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

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polarization state of light on transmission or reﬂec-

tion from the medium and this constitutes a magneto-

optical effect.

For the present case, where the permittivity tensor

is described by Eqn. (1), the magneto-optical be-

havior is said to be gyroelectric. In cases where the

magnetic permeability becomes important, circular

birefringence may also arise out of a similar aniso-

tropy in the permeability tensor [m

] and there may be

a contribution to the magneto-optical behavior which

is gyromagnetic. For further information on gyro-

magnetic modeling see Smith (1965) and Hunt (1966).

2. The Reiterative Reﬂectance Formula

When s or p polarized light of wavelength l

is inci-

dent on the surface of a thin ﬁlm, it is partly reﬂected

and partly transmitted according to the well-known

Fresnel equations. The transmitted light propagates

into the ﬁlm and may be attenuated depending on the

imaginary part of the refractive index. If the ﬁlm is

nonabsorbing or sufﬁciently thin, the light undergoes

multiple reﬂections as it travels back and forth within

the ﬁlm (Fig. 1). Consequently, the calculation of the

net amplitude reﬂectance r

and transmittance t

in-

volves a multiple beam summation that, for either s or

p states, results in (Heavens 1991, pp. 55–9):

þ r

i2r

1 þ r

i2r

1 þ r

i2r

ð2Þ

where r

¼2pn

is the phase thickness of the ﬁlm,

( ¼r

)andt

are the amplitude coefﬁcients

at the two interfaces. Equations (2a) and (2b) may be

applied reiteratively, extending the calculation to the

amplitude reﬂectance and transmittance for multiple

layers. In this case the reﬂectance at the upper bound-

ary of the lth layer is:

l1 ; l

þ r

lþ1

i2r

1 þ r

l1;l

lþ1

i2r

ð3Þ

with a similar expression for transmittance. The cal-

culation begins at the substrate and is applied repeat-

edly, layer by layer, toward the incident medium. At

each stage in the calculation, r

l þ1

is the system re-

ﬂectance calculated in the previous step and r

l1,l

is the

Fresnel amplitude reﬂectance at a boundary between

two semi-inﬁnite media.

The method may be applied to the calculation of

the magneto-optical polar Kerr effects at normal in-

cidence. As mentioned in Sect. 1, the light can be

expected to propagate as two circularly polarized

modes encountering refractive indices n

and n



The latter are related to the Voigt parameter Q, and

the isotropic refractive index n,byn

¼n(17Q/2).

Each of these modes gives rise to separate Fresnel

reﬂection coefﬁcients such that:

7l1;l

l1

 n

l1

þ n

ð4Þ

It follows that for a stack containing several mag-

netic layers, the reiterative scheme must be applied

twice to determine the two amplitude reﬂection co-

efﬁcients r(n

) and r(n



). The magneto-optical Kerr

effect is then determined from the Kerr amplitude

coefﬁcient (see Sect. 3.4), which is related to the dif-

ferential reﬂectance, dr ¼r(n

)r(n



), such that:

k ¼ i

ð5Þ

This approach was applied by Gamble et al. (1985)

to the optimization of polar Kerr effects in multilay-

ers containing a single magnetic ﬁlm. It has particular

relevance to the design of magneto-optical recording

media, which is dealt with in article Magneto-optical

Effects, Enhancement of.

3. The Characteristic Matrix Method

So far the discussion has been limited to the specific

and simple case of light propagating parallel or anti-

parallel to the magnetization, as for the polar con-

ﬁguration at normal incidence. In this case, the light

propagates as two circularly polarized modes with

different velocities and attenuating at different rates

according to the refractive indices n

and n



, giving

rise to magneto-optical effects. A more general solu-

tion to the propagation of radiation in a magnetic

medium requires the general mode structure to be

determined for magnetization in an arbitrary direc-

tion and for any angle of incidence. This is usually

done for a single layer in the three principal conﬁg-

urations and then the analysis is extended to multi-

layered systems. The analysis is outlined below.

All electromagnetic ﬁelds that exist within any me-

dium must satisfy Maxwell’s equations and the 3D

Figure 1

Multiple reﬂections in a single layer.

950

Multilayer Optical Film Modeling

wave equation:

rðr  EÞr

E ¼m

ð6Þ

where

D ¼ e

½e

E ð7Þ

It is assumed that the solutions to the equation are

plane waves of angular frequency, o, described by:

E ¼ E

exp iðp  r  otÞð8Þ

where E

is the electric ﬁeld amplitude, p is the prop-

agation vector, and

r is a position vector. For con-

venience, the Cartesian coordinate system is arranged

so that the plane of incidence is always in the x–z

plane (Fig. 2). The vectors r and

p then reduce to:

r ¼ xi þ zk ð9Þ

and

p ¼ p

i þ p

k ¼

nðai þ gkÞð10Þ

where a and g are direction cosines (a

þg

¼1)as

shown in Fig. 2 and l

is the free space wavelength.

