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Figure 5

Magnetization state of a perpendicular medium after

recording. T is elapsed time, and S is normalized output

signal. The initial state (T ¼0, S ¼1.0) was obtained

from the ideally recorded state using a micromagnetic

simulation (Nakatani et al. 1979). A white circle

indicates that the perpendicular component of the

magnetic moment (M

) of the corresponding grain is

positive, and a black circle means that M

is negative.

¼2 10

erg cm

3

(2 10

3

V ¼10 10 15 nm

,Ms¼400 emu cm

3

(4 10

1

), T ¼350 K. This is a revised version of

Fig. 9 in Uesaka et al. (1997).

Figure 6

Effect of K

V/kT on the thermal stability of

longitudinal and perpendicular media.

¼2 10

erg cm

3

(2 10

3

). The linear

recording density was 423 kfci for both media. Grain

size or thickness of the ﬁlm was varied.

Figure 7

Time dependence of output signal of longitudinal media

with various grain size dispersion s (nm). This is a

revised version of Fig. 3 in Nakatani et al. (1998).

400

Longitudinal and Perpendicular Recording: Thermal Stability

after recording, as a function of the intergranular ex-

change constant A. The thermal stability of longitu-

dinal recording media is degraded with increasing A.

On the other hand, the thermal stability of perpen-

dicular recording media is improved as A is increased.

The reason for these results is understood as follows:

Because the intergranular exchange interaction tends

to align the magnetization, it tends to destroy the

magnetization structure in the transition region and to

support the structure in the saturation region. There-

fore, thermal magnetization change is accelerated in

longitudinal recording media and suppressed in per-

pendicular recording media by the introduction of

exchange coupling between grains.

7. Simulation Model

Lyberatos et al. developed a Monte Carlo simulation

method to investigate the thermal stability problem

of magnetic materials (Lyberatos et al. 1985), and Lu

and Charap used it to investigate the problem of

longitudinal recording media (Lu and Charap 1994,

1995). In the simulations, the frequency factor f

was

ﬁxed at 10

1

. The simulation results mentioned

above also used this value. This is appropriate for a

magnetic recording medium when the ﬁeld (magne-

tostatic, and exchange ﬁeld) is small and parallel to

the easy direction.

However, f

with the ﬁeld applied to a different di-

rection from the easy direction is smaller (1B1/1000)

than f

with the ﬁeld applied parallel to the easy

direction Coffey et al. (1998). In a perpendicular

recording medium, the easy direction of a grain is

almost aligned in the direction perpendicular to the

ﬁlm plane and the magnetostatic ﬁeld is also applied

in this direction, but in a longitudinal recording

medium the direction of the magnetostatic and/or

exchange ﬁeld in a grain does not necessarily coincide

with the easy direction. Therefore, the thermal sta-

bility of longitudinal media could be larger than the

values stated above.

The simulation results mentioned above were ob-

tained assuming coherent rotation of the magnetic

moment of a grain. However, H

and S of a perpen-

dicular medium obtained using this model are larger

than the values obtained experimentally. This may

suggest that each grain does not rotate coherently

and that the thermal stability of a perpendicular

medium is worse than that stated above.

8. Conclusion

Thermal stability of longitudinal recording media

was compared with that of perpendicular recording

media. The decrease of output signal is not similar to

Figure 8

Time dependence of output signal of perpendicular

media with various grain size dispersion s (nm). This is a

revised version of Fig. 5 in Nakatani et al. (1998).

Figure 9

Effect of intergranular exchange constant A on

boundary grain size in which output signal decreases by

10% after three years’ decay. K

¼2 10

erg cm

3

(2 10

3

), V ¼10 10 10 nm

Ms ¼400 emu cm

3

(4 10

1

), T ¼350 K. This is a

revised version of Fig. 6 in Nakatani et al. (1999).

401

Longitudinal and Perpendicular Recording: Thermal Stability

that of H

but rather to that of S. The output signal

of higher recording density (higher frequency) de-

creases faster than that of lower recording density in

the longitudinal medium, but it is opposite in the

perpendicular medium. H

of a longitudinal medium

changes according to the following equation with

n ¼0.5B0.735;

¼ 1 ½kT lnðf

tÞ=ðK

VÞ

ð5Þ

Perpendicular recording will be superior to longi-

tudinal recording from a thermal stability point of

view if much thicker ﬁlm can be used. Reducing the

number of small grains will increase the thermal

stability of longitudinal and perpendicular recording

media. Thermal stability of longitudinal recording

media is degraded with increasing intergranular ex-

change A, but that of perpendicular recording media

is improved as A is increased.

