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)

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?

OBJECTIVESA GENERAL FRAMEWORK FOR

SEMI-EMPIRICAL HAMILTONIAN

Multi-Center Interactions Charge Redistributions

SCED-LCAO

LARGE-SCALE SIMULATIONSO(N)/SCED-LCAO

APPLICATIONS TO CARBON-BASED NANOSTRUCTURES

Self-consistencyEnvironment-Dependent Effects

CURRENT STATUS OF LCAO HAMILTONIANS

Two-center + Henv**

2-center terms explicitly calculated using �pseudo-atomic� orbitals

YesYes**Frauenheim

NOTB**NoNoKaxiras

NOTB**NoNoMenon

NOTB **NoYesNRL

OTB **NoYesAmes

Other FeaturesSelf-consistencyEnvironment Effects

**Environment-dependent effects disappear for systems

with no charge redistributions => Poor bulk phase

diagram for high coordinated bulk phases (FCC, etc)

!

Hiα,jβ = H0iα,jβ +

12· [(Ni − Zi) + (Nj − Zj)] · U

+12·∑

k �=i

Nk · VN (Rik) − Zk · VZ(Rik)

· Siα,jβ

+12·∑

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

TOTAL ENERGY WITHIN SCED-LCAO

Etot = Eband +12·∑

i

(Z2i − N2

i ) · U − 12·

∑i,k : k �=i

NiNk · VN (Rik)

+12·

∑i,k : k �=i

ZiZk · VC(Rik)

VC(R) =e2

4πε0· 1R

=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

BULK PHASE DIAGRAMS FOR SILICON

0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

10.4 0.6 0.8 1 1.2

0

0.2

0.4

0.6

0.8

1

LCAODFT

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

bindin

g ene

rgy pe

r atom

cdia

sc

bccfcc

cdia

sc

fccbcc

cdia

sc

bccfcc

cdia

sc

fccbcc

cdiasc

bcc

fcc

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

Ene

rgy p

er A

tom

(Ry)

LCAODFT

0.5 1 1.5 2Relative 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

fccbcc

graphite

cdia cdia

graphite

sc

fcc

bcc

SCED−LCAO Frauenheim Menon

graphite

BULK PHASE DIAGRAM OF GERMANIUM

0.6 0.7 0.8 0.9 1 1.1relative atomic volume

−4.3

−4.1

−3.9

−3.7

−3.5

−3.3bi

ndin

g en

ergy

per

ato

m (

eV)

LCAOLDA

cdia

sc

bccfcc

TEST OF SCED-LCAO-MD

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

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)

Magnetic Responses of Carbon Tori

Ring Current - Metal

(5,5)/1200

A Simple Picture for Colossal Paramgnetism

mB

-mBB

E

0 100 200 300 400 500Temperature (K)

−0.2

−0.1

0.0

Mag

netic

Mom

ent (

µ B)

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

10−1

100

101

102

(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

FqλpTL ==

Metal Tori

qp =q3p =

Semiconducting q3p ≠

}}

}Formed fromSemiconducting NT

Magnetic Responses of all types of Carbon Nanotori

� 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)T3(L =

Vdc

Bperpendicular Bparallel

Multi-walled nanotube differential resistance measuredin either parallel or perpendicular magnetic field

30x106

25

20

15

10

5

0

dV

/ d

I (o

hm

s)

840-4

Bias ( mV )

B = 3.375 T

30x106

25

20

15

10

5

0

dV

/ d

I (o

hm

s)

840-4

Bias ( mV )

B = 6.75 T

2.5

2.0

1.5

1.0

0.5

V m

ax (

mV

)

86420

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

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?

−5

−4

−3

−2

−1

0

1

2

3

4

5

ε (eV) (12,12)

0.6 0.7 k (π/a)

−0.6

−0.3

0

0.3

0.6

ε (eV)

0 0.2 0.4 0.6 0.8 1 k (π/a)

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

(7,7)

0 0.2 0.4 0.6 0.8 1 k (π/a)

−5

−4

−3

−2

−1

0

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 0.2 0.4 0.6 0.8 1 k (π/a)

−5

−4

−3

−2

−1

0

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 0.2 0.4 0.6 0.8 1 k (π/a)

−5

−4

−3

−2

−1

0

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 0.2 0.4 0.6 0.8 1 k (π/a)

−5

−4

−3

−2

−1

0

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 0.2 0.4 0.6 0.8 1 k (π/a)

−5

−4

−3

−2

−1

0

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 0.2 0.4 0.6 0.8 1 k (π/a)

−5

−4

−3

−2

−1

0

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 0.2 0.4 0.6 0.8 1 k (π/a)

−5

−4

−3

−2

−1

0

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 STRUCTUREOF 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)

0 50 100 150 200H (T)

40

41

42

(EHO

MO+E

LUM

O)/2

23

24

25

26

E F (meV

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

ParallelVertical

0 50 100 150 200H (T)

76

77

78

E LUM

O (m

eV)

2.0

4.0

6.0

8.0

E HOM

O (m

eV)

0 20 40 60 80H (T)

3436384042

(EHO

MO+E

LUM

O)/2

34

35

36

E F (meV

)

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

0 20 40 60 80H (T)

55.6

55.8

56.0

E LUM

O (m

eV)

10

15

20

25

30

E HOM

O (m

eV)

0 20 40 60 80H (T)

54.8

54.9

55.0

55.1

(EHO

MO+E

LUM

O)/2

34

35

36

37

38

E F (meV

)

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

0 20 40 60 80H (T)

57

58

E LUM

O (m

eV)

52

53

54

E HOM

O (m

eV)

0 20 40 60 80H (T)

52

53

54

55

(EHO

MO+E

LUM

O)/2

39

40

41

42

E F (meV

)

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

0 20 40 60 80H (T)

55

56

57

58

E LUM

O (m

eV)

49

50

51

52

E HOM

O (m

eV)

0 20 40 60 80H (T)

54

55

56

(EHO

MO+E

LUM

O)/2

404142434445

E F (meV

)

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

0 20 40 60 80H (T)

55

56

57

58

59

E LUM

O (m

eV)

53

54

E HOM

O (m

eV)

0 10 20 30 40H (T)

52

53

54

(EHO

MO+E

LUM

O)/2

41

42

43

E F (meV

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

0 10 20 30 40H (T)

53

54

55

56

E LUM

O (m

eV)

50

51

52

E HOM

O (m

eV)

Magnetic Field-Dependence of Fermi Energy

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

Significance of our Findings and their Scientific Impact

� The framework of our SCED-LCAO Hamiltonian has a transparent physicalfoundation. 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.

OUTREACH EFFORTS

◆ Collaborations with the Experimental Group of Dr. J. Liu (Duke University) on the � magnetic properties of 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”