The Theoretical Toolbox to Describe the Electronic ...w0.rz-berlin.mpg.de/imprs-cs/download/Rinnke_Th_Toolbox_DFT.pdf · The Theoretical Toolbox to Describe the Electronic Structure

The Theoretical Toolbox to Describe the Electronic Structure of Surfaces

Patrick RinkeFritz-Haber-Institut der Max-Planck-Gesellschaft

Faradayweg 4-6, D-14195 [email protected]

Acknowledgements: Jutta Rogal, Philipp Eggert and Karsten Reuter

10/10/2005 The Theoretical Toolbox to Describe the Electronic Structure of Surfaces 1

mailto:[email protected]

Surfaces of Solids

STM image of “atomic” scriptAFM image of magnetic hard drive(25 µm x 25 µm). Wires are about2000 atoms wide

STM image of electron standing waves ata Ag step

General: - surface is the skin of the solid

Applications: - Microelectronics and semiconductor devicesControlled atom manipulation at surfaces (Nano…)Surface electronic structure and transport at surfacesCrystal growth and epitaxy

- Heterogeneous catalysisChemical bonds at surfaces

- Corrosion / mechanical failureSegregation of minority ingredients Fracture of engineering materialsPassivation, coating layers

Fundamental: Symmetry break (3D → 2D)New localized electronic and vibrational states (surface states & surface phonons)Continuum of states vs. discrete gas particle states


Controlled surface studies: Surface Science

Real vs. single crystal surfaces

- Real surfaces are very complex and often ill defined: polycrystalline materials, disorder, grain boundaries, defects and other irregularities

- Highly dependent on the environment (gas adsorption)- Segregation of impurities depends on sample treatment

⇒ Normal surface experiments often not reproducible (sometimes not even qualitatively!)

⇒ One Solution: the Surface Science Ansatz- Study low-index surfaces of single crystals. - Understand these “idealized” surfaces first, then introduce defects/irregularities in a controlled manner. - Gradually make systems more complex and hope that such systems provide good models to real problems.

(100)

(111) (110)

SEM image of polycrystalline Cu


STM image of GaSb screw dislocations(10 µm x 10 µm)

Experiment and Theory

Experiment

• theory development• first principles simulations

Quantum Mechanics

Physics

• geometric & vibrationalstructure at surfaces

• surface composition• surface electronic structure

Theory

IN OUT Prominent techniques

electrons electrons LEED, RHEED, AES, HREELSphotons photons SXRD, IRAS/RAIRSphotons electrons XPS, UPSelectrons photons IPESions ions ISS/LEIS, SIMS

Special: STM/STS, AFM, TPD


Electronic Structure Methods


quantitative description(as accurately as possible)

∑ ∑ ∑∑∑≠ ≠== −

+−

−−

++=JI Ii ji jiIi

I

JI

JIN

i

iL

I I

I

rre

RreZ

RReZZ

mp

MPH

,

222

1

2

1

2

||21

||||21

22

Free-electron model Indept. el. approximationJellium model Fermi-EnergyDrude/Sommerfeld th. Transport

Band theory Brillouin zoneKronig-Penney Band structure, DOSNearly-free el. model Band gaps, metal/insulator

LCAO Bandwidth↔overlapTight binding s,p,d-bands(Extended) Hückel

Homogeneous electron gas Exchange/correlationThomas-Fermi Theory ScreeningRandom Phase Approx. Quasi-particle concept

(Fermi liquid theory)

Quantum chemistry Hartree-Fock theory- Single reference

Møller-Plesset (MP)Conf. interaction (CI)Coupled cluster (CC) Density-Functional Theory

- Multi reference - LDAMulticonf. SCF (MCSCF) - GGAs (PBE, BLYP)Complete active space - Meta-GGAsSCF (CASSCF) - OEP/EXX (B3LYP)

Quantum Monte-Carlo

Many-Body Perturbation Theory- GW- BSE

Scattering Theory- KKR in LDA, GGA, GW

Tight Binding

Interatomic Potentials- Pair potentials, force fields- Cluster potentials (Stillinger-

Weber, Keating, (M)EAM, BOP…)

Born-Oppenheimer Approximation:

H = Tel + Vnucl-el + Vel-el

Many-body Schrödinger Equation:

qualitative description(conceptual aspects)

somewhere inbetween

Representation of Surface

• Quantum Chemistry

• Quantum Monte Carlo

• Hartree-Fock

• Density-Functional Theory

• GW, BSE

• Tight-Binding

• Interatomic Potentials

• Scattering Theory

• Density-Functional Theory

• GW

• Tight-Binding


DFT - Groundstate

Hohenberg-Kohn Theorem:

