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

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a practical possibility, however. Of course, head

gaps of these dimensions make it necessary to aban-

don some of the major advantages of optical disk

technology. The large separations between head

and media, and between the point of focus on ac-

tual recording media and the disk surface adjacent

to the environment, can no longer be supported.

The SIL must be ﬂown like the head in a conven-

tional magnetic hard drive, but considerably closer

to the recording surface. In these circumstances, in-

tegration of the write coil on the same slider car-

rying the SIL is a natural step. Optical access to the

actual recording medium can no longer be protected

through the substrate and front-surface media a thin

dielectric overcoat must be used. Fortunately, the

refractive indices of the common protective over-

coats such as silicon nitride match reasonably well

to those of SILs. In general, it is also necessary,

because of the extreme intensity in the very small

spot, to desensitize the medium. A fuller treat-

ment of near-ﬁeld optics is given elsewhere in this

encyclopedia.

See also: Magneto-optical Effects: Enhancement of;

Magneto-optic Recording: Overwrite and Associated

Problems; Magneto-optic Recording: Total Film

Stack, Layer Conﬁguration; Optical and Magneto-

optic Data Storage: Channels and Coding

Bibliography

Awano H, Ohnuki S, Shirai H, Ohta N, Yamaguchi A, Sumi S,

Torazawa K 1996 Appl. Phys. Lett. 69 (27), 4257–9

Betzig E, Isaacson M, Lewis A 1987 Appl. Phys. Lett. 51, 2088

Callan H, Josephs R M 1971 J. Appl. Phys. 42, 1977

Honda S, Ueda K, Kusuda T 1981 J. Appl. Phys. 52, 2295

Hosaka S, Shintani T, Kikukawa A, Imura R 1996 J. Magn.

Soc. Jpn. 20 (S1), 79–82

Kaneko M, Nakaoki A 1996 J. Magn. Soc. Jpn. 20 (S1), 7–12

Murakami Y, Maeda S, Takhashi A, Tanaka Y, Watanabe T

1996 J. Magn. Soc. Jpn. 23 (S1), 181–4

News Letter no. 4 (Jan 1998) www.oitda.or.jp/Optoelectronic

Industry and Technology Development Association (OIT-

DA), Japan

Nichia Corporation, Japan, Blue Laser model NLHV500A

October 1999

Nussbaum A, Phillips R A 1976 Contemporary Optics for Sci-

entists and Engineers. Prentice-Hall, Englewood Cliffs, NJ,

pp. 88–90

Ohta M, Fukumoto A, Aratani K, Kaneko M, Watanabe K

1991 J. Magn. Soc. Jpn. 15 (S1), 319–22

Saito J, Sato M, Matsumoto H, Akasaka H 1987 Jpn. J. Appl.

Phys. 26 (suppl. 4), 155

Shono K 1999 J. Magn. Soc. Jpn. 23 (S1), 177–80

Takagi N, Mitani K, Awano H, Shimazaki K, Ohta N 1999 J.

Magn. Soc. Jpn. 23 (S1), 161–4

Tokunaga T, Fujii Y, Yamada K 1999 J. Magn. Soc. Jpn. 23

(S1), 169–72

D. M. Newman

Centre for Data Storage Materials

Coventry University, UK

1220

Super-resolution: Optical and Magnetic Techniques

Textured Magnets: Deformation-induced

There are two different processes for producing

anisotropic rare earth–transition metal–boron

(RE–TM–B) magnets with high energy density. One

is the conventional technique of powder metallurgy

(see Magnets: Sintered). The crystal alignment is

achieved by applying an external magnetic ﬁeld dur-

ing the cold-pressing stage. The alternative processing

route is hot deformation of cast alloys or of ﬁne-

grained powder, the latter consolidated previously by

hot pressing. The hot working process produces grain

alignment along the c -axis perpendicular to the plas-

tic ﬂow direction. The key for understanding the

formability of the inherently brittle alloys is the pres-

ence of the RE-rich grain boundary phase, which is

liquid at the deformation temperature (4700 1C). It

is conceivable that the liquid grain boundary phase

can annihilate microcracks created during the defor-

mation. However, a liquid cannot resist high tensile

stresses. Therefore, the deformation must be per-

formed under compressive stress.

1. Deformation of Cast Alloys

Permanent magnet production by hot deformation of

cast alloys promises to be a potentially simpler and

more economical option in comparison with the sin-

tering route because of the troublesome handling of

magnetic powder that is extremely sensitive to oxygen

contamination.

The simplest deformation method is upsetting of

cylindrical specimens between two punches. Extensive

hot-pressing experiments performed by Shimoda et al.

