J B Stauntonâ , S S A Razeeâ , M F Lingâ¡, D D Johnson§ and. F J Pinski ... Department of Materials Science and Engineering, University of Illinois, Urbana,.
J. Phys. D: Appl. Phys. 31 (1998) 2355–2375. Printed in the UK
PII: S0022-3727(98)72732-9
REVIEW ARTICLE
Magnetic alloys, their electronic structure and micromagnetic and microstructural models J B Staunton†, S S A Razee†, M F Ling‡, D D Johnson§ and F J Pinskik † 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 k 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 for ferromagnetic transition metal alloys. This is work based on the spin density functional theory of the inhomogeneous electron gas which we also discuss briefly. These calculations can be used to provide estimates, from ‘first principles’, of the alloys’ characteristic properties such as the saturation magnetization, Ms , and the exchange, A, and anisotropy, K , constants as well as their Curie temperatures, Tc , which are all important quantities for the micromagnetic modelling of these materials. The electronic reasons for the simple structure of Slater–Pauling curves of Ms versus the number of valence electrons are given. Anisotropy constants, K , can be evaluated only when relativistic effects upon the electronic motion are included. We review the theory of finite-temperature metallic magnetism and highlight how the electronic structure of metals and alloys in their paramagnetic states can still exhibit a local spin polarization originating from the ‘local moment’ spin fluctuations which are excited as the temperature is raised. Finally we show how an alloy’s magnetic state can sharply influence the types of ordered arrangements that the atoms form and conversely how the type of compositional structure can affect Ms and K . We include a discussion of how the compositional structure can be described in terms of static ‘concentration waves’. We illustrate the approach by outlining our recent case studies of two iron-rich alloy systems, FeV and FeAl.
1. Introduction Micromagnetic models of magnetic materials blend classical electrodynamics of continuous media with fundamental aspects of condensed matter theory and are based upon an expression for the free energy, F , of a magnet which is a functional of the magnetization M (r). The magnetization is assumed to vary only its orientation, M (r) = Ms (αx (r), αy (r), αz (r)), Ms being the saturation magnetization and αx , αy and αz its direction cosines. The free energy (in cgs units) is expressed as a sum of four components: Z F [M (r)] = A [(∇αx )2 + (∇αy )2 + (∇αz )2 ] dr Z +
Z Uan (αx , αy , αz ) dr −
H app (r) · M (r) dr
c 1998 IOP Publishing Ltd 0022-3727/98/192355+21$19.50
1 − H 0 (r) · M (r) dr (1) 2 namely an exchange term with exchange constant A, an anisotropy term with Uan = −Kαz2 for uniaxial anisotropy and (2) Uan = K(αx2 αy2 + αy2 αz2 + αz2 αx2 ) for a cubic material (K are magnetocrystalline anisotropy constants), a term describing the interaction with an applied magnetic field and finally a magnetostatic interaction term in which H 0 (r) is the demagnetizing field due to magnetostatic volume and surface charges [1–4]. Most analytical and computational effort in micromagnetics is put into dealing with the last non-local term as accurately as possible and the materials are otherwise characterized by a set of Ms , A and K parameters. There are two important length scales involving these parameters, an ‘exchange length’, lex = (A/Ms2 )1/2 and a domain 2355
J B Staunton et al
wall thickness lw = 2(A/K)1/2 and it is the relative ordering of these and the dimensions of the particles, grains or layers of the ferromagnetic material that are of crucial importance. In this paper we will concentrate on describing how these parameters for magnetic alloys can be determined from ‘first-principles’ condensed matter theory and how they can be represented in terms of the materials’ electronic ‘glue’. These parameters are naturally temperature dependent and the materials’ Curie temperatures, Tc , are also important in setting the scale of this dependence. We will spend some time on this finite-temperature aspect. The properties of magnetic alloys are strongly dependent on their compositional structure. Ms , A, K and Tc all vary when the composition of an alloy is altered and the effective particle or grain size can be set by phase segregation or growth boundaries. Moreover, the magnetic state of an alloy can also have a bearing on the type of compositional structure that can occur. The actual process, namely the kinetics, of ordering and/or phase segregation is itself very important for the design of advanced magnetic materials since it concerns the structural transformations which occur when an alloy is aged. The different structural states which are formed along the transformation path are exploited to achieve special properties. Non-equilibrium systems may undergo several transformations as they relax towards an equilibrium state and it is useful to gain an insight into their dynamics. For example, transient states may appear along the transformation path–states which do not necessarily appear on the equilibrium phase diagram. The diffusion time of atoms in alloys can vary tremendously and the time spent in such metastable states can vary from seconds to geological time scales. Such states can be found by investigating the compositional dependence of the free energy. This paper will also discuss how this can be obtained from ‘first-principles’ electronic structure calculations. Following a selective introduction to the theory of magnetism in metals we begin by discussing how the saturation magnetization, Ms , depends on the composition of binary alloys and then move on to describe how magnetocrystalline anisotropy constants, K, are calculated for these systems. We consider the exchange constants, A, next, together with the determination of Curie temperatures, Tc , and the nature of the paramagnetic state of ferromagnetic metals. We then discuss compositional ordering and phase segregation in alloys which can also be characterized in terms of their electronic structures. We finish the paper with case studies of two iron-rich alloy systems. 2. Metallic magnetism in brief Magnetism and related properties in transition metal alloys are not straightforward to model owing to the electrons’ itinerant nature. States that are derived from the 3d and 4s atomic levels are responsible for the physical properties of the 3d transition metals. Since they are the more spatially extended (of higher principal quantum number) the 4s states determine the metal’s overall size and compressibility whereas the 3d states determine the magnetic properties. 2356
Although, in a sense, they are more localized than the 4s ones, the 3d electrons still propagate throughout the material and the term itinerant magnetism is used. In solids magnetism arises chiefly from electrostatic electron– electron interactions, namely the exchange interactions and in magnetic insulators these can be described rather simply. ‘Spin’ operators, sˆ i , can be specified by associating electrons appropriately with particular atomic sites so that the famous Heisenberg–Dirac Hamiltonian can be used to describe the behaviour of these systems. The Hamiltonian takes the following form: X Hˆ = − Jij sˆ i · sˆj (3) in which Jij is an exchange integral involving atomic sites i and j . In metallic systems it is not possible to distribute their itinerant electrons in this way and such simple pairwise site–site interactions cannot be defined. Instead metallic magnetism is a complicated many-electron effect and has attracted significant effort over a long period to understand and describe it. A widespread approach is to map this problem onto one involving independent electrons moving in the fields set up by all the other electrons. It is this aspect which gives rise to the spin-polarized band structure that is often used to explain the properties of metallic magnets such as the non-integer values of Ms per atom in multiples of the Bohr magneton µB . This picture is not always sufficient and Herring [5], amongst others, pointed out that certain components of metallic magnetism are more naturally described using concepts of localized spins which are strictly relevant only to magnetic insulators. Later on in this paper we will discuss how the two pictures have been combined to explain the temperature dependence of the magnetic properties of bulk transition metals and their alloys and quantify the exchange constants A. In metals, magnetism is intricately connected with other properties via their spin-polarized electronic structures. For example, some materials exhibit a small thermal expansion coefficient below the Curie temperature, Tc , a large forced increase in volume when an external magnetic field is applied, sharp decreases of spontaneous magnetism and of Tc when pressure is applied and large changes in the elastic constants on going through Tc . These are the famous ‘Invar’ effects, so called because they occur in the FCC Invar alloys Fe–Ni (65% Fe), Fe–Pd and Fe–Pt [6]. As mentioned in the introduction the compositional order of an alloy is often subtly linked with its magnetic state. Ni75 Fe25 (Permalloy) is paramagnetic at high temperatures, it becomes ferromagnetic at about 900 K and then, when just 100 K cooler, it chemically orders into the LI2 Cu3 Au-like phase. When it is annealed in a magnetic field it develops directional chemical order [7]. Magnetic short-range correlations above the Curie temperature Tc and magnetic order below weaken and alter the chemical ordering in iron-rich Fe–Al alloys so that a ferromagnetic Fe80 Al20 alloy forms a DO3 ordered structure at low temperatures whereas paramagnetic Fe75 Al25 forms a B2 ordered 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
Magnetic alloys
are sensitive to the compositional structure. For example, ordered Ni–Pt (50%) is an anti-ferromagnetic alloy [8] whereas its disordered counterpart is ferromagnetic. We discuss such magneto-compositional effects later on in the paper. Now the fundamental electrostatic interactions do not couple the direction of magnetization to any spatial direction (see equation (3)), and they therefore fail to give a basis for a description of magnetic anisotropic effects which define the anisotropy constants K. A description of these effects requires a relativistic treatment of the electrons’ motions and a part of this paper is assigned to this topic in relation to transition metal alloys. For the two decades since the ‘landmark’ papers of Callaway and Wang [9], there has been a consensus that spin-polarized band theory, established within the spin density functional (SDF) formalism (see reviews [10–13]) provides a reliable description of magnetic properties of transition metal systems at low temperatures [14–16]. (In fact, all magnetic interactions can be incorporated into the same framework, including magnetic anisotropic effects and magnetostatic effects, by considering the more general relativistic, current density functional theory.) This is essentially the modern version of the Stoner–Wohlfarth theory [17] in which the magnetic moments are assumed to originate from itinerant d electrons whose spins are aligned by exchange interactions. It provides a mechanism for generation of non-integer moments, Ms , together with a plausible account of the many-electron nature of magneticmoment formation at T = 0 K. This theory transforms the many-electron problem into one of independent electrons moving in effective potentials v eff (r) and magnetic fields B eff (r) which themselves depend on all the other electrons. The consequent electronic structure is spin polarized by the effective magnetic fields which themselves are set up self-consistently by the magnetization density produced by this spin-polarized electronic structure. A simple illustration of the spin-polarized band structure picture can be given by using the Hubbard Hamiltonian together with the Hartree–Fock approximation. The Hubbard Hamiltonian has the form X 1 X † † † (0 δij +tij )ai,σ aj,σ + I ai,σ ai,σ ai,−σ ai,−σ (4) Hˆ = 2 ij i,σ †
in which ai,σ and ai,σ are the creation and annihilation operators for electrons with spin σ (= ↑ (↓)) at a site i, 0 a site energy related to an atomic-like d energy level, tij a hopping parameter defining the d band width and I the many-body Hubbard parameter representing the intra-site Coulomb interactions. By making the approximation †
†
†
†
ai,σ ai,σ ai,−σ ai,−σ ' ai,σ ai,σ hai,−σ ai,−σ i †
(5)
where hai,σ ai,σ i is the thermodynamic average of the † number operator ai,σ ai,σ and writing the number of −σ electrons as †
hai,−σ ai,−σ i = 12 n¯ i − 12 µ¯ i σ
in terms of the total number of electrons and magnetization on a site i n¯ i = n¯ i,↑ + n¯ i,↓ µ¯ i = n¯ i,↑ − n¯ i,↓ we can obtain the Hartree–Fock single electron Hamiltonian below, where the fields I n¯ i and I µ¯ i are determined selfconsistently from its eigenstates: X 1 1 † Hˆ H F = 0 + I n¯ i − I µ¯ i σ δij + tij ai,σ aj,σ . 2 2 ij (6) Evidently this approach provides a picture of a σ = ↑ electron ‘seeing’ a different potential associated with a lattice site i from that seen by a σ = ↓ electron so that the bands are ‘exchange split’. When the ‘first-principles’ SDF version of this picture is straightforwardly extended to higher temperatures, it fails severely. For example, the Curie temperature Tc is much too high, there are no ‘local’ moments above Tc which are indicated from neutron scattering measurements and the paramagnetic susceptibility does not easily follow a Curie–Weiss law which is nonetheless commonly observed. Evidently the interaction between the thermally induced spin-wave excitations together with its effect upon the underlying electronic structure has been omitted. We address this problem in the context of magnetic alloys later on in this paper. In the next section we show how the famous Slater– Pauling curve [18] which describes the simple relation of the average magnetic moment per site, Ms , to the number of valence electrons can be understood in terms of simple features from the alloys’ spin-polarized electronic structures. The ground state properties of ferromagnetic, randomly disordered alloys can be described quantitatively by SDF calculations. We then move on to the relativistic generalization, discuss magnetic anisotropic effects in disordered alloys and outline calculations of their anisotropy constants, K. 3. T = 0 K magnetism in metals and alloys Over 30 years ago, Hohenberg and Kohn [19] proved a remarkable theorem, namely that the ground state energy of a many-electron system is a unique functional of the electron density n(r) and is a minimum when it is evaluated for the true ground state density. Later Kohn and Sham [20] developed various aspects of this theorem and provided the important basis for further practical applications of the density functional theory. In particular, they showed how a set of single-particle equations could be written down to include, in principle, all the effects of the correlations among the electrons of the system. These theorems provide the basis of the modern theory of the electronic structure of solids. The general idea behind this scheme is to approach the ground state of the many-electron problem via an effective one-electron picture similar in spirit to the Hartree–Fock scheme which we outlined earlier for the Hubbard model. 2357
The theory proves that the relevant one-body effective potential can, in principle, contain all the effects of electron correlations although, in practice, approximations must be made. All the theorems and methods of the density functional (DF) formalism were soon generalized [21, 22] to deal with cases in which spin-dependent properties play an important role, such as in magnetic systems (SDF). The energy thus becomes a functional both of the density and of the local magnetic density m(r). The proofs of these theorems are provided in the original details and there have been many developments of a formal nature [12, 23]. The many-body effects of the complicated quantummechanical problem are buried in the so-called exchangecorrelation part of the energy functional Exc [n(r), m(r)]. The exact solution is intractable for macroscopic systems and some approximation must be made. The so-called ‘local approximation’ is the most widely used owing to its simplicity and its success in describing the ground state and equilibrium properties of a great many different types of materials. The local approximation (LSDA) takes as its starting point the energy of a uniformly spin-polarized homogeneous electron gas εxc (n(r), |m(r)|) [21, 24] so that Exc can be written in the form Z (7) Exc [n, m] ≈ dr n(r)εxc (n(r), |m(r)|). The functional derivative of this quantity with respect to m(r) provides the effective magnetic fields for the singleelectron equations, namely the spin-polarized band structure discussed in the introduction: Z n(r) v eff [n, m; r] = V ext (r) + e2 dr 0 |r − r 0 | xc δE [n, m] (8) + δn(r) δE xc [n, m] (9) B eff [n, m; r] = B ext (r) + δm(r) where V ext describes an external potential such as a lattice array of nuclei and B ext an external magnetic field. The electron density and magnetization are given by Z εF X n(r) = dε tr(φi∗ (r, ε)φi (r, ε)) (10) i
Z m(r) =
εF
dε
X
tr(φi∗ (r, ε)σφ ˜ i (r, ε))
(11)
i
where the φi (r, ε) obey the Schr¨odinger–Pauli (Kohn– Sham) equation (−∇ 2 + v eff (r)1˜ − σ˜ · B eff (r))φi (r, ε) = εφi (r, ε) (12) where εF is the system’s Fermi energy. 3.1. Elemental metals Electronic structure (band theory) calculations using the LSDA for the pure crystalline state are routinely performed these days. For the elemental magnetic, transition metals, nearly rigidly exchange-split, spin-polarized bands are 2358
Density of States (states per Ry per spin)
J B Staunton et al 30 20 Majority Spin 10 0 10 Minority Spin 20 30 -0.6
-0.4
-0.2
0
0.2
0.4
Energy (Ry) Figure 1. The spin-polarized electronic density of states of BCC Fe in units of states Ry−1 using the one-electron potentials from [15].
