Parallel Quadratic Selected Inversion in Quantum Transport Simulation

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Driven by Moore’s law, the dimensions of transistors have been pushed down to the nanometer scale so that advanced quantum transport (QT) solvers are nowadays required to reliably design such nano-devices. The non-equilibrium Green’s function (NEGF) formalism is suited to this task but is computationally intensive, involving the selected inversion (SI) and the selected solution of quadratic matrix (SQ) equations. Existing algorithms to tackle these numerical problems are ideally suited to GPU acceleration, e.g., the recursive Green’s function (RGF) technique. However, they are typically sequential, limited to block-tridiagonal (BT) matrices, and their implementation has been restricted so far to shared-memory parallelism, limiting the achievable device sizes. To address these shortcomings, we introduce distributed methods that build on RGF and enable parallel SI and SQ. We further extend them to handle BT matrices with arrowhead, allowing for the inclusion of gate leakage currents, a major limiting factor at ultra-scaled device dimensions. We evaluate the performance of our approach on a real dataset from the QT simulation of a nano-ribbon field-effect transistor and perform a comparison with the sparse direct solvers PARDISO and cuDSS. Our SI solver is at least one order of magnitude faster than PARDISO (cuDSS) on CPUs (GPUs), regardless of the system size. When fused, our SI+SQ implementation outperforms the SI-only module of PARDISO by a factor of 1.56 × for the same device dimensions. Performing weak scaling up to 8 CPUs (GPU), our SI+SQ solver achieves a parallel efficiency of \(\eta = 17.1\%\) (\(\eta = 18.5\%\)), thus enabling distributed memory nano-device simulations.

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