(1989) revealed that Pr–Fe–B alloys with the compo-

sition Pr

are particularly suitable for hot de-

formation. Hot pressing at 1000 1C with strain rates of

4

–10

3

1

introduces a preferred magnetization

direction parallel to the press direction. Additionally,

the magnetic alignment is closely related to the struc-

ture of the cast ingots and the direction of principal

stress. The appropriate structure is a columnar struc-

ture in which the c-axis of the hard magnetic Pr

phase (2:14:1 or F phase) is in the plane perpendicular

to the crystal growth direction. Therefore, the prin-

cipal stress during deformation should also be per-

pendicular to the growth direction. Inserting the

cylindrical specimens into an iron ring prevents crack-

ing up to strain rates of 10

1

as well as squeezing

out the liquid intergranular phase. Additionally, in-

creasing the strain rate results in a grain reﬁnement

and with a consequent enhancement of the coercivity.

Studies concerning additive elements show that cop-

per addition is effective in increasing coercivity and

in improving the magnetic alignment. Microstructural

studies reveal that copper (i) reﬁnes the as-cast grain

size and (ii) reduces the melting point of the praseo-

dymium-rich phase. Copper probably leads to a better

wetting behavior and separation of the hard magnetic

grains. In the studies of Shimoda et al.(1989)the

magnetic properties of hot-pressed and annealed

76.5

1.5

magnets were as follows: remanence,

¼1.25 T; intrinsic coercivity,

¼796 kA m

1

;and

maximum energy product, (BH)

max

¼288 kJ m

3

Hot rolling of RE–TM–B alloys has been studied

by Ohki et al. (1989). The essential feature of the hot

rolling is a high strain rate (1–10 s

1

), which makes this

process very suitable for mass production. The opti-

mum rolling temperature is about 950 1C. Regarding

the optimum structure, it is found that the primary

crystallization of iron is necessary for the formation

of ﬁne-grained F crystallites during the following pe-

ritectic reaction. Optimum magnetic properties de-

mand a postdeformation heat treatment that is

beneﬁcial to both the coercivity and the remanence.

Regarding Pr–Fe–B alloys, the following maximum

values have been achieved: (BH)

max

¼340 kJ m

3

and

¼1194 kA m

1

. Concerning neodymium-based al-

loys, the maximum published values are: (BH)

max

269 kJm

3

and

¼525 kA m

1

. Hot rolling of ingots

is also used for the preparation of anisotropic powder

suited for bonded magnets (Hinz et al. 1994).

Extrusion of RE–TM–B alloys results in magnets

with radial c-axis orientation. Nozieres et al. (1988)

carried out extrusion experiments with large ingots,

cast in an iron sheath. Energy products in the range

120–160 kJ m

3

have been obtained.

Regarding the mechanism of deformation and tex-

turing, the microstructural feature of the hot defor-

mation of cast alloys is the marked grain reﬁnement.

It is assumed that the crystal alignment is related to

rotation of the grain debris in the liquid intergranular

phase. The start of the alignment is supported by the

plate-like shape anisotropy with its c-axis perpendic-

ular to the larger and ﬂat surface of the dendritic

grains (Yuri and Ohki 1994). During the deformation

process the cracking occurs preferentially along the

c-plane of the main phase, resulting in an enhance-

ment of the plate-like shape anisotropy and the ro-

tation of the grains. In a later stage the rotation of the

grains is accompanied by an irregular cracking at the

surface of the grains. This generates significant

amounts of small and unaligned grain debris. Post-

deformation heat treatment results in grain growth

with dissolution of the small and unaligned grains

and in blunting of the sharp edges of the large grains.

2. Deformation of Powder Precursors

The possibility of an alignment of compacted

melt-spun RE–TM–B powder precursors by hot

1221

deformation was discovered by Lee et al. (1985). In

the ﬁrst step a fully dense isotropic precursor must be

produced by hot pressing. The high densiﬁcation is

possible because the RE-rich grain boundary phase is

liquid in the applied temperature range from 700 1C

to 750 1C. In the second step the deformation of the

precursor takes place at temperatures between 750 1C

and 800 1C. In the case of the die-upset method, this

step can be accomplished by transferring the sample

to an oversized die cavity (Fig. 1). The deformation

leads to an anisotropic magnet with the alignment

parallel to the pressing direction.

Optimizing the composition in the Nd–Fe–B ter-

nary system involves a compromise between maxi-

mizing the volume fraction of the Nd

B phase

and providing excess neodymium for the grain

boundary phase. According to Nozawa et al. (1988)

the optimum composition of die-upset magnets is in

the range Nd

13.5–14

80.5–80

. Because the coercivity

is reduced after hot deformation of ternary alloys,

the effect of additional elements has been investi-

gated. A 0.5 at.% gallium addition proves to be fa-

vorable leading to a coercivity enhancement of up to

1600 kA m

1

and also a relatively small reduction in

remanence (Nozawa et al. 1988). The partial substi-

tution of cobalt for iron leads to an enhancement of

the Curie temperature of the magnetic main phase

and, therefore, to better thermomagnetic stability of

the magnets. According to Yoshida et al. (1991) the

addition of cerium or silicon has a positive effect on

the workability.

