Cubic antiferromagnets (from a book)

A feature of many crystal lattices is that there can be several different ways of constituting the two equal antiferromagnetic sublattices. Different spin configurations have different topology. The orientation of the spins relative to the crystal axes is a separate issue, determined by magnetocrystalline anisotropy.

Heisenberg exchange is isotropic, so it imposes no particular antiferromagnetic axis.

Simple cubic

 Four possible antiferromagnetic modes for a simple cubic lattice are shown in Fig. 6.8. Two possible superexchange paths J1 and J2 exist for nearest-neighbour and next-nearest-neighbour interactions. Longer-range interactions are possible, especially in metals. Although no simple cubic elements exist, except polonium, the magnetic ions in a compound often form a simple-cubic sublattice: the ‘B’ sites in perovskite, ABO3 are an example. If J1 is the only antiferromagnetic interaction, the bonds are unfrustrated and a G mode is adopted, where all the six nearest neighbours of a particular atom lie on the opposite sublattice. The structure is composed of alternate ferromagnetic [111] planes. If J2 is the only antiferromagnetic interaction, it becomes impossible to satisfy all twelve next-nearest-neighbour bonds simultaneously, and the best solutions have eight of them on the opposite sublattice and four on the same sublattice, as in the A and C modes. There are four magnetic sublattices, and the equation for TN involves a fourth-order determinant. The fourth, F, mode is ferromagnetic. More general magnetic structures can be generated by combining different modes with components along the Cartesian axes. Small distortions of the lattice may favour particular orbital occupancy for Jahn-Teller ions, which modifies orbital overlap, and hence the exchange.

Body-centred cubic

 Here, there exist unfrustrated structures where either nearest-neighbour interactions J1 or next-nearest-neighbour interactions J2 can be completely satisfied. In the latter case, there are two completely decoupled simple-cubic antiferromagnetic structures. The structures are known as type I and type II bcc order, respectively. If both antiferromagnetic interactions are present, conflicts arise which may have to be accommodated in a partly frustrated, compromise ground state. If the two exchange interactions J1 and J2 are represented by molecular field coefficients, the regions of stability of the ferromagnetic and antiferromagnetic structures are as shown in Fig. 6.9.

Face-centred cubic

 The best-known frustrated antiferromagnet has an fcc lattice. Here the nearest-neighbour exchange J1 is always frustrated, just like J2 in the simple-cubic lattice. The fcc lattice divides into four simple-cubic sublattices with magnetizations MA, MB, MC and MD. Each atom has four nearest neighbours on each of the other three sublattices.At best two out of three neighbours of a given spin can be aligned antiparallel. In zero applied field, the molecular field equations are MA = M0BJ (xA),where xA = µ0mA[nAAMA + nAB(MB + MC + MD) + H]/kBT ,etc.

Considering only the intersublattice interaction nAB, it follows that TN = C nAB, θp = 3C nAB, (6.19) where C = µ0NAm2 eff/3kB. Note that θp = 3TN. We can read the strength of the individual exchange interactions in extrapolating to zero the susceptibility measured at high temperature. The N´ eel temperature, however, reflects the extent to which it is possible to satisfy all the exchange bonds simultaneously. When the magnetic structure is known, it may be possible to deduce the two

molecular field coefficients, and hence the exchange interactions J1 and J2 from knowledge of TN and θp.

There are three possible magnetic modes for the fcc lattice, illustrated in Fig. 6.10. The structure with alternating ferromagnetic [001] planes is type I. That with a structure of alternating ferromagnetic [111] planes is type II. Type III is made up of alternating antiferromagnetic [001] planes. The transition-metal monoxides MnO, FeO, CoO and NiO all have type II order. MnTe2 and MnS2 are examples of type I and type III order, respectively. The regions of stability for different values of the first and second neighbour molecular field coefficients are shown in Fig. 6.10. A fourth type of order, consisting of alternating ferromagnetic [110] planes, can be stabilized by interactions extending beyond second neighbours, as in CrN.

Just as there can be no strictly cubic ferromagnet, because of magnetostriction, so there can be no strictly cubic antiferromagnet for the same reason. Magnetostriction produces a deformation of the structure along the antiferromagnetic axis. A slight tetragonal or rhombohedral distortion arises, depending on whether the axis is [100] or [111].


Reference:     Magnetism and Magnetic Materials

  http://bookzz.org/dl/1021010/795f32