Quantum simulations of nanostructured materials: Rh nanoclusters supported on graphene P. Giannozzi Universit` a di Udine and CNR-IOM Democritos, Trieste, Italy Group seminars, Physics section, Udine 2012/02/9 Actual work done by Sara Furlan – Typeset by Foil T EX –
New Carbon-based materials: Fullerenes Highly symmetric and stable C 60 molecule, with icosahedral symmetry, 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 Fullerenes form a new class of cage-like materials composed by elemental Carbon (in the picture, C 70 ) opening new possibilities of functionalization, formation of solids and new compounds with interesting properties
New Carbon-based materials: 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 in K 3 C 60 and similar compounds Solid Polymeric C 60 RbC 60
New Carbon-based materials: Nanotubes • Produced from arc discharge under an electric field (Iijima 1991), or by Chemical Vapor Deposition in presence of Fe or Co catalyzer • Formed by rolled-up graphene sheets, typically 1 to 30 nm diameter, micron lenght (very high aspect, i.e. length to diameter, ratio) • 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, chemical, mechanical properties • Rather stable in air; various methods allow to modify nanotube properties: chemical attack, functionalization, doping, ...
Not-so-new carbon-based materials: Graphene • Graphene – a single layer of graphite – used to be just a good model for graphite surfaces and large-diameter Single-Walled nanotubes... • ...until the day it was produced by a simple exfoliation (“peeling”) technique (Novoselov, Geim 2004) • Graphene has 2D character, very peculiar electronic properties, high electron mobility: a serious candidate for post-Si electronics
(Potential) Applications A number of potential applications have been proposed for graphene and nanotubes, ranging from the obvious to the far-fetched: • field emitters, new electronic devices, gas sensors, gas (hydrogen) storage, exceptionally strong fibers, ... • nanocatalysis , either via functionalization, doping (controlled insertion of impurities), or acting as a support for metal nanoparticles acting as catalyzers. Advantages: – large exposed surface – ample possibility to tune properties – cheap: very little precious metal needed The computational study of a model system for nanocatalysis: Rh nanoclusters supported on graphene , is the main topic of this talk.
Goals of this work • To verify whether pure graphene can actually bind gases like NO 2 as often reported in the literature • To understand the morphology of small Rh nanoclusters (1 to ∼ 20 atom) supported on graphene • To understand if and how the reactivity with gases like NO 2 and CO changes as Rh nanoclusters are supported on graphene Much more is left to be done: more realistic models for carbon nanostructures; effects of curbature, defects, impurities; more realistic study of the catalytic activity Quantum , or first-principles , simulations: from electronic structure.
First-principles simulations Time-dependent Schr¨ odinger equation for electrons and nuclei: � � h 2 h 2 h∂ Φ( r, R ; t ) ¯ ¯ � � ∇ 2 2 m ∇ 2 i ¯ = − R I − r i + V ( r, R ) Φ( r, R ; t ) � � 2 M I ∂t I i Born-Oppenheimer approximation ( M >> m ): � � h 2 − ¯ � ∇ 2 r i + V ( r, R ) Ψ( r | R ) = E ( R )Ψ( r | R ) � 2 m i where the potential felt by electrons is e 2 Z I e 2 + e 2 1 Z I Z J � � � V ( r, R ) = − | � R I − � r I − � 2 2 | � r i − � r j | R J | | � R I | I � = J i,I i � = j r n ) (electrons); R ≡ ( � R 1 , . . . , � Notation: r ≡ ( � R N ) (nuclei). r 1 , . . . ,�
Density-Functional Theory Transforms the many-electron problem into an equivalent problem of (fictitious) non-interacting electrons, the Kohn-Sham Hamiltonian : � � h 2 − ¯ 2 m ∇ 2 Hφ v ≡ r + V R ( � r ) φ v ( � r ) = ǫ v φ v ( � r ) � The effective potential is a functional of the charge density: Z I e 2 � � r ) | 2 V R ( � r ) = − + v [ n ( � r )] , n ( � r ) = | φ v ( � r − � | � R I | v I (Hohenberg-Kohn 1964, Kohn-Sham 1965). Exact form is unknown, but simple approximate forms yielding useful results are known. DFT is in principle valid for ground-state properties only.
