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

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commonly used AMR material for applications.

According to this situation, industrial research and

development aims to optimize the geometry and

readout technique of AMR devices instead of search-

ing for more efﬁcient AMR materials (Eijkel and

Fluitman 1990).

4. Phenomenological Theories

Transport phenomena in magnetic materials are in

general discussed in the framework of two simplifying

models (Rossiter 1987). The ﬁrst is Matthiessen’s rule

that states that thermal and impurity scattering con-

tribute independently to the electrical resistivity (see

Boltzmann Equation and Scattering Mechanisms). The

two-current model of Mott assumes that the scatter-

ing of electrons at impurities in a magnetized sample

does not change the electron spin orientation relative

to the magnetization, and that the scattering is dif-

ferent for majority and minority spin electrons. This

implies that there are two independent currents in a

magnetic material with two corresponding subband

resistivities, r

and r

, that add up to the total

resistivity according to

ð9Þ

This approach clearly ignores spin-mixing, e.g.,

due to ﬁnite temperature magnon scattering. This

effect is sometimes considered by introducing a

parameter r

that represents the rate of spin-ﬂip

transitions leading to (Fert and Campbell 1968)

r ¼

þ r

ðr

þ r

þ 4r

ð10Þ

The AMR originates from the different scattering

rate of electrons that travel parallel or perpendicular

to the direction of the magnetization. This is ascribed

in general primarily to the spin–orbit coupling.

Accordingly, it has been suggested that the AMR

ratio can be expressed by (Campbell 1970)

¼ gða  1Þ with a ¼

ð11Þ

where g, primarily determined by spin–orbit coupling,

describes the anisotropy of the spin-resolved resist-

ivities according to

¼ r

þ gr

; r

¼ r

þ gr

ð12Þ

In practice one determines the subband resistivities

m(k)

and their ratio a indirectly by measuring devi-

ations from Matthiessen’s rule for the temperature

dependence of the resistivity or for ternary alloys.

The applicability of this model is shown in Fig. 3.

Here experimental AMR ratios for various dilute

nickel alloys are plotted as a function of experimental

values for a. Obviously, the data approximately lie on

a straight line given by Eqn. (11), if g is chosen ap-

propriately. However, for alloys for which a is close

to unity, pronounced deviations from the behavior

shown in Fig. 3 occur. Moreover, ﬁrst-principles cal-

culations have shown that the measured a is greatly

reduced in comparison with values calculated on the

basis of the two-current model because spin-mixing

mechanisms are always present in real materials.

These have been sometimes accounted for by using

an extended version of Eqn. (11) (Campbell et al.

1970):

¼ g

ðr

 r

Þr

þ r

ðr

þ r

ð13Þ

5. Quantitative Models

The ﬁrst step towards a quantitative description of

AMR was the combination of the model described

above with a calculation of the residual electrical re-

sistivity of disordered alloys on the basis of Mott’s

two-current model (Akai 1977). This questionable

approximation could be avoided by the development

of a fully relativistic description for the AMR effect

(Banhart and Ebert 1995). The starting point of this

approach is the formulation of the conductivity ten-

sor as a current–current–correlation function within

the Kubo–Greenwood response formalism (Edwards

1958):

cryst

Tr/j

ImG

ðE

Þj

ImG

ðE

ÞS

conf

ð14Þ

Figure 3

AMR ratio for various dilute nickel alloys NiX, with

X ¼Pd, Au, Cu, Fe, Co (Campbell et al. 1970).

820

Magnetoresistance, Anisotropic

A similar but somewhat more complex expression

can be given for the off-diagonal elements s

.In

Eqn. (14) the underlying electronic structure of the

system is described by the retarded electronic Green’s

function G

) for the Fermi energy E

(Butler

1985). In practice, this is calculated by means of

multiple scattering theory.

Model calculations based on Eqn. (14) with the

spin–orbit coupling manipulated reveal the role of this

relativistic effect for AMR. Its major effect is to mix

the minority and majority spin systems. As a conse-

quence the two-current model in its strict sense is

obviously not applicable. For the isotropic resistivity 

it nevertheless gives reasonable results if the spin

polarization at the Fermi level is not too high. This

applies, for example, for the alloy systems Co

1x

and Co

1x

, while for Fe

1x

inclusion of the

spin–orbit coupling increases



 by up to an order of

magnitude compared to a calculation based on the

two-current model.

The model calculations also demonstrate that the

AMR effect is indeed caused by the spin–orbit cou-

pling with AMR increasing quadratically with its

strength. In addition it is found that the spin–orbit

coupling causes AMR nearly exclusively via the mix-

ing of the spin systems. This implies in particular that

contributions to the AMR due to the spin-diagonal

part of the spin–orbit coupling can be neglected.

