Modeling Quasi-Static Characteristics of Devices Consisting of Silicon, Dielectrics, and Conductors Based on Their Helmholtz Free Energy

Devices consisting of multiple distinct regions of semiconductor, dielectric, and conductor present special challenges to standard analysis methods. It is imperative to understand the interactions among these multiple regions because of their direct implication in device switching speed and power loss. We present a novel thermodynamic approach to modeling the quasi-static behavior of several devices including the trench-insulated gate bipolar transistor at a level of detail not otherwise available either experimentally or by standard methods. Our model is based on evaluating the thermodynamic Helmholtz free energy (F) of the device using parameterized trial functions for the electrostatic potential in each of its active regions. The resulting closed-form expressions for F are minimized using appropriate free parameters, yielding a variational solution of the system including the nonlinear effect of mobile charges constrained by either Boltzmann or Fermi-Dirac statistics. This solution is employed to extract the threshold voltage and to construct a terminal capacitance model by combining the internal capacitances distributed throughout the device. The model is then compared with measured terminal capacitance-voltage characteristics of some real devices to identify and interpret individual contributions. Our analysis reveals some characteristics of the interiors of the devices, which are not physically measurable.