Technische Universität München

Part III: Theoretical Surface Science

Clean Surfaces

Karsten Reuter

Lecture course: Solid State Theory

From ideal to real

Up to now: - infinite and perfectly homogeneous solid (ideal lattice) - harmonic lattice vibrations (phonons) - single electron picture (band theory) - electronic and phononic system completely separated (BO approx.)

- beyond the harmonic crystal approximation (phonon-phonon scattering, phonon lifetimes, thermal expansion…)

- beyond mean-field electron correlation („correlated materials“, Mott insulators, Hubbard models,…) - electron-phonon coupling (electron-phonon scattering, Peierls distortions, polarons, low-T supraconductivity/BCS theory, …) - beyond the ideal lattice (point defects, impurities, vacancies, localized states, F centers, …) - beyond the infinite crystal (surface/interface states, band bending, …)

Get real!

Surfaces everywhere…

Phase I

Phase II (T,p) solid – gas solid – liquid solid – solid (“interface”) …

„Skin of a solid“

− N.W. Ashcroft and N.D. Mermin, “Solid State Physics”, Saunders College

(Philadelphia, 1976) − A. Zangwill, “Physics at Surfaces”, Cambridge Univ. Press (Cambridge, 1988) − M. Henzler and W. Göpel, “Oberflächenphysik des Festkörpers”, Teubner (Stuttgart,

1994) − A. Gross, “Theoretical Surface Science – a Microscopic Perspective”, Springer (Berlin,

2002)

− R. Masel, ”Principles of Adsorption and Reaction on Solid Surfaces”, Wiley (New York,

1996) − D.P. Woodruff and T.A. Delchar, “Modern Techniques of Surface Science”,

Cambridge Univ. Press (Cambridge, 1994)

Further reading

Z = p

(2πmkT)½

(N/V) v

1015 sites/cm2, T=300 K, p = 1 atm: Z ~ 108 site-1 s-1 !!

Requires p ~ 10-12 atm to keep a “clean” surface clean for t ~ 1h

Surface Science:

- T→ 0K, UHV - no equilibrium with gas phase - deposition at fixed dosage (1 Langmuir = 10-6 Torr s ≈ 10-9 atm s)

Impingement and the Surface Science ansatz

Complexity vs. atomic-scale understanding

SEM image of polycrystalline Cu

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

Simplification needed…

Which surfaces to study I?

Phase I

Phase II

Phase I / phase II alone (bulk):

GI = NI µI GII = NII µII Total system (with surface):

GI+II = GI + GII + ∆Gsurf

(T,p)

γA

γ = 1/A ( GI+II - Σi Ni µi ) Surface tension (free energy per area)

Surface thermodynamics

Which surfaces to study II?

γ(T,p) = 1/A ( GI+II - Σi Ni µi ) γ ≈ 1/A ( EI+II – N Ebulk ) atom

p,T → 0

Assume: Only M atoms in topmost layer affected by surface, and energy of every atom simply scales with coordination → Eatom= (Zbonds/Zbulk) E coh :

γ ~ (M/A) Ecoh (Zsurf – Zbulk)/Zbulk ~ 100 meV/Å2

~ 0.1 ~ 4 eV ~ ¼ atom/Å2 (TMs)

From surface tension to surface energy

Which surfaces to study III?

Crystals will seek to minimize the surface free energy

∫dA γ = min.

under the constraint of fixed volume:

Wulff construction

… but be aware of kinetics !!!

