Donor ordering and the statistics of quantum dot level spacings and widths from full 3D self-consistent electronic structure

Using full 3D self-consistent electronic structure calculations of small (electron number N similar to 100) lateral quantum dots formed on GaAs-AlGaAs HEMT devices we calculate the statistics of level spacings Delta epsilon(p) and tunneling coefficients Gamma(p) between leads and confined states of the dot. We employ random and ordered donor layer charge distributions, the latter generated through Monte Carlo variable range hopping simulations, as well as a homogeneous (jellium) ionic charge distribution, and examine the effects on these statistics. It has recently been argued that the statistics of the level spacings and widths follow from random matrix theory when the Hamiltonian is described by the Gaussian orthogonal ensemble (GOE) for zero magnetic field B, and by the Gaussian unitary ensemble (GUE) for B sufficiently large to break time reversal symmetry. Specifically it is argued that when the dot wave functions are expanded in an arbitrary basis the expansion coefficients, according to the postulate of Porter and Thomas, are uniformly distributed in Hilbert space. In our calculation we obtain statistics of level spacings and widths by generating many configurations of disordered and ordered donor charge. This corresponds to the experimental situation of thermal cycling of the device. We find that a pronounced transition occurs in the level spacing statistics between the completely disordered donor layer ensemble, which seems to be well described by random matrix theory, and the ordered ensemble which is dominated by secular variations in the coefficients. In particular, a shell structure in the levels, which results from approximate parabolicity in the self-consistent confining potential, is observed. This, and the effects of symmetry under inversion and azimuthal symmetry, are speculated to undermine level repulsion and result in Poisson statistics for the levels here at the band edge. Finally we find that distortions in the dot shape are markedly less significant in varying the widths (and level spacings) than calculations based on a hard wall potential for the dot predict. This suggests that the notion of invariant atomic structure for sufficiently small dots is not invalidated by the randomness inherent in donor positions and shape distortion but, on the contrary, a systematic study of dot structure is possible. (C) 1997 Academic Press Limited.