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