Ab Initio Approach for Thermodynamic Surface Phases with Full Consideration of Anharmonic Effects: The Example of Hydrogen at Si(100)

A reliable description of surfaces structures in a reactive environment is crucial to understand materials’ functions. We present a first-principles theory of replica-exchange grand-canonical-ensemble molecular dynamics and apply it to evaluate phase equilibria of surfaces in a reactive gas-phase environment. We identify the different surface phases and locate phase boundaries including triple and critical points. The approach is demonstrated by addressing open questions for the Si(100) surface in contact with a hydrogen atmosphere. In the range from 300 to 1000 K, we find 25 distinct thermodynamically stable surface phases, for which we also provide microscopic descriptions. Most of the identified phases, including few order-disorder phase transitions, have not yet been observed experimentally. Furthermore, we show that the dynamic Si-Si bonds forming and breaking is the driving force behind the phase transition between $3\times 1$ and $2\times 1$ adsorption patterns.

Figure 1

Phase diagram of the Si(100) surface in a ${\mathrm{D}}_2$ gas phase. Each color represents a thermodynamic phase with a certain deuterium coverage. The white area is the stability region of the pristine surface. The dark blue area surrounded by the orange frame is the stability region of the phase with maximum coverage of $4/3$, sampled in the $3\times 3$, and its the neighbor phases are sampled in the $4\times 4$ supercell. The white lines indicate the phase boundaries, identified by the analysis of the heat capacity. The critical points are marked as blue dots while the triple points are marked as black dots. The atomic-structure images show the top view of eight representative phases with diverse deuterium coverage $\theta$. The golden spheres are the top silicon atom, the green spheres are deuterium atoms, and the dark spheres are Si atoms in deeper layers. The five cyan dashed lines mark the boundaries of order-disorder phase transitions. The red dashed line indicates ${\mu }_{\mathrm{D}}=-0.1\mathrm{eV}$, which marks the ($T,p_{{\mathrm{D}}_2}$) path analyzed in Fig. 2. The grey area at high pressures corresponds to the region at larger chemical potentials than those sampled in this Letter.

Figure 2

Surface free energies of the Si(100) with different D coverage $\theta =4/3$ (red lines) and $\theta =1.0$ (blue lines) as a function of temperature at ${\mu }_{\mathrm{D}}=-0.1\mathrm{eV}$. ${\mu }_{\mathrm{D}}$ is chosen inside the range where both coverages are stable across a range of relevant $T$. This requirement is met with ${\mu }_{\mathrm{D}}$ in the range between $-0.1\mathrm{eV}$ and $-0.01\mathrm{eV}$. The red or blue dashed lines are the surface free energy calculated by the ab initio thermodynamic method at harmonic approximation. The red or blue solid lines are the surface free energy calculated by the REGC method. The reference is the unreconstructed bare surface.