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Spin Fluctuation Theory in Weak Ferromagnets

The single-site theory of magnetism allows us to understand the finite temperature magnetism qualitatively or semiquantitatively, starting from the microscopic Hamiltonian. We can analyze the magnetic properties from metals to insulators on the basis of the theory. The first-principles dynamical CPA can explain quantitatively high-temperature properties such as the Curie constant in the paramagnetic susceptibility. The single-site theory, however, neglects inter-site spin correlations. The latter influences the Curie temperature and the other quantities related with magnetic short range order. In particular, long-wave spin fluctuations are indispensable for understanding the weak ferromagnetism in ZrZn2 , Sc3 In, and Ni3 Al, which is characterized by a small Curie temperature (∼10 K) and a large Rhodes–Wohlfarth ratio (1). Here the Rhodes–Wohlfarth ratio is defined by the ratio of the observed Curie constant to the one based on the local moment model. In this chapter, we present a method which takes into account long-range intersite spin correlations at finite temperatures in the weak ferromagnets [91–96].

5.1 Free-Energy Formulation of the Stoner Theory We have described in Sect. 3.1 the Stoner theory in the weak Coulomb interaction region on the basis of the Hartree–Fock self-consistent equations, as well as the single-site spin fluctuation theories which go beyond the Stoner theory. In this section, we rederive the Stoner theory using the free energy at finite temperatures. We have constructed the dynamical CPA on the basis of the grand canonical ensemble in Chap. 3. There we treated the free energy F (μ, H, T ) for given chemical potential μ, external magnetic field H , and temperature T . Magnetization is then obtained from the thermodynamic relation as M =−

∂F (μ, H, T ) . ∂H

Y. Kakehashi, Modern Theory of Magnetism in Metals and Alloys, Springer Series in Solid-State Sciences 175, DOI 10.1007/978-3-642-33401-6_5, © Springer-Verlag Berlin Heidelberg 2012

(5.1) 135

136

5

Spin Fluctuation Theory in Weak Ferromagnets

The uniform susceptibility is therefore obtained from the relation, χ =−

∂ 2 F (μ, H, T ) . ∂H 2

(5.2)

We can also make use of the free energy F (N, M, T ) for given electron number N , magnetization M, and temperature T when we discuss the thermodynamics of the system. In fact, the free energy F (μ, H, T ) is written as follows. e−βF (μ,H,T ) =



e−β(Eα (M, N )−μN−MH ) =



αNM

e−βF (N,H,T )+βμN .

(5.3)

N

Here we assumed that magnetization Mˆ commutes with the Hamiltonian Hˆ . Eα (M, N ) is the energy eigen value when the electron number N and magnetization M are given. The free energy F (N, H, T ) is defined by e−βF (N, H, T ) =



e−β(Eα (M, N )−MH ) =

αM



e−βF (N, M, T )+βMH .

(5.4)

M

In the same way, the free energy F (N, M, T ) is given by e−βF (N,M,T ) =



e−βEα (M,N) .

(5.5)

α

We can omit fluctuations of the macroscopic variables N and M, so that we obtain from (5.3) and (5.4) the relations F (μ, H, T ) = F (N, H,