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

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Magnetic Fields: Thermoelectric Power

The coefﬁcient of the thermoelectric power (S) meas-

ured in variable magnetic ﬁelds (B) is called magne-

tothermopower. This physical quantity can provide

very valuable complementary information concerning

details of electronic band structure. In comparison

with the magnetoresistivity, the magnetothermo-

power is not sensitive to a uniform change of mobil-

ity of conduction electrons by the Lorentz force. The

magnetothermopower of metallic materials basically

depends on a relative change of the mobility of

conduction electrons with different energy near the

Fermi surface.

The effect of magnetic ﬁelds on thermopower

is important also from a practical point of view.

Thermocouples are widely used as temperature sen-

sors. Therefore, in low-temperature thermometry in

equipment with strong magnetic ﬁelds, such as re-

search cryostats with superconducting magnets, a

knowledge of the sensitivity of a thermocouple to the

magnetic ﬁeld is essential. Another significant and

expanding ﬁeld for the application of thermoelectric

phenomena is thermoelectric heat–energy conversion

and thermoelectric cooling. It is known that the

thermoelectric efﬁciency of some materials can be

considerably enhanced in a magnetic ﬁeld (examples

are bismuth-based alloys which are used for low-

temperature refrigeration).

1. Theoretical Aspects of Magnetothermopower

The thermopower, or Seebeck, coefﬁcient S is deﬁned

as the coefﬁcient in the relation: E ¼S rT between

420

Magnetic Fi elds: Thermoelectric Power

the temperature gradient rT and the electrical ﬁeld E

in a conducting material, where the net current

through the sample is zero (see Boltzmann Equation

and Scattering Mechanisms). There are two other

thermoelectric phenomena: the Peltier heat (charac-

terized by the coefﬁcient P) and the Thomson heat

(characterized by the coefﬁcient m

), which can be

observed under different conditions. However, these

three coefﬁcients S, P, and m

are not independent;

they are connected to each other by the Thomson

relations. For an isotropic medium, without a mag-

netic ﬁeld, the relations read: P ¼TS and m

¼TdS/

dT (Barnard 1972). In a general case, S, P, and m

are tensors of second rank, which can have as

many as nine independent components. The most

general form of the Thomson relation is given by the

expression: P

(B) ¼TS

(B) (Landau and Lifshitz

1977).

If there is, in addition to the temperature gradient,

an external magnetic ﬁeld applied to the sample,

the thermoelectric ﬁeld can be written as:

E ¼SrT þNB rT (Abrikosov 1972). The second

term in this equation is the so called Nernst–

Ettingshausen (N–E) effect, with N the Nernst–

Ettingshausen coefﬁcient. However, apart from the

N–E effect, the coefﬁcient S can also be dependent on

the magnetic ﬁeld. The change of S in an external

magnetic ﬁeld is called magnetothermopower:

DS ¼S(B, T )S(0, T ). In the weak ﬁeld limit, this

dependence is a second-order correction of the ther-

moelectric ﬁeld E in the presence of a magnetic ﬁeld.

With respect to experimental conditions, two dif-

ferent situations are possible, depending on the mu-

tual orientation of the magnetic ﬁeld and the

temperature gradient: the magnetic ﬁeld parallel to

the temperature gradient, and the magnetic ﬁeld

perpendicular to the temperature gradient. One dis-

tinguishes, respectively, longitudinal magnetothermo-

power DS

and transversal magnetothermopower

. In the case of anisotropic materials, special care

should be taken to separate DS

and the inﬂuence of

the Nernst–Ettingshausen effect. According to the

mechanism of the change of thermopower in a mag-

netic ﬁeld, the effect can be subdivided into two main

categories. In the ﬁrst category, the effect is connect-

ed to the orbital motion of the conduction electrons

in a magnetic ﬁeld caused by the Lorentz force. In the

second category, the effect is related to a change in

the electronic structure and/or scattering mechanism

in a magnetic ﬁeld.

With regard to the Lorentz force-driven magneto-

thermopower, two different regimes are distinguish-

able, depending on the strength of the magnetic ﬁeld.

This ﬁeld strength can be characterized by the prod-

uct o

t, where o

¼eB/m*c is the classical cyclotron

frequency, t is the average time between collisions of

a conduction electron with scattering centers, m*is

the effective electron mass, and c is the speed of light.

The low-ﬁeld regime corresponds to o

t51, while

tX1 is the condition for the high-ﬁeld range. From

a theoretical point of view, the high-ﬁeld case has

been much better understood than the low-ﬁeld one.

In the high-ﬁeld regime, the scattering processes

become unimportant, and galvano- and thermomag-

netic phenomena are determined by the topological

properties of the Fermi surface. In the high-ﬁeld limit

tb1, the longitudinal magnetothermopower DS

is independent of the magnetic ﬁeld for any Fermi

surface geometry.