Where there is no magneto-optical behavior, p

has

two values, associated with positive and negative

going waves with respect to the z-direction of the

coordinate system. If magneto-optical activity exists,

then for the polar and longitudinal conﬁgurations

each of these waves propagates as two distinct modes

that are convenient to label p

. For each of these

modes there are corresponding refractive indices n

and direction cosines g

. In the polar case, for

example, these are given by:

¼ nð17gQ=2Þð11Þ

¼ gð 17a

Q=2gÞ

where a, g, and n are the isotropic values that would

apply if the magnetization were reduced to zero.

Similar expressions to Eqns. (11a) and (11b) also exist

for the longitudinal case.

For the transverse conﬁguration, the magnetiza-

tion is in the y direction (Fig. 2), perpendicular to the

direction in which the light is traveling. The result is

that the light propagates as a single mode peculiar

to the p polarization state and consequently the

magneto-optical effects are different from the other

conﬁgurations.

The electric and magnetic ﬁelds,

E and H, corre-

sponding to these modes of propagation in the me-

dium are determined using Eqn. (6) and Maxwell’s

equations, which are common to all electromagnetic

phenomena. Generally, each ﬁeld has three compo-

nents corresponding to the three principal directions

x, y, and z. A full appreciation of the form of these

ﬁelds requires knowledge of all three components.

However, for the purposes of thin ﬁlm and multilayer

calculations, it is only necessary to determine those

components lying in the plane of the ﬁlm (i.e., the x–y

plane). The laws of electromagnetism require these

electric and magnetic ﬁeld components, that are tan-

gential to the layer boundaries, to be continuous

across the interfaces. This very important boundary

condition means that the tangential ﬁeld components

provide a correlation between electromagnetic ﬁelds

in adjacent materials and will be discussed further in

Sect. 3.3. Again, taking the polar case as an example,

the tangential ﬁelds are:

¼Ae

þ Be



þ Ce

id

þ De

id



Z g

ðAV

þ BV



 CV

id

 DV



id



ðAe

þ Be



þ Ce

id

þ De

id



¼

ðAV

 BV



 CV

id

þ DV



id



ð12Þ

where

¼ 17



and

ngV

and A, B, C , and D are arbitrary constants. The ex-

ponents in t and x have been omitted from the above

expressions since the conditions must be satisﬁed for

all time and every point on the interface. This leads to

the well-known Snell’s law and a common term that

may be removed from each expression in Eqns.

(12a)–(12d). The sign of the exponents of the partial

Figure 2

Conventions applied in the characteristic matrix model.

951

Multilayer Optical Film Modeling

waves indicate the direction of propagation with the

positive and negative signs indicating positive and

negative going waves, respectively. The subscript

signs correspond to the two magneto-optical modes

that, of course, are magnetization dependent.

It is worth pointing out that in the polar case at

normal incidence (a ¼0, g ¼1, E

¼0) the electric

ﬁeld (E

, E

), reduces to two counter-rotating circu-

larly polarized modes (Ae

, iAe

) and (Be



iBe



). Furthermore, it is clear from Eqn. (11b) that

¼1 and the two corresponding refractive indices

are n

¼n (17Q/2). This, of course, is the basis for

the differential reﬂectance approach to the calcula-

tion of the polar Kerr effect.

At oblique incidence, for both polar and longitu-

dinal conﬁguration, the modes propagate in two di-

rections g

and are no longer circular but elliptical.

In addition, each has ﬁeld components in the direc-

tion of propagation and so can no longer be regarded

as transverse electromagnetic (TEM) waves. In the

transverse case the mode propagates in a single di-

rection g and is linearly polarized in the p plane.

Again, there are ﬁeld components both perpendicular

and parallel to the direction of propagation in the

magnetic medium and the mode of propagation is not

a simple TEM mode.

The tangential ﬁeld components are determined as

a function of distance along the z direction, as demon-

strated by Eqns. (12a)–(12d) specifically for the polar

conﬁguration. Consequently, the equations relate the

tangential ﬁeld components at one boundary of a thin

ﬁlm, placed at z ¼0 for example, to those at the other

boundary at the ﬁlm thickness z ¼d. It is possible to

express this relationship conveniently in matrix form

such that:

ðF

¼ 0Þ¼½C ðF

Þð13Þ

where (F) is a column matrix whose elements are

, H

, E

, and H

and [C]isa44 matrix that is

characteristic of the ﬁlm.

Each of the three principal magneto-optical con-

ﬁgurations requires a different characteristic matrix.

