1. Surface Electronic Properties
PC4259 Chapter 3
Surface Electronic Properties
Electronic properties critical to most surface functionalities
Surface potential & work function
Electronic states at surface: Intrinsic & extrinsic
Electronic properties of semiconductor surfaces & interfaces
Surface excitation: Plasmon & phonon
Surface magnetism

2. Work function , Vacuum level & Fermi level
Vacuum level Evac
Work function , Vacuum level & Fermi level
Work functions of simple metals
Electron density parameter rs

3. Image potential: A simple model of surface potential
Image charge
An electron is attracted to metal surface by the image charge: + -z - z
Potential energy of electron:

4. Jellium model: electrons in a metal with uniform positive background charge terminating at surface
Friedel oscillation at  =
Electron density decreases smoothly and spill into vacuum
A dipole moment at surface keeps electrons from escaping into vacuum

5. Friedel oscillation at  =
Cut-off of electron waves at the Fermi wavelength:
STM image of Cu(111) at ~ 4 K

6.  measured from field emission
Fowler-Nordheim equation:

7. Work function from thermionic emission
Richardson-Dushman equation:
Richardson constant: A = 120 A/(cm2 K2)
Work function from Kelvin probe

8. Reviews of Solid State Physics (1)
Electronic states in a perfect crystal : Bloch wave function
with:
T: lattice translation vector
Periodically modulated plane wave, ħk = crystal momentum
& E(k) are also periodic in k, the periods are
where G is a reciprocal lattice vector

9. Reviews of Solid State Physics (2)
Effect of periodic potential is most dramatic on states of k = Gn/2 (the boundaries of Brillouin zones): It opens energy bandgaps at k = Gn/2 separating allowed energy bands
1st Brillouin zone
Only need to consider states in 1st Brillouin zone
Effective mass of electrons (& holes) m*:
Si: m*/m0 = 0.98, GaAs: m*/m0 = 0.067
At the top of energy bands, m* is negative!

10. Energy bands in Si crystal
1st Brillouin zone
fcc
bcc
Energy bands in Si crystal
Ei(k) along some axes in 1st Brillouin zone

11. Insulators, Conductors, Semiconductors from energy band structures
Number of electrons to fill an energy band = 2/a = 2 × Number of unit cells. The filling of bands determines electronic properties E E E
conduction band
empty
conduction
band -
Band
gap
partially-filled
band
electron
hole
Forbidden
region
Band
gap
Eg < 5eV
Eg > 5eV +
valence
band
valence band
filled
Insulator Semiconductor Conductor
Si: Eg = 1.1 eV
GaAs: Eg = 1.42 eV
ZnO: Eg = 3.4 eV
SiO2: Eg = 9 eV

12. A model pseudopotential:
Bulk states
vs.
Surface states

13. 1-D semi-infinite chain model:
For z < 0, weak potential (small ):
For z  0, V(z) = V0
Solve Schrödinger equation:
Mostly free-electron-like states not affected significantly:
But states near Brillouin zone boundaries are strongly scattered by periodic potential
&

14. States near Brillouin zone boundary
Coefficients A and B satisfy:
Use a small variable ,
Opening a gap of at

15. Wave functions near zone boundary but inside crystal:
Wave function for E < V0 outside surface:
Wave function matching at z = 0 requires a standing wave in crystal:
Such a matching can always be accomplished for bulk states

16. Surface States: states with imaginary  values are allowed near surface, set
E(q) is real and falls in the bandgap of bulk states if q is not very large
A decaying standing wave in crystal
Only one value of E within the bandgap, thus only one surface state is allowed in 1-D chain

17. Types of Surface States
Shockley states: generated by a bulk periodic potential terminating at surface without other deviation from bulk, free-electron-like, suitable for normal metals and some narrow-gap semiconductors
Tamm states: generated from dangling bonds or significantly reconstructed structures, the tight-binding wave functions derived from atomic orbitals
Extrinsic surface states: defects (including vacancies, steps, impurities) often result in additional states localized around them

18. Surface States of 3-D Crystals
A bulk-terminated or reconstructed surface generally has 2-D periodic order on surface, so wave functions of intrinsic surface states are 2-D Bloch-wave:
and are co-ordinate and wave vector in surface plane
: a decaying function in crystal
In 2-D k-space:
1st Brillouin zones of 2-D lattices
But bulk states also exist near surface, and need to be considered!

