Computational studies of Carbon Nanostructures

P. Giannozzi

Scuola Normale Superiore di Pisa and DEMOCRITOS-INFM

Seminario all’Universita di Udine, 2005/11/28

Fullerenes

Highly symmetric and stable C60 molecule,

observed in 1985 by Kroto and Smalley in

mass spectroscopy of carbon clusters jet

Produced in sizable quantities in 1990 by

Kratschmer and Huffman with arc discharge

in Helium flow

A new form of elemental Carbon with many

variants (such as C70) and possibility of

functionalization, formation of solids, new

compounds, ...

Fullerites

• Solid compounds formed by C60 with other atoms (“dopants”) present a variety

of structures and exotic properties, such as high-Tc superconductivity

Solid

C60

Polymeric

RbC60

Nanotubes

• Produced from arc discharge under an electric field (Iijima 1991), or by Chemical

Vapor Deposition in presence of a catalizer

• Formed by rolled-up graphene sheets, 1 to 30 nm diameter, micron length

• May be “single-walled” or “multiple-walled” (0.34 nm interlayer spacing)

depending on growth conditions

Geometry of Single-Wall Nanotubes (SWNT)

• Large variety of possible geometries – not taking into account defects and

imperfect nanotubes – leading to a large variety of electronic properties

Nanotube engineering

• More methods to modify nanotube properties:

chemical attack, functionalization, doping with atoms, ...

• Potential applications:

field emitters, electronic devices, gas sensors, gas storage, exceptionally strong

fibers, nanomechanics, ...

Role of Computation

Goals of computation at the atomistic level in Carbon Nanostructures:

• to understand and interpret experimental results

• as a basis for further modeling of mesoscopic and macroscopic properties

• to give directions and hints for new experiments

• to access data that would be difficult or impossible to measure

• to simulate situations that would be difficult or impossible to produce

experimentally (computer experiments)

First-principle calculations

Calculations/simulations based on first principles, i.e. on the electronic structure,

are especially accurate, and even predictive.

Density-Functional Theory (DFT) has a very favorable quality/computer time ratio

for calculation of

• ground-state electronic properties

• chemical bonding, atomistic structure

• vibrational and mechanical properties

• dielectric properties

...and much more

Density-Functional Theory

DFT is a ground-state theory, using the charge density as fundamental quantity.

Hohenberg-Kohn theorem (1964): the energy is a universal functional of the charge

density.

Energy as a functional of the density n(r):

EDFT = Ts[n(r)] +EH[n(r)] + Exc[n(r)] +

∫n(r)V (r)dr

is minimized by the ground-state charge density

Ts = kinetic energy, EH = Hartree energy, V (r) = nuclear potential

Exc = exchange-correlation energy (unknown!)

Density-Functional Theory (2)

Kohn-Sham (KS) equations for one-electron orbitals:

(HKS − εi)ψi(r) = 0, HKS = −~

2

2m∇2 + VH(r) + Vxc(r) + V (r)

are solved self-consistently and the charge density is given by

n(r) =∑

v

|ψv(r)|2

(the sum is over occupied states). Hartree and exchange-correlation potentials:

VH(r) =δEH[n(r)]

δn(r)= e2

∫n(r′)

|r − r′|dr′, Vxc(r) =

δExc[n(r)]

δn(r)

Energy and Forces in DFT

Total Energy as a function of nuclear positions {R}

Etot({R}) = EDFT ({R}) + EII({R})

where EDFT depends on {R} via the nuclear potential V = V ({R})

Hellmann-Feynman forces:

Fi = −dE

dRi

= −

∫n(r)

∂V (r)

∂Ri

dr −∂EII

∂Ri

Second derivatives (force constants) can be calculated using Density-Functional

Perturbation Theory (DFPT)

Practical DFT calculations

Among the many possible implementations of DFT, the Plane-Wave

Pseudopotential approach stands for its convenience:

• easy to implement and to check for convergence

• easy to calculate forces: first-principle Molecular Dynamics is possible

• easy to calculate vibrational properties and response functions via DFPT

Software: Quantum-ESPRESSO

The Quantum-ESPRESSO Software Distribution

Quantum-ESPRESSO stands for Quantum opEn-Source Package for Research in

Electronic Structure, Simulation, and Optimization

An initiative by DEMOCRITOS, in collaboration with CINECA Bologna, Princeton

University, MIT, and many other individuals, aimed at the development of high-

quality scientific software

• Released under a free license (GNU GPL)

• Written in Fortran 90, with a modern approach

• Efficient, Parallelized, Portable

Sensitivity of transport properties of Single-Wall Nanotubes

(SWNT) upon exposure to gases

Effect of O2 on resistivity R and thermoelectric power S:

P.G. Collins et al., Science 287, 1801 (2000)

Upon exposure to NH3 and NO2:

J. Kong et al., Science 287, 622

(2000)

Early theoretical results for O2 on nanotubes

• Weak chemisorption (Eb ∼ 0.25 eV)

• Small but sizable charge transfer, ∼ 0.1e

• Transport properties are affected by

induced change of Density of States at

Fermi energy

S.-H. Jhi et al., PRL 85, 1710 (2000)

Early theoretical results for other gases

• NO2: same picture as for O2

• NH3: weak binding, or no binding at all

Open questions:

• Is the “weak chemisorption” picture for real ?

• Does DFT (and in particular LDA) correctly describe gas adsorption on

nanotubes?

• What about alternative explanations:

binding to defects, impurities, contacts ...

Toy model: O2 on a graphene sheet

Energy as a function of C-O distance:

2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4d(C−O)

−0.12

−0.10

−0.08

−0.06

−0.04

−0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

E (

eV

)

PBE−hexLDA−hexPBE−bondLDA−bond

“Equilibrium” structure, LSDA: d(C −O) = 3.00− 3.05 A, Eb ' 0.09− 0.11 eV

“Equilibrium” structure, GGA: d(C −O) = 3.93− 4.01 A, Eb ' 0.009− 0.01 eV

Electronic states and Density of States at LSDA “equilibrium”

-4

-3

-2

-1

0

1

E (

eV)

q XΓ

Electronic band structure

−4 −3 −2 −1 0 1E (eV)

0

DOS

(arb

itrar

y un

its)

Density of States

O2 on a perfect nanotube (8,0) wall

LSDA: d(C −O) = 2.92 − 2.94 A

Eb ' 0.08 eV

GGA: d(C −O) = 3.68 − 3.70 A

Eb ' 0.004 eV

-3

-2

-1

0

1

E (

eV)

q XΓ

Band structure along the axis

−3 −2 −1 0 1E (eV)

DOS

(arb

itrar

y un

its)

Electronic Density of states

Effect of spin polarization

• Graphene

LSDA: dC−O = 2.92 − 2.94 A

Eb ' 0.08 eV

LDA: dC−O = 2.71 − 2.76 A

Eb ' 0.23 eV

• Nanotube:

LSDA: dC−O = 2.92 − 2.94 A

Eb ' 0.08 eV

LDA: dC−O = 2.70 − 2.71 A

Eb ' 0.22 eV

Spin-unpolarized electronic states

-3

-2

-1

0

1

E (

eV)

q XΓ

Results for other gases and other configurations

•

O2 on coupled 5-7 defects (Stone-Wales):

no evidence for stronger binding than on

perfect systems

• NH3: LDA yields “usual” results

GGA yields no binding, no effect on electronic states

• SO2: no evidence of binding found

chemisorbed state found at very high energy (∼ 3.5 eV)

NO2: same picture as for oxygen, binding (∼ 0.4 eV) is an artifact of spin-

unpolarized calculations; binding very weak (∼ 0.04 eV) with spin polarization

More recent experimental results

• Thermal desorption spectra:

O2 physisorbed on SWNT with Eb ∼ 0.19 eV

• FET devices with SWNT:

doping with O2 has different effect from doping with K. Suggestion: O2 binds

to metal contacts

• Photoelectron spectroscopy:

highly purified SWNT, with all impurities removed, are insensitive to O2!

Sensitivity to NO2 and SO2 is intrinsic.

Evidence for different chemisorbed species:

NO2 → NO, NO3 ; SO2 → SO3, HxSO4

“Noncovalent” functionalization?

Strong effect of DDQ (dichlorodicyanoquinone) on trasport properties of SWNT

observed. Less strong effect observed for benzene as well. “Noncovalent”

functionalization with π orbitals postulated:

O

N

Cl

CC

CC

CC

CC

Cl

N

O

...but there is no more evidence of binding in Benzene than in Hexane. The

difference between the top valence band of the Carbon nanotube and the LUMO

of DDQ is just ∼ 0.1 eV at large distance: how reliable is this number?

Summary and conclusions

• DFT results depend on exchange-correlation functionals: LSDA gives weak

(fictitious?) binding, GGA almost no binding

• No evidence for charge transfer or finite DOS at the Fermi energy induced by

O2 in perfect nanotubes

• same picture for O2 in presence of coupled 5-7 (Stone-Wales) topological defects

• same picture for NH3, SO2 (and NO2)

• evidence for “noncovalent” functionalization dubious