Use of the formalism described above leads for the

conductivity tensor automatically to the structure

given by Eqn. (1) that is required from symmetry

considerations. As an example of its application,

Fig. 4 shows results for the AMR ratio obtained for

the alloy system Co

1x

and Co

1x

(without

manipulating the spin–orbit coupling). The satisfying

agreement implies that other possible sources of

AMR, e.g., magnon scattering processes, can be ne-

glected. Equation (14) has been exploited only to ac-

count for the residual resistivity caused by the

chemical disorder in alloys. However, it also allows

the incorporation of other mechanisms and supplies

an appropriate basis to deal with the temperature

dependency of AMR.

6. Summary

The AMR effect is a galvanomagnetic effect, which

can be observed in any spontaneously magnetized

material. Most experimental work that aimed to opt-

imize the AMR ratio of materials applied in sensor

and storage technology has been focused on transi-

tion metal alloys. For these systems, values for the

AMR ratio of up to 30% have been found at low

temperatures, with only a few percent remaining at

room temperature.

A phenomenological description of the AMR ef-

fect is obtained by assuming an anisotropy in the

transport properties, represented by a corresponding

conductivity tensor. This anisotropy can primarily be

ascribed to the presence of the spin–orbit coupling

and the magnetic ordering. Starting from this, mi-

croscopic descriptions of the AMR effect have been

developed. Most of this theoretical work has been

based on Mott’s two-current model together with the

use of some parameters. Modern ab initio band

structure calculations allow parameter-free calcula-

tions of AMR. Corresponding investigations essen-

tially conﬁrm the ideas of the previous more

phenomenological work and supply a very detailed

understanding for the physical mechanisms giving

rise to AMR.

See also: Amorphous Intermetallic Alloys: Resistivity;

Giant Magnetoresistance; Giant Magnetoresistance:

Metamagnetic Transitions in Metallic Antiferromag-

nets; Magnetoresistance: Magnetic and Nonmagnetic

Intermetallics

Bibliography

Akai H 1977 Physica B 86–88, 539–40

Banhart J, Ebert H 1995 Europhys. Lett. 32 , 517–22

Becker R, Do

ring W 1939 Ferromagnetismus. Springer-Verlag,

Berlin

Butherus A D, Nakahara S 1985 IEEE Trans. Magn. 21, 1301–5

Butler W H 1985 Phys. Rev. B 31, 3260–77

Campbell I A 1970 Phys. Rev. Lett. 24, 269–71

Campbell I A, Fert A, Jaoul O 1970 Physica C 3, S95–101

Cracknell A P 1973 Phys. Rev. B 7, 2145–54

Dorleijn J W F 1976 Philips Res. Rep. 31, 287–410

Ebert H, Banhart J, Vernes A 1996 Phys. Rev. B 54, 8479–86

Edwards S F 1958 Phil. Mag. 3, 1020–31

Figure 4

AMR ratio Dr/



 for the alloy systems Co

1x

and

1x

at T ¼0 K calculated by means of Eqn. (14)

(ﬁlled symbols) and compared to experimental data

(Ebert et al. 1996).

821

Magnetoresistance, Anisotropic

Eijkel K J M, Fluitman J H J 1990 IEEE Trans. Magn. 26,

311–21

van Elst H C 1959 Physica 25, 708–20

Fert A, Campbell I A 1968 Phys. Rev. Lett. 21, 1190–2

Freitas P P, Plaskett T S 1990 Phys. Rev. Lett. 64, 2184–7

pel W, Hesse J, Zemel J N (eds.) 1989 Magnetic Sensors.

VCH, Weinheim, Germany

Karplus R, Luttinger J M 1954 Phys. Rev. 95, 1154–60

Kleiner W H 1966 Phys. Rev. 142, 318–26

McGuire T R, Potter R I 1975 IEEE Trans. Magn. 11, 1018–38

Rossiter P L 1987 The Electrical Resistivity of Metals and

Alloys. Cambridge University Press, Cambridge, UK

Smit J 1951 Physica 16, 612–27

Smit J 1955 Physica 21, 877–87

Wijn H P J (ed.) 1991 Magnetic Properties of Metals d-Ele-

ments, Alloys and Compounds. Springer-Verlag, Berlin

H. Ebert and A. Vernes

Universita

tMu

nchen, Munich, Germany

J. Banhart

Fraunhofer-Institut fu

r Angewandte

Materialforschung, Bremen, Germany

Magnetoresistance: Magnetic and

Nonmagnetic Intermetallics

The theory of the magnetoresistance of metals is

concerned with the description of the motion of an

electron under the combined inﬂuence of externally

applied electric and magnetic ﬁelds (assuming ther-

mal equilibrium, i.e., rT ¼0) in the internal periodic

potential owing to the ionic lattice. As is discussed in

Boltzmann Equation and Scattering Mechanisms,an

external magnetic ﬁeld inﬂuences the transport phe-

nomena but does not contribute to a current in the

same way as an electric ﬁeld.

