Buschow K.H.J. (Ed.) Concise Encyclopedia of Magnetic and Superconducting Materials

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determined by Dirac–Fock-type calculations, has to

be replaced by the value reﬂecting the self-consistent

DFT crystal potential.

DFT-based methods have been used to calculate

the CF parameters in many metallic RE and An

systems. As is shown in Table 2 this method inter-

prets well the experimentally observed inﬂuence of

the chemical composition on magnetic properties

(Richter 1998).

The only DFT-based CF data available in insulators

are those for cuprates (Divis

et al.1998).Itappears

that while the sign and magnitude of second-order CF

parameters are obtained correctly, those of fourth-

and sixth-order CF parameters are only poorly des-

cribed. This inadequacy is tentatively associated with

the underestimation of the strong correlation effects

related to the local approximation for the exchange

correlation part of the effective crystal potential (see

Density Functional Theory: Magnetism).

Ab initio calculations of CF parameters can also

be performed by using a many-body approach

(Newman 1971). Its principal idea is to conﬁne all

the interactions in a many-electron system to the

effective, one-electron potentials by using standard

renormalization and averaging procedures within

MBPT. The most important assumptions associated

with these procedures are (Gajek 1995):

(i) The many-electron wave functions are replaced

by determinants built from free-ion orbitals. Only

the electrons from outer shells are considered. For

instance, in the case of PrCl

the 4f,5s, and 5p

orbitals of the Pr

3 þ

ion as well as the 3s and 3p

orbitals of the six Cl



ions are taken into account.

(ii) The corrections due to the nonorthogonality of

RE (An) and ligand orbitals and conﬁguration inter-

action effects are described by the second order of the

perturbation series.

(iii) The Hamiltonian describing the interactions

in the crystal lattice is divided into two parts. The ﬁrst

part represents the interactions within the cluster

formed by the central f ion and ligands; the second

part corresponds to the interactions with the rest of

the crystal lattice. It is usually assumed that the in-

teractions within a cluster can be regarded as a sum

of axially symmetric interactions calculated for the

individual paramagnetic ion–ligand pairs. This sim-

pliﬁcation, identical with postulates (i) and (ii) of

the SM (see above), allows one to neglect the polar-

ization effects. The interaction with the rest of the

crystal lattice is approximated by the Madelung-like

potential.

In contrast with DFT methods, applied mainly to

metallic compounds, the MBPT in its form described

above is suitable for f ions in insulators (Gajek 1995).

3. Ions in Applied Magnetic Fields

3.1 Zeeman Effect

The CF-split f-electron spectra of the ionic or atomic

J multiplets may be further perturbed by applying an

external magnetic ﬁeld, H

. Such a perturbation,

called the Zeeman effect, can be described by the

Hamiltonian H

Zeem

¼l  H

ð20Þ

where

l ¼ m

ðg

L þ g

SÞð21Þ

is the magnetic moment and g

and g

are the orbital

( ¼1) and spin ( ¼2.0023) gyromagnetic factors. For

the calculation of the matrix elements of the Zeeman

term (Eqn. (20)), it is useful to express l by means of

the tensor operator T

(1)

(Wybourne 1965):

ð1Þ

ﬃﬃﬃ

½ðL

þ 2S

Þ7iðL

þ 2S

Þ

ð1Þ

¼L

þ 2S

ð22Þ

/SLJJ

ð1Þ



SLJ

S ¼

ð1Þ

JJ

J 1 J

J

/SLSJ 8T

ð1Þ

8 SLJ

The reduced matrix elements of T

(1)

can be obtained

by the tensor operator method (Wybourne 1965).

Within a given J multiplet one can write:

L þ g

S ¼ g

J ð23Þ

Zeem

¼g

J  H

ð24Þ

where g

¼1 þ(g

1)[J(J þ1) þS(S þ1)L(L þ1)]/

2J(J þ1) is the Lande

factor.

Note that in the absence of the CF the J multiplet

is split by a magnetic ﬁeld into 2J þ1 equidistant

levels each corresponding to a different value of J

where z is the direction of the applied magnetic ﬁeld.

Two neighboring levels are separated by the energy

gap m

Table 2

CF parameters, B

(in cm

1

), of SmCo

(1cm

1

¼0.123

meV).

