Scalable binary-to-gray and Gray-to-Binary code converters using atomic silicon dangling bonds: hierarchical physical design, complexity analysis, and robustness evaluation

The atomic silicon dangling bond (ASDB) technology platform holds promise as an efficient solution for atomically precise digital circuits with extremely high integration density, low energy operation, and compatibility with silicon technology manufacturing processes. However, scalable and functional code conversion systems based on ASDB technology are still unknown. In this work, scalable architectures for binary-to-gray (B2G) and Gray-to-Binary (G2B) code converters are suggested based on hierarchical physical design approach within the SiQAD simulation framework. Physical layouts are synthesized and verified through SimAnneal simulations for 3-bit, 4-bit, and 8-bit converters, while the analytical scalability model is built to evaluate the physical resource requirements of large scale 16-bit, 32-bit, 64-bit, 128-bit, and 256-bit converters. Performance evaluation is performed with respect to the number of dangling bonds, physical size, propagation delay, area-delay product (ADP), computational complexity, and energy consumption of ground state configurations. For the directly simulated 8-bit converters, the B2G and G2B architectures require 300 and 418 dangling bonds, respectively, with G2B requiring approximately 39.3% more dangling bonds than B2G. Binary-wire routing accounts for 66.3% of the total dangling bonds in the 8-bit B2G converter and 73.4% in the corresponding G2B converter, confirming routing as the dominant physical resource. The analytical scalability model further projects 9461 dangling bonds for the 256-bit B2G converter and 13175 dangling bonds for the corresponding G2B converter, indicating a consistent increase in physical resource requirements with converter width. It is shown that there exists a fundamental computational asymmetry between the two code conversion architectures. The proposed B2G converter relies on parallel XOR operations and its computational depth remains constant with O (1) complexity. In contrast, due to the recursive XOR relationship, the propagation delay of G2B converter increases linearly with O ( N ) computational complexity. Dangling bond decomposition demonstrates that binary-wire routing becomes the scalability bottleneck for larger code conversion architecture due to dominant contribution to the physical realization cost. Analysis of fabrication-induced geometric variations, including dangling bond misalignments, proves the stability of logical function and electrostatic charge configuration. The presented hierarchical design methodology provides a scalable approach for atomic code converters and sets practical design guidelines for ASDB-based digital systems in beyond-CMOS computing.