Collective-electron theory of ferromagnetism

Collective-electron theory of ferromagnetism

We have seen that Weiss idea of the molecular field, combined with the Langevin theory of localized moments, gives a rather good description of many properties of ferromagnetic materials. The temperature dependence of the spontaneous magnetization compares favorably with the observed values, and the existence of a phase transition to a paramagnetic state is explained. However, the localizedmoment theory breaks down in one important respect ¨C it is unable to account for the measured values of the magnetic moment per atom in some ferromagnetic materials, particularly in ferromagnetic metals. There are two significant discrepancies.

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First, according to the Weiss theory, the magnetic dipole moment on each atom or ion should be the same in both the ferromagnetic and paramagnetic phases.Experimentally this is not the case. Second, in the localized-moment theory, the magnetic dipole moment on each atom or ion should correspond to an integer number of electrons. Again this is not observed experimentally. To explain the data we need to use the band theory, or collective-electron theory, which we introduced earlier in our discussion of Pauli paramagnetism.

The mechanism producing magnetism in ferromagnetic metals is ultimately the same exchange energy that gives rise to Hund¡¯s rules in atoms and the Weiss molecular field we discussed above. This exchange energy is minimized if all the electrons have the same spin. Opposing the alignment of spins in metals is the increased band energy involved in transferring electrons from the lowest band states (occupied with one up- and one down-spin electron per state) to band states of higher energy. This band energy cost prevents simple metals from being ferromagnetic.
In the elemental ferromagnetic transition metals, Fe, Ni, and Co, the Fermi energy lies in a region of overlapping 3d and 4s bands, as shown schematically in Fig. 6.5. We will assume that the structures of the 3d and 4s bands do not change markedly across the first transition series, and so any differences in electronic structure are caused entirely by changes in the Fermi energy. This approximation is known as the rigid-band model, and detailed band structure calculations have shown that it is a reasonable assumption.

As a result of the overlap between the 4s and 3d bands, the valence electrons only partially occupy each of these bands. For example, Ni, with 10 valence electrons per atom, has 9.46 electrons in the 3d band and 0.54 electrons in the 4s band. The 4s band is broad, with a low density of states at the Fermi level. Consequently, the energy which would be required to promote a 4s electron into a vacant state so that it could reverse its spin is more than that which would be gained by the resulting decrease in exchange energy. By contrast, the 3d band is narrow and has a much higher density of states at the Fermi level. The large number of electrons near the Fermi level reduces the band energy required to reverse a spin, and the exchange effect dominates. If you don¡¯t find it intuitive to think in terms of densities of states, Fig. 6.6 might be useful. Here, instead of drawing the density of states as a continuum it has been approximated as a series of discrete levels. The s band (on the left) has only one level per atom and the band is very broad. Therefore the levels are widely spaced and the band energy E required to promote an electron to the next available level is large. In contrast, the d band (on the right) has five levels to fit in for each atom, and the band itself is very narrow. Therefore the levels are close together and the band energy to promote an electron is much smaller.

It is useful to picture the exchange interaction as shifting the energy of the 3d band for electrons with one spin direction relative to the band for electrons with the opposite spin direction. The magnitude of the shift is independent of thewavevector, giving a rigid displacement of the states in a band with one spin direction relative to the states with the opposite spin direction. If the Fermi energy lies within the 3d band, then the displacement will lead to more electrons of the lower-energy spin direction and hence a spontaneous magnetic moment in the ground state.

The resulting band structure looks similar to that of a Pauli paramagnet in an external magnetic field. The difference is that in this case the exchange interaction causes the change in energy, and an external field is not required to induce the magnetization.

Figure 6.7 shows the 4s and 3d densities of states within this picture. The exchange splitting is negligible for the 4s electrons, but significant for 3d electrons. In Ni, for example, the exchange interaction displacement is so strong that one 3d sub-band is completely filled with five electrons, and all 0.54 holes are contained in the other sub-band. So the saturation magnetization of Ni is Ms = 0.54N¦ÌB, where N is the number of Ni atoms per unit volume.We now see why the magnetic moments of the transition metals do not correspond to integer numbers of electrons!

This model also explains why the later transition metals, Cu and Zn, are not ferromagnetic. In Cu, the Fermi level lies above the 3d band. Since both the 3d sub-bands are filled, and the 4s band has no exchange-splitting, then the numbers of up- and down-spin electrons are equal. In Zn, both the 3d and 4s bands are filled and so do not contribute a magnetic moment.

For the lighter transition metals, Mn, Cr, etc., the exchange interaction is less strong, and the band energy is larger, so the energy balance is such that ferromagnetism is not observed. In fact both Mn and Cr actually have rather complicated spin arrangements which are antiferromagnetic in nature. More about that later!

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