School of Mathematical Sciences

Life Impact The University of Adelaide

Nanocomputing memory Nanocomputing memory devices and logic gates formed devices and logic gates formed

from carbon nanotubes and from carbon nanotubes and metallofullerenesmetallofullerenes

Nanomechanics Group, Nanomechanics Group,

School of Mathematical Sciences, School of Mathematical Sciences,

The University of Adelaide,The University of Adelaide,

Adelaide, SA 5005, AustraliaAdelaide, SA 5005, AustraliaRichard K. F. Lee and James M. Hill

5th – 9th February 2012ICONN 2012, Perth, Western Australia

Overview

2

• Trends in computer requirements:– Smaller in size,– Faster processing,– Increased data capacity.

• Nano memory devices and logic gates:– Continuous approximation,– Lennard-Jones potential,– Memory devices and logic gates.

• Conclusion

Computer size and speed

3

Data storage

4

Punch Card

Magnetic Tape

Floppy Disk

Hard Disk

Media (Data Size):Floppy Disk (360KB ~ 1.44MB)ZIP Disk (100MB ~ 750MB)CD/DVD/Blue-Ray (640MB ~ 50GB)Hard Disk (30MB ~ 3TB)

1TB=1024GB1GB=1024MB1MB=1024KB1KB=1024B

Interaction energy between two molecules

• The non-bonded interaction energy is obtained by summing the interaction potential energy for each atom pair

• In continuum models, the interaction energy is obtained by averaging over the surface of each entity.

where 1 and 2 are the mean atomic surface densities for each molecule,

and is the distance between two surface elements dS1 and dS2 on two different molecules.

€

E = Φ(ρ ij )j

∑i

∑

€

E = η1η2 Φ(ρ )dS1dS2∫∫

Lennard-Jones Potential

• The repulsive term 1/12, dominates at short distances,

• The attractive term 1/6, dominates at large distances (weak interaction),

• Each atom-atom interaction is characterised by two Lennard Jones constants, A=46 and B=412 determined experimentally, and using empirical combining rules, 12=(12)1/2, 12=(1+2)/2,

• Force: F=-d/d

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

+−=

126

126

4

)(

ρ

σ

ρ

σε

ρρρ

BA

: well depth, : van der Waals distance

min = 21/6, min = -

• Mathematician who held a chair of Theoretical Physics at Bristol University (1925-32)

• Proposed Lennard-Jones potential (1931)

(October 27, 1894 – November 1, 1954)

“Father of modern computational chemistry”

Lennard-Jones sphere-point interaction

€

(ρ) = −A

ρ 6 +B

ρ12

Φ f (ρ) =η f πb

ρ

A

2

1

(ρ + b)4 −1

(ρ − b)4

⎡

⎣ ⎢ ⎤

⎦ ⎥

−B

5

1

(ρ + b)10 −1

(ρ − b)10

⎡

⎣ ⎢ ⎤

⎦ ⎥

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

⎫

⎬ ⎪ ⎪

⎭ ⎪ ⎪

Nano memory devices

(1)

(2)

Large energy

gap (~7eV)

Small energy

gap (~1.1eV)

(2) Originally proposed by Y-K Kwon, D Tománek and S Iijima (1999) using MD Simulations

Nano memory device (1)

External E field

Changing State

Y. Chan, R. K. F. Lee, and J. M. Hill, “Metallofullerenes in composite carbon nanotubes as a nanocomputing memory device”, IEEE Transactions on Nanotechnology, 10 (2011) 947-952.

Nano memory device (1)

Metallofullerene (0, 0, Z)

Smaller Nanotube (rcos, rsin, z)

Larger Nanotube (Rcos, Rsin, z)

Distance for the center of the metallofullerene and

Smaller Tube: t2=r2+(Z-z)2

Larger Tube: T2=R2+(Z-z)2

• E = Em-T1+Em-t+Em-T2 + Ef-T1+Ef-t+Ef-T2

• F = Fm-T1+Fm-t+Fm-T2 + Ff-T1+Ff-t+Ff-T2

Energy

Egap

Emin

State |0> State |1>

Detail:

K+@C60

L1=20År=6.093ÅR=6.766Å

Force

Fcritical

Fcritical

State |0> State |1>

Detail:

K+@C60

L1=20År=6.093ÅR=6.766Å

Nano memory device (2)

External E fieldChanging State

R. K. F. Lee, and J. M. Hill, “Design of a two-state shuttle memory device”, CMC: Computers, Materials and Continua, 20 (2010) 85-100.

