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NSF/ITR: LARGE-SCALE QUANTUM- MECHANICAL MOLECULAR DYNAMICS SIMULATIONS C. S. Jayanthi and S.Y. Wu (Principal Investigators) Lei Liu (Post-doc) Ming Yu (Post-doc) Chris Leahy (Graduate Student) Alex Tchernatinsky (Graduate Student) Kevin

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SLIDE 1

NSF/ITR: LARGE-SCALE QUANTUM- MECHANICAL MOLECULAR DYNAMICS SIMULATIONS

  • C. S. Jayanthi and S.Y. Wu (Principal Investigators)

Lei Liu (Post-doc) Ming Yu (Post-doc) Chris Leahy (Graduate Student) Alex Tchernatinsky (Graduate Student) Kevin Driver (Undergraduate Student) University of Louisville

Work supported by: NSF (DMR-0112824) DoE/EPSCoR (DE-FG02-00ER45832)

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SLIDE 2

Motivation and Objective

  • The objective of our research is to develop transferable and reliable

semi-empirical LCAO Hamiltonians and the O(N) molecular dynamics corresponding to this Hamiltonian so that properties of complex systems with reduced symmetry may be studied.

  • Why develop another LCAO Hamiltonian ?
  • Will it have the predictive power of a first-principles calculation?
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SLIDE 3

OBJECTIVES

A GENERAL FRAMEWORK FOR SEMI-EMPIRICAL HAMILTONIAN Multi-Center Interactions Charge Redistributions SCED-LCAO LARGE-SCALE SIMULATIONS O(N)/SCED-LCAO APPLICATIONS TO CARBON-BASED NANOSTRUCTURES Self-consistency Environment- Dependent Effects

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SLIDE 4

CURRENT STATUS OF LCAO HAMILTONIANS

Two-center + Henv

**

2-center terms explicitly calculated using “pseudo-atomic”

  • rbitals

Yes Yes** Frauenheim

NOTB**

No No Kaxiras

NOTB**

No No Menon

NOTB **

No Yes NRL

OTB **

No Yes Ames Other Features Self-consistency Environment Effects **Environment-dependent effects disappear for systems with no charge redistributions => Poor bulk phase diagram for high coordinated bulk phases (FCC, etc)

!

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SLIDE 5

Hiα,jβ = H0

iα,jβ + 1

2 · [(Ni − Zi) + (Nj − Zj)] · U +1 2 ·  

k=i

Nk · VN(Rik) − Zk · VZ(Rik)   · Siα,jβ +1 2 ·  

k=j

Nk · VN(Rjk) − Zk · VZ(Rjk)   · Siα,jβ

SCED-LCAO HAMILTONIAN

! Multi-Center Interactions and Environment-Dependence included ! On-Site Electron-Electron Correlations modeled via Hubbard-like Term ! Inter-site Electron-Electron Correlations modeled via a parameterized function VN ! Electron-Ion interactions modeled via a parameterized function VZ ! Charge distribution Nk at a given site (k) depends on its environment and is determined Self-Consistently

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SLIDE 6

TOTAL ENERGY WITHIN SCED- LCAO

Etot = Eband + 1 2 ·

  • i

(Z2

i − N 2 i ) · U − 1

2 ·

  • i,k : k=i

NiNk · VN(Rik) +1 2 ·

  • i,k : k=i

ZiZk · VC(Rik) VC(R) = e2 4πε0 · 1 R = E0 R

I: Band Structure Energy (Sum over eigenenergies of occupied states) II and III: Double-counting corrections IV: Repulsive ion-ion interaction energy

Coulomb Term

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SLIDE 7

BULK PHASE DIAGRAMS FOR SILICON

0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 10.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 LCAO DFT 0.4 0.6 0.8 1 1.2 0.4 0.6 0.8 1 1.2

Phase diagrams of bulk Si

0.4 0.6 0.8 1 1.2 0.4 0.6 0.8 1 1.2 Wu Menon Kaxiras Frauenheim Wang and Ho NRL relative atomic volume b i n d i n g e n e r g y p e r a t

  • m

cdia sc bcc fcc

cdia sc fcc bcc cdia sc bcc fcc cdia sc fcc bcc cdia sc bcc fcc

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SLIDE 8

BULK PHASE DIAGRAMS FOR CARBON

0.5 1 1.5 2 −0.02 0.03 0.08 0.13 0.18 0.23 0.28 0.33 0.38

Energy per Atom (Ry)

LCAO DFT 0.5 1 1.5 2

Relative Atomic Volume

−0.02 0.08 0.18 0.28 0.38 0.5 1 1.5 2 −0.02 0.08 0.18 0.28 0.38 sc graphite cdia fcc bcc sc fcc bcc graphite cdia cdia graphite sc fcc bcc

SCED−LCAO Frauenheim Menon

graphite

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SLIDE 9

BULK PHASE DIAGRAM OF GERMANIUM

0.6 0.7 0.8 0.9 1 1.1 relative atomic volume −4.3 −4.1 −3.9 −3.7 −3.5 −3.3 binding energy per atom (eV) LCAO LDA cdia sc bcc fcc