obtained which are expected from the simpler Hubbardmodel treatment described in the last section. Examples of band theory calculations for the magnetic 3d transition metals BCC iron and FCC nickel can be found in the book by Moruzzi et al [15]. The figure on p 170 of their book (we show a similar figure in figure 1) shows the density of states of BCC iron as a function of energy. The densities of states for the two spins are almost (but not quite) rigidly shifted. As is typical for BCC structures, the d band has three major peaks. The Fermi energy resides in the top of the d bands for the so-called majority spins between the 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 are completely occupied and the Fermi energy lies in the prominent peak in the minority spin d density of states. The d band width has been a topic for close scrutiny over the years owing to the width extracted from photoemission measurements being much smaller than that of band structure calculations. This serves to emphasize the fact that the SDF theory is a theory for the ground state whereas excited states are probed by spectroscopic measurements. In particular the theory does not correctly describe the correlated motion of the electrons as they are excited into states necessary for them to leave the metal. On the other hand all the ground state properties such as Ms and the lattice spacing are in very good agreement with experimental values. The nearly rigidly split spin-polarized bands of these 3d elemental transition metal magnets are actually special cases. We now show how this simple picture is lost as soon as the electronic structures of ferromagnetic alloys are considered. We focus on compositionally disordered alloys for this purpose, since these provide the starting point for much of the discussion in this article.
Magnetic alloys
3.2. Compositionally disordered alloys The self-consistent Korringa–Kohn–Rostoker coherent potential approximation (SCF-KKR-CPA) [25] is a mean field adaptation of the SDF-LSDA to systems with substitutional disorder such as solid solution alloys. For these systems, the obvious but computationally intractable procedure for finding their ground state properties would involve solving self-consistently the one-electron Kohn– Sham equations for all possible nuclear configurations followed by the averaging of the observables of interest (particle and magnetization density, total energy and so on) over the appropriate ensemble of configurations. For a binary alloy Ac B1−c we introduce an occupation variable ξi which takes on the value 1 if the lattice site i is occupied by an A nucleus and 0 if the site is occupied by a B nucleus. A configuration is specified by assigning values to these variables for each site, namely {ξi }. For an atom of type α = A(B) on site k, the effective one-electron potential, vk,α (r, {ξi }), and magnetic field, Bk,α (r, {ξi }), which enter the Kohn–Sham equations are not independent of the surroundings of site k and depend on all the occupation variables. To find an ensemble average of an observable, we must first find the self-consistent solution to the Kohn–Sham equations for each configuration. By summing the results weighted by the correct probability factor the appropriate ensemble average is obtained. The KKR-CPA was invented to bypass the computational difficulties that would be encountered if this sequence of calculations were to be carried out. The first simplifying assumption of the KKR-CPA is that the occupation of a site either by an A or by a B nucleus is independent of the occupancy of the surrounding sites. This means that shortrange order is neglected for the purposes of calculating the electronic structure and the solid solution is approximated by a random substitutional alloy. The second step is to invert the order of solving the Kohn–Sham equations and the configurational averaging. Consequently one finds a set of Kohn–Sham equations that describe an ‘average’ medium. In the spirit of a mean field theory, the local potential and magnetic field are replaced by vk,α (r) and Bk,α (r) which depend upon the average particle densities and magnetizations on all surrounding sites and depend explicitly only on the occupancy of site k which is of type α. The motion of an electron, on the average, through a lattice of these potentials, which are randomly distributed with the probability of c that a site is occupied by an A atom and of 1 − c that it is occupied by a B atom, is obtained from the solution of the Kohn–Sham equations using the CPA [26]. In this approximation a lattice of identical effective potentials is found such that the motion of an electron through this ordered array closely resembles the motion of an electron on the average through the disordered alloy. The CPA is the requirement that the substitution of a single site of the lattice of these effective potentials either by an A or by a B atom produces no further scattering of the electron on the average. It is then possible to develop a spin density functional theory and calculational scheme in which the partially averaged electronic densities, nA (r) and nB (r), magnetization densities, µA (r) = |mA (r)| and µB (r) = |mB (r)|, associated with the A and B sites are determined self-consistently.
The average Ms , total energies and other equilibrium quantities are also evaluated [25]. Both x-ray and neutron scattering data from solid solutions show the existence of Bragg peaks which define an underlying ‘average’ lattice. This symmetry is apparent in the average electronic structure given by the CPA. The Bloch wavevector is still a useful quantum number but the average Bloch states also have a finite lifetime as a consequence of the disorder. This leads to changes in the resistivity, broadening of spectra and so on. 3.3. Alloy electronic structure and Slater–Pauling curves: Ms versus concentration Before the reasons for the loss of the conventional Stoner picture of rigidly exchange-split bands can be set out, we must first describe some typical features of alloy electronic structure. Much has been written on this subject which has demonstrated clearly how closely these features are connected with the phase stability of the system. An overview of this subject can be found in the books by Pettifor [27] and Ducastelle [28] and articles by Gyorffy et al [29, 30], Connolly and Williams [31], Zunger [32], Staunton et al [33] and many others. Let us consider two elemental d electron densities of states, each with approximate width W , say, with one centred on an energy εA and the other at εB . εA and εB are related to atomic-like d energy levels. If εA − εB W then the alloy’s densities of states will be a ‘split-band’like one [25] and, in Pettifor’s language, an ionic bond is established as charge flows from the A atoms to the B atoms in order to equilibrate the chemical potentials. The virtual bound states associated with impurities in metals are rough examples of split-band behaviour. If, however, εA − εB W , then the alloy’s electronic structure is now classed as being a ‘common-band’-like one. There is now large-scale hybridization between states associated with the A and B atoms and each site in the alloy is nearly charge neutral insofar as an individual ion is efficiently screened via the metallic response function of the alloy [34]. Of course, the actual interpretation of the detailed electronic structure of an alloy involving many bands is always a complicated mixture of these two models. In both cases, half filling of the bands reduces the total energy of the system with respect to the phase-separated case [27, 28, 35] and an ordered alloy will form at low temperatures. When magnetism is added to the problem, the difference between the exchange fields associated with each type of atomic species must also be considered. For majority spin electrons, the rough measure of the degree of ‘split-band’ or ‘common-band’ nature of the density ↑ ↑ of states is governed by (εA − εB )/W and a similar ↓ ↓ measure (εA − εB )/W for the minority spin electrons. If the exchange fields are rather different then the bands are common-band-like ones for electrons of one spin polarization, whilst the bands for the others will be more ‘split-band’-like ones. The upshot is a spin-polarized electronic structure which is not set up by a rigid exchange splitting. Figures 2(a) and 3(a) show schematic energy level diagrams for Ni–Fe and Fe–V alloys respectively. 2359
J B Staunton et al
Atoms
Energy
Alloy
Atoms
Energy
Fe
Alloy
Fe Fe, Ni Antibonding
Ni
Antibonding Fe, V
Fe
V
V
Ni Fe, Ni
Fe, V Bonding
Fe, Ni Bonding
Fe
Fe
(a)
30
Density of States (States per Ry per spin)
Density of States (States per Ry per spin)
(a)
20 10
Majority Spin
0 10 Minority Spin
20 30 -0.6
-0.2 -0.4 Energy(Ry)
0
0.2
30 20 10
Majority Spin
0 10
Minority Spin
20 30 -0.6
-0.2 -0.4 Energy(Ry)
0
0.2
(b)
(b)
Figure 2. (a) A schematic energy level diagram for Ni–Fe alloys. (b) The majority- and minority-spin, compositionally averaged densities of states of ferromagnetic Ni75 Fe25 and its resolution into components weighted by concentration, in units of states per atom Ry−1 per spin. The full line represents the density of states associated with the nickel sites and the dotted line the iron sites.
Figure 3. (a) A schematic energy level diagram for Fe–V alloys. (b) The majority- and minority-spin, compositionally averaged densities of states of ferromagnetic Fe87 V13 and its resolution into components weighted by concentration, in units of states per atom Ry−1 per spin. The full line represents the density of states associated with the iron sites and the dotted line the vaadium sites.
According to Hund’s rules it is often energetically favourable for the majority spin d states to be fully occupied and there are many examples for which, at the cost of a 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 electronic structure must accomplish is to explain simply why the average magnetic moments per atom of so many alloys, Ms , fall on the famous Slater–Pauling curve, when plotted against the alloys’ valence electron per atom ratios. The usual Slater–Pauling curve for the 3d row [18] consists of two straight lines. The plot rises linearly from the beginning of the 3d row, abruptly changes the sign of its gradient and then drops linearly to the end of the row. There are some important groups of compounds and alloys whose parameters do not fall on this line but, for these systems also, there is often some simple pattern. It is easy to see why ferromagnetic alloys of late transition metals, which are characterized by completely filled majority spin d states (such as figure 2(b)), are located
on the negative-gradient straight line. Their magnetization per atom, M = N↑ − N↓ , where N↑(↓) describes the occupation of the majority (minority) spin states, can be expressed in terms of the number of electrons per atom Z so that M = 2N↑ − Z. The occupation of the s and p states scarcely changes across the 3d row and M = 2Nd↑ − Z + 2Nsp↑ which equals 10 − Z + 2Nsp↑ . There are many other systems, most commonly BCCbased alloys, which are not strong ferromagnets in this sense of having filled majority spin d bands but possess a similar quality. The chemical potential (or Fermi energy at T = 0 K) is pinned in a deep trough in the minority spin density of states [37, 38]. Figure 1 shows this for pure BCC iron, the chemical potential sitting in a trough in the minority spin density of states. Figure 3(b) shows another example in an iron-rich iron–vanadium alloy. The other major portion of the Slater–Pauling curve of a positivegradient straight line can be explained by using this aspect of the electronic structure. The pinning of the chemical potential in a valley of the minority spin d density of states constrains Nd↓ to be roughly 3 in all these alloys.