• Ground state energy is unique functional of the density n(r)

• universal functional:

• variational:

• Exchange-correlation:

• Hartee Energy:

• exact unknow fl suitable approximations:- local density approximation (LDA), gradient corrected (GGA)

[ ] [ ] ( ) ( ) rrr dnvnFnE exttot ∫+=

[ ] 00ˆˆ Ψ+Ψ= UTnF

[ ] 0=nnE

δδ

[ ] [ ] [ ] [ ]nEnETnEnE Hexttotxc −−−=

[ ]nExc

[ ] ( ) ( ) ( )∫∫ −= r'rr'rr'r ddvnnnEH 21

: kinetic energy

: electron-electron interaction

T

U

extvn ⇔

minimum at exact density


DFT – Kohn-Sham Scheme

Kohn-Sham:

map system of interacting electrons onto fictitious system of non-interacting electrons that reproduce the exact density

• Kohn-Sham equation:

• Density:

• Hartee potential:

• Exchange-correlation potential:

• in practice: start with trial density and then iterate to self-consistency

( ) ( ) ( ) ( ) ( )rrrrr iiixcHext vvv φεφ =⎥⎦

⎤⎢⎣

⎡+−+

∇−

2

2

( ) ( )∑=occ

iin 2rr φ

( ) ( ) ( )∫ −= rr'rrr dvnvH

( ) [ ]( )r

rn

nEv xcxc δ

δ=


Surfaces in DFT – Repeated Slab Approach – Vacuum Convergence


Density and Vxc in LDA/GGA decay exponentially outside the surface

slabs decoupledonly very short ranged interactions in LDA/GGA z-direction

Ele

ctro

n de

nsity

SlabVacuum

hydrogen passivated Si(001) film

Surfaces in DFT – Repeated Slab Approach – Slab Convergence

hydrogen passivated Si(001) film

finite size effectsslab convergence canbe slow


DFT Total Eneregies - Potential Energy Surfaces

Schrödinger Equation:

• Potential Energy Surface (PES) or also Born-Oppenheimer Surface:

( ) ( ) ( ) ( )}{}{}{}{ˆ RRRR Ψ=Ψ totEH

( )}{RtotE

GaAs(001) ζ(4×2)As Ga

Potential-energy surface for the adsorption of As (left panel) and Ga (right panel) on the Ga-rich GaAs(001)(4×2) surface. The contour spacing is 0.15 eV. Light regions indicate low-energy adsorption positions.

• As prefers site with 3-fold Ga coordination

• Ga prefers the trenchTop and side view of the relaxed GaAs(001) ζ(4×2) surface. Light (dark) balls represent Ga (As) atoms.

K. Seino et. al., Surf. Sci. 507, 406 (2002)


Forces in DFT

Schrödinger Equation:

• Potential Energy Surface (PES) or also Born-Oppenheimer Surface:

• Assume motion of nuclei as classic:

• Hellmann-Feynman force:

• Forces in DFT:

( ) ( ) ( ) ( )}{}{}{}{ˆ RRRR Ψ=Ψ totEH

( )}{RtotE

ii FR =dtdM i

( ) ( ) 00 }{ˆ}{ Ψ∂∂

Ψ−=∂∂

−= RR

RR

Fii

i HEtot

( ) ( ) ( ) ( )444 3444 21444 3444 21

part electronicpartnuclear

21}{ 3

23

2

rRrRrrRR

RRR

R ii

ji

jii

d--neZ-

-

eZZE i

ij

jitot ∫∑ −−=

∂∂

≠


Reconstruction at Si(001) surface

surface cuts two bonds per atom• lone pairs (dangling bonds)• metallic surface• high surface energy

dangling bonds


DFT force minimisation• surface atoms pair up• dimers form• semiconduction state • surface energy lowered

dimers

Phase diagram – ab initio thermodynamics

surface free energy γ:

• Gibbs free energy G:

• Helmholtz free energy F:

• For solids pV and Fvib are typically small:

γ =1A

G T, p,{Ni},{Ri}( )− Niµii

∑⎡

⎣ ⎢

⎤

⎦ ⎥

A : surface area Ni : number of species i µi : chemical potential of species i

G T, p,{Ni},{Ri}( )= F T,V ,{Ni},{Ri}( )+ pV T, p,{Ni},{Ri}( )

F T,V ,{Ni},{Ri}( )= E V ,{Ni},{Ri}( )+ Fvib T,V ,{Ni},{Ri}( )

