Parallelization of NEGF

In the current implementation the NEGF calculation is parallelized by MPI. In addition to the MPI parallelization, if you use MKL, the matrix multiplication and the inverse calculation of matrix in the evaluation of the Green function are also parallelized by OpenMP. In this case, you can perform a hybrid parallelization by MPI/OpenMP which may lead to shorter computational time. The way for the parallelization is completely same as before.

In Fig. 45 we show the speed-up ratio in the elapsed time for the evaluation of the density matrix of 8-zigzag graphene nanoribbon (ZGNR) under a finite bias voltage of 0.5 eV. The energy points of 197 (101 and 96 for the equilibrium and nonequilibrium terms, respectively) are used for the evaluation of the density matrix. Only the $\Gamma$ point is employed for the k-point sampling, and the spin polarized calculation is performed. Thus, the combination of 394 for the three indices are parallelized by MPI. It is found that the speed-up ratio of the flat MPI parallelization, corresponding to 1 thread, reasonably scales up to 64 processes. Furthermore, it can be seen that the hybrid parallelization, corresponding to 2 and 4 threads, largely improves the speed-up ratio. By fully using 64 quad core processors, corresponding to 64 processes and 4 threads, the speed-up ratio is about 140, demonstrating the good scalability of the NEGF method. For the details see also Ref. [73]. It should be also noted that the number of processes in the MPI parallelization can exceed the number of atoms in OpenMX Ver. 3.9.

Figure 45: Speed-up ratio in the parallel computation of the calculation of the density matrix for the FM junction of 8-zigzag graphene nanoribbon (ZGNR) by a hybrid scheme using MPI and OpenMP. The speed-up ratio is defined by $T_1/T_{p}$, where $T_1$ and $T_p$ are the elapsed times by a single core and a parallel calculations. The cores used in the MPI and OpenMP parallelizations are called process and thread, respectively. The parallel calculations were performed on a CRAY-XT5 machine consisting of AMD opteron quad core processors (2.3GHz). In the benchmark calculations, the number of processes is taken to be equivalent to that of processors. Therefore, in the parallelization using 1 or 2 threads, 3 or 2 cores are idle in a quad core processor.