PC4259 Chapter 3Surface Electronic Properties
Surface potential & work function
Electronic states at surface: Intrinsic & extrinsic
Electronic properties of semiconductor surfaces & interfaces
Surface excitation: Plasmon & phonon
Surface magnetism
Electronic properties critical to most surface functionalities
Vacuum level Evac
Work function , Vacuum level & Fermi level
Work functions of simple
metals
Fvac EE
nars1
3
4 30
Electron density parameter rs
Image potential: A simple model of surface potential
-
z
+
-z
An electron is attracted to metal surface by the image charge:
zf ˆ)2(4 2
0
2
z
ei
z
edzzfzU
z ii0
2
16')'()(
Potential energy of electron:
Image charge
Jellium model: electrons in a metal with uniform positive background charge terminating at surface
0,0
0,)(
z
znn r
Electron density decreases smoothly
and spill into vacuum
Friedel oscillation
at = Fk/
A dipole moment at surface keeps electrons from escaping into vacuum
dzznznze
)]()([
)()(
0
Friedel oscillation at = Fk/
STM image of Cu(111) at ~ 4 K
Cut-off of electron
waves at the Fermi
wavelength:
FF k/2
measured from field emission
z
eFez
0
2
16
Fowler-Nordheim equation:
F
f
t
Fj
)(108.6exp
)(
105.1 2/37
2
26
Work function from
thermionic emission
Richardson-Dushman equation:
TkATj
B
exp2
Work function from Kelvin probe
eUcomp /12
Richardson constant: A = 120 A/(cm2 K2)
Reviews of Solid State Physics (1)
Electronic states in a perfect crystal : Bloch wave function
)exp()()( rkrrkk
iu )()( rTrkk
uu with:
Periodically modulated plane wave, ħk = crystal momentum
)(rk
& E(k) are also periodic in k, the periods are
)()( kGk where G is a reciprocal lattice vector
*ia
T: lattice translation vector
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
Effective mass of electrons (& holes) m*:
1221* )]([)()( kk kvk Em
Si: m*/m0 = 0.98, GaAs: m*/m0 = 0.067
At the top of energy bands, m* is negative!
1st Brillouin zone
Only need to consider states in 1st Brillouin zone
1st Brillouin zone
fcc bcc
Energy bands in Si crystal
Ei(k) along some axes in 1st
Brillouin zone
Insulators, Conductors, Semiconductors from energy band structures
E
valence band filled
conduction band empty
Forbiddenregion Eg > 5eV
Bandgap
E
conduction band
Eg < 5eVBandgap
+
-electronhole
E
valence band
partially-filledband
Insulator Semiconductor ConductorSi: Eg = 1.1 eVGaAs: Eg = 1.42 eVZnO: Eg = 3.4 eV
SiO2: Eg = 9 eV
Number of electrons to fill an energy band = 2/a = 2 × Number of
unit cells. The filling of bands determines electronic properties
A model pseudopotential:
c
cps rrZ
rrV
r,/*,0
)(r
Bulk states vs.
Surface states
1-D semi-infinite chain model:
a
zi
a
ziV
a
zVzV
2exp
2expˆ2
cosˆ2)(
For z < 0, weak potential (small ):
For z 0, V(z) = V0
Solve Schrödinger equation:
)()()(2 2
22
zEzzVdz
d
m
Mostly free-electron-like states not affected significantly:
V̂
)exp()( zikAz
mkkE 2/)( 22 &
But states near Brillouin zone boundaries are strongly scattered by periodic potential
ak
States near Brillouin zone boundary
ak /
zakiBzikAz )/2(exp)exp()(
Coefficients A and B satisfy: 0
)(2
2ˆ
ˆ)(2
22
22
B
A
kEa
km
V
VkEkm
Use a small variable , ak /
22222
ˆ1
ˆˆ
2)(
VmaVmaV
amE
Opening a gap of at . V̂2 ak /
Wave functions near zone boundary but inside crystal:
aziazizi
i eVmaVma
eCe /
222/
ˆ1
ˆ
Wave function for E < V0 outside surface:
)(
2exp 02
EVm
zDo
Wave function matching at z = 0 requires a standing wave in crystal: ),0(),0()0( zzz iio
Such a matching can always be accomplished for bulk states
Surface States: states with imaginary values are allowed
near surface, set iq22
222
ˆ1ˆ
2)(
Vma
qVq
amqE
E(q) is real and falls in the bandgap of
bulk states if q is not very large
)exp(~)0( qzzi
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
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
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:
Surface States of 3-D Crystals
)exp(),(),(ss //////
//// rkrr
k izuz
and are co-ordinate and wave vector in surface plane //r//k
: a decaying function in crystal ),( zu ////
rk
1st Brillouin zones
of 2-D lattices
)()( ////// kGkssss
EE In 2-D k-space:
But bulk states also exist near surface, and need to be considered!
