Patterson, Bailey - Solid State Physics Introduction to theory

368 7 Magnetism, Magnons, and Magnetic Resonance

and after canceling,

which is linear in T (antiferromagnetism).

This equation is valid for T > Tc+ = μ0(α1+ω)C1 ≡ TN, the Néel temperature. Thus, if we define

We can also easily derive results for the ferromagnetic case. We choose to drop out one sublattice and in effect double the effect of the other to be consistent with previous work.

The reality of antiferromagnetism has been absolutely determined by neutron diffraction that shows the appearance of magnetic order below the critical temperature. See Fig. 7.3 and Fig. 7.4. Figure 7.5 summarizes our results.

6 2C1μ0 = C of (7.27).

Fig. 7.3. Neutron-diffraction patterns of MnO at 80 K and 293 K. The Curie temperature is 120 K. The low-temperature pattern has extra antiferromagnetic reflections for a magnetic unit twice that of the chemical unit cell. From Bacon GE, Neutron Diffraction, Oxford at the Clarendon Press, London, 1962 2nd edn, Fig. 142 p.297. By permission of Oxford University Press. Original data from Shull CG and Smart JS, Phys Rev, 76, 1256 (1949)

Fig. 7.4. Neutron-diffraction patterns for α-manganese at 20 K and 295 K. Note the antiferromagnetic reflections at the lower temperature. From Bacon GE, Neutron Diffraction, Oxford at the Clarendon Press, London, 1962 2nd edn, Fig. 129 p.277. By permission of Oxford University Press. Original data from Shull CG and Wilkinson MK, Rev Mod Phys, 25, 100 (1953)

370 7 Magnetism, Magnons, and Magnetic Resonance

+[4ω2CACB+(αACA – βBCB)2]1/2}

TN = μ0(α1+ω)C1

Fig. 7.5. Schematic plot of reciprocal magnetic susceptibility. Note the constants for the various cases can vary. For example α1 could be negative for the antiferromagnetic case and αA, βB could be negative for the ferrimagnetic case. This would shift the zero of χ–1

The above definitions of antiferromagnetism and ferrimagnetism are the old definitions (due to Néel). In recent years it has been found useful to generalize these definitions somewhat. Antiferromagnetism has been generalized to include solids with more than two sublattices and to include materials that have triangular, helical or spiral, or canted spin ordering (which may not quite have a net zero magnetic moment). Similarly, ferrimagnetism has been generalized to include solids with more than two sublattices and with spin ordering that may be, for example, triangular or helical or spiral. For ferrimagnetism, however, we are definitely concerned with the case of nonvanishing magnetic moment.

It is also interesting to mention a remarkable theorem of Bohr and Van Leeuwen [94]. This theorem states that for classical, nonrelativistic electrons for all finite temperatures and applied electric and magnetic fields, the net magnetization of a collection of electrons in thermal equilibrium vanishes. This is basically due to the fact that the paramagnetic and diamagnetic terms exactly cancel one another on a classical and statistical basis. Of course, if one cleverly makes omissions, one can discuss magnetism on a classical basis. The theorem does tell us that if we really want to understand magnetism, then we had better learn quantum mechanics. See Problem 7.17.

It might be well to learn relativity also. Relativity tells us that the distinction between electric and magnetic fields is just a distinction between reference frames.

372 7 Magnetism, Magnons, and Magnetic Resonance

where E0 is the ground-state energy of the hydrogen atom. The zeroth-order hydrogen molecular wave functions may be written

where dτi = dxidyidzi and we have used that wave functions for stationary states can be chosen to be real. In (7.69),

Working out the details when (7.68) is put into (7.69) and assuming ψa(1) and ψb(2) are normalized we find

374 7 Magnetism, Magnons, and Magnetic Resonance

Energy – 2E0

Fig. 7.7. Sketch of results of the Heitler–London theory applied to two hydrogen atoms (R/R0 is the distance between the two atoms in Bohr radii). See also, e.g., Heitler [7.26].

In the singlet (or antiparallel spin) state, the eigenvalue of (Sl + S2)2 is 0, so

Comparing these results to Fig. 7.7, we see we can formally write an effective spin Hamiltonian for the two electrons on the two different atoms:

where J is often simply called the exchange constant and J = J(R), i.e. it depends on the separation R between atoms. By suitable choice of J(R), the eigenvalues of H − 2E0 can reproduce the curves of Fig. 7.7. Note that J > 0 gives the parallelspin case the lowest energy (ferromagnetism) and J < 0 (the two-hydrogen-atom case – this does not always happen, especially in a solid) gives the antiparallelspin case the lowest energy (antiferromagnetism). If we have many atoms on a lattice, and if there is an exchange coupling between the spins of the atoms, we assume that we can write a Hamiltonian:

(electrons)

If there are several electrons on the same atom and if J is constant for all electrons on the same atom, then we assume we can write

= ∑k,l Jk,l SkT SlT ,

where SkT and SlT refer to the spin operators associated with atoms k and l. Since

∑′ Sα·SβJαβ differs from ∑ Sα·SβJαβ by only a constant and ∑′k,l JklSkT SlT differs from ∑k,l JklSkT SlT by only a constant, we can write the effective spin Hamiltonian

as

here unimportant constants have not been retained. This last expression is called the Heisenberg Hamiltonian for a system of interacting spins in the absence of an external field.