19. Projection of bulk states in surface Brillouin zone
2-D Brillouin zones
3-D Brillouin zones

20. True Surface States & Surface Resonances

21. Work function measurement with ultra-violet photoelectron (UPS)
 = h - W
W: energy width of PE spectrum
EVac  EF
UV photon: h ~10-50 eV, synchrotron radiation, noble-gas lamp
(He I: 21.2 eV; He II: 40.8 eV; Ne I: 16.8 eV; Ar I: 16.8 eV)

22. UV-photoelectron spectroscopy (UPS)
Excitation radiation: UV (He I: h = 21.2 eV)
Measure DOS near EF (valance band)
Measure work function
Chemical information
Oxidation of Ni(111)

23. Angle-resolved UPS (ARUPS)
Lateral k component is conserved:
Dispersion relation of surface states can be mapped out with ARUPS

24. Surface States on Cu(111) by ARUPS
Near
E0 = eV

25. IPES: to map out DOS of un-occupied states
Inverse-photoemission spectroscopy (IPES)
PES
IPES
IPES: to map out DOS of un-occupied states

26. isochromate mode:
Angle-resolved (or k-resolved) IPES
Bremsstrahlen mode:
Dispersion relation of surface states above EF

27. Surface states near EF by STM/STS
LDOS oscillation of period at different sample bias
Surface states on Cu(111) probed by different methods

28. Computational Studies of Surfaces
Born-Oppenheimer approximation: heavy ions are treated classically and statically (without kinetic energy), while a valence electron moves under the actions from ion cores and other valence electrons.
System energy at 0 K expressed as:
NA, NB,…: numbers of different types of atoms, {Ri}: atomic coordinates
Density functional theory (DFT): a general method, particularly suitable for metals
Tight-binding approach: more suitable for covalent and ionic solids

29. Density functional theory (DFT)
Theorem (Hohenberg & Kohn): the total energy E of an electronic system is completely specified by the ground-state electron density n(r), in a functional form
E takes a minimum when n(r) is the ground state density.
3 parts in total energy: kinetic energy T, electrostatic potential energy U, exchange-correlation term Eex
T = ground-state kinetic energy of a non-interacting inhomogeneous electron gas

30. Electrostatic potential energy U
Eex: accounts for quantum mechanical exchange and correlation effects in a many-body system due to Pauli exclusion between fermions
Non-local effect, complicated functions

31. Kohn-Sham equations
The many-interacting-electrons problem can be converted to many-noninteracting-electrons problem, so the electron density is found by solving a set of Schrödinger-like one-electron equations
Effective one-electron potential:
Electron density from one-electron wave functions:

32. Local density approximation (LDA)
ex(r) is approximately as the exchange energy density of a homogenous electron gas with density n(r), so the total exchange energy
Exchange potential:
Generalized gradient approximation (GGA):
Improving LDA by considering effect of local density gradient

33. Localized bonds in Ge crystal
Tight-binding Computation
Linear combination of atomic orbitals (LCAO):
Localized bonds in Ge crystal

34. solutions of one-electron Schrödinger equation
LCAO wave functions:
solutions of one-electron Schrödinger equation
LCAO expansion coefficients:
Hamiltonian matrix elements:
Inter-atomic overlap integrals:
Only H & S of nearest-neighbors & next nearest-neighbors are significant, most other H & S are 0, tight-binding calculations for covalent or ionic solids are less demanding in computation power than DFT-LDA.

35.  bonds
 bonds
Two  bonds formed by a pair of p orbitals perpendicular to r0. One at a lower energy:  bonding, at a higher level: * antibonding

36. Hybridized orbitals
In diamond or zincblende crystals, four sp3 hybrids form tetrahedrally oriented bonds at 109.5 from one another
Orthonormal quantum states:
sp2 hybrids formed by s, px, py in three-fold coordinated planar crystals (e.g. graphite), the remaining pz forms  bond

37. Computational Surface Studies: Slab arrays
3-D periodic structure formed by slab array. Slabs are separated from one another with sufficient vacuum spacing (~ Å)
Slabs should be thick enough (~ 5-20 atomic layers) to approximate for a surface of a semi-infinite crystal.
N atomic layers
Convergence tests:
: cohesive energy of a bulk atomic layer

38. Surface energies of 4d transition metals calculated using DFT-LDA
Surface energy:
Surface energies of 4d transition metals calculated using DFT-LDA
Parabolic dependence of  on d-band occupation

39. Identify surface states & their decaying from layer-resolved LDOS
Local density of electrons: n(r)
Local density of states (LDOS):
Global density of states (DOS):
Identify surface states & their decaying from layer-resolved LDOS
layer-resolved LDOS for W(100)