The transversal magnetothermopower DS

is pro-

portional to B for closed electron orbits in compen-

sated metals, and is independent of the ﬁeld in other

cases (Blatt et al. 1976). A special case of the high-

ﬁeld regime is realized under the additional condi-

tion _o

T (k

¼1.3806568 10

23

1

is the

Boltzmann constant, _ ¼1.05457266 10

34

Js is

the Planck constant). Under this condition, Landau

quantization of the orbital motion of the conduction

electrons in the plane, normal to the magnetic ﬁeld,

becomes important and all electronic properties of

metals, including thermopower, show an oscillatory

dependence on B

1

with a period DB

1

. The period

depends on the extreme cross-sectional area A

of the

Fermi surface in a plane normal to the magnetic ﬁeld:

1

¼(2pe/_c)(1/A

) (Landau and Lifshitz 1977).

These oscillations can best be observed in a very

pure single crystal at low temperatures: To(_e/

m*ck

)BE1.34B, where B is the ﬁeld in teslas.

In the low-ﬁeld regime, the scattering mechanism is

essential and this makes a theoretical description

extremely difﬁcult. Within this limit, the Lorentz

force driven magnetothermopower is expected to be

proportional to B

(Abrikosov 1972, Landau and

Lifshitz 1977, Ri et al. 1994).

The magnetothermopower due to a change of elec-

tronic structure can be dominant in non-simple met-

als, magnetic or nearly magnetic materials, strongly

correlated systems, and in conductors with magnetic

impurities. The effect essentially depends on details of

the electronic structure of a particular system. No

general theory has been developed up to now. The

inﬂuence of a magnetic ordering on the thermopower

has been discussed and investigated both theoretically

and experimentally. On a qualitative level, the var-

iation of thermopower of itinerant ferromagnets in

the vicinity of Curie temperature has been explained

by Mott, based on his s–d scattering model (Blatt

et al. 1976). A summary of theoretical and experi-

mental results can be found in Blatt et al. 1976, and in

Yagasaki et al. 1996).

A somewhat special ﬁeld represents the thermo-

magnetic phenomena in superconductors. This ﬁeld

has experienced a burst of theoretical and experi-

mental activities in connection with the discovery of

high-temperature superconductors. Large thermo-

magnetic effects were observed in the mixed state of

high-T

materials. These effects are related to the

magnetic ﬂux ﬂow driven by a temperature gradient.

421

Magnetic Fields: Thermoelectric Power

For a review, see Huebener (1979) and Ri et al .

(1994).

A review of the theory of galvanomagnetic and

thermomagnetic effects in metals and of experimental

results prior to 1957 is given in Jan (1957).

2. Details of the Measur ement

Compared to measurements of the thermopower

without a magnetic ﬁeld, magnetothermopower meas-

urements require several additional considerations:

(i) In the high-ﬁeld limit, thermopower can be an-

isotropic even for crystals of cubic symmetry.

(ii) Contributions from galvano- and thermomag-

netic effects, different from thermopower, should be

carefully eliminated (the Nernst–Ettingshausen effect

particularly).

(iii) Calibration of temperature sensors and refer-

ence electrodes in a magnetic ﬁeld is important.

(iv) Care should be taken to avoid a magnetic ﬁeld

gradient.

Several different designs have been introduced for

measurements of magnetothermopower. As an ex-

ample, we mention the apparatus described by Resel

et al. (1996). With this setup, the thermopower can be

measured in magnetic ﬁelds from 0 T to 17 T in the

temperature range from 3 K to 300 K. The main part

of the sample holder is shown in Fig. 1. A special

modulation technique for the temperature gradient is

used in this setup to eliminate spurious voltages. Two

gradient heaters working in opposite directions are

utilized to generate an alternating temperature gra-

dient in the sample. The total heating power pro-

duced by the two heaters is constant. Therefore, the

average temperature of the sample is not affected by

the gradient modulation. In order to avoid the N–E

effect contribution, a parallel alignment of the tem-

perature gradient and magnetic ﬁeld direction is cho-

sen. The temperature difference across the sample is

measured with the help of two Chromel–Constantan

thermocouples.

The sensitivity of this thermocouple is independent

of the magnetic ﬁeld within an error of 5%, in the

ﬁeld range from 0 T to 17 T at temperatures from 3 K

to 300 K. For the reference electrode a Chromel wire

is used, since the thermopower of this alloy has a

comparatively weak and simple dependence on the

external magnetic ﬁeld and the temperature. How-

ever, for precise measurements of magnetothermo-

power of metals, where the effect is small, it is

necessary to make corrections in the thermopower of

Chromel, so that S

(B, T) ¼S

(0, T) þDS(B, T). It

was found that DS(B, T) can be approximated by an

empirical function f(B, T) with an accuracy of

0.1 mVK

1

. This function reads:

f ðB; TÞ¼aT expðbT

ÞþcT

expðdTÞ

Figure 1

A part of the sample holder for thermopower measurement in a magnetic ﬁeld. (1) and (6) sample-pressing and

sample-supporting plates (glass epoxy); (2) sample; (3) AlN plates; (4) copper layers; (5) gradient heater; (7) heat sink

(copper-beryllium bronze); (8) temperature sensor; 9) Al

tubes holding thermocouples; (10) spring (copper–

beryllium bronze).

