Quantum simulations of nanostructured materials: Rh nanoclusters - - PowerPoint PPT Presentation

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 FoilT EX –

New Carbon-based materials: Fullerenes

Highly symmetric and stable C60 molecule, with icosahedral symmetry, observed in 1985 by Kroto and Smalley in mass spectroscopy

• f 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, C70) opening new possibilities

• f functionalization, formation of solids and

new compounds with interesting properties

New Carbon-based materials: Fullerites

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

present a variety of structures and exotic properties, such as high- Tc superconductivity in K3C60 and similar compounds Solid C60 Polymeric RbC60

New Carbon-based materials: Nanotubes

• Produced from arc discharge under an electric field (Iijima 1991),
• r 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),

• r 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 NO2

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 NO2 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¨

• dinger equation for electrons and nuclei:

i¯ h∂Φ(r, R; t) ∂t =

• −
• I

¯ h2 2MI ∇2

• RI −
• i

¯ h2 2m∇2

• ri + V (r, R)
• Φ(r, R; t)

Born-Oppenheimer approximation (M >> m):

• − ¯

h2 2m

• i

∇2

• ri + V (r, R)
• Ψ(r|R) = E(R)Ψ(r|R)

where the potential felt by electrons is V (r, R) =

• I=J

e2 2 ZIZJ | RI − RJ| −

• i,I

ZIe2 | rI − RI| + e2 2

• i=j

1 | ri − rj|

Notation: r ≡ ( r1, . . . , rn) (electrons); R ≡ ( R1, . . . , RN) (nuclei).

Density-Functional Theory

Transforms the many-electron problem into an equivalent problem of (fictitious) non-interacting electrons, the Kohn-Sham Hamiltonian: Hφv ≡

• − ¯

h2 2m∇2

• r + VR(

r)

• φv(

r) = ǫvφv( r) The effective potential is a functional of the charge density: VR( r) = −

• I

ZIe2 | r − RI| + v[n( r)], n( r) =

• v

|φv( r)|2 (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: E ⇒ E[φ, R] = − ¯ h2 2m

• v
• φ∗

v(

r)∇2φv( r)d r +

• VR(

r)n( r)d r + + e2 2 n( r)n( r′) | r − r′| d rd r′ + Exc[n( r)] +

• I=J

e2 2 ZIZJ | RI − RJ| Kohn-Sham equations from the minimization of the energy functional: E(R) = min

φ E[φ, R],

• φ∗

i(

r)φj( r)d r = δij Hellmann-Feynman theorem holds. Forces on nuclei:

• FI = −∇

RIE(R) = −

• n(

r)∇

RIVR(

r)d r

Plane-Wave Pseudopotential method

• The

introduction

• f

pseudopotentials allows

• ne

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 φv: L = µ

• v
• | ˙

φv( r)|2d r + 1 2

• I

MI∇2

• RI − E[φ, R]

(µ = fictitious electronic mass), subject to orthonormality constraints

• n the orbitals, implemented via Lagrange multipliers Λij. The above

Lagrangian generates the following equations of motion: µ¨ φi = −δE δφi +

• ij

Λijφj MI ¨

• RI = −∇

RIE[φ, R]

(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 pseudopotentials, periodic boundary conditions in 12.3×12.78×20 ˚ A3 supercell (60 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: NO2 on perfect graphene

Concurrent binding of two NO2 molecules on perfect graphene surface is found to be possible, but it is just metastable with a low barrier.

Bound metastable state of two NO2 molecules separated by 3 C-C bonds Same as on the left, 5 C-C bonds separation

The bound C atoms turn from perfect sp2 bonding to sp3-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:

“Hole” position, Ea = −39 kcal/mol Rh-C distance 2.30 ˚ A “Bridge” position, 7.8 kcal/mol higher in energy, Rh-C distance 2.18 ˚ A

(23 kcal/mol=1 eV). The Rh atom is very mobile already at 200 K.

Rh trimer on perfect graphene

Rh may form Rh3 trimers with a perfect triangular shape (dRh−Rh = 2.46 ˚ A). On graphene, Rh3 binds with Ea = −36kcal/mol and all atoms in bridge position, forming an almost perfect isosceles triangle dRh−Rh = 2.63, 2.54 ˚ A) The Rh3 trimer is also practically free to move on the surface at room temperature. It can bind up to four CO or NO2 molecules

Rh10 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

Rh10 cluster with two NO2 molecules

. The flat structure has a stronger bond with NO2 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?

Another case: Rh20 clusters

We start from an hexagonal model for Rh20. The isolated cluster is stable in this configuration, but on the surface, it quickly transforms into a less symmetric structure (adsorption energy: Ea = −35 kcal/mol vs −11 kcal/mol for hexagonal model)

NO2 binding on cluster vs at interface

Adsorption energy far from Rh-C contact zone: Ea = −59 kcal/mol Close to the Rh-C contact zone: Ea = −56 kcal/mol Adsorption energy not very sensitive to adsorption site.

Conclusions

Using first-principles molecular dynamics we have shown that

• Perfect graphene doesn’t bind most gases,

not even NO2 (metastable bound states may exist but dissociate at low temperature)

• Rh atoms and trimers bind several molecules of NO2 and CO at

room temperature; they are very mobile and the gas molecules can remove them from the surface

• The Rh10 cluster has, in addition to three-dimensional forms that

are also quite mobile, a flat metastable form that is less mobile and more reactive, but only kinetically stable