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

magnitude of the vector Sq to be S, and assuming that the strength of the exchange interaction is J1 between first and J2 between second neighbors, the energy per spin is

The energy expression takes its minima for vectors q whose components are related by

provided |J2| > |J1|/4. The latter would lead to an incommensurate structure, however the energy associated with such a structure is never an absolute minimum. Apart from the vector q0 = (0, 0, 0), which corresponds to the ferromagnetic state, the inequivalent solutions of interest correspond to the vectors

q1 = (π/a)(0, 0, 1), q2 = (π/a)(1, 1, 0), q3 = (π/a)(1, 1, 1) . (14.4.31)

These are just the vectors given in (14.1.3); they correspond to the structures shown in Fig. 14.2. From the minimum of the energy it can be determined which structure is stable for given signs and ratio of J1 and J2. This is shown in Fig. 14.13.

Fig. 14.13. The range of stability for magnetic structures in a simple cubic crystal associated with di erent vectors q in the space of the couplings J1 and J2

The stability condition for various antiferromagnetic structures can be determined along the same lines in other crystal lattices, too.

In some of the presented antiferromagnetic structures upwardand downward-pointing atomic spins are arranged in such a way as to make up

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two simple interpenetrating sublattices in such a way that the nearest neighbors of each atom with an up spin are atoms with a down spin, and vice versa. This situation is observed in the antiferromagnetic structure formed in a simple cubic lattice – shown in Fig. 14.2(c) – in which up and down spins alternate along the edges, forming face-centered cubic sublattices. Another example that occurs in body-centered cubic lattices is shown in Fig. 14.3(a); here the spins at the vertices of the cubic primitive cell point in the opposite direction as spins at the cell centers. Up and down spins now make up simple cubic sublattices. Other antiferromagnetic structures, especially those formed in face-centered cubic lattices, are more complicated. Whichever atom is chosen, among its neighbors there are some with parallel and some with antiparallel spin, and so the decomposition requires at least four sublattices: two with spins up, and two with spins down.

For simplicity, we shall deal only with the properties of antiferromagnetic materials that can be decomposed into two sublattices, and in which each of the N atoms in volume V is surrounded by z first neighbors located on the other sublattice. We shall denote the sublattices by A and B, and their lattice points by i and j, respectively. We shall also assume that the exchange interaction acts only between nearest neighbors: Jij = J < 0, where i and j denote nearest neighbor lattice points, which are necessarily on opposite sublattices. The Hamiltonian can be written as

where label i runs over sublattice A, and label j over sublattice B, and i, j in the first term denotes the constraint that i and j should be nearest neighbors. Since each pair occurs only once in the sum, a factor of two appears in front of the first term compared to the ferromagnetic case.

The mean-field approximation is introduced in the same way as for ferromagnets. By neglecting second-order terms in the deviation from the mean value,

This expression can be interpreted again by saying that spins feel an e ective field Be . However spins on opposite sublattices are aligned in di erent directions, so the e ective fields are di erent on the two sublattices – but they are independent of the lattice point on each sublattice:

482 14 Magnetically Ordered Systems

This leads to the following temperature dependence of the magnetic susceptibility in the paramagnetic phase above the Néel temperature:

This expression is analogous to the Curie–Weiss law, which gives the susceptibility of ferromagnets above the Curie temperature; the only di erence is that TC is now replaced by −TN. For this reason, susceptibility remains finite at the critical temperature of the magnetic transition, that is to say the divergence observed in the Curie point of ferromagnets does not appear. This might be surprising at first, since both the ferromagnetic and antiferromagnetic phase transitions are of second order, and fluctuations are expected to play equally important roles at the critical temperature in the two cases. However, the order parameters are di erent, so the critical behavior manifests itself in di erent quantities.