422

Magnetic Fi elds: Thermoelectric Power

where

a ¼

2:110

2

51:4 þ 49:1 expð2:91  10

2

BÞ

b ¼2:2  10

4

21:02  10

5

expð0:187BÞ

c ¼4:18  10

5

d ¼7:47  10

5

þ 2:33  10

3

B  4:38  10

2

(where B is the magnetic ﬁeld in teslas).

3. Experimental Results

Most systematic and comprehensive investigations of

the magnetothermopower have been performed for

simple monovalent and divalent metals as well as for

noble metals. It was found that in many cases

DS ¼S(B, T) S(0, T) is large at low temperatures.

In noble and divalent metals, a large contribution to

DS is referred to as an increase of the phonon drag

thermopower in a magnetic ﬁeld. A summary of these

studies is given in Blatt et al. (1976). A comprehensive

investigation of DS in the high-ﬁeld regime was made

Figure 3

Quantum oscillations of the thermoelectric voltage

versus the inverse magnetic ﬁeld for Bi are shown. *:

total thermoelectric voltage; J: oscillating part (broken

line shows the envelope curve).

Figure 2

Magnetothermopower of silver against a magnetic ﬁeld

(Stanley 1974). Results for single crystal specimens of

silver of different purity and crystallographic orientation

are shown. Thermopower was measured with the

longitudinal orientation of the magnetic ﬁeld with

respect to the temperature gradient. &: RRR ¼1380,

along [741] crystallographic direction, T ¼2.5 K; J: the

same sample, T ¼4.4 K; *: RRR ¼2280, along [111],

T ¼4.9 K; and þ: the same sample at T ¼6.2 K.

RRRr(300)/r(4.2).

Figure 4

Thermopower of heavy-fermion compounds against a

magnetic ﬁeld (Marcenat et al. 1990). J: CeAl

T ¼0.58 K; *: CeAl

at T ¼0.5 K; þ:(LaCe)Al

T ¼0.58 K; &: CeRu

at T ¼1.5 K.

423

Magnetic Fields: Thermoelectric Power

for silver (Stanley 1974). Some results for single silver

crystals are presented in Fig. 2.

The quantum oscillations of the magnetothermo-

power and the Peltier coefﬁcient were measured for

bismuth, tin, antimony, aluminum, and zinc single

crystals. The results of these studies are summarized

by Blatt et al. (1976). The quantum oscillations of the

thermoelectric voltage versus the inverse magnetic

ﬁeld at T ¼4.2 K for bismuth are presented in Fig. 3.

Very little has been done with respect to the

magnetothermopower of magnetic materials and in-

termetallic compounds. Some results on DS of heavy-

fermion compounds can be found in Marcenat

(1990): see Fig. 4. Results for magnetic and nearly

magnetic compounds are shown in Fig. 5.

Applications of thermomagnetic effects for thermo-

electric energy conversion form a separate and large

ﬁeld. Semimetals and heavily doped semiconductors

have mainly been used in these applications. Theoret-

ical basics and a review of important experimental

results can be found in Hartman and Honig (1967).

See also: Rare Earth Intermetallics: Thermopower of

Cerium, Samarium, and Europium Compounds

Bibliography

Abrikosov A A 1972 Introduction to the Theory of Normal

Metals. Academic Press, New York

Barnard R D 1972 Thermoelectricity in Metals and Alloys.

Taylor and Francis, London

Blatt F J, Schroeder P A, Foiles C L 1976 Thermoelectric Power

of Metals. Plenum, New York

Foiles C L 1985 Thermopower of Pure Metals In: Hellwege

K-H, Olsen J L (eds.) Landolt-Bo

rnstein. Numerical Data and

Functional Relationships in Science and Technology: new se-

ries, Group III–Crystal and Solid State Physics. Metals: Elec-

tronic Transport Phenomena. Subvolume b. Springer, Berlin,

Vol. 15., pp. 48–104

Hartman T C, Honig J M 1967 Thermoelectric and Thermo-

magnetic Effects and Applications. McGraw-Hill, New York

Huebener R P 1979 Magnetic Flux Structures in Superconduc-

tors. Springer, Berlin

Jan J-P 1957 Galvanomagnetic and Thermomagnetic Effects in

Metals. In: Seitz F, Turnbull D 1957 (eds.) Solid State Phys-

ics. Vol. 5. Academic Press, New York, pp. 1–96

Landau L D, Lifshitz E M 1977 In: Course of Theoretical

Physics: Quantum Mechanics. (3rd edn) Pergamon, Oxford,

Vol. 3

Marcenat C, Jaccard D, Sierro J, Flouquet J, Onuki Y,

Komatsubahara T 1990 Extended transport measurements

on high-purity CeB

. Journal of Low Temperature Physics 78,

261–85

Resel R, Gratz E, Burkov A T, Nakama T, Higa M, Yagasaki

K 1996 Thermopower measurements in magnetic ﬁelds up to

17 tesla using the toggled heating method. Rev. Sci. Instrum.

