Computing with Defects Ideally, if the applied voltage is 0, then all the crosspoints are OFF and so there is no connection between any of the plates. Ideally, If the applied voltage is VDD, then all the crosspoints are ON and so the plates are connected. With defect in nanowires, not all crosspoints will respond this way. 1

Implementing Boolean functions signals in: Xij’s signals out: connectivity top-to-bottom / left-to-right. 2

An example with 16 Boolean inputs A path exists between top and bottom, f = 1 3

Non-Linearities From vacuum tubes, to transistors, to carbon nanotubes, the basis of digital computation is a robust non-linearity. signal in signal out Holy Grail

Broadbent & Hammersley (1957); Kesten (1982); and Grimmett (1999). Percolation Theory Rich mathematical topic that forms the basis of explanations of physical phenomena such as diffusion and phase changes in materials. Sharp non-linearity in global connectivity as a function of random local connectivity. Random Graphs 5 Broadbent & Hammersley (1957); Kesten (1982); and Grimmett (1999).

Percolation Theory Poisson distribution of points with density λ Points are connected if their distance is less than 2r S D Study probability of connected components

There is a phase transition at a critical node density value. Percolation Theory There is a phase transition at a critical node density value. 7

Non-Linearity Through Percolation p2 versus p1 for 1×1, 2×2, 6×6, 24×24, 120×120, and infinite size lattices. Each square in the lattice is colored black with independent probability p1. p2 is the probability that a connected path exists between the top and bottom plates. 8

Margins correlate with the degree of defect tolerance. One-margin: Tolerable p1 ranges for which we interpret p2 as logical one. Zero-margin: Tolerable p1 ranges for which we interpret p2 as logical zero. Margins correlate with the degree of defect tolerance. 9

Margin performance with a 2×2 lattice X11 X21 X12 X22 f Margin g 40% 1 25% 14% 23% 0% f =X11X21+X12X22 g =X11X12+X21X22 Different assignments of input variables to the regions of the network affect the margins. 10

One-margins (always good) f =1 f =0 Defect probabilities exceeding the one-margin would likely cause an (1→0) error. 11

Good zero-margins f =1 f =0 Defect probabilities exceeding zero-margin would likely cause an (0→1) error. 12

Poor zero-margins f =1 f =0 Assignments that evaluate to 0 but have diagonally adjacent assignments of blocks of 1's result in poor zero-margins 13

Lattice duality Note that each side-to-side connected path corresponds to the AND of the inputs; the paths taken together correspond to the OR of these AND terms, so implement a sum-of-products expression. A necessary and sufficient condition for good error margins is that the Boolean functions corresponding to the top-to-bottom and left-to- right plate connectivities f and g are dual functions.

Lattice duality