The Boltzmann theory will be used for the discus-

sion of the general tendencies of the magnetoresistance.

Ohm’s law for the electrical transport in metals in

magnetic ﬁelds is given by

¼ s

ðT; B

ÞE

ð1Þ

where I

is the electrical current density, E

the electric

ﬁeld, and s

the temperature and magnetic ﬁeld-

dependent conductivity tensor. In the absence of an

external magnetic ﬁeld the conductivity becomes a

scalar for a cubic lattice symmetry. If a magnetic ﬁeld

is applied this symmetry is broken, but the Onsager

relationships still hold:

ðB

Þ¼s

ð B

Þð2Þ

The conductivity tensor, s

ðB

Þ, for an isotropic

(cubic) material with B

¼ð0; 0; Be

Þ is given below

from which the corresponding resistivity tensor,

ðBÞ, follows:

ðBÞ¼

ðBÞ s

ðBÞ 0

s

ðBÞ s

ðBÞ 0

00s

ðBÞ

ðBÞ¼

ðBÞ r

ðBÞ 0

r

ðBÞ r

ðBÞ 0

00r

ðBÞ

ð3Þ

where r

and r

are the resistivity in a transverse and

a longitudinal external magnetic ﬁeld, respectively,

and r

is the Hall resistivity.

The transverse and the longitudinal magnetoresist-

ance are deﬁned by (respectively):

ðT; BÞrðT; B ¼ 0Þ

rðT; B ¼ 0Þ

and

ðT; BÞrðT; B ¼ 0Þ

rðT; B ¼ 0Þ

ð4Þ

1. Magnetoresistance in the Scope of the

Boltzmann Formalism

In accordance with Boltzmann Equation and Scatter-

ing Mechanisms we assume that the electrons, moving

through the periodic array of the ions in the lattice,

can be described by Bloch waves. The semiclassical

description of the movement of these packets is given

by (see Eqn. (6) of Boltzmann Equation and Scattering

Mechanisms):

ðk

Þ;

d k

¼þ

½E

þðv

 B

Þ ð5Þ

where E

and B

are the electric and the magnetic

ﬁelds, respectively, and v

is the velocity of a con-

duction electron.

We are looking for a distribution function as a

solution of the Boltzmann equation under the simul-

taneous inﬂuence of an electric and a magnetic ﬁeld.

The Boltzmann equation can be written as:

½E

þðv

 B

Þ  r

f þ v

r

f ¼



ð6Þ

The term on the right-hand side of Eqn. (6) is the

collision term, which is given by:



f ðg

Þ½1  f ðgÞP

-g



 f ðgÞ½1  f ðg

ÞP

g-g



ð7Þ

where P

-g

is the quantum mechanical transition

rate of an electron per unit time from state g

to state

822

Magnetoresistance: Magnetic and Nonmagnetic Intermetall ics

g. Focusing on stationary solutions only and consid-

ering small deviations from the equilibrium, a new

function, F

, is then deﬁned through:

f ¼ f

þ f

with f

¼



ð8Þ

Equation (6) can be linearized, which then reads (see

also Eqn. (36) of Boltzmann Equation and Scattering

Mechanisms):

½C þ OF

¼e



 E

ð9Þ

where C is the collision operator and O the magnetic

operator. These operators are deﬁned by (respectively):

C F

½F

 F

ð10Þ

¼

ðv

 B

Þr

ð11Þ

where

¼ f

ðg

Þ½1  f

ðgÞP

-g

¼ W

ð12Þ

is the symmetric equilibrium transition probability in

ﬁrst order.

1.1 Solution of the Boltzmann Equation

A formal solution for the function F

of the line-

arized Boltzmann equation (Eqn. (9)) is given by:

¼e½C þ O

1



 E

ð13Þ

For the current density it follows that:

¼e



¼e



½C þ O

1





 E

¼ s

E

ð14Þ

where e is the charge, v

the drift velocity of

the conduction electrons, and s

is the conductivity

tensor.

1.2 Relaxation Time Approximation

In the scope of this concept the collision term is re-

placed by a function that is proportional to f

. This

function is a measure of the deviation from the ther-

mal equilibrium in the conduction electron distribu-

tion weighted by the inverse relaxation time, t



¼

f  f

¼

ð15Þ

which is equivalent to setting the collision operator of

Eqn. (10) to:



ð16Þ

The relaxation time, t

, can be anisotropic in k-space

and different for each band.

