Diffuse electric double layer in planar nanostructures due to Fermi-Dirac statistics

Abstract

A double nanocapacitor modelled by two equally charged planar surfaces that confine oppositely charged quanta subjected to Fermi-Dirac statistics is considered theoretically. A global thermodynamic equilibrium was found by minimization of the Helmholtz free energy satisfying constraints that require electroneutrality and fixed total number of confined quanta. The solution obtained by using the Euler–Lagrange method yields self–consistent quantities: distribution of quanta within the pore, electric potential, equilibrium free energy and differential capacitance. Within real values, a rigorous numerical solution and an approximate analytical solution for electrons in the low temperature limit was found. The Fermi–Dirac constraints on the wave functions in the nanopore induced an effect of a diffuse electrical double layer near both charged surfaces. This effect is comparable to the corresponding effect of entropy at finite temperatures and for classical particles, as described by the acknowledged Poisson–Boltzmann theory. At small distances and small surface charges, the electrons are almost evenly distributed within the pore, while at larger distances they condense to the charged surfaces, shielding the electric field. The force between the charged surfaces is repulsive and monotonously decreases with increasing distance between surfaces. The energies stored in the nanocapacitor are up to ≃ 50eV/nm2.