Dynamics of clusters and molecules in contact with an environment

Abstract

We present recent theoretical investigations on the dynamics of metal clusters in contact with an environment, deposited or embedded. This concerns soft deposition as well as irradiation of the deposited/embedded clusters by intense laser pulses. The description of these complex and demanding compounds employs a hierarchical model in an extension of a Quantum-Mechanical/Molecular-Mechanical (QM/MM) approach where the cluster electrons are described by Time-Dependent Density-Functional Theory (TDDFT) and the constituents of the more inert environment by classical equations of motion. Key ingredients are the polarization potentials where, in particular, our QM/MM implementation takes care to include the full dynamical polarizability of the substrate. This is crucial for an appropriate modeling of dynamical scenarios. We discuss the observables accessible in that model, from quantum-mechanical cluster electrons, from classical cluster ions and from the degrees of freedom of the environment (positions, dipole polarizabilities).

We discuss examples of applications for two typical test cases, Na clusters deposited on a MgO(001) surface and Na clusters in/on an Ar substrate. Both environments are insulators with sizeable polarizability. They differ in their geometrical and mechanical properties. We first survey the low-energy properties of these compounds, structure and optical response. We work out the impact of surface corrugation and of polarizability. We analyze the difference between deposited and embedded clusters.

The second part discusses the dynamics of soft deposition processes, for Na clusters impinging on Ar(001) or MgO(001) surfaces. We analyze charge and size effects, and details of energy transfer to the environment. We show how the deposition process can create “hot spots” in the surface where sizeable amounts of energy are stored in internal excitations of the substrate atoms.

Finally, we consider laser irradiation of embedded/deposited Na clusters. These systems serve as generic test cases for chromophore effects. We discuss a broad range of scenarios, from “gentle” to “strong” irradiation processes. The key effect is ionization through the laser pulse. We analyze the effect of the substrate on the angular distribution of emitted electrons and the effect of ionization on the substrate and the interface interaction. The case of strong excitations shows a dramatic change of cluster dynamics due to the environment, in particular hindered (or delayed) Coulomb explosion.

Introduction

This review deals with the dynamics of metal clusters in contact with inert environments, either deposited on a surface or embedded inside a medium. The study of clusters is a rather recent branch of physics, developing with the steadily improving preparation methods and laser analysis. The case of free clusters has been extensively studied in the past and there exists a broad literature on that topic, for books and reviews see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. The progress of the field is also well documented in the impressive series of ISSPIC proceedings [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. Metal clusters play a special role in that field because of their remarkable electronic shell structure [2], [3], [7] and pronounced Mie plasmon resonance which provides a well defined and strong coupling to light [1], [6], [9]. Bi- or trimetallic cluster design also constitutes a subject of great interest in material science, for a recent review, see [24].

The case of clusters in contact with an environment is more involved and covers an extremely large range of physical and chemical situations. The field is still very much under development, see e.g. the collections [25], [26], [27], [28]. One motivation to deal with these compounds is that many experiments can be better performed for non-isolated systems. Substrates serve to prepare well defined conditions of temperature and orientation, they help to hold neutral clusters, and they allow one to gather higher cluster densities. Even the analysis of small molecules can take advantage of handling in an environment as experiments in He droplets show [29], [30], [31], [32]. A further important aspect is that contact with an environment is a realistic scenario in applications. For example, there are promising attempts to employ clusters in the dedicated shaping of nano-scaled devices [33], [34], [35], [36], [37], small Au clusters on surfaces are found to be efficient catalysts [38], metal clusters are considered as nano-junctions in electrical circuitry [39], and the coupling to light is exploited in producing an enhanced photo-current by depositing Au_N on a semi-conductor surface [40]. Metal clusters in an inert substrate are also a simple model system for chromophores where the field amplification effect has large impact on the environment, see e.g. the study of localized melting for the generic combination of Au clusters embedded in ice [41]. Such combinations can be used as a test system to understand the first stages of radiation damage starting with defect formation in solids [42], [43], [44]. Furthermore, there are promising applications in medicine where the frequency selective optical coupling of organically coated metal clusters attached to biological tissue may by used as a tool for diagnosis [45], [46], [47], [48] and for stronger laser fields for localized heating in therapy [49].

