Three-dimensional relaxation of small crystallites was imaged in real time using variable-temperature scanning tunneling microscopy. The micron-sized Pb crystallites, supported on Ru(0001), were equilibrated at 500-550 K, and the volume-preserving shape relaxation was induced by a rapid temperature decrease to 353-423 K. The (111) facet at the top of the crystallite grows by sequential peeling of single atomic layers, which shrink like circular islands. The rate of layer peeling slows dramatically as a new final state is reached. DOI: 10.1103/PhysRevLett.87.186102 PACS numbers: 68.65. -k, 68.35.Fx, 68.35.Md, 68.37.Ef With the relentless drive toward solid state structures of ever decreasing size scales, the issue of the stability of such structures, especially in response to external perturbations, becomes increasingly important. Real-time experimental observations of the complete decay of unstable structures, e.g., patterned Si substrates [1], nanofabricated Si mounds [2], and homoepitaxial islands [3][4][5][6] have clearly shown the discrete single-layer mode of decay. However, in complete contrast to these examples, we discuss for the first time the volume-preserving relaxation of a heteroepitaxial 3D crystallite from one well-defined state to another stable state, in response to a sudden change in chemical potential, here induced by an abrupt change in temperature. In relaxation, the ratio of forward and reverse flux is constantly decreasing, reaching a value of one in the final, equilibrium structure. This results in qualitatively different behavior than a decay process, where the forward flux is strongly dominating. Figure 1 illustrates schematically the evolution of a facetted crystal from a stable high temperature state towards a low temperature state, causing the facet to grow. The thermodynamic driving force [7 -12] for this process is well understood in terms of the increasing density of monatomic steps in the rounded regions of the crystallite near the facet [13] (a step is the boundary of a change in height by an atomic layer). In thermodynamic equilibrium, the ratio between facet radius r and the distance h of the facet from the center of the crystal is equal to the ratio of step free energy b to surface free energy g of the facet [3,14]. With increasing temperature, steps lower their free energy by gaining configurational entropy due to kink formation [15] and by an excess vibrational free energy [5,16]. Since free energies for singular surfaces change much more slowly with temperature than step free energies [17], a temperature increase corresponds to shrinking (and decrease to expansion) of the equilibrium facet diameter. Using linear kinetics, the rate of motion of a step will be proportional to the chemical potential change involved in removing an atom from the step's edge [18,19]. Using this approach, we define the chemical potential of the step bounding the top layer of a facet of radius r, as illustrated in Fig. 1, in terms of the curvature FIG. 1. Schematic crystal shape evolution upo...

It is shown that exact images of the three-dimensional equilibrium shape of crystallites (ECS), recorded at several temperatures between 0.3 and 0.8 of the melting temperature of a solid, can be evaluated to yield absolute values of the surface and step free energies versus temperature, in addition to the formation energy of kinks. The essential input for this novel approach is the temperature variation of the size of a facet on the ECS and of the separation between the Wulff point and that particular facet. This approach promises access to surface free energies over a large temperature range and for well-defined low-index surface orientations. PACS numbers: 68.35.Md, 68.35.Bs, 82.65.Dp The absolute value of the surface free energy of solid materials is a fundamentally important energetic quantity which is needed for the understanding of a large number of basic and applied phenomena, such as crystal growth, surface faceting, growth and stability of thin films, the shape of small crystallites in a supported catalyst, and many general materials science applications. Despite its wellrecognized significance, there are relatively few reliable primary data of experimental surface free energies because they are very difficult to measure. Only a few techniques, such as the zero-creep [1-4] and the cleavage techniques [5], have been used repeatedly in the past to obtain quantitative values for a limited number of solids, mostly metals. "Recommended" values of surface energies of metals have been derived in a phenomenological systematic comparison of some experimental solid surface free energies and liquid metal surface energies for a large number of elements [6]. All of the known values are considered to be averaged over a range of crystallographic orientations. Finally, very little is known about the temperature dependence of the surface free energy of solids since most experimental data were obtained at temperatures near the melting point.In this Letter we propose to utilize the equilibrium crystal shape (ECS) of a solid crystallite for the quantitative determination of absolute surface energies as well as step free energies for a range of temperatures well below the melting temperature. The starting point is an exact image of portions of the ECS at a given temperature, obtained, for example, by high resolution scanning tunneling microscopy (STM) [7,8]. In previous work we have demonstrated how, at a single temperature, the relative formation energies of kinks and steps as well as the step interaction energies of vicinal (111) surfaces can be determined from the regular ECS of small Pb crystallites [9]. We propose now, that analyzing such a crystallite at several temperatures by STM and measuring the facet and crystal size versus temperature, the absolute values of kink formation, step formation, and surface free energy of the low-index facet orientation can be evaluated, as long as the ECS is continuously differentiable, i.e., exhibiting all crystallographic orientations.The ECS of a solid is governed by the temper...

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The shapes of ͑111͒ oriented two-dimensional ͑2D͒ islands and facets, the latter being part of threedimensional ͑3D͒ crystallites of Pb, were equilibrated at 104 -520 K. Island sizes were in the range of 15-90 nm radius, facets typically at 100-270 nm radius. They were imaged by scanning tunneling microscopy to provide the exact outline of the bounding step. Increased step roughening with increasing temperature decreases the radius anisotropy of islands and facets in a consistent manner. Products of island/facet radius times local step curvature versus temperature were obtained experimentally, serving as the basis of absolute step and kink energies at 0 K. They are f 1A (0)ϭ128.3Ϯ0.3 meV, f 1B (0)ϭ115.7Ϯ5.8 meV, and kA ϭ42.5 Ϯ1.0 meV, kB ϭ60.6Ϯ1.6 meV, respectively. The combination of studying small 2D islands ͑unstable at high temperature͒ and large 2D facets allows measurements over a very large range of temperatures.