DFT

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DFT PracticeSurface Sciencebased on Chapter
4, Sholl Steckel

• Recap
• Supercells for surfaces
• Surface relaxation, Surface energy and Surface
  reconstruction
• More advanced topics
• Also see following article

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Required input in typical DFT calculations
VASP input files

• Initial guesses for the unit cell vectors (a1,
  a2, a3) and positions of all atoms (R1, R2, ,
  RM)
• k-point mesh to sample the Brillouin zone
• Pseudopotential for each atom type
• Basis function information (e.g., plane wave
  cut-off energy, Ecut)
• Level of theory (e.g., LDA, GGA, etc.)
• Other details (e.g., type of optimization and
  algorithms, precision, whether spins have to be
  explicitly treated, etc.)

POSCAR
KPOINTS
POTCAR
INCAR
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The DFT prescription for the total
energy(including geometry optimization)
Guess ?ik(r) for all the electrons
Self-consistent field (SCF) loop
Solve!
No
Is new n(r) close to old n(r) ?
Yes
Geometry optimization loop
Calculate total energy E(a1,a2,a3,R1,R2,RM)
Eelec(n(r) a1,a2,a3,R1,R2,RM) Enucl
Calculate forces on each atom, and stress in unit
cell
Move atoms change unit cell shape/size
No
Are forces and stresses zero?
Yes
DONE!!!
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Approximations
Approximation 1 finite number of k-points
Approximation 2 representation of wave functions
Approximation 3 pseudopotentials
Approximation 4 exchange-correlation
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The general supercell

• Initial geometry specified by the periodically
  repeating unit ? Supercell, specified by 3
  vectors a1, a2, a3
• Each supercell vector specified by 3 numbers
• Atoms within the supercell specified by
  coordinates R1, R2, , RM

a3 a3xi a3yj a3zk
a1 a1xi a1yj a1zk
a2 a2xi a2yj a2zk
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More on supercells (in 2-d)
Primitive cell
Wigner-Seitz cell
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The simple cubic supercell

• Applicable to real simple cubic systems, and
  molecules
• May be specified in terms of the lattice
  parameter a

a3 ak
a1 ai
a2 aj
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The FCC supercell

• The primitive lattice vectors are not orthogonal
• In the case of simple metallic systems, e.g., Cu
  ? only one atom per primitive unit cell
• Again, in terms of the lattice parameter a

a3 0.5(i k)
a2 0.5a(j k)
a1 0.5a(i j)
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Supercell for surface calculations

• (001) slab
• Note periodicity along x, y, and z directions
• Two (001) surfaces
• Vacuum and slab thicknesses have to be large
  enough to minimize interaction between 2 adjacent
  surfaces

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Side view of slab supercell
Slab
Supercell
Vacuum
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Yet another view
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Atomic coordinates in supercell

• The atomic positions, in terms of fractional
  coordinates, i.e., in the units of the lattice
  vector lengths are

• k-point mesh M x M x 1 (where M is determined
  from bulk calculations)
• The lattice vectors are fixed (only atomic
  positions within the supercell are optimized)
• Lattice parameter along surface plane fixed at
  DFT bulk value

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Other surfaces
Top views
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Surface unit cells

• Smallest possible surface unit cells preferred,
  but gives atoms less freedom

Smaller unit cell
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Surface unit cells
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Surface relaxation

• Once the initial slab geometry is set, the system
  is then subjected to geometry optimization, i.e.,
  the atoms within the supercell are allowed to
  adjust their positions such that the atomic
  forces are close to zero
• Surface relaxation a general phenomenon, in
  which the interplanar distances normal to the
  free surface change with respect to the bulk
  value. How? And, why?

