Tutorial II:

- Periodic Density Functional Theory Simulations

Solids and Surfaces

FHI-aims workshop

June 24, 2009

Paula Havu and Mina Yoon

Objectives of This Tutorial

o Periodic boundary conditions with DFT

o Cohesive properties of solids

o Basic properties of surfaces

o Electronic band structures & density of states

�r�R

exp(i�G ·�R) = 1

DFT with Periodic Boundary Conditions

Bravais lattice, a1,2,3: primitive vectors : integers

o Bloch theorem:

a1

a2

o Reciprocal lattice:

, where

Real space Reciprocal space

�k�G

�r�R

DFT with Periodic Boundary Conditions

Bravais lattice, a1,2,3: primitive vectors : integers

o Bloch theorem:

a1

a2

o Reciprocal lattice:

, where

Real space Reciprocal space

First Brillouin Zone

�k�G

exp(i�G ·�R) = 1

DFT with Periodic Boundary Conditions

Bravais lattice, a1,2,3: primitive vectors : integers

oBloch theorem:

a1

a2

o Reciprocal lattice:

, where

o KS equation : k-dependento Block like generalized basis function:

o ex: electron density

exp(i�G ·�R) = 1

Primitive Unit Cell vs. Brillouin Zone

Primitive fcc unit cellReal Space

Brillouin ZoneReciprocal Space

geometry.in (fcc Si, a=3.8Å):

lattice_vector 0.0 1.9 1.9lattice_vector 1.9 0.0 1.9lattice_vector 1.9 1.9 0.0

a

Silicon: fcc vs. diamond

•Diamond (=fcc with a basis)

Primitive unit cellReal Space

geometry.in (fcc Si, a=3.8Å):

lattice_vector 0.0 1.9 1.9lattice_vector 1.9 0.0 1.9lattice_vector 1.9 1.9 0.0 atom 0.0 0.0 0.0 Si

geometry.in (diamond Si a=5.4Å):

lattice_vector 0.0 2.7 2.7lattice_vector 2.7 0.0 2.7lattice_vector 2.7 2.7 0.0

atom 0.0 0.0 0.0 Siatom 1.35 1.35 1.35 Si

aa

•fcc

a

Convergence Test: Number of k-points

•Metallic example: fcc Si

•Semiconductor example: Diamond Si

Monkhorst-Pack grid(shift of k-grid by 0.5,0.5,0.5)

Monkhorst-Pack grid(shift of k-grid by 0.5,0.5,0.5)

(shift of k grid by 0.5,0.5,0.5)

(shift of k grid by 0.5,0.5,0.5)

Energy Convergence: Structural Stability

Small number of k-points:

Fcc< bcc < diamond

Sufficient number of k-points:

Diamond < bcc < fcc

Cohesive properties of solids

How to evaluate cohesive properties of solids?Birch-Murnaghan equation of states

Unit cell volume (V)

Tota

l ene

rgy

(E)

From elasticity theory, deformation of an isotropic medium

V: volumeV0: optimum volumeB0: bulk modulusB0’: derivative of B0 (w.r.t. pressure)E0: equilibrium energy

Cohesive energy:

E(V ) = E0 +B0VB′

0

((V0/V )B′

0

B′0−1

+1

)− B0V0

B′0−1

Ecoh = −Ebulk−N ×Eatom

N

Formation of Electronic Bands: Si (Diamond)

1s

2s

2p

3s

3p

4s3d

Ener

gy le

vel

atom (dSi-Si >> 1)

hybridization

solid

antibonding

bonding

Egap

Electronic Band of Diamond Si

[001]

[100]

[111]

[010]

[111] [100]

Valence band

Conduction band

Note: This is LDA results, which underestimatethe experimental gap of 1.17 eV,

Si is an indirect gap semiconductor

X

X

�: Gaussian broadening w= k-point weight

Electronic Structure: Density of States

K

DOS

Density of States: Broadening & k-points

E-EF (eV)

45�45�45, �=0.1

DO

S

0 5-5-10

DO

S

E-EF (eV) 0 5-5-10

3�3�3

12�12�12

30�30�30

�=1.0

�=0.3

�=0.001

E-EF (eV) 0 5-5-10

DO

S

Surface structure

truncated bulk geometry

formation of dimers

dimer buckling

alternating buckling

Example: different Si(100) surface geometries

0

4

2

-2

-4

0

4

2

-2

-4

Energ

y (

eV

)

� �J JK K

Surface band structure

Si truncated bulk geometry Si 2x2 surface reconstruction

Construction of Surfaces: Si(100) surface

Surface: Semi-infinite solid

Slab

supercell

(100)

(010)

(111)

(000)

Construction of “slab”

Vacuum

Convergence of Supercell Parameters

o Two new convergence parameters: vacuum length & number of atom layerso Reconstruction of Si(100) will be discussed in the Tutorial V.

Tips & Tricks: o Unphysical interactions between unsaturated charges on the top & bottom surface can be prevented by saturating with H (Tutorial V).o Dipole layer correction

Number of atom layersin slab Vacuum layer thickness (Å)

2 6 10 16

E / a

tom

[eV

]

-7847.104

-7847.112

-7847.108

Here: Basis set overlap

bulk

Esur f =Eslab−N ×Ebulk

2A

Eslab(N) = 2A×Esur f −N ×Ebulk

Energy required to create an unit area of surface:

How to calculate the reference energy (bulk energy)?

3) Apple to apple:

k x k x 1 k x k x 4

Different symmetry from surfaceBulk+surface need to be convergedWith k-mesh

1) bulk:

2) fitting:

Surface Energy

Overview

Part III: Basic properties of Si(100) surfaces

oGeneration & visualization of surfaces

oSurface band structures & surface energy

Part II: Electronic properties of bulk Si

oElectronic band structure& density of states

Part I: Basic properties of bulk Si & convergence test

oGeneration & visualization of bulk structures

oEnergy convergence tests

oRelative structural stability of different phases

oCohesive properties of bulk structures

Bonus problem!

Surface electronic structure of quantum thin films

2:30-3:10 (40min)

3:10-3:40 (30min)

3:40-4:10 (30min)

4:10-4:30 (20min)

5:00-5:20 (20min)

5:20-6:00 (40min)

2:30-4:30 (120min)

4:30-4:30 (30min)

5:00-6:00 (60min)