2. Normal Magnetoresistance

The effect of an external magnetic ﬁeld on the elec-

trical resistivity can be considered as consisting of

two parts. The ﬁrst originates from the inﬂuence of

the magnetic ﬁeld on the conduction electron trajec-

tories. In between collision events the Lorentz force

deﬂects the electrons on their way through the spec-

imen. This mechanism always increases the resistivity

with the ﬁeld. In the following it will be called ‘‘nor-

mal magnetoresistance.’’ In practice one distinguishes

between two limiting cases known as the ‘‘low’’- and

the ‘‘high’’-ﬁeld limit. Under usual experimental con-

ditions with polycrystalline sample material one is

always in the low-ﬁeld limit where the conduction

electron traverses a path in a plane perpendicular to

the B

-ﬁeld and completes only a small arc before

being scattered into another state in k-space. In order

to provide an example of the inﬂuence of a magnetic

ﬁeld on the resistivity, r(T) curves of YAl

at differ-

ent magnetic ﬁelds are shown in Fig. 1. In inset (a)

the temperature dependence of the resistivity at low

temperatures in different external ﬁelds is given. Inset

(b) shows the ﬁeld dependence of the resistivity at

different temperatures.

Figure 1

Temperature dependence of the resistivity of YAl

from

4 K to 300 K in transverse external magnetic ﬁelds of

various strengths. Inset (a) shows details of the r(B,T)

curves at low temperatures. Inset (b) shows the ﬁeld

dependence of the resistivity for different temperatures.

823

Magnetoresistance: Magnetic and Nonmagnetic Intermetallics

2.1 Kohler Rule

Before discussing the temperature variation of the

normal magnetoresistance an important and useful

rule, the Kohler rule, is discussed. This states that the

magnetoresistance can be expressed as

¼ C

rðB ¼ 0; TÞ



ð17Þ

where C is a general function characteristic of a par-

ticular sample material. Rigorously, Eqn. (17) can be

justiﬁed only if there is a single species of charge car-

rier (i.e., one conduction band model) and the relax-

ation time is the same at all points on the Fermi

surface. Fortunately the Kohler rule has been found

to be a good approximation in many cases where a

single scattering mechanism dominates. In the inset

of Fig. 2 a so-called Kohler plot of YAl

is shown,

and it is clear that there are only small deviations

from a universal curve at very low temperature rang-

es. Kohler plots for the two isostructural compounds

YAl

and YNi

have been compared (Gratz et al.

1995). In YNi

a considerable deviation from the

universal curve has been found at lower tempera-

tures for all ﬁelds, this being associated with its high

residual resistivity. This derives from the impurity

scattering, which at low temperature is the dominant

relaxation mechanism. However, as the tempera-

ture increases the phonon scattering with another

relaxation time becomes the dominant scattering

mechanism.

The temperature variation of the transverse mag-

netoresistance (Dr

/r) of YAl

is shown in Fig. 2.

YAl

is a representative example of many other non-

magnetic compounds, such as YAg, LuAg, YCu

and LuCu

, which show qualitatively a similar

temperature variation of Dr

/r as YAl

(Gratz et al.

2001). Figure 2 shows that as the temperature in-

creases the inﬂuence of the external magnetic ﬁeld on

the resistivity decreases. The increase of Dr

/r with

increasing ﬁeld strength, B, at low temperatures is

closely related to the r

value of the sample, and this

may be highlighted with an example. The value of

/r at 4.2 K and 10 T for a YAg sample (cubic

CsCl structure, r

¼1.8 mOcm) is 18%. The Dr

value for an isostructural sample where 20% of

yttrium has been replaced by lutetium (Y

0.8

0.2

Ag,

i.e., r

has been increased to 3.6 mOcm) is only about

6% at 10 T.

2.2 Temperature and Field Dependence of the

Normal Magnetoresistance

There have been several successful attempts to cal-

culate the temperature dependence of Dr

/r (e.g.,

Blatt 1968). In the following a short outline of the

model and the underlying assumptions on which such

a calculation is based are given (we make use of what

is discussed in Boltzmann Equation and Scattering

Mechanisms).

In accordance with Eqn. (36) of Boltzmann Equa-

tion and Scattering Mechanisms, we assume an exter-

nal force, X

, in the direction X

== e

with i ¼x,y,z:

ðC þ OÞf

¼ Z

¼



ð18Þ

Assuming for the moment that O ¼0, multiplying

from the left by f

, and summing over g we obtain:

C f

ð19Þ

(where the symmetry of the operator C has been

used).

The components of the conductivity tensor are

given by:

¼

VolX

ðZ

þ Z

 f

C f

Þð20Þ

It can be shown that a functional of the form:

½F¼

VolX

ðZ

þ Z

 F

C F

Þð21Þ

takes an extremal value for F ¼f and gives s

(see

Eqn. (20)). Therefore, one can determine F using the

variational principle with the functional given by

Eqn. (21). Searching for the extremal value of this

functional by varying F in a subspace one ﬁnally ob-

tains an approximation for the conductivity tensor.

