3D equilibrium crystal shapes in the new light of STM and AFM
Introduction
The field of surface thermodynamics and in particular the concept of the surface free energy was introduced by Gibbs [1], [2]. The surface free energy, γ, for a one-component material is here defined as the work dW required to create new surface area dA at a constant temperature and chemical potential, γ=dW/dA. For crystalline materials it is understood that this work depends on the atomistic structure of the surface. Hence γ is in general an orientation dependent quantity for crystalline solids. For multi-component crystals the surface free energy may also be a complex function of composition. It is therefore not simply equal to γ but to the sum of γ plus the products of chemical potential and surface excess concentrations, the latter summed over all chemical components. The surface free energy and its dependence on temperature, crystallographic orientation and chemical composition is one of the most fundamental quantities in the areas of surface physics and chemistry. Likewise, it also influences processes in applied areas, e.g. those involving thin film deposition, crystal growth, or the morphological stability of nano- or microscopic artificial structures. Hence it is an important task to fully understand its fundamental physics as well as improve and develop the methods for measuring it accurately.
Historically, one of the fundamental problems in the general context of surface free energy was to find the shape of a crystal, for a given amount of solid matter, where the total free energy would be minimal. The problem was formulated mathematically by Gibbs and independently by Curie [3] as a surface area integral over the orientation dependent surface free energy. The solution was found by Wulff [4], proven by Dinghas [5] and generalized by Herring [6] and Landau and Lifshitz [7]. A brief account of the history of the equilibrium shape of a crystal is given by Herring [6]. The current review builds on the rigorous connection between the orientation and temperature dependent surface free energy of a (one-component) crystal and its equilibrium crystal shape (ECS). It describes the path from quantitative experimental images of parts of the ECS to important fundamental surface energetic quantities. While the functional description of the orientation dependent surface free energy is quasi-continuous, the energetic quantities are related to fundamental excitations on an atomic scale, such as vacancies, adatoms and kinks.
To relate expressions of continuum thermodynamics to crystallographic features of atomic dimensions, it is useful to view the orientation dependent surface free energy of crystals in the framework of the terrace-ledge-kink model of a surface, illustrated in Fig. 1 [8], [9], [10]. The basic structural features are steps of monatomic height, kinks, single adatoms and terrace vacancies. Steps and kinks may be intrinsic crystallographic defects as well as thermally excited defects at elevated temperature, in the latter case causing increasing surface roughness. A first systematic study of step interactions on model single crystal surfaces led to an expression for the surface free energy, f(p),where p is the local step density of the surface, f0(T) the surface free energy of a flat terrace (facet), f1(T) the free energy of an isolated step, and f3(T) a step interaction energy, respectively [11]. Further terms characterizing step interaction, such as f4(T)p4 and f5(T)p5, have also been proposed [12], [13]. The temperature dependence of the step free energy results from kink excitations as well as vibrational motion of step edge and kink atoms. Later theoretical work confirmed the third power dependence of f(p) on p for the step interaction term, physically due to entropic and elastic/electric dipole–dipole interactions [12].
The temperature dependence of equilibrium crystal shapes has also been calculated for 3D Ising models with nearest and next-nearest neighbor atomic interactions, leading to fully faceted crystals at T=0 and crystals with shrinking facets and rounded vicinal regions at T>0 [14], [15], [16], [17]. Each type of facet vanishes at a characteristic roughening temperature. Equivalent work dealt with the two-dimensional ECS [18], [19], [20], [21]. There is much less known about the dynamics of achieving the ECS from an arbitrary non-equilibrium shape, or the dynamics of shape changes from one ECS to another, after the conditions have been altered, e.g. the temperature. Only a few studies have attempted to describe the time dependent shape changes in 3D [22], [23], [24], [25], [26]. A number of reviews of anisotropic surface free energy and ECS have appeared in the past [6], [27], [28], [29], [30], [31], [32], [33], [34], some in close relationship to the effect of adsorption on the ECS [35], [36].
