J. Phys. D: Appl. Phys. 31 (1998) 2355–2375. Printed in the UK PII: S0022-3727(98)72732-9

REVIEW ARTICLE

Magnetic alloys, their electronicstructure and micromagnetic andmicrostructural models

J B Staunton †, S S A Razee†, M F Ling ‡, D D Johnson § andF J Pinski ‖† Department of Physics, University of Warwick, Coventry CV4 7AL, UK‡ Department of Physics, Monash University, Clayton, Victoria 3168, Australia§ Department of Materials Science and Engineering, University of Illinois, Urbana,IL 61801, USA‖ Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221-0011,USA

Received 17 July 1998

Abstract. We present a selective review of electronic structure calculations forferromagnetic transition metal alloys. This is work based on the spin densityfunctional theory of the inhomogeneous electron gas which we also discuss briefly.These calculations can be used to provide estimates, from ‘first principles’, of thealloys’ characteristic properties such as the saturation magnetization, Ms , and theexchange, A, and anisotropy, K , constants as well as their Curie temperatures, Tc ,which are all important quantities for the micromagnetic modelling of thesematerials. The electronic reasons for the simple structure of Slater–Pauling curvesof Ms versus the number of valence electrons are given. Anisotropy constants, K ,can be evaluated only when relativistic effects upon the electronic motion areincluded. We review the theory of finite-temperature metallic magnetism andhighlight how the electronic structure of metals and alloys in their paramagneticstates can still exhibit a local spin polarization originating from the ‘local moment’spin fluctuations which are excited as the temperature is raised. Finally we showhow an alloy’s magnetic state can sharply influence the types of orderedarrangements that the atoms form and conversely how the type of compositionalstructure can affect Ms and K . We include a discussion of how the compositionalstructure can be described in terms of static ‘concentration waves’. We illustratethe approach by outlining our recent case studies of two iron-rich alloy systems,FeV and FeAl.

1. Introduction

Micromagnetic models of magnetic materials blendclassical electrodynamics of continuous media withfundamental aspects of condensed matter theory and arebased upon an expression for the free energy,F , of amagnet which is a functional of the magnetizationM (r).The magnetization is assumed to vary only its orientation,M (r) = Ms(αx(r), αy(r), αz(r)), Ms being the saturationmagnetization andαx, αy andαz its direction cosines. Thefree energy (in cgs units) is expressed as a sum of fourcomponents:

F [M (r)] = A∫

[(∇αx)2+ (∇αy)2+ (∇αz)2] dr

+∫Uan(αx, αy, αz) dr −

∫Happ(r) ·M (r) dr

−1

2H ′(r) ·M (r) dr (1)

namely an exchange term with exchange constantA, ananisotropy term withUan = −Kα2

z for uniaxial anisotropyand

Uan = K(α2xα

2y + α2

yα2z + α2

zα2x) (2)

for a cubic material (K are magnetocrystalline anisotropyconstants), a term describing the interaction with an appliedmagnetic field and finally a magnetostatic interactionterm in whichH ′(r) is the demagnetizing field due tomagnetostatic volume and surface charges [1–4].

Most analytical and computational effort in micromag-netics is put into dealing with the last non-local term asaccurately as possible and the materials are otherwise char-acterized by a set ofMs , A and K parameters. Thereare two important length scales involving these parame-ters, an ‘exchange length’,lex = (A/M2

s )1/2 and a domain

0022-3727/98/192355+21$19.50 c© 1998 IOP Publishing Ltd 2355

J B Staunton et al

wall thicknesslw = 2(A/K)1/2 and it is the relative order-ing of these and the dimensions of the particles, grains orlayers of the ferromagnetic material that are of crucial im-portance. In this paper we will concentrate on describinghow these parameters for magnetic alloys can be deter-mined from ‘first-principles’ condensed matter theory andhow they can be represented in terms of the materials’ elec-tronic ‘glue’. These parameters are naturally temperaturedependent and the materials’ Curie temperatures,Tc, arealso important in setting the scale of this dependence. Wewill spend some time on this finite-temperature aspect.

The properties of magnetic alloys are stronglydependent on their compositional structure.Ms , A, K andTc all vary when the composition of an alloy is alteredand the effective particle or grain size can be set by phasesegregation or growth boundaries. Moreover, the magneticstate of an alloy can also have a bearing on the type ofcompositional structure that can occur. The actual process,namely the kinetics, of ordering and/or phase segregation isitself very important for the design of advanced magneticmaterials since it concerns the structural transformationswhich occur when an alloy is aged. The different structuralstates which are formed along the transformation path areexploited to achieve special properties. Non-equilibriumsystems may undergo several transformations as they relaxtowards an equilibrium state and it is useful to gain aninsight into their dynamics. For example, transient statesmay appear along the transformation path–states which donot necessarily appear on the equilibrium phase diagram.The diffusion time of atoms in alloys can vary tremendouslyand the time spent in such metastable states can vary fromseconds to geological time scales. Such states can befound by investigating the compositional dependence ofthe free energy. This paper will also discuss how thiscan be obtained from ‘first-principles’ electronic structurecalculations.

Following a selective introduction to the theory ofmagnetism in metals we begin by discussing how thesaturation magnetization,Ms , depends on the compositionof binary alloys and then move on to describe howmagnetocrystalline anisotropy constants,K, are calculatedfor these systems. We consider the exchange constants,A, next, together with the determination of Curietemperatures,Tc, and the nature of the paramagnetic stateof ferromagnetic metals. We then discuss compositionalordering and phase segregation in alloys which can alsobe characterized in terms of their electronic structures. Wefinish the paper with case studies of two iron-rich alloysystems.

2. Metallic magnetism in brief

Magnetism and related properties in transition metal alloysare not straightforward to model owing to the electrons’itinerant nature. States that are derived from the 3d and 4satomic levels are responsible for the physical properties ofthe 3d transition metals. Since they are the more spatiallyextended (of higher principal quantum number) the 4sstates determine the metal’s overall size and compressibilitywhereas the 3d states determine the magnetic properties.

Although, in a sense, they are more localized than the4s ones, the 3d electrons still propagate throughout thematerial and the term itinerant magnetism is used. Insolids magnetism arises chiefly from electrostatic electron–electron interactions, namely the exchange interactionsand in magnetic insulators these can be described rathersimply. ‘Spin’ operators,si , can be specified by associatingelectrons appropriately with particular atomic sites so thatthe famous Heisenberg–Dirac Hamiltonian can be used todescribe the behaviour of these systems. The Hamiltoniantakes the following form:

H = −∑

Jij si · sj (3)

in which Jij is an exchange integral involving atomic sitesi andj . In metallic systems it is not possible to distributetheir itinerant electrons in this way and such simple pairwisesite–site interactions cannot be defined. Instead metallicmagnetism is a complicated many-electron effect and hasattracted significant effort over a long period to understandand describe it.

A widespread approach is to map this problem ontoone involving independent electrons moving in the fieldsset up by all the other electrons. It is this aspect whichgives rise to the spin-polarized band structure that is oftenused to explain the properties of metallic magnets suchas the non-integer values ofMs per atom in multiplesof the Bohr magnetonµB . This picture is not alwayssufficient and Herring [5], amongst others, pointed outthat certain components of metallic magnetism are morenaturally described using concepts of localized spins whichare strictly relevant only to magnetic insulators. Later onin this paper we will discuss how the two pictures havebeen combined to explain the temperature dependence ofthe magnetic properties of bulk transition metals and theiralloys and quantify the exchange constantsA.

In metals, magnetism is intricately connected with otherproperties via their spin-polarized electronic structures. Forexample, some materials exhibit a small thermal expansioncoefficient below the Curie temperature,Tc, a large forcedincrease in volume when an external magnetic field isapplied, sharp decreases of spontaneous magnetism andof Tc when pressure is applied and large changes in theelastic constants on going throughTc. These are the famous‘Invar’ effects, so called because they occur in the FCCInvar alloys Fe–Ni (65% Fe), Fe–Pd and Fe–Pt [6].

As mentioned in the introduction the compositionalorder of an alloy is often subtly linked with its magneticstate. Ni75Fe25 (Permalloy) is paramagnetic at hightemperatures, it becomes ferromagnetic at about 900 K andthen, when just 100 K cooler, it chemically orders into theLI 2 Cu3Au-like phase. When it is annealed in a magneticfield it develops directional chemical order [7]. Magneticshort-range correlations above the Curie temperatureTcand magnetic order below weaken and alter the chemicalordering in iron-rich Fe–Al alloys so that a ferromagneticFe80Al 20 alloy forms a DO3 ordered structure at lowtemperatures whereas paramagnetic Fe75Al 25 forms a B2ordered phase at comparatively higher temperatures. (See[28] for specification of these ordered binary alloy LI2, DO3

and B2 structures.) The magnetic properties of many alloys

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are sensitive to the compositional structure. For example,ordered Ni–Pt (50%) is an anti-ferromagnetic alloy [8]whereas its disordered counterpart is ferromagnetic. Wediscuss such magneto-compositional effects later on in thepaper.

Now the fundamental electrostatic interactions do notcouple the direction of magnetization to any spatialdirection (see equation (3)), and they therefore fail to give abasis for a description of magnetic anisotropic effects whichdefine the anisotropy constantsK. A description of theseeffects requires a relativistic treatment of the electrons’motions and a part of this paper is assigned to this topic inrelation to transition metal alloys.

For the two decades since the ‘landmark’ papers ofCallaway and Wang [9], there has been a consensus thatspin-polarized band theory, established within the spindensity functional (SDF) formalism (see reviews [10–13])provides a reliable description of magnetic properties oftransition metal systems at low temperatures [14–16]. (Infact, all magnetic interactions can be incorporated into thesame framework, including magnetic anisotropic effectsand magnetostatic effects, by considering the more generalrelativistic, current density functional theory.) This isessentially the modern version of the Stoner–Wohlfarththeory [17] in which the magnetic moments are assumed tooriginate from itinerant d electrons whose spins are alignedby exchange interactions. It provides a mechanism forgeneration of non-integer moments,Ms , together with aplausible account of the many-electron nature of magnetic-moment formation atT = 0 K. This theory transforms themany-electron problem into one of independent electronsmoving in effective potentialsveff (r) and magneticfields Beff (r) which themselves depend on all the otherelectrons. The consequent electronic structure is spinpolarized by the effective magnetic fields which themselvesare set up self-consistently by the magnetization densityproduced by this spin-polarized electronic structure.

A simple illustration of the spin-polarized bandstructure picture can be given by using the HubbardHamiltonian together with the Hartree–Fock approximation.The Hubbard Hamiltonian has the form

H =∑ij

(ε0δij+tij )a†i,σ aj,σ+1

2I∑i,σ

a†i,σ ai,σ a

†i,−σ ai,−σ (4)

in which a†i,σ and ai,σ are the creation and annihilation

operators for electrons with spinσ (=↑ (↓)) at a sitei,ε0 a site energy related to an atomic-like d energy level,tija hopping parameter defining the d band width andI themany-body Hubbard parameter representing the intra-siteCoulomb interactions. By making the approximation

a†i,σ ai,σ a

†i,−σ ai,−σ ' a†i,σ ai,σ 〈a†i,−σ ai,−σ 〉 (5)

where 〈a†i,σ ai,σ 〉 is the thermodynamic average of the

number operatora†i,σ ai,σ and writing the number of−σelectrons as

〈a†i,−σ ai,−σ 〉 = 12ni − 1

2µiσ

in terms of the total number of electrons and magnetizationon a sitei

ni = ni,↑ + ni,↓µi = ni,↑ − ni,↓

we can obtain the Hartree–Fock single electron Hamiltonianbelow, where the fieldsI ni and I µi are determined self-consistently from its eigenstates:

HHF =∑ij

[(ε0+ 1

2I ni − 1

2I µiσ

)δij + tij

]a†i,σ aj,σ .

