1

UPS=Ultraviolet Photoemission Spectroscopy

XPS=X-Ray Photoemission Spectroscopy

AES=Auger Electron Spectroscopy

ARUPS= Angular Resolved Ultraviolet Photoemission Spectroscopy

APECS=Auger-Photoelectron Coincidence Spectroscopy

Electron Spectroscopy for Chemical Analysis

(ESCA)

BIS= Bremsstrahlung Isocromat Spectroscopy

………………………………….. 1

It is the collective name of a series of techniques of surface analysis

2

Vacuum level

Fermi level

Free electrons

Ener

gy

kk

,

Filled bands

Core levels h

Photoelectron

J

k

Photoemission spectrum (XPS;UPS):filled states

Empty states

2

3 3

4

Fast photoelectrons: no post-collisional interactions

Photoemission cross section: golden rule expression

†

mn

N3

i ii

Interacting system hailtonian H Perturbation: H'=

' [ ( ). . ( )]2

Equivalent alternative formulation, directly from the relativistic

theory,

H' - d xA(x ) · j(x )

mn m n

N

i i i ii

M a a

eH A x p p A x

mc

The photoemission cross section Dσ(w) can be worked out starting from the Fermi golden rule; the photoelectron is in |f>

wD 22| ' |

i Ff

f H i E E

info on ion left behind from energy conservation 4

5

2 2 2 2

† †

†

Hamiltonian after photoionization

H=H , photoelectron KE2 2

H describes final state ionized solid (set of ion states f )

H f f , , f ion,f f

cruci photoelectron andal: io

k k k kk k

f ka

k ka a a a

m m

E

n do not interact

w

D

D

D

22cross section: | ' |

, solid angle accepted by detector

i Ff

f kf

f H i E E

Basic Theoretical framework

5

†

kn

H'= , k = photoelectron momentumkn k n

M a a

†

n

H'= , creates photoelectron

with momentum k

kn k nM a a

w

D

D

D

22cross section: | ' |

, solid angle accepted by detector

i Ff

f kf

f H i E E

†' '

hole state in solid

k km k k m km mm m

f H i f a H i M f a a a i M f a i

m

†

kn

H'= , k = photoelectron momentumkn k n

M a a

7

D

D D

If detector accepts a small ,

density of final states for photoelectron;

I am neglecting for simpicity some further dependence on angles

kk

k

w

D

D

2

2 * †

,

2| |

final hole state. Trick:

| |

k km m i kfnf

km mn

km kn n mm n

M f a i E E

m

M f a i M M i a f f a i

7

differential cross section:

8

w

w

D

D

* †

,

22| |

2k km

k km m i k

kn n

fmf

m i kfm nf

M M i a f f a i E E

M f a i E E

† † 1Imn i k m n m

i k

i a E H a i i a a iE H

w w

differential cross section:

* †

,

* †

,

2

2

k km kn n i k mm nf

k km kn n i k mm n

M M i a E H f f a i

M M i a E H a i

w

w

D

D

sum over final states using closure:

8 A hole Green’s function is involved.

9

spectroscopic notation KLMNO,...

n=1,2,3,4,5,...

guscio N

4s1/2 N1

4p1/2,3/2 N2,N3

4d3/2,5/2 N4,N5

4f5/2,7/2 N6,N7

XPS from Hg vapour using Al Ka h=1486.6 eV. Lines are labelled by final core hole state of Hg+

9

10

The same core lines can be observed by X-Ray emission

Dirac-Fock codes do NOT grant good agreement with experiment

Chemical analysis: tiny amounts suffice to recognize elements (binding energies are well known)

11

12

solid Si 2p XPS core spectrum Si configuration: [Ne]3s23p2

2p is core

12

13

Milano 4 Luglio 2006

Al valence band Ag valence band

Fermi d band

s band

13

14

UPS (ultraviolet photoemission) produces slow electrons- excape probability strongly depends on energy and on angles