See also: Longitudinal Media: Fast Switching; Mag-

netic Recording: Rigid Media, Recording Properties;

Magnetic Recording Media: Advanced; Magnetic

Recording Technologies: Overview; Metal Particle

versus Metal Evaporated Tape; Micromagnetics:

Basic Principles

Bibliography

Abarra E N, Phillips G N, Okamoto I, Suzuki T 1997 Magnetic

force microscopy analysis of thermal stability in 8 Gbits/in

longitudinal recording media. J. Magn. Soc. Jpn. 21 (S2),

291–6

Coffey W T, Crothers D S F, Dormann J L, Geoghegan L J,

Kennedy E C 1998 Effects of an oblique magnetic ﬁeld on

superparamagnetic relaxation time. II. Inﬂuence of the gyro-

magnetic term. Phys. Rev. B. 58 (6), 3249–66

Flanders P J, Sharrock M P 1987 An analysis of time-depend-

ent magnetization and coercivity and of their relationship

to print-through in recording tapes. J. Appl. Phys. 62 (7),

2918–28

He L, Wang D, Doyle W D 1995 Remanence coercivity of

recording media in the high-speed regime. IEEE Trans.

Magn. 31 (6), 2892–4

Hirayama Y, Itoh K, Honda Y, Inaba N, Futamoto M 1996

Thermal stability of CoCr-alloy perpendicular magnetic re-

cording media. J. Magn. Magn. Mater. 193, 253–7

Lu P, Charap S H 1994 Thermal instability at 10 Gbits/in

magnetic recording. IEEE Trans. Magn. 30 (6), 4230–2

Lu P, Charap S H 1995 High-density magnetic recording media

design and identiﬁcation: susceptibility to thermal decay.

IEEE Trans. Magn. 31 (6), 2767–9

Lyberatos A, Wohlfarth E P, Chantrell R W 1985 Simulated

annealing: an application in ﬁne particle magnetism. IEEE

Trans. Magn MAG-21 (4), 1277–82

Nakatani Y, Uesaka Y, Hayashi N 1979 Direct solution of the

Landau–Lifshitz–Gilbert equations for micromagnetics. Jpn.

J. Appl. Phys. 28 (12), 2485–507

Nakatani Y, Hayashi N, Uesaka Y, Fukushima H 1998 Effect

of grain-size dispersion on read/write properties in thin ﬁlm

recording media affected by thermal ﬂuctuation. IEEE Trans

Magn. 34 (4), 1618–20

Nakatani Y, Hayashi N, Uesaka Y, Fukushima H 1999 Effect

of inter-granular exchange coupling on read/write properties

in thin ﬁlm recording media affected by thermal ﬂuctuation.

J. Appl. Phys. 85 (8), Part 2, 4726–8

el L 1955 Some theoretical aspects of rock magnetism. Adv.

Phys. 4, 191

Sato K, Yoshida Y, Okuyama C, Okamoto I, Shinohara M

1997 Thermal stability of perpendicular and longitudinal

recorded information. J. Magn. Soc. Jpn. 21(S2), 321–4

Sharrock M P 1984 Particle-size effects on the switching be-

havior of uniaxial and multiaxial recording materials. IEEE

Trans. Magn. MAG-20(5), 754–6

Uesaka Y, Takahashi Y, Nakatani Y, Hayashi N, Fukushima

H 1997 Monte Carlo simulation of thermal ﬂuctuation in

magnetization of longitudinal and perpendicular recording

media. J. Magn. Magn. Mater. 174, 203–18

Victora R H 1989 Predicted time dependence of the switching

ﬁeld for magnetic materials. Phys. Rev. Lett. 63 (4), 457–60

Yoshida K, Hirayama Y, Igarashi M, Futamoto M, Sugita Y

1999 Switching mechanism of a perpendicular magnetic re-

cording medium at very low temperature. J. Magn. Magn.

Mater. 193, 462–5

Yu M, Doener M F, Sellmyer D J 1998 Thermal stability and

nanostructure of CoCrPt longitudinal recording media. IEEE

Trans. Magn. 34 (4), 1534–6

Y. Uesaka

Nihon University, Fukushima, Japan

Y. Nakatani, N. Hayashi

University of Electrocommunications, Tokyo, Japan

H. Fukushima

Honda-cho, Chiba, Japan

Longitudinal Media: Fast Switching

Every modern computer has several devices that per-

manently store information. Of particular interest are

those storage devices that enable the user not only to

read back information but also to write information.

In general, useful storage devices must have sufﬁcient

storage capacity with short access time. For fast and

nonvolatile mass data storage, as is required in a

personal computer or in a server system, magnetic

hard disk recording is exclusively in use.