67, 1970–5

Ri H-C, Gross R, Gollnik F, Beck A, Huebener R P 1994

Nernst, Seebeck, and Hall effects in the mixed state of

YBa

7d

and Bi

CaCu

8 þ x

thin ﬁlms: a compar-

ative study. Phys. Rev. B 50 , 3312–29

Stanley D J 1974 Low-temperature magnetothermopower of

silver. Proc. R. Soc. London A 339, 97–117

Yagasaki K, Nakama T, Higa M, Sakai E, Burkov A T, Gratz

E, Resel R 1996 Thermopower of GdAl

with see-saw heat-

ing system. J. Phys. Soc. Jpn. 65 Suppl. B, 181–7

K. Yagasaki

University of the Ryukyus, Okinawa, Japan

A. T. Burkov

Russian Academy of Sciences, St Petersburg, Russia

Magnetic Films: Anisotropy

Magnetic anisotropy is the coupling of the magnet-

ization of a material to particular directions, either at

a local or a macroscopic level. It means that the

measured magnetic properties depend on the direc-

tion in which the magnetic ﬁeld is applied, and that in

zero applied ﬁeld, the magnetization will lie along

preferred direction(s). Without magnetic anisotropy,

there can be no coercivity and no remanent magnetic

moment and hence no permanent magnets or mag-

netic recording media. For these technologies, high

anisotropy is generally advantageous, as it causes the

moment to remain ﬁxed in some desired direction.

Figure 5

Thermopower of a nearly magnetic YCo

and of a

ferromagnet TmCo

against temperature. *: TmCo

H ¼0T; J: TmCo

, m

H ¼15 T; þ: YCo

H ¼0T; : YCo

, m

H ¼15 T. The arrow indicates

Curie temperature of TmCo

424

Magnetic Fi lms: Anisotropy

For magneto-optic recording and proposed perpen-

dicular recording media, perpendicular anisotropy is

essential. For present-day longitudinal magnetic re-

cording media, an anisotropy axis along the track

direction allows greater thermal stability of magnetic

bits, and hence smaller bit sizes. Minimizing anisot-

ropy is equally important for technologies that

require high susceptibility, such as transformers. Un-

derstanding what causes anisotropy and how to

control it is thus crucial to the technological use of

nearly all magnetic materials including ﬁlms.

There are many sources of magnetic anisotropy in

solids; for a description (see Magnetism in Solids:

General Introduction; Localized 4f and 5f Moments:

Magnetism; Transition Metal Oxides: Magnetism;

and Density Functional Theory: Magnetism). All

sources of anisotropy are fundamentally linked to

the structure of the material, and therefore must have

the symmetry of that structure. However, the sym-

metry as measured, for instance, by x rays is not

necessarily the same as the local symmetry, and some

substantial and technologically important anisotropy

is found in ﬁlms that superﬁcially seem to be of a

different symmetry. In this case, the material must

have a subtle structural anisotropy that is detectable

only by the most careful measurements, if at all.

Much effort has been devoted to creating anisotropy

in ﬁlms by deliberating making materials whose

structure has the desired symmetry, such as multi-

layers or hexagonal or tetragonal structures with

c-axes either perpendicular to or along a particular

in-plane direction. However, the growth process itself

creates in many cases quite a large perpendicular or

in-plane axis anisotropy even in amorphous and in

structurally cubic materials.

While there can be a wide variety of symmetries,

the most important for technologically interesting

ﬁlms are uniaxial and cubic. In all cases, we speak of

an easy axis or axes along which the magnetization

lies in zero ﬁeld; it is also possible to have an easy

plane of magnetization, which is equivalent to a hard

magnetic axis for which the energy would be high if

the magnetization were forced in that direction. Films

of a wide variety of materials, as well as bulk hexa-

gonal and tetragonal materials, can be approximated

quite well with a uniaxial anisotropy. The anisotropy

energy density E

in a uniaxial material is written as

¼ K

sin

y þ K

sin

y þ y ð1Þ

where y is the angle between the magnetization and the

easy axis and K

is the anisotropy constant for the

material in ergs cm

3

(J m

3

), and depends on tem-

perature. Often the higher order terms (K

) can be ne-

glected. Figure 1 shows classical magnetic hysteresis

curves M(H) for uniaxial magnetic ﬁlms, one with

o0 and one with K

40 (and substantial coercivity).