Moreover, each matrix may be decomposed into

separate isotropic and anisotropic parts such that:

½C

P;L;T

¼½C

þ g

P;L;T

½G

P;L;T

ð14Þ

where the subscripts refer to the principle conﬁgura-

tions. The isotropic matrix [C]

, corresponding to

optical behavior of the ﬁlm, would completely

characterize the ﬁlm if the relative permittivity e

was a scalar quantity, i.e., if there was no magneto-

optical activity. Hence, it remains the same for all

three magneto-optical conﬁgurations. The aniso-

tropic matrix accounts for the gyroelectric properties

of the medium and leads to calculations of observed

magneto-optical effects. To completely characterize

the ﬁlm optically and magneto-optically four

characteristic matrices must be determined in all;

an isotropic matrix and three anisotropic matrices

corresponding to each of the three magneto-optical

conﬁgurations. In the proceeding sections the details

for each of these matrices will be given.

3.1 The Isotropic Matrix

The isotropic matrix has the form:

½C

00C

ð15Þ

So there are ﬁve different elements, which are:

¼cosðrÞ

¼iZg sinðrÞ

¼i sinðrÞ=Zg

¼Z sinðrÞ=gC

¼ g sinðrÞ=Z ð16Þ

where r is the phase thickness:

r ¼ p

d ð17Þ

and the optical impedance Z, has units of ohms and

relates E and H such that E ¼ZH. The optical im-

pedance may be related to the refractive index of the

material through

Z ¼ Z

=n ð18Þ

where Z

is the optical impedance of free space and

has the value 376.7 ohms. The direction cosines, a

and g, are related to the angle of incidence, j,

through Snell’s law.

An important feature of the isotropic matrix is

that it comprises two 2 2 submatrices [I]

and [I]

that:

½C

½I

½0

½0½I 

ð19Þ

These are associated specifically with either p polar-

ized light or s polarized light. Consequently the p and

s modes do not interact and can be treated separately.

Note that Eqn. (2a) for single layer reﬂectance may

be derived using the isotropic matrix, the summation

and matrix methods being completely consistent

(Lissberger 1970). This is to be expected as both

techniques are based on Maxwell equations and em-

ploy the same assumptions and boundary conditions

(Heavens 1991, pp. 49–53).

952

Multilayer Optical Film Modeling

3.2 The Gyroelectric Matrices

The characteristic matrices for the polar and longi-

tudinal conﬁgurations have the same form:

½C

P;L

¼½C

þ g

P;L

00G

ð20Þ

Thus, there are only four different gyroelectric el-

ements for each of the conﬁgurations and these are

summarized in Table 1, where the parameters r and

Z are calculated according to Eqns. (17) and (18).

The magneto-optical behavior for the transverse

conﬁguration is different to the other two cases. For a

s polarized excitation, where the electric ﬁeld oscil-

lates along the direction of magnetization, there is no

gyroelectric effect. Thus the transverse effect is in-

duced by p polarized light only. Like the isotropic

case, there is no interaction between p and s modes;

so, the gyroelectric effect is a linear p polarized effect

stimulated by a p polarized excitation. It is, therefore,

not surprising that the transverse characteristic ma-

trix has a much closer resemblance to the character-

istic matrix for isotropic media:

½C

¼½C

þ g

000

0 G

0000

ð21Þ

The gyroelectric part of the transverse matrix is

much simpler than for the longitudinal and polar

conﬁgurations, there being only two diagonal elements

with the same magnitude but opposite sign (Table 1).

3.3 The Characteristic Matrix Method and

Multilayers

Up to now the discussion has dealt with the charac-

teristic matrix and how it relates the tangential ﬁeld

components at the two extremities of a ﬁlm (Eqn.

(13)). It is appropriate to remember that the tangen-

tial ﬁeld components only are being considered

because at an interface they are continuous. Conse-

quently, it is possible to relate the tangential electric

and magnetic ﬁelds between adjacent layers and ulti-

mately between the layers in a multilayer.

This may be illustrated by considering Fig. 3,

which shows how the boundary conditions effect the

ﬁeld matrix (F). Considering layer one ﬁrst, at the

z ¼0 boundary between medium zero and layer one,

z ¼0

) ¼(F)

. By renaming the ﬁeld matrix at the

second interface (F)

, Eqn. (13) may be rewritten:

ðFÞ

¼½C

ðFÞ

ð22Þ

The tangential ﬁelds in medium zero and layer one

have been related using the characteristics matrix for

the layer [C]

. Considering layer two, it is clear that a

similar analysis may be applied to relate the tangen-

tial ﬁelds in layers one and two so that (F )

¼[C]

(F)

This leads to the conclusion that the ﬁelds in medium

zero and layer two may also be related by multiplying

the characteristic matrices of the two layers:

ðFÞ

¼½C

½C

ðFÞ

ð23Þ

Therein lies the basis for multilayer matrix calcu-

lations in that for a stack of L layers deposited on a

substrate S in an ambient 0, the tangential ﬁelds at

Table 1

The elements of the gyroelectric characteristic matrices where c ¼cos(r) and s ¼sin(r). The parameters r and Z are

given by Eqns. (17) and (18), respectively. The direction cosines g and a are determined using Snell’s law.

