Lingjie Zhou

Introduction

Silicon(Si) is a widely used substrate for molecule beam epitaxy growth due to its high quality and low cost. However, the real surface of Si doesn’t resemble the plane directly cut from bulk material. Due to the recombination of dangling bonds, usually there will be reconstruction patterns formed at the surface. Such phenomenon have been broadly investigated and confirmed through Scanning Tunneling Microscope. To explain and predict such phenomena, we used plane-wave basis sets with ultrasoft pseudopotentials to perform DFT methods to calculate the energy of the surface with and without reconstruction.

Method

The CASTEP [1] package is used to carry out the DFT calculation. The exchange and correlation function were calculated using the Perdew, Burke and Ernzerhof(PBE) functional described within the generalized gradient approximation(GGA) [3]. The ‘on the fly’ generated ultrasoft pseudopotential for Si has a core radius of 1.8 Bohr(0.95 Angstroms) and was generated with 4 electrons in the valence panel with (3s2 3p2) as the electronic configuration.

First, the optimization of bulk material is done to make sure there is no artificial stress in the model [2]. The calculation starts from the experimentally reported result with a=5.381 A. 700eV cutoff energy and 7×7×7 kpoint set is used and the optimized value is a=5.468 Å. The number of kpoint and cutoff is initially hypothesized as sufficient before checking convergence and the convergence will be further checked.

Energy cutoff convergence

The convergence of energy cutoff is first checked by carrying out the optimization of structures with a=5.381 Å and 7×7×7 k points but different cutoff energy. The optimized result (a=5.481) is well converged if we use cutoff energy higher than 600 eV. For the rest of the calculation we would use 700 eV as cutoff energy.

Fig 1 Convergence of energy cutoff

Kpoint convergence

Here is the result if the optimization all start from experimental result(a=5.381 Å) and 700eV cutoff but use the different number of kpoints(6×6×6, 8×8×8, 10×10×10, 15×15×15). The result is well-converged so that we will pick 6×6×6 for our later kpoint mesh.

Fig 2 Convergence of number of k points

Si(001) surface

First, we calculate the surface of Si(001) without reconstruction. Slabs of 3 layers, 4 layers and 5 layers of Si atoms and 10 Å vacuum is used for the calculation. During the calculation, only the atoms of the top layer are allowed to move while the rest are fixed. The optimized structure is very similar to the bulk. To calculate the reconstruction pattern, top two atoms are shifted towards each other. The optimized structure is similar to the experimentally verified reconstruction pattern.

Fig 3 Surface without reconstruction

Fig 4 Surface with reconstruction

The layer dependence of surface energy is plotted below. Due to the limitation of time and computational resources, it’s hard to reach a well-converged result. But the surface energy of reconstructed structure is always smaller than the one without reconstruction. Thus, the reconstructed pattern is preferred. With limited time and computation resources, only one reconstruction pattern is checked and other reconstruction patterns remain to be checked.

Fig 5 layer dependence of surface energy

Reference