(6)Evidently this approach provides a picture of aσ =↑electron ‘seeing’ a different potential associated with alattice site i from that seen by aσ =↓ electron so thatthe bands are ‘exchange split’.

When the ‘first-principles’ SDF version of this pictureis straightforwardly extended to higher temperatures, it failsseverely. For example, the Curie temperatureTc is muchtoo high, there are no ‘local’ moments aboveTc whichare indicated from neutron scattering measurements andthe paramagnetic susceptibility does not easily follow aCurie–Weiss law which is nonetheless commonly observed.Evidently the interaction between the thermally inducedspin-wave excitations together with its effect upon theunderlying electronic structure has been omitted. Weaddress this problem in the context of magnetic alloys lateron in this paper.

In the next section we show how the famous Slater–Pauling curve [18] which describes the simple relationof the average magnetic moment per site,Ms , tothe number of valence electrons can be understood interms of simple features from the alloys’ spin-polarizedelectronic structures. The ground state properties offerromagnetic, randomly disordered alloys can be describedquantitatively by SDF calculations. We then move on tothe relativistic generalization, discuss magnetic anisotropiceffects in disordered alloys and outline calculations of theiranisotropy constants,K.

3. T = 0 K magnetism in metals and alloys

Over 30 years ago, Hohenberg and Kohn [19] proved aremarkable theorem, namely that the ground state energyof a many-electron system is a unique functional of theelectron densityn(r) and is a minimum when it is evaluatedfor the true ground state density. Later Kohn and Sham[20] developed various aspects of this theorem and providedthe important basis for further practical applications of thedensity functional theory. In particular, they showed howa set of single-particle equations could be written downto include, in principle, all the effects of the correlationsamong the electrons of the system. These theorems providethe basis of the modern theory of the electronic structureof solids.

The general idea behind this scheme is to approach theground state of the many-electron problem via an effectiveone-electron picture similar in spirit to the Hartree–Fockscheme which we outlined earlier for the Hubbard model.

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J B Staunton et al

The theory proves that the relevant one-body effectivepotential can, in principle, contain all the effects of electroncorrelations although, in practice, approximations mustbe made. All the theorems and methods of the densityfunctional (DF) formalism were soon generalized [21, 22]to deal with cases in which spin-dependent properties playan important role, such as in magnetic systems (SDF). Theenergy thus becomes a functional both of the density andof the local magnetic densitym(r). The proofs of thesetheorems are provided in the original details and there havebeen many developments of a formal nature [12, 23].

The many-body effects of the complicated quantum-mechanical problem are buried in the so-called exchange-correlation part of the energy functionalExc[n(r),m(r)].The exact solution is intractable for macroscopic systemsand some approximation must be made. The so-called‘local approximation’ is the most widely used owing toits simplicity and its success in describing the ground stateand equilibrium properties of a great many different typesof materials. The local approximation (LSDA) takes asits starting point the energy of a uniformly spin-polarizedhomogeneous electron gasεxc(n(r), |m(r)|) [21, 24] sothatExc can be written in the form

Exc[n,m] ≈∫

dr n(r)εxc(n(r), |m(r)|). (7)

The functional derivative of this quantity with respect tom(r) provides the effective magnetic fields for the single-electron equations, namely the spin-polarized band structurediscussed in the introduction:

veff [n,m; r] = V ext (r)+ e2∫

dr′n(r)

|r − r′|+ δE

xc

δn(r)[n,m] (8)

Beff [n,m; r] = Bext (r)+ δExc

δm(r)[n,m] (9)

whereV ext describes an external potential such as a latticearray of nuclei andBext an external magnetic field. Theelectron density and magnetization are given by

n(r) =∫ εF

dε∑i

tr(φ∗i (r, ε)φi(r, ε)) (10)

m(r) =∫ εF

dε∑i

tr(φ∗i (r, ε)σφi(r, ε)) (11)

where theφi(r, ε) obey the Schrodinger–Pauli (Kohn–Sham) equation

(−∇2+ veff (r)1− σ ·Beff (r))φi(r, ε) = εφi(r, ε) (12)

whereεF is the system’s Fermi energy.

3.1. Elemental metals

Electronic structure (band theory) calculations using theLSDA for the pure crystalline state are routinely performedthese days. For the elemental magnetic, transition metals,nearly rigidly exchange-split, spin-polarized bands are

0

10

10

20

20

30

30

0-0.2-0.4-0.6 0.2 0.4

Energy (Ry)

Den

sity

of

Stat

es (

stat

es p

er R

y pe

r sp

in)

Majority Spin

Minority Spin

Figure 1. The spin-polarized electronic density of states ofBCC Fe in units of states Ry−1 using the one-electronpotentials from [15].

obtained which are expected from the simpler Hubbard-model treatment described in the last section. Examplesof band theory calculations for the magnetic 3d transitionmetals BCC iron and FCC nickel can be found in the bookby Moruzzi et al [15]. The figure on p 170 of their book(we show a similar figure in figure 1) shows the density ofstates of BCC iron as a function of energy. The densitiesof states for the two spins are almost (but not quite) rigidlyshifted. As is typical for BCC structures, the d band hasthree major peaks. The Fermi energy resides in the topof the d bands for the so-called majority spins betweenthe upper two peaks. The saturation magnetization,Ms ,that results from this ‘self-consistent’ calculation is 2.2µB ,which is in good agreement with experiment.

In nickel and cobalt the majority spin d bands arecompletely occupied and the Fermi energy lies in theprominent peak in the minority spin d density of states.The d band width has been a topic for close scrutiny overthe years owing to the width extracted from photoemissionmeasurements being much smaller than that of bandstructure calculations. This serves to emphasize the factthat the SDF theory is a theory for the ground state whereasexcited states are probed by spectroscopic measurements.In particular the theory does not correctly describe thecorrelated motion of the electrons as they are excited intostates necessary for them to leave the metal. On theother hand all the ground state properties such asMs

and the lattice spacing are in very good agreement withexperimental values.

The nearly rigidly split spin-polarized bands of these3d elemental transition metal magnets are actually specialcases. We now show how this simple picture is lost assoon as the electronic structures of ferromagnetic alloys areconsidered. We focus on compositionally disordered alloysfor this purpose, since these provide the starting point formuch of the discussion in this article.

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Magnetic alloys

3.2. Compositionally disordered alloys

The self-consistent Korringa–Kohn–Rostoker coherentpotential approximation (SCF-KKR-CPA) [25] is a meanfield adaptation of the SDF-LSDA to systems withsubstitutional disorder such as solid solution alloys. Forthese systems, the obvious but computationally intractableprocedure for finding their ground state properties wouldinvolve solving self-consistently the one-electron Kohn–Sham equations for all possible nuclear configurationsfollowed by the averaging of the observables of interest(particle and magnetization density, total energy and soon) over the appropriate ensemble of configurations. For abinary alloy AcB1−c we introduce an occupation variableξiwhich takes on the value 1 if the lattice sitei is occupiedby an A nucleus and 0 if the site is occupied by a Bnucleus. A configuration is specified by assigning valuesto these variables for each site, namely{ξi}. For an atomof type α = A(B) on site k, the effective one-electronpotential, vk,α(r, {ξi}), and magnetic field,Bk,α(r, {ξi}),which enter the Kohn–Sham equations are not independentof the surroundings of sitek and depend on all theoccupation variables. To find an ensemble average of anobservable, we must first find the self-consistent solutionto the Kohn–Sham equations for each configuration. Bysumming the results weighted by the correct probabilityfactor the appropriate ensemble average is obtained.

The KKR-CPA was invented to bypass the computa-tional difficulties that would be encountered if this sequenceof calculations were to be carried out. The first simplifyingassumption of the KKR-CPA is that the occupation of asite either by an A or by a B nucleus is independent of theoccupancy of the surrounding sites. This means that short-range order is neglected for the purposes of calculating theelectronic structure and the solid solution is approximatedby a random substitutional alloy. The second step is to in-vert the order of solving the Kohn–Sham equations and theconfigurational averaging. Consequently one finds a set ofKohn–Sham equations that describe an ‘average’ medium.In the spirit of a mean field theory, the local potential andmagnetic field are replaced byvk,α(r) andBk,α(r) whichdepend upon the average particle densities and magnetiza-tions on all surrounding sites and depend explicitly only onthe occupancy of sitek which is of typeα. The motion ofan electron, on the average, through a lattice of these poten-tials, which are randomly distributed with the probability ofc that a site is occupied by an A atom and of 1−c that it isoccupied by a B atom, is obtained from the solution of theKohn–Sham equations using the CPA [26]. In this approx-imation a lattice of identical effective potentials is foundsuch that the motion of an electron through this orderedarray closely resembles the motion of an electron on theaverage through the disordered alloy. The CPA is the re-quirement that the substitution of a single site of the latticeof these effective potentials either by an A or by a B atomproduces no further scattering of the electron on the aver-age. It is then possible to develop a spin density functionaltheory and calculational scheme in which the partially aver-aged electronic densities,nA(r) andnB(r), magnetizationdensities,µA(r) = |mA(r)| andµB(r) = |mB(r)|, associ-ated with the A and B sites are determined self-consistently.

The averageMs , total energies and other equilibrium quan-tities are also evaluated [25].

Both x-ray and neutron scattering data from solidsolutions show the existence of Bragg peaks which definean underlying ‘average’ lattice. This symmetry is apparentin the average electronic structure given by the CPA. TheBloch wavevector is still a useful quantum number butthe average Bloch states also have a finite lifetime as aconsequence of the disorder. This leads to changes in theresistivity, broadening of spectra and so on.

3.3. Alloy electronic structure and Slater–Paulingcurves: Ms versus concentration

Before the reasons for the loss of the conventional Stonerpicture of rigidly exchange-split bands can be set out, wemust first describe some typical features of alloy electronicstructure. Much has been written on this subject whichhas demonstrated clearly how closely these features areconnected with the phase stability of the system. Anoverview of this subject can be found in the books byPettifor [27] and Ducastelle [28] and articles by Gyorffyet al [29, 30], Connolly and Williams [31], Zunger [32],Stauntonet al [33] and many others.

Let us consider two elemental d electron densities ofstates, each with approximate widthW , say, with onecentred on an energyεA and the other atεB . εA and εBare related to atomic-like d energy levels. IfεA− εB � W

then the alloy’s densities of states will be a ‘split-band’-like one [25] and, in Pettifor’s language, an ionic bondis established as charge flows from the A atoms to the Batoms in order to equilibrate the chemical potentials. Thevirtual bound states associated with impurities in metalsare rough examples of split-band behaviour. If, however,εA − εB � W , then the alloy’s electronic structure is nowclassed as being a ‘common-band’-like one. There is nowlarge-scale hybridization between states associated with theA and B atoms and each site in the alloy is nearly chargeneutral insofar as an individual ion is efficiently screenedvia the metallic response function of the alloy [34]. Ofcourse, the actual interpretation of the detailed electronicstructure of an alloy involving many bands is always acomplicated mixture of these two models.