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Surface relaxation

• Results have to be converged with respect to the
  number of layers
• Remember, larger the number of layers, more
  accurate the result, but longer is the
  computational time

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Surface relaxation Convergence

• Relaxation change in the interplanar spacing
  normal to the surface plane with respect to the
  corresponding bulk value
• Note the convergence of interplanar spacings as
  the number of layers is increased
• Also note the oscillations in the sign of the
  change in the interplanar spacing with respect to
  bulk

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Asymmetric vs. symmetric slabs

• If symmetry is exploited, symmetric slabs are
  better
• The bottom or central layers are fixed to ensure
  a bulk-like region
• The lattice vectors are fixed (only atomic
  positions within the supercell are optimized)
• Lattice parameter along surface plane fixed at
  DFT bulk value

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Surface energy

• Energy needed to create unit area of a surface
  from the bulk material
• The surface energy is an anisotropic quantity,
  being smaller for the more stable closer-packed
  surfaces
• Can be computed as

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Surface energy

• Note the quicker convergence with respect to the
  number of layers
• Experimental value is an average over a number of
  surfaces also, experimental value is surface
  free energy, while DFT value is the surface
  internal energy (i.e., DFT results are at 0 K and
  entropic effects are not taken into account)

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Surface energy
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Surface energy the Wulff constructionThe
surface energy as a polar plot
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Surface reconstruction

• Relaxation movement of atoms normal to the
  surface plane
• Reconstruction movement of atoms along the
  surface plane (what do we need to do to allow
  this?)

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The unreconstructed Si (001) surface
Surface unit cell
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The 2x1 reconstruction
Unreconstructed (001) surface
Reconstructed (001) surface

• To see this reconstruction, the surface unit cell
  has to be twice as large as the primitive cell
• Why does this reconstruction happen? To
  passivate dangling bonds

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The (7x7) Si(111) reconstruction

• When heated to high temperatures in ultra high
  vacuum the surface atoms of the Si (111) surface
  rearrange to form the 7x7 reconstructed surface

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Multi-element systems
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CdSe surfaces

• The 0001 family of surfaces are polar (i.e.,
  surface plane does not have bulk stoichiometry)
• Most of the other surfaces are nonpolar

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CdSe nonpolar surfaces Reconstructions
relaxation
Top view
Side view
Before reconstruction
After reconstruction

• The already stable nonpolar surfaces undergo a
  lot of reconstruction, and become even more stable

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CdSe polar surfaces
Top view
Side view

• 4 types of 0001 surfaces
• (0001) Cd-terminated
• (0001) Se-terminated
• (000-1) Cd-terminated
• (000-1) Se-terminated
• Display hardly any relaxation or reconstruction

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Complications Surface energy

• The fundamental difficulty If a surface plane
  does not have the same stoichiometry of the bulk
  material (e.g., polar surfaces), its surface
  energy cannot be uniquely determined! Why?
• The above formula is inadequate, as slab will
  either not have an integer number of CdSe units,
  or will not have identical top and bottom
  surfaces
• However, the following formula will work, but the
  surface energy will be dependent on the elemental
  chemical potential

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CdSe surface energies
Bare surfaces
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CdSe surface energies

• O passivation ? only the 2 (0001) surfaces are
  unstable and hence prone to growth ? nanorods

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• Rock salt (NaCl) crystal structure for all alloy
  compositions

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TiCxN1-x alloy surfaces

• Surfaces may be polar depending on orientation
  and composition

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TiCxN1-x alloy surface energies

• As with CdSe, surface energies are dependent on
  elemental (C and N) chemical potentials
• Most stable surface for a given choice of C and N
  chemical potentials can be determined
• Moreover, the allowed values of C and N
  chemical potentials to maintain a stable bulk
  alloy may be determined (hatched regions)

Conclusion (001) surfaces are the most stable,
regardless of alloy composition
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Key Dates/Lectures

• Oct 12 Lecture
• Oct 19 No class
• Oct 26 Midterm Exam
• Nov 2 Lecture
• Nov 9 Lecture
• Nov 16 Guest Lectures
• Dec 7 In-class term paper presentations