Computational studies of Carbon Nanostructures P. Giannozzi Scuola Normale Superiore di Pisa and DEMOCRITOS-INFM Seminario all’Universit` a di Udine, 2005/11/28
Fullerenes Highly symmetric and stable C 60 molecule, observed in 1985 by Kroto and Smalley in mass spectroscopy of carbon clusters jet Produced in sizable quantities in 1990 by Kr¨ atschmer and Huffman with arc discharge in Helium flow A new form of elemental Carbon with many variants (such as C 70 ) and possibility of functionalization, formation of solids, new compounds, ...
Fullerites • Solid compounds formed by C 60 with other atoms (“dopants”) present a variety of structures and exotic properties, such as high- T c superconductivity Solid Polymeric C 60 RbC 60
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 ) : � E DF T = T s [ n ( r )] + E H [ n ( r )] + E xc [ n ( r )] + n ( r ) V ( r ) d r is minimized by the ground-state charge density T s = kinetic energy, E H = Hartree energy, V ( r ) = nuclear potential E xc = exchange-correlation energy (unknown!)
Density-Functional Theory (2) Kohn-Sham (KS) equations for one-electron orbitals: H KS = − � 2 2 m ∇ 2 + V H ( r ) + V xc ( r ) + V ( r ) ( H KS − ǫ i ) ψ i ( r ) = 0 , are solved self-consistently and the charge density is given by � | ψ v ( r ) | 2 n ( r ) = v (the sum is over occupied states). Hartree and exchange-correlation potentials: n ( r ′ ) V H ( r ) = δE H [ n ( r )] V xc ( r ) = δE xc [ n ( r )] � = e 2 | r − r ′ | d r ′ , δn ( r ) δn ( r )
Energy and Forces in DFT Total Energy as a function of nuclear positions { R } E tot ( { R } ) = E DF T ( { R } ) + E II ( { R } ) where E DF T depends on { R } via the nuclear potential V = V ( { R } ) Hellmann-Feynman forces: n ( r ) ∂V ( r ) F i = − dE d r − ∂E II � = − d R i ∂ R i ∂ R i 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 O 2 on resistivity R and thermoelectric power S : P.G. Collins et al., Science 287, 1801 (2000)
Upon exposure to NH 3 and NO 2 : J. Kong et al., Science 287, 622 (2000)
Early theoretical results for O 2 on nanotubes • Weak chemisorption (E b ∼ 0 . 25 eV) • Small but sizable charge transfer, ∼ 0 . 1 e • 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 • NO 2 : same picture as for O 2 • NH 3 : 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: O 2 on a graphene sheet Energy as a function of C-O distance: 0.12 PBE−hex 0.10 LDA−hex 0.08 PBE−bond 0.06 LDA−bond 0.04 0.02 E (eV) 0.00 −0.02 −0.04 −0.06 −0.08 −0.10 −0.12 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 d(C−O) “Equilibrium” structure, LSDA: d ( C − O ) = 3 . 00 − 3 . 05 ˚ A, E b ≃ 0 . 09 − 0 . 11 eV “Equilibrium” structure, GGA: d ( C − O ) = 3 . 93 − 4 . 01 ˚ A, E b ≃ 0 . 009 − 0 . 01 eV
Electronic states and Density of States at LSDA “equilibrium” 1 0 DOS (arbitrary units) -1 E (eV) -2 -3 -4 Γ X q 0 −3 −2 −1 0 1 −4 E (eV) Density of States Electronic band structure
O 2 on a perfect nanotube (8,0) wall LSDA: d ( C − O ) = 2 . 92 − 2 . 94 ˚ A E b ≃ 0 . 08 eV GGA: d ( C − O ) = 3 . 68 − 3 . 70 ˚ A E b ≃ 0 . 004 eV 1 0 DOS (arbitrary units) E (eV) -1 -2 -3 X Γ q −3 −2 −1 0 1 E (eV) Electronic Density of states Band structure along the axis
Effect of spin polarization • Graphene LSDA: d C − O = 2 . 92 − 2 . 94 ˚ A Spin-unpolarized electronic states E b ≃ 0 . 08 eV LDA: d C − O = 2 . 71 − 2 . 76 ˚ A 1 E b ≃ 0 . 23 eV 0 E (eV) • Nanotube: -1 LSDA: d C − O = 2 . 92 − 2 . 94 ˚ A -2 E b ≃ 0 . 08 eV LDA: d C − O = 2 . 70 − 2 . 71 ˚ A -3 X Γ E b ≃ 0 . 22 eV q
Results for other gases and other configurations O 2 on coupled 5-7 defects (Stone-Wales): • no evidence for stronger binding than on perfect systems • NH 3 : LDA yields “usual” results GGA yields no binding, no effect on electronic states • SO 2 : no evidence of binding found chemisorbed state found at very high energy ( ∼ 3 . 5 eV) NO 2 : 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: O 2 physisorbed on SWNT with E b ∼ 0 . 19 eV • FET devices with SWNT: doping with O 2 has different effect from doping with K. Suggestion: O 2 binds to metal contacts • Photoelectron spectroscopy: highly purified SWNT, with all impurities removed, are insensitive to O 2 ! Sensitivity to NO 2 and SO 2 is intrinsic. Evidence for different chemisorbed species: NO 2 → NO, NO 3 ; SO 2 → SO 3 , H x SO 4
“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: N N C C C C O C C O C C Cl Cl ...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 O 2 in perfect nanotubes • same picture for O 2 in presence of coupled 5-7 (Stone-Wales) topological defects • same picture for NH 3 , SO 2 (and NO 2 ) • evidence for “noncovalent” functionalization dubious
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