Equilibrium shape of crystals

Lead crystal at 473 K J.C. Heyraud and J.J. Metois, Surf. Sci. 128, 334 (1983)

Surface Science part II: “Ideal” surfaces

[001] n

α

steps

kinks

(335) (11 13 19)

vicinal surfaces

(100)

(111) (110)

low-index surfaces

Cutting close-packed materials (fcc/hcp/bcc)

Point: P = h a + k b + l c Direction: Q = OP = [ nh : nk : nl ] with n smallest integer to yield three integer indices

e.g. Q = [1:1:0.5] = [221]

h k

l

[square brackets] single direction (regular brackets) one plane <pointed brackets> set of parallel directions {curly brackets} set of equivalent planes

Miller indices

Describing points, directions and planes in crystals

Plane: R = ( n/h : n/k : n/l ) with n smallest common denominator (to yield integer indices)

Infinities are set to zero. Negative indices get a “-” on top: 1

e.g. R = ( n/1 : n/3 : n/2 ) = (623)

Cubic systems:

(100), (010) and (001) are orthogonal (and in case of monatomic basis: identical) [hkl] is orthogonal to (hkl) Hexagonal systems:

Four index Miller notation ( a1 a2 a3 c ) ( n/h : n/k : n/i : n/l ) Index i is redundant: n/h + n/k = -n/i hcp(0001) and fcc(111) differ only in registry of third-layer

Specialities for cubic and hexagonal lattices

fcc bcc

(100)

(110)

(111)

fcc(100)

con- ventional

bulk unit cell

primitive surface unit cell

Common low-index planes

Atoms at the surface experience a different bonding environment than in the bulk. Adapting to this leads to:

Relaxation: change in atomic positions, but leaving surface unit cell unchanged

Reconstruction: changes of the unit cell/surface periodicity Open surfaces have much higher tendency to reconstruct than close-packed ones

From bulk-truncated to real surface geometries

Wood notation: Primitive lattice a1, a2 , reconstructed lattice b1, b2

If (after rotation by φ) b1 = m a1 b2 = n a2

then M{hkl}-(m x n)Rφo

e.g. (2x1), (√2 x √2)R45o = c(2x2)

Examples of relaxation and reconstruction: Metals

Layer relaxation Smoluchowski smoothing

+ + + - - - -

fcc(110)-(1x2) missing row reconstruction Au(111)-(22x√3) herringbone structure

280 Å

Dimerization at (001) surface of group IV elements

20 Å

Si(111)-(7x7) DAS-model

“Dangling bond minimization”

Examples of relaxation and reconstruction: Semiconductors

Electronic structure of surfaces: Jellium model

Ion cores approximated by uniform, positive background

charge (spatial average), confined to semi-infinite space.

Self-consistent electronic structure calculation

rs = 2 rs = 5

π/kF

N.D. Lang and W. Kohn, Phys. Rev. B 1, 4555 (1970)

Failures: - exponential instead of power law image potential (LDA-problem) - breaks down for TMs (localized d-electrons)

Achievements: - electron spill out (dipole layer, contributes to work function) - Friedel oscillations - oscillatory multilayer relaxation - qualitative description of simple metals (Na, Rb): work function, surface energy…

Projected surface band structure I

- For more detailed insights into the electronic structure, need to explicitly consider ionic lattice - Electronic structure calculation yields bulk electronic band structure - Even if there was no effect of the surface on the actual electronic structure, would still need to describe it differently: only 2D periodicity left, Brillouin zone collapses to 2D sheet!

M K

Γ

(111) surface Brillouin zone

Projected surface band structure II

k┴

k║

E

- Bulk bands project onto continuous regions in projected surface band structure - However, there are additional states that do not derive from this: Surface states (simply new solutions of the electronic problem due to finite crystal)

Projected surface band

structure M M Γ

Cu(111)

Shockley state

Tamm state

Ene

rgy

(eV

)

Tamm surface state

Complex band structure

gap

π/a k⊥

iκ

ε(k⊥)

matching condition potential

exp[ i(k⊥+ iκ) z] ~exp[λz]

Shockley surface state

Classification of surface states

Effects of surface states in semiconductors

Additional surface states in the band gap can be partially filled and then lead to charge imbalances that have electrostatic effects that extend deeply into semiconductors/insulators: - space charge region /band bending - metal-induced gap states (Schottky

barrier) - …