1

• SUPERCELLS
• So far we have seen only 3-D crystalline systems.
  Nature is more complex than this.
• Non-periodic systems (e.g. defects, amorphous
  solids, liquids, etc)
• Problem how can we simulate them with a finite
  number of atoms?
• Solution(s) i) Clusters ? ii) Periodic
  boundary conditions ?
• B) Periodic, but not 3-D (e.g. 0-D (molecules,
  nanostructures), 1-D (quantum wires), 2-D
  (surfaces, graphene)
• Problem plane wave basis set is periodic in 3D
• Solution Add fictitious periodicity in
  non-periodic directions
• The supercell is an artificially constructed,
  large, periodically repeated, simulation cell,
  that allows the simulation of non-3D-periodic
  systems. Special care has to be paid to the
  choice of the supercell lattice parameters

2

• SUPERCELLS
• Examples
• Defects
• How to construct the supercell
• Begin by replicating the crystal unit cell a
  sufficiently large number times in all 3
  directions
• Introduce the defect (e.g., for a vacancy, remove
  one atom)
• Check convergence of properties with size of
  supercell by increasing its size (number of
  repeat units)
• NB1 charged defects may be a challenge
• NB2 defect formation energy calculated wrt bulk
  with same supercell
• Surface
• Begin by replicating the crystal unit cell a
  sufficiently large number times in the direction
  perp. to the surface.
• Add vacuum spacing (create a slab)
• Check convergence with respect to vacuum size and
  slab thickness
• NB surface formation energy calculated wrt bulk
  with same supercell
• NEED TO RELAX!