For the case with σ T c = 2%, (red curve in the figure) increasing TG yields a monotonic increase of SNR value as stated before.For the case without grain-to-grain Curie temperature variation, σ T c = 0 (blue curve), the recording SNR is significantly higher than that in the case σ T c = 2%, however, the SNR shows a peak value around 9 K/nm.Below this optimal TG value, erasureafter-write is the main cause for the SNR reduction due to relatively poor TG.For TG greater than the optimum, the RTW becomes insufficiently short, and reduction of the RTW causes  the degradation of SNR.A higher disk speed will move the optimum TG value toward a lower value.A higher recording field amplitude would require a higher TG to avoid worsening of erasure-after-write.
Before ending this section, we would like to address the choice of the effective exchange stiffness constant, A * , which measures the ferromagnetic exchange coupling between the adjacent Fe atomic monolayers within a L1 0 ordered FePt grain.Fig. 15 shows calculated recording SNR as a function of the effective exchange stiffness constant for three different recording speeds.The head field amplitude for the recording is H head = 0.64 Tesla and the linear density for the recording simulation is D ≈ 1.7 MFCI (bit length B = 15 nm).(In reference that the exchange stiffness constant for bcc Fe is A = 1.3 × 10 −6 erg/cm.)The pink-shaded value A * 0 = 0.45 × 10 −6 erg/cm is the value that used for all the calculations presented in article and is chosen based on comparison with experimental measurements [26].Note that the effective exchange stiffness constant for adjacent Fe monolayers should be dependent on the L1 0 order parameter since any substitution with Fe atoms in a Pt monolayer should increase the exchange coupling between two adjacent Fe monolayers, hence likely to increase the value of A * .At the chosen A * 0 value, all the macro-spins within a grain are oriented in the same direction at the end of a recording process.Even though during recording, magnetization switching could be incoherent, and it usually are, after temperature returns to ambient, every grain is essentially in a single domain state with all 30 macro-spins oriented in the same direction with no residual domain wall left within a grain stack.

IV. CONCLUSION AND REMARKS
A novel micromagnetic model developed based on the atomic structure of L1 0 ordered FePt grains is employed to study the HAMR in granular FePt-L1 0 thin film media.The study found that the medium noise arising from grain-to-grain Curie temperature variation can be completely suppressed with sufficient TG and sufficiently high head field amplitude.The broader the T c distribution requires the higher TG and the greater field amplitude.
The other dominant noise source in HAMR is the thermal noise caused by the magnetization of the grains in the recording zone frozen to the opposition direction of the head field during recording processes.The thermal noise could dominate if the head field amplitude is insufficient and disk speed is high which leads to insufficient RTW.Raising head field amplitude or slowing down disk speed during recording could sufficiently suppress the noise and achieving grain pitch limited recording SNR as long as recording TG is sufficiently high to prevent any erasure-after-write.
The strong head field amplitude dependence of the thermal noise points at a possible cause for the broad variation of head-to-head SNR performance widely observed in practice.In HAMR, the footprint of write-head main pole is much larger than that in conventional perpendicular magnetic recording.The large dimension of the main pole gives rise to the possibility of complex domain formations during recording process which is a very plausible reason for the observed headto-head performance variation as well as dynamic variations of the head field amplitude during recording.
Incorporating the essential physics based on the atomic structure of L1 0 FePt grain is evidently critical for correctly modeling the physical behavior of the granular media during recording.Since the ferromagnetic exchange coupling between the adjacent Fe monolayers (interlayer coupling) and within each Fe monolayer (intra-layer coupling) is significantly different, Curie temperature of perpendicularly ordered L1 0 granular FePt films should have a significantly stronger dependence on grain size than that on grain height.Specific experimental measurements for quantitatively determining the interlayer exchange coupling between adjacent Fe monolayers within a highly L1 0 ordered FePt grains are needed.
The relatively weak ferromagnetic exchange coupling in the ordering direction for L1 0 FePt grains would also imply that increasing grain height is not as effective as increasing grain diameter in terms of having sufficient grain volume.Even greater grain height might be needed as grain diameter reduces for higher area density capabilities [37].