2360
Magnetic alloys
In this circumstance the magnetization per atom M = Z − 2Nd↓ − 2Nsp↓ = Z − 6 − 2Nsp↓ . Further discussion can be found in [38–41]. Here we illustrate some of the remarks made by describing briefly electronic structure calculations of two compositionally disordered alloys, one from each side of the Slater–Pauling plot. We begin with Ni–Fe. The d band widths of iron and nickel are comparable and in both elements the Fermi energy is placed near or at the top of the majority-spin d bands. That there is a larger moment in Fe than there is in Ni, however, is shown via the exchange splitting for Fe being larger. To obtain a rough idea of the electronic structures of Nic Fe1−c alloys we imagine aligning the Fermi energies. The atomic-like d levels of the two which mark the centre of the bands would be at the same energy for the majority spin electrons whereas for the minority spin electrons the levels would be rather different, reflecting the differing 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 of Ni75 Fe25 calculated by the SCF-KKR-CPA. The majority spin density of states is very sharply structured which is evidence that, in this compositionally disordered alloy, the majority spin electrons ‘see’ very little difference between the two types of atom. For the minority spin electrons the situation is to the contrary. In this case the density of states is a ‘split-band’-like one owing to the separation of energy levels (from the different exchange splittings of Ni and Fe) and the compositional disorder. As pointed out earlier, the majority spin d states are fully occupied and this feature persists for all FCC Nic Fe1−c alloys with c greater than 0.4 and has the consequence that the alloys’ average saturation magnetizations, Ms , fall nicely on the negative-gradient slope of the Slater–Pauling curve. For concentrations less than 35% and prior to the martensitic transition into the BCC structure at around 25% (the famous ‘Invar’ alloys), the Fermi energy is forced into the top of majority spin d states and the alloys deviate from the Slater–Pauling curve. Figure 3(a) for BCC Fec V1−c shows another simple schematic energy level diagram. As before the d energy levels of Fe are exchange split, showing that it is energetically favourable for pure BCC Fe to have a net magnetization. There is no exchange splitting in pure vanadium. Just like in the case of the Nic Fe1−c alloys we assume charge neutrality and align the two Fermi energies. Now the vanadium d levels are much closer in energy to the minority spin d levels of iron than they are to its majority-spin ones. On alloying the two metals in a BCC structure, the bonding interactions have a larger effect on the minority-spin levels than they do on those of the majority spin owing to the smaller energy separation. In other words Fe induces an exchange splitting on the V sites, decreasing the kinetic energy, which results in the formation of bonding and anti-bonding minority-spin alloy states. Fewer majority-spin V-related d states than minority-spin d states are occupied so that the moments on the vanadium sites are anti-parallel to the larger ones on the Fe sites. The moments cannot persist for concentrations of iron less than 30% since the Fe-induced exchange splitting on the vanadium sites decreases together with the average
number of Fe atoms surrounding a vanadium site in the alloy. As for the majority-spin levels, well separated in energy, ‘split bands’ form, namely states which reside mostly on one constituent or the other. Figure 3(b) shows the spin-polarized density of states of an iron-rich Fe–V alloy determined by the SCF-KKR-CPA method whereby it is possible to identify all these features. So far everything has been discussed with respect to a spin-polarized but non-relativistic electronic structure. We now briefly examine the relativistic extension to this approach and show how it can be used to describe the important magnetic property magnetocrystalline anisotropy and to calculate values of the anisotropy constants, K. 3.4. Relativistic effects and magnetocrystalline anisotropy: the anisotropy constants K In recent years, magnetocrystalline anisotropy of ferromagnetic transition metal materials has become the subject of intensive theoretical and experimental study because of the technological implications for high-density magneto-optical storage media [42] as well as for the understanding of magnetic properties in general that arises from micromagnetic modelling. A detailed understanding of the mechanism of this anisotropy is needed. Several experimental studies have been devoted both to understanding its origin and also to correlating it to other physical properties of the materials [43–51]. On the other hand, ab initio theoretical approaches which can explain the underlying physics are now only just becoming feasible. Over the past few years, considerable progress has been made in this direction [52–61] and here we will describe our work in this sphere on disordered alloys. In a crystalline solid, the equilibrium direction of the magnetization is along one of the crystallographic directions. The energy required to alter the magnetization direction is called the magnetocrystalline anisotropy energy (MAE). Micromagnetic theory uses an expansion of the MAE which satisfies the symmetry properties of the underlying crystalline lattice, has the correct set of easy and hard directions and is expressed in terms of a small number of anisotropy constants K. Brooks [62] first suggested that the origin of this anisotropy is the interaction of magnetization with the crystal field, namely the spin–orbit coupling. These days both the origin of this effect and the magnetostatic effects which determine domain structure can be shown from the relativistic generalization of the SDF theory. The formal starting point is the fundamental quantum electrodynamics of an electron interacting with an electromagnetic field. The ground state energy is now the minimum of a functional of the charge and current densities—a minimization which is achieved, in principle, by the self-consistent solution of a set of Kohn–Sham Dirac equations for electrons moving in fields dependent on the charge and current densities. Once again approximations for the exchange-correlation part of the functional have to be made. By the formal trick of carrying out a Gordon decomposition of the current into what are orbital and spin components, the effects of spin–orbit coupling upon the electronic structure can 2361
be represented and magnetostatic shape anisotropy also described from within the same theoretical framework [63]. Most theoretical investigations of magnetocrystalline anisotropy and calculations of the anisotropy constants K place their emphasis on spin–orbit coupling effects using either perturbation theory or a fully relativistic theory. Examples of such calculations are described in [52, 59] for transition metals, [55, 56] for ordered transition metal alloys and [53, 54, 57] for layered materials, with varying degrees of success. Typically the total energy or the single-electron contribution to it (if the force theorem is used [64]) is calculated for two magnetization directions separately and then the MAE is obtained by subtracting one from the other: Z
εF1
Z εne1 (ε) dε −
3.5 x=0.50 x=0.25 x=0.75
3.0 2.5 2.0
Co25Pt75
Co50Pt50
1.5 1.0
Co75Pt25
0.5 0 0
300
600
900
1200
1500
Temperature (K)
εF2
εne2 (ε) dε
(13)
where εF1 and εF2 are the Fermi energies when the system is magnetized along the directions e1 and e2 respectively and ne1(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 energy of the system. Therefore, it is numerically more precise to calculate the difference directly [65]. Strange et al [66, 67] have developed a relativistic spin-polarized version of the Korringa–Kohn–Rostoker (SPR-KKR) formalism to calculate the electronic structure of solids and Ebert and Akai [68] have extended this formalism to disordered alloys by incorporating a coherent-potential approximation (SPRKKR-CPA) and used it to describe the electronic structure and other related properties such as magnetic circular x-ray dichroism, hyperfine fields and the magneto-optical Kerr effect [69]. Strange et al [61] and more recently we [65] formulated a theory to calculate the MAE of elemental solids within the SPR-KKR scheme and applied it to Fe and Ni [61]. We subsequently extended this work to disordered alloys and details can be found in [70, 71]. Here we summarize our findings for two alloy systems, Co–Pt and Ni–Pt. Coc Pt1−c alloys are important from the point of view of the fundamental physics of magnetic anisotropy; a large spin magnetic moment is associated with the Co sites, spin polarizing the electronic structure, whereas spin–orbit coupling is stronger on Pt sites. Most experimental work on Co–Pt has been on the ordered tetragonal phase, which has a very large magnetic anisotropy ('400 µeV) and the magnetic easy axis is along the c axis [45, 46]. Although there does not seem to have been any experimental work on the bulk disordered FCC phase of these alloys, some results have been reported for thin films [47–51]. It is found that the magnitude of the MAE is more than one order of magnitude smaller than that of the bulk ordered phase and that the magnetic easy axis varies with the film’s thickness. A study of the MAE of the bulk disordered alloys provides some insight into the mechanism of magnetic anisotropy in the ordered phase as well as in thin films. Figure 4 shows the energy difference of disordered FCC Coc Pt1−c alloys for c = 0.25, 0.5 and 0.75 along the h0, 0, 1i and h1, 1, 1i directions as a function of temperature in the range 0–1500 K. This energy difference, the MAE, is 2362
Magnetocrystalline Anisotropy (µ eV)
J B Staunton et al
Figure 4. The magnetocrystalline anisotropy energy (K /3) defined as the difference in energy between systems magnetized along the h0, 0, 1i and h1, 1, 1i directions of the crystal for Coc Pt1−c alloys for c = 0.25, 0.50 and 0.75, as a function of the temperature.
equal to K/3 if the anisotropy energy is assumed to have the simple form assumed in equation (2) in the introduction for a cubic system. We note that, for all the three compositions, the MAE is positive at all temperatures, implying that the magnetic easy axis is always along the h1, 1, 1i direction of the crystal, although the magnitude of the MAE decreases with increasing temperature. The magnetic easy axis of FCC Co is also along the h1, 1, 1i direction. Thus, alloying with Pt does not alter the magnetic easy axis, namely the sign of K. The equiatomic composition has the largest MAE, which is about 3.0 µeV at 0 K. Addition of Pt to Co results in a monotonic decrease in the average magnetic moment, Ms , of the system with the spin–orbit coupling becoming stronger. This trade-off between spin polarization and spin–orbit coupling is the main reason for the MAE being largest for the equiatomic composition. The magnetocrystalline anisotropy of a system can be understood in terms of its electronic structure. Figure 5(a) shows the spin-resolved density of states on Co and Pt atoms in Co50 Pt50 magnetized along the h0, 0, 1i direction. The Pt density of states is rather structureless except around the Fermi energy where there is spin-splitting due to hybridization with Co d bands. When the direction of magnetization is orientated along the h1, 1, 1i direction of the crystal the electronic structure changes, although the difference is quite small in comparison with the overall density of states. Figure 5(b) depicts this density-of-states difference. In the lower part of the band, dominated by Pt, the difference between the two is small, whereas it becomes quite oscillatory at higher energies dominated by the Co d band complex. There are spikes in this difference at energies at which there are also peaks in the Co-related part of the density of states. Owing to the oscillatory nature of this curve, calculation of the magnitude of the MAE involves taking an integral over the product of the energy and this difference up to the Fermi energy (equation (13)) and is quite small; the two large peaks around 2 and 3 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
Magnetic alloys 3
Co-Majority Spin Co-Minority Spin Co 50 Pt Pt-Majority Spin 50 Pt-Minority Spin
Theory Experiment
Co-Minority Spin 2
2 Co-Majority Spin
1
Pt-Maj. Pt Pt-Min.
0 -8
-10
-6
-4
0
-2
2
Difference Density of States (states/eV)
Energy (eV) (a)
Magnetocrystalline Anisotropy Energy
Density of States (states/eV)
3
1
0
-1
-2
-3
0.010
-4 0
0.005
0.1
0.2 Concentration (Pt)
0.3
0.4
Figure 6. Experimental and theoretical values of the magnetocrystalline anisotropy energy (K /3) of Nic Pt1−c as a function of concentration. The lines are guides to the eye.
0
-0.005
-0.010 -6
-2
-4
0
2
Energy (eV) (b) Figure 5. (a) Spin-resolved densities of states for Co and Pt in FCC Co0.5 Pt0.5 alloy for magnetization along the h0, 0, 1i direction of the crystal. The vertical dotted line indicates the Fermi level. (b) The difference between the densities of states of FCC Co0.5 Pt0.5 alloy for magnetization along the h0, 0, 1i and h1, 1, 1i directions of the crystal. The vertical dotted line indicates the Fermi level. The separation between the two Fermi levels is less than the thickness of this line.
