The first term is the Hamiltonian of the electrons in the band states with energy dispersion The second term corresponds to the electron localized on the jth impurity site with energy The third term describes the change in the single electron energy due to the dynamic mixing between the band states and the localized states. Following Anderson's scheme, the hybridization strength is characterized by the parameter Vkj defined by Anderson.
Where a(r−j) and φd(r−j) are the Wannier function belonging to the band and the localized wavefunction of the impurity on the jth site, respectively. HHF(r) is the singleelectron energy described in the Hartree-Fock approximation.
The hybridization term produces a profound effect on the electronic structure of the system. In general, one shall consider finite but dilute concentrations of impurities, 0<x<<1. In this case, the single-site coherent potential approximation (CPA) is adequate for the manyimpurity system. In the CPA treatment, a configurational averaging is performed, neglecting correlations between positions of the impurities. Consequently, the spacetranslational invariance of the average Green's function is partially restored, and k resumes its well-defined properties as a good quantum number. In momentum space, the diagonal Green's function in CPA can be written as
The integration converges rapidly with in a small range that is proportional to x. The calculated perturbed DOS for GaNxAs1−x with several small values of x is shown in Figure 2.3. Note that the anticrossing interaction leads to a dramatic redistribution of the electronic states in the conduction band. The most striking feature of the DOS curves is the clearly seen gap between E+ and E− that evolves with increasing N content. In the Green's function calculation, the k dependence of Vkj is assumed to be weak on the momentum scale we are interested in.
There is experimental evidence indicating that the values of Vk at the L point in GaNxAs1−x and at the X point in GaNxP1−x are about three to four times smaller than the Vk at the point. This ratio corresponds to a localized wave function decay length (ld) on the order of the lattice constant. This result indicates that the off-zone-center conduction-band minima are affected by the anticrossing interaction only when their energies are close to the localized level. This is consistent with recent measurements of the optical properties of InyGa1−yNxAs1−x alloys, which have shown that alloying with N has only very small effects on the high energy transitions at large k vectors.
ROSSANA HERNANDEZ
ELECTRONICA DEL ESTADO SOLIDO
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