A posteriori error control in numerical simulations of semiconductor nanodevices

A posteriori error estimation and control methods are proposed for a quantum corrected energy balance (QCEB) model that describes electron and hole flows in semiconductor nanodevices under the influence of electrical, diffusive, thermal, and quantum effects. The error estimation is based on the maximum norm a posteriori error estimate developed by Kopteva (2008) for singularly perturbed semilinear reaction–diffusion problems. The error estimate results in three error estimators called the first-, second-, and third-order estimators to guide the refinement process. The second-order estimator is shown to be most effective for adaptive mesh refinement. The QCEB model is scaled to a dimensionless coupled system of seven singularly perturbed semilinear PDEs with various perturbation parameters so that the estimator can be applied to each PDE on equal footing. It is found that the estimator suitable for controlling the approximation error of one PDE (one physical variable) may not be suitable for another PDE, indicating that different parameters account for different boundary or interior layer regions as illustrated by two different semiconductor devices, namely, a diode and a MOSFET. A hybrid approach to automatically choosing different PDEs for calculating the estimator in the adaptive mesh refinement process is shown to be able to control the errors of all PDEs uniformly.