In both cases, half filling of the bands reduces the totalenergy of the system with respect to the phase-separatedcase [27, 28, 35] and an ordered alloy will form at lowtemperatures. When magnetism is added to the problem,the difference between the exchange fields associated witheach type of atomic species must also be considered. Formajority spin electrons, the rough measure of the degreeof ‘split-band’ or ‘common-band’ nature of the densityof states is governed by(ε↑A − ε

↑B)/W and a similar

measure(ε↓A − ε↓B)/W for the minority spin electrons.

If the exchange fields are rather different then the bandsare common-band-like ones for electrons of one spinpolarization, whilst the bands for the others will be more‘split-band’-like ones. The upshot is a spin-polarizedelectronic structure which is not set up by a rigid exchangesplitting. Figures 2(a) and 3(a) show schematic energy leveldiagrams for Ni–Fe and Fe–V alloys respectively.

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J B Staunton et al

Ni

Energy Atoms Alloy

Fe, NiBonding

Fe, Ni

Fe

Fe, NiAntibonding

Fe

Ni

(a)

0 0.2-0.2-0.4-0.6

30

30

20

20

0

10

10

Den

sity

of

Stat

es (

Stat

es p

er R

y pe

r sp

in)

Energy(Ry)

Majority Spin

Minority Spin

(b)

Figure 2. (a) A schematic energy level diagram for Ni–Fealloys. (b) The majority- and minority-spin, compositionallyaveraged densities of states of ferromagnetic Ni75Fe25 andits resolution into components weighted by concentration,in units of states per atom Ry−1 per spin. The full linerepresents the density of states associated with the nickelsites and the dotted line the iron sites.

According to Hund’s rules it is often energeticallyfavourable for the majority spin d states to be fully occupiedand there are many examples for which, at the cost ofa small amount of charge transfer, this is accomplished.Nickel-rich nickel–iron alloys provide such examples [36],as shown in figure 1(b).

The first task which observations of the electronicstructure must accomplish is to explain simply why theaverage magnetic moments per atom of so many alloys,Ms , fall on the famous Slater–Pauling curve, when plottedagainst the alloys’ valence electron per atom ratios. Theusual Slater–Pauling curve for the 3d row [18] consists oftwo straight lines. The plot rises linearly from the beginningof the 3d row, abruptly changes the sign of its gradientand then drops linearly to the end of the row. There aresome important groups of compounds and alloys whoseparameters do not fall on this line but, for these systemsalso, there is often some simple pattern.

It is easy to see why ferromagnetic alloys of latetransition metals, which are characterized by completelyfilled majority spin d states (such as figure 2(b)), are located

V

Energy Atoms Alloy

Bonding

Fe

Fe Fe, VAntibonding

Fe

Fe, V

V

(a)

0 0.2-0.2-0.4-0.6

30

30

20

20

0

10

10

Energy(Ry)

Den

sity

of

Stat

es (

Stat

es p

er R

y pe

r sp

in)

Majority Spin

Minority Spin

(b)

Figure 3. (a) A schematic energy level diagram for Fe–Valloys. (b) The majority- and minority-spin, compositionallyaveraged densities of states of ferromagnetic Fe87V13 andits resolution into components weighted by concentration,in units of states per atom Ry−1 per spin. The full linerepresents the density of states associated with the ironsites and the dotted line the vaadium sites.

on the negative-gradient straight line. Their magnetizationper atom,M = N↑ − N↓, where N↑(↓) describes theoccupation of the majority (minority) spin states, can beexpressed in terms of the number of electrons per atomZ so thatM = 2N↑ − Z. The occupation of the s andp states scarcely changes across the 3d row andM =2Nd↑ − Z + 2Nsp↑ which equals 10− Z + 2Nsp↑.

There are many other systems, most commonly BCC-based alloys, which are not strong ferromagnets in thissense of having filled majority spin d bands but possessa similar quality. The chemical potential (or Fermi energyat T = 0 K) is pinned in a deep trough in the minorityspin density of states [37, 38]. Figure 1 shows this for pureBCC iron, the chemical potential sitting in a trough in theminority spin density of states. Figure 3(b) shows anotherexample in an iron-rich iron–vanadium alloy. The othermajor portion of the Slater–Pauling curve of a positive-gradient straight line can be explained by using this aspectof the electronic structure. The pinning of the chemicalpotential in a valley of the minority spin d density ofstates constrainsNd↓ to be roughly 3 in all these alloys.

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In this circumstance the magnetization per atomM =Z−2Nd↓−2Nsp↓ = Z−6−2Nsp↓. Further discussion canbe found in [38–41]. Here we illustrate some of the remarksmade by describing briefly electronic structure calculationsof two compositionally disordered alloys, one from eachside of the Slater–Pauling plot.

We begin with Ni–Fe. The d band widths of ironand nickel are comparable and in both elements the Fermienergy is placed near or at the top of the majority-spin dbands. That there is a larger moment in Fe than there isin Ni, however, is shown via the exchange splitting forFe being larger. To obtain a rough idea of the electronicstructures of NicFe1−c alloys we imagine aligning the Fermienergies. The atomic-like d levels of the two which markthe centre of the bands would be at the same energy forthe majority spin electrons whereas for the minority spinelectrons the levels would be rather different, reflecting thediffering exchange fields associated with each sort of atom.Figure 2(a) shows this picture schematically and figure 2(b)shows it in detail by depicting the density of states ofNi75Fe25 calculated by the SCF-KKR-CPA. The majorityspin density of states is very sharply structured which isevidence that, in this compositionally disordered alloy, themajority spin electrons ‘see’ very little difference betweenthe two types of atom. For the minority spin electrons thesituation is to the contrary. In this case the density of statesis a ‘split-band’-like one owing to the separation of energylevels (from the different exchange splittings of Ni and Fe)and the compositional disorder. As pointed out earlier, themajority spin d states are fully occupied and this featurepersists for all FCC NicFe1−c alloys withc greater than 0.4and has the consequence that the alloys’ average saturationmagnetizations,Ms , fall nicely on the negative-gradientslope of the Slater–Pauling curve. For concentrations lessthan 35% and prior to the martensitic transition into theBCC structure at around 25% (the famous ‘Invar’ alloys),the Fermi energy is forced into the top of majority spin dstates and the alloys deviate from the Slater–Pauling curve.

Figure 3(a) for BCC FecV1−c shows another simpleschematic energy level diagram. As before the d energylevels of Fe are exchange split, showing that it isenergetically favourable for pure BCC Fe to have a netmagnetization. There is no exchange splitting in purevanadium. Just like in the case of the NicFe1−c alloys weassume charge neutrality and align the two Fermi energies.Now the vanadium d levels are much closer in energyto the minority spin d levels of iron than they are toits majority-spin ones. On alloying the two metals ina BCC structure, the bonding interactions have a largereffect on the minority-spin levels than they do on those ofthe majority spin owing to the smaller energy separation.In other words Fe induces an exchange splitting on theV sites, decreasing the kinetic energy, which results inthe formation of bonding and anti-bonding minority-spinalloy states. Fewer majority-spin V-related d states thanminority-spin d states are occupied so that the moments onthe vanadium sites are anti-parallel to the larger ones on theFe sites. The moments cannot persist for concentrations ofiron less than 30% since the Fe-induced exchange splittingon the vanadium sites decreases together with the average

number of Fe atoms surrounding a vanadium site in thealloy. As for the majority-spin levels, well separated inenergy, ‘split bands’ form, namely states which residemostly on one constituent or the other. Figure 3(b) showsthe spin-polarized density of states of an iron-rich Fe–Valloy determined by the SCF-KKR-CPA method wherebyit is possible to identify all these features.

So far everything has been discussed with respect toa spin-polarized but non-relativistic electronic structure.We now briefly examine the relativistic extension to thisapproach and show how it can be used to describe theimportant magnetic property magnetocrystalline anisotropyand to calculate values of the anisotropy constants,K.

3.4. Relativistic effects and magnetocrystallineanisotropy: the anisotropy constantsK

In recent years, magnetocrystalline anisotropy of ferromag-netic transition metal materials has become the subject ofintensive theoretical and experimental study because of thetechnological implications for high-density magneto-opticalstorage media [42] as well as for the understanding of mag-netic properties in general that arises from micromagneticmodelling. A detailed understanding of the mechanismof this anisotropy is needed. Several experimental studieshave been devoted both to understanding its origin and alsoto correlating it to other physical properties of the materials[43–51]. On the other hand,ab initio theoretical approacheswhich can explain the underlying physics are now only justbecoming feasible. Over the past few years, considerableprogress has been made in this direction [52–61] and herewe will describe our work in this sphere on disordered al-loys.

In a crystalline solid, the equilibrium direction ofthe magnetization is along one of the crystallographicdirections. The energy required to alter the magnetizationdirection is called the magnetocrystalline anisotropy energy(MAE). Micromagnetic theory uses an expansion of theMAE which satisfies the symmetry properties of theunderlying crystalline lattice, has the correct set of easyand hard directions and is expressed in terms of a smallnumber of anisotropy constantsK.

Brooks [62] first suggested that the origin of thisanisotropy is the interaction of magnetization with thecrystal field, namely the spin–orbit coupling. These daysboth the origin of this effect and the magnetostatic effectswhich determine domain structure can be shown from therelativistic generalization of the SDF theory. The formalstarting point is the fundamental quantum electrodynamicsof an electron interacting with an electromagnetic field. Theground state energy is now the minimum of a functional ofthe charge and current densities—a minimization which isachieved, in principle, by the self-consistent solution of aset of Kohn–Sham Dirac equations for electrons movingin fields dependent on the charge and current densities.Once again approximations for the exchange-correlationpart of the functional have to be made. By the formaltrick of carrying out a Gordon decomposition of the currentinto what are orbital and spin components, the effectsof spin–orbit coupling upon the electronic structure can

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J B Staunton et al

be represented and magnetostatic shape anisotropy alsodescribed from within the same theoretical framework [63].

Most theoretical investigations of magnetocrystallineanisotropy and calculations of the anisotropy constantsK

place their emphasis on spin–orbit coupling effects usingeither perturbation theory or a fully relativistic theory.Examples of such calculations are described in [52, 59] fortransition metals, [55, 56] for ordered transition metal alloysand [53, 54, 57] for layered materials, with varying degreesof success. Typically the total energy or the single-electroncontribution to it (if the force theorem is used [64]) iscalculated for two magnetization directions separately andthen the MAE is obtained by subtracting one from the other:∫ ε1

F

εne1(ε) dε −∫ ε2

F

εne2(ε) dε (13)

whereε1F andε2

F are the Fermi energies when the system ismagnetized along the directionse1 ande2 respectively andne1(2) is the electronic density of states. However, the MAE,which in many cases is of the order of micro-electron-volts,is several orders of magnitude smaller than the total energyof the system. Therefore, it is numerically more preciseto calculate the difference directly [65]. Strangeet al[66, 67] have developed a relativistic spin-polarized versionof the Korringa–Kohn–Rostoker (SPR-KKR) formalism tocalculate the electronic structure of solids and Ebert andAkai [68] have extended this formalism to disordered alloysby incorporating a coherent-potential approximation (SPR-KKR-CPA) and used it to describe the electronic structureand other related properties such as magnetic circular x-raydichroism, hyperfine fields and the magneto-optical Kerreffect [69]. Strangeet al [61] and more recently we [65]formulated a theory to calculate the MAE of elementalsolids within the SPR-KKR scheme and applied it to Fe andNi [61]. We subsequently extended this work to disorderedalloys and details can be found in [70, 71]. Here wesummarize our findings for two alloy systems, Co–Pt andNi–Pt.

