engleski

Nanometre resolution investigations of facet displacement during the single crystal growth using the real-time analysis of full two-dimensional digitised interferograms

A growth of nonstoichiometric cuprous selenide spherical single crystals of equilibrium-like shape at around 800 K has been monitored in situ. At these temperatures the equilibrium crystal shape (ECS) is characterized by coexistence of facets (atomically smooth oriented in accordance with the intrinsic cubooctahedral symmetry) and rounded parts (atomically rough). Recently, during the growth of ECS of 4He surrounded by superfluid He at 20&#8211 ; ; 250 mK in the wide range of average growth rates two new growth modes of (0001) facet has been reported. Actually, the facet displacement along its normal in time, measured by optical interferometry accompanied with the particular analysis of interference fringes, reveal two superimposed growth modes: a linear facet displacement in time and a burst-like facet growth. In fact, the optical interferometry method has been restricted only to a very small growth rates (below 1 nm/s) while the burst-like mode is obtained using He pressure monitoring method. The optical interference method as a whole has provided the displacement resolution of about 5 nm. Following the similar method of displacement measurements applied on our spherical ECS crystal at the growth temperature of 800 K (temperature stability better than 1 K) and the method of analysis of interference fringes we have searched for the evidence of the two growth modes existence in the ECS cuprous selenide. There are expected differences in the experimental set-up for crystal facet illumination but the rest of set-up is very similar. The interference fringes placed on the facet have been monitored by CCD camera (512*512 with 256 gray scale) with appropriate optic lenses system. According to different type camera we were able to extend our taking frame rate up to a 25 frames per second. The high frame rate has been forced by the mechanical vibrations present in system as well as the wish to increase the accessible growth rate range. In our analysis of the full two-dimensional digitized interferograms we follow and refine the Fourier-transform method of analysing fringe-patterns of the form: g(r) = a(r) + b(r) cos [ 2&#960 ; ; fr + &#934 ; ; (r)] The method is based upon the fact that the spatial variations of the functions a(r), b(r) and &#934 ; ; (r) are slow compared with the spatial frequency f, which is given by the number of pixels per interference line. Then the peak features in the Fourier spectrum are well separated, and by shifting one of the first order maxima, which is centered at f, to the origin of the inverse space, followed by the inverse Fourier transform, one is able to extract the desired phase field &#934 ; ; (r). The procedure is pretty straightforward in the case when the components of frequency f are integers (in pixels), f = f_0. The experimental setup usually does not yield integer components of f, which is rather given by: f = f_0 + &#948 ; ; . In order to achieve high accuracy in determination of the phase field, one has to find the deviation &#948 ; ; of the carrier frequency from the nearest pixels in the inverse space. We apply the sampling theorem to the amplitude of the Fourier transform g(q), and this procedure yields. We then proceed by shifting the chosen first order maximum to the origin of the inverse space by the integer value f_0, and by subsequent inverse Fourier transform we obtain the phase field &#934 ; ; _0(r). This value is corrected by the contribution &#916 ; ; &#934 ; ; (r) arising from the subpixel correction &#948 ; ; to f. We have tested our procedure on test pattern function with non-integer components of f, and we obtained the phase field with the accuracy of better than 1%. Such an accuracy in determination of the phase field &#934 ; ; (r) gives us the resolution of 2 nm in determination of the position of the facet along its normal during the crystal growth.

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