Projection of bulk states in surface Brillouin zone
2-D Brillouin zones
3-D Brillouin zones
True Surface States & Surface Resonances
Work function measurement with ultra-violet photoelectron (UPS)
EF
EVac
= h - W
W: energy width of PE spectrum
UV photon: h ~10-50 eV, synchrotron radiation, noble-gas lamp
hEE kinB
(He I: 21.2 eV; He II: 40.8 eV; Ne I: 16.8 eV; Ar I: 16.8 eV)
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)
Angle-resolved UPS (ARUPS)
m
kkE
exex
kin 2
)( 2)(//
2)(2
sin2
sin2
)()(//
kinexex mEkk
Lateral k component is conserved:
Gkk )(//
)(//
inex
Dispersion relation
of surface states can be
mapped out with ARUPS
)( )(//inEE k
Surface States on Cu(111) by ARUPS
Near
*
2//
2
0// 2)(
m
kEE
k
emm 46.0*
E0 = - 0.39 eV
Inverse-photoemission spectroscopy (IPES)
PES IPES
IPES: to map out DOS of un-occupied states
221 EeUEE
Angle-resolved (or k-resolved) IPES
Dispersion relation
of surface states above EF
)( )(//inEE k
isochromate mode:
012 EE
Bremsstrahlen mode:
02 eUE
Surface states near EF by STM/STS
Surface states on Cu(111) probed by different methods
LDOS oscillation of period at different sample bias
/// k
*
2//
2
0// 2)(
m
kEE
k
emm 46.0* eV42.00 E
Computational Studies of SurfacesBorn-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.
Density functional theory (DFT): a general method,
particularly suitable for metals
Tight-binding approach: more suitable for covalent and
ionic solids
System energy at 0 K expressed as:
NA, NB,…: numbers of different types of atoms, {Ri}: atomic coordinates
}]{,...;,[ iBA NNEE R
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.
)]([ rnEE
3 parts in total energy: kinetic energy T, electrostatic
potential energy U, exchange-correlation term Eex
exEUTnEE )]([ r
T = ground-state kinetic energy of a non-
interacting inhomogeneous electron gas
jii
jii
ccecee
ZZd
nZdd
nne
UUUU
RR jiR i R-Rr
R-r
rr'r
r'-r
r'r
,
3332
2
1)()()(
2
1
Electrostatic potential energy U
Eex: accounts for quantum mechanical exchange and
correlation effects in a many-body system due to
Pauli exclusion between fermions
rrrr 3)()()]([ dnnnE exex
Non-local effect, complicated functions
Kohn-Sham equationsThe 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
)()()(2
22
rrr iieffvm
Effective one-electron potential:
)]([)(
)( 322 rr'r'-r
r'
R-rr
R j
nvdn
eZ
ev exj
eff
j
Electron density from one-electron wave functions:
2)()( rr in
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
rrrr 3(hom))( ))(()()]([ dnnnE exLDA
ex
Exchange potential: )(
(hom) )()(r
rnn
exex nndn
dv
Generalized gradient approximation (GGA):
rrrrr 3)( )](),([)()]([ dnnnnE exGGA
ex
Improving LDA by considering effect of local density gradient
Tight-binding Computation
Linear combination of atomic orbitals (LCAO):
)()(,
, iRrr i
ic
Localized bonds in Ge
crystal
LCAO wave functions:
solutions of one-electron Schrödinger equation
)()()()(2
22
rrrr
HV
m
0)](),([ ,,
jj
cSH
jiji RRRR
)()(,
, iRrr i
ic
LCAO expansion
coefficients: Hamiltonian
matrix elements:
)()(),( *3jiji RrRrrRR HdH
Inter-atomic overlap
integrals:
)()()( *3jiji RrRrrRR dS
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.