CocPt1−c alloys are important from the point of viewof the fundamental physics of magnetic anisotropy; a largespin magnetic moment is associated with the Co sites,spin polarizing the electronic structure, whereas spin–orbitcoupling is stronger on Pt sites. Most experimental workon Co–Pt has been on the ordered tetragonal phase, whichhas a very large magnetic anisotropy ('400µeV) and themagnetic easy axis is along thec axis [45, 46]. Althoughthere does not seem to have been any experimental workon the bulk disordered FCC phase of these alloys, someresults have been reported for thin films [47–51]. It is foundthat the magnitude of the MAE is more than one order ofmagnitude smaller than that of the bulk ordered phase andthat the magnetic easy axis varies with the film’s thickness.A study of the MAE of the bulk disordered alloys providessome insight into the mechanism of magnetic anisotropy inthe ordered phase as well as in thin films.

Figure 4 shows the energy difference of disorderedFCC CocPt1−c alloys for c = 0.25, 0.5 and 0.75 along the〈0, 0, 1〉 and〈1, 1, 1〉 directions as a function of temperaturein the range 0–1500 K. This energy difference, the MAE, is

0 300 600 900 1200 15000

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Mag

neto

crys

talli

ne A

niso

trop

y (

µeV

)

Temperature (K)

x=0.50x=0.25x=0.75

Co50Pt50

Co25Pt75

Co75Pt25

Figure 4. The magnetocrystalline anisotropy energy (K /3)defined as the difference in energy between systemsmagnetized along the 〈0, 0, 1〉 and 〈1, 1, 1〉 directions of thecrystal for CocPt1−c alloys for c = 0.25, 0.50 and 0.75, as afunction of the temperature.

equal toK/3 if the anisotropy energy is assumed to have thesimple form assumed in equation (2) in the introduction fora cubic system. We note that, for all the three compositions,the MAE is positive at all temperatures, implying that themagnetic easy axis is always along the〈1, 1, 1〉 direction ofthe crystal, although the magnitude of the MAE decreaseswith increasing temperature. The magnetic easy axis ofFCC Co is also along the〈1, 1, 1〉 direction. Thus, alloyingwith Pt does not alter the magnetic easy axis, namely thesign of K. The equiatomic composition has the largestMAE, which is about 3.0µeV at 0 K. Addition of Ptto Co results in a monotonic decrease in the averagemagnetic moment,Ms , of the system with the spin–orbitcoupling becoming stronger. This trade-off between spinpolarization and spin–orbit coupling is the main reason forthe MAE being largest for the equiatomic composition.

The magnetocrystalline anisotropy of a system can beunderstood in terms of its electronic structure. Figure 5(a)shows the spin-resolved density of states on Co and Ptatoms in Co50Pt50 magnetized along the〈0, 0, 1〉 direction.The Pt density of states is rather structureless exceptaround the Fermi energy where there is spin-splitting dueto hybridization with Co d bands. When the direction ofmagnetization is orientated along the〈1, 1, 1〉 direction ofthe crystal the electronic structure changes, although thedifference is quite small in comparison with the overalldensity of states. Figure 5(b) depicts this density-of-statesdifference. In the lower part of the band, dominated byPt, the difference between the two is small, whereas itbecomes quite oscillatory at higher energies dominated bythe Co d band complex. There are spikes in this differenceat energies at which there are also peaks in the Co-relatedpart of the density of states. Owing to the oscillatory natureof this curve, calculation of the magnitude of the MAEinvolves taking an integral over the product of the energyand this difference up to the Fermi energy (equation (13))and is quite small; the two large peaks around 2 and3 eV below the Fermi energy almost cancel each other out,leaving only the smaller peaks to contribute to the MAE.This curve also tells us that states far removed from the

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Magnetic alloys

0

1

2

3

-10 -8 -6 -4 -2 0 2

Den

sity

of

Stat

es (

stat

es/e

V)

Energy (eV)

Co-Majority SpinCo-Minority SpinPt-Majority SpinPt-Minority Spin

Co-Majority Spin

Co-Minority Spin

PtPt-Maj.

Pt-Min.

Co Pt50 50

(a)

-6 -4 -2 0 2

Energy (eV)

-0.010

-0.005

0

0.005

0.010

Dif

fere

nce

Den

sity

of

Stat

es (

stat

es/e

V)

(b)

Figure 5. (a) Spin-resolved densities of states for Co andPt in FCC Co0.5Pt0.5 alloy for magnetization along the〈0, 0, 1〉 direction of the crystal. The vertical dotted lineindicates the Fermi level. (b) The difference between thedensities of states of FCC Co0.5Pt0.5 alloy for magnetizationalong the 〈0, 0, 1〉 and 〈1, 1, 1〉 directions of the crystal. Thevertical dotted line indicates the Fermi level. Theseparation between the two Fermi levels is less than thethickness of this line.

Fermi energy (in this case, 4 eV below) can also contributeto the MAE, not just those around the Fermi surface.

In contrast to what we have found for the disorderedFCC phase of CocPt1−c alloys, the MAE of orderedtetragonal Co–Pt alloy is large ('400 µeV), some twoorders of magnitude greater than what we find for thedisordered Co50Pt50 alloy. Moreover, the magnetic easyaxis is along thec axis [45]. Theoretical calculationsof the MAE for ordered tetragonal CoPt alloy [55, 56]based on scalar relativistic methods and perturbation theoryfor the spin–orbit coupling do reproduce the correct easyaxis but overestimate the MAE by a factor of two. Itis not clear whether it is the atomic ordering or the lossof cubic symmetry of the crystal in the tetragonal phaseor both which is responsible for the altogether different

-4

-3

-2

-1

0

1

2

3

0 0.1 0.2 0.3 0.4

Mag

neto

crys

talli

ne A

niso

trop

y E

nerg

y

Concentration (Pt)

TheoryExperiment

Figure 6. Experimental and theoretical values of themagnetocrystalline anisotropy energy (K /3) of NicPt1−c asa function of concentration. The lines are guides to the eye.

magnetocrystalline anisotropies in disordered and orderedCo–Pt alloys. It is likely to be a combined effect of the twoand we are studying currently the effect of atomic short-range order (ASRO) on the magnetocrystalline anisotropyand anisotropy constants,K, of alloys as the next step. Ourpreliminary results show that an alloy with atomic short-range order of the type which would lead to the orderedCo–Pt alloy found at low temperatures has a greater MAE.Interestingly we also find that a Co–Pt alloy grown with alayering SRO has an easy axis perpendicular to the layersand a much greater magnitude of the MAE still.

Experimental results on ferromagnetic Ni1−cPtc alloys(which are disordered and FCC) reveal that the sign ofK changes, with the easy axis of magnetization changingfrom 〈1, 1, 1〉 to 〈0, 0, 1〉 with the addition of Pt, so thatthe magnitude of the MAE is greatest for around 15% Pt[72]. In figure 6 we show our calculations together withthe experimental results; both show that addition of Pt toNi alters the easy magnetic axis, although the minimumin the MAE at around 15% Pt is less pronounced in thecalculations than it is in the experimental results. Thesignificant underestimate of the small MAE for smallc ispartly due to our neglect of orbital polarization, althoughthe calculation of such tiny energy differences leading to theMAE of these dilute alloys and pure Ni is also sensitive toseveral technical subtleties. In line with other calculationsfor Ni, we find the correct easy axis of magnetization butthe magnitude of the MAE is too small. It is worth pointingout that, although both FCC Ni and FCC Co have negative-signKs so that the magnetic easy axes are〈1, 1, 1〉, additionof Pt to Ni changes the sign ofK but this does not occurwhen Pt is added to Co and only the magnitude ofK isaltered.

In this section we have attempted to show how spindensity functional theory, with its emphasis on spin-

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polarized electronic structure, provides a useful descriptionof magnetic properties of transition metal systems at lowtemperatures and can be used to estimate the saturationmagnetizationMs and also the anisotropy constantsK. Itsmajor approximation is the way in which the exchange andcorrelation part of the energy of these interacting electronsystems is obtained from examination of the homogeneousinteracting electron gas. At finite temperatures this localapproximation neglects the effects of thermally excitedspin-wave excitations and fails to provide even a qualitativedescription of the magnetic properties. The next sectionshows how this shortcoming is redressed. We also pointout how the exchange constantsA can be determined fromthe same general framework.

4. Finite-temperature metallic magnetism: Tc andexchange constants A

Soon after Hohenberg and Kohn, Kohn and Sham publishedtheir papers which launched density functional theoryand then Mermin devised the formal structure of thegeneralization of DFT to finite temperatures [73]. Formagnetic systems in an external potential,V ext , andexternal magnetic field,Bext , it can be proved that, inthe grand canonical ensemble at a given temperatureT

and chemical potentialν, the equilibrium particle densityn(r) and magnetization densitym(r) are determined bythe external potential and magnetic field. The correctequilibrium particle and magnetization densities minimizethe Gibbs grand potential�:

�[n,m] =∫

dr V ext (r)n(r)−∫

drBext (r) ·m(r)

+e2

2

∫ ∫dr dr′

n(r)n(r′)|r − r′|

−ν∫

dr n(r)+G[n,m] (14)

whereG is a unique functional of charge and magnetizationdensities at a givenT andν. The variational principle nowstates that� is a minimum for the equilibriumn andmandG can be written

G[n,m] = Ts [n,m] − T Ss [n,m] +�xc[n,m] (15)

with Ts and Ss being respectively the kinetic energy andentropy of a system of non-interacting electrons withdensitiesn andm at a temperatureT . �xc is the exchangeand correlation contribution to the Gibbs free energy. Theminimum principle can be shown to be identical to thecorresponding equation for a system of non-interactingelectrons subject to an effective potentialV eff :

V eff [n,m] = V ext 1−Bext · σ + e21∫

dr′n(r′)|r − r′|

+1δ�xc

δn(r)+ δ�xc

δm(r)· σ (16)

which satisfy the following set of equations[(− h

2

2m∇2

)1+ V eff

]φi(r, ε) = εφi(r, ε)r′ (17)

n(r) =∫

dε f (ε − ν)∑i

tr(φ∗i (r, ε)φi(r, ε)) (18)

m(r) =∫

dε f (ε − ν)∑i

tr(φ∗i (r, ε)σφi(r, ε)) (19)

where f (ε − ν) is the Fermi–Dirac function andν thechemical potential.

The next logical step is to extend the localapproximation (LDA) to finite temperatures, (LDA), andwrite down�xc in terms of the exchange-correlation partof the Gibbs free energy of a homogeneous electrongas. If this is done the effects of the thermally inducedspin-wave excitations are severely underestimated. Thecalculated Curie temperatures are much too high [14],there are no local moments in the paramagnetic state andthe uniform static paramagnetic susceptibility does notfollow a Curie–Weiss behaviour as is found for manymetallic systems. These spin fluctuations interact as thetemperature is increased and therefore�xc[n,m] shoulddeviate significantly from the local approximation with aconsequent effect upon the form of the effective single-electron states.