The application of the variational principle is more

Figure 2

Transverse magnetoresistance of YAl

for various ﬁeld

strengths. Inset shows the Kohler plot for YAl

824

Magnetoresistance: Magnetic and Nonmagnetic Intermetall ics

difﬁcult if there exists simultaneously an external

magnetic ﬁeld. However, one can still ﬁnd a func-

tional which shows an extremal value. In this case

one has to make use of the solution of the Boltzmann

equation for the two opposite B-ﬁeld directions B

and B



(see Eqn. (18)):

ðC þ OÞf

¼



¼ Z

ðC  OÞf



¼



¼ Z

The conductivity tensor in the presence of an ex-

ternal magnetic ﬁeld is similar to that in Eqn. (20)

and is given by:

ðB

Þ¼

VolX



½Z

þ Z



 f



ðC þ OÞf

ð23Þ

Equation (23) leads to the corresponding functional

(see Eqn. (21)):

½F

; F



¼

Vol X



½Z

þ Z



 F



ðC þ OÞF



ð24Þ

We use for the trial functions the ansatz:

m¼1

; F



m¼1

ð25Þ

where v

is a velocity component and the subscripts

n and s are the band index and the spin direction,

respectively. A function depending on the parameters

a and b follows from the functional. These para-

meters are determined by:

½F



; F



¼ 0;

½F



; F



¼ 0 ð26Þ

Under the assumption of cubic symmetry and two

types of charge carriers (two-band model) and a ﬁeld

with an orientation

B ¼

B, one obtains the follow-

ing for the transverse resistivity:

ðB; T Þ¼

VolðX



½ð

Þþð

Þþ

½ð

Þþð

ÞB

()

 f½ð

Þþð

Þ

þ½ð

þð



1

ð27Þ

where the superscripts I and II denote the two bands

for which the calculation must be performed.

The following abbreviations have been used:

¼ Vol

ð2pÞ

ð28Þ

¼ Vol

ð2pÞ

ð29Þ

xyxy



xxyy

ð30Þ

irsj

¼Vol



ð31Þ

The resistivity without an external magnetic ﬁeld

(B ¼0) in a two-band model (n ¼I,II) is then given

by:

rðB ¼ 0; TÞ¼

VolðX



½ð

Þþð

Þ

ð32Þ

Accordingly, the resistivity for only one conduction

band (n ¼1,

-N) is given by:

rðB ¼ 0; TÞ¼

VolðX



ð2pÞ

ð33Þ

For the calculation of the transverse magnetore-

sistance deﬁned by Eqn. (4), we combine Eqns. (27)

and (32) and obtain (after extensive calculations, the

details of which and a critical discussion of the un-

derlying assumptions are given in Gratz et al. (2001))

the following simple relationship:

a½rðB ¼ 0; TÞ

þ bB

ð34Þ

where a and b are ﬁeld- and temperature-independent

parameters, depending on the properties of the two

different conduction bands.

2.3 Analysis of Magnetoresistance Data of

Nonmagnetic Compounds

From Eqn. (34) it follows that in order to calculate

the temperature variation of the transverse classical

magnetoresistance one simply needs to know the

temperature dependence of the resistivity in zero

magnetic ﬁeld. In order to ﬁt Dr

/r for YAl

(shown

825

Magnetoresistance: Magnetic and Nonmagnetic Intermetallics

in Fig. 2) one can use the Bloch–Gru

neisen relation-

ship (Eqn. (9) of Intermetallic Compounds: Electrical

Resistivity):

rðB ¼ 0; TÞ¼r

þ r

¼ r

þ 4R





Y=T

ðe

 1Þð1  e

x

ð35Þ

The result of a ﬁt of Eqn. (34) to the experimental

data for YAl

is given in Fig. 3 by the solid lines. The

residual resistivity of the YAl

sample is 4.4 mOcm

(see inset (a) of Fig. 1), and the two ﬁt parameters in

the Bloch–Gru

neisen law are R

¼35 mOcm (this val-

ue represents the resistivity at the Debye temperature)

and Y ¼310 K (the Debye temperature of YAl

). The

two ﬁt parameters in Eqn. (34) are a ¼7.6 T

(mOcm)

2

and b ¼1.5.

(a) Anisotropy of the normal magnetoresistance

The above calculations concern the transverse mag-

netoresistance; however, in all of the compounds

studied, there is always a smaller but nonvanishing

longitudinal magnetoresistance observed. In the inset

of Fig. 4 the longitudinal magnetoresistance (Dr

/r)

is shown. The anisotropy of the magnetoresis-

tance, deﬁned as (Dr

– Dr

)/r, of the YAl

sample

under consideration is shown in the main part of

Fig. 4.

