Nanoscience II: Nanoclusters in ‘vacuum’ · form a nanocluster ‘bottom-up’ A liquid or solid is a very effici However, there are crucial differences in the formation process:

Kai Nordlund, Department of Physics, University of Helsinki

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Nanoscience II: Nanoclusters in ‘vacuum’

Kai Nordlund2.11.2010

Matemaattis-luonnontieteellinen tiedekunta

Fysiikan laitos

Materiaalifysiikan osasto

Muokkaa otsikon perustyyliänapsauttamalla

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1. Nanoclusters in “vacuum” vs. in condensed states

Nanoclusters can be manufactured in two distinct kind of environments:

In ‘vacuum’ or strictly speaking in low-density gases or plasmas

In liquids or solids

In both cases atoms aggregate one or a few by time to slowly form a nanocluster ‘bottom-up’

However, there are crucial differences in the formation process:

A liquid or solid is a very efficient heat bath: formation at ambient temperature

There are no permanent cluster-surroundings interactions in a gas

Nevertheless, the final structure may be the same!

Hence it is useful to understand the properties of ‘free’ nanoclusters as a starting point for liquid/solid applications

Single ‘free’ nanoclusters are a pure prototypical nanoscience system


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2. Background: common metallic bonding

Metals in the most common elements have 8-12 bonds

A “metal bond” is not really a covalent chemical bond, but can be understood in terms of the attraction between negative free electrons and positive ions embedded in it

In this case it is energetically favourable for atoms to have many bonds

In the so called

FCC- and HCP-structures: 12 neighbours

BCC: 8 neighbours

Almost all elemental metals have one of these 3 structures

FCC structure


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2. Background: crystal structures

The FCC and HCP structures are actually quite similar from an atomic viewpoint

Both can be obtained by stacking of close-packed hard spheres=> local atomic environment similar => energy difference small

FCC structure HCP structureHCP structure Stacking of hard spheres


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2. Background: FCC surfaces

A crystal can be cut in different ways to produce different surfaces

These are denoted by their Miller indices, which is the crystal direction which is perpendicular to the surface plane

In cubic crystals Miller indices hkl are simply the vectors formed using the cube sides as the x y z axes.

Two important ones:

100 is a bit open

111 is the close-packed one

Has less missing bonds => less energy needed to form it

FCC 100 surface FCC 111 surface


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3. Manufacturing of nanoclusters

Manufacturing of nanoclusters in ‘vacuum’ usually starts from single atoms or very small molecules

These are produced in some sort of an atom/ion source

At its simplest thermal evaporation or a gas bottle

Laser evaporation

Magnetron sputtering

The atoms are then led into a gas or plasma where the condensation occurs


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3. Manufacturing of nanoclusters

In the gas the initially energetic atom are thermalized by collisions with the gas atoms

They occasionally also collide with each other, starting cluster nucleation

Important point to remember: a two-atom collision can not initiate nucleation: a three-body collision is needed

Freshman physics of energy conservation

Example: Cu in an Ar gas

[E. Kesälä, A.Kuronen and K. Nordlund (2005) ]


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3. Energy considerations

The major difference between vacuum/inert gas and liquid/atmospheric cluster growth comes is the role of energy or free energy

The cluster free energy may be written as

where Δg

is negative but γ

positive. Small particles have large surface curvature and hence large γ

During inert gas condensation there are no (or very weak) cluster- surroundings interactions

Hence cluster growth is usually energetically favourable at all sizes: Δg

dominates over γ

Another way of stating the same thing is to say that the metal-inert gas system is supersaturated in the metal vapor

In liquids and the atmosphere the system is in or close to thermodynamic equilibrium and γ

leads to a nucleation barrier

3 24 43Particle Bulk Surface BulkG G G r g r

r

Gparticle

Critical radius

Nucle ation

energy barrier

Equilibrium

Supersaturation


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3. Temperature profile

During inert gas kind of a formation the clusters are initially extremely hot

This is easy to understand: the cohesive energy of metals is of the order of 3 eV

Consider e.g. a 10-atom cluster at 300 K which takes in 1 more atom:

The cluster then gains suddenly 3 eV of energy

This is converted to kinetic energy in the cluster. Hence the cluster is heated by:

Thus during the initial stages of the formation the cluster is at least occasionally very hot

The gas/plasma acts as a heat bath cooling the cluster

Also radiative cooling may be important

3 32 2 10

3 3eV 3eV 2300 K2 B B

B Nk kE Nk T T


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3. Temperature profile, contd.

