# Presentation: 205911

## M. Meyyappan

- Modeling and Design of
- Carbon Nanotube Materials
- and Devices

- NASA Ames Research Center

- Contributors: Deepak Srivastava, M. Anantram

## Simulation Techniques

**Memory Architecture Computer**

** ****Tersoff-Brenner reactive many-body potential**** for hydrocarbons**
**   ****with long range LJ(6-12) Van der Walls interactions**
** ****Parallel implementation on a shared memory Origin2000**

** ****Quantum Molecular Dynamics Simulations**

** ****Tight-binding MD in a non-orthogonal atomic basis**
** ****Previous parametrization: silicon and carbon ****(M. Menon and K. R**
**   ****Subbaswami, Phys. Rev. B 1993-94.**
** ****Extended to heteroatomic systems including C, B, N, H**

**Finite element for homogeneous, **
**Continuum description**

**Mesoscopic dynamics for non-homogeneous**

**Atomistic MD, many-body force fields**

**Semi-emirical, tight-binding MD**

**ab-initio, structure, energetics**

**~ 100 atoms**

**~ 1000 atoms**

**~ **1000,000 atoms

- ~ 1000,000,000 atoms or grid

- ~ bulk continuous media

- Molecular Dynamics

- ~ up to 100s of ns

- Experiments

- ~ up to sec, hours

- Long time structural

- KMC, TDMC

- Hyperdynamics

- Spatio-Temporal Resolution

## Modulus of Carbon Nanotubes

- •	It is a measure of the strength and stiffness against small axial 		stretching and compression strains as well as nonaxial bending 		and torsion strains
- •	Observed properties of CNTs come from the strength of in-plane 	C-C bonds in graphene sheet and facile out-of-plane deformation 	of the structure
- •	For large diameter of small curvature CNT, the modulus, strength 	and stiffness should be comparable with the in-plane modulus & 	strength of graphene sheet
- •	In tubular geometry, elastic strain energies are also affected by the 	instrinsic curvature of the surface.  The elastic energy of a SWNT 	scales as 1/(Radius)2

## Modulus of CNTs (Continued)

- •	For axial strains, the Young’s modulus of SWNT is
- 	where E is strain energy, V is the nanotube volume
- •	Bending stiffness is
- 	where L is the length, C is the curvature of the bent nanotube
- 	(= /L where  is bending angle)
- •	Torsion stiffness is
- 	here  is the torsion angle; shear strain  = R/L

##  Nanotubes are extremely strong highly elastic nanofibers 

-  Nanotubes are extremely strong highly elastic nanofibers

**~ High value of Young’s Modulus    (1.2 -1.3 T Pa for SWNTs)**

**~ Elastic limit up to 10-15% strain**

-  Dynamic response under axial compression, bending torsion

* **redistribution of strain*
* **sharp buckling leading to bond rupture*
* **SWNT is stiffer than MWNT*

- Nanomechanics of Nanomaterials

## Nanomechanics of Nanomaterials

## Nanomechanics of Nanotubes -

- Classical MD (II)

## •	Experiment: buckling and collapse of nanotubes 	embedded in polymer composites.

- •	Experiment: buckling and collapse of nanotubes 	embedded in polymer composites.

- Buckle, bend and
- loops of thick
- tubes..

- Local collapse or
- fracture of thin tubes.

- Nanotubes in Composites

## Stiffness and Plasticity of Compressed

- C Nanotubes

## Plastic Collapse of an (8,0) Carbon Nanotube

**D. Srivastava, M. Menon and K. Cho, Phys. Rev. Lett. (1999)**

##  Tube plastically collapses at the location of the defect

-  Tube plastically collapses at the location of the defect
-  New types of heterojunctions can be created
-  Quantum dot effect in one dimensional system

- Plastic Collapse by Design

## Carbon-based Electronics

## Nanotube Electronics (Basics)

## Trends in Device Miniaturization

- Molecular devices

- Smaller and Faster Devices

- Conventional

- New

- Hybrid

- - Ultra Small MOSFET
- - Interference based devices
- - Resonant tunneling diodes
- - Single electron transistors

- - Carbon nanotube
- - Molecular diodes
- - DNA?