Last but not least, compounds of two different materials are a research topic in its own right. It is interesting and it may even be crucial to watch modifications within each species if the two come into contact. This is typically the case of biological molecules whose properties and behaviors are strongly linked to the (often water) environment. Moreover, the mutual influence of the two species can create new effects which were not possible for the isolated species. This aspect is important as it indirectly points out the key role of interactions between the species and its environment and, correspondingly, the importance of a proper description of such couplings. Thereby, it is essential for a proper description to account for the possible response of both, species and environment. But the combination of different materials and the typically large sizes of the environments pose a very demanding problem for a theoretical description. One needs to find a good compromise between simplification and yet proper inclusion of the environment dynamical response and relevant coupling mechanisms, see Section 1.3.

Even with a simplified account for the environment, the effort remains huge. As a first step, we shall focus on rather simple cases of especially optically active metal clusters in contact with insulators, taking as examples the Ar material and the MgO(001) surface. Both environments are insulators with a large band gap. The metal cluster serves as a chromophore which opens the road to many interesting dynamical scenarios. Deposition processes are also offering surprising scenarios which we shall equally discuss. We will thus discuss both applications, clusters embedded in a “matrix” and clusters deposited on a “surface”.

Fig. 1 provides a few illustrative examples of studies on mixed cluster-environment systems. Let us first focus on the upper right panel which displays the optical response (“color”) of a Ag₈ cluster embedded in a finite size rare gas matrix [50]. The evolution of the peak position with “matrix” size is plotted. It provides an example of how the response of a given species (here the Ag₈ cluster) is affected by its surroundings. The effect is admittedly subtle (see the ordinate scale) but mind that the “matrix” is composed of rare gas atoms, supposed to be extremely inert. One can actually spot differences and qualitatively different trends when considering different rare gases, the different trends being closely related to the different rare gas polarizabilities. The lower right panel focuses on the analysis, not on the cluster, but on the surface itself. The case is a deposit of Pd on an insulating MgO surface and the energies of electronic levels of MgO are recorded as a function of Pd coverage [51]. The interesting point here is that the MgO levels are significantly affected by the deposition process, in spite of the fact that MgO is a well bound insulator. This points out the fact that subtle interaction effects enter the picture as soon as two materials are put in contact. Both the optical response of the embedded Ag₈ cluster and the photoelectron spectra of MgO concern low energy phenomena, close to the ground state of the system. Experiments on (possibly violent) dynamical scenarios have also been performed. We illustrate them in the left column of Fig. 1. The left bottom panel shows again an optical response of an embedded cluster (Ag cluster inside a bulk glass matrix) but this time, in relation to a violent laser irradiation [52]. The optical responses prior and after irradiation with a strong laser pulse are plotted. The spectra before and after irradiation show significant differences, indicating that the irradiation provoked a sizeable shape variation of the embedded cluster (see also the discussion at the beginning of Section 5). Finally we consider in the upper left panel an even more violent scenario but in a somewhat different context. The system under study is an adenosine monophosphate nucleotide molecule coated by a finite number of water molecules [53]. Collisions with neutral atoms provoke the fragmentation of the complex. The fragmentation spectrum, plotted as usual as a function of mass over charge ratio, exhibits a sizeable dependence on the number $m$ of coating molecules. The example thus demonstrates the intricate relation between system and “matrix” in this example of biomolecular systems, even in the course of violent disintegration. This experiment takes care to control and vary systematically the number of embedding water molecules. The study of such model systems eventually allows one to decipher elementary mechanisms responsible for DNA damage by irradiation.

