Absolute orientation-dependent anisotropic TiN(111) island step energies and stiffnesses from shape fluctuation analyses

Abstract

In situ high-temperature (1165--1248 K) scanning-tunneling microscopy was used to measure temporal fluctuations about the anisotropic equilibrium shape of two-dimensional TiN(111) adatom and vacancy islands on atomically smooth TiN(111) terraces. The equilibrium island shape was found to be a truncated hexagon bounded by alternating 〈110〉 steps, which form [100] and [110] nanofacets with the terrace. Relative step energies \ensuremath{\beta} as a function of step orientation \ensuremath{\varphi} were obtained from the inverse Legendre transformation of the equilibrium island shape to within an orientation-independent scale factor \ensuremath{\lambda}, the equilibrium chemical potential of the island per unit TiN area. We find that for alternating ${S}_{1}$ and ${S}_{2}$ 〈110〉 steps, the ratio ${\ensuremath{\beta}}_{1}/{\ensuremath{\beta}}_{2}=0.72\ifmmode\pm\else\textpm\fi{}0.02.$ The parameter \ensuremath{\lambda} and, hence, absolute orientation-dependent values of \ensuremath{\beta}(\ensuremath{\varphi}) and step stiffnesses $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\beta}}(\ensuremath{\varphi})$ were extracted from quantitative shape fluctuation data using an exact theoretical approach valid for anisotropic islands. For the two 〈110〉 steps, we obtain ${\ensuremath{\beta}}_{1}=0.23\ifmmode\pm\else\textpm\fi{}0.05$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\beta}}}_{1}=1.9\ifmmode\pm\else\textpm\fi{}1.1\mathrm{eV}/\AA{}$ with ${\ensuremath{\beta}}_{2}=0.33\ifmmode\pm\else\textpm\fi{}0.07$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\beta}}}_{2}=0.08\ifmmode\pm\else\textpm\fi{}0.02\mathrm{eV}/\AA{}$ over the observed temperature range. Due to the correspondingly high kink energies, TiN(111) step energies exhibit only a very weak temperature dependence between 0 K and the maximum measurement temperature 1248 K.