Model Calculation Co nsidering Recording Time Window
for Heat-Assiste d Mag netic Rec ording
T. Kobayashi and Y. Fujiwara
Graduate School of Engineering, Mie Univ., 1577 Kurimamachiya-cho, Tsu 514-8507, Japan

We improve our model calculation for heat-assisted magnetic recording (HAMR) by introducing the concept of
the recording time window proposed in the micromagnetic calculation. The improved model calculation includes all
the equations for the HAMR conditions used in the previous model. The difference is the introduction of therecording time window to determine the composition of the medium and the writing field. This improvement meansthat the results obtained using the model calculation become consistent with those obtained using a micromagnetic
calculation. The minimum anisotropy constant ratio of the medium at 2, 3, and 4 Tbpsi can be determined using theimproved model calculation.
Key wor ds : heat-assisted magnetic recording, model calculation, recording time window, areal density, anisotropy
constant ratio
1. I ntro duction
Heat-assisted magnetic recording (HAMR) is arecording method in which the medium is heated toreduce coercivity during the writing period. HAMR has
been studied with the aim of solving the trilemma
problem1) of magnetic recording (MR). In most cases, amicromagnetic calculation is used for the HAMR design.
A feature of the micromagnetic calculation is its precise
simulation based on the actual situation. On the other
hand, this calculation requires a long time, and it issometimes difficult to grasp the physical implications of
the obtained results.
We have reported a design method that uses amodel calculation for the HAMR design2) of 4 Tbpsi inorder to shorten the calculation time and grasp thephysical implications. In that paper, we revealed thecomplexrelationshipbetweencertaindesignparameters and the anisotropy constant ratio K u / K bulk
that we introduced. K u / K bulk is the intrinsic ratio of
medium anisotropy constant K u to bulk K u . It isnecessary to design a medium with a smaller K u / K bulk
in terms of achieving good media productivity. We havesubsequently improved our design method, andrevealed the relationships between the many design
parameters and K u / K bulk 3). Then, we applied thephysical implications to the many design parameters
using our model calculation.
Our model calculation has a problem in that theminimum K u / K bulk value cannot be obtained at 2
Tbpsi. The reason is that the Curie temperature T capproaches the writing temperature T w as the areal
density decreases in previous calculations2, 3) since thegrain volume increases. A certain time is necessaryduring cooling from T c to T w . The recording time
window" RW proposed in the micromagnetic
calculation4) is a time during cooling from T c to T w . It
is reported that a " RW value of around 0.1 ns issuitable for the micromagnetic calculation4). In thisstudy, we improve our model calculation by introducing" RW , and we fix " RW to 0.1 ns. This improvementmeans that the results obtained using the model
calculation become consistent with those obtained using
the micromagnetic calculation. Then, we provide thedependence of K u / K bulk on the areal density.
2. Previous Model Calculation
The medium was assumed to be granular. The
arrangement of the grains was not considered.
The HAMR design procedure for obtaining theminimum K u / K bulk value using the previous model
calculation is shown in Fig. 1. First, K u / K bulk = 1 andthe design parameters are set. Then, the composition of
the medium and the writing field H w are determinedusing the equation:

K " + (T w , H w ) = TSFw ,

(1)

where

K " + (T w , H w ) =2
K um (T w )Vm #
Hw &
%1+
(
kT w
$ H cm (T w ) '(2)is the medium thermal stability factor3) ( K um : meananisotropy constant, Vm : grain volume for mean grain
size Dm , k : Boltzmann constant, H cm : mean
coercivity assumed to be equal to mean anisotropy field
2K um / M s , M s : magnetization), andTSFw = TSF(" w , n, # D , # K )(3)is the statistical thermal stability factor5) ( " w = d B / v :writing period, d B : bit pitch, v : linear velocity n :grain number per bit, " D : standard deviation of grainsize, " K : standard deviation of anisotropy). TSFw iscalculated statistically using many bits and grain-error
probability%%% D (2 K ((P = 1" exp'" f 0# w exp'"TSFw $ '* $ u *** (4)''D& m ) K um )*)&&( f 0 = 1011 s-1: attempt frequency, D : grain size, K u :anisotropy constant) with a 10-3 bit error rate.
The compositions of the medium and H w are
determined using Eq. (1). This means that writing
completion is defined as the state in which the written
bit is stable at T w under H w during " w for the
medium with n , " D , and " K .
Next, four HAMR conditions I, II, III, and IV are
examined. If there are margins for all four conditions,
K u / K bulk can be reduced. When one of the four
conditions reaches the limit, the minimum K u / K bulk
value can be determined, and that condition becomes alimiting factor3).
Condition I, which is the information (written bits)
stability during 10 years of archiving, is expressed by