From a practical point of view, it is highly desir-

able that the data transfer rate is increased together

with the storage capacity. Increasing the linear bit

density in a hard drive (i.e., the number of bits per

unit length in the down-track direction) has the de-

sired effect of increasing storage density and transfer

rate. However, the performance of a hard disk drive

not only depends on the maximum transfer rate but

also on access time. Since the access time includes the

latency until the transducer reads a specific sector of

402

Longitudinal Media: Fast Switching

a track, it is controlled by the rotational speed. For

high-performance drives, therefore, very high rota-

tional speeds become attractive. Disk drives spin-

ning the disk as fast as 15000 rpm are commercially

available.

The practically realizable writing speed depends on

the design of the writing head and the driving elec-

tronics. Traditionally, the contribution of the medi-

um to the speed of recording has been neglected. For

modern high-performance drives, however, the con-

tributions of the media can no longer be disregarded.

This article reviews the fundamental parts of the

theory of fast switching in media as well as experi-

mental information available in the literature.

1. The Different Time Regimes in Magnetization

Reversal

Figure 1 shows the time axis involved in magnetic

recording. For ‘‘long times,’’ referred to as ‘‘viscosity

regime’’, say at times larger than 100 ns, the magnet-

ization reversal behavior is adequately described by

the theory of thermally activated magnetization re-

versal over ﬁnite energy barriers (Arrhenius–Ne

formalism, see Sect. 1.1). At very short times, intrin-

sic dynamics lead to precessional motion that deter-

mines the reversal behavior, which is reviewed in

Sect. 1.2. As a numerical example, consider magnetic

recording in a hard disk drive of relatively low per-

formance (1999). Assuming a bit length of 200 nm at

a speed of 10 ms

1

, the time for one bit is roughly

20 ns. In order to achieve high recording perform-

ance, the switching process needs to be completed

within a small enough fraction of the bit cell dura-

tion, e.g., 4–5 ns. As Fig. 1 illustrates, this time falls

between the viscosity regime and the precessional re-

gime, in a timescale in which the dynamic behavior is

extremely complex.

1.1 Switching in the ‘‘Viscosity Regime’’

The theory of thermally activated magnetization

reversal is now reviewed. At ﬁrst sight, thermally

activated magnetization reversal does not seem to be

related to the switching speed at all, but a closer ex-

amination shows that this is not true.

Street and Woolley (1949) ﬁrst discussed thermally

activated magnetization reversal and pointed out

that every magnetization reversal process is initiated

by the thermal energy k

T (k

¼Boltzmann’s con-

stant ¼1.38 10

23

1

, T ¼temperature in K).

el (1949) expressed the probability of magnetiza-

tion reversal as

r ¼ f

exp 



ð1Þ

where r is the reversal rate, DE is the magnetic energy

barrier to be surmounted, and f

is a frequency factor

of the order of 10

Hz. Equation (1) has the same

form as it appears in the theory of semiconductors;

therefore, it is also referred to as the ‘‘Arrhenius–

el’’ approach. In a simple phenomenological in-

terpretation, Eqn. (1) means that the thermal energy

makes f

trials per second to reverse the magnetiza-

tion. The magnetic energy, DE, and the thermal en-

ergy, k

T, determine the probability of success of

these trials; for DEbk

T, the probability of success is

virtually zero, while it is large for DE 5k

T. Thus,

Eqn. (1) predicts a time-dependent magnetiza-

tion. Clearly, the concept breaks down when f

approached.

Magnetic recording media consist of very ﬁne

grains or particles. In this case, the (short-range)

quantum mechanical exchange forces are sufﬁciently

strong to keep the magnetization in these particles

uniform (Brown 1969).

For the case of a single-domain particle, the energy

barrier entering in Eqn. (1) can be evaluated easily

using the model of coherent rotation (Stoner and

Wohlfarth 1948). Consider a single-domain particle

with uniaxial anisotropy that can be either of shape

or magnetocrystalline origin. As indicated in Fig. 2,

the easy axis (dashed line) forms an angle y

with the

applied ﬁeld,

H. The magnetization,

M, makes an

angle y with the ﬁeld axis. The magnetic energy, E(y),

of the particle in the applied ﬁeld is then given by

Figure 1

Time axis in magnetic recording. For long times (X10

7

s) the magnetization reversal is governed by thermally

activated reversal, while at short times (p10

11

s) it is governed by spin precession dynamics.