40. Layer- & band-resolved LDOS:
Band narrowing at surfaces: a quite common trend induced by lower coordination of surface atoms
layer-resolved d-band LDOS for Pd(210)

41. Smoothening of electron distribution
Lateral distribution of electrons at metal surface is much smoother than in bulk
Valence electron n(r) near Cu(001) surface
Contraction of first interlayer spacing
effectively positive charge region above ion, so the top-layer atoms are pushed towards the bulk

42. Semiconductor Surface & Interface States
Importance: they can induce band bending over significant region away from surface/interface due to much lower free carrier density in semiconductor than that in metal
Work function:
 = Evac – EF(at surface)
Electron affinity:
 = Evac - EC(at surface)
Band bending: eVS

43. Surface & Interface States in Bandgap
Valence & conduction bands in tight-binding model: as bands of states derived from atomic orbital A and B
Lower coordination at surface leads to wave functions with less overlapping and interactions, so less splitting and shift of energy levels than in bulk, yielding surface states in bandgap
Surface donor-type state: neutral when fully occupied, positive when empty
Surface acceptor-type state: neutral when fully empty, negative when occupied

44. Intrinsic & Doped Semiconductors
Intrinsic Fermi level: E
conduction
band
Band
gap + -
valence EV EC ED EA Ei
Intrinsic carrier density:
Fermi-Dirac distribution:
If doped with donors or acceptor at density ND or NA >> ni, and EC – ED or EA – EV  3kT  eV, dopants fully ionized, so:
ND  n, EF  ED for n-type; or NA  p, EF  EA for p-type

45. Band Bending at n-type Semiconductor Surface with acceptor-type surface states near mid-gap
To achieve equilibrium between bulk & surface, EF must be leveled throughout the material
Charge balance
Negative charge layer of density QSS at surface must be balanced with an equal amount of positive charges, which are the ionized donors in the depletion layer or space charge layer:
(in Schottky approximation)

46. Band Bending Analysis
Define electric potential  and electron potential V as:
Total band bending:
Carrier densities: and
Poisson’s equation:

47. Charge balance & band bending
In full-depletion approximation
Poisson’s equation:
Total band bending:
Depletion layer thickness typically ~ 102 Å

48. Inversion: Formation of a layer with in n-type semiconductor
In this inversion layer, p > n
Accumulation: with donor-type surface states near EC, the band bending is down-ward, and n > nb in near-surface region

49. Strong influence factors for reconstructions observed on various semiconductor surfaces
Surface structure observed is the lowest free-energy one kinetically accessible under preparation conditions (e.g. temperature and gas phase environment)
Semiconductors are most covalent or ionic-bonding. The surfaces tend to minimize the dangling bond density by reconfiguration. The remaining dangling bonds tend to be either fully occupied (saturated) or completely empty
Semiconductor surface tends to be insulating (or semiconducting) by maintaining a gap between occupied and empty surface states
Semiconductor surface tends to maintain charge neutrality for reducing electrostatic energy

50. Si(100) dehybridization: sp3  spx + py/z 2) dimerization:
py/z   bond
spx   & * bonds
Symmetric dimer model has no band gap in surface states
Asymmetric dimer model has a band gap in surface states, agrees with experimental results
Surface remains semiconducting

51. A metastable structure obtained by cleavage at RT
Cleavage Si(111)-21
A metastable structure obtained by cleavage at RT
Significant re-bonding to form  chains running along
Atoms along  chains are nearly sp2 coordinated, and the pz orbitals form  and * bands

52. Si(111)-21
The -band fully occupied and *-band totally empty, with a gap of ~ 0.3 eV between them
Surface remains semiconducting

53. Si(111)-77
Rest atoms
Dimers

54. S1 band intersects with EF, so Si(111)-77 is a metallic surface
Surface states on Si(111)-77
S1 band intersects with EF, so Si(111)-77 is a metallic surface

55. (110) Surfaces of III-V Semiconductors
Maintaining charge neutrality naturally, no reconstruction
STM image of InP(110)
Buckling of zig-zag chains,  ~ 30 Group V atoms move outwards

56. Surface States on III-V(110)
Surface remains semiconducting after relaxation

57. GaAs(100)
Variation of atomic structures with preparation conditions, in particular the ratio of Ga and As fluxes
GaAs(100)-24
Number of valence electrons is just enough to fill all  bonds between neighboring atoms and the dangling bonds at As atoms, while all dangling bonds at Ga atoms are totally empty.
This is a surface structure satisfying the electron counting rule, so it is semiconducting
4a2

