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Th. 3e cycle--Math. pures--Besançon, 1978. N°: 293.

No description available.

Nous montrons de quelle manière il est possible de surmonter de sérieux problèmes de compréhension et de classification du comportement phonon des alliages semiconducteurs usuels, qui découlent de l'utilisation classique de l'Approximation du Cristal Virtuel (ACV) pour décrire le désordre d'alliage. Ainsi, des systèmes aussi différents que In- GaAs (1-liaison!1-phonon), InGaP (2-phonon modifé) et ZnTeSe (2-liaison!1-phonon) trouvent une unité de comportement - vraisemblablement universelle - dans le cadre d'un modèle de 'Percolation' (1-liaison!2-phonon) développé sur site. Le changement de paradigme de la VCA (échelle macroscopique) vers la Percolation (échelle mésoscopique) révèle une authentique spécificité des phonons du centre de la zone de Brillouin - qui est d'apporter une information naturelle sur le désordre d'alliage à cette échelle inhabituelle qu'est le mésoscopique. Nous irons jusqu'à introduire une terminologie propre, et dire que ces phonons se comportent en véritable MESOSCOPE'. En particulier, cela ouvre des perspectives nouvelles pour l'étude des effets d'organisation spontanée à longue portée dans les cristaux mixtes, qu'il s agisse de ségrégation (InGaP2) ou d'anti-ségrégation (GaInAsN). / We demonstrate how to overcome serious problems of understanding and classifcation of vibration spectra in semiconductor alloys, following from traditional use of the Virtual Cristal Approximation (VCA). We show that such different systems as InGaAs (1- bond!1-phonon behavior), InGaP (modifed 2-phonon) and ZnTeSe (2-bond 1-phonon) obey, in fact, the same phonon mode behavior, hence probably a universal one, of a percolation type (1-bond!2-phonon, developed in our group). The change of paradigm from the VCA (macroscopic scale) to the Percolation (mesoscopic scale) reveals a specificity of zone-center phonons, that is to provide natural insight into the alloy disorder at the unusual mesoscopic scale. In fact, we introduce a terminology, and say that these phonons behave as a true 'MESOSCOPE'. In particular, this opens up new perspectives for studying the effects of long-range spontaneous organization in mixed crystals, ranging from segregation (InGaP2) to anti-segregation (GaInAsN).

Doctor of Philosophy / Department of Mathematics / Pietro Poggi-Corradini / This thesis contains four chapters. In the first chapter, the theory of continuous p-modulus in the plane is introduced and the background p-modulus properties are provided. Modulus is a minimization problem that gives a measure of the richness of families of curves in the plane. As the main example, we compute the modulus of a 2-by-1 rectangle using complex analytic methods. We also introduce discrete modulus on a graph and its basic properties. We end the first chapter by providing the relationship between connecting modulus and harmonic functions. This is the fact that computing the modulus of the family of walks from a to b is equivalent to minimizing the energy over all potentials with boundary values 0 at a and 1 at b. In the second chapter, we are interested in the connection between the continuous and the discrete modulus. We study the behavior of side-to-side modulus under some grid refinements and find an upper bound for the discrete modulus using the concept of Fulkerson duality between paths and cuts. These calculations show that the refinement will lower the discrete modulus. Since connecting modulus can also be computed by minimizing the Dirichlet energy of potential functions, we recall an argument of Jacqueline Lelong-Ferrand, that shows how refining a square grid in a ``geometric'' fashion, naturally decreases the 2- the energy of a potential. This monotonicity can be used to prove the convergence between continuous and discrete modulus. We first review the linear theory of discrete holomorphicity and harmonicity as provided by Skopenkov and Werness. Instead of reviewing their work in full generality, we present the outline of their arguments in the special case of square grids. Then use these results to prove the convergence between the continuous and discrete case. We believe that our method of proof generalizes to the full case of quadrangular grids that Werness studies. In the third chapter, we show how to generalize all our proofs for 2-modulus to the case of quadrangular grids with some geometric conditions on the lengths of edges and the angles between them. In the last chapter, a connection with potentials when p is not 2 is discussed in the square grid case. We study the behavior of side-to-side p-modulus under the same refinements as before and we find upper bound for the p-modulus, but only when p > 2. The rest of the chapter is dedicated to generalizing the results from Chapter 2 to the case 2 < p.