Transcript Defense Presentation.. - UB Electronic Packaging Laboratory
A Multi-scale Electro-Thermo-Mechanical Analysis of Single Walled Carbon Nanotubes
PhD defense Tarek Ragab
Electronic Packaging Laboratory CSEE department, University at Buffalo
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Objectives
1.
Use MD simulations for simulating a (10, 10) armchair (SWCNT) under uniaxial tension till failure, and calculate the stresses using an approach based on virial stress theorem and compare the results with stresses calculated by the widely used method based on engineering stresses.
2.
Study the effect of the boundary conditions, displacement increment in MD simulations on the calculated stress values.
3.
Study the effect of the length and strain rate on the stress strain behaviour of perfect SWCNTs under uniaxial tension.
4.
Study the mechanism of unravelling in carbon nanotubes during field emission .
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5.
Objectives
Formulate a quantum mechanical model based on the relaxation time approximation for calculating the joule heating in metallic SWCNTs and use this model to study the effect of the temperature and the electric field on the joule heating power generated.
6.
Formulate a similar model to calculate the electron-induced wind forces in metallic SWCNTs at different temperatures and under different values of electric filed.
7.
Develop an Ensemble Monte Carlo (EMC) simulator for calculating the joule heating and the electron-induced wind forces semi-classically directly without approximations and compare the results to that obtained using the relaxation time approximation to asses its limitation.
8.
Extract the values of the effective charge number in metallic SWCNTs under different temperatures.
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Quantum physics of joule heating and induced force • Generally, the power generated is: •
w
1 ( 4 3 ) 2
k
E
S m
And the induced forces generated is: )
f
w
1 ( 4 3 ) 2
(
k
k
)
S m f
Where,
f
0
e
k
E
) )
f
k
S m
1 )
f f
k
d d
4
Monte Carlo simulations for joule heating and forces 5
What is needed?
• • • • • Understand the structure of the Carbon Nanotubes (CNT) Geometry and Periodicity Reciprocal lattice and BZ • • • Calculate the scattering rate for each scattering events.
Energy dispersion relation Phonon dispersion relation Scattering rates Develop MC Simulator 6
What are Carbon Nanotubes (CNTs) • • • Carbon Nanotubes (CNTs) were first manufactured in the laboratory in 1991 by Sumio Iijima.
CNTs are the tubes formed of folding a graphite layer. CNTS have radii ranging from 1 to 50 nanometers, and lengths that can reach a millimeter long.
Geometry of a single graphite sheet. 7
Example 8
What are Carbon Nanotubes (CNTs)
Chirality of Carbon nanotubes
• CNTs are normally defined by the Chirality vector Ch (n,m) of the revolved graphite sheet, where the two main categories of carbon nanotubes according to chirality are Chiral and achiral (Zigzag and armchair) CNTs.
Y X a1 30° Ch a2 A. (3,3) Armchair nanotube Y X a1 Ch B. (5,0) Zigzag nanotube Y X a1 Ch a2 C. (4,2) Chiral nanotube 9
What are Carbon Nanotubes (CNTs)
Brillouin Zone for graphene
a b i
j b i
2
a i
2
ij a i
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Brillouin Zone for CNT
What are Carbon Nanotubes (CNTs)
K
K
| 2
C h
2 | 2
T
|
T
| 2
C h
Reciprocal lattice and first and second Brillouin zone for (10,10)CNT.
h
2 11
Energy Dispersion relation The electrical conductance of graphite or carbon nanotubes is attributed to the 2p transport process.
z electron, while the other three valence electrons play no role in the electrical 12
Energy dispersion relation 2 [
2
m
i
Tight Binding Method
E i
i
i
j n
1
ij
( )
j
j
1
N l N
1
e i
i i
i i
i i i dr i dr
l
j
(
r
R l
)
2
p z
t
3
k a x o
) cos( 2 3
k a o
2
2
3
k a o
) 2
13
Energy dispersion relation of graphite
Tight Binding Method
• The figure shows the energy dispersion relation of graphite based on the 2p z orbital as a basis function for Bloch’s function, showing zero energy gap at the “K” points (after (Minot 2004)).
• Armchair CNTs are always metallic.
• Zigzag and chiral CNTs are metallic if n-m is a multiple of3.
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Energy Dispersion relation for (10,10) CNT 15
Phonon dispersion relation In physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice.
Phonons are a quantum mechanical version of normal modes in classical mechanics.
