The calculation of the Stoner exchange parameter I for metals within the itinerant model

The Stoner Model of Itinerant Electron Magnetism

Stoner (1936, 1938) formulated his model of itinerant electron magnetism which can be seen as a transcription of the Weiss model for metallic solids. Stoner replaced the local ‘‘free atom-like’’ energy levels in the Weiss model, which are characterized by their respective magnetic quantum number, by the electronic bandstructure. Due to the crystal field interaction in solids the angular momentum appears to be quenched (angular quantum numbers are no longer relevant), so that only the k vector and the spin of the valence electrons remains in describing the state of the electrons. The model is based on the following three postulates: (i) the carriers of magnetism are the electrons in the d band, (ii) effects of exchange and correlation are treated within a molecular field term, and (iii) it must conform to Fermi statistics.

It is assumed that for an applied magnetic field, which is the molecular field (eventually including an external field Hex) the Fermi energy of the paramagnetic state is shifted to new values e þ f for spin-up and e f for spin-down which leads to new occupation numbers n þ and n (Fig. 1, left panel). This picture is identical to the actual shifting of the two spindependent bands by the spin splitting DE (Fig. 1, right panel).

Since the Fermi energy is the energy of the highest occupied state it is equal to the chemical potential m; and consequently the new Fermi energies, caused by the field, can be seen as chemical potentials for spinup ( þ ) and spin-down ( ) as m þ and m . The Stoner equations can now be formulated:

The number of spin-up and spin-down electrons n7 is given by the integral over all occupied states, where for T ¼ 0 K the upper limit can be taken as the Fermi energy eF. In its original formulation, Stoner assumed that the spin-up and spin-down bandstructure were equal so that the density of states N(e) are the same for both spin directions. This treatment is referred to as the rigid band picture, which assumes that during the transition from a nonmagnetic to a ferromagnetic state the spin-up and spin-down bands do not change their shape and that they only become shifted energetically by the spin splitting DE (c.f. Table 1) given by:

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In Eqns. (5) and (6), the quantity I is the Stoner exchange factor and M is the magnetic moment. From the Stoner equations the susceptibility can be calculated which becomes:

where df/de is the energy derivative of the Fermi– Dirac distribution. Equation (7) is the most general form of the Stoner susceptibility which for T ¼ 0 K reduces to:

which is the Pauli susceptibility 2m2 BN(eF) enhanced by the exchange interaction via a factor S (Stoner enhancement factor). If the term IN(eF) becomes larger than 1, w becomes negative, which means that the interacting electron gas is unstable against the formation of a magnetic moment (spontaneous magnetic order). This gives rise to the formulation of the famous Stoner criterion for the onset of magnetism which happens if:

Since the Stoner exchange factor I is an atom specific constant, a possible transition to a magnetic state depends only on the density of states at the Fermi energy N(eF). This in turn means that bands are needed which are narrow enough to create a large density of states. This is not only the explanation why band magnetism occurs only in the 3d transition metals (Cr, Mn, Fe, Co, and Ni) and the 5f-actinides starting with curium. The band width of the 4d and 5d series is already too large, which implies that a certain degree of localization is necessary.

If a system fulfills the Stoner criterion and thus has a magnetic ground state, Eqn. (7) also allows to calculate the Curie temperature by requiring that Tc is given by the temperature for which the denominator of Eqn. (7) becomes zero:

Eqn. (10) is a temperature dependent Stoner criterion. Due to the convolution with the function df/de, the effective density of states becomes progressively smaller when the temperature rises until the relation given above is fulfilled. Once this happens, the Stoner theory does no longer see any magnetic moment so that the paramagnetic state (the state above Tc) is the nonmagnetic state. This is a new mechanism compared to the Weiss model, where the paramagnetic state was given by complete disorder of the local moments, or to the Heisenberg model within the mean field approximation, where the thermal agitation leads to a breakdown of the interaction aligning neighboring spins. Unfortunately the Curie temperatures calculated within the Stoner model are too high by a factor of 4–8 (Gunnarsson 1976). The thermal excitation mechanism is called single-particle excitations or simply Stoner excitations. At T ¼ 0 K all states below the Fermi energy are occupied and all states above are empty. With rising temperature the softening of the Fermi–Dirac distribution creates holes below the (T ¼ 0 K) Fermi energy and creates occupied states above.

Reference: Itinerant Electron Systems: Magnetism (Ferromagnetism)

P. Mohn 

Technische Universita¨t Wien, Austria