Experiment 250 to 584 78 þ100 þ168

Nova

k96 485 205 þ79 þ231

Hummler 96 706 111 þ21 ()40

Daalderop 96 1061 150 þ40 þ72

Richter 95 1056 206 þ122 þ212

Source: Richter (1998).

390

Localized 4f and 5f Moments: Magnetism

3.2 Paramagnetism

A paramagnetic system can be treated as an assembly

of N noninteracting f-electron ions in magnetic ﬁeld

. The thermal average of ionic magnetic moments

is given by the Boltzmann statistics formula:

M ¼ð1=ZÞ

expðE

TÞð25Þ

where l

is the magnetic moment of the ith energy

level:

¼m

/i7L þ g

S7iS ð26Þ

and where Z is the partition function:

Z ¼

expðE

TÞð27Þ

and 7 iS are, respectively, eigenvalues and eigen-

vectors of the total perturbation Hamiltonian H

which includes the CF (Eqn. (12)) and Zeeman terms

(Eqn. (20)):

ðkÞ

 l  H

ð28Þ

Note that E

denotes the energy of the levels with

respect to the ground state.

The calculation of the magnetic moment using

Eqns. (25)–(28) is straightforward provided that the

CF parameters for the given compound are known.

The magnetic moment is in general anisotropic, as

besides the magnitude of H

it also depends on its

orientation with respect to the crystallographic axes

(see Fig. 3). In the absence of the CF and neglecting

the higher excited multiplets, the right-hand side of

Eqn. (25) can be expressed in terms of the Brillouin

function (Gignoux 1991).

The tensor of magnetic susceptibility v is given by

the formula:

-0

ð29Þ

An alternative perturbation approach to the Zeeman

effect allows one to write the susceptibility per ion

along the principal axes (a ¼b ¼a, b, c) as (Luong

and Franse 1995):



7ði7J

7iÞ7

þ 2

jai

7ðj7J

7iÞ7

 E

()

expðE

TÞ

ð30Þ

where the ﬁrst term in the braces, temperature de-

pendent, is called the Curie term and the second,

temperature independent, is the Van Vleck term. For

a polycrystalline sample the susceptibility w

(p)

is ob-

tained by the approximation:

ðpÞ

¼ðw

ðaÞ

þ w

ðbÞ

þ w

ðcÞ

Þ=3 ð31Þ

From symmetry considerations it follows that: (a) w is

isotropic in cubic compounds; (b) w

(c)

in hexa-

gonal and tetragonal systems; and (c) w

(a)

(b)

(c)

in orthorhombic systems. At high temperatures the

inverse susceptibility along each axis tends to be lin-

ear with the same slope, giving rise to the same ef-

fective moment g

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðJðJ þ1ÞÞ

. For a noncubic

compound, the shifts between the high-temperature

susceptibilities are proportional to the second-order

CF parameters. The low-temperature behavior of the

susceptibility is strongly inﬂuenced by the CF. In

particular the Van Vleck contribution to the suscep-

tibility tends to dominate in the case of non-Kramers

ions in which the lowest energy state is a singlet or

nonmagnetic doublet. Such compounds are called

Van Vleck paramagnets (Gignoux 1991, Aminov

et al. 1996). As an example of such system the sus-

ceptibility of the Pr

3 þ

ion in a cubic CF is shown in

Fig. 4.

4. Cooperative Effects

A magnetic state arising from localized moments in a

solid is the net effect of competing inﬂuences: the

Figure 3

Magnetization of ErGa

as a function of the external

magnetic ﬁeld. In (001) direction, only data for B44T

are shown since data below 4T are strongly inﬂuenced

by exchange interaction inducing two metamagnetic

transitions. The lines are obtained from calculations

based on CF and Zeeman Hamiltonian (Eqn. (28)), with

CF parameters obtained from density functional theory

(Divis

et al. 1997).

391

Localized 4f and 5f Moments: Magnetism

ordering resulting from a magnetic coupling is per-

turbed by the thermal energy which tends to random-

ize the orientation of each moment. The cooperative

effects are primarily due to pair exchange coupling,

derived from the inﬂuence of the Pauli principle on

the Coulomb repulsion between electrons. One can

distinguish two classes of exchange coupling (Hurd

1982). Direct exchange operates between moments on

ions whose magnetic orbitals have significant overlap.

