Saraga, Daniel; (1999) Quantum chaos in the resonant tunnelling diode. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

The resonant tunneling diode in tilted fields was introduced recently as a new experimental probe of quantum chaos. However, its semiclassical interpretation in terms of classical structures (e.g., periodic orbits) remained controversial. The aim of this work is to achieve a better understanding of this system within the framework of quantum chaos. Our main finding is that the correct semiclassical description involves orbits of a new type (saddle orbits), which are complex and non-periodic. The experimental data is analyzed in order to single out the oscillatory structure of the current. We use a simple theoretical model of the current across the diode, based on the Bardeen weak tunneling matrix element. Quantum calculations are performed, taking advantage of the scaling property of the Hamiltonian. We compare current amplitudes with experiments, and study scarring of wave functions as well as Wigner distributions. We propose a model describing the effects of tori quantization around a stable periodic orbit (PO). The classical dynamics are analyzed through Poincare surfaces of section and a study of the relevant POs. We develop a general semiclassical formalism for matrix elements given by a projection operator on a Gaussian. This allows us to carry out a detailed comparison of several semiclassical formulae, and to situate them within the context of general semiclassical theories of matrix elements. In particular, we discuss the different hypotheses one can make on the smoothness of the observable, as well as three levels of approximation. PO theories are shown to be only partially successful in the description of the current across the diode. In addition to the saddle orbits, we propose another type of orbits, which are real and minimize the gradient of the phase. We show that they also give very accurate semiclassical contributions. Complex dynamics are investigated in order to calculate the contribution of complex POs and saddle orbits.

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