Fundamentals of the Physics of Solids / 14-Magnetically Ordered Systems

Fig. 14.10. The d electrons of Mn3+ and Mn4+ ions on the t2g and eg levels

takes place with high probability only when the spins of neighboring ions are aligned ferromagnetically. Since hopping lowers the total energy by reducing the kinetic energy, ferromagnetic alignment of the spins is energetically favored. Thus, in contrast to superexchange, which is often antiferromagnetic, double exchange is always ferromagnetic. The propagation (hopping) of the electrons in the crystal indicates that such systems are metallic.

Such a double exchange plays an important role in magnetite, which contains divalent (Fe2+, 3d6) and trivalent (Fe3+, 3d5) iron ions. The spins of the Fe2+ and Fe3+ ions at octahedral sites are aligned by double exchange interactions. On the other hand, the spins of the Fe3+ ions at tetrahedral sites are antiparallel to the spins at octahedral sites because of antiferromagnetic superexchange interactions.

14.3 Simple Models of Magnetism

In the previous section we saw that, up to an additive constant, the interaction between two localized spins S1 and S2 can always be written as

independently of the specific mechanism of the exchange – although the coupling strength J depends on the character of the exchange and on the separation of atomic spins. We now have to generalize this formula to the case when atomic spins are arranged in a regular crystalline structure.

14.3.1 The Isotropic Heisenberg Model

It is not easy to generalize the exchange interaction between two spins to a solid that contains a large number of magnetic atoms. For simplicity we shall assume that a net spin Si is located at each lattice point Ri of the crystal. It is known that the atomic magnetic moment can also come from orbital motion; indeed, one should therefore deal with the total angular momentum J i and the corresponding magnetic moment μi = −gJ μBJ i. However, the microscopic origin of the moments is of little importance for the phenomena

470 14 Magnetically Ordered Systems

discussed here, therefore we shall continue to speak about spins and use the notation Si – although by this we shall invariably mean the total localized angular momentum of the ion. The only notational hint for this will be using g instead of ge or gJ for the g-factor.

The net spin Si can be the resultant of several electronic spins, and electrons may be exchanged with the electrons of several neighboring atoms. The problem is greatly simplified by the assumption that the total interaction can be written as the sum of pairwise interactions. If it is also assumed that for each atom it is su cient to consider only the ground-state configuration as determined by Hund’s rules (or by the splitting due to the crystal field), and higher-lying excited states can be neglected, then using some simple considerations the results obtained for two electrons are fairly straightforwardly generalized without a full quantum mechanical treatment. In the subspace of the Hilbert space that contains the lowest-lying states the magnitude of each atomic spin (and therefore of each atomic magnetic moment) is constant. Interactions may at most change their orientations. If the interaction between the atomic spins at lattice points Ri and Rj is given by a term of the form of the Heisenberg exchange interaction, Si · Sj , and the strength of the interaction can be characterized by the exchange coupling – i.e., the amplitude Jij = J(Ri − Rj ), which depends on the separation of the spins –, then the

Since the indices i and j both run over the entire lattice, each pair of spins occurs with the correct weight factor. It was first pointed out by W. Heisenberg (1928), and independently of him by J. Frenkel that ferromagnetism can be explained in terms of such pairwise interactions between localized spins. For this reason the model described by the above Hamiltonian is called the Heisenberg model of magnetism. According to the foregoing, states in which electrons are transferred from one atom to the other (for no matter how short a period) are not present explicitly in this model. Only the orientation of atomic moments is considered.

At low temperatures the exchange interaction does not usually leave atomic spins independent: they tend to align each other into a preferred direction, giving rise to an ordered structure. The simplest of them is ferromagnetic order.This is observed in materials in which the sign of the exchange interaction is positive, hence it lines up neighboring spins in the same direction. However, the exchange interaction is not necessarily of ferromagnetic character. As it was discussed at the beginning of this chapter, ferromagnets form a relatively small group of magnetically ordered materials. More numerous are antiferromagnets, in which the dominant nearest-neighbor coupling is antiferromagnetic (Jij < 0), which gives rise to an ordered array of alternately

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since the only bilinear expression in the spin components that is invariant under the symmetries of a cubic crystal is the isotropic combination

i,j

In tetragonal crystals one axis has a privileged status over the two others. This uniaxial anisotropy has to be reflected in the magnetic Hamiltonian, too. Choosing the z-axis along this direction, the Hamiltonian that possesses the symmetries of the crystal is

i,j

When > 1, the z components of the spins are coupled most strongly, so a nonvanishing magnetization appears preferentially along this direction. In this case the magnetic material has an easy axis of magnetization.When < 1, the spin components in the (x, y) plane are coupled strongly, and so magnetization appears in this plane – the easy plane of magnetization. In an extreme case of uniaxial anisotropy only the z components of the spins are coupled:

i,j

This is the Ising model,6 a fundamental model of statistical physics, since the partition function can be calculated as a sum over all possible classical configurations of the spin projections. On the other hand, when = 0, the XY model is recovered:

i,j

In tetragonal crystals, in addition to the contribution of ion pairs to uniaxial anisotropy, a term that corresponds to single-ion anisotropy,

i

may also appear in the Hamiltonian. In cubic crystals the first anisotropic contribution contains the fourth-order product of spins. The single-ion anisotropy term is

We shall choose the z-axis as the quantization direction, and instead of the operators Sx and Sy we shall repeatedly use the spin-projection-lowering and -raising operators

6E. Ising, 1925. It would be more appropriate to call it the Lenz–Ising model, since it was proposed by Ising’s supervisor, W. Lenz, in 1920.