Such interaction, which occurs in 3d intermetallic

compounds and is responsible for the high ordering

temperatures in ferromagnets used for technical ap-

plications, gives a strong but short-range coupling (see

Itinerant Electron Systems: Magnetism (Ferromagnet-

ism)). Weaker, indirect exchange, coupling moments

over relatively large distances, occurs in f-electron

compounds. It is mediated by itinerant electrons in

metals and by nonmagnetic ions (ligands) in insula-

tors. These couplings are known as the RKKY (after

Ruderman, Kittel, Kasuya, and Yosida) and super-

exchange effects, respectively. The physical interaction

mechanisms which occur in f-electron compounds

also include electric multipole interactions and mag-

netic dipole–dipole coupling (Wolf 1971, Aminov et al.

1996, Jensen and Mackintosh 1991).

4.1 Form of Interaction Hamiltonian

The exchange interaction between two S state ions

(orbitally nondegenerate) i and j, having spins S

and S

, respectively, can be described by an isotropic

Heisenberg Hamiltonian:

¼J

ð32Þ

where J

is called the exchange parameter which can

be positive (ferromagnetism), negative (antiferromag-

netism), or oscillatory (e.g., helical structures) as in

the case of an RKKY-type interaction. Within the

ground state multiplet it is often convenient to ex-

press H

in terms of J rather than S (see Sect. 3.1):

¼J

ðg

 1Þ

ð33Þ

In systems containing f-electron ions the isotropic

exchange operator (Eqns. (32) and (33)) has to be in

general extended by anisotropic terms. The aniso-

tropy of the coupling is associated with the orbital

moment of the electrons involved. The exchange

Hamiltonian then takes the form (Wolf 1971, Baker

1971, Levy 1975, Jensen and Mackintosh 1991):

ðijÞT

ðmÞ

ðJ

ÞT

ðm

ðJ

Þð34Þ

where the terms that appear in the sum are restricted

by symmetry, but otherwise may exhibit a large va-

riety, depending on their origin. The matrix elements

of the operator (Eqn. (34)) can be calculated using the

methods of Racah algebra (Levy 1975). The general

exchange operator (Eqn. (34)) contains, besides the

bilinear isotropic term (Eqn. (32)), also higher order

isotropic interactions and all types of anisotropic

exchange, e.g., the pseudodipole exchange and the

antisymmetric Dzyaloshinskii–Moriya exchange (see

Transition Metal Oxides: Magnetism). The effective

spin–spin coupling due to the electric multipole in-

teraction produced by aspherical charge distributions

in solids can be also be written like Eqn. (34) (Wolf

and Birgeneau 1968, Levy 1975). For f electrons there

are up to 4096 exchange parameters K

of which

1954 are independent.

These numbers further increase in cases when the

interacting RE ions are not identical (Levy 1975). The

exchange anisotropy may be very pronounced:there

are systems in which the isotropic Heisenberg-type

exchange (Eqn. (32)) alone appears to be completely

incapable of explaining the observations (Cone and

Wolf 1978, Jandl et al. 1999, Nugroho et al. 2001).

Consequently, it is meaningless to apply the operator

(Eqn. (34)) in an arbitrarily simpliﬁed form. One of

the most important practical roles of the theory is

thus the identiﬁcation of special cases where the real

interaction can be described by only a few parame-

ters. This is the case, for example, in the interaction

of Er

3 þ

impurities with the Tb

3 þ

magnetic sublattice

in Tb(OH)

where the number of parameters K

can be reduced from the very large number of 21 736

to 8 (Cone and Wolf 1978). In some cases, one can

lower the number of parameters required to describe

the anisotropic exchange coupling by projection of

the operator (Eqn. (34)) onto the states of an effective

total angular momentum (ﬁctitious spin) s

Figure 4

Splitting of the ground state multiplet (J ¼4) of the

3 þ

ion in Pr

(SO

)

8H

O (inset). The susceptibility is

determined by the temperature independent Van Vleck

term in Eqn. (30) (full lines) at low temperatures. The

dashed line is the high-temperature asymptote of the

thermal variation of the reciprocal susceptibility.

392

Localized 4f and 5f Moments: Magnetism

The most general operator H

coupling two s

1/2 (Kramers) systems is:

ð35Þ

where the second-order tensor K has only nine con-

stants. This number can usually be further reduced.