K um (T a )Vm" TSF10 ,kT awhere H adj is the maximum head field that can holdthe information under the main pole during rewriting.
Conditions II and III can be combined as"T "T $T $T
,
=
#
=
"x
"y $x $y(9)

since "T /"x # "T /"y . Condition IV has margins for all
the cases we examined. Therefore, the major limiting
factors in the design are condition I given by Eq. (5) (I.
K um (T a )Vm / kT a " TSF10 ) and conditions II and III given
by
Eq.
(9)
(hereafter,"T / "x = "T / "y ,"T /"x = "T /"y , and Eq. (9) are expressed as"T / "x(y) , "T /"x(y) , and "T / "x(y) # $T /$x(y) ,respectively).
When the areal density is 2 Tbpsi, Vm becomeslarge, and the Curie temperature T c approaches T w .
Then, the calculation cannot be carried out, and thisalso arises a problem from a physical point of view. A
certain time is necessary during cooling from T c to T w .

(5)where T a is the maximum working temperature of thehard drive, and TSF10 is the statistical thermal
stability factor during 10 years of archiving
TSF(10 years, n, " D , " K ) .
Condition II, which is the information stability on
the trailing side located 1 bit from the writing position
during writing, is expressed by"T T w # T rec %T
,
=
$
"x
"x
%x

(6)

where "T / "x is the medium thermal gradient for the
down-track direction, which is the minimum thermal
gradient required by the medium for information
stability, T rec is the maximum temperature at which
the information on the trailing side located 1 bit fromthe writing position can be held during writing, and"T /"x is the heat-transfer thermal gradient for the
down-track direction, which is calculated by aheat-transfer simulation.
Condition III, which is the information stability inadjacent tracks during rewriting, is expressed by"T T w # T adj %T
,
=
$
"y
%y
"y

(7)

where "T / "y is the medium thermal gradient for the
cross-track
direction,
T adj is the maximum
temperature at which the information in adjacent
tracks can be held during rewriting, and "T /"y is theheat-transfer thermal gradient for the cross-track
direction.
Condition IV, which is the information stability
under the main pole during rewriting, is expressed by

H adj " H w ,(8)

Fig. 1 HAMR design procedure for obtaining theminimum anisotropy constant ratio K u / K bulk using aprevious model calculation.
3. I m proved Mo del Calcula tion
3.1 Recording time win dow
We introduce the concept of the recording time
window4) " RW proposed in the micromagnetic
calculation for the purpose of improving our model
calculation.
First, we examine the physical implication of " RW .
The magnetization M s reversal number during a time
" is given by()

f 0" exp #K $ ,(10)

where K " is the medium thermal stability factor.
When " = 1 / f 0 = 10-11 s = 0.01 ns, Eq. (10) becomes()

exp "K # .

exp("K # " ) are better around the time corresponding to
T rec in terms of information (written bit) stability at

the position 1 bit before the writing position.

(11)

Equation (11) is the M s reversal probability for eachattempt. For example, when K " = 0, exp "K #becomes one, where the M s reversal always occurs foreach attempt. K " + where M s is parallel to H w , andK " # where M s is antiparallel to H w are expressedby(

K " + (T , H w ) =2
K um (T )Vm #
Hw &
(,
%1+
kT
$ H cm (T ) '

)

(12)
(a)

and
2
K um (T )Vm $Hw 'K " #(T , H w ) =&1#) ( H w " H cm (T )) ,kT
% H cm (T ) (K " #(T , H w ) = 0( H cm (T ) < H w ) ,(13)

respectively. Therefore, the probability for each attemptwhere M s and H w change from parallel toantiparallel is expressed byexp("K # + ) .