Polar Q/2 0 irs (Z/g)(rcs) (rc þs)/(Zg) i (rs/g

)

Longitudinal (Q/2)(g/a)0(rc þs) i(Z/g) rsi(rs/Zg)(rcs)/g

Transverse Q (g/a) s 00 0 0

Figure 3

The concept behind the characteristic matrix model.

953

Multilayer Optical Film Modeling

the interfaces with the ambient and substrate are re-

lated by multiplying together all the characteristics

matrices of the layers comprising the stack:

ðFÞ

i¼1

½C

ðFÞ

ð24Þ

Note that Eqn. (24) is of the same form as Eqn.

(22) and consequently describes the multilayer as a

single layer of characteristic matrix [C]

½C

.To

the ﬁrst order, [C]

has the same form as the char-

acteristic matrix for the single layers comprising the

multilayer. For magneto-optic calculations, multiply-

ing together the single layer matrices results in higher

order terms in Q appearing in the isotropic part of

[C]

which are usually negligible. However, it is bet-

ter to remove these terms completely and this may

be done using a relatively simple scheme devised by

Atkinson and Lissberger (1993).

3.4 The Calculation of Optical and Magneto-optical

Effects

It remains to calculate the optical and magneto-

optical amplitude coefﬁcients. The amplitude coefﬁ-

cients are related to the s and p polarized electric

ﬁelds of the incident, reﬂected, and transmitted light

(Fig. 4) such that:

p;s



Op;s

p;s

Sp;s

Op;s

p;s



Op;s

Os;p

p;s

Sp;s

Os;p

ð25Þ

the O and S subscripts indicating the ﬁelds in the

incident and substrate media, respectively, the þ and

 superscripts indicating direction of propagation.

Consequently, it is necessary to express the tangential

ﬁelds as p and s electric ﬁeld components. It should

be remembered that the net tangential ﬁeld is a

summation of the negative and positive going waves

at the interface. With reference to Fig. 4, and using

E ¼ZH, the boundary ﬁeld matrix may be expressed

in terms of the s and p components of the electric

ﬁelds:

ðFÞ¼

ðE

þ E



Þg

ðE

 E



Þ=Z

þ E



ðE

 E



Þg=Z

ð26Þ

The characteristic matrix equation, may then be

related to the p and s electric ﬁeld components in the

incident medium and the substrate:

ðE

þ E



Þg

ðE

 E



Þ=Z

þ E



ðE

 E



Þg

¼½C

E

ð27Þ

The optical and magneto-optical coefﬁcients, as de-

ﬁned by Eqns. (25a)–(25d), are determined by solving

the relatively simple set of simultaneous equations

represented by Eqn. (27).

The polar and longitudinal effects take the form of

a change in the state of polarization known as the

complex Kerr rotation in reﬂection and complex

Faraday rotation in transmission. In both cases, the

induced magneto-optical components, orthogonal to

the incident linear polarization state, result in ellip-

tically polarized light. The complex rotations are re-

lated to the amplitude coefﬁcients ratios k

p,s

s,p

and

p,s

s,p

as previously mentioned. The transverse effect

is an exception in that it is a p polarized effect in-

duced by a p polarized excitation. In this case, the

effect is detected as a change in reﬂectance or trans-

mittance. For details on how to determine the mag-

neto-optical effects from the amplitude coefﬁcients

see article Faraday and Kerr Rotation: Phenomeno-

logical Theory .

4. Concluding Remarks

Accurate modeling of the interaction of light with

multilayered recording media is important for the

purposes of characterization and performance opti-

mization. The two methods described are well-estab-

lished and founded on Maxwell’s equations and are

consistent with each other. Both methods calculate

the optical and magneto-optical amplitude coefﬁ-

cients, which are related to observable effects. The

material parameters required for the calculations are

the thickness, refractive index, and magneto-optical

parameter for each layer constituting the multilayer.

The ﬁrst method is a phenomenological approach

where light undergoes a series of multiple reﬂections

Figure 4

The s and p polarized electric and magnetic ﬁeld

components on reﬂection and transmission.

954

Multilayer Optical Film Modeling

within the media. It provides a relatively uncompli-

cated way to perform optical calculations but for

the magneto-optical case is limited to the polar Kerr

effect at normal incidence. The second, a matrix

method, is a more general approach and is conven-

iently used for multilayer systems.

See also: Magneto-optic Effects, Enhancement of;

Magneto-optic Multilayers; Magneto-optic Record-

ing: Total Film Stack, Layer Conﬁguration

Bibliography

Atkinson R, Lissberger P H 1993 Correct formulation of ﬁrst-

order magneto-optical effects in multilayer thin ﬁlms on

terms of characteristic matrices and derivation of a related

superposition principle. J. Magn. Magn. Mater. 118, 271–7

Gamble R, Lissberger P H, Parker M R 1985 A simple analysis

for the optimization of normal Polar magneto-optical Kerr

effect in multilayer coatings containing a magnetic ﬁlm. IEEE

Trans. Magn. 21, 1651–3

Heavens O S 1991 Optical Properties of Thin Solid Films. Dover

Publications Inc, New York

Hunt R P 1966 Magneto-optic scattering from thin solid ﬁlms.