Fermi energy (in this case, 4 eV below) can also contribute to the MAE, not just those around the Fermi surface. In contrast to what we have found for the disordered FCC phase of Coc Pt1−c alloys, the MAE of ordered tetragonal Co–Pt alloy is large ('400 µeV), some two orders of magnitude greater than what we find for the disordered Co50 Pt50 alloy. Moreover, the magnetic easy axis is along the c axis [45]. Theoretical calculations of the MAE for ordered tetragonal CoPt alloy [55, 56] based on scalar relativistic methods and perturbation theory for the spin–orbit coupling do reproduce the correct easy axis but overestimate the MAE by a factor of two. It is not clear whether it is the atomic ordering or the loss of cubic symmetry of the crystal in the tetragonal phase or both which is responsible for the altogether different
magnetocrystalline anisotropies in disordered and ordered Co–Pt alloys. It is likely to be a combined effect of the two and we are studying currently the effect of atomic shortrange order (ASRO) on the magnetocrystalline anisotropy and anisotropy constants, K, of alloys as the next step. Our preliminary results show that an alloy with atomic shortrange order of the type which would lead to the ordered Co–Pt alloy found at low temperatures has a greater MAE. Interestingly we also find that a Co–Pt alloy grown with a layering SRO has an easy axis perpendicular to the layers and a much greater magnitude of the MAE still. Experimental results on ferromagnetic Ni1−c Ptc alloys (which are disordered and FCC) reveal that the sign of K changes, with the easy axis of magnetization changing from h1, 1, 1i to h0, 0, 1i with the addition of Pt, so that the magnitude of the MAE is greatest for around 15% Pt [72]. In figure 6 we show our calculations together with the experimental results; both show that addition of Pt to Ni alters the easy magnetic axis, although the minimum in the MAE at around 15% Pt is less pronounced in the calculations than it is in the experimental results. The significant underestimate of the small MAE for small c is partly due to our neglect of orbital polarization, although the calculation of such tiny energy differences leading to the MAE of these dilute alloys and pure Ni is also sensitive to several technical subtleties. In line with other calculations for Ni, we find the correct easy axis of magnetization but the magnitude of the MAE is too small. It is worth pointing out that, although both FCC Ni and FCC Co have negativesign Ks so that the magnetic easy axes are h1, 1, 1i, addition of Pt to Ni changes the sign of K but this does not occur when Pt is added to Co and only the magnitude of K is altered. In this section we have attempted to show how spin density functional theory, with its emphasis on spin2363
J B Staunton et al
polarized electronic structure, provides a useful description of magnetic properties of transition metal systems at low temperatures and can be used to estimate the saturation magnetization Ms and also the anisotropy constants K. Its major approximation is the way in which the exchange and correlation part of the energy of these interacting electron systems is obtained from examination of the homogeneous interacting electron gas. At finite temperatures this local approximation neglects the effects of thermally excited spin-wave excitations and fails to provide even a qualitative description of the magnetic properties. The next section shows how this shortcoming is redressed. We also point out how the exchange constants A can be determined from the same general framework. 4. Finite-temperature metallic magnetism: Tc and exchange constants A Soon after Hohenberg and Kohn, Kohn and Sham published their papers which launched density functional theory and then Mermin devised the formal structure of the generalization of DFT to finite temperatures [73]. For magnetic systems in an external potential, V ext , and external magnetic field, B ext , it can be proved that, in the grand canonical ensemble at a given temperature T and chemical potential ν, the equilibrium particle density n(r) and magnetization density m(r) are determined by the external potential and magnetic field. The correct equilibrium particle and magnetization densities minimize the Gibbs grand potential : Z Z [n, m] = dr V ext (r)n(r) − dr B ext (r) · m(r) ZZ e2 n(r)n(r 0 ) + dr dr 0 2 |r − r 0 | Z −ν dr n(r) + G[n, m] (14) where G is a unique functional of charge and magnetization densities at a given T and ν. The variational principle now states that is a minimum for the equilibrium n and m and G 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 and entropy of a system of non-interacting electrons with densities n and m at a temperature T . xc is the exchange and correlation contribution to the Gibbs free energy. The minimum principle can be shown to be identical to the corresponding equation for a system of non-interacting electrons subject to an effective potential V˜ eff : Z n(r 0 ) V˜ eff [n, m] = V ext 1˜ − B ext · σ˜ + e2 1˜ dr 0 |r − r 0 | δxc δxc + · σ˜ (16) +1˜ δn(r) δm(r) which satisfy the following set of equations h ¯2 2 ˜ ∇ 1 + V˜ eff φi (r, ε) = εφi (r, ε)r 0 − 2m 2364
(17)
Z n(r) =
dε f (ε − ν)
tr(φi∗ (r, ε)φi (r, ε))
(18)
tr(φi∗ (r, ε)σφ ˜ i (r, ε))
(19)
i
Z m(r) =
X
dε f (ε − ν)
X i
where f (ε − ν) is the Fermi–Dirac function and ν the chemical potential. The next logical step is to extend the local approximation (LDA) to finite temperatures, (LDA), and write down xc in terms of the exchange-correlation part of the Gibbs free energy of a homogeneous electron gas. If this is done the effects of the thermally induced spin-wave excitations are severely underestimated. The calculated Curie temperatures are much too high [14], there are no local moments in the paramagnetic state and the uniform static paramagnetic susceptibility does not follow a Curie–Weiss behaviour as is found for many metallic systems. These spin fluctuations interact as the temperature is increased and therefore xc [n, m] should deviate significantly from the local approximation with a consequent effect upon the form of the effective singleelectron states. The idea is therefore to model the effects of the spin fluctuations whilst still maintaining the spin-polarized single-electron basis. Thus the properties of magnetic metals at finite temperatures can be described; that is, Ms , A and K can be determined as functions of temperature. The straightforward extension of spin-polarized band theory to finite temperatures misses the dominant thermal fluctuation of the magnetization since the thermally ¯ can only vanish together with averaged magnetization, M, the ‘exchange splitting’ of the electronic bands which is destroyed by particle–hole ‘Stoner’ excitations across the Fermi surface. An important piece of this neglected component can be pictured as orientational fluctuations of ‘local moments’ which are the magnetizations, within each unit cell of the underlying crystalline lattice, formed by the collective behaviour of all the electrons. The picture now involves the propagation of independent electrons through effective magnetic fields whose orientations fluctuate. M¯ can now vanish as the disorder of the ‘local moments’ grows. From this broad consensus [74], there are several approaches which differ according to which aspects of the fluctuations are deemed to be the most important for the materials under scrutiny. 4.1. Fluctuating ‘local moments’ At the outset of this work some 15 years ago, models of the ferromagnetic 3d transition metals Fe, Co and Ni at finite temperatures fell into one of two camps. By and large the Stoner excitations were neglected and the orientations of the ‘local moments’, which were assumed to have fixed magnitudes independent of their orientational environment, were the degrees of freedom over which thermal averaging was to be performed. The fluctuating local band (FLB) theory [75–77] was based on the assumption that there is a large amount of short-range magnetic order even in the paramagnetic phase. There are large spatial regions in which the ‘local moments’, with magnitudes equivalent
Magnetic alloys
to the magnetization per site in the ferromagnetic state at T = 0 K, are nearly aligned, namely where the orientations vary gradually. In these regions, the usual spin-polarized band theory can be applied and perturbations to it made. The results of quasi-elastic neutron scattering experiments of Ziebeck et al [78] on the paramagnetic phases of Fe and Ni, later confirmed by Shirane et al [79] are given a simple, though not uncontroversial [80], interpretation by this picture. In the case of inelastic neutron scattering, however, even the basic observations are controversial, let alone their interpretations in terms of ‘spin waves’ above Tc which may feature in such a model. It is rather difficult to carry out realistic calculations [81] in which the magnetic and electronic structures are mutually consistent and consequently to examine the full implications of the FLB picture and to improve it systematically. The second type of approach is the ‘disordered local moment’ (DLM) picture [82–85]. Here, the local moment entities associated with each lattice site are commonly thought to fluctuate fairly independently and an apparent total absence of magnetic short-range order (SRO) is assumed at the outset. Early work was based on the Hubbard Hamiltonian. The procedure had the advantage of being fairly straightforward and more specific than that in the case of FLB theory and various calculations which gave a reasonable description of experimental data were performed. The model Hamiltonian’s drawback was its simple parameter-dependent basis which could not provide a realistic description of the electronic structure which must support the important magnetic fluctuations. The dominant mechanisms might therefore be missed. It is also difficult to improve this work systematically. The paramagnetic state of body-centred cubic iron has attracted much attention and it is generally agreed that there are ‘local moments’ in this material for all temperatures, although the relevance of a Heisenberg Hamiltonian to a description of their behaviour has been debated in depth. In suitable limits, both the FLB and DLM approaches can be cast into a form from which an effective classical Heisenberg Hamiltonian can be extracted. The ‘exchange-interaction’ parameters Jij are specified in terms of the electronic structure owing to the itinerant nature of the electrons and the small-wavevector limit, if the lattice Fourier transform is taken, gives an estimate of the exchange constant A. Unfortunately the Jij terms determined from the former FLB approach turned out to be short ranged and therefore inconsistent with the initial picture of substantial magnetic SRO above Tc [81]. From the DLM perspective, the interactions, Jij , of paramagnetic iron can be obtained from consideration of the energy of an interacting electron system in which the local moments are constrained to be orientated along directions eˆ i and eˆ j on sites i and j whilst an approximate average is taken over all the possible orientations on the other sites [86, 87]. The Jij terms calculated in this way are suitably short ranged. A scenario in between these two limiting cases has been proposed by Heine and Joynt [88] and Samson [89]. They too were guided by the apparent substantial magnetic SRO above Tc in Fe and Ni deduced from neutron scattering data and they emphasized how the orientational magnetic
disorder involves a balance in the free energy between energy and entropy. This balance is delicate and they showed that it is possible for the system to disorder on a coarser than atomic length scale and consistency between the magnetic and electronic structures to be maintained. The length scale is, however, not as coarse as that initially proposed when the FLB theory was set up [75, 76]. 4.2. ‘First-principles’ theories These models can be made ‘first-principles’ ones by grafting the effects of these orientational spin fluctuations onto SDF theory [87, 90, 91]. It is assumed that fast and slow motions can be identified and separated. On a time scale long in comparison with an electronic hopping time, but short when compared with a typical spin fluctuation time, the spin polarizations of the electrons leaving a site are sufficiently correlated to those of electrons arriving that a non-zero magnetization exists when the appropriate quantity is averaged on this time scale. These are the ‘local moments’ which can change their orientations slowly with respect to the time scale whereas their magnitudes fluctuate rapidly. The standard SDF theory which we have discussed already can be adapted to describe the states of the system for each orientational configuration {eˆ i } in a similar way to that in which Uhl et al [93], Sandratskii and Kubler [94, 95] and others have tackled non-collinear magnetic systems. In principle such a description yields the magnitudes of the local moments, which can depend on the orientational environment µk ({eˆ i }), and the electronic grand potential for the constrained system {eˆ i }. This is the framework from which exchange constants A for the micromagnetic free energy can also be calculated [96, 97] by considering grand-potential differences for various orientational configurations. For BCC Fe and fictitious BCC Co the moments are fairly independent of their orientational environment whereas those in FCC Fe, Co and Ni are further away from being local quantities. Thus the long-time averages can be replaced by ensemble averages with the Gibbsian −1 measure P {eˆ i } = Q ZR exp(−β{eˆ i }), where the partition function Z = deˆ i exp(−β{eˆ i }) (β = 1/(kB T )). i The thermodynamic free energy, which accounts for the entropy associated with the orientational fluctuations as well as creation of electron–hole pairs, is given by F = −kB T log Z. The role of a classical ‘spin’ (local moment) Hamiltonian, albeit a highly complicated one, is played by {eˆ i }. By choosing a suitable reference ‘spin’ Hamiltonian 0 {eˆi } and expanding about it using the Feynman–Peierls’ inequality [98], an approximation to the free energy is obtained: F ≤ F0 + h − 0 i0 = F˘ (20) with
Z 1 Y ln (21) deˆ i exp(−β0 ) β i YZ YZ hXi0 = deˆ i X exp(−β0 ) deˆ i exp(−β0 ) F0 = −
i
i
2365
J B Staunton et al
=
YZ
deˆ i P0 {eˆ i }X{eˆ i }.
