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