Magnetism of Ferromagnetic Metals above their Curie Temperature P.J. Brown, Grenoble (Institut Laue-Langevin) Although the 3d metals: iron, cobalt pond to integral or half integral S and are tations for a local moment (Heisenberg) and nickel were amongst the first ferro­ generally not consistent with the mo­ and a Stoner system are shown schema­ magnetic materials to be recognised, full ments per atom obtained from the satu­ tically in Fig. 1. understanding of their magnetic proper­ ration magnetisation. The occurrence of For the local moment system, the ma­ ties has lagged far behind that of the less non-integral moments does not pose a gnetic electrons are contained in narrow widely known magnetic insulators such problem If it is accepted that the magne­ bands separated by an amount Eu which as europium oxide. The magnetic pro­ tic electrons belong to a partially filled is the intra-atomic exchange, or Hund's- perties of insulators are well accounted band and thus contribute to the Fermi rule, energy; one spin band is full and the for by the simple Curie Weiss theory in surface. The saturation magnetisation other empty. The interatomic coupling which magnetic moments localised on then measures the mean number of un­ energy is Ej = ΣJijSi•Sj « Eu being the the magnetic ions respond both to exter­ paired electrons per atom, and individual sum of pairwise interactions between nal fields and to an internal field arising atomic moments fluctuate on a time atomic spins Si, Sj. The magnetic excita­ from inter-atomic exchange interactions scale of the order h/W where W is the tions of such a system are collective of the Heisenberg type. This theory band width. transverse fluctuations of the atomic leads to the well known linear depen­ The first collective electron or band moments (spin waves). They have a dence of the inverse magnetic suscepti­ model of ferromagnetism was given by dispersion for T « Tc as indicated in bility on temperature — the Curie Weiss Stoner 1). As in the Curie Weiss theory Fig. 1a with the energy of the spin-wave law. exchange is introduced as a molecular at the Brillouin zone-boundary being = 1 / x = (T-θ)/P2eff field proportional to the mean magneti­ kTc. As the temperature is raised to­ where θ is a temperature related to the sation. This exchange field produces an wards Tc the spin-wave spectrum is Curie temperature, P eff is the effective energy difference between the spin-up thermally populated resulting in a reduc­ local moment. If the ionic moments are and spin-down electron bands causing tion in the ordered component of ma­ due to electron spin only, as is the case electrons to move from one band to the gnetisation and a consequent lowering for many transition metal salts, and the other to maintain a constant electrosta­ (renormalisation) of the spin-wave ener­ total ionic spin is S tic potential. The imbalance in the popu­ gies. At the transition temperature, the P2eff = 4S (S +1) Bohr magnetons2 lation of the bands gives rise to the ma­ whole spin-wave spectrum is thermally Values of Peff obtained for transition gnetisation. Stoner showed that, with a populated resulting in the complete loss metal salts lead to integer or half integer positive exchange parameter greater of long-range order. The individual values of S which agree with those than a certain critical value the magneti­ atomic moments point in random direc­ found in the ordered magnetic state. sation-induced band splitting would be tions, but retain the same magnitudes Although the paramagnetic suscepti­ self sustaining below a critical tempera­ since thermal energies are not sufficient bility of many ferromagnetic metals and ture, giving rise to a ferromagnetic state. to excite the majority carriers into the alloys follows a Curie Weiss law over The band structures for the magnetic minority band. some range of temperature, the effec­ electrons and the frequency ω versus The Stoner picture of ferromagnetism tive moments deduced do not corres­ wave-vector q spectra of magnetic exci­ is illustrated in Fig. 1b. At T « Tc the majority and minority spin-bands are split by an amount Eu which in this case is only of order kTc and less than the band width. Although at q = 0 a finite energy Eu is needed to promote an elec­ tron from the majority to the minority band, at q = q0 such a transfer can take place without the expenditure of energy. In the Stoner system the magnetic ex­ citations are of two kinds: at low ener­ gies and low q, collective spin wave exci­ tations can occur as indicated by the dashed curve in Fig. 1b, but over the whole of the hatched area the excita­ tions correspond to transfer of electrons from one band to another (Stoner excita­ Fig. 1 — Distribution in q ω space of electron states (dotted) and magnetic excitations. The tions). These Stoner excitations do not dashed lines represent the spin waves, and the hatched areas