[
M u t n
N
n
1
[
K
]{
n
( )}
( 2 [
M
0 where
n N
1
[
n
]
iq r n
16
Associate cell of graphite using 4 th nearest atoms 17
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0 0 0.1
0.2
0.3
0.4
0.5
q/qmax 0.6
0.7
Phonon Dispersion relation for (10,10) CNT 0.8
0.9
1 18
E
5 4 3 2 1 0 0 Electron-phonon scattering rates Conservation of energy and momentum 10 9 8 7
E e f
f
E e i
E
p k f k q
6 0.1
0.2
Ep 0.3
0.4
q 0.5
k/kmax 0.6
0.7
0.8
0.9
1 19
Electron-phonon scattering rates Scattering mechanisms LA phonons Emission Absorption LO phonons Emission Absorption Forward scattering Backward scattering 20
Electron-phonon scattering rates • Scattering Probability According to Fermi golden rule
S m
2
k
ep
k
, 2
S k i
i
)
f
,
f
2
D LA
E p
(
q
2 ( 2
d
2
N E p
) 2 1 2 )
dE f
1
dk S k i
i
f
,
f
2
D LO
E p
p
) 2 1 2 )
dE f
1
dk
21
LA phonons LO phonons Electron-phonon scattering rates • Scattering Probability According to Fermi golden rule 22
I
e
10 9 I-V curve 1
E
k dk
23
I-V curve 24
w
1 160
m
1 10 9 Power generated along the length of CNT
m
S m k
k
m
f k
m dk
25
Temperature Effect (Heat) 26
Force generated along the length of CNT
w
1 160
m
1 10 9 (
k
k
)
S m k
k
m
f k
m dk
27
Monte Carlo simulation results 28
Monte Carlo simulation results 29
Monte Carlo simulation results (force) 30
Monte Carlo simulation results (force and effective charge number)
F
*
Z eE
3.465E-3 , 9.186E-3, 0.0127 and 0.015 Å 1 for 300, 600, 900, 1200K 31
Molecular dynamics simulation for the failure of carbon nanotubes under Uniaxial tension • • • • • NVT ensemble…..Brendsen thermostat atom, T=300K Integration algorithm….. third order predictor-corrector algorithm Time step….. 0.5 Fs <10% Vibration period of the carbon atom Boundary Conditions…..Fixed boundary conditions Armchair (10,10) CNTs are only simulated 32
Molecular dynamics simulation for the failure of carbon nanotubes under uniaxial tension
Potential
• Uses the 2nd generation Reactive Empirical Bond Order (REBO) potential (Brenner et. al. 2002) to model the carbon-carbon bond and to allow for bond breaking and formation .
Stress Calculations
• Engineering Stress : • Virial stress: Forces on the end atoms are added up and divided by the area int 1 2
V
(
j f ij
r ij
) • With some simplifications this can be written as: int
V
1 (
f i
r i
) 33
Stress calculation using the two approaches for different strain rates
A: Virial stress B: Engineering stress
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• Displacement increment study •
Objectives:
Try to find the magnitude of the displacement increment beyond which the change in mechanical behavior can be neglected Study the effect of the length on the maximum stress level • • • Study the effect of the strain rate on the maximum stress level •
Changing Parameters:
Length: 12.3 A~1180.8A
Strain rate: 1.69E+8~1.69E+11 sec -1 Displacement increment: 0.00025A~0.25A
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Effect of the displacement increment on the maximum stress in the simulated CNTs 36
Unraveling of the end of CNTs • • • (18, 0) Zigzag and (10, 10) Armchair CNTs Two Kinematic schemes Maximum force in atomic chain is 18eV/angstroms 37
Unraveling of the end of CNTs • • (10, 10) Armchair CNTs Unraveling Force= 15eV/angstroms • • (18, 0) Zigzag CNTs Unraveling Force= 10eV/angstroms 38
Conclusions
1.
2.
3.
4.
5.
6.
Engineering stresses can underestimates the stresses in CNTs by 35% A value of 1.76% of the unrestrained bond length is required for displacement increment.
Integral form using relaxation time approximation can only be used to find a rough estimate of the joule heating and electron induced wind forces.
A semi-classical transport model using Ensemble Monte Carlo simulation model is developed for calculating the joule heating in carbon nanotubes and can be used to calculate the joule heating in any other nanoscale material.
Applying Joule’s law in CNT under high current densities is not appropriate.
Values of effective charge number Z * 15E-3 according to the temperature. in CNTs can vary from 4.65E-3 to 39
Original contributions
1.
2.
3.
4.
5.
A Simplification for the virial stress formula is derived to ease the calculations of virial stresses in multibody potentials.
A parametric study was performed for molecular dynamics simulations of carbon nanotubes to quantify the threshold value for the displacement increment used for carbon nanotubes. This can be used in any other study.
A method is proposed to compute the current-voltage relation of carbon nanotubes based on the relaxation time approximation and gives satisfactory results in comparison with experimental data.
A semi-classical transport model using Ensemble Monte Carlo simulation model is developed for calculating the joule heating in carbon nanotubes and can be used to calculate the joule heating in any other nanoscale material.
A new method for calculating the electron-induced wind forces and effective charge number is formulated and used to calculate the effective charge number in armchair single-walled carbon nanotubes numerically for the first time. This method is not limited to carbon nanotubes and can be used for any material. 40
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Publications
“Joule heating in single-walled carbon nanotubes”.
Journal of Applied Physics
, Vol. 106, Issue 6, pp 63705, (2009).
Science & Technology
, Vol. 20, Issue 14.
Selected for simultaneous publication in the Virtual Journal of Nanoscale
“A framework for stress computation in Single-walled carbon nanotubes under uniaxial tension”.
Computational Materials Science
, Vol. 46, Issue 4, pp 1135, (2009). “A quantum mechanical formulation of electron transport induced wind forces in metallic single walled carbon nanotubes”.
Carbon
, Vol. 48, Issue 1, pp 47, (2010). “Semi-classical transport for predicting joule heating in carbon nanotubes”.
physics letters A
, Vol. 374, Issue 24, pp 2475, (2009). Ragab, T., Basaran, C., “The prediction of the effective charge number in single walled carbon nanotubes using Monte Carlo simulation”.
Carbon
. Under review.
Ragab, T., Basaran, C., “The unravelling of open-ended single walled carbon nanotubes using molecular dynamics simulations”.
ASME Journal of Electronic Packaging
. Submitted for publication. 41
Recommendations for future research
1.
hot phonon effect 2.
phonon-phonon scattering 3.
semiconducting carbon nanotubes and nanotubes with different chiralities.
4.
Material properties for carbon nanotubes to formulate a complete constitutive model for simulating carbon nanotubes at a larger macroscopic scale in composites using finite element method.
5.
New phonon dispersion relation 6.
Defects and impurities 42
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Thank You
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