For melt spinning, a high quench rate should pre-

vent the formation of coarse grains on the wheel-free

surface of the ribbons. Surface velocities of

30–34 m s

1

lead to an optimum alignment after up-

setting (Fuerst and Brewer 1993). Concerning the

deformation conditions, increasing the die-upset level

results in an enhancement of the alignment up to a

height reduction of about 75%. The optimum defor-

mation temperature is in the range 750–800 1C and

the optimum strain rate is 10

2

–10

1

. Die-upset

magnets are commercially available. The energy

product of the different grades ranges from about

255 kJ m

3

to 340 kJ m

3

and the intrinsic coercivity

ranges from 955 kA m

1

to 1670 kA m

1

(Panchana-

than 1995). The published maximum energy product

is about 400 kJ m

3

. Saito et al. (1998) investigated

the magnetic properties of amorphous powder con-

solidated by shock compression. Die-upset magnets

with the composition Nd

13.5

(Fe

0.975

0.025

)

0.5

show a maximum energy product of 433 kJ m

3

and

an intrinsic coercivity of 995 kA m

1

offer optimal

deformation.

Mechanically alloyed powder (see Magnets: Me-

chanically Alloyed) and HDDR powder (see Magnets:

HDDR Processed) compacts can also be used as pre-

cursors for the successful production of die-upset

magnets (e.g., Schultz et al. 1991). A comparison of

deformation and magnetic properties has been given

by Kirchner et al. (1998) (Fig. 2).

The hot deformation of ﬁne-grained powder pre-

cursors is also suited to produce radially oriented ring

magnets by backward extrusion (Croat 1989). Mag-

nets with different dimensions and energy products of

more than 280 kJ m

3

are commercially available.

The process seems to be especially attractive because

the magnetic properties of rings surpass those of ring

Figure 1

Schematic representation of the preparation of

RE–TM–B magnets from ﬁne-grained powder by hot

pressing (isotropic magnet) and subsequent hot die-

upsetting (anisotropic magnet).

Figure 2

Demagnetization curves measured along the texture axis

of die-upset Nd–Fe–B magnets produced from melt-

spun (m), mechanically alloyed ( ), and HDDR

powders () (after Kirchner et al. 1998).

1222

Textured Magnets: Deformation-induced

magnets made from sintered material, especially at

small dimensions. Backward extruded ring magnets

do not quite reach the maximum properties of die-

upset magnets. The reason is the inherent inhomoge-

neity of the deformation process, which results in

inhomogeneous properties in the cross-section of the

rings (Gru

nberger et al. 1996).

Contrary to cast ingots, microstructural investiga-

tions in ﬁne-grained alloys show the formation of ﬂat

platelet-shaped grains which are larger than those of

the starting material (e.g., Lee et al. 1985). Therefore,

it is consequent to discuss diffusion-supported mech-

anisms for deformation and texturing. It is believed

that deformation and alignment are produced by a

combination of grain boundary sliding and aniso-

tropic grain growth caused by the inherent aniso-

tropy of the crystal growth of the tetragonal F phase

or by the anisotropy of the elastic properties. Lewis

et al. (1997) published a survey of the existing liter-

ature in this ﬁeld. Gru

nberger (1998) investigated the

applicability of the model of solution precipitation

creep. This is based on the transport of matter in the

liquid intergranular phase caused by the local de-

pendence of the solution equilibrium between solid

and liquid on the local stress state. Because the local

stress state depends on the elastic properties, the an-

isotropy of the latter can contribute to the develop-

ment of texture. The kinetic analysis of the strain rate

stress relationships of die-upset experiments indicates

an interface-controlled process.

See also: Magnetic Materials: Hard; Magnets:

Bonded Permanent Magnets; Magnets: Rema-

nence-enhanced; Magnets: Sintered

Bibliography

Croat J J 1989 Current status of rapidly solidiﬁed Nd–Fe–B

permanent magnets. IEEE Trans. Magn. 25, 3550–4

Fuerst C D, Brewer E G 1993 High-remanence rapidly solid-

iﬁed Nd–Fe–B: die-upset magnets. J. Appl. Phys. 73, 5751–6

Gru

nberger W 1998 The solution-precipitation creep—a model

for deformation and texturing mechanisms of nanocrystalline

NdFeB alloys. In: Proc. 15th Int. Workshop on Rare-Earth

Magnets and Their Applications. Dresden, Germany, pp.

333–48

Gru

nberger W, Hinz D, Schla

fer D, Schultz L 1996 Microstruc-

ture, texture, and magnetic properties of backward extruded

NdFeB ring magnets. J. Magn. Magn. Mater. 157/158,41–2

Hinz D, Handstein A, Harris I R 1994 Microstructure and

magnetic properties of anisotropic NdFeB powders from hot

rolled ingots by HD process. IEEE Trans. Magn. 30, 601–3

Kirchner A, Gru

nberger W, Gutﬂeisch O, Neu V, Mu

ller K-H,

Schultz L 1998 A comparison of the magnetic properties and

deformation behaviour of Nd–Fe–B magnets made from

melt-spun, mechanically alloyed and HDDR powders. J.

