A Multi-scale Electro-Thermo-Mechanical Analysis of
Single Walled Carbon Nanotubes
PhD defense
Tarek Ragab
Electronic Packaging Laboratory
CSEE department, University at Buffalo
1
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 .
2
Objectives
5.
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.
3
Quantum physics of joule heating and induced force
• Generally, the power generated is:
w(
1
4
)
3
2
  ( E ( k )  E ( k ))  S
m
( k , k )  f ( k )  (1  f ( k )) d k d k
• And the induced forces generated is:
w(
1
4
)
3
2
(
k   k )  S m ( k , k )  f ( k )  (1  f ( k )) d k d k
Where,
f ( k )  f ( E ( k )  e ( k ) v k  E )
0
 (k ) 
1
S
k
m
( k , k )
4
Monte Carlo simulations for joule heating and forces
5
What is needed?
•
Understand the structure of the Carbon Nanotubes (CNT)
•
•
•
Calculate the scattering rate for each scattering events.
•
•
•
•
Geometry and Periodicity
Reciprocal lattice and BZ
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.
a2
30°
Ch
a1
Y
X
A . (3,3) A rm chair nanotube
Ch
Y
a1
X
B . (5,0) Z igzag nanotube
a2
a1
Ch
Y
X
C . (4,2) C hiral nanotube
9
What are Carbon Nanotubes (CNTs)
Brillouin Zone for graphene
a i  b j  2 ij
bi 
2
| ai |
10
ai
What are Carbon Nanotubes (CNTs)
Brillouin
Zone for
CNT
K 
K 
Reciprocal
lattice and
first and
second
Brillouin
zone for
(10,10)CNT.
2
| Ch |
2
|T |
2
2
Ch
T
K  C h  2 
11
Energy Dispersion
relation
The electrical conductance of
graphite or carbon nanotubes
is attributed to the 2pz
electron, while the other
three valence electrons play
no role in the electrical
transport process.
12
Energy dispersion relation
2
[
  U ( r )] i ( r )  E i i ( r )
2
2m
Tight Binding Method
n
 i (k , r ) 
C
ij
 j (k , r ) 
(k ) j (k , r )
j 1
Ei (k ) 
i  i
i i
1
N
N

e
ik  R l
 j ( r  Rl )
l 1
  i   i dr
*

E ( k )   2 p z  t 1  4 cos(
  i  i dr
*
3k x ao
2
) cos(
3k y ao
)  4 cos (
2
2
3k y ao
2
13
)
Energy dispersion relation of graphite
Tight Binding Method
• The figure shows the energy
dispersion relation of graphite based
on the 2pz 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.
14
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.
N
[ M ]{u n ( t )} 
 [K
n  n
]{u n  ( t )}
n  1
2
( [ M ]  [ Kˆ ( q )]) uˆ ( q ,  )  0
where
[ Kˆ ( q )] 
N
 [ K n ]e
 iq rn
n 1
16
Associate cell of graphite using 4th nearest atoms
17
0.2
E 
0.18

0.16


0.14


Energy (ev)
0.12
0.1 

0.08

0.06

0.04

0.02

0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
q/qmax
Phonon Dispersion relation for (10,10) CNT
18

Energy(ev)
Electron-phonon scattering rates
Conservation of energy and momentum
10
Ee  Ee   E p
f
9

8
f
i
   
i
k  k  q
f
7
i
6
5
4
3
2
Ep
1
q 
0
0
0.1
0.2
0.3
0.4
0.5
k/kmax
0.6
0.7
0.8
0.9
1
19
Electron-phonon scattering rates
Scattering mechanisms
LA phonons
Emission
Absorption
Forward scattering
LO phonons
Emission
Absorption
Backward scattering
20
Electron-phonon scattering rates
• Scattering Probability
According to Fermi golden rule
S m (( k , ), ( k , )) 
S ( k i , i ) 
2
S ( k i , i ) 
D LA

all  possible  k f ,
 k  , 
f
2 Ep
f
2 Ep
2
 k ,
ep
 (q  (
D LO

all  possible  k f ,
Hˆ
2
2
d
 ( E ( k ,   )  E ( k ,  )   )
1
) )  (N (E p ) 
2
 (N (E p ) 
1
2

1
2
)

2
1
2
1
dE f
dk
21
)
1
dE f
dk
Electron-phonon
scattering rates
LA phonons
• Scattering
Probability
According to Fermi
golden rule
LO phonons
22
I-V curve
I 
e
10

  
 9
f ( k , ) 
1 E
k
dk
23
I-V curve
24
Power generated along the length of CNT
w
1

160
10
   ( E ( k ,   )

m 1
m
 E ( k , ))  S m (( k , ), ( k , ) m )  f ( k , )  (1  f (( k , ) m )) dk
 9
25
Temperature Effect (Heat)
26
Force generated along the length of CNT
w
1

160
10
  (

m 1
k   k )  S m (( k , ), ( k , ) m )  f ( k , )  (1  f (( k , ) m )) dk
 9
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,
for
300,
600,
1
0.0127 and 0.015 Å
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 :
Forces on the end atoms are added up and
divided by the area
1
int


  
(   f ij rij )
2V i  V j
•
Virial stress:
•
With some simplifications this can be written as:
1
int


  
(
f
r
)

i
i
V i V
33
Stress calculation using the two approaches for different
strain rates
A: Virial stress B: Engineering stress
34
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
35
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.
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.
5.
6.
Applying Joule’s law in CNT under high current densities is not
appropriate.
Values of effective charge number Z* in CNTs can vary from 4.65E-3 to
15E-3 according to the temperature.
39
Original contributions
1.
2.
3.
4.
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.
5.
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
40
nanotubes and can be used for any material.
Publications
1.
2.
3.
4.
5.
6.
“Joule heating in single-walled carbon nanotubes”. Journal of
Applied Physics, Vol. 106, Issue 6, pp 63705, (2009). Selected for
simultaneous publication in the Virtual Journal of Nanoscale
Science & Technology, Vol. 20, Issue 14.
“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
43
Thank You
44