regions dominated by the conserve the value of individual atomic Stoner excitations. The Heisenberg and Stoner limits are represented in (a) and (b) respec­ moments. In the Stoner system, thermal tively. Efis the Fermi energy and the wave-vector q is normalised to unity at the Brillouin zone population of the Stoner excitations boundary. which takes place as the temperature 25 rises leads to a reduction in the band can then be used to relate the scattering short-range ferromagnetic order with an splitting which goes to zero at the Curie function for magnetic scattering to the inverse correlation range proportional to temperature so that in the paramagnetic imaginary part of the generalised ma­ [T/(T- Tc)]1/2 and a paramagnetic scat­ phase no atomic moments exist. gnetic susceptibility x"(qω) 2,3> tering function with the double Lorent- zian form Itinerant versus Local Moment Models S(q,ω) = M/(K2 + q2) • [Aq2/(A2q4 +ω2)] One may ask to what extent does which corresponds to magnetic excita­ either of these extreme models account In addition the integral of S(q,co) over all tions of a diffusive rather than a pro­ ω is proportional to the Fourier transform pagating type. for the properties of metallic ferroma- of the instantaneous spin-density spin- gnets at finite temperatures. Important There have been a number of different bulk properties which have to be ex­ density correlation function (SDSDCF) experimental neutron scattering studies which gives a snap-shot picture of the of the nature of the magnetic excitations plained are the magnetic susceptibility, average magnetisation around any atom the Curie temperature, the temperature in iron and nickel above Tc . Those which dependence of the atomic volume, and at a given instant in time. have perhaps excited the most wide­ the temperature variation of the specific spread interest presented evidence for heat. Analysis of the variation of many Magnetic Excitation Below Tc the persistence of propagating spin properties through the Curie tempera­ Triple axis neutron spectrometry waves at temperatures well above Tc, ture gives strong evidence for the per­ allows the scattering function to be 5, 6,7) and have become the object of sistence of long-lived magnetic mo­ determined over a range of q and to some controversy. One should point out ments in the paramagnetic phase. Im­ limited by practical considerations to be that although the evidence on magnetic portant among these phenomena are of the same order or less than the inci­ excitations provided by neutron scatter­ the susceptibility, the resistivity, and the dent neutrons' wavelength and frequen­ ing is rather direct, the difficulties in specific heat. The temperature depen­ cy. The results obtained for iron and determining the purely magnetic cross- dence of the atomic volume and the nickel 4,5) are illustrated in Fig. 2 where sections are formidable because the compressibility of iron and nickel sug­ they are compared with those for the nuclear scattering is in general conside­ gest that no dramatic change, which Heisenberg-like ferromagnet EuO. The rably stronger than the magnetic scat­ would be consequent on a total loss of figure shows the dispersion curve for the tering. moment, occurs at Tc . acoustic mode of collective magnetic These bulk measurements indicate excitations (spin-waves). It can be seen Neutron Scattering with Polarisation that the simple Stoner picture cannot be that whereas in EuO excitations pro­ Analysis appropriate to ferromagnets such as iron pagate at all wave-vectors out to the An experimental approach to the pro­ and nickel. On the other hand, whilst the boundary of the Brillouin zone in accor­ blem of separating the magnetic compo­ local moment theory can account for dance with the expectations for a local nent from other contributions to the many of the bulk properties of metallic moment system (Fig. 1a), the results for neutron scattering is to use a polarised magnets it fails to explain satisfactorily the metallic magnets is quite different. neutron beam — one in which all the difference in the ordered and effec­ The figure shows that the 'stiffness' of neutron spins point in a given direction tive moments. Non-integer moments as the spin waves is much greater than for — and to analyse the polarisation of the well as spectroscopic studies show that EuO, even allowing for the considerably scattered neutrons. In magnetic scatter­ magnetic electrons contribute to the higher Curie temperatures. The energy ing as opposed to nuclear scattering, the Fermi surface and since the bands are of the collective mode rises very sharply cross-section depends on the relative not narrow make the existence of local with increasing wave-vector and in iron orientations of the neutron spins and the moments in the paramagnetic phase dif­ a well defined spin-wave is still apparent wave-vector of the momentum transfer.