Virtual crystal

Temary and quaternary semiconductors are theoretically described by the virtual crystal approximation (VGA) [7], Within the VGA, ternary alloys with the composition AB are considered to contain two sublattices. One of them is occupied only by atoms A, the other is occupied by atoms B or G. The second sublattice consists of virtual atoms, represented by a weighted average of atoms B and G. Many physical properties of ternary alloys are then expressed as weighted linear combinations of the corresponding properties of the two binary compounds. For example, the lattice constant d dependence on composition is written as ... 

To further discuss the underlying mechanisms that forces the phase stabilities we also did calculations where the alloying effects were treated within the so-called virtual crystal approximation (VGA) where the real alloy constituents are replaced by an atom with an average (noninteger) atomic number. 

In a previous work we showed that we could reproduce qualitativlely the LMTO-CPA results for the Fe-Co system within a simple spin polarized canonical band model. The structural properties of the Fe-Co alloy can thus be explained from the filling of the d-band. In that work we presented the results in canonical units and we could of course not do any quantitative comparisons. To proceed that work we have here done calculations based on the virtual crystal approximation (VGA). In this approximation each atom in the alloy has the same surrounding neighbours, it is thus not possible to distinguish between random and ordered alloys, but one may analyze the energy difference between different crystal structures. 

Figure 2.Virtual crystal approximation calculations (solid line) compared with coherent potential approximation calculations for Fe-Co (longdashed line), Fe-Ni (dot-dashed line) and Fe-Cu (dashed line). The fcc-bcc energy difference is shown as a function of the atomic number.

Cathodic electrodeposition of microcrystalline cadmium-zinc selenide (Cdi i Zn i Se CZS) films has been reported from selenite and selenosulfate baths [125, 126]. When applied for CZS, the typical electrocrystallization process from acidic solutions involves the underpotential reduction of at least one of the metal ion species (the less noble zinc). However, the direct formation of the alloy in this manner is problematic, basically due to a large difference between the redox potentials of and Cd " couples [127]. In solutions containing both zinc and cadmium ions, Cd will deposit preferentially because of its more positive potential, thus leading to free CdSe phase. This is true even if the cations are complexed since the stability constants of cadmium and zinc with various complexants are similar. Notwithstanding, films electrodeposited from typical solutions have been used to study the molar fraction dependence of the CZS band gap energy in the light of photoelectrochemical measurements, along with considerations within the virtual crystal approximation [128]. 

In the virtual-crystal approximation (VCA) (Nordheim 1931), the site energy of an alloy atom is taken to be... 

ANG AO ATA BF CB CF CNDO CPA DBA DOS FL GF HFA LDOS LMTO MO NN TBA VB VCA WSL Anderson-Newns-Grimley atomic orbital average t-matrix approximation Bessel function conduction band continued fraction complete neglect of differential overlap coherent-potential approximation disordered binary alloy density of states Fermi level Green function Flartree-Fock approximation local density of states linear muffin-tin orbital molecular orbital nearest neighbour tight-binding approximation valence band virtual crystal approximation Wannier-Stark ladder... 

The first contribution inside the curly brackets represents the change in the bond energy within the virtual crystal approximation, that is... 

In order to take into account the complex structure of the valence band, Dietl et al. (2000, 2001c) and Abolfath et al. (2001) have computed hole energies by diagonalizing the 6 x 6 k p Luttinger matrix together with the p-d exchange contribution taken in the virtual crystal and molecular field approximation,... 

The CPA method has important properties apart from its relative simplicity. It is analytic in z,158 and thus respects the elementary physical constraints causality, Kramers-Kronig relations, sum rules, positive definite spectrum, etc. What is more, it is universal this method describes the virtual-crystal limit A W, the isolated-impurity limit cA - 0 or cA - 1, and the isolated-molecule limit W- 0, with the correct contribution of each molecular level.122 Indeed, the CPA may be derived159 from this last limit, as well as from that in the locator formalism.122... 

Thus, we study the binary alloy (cf. Section IV.B) with the intermolecular interactions W of Section I. We are in the case of weak disorder A W, e.g. of a virtual crystal with one polariton band, and for A W we have separation into two bands, around vA and vB, each one embodying a polariton broadened by the disorder. 

Figure 4.22. Polariton solutions for a 3D mixed crystal in the mean-polarizability approximation (4.117). In strong local field (A3), one obtains a resonance of the virtual crystal cAwA + cBwB another solution, strongly shifted, exists at low frequencies. On the contrary, in weak local fields (A,), the frequencies of the pure A and B crystals, slightly shifted, are solutions. We note that for cB - 0, one of the solutions tends, for any strength of the local field A, to ojb, which is the frequency of B unshifted by the interaction with the lattice A.

Note A/B implies A grown or strained to B and vice versa. A B implies no growth direction or explicit strain dependence, i.e. natural. ) T = theoretical E = experimental AVL = average lattice XPS = X-ray photoelectron spectroscopy PL = photoluminescence CL = cathodoluminescence UPS = ultraviolet photoelectron spectroscopy LMTO = linear muffin tin orbital method LAPW = linearised augmented plane wave method PWP = plane wave pseudopotential method VCA = virtual crystal approximation. 

Impurity and Aperiodicity Effects in Polymers.—The presence of various impurity centres (cations and water in DNA, halogens in polyacetylenes, etc.) contributes basically to the physics of polymeric materials. Many polymers (like proteins or DNA) are, however, by their very nature aperiodic. The inclusion of these effects considerably complicates the electronic structure investigations both from the conceptual and computational points of view. We briefly mentioned earlier the theoretical possibilities of accounting for such effects. Apart from the simplest ones, periodic cluster calculations, virtual crystal approximation, and Dean s method in its simplest form, the application of these theoretical methods [the coherent potential approximation (CPA),103 Dean s method in its SCF form,51 the Hartree-Fock Green s matrix (resolvent) method, etc.] is a tedious work, usually necessitating more computational effort than the periodic calculations... 

We make the ensemble average of Eq. (6.7) and suppose that t are decoupled so that each t is replaced by i. In reality, in Eq. (6.7) only immediately successive indices caimot repeat, and the first corrections are of third order in the I matrix this is an advantage with respect to the ensemble average of Eq. (6.4) in the virtual crystal approximation (VCA), because first corrections would be of second order in the w matrix. In the ATA we replace each by I in Eq. (6.7) the result corresponds to Eq. (6.4) with w = t/(l -F GooO- We obtain... 

Early on, the Virtual Crystal Approximation (VCA) was tried. Here one just puts... 

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