For a ﬁlm with a perpendicular easy axis, y would

be measured with respect to the perpendicular and K

would be40. In zero applied ﬁeld, for a uniaxial

material, if there were no dipolar coupling, the sam-

ple would be completely magnetized with M lying

along the perpendicular, either up or down. The long-

range nature of magnetic dipolar coupling causes a

mixture of up and down domains to form in ﬁlms

with perpendicular anisotropy in the equilibrium

state; this state is easily imaged in MFM (see Mag-

netic Force Microscopy) and has a classical stripe

domain pattern. Like any other magnetic domain

structure, hysteresis is important, so that remanent

states with a single domain structure can be created

following application of a saturating magnetic ﬁeld

(see Fig. 1(b)), essential to most technological uses

(e.g., magnetic and magneto-optic recording). Equa-

tion (1) assumes isotropy relative to other directions,

e.g., that for y ¼901, the energy is equal in all direc-

tions in plane; this is not always a good assumption.

The anisotropy energy density in a material with

cubic symmetry requires two constants and more

Figure 1

M(H) curves for ﬁlms with (a) a hard perpendicular axis, i.e., easy plane, owing to shape anisotropy; (b) easy

perpendicular axis owing to growth-induced anisotropy. M(H) is shown forH applied parallel and perpendicular to

the plane of the ﬁlm. H

¼4pM

¼2K

is shown in (a); for (b), H

480 kOe (8 T). In (b), the large coercive ﬁeld

is shown; for (a) H

¼25 Oe. The ﬁlm in (a) is f.c.c. (111) CoPt

grown at 800 1C; (b) is amorphous TbFe

ﬁlm

grown at 45 1C, both by e-beam co-evaporation. Data taken at room temperature.

425

Magnetic Films: Anisotropy

angles to describe it:

¼ K

ða

þ a

ÞþK

ða

Þþ:y ð2Þ

where K

and K

are constants for the material, de-

pendent on temperature, and a

are the cosines of the

angles M makes with the crystal axes. Often, K

E0.

In this case, if K

40 (as in bulk b.c.c. iron), the set of

/100S directions are the easy axes, the set of /111S

directions are the hard axes, and the /110S are

intermediate. If K

o0 (as in bulk f.c.c. nickel), the

/111S are easy directions and the /100S hard. When

a 0, the easy axes depend on the K

ratio.

1. Anisotropy in Films

Starting from the basic principles of anisotropy, it is

possible to distinguish a number of different contri-

butions in ﬁlms, at least theoretically. In addition

to the magnetocrystalline anisotropy expected from

measurements on equivalent bulk materials, there are

a number of contributions that are unique to ﬁlms.

The simplest to understand is shape anisotropy,

which is due to the difference in demagnetization

factor in the directions parallel and perpendicular to

the ﬁlm plane. This contribution causes a hard per-

pendicular axis (an easy plane). Other contributions

result from the modiﬁed nature of the atomic ar-

rangements in ﬁlms. The broken/altered bonds at the

top and bottom surfaces of a ﬁlm or interface in a

multilayer cause a change in the crystal ﬁelds and

dipole energies of the atoms/ions at these surfaces

and interfaces.

These interfaces will usually result in uniaxial

anisotropy, which may have K

greater or less than

0 (easy perpendicular axis or easy plane). In addition,

surfaces and interfaces may cause reconstructions

and relaxations of the atomic arrangement such that

the atomic structure at top and/or bottom may be

quite different from a simple termination of the bulk

structure, leading to additional anisotropy. Strain in

a material due to intrinsic growth phenomena, epi-

taxy, or differential thermal contraction will generally

give rise to a uniaxial anisotropy.

Experimentally, it is not always easy to separate

these contributions. For ﬁlms with uniaxial anisot-

ropy (the most common), K

is usually written as a

sum of the various contributions (discussed in more

detail below):

¼2pM

þ K

=t ð3Þ

where 2pM

is the shape contribution, K

is the

stress-induced contribution, K

the growth-induced

or intrinsic magnetocrystalline term for uniaxial sym-

metry materials, and K

/t due to the surfaces, where

t is the ﬁlm or layer thickness. If cubic terms are

needed, or if the angles about which the uniaxial an-

isotropy occurs for each contribution (e.g., growth

induced terms can be tilted away from the ﬁlm nor-

mal due to incident atomic beam effects, while shape

anisotropy is strictly relative to the ﬁlm plane), care

must be taken to add these contributions more care-

fully (see, e.g., Hellman et al. 1989).

In addition, many magnetic materials of interest

are multi-component alloys, amorphous, or nano-

crystalline composites. In such materials, local mag-

netic anisotropy may be large but is expected to be

randomly oriented at different sites, possibly giving

rise to coercivity but no net macroscopic anisotropy.

However, for ﬁlms made from these materials, strain

or local chemical or structural ordering can lead to an

alignment of the local anisotropy axes, causing large

macroscopic anisotropy. The source of this seemingly

extrinsic anisotropy may be quite fundamental: e.g.,

the growth process of a ﬁlm, or the epitaxial strain of

the magnetic layer on a substrate.