(22)
i
With 0 in the form X (1) X (2) ωi (eˆ i ) + ωij (eˆ i , eˆ j ) + · · · 0 = i
(23)
i,j,i6=j
a scheme is set up which can in principle be systematically improved. Minimizing F˘ to obtain the best estimate of the free energy gives ωi(1) , ωij(2) and so on as expressions involving resticted averages of {eˆ i } over the orientational configurations [87, 90]. A type of mean-field theory, which turns out to be equivalent to a ‘first-principles’ formulation of the DLM picture, is produced by taking the first term only in the equation above. Using a generalization of the SCF-KKR-CPA method [25] discussed earlier in the context of compositional disorder in alloys (section 3.3), explicit calculations were performed for BCC Fe and FCC Ni. The average magnitude of the local moments, hµi ({eˆ j })ieˆ i = µi (eˆ i ) = µ, ¯ in the paramagnetic phase of iron was 1.91µB [87] (the total magnetization is zero since hµi ({eˆ j })eˆ i i = 0). This value is of roughly the same magnitude as the magnetization per atom in the low-temperature ferromagnetic state. The uniform, paramagnetic susceptibility, χ (T ), followed a Curie–Weiss dependence upon temperature as observed experimentally and the estimate of the Curie temperature Tc was found to be 1280 K, also comparing well with the experimental value of 1040 K [92]. In nickel, however, µ¯ was found to be zero and the theory reduced to the conventional SDF version of the Stoner model with all its shortcomings [14, 91]. This mean-field DLM picture of the paramagnetic state has been improved by including the effects of correlations between the local moments to some extent. This has been achieved by integrating the consequences of Onsager cavity fields into the theory [91, 99]. Tc for Fe is now 1015 K and a 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 Cu85 Mn15 showed that the magnetic interactions between the Mn local moments fell off exponentially with distance and were oscillatory. This was in agreement with extensive neutron scattering data and the underlying electronic mechanisms were also identified [100]. An earlier application to FCC Fe showed how the magnetic correlations change from anti-ferromagnetic to ferromagnetic as the lattice is expanded [101]. This study complemented total energy calculations for FCC Fe both for ferromagnetic and for anti-ferromagnetic states at absolute zero for a range of lattice spacings. This effect is also relevant for the understanding of the properties of FCCbased Fe alloys, in particular, the Invar alloys [102]. For nickel, the theory resembles the static, hightemperature limit of the theory of Murata and Doniach [103], Moriya and co-workers [74, 104], Lonzarich and Taillefer [105] and others to describe weak itinerant ferromagnets. Nickel is consequently described in terms of a Stoner theory but one in which the spin fluctuations have drastically renormalized the exchange interaction 2366
and lowered Tc from about 3000 K [14] to 450 K [91]. The neglect of the dynamical aspects of these spin fluctuations has over-estimated this renormalization somewhat but χ(T ) again exhibits Curie–Weiss behaviour as is found experimentally and an adequate description of neutron scattering data is also provided. Recent inverse photoemission measurements by von der Linden et al [106] have confirmed the collapse of the ‘exchange-splitting’ of the electronic bands of nickel as the temperature is raised towards the Curie temperature in accord with this Stoner-like picture, although spin-resolved, resonant photoemission measurements by Kakizaki et al [107] indicate the presence of spin fluctuations. This approach has no parameters, everything being set up from SDF theory. It thus represents a well-defined stage of approximation. However, there are still some obvious shortcomings in this work (exemplified by the discrepancy between the theoretically determined and experimentally measured Curie constants) and it is worth highlighting the key one, which is the neglect of the dynamical effects of the spin fluctuations that has been emphasized by Moriya [104, 105] and others. Uhl and Kubler [108] have also set up an ab initio approach for dealing with the thermally induced spin fluctuations and they also treat these excitations classically. They calculate total energies of systems constrained to have spin-spiral configurations with a range of different propagation vectors q of the spiral, polar angles and spiral magnetization magnitudes using the non-collinear fixed spin moment method. A fit of the energies to an expression involving q, polar angles and magnetization magnitudes is then made in order to set up another ‘local moment Hamiltonian’. The Feynman–Peierls inequality is also used when a quadratic form is used for the ‘reference Hamiltonian’. Stoner particle–hole excitations are neglected. Similar results to those in reference [91] have been obtained for BCC Fe and FCC have also studied Co and have recently generalized the theory to describe magnetovolume effects. FCC Fe and Mn have been studied alongside the ‘Invar’ ordered alloy [109]. Once again exchange constants A can be extracted from this work.
4.3. ‘Local exchange splitting’ The range of validity of these sorts of ab initio theoretical approaches and the severity of the approximations employed can be found by comparing their underlying electronic bases with suitable spectroscopic measurements. An early prediction from a ‘first-principles’ implementation of the DLM picture was that a ‘local exchange’ splitting should be evident in the electronic structure of the paramagnetic state of BCC iron [90, 110]. In addition, the size of this splitting was expected to vary sharply as a function of the wavevector and energy. In fact at some points, if the ‘bands’ were flat in that region of wavevector and energy space, the local exchange splitting would be roughly of the same magnitude as the rigid exchange splitting of the electronic bands of the ferromagnetic state whereas at other points, where the ‘bands’ are broader, the splitting would vanish completely. It is this local exchange splitting that causes the local moment to be established.
Magnetic alloys
The photoemission (PES) experiments of Kisker et al [111] and inverse photoemission (IPES) experiments performed by Kirschner et al [112] found these features. The experiments essentially focused on the electronic structure around the centre of the Brilliouin zone and at a point on the zone boundary for a range of energies. In the first region states were interpreted as being exchange split whereas in the latter region they were not, though all were broadened by the magnetic disorder. Later Haines et al [113] used a tight-binding model to describe the electronic structure and employed the recursion method to average over various orientational configurations. They concluded that a modest degree of magnetic SRO is compatible with the limited spectroscopic measurements available for paramagnetic iron. More extensive spectroscopic data on the paramagnetic states of the ferromagnetic transition metals would be invaluable in developing the theoretical work on the important spin fluctuations in these systems. Having outlined how one can deal with the elemental magnetic metals at finite temperatures from a ‘firstprinciples’ electronic structure basis, we will now consider how this is carried out for alloys. A longstanding issue in the study of metallic alloys is the interplay between compositional order and magnetism and the dependence of magnetic properties on the local chemical environment. Magnetism is frequently connected to the overall compositional ordering as well as the local environment in a subtle and complicated way. 5. Magnetism and compositional order In order to examine the interrelation between magnetism and compositional order it is necessary to deal with the statistical mechanics of thermally induced compositional fluctuations. References [29, 30, 33, 100] describe this in some detail so here we will give an outline and show how magnetic effects are incorporated. In particular we show how to set up an expression for the free energy dependent on the composition of the alloy, which can be used to find metastable states and stable states of alloys [114]. 5.1. The grand potential and equation of state of a binary alloy Atomic ordering in alloys is usefully described in terms of ‘concentration waves’ [115]. As the atoms begin to arrange themselves while the alloy cools, the probability of finding a particular atomic species occupying a lattice site, namely the site-dependent concentration, varies from site to site and traces out a static concentration wave. For example, the well-known Cu3 Au and CuZn ordered alloys are characterized by a concentration wave with wavevector q = {0, 0, 1} on FCC and BCC crystal lattices, respectively. On the other hand, an infinitely longwavelength concentration wave, q → 0, describes an alloy which undergoes phase segregation at low temperatures. One approach to describe such phenomena quantitatively is to follow the formalism of Evans [116] for a density functional theory of classical fluids and adapt it to the binary alloy lattice model relevant here. This formalism formed
the basis for the work by Gyorffy and Stocks [29, 30] and subsequent developments by ourselves [33, 100] and we review it briefly below. We consider the grand canonical ensemble of NA A and NB B nuclei occupying N (= NA +NB ) locations referred to a fixed lattice position and define an equilibrium probability density P0 {ξi } = P0 (ξ1 , ξ2 , . . . , ξN ) of occupation of sites by A nuclei at a temperature T (the fraction c = NA /N ). ξi = 1 (0) if an A (B) nucleus occupies site i: P0 {ξi } =
1 exp[−β(H {ξi } − ν¯ NA )] Z
(24)
where H {ξi } is the Hamiltonian when there are NA nuclei present and ν¯ is the chemical potential for the nuclei. The grand partition function Z is Y X exp[−β(H {ξi } − ν¯ NA )]. (25) Z= l
ξl =1,0
Following Evans [116], we consider the functional Y X 1 P {ξi } H {ξi } − ν¯ NA + log P {ξi } . [P ] = β l ξl =1,0 (26) For the equilibrium probability density we have [P0 ] = −(1/β) log Z which is equivalent to the grand potential. [PP ] > [P0 ] (P 6= P0 ) for all probability densities with Q l ξl =1,0 P = 1. P We take Hamiltonians of the form H {ξi } = W {ξi } − i νiext ξi . W {ξi } describes the ‘internal energy’ of the system which has the NA nuclei P arranged on some of the N sites according to {ξi } and i νiext ξi is an arbitrary external potential. (Note that W is not restricted to a pairwise form.) Thermal averages are defined by Y X P0 {ξi }X{ξi } (27) hXi = l
ξl =1,0
and ci0 = hξi i is the equilibrium probability of finding an A nucleus on lattice site i, namely a ‘local’ concentration when an external potential {νiext } is applied to the system. P0 is evidently a function of the νiext terms and therefore so are the local concentrations. It is possible to show [116] that, for a given form of W , only one particular set of νiext terms, {νiext }, can determine a given {ci0 }. Since P0 = P0 {νiext }, P0 is also determined by a given set of ci , {ci }. It follows that, for a given W , we can specify Y X 1 P0 {ξi } W {ξi } + log P0 {ξi } . (28) F {ci } = β l ξl =1,0 That is, F is a unique function of the ci , which may be nonequilibrium ones (this is the case for any {νiext }). Similarly, X X ˜ i } = F {ci } − ν¯ {c ci − νiext ci . (29) i
i
˜ i } = , When the ci are equal to the equilibrium values {c the grand potential, ˜ δ =0 (30) δcl {ci =ci0 } 2367
J B Staunton et al
for all l. Thus the correct equilibrium concentrations {ci0 } ˜ to give . minimize F {ci } is an ‘internal’ Helmholtz free energy. We have δF{ci } − νlext = ν¯ (31) δcl {ci0 } in which the first term on the left-hand side defines an intrinsic chemical potential νlin . Equation (31) is the important equation of state. For example for a non-interacting (NI) system (W = 0), FNI {ci } = P β −1 i [ci log ci + (1 − ci ) log(1 − ci )], equal to the product of kB T and the ‘point entropy’.
it at the equilibrium concentrations and recalling that the {ci } depend on {νiext } gives ˜ i} δ {c = −c0 . l δνlext ci0 A second differentiation gives a linear response function, 0 ˜ δ2 = − δcl . ext ext ext δνk δνl {ci0 } δνk This can easily be shown to be proportional to the atom– atom correlation function hξl ξk i − hξl ihξk i by differentiating the equilibrium grand potential twice:
5.2. Correlation functions and atomic short-range order in alloys The effects of interactions are most readily included via a hierarchy of direct correlation functions (so called by way of the close analogy with similar quantities defined for classical fluids [116, 117]). Let us define 8{ci }, the interaction part of F, as −(F − FNI ) and we can write the intrinsic chemical potential cl 1 δ8{ci } in (32) − νl {ci } = log β 1 − cl δcl and δ8/δcl = Sl(1) is a ‘self-energy’, the contribution to the internal local chemical potential from all the interactions. It is an additional single-site effective potential which determines self-consistently via equation (31) the equilibrium concentrations†. From equation (31) 0 cl 1 log − Sl(1) {ci0 } − νlext = ν¯ β 1 − cl0 which can be arranged to give cl0 =
1
exp[β(Sl(1) − ν¯ − νlext )] . + exp[β(Sl(1) − ν¯ − νlext )]
Sl(1) is in fact the first member of a hierarchy of correlation functions generated by 8{ci }, the higher order functions being obtained by further differentiation; for example Skl(2) = (3) Sklm
δ28 δck δcl 3
δ 8 = ···. δck δcl δcm
Following Gyorffy and Stocks [29, 30] we note that the second derivative evaluated at the equilibrium concentrations can be referred to as the Ornstein–Zernike direct correlation function by analogy with its designation in the theory of non-uniform classical fluids. ˜ i } can also be used to generate another hierarchy {c of correlation functions. Differentiating it once, this time with respect to the ‘external’ chemical potential, evaluating † It is analogous to the effective potential that appeared in the singleelectron Kohn–Sham equations of the density functional theory of the inhomogeneous electron gas.
