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Nanoscience II: Nanoclusters in vacuum Kai Nordlund 2.11.2010 Matemaattis-luonnontieteellinen tiedekunta Fysiikan laitos Materiaalifysiikan osasto Muokkaa 1.

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

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

nanoscience system 2 Muokkaa 2. Background: otsikoncommon perustyyliä metallic bonding Muokkaa 2.

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

1 Nanoscience II: Nanoclusters in vacuum Kai Nordlund Matemaattis-luonnontieteellinen tiedekunta Fysiikan laitos Materiaalifysiikan osasto Muokkaa 1. Nanoclusters otsikonin perustyyliä 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 2 Muokkaa 2. Background: otsikoncommon perustyyliä metallic bonding Muokkaa 2. Background: otsikoncrystal perustyyliä structures 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 FCC structure In the so called FCC- and HCP-structures: 12 neighbours BCC: 8 neighbours 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 structure Stacking of hard spheres Almost all elemental metals have one of these 3 structures 3 4

Muokkaa 2. Background: otsikonfcc perustyyliä surfaces Muokkaa 3.

small molecules 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

Two important ones: 100 is a bit open FCC 100 surface FCC 111 surface 111 is the These are produced in some sort of an atom/ion source At its simplest thermal evaporation or a gas bottle Laser

Manufacturing otsikon of perustyyliä nanoclusters In the gas the initially energetic atom are thermalized by collisions with the gas atoms They occasionally also collide with each other, starting

2 Muokkaa 2. Background: otsikonfcc perustyyliä surfaces Muokkaa 3. Manufacturing otsikon of perustyyliä nanoclusters A crystal can be cut in different ways to produce different surfaces Manufacturing of nanoclusters in vacuum usually starts from single atoms or very small molecules 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 FCC 100 surface FCC 111 surface 111 is the 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 close-packed one Has less missing bonds => less energy needed to form it 5 6 Muokkaa 3. Manufacturing otsikon of perustyyliä 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) ] 7 Muokkaa 3. Energy otsikon considerations perustyyliä 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 GParticle GBulk GSurface r gbulk 4r 3 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 G particle Hence cluster growth is usually energetically favourable at all sizes: Δg dominates over γ Nucle ation Equilibrium energy Critical radius Another way of stating the same thing is to barrier r say that the metal-inert gas system is Supersaturation 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 8

atoms forming clusters in an Ar gas at two different gas pressures From MD simulations similar to the one just shown 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.

heat bath cooling the cluster Also radiative cooling may be important 9 [E. Kesälä, A.Kuronen and K. Nordlund Phys. Rev. B 75, 174121 (2007)] 10 Muokkaa 3.

3 Muokkaa 3. Temperature otsikonprofile perustyyliä Muokkaa 3. Temperature otsikon profile, perustyyliä contd. 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 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 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: 3 3eV 3eV E NkBT T K 2 Nk 10 k 2 B 2 B 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 9 [E. Kesälä, A.Kuronen and K. Nordlund Phys. Rev. B 75, (2007)] 10 Muokkaa 3. Nozzle-type otsikon cluster perustyyliä sources Muokkaa 3. Nozzle-type otsikon sources perustyyliä 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 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 Ar 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] 11 12

Muokkaa 3. Nozzle-type otsikon sources perustyyliä Muokkaa 3.

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

Smoother www.epion.com 13 14 Muokkaa 3. Gas-aggregation otsikon perustyyliä sources Muokkaa 4.

$can be made E.g. TiN has been demonstrated [Qiang et al, Surf. Coat. Techn. 100 (1998) 27 ] Large fraction of clusters is ionized (1/3 q=+1, 1/3 q=0, 1/3 q=-1) Upscaling =?$ ? 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

? 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

4 Muokkaa 3. Nozzle-type otsikon sources perustyyliä Muokkaa 3. Gas-aggregation otsikon perustyyliä 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 Another important source type is the gas aggregation source Vaporized atoms are introduced into a flow of cold gas No nozzle involved Epion Ultra Smoother Muokkaa 3. Gas-aggregation otsikon perustyyliä sources Muokkaa 4. Cluster otsikon sizes perustyyliä 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 =?? 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 % 15 16

Muokkaa 4. Cluster otsikon sizes: magic perustyyliä numbers Muokkaa 4.

clusters and ones without electronic effects purely ge

5 Muokkaa 4. Cluster otsikon sizes: magic perustyyliä numbers Muokkaa 4. Cluster otsikon sizes: magic perustyyliä numbers cont d Measurements of cluster sizes tend to show an unsmooth distribution of cluster sizes There are actually 2 entirely different explanations to why clusters have magic numbers! Some clusters are produced preferentially to others! Example: Pb clusters 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 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) vibrational modes may differ with different cluster sizes: F ET( S S ) config For small clusters of certain elements electronic effects dominate stability E.g. alkali metals vib Muokkaa 4. Magic numbers otsikon perustyyliä due to geometry Muokkaa 4. Magic numbers otsikon perustyyliä due to geometry 1: 1 at 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: 2: 13 at 0: 55 at 0: 147 at 5: 309 at 6: 561 at This kind of behaviour has been observed experimentally. For Xe + clusters the magic numbers 13, 55 and 147 clearly stand out: [S. Prasalovich, PhD thesis, Univ. Gothenburg 2005] However, for Ar + clusters only very weak maxima are visible: Elements with other structures and larger clusters may obtain different magic numbers than these [S. Prasalovich, PhD thesis, Univ. Gothenburg 2005] Explanation not certain, but attributed to entropy effects by authors 19 20