403

Longitudinal Media: Fast Switching

(assuming uniaxial anisotropy)

EðyÞ¼m

HVcosy þ K

Vsin

ðy  y

Þð2Þ

where M

is the saturation magnetization of the

material, K

is the anisotropy constant, V is the par-

ticle volume, and m

¼4p 10

7

Vs A

1

. From

the anisotropy constant, the anisotropy ﬁeld, H

, can

be derived as H

¼2K

. The free parameter of

the system is the angle y between magnetization and

ﬁeld. As Fig. 2 illustrates, there are two local energy

minima in the general case. The local minimum hav-

ing the lower energy corresponds to the case of align-

ment between the magnetization and the applied

ﬁeld, whereas the other minimum represents oppos-

ing magnetization and ﬁeld. The latter state is stable

if the energy barrier, DE, which has to be overcome at

magnetization reversal is larger than zero. Note that

the energy barrier is ﬁeld dependent, as illustrated in

Fig. 2. The ﬁeld strength at which the energy barrier

becomes zero is the maximum attainable switching

ﬁeld (Stoner and Wohlfarth 1948):

ðy

Þ7 ¼

ðcos

2=3

ðy

Þþsin

2=3

ðy

ÞÞ

3=2

ð3Þ

Most theoretical analyses have considered the case

¼0, i.e., alignment of the applied ﬁeld with the

easy axis. The energy barrier then simpliﬁes to

DEðy

¼ 0; hÞ¼K

Vð1 þ hÞ

ð4Þ

If there are N

grains, the number of grains not yet

switched, n(t), is given by the exponential n(t) ¼N

exp(– rt). Together with the rate equation (Eqn. (1))

and the condition that n(t) ¼N

/2 at remanent co-

ercivity (the remanent coercivity is the ﬁeld which

switches 50% of the magnetization), one can

derive (Kneller and Luborsky 1963, Sharrock and

McKinney 1981) the following:

ðy

¼ 0; tÞ7 ¼ H

1 

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ln 2

ð5Þ

Equation (5) gives the ﬁeld that is required to re-

verse 50% of the magnetization of a previously sat-

urated magnetic sample subjected to a ﬁeld pulse of

duration t.

For thin-ﬁlm media, the grains are oriented at

random in the ﬁlm plane, so that the case of a mag-

netically easy axis aligned with the ﬁeld is highly

exceptional. Victora (1989) gave an approximation

for the energy barrier for the general case y ¼0:

DEðy

; hÞD 2K



3=2

1 þ

ðy



3=2

sin½2ðy

ðy

Þy

Þ ð6Þ

where y

ðy

Þ¼y

þ tan

1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

tany

Bertram and Richter (1999) and Richter and

Ranjan (1999) have shown that for thin-ﬁlm media

consisting of noninteracting grains with their easy

Figure 2

Thermally activated magnetization reversal. In the general case, the magnetization can have two local energy minima.

The thermal energy can push the magnetization over the energy barrier DE.

404

Longitudinal Media: Fast Switching

axes oriented at random in the plane, the time-

dependent coercivity of the entire ﬁlm can be appro-

ximated by

ðtÞ7 ¼ 0:566H

1  0:977

ln 2



2=3

ð7Þ

Equation (7) assumes that there is no magnetic in-

teraction between the grains. Figure 3 illustrates the

time dependence of the remanent coercivity for a two-

dimensional, isotropic, noninteracting particle en-

semble. As can be seen, for low values of the ratio

V/k

T, the coercivity at long times (storage) is

significantly lower than at short times (writing). Since

very small grains are required to achieve high signal-

to-noise ratios (Richter 1999), it is important to

note that, especially in high-density recording, the

coercivity obtained for typical magnetometer meas-

urements is considerably lower than that relevant for

the writing process.

1.2 Precession Dynamics

According to quantum theory, a magnetic moment,

, is accompanied by an angular momentum,

L . The

ratio between the magnetic moment and the angular

momentum is called the gyromagnetic ratio, g:

¼ g L

ð8Þ

where g ¼1.761 10

1

2

Since magnetization is created by electrons that

have the electrical charge e, g in Eqn. (8) is written

as a negative value. If the magnetic dipole is subjected

to a homogeneous magnetic ﬁeld,

H, it experiences a

torque,

, which seeks to orient it with the ﬁeld

direction:

¼ m

H



ð9Þ

Since T

¼ dL

=dt, Eqns. (8) and (9) give



¼m

7g7ðm

H

Þð10Þ

Equation (10) is the magnetic torque equation and

means that the magnetization precesses counterclock-

wise around the direction of the ﬁeld, as illustrated in

Fig. 4. Since there is no damping considered, the

precession lasts forever.