2368
δcl0 = β(hξl ξ ki − hξl ihξk i) = βαij . δνkext These correlation functions are directly related to the Warren–Cowley parameters and are measured in diffuse scattering (such as x-ray, electron and neutron) experiments. (Once again, further correlation functions can be found by subsequent differentiation.) Returning to equation (31) and taking the derivative with respect to the local concentration ck on a site k, we obtain an expression for the linear response function and hence the pair correlation function. For an alloy with a homogeneous concentration distribution {ci0 = c} which is then perturbed by a small, inhomogeneous change to the external potential {δνi } with the result that a new inhomogeneous 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 alloy can thus be described fully in terms of the lattice Fourier transform of the ‘direct correlation function’ Skl(2) and provides direct information on the stability of the randomly disordered alloy against concentration fluctuations at a given temperature T [29]. An expression for the Helmholtz free energy can also be constructed in terms of the Skl(2) . In terms of a set of local concentrations {ci }, we can write X F {ci } ≈ β −1 [ci log ci + (1 − ci ) log(1 − ci )] − 8(c) −
X i
=β
−1
i
Si(1) (ci − c) −
X
Sij(2) (ci − c)(cj − c)
ij
X [ci log ci + (1 − ci ) log(1 − ci )] − 8(c) i
−
X
Sij(2) (ci − c)(cj − c)
(34)
ij
by expanding the interaction part of the free energy 8 around that of a reference homogeneously disordered system with the same average concentration c. By calculating Skl(2) for a disordered alloy magnetized along a given direction and then comparing it with the value obtained when the system is magnetized along another direction, the dependence of the anisotropy constants K
Magnetic alloys
upon the locally varying composition can be extracted, together with a handle upon the directional compositional order which can be set up when a magnetic alloy is annealed in a magnetic field. We are working on this topic. The above equation (34) can be used in a simple model of the kinetics of diffusional transformations as introduced by Cahn [118] and used by Khachaturyan [115, 119]. It can be used to find equilibrium and quasi-equilibrium states. The model is based on nonlinear Onsager microscopic diffusion equations. The rate of change of the relaxation parameters, the local concentrations ci (t), with time is directly proportional to the thermodynamic driving force, δF dci (t) X = Lij dt δc j (t) j
(35)
where the Lij are kinetic coefficients proportional to the probability of an atom undergoing a diffusional hop between sites i and j in unit time. We have recently used this model, making the simplifying assumption that the kinetic coefficients are independent of the environment, namely the ci (t) terms, just like Cahn and Khachaturyan did and investigated the metastable and stable ordered phases AuFe and ferromagnetic FeAl alloys [114]†. We refer to the latter in one of our case studies later on. Elsewhere [33] we have described in detail how S (2) (q, T ) can be calculated from a ‘first-principles’, meanfield approach. Here the interacting part of the Helmholtz free energy 8 ≈ −hH {ξi }i in which the angular brackets denote the average with respectQto the inhomogeneous distribution function P {ξi } = i pi (ξi ) with pi (1) = We start from the SCFci and p(0) = 1 − ci . KKR-CPA description of the electronic structure of the high-temperature, compositionally disordered state [25]. An important ingredient for the statistical averaging is the replacement of the mean fields (Weiss fields) by Onsager cavity fields and the calculated spinodal ordering temperatures which we obtain are consequently closer to agreement with experiment than would be expected from a standard mean-field estimate. All electronic effects can be treated. In particular, the alteration of charge that occurs as the atoms are rearranged is fully incorporated; that is, charge-transfer effects are included. At present our calculations do not include lattice-displacement effects— the nuclei can occupy only ideal crystal lattice positions— but the formalism and techniques are sufficiently flexible to allow their incorporation and work to carry this out is in progress. From S (2) (q, T )) it can be determined whether an alloy’s tendency to order depends on the magnetic state of the system and the spin polarization of its electronic structure. 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 describe two case studies in which we show the extent to which an alloy’s compositional structure is dependent on whether the underlying electronic structure is ‘globally’ or ‘locally’ spin polarized, 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-rich Fe–V and Fe–Al alloys. S (2) (q) peaks at those q values which identify the static concentration wave for which the system is unstable at low enough temperatures. An important part of S (2) (q) involves an electronic state-filling effect and links neatly with the observation noted earlier that half-filled bands promote ordered structures whilst nearly filled or nearly empty states are compatible with systems that cluster when they are cooled [28, 35]. This propensity can be totally different depending on whether the electronic structure is spin polarized and hence whether the compositionally disordered alloy is ferromagnetic or paramagnetic, as is the case for nickel-rich Ni75 Fe25 , for example [36]. Earlier remarks in this paper about bonding in alloys and spin polarization are pertinent in this context. For example in strongly ferromagnetic alloys like Ni75 Fe25 , majorityspin electrons which completely occupy the d states ‘see’ very little difference between the two types of atomic site and hence contribute little to S (2) (q) and it is the nature of the minority-spin states which determines the eventual compositional structure. On the other hand for those alloys, usually BCC based, in which the Fermi energy is pinned in a valley in the minority density of states the ordering tendency is mostly set by the majority-spin electronic structure [37]. For a ferromagnetic alloy, an expression for the magneto-compositional cross correlation function υ(q) = P (1/N ) j (hµi ξj i − hµi ihξj i) exp[iq · (Ri − Rj )] can be derived and evaluated [120, 121]. This turns out to be a simple product υ(q) = γ (q)α(q), where υij is a convolution of γik = δhµi i/δck and αkj . γik has components γik = (µA − µB )δik + c
δµA δµB i + (1 − c) i . δck δck
B The last two quantities, δµA i /δck and δµi /δck , which can also be related to the spin-polarized electronic structure of the disordered alloy, describe the changes to the magnetic moment on a site i in the lattice if it is occupied by an A and a B atom respectively as the probability of occupation is altered on another site k. In other words, γik quantifies how the sizes of the magnetic moments depend upon their chemical environment. It can show how the saturation magnetization Ms depends on compositional microstructure, Ms {ci }. We have studied how the magnetic moments in Fe– V and Fe–Cr alloys depend on their local environments from this framework [121]. Furthermore, if the application of a small external magnetic field along the direction of the magnetization is considered, expressions dependent upon the electronic structure for the static longitudinal susceptibility χL (q) can be found. α(q), υ(q) and χL (q) can be straightforwardly compared with x-ray [122] and neutron scattering [123], nuclear magnetic resonance and M¨ossbauer spectroscopy data. In particular, the cross sections obtained from diffuse polarized neutron scattering can be expressed dσ dσ N M dσ M dσ N +ζ + = dω ζ dω dω dω
2369
J B Staunton et al
where ζ = +1 (−1) if the neutrons are polarized (anti-) parallel to the magnetization. The nuclear component dσ N /dω is proportional to the atomic shortrange order, α(q) (closely related to the Warren–Cowley short-range order parameters). The magnetic component dσ M /dω is proportional to χL (q). Finally dσ N M /dω describes the magneto-compositional correlation function γ (q)α(q) [124, 125]. By interpreting such experimental measurements by these sorts of calculations and then unravelling the calculations, the electronic mechanisms which underlie the correlations can be extracted [120, 126]. 5.3. The ASRO and magnetism in iron-rich Fe–V and Fe–Al alloys In this subsection we describe our investigation of the Curie temperatures, Tc , magnetic correlations, from which exchange constants A at high temperatures can be extracted, and atomic short-range order present in two iron alloy systems, Fe–Al and Fe–V, at (or rapidly quenched from) temperatures T0 above any compositional ordering temperature [127]. We find the ASRO to be sensitive to whether T0 is above or below the alloys’ magnetic Curie temperatures Tc and also to the details of the paramagnetic state. 5.3.1. FeV. Iron-rich Fe–V alloys have several qualities which make them suitable for an investigation of both ASRO and magnetism. The large difference in the coherent neutron scattering lengths, bF e − bV ≈ 10 fm, and a small size effect associated with alloying make them good candidates for neutron diffuse scattering experimental analyses and also their Curie temperatures (≈1000 K) lie in a range which makes it possible to compare and contrast the ASROs set up both by the ferromagnetic and by the paramagnetic states. Following some early neutron data from powdered crystal samples [128], Cable et al [126] carried out some measurements at room temperature of the scattering from a single crystal of Fe87 V13 along the {1, 0, 0}, {1, 1, 0} and {1, 1, 1} symmetry directions. A saturating magnetic field was applied parallel to the scattering vector which extinguished the magnetic scattering, allowing the nuclear scattering to be extracted from the data. The structure of the curves is attributed to nuclear scattering connected with ASRO, c(1 − c)(bF e − bV )2 α(q). Although the sample had been annealed at and quenched rapidly from 1270 K, above the alloy’s Curie temperature of 1180 K, the effective temperature for the ASRO was estimated to be 900 K. Pierron-Bohnes et al [129] overcame the problem of making such an estimate by carrying out experiments on a single crystal of Fe80 V20 in situ. The measurements were carried out at temperatures above the sample’s Curie temperature Tc . We calculated the ASROs both of ferromagnetic and paramagnetic Fe87 V13 and also paramagnetic Fe80 V20 . We also characterized the magnetic correlations present in the paramagnetic alloys together with estimates of the Curie temperatures. (We had previously determined the saturation magnetic moments, Ms (c), in the disordered ferromagnetic 2370
alloys [37] in excellent agreement with the values found experimentally [128].) An interpretation of both groups’ experiments was found as well as an investigation of the effect of magnetic order upon ASRO in these systems [127]. The most intense peaks in the data from the ferromagnetic Fe87 V13 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 double peak structure around { 12 , 12 , 12 }. We showed [120, 127] how our ASRO calculations for ferromagnetic Fe87 V13 could reproduce all the details of the data. This alloy is of the type in which the chemical potential is pinned in a trough of the minority-spin density of states so that there is substantial hybridization between states associated with the two different atomic species (figure 3(b)) and the ordering tendency is governed primarily by the majorityspin electrons. These states are roughly half filled to produce the strong ordering tendency [28, 35]. The calculations also showed that part of the structure around { 12 , 12 , 12 } can be traced back to the majority-spin Fermi surface of the alloy. By fitting the direct correlation function S (2) (q) in of real-space parameters, S (2) (q) = S0(2) + P terms P (2) n i∈n Sn exp(iq · Ri ) we found the fit to be dominated by the first two parameters which determine the large peak at {1, 0, 0}. However, the fit also showed that there is a long-range component that we showed to come from the Fermi-surface effect. The real-space fit of their data produced by Cable et al also showed large negative values for the first two shells followed by a weak long-ranged tail. In their paper Cable et al maintained that the effective temperature for at least part of the sample was indeed below its Curie temperature. We checked this by carrying out calculations for the ASRO of paramagnetic (DLM) Fe87 V13 . Once again, we found the largest peaks to be located at {1, 0, 0} and {1, 1, 1} but a careful scrutiny found that there was less structure around { 12 , 12 , 12 } than there was for the ferromagnetic alloy. The ordering correlations are also weaker in this state. For the paramagnetic DLM state, the local exchange splitting also pushes many anti-bonding states above the Fermi energy, although this is no longer wedged in a valley in the density of states, see figure 7. This produces a similar, although weaker, compositional ordering mechanism to that of the ferromagnetic alloy. The real-space fit of S (2) (q) also showed the long-ranged tail to be reduced. Evidently the ‘local-moment’ spin-fluctuation disorder has smeared the alloy’s Fermi surface and lessened its effect upon the ASRO. Pierron-Bohnes et al [129] measured neutron diffuse scattering intensities from Fe80 V20 in its paramagnetic state at 1473 and 1133 K (the Curie temperature is 1073 K) for scattering vectors both in the (1, 0, 0) plane and in the (1, −1, 0) plane, following a standard correction for instrumental background and multiple scattering. They also found maximal intensity about {1, 0, 0} and {1, 1, 1} but no additional structure about { 12 , 12 , 12 }. Our calculations of the ASRO of paramagnetic Fe80 V20 , very similar to those of Fe87 V13 , are consistent with these features. We also studied the type and extent of magnetic correlations in the paramagnetic state. We found
Density of States (States per Ry per spin)
Magnetic alloys
30 20 10 0
-0.6
-0.2 -0.4 Energy(Ry)
0
Figure 7. The compositionally averaged density of states of paramagnetic (DLM) Fe87 V13 and its resolution into components associated with Fe (full line) and Al (dotted line) sites weighted by concentration, in units of states per atom Ry−1 per spin.
