Hybrid Discontinuous Galerkin Approach for the Solution of Quantum Liouville-Type Equations

For the evaluation of charge carrier transport in quantum devices, quantum Liouville-type equations (QLTE) have proven to be powerful as a starting point for the numerical analysis of transport characteristics. Due to the complexity of quantum devices, the computational time is to be further reduced using high-performance computing methods. Therefore, a hybrid discontinuous Galerkin (DG) approach is proposed to solve quantum Liouville-type equations (QLTE) efficiently. Due to resulting block diagonal matrices occurring after an approximation, the system matrix can be computed efficiently without an inversion of large matrices when considering time-dependent analyses. As a result, the time-dependent exponential integrator can be represented as a polynomial of the resulting system matrix and the time evolution can be explicitly determined via matrix-vector multiplications. Most importantly, along with the discontinuous Galerkin approach, the accuracy via an appropriate choice of elements can be improved. The general approach is presented and validated.