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Abstract
The semiclassical Bloch–Boltzmann theory is at the heart of our understanding of conduction in solids, ranging from metals to semiconductors. Physical systems that are beyond the range of applicability of this theory are thus of fundamental interest. It appears that in quasicrystals and related complex metallic alloys, a new type of breakdown of this theory operates. This phenomenon is related to the specific propagation of electrons. We develop a theory of quantum transport that applies to a normal ballistic law, and also to these specific diffusion laws. As we show, phenomenological models based on this theory describe correctly the anomalous conductivity in quasicrystals. Ab initio calculations performed on approximants also confirm the validity of this anomalous quantum diffusion scheme. This provides us with an ab initio model of transport in approximants such as α-AlMnSi and AlCuFe 1/1 cubic approximant.
Acknowledgements
The work presented in this paper has been ongoing since the 1990s. Our work owes much to discussions with T. Fujiwara, J. Bellissard, J. Friedel, and N.W. Ashcroft. We are very grateful to many colleagues with whom we had collaborations during this time: C. Berger, F. Cyrot-Lackmann, J. Delahaye, T. Grenet, F. Hippert, T. Klein, L. Magaud, J.J. Préjean, P. Quémerais, S. Roche, and F. Triozon. The computations were performed at the Service Informatique Recherche (SIR), Université de Cergy-Pontoise. Part of the numerical results were obtained by using the Condor Project (http://www.condorproject.org/). GTL thanks Y. Costes, SIR, for computing assistance. JPJ thanks the LANL group for their warm hospitality and DGA under contract 07.60.028.00.470.75.01.