The idea is therefore to model the effects of thespin fluctuations whilst still maintaining the spin-polarizedsingle-electron basis. Thus the properties of magneticmetals at finite temperatures can be described; that is,Ms ,A andK can be determined as functions of temperature.The straightforward extension of spin-polarized bandtheory to finite temperatures misses the dominant thermalfluctuation of the magnetization since the thermallyaveraged magnetization,M, can only vanish together withthe ‘exchange splitting’ of the electronic bands whichis destroyed by particle–hole ‘Stoner’ excitations acrossthe Fermi surface. An important piece of this neglectedcomponent can be pictured as orientational fluctuations of‘local moments’ which are the magnetizations, within eachunit cell of the underlying crystalline lattice, formed by thecollective behaviour of all the electrons. The picture nowinvolves the propagation of independent electrons througheffective magnetic fields whose orientations fluctuate.M

can now vanish as the disorder of the ‘local moments’grows. From this broad consensus [74], there are severalapproaches which differ according to which aspects of thefluctuations are deemed to be the most important for thematerials under scrutiny.

4.1. Fluctuating ‘local moments’

At the outset of this work some 15 years ago, models of theferromagnetic 3d transition metals Fe, Co and Ni at finitetemperatures fell into one of two camps. By and large theStoner excitations were neglected and the orientations ofthe ‘local moments’, which were assumed to have fixedmagnitudes independent of their orientational environment,were the degrees of freedom over which thermal averagingwas to be performed. The fluctuating local band (FLB)theory [75–77] was based on the assumption that thereis a large amount of short-range magnetic order even inthe paramagnetic phase. There are large spatial regionsin which the ‘local moments’, with magnitudes equivalent

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to the magnetization per site in the ferromagnetic state atT = 0 K, are nearly aligned, namely where the orientationsvary gradually. In these regions, the usual spin-polarizedband theory can be applied and perturbations to it made.The results of quasi-elastic neutron scattering experimentsof Ziebeck et al [78] on the paramagnetic phases of Feand Ni, later confirmed by Shiraneet al [79] are given asimple, though not uncontroversial [80], interpretation bythis picture. In the case of inelastic neutron scattering,however, even the basic observations are controversial,let alone their interpretations in terms of ‘spin waves’aboveTc which may feature in such a model. It is ratherdifficult to carry out realistic calculations [81] in which themagnetic and electronic structures are mutually consistentand consequently to examine the full implications of theFLB picture and to improve it systematically.

The second type of approach is the ‘disordered localmoment’ (DLM) picture [82–85]. Here, the local momententities associated with each lattice site are commonlythought to fluctuate fairly independently and an apparenttotal absence of magnetic short-range order (SRO) isassumed at the outset. Early work was based on theHubbard Hamiltonian. The procedure had the advantageof being fairly straightforward and more specific than thatin the case of FLB theory and various calculations whichgave a reasonable description of experimental data wereperformed. The model Hamiltonian’s drawback was itssimple parameter-dependent basis which could not providea realistic description of the electronic structure which mustsupport the important magnetic fluctuations. The dominantmechanisms might therefore be missed. It is also difficultto improve this work systematically.

The paramagnetic state of body-centred cubic ironhas attracted much attention and it is generally agreedthat there are ‘local moments’ in this material for alltemperatures, although the relevance of a HeisenbergHamiltonian to a description of their behaviour has beendebated in depth. In suitable limits, both the FLB andDLM approaches can be cast into a form from which aneffective classical Heisenberg Hamiltonian can be extracted.The ‘exchange-interaction’ parametersJij are specified interms of the electronic structure owing to the itinerantnature of the electrons and the small-wavevector limit, ifthe lattice Fourier transform is taken, gives an estimateof the exchange constantA. Unfortunately theJij termsdetermined from the former FLB approach turned out tobe short ranged and therefore inconsistent with the initialpicture of substantial magnetic SRO aboveTc [81]. Fromthe DLM perspective, the interactions,Jij , of paramagneticiron can be obtained from consideration of the energy of aninteracting electron system in which the local moments areconstrained to be orientated along directionsei and ej onsites i and j whilst an approximate average is taken overall the possible orientations on the other sites [86, 87]. TheJij terms calculated in this way are suitably short ranged.

A scenario in between these two limiting cases has beenproposed by Heine and Joynt [88] and Samson [89]. Theytoo were guided by the apparent substantial magnetic SROaboveTc in Fe and Ni deduced from neutron scatteringdata and they emphasized how the orientational magnetic

disorder involves a balance in the free energy betweenenergy and entropy. This balance is delicate and theyshowed that it is possible for the system to disorder ona coarser than atomic length scale and consistency betweenthe magnetic and electronic structures to be maintained.The length scale is, however, not as coarse as that initiallyproposed when the FLB theory was set up [75, 76].

4.2. ‘First-principles’ theories

These models can be made ‘first-principles’ ones bygrafting the effects of these orientational spin fluctuationsonto SDF theory [87, 90, 91]. It is assumed that fast andslow motions can be identified and separated. On a timescale long in comparison with an electronic hopping time,but short when compared with a typical spin fluctuationtime, the spin polarizations of the electrons leaving a siteare sufficiently correlated to those of electrons arrivingthat a non-zero magnetization exists when the appropriatequantity is averaged on this time scale. These are the ‘localmoments’ which can change their orientations slowly withrespect to the time scale whereas their magnitudes fluctuaterapidly.

The standard SDF theory which we have discussedalready can be adapted to describe the states of the systemfor each orientational configuration{ei} in a similar wayto that in which Uhl et al [93], Sandratskii and Kubler[94, 95] and others have tackled non-collinear magneticsystems. In principle such a description yields themagnitudes of the local moments, which can depend onthe orientational environmentµk({ei}), and the electronicgrand potential for the constrained system�{ei}. Thisis the framework from which exchange constantsA forthe micromagnetic free energy can also be calculated[96, 97] by considering grand-potential differences forvarious orientational configurations.

For BCC Fe and fictitious BCC Co the momentsare fairly independent of their orientational environmentwhereas those in FCC Fe, Co and Ni are further awayfrom being local quantities. Thus the long-time averagescan be replaced by ensemble averages with the GibbsianmeasureP {ei} = Z−1 exp(−β�{ei}), where the partitionfunction Z = ∏

i

∫dei exp(−β�{ei}) (β = 1/(kBT )).

The thermodynamic free energy, which accounts for theentropy associated with the orientational fluctuations aswell as creation of electron–hole pairs, is given byF =−kBT logZ. The role of a classical ‘spin’ (local moment)Hamiltonian, albeit a highly complicated one, is played by�{ei}.

By choosing a suitable reference ‘spin’ Hamiltonian�0{ei} and expanding about it using the Feynman–Peierls’inequality [98], an approximation to the free energy isobtained:

F ≤ F0+ 〈�−�0〉0 = F (20)

with

F0 = − 1

βln∏i

∫dei exp(−β�0) (21)

〈X〉0 =∏i

∫dei X exp(−β�0)

/∏i

∫dei exp(−β�0)

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J B Staunton et al

=∏i

∫dei P0{ei}X{ei}. (22)

With �0 in the form

�0 =∑i

ω(1)i (ei )+

∑i,j,i 6=j

ω(2)ij (ei , ej )+ · · · (23)

a scheme is set up which can in principle be systematicallyimproved. Minimizing F to obtain the best estimate ofthe free energy givesω(1)i , ω(2)ij and so on as expressionsinvolving resticted averages of�{ei} over the orientationalconfigurations [87, 90].

A type of mean-field theory, which turns out tobe equivalent to a ‘first-principles’ formulation of theDLM picture, is produced by taking the first term onlyin the equation above. Using a generalization of theSCF-KKR-CPA method [25] discussed earlier in thecontext of compositional disorder in alloys (section 3.3),explicit calculations were performed for BCC Fe andFCC Ni. The average magnitude of the local moments,〈µi({ej })〉ei = µi(ei ) = µ, in the paramagnetic phaseof iron was 1.91µB [87] (the total magnetization is zerosince 〈µi({ej })ei〉 = 0). This value is of roughlythe same magnitude as the magnetization per atom inthe low-temperature ferromagnetic state. The uniform,paramagnetic susceptibility,χ(T ), followed a Curie–Weissdependence upon temperature as observed experimentallyand the estimate of the Curie temperatureTc was found tobe 1280 K, also comparing well with the experimental valueof 1040 K [92]. In nickel, however,µ was found to be zeroand the theory reduced to the conventional SDF version ofthe Stoner model with all its shortcomings [14, 91].

This mean-field DLM picture of the paramagnetic statehas been improved by including the effects of correlationsbetween the local moments to some extent. This has beenachieved by integrating the consequences of Onsager cavityfields into the theory [91, 99].Tc for Fe is now 1015 K anda reasonable description of neutron scattering data is given.This approach has recently been generalized to alloys [100].

A first application to the paramagnetic phase of the‘spin glass’ alloy Cu85Mn15 showed that the magneticinteractions between the Mn local moments fell offexponentially with distance and were oscillatory. This wasin agreement with extensive neutron scattering data andthe underlying electronic mechanisms were also identified[100]. An earlier application to FCC Fe showed how themagnetic correlations change from anti-ferromagnetic toferromagnetic as the lattice is expanded [101]. This studycomplemented total energy calculations for FCC Fe both forferromagnetic and for anti-ferromagnetic states at absolutezero for a range of lattice spacings. This effect is alsorelevant for the understanding of the properties of FCC-based Fe alloys, in particular, the Invar alloys [102].

For nickel, the theory resembles the static, high-temperature limit of the theory of Murata and Doniach[103], Moriya and co-workers [74, 104], Lonzarich andTaillefer [105] and others to describe weak itinerantferromagnets. Nickel is consequently described in termsof a Stoner theory but one in which the spin fluctuationshave drastically renormalized the exchange interaction

and lowered Tc from about 3000 K [14] to 450 K[91]. The neglect of the dynamical aspects of thesespin fluctuations has over-estimated this renormalizationsomewhat butχ(T ) again exhibits Curie–Weiss behaviouras is found experimentally and an adequate description ofneutron scattering data is also provided. Recent inversephotoemission measurements by von der Lindenet al [106]have confirmed the collapse of the ‘exchange-splitting’of the electronic bands of nickel as the temperatureis raised towards the Curie temperature in accord withthis Stoner-like picture, although spin-resolved, resonantphotoemission measurements by Kakizakiet al [107]indicate the presence of spin fluctuations.

This approach has no parameters, everything being setup from SDF theory. It thus represents a well-defined stageof approximation. However, there are still some obviousshortcomings in this work (exemplified by the discrepancybetween the theoretically determined and experimentallymeasured Curie constants) and it is worth highlighting thekey one, which is the neglect of the dynamical effectsof the spin fluctuations that has been emphasized byMoriya [104, 105] and others. Uhl and Kubler [108] havealso set up anab initio approach for dealing with thethermally induced spin fluctuations and they also treat theseexcitations classically. They calculate total energies ofsystems constrained to have spin-spiral configurations witha range of different propagation vectorsq of the spiral,polar angles and spiral magnetization magnitudes using thenon-collinear fixed spin moment method. A fit of theenergies to an expression involvingq, polar angles andmagnetization magnitudes is then made in order to set upanother ‘local moment Hamiltonian’. The Feynman–Peierlsinequality is also used when a quadratic form is used for the‘reference Hamiltonian’. Stoner particle–hole excitationsare neglected. Similar results to those in reference [91]have been obtained for BCC Fe and FCC have also studiedCo and have recently generalized the theory to describemagnetovolume effects. FCC Fe and Mn have been studiedalongside the ‘Invar’ ordered alloy [109]. Once againexchange constantsA can be extracted from this work.