Phys. D: Appl. Phys. 31, 1660–6

Lee R W, Brewer E G, Schaffel N A 1985 Processing of neo-

dymium–iron–boron melt-spun ribbons to fully dense mag-

nets. IEEE Trans. Magn. 21, 1958–63

Lewis L H, Thurston T R, Panchanathan V, Wildgruber U,

Welch D O 1997 Spatial texture distribution in thermome-

chanically deformed 2-14-1-based magnets. J. Appl. Phys. 82,

3430–41

Nozawa Y, Iwasaki K, Tanigawa S, Tokunaga M, Harada H

1988 Nd–Fe–B die-upset and anisotropic bonded magnets.

J. Appl. Phys. 64, 5285–9

Nozieres J P, Perrier de la Bathie R, Gavinet J 1988 A novel

process for rare earth–iron–boron permanent magnets prep-

aration. J. Physique 49 (C8), 667–8

Ohki T, Yuri T, Miyagawa M, Takahashi Y, Yoshida C,

Kambe S, Higashi M, Itayama K 1989 A new method of

producing Pr–Fe–B base hard magnetic materials. In: Proc.

10th Int. Workshop on Rare-Earth Magnets and Their Appli-

cations. pp. 399–408 Kyoto, Japan

Panchanathan V 1995 Magnequench magnets status overview.

J. Mater. Eng. Perform. 4, 423–9

Saito T, Fujita, M, Kuji T, Fukuoka K, Syono Y 1998 The

development of high performance Nd–Fe–Co–Ga–B die up-

set magnets. J. Appl. Phys. 83, 6390–2

Schultz L, Schnitzke K, Wecker J, Katter M, Kuhrt C 1991

Permanent magnets by mechanical alloying. J. Appl. Phys.

70, 6339–44

Shimoda T, Akioka K, Kobayashi O, Yamagami T, Ohki T,

Miyagawa M, Yuri T 1989 Hot-working behavior of cast

Pr–Fe–B magnets. IEEE Trans. Magn. 25, 4099–104

Yoshida Y, Kasai Y, Watanabe T, Shibata S, Panchanathan V,

Croat J J 1991 Hot workability of melt-spun Nd–Fe–B mag-

nets. J. Appl. Phys. 69, 5841–3

Yuri T, Ohki T 1994 In: Crystal alignment in Pr–Fe–B hot-rolled

magnet. Proc. 14th Int. Workshop on Rare-Earth Magnets and

Their Applications, pp. 645–54 Birmingham, UK

W. Gru

nberger

Institut fu

r Festko

rper- und Werkstofforschung

Dresden, Germany

Thin Film Magnetism: Band Calculations

The understanding and design of complex magnetic

nanostructures is one of the current frontiers in mag-

netism. Research in this ﬁeld is stimulated by its in-

teresting fundamental physics as well as applications

today and in the future in the ﬁeld of data storage

and magnetoelectronics. New experimental tech-

niques, like molecular beam epitaxy and spin-polar-

ized electron spectroscopy techniques, for growth and

characterization of the nanostructures go hand-in-

hand with powerful theoretical methods and provide

a great deal of insight into these phenomena. Ab in-

itio calculations based on density functional theory

play a crucial role for the microscopic understanding

of fundamental magnetic properties, such as magnet-

ization and magnetic order.

1. Spin-density Functional Theory

Ab initio calculation means that the system under

consideration is described without any free parameter.

1223

Thin Film Magnetism: Band Calculations

The problem that needs to be solved is the many-body

problem of a system formed by interacting electrons

and nuclei. With the Born–Oppenheimer approxima-

tion (Born and Oppenheimer 1927) the problem can

be reduced to a system of N interacting electrons

moving in the electrostatic potential V

ext

caused by

the nuclei

H ¼

i¼1





i¼1

ext

ðr

Þþ

iaj

 r

ð1Þ

where e

¼e

/4pe

. e

is the dielectric constant. r

the position of electron i while all other constants

have their traditional meaning. Depending on the

boundary conditions used for the external potential,

systems with arbitrary dimensions like atoms or clus-

ters, wires, slabs, or bulk materials can be described

(Fig. 1) Atoms and clusters are considered to be quasi

one-dimensional and are ﬁnite in all spatial direc-

tions. Wires are conﬁned in two directions and are

translational invariant in the third, slabs are ﬁnite in

one direction and periodically repeated in the other

two directions, and bulk systems are translationally

invariant in all spatial directions.

Density functional theory (Hohenberg and Kohn

1964, Kohn and Sham 1965, Sham and Kohn 1966)

constitutes the main underlying basis for the solution

of this problem. In the generalization of density

functional theory to magnetic systems (Rajagopal

1980, Hedin and Lundqvist 1971, Barth and Hedin

1972, Moruzzi et al. 1978, Gunnarsson and Lundq-

vist 1976, Gunnarsson et al. 1972), the so-called spin

density functional theory, one distinguishes between

two types of electrons, namely the majority electrons

(s ¼þ) and the minority electrons (s ¼) whose

spin is in and opposite to the direction of magneti-

zation, respectively. In addition to the Coulomb po-

tential of the nuclei the electrons experience an extra

contribution 7m

H due to an external ﬁeld H. the

spin degeneracy is lifted. As a consequence the ma-

jority electrons are more attracted than the minority

electrons. Contributions from orbital momenta and

from spin–orbit coupling are neglected, giving an

adequate approximation for itinerant magnetism.

