Address Generation for Nanowire Decoders Jia Wang, Ming-Yang Kao, Hai Zhou Electrical Engineering & Computer Science Northwestern University U.S.A.

Mar. 2007Northwestern University, U.S.A.2 Emerging Technologies for Computing Achieving high computing power at a low cost is essential to the success of our modern civilization. –CMOS technology scaling – Moore’s Law: exponential increase in computing complexity at lowered cost. –Advances in design methodologies: ability to harness the computing complexity. ITRS predicts CMOS is approaching the physical limit. –Research new mechanism for information storage/processing. –Research appropriate design methodologies. Nanowire crossbars –Demonstrated to be feasible. –Bi-stable molecular at crossbar junctions for information storage. –FPGA-like mechanism (LUT) for information processing. –High density and compatible with conventional CMOS Boolean logic.

Mar. 2007Northwestern University, U.S.A.3 Hybrid System with Nanowire Crossbar Fabricating perfect pattern is almost impossible at nanoscale. –Hard to achieve precise interconnects. –High defect rate. Require post-fabrication configuration –Correct functionality. –Defect tolerance. Hybrid system: nanowire crossbar and CMOS –Nanowire crossbar: high density of functionalities. –CMOS: reliable for configuration and interfacing to conventional systems. Nanowire decoders –One mechanism to interface nanowire crossbars with CMOS circuits. –Demonstrated to be feasible.

Mar. 2007Northwestern University, U.S.A.4 Nanowire Decoders Pass-transistor-like structures formed randomly at the contacts of mesowires (CMOS) and nanowires allow mesowires to control the resistance of nanowires. Once a proper internal address is applied, one nanowire connects to external circuits (individually addressable). More nanowires can be individually addressable with more mesowires. –However, more mesowires require more interfacing CMOS circuits and thus reduce density. –Finding the majority, if not all, of the proper addresses is necessary.

Mar. 2007Northwestern University, U.S.A.5 Motivation Previous works –Rachlin et al. [6] provided a theoretical bound on the necessary number of mesowires for the existence of a given number of individually addressable nanowires at a given probability. –Chen et al. [7] proposed a heuristic approach for address generation by performing random testing trails. Need special configurable junctions to generate all the addresses. Cannot determine in finite time if the required number of the addresses are presented without the above special junctions. Post-fabrication configuration revisited. –Testing provides information for configuration generating. –Generate configuration for each fabricated decoder. –Dedicated machines vs. built-in self-configuration

Mar. 2007Northwestern University, U.S.A.6 Motivation (Cont.) Dedicated machines –Allow sophisticated testing mechanisms and configuration algorithms. –Time consuming and costly. Built-in self-configuration –Allow configurations generation “in parallel”. –Complexities of testing mechanisms and configuration algorithms are limited. Research built-in self-configuration for address generation. –Formal modeling of nanowire decoders. –Formal modeling of testing mechanisms for information collecting purpose. –Design algorithms generate all the addresses in finite time if required generate most addresses efficiently

Mar. 2007Northwestern University, U.S.A.7 Modeling Nanowire Decoder An address w is a subset of the mesowires where each mesowire in it is activated when the address w is applied. For one contact group, K(w) is the set of the nanowires whose resistance remain 0 when the address w is applied. –K(empty set) is the set of all the nanowires. –K(x  y) = K(x)  K(y). For one contact group, a set of addresses W is proper iff –For all w  W, K(w) is not empty. –For all x and y  W, K(x)  K(y) is empty. Proper addresses can be determined if K({m}) is known for every mesowire m.

Mar. 2007Northwestern University, U.S.A.8 Modeling Testing Mechanism To know every K({m}) is not cost-effective. On-off measurement: f(w) = 0 or 1. –f(w) = 1 iff K(w) is the empty set. –Easy to implement – could be the simplest one. –f is non-decreasing: f(x)  f(y) if x  y. A set of addresses W is proper iff –For all w  W, f(w) = 0. –For all different x and y  W, f(x  y) = 1. Algorithms inspect decoders through the measurements. Measure efficiency of algorithms by # of measurements performed.

Mar. 2007Northwestern University, U.S.A.9 Nano Address Generation Given one nanowire decoder. Suppose the non-decreasing f(w) can be evaluated for every address w for each contact group. Generate maxA proper addresses or generate as many proper addresses as possible if there are less than maxA proper addresses. To be exploited for efficient algorithm: –Most probably there will be more than maxA proper addresses in the decoder to ensure a specific yield of maxA addresses.

Mar. 2007Northwestern University, U.S.A.10 Maximal Addresses Maximal address w: –f(w) = 0. –For all w  x, f(x) = 1. Maximal addresses are proper addresses. –Suppose x and y are two different maximal addresses. Then x  x  y. Thus f(x  y) = 1. # of proper addresses is no more than # of maximal addresses. –Otherwise there are two different proper addresses x and y and a maximal address w such that x  w and y  w. Then x  y  w. Thus f(x  y) = 0 which is not possible.

Mar. 2007Northwestern University, U.S.A.11 Maximal Address Generation Single contact group: –For any address w, greedily expand w to a maximal address. –Heuristics to reduce the number of measurements performed. Skip addresses which is a subset of a known maximal address or of which a known maximal address is a subset. Start from addresses with less elements. Multiple contact groups? –PMAG: apply the above algorithm to find all the maximal addresses in each contact group until maxA addresses are generated. –Not efficient if not all the maximal addresses are required. Experimental show generating all the maximal addresses for one contact group requires significantly more measurements than generating most of them. –# measurements won’t change if more contact groups are added. But adding contact groups makes the problem “easier” since there will be more maximal addresses.

Mar. 2007Northwestern University, U.S.A.12 Maximal Address Generation (Cont.) Maximal Address Generation Algorithm (MAG): –For any address w, greedily expand w to a maximal address in every contact group. Would result in multiple maximal addresses at a time. –When performing expanding in one contact group, skip addresses which is a subset of a known maximal address or of which a known maximal address is a subset. –Start from addresses with less elements. Efficient when there are more than maxA maximal addresses. –Won’t stuck at one contact group. –If there are more contact groups, more maximal addresses are generated through expanding one address – measurements can be saved!

Mar. 2007Northwestern University, U.S.A.13 Experiments Generate K({m}) for each contact group and each mesowire m randomly. Nanowires are assigned to K({m}) independently at probability p = 0.5. # of measurements vs. # of proper addresses generated for single contact group, 512 nanowires, 30 mesowires. Compare the MAG algorithm to the PMAG algorithm for multiple contact groups. 16 mesowires. There are 8 nanowires in each contact group addresses are required.