This form of the Heisenberg Hamiltonian already tells us two important things:

1.It is applicable to atoms with arbitrary spin.

2.Closed shells contribute nothing to the Heisenberg Hamiltonian because the spin is zero for a closed shell.

Our development of the Heisenberg Hamiltonian has glossed over the approximations that were made. Let us now return to them. The first obvious approximation was made in going from the two-spin case to the N-spin case. The presence of a third atom can and does affect the interaction between the original pair. In addition, we assumed that the exchange interaction between all electrons on the same atom was a constant.

Another difficulty with the extension of the Heitler–London method to the n- electron problem is the so-called “overlap catastrophe.” This will not be discussed here as we apparently do not have to worry about it when using the simple Heisenberg theory for insulators.7 There are also no provisions in the Heisenberg Hamiltonian for crystalline anisotropy, which must be present in any real crystal. We will discuss this concept in Sects. 7.2.2 and 7.3.1. However, so far as energy goes, the Heisenberg model does seem to contain the main contributions.

But there are also several approximations made in the Heitler–London theory itself. The first of these assumptions is that the wave functions associated with the electrons of interest are well-localized wave functions. Thus we expect the Heisenberg Hamiltonian to be more nearly valid in insulators than in metals. The assumption is necessary in order that the perturbation approach used in the Heit- ler–London method will be valid. It is also assumed that the electrons are in nondegenerate orbital states and that the excited states can be neglected. This makes it

7 For a discussion of this point see the article by Keffer, [7.37].

376 7 Magnetism, Magnons, and Magnetic Resonance

harder to see what to do in states that are not “spin only” states, i.e. in states in which the total orbital angular momentum L is not zero or is not quenched. Quenching of angular momentum means that the expectation value of L (but not L2) for electrons of interest is zero when the atom is in the solid. For the nonspin only case, we have orbital degeneracy (plus the effects of crystal fields) and thus the basic assumptions of the simple Heitler–London method are not met.

The Heitler–London theory does, however, indicate one useful approximation: that J 2 is of the same order of magnitude as the electrostatic interaction energy between two atoms and that this interaction depends on the overlap of the wave functions of the atoms. Since the overlap seems to die out exponentially, we expect the direct exchange interaction between any two atoms to be of rather short range. (Certain indirect exchange effects due to the presence of a third atom may extend the range somewhat and in practice these indirect exchange effects may be very important. Indirect exchange can also occur by means of the conduction electrons in metals, as discussed later.)

Before discussing further the question of the applicability of the Heisenberg model, it is useful to get a physical picture of why we expect the spin-dependent energy that it predicts. In considering the case of two interacting hydrogen atoms, we found that we had a parallel spin case and an antiparallel spin case. By the Pauli principle, the parallel spin case requires an antisymmetric spatial wave function, whereas the antiparallel case requires a symmetric spatial wave function. The antisymmetric case concentrates less charge in the region between atoms and hence the electrostatic potential energy of the electrons (e2/4πε0r) is smaller. However, the antisymmetric case causes the electronic wave function to “wiggle” more and hence raises the kinetic energy T (Top 2). In the usual situation (in the two- hydrogen-atom case and in the much more complicated case of many insulating solids) the kinetic energy increase dominates the potential energy decrease; hence the antiparallel spin case has the lowest energy and we have antiferromagnetism (J < 0). In exceptional cases, the potential energy decrease can dominate the kinetic energy increases, and hence the parallel spin case has the least energy and we have ferromagnetism (J > 0). In fact, most insulators that have an ordered magnetic state become antiferromagnets at low enough temperature.

Few rigorous results exist that would tend either to prove or disprove the validity of the Heisenberg Hamiltonian for an actual physical situation. This is one reason for doing calculations based on the Heisenberg model that are of sufficient accuracy to yield results that can usefully be compared to experiment. Dirac8 has given an explicit proof of the Heisenberg model in a situation that is oversimplified to the point of not being physical. Dirac assumes that each of the electrons is confined to a different specified orthogonal orbital. He also assumes that these orbitals can be thought of as being localizable. It is clear that this is never the situation in a real solid. Despite the lack of rigor, the Heisenberg Hamiltonian appears to be a good starting place for any theory that is to be used to explain experimental magnetic phenomena in insulators. The situation in metals is more complex.

8 See, for example, Anderson [7.1].