In ferromagnets the magnetic order parameter is the uniform magnetization. Its long-wavelength fluctuations govern the critical behavior around the critical point TC, and give rise to the divergence of susceptibility in a uniform magnetic field. In contrast, the order parameter in antiferromagnets is the sublattice magnetization, more precisely MA −MB. According to the Fourier representation (14.1.2), when the magnetic moment varies periodically in space this di erence is proportional to the Fourier coe cient associated with the wave vector q0 of the antiferromagnetic structure. Therefore fluctuations do not become critical in the long-wavelength limit either but at wavelengths that correspond to the periodicity of the antiferromagnetic structure. By considering the wave-number-dependent susceptibility – i.e., the magnetic response to finite-wavelength perturbations – as a function of k, no divergencies are observed at k = 0 (which is associated with a uniform field), only around k = q0. This component of the susceptibility corresponds to the response to an alternating magnetic field (staggered field) that is opposite on the two sublattices. It is therefore called staggered susceptibility.

Below the Néel temperature the net magnetization vanishes. If the orientations of the moments on the two sublattices would remain each other’s opposite in an external magnetic field, energy could be reduced only by making the lengths of the moments unequal on the two sublattices. In fact in the energetically most favorable arrangement the sublattice magnetizations are not aligned with the field lines: they are tilted symmetrically and make an angle θ with the direction perpendicular to the field, as shown in Fig. 14.14(a).

isotropic antiferromagnet in a magnetic field. (b) The temperature dependence of susceptibility

In the mean-field theory the sublattice magnetization M A of sublattice A must be aligned with the e ective field acting on the sublattice,

that is, just like M A, Be , A also has to make an angle θ with the direction perpendicular to the magnetic field H. Writing this condition in terms of the components of Be , A, and bearing in mind that the Weiss coe cient λ is now negative,

In weak magnetic fields θ is small, so the moments are almost perpendicular to the field. Expanding the trigonometric functions to leading order,

A similar expression applies to the magnetization of the other sublattice, too. Expressing the component of the induced magnetization in the direction of the magnetic field,

Substitution of the expression for λ into this formula gives the susceptibility, which is independent of the sublattice magnetization and therefore of temperature as well:

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The symbol in the subscript refers to the almost perpendicular orientation of the moments with respect to the magnetic field.

Comparing this result with (14.4.43), the expression for the susceptibility above the Néel temperature, the same value is obtained at the critical point. Thus in an isotropic antiferromagnet susceptibility is constant below the Néel temperature, and starts to decrease above it. As illustrated in Fig. 14.14(b), at very high temperatures the same 1/T dependence is observed as in paramagnets (see Curie’s law on page 52).

However, there are no perfectly isotropic materials in which this behavior could be observed. The inevitable anisotropy stemming from the crystalline structure gives rise to interactions that will keep spins along a preferred crystallographic direction. If the weak external magnetic field also acts in this direction, then anisotropy does not let spins rotate away from this direction and be almost perpendicular to the field. The susceptibility measured under such circumstances is the so-called parallel susceptibility, χ . To determine it, one should start with (14.4.36), but assume that the magnetization of one sublattice is along the external magnetic field, while that of the other is along the opposite direction. At finite temperatures, where the magnitude of the sublattice magnetization is smaller than the saturation value, the magnetic field increases the magnetization on the sublattice where it is aligned with the field, and decreases it on the other sublattice. The result is a highly temperature dependent net magnetization and susceptibility – which vanish in the T → 0 limit. The temperature dependence of parallel susceptibility is shown in Fig. 14.14(b). In macroscopic samples the direction of sublattice magnetization may vary from domain to domain. Therefore measurements of the susceptibility give a weighted average of χ and χ , specifically

According to the foregoing, in antiferromagnets that exhibit su ciently strong uniaxial anisotropy sublattice magnetization is along the easy axis of magnetization even in the presence of a magnetic field. Spins will rotate away from this direction – that is, the magnetic field will overcome anisotropy – only when the magnetic field applied in the direction of easy magnetization exceeds a threshold value. In such a field spins will suddenly turn from the parallel direction to a configuration in which the sublattice magnetizations are oriented symmetrically and make an angle θ with the easy axis of magnetization. This first-order phase transition is therefore called spin-flop transition, and the resulting phase is the spin-flop phase. When the field is increased even further, the sublattice magnetizations become more and more aligned with the field, and finally a perfectly collinear structure arises through a second-order phase transition. The theoretically predicted and experimentally measured phase diagrams are shown in Fig. 14.15.

