DFT – Practice Surface Science [based on Chapter 4, Sholl & Steckel]
description
Transcript of DFT – Practice Surface Science [based on Chapter 4, Sholl & Steckel]
DFT – PracticeSurface Science
[based on Chapter 4, Sholl & Steckel]
• Recap• Supercells for surfaces• Surface relaxation, Surface energy
and Surface reconstruction• More advanced topics• Also see following article:
Required input in typical DFT calculations
• 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
VASP input files
The DFT prescription for the total energy(including geometry optimization)
)()()(2
22
rrrvm iiieff
Guess ψik(r) for all the electrons
Is new n(r) close to old n(r) ?
Calculate total energyE(a1,a2,a3,R1,R2,…RM) = Eelec(n(r); {a1,a2,a3,R1,R2,…RM}) + Enucl
Yes
NoSolve!
occ
iik
BZ
kk rwrn
2)(2)(
Calculate forces on each atom, and stress in unit cell
Move atoms; change unit
cell shape/sizeYes
DONE!!!
NoAre forces and stresses zero?
Self-consistent field (SCF) loop
Geometry optimization loop
Approximations
)()()(k2
22
rururvim ikikikeff
veff (r) v(r)e2 n(r')
r r'd3r'
Exc[n(r)]n(r)
Approximation 1: finite number of k-points
Approximation 2: representation of wave functions
Approximation 3: pseudopotentials
Approximation 4: exchange-correlation
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}
a1 = a1xi + a1yj + a1zk
a2 = a2xi + a2yj + a2zk
a3 = a3xi + a3yj + a3zk
More on supercells (in 2-d)
Primitive cellWigner-Seitz cell
The simple cubic “supercell”
• Applicable to real simple cubic systems, and molecules• May be specified in terms of the lattice parameter a
a1 = ai
a2 = aj
a3 = ak
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
a1 = 0.5a(i + j)
a2 = 0.5a(j + k)a3 = 0.5(i + k)
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
Side view of slab supercell
Supercell
Slab
Vacuum
Yet another view
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
Other surfaces
Top views
Surface unit cells
• Smallest possible surface unit cells preferred, but gives atoms less “freedom”
Smaller unit cell
Surface unit cells
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?
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
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
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
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
bulkslab nEEA
2
1
Energy of entire supercell containing n atoms
Energy per atom of bulk material
Surface unit cell area
Two surfaces per supercell
Units: eV/A2 = 16.02 J/m2
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)
Surface energy
Surface energy – the Wulff constructionThe surface energy as a polar plot
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?)
The unreconstructed Si (001) surface
Surface unit cell
The 2x1 reconstruction
• 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
Unreconstructed (001) surfaceReconstructed (001) surface
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
Multi-element systems
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
CdSe nonpolar surfaces Reconstructions & relaxation
• The already stable nonpolar surfaces undergo a lot of reconstruction, and become even more stable
Top view Side view Top view Side view
Before reconstruction
After reconstruction
CdSe polar surfaces
• 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
Top view Side view
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
bulkslab nEEA
2
1
bulkSeCd
SeSeCdCdslab
E
nnEA
2
1
CdSe surface energiesBare surfaces O covered surfaces
CdSe surface energies
• O passivation only the 2 (0001) surfaces are unstable and hence prone to growth nanorods
• Rock salt (NaCl) crystal structure for all alloy compositions
Ti
N (or C)
TiCxN1-x alloy surfaces• Surfaces may be polar depending on orientation and composition
<001>
TiC0.5N0.5
TiC0.5N0.5
TiN
TiN
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
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