Critical behaviour of the curved region near 111-facet edge of equilibrium shape cuprous selenide large single crystals

Abstract

Several millimetres large spherical cuprous selenide single crystals with well developed (1 1 1) facets grown at about 30 K below the roughening temperature ( T R≈830 K) and rapidly cooled to room temperature were used to test the universality and value of critical exponent describing the surface profile behaviour near the facet edge. Enlarged photographs (52.5 times) of part of the crystal profile were digitised with resulting spatial resolution of 0.1904±0.0001 μm. After FFT low pass filtering, the position of crystal silhouette edge was determined as the loci of the extremes in the first derivative of each image row intensity profile. For assumed critical dependence z= A( x− x 0) θ , the inverse logarithmic derivative applied to crystal profile data points disclosed the extent of intervals of different behaviour, giving independently the respective indicative values of fitting parameters θ and x 0. In three distinct regions non-linear Levenberg–Marquardt fitting was applied to original data sets. In the region farthest away from the facet, the behaviour is well described by θ≈2.5 or by Andreev formula z= A( x 0− x) 2+ B( x 0− x) 4 . In the stepped region, for ϕ=13.98–17.12° (tilt angle relative to facet plane), the critical exponent θ=1.499±0.003 is found, in agreement with Pokrovsky–Talapov universality class predicted value of θ= 3 2 . The step interaction energy, step free energy and facet free energy ratios obtained from data fitting parameters only, are compared to published values for 4He, Si and Pb single crystals. The behaviour in the immediate vicinity of the facet edge is discussed in the light of dynamics features recently observed on different single crystals during growth (cuprous selenide, 4He) and equilibration (Pb).