Diffusion Coefficient for Electron Transport Simulation

The Monte Carlo method can be used to directly solve the Boltzmann equation by randomly producing trajectories to obtain a distribution of solutions. The group wrote a simulation which intends to accomplish this for the travel of electrons through an Arsenic-doped Silicon lattice. This information can be used to learn more about the behavior of charge carriers as they travel through the lattice, to study the effect of energy band warping on transport, and to help in potential applications for semiconductor devices. Verification of the simulation’s success includes comparing with known constants such as drift velocity, free flight time, and diffusion coefficient. We calculated the diffusion coefficient, an important physical quantity that describes the way charge carriers move through a material, in Mathematica using the positions generated by this simulation to determine spatial concentrations to compare with solutions to Fick’s second law. Testing of the method was performed using positions generated by a Random Walk. After producing diffusion coefficients, we studied the relationship with temperature at 50K, 300K, 550K, and 800K as well as change in ion impurity concentration at 10¹³, 10¹⁵, and 10¹⁷ ions/cm³ over electric field strengths of 10 000, 510 000, and 1 010 000 V/m.