γ ≈1A

E V ,{Ni},{Ri}( )− Niµii

∑⎡

⎣ ⎢

⎤

⎦ ⎥

DFT


First-principles atomistic thermodynamics applied to oxidation of Pd(100)

µO2(T, p)

G(T, p) = Etot + Fvib – TSconf + pV

DFTµΟ (T, p) = ½ µΟ (T, p0) + ½ kT ln(p/p0)

2

FP-(L)APW+loGGASupercell-Approach


C.M. Weinert and M. Scheffler, Mater. Sci. Forum 10-12, 25 (1986);

E. Kaxiras et al., Phys. Rev. B 35, 9625 (1987)

K. Reuter and M. Scheffler, Phys. Rev. B 65, 035406 (2002);

Phys. Rev. Lett. 90, 046103 (2003)

Stability of different phases on Pd(100)

∆µO (eV)

pO (atm)

(√5 × √5)R27°

p(2 × 2)

c(2 × 2) clean Pd(100)

T=300 K

T=600 K

∆G

(meV

/Å2 )

metal adla

yer

surf

. oxi

de

bulk oxide

∆G ∆µ0( ) ≈ −1A

EO@Mtot − EM

tot − NO12

EO2

tot + ∆µ0

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡

⎣ ⎢ ⎤

⎦ ⎥ = −1A

NO Eb + NO∆µ0[ ]


Stability of different phases on Pd(100)

Experiment Theory

E. Lundgren et al., Phys. Rev. Lett. 92, 046101 (2004).

µΟ (T, p) = ½ µΟ (T, p0) +½ kT ln(p/p0)

2


Introducing CO

µO2(T, p) µCO(T, p)X

equilibrium(“constrained”)

G(T, p) = Etot + Fvib – TSconf + pV

DFTFP-(L)APW+loGGASupercell-Approach


Introducing CO - Structures

hollow bridge on-top

• (√5 × √5)R27° surface oxide:

• 4 top, 2 bridge, 6 hollow, 2 hollow-substitutional sites • adsorption of O, CO, vacancies, mixed phases ...

⇒ close to 200 structures considered !!10/10/2005 The Theoretical Toolbox to Describe the Electronic Structure of Surfaces 19

3D-Theoretical “phase diagram”

00

-0.5

-1.0

-1.5

-1.0

-0.5

-1.5-2.0

-2.5200

100

0

-100

-200

∆µ O (eV)∆µ

CO (eV)

∆G(m

eV/Å

2 )


Theoretical “phase diagram” in an O2 and CO environment (constrained equilibrium)

∆µO (eV)

∆µ C

O(e

V)

p CO

(atm

)

pO (atm)surface oxide + 2CO bridge

300 K600 K

surface oxide + CO bridge

PdO bulksurface oxide(√5 × √5)R27°p(2 × 2)-O/Pd(100)

clean Pd(100)

c(2√2 × √2)R45°CO/Pd(100)

(1 × 1)-CO bridge/Pd(100)


Wavefunctions and Density - Insights into the Spatial Distribution of the Electrons

( ) ( ) ( ) ( ) ( )rrrrr iiixcHext vvv φεφ =⎥⎦

⎤⎢⎣

⎡+−+

∇−

2

2

Kohn-Sham equation:

• Electron density:

• Density difference:(adsorption, desorption, adlayers, defects)

• Difference density:(adsorption)

wavefunctions

single particle energies(atomic/molecular levels, bandstructure)

( ) ( )∑=occ

iin 2rr φ

∆n r( )= n r( )− nref

surface r( )

n∆ r( )= n r( )− nref

surface r( )− nrefadsorbate r( )


CO Adsorption at Transition Metal Surfaces - A Model System

Wavefunctions and energies: three outer valence orbitals of molecular CO

Electron density of the valence molecular orbitals of a free CO molecule and their DFT-GGA Kohn-Sham eigenvalues (far left) with respect to the vacuum level. The lower and upper small black dots represent the positions of the C and O atoms, respectively. The first contour lines are at 8 x 10-3 bohr-3, except for the 2π∗ orbital where it is 15 x 10-3 bohr-3, and the highest- valued contour lines are at 0.5, 0.3, 0.2, 0.15, and 0.15 bohr-3 for the 3σ, 4σ, 1π, 5σ, and 2π∗ orbitals, respectively.