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SLIDE 10

TEST OF SCED-LCAO-MD

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SLIDE 11

A MOLECULAR DYNAMICS STUDY OF THE LOCAL BENDING OF A SWCNT BY AN AFM TIP

The buckle region behaves as a nanoscale potential barrier sp2

sp3

Buckle

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SLIDE 12

Magnetic Moments for Carbon Structures

Magnetic Moment (emu/g) B=0.1 Tesla Benzene 0.0017 C60

  • 0.00036

Graphite

  • 0.03 ( T < 100 K)

Carbon Nanotube -0.025 Carbon Nanotori ~ 27

V.Elser and R.C. Haddon, Nature 325, 792 (1987). R.C. Haddon et al., Nature 350, 46 (1991).

  • J. Heremans, C.H. Olk, and D.T. Morelli, Phys. Rev. B49, 15122 (1994)

R.S. Ruoff et al, J. Phys. Chem. 95, 3457 (1991) A.P. Ramirez et al, Science 265, 84 (1994)

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SLIDE 13

Magnetic Responses of Carbon Tori

Ring Current - Metal (5,5)/1200

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SLIDE 14

A Simple Picture for Colossal Paramgnetism

mB

  • mB

B E

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SLIDE 15

100 200 300 400 500 Temperature (K) −0.2 −0.1 0.0 Magnetic Moment (µB)

(5,5)/1480 L=74T (7,4)/1364 L=11T (7,3)/1580;(10,0)/1600

10

−1

10 10

1

10

2

(5,5)/1500 L=75T (5,5)/1200 L=60T (7,4)/1860 L=15T (9,0)/1332 L=37T (9,0)/1296 L=36T

F

qλ pT L = =

Metal Tori

q p =

q 3 p =

Semiconducting

q 3 p ≠

}

}

}

Formed from Semiconducting NT

Magnetic Responses of all types of Carbon Nanotori

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SLIDE 16
  • Nanotori formed from Metal-I Carbon Nanotubes exhibit

Giant Paramagnetic moments at any Radius

  • Nanotori formed from Metal-II Carbon Nanotubes exhibit

Giant Paramagnetic moments at Selected Radii or "Magic Radii", as dictated by the relation

  • The enhanced magnetic moment has been explained in

terms of the interplay between the geometric structure and the ballistic motion of de-localized π-electrons in the metallic nanotube.

q)T 3 ( L =

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SLIDE 17

Vdc Bperpendicular Bparallel Multi-walled nanotube differential resistance measured in either parallel or perpendicular magnetic field

30x10

6

25 20 15 10 5 dV / dI (ohms) 8 4

  • 4

Bias ( mV ) B = 3.375 T 30x10

6

25 20 15 10 5 dV / dI (ohms) 8 4

  • 4

Bias ( mV ) B = 6.75 T 2.5 2.0 1.5 1.0 0.5 V max ( mV ) 8 6 4 2 B ( T ) B parallel B perpendicular

Splitting reflects Fermi energy shift between two field orientations Bias for maximum resistance (Vmax) shifts with increasing magnetic field.

  • S. Chakraborty, B.W. Alphenaar, and K. Tsukagoshi

Magnetic Field Orientation Dependence in MWNT Resistance

EXPERIMENTAL MOTIVATION FOR STUDYING THE ELECTRONC STRUCTURE OF A MWCNT

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SLIDE 18

Questions

  • How does the band structure evolve as additional shells

are added to the MWNT?

  • How does the Fermi Energy Change as a function of the

applied magnetic field?

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SLIDE 19

−5 −4 −3 −2 −1 1 2 3 4 5 ε (eV) (12,12) 0.6 0.7 k (π/a) −0.6 −0.3 0.3 0.6 ε (eV) 0.2 0.4 0.6 0.8 1 k (π/a)

Single Wall: (7,7) and (12,12)

(7,7) 0.2 0.4 0.6 0.8 1 k (π/a) −5 −4 −3 −2 −1 1 2 3 4 5 ε (eV)

Double Walls: (7,7)@(12,12)

0.6 0.7 k (π/a) −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 ε (eV) 0.2 0.4 0.6 0.8 1 k (π/a) −5 −4 −3 −2 −1 1 2 3 4 5 ε (eV)

Three Walls: (7,7)@.@(17,17)

0.6 0.7 k (π/a) −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 ε (eV) 0.2 0.4 0.6 0.8 1 k (π/a) −5 −4 −3 −2 −1 1 2 3 4 5 ε (eV)

Four Walls: (7,7)@..@(22,22)

0.6 0.7 k (π/a) −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 ε (eV) 0.2 0.4 0.6 0.8 1 k (π/a) −5 −4 −3 −2 −1 1 2 3 4 5 ε (eV)

Five Walls: (7,7)@...@(27,27)

0.6 0.7 k (π/a) −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 ε (eV)

0.2 0.4 0.6 0.8 1 k (π/a) −5 −4 −3 −2 −1 1 2 3 4 5 ε (eV)