(14)

(b)

On the other hand,exp("K # " )(15)is the probability for each attempt where M s and H wchange from antiparallel to parallel.
In this paper, " RW is defined by" RW =

Tc # Tw,$( T / $x ) % v(16)v is the linear velocity. Since v = "x /"t ,
"
T
/
"
x
(
) # v is the cooling rate "T /"t . Therefore, " RW
is the cooling time from T c to T w . Then, therelationship between H w and T w is defined by

where

H w = H cm (T w ) =2K um (T w ).
M s (T w )

(17)

(

)

From this definition, the probability exp "K # " isalways equal to one during the cooling time " RW .
Figure 2 shows the dependence of themagnetization reversal probability on time. The
calculation conditions and parameters are the same as
those reported elsewhere2, 3). The closed circles are theprobabilities for each attempt. T rec is the temperature
at the position 1 bit before the writing position. A lower
exp "K # + and a higher exp "K # " are better during
the cooling time " RW from T c to T w in terms ofstable writing, and both lower exp "K # +
and

(

)

(

)

(

)

(c)
Fig. 2 Dependence of magnetization reversal
probability on time for (a) H w = H cm (T w ) , (b)H w = 0.5H cm (T w ) , and (c) H w = 1.5H cm (T w ) .
The result is shown in Fig. 2 (a) when T w = 500 K
and H w = H cm (T w ) = 13.4 kOe according to Eq. (17).
The time corresponding to T c is 0 ns, that
corresponding to T w is 0.1 ns, and that corresponding
to T rec is 0.67 ns. The resultant " RW value is 0.1 ns. It
is reported that a " RW value of around 0.1 ns issuitable for the micromagnetic calculation4). The
exp "K # + and exp "K # " values are both one at thetime corresponding to T c since K " ± = 0. The
exp "K # + values are almost zero, and the attempt
number is ten during the cooling time " RW , which isandsuitable for stable writing. The exp "K # +exp "K # " values are both almost zero around the

(

)

(

)

(

)

(

)

(

)

time corresponding to T rec , which is suitable for
information stability at the position 1 bit before thewriting position.
Figure 2 (b) shows the result when T w = 500 K
and H w = 0.5H cm (T w ) = 6.7 kOe instead of Eq. (17)where the composition and K u / K bulk are the same as
those in Fig. 2 (a). The resultant " RW value is 0.02 ns.
exp "K # + has a non-zero value, and the attempt
number is only two during " RW , which is not suitable
for stable writing. This corresponds to “write-error”.
On the other hand, Fig. 2 (c) shows the result whenT w = 500 K and H w = 1.5H cm (T w ) = 20.1 kOe insteadof Eq. (17) where the composition and K u / K bulk are
the same as those in Fig. 2 (a). The resultant " RW
value is 0.22 ns. In this case, exp "K # " has a
non-zero value around the time corresponding to T rec ,which is unsuitable as regards the information stability
at the position 1 bit before the writing position. This
corresponds to “erasure-after-write”.

(

)

(

)

becomes a limiting factor.
The improved model calculation includes Eq. (20)instead of Eq. (1), and Eqs. (5), (6), (7), and (8) in theprevious model calculation. Therefore, this calculationis almost the same as the previous model calculation.
The difference is the introduction of the time " RW .
" RW (Eqs. (16) and (17)) is the time from T c to T w foraligning M s with H w (writing bit), and " w (Eq. (20))
is the time from T w to T rec for the information(written bit) stability during the writing process.
“Write-error” as regards " RW (writing bit) and " w(the written bit stability during the writing process) canbe suppressed by Eqs. (16), (17) and Eq. (20),respectively. “Erasure-after-write” as regards the timeafter " w (the written bit stability after the writing
process) can also be suppressed by Eq. (6).