In the case of free electrons occupying a spherical

Fermi surface, both the transverse and the longitudi-

nal magnetoresistance vanish identically. For Dr

simple arguments can be given, the transverse elec-

tric Hall ﬁeld is just sufﬁcient to compensate for the

deﬂection produced by the magnetic ﬁeld. When there

are two types of carrier, the resulting Hall ﬁeld is an

average of the two, and therefore cannot exactly

compensate for the external magnetic ﬁeld. For Dr

it can be argued that the electrons travel down the

magnetic lines of force, under the inﬂuence of the

magnetic ﬁeld, as if the magnetic ﬁeld does not exist.

It is clear from experiments that in order to account

for the nonvanishing longitudinal magnetoresistance

a more complicated theory must be developed than

that provided by Eqn. (27).

3. Magnetoresistance in Ferromagnetic

Rare-earth (RE) Compounds

The situation is even more complex in compounds

where the RE ions bear magnetic moments, since

spin-dependent scattering processes are also involved,

below as well as above the magnetic ordering tem-

perature. The resistivity of GdAl

for various trans-

verse magnetic ﬁelds is presented in Fig. 5. The

change of the r(T) curves with the magnetic ﬁeld is

typical of ferromagnetic compounds. In inset (a) of

Fig. 5, the transverse magnetoresistance of GdAl

the vicinity of the Curie temperature (T

¼168 K) is

shown.

One of the ﬁrst studies of the magnetic contribu-

tion to the resistivity using a mean ﬁeld theory was

made by Kasuya (1968). A comprehensive theoretical

treatment of the magnetoresistance in the ferromag-

netic and antiferromagnetic state was carried out by

Yamada and Takada (1973a, 1973b). In the following

we provide a brief calculation of the magnetoresist-

ance in a ferromagnet.

Figure 3

Solid lines are the result of ﬁt procedures of Eqn. (34)

(a and b are the ﬁt parameters in Eqn. (34), see text) to

the experimentally determined transverse

magnetoresistance data of YAl

(symbols).

Figure 4

Anisotropy of the normal magnetoresistance

(Dr

– Dr

)/r. Inset shows the temperature variation of

the longitudinal magnetoresistance of YAl

for various

ﬁeld strengths.

826

Magnetoresistance: Magnetic and Nonmagnetic Intermetall ics

3.1 Magnetic Scattering in the Molecular Field

Approximation

For this calculation we make use of the correspond-

ing equations in Boltzmann Equation and Scatter-

ing Mechanisms. Combining Eqns. (81a), (84a), and

(85a) in a molecular ﬁeld approximation for a RE

element with a total quantum number J, we obtain

the following expression for the resistivity owing to

spin-dependent scattering:

ðB; T Þ¼C

JðJ þ 1Þ

S  /J

Z/J

sinh



ð36Þ

where /yS denotes the thermodynamical expecta-

tion value in the molecular ﬁeld approximation. Here

S ¼ JðJ þ 1Þ/J

ScothZ ð37Þ

and

S ¼ JB

ð2JZÞð38Þ

where B

(2JZ) is the Brillouin function and Z is a

function of the reduced temperature, T/T

, given by:

Z ¼

2JðJ þ 1Þ

S þ



ð39Þ

Note that Eqn. (39) differs from Eqn. (83c) of Boltz-

mann Equation and Scattering Mechanisms since an

external magnetic ﬁeld, B, is now taken into consid-

eration.

The prefactor in Eqn. (36) is given by:

Volv

ðe

ðg  1Þ

JðJ þ 1Þð40Þ

In Fig. 7 of Intermetallic Compounds: Electrical

Resistivity the experimentally determined spin disor-

der resistivity, r

spd

( ¼C

, see Eqn. (82a) of Boltz-

mann Equation and Scattering Mechanisms), is shown

for some REAl

and REPt compounds. As can be

seen from this ﬁgure, r

spd

( ¼ r

(T4T

)) is propor-

tional to the deGennes factor, (g1)

J(J þ1), which

means that 7

and v

) are approximately the

same across these series.

By combining Eqns. (36)–(40), Dr

spd

can be

calculated, which describes the difference in the mag-

netic (spin-dependent) resistivity with and without an

external magnetic ﬁeld, normalized to r

spd

( ¼r

(T -N)). The formula obtained in this way covers

the temperature range below and above T

and can

be calculated for different external magnetic ﬁeld

strengths.

Calculated Dr

spd

vs. T curves for GdAl

(J ¼

7/2, r

spd

¼72 mOcm, T

¼168 K) for various magnet-

ic ﬁeld strengths are shown in Fig. 6.