1.1 Shape Anisotropy

The source of shape anisotropy lies in the difference

in demagnetization factors in the directions parallel

and perpendicular to the ﬁlm plane; this energy is

fundamentally a summation of the magnetic dipole–

dipole coupling of all the moments in the ﬁlm. For a

ﬁlm whose thickness is much less than its other di-

mensions, shape anisotropy gives an easy plane,

which is equivalent to a hard perpendicular axis:

¼2pM

(m

/2 in SI units) for ﬁlms with

thickness that is small compared to their other two

dimensions. Uniaxial shape anisotropy (or other ﬁlm-

related uniaxial contributions) often dominate an

intrinsic cubic magnetocrystalline anisotropy, as in

Fig. 1(a) for f.c.c. CoPt

. There is also a small con-

tribution for antiferromagnets; here the dipolar in-

teraction causes the moments to align side-by-side,

which, summed up over the moments in a thin ﬁlm,

means K

40. This K

is however small (compared to

2pM

where M

refers to a sublattice magnetization)

and is in essence a surface contribution, hence it

varies inversely with ﬁlm thickness.

Columnar microstructures are often found in ﬁlms.

It is important to note that unless the areal density of

columns in a ﬁlm is extremely low (less than 0.5 ﬁll-

ing), the shape anisotropy of these columns alone will

not produce a net perpendicular anisotropy (Yafet

et al. 1986). However, effects such as oxidation and

the broken/altered bonds at the surfaces of these col-

umns, like the surfaces and interfaces of ﬁlms dis-

cussed below, could lead to anisotropy via the other

mechanisms described in this article.

1.2 Intrinsic Magnetocrystalline Anisotropy

If the lattice constant and structure of a material

is the same for a ﬁlm as for a bulk material, the in-

trinsic magnetocrystalline anisotropy is the same. By

426

Magnetic Fi lms: Anisotropy

manipulating the growth conditions, different types

of anisotropy can be created, even for the same crys-

tal structure. Examples of this are growth of h.c.p.

cobalt with its c axis perpendicular, or using epitaxial

relationships to cause the c-axis of tetragonal CoPt or

FePt to grow along a particular in-plane direction

(Coffey et al. 1995, Farrow et al. 1996). In each case,

the easy magnetic axis will lie along the c-axis, but the

net effect will be perpendicular or in-plane anisotro-

py. This contribution to anisotropy is independent of

thickness as long as the crystal quality is maintained

and strain is not a factor.

1.3 Strain-induced Anisotropy

In general, ﬁlms prepared by vapor deposition proc-

esses are not structurally identical to their bulk coun-

terparts. It is of course possible to create structures not

observed in bulk materials, e.g., f.c.c. cobalt, which

then have the symmetry of this new material. But even

if the structure is nominally the same as a known bulk

material, vapor deposition produces strains and de-

fects that inﬂuence the magnetic properties. There are

many sources of strain in vapor-deposited materials,

discussed more completely in Stress Coupled Phenom-

ena: Magnetostriction and Magnetoelasticity in Nano-

scale Heterogenous Materials. Differential thermal

contraction and epitaxy with a substrate of a differ-

ent lattice constant are sources of stress produced by

the substrate (or by an overlayer, although these are

usually thin and hence produce less strain in the ﬁlm

than the substrate). There are also a number of sources

of stress intrinsic to the growth process, e.g., bom-

bardment by energetic ions/atoms in sputtering tends

to produce compressive strain; early stage coalescence

phenomena tend to produce tensile strain. These types

of strain tend to be uniaxial relative to the ﬁlm plane

and can be either compressive or tensile.

The link between strain and anisotropy is through

magnetostriction. Magnetostriction is fundamentally

linked to spin-orbit coupling and to crystal ﬁelds as

discussed in Magnetism in Solids: General Introduc-

tion and Electron Systems: Strong Correlations.

Strain could in fact be regarded to result in anisot-

ropy by simply being magnetocrystalline anisotropy

but for an e.g., tetragonal structure instead of the

intrinsic cubic structure. However, it is not usually

calculated in this way, in part because the strain and

hence the tetragonal distortion is typically a few parts

in 10

. Instead it is treated as a separate contribution

to the anisotropy. A uniaxial strain in an isotropic

(or nearly isotropic) material results in a uniaxial

contribution to the anisotropy:

¼3ls=2 ð4Þ

where l is the magnetostrictive constant and s is the

strain. The sign convention for strain is s40 is ten-

sile, and so0 is compressive.

For materials with positive magnetostrictive con-

stant l, this means compressive strain causes a per-

pendicular easy axis. For cubic materials, or even

hexagonal materials since the plane cannot be assumed

to be isotropic, the anisotropy produced by various

types of strain is more complex (see, e.g., Cullity 1972,

p. 250 forward). Magnetostriction is typically meas-

ured by measuring the lattice distortion on applying

a magnetic ﬁeld, which eliminates domains. Since

most ﬁlms are on substrates, this is most easily done

by measuring deﬂection (i.e., bending) of the subst-

rate/ﬁlm combination on magnetizing the ﬁlm. In this

case, a knowledge of the mechanical properties of

substrate and ﬁlm (specifically, Young’s modulus

and Poisson’s ratio) is needed to make quantitative

calculations.