3.2 De sign procedure
The improved design procedure for obtaining theminimum K u / K bulk value is shown in Fig. 3. We fix" RW to 0.1 ns. First, " RW = 0.1 ns, K u / K bulk = 1, andthe design parameters including T w , "T /"x and v
are set. T c is determined from Eq. (16) asT c = T w + " RW #

$T
#v.
$x

(18)

Then, the Cu composition z in (Fe 0.5 Pt 0.5 )1"z Cu z of themedium is determined using the equation6):

Tc =2J (4(1" z))s(s + 1)
,
3k

(19)

where J is the exchange integral and s is the spin.
The temperature dependence of the magnetic properties
is determined by z and K u / K bulk . The composition is
independent of K u / K bulk . The H w value isdetermined using Eq. (17), which is dependent on
K u / K bulk . The above means that the " RW of thecooling time from T c to T w is necessary during thewriting process at which M s aligns with the directionof H w .
Next, new condition, which is the information
stability at the writing position during " w = d B / v ( d B :bit pitch) expressed by

K " + (T w , H w ) # TSFw ,(20)

is added instead of Eq. (1). Then, the four HAMR
conditions I, II, III, and IV mentioned above, that isEqs. (5), (6), (7), and (8), respectively, are examined. If
there are margins for all five conditions, K u / K bulk canbe reduced. Since H w is a function of K u / K bulk , H w
must be recalculated for reducing K u / K bulk . When oneof the five conditions reaches the limit, the minimum
K u / K bulk value can be determined. That condition

Fig. 3 HAMR design procedure for obtaining theminimum anisotropy constant ratio K u / K bulk using an
improved model calculation.
3.3 Calcula tion resul ts
The calculation conditions and parameters are thesame as those reported elsewhere3, 4).
The dependences of the minimum K u / K bulk value
on T w are shown in Tables 1, 2, and 3 for user areal
densities of 2, 3, and 4 Tbpsi, respectively. The areal
density calculated from the bit area S is larger thanthe user areal density. The difference is for the code oferror correction, etc. The S value is inversely
proportional to the areal density, and the heat-spot
diameter d w is inversely proportional to the squareroot of the areal density. The mean grain size Dm iscalculated using S / n " # where n = 4 is the grain
number per bit, and " = 1 nm is the non-magnetic
spacing between grains.
The Curie temperature T c is " RW # ($T /$x ) # v ( v= 10 m/s) higher than T w . T w is determined by the T cof the medium and not by the light power used forheating. If the light power alone is increased for a
medium with the same T c , the written bits will be
spread in the cross-track direction, and it becomes
impossible to keep the track pitch constant. Therefore,
T c must be increased to increase T w .
The tables also show the magnetization M s , themean anisotropy constant K um , the mean coercivity
H cm , and K um Vm / kT at 300 K.
TSFw under the condition K " + (T w , H w ) # TSFw isconstant for T w , and is dependent on the areal density
since the bit pitch decreases as the areal density
increases2, 3). TSF10 under condition I is constant for
T w and the areal density, and K um (T a )Vm / kT a
increases as T w increases since K um (T a ) increases7).
"T /"x(y) under conditions II and III also increases as
T w increases3). H adj under condition IV is sufficiently
larger than H w . The optimum bit pitch d B , track
pitch d T , and d T / d B values are shown in the table.

Table 2 Calculation results of HAMR design for 3
Tbpsi and various writing temperatures T w .

Table 1 Calculation results of HAMR design for 2
Tbpsi and various writing temperatures T w .

Table 3 Calculation results of HAMR design for 4
Tbpsi and various writing temperatures T w .

The minimum anisotropy constant ratio K u / K bulkof the medium at 2, 3, and 4 Tbpsi can be obtainedusing the improved model calculation. K u / K bulk and/or
the writing temperature must be increased to realize ahigh areal density.
Acknowledge ments We acknowledge the support of
the Advanced Storage Research Consortium (ASRC),
Japan.
ReferencesFig.

4

Dependence of anisotropy constant ratio
K u / K bulk on user areal density for various writing
temperatures T w . Dotted lines are results calculatedusing a previous model.
The dependence of K u / K bulk on the user areal
density for various T w values is summarized in Fig. 4.
The dotted lines show results calculated using theprevious model. K u / K bulk at 2 Tbpsi can be obtainedusing the improved model calculation. K u / K bulk and/or
T w must be increased to achieve a high areal density.
4. Conclusion s
We
improved our
modelcalculationforheat-assisted
magnetic
recording(HAMR)
byintroducing the concept of the recording time window
proposed in the micromagnetic calculation. This
improvement means that the results obtained using themodel calculation become consistent with thoseobtained using a micromagnetic calculation.

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Rece ived No v. 10 , 20 15; Ac cepted Ma r. 30, 201 6