The problem with Eqn. (39) is that a value for Z is

not easily obtained, since Z appears also in the argu-

ment of the Brilloun function (Eqn. (39) represents an

implicit function for Z). Furthermore, Eqns. (36) and

(37) include Z as sinh Z and coth Z, respectively.

For limited temperature regions below and above

and at T ¼T

, simple expressions can be obtained

Figure 5

Temperature dependence of the resistivity of GdAl

from 4 K to 300 K in transverse magnetic ﬁelds of

various strengths. Inset (a) shows details of the

transverse magnetoresistance in the vicinity of the Curie

temperature of GdAl

¼168 K). Inset (b) shows the

2/3

dependence of the transverse magnetoresistance of

GdAl

at the Curie temperature.

Figure 6

Calculated temperature dependence of the spin-

dependent magnetoresistance (Dr

r

(B,T)–r

(0,T))

for different ﬁeld strengths of GdAl

plotted as Dr

spd

vs. T (r(GdAl

)

spd

¼72 mOcm, J ¼7/2).

827

Magnetoresistance: Magnetic and Nonmagnetic Intermetallics

for Dr

spd

. There are three basic regions:

(i) Paramagnetic region for TbT

spd

¼C

ðT=T

 1Þ

ð41Þ

where

4ðJ

þ J þ 1Þ

; h ¼

B ð42Þ

and under the condition (T/T

1)bh.

(ii) Magnetically ordered region for ToT

spd

¼C

1  T=T



1=2

ð43Þ

where

2ð2J

þ 2J þ 1Þ

ð44Þ

and under the condition 1b[1–(T/T

)]

(iii) For the increase of the minimum at TDT

with increasing magnetic ﬁeld:

spd

¼C



2=3

ð45Þ

under the condition [(T/T

)1]

. From Eqn. (45)

it follows that 7Dr

spd

7 at T

increases propor-

tional to B

2/3

3.2 Analysis of the Magnetoresistance in

Ferromagnetic Compounds

A conﬁrmation of Eqn. (45) is shown in inset (b) of

Fig. 5 for GdAl

. It is found that a ﬁt of Eqn. (36) to

the experimental data for GdAl

gives good agree-

ment in the paramagnetic temperature range and in

ﬁelds up to about 6 T, and below T

( ¼168 K) down

to roughly 80 K. In order to subtract the normal

magnetoresistance from the experimental data, the

magnetoresistance of YAl

(see Fig. 2) is used, which

is normalized to the residual resistivity of GdAl

the corresponding ﬁeld strength. (The normalization

is necessary, since GdAl

and YAl

do not show the

same residual resistivity.)

Not such a good agreement is found for other

REAl

compounds, because of the following:

(i) For lower values of T

in REAl

compounds,

the correction for the normal magnetoresistance be-

comes more important. In order to demonstrate that

the normal magnetoresistance gives rise to positive

Dr/r values at low temperatures, data for other mag-

netic REAl

compounds (with R ¼Tb, Nd, or Ho)

with decreasing T

values are presented in Fig. 7.

(ii) Equation (36) is based on the molecular ﬁeld

concept, which is not valid in the spin wave region at

low temperatures.

(iii) An inﬂuence of the crystal electric ﬁeld might

also play a role in the temperature dependence of

spd

and this is not accounted for in Eqn. (36).

A comprehensive discussion of the magnetoresist-

ance in ferromagnetic iron, cobalt, and nickel com-

pounds is given by Campbell and Fert (1982).

(a) Anisotropy of the magnetoresistance in

ferromagnetic compounds

In all these compounds the anisotropy curves

((Dr

Dr

)/r vs. T) show a sharp change at the on-

set of the ferromagnetic order. It is interesting to note

that (Dr

Dr

)/r for ToT

is positive for the light

REAl

compounds and mainly negative for the heavy

REAl

compounds (Gratz et al. 2001). The problem

of the anisotropy of RE impurities in a noble metal

matrix has been attributed to hybridization of the

valence electron states of the RE impurities with

conduction electron states (Lacueva et al. 1982).

4. Magnetoresistance in Antiferromagnetic RE

Compounds

It is obvious that the situation is more complex in the

case of an ‘‘antiferromagnetic’’ spin arrangement,

since there are a very large number of possible spin

Figure 7

Temperature variation of the transverse

magnetoresistance of REAl

compounds, with

decreasing magnetic ordering temperatures and at

various ﬁeld strengths. The positive Dr

/r values

towards the low-temperatures region are caused by the

normal magnetoresistance, which is superimposed on

the negative spin-dependent magnetoresistance.