The inclusion of damping leads to the Landau–

Lifshitz equation (Landau and Lifshitz 1935), which

was later modiﬁed by Gilbert (1955). The Landau–

Lifshitz–Gilbert equation for the magnetization is

¼m

7g7ðM

H

Þþ



ð11Þ

where a is a dimensionless phenomenological damp-

ing constant. Landau and Lifshitz (1935) considered

only very small damping and gave an expression that

can be written as

¼

7g7

1 þ a

ðM

H

Þ

7g7a

ð1 þ a

ÞM

½M

ðM

H

Þ

¼m

7ðM

H

Þm

l½M

ðM

H

Þ ð12Þ

It is customary to write the Landau–Lifshitz equa-

tion in the form given on the right-hand side of Eqn.

(12). As shown in Fig. 4(b), the vector 

m ð

m 

HÞ

(or 

M ð

M 

HÞ, respectively) points towards the

direction of

H (as it should) and the magnetization

aligns with the ﬁeld for t-N. However, Eqn. (12)

implies that only the precessional part of the motion

of the magnetization vector, namely ð

M 

HÞ,

Figure 4

Precession. An undamped system would precess forever,

while in a damped system (right) the magnetization

approaches equilibrium asymptotically.

Figure 3

Time-dependent coercivity in the ‘‘viscosity regime.’’

The curves hold for an assembly of single-domain

particles with their easy axes in plane. The lines are

calculated using Eqn. (7) and the dots are exact solutions

for the entire particle assembly.

405

Longitudinal Media: Fast Switching

experiences damping. This means that losses owing to

any nonprecessional magnetization changes are ne-

glected—which is only true for small damping.

Although the two equations can formally be trans-

ferred into one another, they describe different phys-

ics and the Gilbert form should be preferred (Kikuchi

1956, Mallinson 1987). Safonov has since pointed out

that both the Landau–Lifshitz as well as the Gilbert

form are compatible with thermodynamics only for

very small damping (Safonov 1999).

In the following, the magnetization reversal of a

single spin, or, alternatively, that of a homogeneously

magnetized single-domain particle i s considered. For

many applications, it is useful to write the Landau–

Lifshitz–Gilbert equation in a parametric form in which

the magnetization vector is described by the polar angle

y(t) and the azimuth angle f(t), as indicated in Fig. 4.

In this case, the following pair of equations has to be

solved (Bertram and Mallinson 1970):

7g7

Vð1 þ a



siny



ð13aÞ

7g7

Vð1 þ a

siny

sin



ð13bÞ

Evaluation of the partial derivatives of the magnetic

energy, E, yields

 g

1 þ a



h cosy

siny  siny

cosy cosfðÞ

þsiny cosy 

siny

sinf

ð14aÞ

7g7m

1 þ a

cosy

 siny

coty cosfðÞ



cosy

þ h siny

sinf

siny



ð14bÞ

It is useful to deﬁne the natural precession frequency,

nat

(Hempel 1962):

nat

7g7m

ð15Þ

At zero applied ﬁeld, the precession of the magnet-

ization is driven only by the anisotropy ﬁeld which gives

nat

its name. For modern hard disk media, H

E800 kA m

1

(10 k Oe) and f

nat

E28 GHz.

For y

¼0, which represents alignment of the easy

axis with the applied ﬁeld, Eqns. ((14a) and (14b))

become

¼ O

nat

1 þ a

sinyðh þ cosyÞð16aÞ

¼ O

nat

1 þ a

h þ cosyðÞ ð16bÞ

Consider ﬁrst the case of vanishing damping, a-0.

According to Eqns. (16a) and (16b), the polar angle y

remains constant and the magnetization continues

with its precession movement—regardless of the ap-

plied ﬁeld value. The magnetization cannot switch,

because there is no mechanism by which the spin

system can transport energy to the lattice. For small

damping, however, the polar angle of the magnetiza-

tion can change with time and the applied ﬁeld can

switch the magnetization (see Fig. 5). Equations (16a)

and (16b) show that for small damping the precession

movement, df/dt, depends only very weakly on the

damping, but the rate of change of the polar angle,

dy/dt, is directly proportional to the damping con-

stant. Larger damping therefore accelerates the

switching speed, as long as the system does not

become critically damped. Kikuchi (1956) ﬁrst analy-

zed the switching problem and showed that the

switching time is proportional to (1 þa

)/a. The

switching time is therefore minimized for a ¼1, which

corresponds to the critically damped case.

For the case of alignment of the easy axis with the

ﬁeld (y

¼0, see Eqns. (16a) and (16b)), the diffe-

rential equation for the polar angle y can be solved

analytically:

t ¼

1 þ a

nat

cos yðtÞ=2ðÞ

cos yð0Þ=2ðÞ



1  h

h þ cosyðtÞ

h þ cosyð0Þ



 1

sin yðtÞ=2ðÞ

sin yð0Þ=2ðÞ



1 þ h

;

ð17Þ

Note that the initial value of the polar angle, y(0),

determines the reversal time. For y(0) ¼0, the solution

degenerates and the magnetization never switches. In

order to reverse, the magnetization needs an ‘‘initial

Figure 5

Trajectory of the tip of the magnetization vector during

reversal. The ﬁeld is applied along the easy axis.

