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Metal atom clusters: an exotic example of exotic ions
Nuclear Inst~ments North-Holland
and Methods in Physics Research B53 (1991) 395-403
395
Metal atom clusters: an exotic example of exotic ions X3.0. Lutz and K.H. Meiwes-Broer FakuMiifGr Physik, iJnioersi!&Bielefefcj,W-4800 Bielefeid I, Gemany
We review some of the phenomena which are encountered when producing and studying free metal atom clusters. This concerns, in particular, some aspects of the stability of “hot” clusters as well as models which can be constructed from experimental data. Furthermore. some exuerimental tecbniaues and corresoonding results are discussed which allow the electronic structure of size-selected free metal-clusters to be probed.
If a piece of metal is divided into ever smaller pieces, eventually it will lose its properties as a solid, and this will presumably happen long before it has been chopped into individual atoms. There are many interesting examples of size dependencies: the ionization potentials and electron affinities of free Al clusters; the melting temperature of surface-deposits Au particles (fig. 1); the chemical reactivity which is of great importance in heterogenous catalysis (fig. 2). An important influence of the number N of atoms in a cluster on its properties can be found when investigating the geometric structure. Quantum chemistry calculations reveal that clusters possess in general many energetically close-lying geometries (isomers). Fig. 3 shows as an example the two lowest-energy structures of Na,; they display residues of tetrahedral elements, but they do not resemble the bulk bee structure. Therefore, size dependent, there must be a restructuring of such small particles as N is changed. This is what a solid state physicist would also expect: different crystal structures such as hcp and fee are closely related in that they can be created by stacking close-packed (111) planes on top of each other simply in a different order: hcp corresponds to a sequence of 1, 2,1, 2, .... fee corresponds to 1, 2, 3, 1, 2, 3, ... Therefore, the crystal is determined by rather small differences in the (long range) atomic interaction potential, and it is easy to accept that the structure of small clusters will in general not be identical to the structure of the condensed matter. What is more, the structural isomers are related to different minima in the potential hypersurface of the cluster which may be of different depth and occupy different phase space volume; which one the cluster assumes in its creation process is a priori not certain. A cluster may even jump from one minimum to another, the energy difference between the total energy and the bottom of the minimum appearing as excitation energy. A metal cluster of finite temperature may thus have an ill-defined and time-varying geometric structure; this tendency is sup0168-583X/91/$03.50
ported by a rather non-directional type of bonding (which will be further discussed below), in contrast, for example, to molecules with strongly hybridized covalent bonds **. This “fragility” of clusters is not only manifest in their structure, but also in their stability: let us add a small amount of energy (e.g. of the order of the sublimation energy per atom, typically a few eV) to the surface of a particle (large number N of atoms); within atomic vibration times of lo-l4 to 10-r’ s this may result in the evaporation of an atom. However, it is much more probable that phonons will distribute this energy throughout the lattice, causing only a very slight temperature increase. It will then take a long time before statistical fluctuations again concentrate the appropriate energy portion in a (surface) bond to evaporate an atom. By simple statistical arguments this time decreases if N decreases. Transition state theory is able to describe this process and calculate size-dependent decay probabi~ties. It should be noted that the heat of sublimation is size-dependent, too (generally smaller for smaller N). Altogether, therefore, clusters will react quite sensitively to transfer of even small amounts of energy. This phenomenon has consequences for cluster studies. Suppose, for example, that we want to study the size dependence of a certain physical or chemical property. We then have to prepare an ensemble of clusters, ideally having identical masses, and not being disturbed by
*’ We might mention here that the distinction between clusters and molecules is, in some way, rather arbitrary; it would be difficult to rigorously justify it on the basis of purely physical arguments, e.g., the type of bonding: for example, the near-neighbor interaction in an atomic aggregate causes a splitting and shifting of energy levels which may, size-dependent, change the s-p energy distance in s/p type metal atoms; as a consequence, such a cluster (e.g. Hg,) may with increasing size experience different types of bonding, from van der Waals, to covalent, to metallic [6].
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n Fig. 2. Chemical reactivity of positively charged iron clusters with hydrogen. After ref. [4].
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Fig. 1. Ionization potentials (a) [l] and photodetachment threshold energies (electron affinities) [2] of Al clusters (b) in a molecular beam. c) Melting temperatures of gold particles deposited on a carbon film [3]. Note the different particle size scales.
secondary undesired phenomena. These requirements cause problems with such established tcchuiques as, e.g., matrix isolation or surface deposition of clusters: although it appears now to be possible to prepare such mass-defined ensembles by deposition of intense massselected cluster ion beams, the effect of cluster-surface or cluster-matrix interaction may influence and even dominate the studied properties (although these interactions may be quite interesting by themselves). Therefore, in order to isolate properties of the free clusters
themselves we should resort to the use of beams of free clusters, if possible mass selected. Such free massselected cluster beams can, for example, be prepared by surface sputtering with energetic ions, or (laser) discharge evaporation of surfaces. In such processes, a fraction of (positively or negatively) charged clusters is created in addition to the usually much more abundant production of neutral individual atoms and clusters. This charged component can be used to prepare an ion beam which by conventional ion-optical methods (magnetic separation, time-of-flight techniques) is then mass-separated. Alternatively, instead of using the charged component in the original production process, the neutral component can be post-ionized (e.g. by electron or photon impact, electron attachment, or charge exchange) and then again shaped into an ion beam. Such experiments, of course, have problems of their own: in an experiment involving a cluster beam of well-defined mass we must be certain that at least this mass is not changing during the experiment. As outlined above, though, this can happen if the clusters are not produced with low internal energy. This effect can be demonstrated for example with sputtered cluster beams. A sputter process is quite brutal; much energy will be deposited in the surface. A sputtered cluster, therefore, will possess much internal energy; it can hardly get rid of this internal energy in free. flight except by evaporation of atoms or larger fragments. In a typical 6.3
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Fig. 3. Energy optimized geometries of two neutral Na, isomers. Distances in atomic units. After ref. [5].