828

Magnetoresistance: Magnetic and Nonmagnetic Intermetall ics

conﬁgurations. The spin arrangements, which in ad-

dition depend on the strength (and direction; how-

ever, only in a single-crystal sample, see Elemental

Rare Earths: Magnetic Structure and Resistance,

Correlation of ) of the applied external ﬁeld, are most

important for the sign and the value of this spin-de-

pendent magnetoresistance. Furthermore, there is an

effect associated with the shape of the Fermi surface,

which alters in an antiferromagnet, since the spin or-

dering has a periodicity different from that of the lat-

tice and additional (magnetic) Brillouin zones appear

below T

. This effect is sometimes called the ‘‘super-

zone boundary effect’’ (Elliott and Wedgwood 1963).

In the paramagnetic state (T4T

) there is, in

principle, no difference to that which occurs in a

ferromagnetic material with a negative magneto-

resistance at a given temperature above T

, Dr

spd

p– B

(Eqn. (41)).

Yamada and Takada (1973a, 1973b) discussed the

ﬁeld and temperature dependence of Dr

spd

for

two simple magnetic moment conﬁgurations in the

antiferromagnetic ordered state. The assumption is

made that there are two magnetic sublattices with

opposite magnetic moments. These sublattices can

have moments aligned either parallel or perpendicu-

lar to the external B-ﬁeld. In the latter case there is no

inﬂuence of B on Dr

spd

if the moments in both

sublattices are perpendicular to B (e.g., see the trans-

verse c-axis magnetoresistance in erbium for T ¼60 K

shown in Fig. 5 of Elemental Rare Earths: Magnetic

Structure and Resistance, Correlation of ). In the

former case there is an inﬂuence on Dr

spd

, since

the external ﬁeld creates disorder in the sublattice

with the opposite orientated spin arrangement (see

the transverse a-axis magnetoresistance in erbium for

T ¼64 K in Fig. 4 of Magnetic Structure and Resist-

ance in Elemental Rare Earths, Correlation of ). Thus,

spd

is positive. If the B-ﬁeld strength (necessary

to induce a ferromagnetic spin arrangement) exceeds

the critical ﬁeld, Dr

spd

starts to decrease as in a

ferromagnet. The corresponding formula derived by

Yamada and Takada (1973a) for T oT

spd

¼ w

ðT=T

Þh

ð46Þ

where h ¼(gm

/2k

)B. The function w

includes an

important parameter a ( ¼V(0)/V(Q), where Q is the

wave vector of the antiferromagnetic spin arrange-

ment and V(0)/V(Q) describes the relative strength of

the nearest-neighbor interaction in the magnetic lat-

tice). The condition for antiferromagnetism is V(Q)4

V(0) (ao1) and for ferromagnetism V(Q)oV(0)

(a41). The calculated temperature variation of the

function w

(T/T

) for different a values (according to

Eqn. (3.16) in Yamada and Takada (1973a)) is shown

in Fig. 8. It can be seen that the discontinuity at T

increases with increasing a.

4.1 Analysis of the Magnetoresistance in

Antiferromagnetic Compounds

In order to give an example of the magnetoresistance

of an antiferromagnetic moment conﬁguration,

DyAg (T

¼56 K) will be considered. The REAg

compounds crystallize in the cubic CsCl-type struc-

ture. Neutron diffraction studies show that DyAg has

an antiferromagnetic spin arrangement of the (p,p,0)

type below T

¼46.5 K. At temperatures T

oToT

a sinusoidally modulated spin wave propagation

exists along the /110S direction, polarized in the

/001S direction (Kaneko et al. 1987). Figure 9

shows Dr

/r vs. T for DyAg, which is typical of

many of these polycrystalline cubic REAg antiferro-

magnetic compounds. The analysis of these magneto-

resistance data, using the above-outlined concept, is

shown in the following. For the determination of the

function w

(T/T

) from the Dr

/r data, the magne-

toresistance of LuAg is used to correct for the normal

magnetoresistance. The function w

thus obtained

(using Eqn. (46)) for DyAg is shown in the inset of

Fig. 9. Except for the temperature region T

oToT

where there is a sinusoidally modulated spin conﬁg-

uration, the function w

(T/T

) is basically independ-

ent of the magnetic ﬁeld strength, as it should be.

This result exempliﬁes the correspondence with the

calculation shown in Fig. 8 (especially in the para-

magnetic range).

For ﬁelds smaller than the critical ﬁeld, in many

antiferromagnetic compounds a step-like change of

Dr/r at T

has been observed. The experimental

evidence of a discontinuity can serve as a ﬁrst proof

as to whether a noncollinear spin structure appears at

the magnetic ordering temperature.

Figure 8

Temperature variation of the function w

(T/T

)

calculated for different a values and for a Neel

temperature of 50 K (according to Eqn. (3.16) in

Yamada and Takada (1973a)).

829

Magnetoresistance: Magnetic and Nonmagnetic Intermetallics