Electrostatics in semiconducting devices I : The Pure Electrostatics Self Consistent Approximation

In quantum nanoelectronics devices, the electrostatic energy is the largest energy scale at play and, to a large extend, it determines the charge distribution inside the devices. Here, we introduce the Pure Electrostatic Self consistent Approximation (PESCA) that provides a minimum model that describes how to include a semiconductor in an electrostatic calculation to properly account for both screening and partial depletion due to e.g. field effect. We show how PESCA may be used to reconstruct the charge distribution from the measurement of pinch-off phase diagrams in the gate voltages space. PESCA can also be extended to account for magnetic field and calculate the edge reconstruction in the quantum Hall regime. The validity of PESCA is controlled by a small parameter $\kappa = C_g/C_q$, the ratio of the geometrical capacitance to the quantum capacitance, which is, in many common situations, of the order of 1%, making PESCA a quantitative technique for the calculation of the charge distribution inside devices.