It follows from the foregoing that a systematic study of 3D equilibrium crystal shapes can yield important surface energetic quantities, such as step, kink, surface and step–step interaction free energies [30], [37], [38], [39], [40], [41], [42], [43], [44]. Exact 3D images of well equilibrated crystallites are needed for the evaluation of fundamental energies of forming these defects and of their interaction. A beautiful example of an equilibrated Pb crystallite, imaged by scanning electron microscopy (SEM), is shown in Fig. 2a [37]. The crystal of diameter is viewed along a 〈110〉 direction, such that flat (111) and (100) oriented facets appear on the periphery. The transitions between facets and rounded regions (vicinal surfaces) is reported to be continuous for clean Pb [37], at least below about [45], [46]. A more recent example of an equilibrated Pb crystallite, supported by a Ru(001) crystal and imaged by scanning tunneling microscopy (STM) at , is presented in Fig. 2b. The high resolution of STM allows the clear observation of the step bounding the (111) facet as well as small side facets which are not visible on the ECS imaged by SEM. Partial images of the (111) facet-to-vicinal transition region show also individual monatomic steps of the vicinal surface, Fig. 2c [47], [48]. Observations of the ECS, especially of flat facets and adjacent vicinal regions, will at first provide relative step and surface free energies. By contrast, the experimental determination of absolute step free energies, kink formation and step interaction energies is more difficult and requires considerable effort. However, absolute values of all these energies play an important role in governing crystal growth morphologies as well as kinetic processes associated with shape changes [48]. Furthermore, absolute step and kink energies are considered to be the key to absolute surface free energies of well defined low-index orientations [42], [43], [49]. Hence it is imperative to utilize the full potential of analyzing 3D equilibrium crystal shapes for obtaining these fundamentally important physical quantities. In general, it is necessary to monitor the ECS as a function of temperature and to extract temperature dependent characteristic morphological parameters [43], [50]. In addition one has to ascertain that true 3D equilibrium of an ensemble of crystallites is achieved [51], [52]. One obstacle in reaching the true ECS can be the activation barrier for growing (or dissolving) new layers on facets [53], [54], [55]. This problem can be overcome through the study of dislocated crystallites [51].
In the general context of 3D crystallites, the systematic study of 2D nano-crystals, in the form of adatom or vacancy islands on extended flat surfaces, is also important [56], [57], [58], [59]. In particular, the temperature dependence of the shape of 2D islands yields information on the absolute step and kink formation energies of the bounding steps [50], [58], [59], complementary to that derived from facets of 3D crystallites [60]. The absolute step free energies serve then to calibrate other surface energetic quantities.
Section snippets
Basic physics of 3D crystallites
The structure of free solid crystallites cannot be studied because experimentally the crystallites need to be at a fixed location and in a known orientation relative to the probe. Hence all experiments are carried out for supported crystallites. The interaction between the crystallite and the support itself may disturb the shape and/or the structure of the crystallite, especially at the interface. So, if the crystallite is very small, e.g. less than in diameter, the influence of the
Unsolved problems in the past and their solution
For a complete quantitative analysis of 3D equilibrated crystallites one needs a high resolution microscopy capable of providing a true 3D image with proper scaling. The techniques used in the past (with the exception of crystals [76]) were mostly scanning electron (SEM) and reflection electron microscopies (REM). The resolution of SEM is at best 3– and thus incapable of resolving steps of monatomic height. The latter can be resolved by REM but this technique suffers from severe image
New results from equilibrated 3D crystallites
In this review we focus on new studies based on STM (AFM) images of metal crystallites under UHV conditions. Most examples presented will be 3D Pb crystallites imaged at temperature by STM. We also review a re-evaluation of the ECS of Au [40]. Other recent studies have dealt with 3D Pd [98], [99] and Au [100] crystallites. In the first case the interfacial energy of Pd/Al2O3 [99] was determined while in the second the free energies of the steps bounding the highly anisotropic, lenticular (110)
General discussion of new results and outlook
The current overview of recent theoretical and experimental work on metal crystallites and their 3D ECS shows that significant advances in the study of the kinetics of approach of the final shape and of the detailed evaluation of the ECS have been made. An important prerequisite is the atomic step resolution in images of the ECS, provided by the new microscopies of STM and AFM. These techniques allow true 3D imaging of the ECS and hence an analysis of all aspects of shape, in particular of 2D
Acknowledgements
I would like to take this opportunity to thank a number of former colleagues who introduced me to various aspects relevant to the present review: Norm A. Gjostein, Malcolm McLean and Paul Wynblatt in the period of 1965–73. Later I benefitted from discussions with Jacques Villain, Bill Mullins, Michael Wortis, Henk van Beijeren, Walter Selke, Paul Wynblatt, George Comsa and Harald Ibach. None of this would have materialized without the excellent collaboration with my students, postdocs and
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