Density-Functional Theory II The total energy is also a functional of the charge density: h 2 − ¯ � � � φ ∗ r ) ∇ 2 φ v ( � E ⇒ E [ φ, R ] = v ( � r ) d� r + V R ( � r ) n ( � r ) d� r + 2 m v � n ( � r ) n ( � e 2 e 2 r ′ ) Z I Z J r ′ + E xc [ n ( � rd� � + d� r )] + r − � | � R I − � 2 2 r ′ | | � R J | I � = J Kohn-Sham equations from the minimization of the energy functional: � φ ∗ E ( R ) = min φ E [ φ, R ] , i ( � r ) φ j ( � r ) d� r = δ ij Hellmann-Feynman theorem holds. Forces on nuclei: � � F I = −∇ � R I E ( R ) = − n ( � r ) ∇ � R I V R ( � r ) d� r
Plane-Wave Pseudopotential method • The introduction of pseudopotentials allows one to ignore chemically inert core states and to use a plane waves basis set • Plane waves are orthogonal and easy to check for completeness ; they allow to efficiently calculate the needed Hφ products and to solve the Poisson equation using Fast Fourier Transforms (FFTs) • Supercells allow to study systems in which perfect periodicity is broken (surfaces, defecs) or absent (amorphous, liquids) • Iterative techniques like Car-Parrinello Molecular Dynamics allow to treat rather big systems with affordable computational effort
Car-Parrinello Molecular Dynamics Introduce fictitious dynamics on the electronic orbitals φ v : r + 1 � | ˙ � � r ) | 2 d� M I ∇ 2 L = µ φ v ( � R I − E [ φ, R ] � 2 v I ( µ = fictitious electronic mass), subject to orthonormality constraints on the orbitals, implemented via Lagrange multipliers Λ ij . The above Lagrangian generates the following equations of motion: φ i = − δE M I ¨ µ ¨ � � + Λ ij φ j R I = −∇ � R I E [ φ, R ] δφ i ij (nuclear motion is classical). These equations can be integrated (i.e. solved) for both electrons and nuclei using classical Molecular Dynamics algorithms. The combined electronic and nuclear dynamics keeps electrons close to the ground state.
Typical Simulation Protocol All simulations consists of several steps, following a general scheme: • Electrons are brought to the ground state (“electronic minimization”) at fixed nuclei, e.g. using “damped” dynamics • The dynamics is started, gradually increasing the nuclear temperature to the desired value (“thermalization”) • The system is left free to evolve for a few pico-seconds, with the temperature controlled by a “thermostat” (e.g. Nos´ e); minimum- energy structures can be found by slowly lowering the temperature Technical details: spin-polarized PBE exchange-correlation functional, Ultrasoft A 3 supercell (60 pseudopotentials, periodic boundary conditions in 12.3 × 12.78 × 20 ˚ graphene atoms), Γ point only ( k = 0) for Brillouin Zone sampling, plane-wave/ charge-density cutoff 25 / 250 Ry, time step 0.12 fs, µ = 400 a.u., Verlet algorithm
Test: NO 2 on perfect graphene Concurrent binding of two NO 2 molecules on perfect graphene surface is found to be possible, but it is just metastable with a low barrier. Bound metastable state of two NO 2 Same as on the left, 5 C-C bonds molecules separated by 3 C-C bonds separation The bound C atoms turn from perfect sp 2 bonding to sp 3 -like bonding. At as little as 150 K both molecules fly away
Test: Rh atom on perfect graphene Optimized geometries for a single Rh atom on perfect graphene: “Bridge” position, 7.8 kcal/mol higher “Hole” position, E a = − 39 kcal/mol in energy, Rh-C distance 2.18 ˚ Rh-C distance 2.30 ˚ A A (23 kcal/mol=1 eV). The Rh atom is very mobile already at 200 K.
Rh trimer on perfect graphene Rh may form Rh 3 trimers with a perfect triangular shape ( d Rh − Rh = 2 . 46 ˚ A). On graphene, Rh 3 binds with E a = − 36 kcal/mol and all atoms in bridge position, forming an almost perfect isosceles triangle d Rh − Rh = 2 . 63 , 2 . 54 ˚ A) The Rh 3 trimer is also practically free to move on the surface at room temperature. It can bind up to four CO or NO 2 molecules
Rh 10 cluster, stable and metastable form Let us look for clusters with a stronger bond to the surface and a with a large exposed surface. Starting from a fragment of crystal Rh: The low-energy structure is three-dimensional. The flat structure is metastable, but kinetically stable at room temperature
Rh 10 cluster with two NO 2 molecules . The flat structure has a stronger bond with NO 2 molecules than the three-dimensional one, and it is still kinetically stable
Some details on geometry molecules What is the effect of the continuous ”waving” of the graphene surface at root temperature?
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