1. 1 A Multi-scale Electro-Thermo-Mechanical Analysis of Single Walled Carbon Nanotubes PhD defense Tarek Ragab Electronic Packaging Laboratory CSEE department, University at Buffalo

2. 2 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.

3. 3 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.

4. 4 Quantum physics of joule heating and induced force Generally, the power generated is: Where, And the induced forces generated is:

5. 5 Monte Carlo simulations for joule heating and forces

6. 6 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

7. 7 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.

8. 8 Example

9. 9 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. Chirality of Carbon nanotubes What are Carbon Nanotubes (CNTs)

10. 10 Brillouin Zone for graphene What are Carbon Nanotubes (CNTs)

11. 11 Brillouin Zone for CNT What are Carbon Nanotubes (CNTs) Reciprocal lattice and first and second Brillouin zone for (10,10)CNT.

12. 12 Energy Dispersion relation The electrical conductance of graphite or carbon nanotubes is attributed to the 2p z electron, while the other three valence electrons play no role in the electrical transport process.

13. 13 Tight Binding Method Energy dispersion relation

14. 14 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. Energy dispersion relation of graphite Tight Binding Method

15. 15 Energy Dispersion relation for (10,10) CNT

16. 16 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. where

17. 17 Associate cell of graphite using 4 th nearest atoms

18. 18 Phonon Dispersion relation for (10,10) CNT

19. 19 Electron-phonon scattering rates Conservation of energy and momentum

20. 20 Electron-phonon scattering rates Scattering mechanisms Backward scatteringForward scattering EmissionAbsorption LA phononsLO phonons EmissionAbsorption

21. 21 Electron-phonon scattering rates Scattering Probability According to Fermi golden rule

22. 22 Electron-phonon scattering rates LA phonons LO phonons Scattering Probability According to Fermi golden rule

23. 23 I-V curve

24. 24 I-V curve

25. 25 Power generated along the length of CNT

26. 26 Temperature Effect (Heat)

27. 27 Force generated along the length of CNT

28. 28 Monte Carlo simulation results

29. 29 Monte Carlo simulation results

30. 30 Monte Carlo simulation results (force)

31. 31 Monte Carlo simulation results (force and effective charge number) 3.465E-3, 9.186E-3, 0.0127 and 0.015 for 300, 600, 900, 1200K

32. 32 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 Molecular dynamics simulation for the failure of carbon nanotubes under Uniaxial tension

33. 33 Molecular dynamics simulation for the failure of carbon nanotubes under uniaxial tension 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. Potential Engineering Stress : Forces on the end atoms are added up and divided by the area Virial stress: Stress Calculations With some simplifications this can be written as:

34. 34 Stress calculation using the two approaches for different strain rates A: Virial stress B: Engineering stress

35. 35 Displacement increment study Changing Parameters: Length: 12.3 A~1180.8A Strain rate: 1.69E+8~1.69E+11 sec -1 Displacement increment: 0.00025A~0.25A 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 Objectives:

36. 36 Effect of the displacement increment on the maximum stress in the simulated CNTs

37. 37 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

38. 38 Unraveling of the end of CNTs (10, 10) Armchair CNTs Unraveling Force= 15eV/angstroms (18, 0) Zigzag CNTs Unraveling Force= 10eV/angstroms

39. 39 Conclusions 1.Engineering stresses can underestimates the stresses in CNTs by 35% 2.A value of 1.76% of the unrestrained bond length is required for displacement increment. 3.Integral form using relaxation time approximation can only be used to find a rough estimate of the joule heating and electron induced wind forces. 4.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.Applying Joule’s law in CNT under high current densities is not appropriate. 6.Values of effective charge number Z * in CNTs can vary from 4.65E-3 to 15E-3 according to the temperature.

40. 40 Original contributions 1.A Simplification for the virial stress formula is derived to ease the calculations of virial stresses in multibody potentials. 2.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. 3.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. 4.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 nanotubes and can be used for any material.

41. 41 Publications 1.“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. 2.“A framework for stress computation in Single-walled carbon nanotubes under uniaxial tension”. Computational Materials Science, Vol. 46, Issue 4, pp 1135, (2009). 3.“A quantum mechanical formulation of electron transport induced wind forces in metallic single walled carbon nanotubes”. Carbon, Vol. 48, Issue 1, pp 47, (2010). 4.“Semi-classical transport for predicting joule heating in carbon nanotubes”. physics letters A, Vol. 374, Issue 24, pp 2475, (2009). 5.Ragab, T., Basaran, C., “The prediction of the effective charge number in single walled carbon nanotubes using Monte Carlo simulation”. Carbon. Under review. 6.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.

42. 42 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

43. 43

44. 44 Thank You