where V schr is the quantum correction potential, z is the direction perpendicular to the interface, n q is the quantum density from the Schrödinger equation which is equivalent to the converged Monte Carlo concentration, V p is the potential from the Poisson solution, V 0 is the arbitrary reference potential far away from the quantum region such that the correction goes to null in the region of semi-classical behavior. Even though the above-mentioned potentials for quantum correction differ in their method of calculation and their basic assumptions, yet when it comes to their inclusion into Monte Carlo simulation they are all incorporated the same way.