58. O atoms are more protrude than Zn atoms in surface dimers
O atoms are more protrude than Zn atoms in surface dimers. The situation is similar to the buckling case on GaAs(110)
The intrinsic surface states are far from midgap

59. Si-SiO2 Interfaces
Interface state density can be down to 108 cm-2 eV-1 near midgap
Flat band at Si-SiO2 interfaces
Si(100) is preferred

60. MOSFET
Interface quality is important to a sensitive gate control and high carrier mobility in the channel

61. Contact Between Metals
Thomas-Fermi screening
with
For Cu: n = 8.51022 cm-3, Å
Metal-Semiconductor Contact
Interface dipole layers in metals much thinner than Space charge layer in semiconductors

62. Metal-semiconductor Contact
Band bending (in Schottky model):
Schottky barrier height
Schottky diode
Schottky contact vs. Ohmic contact

63. Deviation from Schottky Model
For real metal-semiconductor contact, Schottky barrier height often deviates significantly from Schottky-model value
eVSB varies much less dramatically than M
Metal-induced Gap States (MIGS)
Effect of MIGS: Interface dipole of energy  between MIGS and metal

64. Fermi Level Pinning by MIGS
With a high of MIGS density (> 1012 cm-2), EF is pinned, M basically has no effect on the interface EF and band bending
EF pinning may occur after only ML of metal is deposited on semiconductor

65. localized at the surface and its amplitude decays with the depth
Bulk plasmon: quantization of collection valence electron density oscillation at frequency
ħp is typically ~ eV
Surface plasmon
localized at the surface and its amplitude decays with the depth

66. Surface phonon: collective lattice vibrations localized near surface, amplitude attenuates normal to the surface
Phonon dispersion relation plot in surface Brillouin zone
ħ < 100 meV
Rayleigh wave: sound wave with a constant velocity vRW slightly below the speed of bulk transverse wave
Rayleigh wave

67. Electron energy loss spectroscopy (EELS)
Measure the spectrum of primary electrons with characteristic energy losses:
Excitation of core electrons (Eloss ~ eV)
Excitation of valance electrons or plasmons (Eloss~1-20 eV)
Phonon and adsorbate vibration excitation (Eloss < 100 meV)
A primary electron may go through a single loss scattering or multiple loss events

68. Difficulty of EELS Strong background near elastic peak
Other secondary electron characteristic peaks
Spread in primary beam energy
Primary electrons with specific energy loss
Elastic peak

69. Core-level EELS

70. Multiple plasmon loss peaks
Plasmon Detection with Normal EELS
satellite peaks near elastic or co-level loss peaks
Multiple plasmon loss peaks
Bulk plasmon
normal emission
Surface plasmon
grazing emission to enhance sensitivity

71. High-resolution EELS (HREELS)
Phonon detection
Measure the adsorption configurations of atoms & molecules on surface based on the characteristic vibration modes of a particular bonding
High energy resolution ( 5 meV or 40 cm-1)
Field-emitter cathode
Primary E ~ 5 eV
Precision (127°) monochromators

72. HREELS: for adsorption configurations of atoms and molecules

73. Ferromagnetic Ordering in Solid
Quantum exchange interaction:
Jij: exchange constant between electrons at atoms i and j
Jij > 0: ferromagnetic magnetic moments tend to align, e.g. in Fe, Co, Ni
Jij < 0: anti-ferromagnetic neighboring moments anti-parallel
Ferromagnetic order is destroyed above Curie temperature TC
Excess of up-spin density n over n:
 band shifted from  by :
 = IR
I: Stoner parameter

74. layer-resolved d-band LDOS for Pd(210)
Stoner criterion: occurrence of FM order requires
Band narrowing at surface leads to an increase in LDOS at EF
  is enhanced at surface region
layer-resolved d-band LDOS for Pd(210)

75.  of isolated 3d atoms determined by Hund’s rule:
Low-dimensional system magnetism
 of isolated 3d atoms determined by Hund’s rule:
Electron spins in an atom are aligned as much as allowed by Pauli exclusion principle
A 3d-metal monolayer on Ag(001) remains ferromagnetic (at low T) with a quite large 

76. Effect of T in Low-dimensional System Magnetism
Ferromagnetic ordering is stabilized by collective exchange coupling between neighboring atoms
TC also depends on nnn:
TC is lower in a thinner free-standing film of FM metal