bonds
)()(),( 03 * rrrrRR ji ssss HdH
bonds
Two bonds formed by a pair of
p orbitals perpendicular to r0.
One at a lower energy: bonding,
at a higher level: * antibonding
0rRR ji
Hybridized orbitals
In diamond or zincblende crystals, four sp3 hybrids form tetrahedrally oriented bonds at 109.5 from one another
zyx pppssp 2131
Orthonormal quantum states:
sp2 hybrids formed by s, px, py in three-fold coordinated planar crystals (e.g. graphite), the remaining pz forms bond
xpssp 23
112
3-D periodic structure formed by slab array. Slabs are separated from one another with sufficient vacuum spacing (~ 10-20 Å)
Computational Surface Studies: Slab arrays
Slabs should be thick enough (~ 5-20 atomic layers) to approximate for a surface of a semi-infinite crystal.
Convergence tests: N atomic
layers)()()( )1()( bulk
cohslab
cohslab
coh NENE
)(bulkcoh : cohesive energy of a bulk atomic layer
Surface energy: )(2
1 )()( NENA
slabcoh
bulkcoh
Surface energies of 4d transition metals calculated using DFT-LDA
Parabolic dependence of on d-band occupation
Local density of states (LDOS):
)()(),(2
in r r i
Local density of electrons: n(r)
Global density of states (DOS):
r r 3),()( dnn
layer-resolved LDOS for W(100)
Surface states
Identify surface states & their decaying from layer-resolved
LDOS
Layer- & band-resolved LDOS:
layer-resolved d-band LDOS for Pd(210)
)()(),(2
in r r i
Band narrowing at surfaces: a quite common trend induced
by lower coordination of surface atoms
Valence electron n(r) near Cu(001) surface
Lateral distribution of electrons at metal surface is much smoother than in bulk
Smoothening of electron distribution
Contraction of first interlayer spacing
effectively positive charge region above ion, so the top-layer atoms are pushed towards the bulk
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
)( FCs EEeVΦ
Valence & conduction bands in tight-binding model: as bands of states derived from atomic orbital A and B
Surface & Interface States in Bandgap
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
E
conduction band
Bandgap
+
-
valence band
EV
EC
ED
EA
Ei
)/ln()( 21
21
VCVCi NNkTEEE
Intrinsic Fermi level:
Intrinsic & Doped Semiconductors
]/)exp[(1
1)(
kTEEEf
FFD
Fermi-Dirac distribution:
kT
ENNpn g
VCii 2exp)( 21
Intrinsic carrier density:
If doped with donors or acceptor at density ND or NA >> ni, and EC – ED or EA – EV 3kT 0.078 eV, dopants fully ionized, so:
ND n, EF ED for n-type; or NA p, EF EA for p-type
Band Bending at n-type Semiconductor Surface with acceptor-type surface states near mid-
gapTo 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:
deNQQ DscSS (in Schottky approximation)
Define electric potential and electron potential V as:
Band Bending Analysis
)()( zEEze iF
Total band bending:
)]0([ bS eeV
Carrier
densities: and
]/)(exp[)( kTzenzn i
]/)(exp[)( kTzenzp i
Poisson’s equation: 0
2
2
2
2
rdz
d
dz
Vd
dz
d
E
)()( zezeV b
Charge balance & band bending
deNQQ DscSS
In full-depletion approximation
002
2
2
2
r
D
r
eN
dz
d
dz
Vd
dz
d
E
Poisson’s equation:
dzdzeN
zr
D 0),()(0
E
dzdzeN
zr
Db 0,)(
2)( 2
0
Total band bending:
0
22
2)]0([
r
DbS
dNeeeV
Depletion layer thickness typically ~ 102 Å
Inversion: Formation of
a layer with in
n-type semiconductoriF EE
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
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
Si(100)
1) dehybridizatio
n:
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
Cleavage Si(111)-21
A metastable structure obtained by cleavage at RT
Significant re-bonding to form chains running along ]101[
Atoms along chains are nearly
sp2 coordinated, and the pz
orbitals form and * bands
