Ferromagnetic metals and Half-metals

As discussed in the earlier sections, magnetism in materials mainly arises due to the spin and orbital motion of electrons. It has also become obvious that interaction of electrons with the ions cannot be ignored for understanding the electronic properties of solids. This led to the modeling of electron motion in a periodic square potential created by the regular arrangement of positive ions in a crystal. When atoms are brought together to form a solid, their atomic orbitals overlap leading to closely spaced electron energy levels. The separation between the electron energy levels decrease, leading to groups of very closely spaced (almost continuous energy states) or allowed energy bands separated by unallowed energy gaps or forbidden energy gap. The closely spaced electron states within a band can be filled with electrons according to Pauli's principle of occupancy of electrons in each closely space energy level. Depending upon the number of electrons available in the solid, one would end up filling several of these bands, fully in some cases and partially in others. Electrons in highest filled/partially filled bands can only contribute to conduction. The highest filled band is called the valance band and the highest partially filled band is called the conduction band. As the name suggests electrons in conduction band contribute to the conduction process. Metals are solids with partially filled conduction band and hence are good conductions. Band structure of metals and semiconductors are schematically depicted in Fig. 7.1. This is the essence of the band theory of solids. The localized or atomic theory and the itinerant or band theory are the two extreme case models used for understanding the magnetization of solids. The itinerant model, in which the magnetic moments are considered to be due to conduction band electrons is considered to be appropriate for 3d transition elements iron, cobalt and nickel, which are ferromagnetic elements [1].

(a) (b) Figure 7.1: Typical band structure of (a) metals showing partially filled conduction band (CB), and (b) semiconductors, showing a small energy gap (Eg = 1 to 2 eV ) separating the completely filled (valence band, VB) and empty (conduction band, CB)

The band theory of ferromagnetism is an extension of Pauli's itinerant theory of paramagnetism to ferromagnetism with the inclusion of an exchange interaction (internal effective magnetic field) to align the electrons in a cooperative manner in the absence of an external applied field. This causes a relative displacement of the spin-up and spin-down half-bands called the exchange splitting. This is qualitatively similar to the scenario encountered in paramagnetism under applied field. However, in the case of ferromagnetism, the shift in energy is much larger and it occurs in the absence of the applied field! The net spontaneous magnetization is determined by the difference in the occupancy between the spin-up and spin-down states.

References:

[1]. D. Jiles, Introduction to the electronic properties of materials, 2e, Nelson Thomas Ltd., Cheltenham, UK, 2001.