As was seen in Section 1.1, clusters in contact with an environment comprise a world of different situations and systems. It isthus important to clarify the situation by some classification and by considering limiting cases. The cluster- environment compounds cover very different systems, from small mixed molecules up to nanometer scale clusters deposited on an infinite surface. It is well known that small systems usually exhibit specific size effects which tend to level off for larger size. For clusters in an environment, size effects appear twice, in terms of the cluster size and in terms of the environment size. That is illustrated schematically in Fig. 2 in the case of simple metal clusters (Na) in contact with insulator environments (MgO surface and Ar surface and/or matrix), which represent typical systems discussed in this paper. The transition between size-specific and generic behaviors depends to a large extent on the considered observable. Size specific effects appear as fluctuations on the observed values of a given observable while the trend at larger sizes is more monotonous and exhibits a slow convergence towards the bulk value. This question has been addressed for a long time in the case of free clusters, see e.g. [2], [3], [54]. The impact of environment size on cluster properties was also considered in a few experiments as for example [55].

The tour from small to large systems concerns also the way such composite objects can be described theoretically. Very small systems, in practice mixed clusters, can be treated by sophisticated quantum chemistry methods, while bulk materials call for techniques from solid state or surface science sacrificing some details. Treating the mixed system of a cluster in contact with an environment thus corresponds to an intermediate situation in which one would like to combine the advantages of these two extremes: a detailed treatment of the cluster with a less detailed description of the environment. This calls for hierarchical methods, see Sections 1.3 Description of cluster and environment, 2 Model.

Fig. 2 does also show typical cluster-environment configurations. Three free Na clusters are shown along the axis “cluster size”, Na₇⁺, Na₂₁⁺ and Na₉₂. The first insert along the axis “Matrix size” shows the very small compound NaAr₆. The further figures along that axis represent a medium size ($N=55$) and large Ar cluster $N=561$. The plane between the two axes shows the two typical test cases which we will consider in the following, down to the left a sketch of Na₈Ar₄₃₄ as an example of an embedded cluster and to the right of that Na₈@MgO(001) for a deposited cluster, while a small mixed cluster Na₈Ar₄₂ is also indicated close to the cluster axis.

The mixing of two different systems, a metal cluster with an insulator material, also induces a larger span in energy and length scales. The metal has strongly delocalized electrons with a large mean free path and small energy differences. On the other hand, the electrons in the insulating materials remain tightly bound to atoms and involve large electronic energies. This holds for rare gases as well as for the ionic crystals in our sample (NaCl and MgO). Thus the description of such mixed systems has to accommodate larger range of energies (with corresponding time scales) and lengths, which complicates matters as compared to free clusters.

As we are primarily interested in dynamical scenarios, we briefly recall key time scales for the systems which we are considering in this paper. These are sketched in Fig. 3, including both “intrinsic” times of the system itself and “external” time scales associated to the excitation process. To have a specific example, we consider the system times associated to Na clusters and for the excitation process with an optical laser. We ignore the extremely short times associated with Na core electrons, which will play no role in the following.

The pulse duration of optical lasers may be varied over a wide range and extends in principle from fs to ps or even ns. We focus here on a fs laser with pulse widths of order a few tens to a few hundreds of fs. The fastest cluster time scales concern the motion of the valence electrons. As already mentioned, metal clusters act as excellent chromophores. Coupling to light is predominantly mediated through the Mie plasmon, whose period lies in the fs range. It corresponds to the collective oscillations of the electron cloud with respect to the ionic background, triggered by an external electric field. Other single-particle excitations and direct electron escape, i.e. single-particle excitation into the continuum, lie in the same range, but with a wider span from sub-fs to several fs.

