Formulation of the Boltzmann equation as a multi-mode drift-diffusion equation

We present a multi-mode drift-diffusion equation as reformulation of the Boltzmann equation in the discrete momentum space. This is shown to be similar to the conventional drift-diffusion equation except that it is a more rigorous solution to the Boltzmann equation because the current and carrier densities are resolved into Mx1 vectors, where M is the number of modes in the discrete momentum space. The mobility and diffusion coefficient become MxM matrices which connect the M momentum space modes. This approach is demonstrated by simulating electron transport in bulk silicon.