Post on 16-Mar-2022
transcript
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)
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