Scalable Binary-to-Gray and Gray-to-Binary Converters in pNML: Analysis of Latency, Energy–Delay Tradeoffs, and Computational Asymmetry

Abstract

Perpendicular Nanomagnetic Logic (pNML) has emerged as a promising beyond-CMOS computing technology due to its non-volatility, near-zero leakage power consumption, and capability for dense three-dimensional integration. However, the realization of scalable functional subsystems using field-clocked dipole-coupled nanomagnets remains a significant research challenge. This paper presents scalable implementations of Binary-to-Gray (B2G) and Gray-to-Binary (G2B) code converters using perpendicular nanomagnetic logic (pNML). The proposed architectures are realized using existing minority voter-based logic primitives and synthesized through the MagCAD nanomagnetic design framework, with functional validation performed via automatically generated VHDL models in the Xilinx Vivado environment. Complete physical-layout implementations are demonstrated for 3-bit, 4-bit, 8-bit, and 16-bit converter architectures. The results reveal a fundamental computational asymmetry between B2G and G2B conversion mechanisms. The B2G converters exhibit constant latency due to parallel XOR operations, achieving <inline-formula> <tex-math notation="LaTeX">$O(1)$ </tex-math></inline-formula> delay complexity, whereas G2B converters demonstrate linearly increasing delay with bit-width due to cascaded dependencies, resulting in <inline-formula> <tex-math notation="LaTeX">$O(N)$ </tex-math></inline-formula> complexity. This asymmetry is further reflected in the area–delay product (ADP), where B2G designs show moderate growth from 13.77 to <inline-formula> <tex-math notation="LaTeX">$92.92~\mu $ </tex-math></inline-formula>m<inline-formula> <tex-math notation="LaTeX">${}^{2}\cdot \mu $ </tex-math></inline-formula>s for 3-bit to 16-bit configurations, while G2B implementations exhibit a substantially larger increase from 30.84 to <inline-formula> <tex-math notation="LaTeX">$1349.42~\mu $ </tex-math></inline-formula>m<inline-formula> <tex-math notation="LaTeX">${}^{2}\cdot \mu $ </tex-math></inline-formula>s over the same range. The switching energy of CoFeB/MgO, Co/Pt, and DMI-enhanced Co/Pt/Ir nanomagnetic materials is found to scale with magnet count based on experimentally validated switching times. The switching power is determined by material switching time and arises from the cumulative switching of individual nanomagnets under field-clock excitation. When compared to the CoFeB baseline, the Co/Pt system reduces energy consumption by approximately 32%, while a Co/Pt/Ir multilayer system results in significant performance gains, achieving an average energy–delay product (EDP) improvement of approximately 82%. Furthermore, fabrication-variation analysis under ±5% and ±10% geometric perturbations confirms the robustness of the proposed architectures against practical manufacturing variations. The results indicate that pNML is a promising platform for low-power, non-volatile nanoscale computing and identify G2B conversion as the primary latency and efficiency bottleneck in scalable code conversion architectures.