406

Longitudinal Media: Fast Switching

kick’’ away from the metastable orientation y ¼0,

which can, for instance, be provided thermally. In

Sect. 1.3, the combination of thermal activation and

precession dynamics is discussed.

1.3 Transition between Viscosity and Precession

Regime

In Sect. 1.1, the rate equation (Eqn. (1)) is introduced

phenomenologically. Brown (1963) ﬁrst treated ther-

mal ﬂuctuations of a single-domain particle in depth

and derived the rate equation as well as an expression

for the frequency factor, f

. Brown treated the prob-

lem in an analogous way to that of a particle under-

going Brownian motion. This approximation is valid,

because the correlation time of the thermal energy

(E10

13

s) is much shorter than the response time of

a spin system (E1/O

nat

E10

11

s).

The starting point for further calculation is the

Landau–Lifshitz–Gilbert equation (Eqn. (11)) in

which the thermal forces are considered by adding a

random ﬁeld term to the effective ﬁeld. This extra

ﬁeld term has a statistical average of zero and is

normally distributed with a variance given by the

ﬂuctuation dissipation theorem (Brown 1963). The

modiﬁed Landau–Lifshitz–Gilbert equation is called

the ‘‘Langevin equation’’. For the simplest case in

which the easy axis is aligned with the ﬁeld, Brown

solved the Langevin equation and obtained a relax-

ation rate of the form given in Eqn. (1). An expres-

sion was obtained for the frequency, f

, which

typically yields values of the order of 10

Hz. Dur-

ing the derivation, Brown used the approximation

that the magnetic energy barrier is much greater than

the thermal energy. Note that the energy barrier is

ﬁeld dependent and that large reverse ﬁelds inevitably

make the energy barrier so small that the assumption

DEbk

T is no longer valid.

The general solution for the aligned case including

energy barriers of arbitrary height has been obtained

(Safonov 1999, Safonov and Bertram 2000). When

the assumption DEbk

T is dropped it is not legit-

imate to neglect the occupancies of the magnetization

states away from the energy minimum. Therefore,

one deals with a variety of magnetization orientations

in each state as opposed to a discrete orientation

model.

In the classical Arrhenius–Ne

el case, the relaxation

time, t

, is governed by the time that it takes the

magnetization to ﬁnd its way to the top of the energy

barrier. For the general case, however, the time that

the magnetization needs to reach the lower energy

minimum, t

, cannot be neglected. In addition, one

has to consider a third relaxation time, t

, which

reﬂects that the magnetization does not necessarily

switch to a new state once it has reached the energy

maximum; it can also fall back into its original

state. The three relaxation times add up to the total

relaxation time, t (Safonov and Bertram 2000):

t ¼ t

þ t

ð18Þ

Therefore, the relaxation time is longer than that

from the simple Arrhenius–Ne

el theory, or, equiva-

lently, the faster one desires to switch the magneti-

zation, the higher the required ﬁeld. Figure 6

compares the time dependence of the coercivity for

Arrhenius–Ne

el behavior with the exact solution,

whereby the frequency, f

, (Safonov and Bertram

2000)

¼ 2

ﬃﬃﬃ

1 þ a

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

9g9H

ð19Þ

It is important to realize that the precession fre-

quency of the magnetization changes as a function of

magnetization orientation. When the magnetization

moves from one energy minimum to another, the

precession frequency ﬁrst decreases, then becomes

zero at the energy maximum, and ﬁnally starts to

precess again in the opposite sense. Owing to the low

precession frequencies, the magnetization behaves

‘‘sluggishly,’’ which explains that deviations from

Eqn. (5) are observed at surprisingly long times E

100/f

y1000/f

. The problem can be regarded as a

‘‘random walk’’ of a nonlinear oscillator (Safonov

1999). For a typical anisotropy ﬁeld of a recording

medium, H

¼800 kA m

1

(10 k Oe), one ﬁnds f

100 GHz (a ¼0.04). The steep increase of coercivity is

found at tE1/f

, which then amounts to 0.01ns. The

phenomenological damping parameter, a, which has

been set to 0.04 without further consideration, is

discussed below.

Figure 6

Time dependence of coercivity for various energy

barriers K

V. The solid lines give the exact solutions.

The dashed lines correspond to purely thermal reversal.

407

Longitudinal Media: Fast Switching

1.4 Switching Dynamics of Recording Media

The previous sections have dealt with simple systems.

