Approximate analytic solution to electrical potential and space charge distribution in two carrier conductors under equilibrium
Abstract A one dimensional Poisson–Boltzmann equation describes various systems in different areas of physics and chemistry. In particular, an electrical potential in a conducting media with two types of mobile carriers (obeying Boltzmann statistics), confined in between two planar electrodes, is given by a solution of Poisson–Boltzmann equation. Usually, the Poisson–Boltzmann equation is solved numerically. In this work an approximate analytic solution to the one dimensional Poisson–Boltzmann equation with arbitrary boundary conditions is presented. The approximate solution is an analytic compact expression in terms of elementary functions only. The maximum error in the approximation is calculated analytically. It is shown that the error depends on the magnitude of the contact potentials, decreasing for low contact potentials but reaching a maximum for the high ones. Thus, the approximation holds in the cases were the Debye–Huckel linear approximation does not. The precision depends on the ratio of the device length to the space charge region, improving exponentially with increase of device length. Using the approximate solution, we calculate the differential capacitance of a metal 1 |conducting-media|metal 2 device and show that the Gouy–Chapman differential capacitance is a special case of a more general solution.