ferromagnetic correlations which grow in intensity as T is reduced. These lead to an estimate of the Curie temperature Tc = 980 K, which agrees well with the measured value of 1073 K. (Tc for Fe87 V13 of 1075 K also compares well with the measured value of 1180 K.) This 10% level of disagreement is what might be expected from a mean-field theory. By modelling the paramagnetic alloy subsequently as a Stoner paramagnet with no local moments and hence zero exchange splitting of the electronic structure, local or otherwise, and repeating the calculations of ASRO we found the maximum intensity at about { 12 , 12 , 0} and equivalent points. This is in striking contrast both to the DLM calculations and to the experimental data. In summary, we concluded that experimental data on Fe–V alloys are well interpreted by our calculations of ASRO and magnetic correlations. ASRO and hence the alloys’ ordering tendencies are evidently strongly affected by the local moments associated with the iron sites in the paramagnetic state so that there are only small differences between the topologies of the ASRO of samples quenched from above and below Tc . The main difference is the growth of structure in the ASRO around { 12 , 12 , 12 } for the ferromagnetic state. The ASRO strengthens quite sharply as the system becomes magnetically ordered and it would be interesting if an in situ, polarized-neutron, scattering experiment were performed to investigate this. 5.3.2. FeAl. Iron-rich Fe–Al alloys are promising candidates for the basis of a new family of superalloys and have therefore attracted the attention of material scientists [131]. McKamey et al [132] and others have reviewed their properties together with the effects of the addition of impurities. One factor limiting the use of these alloys seems to be their poor ductility above 900 K. It is therefore appropriate to understand the processes which govern how the atoms order. Their high Curie temperatures (≈900 K) imply that the degree of magnetic order is liable to be an important factor. These alloys are consequently also interesting test beds for theories such as ours which treat the interplay between magnetic and chemical ordering. The phase diagram [130] shows that, on cooling from the melt, paramagnetic Fe80 Al20 forms a solid solution. The alloy
becomes ferromagnetic at 935 K and then forms an apparent DO3 ordered phase at about 670 K. An alloy with just 5% more aluminium orders into a B2 phase from the paramagnetic state at roughly 1000 K and orders into a DO3 phase at lower temperatures. Iron-rich Fe–Al alloys and Fe80 Al20 , in particular, have been subjects of a series of x-ray and neutron diffuse scattering experiments thanks to the favourable differences in valence and neutron scattering length of Fe and Al and recently we compared our ASRO calculations with these data [127]. Epperson and Spruiell [133] investigated the ASRO via x-ray studies on polycrystalline samples with around 25% aluminium, heat treated in a variety of ways and quenched from a range of temperatures. They provided evidence that the ordering tendency was towards the DO3 superstructure. This is characterized by concentration waves with q = {1, 0, 0} and { 12 , 12 , 12 }. Schweika et al [134] carried out an in situ, unpolarized, neutron experiment on a single crystal of Fe80 Al20 at temperatures in the range 823–1073 K. They made an assessment of the magnetic scattering in the ferromagnetic regime so that they could attempt to isolate the nuclear scattering. They presented data for the ferromagnetic alloy from the {0, 0, 1} and {2, 1, 1} planes at T = 823 K and the {0, 0, 1} plane at 923 K, as well as the {2, 1, 1} plane for the paramagnetic alloy at 1073 K. For the ferromagnetic alloy, substantial intensity was found around {0, 0, 1} equivalent q points, which was skewed in the {1, 1, 1} directions, with a possible subsidiary peak at { 12 , 12 , 12 }. A peculiar feature noted was that the short-range order peak at {1, 0, 0} decreased only marginally with increasing temperature in the range T = 823–923 K, which the authors put down to a competition between a Fe–Al chemical interaction and the ferromagnetic coupling between Fe moments. From our calculations of the ASRO we were able to describe this effect in terms of the electronic structure. A later paper by Schweika [135] extended the experimental study further to the paramagnetic state, just above the Curie temperature 935 K. At 1013 K, ¯ 0} plane was found the scattering intensity from the {1, 1, to peak around {1, 0, 0} equivalent points only, but there was also a distortion along {1, 1, 1}. Evidence for a phase transition into an ordered phase characterized by { 12 , 12 , 12 } concentration wavevectors only at approximately 650 K was also presented. This is consistent with a B32 transient ordered phase observed by Gao and Fultz [136] in Fe75 Al25 . Pierron-Bohnes et al [137] carried out x-ray diffuse scattering measurements on Fe80 Al20 quenched from 772 K in the ferromagnetic state. They found maximum intensity at {1, 0, 0} and subsidiary peaks at { 12 , 12 , 12 } in accord with Schweika et al. They complemented this study [138] by diffuse, unpolarized neutron scattering measurements on a single crystal in situ for several temperatures in the range 973–1573 K in the paramagnetic regime. Peaks were observed at {1, 0, 0} points which were slightly elongated along the {1, 1, 1} direction for the lowest temperature of 973 K. No subsidiary peaks around { 12 , 12 , 12 } were noted. All the data seem to show that there is an apparent weakening of the chemical interactions in the ferromagnetic state with a reduced propensity towards B2-type ordering and an increased tendency towards DO3 -type order. 2371
J B Staunton et al (1,1,1)
(0,1,0)
(1,1,1)
(0,1,0)
9.0
9.0 3.0
3.0 6.0
4.8
4.8
0.8
6.0 3.2
3.2 2.4
2.4 1.6
1.6
0.8
(1,0,1)
(0,0,0)
(0,0,0)
(1,0,1)
(a)
(b)
(0,1,0)
(1,1,1) 4.0
4.0 16
16
8.0
8.0 4.0
4.0
4.0
4.0
(0,0,0)
(1,0,1)
(c) Figure 8. (a) The ASRO (α(q, T )) for the {1, 1¯ , 0} plane for paramagnetic (DLM) Fe80 Al20 at 200 K above the theoretical spinodal temperature of 2700 K in Laue units. (b) The ASRO (α(q, T )) for the {1, 1¯ , 0} plane for ferromagnetic Fe80 Al20 at 1000 K (the theoretical spinodal temperature is 485 K) in Laue units. (c) The same as for (b) but for a lower temperature of 500 K.
We carried out calculations of the ASRO both in ferromagnetic and in DLM-paramagnetic states of Fe80 Al20 . Parallel calculations of χ (q, T ), based on the DLM model, indicated that there are strong ferromagnetic correlations in paramagnetic Fe80 Al20 and a Tc of 1130 K, in fair agreement with the experimental value of 935 K. Evidently this treatment captures the correct order of magnitude 2372
for the energies of the spin fluctuations, just as it does for the elemental metals and FeV. Figures 8(a)–(c) show the ASRO for both the paramagnetic state (α DLM (q, T )) and the ferromagnetic (α F M (q, T )) state of Fe80 Al20 for ¯ 0) plane. α DLM (q, T ) peaks at {1, 0, 0} exhibit the (1, 1, the same B2-type ASRO as that of the high-temperature experimental data [138].
Magnetic alloys
α F M (q, T ) is presented at two temperatures, one slightly below the Curie temperature at T = 1000 K (figure 8(b)) and the other at the lower temperature of 500 K (figure 8(c)). For the higher temperature, α F M (q, T ) peaks at {1, 0, 0} with a strong skewing along the {1, 1, 1} direction. This feature is also observed in the experimental data [134]. For the lower temperature, this skewing turns into an incommensurate peak at {0.8, 0.8, 0.8} and large intensity from {0.5, 0.5, 0.5} to {1, 1, 1}. The latter feature in the calculated α F M (q, T ) compares well with that observed in the experimental data [134, 137]. We have also investigated the nature of the stable and/or metastable states of the ferromagnetic alloy when it is cooled to just below the ordering spinodal temperature [114] using the equation for the free energy (34), namely in terms of S (2) (q) which specifies the ASRO above. We find that the alloy orders into domains of B32 structure consistent with the findings of Schweika et al and Gao and Fultz [135, 136]. The structure along {1, 1, 1} owes its origin to a similar Fermi surface effect to that which caused the structure around { 12 , 12 , 12 } in ferromagnetic Fe87 V13 . It may, therefore, be responsible for the tendency of ironrich ferromagnetic Fe–Al alloys to form DO3 ordered alloys, whereas only B2-type correlations are seen in their paramagnetic states, where local-moment disorder has removed the remnants of sharp structure from the electronic structure around the Fermi energy. In this section, we have reviewed both our calculations of ASROs of Fe–V and Fe–Al alloys and diffuse neutron and x-ray scattering data. Fair agreement is obtained, although, for FeAl, the strength of the calculated ASRO for the paramagnetic alloys is somewhat larger than that deduced from experiment. This discrepancy is possibly due to the theoretical constraint of a rigid lattice. On the other hand, the estimates of the Curie temperatures agree well with empirical values. The topology of the ASRO, α(q), in q space is accurately described by the theory together with trends which accompany changes in magnetic state and temperature and can be understood from the electronic structure. The relative occupation of bonding and antibonding states determined by the ‘global’ or ‘local’ exchange splitting is an important consideration. In both these iron-alloy systems, Fermi-surface effects lead to intensity in α(q) around { 12 , 12 , 12 } for the ferromagnetic state which diminishes in the paramagnetic state. We have postulated that this is the reason for the tendency for ferromagnetic Fe80 Al20 alloys to form DO3 ordered alloys, whilst their paramagnetic counterparts exhibit B2 correlations. These encourage the formation of B2 ordered phases for alloys with more aluminium and therefore lower Curie temperatures. 6. Summary Recent improvements in the numerical algorithms and available computational power have led to rapid developments in the field of micromagnetics [3, 4]. Magnetic properties of alloys can now be modelled in detail. It is timely for ‘first-principles’ theories of the electronic structure of
such alloys to feed into these models and to give guidance with regard to how the saturation magnetization, Ms , and the exchange, A, and anisotropy, K, constants depend both on composition and on temperature. This paper has attempted to show the extent to which this can be achieved at present and has emphasized the importance of basing this work upon a clear picture of the underlying spin-polarized electronic ‘glue’ of these systems. This picture can itself be tested independently by a range of spectroscopic experiments. Spin density functional theory of the inhomogeneous electron gas provides the formal basis for carrying this out and part of this paper has described how the relativistic version of this theoretical framework is necessary in order to describe magnetocrystalline anisotropy. Magnetic order can also have a marked influence on the compositional structure in metallic alloys and we have also devoted a portion of the paper to this issue and have included a brief review of the theory of compositional ordering in alloys. We look forward to an increasingly important role for electronic structure calculations both in the modelling of the magnetic properties of existing magnetic alloys and in the design of new magnetic materials. Acknowledgments Some of the work reviewed in this paper was supported in part by the EPSRC in the UK and by the USA Department of Energy, Office of Basic Energy Sciences, Division of Materials Science grant DEFG02-96-ER45439 through the University of Illinois. It is a pleasure to acknowledge many discussions and collaborations with B L Gyorffy, G M Stocks and B Ginatempo. References [1] Brown W F Jr 1962 Magnetostatic Principles in Ferromagnetism (Amsterdam: North-Holland) [2] Craik D J and Tebbell R S 1965 Ferromagnetism and Ferromagnetic Domains (Amsterdam: North-Holland) [3] Schabes M E 1991 J. Magn. Magn. Mater. 95 249–88 [4] Wongsam M A and Chantrell R W 1996 J. Magn. Magn. Mater. 152 234–42 [5] Herring C 1966 Magnetism vol 4, ed G T Rado and H Suhl (New York: Academic) [6] Wassermann E F 1991 J. Magn. Magn. Mater. 100 346–62 [7] Chikazurin S and Graham C D 1969 Magnetism and Metallurgy vol 2, ed A E Berkowitz and E Kneller (New York: Academic) p 577 [8] Kuentzler R 1980 Physics of Transition Metals 1980 ed P Rhodes (Bristol: Institute of Physics) pp 397–400 [9] Wang C S and Callaway J 1977 Phys. Rev. B 15 298–306 Callaway J and Wang C S 1977 Phys. Rev. B 16 2095–105 [10] Rajagopal A K 1980 Adv. Chem. Phys. 41 59 [11] Kohn W and Vashishta P 1982 Physics of Solids and Liquids ed B I Lundqvist and N March (New York: Plenum) [12] Dreizler R M and da Providencia J (eds) 1985 Density Functional Methods in Physics (New York: Plenum) [13] Jones R O and Gunnarsson O 1989 Rev. Mod. Phys. 61 689–746 [14] Gunnarsson O 1976 J. Phys. F: Met. Phys. 6 587–606 [15] Moruzzi V L, Janak J F and Williams A R 1978 Calculated Electronic Properties of Metals (Oxford: Pergamon) 2373