The two types of particles are represented by spin-

dependent densities n

(r) and n



(r), leading accord-

ingly to the following functional for the ground state

energy

E½n

; n



¼T½nþ

ext

ðrÞnðrÞ

þ U½nþE

½n

; n



ð2Þ

where T[n] is the functional of the kinetic energy for a

set of N noninteracting electrons of density n(r). U[n]

is the classical Hartree energy

U½n¼

rnðrÞ

nðr

7r  r

ð3Þ

and E

, n



] the exchange-correlation energy

which includes all many-body interactions, exchange

and correlation, in an effective way. The total density

is given by

nðrÞ¼n

ðrÞþn



ðrÞð4Þ

and the magnetization density is given by

mðrÞ¼n

ðrÞn



ðrÞð5Þ

Clearly, the ground state energy is a functional of

both spin-dependent densities, E[n

, n



]. Using spin-

dependent densities in terms of single-particle wave

functions j

(r)

ðrÞ¼

ðrÞ7

ð6Þ

the variation of the ground state energy yields the

spin-dependent Kohn–Sham equations



þ V

ðrÞ



ðrÞ¼E

ðrÞð7Þ

and, by analogy with the spin-independent case an

effective though spin-dependent potential follows as

¼ V

ext

ðrÞþ

nðr

7r  r

þ V

ðrÞð8Þ

with

dn7ðrÞ

Under the local spin density approximation (LSDA),

the exchange-correlation energy (Barth and Hedin

1972, Moruzzi et al. 1978) is written as Eqn. (9)

½n

; n



E

rðn

ðrÞþn



ðrÞÞ

 e

hom

ðn

ðrÞ; n



ðrÞÞ ð9Þ

and

ðn

þ n



Þ e

hom

ðn

; n





¼n

ðrÞ

ð10Þ

Figure 1

Dimension of the systems under consideration. Zero-

dimensional clusters, one-dimensional wires, two-

dimensional slabs, and three-dimensional bulk systems.

1224

Thin Film Magnetism: Band Calculations

Different approximations of the local density and the

local spin density functional (Hedin and Lundqvist

1971, Barth and Hedin 1972, Moruzzi et al. 1978,

Vosko et al. 1980) have been used. They lead to

variations within the error of the local density or spin

density approximation itself.

2. Stoner Model of Ferromagnetism

The one-particle nature of the Kohn–Sham equation

(Eqn.(7)) allows a Stoner-like model to be derived for

ferromagnetism (Vosko and Perdew 1975, Gunnars-

son 1976, Janak 1977, Stoner 1938, Blu

gel 1995).

Starting from the assumption that the magnetization

density m(r) is a small parameter compared to the

density n(r), the exchange and correlation potential

can be expanded in powers of m(r). The linear

approximation yields

¼ V

8mðrÞ

VðnðrÞÞ;

V40 ð11Þ

where V

is the exchange-correlation potential for

the nonmagnetic case. As a consequence, the majority

of electrons experience a more attractive potential as

minority electrons. In the Stoner model this potential

shift is expressed in terms of a constant

¼ V

IM; M ¼

atom

drmðrÞð12Þ

with the local atomic moment M and the Stoner pa-

rameter I. In the case of ferromagnetism M is the

same for all atoms. As a consequence of the constant

potential shift in Eqn. (12), the single particle wave

functions j

(r) (Eqn. (7)) do not change with respect

to the nonmagnetic wavefunctions, while the eigen-

values E

are shifted rigidly downwards or upwards

ðrÞ¼j

ðrÞ and E

¼ E

IM ð13Þ

This is the so-called spin–split bandstructure (Fig. 2).

The symmetric energy shift is transferred to the

spin-dependent density of states

ðEÞ¼nðE7

IMÞð14Þ

with respect to the density of states of the nonmagnetic

system n(E) (Fig. 2). Integration over all occupied

states up to the Fermi energy E

gives the number of

electrons

N ¼

dEðn

ðE þ

IMÞþn

ðE 

IMÞð15Þ

and the moment per atom

M ¼

dEðn

ðE þ

IMÞn

ðE 

IMÞÞ ð16Þ

Since the density of the nonmagnetic system n

and the

number of electrons N are known, the Fermi energy

and the moment M can be obtained from the

self-consistent solution of Eqns. (15) and (16). As a

sufﬁcient condition for a ferromagnetic solution the

Stoner criterion

ðE

Þ41 ð17Þ

can be derived. Large Stoner parameters or exchange

integrals I and large density of states at the Fermi en-

ergy n

) favor ferromagnetism. The corresponding

parameters deduced from a nonspin-polarized calcula-

tion are shown in Table 1 (Gunnarsson 1976, Janak

1977). Obviously, the Stoner criterion is only fulﬁlled

for Fe, Co, and Ni, precisely those metals that show

ferromagnetism. Fig. 3 shows the densities of states for

Fe, Co, and Ni computed by means of a spin-density

functional calculation. The results show the dominat-

ing contribution of the 3d electrons. The density of

states is exchange split and otherwise very complicated.

