Electric leakage current density in phase field simulations for nanogenerator concepts

For the purpose of energy harvesting on the nanoscale with ferroelectric barium titanate (BaTiO₃), our nanogenerator concept transforms parasitic mechanical oscillations into usable electric energy. Experimental difficulties occur in sample preparation, e.g., surface roughness, bonding contact, and leakage currents. Latter can be considered in our finite element phase field model, such that the nanogenerator concept can be optimized. Leakage current density is implemented from different phenomenological approaches. First, Ohm's law represents a linear relation between leakage and electric field. Second, the Space-Charge-Limited Current (SCLC) relation assumes a quadratic dependency on the electric field. So far, SCLC is suitable for one-dimensional problems, however, in two or three dimensions it is not found in literature. Therefore, we discuss a reformulation to bring into account that the electrical field E is a vector out of ℝ³ in our model. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)