J B Staunton et al [16] Koelling D D 1981 Rep. Prog. Phys. 44 144–212 [17] Stoner E C 1939 Proc. R. Soc. A 139 339 Wohlfarth E P 1953 Rev. Mod. Phys. 25 211 [18] Chikazumi S 1964 Physics of Magnetism (New York: Wiley) [19] Hohenberg P and Kohn W 1964 Phys. Rev. B 136 864–72 [20] Kohn W and Sham L J 1965 Phys. Rev. A 140 1133–8 [21] von Barth U and Hedin L 1970 J. Phys. C: Solid State Phys. 5 1629–42 [22] Rajagopal A K and Callaway J 1973 Phys. Rev. B 7 1912 [23] Lieb E 1983 Int. J. Quant. Chem. 24 243 [24] Hedin L and Lundqvist B I 1971 J. Phys. C: Solid State Phys. 4 2064–83 Gunnarsson O and Lundqvist B I 1976 Phys. Rev. B 13 4274–98 Ceperley D M and Alder B J 1980 Phys. Rev. Lett. 45 566–9 Vosko S H, Wilk L and Nusair M 1980 Can. J. Phys. 58 1200–11 [25] Stocks G M, Temmerman W M and Gyorffy B L 1978 Phys. Rev. Lett. 41 339 Stocks G M and Winter H 1982 Z. Phys. B 46 95–8 Johnson D D, Nicholson D M, Pinski F J, Gyorffy B L and Stocks G M 1990 Phys. Rev. B 41 9701–16 [26] Soven P 1967 Phys. Rev. 156 809–13 [27] Pettifor D G 1995 Bonding and Structure of Molecules and Solids (Oxford: Oxford University Press) [28] Ducastelle F 1991 Order and Phase Stability in Alloys (Amsterdam: North-Holland) [29] Gyorffy B L and Stocks G M 1983 Phys. Rev. Lett. 50 374 [30] Gyorffy B L, Johnson D D, Pinski F J, Nicholson D M, and Stocks G M 1989 Alloy Phase Stability ed G M Stocks and A Gonis (Dordrecht: Kluwer) pp 421–68 [31] Connolly J W D and Williams A R 1983 Phys. Rev. B 27 5169–72 [32] Zunger A 1993 Statics and Dynamics of Alloy Phase Transformations ed A Gonis and P E A Turchi (New York: Plenum) [33] Staunton J B, Johnson D D and Pinski F J 1994 Phys. Rev. B 50 1450–72 [34] Ziman J M 1964 Adv. Phys. 13 89 [35] Heine V and Samson J H 1983 J. Phys. F: Met. Phys. 13 2155–68 [36] Staunton J B, Johnson D D and Gyorffy B L 1987 J. Appl. Phys. 61 3693–6 [37] Johnson D D, Pinski F J and Staunton J B 1987 J. Appl. Phys. 61 3715–17 [38] Kubler J 1984 Physica B 127 257–63 [39] Malozemoff A P, Williams A R and Moruzzi V L 1984 Phys. Rev. B 29 1620–32 [40] Williams A R, Moruzzi V L, Malozemoff A P and Terakura K 1983 IEEE Trans. Magn. 19 1983–8 Williams A R, Malozemoff A P, Moruzzi V L and Matsui M 1984 J. Appl. Phys. 55 2353–5 [41] Gubanov V A, Liechtenstein A I and Postnikov A V 1992 Magnetism and Electronic Structure of Crystals (Berlin: Springer) [42] Falicov L M et al 1990 J. Mater. Res. 5 1299 Bader S D 1990 Proc. IEEE 78 909 Wakiyama T 1991 Physics and Engineering Applications of Magnetism ed Y Ishikawa and N Miura (Berlin: Springer) p 133 [43] St¨ohr J and K¨onig H 1995 Phys. Rev. Lett. 75 3748 and references therein [44] Weller D, St¨ohr J, Nakajima R, Carl A, Samant M G, Chappert C, Megy R, Beauvillain P, Veillet P and Held G A 1995 Phys. Rev. Lett. 75 3752 and references therein [45] Hadjipanayis G and Gaunt P 1979 J. Appl. Phys. 50 2358 [46] Lin C J and Gorman G L 1992 Appl. Phys. Lett. 61 1600 2374
[47] Weller D, Brandle H, Lin C J and Notary H 1992 Appl. Phys. Lett. 61 2726 [48] Weller D, Brandle H and Chappert C 1993 J. Magn. Magn. Mater. 121 461 [49] Suzuki T, Weller D, Chang C A, Savoy R, Huang T C, Gurney B and Speriosu V 1994 Appl. Phys. Lett. 64 2736 [50] Maret M, Cadeville M C, Staiger W, Beaurepaire E, Poinsot R and Herr A 1996 Thin Solid Films 275 224 [51] Tyson T A, Conradson S D, Farrow R F C and Jones B A 1996 Phys. Rev. B 54 R3702 [52] Gay J G and Richter R 1986 Phys. Rev. Lett. 56 2728 [53] Guo G Y, Temmerman W M and Ebert H 1991 J. Phys.: Condens. Matter 3 8205 [54] Victora R H and MacLaren J M 1993 Phys. Rev. B 47 11 583 [55] Sakuma A 1994 J. Phys. Soc. Japan 63 3053 [56] Solovyev I V, Dederichs P H and Mertig I 1995 Phys. Rev. B 52 13 419 and references therein [57] Daalderop G H O, Kelly P J and Schuurmans M F H 1993 Phys. Rev. Lett. 71 2165 [58] Wu R and Freeman A J 1996 J. Appl. Phys. 79 6209 and references therein [59] Trygg J, Johansson B, Eriksson O and Wills J M 1995 Phys. Rev. Lett. 75 2871 [60] Strange P, Ebert H, Staunton J B and Gyorffy B L 1989 J. Phys.: Condens. Matter 1 3947 [61] Strange P, Staunton J B, Gyorffy B L and Ebert H 1991 Physica B 172 51 [62] Brooks H 1940 Phys. Rev. B 58 909 [63] Jansen H J F 1988 Phys. Rev. B 38 8022 [64] Mackintosh A R and Andersen O K 1980 Electrons at the Fermi Surface ed M Springfold (Cambridge: Cambridge University Press) Weinert M, Watson R E and Davenport J W 1985 Phys. Rev. B 32 2115 [65] Staunton J B, Matsumoto M and Strange P 1992 Applications of Multiple Scattering Theory to Materials Science ed W H Butler et al Mater. Res. Soc. Symp. Proc. 253 309 [66] Strange P, Staunton J B and Gyorffy B L 1984 J. Phys. C: Solid State Phys. 17 3355 [67] Strange P, Ebert H, Staunton J B and Gyorffy B L 1989 J. Phys.: Condens. Matter 1 2959 [68] Ebert H and Akai H 1992 Applications of Multiple Scattering Theory to Materials Science ed W H Butler et al Mater. Res. Soc. Symp. Proc. 253 329 [69] For a recent review see Ebert H 1996 Rep. Prog. Phys. 59 1665 [70] Razee S S A, Staunton J B and Pinski F J 1997 Phys. Rev. B 56 8082 [71] Razee S S A, Staunton J B, Pinski F J, Ginatempo B and Bruno E 1998 J. Appl. Phys. 83 7097 [72] Kadomatsu H, Tokunaga T and Fujiwara H 1981 J. Phys. Soc. Japan 50 3 [73] Mermin N D 1965 Phys. Rev. A 137 1441 [74] Moriya T (ed) 1981 Electron Correlations and Magnetism in Narrow Band Systems (New York: Springer) [75] Korenman V, Murray J L and Prange R E 1977 Phys. Rev. B 16 4032 Korenman V, Murray J L and Prange R E 1977 Phys. Rev. B 16 4048 Korenman V, Murray J L and Prange R E 1977 Phys. Rev. B 16 4058 [76] Capellman H 1977 Z. Phys. B 34 29 [77] Korenman V 1985 J. Appl. Phys. 57 3000–5 [78] Ziebeck K R A, Brown P J, Deportes J, Givord D, Webster P J and Booth J G 1983 Helv. Phys. Acta. 56 117–30 [79] Shirane G, Boni P and Wicksted J P 1986 Phys. Rev. B 33 1881–5 [80] Edwards D M 1984 J. Magn. Magn. Mater. 45 151–6
Magnetic alloys [81] Wang C S, Prange R E and Korenman V 1982 Phys. Rev. B 25 5766–77 [82] Hubbard J 1979 Phys. Rev. B 20 4584 [83] Hasegawa H 1979 J. Phys. Soc. Japan 46 1504 [84] Edwards D M 1982 J. Phys. F: Met. Phys. 12 1789–810 [85] Liu S H 1978 Phys. Rev. B 17 3629–38 [86] Oguchi T, Terakura K and Hamada N 1983 J. Phys. F: Met. Phys. 13 145–60 [87] Gyorffy B L, Pindor A J, Staunton J B, Stocks G M and Winter H 1985 J. Phys. F: Met. Phys. 15 1337–86 [88] Heine V and Joynt R 1988 Europhys. Letts. 5 81–5 [89] Samson J 1989 J. Phys.: Condens. Matter 1 6717–29 [90] Staunton J B, Gyorffy B L, Pindor A J, Stocks G M and Winter H 1985 J. Phys. F: Met. Phys. 15 1387–1404 [91] Staunton J B and Gyorffy B L 1992 Phys. Rev. Lett. 69 371–4 [92] Arajs S and Miller D S 1960 J. Appl. Phys. 31 986 Sucksmith W and Pearce R R 1938 Proc. R. Soc. A 167 189 [93] Uhl M, Sandratskii L M and Kubler J 1992 J. Magn. Magn. Mater. 103 314–24 [94] Sandratskii L M and Kubler J 1993 Phys. Rev. B 47 5854–60 [95] Sandratskii L M 1998 Adv. Phys. 47 91–160 [96] Liu K L and Vosko S H 1978 J. Phys. F: Met. Phys. 8 1539–56 [97] Blackman J A, Morgan T and Cooke J F 1985 Phys. Rev. Lett. 55 2814–17 [98] Feynman R P 1955 Phys. Rev. 97 660 [99] Brout R and Thomas H 1967 Physics 3 317 [100] Ling M F, Staunton J B and Johnson D D 1994 Europhys. Lett. 25 631–6 Ling M F, Staunton J B and Johnson D D 1994 J. Phys.: Condens. Matter 6 5981–6000 [101] Pinski F J, Staunton J B, Gyorffy B L, Johnson D D and Stocks G M 1986 Phys. Rev. Lett. 56 2096–9 [102] Johnson D D, Pinski F J, Staunton J B, Gyorffy B L and Stocks G M 1989 Physical Metallurgy of Controlled Expansion Invar-type Alloys ed K C Russell and D F Smith (Warrendale, PA: The Materials, Metals and Metallurgy Society) Johnson D D and Shelton W A 1997 The Invar Effect—A Centennial Symposium ed J Wittenauer (Pittsburgh, PA: The Minerals, Metals and Materials Society) p 63 [103] Murata K K and Doniach S 1972 Phys. Rev. Lett. 29 285 [104] Moriya T 1979 J. Magn. Magn. Mater. 14 1 [105] Lonzarich G G and Taillefer L 1985 J. Phys. C: Solid State Phys. 18 4339–71 [106] von der Linden W, Donath M and Dose V 1993 Phys. Rev. Lett. 71 899–902 [107] Kakizaki A, Fujii J, Shimada K, Kamata A, Ono K, Park K H, Kinoshita T, Ishii T and Fukutani H 1994 Phys. Rev. Lett. 17 2781–4 [108] Uhl M and Kubler J 1996 Phys. Rev. Lett. 77 334–7 [109] Uhl M and Kubler J 1997 J. Phys.: Condens. Matter 37 7885–902 [110] Gyorffy B L, Kollar J, Pindor A J, Staunton J B, Stocks G M and Winter H 1983 Proc. Workshop on 3d Metallic Magnetism in Institut von Laue-Langevin Grenoble, France in March 1983 pp 121–46 [111] Kisker E, Schroder K, Campagna M and Gudat W 1984 Phys. Rev. Lett. 52 2285–8 Kisker E, Schroder K, Campagna M and Gudat W 1985 Phys. Rev. B 31 329–39
[112] Kirschner J, Globl M, Dose V and Scheidt H 1984 Phys. Rev. Lett. 53 612–15 [113] Haines E M, Clauberg R and Feder R 1985 Phys. Rev. Lett. 54 932–4 Haines E M, Heine V and Ziegler A 1986 J. Phys. F: Met. Phys. 16 1343–64 [114] Bielby R F, Staunton J B, Ling M F and Pinski F J 1998 Phys. Rev. B submitted [115] Khachaturyan A G 1983 Theory of Structural Transformations in Solids (New York: Wiley) [116] Evans R 1979 Adv. Phys. 28 143 [117] Hansen J-P and McDonald I R 1976 Theory of Simple Liquids (New York: Academic) [118] Cahn J 1961 Acta. Metall. 9 795 [119] Khachaturyan A G 1973 Phys. Status Solidi b 60 9 [120] Staunton J B, Johnson D D and Pinski F J 1990 Phys. Rev. Lett. 65 1259–62 [121] Ling M F, Staunton J B and Johnson D D 1995 J. Phys.: Condens. Matter 7 1863–87 Johnson D D, Staunton J B, Gyorffy B L, Pinski F J and Stocks G M 1990 Mater. Res. Soc. Symp. Proc. 166 231 [122] Krivoglaz M A 1969 Theory of X-ray and Thermal Neutron Scattering by Real Crystals (New York: Plenum) [123] Lovesey S W 1984 Theory of Neutron Scattering from Condensed Matter. Volume 1. Nuclear Scattering (Oxford: Clarendon) [124] Marshall W 1968 J. Phys. C: Solid State Phys. 1 88 [125] Cable J W and Medina R A 1976 Phys. Rev. B 13 4868 [126] Cable J W, Child H R and Nakai Y 1989 Physica B 156–157 50 [127] Staunton J B, Ling M F and Johnson D D 1997 J. Phys.: Condens. Matter 9 1281–1300 [128] Mirebeau I, Cadeville M C, Parette G and Campbell I A 1982 J. Phys. F: Met. Phys. 12 25 [129] Pierron-Bohnes V, Kentzinger E, Cadeville M C, Sanchez J M, Caudron R, Solal F and Kozubski R 1995 Phys. Rev. B 51 5760 [130] Massalski T B, Okamoto H, Subramanian P R and Kacprzak L (eds) 1990 Binary Alloy Phase Diagrams 2nd edn (Metals Park, OH: American Society for Metals) [131] Stephens J R 1987 High Temperature Ordered Intermetallic Alloys ed C C Koch et al (Pittsburgh: MRS) [132] McKamey C G, Devan J H, Tortorelli P F and Sikka V K 1991 J. Mater. Res. 6 1779–805 [133] Epperson J E and Spruiell J E 1969 J. Phys. Chem. Solids 30 1733 [134] Schweika W, Monkenbusch M and Ackermann H 1989 Physica B 156–157 78 [135] Schweika W 1990 Mater. Res. Soc. Symp. Proc. 166 249 [136] Gao Z Q and Fultz B 1993 Phil. Mag. B 67 787–800 [137] Pierron-Bohnes V, Lefebvre S, Bessiere M and Finel A 1990 Acta. Met. Mater. 38 2701–10 [138] Pierron-Bohnes V, Cadeville M C, Finel A and Schaerpf O 1991 J. Physique I 1 247 [139] Anthony L, Nagel L J, Okamoto J K and Fultz B 1994 Phys. Rev. Lett. 73 3034 [140] Gelatt C D, Williams A R and Moruzzi V L 1983 Phys. Rev. B 27 2005 Corb B W, O’Handley C and Grant N J 1983 Phys. Rev. B 27 636
2375