For example, a special nature of low-lying 4f states

imprinted by the CF can reduce Eqn. (35) to an Ising-

like form, determined by a single exchange parameter

. Magnetic properties of several dysprosium, ter-

bium, and holmium compounds have been analyzed

using the Ising-like effective spin Hamiltonian (Wolf

1971, Nekvasil and Veltrusky´ 1994).

Data allowing one to determine a fairly large

number of unrelated parameters in Eqn. (34) are ob-

tained using various spectroscopic methods including

EPR, optical absorption, and inelastic neutron scat-

tering as well as by measurements of macroscopic

properties such as susceptibility and specific heat

(Wolf 1971, Jensen and Mackintosh 1991).

Note that although investigators are usually aware

of the anisotropic exchange effects, the simple iso-

tropic form of Eqn. (33) is usually used to examine

the magnetic structures described in the following.

4.2 Magnetic Structures

The magnetic structures assumed by f-electron com-

pounds are usually examined within the mean ﬁeld

theory. The essential feature of this conceptually sim-

ple method is the approximation of the two-ion in-

teraction (Eqn. (33)) by effective single-ion terms in

which the instantaneous values of the J operators in

the surroundings of any particular ion are replaced

by their expectation values (Jensen and Mackintosh

1991). The exchange term then can be written in the

Zeeman-like form:

¼l  H

ð36Þ

where the applied ﬁeld H

appearing in Eqn. (20) is

replaced by a molecular ﬁeld H

¼l

/JS ð37Þ

where the molecular ﬁeld constant l

is determined

by the strength of the exchange interaction and is

calculated from the divergence of magnetic suscepti-

bility w(T) at the ordering temperature T

. According

to the Curie–Weiss law:

wðTÞ

ðTÞ

 l

ð38Þ

leading to:

¼ w

1

ðfB

g; T

Þð39Þ

where w

(T ) is the component of the single-ion sus-

ceptibility parallel to H

(Wang and Cooper 1970):

ðTÞ¼

7/n7J

7nS7

expðE

TÞ



7/n7J

7nS7

expðE

TÞ

þ 2g

nam

7/n7J

7nS7

 E



expðE

TÞ

ð40Þ

where E

and 7nS are, respectively, eigenvalues and

eigenvectors of the Hamiltonian of the form of Eqn.

(28), in which the Zeeman term has been replaced by

the exchange term (Eqn. (36)).

The complete Hamiltonian within the ground state

multiplet is

ðkÞ

 l ðH

þ H

Þð41Þ

where the only unknowns are the CF parameters B

and the molecular ﬁeld constant l

Equation (41) is adequate in explaining the occur-

rence of most observed magnetic structures (Jensen

and Mackintosh 1991) as well as various physical

properties associated with the f electrons (see Sect. 2).

(a) Nonmetallic compounds

Nonmetallic RE compounds are ferro- or antiferro-

magnets (see Magnetism in Solids: General Introduc-

tion) in which the magnetic ordering occurs owing to

the superexchange, the electric multipole, and the

classical dipole–dipole interactions. The ordering

temperatures are typically of the order of 1K. The

magnetic properties have been systematically studied

in number of compounds including RE ethylsulfates,

tetraﬂuorides LiREF

with scheelite structure,

REMO

crystals having zircon structure (M ¼V, P,

As), triﬂuorides REF

, RE elpasolites, and RE

cuprates. The highest value among trivalent RE com-

pounds of 17.9K was reported for the antiferromag-

netic cuprate PrBa

. An even higher ordering

temperature of 34K occurs in the antiferromagnetic

perovskite TbBaO

, where the terbium ions are tetra-

valent. The CF ground state in the above-mentioned

magnetically ordered RE compounds is usually the

Kramers doublet or non-Kramers (quasi)doublet or

(quasi)triplet (for a survey of magnetic ordering data

see Aminov et al. 1996).

An example of localized magnetism in An com-

pounds is UO

(Fournier 1985). In this semiconduc-

tor the special nature of the band structure ensures

good localization of the 5f electrons (Kelly and

Brooks 1980, Amoretti et al. 1989, Divis

et al. 1996).

393

Localized 4f and 5f Moments: Magnetism

orders antiferromagnetically at T

¼31K. An-

other example is UCl

, an antiferromagnet with

¼6.5K (Fournier 1985) (see also 5f Electron

Systems: Magnetic Properties).

