11 Computing with Defects Ideally, if the applied voltage is 0, then all the crosspoints are OFF...
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Transcript of 11 Computing with Defects Ideally, if the applied voltage is 0, then all the crosspoints are OFF...
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