## Electrons are waves. de Broglie wavelength of an electron is,
                                h/p, 
     where p is the momentum
Device dimensions are much larger than the electron wave length
Transit time through the device is much larger than the scattering time
Diffusion equation for semiconductors

**Electrons are waves. de Broglie wavelength of an electron is,**
**                                ****h/p****, **
**     ****where p is the momentum**
**Device dimensions are much larger than the electron wave length**
**Transit time through the device is much larger than the scattering time**
**Diffusion equation for semiconductors**

- Diffusive

- Ballistic         Phase-coherent

- Conventional Methods of
- Device Modeling

## Electrons behave as waves rather than particles
Schrodinger’s wave equation
Poisson equation still important
Landauer-Buttiker Scattering theory
In this theory,
              Current,

           T(E) –     Transmission probability for an electron to traverse       
                           the device at energy E
           fLEFT(E) – occupancy factor / probability for an electron to be
                           incident from the left contact (Fermi- dirac factor)

**Schrodinger’s wave equation**
**Poisson equation still important**
**Landauer-Buttiker Scattering theory**
**In this theory,**
**              ****Current,**
**           ****T(E) –     Transmission probability for an electron to traverse       **
**                           ****the device at ***energy E*
**           ****f**LEFT**(E) – occupancy factor / probability for an electron to be**
**                           ****incident from the left contact (Fermi- dirac factor)**

## Quantum chemistry tools (perform energy minimization) or Molecular Dynamics (MD) are used to find chemically and mechanically stable / preferable structures
Schrodinger equation describes electron flow through the device
Poisson’s equation gives the self-consistent potential profile

- Schrodinger equation describes electron flow through the device
- Poisson’s equation gives the self-consistent potential profile

- Chemically & Mechanically Stable Structures
- Energy Minimization, quantum chemistry (hundred atoms)
- Molecular Dynamics simulations (millions of atoms)
- Number of atoms   accuracy

- Current, Electron Density
- Schrodinger’s equation / non equilibrium Green’s function

- Potential (Voltage) Profile
- Poisson’s equation

- New Devices

- Transport in Molecular Structures -
- Interplay between Chemistry and Physics

## Resonant Tunneling Diode

## Example: Resonant Tunneling Diode

- Current

- Voltage

-   Negative differential resistance
-   Peak to valley ratio should be large

## Applying of Bloch’s theorem:

For graphene, symmetry dictates that t1=t2=t3

**For graphene, symmetry dictates that **t1=t2=t3

- Blue box – unit cell of graphene
- a1 & a2 – lattice vectors
- r1, r2 & r3 - bond vectors
- Two atoms per unit cell

- Graphene

- Graphene

## Graphene

**a****2**

**a****1**

**a****2**

**a****1**

## eik=eik(+2)

****** ****= e****ik****x****x+ik****y****y  ****(u v)**

- Example, (6,0) zigzag tube,

- Graphene to Nanotube

## Nanotube Wavefunction

- p - integer

## 

Metallic nanotubes:
        n-m = 3*integer

Semiconducting tubes:    
     Bandgap  1/Diameter

Armchair tubes are truly metallic

Other metallic tubes have a tiny curvature induced bandgap

**Metallic nanotubes:**
**        ****n-m = 3*integer**
**Semiconducting tubes:    **
**     ****Bandgap ******** 1/Diameter**
**Armchair tubes are truly metallic**
**Other metallic tubes have a tiny curvature induced bandgap**

- Summary of Main Electronic Properties

## Summary of Main Electronic Properties

- Armchair tubes do not develop a band gap

## Shapes in Nature

- Nanohorns
- Torus

## Close to E=0, only two sub-bands, 		(6.5 k)
At higher energies, 		                            	(< 1k) 
Low bias record (multi-wall nanotube)           	(500) 

**Close to E=0, only two sub-bands, ****	****	****(6.5 k********)**
**At higher energies, ****	****	****                            ****	****(< 1k********)**** **
**Low bias record (multi-wall nanotube)           ****	****(500********)**** **

- Can subbands at the higher energies be accessed to drive large currents through these molecular wires?

- Armchair Nanotube: Bands

**Notes:**
Figure below is a pictorial representation of the MIT 25nm MOSFET

## Frank et. al, Science 280 (1998)

- Quantum Conductance Experiment

## E ~ ±120meV, non-crossing bands open 
At E~2eV, electrons are injected into about 80 subbands 

Yet the conductance is ~  5 e2/h

-   VAPPLIED < 200mV, G~2e2/h
-  VAPPLIED > 200mV, slow increase

**E ~ ±120meV, non-crossing bands open**** **
**At E~2eV, electrons are injected into about 80 subbands**** **
**Yet the conductance is ~  5 e****2****/h**

- Quantum Conductance Experiment (cont.)