Six Walls: (7,7)@...@(32,32)

0.6 0.7 k (π/a) −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 ε (eV)

0.2 0.4 0.6 0.8 1 k (π/a) −5 −4 −3 −2 −1 1 2 3 4 5 ε (eV)

Seven Walls: (7,7)@...@(37,37)

0.6 0.7 k (π/a) −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 ε (eV)

0.2 0.4 0.6 0.8 1 k (π/a) −5 −4 −3 −2 −1 1 2 3 4 5 ε (eV)

Eight Walls: (7,7)@...@(42,42)

0.6 0.7 k (π/a) −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 ε (eV)

ELECTRONIC STRUCTURE OF MULTI-WALL CARBON NANOTUBES

(7,7)@(12,12)@(17,17)@(22,22) @(27,27)@(32,32)@(37,37)@(42, 42)

(1) (2) (3) (4) (5) (6) (7) (8)

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SLIDE 20

50 100 150 200 H (T) 40 41 42 (EHOMO+ELUMO)/2 23 24 25 26 EF (meV)

Double Walls: (7,7)@(12,12)

Parallel Vertical 50 100 150 200 H (T) 76 77 78 ELUMO (meV) 2.0 4.0 6.0 8.0 EHOMO (meV) 20 40 60 80 H (T) 34 36 38 40 42 (EHOMO+ELUMO)/2 34 35 36 EF (meV)

Three Walls: (7,7)@.@(17,17)

20 40 60 80 H (T) 55.6 55.8 56.0 ELUMO (meV) 10 15 20 25 30 EHOMO (meV) 20 40 60 80 H (T) 54.8 54.9 55.0 55.1 (EHOMO+ELUMO)/2 34 35 36 37 38 EF (meV)

Four Walls: (7,7)@..@(22,22)

20 40 60 80 H (T) 57 58 ELUMO (meV) 52 53 54 EHOMO (meV) 20 40 60 80 H (T) 52 53 54 55 (EHOMO+ELUMO)/2 39 40 41 42 EF (meV)

Five Walls: (7,7)@...@(27,27)

20 40 60 80 H (T) 55 56 57 58 ELUMO (meV) 49 50 51 52 EHOMO (meV) 20 40 60 80 H (T) 54 55 56 (EHOMO+ELUMO)/2 40 41 42 43 44 45 EF (meV)

Six Walls: (7,7)@...@(32,32)

20 40 60 80 H (T) 55 56 57 58 59 ELUMO (meV) 53 54 EHOMO (meV) 10 20 30 40 H (T) 52 53 54 (EHOMO+ELUMO)/2 41 42 43 EF (meV)

Seven Walls: (7,7)@...@(37,37)

10 20 30 40 H (T) 53 54 55 56 ELUMO (meV) 50 51 52 EHOMO (meV)

Magnetic Field-Dependence of Fermi Energy

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SLIDE 21

PROJECTS COMPLETED

◆ The framework of our SCED-LCAO Hamiltonian is complete and the parameterized semi-empirical Hamiltonian for Si, Ge, and C are available for applications. ◆ Codes for total energy calculations and molecular dynamics, including the O(N) scheme for large-scale simulations are complete. ◆ Bulk phases of Silicon, Carbon, and Germanium have been tested against first- principles calculations and compared with other LCAO Hamiltonians ◆ SCED-LCAO-MD has been applied to study the surface reconstruction of Si(100) and the stable configurations of ad-dimers of silicon on Si(100). ◆ In tandem with the methodology development of SCED-LCAO, studies on the electronic, magnetic, and the mechanical properties of carbon nanostructures have been initiated. Specifically, the projects studied include: * The electronic structure of multi-wall carbon nanotubes * Colossal paramagnetism in metallic carbon nanotori * A molecular dynamics study of the various stages of bending including the cutting of a nanotube by an AFM tip

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SLIDE 22

Significance of our Findings and their Scientific Impact

  • The framework of our SCED-LCAO Hamiltonian has a transparent physical
  • foundation. Our case studies have demonstrated that the inclusion of environment-

dependent and charge redistribution effects in a flexible manner allows the Hamiltonian to predict the properties of complex systems with low-symmetries.

  • Our finding on the “giant paramagnetic moment of metallic carbon nanotori” may

lead to the development of nano-scale ultra-sensitive magnetic sensors.

  • Our simulation study on the deformation of SWCNTs has clarified the nature of

nanoscale potential barriers created by the buckle.

  • Our study on the electronic structure of MWCNTs is expected to provide an

understanding of the quantum transport in these systems.

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SLIDE 23

OUTREACH EFFORTS

◆ Collaborations with the Experimental Group of Dr.

  • J. Liu (Duke University) on the “ magnetic properties
  • f carbon nanotori”

◆ Collaborations with Dr. Alphenaar (EE Department , University of Louisville) on the “Electronic and Transport properties of Multi-wall carbon nanotubes” ◆ Collaborations with Dr. Dai at Stanford University on the “Manipulation of SWCNTs by an AFM tip”