4.3. ‘Local exchange splitting’

The range of validity of these sorts ofab initio theoreticalapproaches and the severity of the approximationsemployed can be found by comparing their underlyingelectronic bases with suitable spectroscopic measurements.An early prediction from a ‘first-principles’ implementationof the DLM picture was that a ‘local exchange’ splittingshould be evident in the electronic structure of theparamagnetic state of BCC iron [90, 110]. In addition,the size of this splitting was expected to vary sharply asa function of the wavevector and energy. In fact at somepoints, if the ‘bands’ were flat in that region of wavevectorand energy space, the local exchange splitting would beroughly of the same magnitude as the rigid exchangesplitting of the electronic bands of the ferromagnetic statewhereas at other points, where the ‘bands’ are broader, thesplitting would vanish completely. It is this local exchangesplitting that causes the local moment to be established.

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The photoemission (PES) experiments of Kiskeret al[111] and inverse photoemission (IPES) experimentsperformed by Kirschneret al [112] found these features.The experiments essentially focused on the electronicstructure around the centre of the Brilliouin zone and at apoint on the zone boundary for a range of energies. In thefirst region states were interpreted as being exchange splitwhereas in the latter region they were not, though all werebroadened by the magnetic disorder. Later Haineset al[113] used a tight-binding model to describe the electronicstructure and employed the recursion method to averageover various orientational configurations. They concludedthat a modest degree of magnetic SRO is compatiblewith the limited spectroscopic measurements available forparamagnetic iron. More extensive spectroscopic dataon the paramagnetic states of the ferromagnetic transitionmetals would be invaluable in developing the theoreticalwork on the important spin fluctuations in these systems.

Having outlined how one can deal with the elementalmagnetic metals at finite temperatures from a ‘first-principles’ electronic structure basis, we will now considerhow this is carried out for alloys. A longstandingissue in the study of metallic alloys is the interplaybetween compositional order and magnetism and thedependence of magnetic properties on the local chemicalenvironment. Magnetism is frequently connected tothe overall compositional ordering as well as the localenvironment in a subtle and complicated way.

5. Magnetism and compositional order

In order to examine the interrelation between magnetismand compositional order it is necessary to deal with thestatistical mechanics of thermally induced compositionalfluctuations. References [29, 30, 33, 100] describe this insome detail so here we will give an outline and show howmagnetic effects are incorporated. In particular we showhow to set up an expression for the free energy dependenton the composition of the alloy, which can be used to findmetastable states and stable states of alloys [114].

5.1. The grand potential and equation of state of abinary alloy

Atomic ordering in alloys is usefully described in termsof ‘concentration waves’ [115]. As the atoms begin toarrange themselves while the alloy cools, the probabilityof finding a particular atomic species occupying a latticesite, namely the site-dependent concentration, varies fromsite to site and traces out a static concentration wave.For example, the well-known Cu3Au and CuZn orderedalloys are characterized by a concentration wave withwavevectorq = {0, 0, 1} on FCC and BCC crystal lattices,respectively. On the other hand, an infinitely long-wavelength concentration wave,q→ 0, describes an alloywhich undergoes phase segregation at low temperatures.One approach to describe such phenomena quantitativelyis to follow the formalism of Evans [116] for a densityfunctional theory of classical fluids and adapt it to the binaryalloy lattice model relevant here. This formalism formed

the basis for the work by Gyorffy and Stocks [29, 30] andsubsequent developments by ourselves [33, 100] and wereview it briefly below.

We consider the grand canonical ensemble ofNA A andNB B nuclei occupyingN (= NA+NB) locations referred toa fixed lattice position and define an equilibrium probabilitydensityP0{ξi} = P0(ξ1, ξ2, . . . , ξN) of occupation of sitesby A nuclei at a temperatureT (the fractionc = NA/N ).ξi = 1 (0) if an A (B) nucleus occupies sitei:

P0{ξi} = 1

Zexp[−β(H {ξi} − νNA)] (24)

whereH {ξi} is the Hamiltonian when there areNA nucleipresent andν is the chemical potential for the nuclei. Thegrand partition functionZ is

Z =∏l

∑ξl=1,0

exp[−β(H {ξi} − νNA)]. (25)

Following Evans [116], we consider the functional

�[P ] =∏l

∑ξl=1,0

P {ξi}(H {ξi} − νNA + 1

βlogP {ξi}

).

(26)For the equilibrium probability density we have�[P0] =−(1/β) logZ which is equivalent to the grand potential.�[P ] > �[P0] (P 6= P0) for all probability densities with∏l

∑ξl=1,0P = 1. We take Hamiltonians of the form

H {ξi} = W {ξi} −∑

i νexti ξi . W {ξi} describes the ‘internal

energy’ of the system which has theNA nuclei arranged onsome of theN sites according to{ξi} and

∑i ν

exti ξi is an

arbitrary external potential. (Note thatW is not restrictedto a pairwise form.) Thermal averages are defined by

〈X〉 =∏l

∑ξl=1,0

P0{ξi}X{ξi} (27)

and c0i = 〈ξi〉 is the equilibrium probability of finding an

A nucleus on lattice sitei, namely a ‘local’ concentrationwhen an external potential{νexti } is applied to the system.P0 is evidently a function of theνexti terms and thereforeso are the local concentrations. It is possible to show[116] that, for a given form ofW , only one particular setof νexti terms, {νexti }, can determine a given{c0

i }. SinceP0 = P0{νexti }, P0 is also determined by a given set ofci ,{ci}.

It follows that, for a givenW , we can specify

F{ci} =∏l

∑ξl=1,0

P0{ξi}(W {ξi} + 1

βlogP0{ξi}

). (28)

That is,F is a unique function of theci , which may be non-equilibrium ones (this is the case for any{νexti }). Similarly,

�{ci} = F{ci} − ν∑i

ci −∑i

νexti ci . (29)

When theci are equal to the equilibrium values�{ci} = �,the grand potential,

δ�

δcl

∣∣∣∣{ci=c0

i }= 0 (30)

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J B Staunton et al

for all l. Thus the correct equilibrium concentrations{c0i }

minimize � to give�.F{ci} is an ‘internal’ Helmholtz free energy. We have

δF{ci}δcl

∣∣∣∣{c0i }− νextl = ν (31)

in which the first term on the left-hand side definesan intrinsic chemical potentialνinl . Equation (31) isthe important equation of state. For example for anon-interacting (NI) system (W = 0), FNI {ci} =β−1∑

i [ci logci+ (1−ci) log(1−ci)], equal to the productof kBT and the ‘point entropy’.

5.2. Correlation functions and atomic short-rangeorder in alloys

The effects of interactions are most readily included viaa hierarchy of direct correlation functions (so called byway of the close analogy with similar quantities definedfor classical fluids [116, 117]). Let us define8{ci}, theinteraction part ofF , as−(F −FNI ) and we can write theintrinsic chemical potential

νinl {ci} =1

βlog

(cl

1− cl

)− δ8{ci}

δcl(32)

and δ8/δcl = S(1)l is a ‘self-energy’, the contribution

to the internal local chemical potential from all theinteractions. It is an additional single-site effective potentialwhich determines self-consistently via equation (31) theequilibrium concentrations†. From equation (31)

1

βlog

(c0l

1− c0l

)− S(1)l {c0

i } − νextl = ν

which can be arranged to give

c0l =

exp[β(S(1)l − ν − νextl )]

1+ exp[β(S(1)l − ν − νextl )].

S(1)l is in fact the first member of a hierarchy of correlation

functions generated by8{ci}, the higher order functionsbeing obtained by further differentiation; for example

S(2)kl =

δ28

δckδcl

S(3)klm =

δ38

δckδclδcm· · · .

Following Gyorffy and Stocks [29, 30] we note thatthe second derivative evaluated at the equilibriumconcentrations can be referred to as the Ornstein–Zernikedirect correlation function by analogy with its designationin the theory of non-uniform classical fluids.

�{ci} can also be used to generate another hierarchyof correlation functions. Differentiating it once, this timewith respect to the ‘external’ chemical potential, evaluating

† It is analogous to the effective potential that appeared in the single-electron Kohn–Sham equations of the density functional theory of theinhomogeneous electron gas.

it at the equilibrium concentrations and recalling that the{ci} depend on{νexti } gives

δ�{ci}δνextl

∣∣∣∣c0i

= −c0l .

A second differentiation gives a linear response function,

δ2�

δνextk δνextl

∣∣∣∣{c0i }= − δc0

l

δνextk

.

This can easily be shown to be proportional to the atom–atom correlation function〈ξlξk〉−〈ξl〉〈ξk〉 by differentiatingthe equilibrium grand potential twice:

δc0l

δνextk

= β(〈ξlξk〉 − 〈ξl〉〈ξk〉) = βαij .

These correlation functions are directly related tothe Warren–Cowley parameters and are measured indiffuse scattering (such as x-ray, electron and neutron)experiments. (Once again, further correlation functions canbe found by subsequent differentiation.)

Returning to equation (31) and taking the derivativewith respect to the local concentrationck on a sitek, weobtain an expression for the linear response function andhence the pair correlation function. For an alloy with ahomogeneous concentration distribution{c0

i = c} whichis then perturbed by a small, inhomogeneous change tothe external potential{δνi} with the result that a newinhomogeneous concentration distribution is set up{c+δci},the lattice Fourier transform of the response function is

α(q) = βc(1− c)1− βc(1− c)S(2)(q, T ) . (33)

The atomic short-range order (ASRO) present in the alloycan thus be described fully in terms of the lattice Fouriertransform of the ‘direct correlation function’S(2)kl andprovides direct information on the stability of the randomlydisordered alloy against concentration fluctuations at agiven temperatureT [29].

An expression for the Helmholtz free energy can alsobe constructed in terms of theS(2)kl . In terms of a set oflocal concentrations{ci}, we can write

F{ci} ≈ β−1∑i

[ci logci + (1− ci) log(1− ci)] −8(c)

−∑i

S(1)i (ci − c)−

∑ij

S(2)ij (ci − c)(cj − c)

= β−1∑i

[ci logci + (1− ci) log(1− ci)] −8(c)

−∑ij

S(2)ij (ci − c)(cj − c) (34)

by expanding the interaction part of the free energy8 around that of a reference homogeneously disorderedsystem with the same average concentrationc.

By calculatingS(2)kl for a disordered alloy magnetizedalong a given direction and then comparing it with the valueobtained when the system is magnetized along anotherdirection, the dependence of the anisotropy constantsK

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Magnetic alloys

upon the locally varying composition can be extracted,together with a handle upon the directional compositionalorder which can be set up when a magnetic alloy is annealedin a magnetic field. We are working on this topic.

