Deformation of Nanotubes

Yang Xu and Kenny Higa

MatSE 385

http://hp720.ceg.uiuc.edu/~rotkin/nanotube/nems-1.html

Introduction

• We have adopted a simple model of a nanoelectromechanical switch [1]

• We have simulated the effect of introducing a defect by deleting atoms in a cylindrical region

Presentation: Pull-in voltages of carbon nanotube-based nanoelectromechanical switches. Marc Dequesnes, et. al

Algorithm overview

Hybrid tight-binding method [2] combines features of both ab initio and MD methods

Schrodinger equation

Wave function

Charge density

Poisson solver

Potential distribution

Initial guess potential

Solve Schrodinger equation self-consistently

Iteration to get self- consistent results

• Tight-binding approximation: consider interactions between layers (cross-sectional slices) of the nanotube

• Interaction potential is non-zero only for nearest neighbors

Copyright V. H. Crespi. Distributed under the Open Content License (http://opencontent.org/opl.shtml).

Schrodinger equation

Wave function

Charge density

Poisson solver

Potential distribution

Initial guess potential

Solve Schrodinger equation self-consistently

schottkyelectronelectronexternal VVVU

* Construct the Potential Matrix

Iteration

• Only considering potential terms of Hamiltonian

• Kinetic part which is relative to temperature

(Ek=3/2nKT) is constant in our model

Image forceem-n

EF

Ec

Metal Nanotube * Schottky barrier potential is included

Schrodinger equation

Wave function

Charge density

Poisson solver

Potential distribution

Initial guess potential

Solve Schrodinger equation self-consistently

schottkyelectronelectronexternal VVVU

),(*),(1

1),()()(

1ii

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ikTE ErEr

edEregEfr

i

* Tight-Binding Approximation

Iteration

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12

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1

2

12

2

1

212,2

2,1212

23221

121

r

r

r

r

E

r

r

r

r

rV

rV

rV

rV

L

L

L

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LactualLL

LLLactual

actual

actual

We got a block-diagonal matrix eigenvalue problem

Solve Schrodinger equation self-consistently

• Finite-element method used to solve Poisson equation to determine potential field

• Charge is non-zero only in nanotube giving sparse matrix system

• Iterate until self-consistent solution is obtained

Schrodinger equation

Wave function

Charge density

Poisson solver

Potential distribution

Initial guess potential

Iteration

Quantum results

Electrostatic Force applied on the nanotube

Molecular dynamics

• Initialize velocities from Maxwell-Boltzmann distribution

• Velocity Verlet algorithm used to update carbon atom positions

• Particle motion influenced by van der Waals interactions, covalent bonding, electric field

Van der Waals interactions

• Nanotube interactions with graphite plane important on nanoscale [1]

• Modeled using Lennard-Jones potential

• Existing code used to calculate van der Waals force per unit length [1]

• Horizontal forces neglected

Tersoff Potential

ij

ijAijijRi rVBrVE )]()([

• Tersoff potential has been successfully for carbon bonding in graphite, diamond [3]

• Realistic model of bond energies and lengths

• Sum over nearest neighbors• Attractive and repulsive forms similar to

Morse potential but considers bond order

Electric field

• Electric charge per length determined from quantum calculations

• Force due to external field calculated from analytical expression

• Force due to induced electric field calculated using image charges

• Horizontal forces neglected

Some pictures

Some pictures

Another picture

One last picture

Conclusion

• Quantum effects are important at the nanotube ends

• Simulating 6e-11 seconds takes around 10 hours• We have not generated enough data for

quantitative conclusions• Nanotube has not made contact with graphite plate• Simulations suggest that nanotubes tend to bend

most near point of attachment

References[1] Desquesnes, M., Rotkin, S. V., and Aluru, N. R. “Calculation of pull-inn

voltages for carbon-nanotube-based nanoelectromechanical switches”, Nanotechnology (13) 120-131, 2002.

[2] Clementi E. “Ab initio computations in atoms and molecules”, (reprinted from IBM Journal of Research and Development 9, 1965), IBM J. Res. Dev. 44 (1-2:228-245, 2000.

[3] Brenner, D. W. “Emperical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films”, Physical Review B. Volume 42, Number 15: 9458-9471, 1990.

Special thanks to Yan Li, Zhi Tang, Rui Qiao, and Marc Dequesnes for their advice and for writing the code that formed the basis for our project.