Si(111)-21
The -band fully occupied
and *-band totally
empty, with a gap of ~
0.3 eV between them
Surface remains semiconducting
Si(111)-77
Rest atomsDimers
Surface states on Si(111)-77
S1 band intersects with EF, so
Si(111)-77 is a metallic surface
STM image of InP(110)
(110) Surfaces of III-V Semiconductors
Maintaining charge neutrality naturally, no reconstruction
Buckling of zig-zag chains, ~ 30 Group V atoms move outwards
Surface States on III-V(110)
Surface remains semiconducting after relaxation
GaAs(100)
Variation of atomic structures with
preparation conditions, in particular
the ratio of Ga and As fluxes
4a2
GaAs(100)-24
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
0)1ZnO(10
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
Si-SiO2 InterfacesInterface state density can be down to 108 cm-2 eV-1 near midgap
Si(100) is preferred
Flat band at Si-SiO2 interfaces
MOSFET
Interface quality is important to a sensitive gate control and high carrier mobility in the channel
Contact Between Metals
Thomas-Fermi screening
6/130 )/(5.0 nalTF
)/exp(~)( TFlrr
qr
with
For Cu: n = 8.51022 cm-3, Å55.0TFl
Interface dipole layers in metals much
thinner than Space charge layer in
semiconductors
Metal-Semiconductor Contact
Metal-semiconductor Contact
SCMBS eVeV Band bending (in Schottky model):
SCMn
SBeV )(
Schottky barrier height
Schottky diode
Schottky contact vs. Ohmic contact
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 SCM
nSBeV )(
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 0.2-0.5 ML of metal is
deposited on semiconductor
Bulk plasmon: quantization of collection valence
electron density oscillation at frequency
m
nep
0
2
Surface plasmon
2p
sp
ħp is typically ~ 10-20 eV
localized at the surface
and its amplitude decays
with the depth
Surface phonon: collective lattice vibrations localized
near surface, amplitude attenuates normal to the surface )exp(),(
,,//
tiiezs z- ////
//// rkAr
kk
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 wavekvRW
Electron energy loss spectroscopy
(EELS)
Measure the spectrum of primary electrons with characteristic energy losses:
Excitation of core electrons (Eloss ~ 50-1000 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
Difficulty of EELS
Elastic peak
Primary electrons with specific energy loss
Strong background near elastic peak
Other secondary electron characteristic peaks
Spread in primary beam energy
Core-level EELS
Plasmon Detection with Normal EELS
satellite peaks near elastic or co-level loss peaks
Multiple plasmon loss peaks
Surface plasmon
grazing emission to enhance sensitivity
Bulk plasmon
normal emission
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
HREELS: for adsorption
configurations of atoms
and molecules
Quantum exchange interaction: ijS
E J i jS S
Jij: exchange constant between electrons at atoms i and j
Ferromagnetic Ordering in Solid
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:
N
nnR
band shifted from by :
= IR I: Stoner parameter
Stoner criterion: occurrence of FM order requires
1)(~
)2/)(( FF EDINVEID
layer-resolved d-band LDOS for Pd(210)
Band narrowing at surface leads
to an increase in LDOS at EF
is enhanced at surface region
of isolated 3d atoms
determined by Hund’s rule:
Low-dimensional system magnetism
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
Effect of T in Low-dimensional System Magnetism Ferromagnetic ordering is stabilized by collective exchange
coupling between neighboring atoms
TC also depends on nnn:
k
JnT nn
C 4
TC is lower in a thinner free-standing film of FM metal
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