The spin-up and spin-down densities of states remain,

however, very similar which conﬁrms the validity of a

Stoner model.

The corresponding spin moments in comparison to

experimental results for spin moments and total mo-

ments are given in Table 2. The exchange integral I is

an interatomic property that depends on the type of

Figure 2

Spin–split band structure and density of states for free electrons; majority and minority bands are shifted downwards

and upwards, respectively.

1225

Thin Film Magnetism: Band Calculations

atom but is independent on structure and environ-

ment. According to Gunnarsson (1976) and Janak

(1977) the exchange integrals of the transition metals

have the following trend

ð18Þ

The density of states at the Fermi level is reciprocal

with the bandwidth

nðE

ÞBW ð19Þ

Table 1

Density of states n(E

) (in eV

1

) at the Fermi energy calculated for nonmagnetic systems, Stoner parameter I (in eV),

and In(E

) (Gunnarsson 1976, Janak 1977).

Metal Na Al Cr Mn Fe Co Ni Cu Pd Pt

n(E

) 0.23 0.21 0.35 0.77 1.54 1.72 2.02 0.14 1.14 0.79

I 1.82 1.22 0.76 0.82 0.93 0.99 1.01 0.73 0.68 0.63

In(E

) 0.41 0.25 0.27 0.63 1.43 1.70 2.04 0.11 0.78 0.50

Figure 3

Spin–polarized density of states of Fe, Co, and Ni in a bulk phase (upper panel) and for the corresponding atom in a

(100) surface (lower panel). The shades denote the contribution of s (darkest), p (lightest), and d electrons (medium

shade).

Table 2

Magnetic moments (m

atom) for Fe, Co, and Ni

calculated in the LSDA approximation (Moruzzi et al.

1978) in comparison to experimental values for the spin

moment and the total moment.

Metal M

LSDA

spin

Fe 2.15 2.12 2.22

Co 1.56 1.57 1.71

Ni 0.59 0.55 0.61

1226

Thin Film Magnetism: Band Calculations

The stronger the electrons are localized, the smaller

the bandwidth, the larger is the density of states. The

bandwidth is zero in the atomic limit. The Stoner

criterion is always fulﬁlled. The magnetic moment is

determined by Hunds rule. If we restrict our further

considerations to the tightly bound d electrons of

transition metals, the bandwidth can be expressed by

means of a tight-binding model

¼ 2

ﬃﬃﬃﬃﬃﬃﬃﬃ

ðR

Þð20Þ

According to this approximation the bandwidth

depends on the overlap integral or hopping matrix

element h

of the d electrons and the number of

nearest neighbor atoms or coordination number N

The overlap integral h

is element specific, is deter-

mined by the overlap of the d wavefunctinos, and, of

course, is reciprocal with the nearest neighbor dis-

tance R

. The trend in the transition metals follows

the relation

) W

) n

ð21Þ

In Fig. 4 the general trend of the bandwidths of

transition metals, lanthanides, and actinides is shown

(Blu

gel 1995).

3. Magnetism in Reduced Dimensions

Understanding magnetism in low-dimensional sys-

tems looks rather straightforward. The simplest low-

dimensional system is the isolated atom, for which it

is known that nearly all transition metal atoms have

magnetic moments. The magnetism is well described

by Hund’s rules. On the other hand, it is known that

among the transition metals only ﬁve remain mag-

netic in their bulk crystalline phase: Co and Ni are

ferromagnetic, Cr is antiferromagnetic, and Mn and

Fe are ferromagnetic or antiferromagnetic depending

on their crystal structure. Low-dimensional transition

metals, that is interfaces, surfaces, ultrathin ﬁlms, and

monolayers, should behave between these two ex-

tremes. As a consequence an enormous number of ab

initio calculations have been performed in this ﬁeld in

the 1990s (Blu

gel 1995, Weinert and Blu

gel 1993,

Asada et al. 1999).

Following the considerations above a reduction of

the coordination number N

would cause band nar-

rowing and would enhance the tendency towards

magnetism due to the Stoner criterion.

The reduction of the coordination number, how-

ever, can be achieved by a reduction in the dimension

of the system under consideration. An fcc crystal has

a coordination number of N

fcc

¼12, an atom of the

(100) surface of the same crystal is surrounded by

eight atoms N

(100)

¼8, and the monolayer atoms have

only four nearest neighbors, N

¼4. Keeping the

nearest neighbor distance and the overlap integrals

ﬁxed, the density of states and the bandwidths would

obey the following relation

fcc

: n

ð100Þ

: n

fcc

ð100Þ

¼1 : 1:22 : 1:73 ð22Þ

The reduction in the coordination number is hence

the origin of the enhancement of magnetism at in-

terfaces, surfaces, and for monolayers or ultrathin

ﬁlms. As a consequence these systems offer the op-

portunity to ﬁnd totally new magnetic materials.

