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IEEE MAGNETICS LETTERS, Volume 11 (2020) 6502604

Information Storage

Analytical Estimation of Transition Jitter for the Heat-Assisted Magnetic Recording

Process

Niranjan A. Natekar and R. H. Victora
Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA

Received 26 Feb 2020, revised 18 Apr 2020, accepted 27 Apr 2020, published 4 May 2020, current version 29 May 2020.
Abstract—In the heat-assisted magnetic recording (HAMR) process, the transition jitter is known to depend on the grain
size as well as the heat spot thermal gradient. We find that the jitter also depends on the Voronoi grain size distribution, as
well as exhibiting a surprising nonlinear dependence on reader width. By combining the noise due to these dependencies
we used an analytical formulation to estimate the jitter in the HAMR process. The accuracy of the analytical formulation
is verified by comparing the predicted jitter to micromagnetic simulations. This simple formulation provides both physical
insight and conserves computational time relative to lengthy (and complex) recording simulations.

Index Terms—Information storage, heat-assisted magnetic recording, jitter, noise, read head, recording media, micromagnetics.

I. INTRODUCTION reduce computational time, we use an analytical formulation that can
help estimate the jitter. The fact that, for HAMR, the jitter is mostly
New magnetic recording technologies, such as shingled magnetic
determined experimentally or via simulations suggests a lack of an
recording [Wood 2009, Amer 2011, Hall 2012], microwave-assisted
analytical technique to estimate its value. This formulation provides a
magnetic recording [Zhu 2010], and heat-assisted magnetic recording
useful cross check for jitter obtained from different techniques.
(HAMR) [Kryder 2008, Greaves 2013, Wang 2013, Wu 2013], are fast
gaining popularity due to the predicted high user areal density (UAD)
values. Single-layer FePt media has been widely studied to understand II. SIMULATION CONDITIONS
the potential and limits of the HAMR process. In order to increase
The formulation estimates jitter (σ) for the single-layer FePt media
the UAD, composite media, such as the FePt/FeRh media [Thiele
(384 nm downtrack (DT) × 48 nm crosstrack (CT) × 9 nm thickness)
2003] (which takes advantage of the sharp FeRh phase transition) and
on which a single-tone sequence is written. The media consist of
FePt/Cr/X/FePt media [Victora 2015] (which enjoys the advantage of
Voronoi grains, and a 1 nm nonmagnetic grain boundary is used
the sharp AFC transition due to the Cr layer) have been modeled. To
around the grains. The bit length used in the simulation is 21 nm,
counter the poor switching performance of these structures, micro-
and the head velocity is 20 m/s. A Gaussian heat spot with DT and CT
magnetic simulations have helped develop novel media designs, such
full-width at half-maximum (FWHM) of 40 nm, and peak temperature
as thermal exchange coupled composite (ECC) media [Liu 2016a,
of 850 K was used to heat the media. Readback is implemented
2016b]. In an effort to reduce the writing and recording temperature in
with a shield-to-shield spacing of 21 nm and a read width (RW)of
the HAMR process, a modified low-temperature thermal ECC media
20 nm. For the switching probability (SP) calculations, the medium
with lower T c for the write and storage layer was recently proposed
is cooled down with a constant cooling rate (CLR) from a peak
[Natekar 2019].
temperature (T peak ) down to room temperature (300 K), and at a certain
One way to characterize the recording performance of the HAMR
temperature, the direction of the uniform applied field (H appl = 8 kOe)
mediaistocalculatethetransitionjitterbyanalyzingthewrittenpattern
is reversed. Differentiating this SP curve yields the SP distribution
during the recording process. To implement the HAMR process, mi-
(SPD). A similar idea of relating the jitter and SPD for different media
cromagnetic simulations with a Gaussian heat spot are implemented.
variationshasbeendescribedpreviously[Liu2017]withoutexplaining
To ensure enough transitions, a single-tone sequence is written on the
the individual factors that contribute to the jitter. Previous references
media using a fixed external field modulated at a certain frequency. To
[Hohlfeld 2019] have also attempted to study the effects of individual
correctly estimate the jitter, we need a significant number of transitions
recording parameters, such as the thermal fluctuations on jitter, but
written on different media with similar average grain size (GS)and
none have attempted to propose a complete analytical technique to
GS distribution (GSD), but different grains. These micromagnetic
calculate jitter. Also, the formulae obtained in these references are
simulations are long (∼300 ns), computationally demanding (often
dependent on calculating several complicated terms. This letter de-
requiring the use of GPUs or supercomputers to calculate the magne-
scribes, in detail, the individual contributions to the jitter of various
tostatic interactions), involve errors (∼5%), and are often dependent
physical characteristics, and provides analytic expressions for them.
on the random number generator used for calculating the thermal field,
which can be different for different systems. To avoid these issues and
III. RESULTS AND DISCUSSION
The first factor to consider is the grain pitch (GP). The transition
Corresponding author: Niranjan A. Natekar (e-mail: [email protected]).
Digital Object Identifier 10.1109/LMAG.2020.2992221 center can lie anywhere within a grain of pitch (GP = D + 1 nm).
1949-307X © 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
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6502604 IEEE MAGNETICS LETTERS, Volume 11 (2020)