C. Stampfl and M. Scheffler in Handbook of Surface Science Vol. 2

CO on Ru(0001)


Electron density distribution of the CO-derived states for CO adsorbed on the on-top site of Ru(0001) and their DFT-GGA Kohn-Sham eigenvalues (far left) with respect to the vacuum level. The lower and upper small black dots represent the positions of the C, O and Ru atoms, respectively.

n r( )

n∆ r( )= nCO @ Ru(0001) r( ) − nRu(0001) r( )− nCO r( )


Co-adsorption - CO and O on Ru(0001)


Perspective and side views of the various phases of O and CO on Ru(0001). Large and small (green and red) circles represent Ru, O and C atoms, respectively. The lower panel shows the electron density of the valence states. The contour lines are in bohr-3 and the distance in Angström.


Bandstructure - Electronic Structure of Bulk Silicon

Kohn-Sham equation: −

∇2

2+ vext r( )− vH r( )+ vxc r( )

⎡

⎣ ⎢

⎤

⎦ ⎥ φnk r( )= εnkφnk r( )

Brillouin zone

fcc crystal structure


Projected Bandstructure - Si -> Si(001)

Broken translational symmetry at surface -> k no longer a good quantum number, but k||

E = En (k|| ,{k⊥}) := E PBS (k|| )

Projected Bandstructure:

Bulk SiBulk Si in Si(001) p1x1 surface Brillouin zone


Surface Bandstructure - Si(001)

dangling bonds

Bulk terminated Si(001) surface:• 2 lone pairs (dangling bonds)• metallic surface

from Schmeidts et al., Phys. Rev. B 27, 5012 (1983)

bridge bond state

dangling bond state


Si(001) - Reconstructions

p1x1

(bulk terminated)

p2x1


c4x2p2x2

up

down

∆Etot /per dimer : p2x1 0,057meV⎯ → ⎯ ⎯ ⎯ p2x2 0,003meV⎯ → ⎯ ⎯ ⎯ c4x2

surface unit cells

Si(001) - ARPES

DFT total energy calculations predict c4x2 as ground state, but p2x2 is only 3 meV/dimer higher in energy -> alternative criterium

ARPES

c4x2 2x1

ARPES : Angle Resolved PhotoEmission Spectroscopy from Enta et al., Phys. Rev. Lett. 65, 2704 (1990)


Si(001) - ARPES: from Spectrum to Bandstructure


from Enta et al., Phys. Rev. Lett. 65, 2704 (1990)

c4x2 2x1

ARPES

Si(001) 2x1 experiment <-> theory

Theory: Rohlfing et al., PRB 52, 1905 (1995)Exp: Uhrbert et al., PRB 24, 4684 (1981)

Johansson et al., PRB 42, 1305 (1990)

c4x2 : open symbols2x1 : solid symbols

Si(001) 2x1 - Surface Bandstructure

Projection onto atomic orbitals of dimer:

φnk (r) ≈ cnkµµ∑ χµ (r) → Nnk (M) = cnkµχµ (r)

µ ∈M∑

2

= cnkµ

* cnkµ χµ χνµ,ν ∈M∑

from DFT code


Si(001) 2x1 - Surface Bandstructure at Γ


black diamonds : projection onto dimersred squares : projection onto surface layer

nP (r) = fnkwk φnk (r) 2

nk ∈P∑ → OP = nP (z)dz

surface state

b

c

∫ nP (z)dza

c

∫

surface resonance

fnk : occupation factor, wk : k-point weight

Si(001) 2x1 - Projected Density of States

Density of states: Projected density of states:

N DOS (ε) = wnkδ(nk∑ ε −εnk ) Nν

PDOS (ε) = wnk χν φnk2δ(

nk∑ ε −εnk )


Si(001) 4x2 - Surface Bandstructure at Γ


Further Reading

• Handbook of Surface Science, ed. S. Holloway and N. V. Richardson, Elsevier Science (Amsterdam, 2000)

• Theoretical Surface Science – a Microscopic Perspective, A. Gross, Springer (Berlin, 2002)

• Principles of Surface Physics, F. Bechstedt, Springer (Berlin Heidelberg 2003)

• Modern Techniques of Surface Science, D.P. Woodruff and T.A. Delchar,Cambridge Univ. Press (Cambridge, 1994)

• Physics at Surfaces, A. Zangwill, Cambridge Univ. Press (Cambridge, 1988)

• Principles of Adsorption and Reaction on Solid Surfaces, R. Masel, Wiley (New York, 1996)

• Solid State Physics, N.W. Ashcroft and N.D. Mermin, Saunders College (Philadelphia, 1976)


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The Theoretical Toolbox to Describe the Electronic ...w0.rz-berlin.mpg.de/imprs-cs/download/Rinnke_Th_Toolbox_DFT.pdf · The Theoretical Toolbox to Describe the Electronic Structure