An example of strain-induced anisotropy in a ﬁlm

of an otherwise cubic material is strained Laves phase

TbFe

, which may be useful for giant magneto-

strictive devices (Wang et al. 1996).

1.4 Growth-induced Anisotropy

This contribution fundamentally comes from the

same sources as magnetocrystalline anisotropy but

arises in structures where the underlying symmetry is

apparently cubic or isotropic. It is a separate contri-

bution from strain-induced anisotropy, and can be

identiﬁed in ﬁlms with extremely low strain, or where

the sign of the strain can be changed with no signif-

icant change in anisotropy. It originates intrinsically

from the growth process producing a subtle aniso-

tropy in short-range order, which can be either struc-

tural or chemical or both. The structural or chemical

anisotropy in turn produces a magnetic anisotropy,

either via crystal ﬁeld effects (often known as single-

ion anisotropy) or via dipolar coupling (sometimes

called pair anisotropy). The latter is generally signif-

icant only when the crystal ﬁeld contribution is small,

e.g., for L ¼0 ions such as gadolinium.

The current magneto-optic recording media, qua-

ternary alloys related to amorphous Tb–Fe, rely

completely on this source of anisotropy to give a

perpendicular easy axis, as in Fig. 1(b) (see, e.g.,

Hellman and Gyorgy 1992 and references given

there). Seemingly cubic (f.c.c.) Co–Pt alloys grown

by vapor co-deposition at deposition temperatures

near 400 1C have anisotropies nearly 10% of that

found in the best Co/Pt multilayers; this effect has

been attributed to growth-induced cobalt clustering

into platelets within a platinum-rich matrix, much

like an incoherent local multilayer (see, e.g., Weller

et al. 1992, Shapiro et al. 1999, and references there-

in). For both a-Tb–Fe and f.c.c. Co–Pt, EXAFS has

shown signs of anisotropic local structure (e.g.,

Harris et al. 1992, Tyson et al. 1996). The character-

istic feature of this source of anisotropy is that it

is produced by the vapor deposition process and

427

Magnetic Films: Anisotropy

vanishes upon annealing of the material, even at

temperatures that produce no obvious structural

change. It is largely deduced by experimentally rul-

ing out other sources of anisotropy, e.g., strain, sur-

face, and intrinsic magnetocrystalline anisotropies.

The growth process can break the symmetry of the

structure by several quite distinct means. First, there

can be a chemical or structural memory of the inci-

dent atomic beam directions. This effect is dominant

at low growth temperatures, where growth phenom-

ena such as kinetic roughening are important. Co-

lumnar microstructure tilted at an angle intermediate

to the incident beam direction and the ﬁlm normal

are an obvious structural manifestation of this effect;

such shadowing-related effects are much studied in

growth simulations. Even in materials where this ob-

vious structure is not seen, the magnetic anisotropy

can still reﬂect the effect of the incident beam angle

(e.g., Hellman et al. 1987). This effect of the incident

beams on anisotropy (and in turn on coercivity) is

quite pervasive, and is seen in nearly all alloys, par-

ticularly those grown at room temperature and be-

low. In some cases, it may be used to advantage, by

giving a preferred in-plane axis, but it can also be

detrimental to desired low coercivity materials.

At higher growth temperature, adatom surface

mobility typically erases this memory of the incident

beam direction(s). However, an even stronger source

of growth-induced anisotropy occurs in this growth

regime. Here, at intermediate growth temperatures,

surface adatom mobility is extremely high compared

to bulk mobility (typical root mean square diffusion

lengths are tens to hundreds of nanometers during

the lifetime of the adatom before it is buried by the

next layer of adatoms at typical growth rates, while

typical r.m.s. diffusion lengths for an atom once bur-

ied in the bulk of the ﬁlm for the duration of a typical

deposition are less than an atomic spacing). As a

result, in this intermediate growth regime, an as-

deposited ﬁlm is really a sequence of surface layers,

trapped in. At still higher temperatures, typically

above approximately one-third of the materials’

melting temperature T

, bulk atomic diffusion be-

comes important and the equilibrium structures are

found. The result (shown in Fig. 2) is that for both

amorphous Tb–Fe and crystalline f.c.c. Co–Pt ﬁlms,

anisotropy increases with increasing growth temper-

ature up to the onset of appreciable bulk atomic

mobility (Hellman et al. 1999).

Defects also affect the anisotropy, although if they

are point defects randomly distributed, their effect

will not add up to a net macroscopic anisotropy.

However, the local anisotropy associated with stack-

ing faults, dislocation layers, and internal interfaces

in a columnar growth structure can, at least in theory,

produce macroscopic anisotropy.