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experiment, designed to study this process, the cluster beam will encounter after some drift distance a repeller grid. If the particle has fragmented along the drift length, it has lost mass and therefore also energy; as a consequence, it will not be able to surmount the repeller barrier of appropriate height. The decay probability of copper clusters as determined in this experiment (fig. 4) shows a strong increase towards larger clusters *‘: The tendency of such data is nicely reproduced by a modified transition state theory; this theory treats the clusters as hot droplets which (because of their internal energy) evaporate individual atoms as they proceed downstreams in the beam [7]. A particularly stable cluster will be particularly resistent against further decay; this will also be apparent in the total mass spectrum usually detected some ns after creation of the cluster. The decay probabilities as well as the total mass spectra will thus display intensity jumps at certain mass numbers (“magic numbers”) and, for monovalent clusters as Ag, and cu,, “even-odd oscillations” (fig. 5). This phenomenon yields interesting information on the cluster. A qualitative picture is easy to obtain if we recall that we are dealing with metal clusters. An encyclopedia defines a metal as an “. . . element with good electrical conductivity at room temperature”. At first glance this would seem to imply that metal clusters are good conductors. This conclusion, however, may be severely in error. Indeed, in a system of finite size quantum mechanics introduces energy states which are clearly separated, with larger separations for smaller cluster dimension R. For R of a few A (as in atoms, molecules or
#’ This finding is not in contradiction to the statistical argument mentioned earlier since each detected cluster N has with some probabilty had a prior history of decays (“evaporative ensemble”)
small clusters), the energy spacings are of the order of eV. Since electrical conductivity requires that electrons can be lifted by an applied (weak) electric field into close-lying unoccupied states, we cannot regard a small cluster as a good conductor. If, however, we take from condensed matter physics the picture of strongly delocalized (valence) electrons, we arrive at a rather simple model [lo]: Apparently, the bonding electrons “see” only a little of the atomic cores. They behave very much like electrons in the electronic band of a solid. Therefore, in a first approximation one can smear out the positive charge of the atomic cores over the entire cluster volume, and then calculate the electronic states in the corresponding effective potential (cf. fig. 6). Evidently, the effective potential resembles a well, with 2.0
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typical dimensions of the cluster size R. Quantum mechanics now arranges distinct energy states (shells) in such a “jellium” potential. In the simplest approximation, this droplet is spherically symmetric. Just as, for example, in nuclei which display qualitatively rather similar effective single-particle potentials, particularly stable configurations are obtained if a shell is filled with the maximum allowable number of electrons (“magic numbers”). A further refinement of this model allows for the deformation of the jelhum and the corresponding effective potential due to the self-consistent interaction of the positive background with anisotropic electron states outside closed-shell configurations. This is analogous to the Nilsson model in nuclear physics; a difference, of course, lies in the fact that in nuclei the nucleons themselves create the effective potential in which they adopt quantum mechanical states, while in the cluster jellium model (due to the very different masses of electrons and ionic cores) only the electrons are treated quantum mechanically. This “deformed jellium model” of clusters [12] is able to explain (at least qualitatively) the above-mentioned magic numbers as well as the even-odd oscillations in mass spectra, fragment stabilities, photoio~~tion thresholds (see below), etc. Note that the discussion followed so far relates metal cluster stabilities to their electronic structure. This can be further verified by comparing clusters of the same number of atoms N but having different charges, i.e., different number of electrons N,. Such experiments which, e.g., compare the mass spectra of positive and negative sputtered cluster ions demonstrate that the dominant structures in the spectra are related to N, and not to N. One of these studies shows a further aspect of the analogy to nuclear physics: the confirmation of magic numbers up to very high values. In nuclear physics, much discussion has occurred around ‘*islands of stabilities” for nuclear charges 2 > 100; in clusters, enhanced stabilities up to N - 1500 have been observed and related to electronic structures, while particular stabilities for larger N can be explained by geometric structures [13] *3. In order to further study the details of cluster structure and characteristics, for example by spectroscopic techniques, the discussion so far suggests that the clusters should be cold, thereby greatly reducing their fragmentation probability. Only in such situations can one
It3 It should be noted that there exists a second kind of “magic numbers” which appear particularly with rare gas clusters. Apparently, in these cases it is the geometric structure which yields special stabilities; whenever a full shell of 11 “tennis ball-like atoms are arranged around an imer core, particular stability is encountered (an icosahedral structure, cf., e.g., ref. [14]).
