Step Dynamics in 3D Crystal Shape Relaxation

Abstract

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...

Facet evolution on supported nanostructures: Effect of finite height

“…Although steps are never perfectly circular, there are many experiments where relaxing nanostructures are well approximated as being axisymmetric. 5,7,19,20 Our results for the straight-step system should be applicable to the special situations where step line tension is unimportant. This system can perhaps be thought of as a model for a set of steps that evolves from a "step bunch" initial configuration.…”

“…Because of the step line tension, the radius of the top step shrinks and undergoes a monotonic collapse, resulting in a successive "peeling" of the top layer. 5,7 The removal of the top layer reveals an underlying one that has a larger radius. 5 Therefore, the facet evolution consists of a radial expansion along with a downward translation.…”

“…5,7 The removal of the top layer reveals an underlying one that has a larger radius. 5 Therefore, the facet evolution consists of a radial expansion along with a downward translation. This evolution has been studied in the case of infinite axisymmetric structures: in particular, for an equally spaced initial step configuration, we have 8,18,26 r͑t͒ = O͑t 1/4 ͒ and h 0 − h͑t͒ = O͑t 1/4 ͒ for t ӷ 1, where r͑t͒ is the facet radius, h͑t͒ is the ͑dimensionless͒ vertical location of the facet relative to some reference height h 0 , and t is ͑dimensionless͒ time.…”

“…As a result, in the absence of material deposition, the perimeter of a single isolated step generally shrinks-a phenomenon that is well understood in terms of a step line tension. 4,5 In addition to step curvature, another quantity that is clearly finite is the number of monolayers that make up the nanostructure. However, the effects of finite height have received much less attention in theoretical treatments compared to infinite and periodic surface features.…”

“…5,6 ͑An extremal step is one that has only one neighboring step.͒ For axisymmetric structures with a ͑circu-lar͒ facet, scaling laws of the form r͑t͒ = O͑t ͒ are often observed. 2,[7][8][9] Here, r͑t͒ is the facet radius, t is time, and the exponent depends on the initial structure's shape and the dominant transport process.…”