Example: temperature profile of Cu atoms forming clusters in an Ar gas at two different gas pressures

From MD simulations similar to the one just shown

[E. Kesälä, A.Kuronen and K. Nordlund Phys. Rev. B 75, 174121 (2007)]


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3. Nozzle-type cluster sources

One of the earliest varieties of cluster sources were those based on supersonic nozzles

On the left side the atoms are in a gas or plasma

This may be a pure gas or a mixture of atoms and an inert (noble) gas

Initial pressure may be several bars

This gas is then allowed to expand into a vacuum

Adiabatic expansion => gas cools strongly

Condensation occurs during cooling


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3. Nozzle-type sources

The original sources of this type produced almost no clusters at all (at most 1 cluster among a million atoms)

But with laval nozzles and improvements quite good cluster beams can be obtained

In a laval nozzle, the sides reflect atoms back allowing for more growth

At least some 10% of atoms in clusters

Middle of beam may be purely clusters

Especially well suited for noble gases like Ar

Initial gas may be at room temperature as it cools down to Ar condensation temperature on expansion

Can also well be upscaled to very high currents

[Seki, Matsuo, Takaoka, IEEE proceedings 2002]

Ar


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3. Nozzle-type sources

Thanks to the upscaling possibility this kind of cluster source has in fact been commercialized!

It turns out that Ar clusters are well suited for smoothing surfaces to an sub-1-atom layer smoothness

A device has been made which can smooth entire 300 mm Si wafers at a time

Needs to be conventionally polished in advance, though

Epion Ultra Smoother®

www.epion.com


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3. Gas-aggregation sources

Another important source type is the gas aggregation source

Vaporized atoms are introduced into a flow of cold gas

No nozzle involved


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3. Gas-aggregation sources

These kind of sources can produce quite ‘pure’ cluster beams with virtually no single atoms

Since atoms are obtained by vaporization/sputtering from a solid source the initial material can be virtually any solid

Well suited for at least metals and semiconductors

By mixing in a reactive gas in addition to the aggregation gas, also compound clusters can be made

E.g. TiN has been demonstrated [Qiang et al, Surf. Coat. Techn. 100 (1998) 27 ]

Large fraction of clusters is ionized (1/3 q=+1, 1/3 q=0, 1/3 q=-1)

Upscaling = ??


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4. Cluster sizes

A central concept in nanocluster sizes is whether they are monodisperse or polydisperse:

Monodisperse: all of some size

Polydisperse: variable size

The kind of sources just described all produce polydisperse cluster distributions

The best gas aggregation ones can give maybe 10% variation around the average size

Nozzle sources much more

Monodisperse clusters can be obtained with a mass filtering system after the source itself

Quadrupole mass spectrometers, typically ~1% resolution

Time of flight equipment: even single-atom resolution

Definition of ‘monodisperse’ a bit vague:

Cluster scientists tend to mean single-atom “true” monodispersity

Chemists often happy with resolution of a few %


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4. Cluster sizes: magic numbers

Measurements of cluster sizes tend to show an unsmooth distribution of cluster sizes

Some clusters are produced preferentially to others!

Example: Pb clusters

Related to cluster stability: the most stable clusters are less likely to break up if they are hot, and (alternatively) one more atom to it is more likely to be re-emitted if the cluster is hot

Directly comparable to nuclear physics: the most abundant elements in the universe are the ones with the stablest isotopes!

Growth conditions “slightly” different, though (stars and supernovas)


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4. Cluster sizes: magic numbers cont’d

There are actually 2 entirely different explanations to why clusters have magic numbers!

For large clusters and ones without electronic effects purely geometric ones dominate

E.g. noble metals at large sizes and noble gases at all sizes

Geometry means here two things: energy at 0 K but also entropy effects at elevated temperature: configurational entropy and vibrational modes may differ with different cluster sizes:

For small clusters of certain elements electronic effects dominate stability

E.g. alkali metals

( )config vibF E T S S


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4. Magic numbers due to geometry

The 0 K energetic geometry effects are easy to understand at least qualitatively

For instance elements with the FCC structure as the ground state tend to be in the form of perfect icosahedra (20-sided polygon with equilateral triangles as sides)

Perfect icosahedra can be formed from atoms only for certain fixed numbers of atoms:

Elements with other structures and larger clusters may obtain different magic numbers than these

6: 561 at5: 309 at0: 147 at0: 55 at2: 13 at1: 1 at


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4. Magic numbers due to geometry

This kind of behaviour has been observed experimentally.