be sure to prepare a certain cluster size and still have it available when doing an experiment with it. Such “cold” clusters can be produced by the wellknown technique of seeded jet-expansion. Tbe principle is simple: material of the desired kind is evaporated (if necessary, by an intense laser pulse, or an electric discharge) and flushed by a high-pressure gas pulse (for example, helium) into vacuum. The expansion occurs in hydrodynamic flow conditions and is isentropic. It results in a dramatic cooling of the jet, including the embedded particles which may even grow further to very large sizes during expansion. This process of cluster growth under high-pressure conditions can, for example, be described by percolation theory 1151. Unfortunately, cooling of supersonic jets to very low temperatures (- 1 K) occurs mainly in the translational degree of freedom. The rotational, and even more so the vibrational degrees of freedom “freeze” at somewhat higher temperatures; for clusters, these temperatures have not been measured yet, although from general principles rough estimates may be obtained. In any case, the according temperatures usually he much lower than those obtained by other techniques (e.g., sputtering, or impact ionization of neutral cluster beams). This “cooling” is also demonstrated by the mass spectra produced by a (properly operated) supersonic jet-expansion; they show a smooth envelope with little or none of that structure (cf. fig. 5) which is familiar from fragmenting “hot” clusters. Let us now come back to the question of how the electronic band structure evolves as the particle becomes bigger. This point is, of course, at the bottom of the problem of the atom-solid transition. One immediately apparent experimental procedure would be to use photon absorption in order to excite an electron from a bound to an empty level in the cluster potential well. Thereby, the dipole convolution of empty and occupied states would be directly accessible to spectroscopy, just as in atoms or molecules. Such experiments are currently performed in several laboratories. Besides sharp resonances corresponding to well-defined single particle excitations, broad absorption maxima with widths of roughly lo-20% of the excitation energy have been found for alkali clusters [16,17]. These maxima can be related to plasmon-like excitations, and apparently represent collective motions of the electrons. Such excitations could be viewed as elementary collective excitations (as e.g., surface plasmons in a solid) or as coherent coupling of single-particle excitations (as e.g., more common in nuclear physics or also in atomic physics). Evidently, at certain energies the single-particle transition strengths are overwhelmed by collective modes. In contrast to nuclear physics where collective modes generally appear in the continuum, the collective structures found so far in clusters he in the bound part of the spectrum. The quantitative explanation of the
H.O. Lutz, K.H. Meiwes-Broer
observed resonances in terms of a many-body quanturn-mechanical treatment is still lacking. Classically, the purely collective aspects are in principle simple: Mie light scattering theory treats the photon-particle interaction in terms of classical electrodynamics [HI. The resulting Mie cross section describes scattering and absorption of light by a cluster which is thought to be homogeneous of (complex) refractive index, and has found application in the explanation of such phenomena as the colour of church window glasses containing disperse metal particles. It should be mentioned that in non-spherical particles there may be different resonance frequencies, corresponding to different restoring forces on the charge cloud along different symmetry directions. Clearly, such a classical treatment is only applicable for fairly large clusters where the electronic states are so closely spaced that they appear as homogeneous bands. For smaller clusters, the quantum mechanical nature of the system introduces increasing “graininess” due to the increasing level separations with decreasing R. As a consequence, the classical Mie resonance “fragments” into ever more individual lines which finally must coincide with the single-particle transitions in the system. The (time dependent) local density formalism appears to be well suited to treat such effects since it directly obtains the (ground state) electron density and, hence, the cluster susceptibility. On the basis of the jellium model, this approach has been used by Ekardt and
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coworkers [19] to qualitatively explain the N-dependent width and behavior of the observed broad resonances in terms of collective modes in clusters of finite temperature. This still leaves the problem of the single-particle states. One important experimental tool for their direct experimental study is ultraviolet photoelectron spectroscopy (UPS) [20]. This technique has actually given very interesting results. The principle of the experiment is shown in fig. 7. The beam of negatively charged clusters is produced in a laser evaporation cluster source and cooled by jet-expansion. The charged clusters are accelerated to a known energy (a few hundred ev) and mass-selected by timeof-flight. After some drift length, a second pulsed UV laser detaches electrons from clusters of selected mass. These electrons are collected in a strong magnetic field and guided in a weak magnetic field through a drift length of a few meters to a detector; their time-of-flight can be converted into kinetic energy of the photoelectrons. The binding energy of the electrons in the cluster (Es) is then given by the difference of photon energy and electron kinetic energy. Note that such photon-induced transitions are fast compared to typical vibration frequencies. The ionization process is, therefore, “ vertical” with respect to the cluster geometry. On the other hand it is probably slow (“adiabatic”) with respect to the electron motion (the energies are not far from elecfrondetector
, detector
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Fig. 7. Schematic experimental setup for UPS measurements on mass-selected cluster anions (from ref. [2]).
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threshold!). The energies E,, therefore, correspond to energy differences between the anion and the neutral cluster in the anion geometry. Fig. 8 shows examples of
photoelectron spectra for a sequence of selected clusters. Clearly, the evolution of the electronic structure of such small aggregates is not continuous, rather it hap-
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pens in distinct steps. Addition of one atom may create considerably different electron spectra (and therefore also reflects different electronic structure).
These spectra contain much information and cannot be discussed here in detail. Only a few aspects will be highlighted. First of all we may point out that the
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Fig. Sb. UPS spectra of various metal clusters for N = 13 to 22 (from ref. [21]).
