deformation of nanotubes yang xu and kenny higa matse 385 rotkin/nanotube/nems-1.html
TRANSCRIPT
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
N
ikTE ErEr
edEregEfr
i
* Tight-Binding Approximation
Iteration
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2
12
2
1
2
12
2
1
212,2
2,1212
23221
121
r
r
r
r
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r
r
r
r
rV
rV
rV
rV
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.