J. Appl. Phys. 38, 1652–71

Lissberger P H 1970 Optical applications of dielectric thin ﬁlms.

Rep. Prog. Phys. 33, 197–268

Macleod H A 1969 Thin-ﬁlm Optical Filters. Hilger, London

Smith D O 1965 Magneto-optical scattering from multilayer

magnetic and dielectric ﬁlms. Part 1. General theory. Opt.

Acta 12, 13–45

Zvezdin A K, Kotov V A 1997 Modern Magneto-optics and

Magneto-optical Materials. IOP Publishing Ltd, Bristol, UK

W. R. Hendren

Queen’s University of Belfast, UK

Multilayers: Interlayer Coupling

The exchange coupling of magnetic ﬁlms across me-

tallic interlayers was ﬁrst observed in 1986 for dys-

prosium and gadolinium ﬁlms separated by yttrium

interlayers, and for iron ﬁlms separated by chromium

interlayers (Salamon et al. 1986, Majkrzak et al. 1986,

Gru

nberg et al. 1986. For ferromagnetic ﬁlms like

those of gadolinium and iron, the coupling leads to

parallel or antiparallel alignment of the magneti-

zations on opposite sides of the interlayer, depending

on the interlayer thickness D, as seen in Figs. 1(a) and

1(b) respectively. For obvious reasons, the coupling

leading to (a) is called ‘‘ferromagnetic’’(F), and to

(b) ‘‘antiferromagnetic’’ (AF). For ﬁlms with helical

magnetic structure like dysprosium, the coupling leads

to an angle, c, between the magnetic moments on

both sides of the yttrium interlayer, which depends on

the yttrium thickness, as seen in Fig. 1(c). The actual

alignment is also affected by other interactions, like

anisotropy or an external ﬁeld, H

ext

. Large enough

ﬁelds, H

ext,

overcome the coupling and align the mag-

netizations parallel.

In 1990, it was established (Parkin et al. 1990) that

the oscillation of the magnetic coupling between F

and AF alignment, as a function of interlayer thick-

ness, is a general phenomenon of transition metal

ferromagnets separated by nonmagnetic interlayers.

Previously oscillations had only been seen for gado-

linium separated by yttrium. The discovery in 1988 of

the giant magnetoresistance (GMR) effect in the

Fe/Cr system (see Giant Magnetoresistance) led to

enhanced interest in the magnetic coupling of tran-

sition metal ferromagnets, because of the many ap-

plications of GMR.

1. Theoretical Models

Interlayer exchange coupling is believed to be an in-

direct exchange interaction mediated by the conduc-

tion electrons of the spacer layer. It is closely related

to the Ruderman–Kittel–Kasuya–Yoshida (RKKY)

interaction between localized moments mediated by

the conduction electrons of a host metal. The essen-

tial ingredients of the RKKY interaction, a localized

spin-polarized perturbation and a sharp Fermi sur-

face, lead to the well-known coupling oscillations.

Like the exchange coupling in the rare earth metals

themselves, the magnetic coupling of gadolinium,

dysprosium, holmium, and erbium through yttrium

and lutetium spacer layers has been described by the

RKKY interaction.

For transition metal ferromagnets separated by

paramagnetic interlayers, the description of the cou-

pling needs to be modiﬁed. A phenomenological

description of the coupling proposed to explain the

Figure 1

Coupling between magnetic ﬁlms across metallic

interlayer (dark shaded). Encircled arrows indicate

moments in monolayer sheets parallel to the interfaces.

The arrows indicate ferromagnetic coupling across the

interlayer in (a) and antiferromagnetic coupling in (b).

The magnetic coupling of ﬁlms with a helical magnetic

structure shown in (c) leads to a phase angle c.

955

Multilayers: Interlayer Coupling

experimental observations (Bu

rgler et al. 2001) gives

the interlayer coupling energy, E

, per unit area as:

¼J

cosðyÞJ

cos

ðyÞð1Þ

Here y is the angle between the magnetizations

and

of the ﬁlms on both sides of the spacer layer.

The parameters J

and J

describe the type and the

strength of the coupling. If the term with J

domi-

nates, then from the minima of Eqn. (1) the coupling

is F (AF) for positive (negative) J

. If the term with J

dominates and is negative, we obtain 901 coupling.

The ﬁrst term of Eqn. (1) is often called the bilinear

coupling, and the second the biquadratic coupling.

The microscopic mechanism leading to the bilinear

exchange coupling J

has essentially the same phys-

ical origin as the RKKY interaction. However, one

must consider the itinerant nature of electrons in

transition metal ferromagnets, which gives rise to the

spin-split band structure and spin-dependent reﬂect-

ivities at the paramagnet/ferromagnet interfaces. The

spin-dependent reﬂectivity is illustrated in Fig. 2(a),

where it is assumed that electrons with their spins

parallel (antiparallel) to the magnetizations

and

are weakly (strongly) reﬂected at these interfaces.