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electron band structure of the solid material is in general quite different from that of the clusters. The monovalent metals Cu and Ag may serve as an example: their bulk UPS spectra feature a quite flat contribution of the s/p-band close to threshold (i.e. around the Fermi edge), in contrast to the strong (and in both elements quite similar!) structure (caused by the quantum mechanical “graininess” of the delocalized electron states) in the corresponding cluster UPS spectra. This difference between bulk and cluster UPS spectra is less evident in the transition metals due to the high m~tip~city and the more localized nature of the d states near the Fermi level 1211(cf. also fig. 8). Furthermore, the ionization thresholds are expected to show an R-dependence. Of course, we have to take into account that the image charge contribution (which is contained in the work function of a flat surface) must be corrected according to the shape of the finite cluster. For a spherical cluster, one obtains a l/R-dependence if a completely delocalized electron cloud is assumed. Deviations from this l/R-scaling reflect the quantum structure of the cluster; they are particularly pronounced in small clusters as well as in those which have rather localized outer electrons (as, e.g., the transition metals) (211. One further feature may be noted: Whenever a “shell” of delocalized electrons (e.g. in the jellium model) is closed and a spherical configuration of high stability is reached, addition of one electron yields particularly low ionization energy (since this electron has to be filled into a new “shell” of low binding energy). This behaviour can be directly seen between, e.g., Ag; and Ag,, or Ag,g and Ag;c (cf. fig. 8). Ag;, for example, contains altogether eight valence electrons, and produces therefore a closed ‘“shell” of electrons having p character. Addition of one more electron leads to occupation of the next higher state with low-binding energy, and a new peak at low E, appears in the UPS spectrum (similarly for Ag,). In chemical language (and in the picture of vertical transitions), the energy difference between this peak and the next one at higher binding energy could be called the “ homo-lumo” gap. Of course, only closed shells exhibit spherical symmetry. Deviations from spherical symmetry break the degeneracies of angular momentum states and cause a splitting of energy levels. This phenomenon and the corresponding general symmetry rules allow us at least qualitatively to explain some of the structures found in UPS spectra from open-shell clusters. Summary: Expe~mental technique in cluster research has rapidly advanced in recent years. Mass-separated cluster beams of sufficiently high intensity are now available to perform spectroscopic experiments which directly reflect the size evolution of the band structure in the particles. Mass separation so far is generally obtained by ion-optical mass separation (time-
of-flight, magnetic separation, etc.). No satisfactory method has been established yet to produce some variety of neutral clusters with defined mass (except an interesting technique relying on the kinematics of cluster-gas atom scattering [22]), although charge changing collisions or photodetachment from negative clusters showed very promising results recently. In addition, a better knowledge of the actual vibrational temperature of jetprepared clusters would be highly desirable. At the moment, it is not clear if these clusters are “crystalline” or “liquid” or in some complicated dynamical state. Also on the theoretical side much progress has been made. Ab initio quantum chemical methods, of course, can give a very detailed insight into the structure and dynamics of clusters; in practice, however, such methods become extremely complicated even for simple (monovalent) clusters with more than, say, ten atoms. A complete treatment of clusters of more complicated elements (e.g., the transition metals) is practically out of reach of standard quantum chemical methods at present. Many more or less sophisticated models and approximations have, therefore, been applied. At the heart of the problem, of course, lies the a priori unknown cluster geometry (cf. the isomer problem mentioned earlier), which requires a high accuracy treatment. This will be particularly evident for clusters bound by more localized electrons (e.g., the transition metals), as well as in forthcoming experiments in which the band structure will be probed ever more deeply by higher photon energies. Since cluster science has raised interest in many other areas of research (chemistry, solid state physics, nuclear physics, atomic and molecular physics, etc.) one may hope, though, that we may witness many interesting developments in the near future.
This work has been supported by the Bundesminister fiir Forschung und Technologie (BMFT) and the Deutsche Forschungsgemeinschaft (DFG).
References [l] K. S&river, J.L. Persson, E.C. Honea and R.L. Whetten, Phys. Rev. Lett. 64 (1990) 2539. [2] G. Gantefijr, M. Gausa, K.H. Meiwes-Broer and H.O. Lutz, 2. Phys. D9 (1988) 253. [3] Ph. Buffat and J.-P. Borel, Phys. Rev. Al3 (1976) 2287. 141 A. Kaidor and D.M. Cox, J. Chem. Sot. Faraday Trans. 86 (1990) 2459. [5] I. Moullet, J.L. Martins, F. Reuse and J. Buttet, Phys. Rev. Lett. 65 (1990) 476. [6] H. Haberland, H. Kommeier, H. Langosch, M. Oschwald and G. Tanner, J. Chem. Sot. Faraday Trans. 86 (3990) 2473;
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[7] [8] [9]
[lo]
[ll] [12] [13] [14]
K. Rademann, B. Kaiser, U. Even and F. Hensel, Phys. Rev. Lett. 59 (1987) 2319. C.E. KIots, J. Chem. Phys. 83 (1985) 5854; C.E. Klots, Z. Phys. D5 (1987) 83. W. Begemann, K.H. Meiwes-Broer and H.O. Lutz, Phys. Rev. Lett. 56 (1986) 2248. W. Begemann, S. Dreihofer, K.H. Meiwes-Broer and H.O. Lutz, in; Physics and Chemistry of Small Clusters, eds. P. Jena, B.K. Rao and S.N. Khanna, NATO AS1 Ser. B158 (1987) 269. a) J.L. Martins, R. Car and J. Buttet, Surf. Sci. 106 (1981) 265. b) W. Ekardt, Phys. Rev. B29 (1984) 1558. c) W.D. Knight, K. Clemenger, W.A. de Heer, W. Saunders, M.Y. Chou and M.L. Cohen, Phys. Rev. Lett. 52 (1984) 2141. W.-D. Schone, thesis, University Bielefeld, (1990). K. Clemenger, Phys. Rev. B32 (1985) 1359. T.P. Martin, T. Bergmann, H. GGhlich and T. Lange, Phys. Rev. Lett. 65 (1990) 748. 0. Echt, D. Kandler, T. Leisner, W. Miehle, E. Recknagel, J. Chem. Sot. Faraday Trans. 86 (1990) 2411.
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[15] 0. Knospe, R. Schmidt and G. Seifert, Phys. Lett. Al29 (1988) 236. [16] K. Selby, M. Vollmer, J. Masui, V. Kresin, W.A. de Heer and W.D. Knight, Phys. Rev. B40 (1989) 5417. [17] C. Brechnignac, Ph. Cahuzac, F. Carlier and J. Leygnier, Chem. Phys. Lett. 164 (1989) 433. [18] G. Mie, Ann. Phys. 25 (1908) 377. [19] W. Ekardt, Phys. Rev. B31 (1985) 6360; Z. Penzar, W. Ekardt and A. Rubio, to be published (1991). [20] a) D.G. Leopold, J.H. Ho and W.C. Lineberger, J. Chem. Phys. 86 (1987) 1715. b) C.L. Pettiette, S.H. Yang, M.J. Craycraft, J. Conceicao, R.T. Laaksonen, 0. Cheshnovsky and R.E. Smalley, J. Chem. Phys. 88 (1988) 5377. c) G. Gantefiir, K.H. Meiwes-Broer and H.O. Lutz, Phys. Rev. A37 (1988) 2716. [21] G. Gantefiir, M. Gauss, K.H. Meiwes-Broer and H.O. Lutz, Faraday Discuss. Chem. Sot. 86 (1988) 197. [22] U. Buck, J. Phys. Chem. 92 (1988) 1023.
