Nanoscale Digital Computation Through Percolation Mustafa Altun Electrical and Computer Engineering DAC, “Wild and Crazy Ideas” Session ─ San Francisco, July 29, 2009 University of Minnesota

Non-Linearities 2 From vacuum tubes, to transistors, to carbon nanotubes, the basis of digital computation is a robust non-linearity. signal in signal out Holy Grail

Randomness at the Nanoscale 3 Probabilistic FET-like connections in a stochastically assembled nanowire array. Self-assembled topologies. High density of bits/ logic/interconnects. High defect and failure rates. Inherent randomness in both interconnects and signal values. General Characteristics of Nanoscale Circuits:

Nanoscale Computation through Percolation Given: Physical structures exhibiting randomness. Want: Robust digital computation. “WACI” idea: Exploit the mathematics of percolation.

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 Broadbent & Hammersley (1957); Kesten (1982); and Grimmett (1999).

Percolation Theory 6 Poisson distribution of points with density λ Points are connected if their distance is less than 2r Study probability of connected components S D

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

88 Nanowire crossbar arrays Suppose that, in this technology, crosspoints are FET-like junctions. When a high or low voltage is applied, these develop low or high impedances, respectively. signal out signal in

99 Crosspoints as squares We model each crosspoint as a square. (Black corresponds to ON; white corresponds to OFF.)

1010 Implementing Boolean functions signals in: X ij ’s signals out: connectivity top-to-bottom / left-to-right.

11 An example with 16 Boolean inputs 11 A path exists between top and bottom, f = 1

1212 An example on 2×2 array Relation between p 1 ─ probability of experiencing ON crosspoint ─ and switch’s behavior. If p 1 is 0.9 then the switch is ON with probability 95%. (The probability of getting an error is 5%.) If p 1 is 0.1, the switch is OFF with probability 95%. (The probability of getting an error is 5%.)

1313 Non-Linearity Through Percolation p 2 versus p 1 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 p 1. p 2 is the probability that a connected path exists between the top and bottom plates. p1p1 p2p2

1414 Defects matter! 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 V DD, then all the crosspoints are ON and so the plates are connected. With defect in nanowires, not all crosspoints will respond this way.

15 Margins 15 One-margin: Tolerable p 1 ranges for which we interpret p 2 as logical one. Zero-margin: Tolerable p 1 ranges for which we interpret p 2 as logical zero. Margins correlate with the degree of defect tolerance.

16 Margin performance with a 2×2 lattice 16 f =X 11 X 21 +X 12 X 22 g =X 11 X 12 +X 21 X 22 Different assignments of input variables to the regions of the network affect the margins. X 11 X 21 X 12 X 22 fMarging % % %023% % % % %1

17 One-margins (always good) 17 Defect probabilities exceeding the one-margin would likely cause an (1 → 0) error. f =1f =0 ONE- MARGIN

18 Good zero-margins 18 Defect probabilities exceeding zero-margin would likely cause an (0 → 1) error. f =0f =1 ZERO- MARGIN

19 Poor zero-margins 19 Assignments that evaluate to 0 but have diagonally adjacent assignments of blocks of 1's result in poor zero-margins f =0 f =1 POOR ZERO-MARGIN

20 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.

21 Lattice duality

22 Further work Solve the logic synthesis problem. (Bring continuum mathematics into the field.) Explore physical implementation in nanowire arrays. Explore percolation as a model for digital computation with DNA and other molecular substrates.

Funding 23 NSF CAREER Award # MARCO (SRC/DoD) Contract #NT-1107