This spin-dependent reﬂection at the interfaces

gives rise to spin-dependent quantum well states that

cause an oscillatory polarization in the interlayer. As

a result, there are spin-dependent interference effects

like the formation of standing electron waves for

certain interlayer thicknesses as indicated in Fig. 2(a).

Due to the similarity of the arrangement in Fig. 2(a)

with an optical Fabry Perot interferometer, this is

sometimes also called the Fabry Perot model of os-

cillatory coupling. Oscillations in the coupling, ob-

served as the thickness of a magnetic layer or other

overlayer is varied, are explained by additional spin-

dependent interference effects.

The predominant contribution to the coupling is

from electrons with wave vectors, Q

, that are critical

spanning vectors of the Fermi surface of the inter-

layer material, i.e., vectors in the direction perpen-

dicular to the interface that connect two sheets of the

Fermi surface parallel to each other. For the Fermi

surface of gold shown in Fig. 2(b), there are two such

vectors in the [100] and one vector in the [111] direc-

tion. The periods of the oscillatory coupling are given

by L ¼2p/Q

and thus are determined solely by the

electronic properties of the interlayer material.

This quantum well model leads to the following

dependence of J

of Eqn. (1), connected with the i

Fermi surface spanning vector Q

, on the interlayer

thickness D:

¼ J

sinðQ

D þ f

Þ=D

ð2Þ

The amplitude J

includes Fermi surface geometry

factors and interface reﬂection probabilities. Good

parallelism of the parts of the Fermi surface at the

endpoints of a critical spanning vector will contribute

to large amplitudes of the corresponding oscillatory

coupling. This so-called ‘‘nesting’’ effect results in a

decrease of the coupling with interlayer thickness as

1

rather than the more typical D

2

If, due to favorable band matching, electrons with

one spin have good transmission at the interfaces,

and with the other spin have good reﬂection, then the

result will be large amplitudes in oscillatory coupling

strength. This is thought to be the case for the cou-

pling of cobalt across rhodium and ruthenium inter-

layers (see Table 1). In principle, the quantum well

model can be extended to nonmetallic spacers using a

complex Fermi surface, but materials difﬁculties have

limited experimental work in this area.

The theoretical description of magnetic coupling

assumes perfect interfaces. Experimentally, there is

always some disorder at the interfaces. The interac-

tion of Eqn. (2) is deﬁned at each discrete interlayer

thickness D ¼nd, where d is the spacing of neighbor-

ing atomic layers in the interlayer. The direction of

magnetization in a magnetic layer can only change

over a ﬁnite lateral distance l, comparable to a

domain wall width. If there are interlayer thickness

ﬂuctuations on a lateral length scale smaller than

this magnetic response length l, the observed coupl-

ing strength is an average over regions of different

thickness:

ðtÞ¼

Pðt; nÞJ

ðnÞð3Þ

where P(t, n) is the fraction of the interlayer area that

is n layers thick when the average thickness is t.

Short-period oscillations of the coupling are averaged

out more readily than long-period oscillations, and

therefore are only observed in samples with relatively

smooth interfaces. Thickness ﬂuctuations, and also

Figure 2

(a) Illustration of spin-dependent reﬂectivity at the

nonmagnetic/magnetic interfaces for the explanation of

oscillatory coupling; (b) a cross section of the gold

Fermi surface with critical spanning vectors in the [100]

and [111] directions.

956

Multilayers: Interlayer Coupling

interfacial disorder on a ﬁner scale due to alloying,

lead to a measured magnetic coupling that is weaker

than that expected theoretically.

Although J

of the biquadratic coupling term in

Eqn. (1) can be of intrinsic origin, most experimental

results to date are well explained by a model in which

the biquadratic coupling is an extrinsic effect due to

the thickness ﬂuctuations of the interlayer. For ex-

ample, if the bilinear coupling varies spatially due to

the thickness ﬂuctuations, the sum of the intralayer

coupling energy (exchange stiffness) and interlayer

coupling energies is minimized when the magnetiza-

tions of the layers ﬂuctuate about their average di-

rections. The energy is lowered the most when the

average magnetizations of the two layers are perpen-

dicular. This biquadratic coupling leads to observa-

tion of 901 coupling when the average bilinear

coupling is small.

The RKKY description of magnetic coupling in

rare-earth multilayers and the quantum well model of

coupling in transition metal multilayers assume that

the electrons in the spacer layers are noninteracting.

At the other extreme, the electrons of the interlayer

may be strongly interacting, as for example local

moments coupled to each other in antiferromagnetic

manganese. A phenomenological model for this sit-

uation, known as the torsion or proximity model, has

been proposed (Slonczewski 1995) that treats the

magnetic nature of the spacer layer and predicts

slightly different observable behavior.

The appropriate description of the coupling may

fall between the assumed noninteracting electrons of

the interlayer in the RKKY and the quantum well

models and the strongly coupled local moments of

the torsion model. A model including an induced spin

density wave was proposed to explain experiments on

the coupling of dysprosium through yttrium and

lutetium. There is also strong evidence from the tem-

perature dependence of the coupling in an iron

whisker-based Fe/Cr/Fe trilayer that the chromium

electrons are in an interacting, itinerant, spin density

wave state (Pierce et al. 1999).

2. Experimental Methods and Examples

Polarized neutron diffraction provided the initial

evidence for the oscillatory exchange coupling of su-

perlattices of gadolinium layers separated by yttrium

interlayers. These measurements were supplemented

by magnetization measurements that show the re-

manent magnetization and the saturation magnetiza-

tion oscillating as a function of yttrium thickness.

The coupling of rare earths with complex spin

structures like dysprosium, holmium, and erbium

through interlayers of yttrium and lutetium can be

studied with unpolarized neutron diffraction, because

there are distinct magnetic scattering peaks deriving

from the incommensurate magnetic order. The phase

and chirality of the helical spin structure of dyspro-

sium is preserved through yttrium spacer layers over

10 nm thick in superlattices grown along the c-axis

direction. The stronger coupling found for c-axis su-

perlattices, compared to a- or b-axis structures, was

attributed to the directional properties of the inter-

layer Fermi surface.

Superlattices, as well as simpler trilayer structures,

have been used to investigate the coupling of tran-

sition-metal ferromagnets. As an alternative to pre-

paring many superlattices or trilayer samples, each

with a specific interlayer thickness, it is possible with

a trilayer to grow the spacer layer in the form of a

wedge with continuously varying thickness. An ex-

ample of such a structure is shown in Fig. 3(a).

The periods of the oscillatory interlayer coupling

can be determined from the variation of the magnetic

domains with changing thickness of the interlayer

wedge. The domains can be detected optically with

the magneto-optic Kerr effect, or by means of spin-

polarized electrons using scanning electron micros-

copy with polarization analysis (SEMPA).

Table 1

Observed coupling strengths and periods.

Sample

Maximum strength 7J

þJ

7 (mJ m

2

)

at (thickness (nm)) Periods in ML (nm)

Co/Cu/Co (100) 0.4 (1.2) 2.6 (0.47), 8 (1.45)

Co/Cu/Co (110) 0.7 (0.85) 9.8 (1.25)

Co/Cu/Co (111) 1.1 (0.85) 5.5 (1.15)

Fe/Au/Fe (100) 0.85 (0.82) 2.5 (0.51), 8.6 (1.75)

Fe/Cr/Fe (100) 41.5 (1.3) 2.1 (0.3), 12 (1.73)

Fe/Mn/Fe (100) 0.14 (1.32) 2 (0.33)

Co/Ru (0001) 6 (0.6) 5.1 (1.1)

Co/Rh/Co (111) 34 (0.48) 2.7 (0.6)

Co/Os (111-text’d) 0.55 (0.9) 7 (1.5)

Co/Ir (111) 2.05 (0.5) 4.5 (1.0)

957

Multilayers: Interlayer Coupling

Figures 3(b)-3(d) displays SEMPA magnetization

images in which the black (white) regions correspond

to magnetization to the left (right). Note that in

part (b), the bare iron whisker displays two domains

with a 1801 wall in between. Parts (c) and (d) show

domains in the upper iron ﬁlm, separated from

the whisker by a chromium spacer grown at an iron

whisker substrate temperature of 30 1C and 350 1C,

respectively. At 350 1C, the chromium grows in a

layer-by-layer mode, forming a very smooth inter-

face, while the lower temperature leads to a rougher

chromium growth.

The striking difference in the magnetic domain

patterns of parts (c) and (d) of Fig. 3 indicates how

thickness ﬂuctuations affect which periods of the

magnetic coupling can be observed. For the near

ideal growth in (d), both a short- and long-period

coupling can be observed, with the short period

clearly dominating. In (c), the short-period coupling

is averaged out by thickness ﬂuctuations as described

by Eqn. (3), and only the long period is observed. The

long and short periods were determined to be 12 ML

and 2.1 ML respectively. The short-period oscilla-

tions derive from a wave vector connecting strongly

nested regions of the chromium Fermi surface. The

slight difference between this Fermi surface critical

spanning wave vector and the lattice wave vector

leads to phase slips in the short-period oscillations at

the chromium thicknesses indicated by arrows.

For a measurement of coupling strengths, it is

necessary to apply a magnetic ﬁeld. The simplest and

most frequently employed method is a measurement

of the remagnetization curve, where the strength of

the coupling is determined from the ﬁeld required to

switch from AF to F coupling. A measurement es-

sentially of this type can be made in parallel with a

sample structure as in Fig. 3(a) by using magneto-

optical microscopy; the coupling strength is deter-

mined from the external ﬁeld required to switch the

antiferromagnetically coupled domains, causing them

to disappear. Alternatively, one can measure the

magnetic coupling by determining the frequency

shifts of coupled spin wave modes, which represent

the restoring forces caused by the coupling.