For the ab initio description of surfaces and ultra-

thin magnetic ﬁlms, different geometries are used. In

a superlattice the metallic ﬁlms are separated by vac-

uum layers but otherwise are periodically repeated in

all spatial directions. Surface geometry consists of

semi-inﬁnite metal and vacuum parts whereas slab

geometry is a freestanding metal ﬁlm embedded in

vacuum layers (Fig. 5).

Figure 3 shows the local densities of states for Fe,

Co, and Ni (100) surfaces computed by means of a

spin-density functional calculation. The comparison

of bulk (upper panel) and surface (lower panel) den-

sity of states illustrates the effect of band narrowing

clearly. Enhancement of the magnetic moments at the

surface in comparison to the bulk material is given in

Table 3.

Figure 5

Superlattice, semi-inﬁnite surface system, and slab.

Figure 4

Trend in the bandwidth of the transition metal,

lanthanides and actinides as a function of the atomic

number.

1227

Thin Film Magnetism: Band Calculations

The magnetic moment of the Ni (110) surface is

higher than for the (100) surface. The results for Fe

are opposite. This behavior is related to the surface

orientation, which determines the coordination num-

ber and the next nearest neighbor distance between

the surface atoms. The largest magnetic moment will

be obtained for the most open surface. Ni is an fcc

system: the low-index surface (110) is the most open,

(100) is intermediate, and (111) is the closed packed

surface. The trend for bcc systems is opposite. Fe is a

bcc system.

It is implied in the discussion so far that the co-

ordination number is the crucial feature in interface

magnetism but this is not the whole truth. The fol-

lowing features have to be taken into account to

achieve a quantitative description. First of all, the

point symmetry of a surface is lower in comparison to

the bulk system. Symmetry reduction causes level

splitting, changes the shape of the density of states,

and is unfavourable for magnetism. Second, the re-

laxation of the charge density at the surface changes

hybridization at the surface and causes a redistribu-

tion of spectral density. As a consequence, a relative

shift of the sp band against the d band and/or a

change of spectral weight is obtained (see Fig. 3). The

number of d states is changed. As a result magnetism

can be enhanced or reduced (see Ni). The neglect of

these interplaying effects have caused confusion in

the past in the ﬁeld of surface and interface magnet-

ism. But they are readily included in self-consistent

electronic structure calculations.

4. Ground State Magnetic Order

Based on the calculation of the total energy (Eqn.

(10)), ab initio electronic structure calculations pro-

vide a perfect tool for investigating the magnetic or-

der. Comparison of the total energy for different

magnetic conﬁgurations DE allows a systematic

search of the magnetic order in the ground state.

As an example, the magnetic order of 3d transition

metal monolayers on top of a Cu substrate was in-

vestigated (Blu

gel 1995). The total energy of a ferro-

magnetic conﬁguration E

is compared to the total

energy of an antiferromagnetic conﬁguration E

(Fig. 6). The energy differences

DE ¼ E

 E

ð23Þ

are shown in Fig. 7. Fe, Co, and Ni layers prefer

ferromagnetic, whereas V, Cr, and Mn prefer an an-

tiferromagnetic order.

This is an arbitrary example chosen to demonstrate

the method. Similar calculations can be applied to

various other problems concerning the magnetic con-

ﬁguration, for example surface steps and edges or

Table 3

Magnetic moments (m

atom) for Fe, Co, and Ni

calculated in LSDA approximation (Blu

gel 1995) for

(100) and (110) surfaces in comparison to bulk moments

(Moruzzi et al. 1978).

Metal M

LSDA

(100)

(110)

Fe 2.15 2.88 2.43

Co 1.56 1.85 —

Ni 0.59 0.68 0.74

Figure 6

Schematic representation of ferromagnetic and c (2 2) antiferromagnetic order of a monolayer ﬁlm grown as an

overlayer on a fcc (001) substrate.

Figure 7

Total energy difference per atom for a ferromagnetic

and antiferromagnetic phase for 3d monolayers on a n,

Cu (100) and *, Ag (100) substrate.

1228

Thin Film Magnetism: Band Calculations

surface alloys (Freyss et al. 1996) also including non-

collinear magnetic order (Sandratskii 1998)

The capability of this method was nicely conﬁrmed

by experimental observation of the local magnetic

order in surface alloys by means of spin-polarized

scanning tunneling spectroscopy (Heinze et al. 2000).

