Quantum chaos in nano-sized billiards in layered two-dimensional semiconductor structures

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Abstract
We consider two-dimensional, electron-rich cavities that can be created at a (AlGa)As-GaAs interface. In the modelling of such cavities we include features that are typical for small semiconductor structures or devices, i.e., soft walls representing electrostatic confinement and disorder due to ionized impurities. The introduction of soft walls is found to have a profound effect on the dynamic behaviour. There are situations in which there is a crossover from a Wigner distribution for the nearest level spacing to an effectively Poisson-like one as the confining walls are softened. The crossover occurs in a region which is accessible experimentally. A mechanism for the crossover is discussed in terms of groups of energy levels being separated from each other as walls become soft. The effects of disorder are found to be negligible for high-mobility samples, i.e., the motion of the particles is ballistic. These findings are of a general nature. Chaotic Robnik dots, circular dots with a special "dent," are also investigated. In this case there is no crossover from Wigner to Poisson distributions. An explanation for this difference is proposed. Finally, the effects of leads are investigated in an elementary way by simply attaching two stubs to a circular dot. For wide stubs, which in our simple model would correspond to open leads, we obtain Wigner statistics indicating a transition to irregular behaviour. A lead-induced transition of this kind appears consistent with recent measurements of the line-shape of the weak localization peak, observed in the low-temperature magnetoresistance of square semiconductor billiards. Finally, implications for conductance fluctuations are briefly commented on. (c) 1996 American Institute of Physics.