1. 1 Jianwei Dai, Lei Wang, and Faquir Jain Department of Electrical and Computer Engineering University of Connecticut Analysis of Defect Tolerance in Molecular Electronics Using Information-Theoretic Measures

2. 2 Computational fabrics beyond CMOS roadmap –Crossbar-based molecular integrated systems –Emerging challenges Our approach –Molecular electronics as an information processing medium –Information processing channel model –Determining the performance limits via information-theoretic concepts Evaluation Conclusion Outline

3. 3 Molecular Electronics Nanowire-based crossbar –High density, massive redundancy –Existing systems: nanoPLA, NASICS, and CMOL,etc. –Excessive defect density: defect rate could reach 10 -3 ~ 10 -1 Existing solutions –Post-fabrication reconfiguration –Error correcting codes –N-modular redundancy (NMR)

4. 4 Research Issues Open problems in molecular electronic computing What are the performance limits imposed by excessive non- idealities inherent in nano/molecular fabrics? How can we achieve reliable computing with performance approaching the fundamental limits? Our approach to address these problems Information Theoretical Analysis Computational fabrics Information Transfer Capacity C Implementation defects, transient errors, variations

5. 5 Quantifying Reliability via Information-Theoretic Measures Entropy Example Consider a 2-input AND gate in conventional CMOS Mutual information Channel capacity  : error probability, e.g.,  = 10 -6

6. 6 Defects in Molecular Electronics Defects in molecular electronics –Crosspoint stuck-at-open –Crosspoint stuck-at-closed –Nanowire open J1J1 J2J2 AB Y1Y0Y1Y0 00 01 00 01 11 10 11 J3J3 A B Y1Y1 Y0Y0 J1J1 AB Y1Y0Y1Y0 10 01 00 01 11 10 11 J3J3 J2J2 Y1Y1 Y0Y0 A B Stuck-at-open J1J1 AB Y1Y0Y1Y0 XX 01 00 XX 10 11 J3J3 J2J2 Y1Y1 Y0Y0 A B Stuck-at-closed AB Y1Y0Y1Y0 X1 01 00 X0 X1 10 11 J1J1 J3J3 J2J2 Y1Y1 Y0Y0 A B Nanowire open

7. 7 Molecular Electronics as An Information Processing Medium 1 nanowire open 1 0 1 0 1 0 0 Undetermined stuck-at-open stuck-at-close d p1p  1 Channel model –Statistical mappings reflect the randomness across different crossbars –Non-symmetric –Scalable to complex systems Consider a single column with N crosspoints implementing M-input AND (N  M) Any M out of N rows Crossbar LogicChannel Model

8. 8 Determining Performance Limits via Information-Theoretic Concepts From the definition of channel capacity, we can get Conditional probability of channel mapping under defects: where Observation: When m i bits in the input X i are 1, the output Y could be wrong (X i being mapped to X e =11…11, thus a 0-to-1 output error) if the number of defect-free crosspoints in this column is no more than m i

9. 9 Case 1: 2-crosspoint column for a 2-input AND gate Case 2: 3-crosspoint column for a 2-input AND gate The reliability of this gate is improved by employing the inherent redundancy in molecular crossbars Case Studies

10. 10 Evaluation Parameters –C ideal = 1 bit / use –N r = 0 ~ 4 –P d = 0.001 ~ 0.1 Any 2 out of N rows 2-input AND gate How much redundancy is needed for a desired level of reliable performance, e.g., matching that of the CMOS technology?

11. 11 Quantitative Analysis of Redundancy vs. Reliability Intrinsic relationship between the fundamental limit on reliability (C u ) and inherent redundancy (N r ) in molecular computing systems

12. 12 Conclusion Excessive defects in molecular electronics raise a question on how to exploit redundancy effectively and efficiently An information-theoretic method is developed for analysis of redundancy-based defect tolerance in molecular integrated systems The proposed method allows quantitative study of the interplay between the fundamental limits on reliability and inherent redundancy in molecular integrated systems Future work –Exploration of defect-tolerant design techniques approaching the fundamental limits