For Xe+ clusters the magic numbers 13, 55 and 147 clearly stand out:

However, for Ar+ clusters only very weak maxima are visible:

Explanation not certain, but attributed to entropy effects by authors

[S. Prasalovich, PhD thesis, Univ. Gothenburg 2005]

[S. Prasalovich, PhD thesis, Univ. Gothenburg 2005]


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4. Magic numbers due to electronic effects

For instance in alkali metals (like sodium, Na) magic numbers are also observed:

This is widely accepted to be due to electronic effects


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The simplest way to treat the electronic effects is in the so called jellium model

In this model the atoms are treated as positive ions forming a constant positive background, the so called ‘jellium’ density

The conduction electrons are thought to move in this background density, described by a single parameter n0

The interaction of the electrons with the jellium is then calculated with e.g. density functional theory

The jellium models are useful in a wide range of systems (and

were to a large extent developed in Finland by Manninen, Nieminen, and Puska).

For alkali metals it is particularly easy to form jellium models, since each atom is easily ionized and contributes almost exactly one electron to the free electron gas


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The electronic structure of nanoclusters can be calculated in the jellium framework by considering the nanocluster as a simple background potential

First approximations e.g. spherical or harmonic wells

Then the Schrödinger equation is solved in this background almost exactly as for atoms

At its simplest the solution can be done for a single electron, leading to quantized energy levels with fixed possible occupations just like for the hydrogen atom

Also modern electronic-structure calculation methods can be used to solve the system for interacting electrons


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However, already the solution for non-interacting electrons gives peaks at numbers of electrons/atoms which agree with experiments:


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4. Magic numbers: electronic vs. geometric eff.

Another interesting question is of course where the cross-over from electronic to structural effects occurs

In Na this cross-over is believed to occur as high as around 2000 atoms!

- “reverse” measurement: dips correspond to magic numbers

[From Baletto and Ferrando,Rev. Mod. Phys. 77 (2005) 371:An excellent review on structuraleffects in nanoclusters]


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5. Structure of nanoclusters

The atomic structure of nanoclusters depends on a multitude of factors:

Bulk crystal structure

Surface energy vs. cohesive energy

Electronic effects

Entropic effects (at higher temperatures)

If a nanocluster would consist of a purely isotropic, homogeneous medium, it would always be purely spherical

Forming a surface requires energy (the surface (free) energy) and for a given volume of material the structure which has a minimum surface area is given by a sphere

- Exception: negative surface energy materials which actually want to be porous

However, all materials of course have an atomic structure, and hence at least small nanoclusters are unlikely to be spheres


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5. FCC nanocluster structures

Instead of giving an overview of different materials, we will now focus on what is probably the most studied case:

Nanoclusters made of materials which in the bulk have the FCC structure at low temperatures (0 K limit)

The equilibrium structure should be given by the balance of the following energy terms:

where r* is some effective radius giving the cluster size

We now assume the electronic effects are negligible (which at least for noble gas clusters certainly is a good approximation)

If the cluster is cut from a FCC single crystal, there should be no strain except that due to the surface (which is included in the surface energy terms)

( *) ( *) ( *) ( *) ( *)TOT cohesion surface strain electronicE r E r E r E r E r


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5. FCC nanocluster structures

Then the energy is given by:

The cohesion term is the same for the same number of atoms N

The surface term can be given as a sum over the individual surfaces over which the crystal is cut

But all cut directions which occur over the same crystal direction hkl have equivalent energies per area A

The sum can be grouped over which surface planes are present

( *) ( *) ( *)TOT cohesion surfaceE r E r E r

independent hkl

1

( *)hkl

N

surface hkl hkli

E r A


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5. Truncated octahedron

The minimum energy structure for a nanocluster of N atoms can then be found by seeking the minimum of the surface energy with respect to the different areas Ahkl :

The surface with the lowest energy tend to be the 111 and 100 surfaces (in this order)