6 Muokkaa 4. Magic numbers otsikon perustyyliä due to electronic effects Muokkaa 4. Magic numbers otsikon perustyyliä due to electronic effects For instance in alkali metals (like sodium, Na) magic numbers are also observed: 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 n 0 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). This is widely accepted to be due to electronic effects 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 Muokkaa 4. Magic numbers otsikon perustyyliä due to electronic effects Muokkaa 4. Magic numbers otsikon perustyyliä due to electronic effects The electronic structure of nanoclusters can be calculated in the jellium framework by considering the nanocluster as a simple However, already the solution for non-interacting electrons gives peaks at numbers of electrons/atoms which agree with background potential First approximations e.g. spherical or harmonic wells experiments: 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 23 24

high as around 2000 atoms! - reverse measurement: dips correspond to magic numbers The atomic structure of nanoclusters depends on a multitude of factors: Bulk crystal structure Surface energy vs.

7 Muokkaa 4. Magic numbers: otsikon perustyyliä electronic vs. geometric eff. Muokkaa 5. Structure otsikon of nanoclusters perustyyliä 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 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 [From Baletto and Ferrando, Rev. Mod. Phys. 77 (2005) 371: An excellent review on structural effects in nanoclusters] - 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 Muokkaa 5. FCC nanocluster otsikon perustyyliä structures Muokkaa 5. FCC nanocluster otsikon perustyyliä 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: Then the energy is given by: E (*) r E (*) r E (*) r TOT cohesion surface 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 E ( r*) E ( r*) E ( r*) E ( r*) E ( r*) TOT cohesion surface strain electronic 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) 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 Nindependent hkl E (*) r A surface hkl hkl ihkl

Muokkaa 5. Truncated otsikon octahedron perustyyliä Muokkaa 5.

min Nindependent hkl ihkl 1 A 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

8 Muokkaa 5. Truncated otsikon octahedron perustyyliä Muokkaa 5. Icosahedron otsikon perustyyliä 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 A hkl : min Nindependent hkl ihkl 1 A 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): and surfaces, close to spherical: almost minimal total A hkl hkl 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 Muokkaa 5. Icosahedron otsikon perustyyliä 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: E (*) r E (*) r E (*) r E (*) r E (*) r TOT cohesion surface strain grainboundary The strain term increases strongly with increasing cluster size because the distance between the mismatch at the outer edge keeps increasing Muokkaa 5. Decahedron otsikon perustyyliä 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-tovolume 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 31 32

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

9 Muokkaa 5. Decahedral otsikon cluster, perustyyliä for real Muokkaa 5. Icosahedra otsikon vs. perustyyliä 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 33 [Baletto and Ferrando, Rev. Mod. Phys 77 (2005) 387] 34 Muokkaa 5. Special otsikon case: Co perustyyliä nanoclusters Muokkaa 5. Very small otsikon nanoclusters perustyyliä 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 For the smallest nanoclusters, electronic effects take over at some point, and all of the previous becomes irrelevant Example: shapes of Au nanoclusters state! They can then have some of the same shapes as those for regular FCC clusters just described 35 36

10 Muokkaa 5. Very small otsikon nanoclusters perustyyliä Muokkaa 6. Melting otsikon of nanoclusters perustyyliä Right now lots of research is going on about the structure of slightly larger Au nanoclusters in Finland: Häkkinen and Manninen predict Au 13 is still flat Johansson and Pyykkö predict Au 32 is a fullerene 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 Muokkaa 6. Melting otsikon of nanoclusters: perustyyliäresults Au nanoclusters, experiment and theory. Smooth behaviour, can be understood with analytical model of interface weakening: Area Tmelt () r Tmelt, bulk C1 Volume 2 r C Tmelt, bulk C2 T 3 melt, bulk r r (C 1, C 2 are undetermined constants) Experimental results: Na clusters Note that not even monotonous with size! No clear correlation to magic numbers: not really understood! 2 [Roy L. Johnston: Atomic and Molecular Clusters. Taylor & Francis 2002] 39

napsauttamalla condensed states Nano-2 Nano-2 Nanoscience II: Nanoclusters in vacuum

napsauttamalla condensed states Nano-2 Nano-2 Nanoscience II: Nanoclusters in vacuum Muokkaa 1. Nanoclusters otsikonin perustyyliä vacuum vs. in condensed states Nanoscience II: Nanoclusters in vacuum Kai Nordlund 2.11.2010 Matemaattis-luonnontieteellinen tiedekunta Fysiikan laitos Materiaalifysiikan