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Metal atom clusters: an exotic example of exotic ions X3.0. Lutz and K.H. Meiwes-Broer FakuMiifGr Physik, iJnioersi!&Bielefefcj,W-4800 Bielefeid I, Gemany
We review some of the phenomena which are encountered when producing and studying free metal atom clusters. This concerns, in particular, some aspects of the stability of “hot” clusters as well as models which can be constructed from experimental data. Furthermore. some exuerimental tecbniaues and corresoonding results are discussed which allow the electronic structure of size-selected free metal-clusters to be probed.
If a piece of metal is divided into ever smaller pieces, eventually it will lose its properties as a solid, and this will presumably happen long before it has been chopped into individual atoms. There are many interesting examples of size dependencies: the ionization potentials and electron affinities of free Al clusters; the melting temperature of surface-deposits Au particles (fig. 1); the chemical reactivity which is of great importance in heterogenous catalysis (fig. 2). An important influence of the number N of atoms in a cluster on its properties can be found when investigating the geometric structure. Quantum chemistry calculations reveal that clusters possess in general many energetically close-lying geometries (isomers). Fig. 3 shows as an example the two lowest-energy structures of Na,; they display residues of tetrahedral elements, but they do not resemble the bulk bee structure. Therefore, size dependent, there must be a restructuring of such small particles as N is changed. This is what a solid state physicist would also expect: different crystal structures such as hcp and fee are closely related in that they can be created by stacking close-packed (111) planes on top of each other simply in a different order: hcp corresponds to a sequence of 1, 2,1, 2, .... fee corresponds to 1, 2, 3, 1, 2, 3, ... Therefore, the crystal is determined by rather small differences in the (long range) atomic interaction potential, and it is easy to accept that the structure of small clusters will in general not be identical to the structure of the condensed matter. What is more, the structural isomers are related to different minima in the potential hypersurface of the cluster which may be of different depth and occupy different phase space volume; which one the cluster assumes in its creation process is a priori not certain. A cluster may even jump from one minimum to another, the energy difference between the total energy and the bottom of the minimum appearing as excitation energy. A metal cluster of finite temperature may thus have an ill-defined and time-varying geometric structure; this tendency is sup0168-583X/91/$03.50
ported by a rather non-directional type of bonding (which will be further discussed below), in contrast, for example, to molecules with strongly hybridized covalent bonds **. This “fragility” of clusters is not only manifest in their structure, but also in their stability: let us add a small amount of energy (e.g. of the order of the sublimation energy per atom, typically a few eV) to the surface of a particle (large number N of atoms); within atomic vibration times of lo-l4 to 10-r’ s this may result in the evaporation of an atom. However, it is much more probable that phonons will distribute this energy throughout the lattice, causing only a very slight temperature increase. It will then take a long time before statistical fluctuations again concentrate the appropriate energy portion in a (surface) bond to evaporate an atom. By simple statistical arguments this time decreases if N decreases. Transition state theory is able to describe this process and calculate size-dependent decay probabi~ties. It should be noted that the heat of sublimation is size-dependent, too (generally smaller for smaller N). Altogether, therefore, clusters will react quite sensitively to transfer of even small amounts of energy. This phenomenon has consequences for cluster studies. Suppose, for example, that we want to study the size dependence of a certain physical or chemical property. We then have to prepare an ensemble of clusters, ideally having identical masses, and not being disturbed by
*’ We might mention here that the distinction between clusters and molecules is, in some way, rather arbitrary; it would be difficult to rigorously justify it on the basis of purely physical arguments, e.g., the type of bonding: for example, the near-neighbor interaction in an atomic aggregate causes a splitting and shifting of energy levels which may, size-dependent, change the s-p energy distance in s/p type metal atoms; as a consequence, such a cluster (e.g. Hg,) may with increasing size experience different types of bonding, from van der Waals, to covalent, to metallic [6].
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n Fig. 2. Chemical reactivity of positively charged iron clusters with hydrogen. After ref. [4].
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I
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NUMBER OF ATOMS
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-
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Fig. 1. Ionization potentials (a) [l] and photodetachment threshold energies (electron affinities) [2] of Al clusters (b) in a molecular beam. c) Melting temperatures of gold particles deposited on a carbon film [3]. Note the different particle size scales.
secondary undesired phenomena. These requirements cause problems with such established tcchuiques as, e.g., matrix isolation or surface deposition of clusters: although it appears now to be possible to prepare such mass-defined ensembles by deposition of intense massselected cluster ion beams, the effect of cluster-surface or cluster-matrix interaction may influence and even dominate the studied properties (although these interactions may be quite interesting by themselves). Therefore, in order to isolate properties of the free clusters
themselves we should resort to the use of beams of free clusters, if possible mass selected. Such free massselected cluster beams can, for example, be prepared by surface sputtering with energetic ions, or (laser) discharge evaporation of surfaces. In such processes, a fraction of (positively or negatively) charged clusters is created in addition to the usually much more abundant production of neutral individual atoms and clusters. This charged component can be used to prepare an ion beam which by conventional ion-optical methods (magnetic separation, time-of-flight techniques) is then mass-separated. Alternatively, instead of using the charged component in the original production process, the neutral component can be post-ionized (e.g. by electron or photon impact, electron attachment, or charge exchange) and then again shaped into an ion beam. Such experiments, of course, have problems of their own: in an experiment involving a cluster beam of well-defined mass we must be certain that at least this mass is not changing during the experiment. As outlined above, though, this can happen if the clusters are not produced with low internal energy. This effect can be demonstrated for example with sputtered cluster beams. A sputter process is quite brutal; much energy will be deposited in the surface. A sputtered cluster, therefore, will possess much internal energy; it can hardly get rid of this internal energy in free. flight except by evaporation of atoms or larger fragments. In a typical 6.3
43- C,”
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Fig. 3. Energy optimized geometries of two neutral Na, isomers. Distances in atomic units. After ref. [5].
