Application of Johnson's approximation in finite element modeling for electric field-dependent materials

Johnson's approximation is implemented in a finite element code to simulate the electric field dependence of a core–shell microstructure material. We show how the microstructure, based here on a 50:50 volume fraction, influences the measured effective permittivity as a function of applied voltage. Using a Johnson's parameter of β = 1.0 × 10¹⁰ Vm⁵/C³, verified from commercial BaTiO₃-based multilayer ceramic capacitors (MLCC), we show how the microstructure and the difference in core and shell conductivities alter the local fields generated and how this influences the voltage dependence of the effective permittivity. Systems that comprise a conductive core-like material surrounded by a resistive shell experience little or modest voltage dependence due to the shell material providing shielding to large electric fields within the cores. Conversely, if the core material is more resistive than the shell material, substantial voltage dependence occurs with simulations showing over a 50% decrease in the effective permittivity. These simulations give improved understanding of voltage dependence and provide a method to help guide the design of future materials for MLCCs with improved performance.