The probability distribution function for the transition center is f(x) =
(1/GP). The corresponding jitter value is Var (f(x)). Normalizing the
jitter in the CT direction with N (number of microtracks) yields the
square of the jitter due to GS [Valcu 2010]
GP 3
σ 2 = . (1)
GS (RW × 12)
The next effect specific to the HAMR process is that of the heat
spot. An ideal heat spot with an infinite thermal gradient (FWHM
∼0) would ensure a sharp transition with no additional jitter. The
finite thermal gradient and FWHM value, however, lead to a positional
gradient in the SP. To find this gradient, we use the chain rule to relate
the SP [P(T)] as a function of position to the SPD where dP(T)/dT
denotes the SPD and where dT/dx is the gradient at write temperature.
Since the SPD calculation already takes into account the effect of
other recording factors, such as cooling rate and thermal fluctuations,
no separate equations need to be derived to relate these factors to
Fig.1. Jitter comparison (σ theory and σ simulation ) for different GP (T peak =
the jitter value. This calculation can also account for variations of 850 K, heat spot FWHM = 40 nm, and RW = 20 nm).
intrinsic magnetic parameters, thus reducing the overall complexity
in calculating the noise in the HAMR process. This calculation is
faster (∼1 ns), does not involve the calculation of magnetostatic
interactions, and is largely system independent. This simplifies the
proposed analytical formulation. Since the SPD is Gaussian in nature
[Liu 2016a, 2016b] and the thermal gradient at the writing temperature
is constant, the positional gradient is Gaussian. Using the chain rule
and normalizing the jitter in the CT direction yield the jitter due to the
heat spot geometry. σ dP/dT denotes the standard deviation of the SPD

GP σ dP/dT
σ dP = × . (2)
dx RW dT
dx
Since the medium consists of Voronoi grains with varying GP, the fi-
nal jitter value experiences an additional jitter due to the nonuniformity
of the Voronoi grains. This nonuniformity is expressed by the GSD
(=δD/D). Mathematically we consider a Taylor expansion for (1)

GP 3 δD 3
2
σ = σ 2 + ×
jit GS Fig. 2. Comparison of jitter (σ theory and σ simulation ) for different T peak
RW × 12 D
(GP = 7 nm and RW = 20 nm).
2
δD δD

+ 3 × + 3 × .
D D heat spot with infinite thermal gradient and jitter calculated for media
with GSD = 0. For the SP simulations, the cooling rate CLR =
Since δD can be positive and negative, we average over all δD values,
(dT/dx) (at writing temperature) × dx/dt (head velocity) = 390 K/ns
leaving behind only the terms with even power
(dT/dx ∼19.5 K/nm and dx/dt = 20 m/s). The close agreement of the
2
2
2

σ δD = σ − σ GS (3) jitter values obtained from the two different techniques helps define
jit
the accuracy of the analytical equations.
GP δD
3 2
σ 2 = × 3 × . (4) The next comparison is done for the heat spot gradient. To do so,
δD
RW × 12 D
simulations are implemented for a media with average GP = 7nmand
Thus, the analytical formulation given by different T peak (peak temperature of heat spot). To maintain the same
2
2
2
2
σ theory = σ GS + σ δD + σ dP (5) width of the write spot, the FWHM and the corresponding CLR value
dx is varied. Fig. 2 shows a comparison of the jitter obtained with the
now predicts the square of the jitter (σ theory ) to be a combination of analytical formulation and the simulations.
(1), (2), and (4). Once again, the values are comparable to each other, showing
To verify the analytical formulation, comparisons are implemented the accuracy of the derived analytical equations. Another aspect of
for different recording conditions. The comparisons are implemented the single-layer FePt media is the effect of media variations on the
for jitter calculated from two sources: (5) (denoted as σ theory )and magnetic parameters of FePt [Natekar 2017]. These media variations
micromagnetic simulations (denoted as σ simulation ). For the simulations, arise due to improper fabrication processes and a variation in the shape
jitter is calculated considering at least 200 zero crossings. The first and size of the grains. Jitter values calculated using micromagnetic
comparison is done to observe the effect of different GP values (see simulations are compared to the analytical estimate for different GSD,
Fig. 1) with GSD = 20%. σ ideal denotes the jitter value for a Gaussian σ Tc and σ Hk . Fig. 3(a)–(c) shows the comparison of jitter for different
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IEEE MAGNETICS LETTERS, Volume 11 (2020) 6502604





























Fig. 4. Comparison of jitter (σ theory and σ simulation ) for different GP and
RW values (ellipse indicates discrepancy of interest).












Fig. 5. Assumed square potential for analytical formulation.


the linear reader detects additional noise due to curved transitions that
is not reflected in the analytical estimate, which assumes a straight
transition. Hence, σ simulation >σ theory . The detection of extra noise due
to the curved nature of the transition in the HAMR process has been
studied in detail previously [Zhu 2017]. The case for small RW (<
RW ideal ) is interesting. From Fig. 4, for RW = 10 nm, the analytical
formulation predicts a jitter value greater than the actual value for
different GP values. This is unexpected for small RW, where the curved
nature of the transition has little effect on the detected noise. For this
RW value, the discrepancy can be explained by understanding the
nonlinear behavior of the playback for small RW (∼GP). Fig. 5 shows
the shape of the potential curve assumed in the analytical formulation.
Fig. 3. Comparison of jitter (σ theory and σ simulation ) for different media The analytical formulation assumed the square potential to be perfectly
variations (FWHM = 40 nm and GP = 7nm).
aligned with respect to the microtracks. In reality though, the head
potential is frequently misaligned, as shown in Fig. 5. To understand
media variations. The analytical formulation predicts jitter values
the corresponding jitter calculation, we need to consider the calculation
close to the values provided by the micromagnetic simulations. The
2
analytical formulation assumes the jitter due to GS (σ ) and the of signal power (SP sq ) and noise power (NP sq ) due to the misaligned
GS
2
thermalgradient (σ dP )tobeindependent.Inrealitythough,theyarenot potential curve. For a microtrack length equal to GP, the RW can be
dx defined as RW = N × GP + F. Thus, for the square pulse, the SP and
completely independent and exhibit a minor correlation. Correlations
NP are, respectively, given by (6) and (7) where n 2 represents the
in grain position can also affect the jitter [Hara 2015]. This can explain DT
normalized NP in the DT direction.
the small discrepancy between the jitter values obtained from the two
sources in Figs. 1–3. 1 GP−F 2
The final factor that alters the eventual jitter value in the recording SP sq = GP × 0 ((GP − x) + (N − 1) GP + (F + x)) dx
processisthereaderwidth(RW).Fig.4showstheeffectonjitteras RW
GP
varies from 10 to 40 nm for different GP values. The ideal RW value 1 2
+ ((GP − x) + (N) GP + (F + x − GP)) dx
(RW ideal ) is about half the heat spot FWHM. RW ideal ∼20 nm when the GP GP−F
heat spot FWHM equals 40 nm and T peak = 850 K. For RW > RW ideal , (6)
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6502604 IEEE MAGNETICS LETTERS, Volume 11 (2020)


Table 1. Corrected jitter values for RW = 10 nm and different GP. and systems. Additionally, it provides a cross-check for verifying the
accuracy of jitter predictions obtained via other techniques, such as
micromagnetic simulations.

ACKNOWLEDGMENT
The authors thank T. Qu, W. H. Hsu, and Z. Liu for their useful discussions. This work
was supported by the Advanced Storage Research Consortium.

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