Parallel Solution of the Schrödinger-Poisson Equation on GPUs

Quantum mechanical effects exhibited in carrier transport must often be accounted for in the development of future electronic devices. To achieve physically reasonable results, the transport equation of the quantum mechanical system and the electrical problem (Poisson equation) have to be solved self-consistently. To calculate IV-characteristics the Newton method has to be applied on a coupled Schrödinger-Poisson system for each bias point, requiring the assembly of the Jacobian with respect to the unknowns. In a typical simulation several millions of Schrödinger-type equations need to be solved for the assembly and a parallelization of the procedure is essential. Special care has to be taken because of the memory limitation of the GPU. To prevent a parallel storage of the system matrices, the discretization is carried out by a reformulation of the problem in terms of one-sided boundary conditions. An explicit scheme can be employed and no individual system matrices need to be assembled. Traditional CPUs are utilized for reference. Benchmarks study the scalability of the approach when using up to several thousands of CUDA cores in parallel.