H.O. Lutz, K.H. Meiwes-Broer
391
/ Metal atom clusters
Decay
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5
10
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Fig. 4. Decay probabilities of sputtered Cu$ clusters; dashed line: evaporative ensemble model [7]. The upper and lower data sets originate from two different time windows. From ref.
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5. Mass spectrtmr of sputtered Cu$ clusters [9].
PI.
experiment, designed to study this process, the cluster beam will encounter after some drift distance a repeller grid. If the particle has fragmented along the drift length, it has lost mass and therefore also energy; as a consequence, it will not be able to surmount the repeller barrier of appropriate height. The decay probability of copper clusters as determined in this experiment (fig. 4) shows a strong increase towards larger clusters *‘: The tendency of such data is nicely reproduced by a modified transition state theory; this theory treats the clusters as hot droplets which (because of their internal energy) evaporate individual atoms as they proceed downstreams in the beam [7]. A particularly stable cluster will be particularly resistent against further decay; this will also be apparent in the total mass spectrum usually detected some ns after creation of the cluster. The decay probabilities as well as the total mass spectra will thus display intensity jumps at certain mass numbers (“magic numbers”) and, for monovalent clusters as Ag, and cu,, “even-odd oscillations” (fig. 5). This phenomenon yields interesting information on the cluster. A qualitative picture is easy to obtain if we recall that we are dealing with metal clusters. An encyclopedia defines a metal as an “. . . element with good electrical conductivity at room temperature”. At first glance this would seem to imply that metal clusters are good conductors. This conclusion, however, may be severely in error. Indeed, in a system of finite size quantum mechanics introduces energy states which are clearly separated, with larger separations for smaller cluster dimension R. For R of a few A (as in atoms, molecules or
#’ This finding is not in contradiction to the statistical argument mentioned earlier since each detected cluster N has with some probabilty had a prior history of decays (“evaporative ensemble”)
small clusters), the energy spacings are of the order of eV. Since electrical conductivity requires that electrons can be lifted by an applied (weak) electric field into close-lying unoccupied states, we cannot regard a small cluster as a good conductor. If, however, we take from condensed matter physics the picture of strongly delocalized (valence) electrons, we arrive at a rather simple model [lo]: Apparently, the bonding electrons “see” only a little of the atomic cores. They behave very much like electrons in the electronic band of a solid. Therefore, in a first approximation one can smear out the positive charge of the atomic cores over the entire cluster volume, and then calculate the electronic states in the corresponding effective potential (cf. fig. 6). Evidently, the effective potential resembles a well, with 2.0
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398
H. 0. Lutz, K H. ~g~w~s-~roer / Metal atom clusters
typical dimensions of the cluster size R. Quantum mechanics now arranges distinct energy states (shells) in such a “jellium” potential. In the simplest approximation, this droplet is spherically symmetric. Just as, for example, in nuclei which display qualitatively rather similar effective single-particle potentials, particularly stable configurations are obtained if a shell is filled with the maximum allowable number of electrons (“magic numbers”). A further refinement of this model allows for the deformation of the jelhum and the corresponding effective potential due to the self-consistent interaction of the positive background with anisotropic electron states outside closed-shell configurations. This is analogous to the Nilsson model in nuclear physics; a difference, of course, lies in the fact that in nuclei the nucleons themselves create the effective potential in which they adopt quantum mechanical states, while in the cluster jellium model (due to the very different masses of electrons and ionic cores) only the electrons are treated quantum mechanically. This “deformed jellium model” of clusters [12] is able to explain (at least qualitatively) the above-mentioned magic numbers as well as the even-odd oscillations in mass spectra, fragment stabilities, photoio~~tion thresholds (see below), etc. Note that the discussion followed so far relates metal cluster stabilities to their electronic structure. This can be further verified by comparing clusters of the same number of atoms N but having different charges, i.e., different number of electrons N,. Such experiments which, e.g., compare the mass spectra of positive and negative sputtered cluster ions demonstrate that the dominant structures in the spectra are related to N, and not to N. One of these studies shows a further aspect of the analogy to nuclear physics: the confirmation of magic numbers up to very high values. In nuclear physics, much discussion has occurred around ‘*islands of stabilities” for nuclear charges 2 > 100; in clusters, enhanced stabilities up to N - 1500 have been observed and related to electronic structures, while particular stabilities for larger N can be explained by geometric structures [13] *3. In order to further study the details of cluster structure and characteristics, for example by spectroscopic techniques, the discussion so far suggests that the clusters should be cold, thereby greatly reducing their fragmentation probability. Only in such situations can one
It3 It should be noted that there exists a second kind of “magic numbers” which appear particularly with rare gas clusters. Apparently, in these cases it is the geometric structure which yields special stabilities; whenever a full shell of 11 “tennis ball-like atoms are arranged around an imer core, particular stability is encountered (an icosahedral structure, cf., e.g., ref. [14]).