The above equation (34) can be used in a simple modelof the kinetics of diffusional transformations as introducedby Cahn [118] and used by Khachaturyan [115, 119]. It canbe used to find equilibrium and quasi-equilibrium states.The model is based on nonlinear Onsager microscopicdiffusion equations. The rate of change of the relaxationparameters, the local concentrationsci(t), with time isdirectly proportional to the thermodynamic driving force,

dci(t)

dt=∑j

LijδF

δcj (t)(35)

where the Lij are kinetic coefficients proportional tothe probability of an atom undergoing a diffusional hopbetween sitesi and j in unit time. We have recentlyused this model, making the simplifying assumption thatthe kinetic coefficients are independent of the environment,namely theci(t) terms, just like Cahn and Khachaturyan didand investigated the metastable and stable ordered phasesAuFe and ferromagnetic FeAl alloys [114]†. We refer tothe latter in one of our case studies later on.

Elsewhere [33] we have described in detail howS(2)(q, T ) can be calculated from a ‘first-principles’, mean-field approach. Here the interacting part of the Helmholtzfree energy8 ≈ −〈H {ξi}〉 in which the angular bracketsdenote the average with respect to the inhomogeneousdistribution function P {ξi} =

∏i pi(ξi) with pi(1) =

ci and p(0) = 1 − ci . We start from the SCF-KKR-CPA description of the electronic structure of thehigh-temperature, compositionally disordered state [25].An important ingredient for the statistical averaging isthe replacement of the mean fields (Weiss fields) byOnsager cavity fields and the calculated spinodal orderingtemperatures which we obtain are consequently closer toagreement with experiment than would be expected froma standard mean-field estimate. All electronic effects canbe treated. In particular, the alteration of charge thatoccurs as the atoms are rearranged is fully incorporated;that is, charge-transfer effects are included. At present ourcalculations do not include lattice-displacement effects—the nuclei can occupy only ideal crystal lattice positions—but the formalism and techniques are sufficiently flexibleto allow their incorporation and work to carry this out is inprogress.

From S(2)(q, T )) it can be determined whether analloy’s tendency to order depends on the magnetic stateof the system and the spin polarization of its electronicstructure. If the system is paramagnetic then the presence of‘local moments’ and therefore a ‘local exchange splitting’will have an effect. In the next subsection we will describetwo case studies in which we show the extent to which analloy’s compositional structure is dependent on whether theunderlying electronic structure is ‘globally’ or ‘locally’ spinpolarized, namely whether the system is quenched from a

† AB denotes an alloy of A and B, which is A-rich.

ferromagnetic or paramagnetic state. We look at iron-richFe–V and Fe–Al alloys.

S(2)(q) peaks at thoseq values which identify the staticconcentration wave for which the system is unstable atlow enough temperatures. An important part ofS(2)(q)involves an electronic state-filling effect and links neatlywith the observation noted earlier that half-filled bandspromote ordered structures whilst nearly filled or nearlyempty states are compatible with systems that cluster whenthey are cooled [28, 35]. This propensity can be totallydifferent depending on whether the electronic structureis spin polarized and hence whether the compositionallydisordered alloy is ferromagnetic or paramagnetic, as is thecase for nickel-rich Ni75Fe25, for example [36]. Earlierremarks in this paper about bonding in alloys and spinpolarization are pertinent in this context. For examplein strongly ferromagnetic alloys like Ni75Fe25, majority-spin electrons which completely occupy the d states ‘see’very little difference between the two types of atomic siteand hence contribute little toS(2)(q) and it is the natureof the minority-spin states which determines the eventualcompositional structure. On the other hand for those alloys,usually BCC based, in which the Fermi energy is pinnedin a valley in the minority density of states the orderingtendency is mostly set by the majority-spin electronicstructure [37].

For a ferromagnetic alloy, an expression for themagneto-compositional cross correlation functionυ(q) =(1/N)

∑j (〈µiξj 〉 − 〈µi〉〈ξj 〉) exp[iq · (Ri − Rj )] can be

derived and evaluated [120, 121]. This turns out tobe a simple productυ(q) = γ (q)α(q), where υij isa convolution of γik = δ〈µi〉/δck and αkj . γik hascomponents

γik = (µA − µB)δik + c δµAi

δck+ (1− c)δµ

Bi

δck.

The last two quantities,δµAi /δck and δµBi /δck, which canalso be related to the spin-polarized electronic structure ofthe disordered alloy, describe the changes to the magneticmoment on a sitei in the lattice if it is occupied byan A and a B atom respectively as the probability ofoccupation is altered on another sitek. In other words,γik quantifies how the sizes of the magnetic momentsdepend upon their chemical environment. It can show howthe saturation magnetizationMs depends on compositionalmicrostructure,Ms{ci}.

We have studied how the magnetic moments in Fe–V and Fe–Cr alloys depend on their local environmentsfrom this framework [121]. Furthermore, if the applicationof a small external magnetic field along the direction ofthe magnetization is considered, expressions dependentupon the electronic structure for the static longitudinalsusceptibilityχL(q) can be found.α(q), υ(q) andχL(q)can be straightforwardly compared with x-ray [122] andneutron scattering [123], nuclear magnetic resonance andMossbauer spectroscopy data. In particular, the crosssections obtained from diffuse polarized neutron scatteringcan be expressed

dσ

dω

∣∣∣∣ζ

= dσN

dω+ ζ dσNM

dω+ dσM

dω

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J B Staunton et al

where ζ = +1 (−1) if the neutrons are polarized(anti-) parallel to the magnetization. The nuclearcomponent dσN/dω is proportional to the atomic short-range order,α(q) (closely related to the Warren–Cowleyshort-range order parameters). The magnetic componentdσM/dω is proportional toχL(q). Finally dσNM/dωdescribes the magneto-compositional correlation functionγ (q)α(q) [124, 125]. By interpreting such experimentalmeasurements by these sorts of calculations and thenunravelling the calculations, the electronic mechanismswhich underlie the correlations can be extracted [120, 126].

5.3. The ASRO and magnetism in iron-rich Fe–V andFe–Al alloys

In this subsection we describe our investigation of theCurie temperatures,Tc, magnetic correlations, from whichexchange constantsA at high temperatures can be extracted,and atomic short-range order present in two iron alloysystems, Fe–Al and Fe–V, at (or rapidly quenchedfrom) temperaturesT0 above any compositional orderingtemperature [127]. We find the ASRO to be sensitive towhetherT0 is above or below the alloys’ magnetic CurietemperaturesTc and also to the details of the paramagneticstate.

5.3.1. FeV. Iron-rich Fe–V alloys have several qualitieswhich make them suitable for an investigation of bothASRO and magnetism. The large difference in the coherentneutron scattering lengths,bFe − bV ≈ 10 fm, anda small size effect associated with alloying make themgood candidates for neutron diffuse scattering experimentalanalyses and also their Curie temperatures (≈1000 K) liein a range which makes it possible to compare and contrastthe ASROs set up both by the ferromagnetic and by theparamagnetic states.

Following some early neutron data from powderedcrystal samples [128], Cableet al [126] carried out somemeasurements at room temperature of the scattering froma single crystal of Fe87V13 along the {1, 0, 0}, {1, 1, 0}and {1, 1, 1} symmetry directions. A saturating magneticfield was applied parallel to the scattering vector whichextinguished the magnetic scattering, allowing the nuclearscattering to be extracted from the data. The structureof the curves is attributed to nuclear scattering connectedwith ASRO, c(1 − c)(bFe − bV )

2α(q). Although thesample had been annealed at and quenched rapidly from1270 K, above the alloy’s Curie temperature of 1180 K,the effective temperature for the ASRO was estimated to be900 K. Pierron-Bohneset al [129] overcame the problemof making such an estimate by carrying out experimentson a single crystal of Fe80V20 in situ. The measurementswere carried out at temperatures above the sample’s CurietemperatureTc.

We calculated the ASROs both of ferromagnetic andparamagnetic Fe87V13 and also paramagnetic Fe80V20. Wealso characterized the magnetic correlations present in theparamagnetic alloys together with estimates of the Curietemperatures. (We had previously determined the saturationmagnetic moments,Ms(c), in the disordered ferromagnetic

alloys [37] in excellent agreement with the values foundexperimentally [128].) An interpretation of both groups’experiments was found as well as an investigation of theeffect of magnetic order upon ASRO in these systems [127].

The most intense peaks in the data from theferromagnetic Fe87V13 alloy occur at{1, 0, 0} and{1, 1, 1},indicative of a β-CuZn (B2) ordering tendency [126].There is also substantial intensity in the form of a doublepeak structure around{ 12, 1

2,12}. We showed [120, 127]

how our ASRO calculations for ferromagnetic Fe87V13

could reproduce all the details of the data. This alloyis of the type in which the chemical potential is pinnedin a trough of the minority-spin density of states so thatthere is substantial hybridization between states associatedwith the two different atomic species (figure 3(b)) and theordering tendency is governed primarily by the majority-spin electrons. These states are roughly half filled toproduce the strong ordering tendency [28, 35]. Thecalculations also showed that part of the structure around{ 12, 1

2,12} can be traced back to the majority-spin Fermi

surface of the alloy.By fitting the direct correlation functionS(2)(q)

in terms of real-space parameters,S(2)(q) = S(2)0 +∑

n

∑i∈n S

(2)n exp(iq ·Ri ) we found the fit to be dominated

by the first two parameters which determine the large peakat {1, 0, 0}. However, the fit also showed that there isa long-range component that we showed to come fromthe Fermi-surface effect. The real-space fit of their dataproduced by Cableet al also showed large negative valuesfor the first two shells followed by a weak long-ranged tail.

In their paper Cableet al maintained that the effectivetemperature for at least part of the sample was indeed belowits Curie temperature. We checked this by carrying outcalculations for the ASRO of paramagnetic (DLM) Fe87V13.Once again, we found the largest peaks to be located at{1, 0, 0} and{1, 1, 1} but a careful scrutiny found that therewas less structure around{ 12, 1

2,12} than there was for the

ferromagnetic alloy. The ordering correlations are alsoweaker in this state. For the paramagnetic DLM state,the local exchange splitting also pushes many anti-bondingstates above the Fermi energy, although this is no longerwedged in a valley in the density of states, see figure 7.This produces a similar, although weaker, compositionalordering mechanism to that of the ferromagnetic alloy. Thereal-space fit ofS(2)(q) also showed the long-ranged tail tobe reduced. Evidently the ‘local-moment’ spin-fluctuationdisorder has smeared the alloy’s Fermi surface and lessenedits effect upon the ASRO.

Pierron-Bohneset al [129] measured neutron diffusescattering intensities from Fe80V20 in its paramagnetic stateat 1473 and 1133 K (the Curie temperature is 1073 K)for scattering vectors both in the(1, 0, 0) plane and inthe (1,−1, 0) plane, following a standard correction forinstrumental background and multiple scattering. They alsofound maximal intensity about{1, 0, 0} and{1, 1, 1} but noadditional structure about{ 12, 1

2,12}. Our calculations of the

ASRO of paramagnetic Fe80V20, very similar to those ofFe87V13, are consistent with these features.