Real recording media consist of particle systems,

which are magnetically coupled by intergranular ex-

change and magnetostatic forces. Real systems also

show a high degree of disorder. The interactions and

the degree of disorder control the details of the

switching process.

So far, the damping has been treated on a phen-

omenological basis. Using ferromagnetic resonance,

the damping parameter has been determined to be

around 0.02 for cobalt-based recording media (Inaba

et al. 1997). The damping parameter describes how

effectively energy can be transported away from the

spin system into the lattice, where it is eventually

dissipated and transformed into heat. Peng and

Bertram (1998) modeled the switching behavior of a

two-dimensional isotropic recording medium using

a regular array of hexagons. They solved the full

Langevin equation and calculated coercivity as func-

tion of ﬁeld pulse time. The result obtained is similar

to that shown in Fig. 6; it also agrees well with the

simple formula of Eqn. (7) for times longer than

B100–1000 f

1

(Bertram and Richter 1999).

In complex interacting systems, such as recording

media, the spin system does not only perform a sim-

ple precession in unison as discussed above. The spin

system can be excited in various modes or ‘‘spin

waves’’ (Chantrell et al. 1998). Irregularities or non-

uniform grain size distributions, which are present in

real recording media, can disturb regular precession

patterns of the magnetization. This means that the

energy inherent in the spin system is transported ﬁrst

into complicated magnetization modes before it is

ﬁnally dissipated in the lattice. Effectively, this

increases the damping parameter, and the overall res-

ponse time of the spin system to an external ﬁeld is

reduced (Hannay et al. 1999). For a discussion of the

physical origin of damping see Suhl (1998).

The role of disorder in the switching response of a

polycrystalline magnetic ﬁlm to ﬁeld pulses of varying

duration was investigated in the experimental work

of Rizzo et al. (1999). Their magnetization data as a

function of time show a distinct kink at 10 ns, indi-

cating a shift from an exponential to a logarithmic

time dependence of the magnetization. Using con-

cepts of statistical physics, they distinguished two re-

gimes, a thermal regime characterized by statistical

relaxation between states of metastable equilibrium

and a nonequilibrium regime described by the driven

response to the applied ﬁeld. The time constant as-

signed to the transition between the two regimes is

5 ns. The media used for this experiment had a very

low coercivity; in a subsequent experiment, however,

no such kink was found for a medium with higher

coercivity down to ﬁeld pulses of 1ns (Rizzo et al.

2000).

2. Experimental Techniques and Data

The ﬁrst experimental investigation of switching

speed in recording media was by Thornley (1975).

Particulate tapes with low coercivity were subjected

to subnanosecond ﬁeld pulses. These ﬁeld pulses were

created using a recording head equipped with a mi-

crostrip-line. Switching speed values of the order of

2 10

5

Asm

1

were assigned to the various media.

Since then switching speed measurements have been

reconsidered and more comprehensive measurements

have been performed (Doyle and He 1993, He et al.

1995, He et al. 1996). It was found—consistent with

Thornley’s early results—that oxide media such as

g-Fe

, CrO

, and barium ferrite switch more slowly

than the more advanced metallic particles and metal-

evaporated tape. Since the ﬁeld created by the strip-

line is insufﬁcient to switch high coercivity media,

such as metallic particles, it is inevitable to use an

additional bias ﬁeld to assist the switching process. It

has also been reported that the damping constant

measured using FMR is inversely correlated with the

switching speed observed (Yu and Harrell 1996).

Microstrip-line measurements have been extended

to thin-ﬁlm media (Stinnett et al. 1998). In order to

achieve higher pulse ﬁelds, the minimum achievable

pulse time is compromised to about 10 ns. The

switching behavior of the thin-ﬁlm media is found

to be similar to that of the particulate media inves-

tigated before.

Apart from microstrip techniques, alternative tech-

niques using spin-stands have been suggested to

measure the time dependence of coercivity (Rubin

et al. 1998, Moser et al. 1999). Di-bits have been

written with pulsed ﬁelds from a recording head. The

coercivity is identiﬁed with the current needed to

achieve 50% of the maximum di-bit signal. The data

obtained have to be corrected for the dynamic be-

havior of the head as well for the demagnetizing ﬁeld

associated with the di-bit. A different spin-stand

technique uses the idea that each pass of a d.c. excited

head presents a ﬁeld pulse of duration DtEg/v to the

media, where v is the linear velocity and g is the gap

length of the writer (Richter et al. 1998). Longer

times can be realized by passing the head repeatedly

over the media, resulting in a longer exposure of the

media to the head ﬁeld. It has been argued that the

repetitive application of a short pulse (‘‘reptation’’) is

very close to exposing the media only once with a

correspondingly longer pulse.