be sure to prepare a certain cluster size and still have it available when doing an experiment with it. Such “cold” clusters can be produced by the wellknown technique of seeded jet-expansion. Tbe principle is simple: material of the desired kind is evaporated (if necessary, by an intense laser pulse, or an electric discharge) and flushed by a high-pressure gas pulse (for example, helium) into vacuum. The expansion occurs in hydrodynamic flow conditions and is isentropic. It results in a dramatic cooling of the jet, including the embedded particles which may even grow further to very large sizes during expansion. This process of cluster growth under high-pressure conditions can, for example, be described by percolation theory 1151. Unfortunately, cooling of supersonic jets to very low temperatures (- 1 K) occurs mainly in the translational degree of freedom. The rotational, and even more so the vibrational degrees of freedom “freeze” at somewhat higher temperatures; for clusters, these temperatures have not been measured yet, although from general principles rough estimates may be obtained. In any case, the according temperatures usually he much lower than those obtained by other techniques (e.g., sputtering, or impact ionization of neutral cluster beams). This “cooling” is also demonstrated by the mass spectra produced by a (properly operated) supersonic jet-expansion; they show a smooth envelope with little or none of that structure (cf. fig. 5) which is familiar from fragmenting “hot” clusters. Let us now come back to the question of how the electronic band structure evolves as the particle becomes bigger. This point is, of course, at the bottom of the problem of the atom-solid transition. One immediately apparent experimental procedure would be to use photon absorption in order to excite an electron from a bound to an empty level in the cluster potential well. Thereby, the dipole convolution of empty and occupied states would be directly accessible to spectroscopy, just as in atoms or molecules. Such experiments are currently performed in several laboratories. Besides sharp resonances corresponding to well-defined single particle excitations, broad absorption maxima with widths of roughly lo-20% of the excitation energy have been found for alkali clusters [16,17]. These maxima can be related to plasmon-like excitations, and apparently represent collective motions of the electrons. Such excitations could be viewed as elementary collective excitations (as e.g., surface plasmons in a solid) or as coherent coupling of single-particle excitations (as e.g., more common in nuclear physics or also in atomic physics). Evidently, at certain energies the single-particle transition strengths are overwhelmed by collective modes. In contrast to nuclear physics where collective modes generally appear in the continuum, the collective structures found so far in clusters he in the bound part of the spectrum. The quantitative explanation of the
H.O. Lutz, K.H. Meiwes-Broer
observed resonances in terms of a many-body quanturn-mechanical treatment is still lacking. Classically, the purely collective aspects are in principle simple: Mie light scattering theory treats the photon-particle interaction in terms of classical electrodynamics [HI. The resulting Mie cross section describes scattering and absorption of light by a cluster which is thought to be homogeneous of (complex) refractive index, and has found application in the explanation of such phenomena as the colour of church window glasses containing disperse metal particles. It should be mentioned that in non-spherical particles there may be different resonance frequencies, corresponding to different restoring forces on the charge cloud along different symmetry directions. Clearly, such a classical treatment is only applicable for fairly large clusters where the electronic states are so closely spaced that they appear as homogeneous bands. For smaller clusters, the quantum mechanical nature of the system introduces increasing “graininess” due to the increasing level separations with decreasing R. As a consequence, the classical Mie resonance “fragments” into ever more individual lines which finally must coincide with the single-particle transitions in the system. The (time dependent) local density formalism appears to be well suited to treat such effects since it directly obtains the (ground state) electron density and, hence, the cluster susceptibility. On the basis of the jellium model, this approach has been used by Ekardt and
/ Metal atom clusters
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coworkers [19] to qualitatively explain the N-dependent width and behavior of the observed broad resonances in terms of collective modes in clusters of finite temperature. This still leaves the problem of the single-particle states. One important experimental tool for their direct experimental study is ultraviolet photoelectron spectroscopy (UPS) [20]. This technique has actually given very interesting results. The principle of the experiment is shown in fig. 7. The beam of negatively charged clusters is produced in a laser evaporation cluster source and cooled by jet-expansion. The charged clusters are accelerated to a known energy (a few hundred ev) and mass-selected by timeof-flight. After some drift length, a second pulsed UV laser detaches electrons from clusters of selected mass. These electrons are collected in a strong magnetic field and guided in a weak magnetic field through a drift length of a few meters to a detector; their time-of-flight can be converted into kinetic energy of the photoelectrons. The binding energy of the electrons in the cluster (Es) is then given by the difference of photon energy and electron kinetic energy. Note that such photon-induced transitions are fast compared to typical vibration frequencies. The ionization process is, therefore, “ vertical” with respect to the cluster geometry. On the other hand it is probably slow (“adiabatic”) with respect to the electron motion (the energies are not far from elecfrondetector
, detector
IO”
pressure stage
Fig. 7. Schematic experimental setup for UPS measurements on mass-selected cluster anions (from ref. [2]).
400
H.O. Lutz, K.H. Meiwes-Broer / Metal atom clusters
threshold!). The energies E,, therefore, correspond to energy differences between the anion and the neutral cluster in the anion geometry. Fig. 8 shows examples of
photoelectron spectra for a sequence of selected clusters. Clearly, the evolution of the electronic structure of such small aggregates is not continuous, rather it hap-
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Fig. 8a. UPS spectra of various metal clusters for N = 3 to 12 from ref. [21].
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401
H.O. Lutz, K.H. Meiwes-Broer / Metal atom clusters
pens in distinct steps. Addition of one atom may create considerably different electron spectra (and therefore also reflects different electronic structure).
These spectra contain much information and cannot be discussed here in detail. Only a few aspects will be highlighted. First of all we may point out that the
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Electron
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2
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Fig. Sb. UPS spectra of various metal clusters for N = 13 to 22 (from ref. [21]).
