Conduction Band Degeneracy

The absolute CB minima of the indirect-band-gap crystals show an orientational degeneracy in k-space (see Table 3.4) which translates on the EM donor levels in these crystals. The donor optical spectra show that in the case of the first ns donor states, this multi-valley degeneracy is broken and the resultant splitting of the levels must be accounted for satisfactorily by theory. This degeneracy is also of fundamental importance in explaining the splitting of the donor spectra when an external uniaxial perturbation, like a magnetic field or a uniaxial stress, is applied to a crystalline sample. 

Is states. The three )(ls(T2)) include only the p wave functions, and the Is (T2) level should be somewhat deeper than the Is (E) state. This is borne out by experiments, with the energy of the Is (E) state not too different from the one calculated using the one-valley approximation, and the Is (T2) ground state slightly higher in energy. 

The value of ip (Is) at r - 0 for a given donor can be estimated from the splitting of the ESR line due to the interaction between the electronic and nuclear spins mentioned in Sect. 1.3.5, and this point is discussed in the review [33], This property distinguishes between donors whose ground state has Is (Ai) symmetry, with values of ip (Is) (0) 2 much larger than those with Is (T2) symmetry. 

The question of the long-range changes in the eigenvalues of the one-valley Hamiltonian (5.5) produced by the lattice deformation arising from 

In the above discussion of the electronic structure of the donor levels, the electron spin has been neglected. It has been, however, proven necessary to introduce the spin-orbit coupling to explain the observation of parity-forbidden transitions for donors with relatively deep ls(Ai) ground states. Using the double group representation of Td, it is found (see Table B.4 of appendix B) that the simple representations Ai and E transform into the T6 and Tg double representations, respectively and that T2 transforms into T7 + Eg. Electric-dipole transitions are symmetry-allowed between A (Tg) and the two T2 (Ty) and T2 (r8) levels. 

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Fig. 6. Self-consistent band structure (48 valence and 5 conduction bands) for the hexagonal II arrangement of nanotubes, calculated along different high-symmetry directions in the Brillouin zone. The Fermi level is positioned at the degeneracy point appearing between K-H, indicating metallic behavior for this tubule array[17. ...

The semiconductor surface where the Fermi level is pinned at a surface state of high density (Fig. 2-31) is in the state of degeneracy of electron levels, because of the high electron state density at the surface Fermi level. Similarly, the surface degeneracy is also established when the band bending becomes so great that the Fermi level is pinned either in the conduction band or in the valence band as shown in Fig. 2-32. 

Such an interfacial degeneracy of electron energy levels (quasi-metallization) at semiconductor electrodes also takes place when the Fermi level at the interface is polarized into either the conduction band or the valence band as shown in Fig. 5-42 (Refer to Sec. 2.7.3.) namely, quasi-metallization of the electrode interface results when semiconductor electrodes are polarized to a great extent in either the anodic or the cathodic direction. This quasi-metallization of electrode interfaces is important in dealing with semiconductor electrode kinetics, as is discussed in Chap. 8. It is worth noting that the interfacial quasi-metallization requires the electron transfer to be in the state of equilibrimn between the interface and the interior of semiconductors this may not be realized with wide band gap semiconductors. 

Here Nc is the density of states in the conduction band, g the level degeneracy factor, n the carrier concentration in the band, A the activation energy of the level, Boltzmann s constant, and T the temperature. Now, in general, except at fairly low temperatures, the occupancy for shallow levels (with/ = /s) will be small, i.e., fs 1, and consequently... 

The parameter (v) is the average thermal velocity of an electron, Nc the density of states in the conduction band, g the degeneracy of the deep level, and A x = EQ — ET the electron transition energy. Equation (9) also relates the capture constant to the emission rate because of the definition... 

For each donor, go/gi is a degeneracy factor, Nc = 2(2nmn k) W is the effective conduction-band density of states at IK, h is Planck s constant, Ed is the donor energy, and Edo and ao are defined by Ed = Edo - otoT. The above equation describes the simplest type of charge balance, in which each of the one or more donors has only one charge-state transition within a few kT of the Fermi energy. An example of such a donor is Ga on a Zn site in ZnO. If there are double or triple donors, or more than one acceptor, proper variations of Eq. 5 can be found in the literature. ... 

Fig. 1 Schematic molecular orbital diagram for Ti02 sixfold-coordinated T + ions, and threefold-coordinated ions with covalent mixing of Ti and O atomic states in SALC s. Symmetry designations for occupied valence band states, the empty conduction band states, and the 0-atom 2s shallow core state. Respective degeneracies for A, E and T states, are 1, 2 and 3 [13]...

The occupied valence band states, and empty conduction band states for nanocrystalline Ti02 films with a physical thickness >4nm, and annealed at a temperature of at least 700 °C (a) in qualitative and quantitative agreement with the ionic energy level approach of Cotton in [13] using SALC s of atomic states as a basis set, and (b) display a complete removal of J-T d-state degeneracies. 

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