Energy bands splittings

Fig. 39. Energy-band splittings for some typical semiconductors, (a) Zinc-blende, s and p states mixed, (b) Fluorite (Mg2Sn has n = 1, CaF2 has n = 2). (c) Rock salt for case where s and p states do not mix appreciably.

There are two significant consequences of the energy band splitting First the populations of spin up and spin down carriers are different. Second if the g factor of the materials is sufficiently high, then in modest magnetic fields there will be significant differ-... 

For atomic (gas) sodium (Na), the electronic configuration is ls 2s 2p 3s, leading to filled electronic energy levels Is, 2s and 2p, while the 3s level is half-filled. The other excited levels, 3p, 4s..., are empty. In the solid state (the left-hand side in Figure 4.6), these atomic energy levels are shifted and split into energy bands bands Is, 2s and 2p are fully occupied, while the 3s (/ = 0) band, the conduction band, is half-filled, so that a large number N 21 + l)/2 = N/2) of empty 3s excited levels is still available. As a result, electrons are easily excited into empty levels by an applied electric field, and so become free electrons. This aspect confers the typical metallic character to solid sodium. 

The behavior of the magnetic susceptibility as a function of temperature is in accord with what one should expect from an array of small isolated metal clusters, with splitting between the filled and the empty intra-cluster electronic energy bands of the order of 30 K. The magnetic moment per cluster of less than one electron spin still requires clarification. 

We consider first (Fig. 14 a) what happens at very large distances. The Hamiltonian (11) (without the correction (35)) would then give rise to a very narrow band (a level). With the correction (35), the band splits into two separate sub-bands (two energy levels) Eo and Eq + Uh (see Fig. 14 a). These two sub-bands containing each M (and not 2M) states, represent, respectively, a state in which each core holds one spin and a state in which half of the cores hold two antiparallel spins, and the others empty (polar states). The two sub-bands are separated by a gap which is exactly Uh- Without excitation to the highest sub-band, in this conditions the lower sub-band is fully occupied. It represents the insulator s state in which all electrons are sitting in the cores, i.e. all electrons are fully localized. 

When the cores are approached, the sub-bands split, acquiring a bandwidth, and decreasing the gap between them (Fig. 14 a). At a definite inter-core distance, the subbands cross and merge into the non-polarized narrow band. At this critical distance a, the narrow band has a metallic behaviour. At the system transits from insulator to metallic (Mott-Hubbard transition). Since some electrons may acquire the energies of the higher sub-band, in the solid there will be excessively filled cores containing two antiparallel spins and excessively depleted cores without any spins (polar states). 

Interaction of two chromophores exhibiting allowed (strong) n-n absorption bands splits the excited state into two energy levels with the energy gap 2VSj (Davydov splitting), Figure 12. 

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