Parallel Quadratic Selected Inversion in Quantum Transport Simulation

Abstract

Driven by Moore's Law, the dimensions of transistors have been pushed down to the nanometer scale. Advanced quantum transport (QT) solvers are required to accurately simulate such nano-devices. The non-equilibrium Green's function (NEGF) formalism lends itself optimally to these tasks, but it is computationally very intensive, involving the selected inversion (SI) of matrices 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 so-called recursive Green's function (RGF) technique, but they are typically sequential, require block-tridiagonal (BT) matrices as inputs, and their implementation has been so far restricted to shared memory parallelism, thus limiting the achievable device sizes. To address these shortcomings, we introduce distributed methods that build on RGF and enable parallel selected inversion and selected solution of the quadratic matrix equation. We further extend them to handle BT matrices with arrowhead, which allows for the investigation of multi-terminal transistor structures. We evaluate the performance of our approach on a real dataset from the QT simulation of a nano-ribbon transistor and compare it with the sparse direct package PARDISO. When scaling to 16 GPUs, our fused SI and SQ solver is 5.2x faster than the SI module of PARDISO applied to a device 16x shorter. These results highlight the potential of our method to accelerate NEGF-based nano-device simulations.

Figures (9)

• Figure 1: (a) Schematic view of a nano-ribbon field-effect transistor with a gate-all-around configuration agrawal_silicon_2024zhang_gate-all-around_2022huang_3-d_2020. Three ribbons are stacked on top of each other, forming the device channel. Electrons (or holes) can be injected into the device structure at the source (left), drain (right), and gate (middle) contacts and then exit at one of these locations. (b)-(d) Typical sparsity patterns of the Hamiltonian matrices that arise in the quantum transport simulation of such devices. (b) Block-tridiagonal matrix describing a classical two-terminal transistor channel (only source and drain). (c) Block-tridiagonal matrix with a large, but sparse central block, further accounting for electron/hole injection and collection at the central gate contact(s). (d) Reordering of the matrix in (c) into a block-tridiagonal with arrowhead sparsity pattern.
• Figure 2: High-level overview of the derivation of the forward pass of the RGF algorithm for a block-tridiagonal (BT) structured sparse matrix. (a) BT sparsity pattern of the input $A$ matrix and blocks involved in the $i$-th step of the forward pass. (b) Slicing of the matrix at the $i$-th step and zoom-in on the block operations involved in the forward update. (c) Reduction of the block operation in (b) by avoiding a priori zero computations.
• Figure 3: Extension of the RGF algorithm toward matrices with a block-tridiagonal with arrowhead sparsity pattern. (a) Slicing of the system matrix at the $i-th$ step of the forward Schur complement operation. (b) Slicing of the system matrix at the $i-th$ step of the backward selected inversion operation. The orange and marine blue arrows showcase the stepping direction and indicate the block operations performed.
• Figure 4: Block operations in the forward update of the extended RGF algorithm ($i$-th step). (a) With all zero operations. (b) After compressing, avoiding a priori zero operations.
• Figure 5: (a) Graphical representation of the permutation $PAP^T$ where the matrix $A$ is distributed over three processes. All partitions have a different color. (b) Result of the permutation and re-ordering of the partitioned matrix $A$ according to the operation in (a). The hatches refer to the fill-in induced by the permutation during the Schur complement operation.