Quantum scattering, resonant states, and conductance fluctuations in an open square electron billiard

Electron transport was studied in an open square quantum dot with a dimension typical for current experiments. A numerical analysis of the probability density distribution inside the dot was performed which enabled us to unambiguously map the resonant states which dominate the conductance of the structure. It was shown that, despite the presence of dot openings, transport through the dot is effectively mediated by just a few (or even a single) eigenstates of the corresponding closed structure. In a single-mode regime in the leads, the broadening of the resonant levels is typically smaller than the mean energy level spacing, $\Delta .$ On the contrary, in the many-mode regime this broadening typically exceeds $\Delta$ and has an irregular, essentially non-Lorentzian, character. It was demonstrated that in the latter case eigenlevel spacing statistics of the corresponding closed system are not relevant to the averaged transport properties of the dot. This conclusion seems to have a number of experimental as well as numerical verifications. The calculated periodicity of the conduction oscillations in the open dot is related to the formation of the global shell structure of the corresponding isolated square. The shell structure reflects periodic clustering of levels on the scale exceeding the mean level spacing separation. Each shell can be ascribed to the certain family of the periodic orbits in the square. However, a particular arrangement of the leads may lead to the selective coupling between them, so that not all shells (or, alternatively, families of periodic orbits) mediate transport through the dot. This selective coupling leading to the suppression of the contribution from some families of orbits can be tested experimentally on the dots with the different arrangements of the leads.