Remagnetization curves from trilayers consisting

of two 6 nm f.c.c. cobalt ﬁlms separated by two dif-

ferent thicknesses of copper are displayed in Fig. 4.

The coupling is ferromagnetic in Fig. 4(a) and the

ﬁlms always reverse their magnetizations in unison.

In Fig. 4(b), the strength of the antiferromagnetic

coupling is determined by the ﬁeld, H

ﬂip

, where the

coupling changes from AF to F. In order to measure

the strength of the coupling in ferromagnetically cou-

pled regions by remagnetization curves, it is necessary

to resort to a technique known as ‘‘spin engineering.’’

To measure the coupling between two ferromagnetic

ﬁlms, F

and F

, across an interlayer I, an additional

thick ferromagnetic ﬁlm, F

, is added, such that it is

strongly antiferromagnetically coupled to F

by using

a proper interlayer, I

, as shown in Fig 5. F

and F

can be held with their magnetizations opposite to the

external ﬁeld, H, until H overcomes the ferromag-

netic coupling and reverses the magnetization of F

Figure 3

Magnetic domains in Fe/Cr/Fe samples seen by means

of SEMPA, which display oscillatory coupling. Sketch

of the sample seen in part (a); domains are observed in

the bare whisker substrate (b); and in the upper iron ﬁlm

grown at 30 1C (c), and at 350 1C (d) (after Unguris et al.

1991).

Figure 4

Remagnetization curves of a Co/Cu/Co trilayer

structure: (a) with ferromagnetic coupling; (b) with

antiferromagnetic coupling.

958

Multilayers: Interlayer Coupling

thereby determining the ferromagnetic coupling

across I.

In Fig. 6(a) we see the result of an evaluation of

remagnetization curves for a Fe/Au

wedge

/Fe structure

grown on a silver-buffered GaAs(100) substrate. The

spin engineering technique has been used to deter-

mine the positive values of the coupling. The coupling

is strongly ferromagnetic in Fig. 6(a) for small d

probably due to pinholes and magnetic bridges. For

increasing d

ferromagnetic coupling quickly de-

creases, until there are oscillations around zero. Two

periods of oscillation are superimposed with an am-

plitude that is attenuated as a function of the inter-

layer thickness.

Measurements of the coupling strength in a Fe/

wedge

/Fe trilayer grown on an iron whisker, by

observing the disappearance of antiferromagnetically

coupled domains in a Kerr microscope, are shown in

Fig. 6(b). Two periods of oscillatory coupling,

2.48 ML and 8.6 ML, were determined from the da-

ta. Both the samples used in Figs. 6(a) and 6(b) were

grown very carefully; the stronger coupling for the

iron whisker sample is indicative of the better growth

occurring naturally on that substrate. The data of

Fig. 6(b) were further analyzed, taking into account

thickness ﬂuctuations to obtain ‘‘unaveraged’’ values

to compare with theory; the coupling strength for the

short- and long-period oscillations was found to be

60% and 15% of that calculated respectively.

In Table 1 we have compiled some measured val-

ues for interlayer coupling strengths and periods. As

shown in Fig. 3, the thickness ﬂuctuations can ob-

scure the short period oscillations by averaging but

do not change the periods, which are determined by

the interlayer Fermi surface. On the other hand, the

thickness ﬂuctuations dramatically affect the cou-

pling strength, as seen from Fig. 6, and thus the val-

ues listed in Table 1 are representative coupling

strengths for specific samples. Table 1 gives the max-

imum observed coupling strength, 7J

þJ

7, and the

spacer layer thickness in ML (in nanometers) at

which it was measured. The coupling strength also

decreases with increasing temperature.

3. Applications

Applications of interlayer exchange coupling have

been suggested. For example, antiferromagnetic cou-

pling can be used in ‘‘artiﬁcial’’ antiferromagnets

(van den Berg et al. 1996), i.e., layered structures of

ferromagnetic ﬁlms, coupled strongly antiferromag-

netically, such that the total moment disappears. The

advantage of such a structure as compared to a

Figure 6

Coupling strength as a function of thickness in Fe/

Au(wedge)/Fe on (a) Ag-buffered GaAs substrate; (b)

on Fe whisker. The inset in (a) includes ranges where the

coupling is ferromagnetic.

Figure 5

Structure (lower right side) and typical remagnetization

curve of a Fe/Au/Fe sample with spin engineering.

Figure 7

Sensor for the measurement of the rotational angle of an

object by means of the GMR effect, which makes use of

an artiﬁcial antiferromagnet on the basis of

antiferromagnetic coupling.

959

Multilayers: Interlayer Coupling