This technique breaks down for very short-time

pulses for which the magnetization reversal process

is dominated by precession dynamics. The magnetic

state is characterized by the integrated noise, whereby

the noise is at a maximum when the medium is

demagnetized; i.e., it is in the remanent coercive state.

Wachenschwanz and Alex (1999) suggested a modi-

ﬁcation of the ‘‘reptation technique’’ where the

408

Longitudinal Media: Fast Switching

amount of erasure of a long wavelength square-wave

recording rather than the d.c. erased noise is used as

an indicator of the magnetization state. Although this

modiﬁcation has practical advantages, the effect of

the demagnetizing ﬁeld associated with the transi-

tions as well as head nonlinearities can alter the

result.

Figure 7 shows dynamic coercivity data obtained

using the reptation technique for about eight orders

of magnitude in time (Richter 1999). As Eqn. (7)

suggests, the data should follow a straight line when

plotted vs. [ln(f

t/ln 2)]

2/3

. Deviations are seen for

[ln(f

t/ln 2)]

2/3

p3.5, corresponding to tp500/f

which is roughly consistent with the results of Sect.

1.3. On the other hand, these deviations are, at least

partially, due to a demagnetization effect inherent to

the measurement. The slope can be used to estimate

the value for K

V/k

The Landau–Lifshitz equation predicts that the

media can, in principle, be switched inﬁnitely fast for

a sufﬁciently large driving ﬁeld. Such an experiment

has indeed been done and it is conﬁrmed that the

magnetization can be switched with ﬁeld pulses as

small as 6 ps (Siegmann et al. 1995), consistent with

the Landau–Lifshitz equation (He and Doyle 1996).

More experiments of this kind have shown that the

damping for recording media is surprisingly high and

can be close to 1, whereas that of perpendicular

media is somewhat smaller, about 0.3 (Tudosa et al.

2004). In these experiments, the spins inside the

grains are excited in a non-uniform manner and the

damping so obtained cannot directly be compared

with that of traditional FMR experiments in which

the magnetization precesses uniformly around the

equilibrium orientation.

The high speed experiments have also revealed that

the ultimate speed limit for the switching of recording

media is of the order of a few pico-seconds which is

slower than previously expected (Tudosa et al. 2004).

For practical applications, this ultimate limit has no

consequence, because the linear densities and spin-

ning speeds that would lead to such high data rates

cannot be achieved. Moreover, the driving ﬁelds used

in these experiments are higher than those that can be

obtained in disk drives.

3. Recording Considerations

The switching speed under recording conditions has

only been addressed theoretically so far. The switch-

ing speed in the dynamical regime (no thermal acti-

vation included) has been calculated (Fang and Zhu

1996, Peng and Bertram 1997). Similarly to the con-

siderations for single grains, small damping para-

meters—which are likely to be applicable to thin-ﬁlm

media—have been reported to increase the switching

time.

The switching speed of longitudinal recording

media has been compared theoretically with that of

vertically oriented media (Peng and Bertram 1997).

Media were compared with equal coercivities, where-

by the longitudinal media were subjected to a ring

head ﬁeld and the vertical media were subjected to a

ﬁeld created by a probe head in conjunction with a

magnetically soft underlayer. Longitudinal media

were found to switch faster, which is fundamentally

owing to the effect of the ﬁlm-demagnetizing ﬁeld.

For longitudinal media, the ﬁlm-demagnetizing ﬁeld

is perpendicular to the reversal direction and thus

creates an extra torque on the magnetization. In the

case of vertical media, there is no extra torque, be-

cause the demagnetizing ﬁeld is parallel to the re-

versal direction.

4. Summary

Data rates in magnetic recording begin to approach

the intrinsic switching speed limits of recording media.

It is thought that precession dynamics limit the switch-

ing speed of longitudinal recording media to times of

the order of 0.1 ns in practice. For times longer than –

100 ns, dynamical effects are described well by the

Arrhenius–Ne

el formalism. The theoretical under-

standing of the transition between the viscosity regime

and the precessional regime is not fully developed.

Acknowledgments

I am greatly indebted to V. Safonov and H. N. Ber-

tram for making their results available to me prior to

publication and to T. Silva and R. W. Chantrell for

helpful comments.

See also: Magnetic Recording Technologies: Over-

view; Magnetic Recording: Rigid Media, Recording

Properties; Magnetization Reversal Dynamics; Mag-

netic Viscosity; Micromagnetics: Basic Principles

Figure 7

Measured time dependence of coercivity for thin-ﬁlm

media (reproduced by permission of IEEE from IEEE

Trans. Magn., 1999, 35, 2790–5).

409

Longitudinal Media: Fast Switching