1.5 Surfaces of Ultrathin Films

The broken bonds of the surface of the ﬁlm (and the

bottom of the ﬁlm, next to the substrate) lead to a

uniaxial anisotropy which can be of either sign, as

originally described by Ne

el in 1954. This topic is

discussed in Monolayer Films: Magnetism, and will be

touched on only briefly here. This contribution arises

fundamentally from the same sources as cause bulk

magnetocrystalline anisotropy. A good example is

that f.c.c. cobalt ﬁlms a few monolayers thick have a

perpendicular easy axis, which drops into the plane as

Figure 2

Dependence of growth-induced anisotropy K

(shape contribution 2pM

removed) on growth temperature for (a)

f.c.c. (as measured by x ray and TEM) CoPt

and (b) amorphous TbFe

, prepared under various growth conditions

(e.g., different epitaxial orientations, sputtered, e-beam co-evaporated, under tensile or compressive strain as grown).

In (a), annealing at 450 1C eliminates K

for all samples; in (b), annealing eliminates K

, but higher annealing

temperatures are required for ﬁlms grown at higher temperatures (see Hellman et al. 1999, Hellman and Gyorgy 1992,

and Shapiro et al. 1999 for details). Data taken at room temperature.

428

Magnetic Fi lms: Anisotropy

the ﬁlm thickness increases. This anisotropy is ex-

tremely dependent on the details of the surface struc-

ture, and hence is affected by impurity adsorption,

surface reconstruction, relaxation, segregation, and

other properties well known to surface scientists. To

the extent that the ﬁlm structure can be assumed to

remain constant, independent of thickness, this con-

tribution to the total anisotropy will fall off as inverse

thickness, i.e., K

¼K

/t where K

is a surface aniso-

tropy in ergs cm

2

(J m

2

) and t is the ﬁlm thickness.

1.6 Interfaces in Multilayers and

Nanocrystalline Films

This contribution is fundamentally the same as the

preceding one for surfaces, except that instead of

broken bonds at a vacuum interface, there are bonds

to a neighboring layer. The anisotropy in multilayers

is generally uniaxial, often quite sizable, and can be of

technological significance (e.g., optimized Co/Pt mul-

tilayers of particular thickness for each layer have

42 10

ergs cm

3

(2 10

3

), making them

possible future perpendicular recording or magneto-

optic recording media; see, e.g., Weller et al. 1993).

Figure 3 shows the dependence of anisotropy on co-

balt layer thickness for different orientations. In gen-

eral, there are contributions both from the strain

induced by the lattice mismatch of the two (or more)

materials, and from an intrinsic interface contribu-

tion, as well as the intrinsic magnetocrystalline an-

isotropy, which may also be uniaxial. In principle,

these can be separated by measuring the dependence

of the anisotropy on layer thickness, as in Fig. 3, since

the interface contribution to the total anisotropy

should drop off as inverse thickness, while the others

may remain constant. However, there are dangers in

the interpretation of this data since both strain and

interface structure may depend on thickness (see

Monolayer Films: Magnetism).

In nanocrystalline ﬁlms, internal interfaces be-

tween magnetic and nonmagnetic (or other magnetic)

materials can also lead to local anisotropy, although

it will tend to be randomly oriented and therefore not

lead to a net macroscopic anisotropy. However, any

alignment of the nanocrystal axes by growth effects

would lead to net anisotropy.

1.7 Unidirectional Anisotropy

In principle, the lowest order symmetry possible for

anisotropy is uniaxial, which has 1801 symmetry. It

was however realized many years ago that by taking

advantage of exchange coupling between the magnetic

material of interest and an antiferromagnet with sub-

stantial anisotropy, such as in a bilayer or multilayer,

a unidirectional anisotropy could be created (see, e.g.,

review by Nogues and Schuller 1999). This effect is

technologically important for biasing of magnetore-

sistive read heads or other sensors. The fundamental

source of this symmetry breaking is the alignment of

the antiferromagnet’s Ne

el vector, most commonly by

cooling through the Ne

el temperature of the anti-

ferromagnet while applying a ﬁeld sufﬁcient to align

the ferromagnet. Exchange coupling then forces the

el vector and hence one sublattice to lie along a

direction set by the cooling ﬁeld. Subsequent appli-

cation of a ﬁeld in another direction is insufﬁcient to

reverse the antiferromagnet, owing to its high mag-

netocrystalline anisotropy and low net moment.

The ferromagnet is exchange coupled to the anti-

ferromagnet and hence does not reverse in low ﬁelds.

The detailed nature of this exchange biasing becomes

Figure 3

(a) K

(i.e., K

with shape contribution 2pM

removed) and (b) d

K

for epitaxial and sputtered Co/Pt

multilayers versus cobalt layer thickness d

. Data taken at room temperature (reproduced by permission of the

Materials Research Society from MRS Symp. Proc., 1993, 313, 791–7).

429

Magnetic Films: Anisotropy