The most widely varying times are related to electron relaxation due to damping from electron–electron collisions and thermal electron evaporation. Both strongly depend on the internal excitation of the cluster which may be characterized by an average excitation energy $E^{\ast }$. That might be expressed alternatively by an electronic temperature $T$ which allows easier comparison between systems of different size. The Fermi gas model then provides a simple connection between temperature and internal excitation energy. For $k_{B}T\ll {ϵ}_F$, $T$ can be estimated as $k_{B}T=2{({ϵ}_F{E}^{\ast }/N)}^{1/2}/\pi$, where ${ϵ}_F={ħ}^2{(9{\pi }^2/4)}^{2/3}/\left(2m_e{r}_s^2\right)$ is the Fermi energy. In the case of Na at bulk density, this amounts to $k_{B}T={\left(1.28\mathrm{eV}{E}^{\ast }\right)}^{1/2}$. Electronic thermalization is mostly mediated by electron–electron collisions. Fermi liquid theory leads to a $T^{-2}$ law for the corresponding collision time [56], [57]. At low temperatures, most collisions are Pauli blocked and relaxation times become comfortably long. Inclusion of electron–electron collisions in dynamical scenarios is at present mostly done in semi-classical treatments at the level of the Vlasov–Ühling–Uhlenbeck approach [58], [59], [60], [61]. A tractable quantum-mechanical description of the collisions has yet to be developed. Thus we do not include electron–electron collisions in the studies presented here. It ought to be kept in mind that they should be included for violent processes in the future stages of the theory. The time scale for electron evaporation depends even more dramatically on temperature (or excitation energy). It is given by the Weisskopf estimate which predicts a trend dominated by the exponential factor $exp(E_{\mathrm{IP}}/\left(k_{B}T\right))$, where $E_{\mathrm{IP}}$ denotes the value of the ionization potential [62], [63]. In general, electron evaporation represents a very efficient cooling mechanism for highly excited clusters [64].

Ionic motion runs at much slower time scales and spans a wide range of times. Vibration frequencies typically lie in the meV regime. That means they have cycle times of order 100 fs to 1 ps which can be measured by Raman scattering (see, e.g., [65]). Strong perturbations (laser, projectile) can lead to large amplitude ionic motion and cluster explosion due to Coulomb pressure generated by ionization and thermal excitation. The coupling between electrons and ions proceeds at an electronic time scale, i.e. within a few fs. But the effects on ions develops at much slower scale, typically around 100 fs, because of the much heavier ion mass. Note that the time scale for an explosion depends on the violence of the process and becomes shorter with a higher initial ionization. Besides ionization effects, the further energy transfer from electrons to ions takes much longer, up to the ns range [66]. Ionic relaxation processes are even slower, e.g. thermal emission of a monomer can easily last μs.

Relevant time scales at the side of the environment lie in the same span. Ionic/atomic times typically scale with the square root of ion/atom masses and come again into the range of hundreds of fs to ps for typical environments studied here. Moreover, it should be noted that the slowest ionic/atomic vibration and relaxation times increase with system size (more precisely $\propto N^{1/3}$). That size effect has to be kept in mind when interpreting results from finite samples. On the other hand, the electronic degrees-of-freedom of the environment show shorter time scales than those in the cluster. We consider inert materials where the electrons remain rather tightly bound to their parent atoms which produces much shorter time scales and which allows a simplified QM/MM treatment. The coupling of the environment electronic degrees-of-freedom with cluster electrons proceeds at the same short time scale. The corresponding typical values lie in the sub-fs range. We thus need to resolve the dynamics at these rather short time scales.

This quick survey shows that relevant dynamical scenarios comprise a large span of time scales which is a great challenge for a theoretical description. Already in free clusters, ionic motion may require a simulation time up to several ps while electronic motion has to be resolved down to a fraction of fs. The effect becomes even more dramatic for embedded/deposited clusters. One aims at covering the possibly very slow relaxation of the large environment while accounting for its especially fast electronic response. This means that one is going to extend the typical range of time scales to be accounted for by 2 orders of magnitude compared to the case of free metal clusters, as illustrated in Fig. 3.

There is a broad range of theoretical approaches to deal with statics and dynamics of free clusters, from macroscopic models over shell models using an educated guess for the cluster mean field up to fully fledged ab initio methods, for an overview see chapter 3 of [9] and Section 2.1.1 here. The higher complexity of mixed systems (cluster/molecule + environment) calls for a re-examination of the modeling.

The first step is the description of static properties. Such static studies have been undertaken for a long time and have allowed one to understand many properties of such “dressed” clusters or molecules [67]. All approaches which are used for free clusters can be used for the whole combined system. One even performs fully quantum-mechanical calculations of adsorbate and substrate [68], [69], [70], [71], [72]. That, however, imposes a heavy restriction on the “substrate” size. That limitation can be overcome by the sophisticated Quantum Mechanical/Molecular Mechanical (QM/MM) approaches of quantum chemistry. These exploit a hierarchy of importance from the active zone of interest down to the farther outskirts of the system and couple a quantum description of the active piece to a classical description of the environment. That method had been developed first for dealing with the very complex systems of bio-chemistry [73] and is often used in that context, see e.g [74], [75], [76], [77]. But it is also extremely useful in surface chemistry [78], [79], [80]. For a detailed description see [81]. For ionic crystals, one even adds a further outer layer with inert ions, i.e. without the internal dipole polarizability, to account for the long range Ewald summation in the material [78]. It is the method of choice, in particular if we have that clear distinction between a metal cluster and an inert, insulating substrate.