It is possible to cut an FCC crystal only along 111 directions

But the surface-to-volume ratio becomes quite high

By cutting a FCC crystal along surfaces in both the 111 and 100 directions one arrives at the truncated octahedron shape (a.k.a. Wulff polyhedron): 8 111 and 6 100 surfaces, close to spherical: almost minimal total A

independent hkl

1min

hkl

N

hkl hkli

A


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5. Icosahedron

The (Mackay) icosahedron shape (already discussed above) can not be obtained by cutting a single FCC crystal

Instead it can be understood to be formed by first cutting 20 identical regular tetrahedral pyramids along 111 facets from an FCC crystal

These 20 pyramids can then be joint so that always one 111 surface is on the outside, and the rest on the inside

This forms a regular icosahedron

Because all surfaces are now 111 and the shape is very close to spherical, this has a low energy


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5. Icosahedron

But there is an important catch: the match of the tetrahedra is actually not perfect, there is a slight mismatch between the tetrahedra which translates into inequal atom bond lengths

In other words the cluster is strained

Moreover, when crossing from one tetrahedron to the next, the crystal structure is not preserved. Instead at each transition point there is a so called twin grain boundary

Because of this, this structure is often called also “multiply twinned icosahedron”

Thus the total energy now has four terms:

The strain term increases strongly with increasing cluster size

because the distance between the mismatch at the outer edge keeps increasing

( *) ( *) ( *) ( *) ( *)TOT cohesion surface strain grainboundaryE r E r E r E r E r


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5. Decahedron

Yet one more class of noncrystalline cluster shapes are the decahedral ones

These can be formed by combining five regular tetrahedra, leaving only 111 surfaces

However, doing this directly leads to a large surface-to- volume ratio

A solution is the Marks decahedron where some atoms are removed from the edges where the tetrahedra meet

This leads to a ‘defect’ energy at the surface raising the energy above that of the icosahedron for the smallest clusters

On the other hand the strain energy increases less with size


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5. Decahedral cluster, for real


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5. Icosahedra vs. truncated octahedra

The icosahedra have little strain and the lowest possible surface energy at the smallest cluster sizes

Hence by geometry alone one would expect that the smallest FCC clusters are icosahedral, while the larger ones are truncated octahedra

Possibly an intermediate regime of Marks decahedra

This is what is observed

Circles icosahedra

Triangles truncated octahedra

Squares decahedra

Also experimentally!

E.g. for Ar crossover at 750 atoms

[Baletto and Ferrando, Rev. Mod. Phys 77 (2005) 387]


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5. Special case: Co nanoclusters

An interesting special case is Co nanoclusters

The ground state crystal structure of Co is HCP

However, the FCC-HCP energy difference is small

It turns out that Co nanoclusters have the FCC structure as the ground state!

They can then have some of the same shapes as those for regular FCC clusters just described


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5. Very small nanoclusters

For the smallest nanoclusters, electronic effects take over at some point, and all of the previous becomes irrelevant

Example: shapes of Au nanoclusters


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5. Very small nanoclusters

Right now lots of research is going on about the structure of slightly larger Au nanoclusters in Finland:

Häkkinen and Manninen predict Au13 is still flat

Johansson and Pyykkö predict Au32 is a fullerene


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6. Melting of nanoclusters

Due to all of this, it is not surprising that nanoclusters melt at lower temperatures than the bulk material:

The surface energy can be considered to weaken the cohesion of the cluster as a whole

Surface melting known to occur below bulk melting also on macroscopic surfaces

The many possible configurations of the clusters may increase the entropy of the disordered state


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6. Melting of nanoclusters: results

Au nanoclusters, experiment and theory.

Smooth behaviour, can be understood with analytical model of interface weakening:

(C1 , C2 are undetermined constants)

Experimental results: Na clusters

Note that not even monotonous with size!

No clear correlation to magic numbers: not really understood!

[Roy L. Johnston: Atomic and Molecular Clusters. Taylor & Francis 2002]

, 1

22

, 2 ,3

( )melt melt bulk

melt bulk melt bulk

AreaT r T CVolume

CrT C Tr r

Documents

Nanoscience II: Nanoclusters in ‘vacuum’ · form a nanocluster ‘bottom-up’ A liquid or solid is a very effici However, there are crucial differences in the formation process: