Introduction

The Pt(110) surface has been experimentally shown to undergo surface reconstruction. The mechanism for surface reconstruction is to have missing alternate rows of the top layer of the surface in a \((2\times1)\) surface unit cell. The aim of this study is to verify that the reconstructed Pt(110) is energetically favorable by computing and comparing the surface energy of Pt(110) surface with and without surface reconstruction

All energy calculations were carried out using plane-wave based DFT. The GGA based PBE functional was used to treat the exchange-correlation effects. The ion and core were treated using ultrasoft pseudopotentials generated on the fly (OTFG ultrasoft) with Koelling-Harmon relativistic treatments. Pseudo atomic calculations for Pt treated the 4f14 5s2 5p6 5d9 6s1 electrons as valence electrons and the electrons in lower levels were treated as part of the frozen core.

Calculation of Lattice constant for bulk Pt

Before attempting to compute surface energies we must verify we have the correct lattice parameters for the conventional unit cells used to build the desired surfaces. By correct lattice parameter(s) we refer to the lattice parameter(s) that minimizes the total energy calculated by our DFT code consistent with our choice of exchange correlation functional and pseudo potentials. Using any other lattice constant will introduce errors in our calculations by introducing artificial stress.

Bulk Pt naturally occurs in the FCC crystal structure. The conventional unit cell and primitive unit cell of bulk Pt are shown in figure 1. The first step of our calculations was to geometry optimize the primitive unit cell of bulk Pt, allowing the lattice parameters to relax. For this calculation we used a cutoff of \(480\;eV\) and \(k\) point separation of \(0.04\;\mathring{A}^{-1}\). These calculation parameters were the default parameters for the ‘ultra-fine’ setting for CASTEP calculations as implemented by Materials Studio™. The geometry optimization yielded a lattice constant of \(3.967\;\mathring{A}\) for the conventional unit cell (\(2.805\;\mathring{A}\) for the primitive unit cell).

The next step in our investigation is to test the convergence of the total energy with respect to the cutoff energy and \(k\) point sampling. Figure 2 shows the plots for the results of the calculations to test for convergence. From these plots we see that total energy of the primitive unit cell converges to a tolerance of \(1\;m\,eV\) for cutoff energy of \(480\;eV\) and k point separation of \(0.025\;\mathring{A}^{-1}\).

Calculation of surface energy of Pt(110) surface without surface reconstruction

The surface energy of Pt(110) was calculated by cleaving the optimized conventional unit cell of bulk Pt along the (110) plane. The cleaved unit cell was repeated infinitely in directions parallel to the cleaved plane and a finite number of times perpendicular to the cleaved plane to obtain a slab of desired thickness (in atomic layers). Both surfaces of the slab have Pt(110) surface. Now to employ plane wave based DFT calculations a vacuum slab unit cell was created containing a unit cell of the infinite slab (in directions parallel to the surface) and a vacuum layer in the direction normal to the surface. Figure 3 shows such a vacuum slab unit cell containing a 5 atomic layer thick slab separated by \(10\;\mathring{A}\) of vacuum from the next parallel slab.

Once we have defined a vacuum slab as depicted in figure 3, we fix the position of the bottom two layers to mimic bulk and allow the remaining layers of atoms to relax. In our DFT calculations for the slab unit cell, a self-consistent dipole correction was introduced to correct for the dipole rising from the asymmetry of the cell due to constraining the two bottom layers.

The surface energy (\(\sigma\)) can be calculated using,

\begin{equation} \sigma=\frac{ E_{slab}-n\,E_{bulk} }{A}. \label{surf_energy}\end{equation}

Where, \(E_{slab}\) is the total energy of the vacuum slab unit cell, \(E_{bulk}\) is the total energy of a primitive unit cell of Pt in bulk, \(A\) is the surface area of the slab in the vacuum slab unit cell and \(n\) is the number of Pt atoms in the vacuum slab unit cell.

First we need to determine the convergence tolerance of our calculation of  vacuum slab unit cell energy with respect to cutoff energy and k point sampling used. Here we chose to use the cutoff energy and k point separation used for the bulk unit cell, i.e. \(480\;eV\) and \(0.025\;\mathring{A}^{-1}\) respectively. Table 01 below lists the values obtained for the total energy of the vacuum slab unit cell (6 atomic layer slabs with vacuum separation of \(8\;\mathring{A}\)) for different cutoff and k point sampling around the chosen values. From this table we see that the total energy of the vacuum slab unit cell has a convergence tolerance of \(\sim\;10\;meV\).

Table 01 : Tests for convergence of total energy with respect to k point separation and cutoff energy

Cutoff Energy (eV)	k point separation (1/Å)	Total energy of vacuum slab (eV)
440	0.025	-78303.015
480	0.025	-78303.017
520	0.025	-78303.017
480	0.020	-78303.009
480	0.025	-78303.017
480	0.030	-78303.003

For a given choice of cutoff energy and k point sampling, the thickness of the vacuum layer and the slab thickness will be the two parameters that effect the convergence of the surface energy. In order to test the convergence of the surface energy with respect to these parameters we compute \(\sigma\) while varying vacuum thickness (for a slab with 5 atomic layers) and also  \(\sigma\) while varying slab thickness (with a vacuum thickness of \(10\;\mathring{A}\)). The plots for these calculations are shown in figure 4. From the plots presented in figure 4 we can say for a vacuum thickness of \(8\;\mathring{A}\) and a slab thickness of 6 atomic layers the surface energy converges to (keeping in mind that the convergence tolerance of the total energy of the vacuum slab with respect to energy cutoff and k point sampling is \(10\;m\,eV\)),

\begin{equation}\sigma_{unreconstructed}=125\pm 1 \;\frac{m\,eV}{\mathring{A}^2}.\label{sigma_unrec}\end{equation}

 

Calculation of surface energy of Pt(110) surface with surface reconstruction

The vacuum slab unit cell containing the reconstructed Pt(110) depicted in figure 5 was geometry optimized and the resulting total energy was used to calculate the surface energy of the reconstructed surface using \eqref{surf_energy}. Ideally we would want to retest the convergence tolerance with respect to cutoff energy, k point sampling, slab thickness and vacuum thickness. However given the computational cost of doing these calculations we chose to assume that the convergence tolerance for the surface energy is the same as surface energy calculation for the unreconstructed surface. With this assumption we get,

\begin{equation}\sigma_{reconstructed}=119\pm 1 \;\frac{m\,eV}{\mathring{A}^2}.\label{sigma_rec}\end{equation}

Conclusion

From the results given in equations\eqref{sigma_unrec} and \eqref{sigma_rec} we  see that,

\begin{equation}\sigma_{reconstructed}<\sigma_{unreconstructed}.\label{result}\end{equation}

This indicates that the “missing row” reconstructed surface of Pt(110) is energetically favorable compared to Pt(110) without surface reconstruction. This finding is in line with experimental evidence supporting this claim [1].

References

[1] H. Niehus, “Analysis of the Pt(110)–(1 × 2) surface reconstruction”, Surf. Sci., 145 (1984), pp. 407-418