(b) Metallic compounds

In general, the magnetic ordering temperatures in

metallic compounds are lower than room tempera-

ture. The most important contribution to the mag-

netic coupling is provided by indirect exchange. Its

origin is due to the polarization of the conduction-

electron gas by the spin on one ion, which inﬂuences

the spin of a second ion.

In the widely used isotropic approximation the ex-

change parameter in Eqn. (33) takes the Fourier

transform form (Jensen and Mackintosh 1991):

JðqÞe

iqðR

R

ð42Þ

where N is the number of ions and

JðqÞ¼ðg

 1Þ

ðqÞ

ðq

ð43Þ

and

ðqÞ¼

7IðqÞ7

wðqÞð44Þ

where I(q) and w(q) are, respectively, Fourier trans-

forms of the exchange parameter between a conduc-

tion-electron spin and the f spins and magnetic

conduction-electron susceptibility, and V is the vol-

ume. The Fourier transform [J(q)J(0)] can be ex-

tracted from elastic neutron scattering data. The

position of maximum on a [J(q)J(0)] vs. q curve,

which in general occurs at nonzero q, determines the

type of magnetic order (see below).

The susceptibility w(q) is given by:

wðqÞ¼

 f

kq

ðk  qÞe

ðkÞ

ð45Þ

where f

is the Fermi–Dirac distribution function. A

large contribution to the sum is made by pairs of

electronic states, separated by q, one of which is oc-

cupied and the other empty, and both of which have

energies very close to the Fermi level. Parallel or

nesting regions of the Fermi surface tend to produce

peaks at the wave vector Q which separates them.

Such peaks are known as Kohn anomalies. The Kohn

anomalies in J(q) Fourier transform into Friedl os-

cillations in J(R). First-principle calculations of J(q)

have not been very successful. While the calculated

w(q) exhibits the expected peaks, the matrix elements

determining I(q ) are much less tractable.

A simple mean ﬁeld calculation shows that spon-

taneous ordering occurs at the wave vector Q for

which J(q) has its maximum value and the Ne

el tem-

perature T

JðQÞJ ðJ þ 1Þ

ð46Þ

Considering Eqn. (46) one ﬁnds that T

is pro-

portional to the de Gennes factor (g

1)

J(J þ1),

provided that the susceptibility of the conduction-

electron gas is constant. As a representative example

of this frequently used approach, the values of T

for antiferromagnetic RCu

displayed in Fig. 5 may

serve. The remaining discrepancy between the exper-

imental and calculated data in Fig. 5 can be ascribed

to variation of J

(q) across the isostructural series

and neglecting of the CF as well as of the aniso-

tropic part of the exchange interaction (Jensen and

Mackintosh 1991).

The long-range and oscillatory indirect exchange

gives arise to large variety of magnetic structures in

RE metals and intermetallic compounds. A compound

is ferromagnetic if a maximum on (q) occurs at Q ¼0

Figure 5

Values of the Curie (Ne

el) temperature in the RCu

and RAl

series and values of the de Gennes factor

 1)

J(J þ1) across the 4f series. The values shown

are for R

3 þ

ions.

394

Localized 4f and 5f Moments: Magnetism

and simple antiferromagnetic at the reduced propaga-

tion vector Q ¼1/2 (0,0,1). Such structures occur,

for example, in metallic gadolinium and most of the

REAl

compounds. For 0oQo1/2 (0,0,1) different

types of magnetic structure occur including the helix,

the cone, and the longitudinal wave structures (see

Fig. 6). In general, the magnetic easy axis may change

from one RE to another owing to the CF interaction.

5. Summary

This article has considered f-electron compounds in

which the RE or An ions are assumed to be present

in well-deﬁned ionic states. Most of the magnetic

properties of these compounds can be described in

terms of two microscopic interactions involving the

f electrons: the CF interaction and the exchange

interaction. Various theoretical and experimental

methods used to study and describe these interac-

tions have been discussed. These methods can be also

applied to 3d–4f alloys (see Alloys of 4f (R) and 3d

(T) Elements: Magnetism, Magnetoelastic Phenome-

na, and Magnetic Systems: External Pressure-induced

Phenomena) and compounds (see Transition Metal

Oxides: Magnetism).