## The strength of the two processes are determined by:
      Tunneling distance, Barrier height (ENC), Scattering

ENC  1/Diameter. So the importance of Zener tunneling increases with increase in nanotube diameter.

**The strength of the two processes are determined by:**
**      ****Tunneling distance, Barrier height (********E****NC****), ****Scattering**
******E****NC**** ******** 1/Diameter. So the importance of Zener tunneling increases with increase in nanotube diameter.**

- Semiclassical Picture

**Notes:**
(r,r’) are indices for the matrix equation
These equations are solved only inside the green box.
Yet Sch Eqn or the above equations are not being solved for a closed system. The open boundaries in the contacts are included via a self-energy.

## Semiclassical Picture (cont.)

- The differential conductance is not comparable to the increase in the number of subbands. (20,20) nanotube – 35 subbands at 3.5eV
- Two classes of experiments with order of magnitude current that differs by a factor of 5!
- Our ballistic calculations agree with increase in conductance

**Notes:**
The first feature is directly a consequence of how the CB bends in poly silicon.

## Nanotubes are the best nanowires, at present!
However, Bragg reflection limits the current carrying capacity of nanotubes

Large diameter nanotubes exhibit Zener tunneling

Conductance much larger than 4e2/h is difficult

**However, Bragg reflection limits the current carrying capacity of nanotubes**
**Large diameter nanotubes exhibit Zener tunneling**
**Conductance much larger than 4e****2****/h is difficult**

- Phys. Rev. B _62_, 4837 (2000)

- Summary
- (Current Carrying Capacity of Nanotubes)

**Notes:**
First term of n(x) is the electron density in the case of 3D bulk and is equal to the doping density. 
First term of V(x) says that the potential at x=0 increases with kf / doping
These expressions are valid only at ZERO temp and are a poor approximation to what happens at ROOM TEMP and these doping densities.
Nevertheless they are useful.

## Band Structure

## 2-Point Nanotube Heterojunctions

## Slide 37

## T-Shaped Nanotubes

## Two Dimensional Molecular "Networks"

## Nanoelectronics with Doping

## Nanoelectronics with Doping

## Multi-wall Y-Junction Carbon Nanotubes

**Experimental Synthesis of  Multi-wall Carbon Nanotube **
**Y-Junctions (2000)**

## Transport in Y-Junction Carbon Nanotubes

**Rectification and Ballistic Quantum**
**Transport in Carbon Nanotube Y junction**

**I**

**I**

**V  =  0**
**V  =  0**

**2**

**3**

**1**

**V   =   0**

**3**

**1**

**A Andriotis, M. Menon, D. Srivastava and, L. Chernozatonski, submitted, Phys. Rev. Lett. (2001)**

## Rectification in Y-junction Nanotube has a Strong

**A Andriotis, M. Menon, D. Srivastava and L. Chernozatonski, submitted, Appl. Phys. Lett. (2001)**

- Dependence on the Strunctural Symmetry

## Nanotube Devices

## Mechanical deformation alter the electronic deformation of nanotubes : effect is chirality dependent

**Mechanical deformation alter the electronic deformation of nanotubes : effect is chirality dependent**

- Nano Electromechanical Effects (NEMS)

## CNT Conductance Decreases with Deformation

- Tombler et al, Nature _405_, 769 (2000)

- Stretching of bonds
- Opens bandgap in most nanotubes
- [Phys. Rev. B, vol. 60 (1999)]

- What is the conductance decrease due to?

- Tombler et. al, Nature 405, 769 (2000) showed that the
- conductance of SWNT decreases when deformed by AFM tip.

## Model to Explain the Experimental Observation

-                                     Structure Relaxation
- Central 150 atoms were relaxed using DFT and the remaining 2000+ atoms were relaxed using a universal force field

- Density of states and conductance were computed using four orbital tight-binding method with various parametrizations

- 1) AFM Deformation

- 2) Bending

## Bond Length Distribution & Conductance

## AFM Deformed versus Stretched

## Metallic zigzag nanotubes develop largest bandgap with tensile strain.

All other chiralities develop bandgap that varies with chirality (n,m).