The coe cient λ specifies the relation between the internal field and magnetization in the Weiss theory. If exchange occurs only with the z nearest neighbors then J0 = J z. When secondand third-neighbor interactions cannot be neglected, J0 is the weighted average of the various coupling constants. The necessary condition for ferromagnetic ordering is that J0 should be positive. This can happen even if some of the latter couplings are antiferromagnetic.

It was derived on page 57 in Section 3.2.6 on paramagnetism that the magnetization due to the magnetic moment of an atom of spin S in a magnetic field B is

where BS (x) is the Brillouin function. In the mean-field theory this expression has to be modified by replacing the magnetic induction B by the e ective field

In contrast to the paramagnetic case, the e ective field itself also depends on magnetization. Therefore the self-consistent equation

has to be solved now.

In the absence of an external magnetic field the argument of the Brillouin function is

Expressing condition (14.4.12) of self-consistency in terms of this variable,

The solutions to this equation are easily obtained graphically by plotting the leftand right-hand sides separately, as in Fig. 14.11. The thermal average of the magnetic moment can be read o from the intersection points.

The slope of the expression on the left-hand side is proportional to temperature. Therefore above a certain critical temperature TC only the trivial solution x = 0 – and consequently M = 0 – exists. The same solution exists also below TC, however it becomes unstable. In this regime there exists another solution with finite magnetization whose free energy is lower.

According to the expansion (3.2.84), the slope of the Brillouin function at x = 0 is (S +1)/3S, thus the equation that determines the critical temperature of the ferromagnetic state, the Curie temperature is

It is immediately seen that the susceptibility diverges as the critical temperature TC of the ferromagnetic transition is approached.

Below but close to TC an explicit formula can be obtained for the temperature dependence of spontaneous magnetization as well by taking into account the first correction beyond the linear term in the expansion of the Brillouin function. The self-consistent solution of the equation then leads to

In the critical point itself magnetization is a nonlinear function of the magnetic field, as (14.4.18) shows that the term which is linear in magnetization drops out at T = TC. This leads to

Fig. 14.12. Comparison of the measured magnetic properties of nickel [P. Weiss and R. Forrer, Ann. de Phys. 5, 153 (1926)] with the results obtained in the mean-field theory for S = 1/2. (a) Magnetization and (b) inverse susceptibility, as functions of temperature

At low temperatures magnetization tends to the saturation value M0 = (N/V )|g|μBS. Using the expansion coth x ≈ 1 + 2e−2x in the form (3.2.79) of the Brillouin function for large values of its argument, the leading-order contribution is

478 14 Magnetically Ordered Systems

In this region the first correction to the temperature dependence of magnetization is obtained by replacing the magnetization by its saturation value on the right-hand side of (14.4.12). Then

At intermediate temperatures no analytical expression can be derived. Figure 14.12 shows the measured temperature dependence of magnetization and inverse susceptibility for nickel, compared with the results obtained in the mean-field approximation for a spin S = 1/2 using the Brillouin function, and from the Curie–Weiss law.

A similar behavior is observed in other ferromagnetic materials, too. This leads to the conclusion that the mean-field theory provides a qualitatively good description for the finite-temperature behavior of the properties of ferromagnetic materials, however, its quantitative predictions are incorrect at low temperatures as well as close to the critical point – where analytical results can be obtained. To improve the description, one has to go beyond the mean-field approximation. At low temperatures a more accurate treatment of the low-energy excited states is required, while in the vicinity of the critical point critical fluctuations need to be taken into account.

14.4.2 The Mean-Field Theory of Antiferromagnetism

We have seen that even in cubic crystals various antiferromagnetic structures may occur. The reason for this is that besides the exchange interaction between nearest-neighbor moments, secondand third-neighbor couplings also play a role in determining the relative orientation of magnetic moments. This can be demonstrated most easily on the example of structures formed in a simple cubic lattice.

We shall consider the spins Si localized at lattice points as classical vectors, expand the spin density

i

into a Fourier series, and assume that the magnetic structure can be represented by a single Fourier component, i.e.,

which is equivalent to saying that localized spins can be given in the form

This choice is justified for collinear structures, where the upward or downward direction of the spin vector is determined by the phase factors. Taking the