DFT – Practice
Surface 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
VASP input files
•
Initial guesses for the unit cell
vectors (a1, a2, a3) and positions of
all atoms (R1, R2, …, RM)
POSCAR
•
k-point mesh to “sample” the
Brillouin zone
KPOINTS
•
Pseudopotential for each atom type
POTCAR
•
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.)
INCAR
The DFT prescription for the total energy
(including geometry optimization)
Guess ψik(r) for all the electrons
BZ
occ
k
i
n(r )  2 wk   ik (r )
2
 

 2  veff (r ) i (r )   i i (r )

 2m

2
Self-consistent
field (SCF) loop
Solve!
Is new n(r) close to old n(r) ?
No
Yes
Calculate total energy
E(a1,a2,a3,R1,R2,…RM) = Eelec(n(r); {a1,a2,a3,R1,R2,…RM}) + Enucl
Geometry
optimization loop
Calculate forces on each atom, and stress in unit cell
Are forces and stresses zero?
Yes
DONE!!!
No
Move atoms;
change unit
cell shape/size
Approximations
 2

2
  ik   veff (r )uik (r )   ikuik (r )

 2m

Approximation 1: finite number of k-points
Approximation 2: representation of wave functions
v eff (r)  v(r)  e
2

n(r') 3 E xc [n(r)]
d r'
r  r'
n(r)
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}
a3 = a3xi + a3yj + a3zk
a1 = a1xi + a1yj + a1zk
a2 = a2xi + a2yj + a2zk
More on supercells (in 2-d)
Primitive cell
Wigner-Seitz cell
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
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)
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
Slab
Supercell
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
Units: eV/A2 = 16.02 J/m2
1
Eslab  nEbulk 

2A
Two surfaces
per supercell
Energy per atom of bulk material
Energy of entire supercell containing n atoms
Surface unit cell area
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 construction
The 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
Reconstructed (001) surface
Unreconstructed (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
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
Top view
Side view
Top view
Side view
Before
reconstruction
After
reconstruction
• The already stable nonpolar surfaces undergo a lot of
reconstruction, and become even more stable
CdSe polar surfaces
Top 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
Side view
Complications: Surface energy

1
Eslab  nEbulk 
2A
• 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
1
Eslab  nCd Cd  nSe  Se 

2A
Cd   Se  Ebulk
CdSe surface energies
Bare surfaces
O covered surfaces
CdSe surface energies
• O passivation  only the 2 (0001) surfaces are unstable
and hence prone to growth  nanorods
Ti
N (or C)
• Rock salt (NaCl) crystal
structure for all alloy
compositions
TiCxN1-x alloy surfaces
• Surfaces may be polar depending on orientation and composition
<001>
TiC0.5N0.5
TiN
TiN
TiC0.5N0.5
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
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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