Method and Theory

First, an initial system is generated. This is accomplished by

  1. Starting with a unit cell system.

  2. Generating a transformed system by rotating the unit cell such that the new system’s box vectors correspond to crystallographic directions, and filled in with atoms to remain a perfect bulk cell when the three boundaries are periodic.

  3. All atoms are shifted by a fractional amount of the box vectors if needed.

  4. A supercell system is constructed by combining multiple replicas of the transformed system.

Two LAMMPS simulations are then performed that apply an energy/force minimization on the system, and the total energy of the system after relaxing is measured, \(E_{total}\). In the first simulation, all of the box’s directions are kept periodic (ppp), while in the second simulation two are periodic and one is non-periodic (ppm). This effectively slices the system along the boundary plane creating two free surfaces, each with surface area

\[A = \left| \vec{a_1} \times \vec{a_2} \right|,\]

where \(\vec{a_1}\) and \(\vec{a_2}\) are the two lattice vectors corresponding to the periodic in-plane directions.

The formation energy of the free surface, \(E_{f}^{surf}\), is computed in units of energy over area as

\[E_{f}^{surf} = \frac{E_{total}^{surf} - E_{total}^{0}} {2 A}.\]

The calculation method allows for the specification of which of the three box dimensions the cut is made along. If not specified, the default behavior is to make the \(\vec{c}\) vector direction non-periodic. This choice is due to the limitations of how LAMMPS defines triclinic boxes. \(\vec{c}\) vector is the only box vector that is allowed to have a component in the Cartesian z direction. Because of this, the other two box vectors are normal to the z-axis and therefore will be in the cut plane.