The next step, namely to consider dynamical situations, requires a much larger effort. One way would be to consider small samples as representative of the environment. However, truly dynamical calculations accounting for all electronic degrees of freedom in such “small” systems are not yet available. They may show up within a few years from now. Even if such calculations could be available in the near future, it is likely that they will be limited to rather small numbers of particles. And there will remain a gap between such small systems and very large (bulk) ones. We shall thus not elaborate further on these approaches and look for a description of dynamical processes with appropriate simplifications. The simple-most and widely used approach is a purely classical molecular dynamical simulation using effective force fields [82], [83], [84], [85], [86]. However, there are many situations where the quantum mechanical aspects of the cluster electrons become important. The QM/MM methods offer here again a powerful tool of description. Still, these methods freeze electronic degrees of freedom of the environment inside phenomenological interaction potentials and thus cannot account for a proper dynamical response of the environment. This, however, becomes a key issue as soon as one considers truly dynamical situations, such as strong irradiation processes in biological systems as well as for clusters in/on a substrate. One thus needs to go even one step further in order to account for the electronic response of the environment. We have developed over the last years such a QM/MM model augmented by a simplified treatment of electronic degrees of freedom of the environment for Ar environments [87], [88], [89] and for MgO(001) surfaces [90]. This dynamical QM/MM model constitutes the basis of results which will be presented in the following.

Even with the enormous savings when using QM/MM, a full dynamical, microscopic treatment of clusters in/on a substrate is not feasible because of the much too large number of degrees of freedom for the environment. There are two complementing directions in which the problem of system size can be attacked. One solution is to simulate the environment in terms of a finite size system. In other words, the environment is modeled by a finite cluster of the environment material. Of course, the convergence of results from such a finite system to bulk values has to be carefully tested. But the “finite environment” as such is an interesting system as it allows one to vary the size of the “environment” and so to analyze theoretically the impact of embedding on the cluster and on the environment. There are, in fact, experiments performed following that strategy in the case of solvated biomolecules, see the example in Fig. 1. The study of such model systems is thus becoming a key issue and they are better accessible to a dynamical QM/MM description. The alternative solution for a simulation of bulk material is to consider a finite piece of the system and to copy it to an infinite number of similar pieces with periodic boundary conditions. This method is well known from simulations of true bulk in solids, liquids, and plasma [91], [92], [93]. Surfaces can be modeled that way with periodic copies in the lateral direction. Lattice translational symmetry is broken in the vertical direction. Thus one uses here the “finite sample” approach in considering only a finite number of layers. A more detailed description of the modeling will be given in Section 2.

The paper is organized as follows. Section 2 provides a detailed presentation of the model used in the following, discussing also its relations to other approaches and pointing out the importance of properly including dynamical effects.Section 3 focuses on structural and low energy aspects for atoms and for clusters in contact with an environment. It allows one to validate our generalized QM/MM approach with respect to experimental results and other theoretical approaches, as structure properties can be accessed at various levels of sophistication. That section also contains a discussion of optical properties which constitute the doorway to dynamical scenarios. Section 4 is devoted to the study of deposited clusters and deposition scenarios. We discuss both the response of the cluster itself and the response at the side of the environment, analyzing in detail the excitation of internal degrees of freedom of the environment. Section 5 discusses the dynamics following irradiation by intense laser pulses, mostly for embedded clusters. We are addressing highly non-linear situations involving sizeable ionization of the cluster. The response of the system is analyzed in terms of all its constituents, the electrons and ions of the cluster, and the atoms of the matrix (including their internal excitations). Finally, Section 6 summarizes our major conclusions and outlines future perspectives suggested by these studies.