See also: Monolayer Films: Magnetism; Density

Functional Theory: Magnetism; Itinerant Electron

Systems: Magnetism (Ferromagnetism); Multilayers:

Interlayer Coupling

Bibliography

Asada T, Bihlmayer G, Handschuh S, Heinze S, Kurz Ph,

Blu

gel S 1999 J. Phys.: Condens. Matter. 11, 9347

Barth U V, Hedin L 1972 J. Phys. C 5, 1629

Blu

gel S 1995 Habilitation. RWTA, Aachen

Born M, Oppenheimer J R 1927 Ann. Phys. 84, 457

Freyss M, Stoefﬂer D, Dreysse

1996 Phys. Rev. B 54, 12677

Gunnarsson O 1976 J. Phys. F: Met. Phys. 6, 587

Gunnarsson O, Lundqvist B I 1976 Phys. Rev. B 13, 4274

Gunnarsson O, Lundqvist B I, Lundqvist S 1972 Solid State

Commun. 11, 149

Hedin L, Lundqvist B 1971 J. Phys. C 4, 2064

Heinze S, Bode M, Kubetzka A, Pietzsch O, Nie X, Bluegel S,

Wiesendanger R 2000 Science 288, 1805

Hohenberg P, Kohn W 1964 Phys. Rev. 136, 864

Janak J F 1977 Phys. Rev. B 16, 255

Kohn W, Sham L J 1965 Phys. Rev. 140, 1133

Moruzzi V R, Janak J F, Williams A R 1978 Calculated Elec-

tronic Properties of Metals. Pergamon Press, New York

Rajagopal A K 1980 Adv. Chem. Phys. 41,59

Sandratskii L M 1998 Adv. Phys. 47,91

Sham L J, Kohn W 1966 Phys. Rev. 145, 561

Stoner E C 1938 Proc. R. Soc. London 165, 372

Vosko S H, Perdew J P 1975 Can. J. Phys. 53, 1385

Vosko S H, Wilk L, Nussair N 1980 Can. J. Phys. 58, 1200

Weinert M, Blu

gel S 1993 In: Bennet L H, Watson R E (eds.)

Magnetic Multilayers. World Scientific, Singapore

I. Mertig

Technische Universita

t Dresden, Germany

Thin Film Magnetism: PEEM Studies

A Photoemission Electron Microscope (PEEM) im-

ages photo and/or secondary electrons, which are

generated at the surface of a solid sample by the ab-

sorption of x-ray or ultraviolet (UV) radiation. De-

pending on the wavelength of the radiation different

contrast mechanisms are present, providing informa-

tion on the elemental and chemical composition, and

the electronic and magnetic properties of the mate-

rial. Magnetic domain imaging is a major application

of the PEEM technique, exploiting strong x-ray mag-

netic dichroism effects, e.g., at the L edges of the

magnetic 3d transition metals (e.g., Cr, Mn, Fe, Co,

Ni). For a comparison with other domain imaging

techniques see Kerr Microscopy, Magnetic Materials:

Transmission Electron Microscopy, Magnetic Force

Microscopy, and references therein.

The PEEM technique stands out against other

magnetic imaging techniques through its surface sen-

sitivity and element speciﬁcity, making PEEM an

ideal tool for the investigation of ultra-thin magnetic

ﬁlms, multi-layers, and alloys. In contrast to other

high resolution, magnetic imaging techniques, PEEM

is also sensitive to antiferromagnetic order. The spa-

tial resolution of PEEM for magnetic domain imag-

ing is typically about 50–100 nm, which positions

PEEM between Transmission Electron Microscopy

and optical techniques, such as Kerr Microscopy.

1. Photoemission Electron Microscopy

PEEM was ﬁrst used in the 1930s and has since then

matured into an established surface science technique

(Sto

hr et al. 1993, Tonner et al. 1995, Sto

hr et al.

1998, Anders et al. 1999). PEEM is closely related to

the Low Energy Electron Microscope (LEEM) and

the Spin-polarized Low-energy Electron Microscope

(SPLEEM), which were pioneered by Bauer (1994)

and Duden and Bauer (1998). All three techniques

utilize low-energy electrons to form an image repre-

senting physical properties of the sample surface.

LEEM and SPLEEM image diffracted low-energy

electrons and thereby provide information on the

local crystallographic (LEEM) and magnetic

(SPLEEM) structure of the surface of a crystalline

sample. PEEM, in contrast, images electrons gener-

ated by photoionization and therefore is not limited

to the study of crystalline samples (see also Photo-

emission: Spin-polarized and Angle-resolved). As light

sources, UV gas discharge lamps, UV lasers, and

synchrotron radiation sources have been used. Syn-

chrotron radiation offers the important advantage of

tunability of the wavelength of the illumination,

thereby allowing a selection between various mech-

anisms of contrast. X-ray PEEM thus combines as-

pects of spectroscopic and microscopic methods and

is called a spectromicroscopy technique.

A typical PEEM setup using synchrotron radiation

from a bending magnet is shown in Fig. 1 (Anders

et al. 1999, Scholl et al. 2002). X rays pass through a

moveable aperture, which selects the polarization of

the radiation. A spherical grating monochromator

and an exit slit monochromatize and focus the beam

onto the sample. A schematic kinetic energy spectrum

of the emitted electrons after x-ray absorption is

shown in Fig. 2(a). Photoemission lines appear at

high kinetic energy, followed by a broad tail of sec-

ondary electrons which peak at low energies close

to the work function cut-off. PEEM microscopes

usually accept the total electron yield without prior

1229

Thin Film Magnetism: PEEM Studie s