Nanoscale Digital Computation Through

Percolation

Mustafa AltunMustafa Altun

Electrical and Computer Engineering

DAC, “Wild and Crazy Ideas” Session ─ San Francisco, July 29, 2009

University of Minnesota

Non-Linearities

22

From vacuum tubes, to transistors, to carbon nanotubes, the basis of digital computation is a robust non-linearity.

signal in

sig

nal out

Holy Grail

Randomness at the Nanoscale

33

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 TheoryRich 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.

RandomGraphs

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

0 .0 0 .2 0 .4 0 .6 0 .8 1 .0

P ro b a b ility o f lo ca l co n n ectiv ity

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Percolation Theory

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Poisson distribution of points with density λPoints are connected if their distance is less than 2r

Study probabilityof connected components

S

D

Percolation Theory

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

8888

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.

METAL INSULATOR

TOP

LEFT RIGHT

BOTTOM

V-Applied

signal out

sig

nal in

9999

Crosspoints as squares

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

10101010

Implementing Boolean functions

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C Columns

LE

FT

RIG

HTR

Row

s

f (X

11,…

,XR

C)

g (X11,…,XRC)

N R

ows

M Columns

INSULATORV-applied (Xij)

BOTTOM

X11

XRCXR1

X12 X1C

XR2

X(R-1)1

X21

X1(C-1)

X2C

X(R-1)C

XR(C-1)

signals in: Xij’ssignals out: connectivity top-to-bottom / left-to-right.

1111

An example with 16 Boolean inputs

1111

A path exists between top and bottom, f = 1

RIG

HT

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BOTTOM

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FT

0

0 1

1

1 0

00

1

1

1

1 0

1

00

RIG

HT

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BOTTOML

EF

T

12121212

An example on 2×2 array

Relation between p1 ─ probability of experiencing ON crosspoint ─ and switch’s behavior.

If p1 is 0.9 then the switch is ON with probability 95%. (The probability of getting an error is 5%.)

If p1 is 0.1, the switch is OFF with probability 95%. (The probability of getting an error is 5%.)

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p14 4p1

3(1-p1) 4p1(1-p1)3 (1-p1)

4 p1=0.9 0.66 0.29 3.6e-3 1e-4 p1=0.1 1e-4 3.6e-3 0.29 0.66

13131313

Non-Linearity Through Percolation

p2 versus p1 for 1×1, 2×2, 6×6, 24×24, 120×120, and infinite size

lattices.

TOP

BOTTOM

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.

0 .0 0 .2 0 .4 0 .6 0 .8 1 .0

0 .0

0 .2

0 .4

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pc

p1p 2

14141414

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 VDD, then all the crosspoints are ON and so the plates are connected.

With defect in nanowires, not all crosspoints will respond this way.

TOP

BOTTOM

BOTTOM

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HTL

EF

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Ideal case

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IGH

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Real case

Real case

V-Applied

Ideal case

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DEFECT

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Margins

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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.

0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 - P ro b a b ility o f lo ca l co n n ectiv ity

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Z E R O -M A R G IN

p1

p2

O N E - M A R G IN

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Margin performance with a 2×2 lattice

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f =X11X21+X12X22

g =X11X12+X21X22

Different assignments of input variables to the regions of the network affect the margins.

LE

FT

RIG

HT

BOTTOM

TOP

X12X11

X22X21

X11 X21 X12 X22 f Margin g Margin

0 0 0 0 0 40% 0 40%

0 0 0 1 0 25% 0 25%

0 0 1 1 1 14% 0 23%

0 1 0 1 0 23% 1 14%

0 1 1 0 0 0% 0 0%

0 1 1 1 1 14% 1 14%

1 1 1 1 1 25% 1 25%

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One-margins (always good)

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Defect probabilities exceeding the one-margin would likely cause an (1→0) error.

LE

FT

RIG

HT

BOTTOM

TOP

1

0 1

0

f =1f =00 .0 0 .2 0 .4 0 .6 0 .8 1 .0

- P ro b a b ility o f lo ca l co n n ectiv ity

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Pro

babi

lity

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loba

l con

nect

ivit

y

p1

p2

ONE-MARGIN

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Good zero-margins

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Defect probabilities exceeding zero-margin would likely cause an (0→1) error.

LE

FT

RIG

HT

BOTTOM

TOP

0

1 1

0

f =0f =10 .0 0 .2 0 .4 0 .6 0 .8 1 .0

- P ro b a b ility o f lo ca l co n n ectiv ity

0 .0

0 .2

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0 .6

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Pro

babi

lity

of g

loba

l con

nect

ivit

y

p1

p2

ZERO-MARGIN

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Poor zero-margins

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

LE

FT

RIG

HT

BOTTOM

TOP

1

1 0

0

f =0f =10 .0 0 .2 0 .4 0 .6 0 .8 1 .0

- P ro b a b ility o f lo ca l co n n ectiv ity

0 .0

0 .2

0 .4

0 .6

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1 .0

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Pro

babi

lity

of g

loba

l con

nect

ivit

y

p1

p2

POOR ZERO-MARGIN

2020

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.

TOP

C Columns

LE

FT

RIG

HTR

Row

s

f (X

11,…

,XR

C)

g (X11,…,XRC)

BOTTOM

X11

XRCXR1

X12 X1C

XR2

X(R-1)1

X21

X1(C-1)

X2C

X(R-1)C

XR(C-1)

2121

Lattice duality

LE

FT

RIG

HT

BOTTOM

TOP

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HTL

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TBOTTOM

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0

0 1

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0 1

01

1

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1 0

),.....,(),.....,( 1111 rcrcD XXgXXfgf

2222

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

2323

NSF CAREER Award #0845650

MARCO (SRC/DoD) Contract #NT-1107