H

P

SF

AF

T

Fig. 14.15. The phase diagram (temperature vs. magnetic field plot) of an anisotropic antiferromagnet with an easy axis of magnetization. Part (a) shows a schematic phase diagram, while part (b) is the measured phase diagram of MnF2 below the Néel temperature [Y. Shapira and S. Foner, Phys. Rev. B 1, 3083 (1970)]

14.4.3 The General Description of Two-Sublattice

Antiferromagnets

The above simple treatment of antiferromagnets led to the result that in the paramagnetic region the magnetic susceptibility can be written as

where TN is the Néel temperature. Experiments, on the other hand, have shown that while in the vast majority of antiferromagnetic materials the temperature dependence of susceptibility is fairly well approximated by the formula

the value of Θ is nevertheless di erent from the Néel temperature. As data in Table 14.6 indicate, the deviation can sometimes be very large.

This deviation is the result of the applied approximation, whereby only the exchange interaction between nearest neighbors was taken into account, and so the behavior of each sublattice was entirely determined by the e ective field due to the other sublattice. To demonstrate this, we shall determine the critical temperature and the Θ parameter of two-sublattice antiferromagnets

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Table 14.6. Néel temperature and the characteristic temperature Θ that appears in the fit of susceptibility data

for the case when the exchange interaction between second neighbors (located on the same sublattice) cannot be neglected.

We shall denote the strength of the exchange interaction between nearest neighbors on opposite sublattices by JAB; the number of nearest neighbors (on sublattice B) of an atom on sublattice A by zAB, and the same for an atom on sublattice B by zBA. Along the same lines, we shall denote the strength of the exchange interaction between second neighbors (located on the same sublattice) by JAA and JBB, and the number of second neighbors by zAA and zBB, respectively. We shall also assume that the two sublattices are of the same structure, that is zAB = zBA, JAA = JBB, and zAA = zBB. In the mean-field theory the sublattice magnetization has to be determined self-consistently from the equations

where the e ective field acting on the spins of sublattice A is given by the generalization of (14.4.37),

The e ective field acting on sublattice B is given by an analogous formula. In the absence of a magnetic field, and in the vicinity of the phase transi-

tion point, where the sublattice magnetization is small, the expansion of the

Nontrivial solutions exist only when the determinant of the coe cients vanishes. When the nearest-neighbor interaction is antiferromagnetic, i.e., JAB < 0, the physically meaningful solution for the transition temperature is

To determine the susceptibility in the disordered phase, a weak external field is turned on. The e ective field on the atoms in sublattice A is then

Using the linear expansion of the Brillouin function, the sublattice magnetization in this field is

(14.4.57) Adding this expression to the analogous one for the other sublattice,

+ 2S(S + 1) (JABzAB + JAAzAA) (MA + MB) .

3kBT

Using the form (14.4.51) for the susceptibility, some rearrangement leads to

Comparison with expression (14.4.55) for the Néel temperature shows that Θ > TN if the second-neighbor interaction is also antiferromagnetic. This applies to the majority of cases.

14.4.4 The Mean-Field Theory of Ferrimagnetism

Within the framework of the mean-field theory, ferrimagnets can be treated much in the same manner as antiferromagnets. In the simplest case the lattice may be decomposed into two sublattices, however, the sublattice magnetizations do not cancel each other now. Either because the magnetic atoms on the two sublattices are unequal in number, or because their magnetic moments are of unequal magnitude. We shall examine the latter case, and generalize the equations derived for the sublattice magnetizations in antiferromagnets by letting the spin magnitudes (SA and SB) as well as the exchange integrals be di erent on the two sublattices:

N

MA = 2V |g|μBSABSA (β|g|μBSABe , A) ,

(14.4.60)

N

MB = 2V |g|μBSBBSB (β|g|μBSBBe , B) .