We also studied the type and extent of magneticcorrelations in the paramagnetic state. We found

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Magnetic alloys

30

20

0

10

0-0.2-0.4-0.6Energy(Ry)

(St

ates

per

Ry

per

spin

)D

ensi

ty o

f St

ates

Figure 7. The compositionally averaged density of statesof paramagnetic (DLM) Fe87V13 and its resolution intocomponents associated with Fe (full line) and Al (dottedline) sites weighted by concentration, in units of states peratom Ry−1 per spin.

ferromagnetic correlations which grow in intensity asT isreduced. These lead to an estimate of the Curie temperatureTc = 980 K, which agrees well with the measured valueof 1073 K. (Tc for Fe87V13 of 1075 K also compares wellwith the measured value of 1180 K.) This 10% level ofdisagreement is what might be expected from a mean-fieldtheory.

By modelling the paramagnetic alloy subsequently asa Stoner paramagnet with no local moments and hencezero exchange splitting of the electronic structure, localor otherwise, and repeating the calculations of ASROwe found the maximum intensity at about{ 12, 1

2, 0} andequivalent points. This is in striking contrast both to theDLM calculations and to the experimental data.

In summary, we concluded that experimental data onFe–V alloys are well interpreted by our calculations ofASRO and magnetic correlations. ASRO and hence thealloys’ ordering tendencies are evidently strongly affectedby the local moments associated with the iron sites in theparamagnetic state so that there are only small differencesbetween the topologies of the ASRO of samples quenchedfrom above and belowTc. The main difference is thegrowth of structure in the ASRO around{ 12, 1

2,12} for the

ferromagnetic state. The ASRO strengthens quite sharplyas the system becomes magnetically ordered and it wouldbe interesting if anin situ, polarized-neutron, scatteringexperiment were performed to investigate this.

5.3.2. FeAl. Iron-rich Fe–Al alloys are promisingcandidates for the basis of a new family of superalloys andhave therefore attracted the attention of material scientists[131]. McKamey et al [132] and others have reviewedtheir properties together with the effects of the additionof impurities. One factor limiting the use of these alloysseems to be their poor ductility above 900 K. It is thereforeappropriate to understand the processes which govern howthe atoms order. Their high Curie temperatures (≈900 K)imply that the degree of magnetic order is liable to bean important factor. These alloys are consequently alsointeresting test beds for theories such as ours which treatthe interplay between magnetic and chemical ordering. Thephase diagram [130] shows that, on cooling from the melt,paramagnetic Fe80Al 20 forms a solid solution. The alloy

becomes ferromagnetic at 935 K and then forms an apparentDO3 ordered phase at about 670 K. An alloy with just5% more aluminium orders into a B2 phase from theparamagnetic state at roughly 1000 K and orders into aDO3 phase at lower temperatures. Iron-rich Fe–Al alloysand Fe80Al 20, in particular, have been subjects of a series ofx-ray and neutron diffuse scattering experiments thanks tothe favourable differences in valence and neutron scatteringlength of Fe and Al and recently we compared our ASROcalculations with these data [127].

Epperson and Spruiell [133] investigated the ASROvia x-ray studies on polycrystalline samples with around25% aluminium, heat treated in a variety of ways andquenched from a range of temperatures. They providedevidence that the ordering tendency was towards the DO3

superstructure. This is characterized by concentrationwaves withq = {1, 0, 0} and { 12, 1

2,12}. Schweikaet al

[134] carried out anin situ, unpolarized, neutron experimenton a single crystal of Fe80Al 20 at temperatures in the range823–1073 K. They made an assessment of the magneticscattering in the ferromagnetic regime so that they couldattempt to isolate the nuclear scattering. They presenteddata for the ferromagnetic alloy from the{0, 0, 1} and{2, 1, 1} planes atT = 823 K and the{0, 0, 1} plane at923 K, as well as the{2, 1, 1} plane for the paramagneticalloy at 1073 K. For the ferromagnetic alloy, substantialintensity was found around{0, 0, 1} equivalentq points,which was skewed in the{1, 1, 1} directions, with a possiblesubsidiary peak at{ 12, 1

2,12}. A peculiar feature noted was

that the short-range order peak at{1, 0, 0} decreased onlymarginally with increasing temperature in the rangeT =823–923 K, which the authors put down to a competitionbetween a Fe–Al chemical interaction and the ferromagneticcoupling between Fe moments. From our calculations ofthe ASRO we were able to describe this effect in terms ofthe electronic structure. A later paper by Schweika [135]extended the experimental study further to the paramagneticstate, just above the Curie temperature 935 K. At 1013 K,the scattering intensity from the{1, 1, 0} plane was foundto peak around{1, 0, 0} equivalent points only, but therewas also a distortion along{1, 1, 1}. Evidence for a phasetransition into an ordered phase characterized by{12, 1

2,12}

concentration wavevectors only at approximately 650 Kwas also presented. This is consistent with a B32 transientordered phase observed by Gao and Fultz [136] in Fe75Al 25.

Pierron-Bohneset al [137] carried out x-ray diffusescattering measurements on Fe80Al 20 quenched from 772 Kin the ferromagnetic state. They found maximum intensityat {1, 0, 0} and subsidiary peaks at{ 12, 1

2,12} in accord with

Schweikaet al. They complemented this study [138] bydiffuse, unpolarized neutron scattering measurements ona single crystalin situ for several temperatures in therange 973–1573 K in the paramagnetic regime. Peaks wereobserved at{1, 0, 0} points which were slightly elongatedalong the {1, 1, 1} direction for the lowest temperatureof 973 K. No subsidiary peaks around{ 12, 1

2,12} were

noted. All the data seem to show that there is an apparentweakening of the chemical interactions in the ferromagneticstate with a reduced propensity towards B2-type orderingand an increased tendency towards DO3-type order.

2371

J B Staunton et al

(0,1,0) (1,1,1)

(1,0,1)(0,0,0)

9.03.0

6.0

9.0

6.03.0 4.8

3.2

2.41.6

0.84.8

3.2

2.4

1.6

0.8

(1,1,1)(0,1,0)

(1,0,1)(0,0,0)

(a) (b)

(0,1,0) (1,1,1)

(1,0,1)(0,0,0)

4.04.0

1616

8.08.0

4.04.0

4.04.0

(c)

Figure 8. (a) The ASRO (α(q,T )) for the {1, 1, 0} plane for paramagnetic (DLM) Fe80Al20 at 200 K above the theoreticalspinodal temperature of 2700 K in Laue units. (b) The ASRO (α(q,T )) for the {1, 1, 0} plane for ferromagnetic Fe80Al20 at1000 K (the theoretical spinodal temperature is 485 K) in Laue units. (c) The same as for (b) but for a lower temperature of500 K.

We carried out calculations of the ASRO both inferromagnetic and in DLM-paramagnetic states of Fe80Al 20.Parallel calculations ofχ(q, T ), based on the DLM model,indicated that there are strong ferromagnetic correlationsin paramagnetic Fe80Al 20 and a Tc of 1130 K, in fairagreement with the experimental value of 935 K. Evidentlythis treatment captures the correct order of magnitude

for the energies of the spin fluctuations, just as it doesfor the elemental metals and FeV. Figures 8(a)–(c) showthe ASRO for both the paramagnetic state (αDLM(q, T ))and the ferromagnetic (αFM(q, T )) state of Fe80Al 20 forthe (1, 1, 0) plane. αDLM(q, T ) peaks at{1, 0, 0} exhibitthe same B2-type ASRO as that of the high-temperatureexperimental data [138].

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αFM(q, T ) is presented at two temperatures, oneslightly below the Curie temperature atT = 1000 K(figure 8(b)) and the other at the lower temperature of500 K (figure 8(c)). For the higher temperature,αFM(q, T )peaks at{1, 0, 0} with a strong skewing along the{1, 1, 1}direction. This feature is also observed in the experimentaldata [134]. For the lower temperature, this skewingturns into an incommensurate peak at{0.8, 0.8, 0.8} andlarge intensity from{0.5, 0.5, 0.5} to {1, 1, 1}. The latterfeature in the calculatedαFM(q, T ) compares well with thatobserved in the experimental data [134, 137]. We have alsoinvestigated the nature of the stable and/or metastable statesof the ferromagnetic alloy when it is cooled to just belowthe ordering spinodal temperature [114] using the equationfor the free energy (34), namely in terms ofS(2)(q) whichspecifies the ASRO above. We find that the alloy ordersinto domains of B32 structure consistent with the findingsof Schweikaet al and Gao and Fultz [135, 136].

The structure along{1, 1, 1} owes its origin to asimilar Fermi surface effect to that which caused thestructure around{ 12, 1

2,12} in ferromagnetic Fe87V13. It

may, therefore, be responsible for the tendency of iron-rich ferromagnetic Fe–Al alloys to form DO3 orderedalloys, whereas only B2-type correlations are seen intheir paramagnetic states, where local-moment disorder hasremoved the remnants of sharp structure from the electronicstructure around the Fermi energy.

In this section, we have reviewed both our calculationsof ASROs of Fe–V and Fe–Al alloys and diffuse neutronand x-ray scattering data. Fair agreement is obtained,although, for FeAl, the strength of the calculated ASROfor the paramagnetic alloys is somewhat larger than thatdeduced from experiment. This discrepancy is possiblydue to the theoretical constraint of a rigid lattice. On theother hand, the estimates of the Curie temperatures agreewell with empirical values.

The topology of the ASRO,α(q), in q spaceis accurately described by the theory together withtrends which accompany changes in magnetic state andtemperature and can be understood from the electronicstructure. The relative occupation of bonding and anti-bonding states determined by the ‘global’ or ‘local’exchange splitting is an important consideration. In boththese iron-alloy systems, Fermi-surface effects lead tointensity in α(q) around { 12, 1

2,12} for the ferromagnetic

state which diminishes in the paramagnetic state. Wehave postulated that this is the reason for the tendencyfor ferromagnetic Fe80Al 20 alloys to form DO3 orderedalloys, whilst their paramagnetic counterparts exhibit B2correlations. These encourage the formation of B2 orderedphases for alloys with more aluminium and therefore lowerCurie temperatures.

6. Summary

Recent improvements in the numerical algorithms andavailable computational power have led to rapid develop-ments in the field of micromagnetics [3, 4]. Magnetic prop-erties of alloys can now be modelled in detail. It is timelyfor ‘first-principles’ theories of the electronic structure of

such alloys to feed into these models and to give guid-ance with regard to how the saturation magnetization,Ms ,and the exchange,A, and anisotropy,K, constants dependboth on composition and on temperature. This paper hasattempted to show the extent to which this can be achievedat present and has emphasized the importance of basing thiswork upon a clear picture of the underlying spin-polarizedelectronic ‘glue’ of these systems. This picture can itselfbe tested independently by a range of spectroscopic exper-iments. Spin density functional theory of the inhomoge-neous electron gas provides the formal basis for carryingthis out and part of this paper has described how the rela-tivistic version of this theoretical framework is necessary inorder to describe magnetocrystalline anisotropy. Magneticorder can also have a marked influence on the composi-tional structure in metallic alloys and we have also devoteda portion of the paper to this issue and have included a briefreview of the theory of compositional ordering in alloys.We look forward to an increasingly important role for elec-tronic structure calculations both in the modelling of themagnetic properties of existing magnetic alloys and in thedesign of new magnetic materials.

Acknowledgments

Some of the work reviewed in this paper was supported inpart by the EPSRC in the UK and by the USA Departmentof Energy, Office of Basic Energy Sciences, Division ofMaterials Science grant DEFG02-96-ER45439 through theUniversity of Illinois. It is a pleasure to acknowledgemany discussions and collaborations with B L Gyorffy,G M Stocks and B Ginatempo.

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