See also: Crystal Field and Magnetic Properties,

Relationship between

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396

Localized 4f and 5f Moments: Magnetism

Longitudinal and Perpendicular Recording:

Thermal Stability

The magnetization, coercivity, and recorded signal of

longitudinal and perpendicular media decrease with

time due to thermal effects. In this section, the effects

of material parameters (average grain size, ﬁlm thick-

ness, grain size dispersion, anisotropy energy con-

stant, exchange energy constant between adjacent

grains) and recording density on thermal stability of

longitudinal and single-layer perpendicular media

will be reported. Most of the results were obtained

using computer simulation, because it is not easy to

change the parameters except ﬁlm thickness inde-

pendently in an experiment.

1. Thermal Stability of Magnetization of Small

Particles

The stability of the magnetization of small particles

or grains, as used in magnetic recording media, is

affected by thermal energy, which causes transitions

over the energy barriers which give rise to the hys-

teresis. The stability of the magnetization (and con-

sequently the recorded signal) is dependent on the

ratio K

V/kT. Here, K

is the uniaxial anisotropy

constant (including shape anisotropy constant), V is

the volume of the particle, k is the Boltzmann con-

stant, and T is the absolute temperature. Therefore,

not only the magnetic properties (magnetization, co-

ercivity, squareness, etc.) but also the recorded signal

of magnetic recording media may change a long time

after recording.

Suppose that a ﬁeld is applied to a particle. If the

ﬁeld is not large (oH

if the ﬁeld direction coincides

with the easy direction), the particle has two mini-

mum energy states, state 1 and state 2, and there is an

energy barrier between the states 1 and 2. Say that the

barrier height from the state 1 is DE

and that from

the state 2 is DE

The changing rate of the probability, N

, in which

a particle exists in the state 1, is given by

¼f

exp 



N

exp 



ð1Þ

where N

is the probability in which the particle exists

in the state 2. Suppose that DE

bDE

, then we have

¼

ð2Þ

where

t ¼

exp



ð3Þ

These relations are called the Arrhenius–Ne

el law

(Ne

el 1955). The stability of the magnetization can be

quantiﬁed by comparing the relaxation time with the

characteristic time of measurement t. Setting t/t ¼1

gives a critical volume in zero ﬁeld V

¼kT ln(f

t)/K.

Because of the rapid variation of t with V, to a good

approximation for V4V

, t5t and the behavior is

thermally stable. However, for VoV

, t bt, and the

behavior is referred to as ‘‘superparamagnetic.’’ Such

grains have zero remanence and cannot be used to

store information. Clearly there is a minimum grain

size for magnetic recording. Although this is easy

to estimate for isolated (noninteracting) grains, it

becomes much more difﬁcult for real media where

intergranular interactions (exchange and magneto-

static) can be very strong. This problem is in the

realm of computer simulation, to which this article is

devoted.

2. Time Dependence of Squareness, Coercivity,

and Output Signal

Figures 1 and 2 show time dependence of squareness

S, coercivity H

, and output signal of longitudinal

Figure 1

Time dependence of squareness, coercivity, and output

signal of a longitudinal medium. An enlarged ﬁgure for

t ¼1–100 s and output ¼0.96–1 is also shown. The

physical parameters of the medium is as follows;

¼2 10

erg cm

3

(2 10

3

V ¼10 10 10 nm

,Ms¼400 emu cm

3

(4 10

1

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

Fig. 6 in Uesaka et al. (1997).

397

Longitudinal and Perpendicular Recording: Thermal Stability

and perpendicular media. The vertical axes are the

values normalized by those just after recording. The

horizontal axes for S and H

mean half of the time to

measure a M–H loop. The results in the ﬁgures are

simulation results at 350 K for 2 D random ﬁlms with

uniform rectangular grains with uniaxial anisotropy

energy constant K

¼2 10

erg cm

3

(2 10

3

volume V ¼10 10 10 nm

, frequency factor

¼1 10

1

, Gilbert’s damping constant a ¼1. In

both ﬁgures, the decrease of output signal is not

similar to that of H

, but rather to that of S.

It has been reported that the H

of a longitudinal

medium changes according to the following equation

(Sharrock 1984);

¼ 1 ½kT lnðf

tÞ=ðK

VÞ

ð4Þ

Here, kT is the thermal energy, f

is a frequency

factor of the order of 10

–10

1

, K

is the aniso-

tropy energy constant of a particle (or a grain) in

the medium, and V is the volume of the particle.