Experiments on a sample of metallic tubes will show varying decrease in conductance.

Some semiconducting tubes will show an increase in conductance upon crushing with an AFM tip.

**All other chiralities develop bandgap that varies with chirality (n,m).**
**Experiments on a sample of metallic tubes will show varying decrease in conductance.**
**Some semiconducting tubes will show an increase in conductance upon crushing with an AFM tip.**

- What Happens to Other Chiralities?

## Metallic nanotubes develop a bandgap upon strain. Detalied simulations show that this is a plausible explanation for the  recent experiment on electromechanical properties by Tombler et al, Nature (2000)

In contrast, we expect nanotube lying on a table to behave differently. A drastic decrease in conductance is expected to occur only after sp3 type hybridization occurs between the top and bottom of the nanotube.

**In contrast, we expect nanotube lying on a table to behave differently. A drastic decrease in conductance is expected to occur only after sp****3**** type hybridization occurs between the top and bottom of the nanotube.**

- Suspended in air

- Table Experiment

- Summary (Electromechanical Switch)

## Mechano-Chemical Effects: Kinky Chemistry

**Cohesive**
** ****Energy**

**Binding**
**Energy**

## Functionalization of Nanotubes

## Mechano-Chemical Effects: Kinky Chemistry

**SEM images of MWNTs dispersed on a V-ridged Formvar substrate**

**D. Srivastava, J. D. Schall, D. W. Brenner, K. D. Ausman, M. Feng and R. Ruoff, J. Phys. Chem. Vol. 103, 4330 (1999).**

##  A. C. Dillon et. al. Nature (1997)

**   ****~ ****fractional weight of H2 – ****5-10%**
** **
** ****Y. Ye et. al. Appl. Phys. Letters (1999)**
**    ****adsorption in crystaline ropes of SWNTs – ****8%**** - at 80K and 12MPa**
** **
** ****Chen et. al. Science (1999)**
**    ****Li doped and K doped MWNTs, ****14-20%**** - ambient press and RT**
** ****C. Liu et. al., Science (1999)**
**    ****storage in SWNTs ****4.2%**** at room temperature and 10MPa**

**Experimental Observations**

- H Storage in Nanotubes

## Modeling and Simulations

-  **Q. Wang and K. Johnson et. al., J. Chem. Phys. (1999)**
- 	-	Monte Carlo methods with quantum effects
- 	-	Results are in agreement with activated carbon data
- 	-	Storage and absorption is much less as reported above by 			experiments
- 	-	Only mechanism is H2 adsorption via physisorption

- H Storage in Nanotubes - Physisorption

## H Storage in Nanotubes - Physisorption

## H Storage in Nanotubes - Physisorption

##  Chemisorption of H/Nanotube			(?)

-  Chemisorption of H/Nanotube			(?)
-  Surface Diffusion of H/Nanotube		(?)
-  Recombination H-Nanotube + H = H2	(?)
-  Catalytic Dissociation of H2			(?)
-  Physi/Chemisorption				(?)
-  Physisorption	(Investigated in Detail)	(?)

- H Storage in Nanotubes - Chemisorption
- and Hybrid Mechanisms

## H Chemisorption on Graphene Sheet

**•****	****Computed on** **a** **grid – with full dynamics of surface C atoms**
**   ****	****and H atom normal to the surface**

- Binding Energy = 1.38 eV
- Diffusion Barrier = 0.50eV

## H Chemisorption and Diffusion on

- H Chemisorbed on the outside:  Barrier = 0.83 eV

- H Chemisorbed on the inside:  Barrier = 0.33 eV

- (10,10) Carbon Nanotube

## Diffusion Rate Constant via Wave-Packet Dynamics

- and Flux-Flux Correlation Function Formalism

## H/(10,10) nanotube (outside)	Diffusion Barrier = 0.83eV

**H/Graphene Sheet****	****	****Diffusion Barrier = 0.50eV**
**(10,10) nanotube (inside)****	****Diffusion Barrier = 0.33eV**

- •	Anisotropy in the diffusion on the in and ouside of a 		nanotube
- •	Magnitude of anisotropy will be inversely 	proportional 	to the diameter of the nanotube

- H Chemisorption and Diffusion: Comparison

##     J. Han, A Globus and R. Jaffe

**    ****J. Han, A Globus and R. Jaffe**

- Molecular Machines and Laser Motor

## Molecular Machines and Laser Motor