1
2
3
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H. 0. Lutz, KH. Me~we~-~~~~/ Metal atom clusters
electron band structure of the solid material is in general quite different from that of the clusters. The monovalent metals Cu and Ag may serve as an example: their bulk UPS spectra feature a quite flat contribution of the s/p-band close to threshold (i.e. around the Fermi edge), in contrast to the strong (and in both elements quite similar!) structure (caused by the quantum mechanical “graininess” of the delocalized electron states) in the corresponding cluster UPS spectra. This difference between bulk and cluster UPS spectra is less evident in the transition metals due to the high m~tip~city and the more localized nature of the d states near the Fermi level 1211(cf. also fig. 8). Furthermore, the ionization thresholds are expected to show an R-dependence. Of course, we have to take into account that the image charge contribution (which is contained in the work function of a flat surface) must be corrected according to the shape of the finite cluster. For a spherical cluster, one obtains a l/R-dependence if a completely delocalized electron cloud is assumed. Deviations from this l/R-scaling reflect the quantum structure of the cluster; they are particularly pronounced in small clusters as well as in those which have rather localized outer electrons (as, e.g., the transition metals) (211. One further feature may be noted: Whenever a “shell” of delocalized electrons (e.g. in the jellium model) is closed and a spherical configuration of high stability is reached, addition of one electron yields particularly low ionization energy (since this electron has to be filled into a new “shell” of low binding energy). This behaviour can be directly seen between, e.g., Ag; and Ag,, or Ag,g and Ag;c (cf. fig. 8). Ag;, for example, contains altogether eight valence electrons, and produces therefore a closed ‘“shell” of electrons having p character. Addition of one more electron leads to occupation of the next higher state with low-binding energy, and a new peak at low E, appears in the UPS spectrum (similarly for Ag,). In chemical language (and in the picture of vertical transitions), the energy difference between this peak and the next one at higher binding energy could be called the “ homo-lumo” gap. Of course, only closed shells exhibit spherical symmetry. Deviations from spherical symmetry break the degeneracies of angular momentum states and cause a splitting of energy levels. This phenomenon and the corresponding general symmetry rules allow us at least qualitatively to explain some of the structures found in UPS spectra from open-shell clusters. Summary: Expe~mental technique in cluster research has rapidly advanced in recent years. Mass-separated cluster beams of sufficiently high intensity are now available to perform spectroscopic experiments which directly reflect the size evolution of the band structure in the particles. Mass separation so far is generally obtained by ion-optical mass separation (time-
of-flight, magnetic separation, etc.). No satisfactory method has been established yet to produce some variety of neutral clusters with defined mass (except an interesting technique relying on the kinematics of cluster-gas atom scattering [22]), although charge changing collisions or photodetachment from negative clusters showed very promising results recently. In addition, a better knowledge of the actual vibrational temperature of jetprepared clusters would be highly desirable. At the moment, it is not clear if these clusters are “crystalline” or “liquid” or in some complicated dynamical state. Also on the theoretical side much progress has been made. Ab initio quantum chemical methods, of course, can give a very detailed insight into the structure and dynamics of clusters; in practice, however, such methods become extremely complicated even for simple (monovalent) clusters with more than, say, ten atoms. A complete treatment of clusters of more complicated elements (e.g., the transition metals) is practically out of reach of standard quantum chemical methods at present. Many more or less sophisticated models and approximations have, therefore, been applied. At the heart of the problem, of course, lies the a priori unknown cluster geometry (cf. the isomer problem mentioned earlier), which requires a high accuracy treatment. This will be particularly evident for clusters bound by more localized electrons (e.g., the transition metals), as well as in forthcoming experiments in which the band structure will be probed ever more deeply by higher photon energies. Since cluster science has raised interest in many other areas of research (chemistry, solid state physics, nuclear physics, atomic and molecular physics, etc.) one may hope, though, that we may witness many interesting developments in the near future.
This work has been supported by the Bundesminister fiir Forschung und Technologie (BMFT) and the Deutsche Forschungsgemeinschaft (DFG).
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H.O. Lutz, K.H. Meiwes-Broer
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[lo]
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K. Rademann, B. Kaiser, U. Even and F. Hensel, Phys. Rev. Lett. 59 (1987) 2319. C.E. KIots, J. Chem. Phys. 83 (1985) 5854; C.E. Klots, Z. Phys. D5 (1987) 83. W. Begemann, K.H. Meiwes-Broer and H.O. Lutz, Phys. Rev. Lett. 56 (1986) 2248. W. Begemann, S. Dreihofer, K.H. Meiwes-Broer and H.O. Lutz, in; Physics and Chemistry of Small Clusters, eds. P. Jena, B.K. Rao and S.N. Khanna, NATO AS1 Ser. B158 (1987) 269. a) J.L. Martins, R. Car and J. Buttet, Surf. Sci. 106 (1981) 265. b) W. Ekardt, Phys. Rev. B29 (1984) 1558. c) W.D. Knight, K. Clemenger, W.A. de Heer, W. Saunders, M.Y. Chou and M.L. Cohen, Phys. Rev. Lett. 52 (1984) 2141. W.-D. Schone, thesis, University Bielefeld, (1990). K. Clemenger, Phys. Rev. B32 (1985) 1359. T.P. Martin, T. Bergmann, H. GGhlich and T. Lange, Phys. Rev. Lett. 65 (1990) 748. 0. Echt, D. Kandler, T. Leisner, W. Miehle, E. Recknagel, J. Chem. Sot. Faraday Trans. 86 (1990) 2411.
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[15] 0. Knospe, R. Schmidt and G. Seifert, Phys. Lett. Al29 (1988) 236. [16] K. Selby, M. Vollmer, J. Masui, V. Kresin, W.A. de Heer and W.D. Knight, Phys. Rev. B40 (1989) 5417. [17] C. Brechnignac, Ph. Cahuzac, F. Carlier and J. Leygnier, Chem. Phys. Lett. 164 (1989) 433. [18] G. Mie, Ann. Phys. 25 (1908) 377. [19] W. Ekardt, Phys. Rev. B31 (1985) 6360; Z. Penzar, W. Ekardt and A. Rubio, to be published (1991). [20] a) D.G. Leopold, J.H. Ho and W.C. Lineberger, J. Chem. Phys. 86 (1987) 1715. b) C.L. Pettiette, S.H. Yang, M.J. Craycraft, J. Conceicao, R.T. Laaksonen, 0. Cheshnovsky and R.E. Smalley, J. Chem. Phys. 88 (1988) 5377. c) G. Gantefiir, K.H. Meiwes-Broer and H.O. Lutz, Phys. Rev. A37 (1988) 2716. [21] G. Gantefiir, M. Gauss, K.H. Meiwes-Broer and H.O. Lutz, Faraday Discuss. Chem. Sot. 86 (1988) 197. [22] U. Buck, J. Phys. Chem. 92 (1988) 1023.