The atomically smooth SrTiO(3) (100) with steps one unit cell in height was obtained by treating the crystal surface with a pH-controlled NH(4)F-HF solution. The homoepitaxy of SrTiO(3) film on the crystal surface proceeds in a perfect layer-by-layer mode as verified by reflection high-energy electron diffraction and atomic force microscopy. Ion scattering spectroscopy revealed that the TiO(2) atomic plane terminated the as-treated clean surface and that the terminating atomic layer could be tuned to the SrO atomic plane by homooepitaxial growth. This technology provides a well-defined substrate surface for atomically regulated epitaxial growth of such perovskite oxide films as YBa(2)Cu(3)O(7-delta).

The atomically ultrasmooth surfaces with atomic steps of sapphire substrates were obtained by annealing in air at temperatures between 1000 and 1400 °C. The terrace width and atomic step height of the ultrasmooth surfaces were controlled on an atomic scale by changing the annealing conditions and the crystallographic surface of substrates. The obtained ultrasmooth surface was stable in air. The topmost atomic structure of the terrace was examined quantitatively by atomic force microscopy and ion scattering spectroscopy as well as a theoretical approach using molecular dynamics simulations.

The terminating atomic plane of SrTiO3 (001) surface was investigated by means of coaxial impact-collision ion scattering spectroscopy (CAICISS). CAICISS spectra proved that SrTiO3 (001) surfaces of as-supplied substrates as well as of O2-annealed substrates were predominantly terminated with TiO2 atomic plane, while the SrO atomic plane came at the topmost surface of SrTiO3 (001) homoepitaxial film. This indicates the structural conversion of the topmost atomic layer from TiO2 to SrO occurred during the SrTiO3 homoepitaxial growth. The azimuth rotational CAICISS spectra exhibited a fourfold symmetry in the surface atom alignments, showing the square lattice structure of a terminating plane.

An exact perturbation expansion, to the order 1/S2, is derived for the Heisenberg ferromagnet. The equivalence of the Dyson-Maleev (DM), Holstein-Primakoff (HP) and matching-of-matrix-element (MME) transformations is proven. They give identical T5I2 and T4 coefficients. For the planar ferromagnet, an exact expansion in anisotropy against exchange field proves the equivalence of the H P and MME transformations. Several explicit results and numerical estimates are given.

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