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Jianwei Dai, Lei Wang, and Faquir JainDepartment of Electrical and Computer Engineering

University of Connecticut

Analysis of Defect Tolerance in Molecular Electronics Using Information-Theoretic

Measures

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• Computational fabrics beyond CMOS roadmap– Crossbar-based molecular integrated systems– Emerging challenges

• Our approach– Molecular electronics as an information processing medium – Information processing channel model– Determining the performance limits via information-theoretic

concepts• Evaluation• Conclusion

Outline

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Molecular Electronics

• Nanowire-based crossbar – High density, massive

redundancy– Existing systems: nanoPLA,

NASICS, and CMOL,etc.– Excessive defect density: defect

rate could reach 10-3 ~ 10-1 • Existing solutions

– Post-fabrication reconfiguration– Error correcting codes– N-modular redundancy (NMR)

Nanowire crossbar structure

Output from OR gate Arrays

AND gate Arrays

Output from AND gate Arrays

Pull-up Arrays

Pull-down Arrays

Input

OR gate Arrays

Pu

ll-up

A

rrays

Pu

ll-dow

n

Arrays

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Research Issues

Open problems in molecular electronic computing• What are the performance limits imposed by excessive non-

idealities inherent in nano/molecular fabrics?• How can we achieve reliable computing with performance

approaching the fundamental limits?

Our approach to address these problems

InformationTheoretical

Analysis

Computational

fabrics

InformationTransfer Capacity C

Implementationdefects, transient errors, variations

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Quantifying Reliability via Information-Theoretic Measures

• Entropy • Example

Consider a 2-input AND gate in conventional CMOS

)|()(

)|()();(

YXHXH

XYHYHXYI

• Mutual information

• Channel capacity

);(max)(

YXICxp

u

x

xpxpXH ))((log)()( 2

)}|()(max{ XYHYHCu

)1(log)1(log1 22

99997.0 usebits /

: error probability, e.g., = 10-6

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Defects in Molecular Electronics

• Defects in molecular electronics – Crosspoint stuck-at-open – Crosspoint stuck-at-closed – Nanowire open

J1 J2

AB

Y1Y0 00

01 00

00 01 11

10 11

J3

A

B

Y1 Y0

J1

AB

Y1Y0 10

01 00

00 01 11

10 11

J3

J2

Y1 Y0

A

B

Stuck-at-open

J1

AB

Y1Y0 XX

01 00

XX XX XX

10 11

J3

J2

Y1 Y0

A

B

Stuck-at-closed

AB

Y1Y0 X1

01 00

X0 X0 X1

10 11J1

J3

J2

Y1 Y0

A

B

Nanowire open

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Molecular Electronics as An Information Processing Medium

1

nanowire open

1p

dpp

dpp 1

0

1

0

1

0 0Undetermined

dpp

dp1p

1p

stuck-at-open

stuck-at-close

X

X YeX

Y

X

Y

X

Y

X eX Ydp1p

1

• Channel model– Statistical mappings reflect the

randomness across different crossbars

– Non-symmetric– Scalable to complex systems

• Consider a single column with N crosspoints implementing M-input AND (N M)

1Mx

Any M out of N rows

0x1x2x Y

Y

Crossbar Logic

1)(21)(2 MMp

0X

1X

2X

12MX

XeX Y

0Y

1Y

11p

1)1(2Mp

1)1(2Mp

Channel Model

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Determining Performance Limits via Information-Theoretic Concepts

• From the definition of channel capacity, we can get );(max

)(YXIC

xpu

),|1(1)|0( ii XXYPXXYP

)|1...11(

)|1(

ie

i

XXXP

XXYP

im

nnn

M

nim P

C

C

0

1

Mmi

Mmi

nNd

nd

nNn ppCP )1(

• Conditional probability of channel mapping under defects:

where

• Observation: When mi bits in the input Xi are 1, the output Y could be wrong (Xi being mapped to Xe=11…11, thus a 0-to-1 output error) if the number of defect-free crosspoints in this column is no more than mi

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Case 1: 2-crosspoint column for a 2-input AND gate

01.0)00|1( XYP

09.0)01|1( XYP09.0)10|1( XYP

1)11|1( XYP

)|1|(1)|0( XYPXYP

95953.0uC

Case 2: 3-crosspoint column for a 2-input AND gate

001.0)00|1( XYP

0145.0)01|1( XYP0145.0)10|1( XYP

1)11|1( XYP

)|1|(1)|0( XYPXYP

99427.0uCusebits / usebits /

The reliability of this gate is improved by employing the inherent redundancy in molecular crossbars

Case Studies

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Evaluation

• Parameters

– Cideal = 1 bit / use

– Nr = 0 ~ 4

– Pd = 0.001 ~ 0.1

1xAny 2 out of N rows

0x

Y

Y

2-inputAND gate

How much redundancy is needed for a desired level of reliable performance, e.g., matching that of the CMOS technology?

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1E-171E-161E-151E-141E-131E-121E-111E-101E-091E-081E-071E-061E-05

0.00010.0010.010.1

1

Defect Rate

Cid

ea

l - C

u

Nr = 0

Nr = 1

Nr = 2

Nr = 3

Nr = 4

Quantitative Analysis of Redundancy vs. Reliability

Intrinsic relationship between the fundamental limit on reliability (Cu) and inherent redundancy (Nr) in molecular computing systems

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Conclusion

• Excessive defects in molecular electronics raise a question on how to exploit redundancy effectively and efficiently

• An information-theoretic method is developed for analysis of redundancy-based defect tolerance in molecular integrated systems

• The proposed method allows quantitative study of the interplay between the fundamental limits on reliability and inherent redundancy in molecular integrated systems

• Future work– Exploration of defect-tolerant design techniques

approaching the fundamental limits