The Theoretical Toolbox to Describe the Electronic...
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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
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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
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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
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STM image of GaSb screw dislocations(10 µm x 10 µm)
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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
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Electronic Structure Methods
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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
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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
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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
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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=
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Surfaces in DFT – Repeated Slab Approach – Vacuum Convergence
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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
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Surfaces in DFT – Repeated Slab Approach – Slab Convergence
hydrogen passivated Si(001) film
finite size effectsslab convergence canbe slow
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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)
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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 ∫∑ −−=
∂∂
≠
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Reconstruction at Si(001) surface
surface cuts two bonds per atom• lone pairs (dangling bonds)• metallic surface• high surface energy
dangling bonds
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DFT force minimisation• surface atoms pair up• dimers form• semiconduction state • surface energy lowered
dimers
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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
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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
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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)
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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[ ]
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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
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Introducing CO
µO2(T, p) µCO(T, p)X
equilibrium(“constrained”)
G(T, p) = Etot + Fvib – TSconf + pV
DFTFP-(L)APW+loGGASupercell-Approach
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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
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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 )
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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)
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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( )
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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.
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C. Stampfl and M. Scheffler in Handbook of Surface Science Vol. 2
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CO on Ru(0001)
C. Stampfl and M. Scheffler in Handbook of Surface Science Vol. 2
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( )
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Co-adsorption - CO and O on Ru(0001)
C. Stampfl and M. Scheffler in Handbook of Surface Science Vol. 2
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.
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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
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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
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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
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Si(001) - Reconstructions
p1x1
(bulk terminated)
p2x1
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c4x2p2x2
up
down
∆Etot /per dimer : p2x1 0,057meV⎯ → ⎯ ⎯ ⎯ p2x2 0,003meV⎯ → ⎯ ⎯ ⎯ c4x2
surface unit cells
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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)
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Si(001) - ARPES: from Spectrum to Bandstructure
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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
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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
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Si(001) 2x1 - Surface Bandstructure at Γ
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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
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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 )
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Si(001) 4x2 - Surface Bandstructure at Γ
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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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