Different Ion @C60 for nano memory device (2)

Ion 0(Å) (meV) (Å) Emin(eV) Egap(eV) Fcritical (eV/Å) Mass(u)

K+ 4.0010 3.0352 7.23235 -4.39478 1.13255 0.46929 39.102

F- 2.495 0.403 7.23482 -4.36394 1.12464 0.46615 19.00

Mg2+ 0.7926 38.798 7.23466 -4.36577 1.12517 0.46635 24.31

Mg2+ 0.9929 37.944 7.23454 -4.36724 1.12556 0.46651 24.31

Mg2+ 1.0600 37.944 7.23449 -4.36783 1.12572 0.46657 24.31

Cl- 2.4192 4.336 7.23422 -4.37120 1.12655 0.46691 35.453

Cl- 4.40 4.332 7.23075 -4.41539 1.13776 0.47136 35.453

Cl- 4.05 6.509 7.23093 -4.41263 1.13713 0.47111 35.453

Cl- 4.45 4.622 7.23045 -4.41925 1.13873 0.47174 35.453

Na+ 3.33 0.124 7.23478 -4.36449 1.12477 0.46620 22.990

Na+ 2.43 2a.031 7.23450 -4.36787 1.12568 0.46656 22.990

Na+ 2.58 0.643 7.23472 -4.36524 1.12498 0.46628 22.990

Li+ 2.224 13.429 7.23381 -4.37614 1.12787 0.46734 6.941

I- 4.286 10.149 7.22895 -4.43797 1.14357 0.47367 126.90

=L+r-Zmin

Transfer time for nano memory device (2)

€

t f ≈m

2

dZ

Fext × (Z + Zmin )−Z min

Z min

∫

=4mZmin

FextExample:

K+@C60

2L=27Å, Fext=0.5eV/Å

tf=2.4933ps (1ps=10-12s)

State Switching Rate ~ 401Gbit/s

Nano logic gate

Input Output

I1 I2 AND OR NAND NOR

T T T T F F

T F F T T F

F T F T T F

F F F F T T

T = TRUEF = FALSE

R. K. F. Lee, and J. M. Hill, “Design of a nanotori-metallofullerene logic gate”, (2011), submitted to IEEE Transactions on Computers.

Nano logic gate

Maximum energy – Minimum energy < 0.011eV

I1 I2 Det.

AND OR

NAND NOR

+ + O4 + + - -

+ - O2 - + + -

- + O3 - + + -

- - O1 - - + +

Conclusion

• Memory devices and logic gates:– Nano size,– Electrical field control.

• For a fast state switching rate / time:– Light Ion,– Large external force,– Short nanotube length,– Around 400 Gbit/s.

20

Acknowledgement

• All colleagues in the Nanomechanics Group• Australian Research Council

http://www.maths.adelaide.edu.au/nanomechanics/

Thank you!http://www.maths.adelaide.edu.au/nanomechanics/http://www.maths.adelaide.edu.au/nanomechanics/

References

• G. E. Moore, Technical Digest International Electron Devices Meeting, 21 (1975) 11-13.

• B. J. Cox, N. Thamwattana and J. M. Hill, Proceedings of The Royal Society A, 463 (2007) 461-476.

• B. J. Cox, N. Thamwattana and J. M. Hill, Proceedings of The Royal Society A, 463 (2007) 477-494.

• Y. Chan, R. K. F. Lee and J. M. Hill, IEEE Transactions on Nanotechnology, 10 (2011) 947-952.

• R. K. F. Lee and J. M. Hill, CMC: Computers, Materials and Continua, 20 (2010) 85-100.

• R. K. F. Lee and J. M. Hill, “Design of a nanotori-metallofullerene logic gate”, (2011), submitted to IEEE Transactions on Computers.