Physical nanoscale systems have been analyzed both from an electrostatic point of view and quantum mechanically with respect to quantum computation. We introduce an elaborate code for the efficient numerical simulation of nanoscale electrostatics via a higher – order relaxation algorithm with a large variety of boundary conditions which then is applied to a set of physically relevant problems. Great emphasis is put on screening effects as well as capacitive coupling between spatially separated conducting regions. Specifically, we analyze the depletion of a two – dimensional electron gas using different methods. The effect of surface charges due to the pinning of the Fermi level at a semiconductor surface is shown to play an important role in that it can shift the whole system characteristics, underlining the importance of chemical potentials and work functions.
The capacitive coupling is further used to model the interactions in an interacting network of quantum dots, and the use of the capacitance formalism in the quantum mechanical context is explicitly justified. Quantum dot arrays are then analyzed on a general footing with respect to quantum computation and charge qubits based on an extended Hubbard Hamiltonian model. For systems with at most two operative electrons, general restrictions apply, introducing certain constraints on what realizations of this type of charge qubit may eventually look like. Furthermore, the interaction of the macroscopic world with the quantum dot network via quantum gates is discussed. Again, general arguments allow us to rule out certain scenarios of quantum gates. For example it turns out that capacitive coupling alone is not sufficient for full single qubit operation. Alternative ways are discussed, and finally, by using an external magnetic field and its resulting Aharonov – Bohm phases on the array, full single qubit operation based on charge is demonstrated.