Several n values have been reported. They are listed

in Table 1. The time dependence of H

in Fig. 1 co-

incides well with n ¼0.735 case which He et al. (1995)

reported for the 2D random case.

3. Recording Density Dependence of Output

Signal

A ﬁgure with enlarged time and output scales is in-

serted in Fig. 1. It is seen from this enlarged ﬁgure

and from Fig. 2 that the rate of decay of the output

signal increases with recording density in the longi-

tudinal medium, whereas the opposite is true in the

perpendicular medium. This was in agreement with

experiment (Hirayama et al. 1996, Sato et al. 1997).

Figure 3 shows linear recording density dependence

of signal output during a time span of 17 h measured

for perpendicular (CoCrPt and CoCrTa) and longi-

tudinal (CoCrPt) media (Hirayama et al. 1996). This

result arises from the fact that switching begins near

the transition region in a longitudinal recording me-

dium, though it begins in the saturation region in a

perpendicular recording medium, as shown in Figs. 4

and 5. The improved thermal stability of perpendic-

ular recording at high densities is the reason for its

continued development.

4. Grain Size and Height of a Grain

Figure 6 shows the dependence of the thermal stabi-

lity of longitudinal and perpendicular media on K

kT obtained using computer simulation. The aniso-

tropy energy constant K

was ﬁxed at 2 10

ergcm

3

(2 10

3

) and the grain size or height of the

magnetic layer of the media was varied (‘‘long., grain

size’’ in the ﬁgure means that the grain size of the

longitudinal media was varied, ‘‘long., height’’ means

Figure 2

Time dependence of squareness, coercivity, and output

signal of a perpendicular medium. The physical

parameters of the medium is as follows;

¼2 10

erg cm

3

(2 10

3

V ¼10 10 10 nm

,Ms¼400 emu cm

3

(4 10

1

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

Fig. 7 in Uesaka et al. (1997).

Table 1

Value of n in Eqn. 4 obtained by various conditions.

Value of n Conditions Method References

0.5 g-Fe

tape experiment Flanders and Sharrock (1987)

0.5 g-Fe

, CrO

, BaFe tapes experiment He et al. (1995)

0.667 distributed easy axis orientation analytical calculation Victora (1989)

0.667 cubic anisotropy paricles analytical calculation Sharrock (1984)

0.735 2 D random computer simulation Lu and Charap (1995)

398

Longitudinal and Perpendicular Recording: Thermal Stability

that the height of the longitudinal media was varied,

and so on). The standard values of the grain size and

height of a grain were 10 nm. The vertical axis is the

time when the output signal decreases 5% from that

just after recording.

This ﬁgure shows that increasing grain size in lon-

gitudinal media and increasing height in perpendic-

ular media give larger thermal stability than the other

two cases. However, increasing the grain size gives

larger noise. Therefore, if much thicker ﬁlms can be

used in perpendicular recording than the ﬁlms in

longitudinal recording, perpendicular recording will

be superior to longitudinal recording from a thermal

stability point of view.

Abarra et al. (1997) and Yu et al. (1998) investi-

gated longitudinal recording ﬁlms with various thick-

nesses and found that thermal stability increases with

increasing thickness.

5. Grain Size Dispersion

Figures 7 and 8 show time dependence of signal out-

puts of longitudinal and perpendicular recording me-

dia with various grain size dispersions. The output

signal of both media decreases faster with increasing

grain size dispersion. This is because the magnetic

moments of small grains switch faster than those of

large grains. This suggests that reducing the number

of small grains will increase the thermal stability of

longitudinal and perpendicular recording media.

6. Exchange Interaction Between Adjacent Grains

Figure 9 shows the grain size for which the output

signal after three years decreases to 90% of that just

Figure 4

Magnetization state of a longitudinal 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 track direction component of the

magnetic moment (M

) of the corresponding grain is

positive, and a black circle means 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. 8 in Uesaka et al. (1997).

Figure 3

Linear recording density dependence of output signal

during time span of 17 h measured for longitudinal and

perpendicular media after Hirayama et al. (1996).

399

Longitudinal and Perpendicular Recording: Thermal Stability