11

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.

ON

OFF TOP

BOTTOM

BOTTOM

TOP

RIG

HTL

EF

TL

EF

T

Ideal case

BOTTOM

TOPR

IGH

TLE

FT

TOP

BOTTOM

RIG

HTL

EF

T

Real case

Real case

V-Applied

Ideal case

RIG

HT

DEFECT

22

Implementing Boolean functions

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

X11

XRCXR1

X12 X1C

XR2

X(R-1)1

X21

X1(C-1)

X2C

X(R-1)C

XR(C-1)

TOP

C Columns

LE

FT

RIG

HTR

Row

s

f (X

11,…

,XR

C)

g (X11,…,XRC)

N R

ows

M Columns

BOTTOM

3

An example with 16 Boolean inputs

3A path exists between top and bottom, f = 1

RIG

HT

TOP

BOTTOM

LE

FT

0

0 1

1

1 0

00

1

1

1

1 0

1

00

RIG

HT

TOP

BOTTOM

LE

FT

Non-Linearities

4

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

signal in

sign

al o

ut

Holy Grail

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

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

Pro

babi

lity

of g

loba

l con

nect

ivit

y

5

Percolation Theory

6

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. 7

88

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

0 .6

0 .8

1 .0

pc

p1p 2

9

Margins

9

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

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

-

Pro

babi

lity

of g

loba

l con

nect

ivit

y

Z E R O -M A R G IN

p1

p2

O N E -M A R G IN

10

Margin performance with a 2×2 lattice

10

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%

11

One-margins (always good)

11

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 =0

ONE-MARGIN

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

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

-

Pro

babi

lity

of g

loba

l con

nect

ivit

y

p1

p2

12

Good zero-margins

12

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

LE

FT

RIG

HT

BOTTOM

TOP

0

1 1

0

f =0f =1

ZERO-MARGIN

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

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

-

Pro

babi

lity

of g

loba

l con

nect

ivit

y

p1

p2

13

Poor zero-margins

13

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 =1

POOR ZERO-MARGIN

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

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

-

Pro

babi

lity

of g

loba

l con

nect

ivit

y

p1

p2

14

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.

X11

XRCXR1

X12 X1C

XR2

X(R-1)1

X21

X1(C-1)

X2C

X(R-1)C

XR(C-1)

TOP

C Columns

LE

FT

RIG

HTR

Row

sf (

X11

,…,X

RC)

g (X11,…,XRC)

BOTTOM

15

Lattice dualityL

EF

T

RIG

HT

BOTTOM

TOP

RIG

HTL

EF

TBOTTOM

TOP

0

0 1

0

0 1

01

1

1

0

1 1

1

11

0

10

0

00

0 1

0

1

1

10

0

1 0

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