Attractive step-step interactions, tricriticality, and faceting in the orientational phase diagram of silicon surfaces between [113] and [114]

A synchrotron x-ray scattering study is presented of the orientational phase diagram of Si surfaces misoriented by up to 6° from the cubic [113] direction towards [001] and for temperatures between 300 and 1500 K. At the highest temperatures (above 1223 K), the surface is uniformly stepped. In this region, the intensity of near-specularly scattered x rays increases with decreasing temperature, suggesting a corresponding increase in the surface roughness. Specifically, the temperature dependence of the intensity of the near-specular diffuse scattering may be described as a power law versus reduced temperature: ${\mathit{I}}_{\mathit{D}}$∼${\mathit{t}}_{\mathit{s}}^{\mathrm{-}\lambda }$, with λ=0.76±0.2 (${\mathit{t}}_{\mathit{s}}$(θ)=[T-${\mathit{T}}_{\mathit{s}}$(θ)]/${\mathit{T}}_{\mathit{s}}$(θ), where ${\mathit{T}}_{\mathit{s}}$(θ) is spinodal temperature for surface misorientation θ). Below ${\mathit{T}}_{\mathit{t}}$=1223±40 K, there is a two-phase region in which (113) facets appear in coexistence with the stepped phase. We identify ${\mathit{T}}_{\mathit{t}}$ as a tricritical point. The misorientation angle at the phase boundary separating one- and two-phase regions may be also described as a power law versus reduced temperature: θ=${\mathrm{\theta }}_0$ ${\mathit{t}}^{\mathrm{\beta }}$, with β=0.42±0.10 [t=(${\mathit{T}}_{\mathit{t}}$-T)/${\mathit{T}}_{\mathit{t}}$]. The behavior of the intensity of the diffuse scattering above ${\mathit{T}}_{\mathit{t}}$ and of the phase boundary below ${\mathit{T}}_{\mathit{t}}$ can be understood qualitatively on the basis of a mean-field theory incorporating a direct attractive interaction between steps. However, the observed tricritical exponents are not given correctly. For temperatures between a triple point at ${\mathit{T}}_3$=1134±40 and 300 K, coexistence between the (113) facet and the (114) facet is found.