Fig. 1 .
Fig. 1.Left: Modeled FePt granular media with temperature profile during a recording.Top Right: Recording thermal profile calculated using COMSOL from top to bottom at different depth of medium grain.Bottom Right: Illustration of each grain is modeled by a stack of macro-spins with each macro-spin modeling a single Fe atomic monolayer.Adjacent Fe monolayers are coupled by the ferromagnetic exchange interaction mediated by the Pt monolyaer in between.

Fig. 2 .
Fig. 2. (a) Simulated recording process with color representing the perpendicular component of magnetization of each grain.Zero magnetization for the region above Curie temperature shown in green.Yellow and blue show opposition magnetization components.(b) Mean readback" magnetization (upper) and corresponding variance (lower) for a read width of 30 nm.

Fig. 3 .
Fig. 3. Calculated SNR as a function of recording TG for a range of head field amplitudes.A standard deviation of σ T C = 2% is assumed for grain Curie temperature distribution.The symbols are actual calculation results and the curves are drawn to guide the eye (for this plot and the rest of the plots in the paper).

Fig. 4 .
Fig.4.Calculated SNR for three media of different values of σ T c .Note for each medium case, head field amplitude is adjusted such that the SNR saturation values at high TG for each case match each other.The results here show with sufficient high TG and raised head field amplitude, the noise arising from grain-go-grain Curie temperature variation can be completely suppressed.

Fig. 6 .
Fig. 6.Simulated recording tracks at three different head field amplitude.The TG is TG = 15 K/nm.Comparing the recorded magnetization patterns between different head field amplitudes, one can see that the recording patterns are incomplete with grains magnetized in wrong directions.Repeating the exact same recording process with the same granular grain structure, the wrongly magnetized grains are different every time since it is caused by the random process, a nature of thermal noise in heat-assisted recording.

Fig. 7 .
Fig. 7. Calculated recording SNR as function of head field amplitude.The recording TG is TG = 15 K/nm.The linear density is D = 1700 KFCI, corresponding to a bit length B = 15 nm.

Fig. 8 .
Fig. 8. Illustration of transient magnetization states for the grains within recording zone during recording.The curve representing energy versus magnetization angle as the medium cools while moving away from the NFT.The difference between the two energy minima is due to the head field and the energy barrier due to the grain magnetocrystalline anisotropy.Upper: The case with a relatively smaller head field and the limited magnetic potential energy was not strong enough such that a number of grains have their magnetization opposite to the field direction after the energy barrier becomes too high.Lower: A stronger head field leads to much reduced the number of wrongly magnetized grans due to higher magnetic potential energy as well as extended RTW.

Fig. 9 .
Fig. 9. Calculated recording SNR as a function of TG for a range of different disk speed during recording, all at the same exact field amplitude.Decreasing disk speed during recording significantly raises the SNR level.

Fig.
Fig.10.Simulated recording tracks at three different disk speed over the same medium.Same head field amplitude H head = 0.64 T is used for all recording simulations shown here.At higher speeds, the number of wrongly magnetized grains is evidently more than that at the lower recording speed.

Fig. 11 .
Fig. 11.Calculate recording SNR as a function of disk speed for a range of different head field amplitude in recording.The linear recording density in all the cases is the same: D = 1.7 MFCI.The same TG, TG = 15 K/nm is used for recording in all cases in this figure.

Fig. 12 .
Fig. 12. Calculated recording SNR as a function of head field amplitude for three difference disk speed during writing.

Fig. 13 .
Fig. 13.Calculated recording SNR contours (in dB) as function of recording linear density and writing data rate for three different head field amplitudes.The recording thermal gradient of TG = 15 K/nm is used.A Tc distribution of σ T c = 2% is assumed over the grains in the medium.The SNR values are also indicated by the color mapping.

Fig. 14 .
Fig. 14.Calculated recording SNR as a function of recording TG for the case of σ T c = 0 (blue) and the case of σ T c = 2% (red).The head field amplitude is 1.2 Tesla, the recording bit length is B = 6 nm and the disk linear speed is v = 10 m/s.

Fig. 15 .
Fig. 15.Calculated recording SNR as a function of the effective Fe-Fe exchange coupling stiffness constant A * for three cases, each at a different disk speed.The head field amplitude is set at H head = 0.64 Tesla and the linear recording density is D = 1.7 MFCI.