14

15

UPS produces slow electrons- excape probability strongly depends on coverage

No Ag

1 monolayer Ag

2 monolayer Ag

background

Background due to incoherent losses: one can measure it by eels

The universal electron mean free path curve. Electron spectroscopies are surface sensitive (because of outgoing electrons, much more than for incoming photon mean free path)

16

Laplace equation for moving electron (constant speed v = l/T= w/k)

17

3

3( )

4

Since v exp( . ) exp( .v ) and

exp( .v ) ( .v), one obtains:

v (2 ) ( .v)(2 )

i t

i kr t

r t d k ik r ik t

ik t d e k

d kdr t e k

w

w

w w

w w

Jean Baptiste Joseph Fourier

The produced by the fast electron is given by: 4 ( ) ( vt)

To Fourier transform we need:

D divD e r

D

David Penn, Phys. Rev B35 (1986)

Hence, ikD=4 e2 ( -k.v) w

We can explain qualitatively the universal mean-free-path curve by a simpliefied model

The electron is treated as a classical point charge moving in the solid with a constant velocity

18

2

2

2

2

scr

8 ( v)( , ) is consistent with the above result.

But ( , ) ( , ) ( , )

8 ( v)( , ) .

( , )

We can obtain the screened potential, si

eened poten

nce ( , ) ( )

t

,

e k kD k

i k

D k k E k

e k kE k

i k k

E k ikV k

ww

w w w

ww

w

w w

2

2

8 ( v)ial ( , )

( , )

e kV k

k k

ww

w

ikD=4 e2 ( -k.v) w

19

23

2

( v) 1decay rate Im( )

2 ( , )

e kd kd

k k

ww

w

2

2

The potential of the screening charges

acting on the electron*electron charge

8 ( v)screened

=self-energy

potential ( , )(

.

But Im(self-energy)=dec

,

ay

)

rate

e kV k

k k

ww

w

Recall: the Dielectric function

1°-order Perturbation theory in exact many-body system

22

0 02

1 4Im 0 ( ) ( )

,k n n

n

en

k k

w w w w

w

20

23

2

( v) 1Im( )

2 ( , )

is proportional to the sum of the Fourier components of the disturbance

at the excitation energies o

Thus the decay r

f the sy

ate

stem.

e kd kd

k k

ww

w

At tens or hundreds of eV all solids are well approximated by

Jellium (gaps are much less) and behave similarly; at low

energy the losses are often small (Landau quasiparticles are

narrow in energy, as we shall see) and this explains

qualitatively the universal curve.The minimum corresponds

to the energy region in which multiple plasmon excitations

occur.

21

Shen (PRB 1990) e Tjernberg (J. Phys. C 1997) note that line shape depends strongly on photon energy, since the O cross section decreases with energy faster.

(compare 777.3 eV and 778.9 eV)

CoO valence band in UPS CoO has an octahedral structure and is an antiferromagnet: strong correlation produces a complex multiplet structure which informs us about the screened interaction.

21

hv=777.3 eV

hv=778.9 eV

22

CoO UPS ultraviolet photoelectron spectroscopy

Still another line shape at 40.8 eV

22

Ag has a surface state at the Fermi energy, s-p states below the Fermi energy and a filled d band 4 eV below.

Exchange splitting in final state ions

2

In the Hartree-Fok picture, NO has a partially filled 2 shell

with spin-orbitals , ; is also paramagnetic.O

NO

N2

N1s

O2

545 540

binding energy

415 425

binding energy (eV)

O1s

NO

545 540

1.2 eV

0.9 eV 0.9 eV

0.9 eV

25

Ratio 2:1 Ratio 3:1

Core level XPS

26

+

z

π s π sNO : 4 determinants all with Λ=|L |=1.

π s π s

We must account for singlet-triplet splitting.

1 1The singlet is :

2s s

1 1readily seen to be a singlet since 0

2S s s

3 1

2

s

s s

s

+The partially filled shells of NO include O1s denoted by s below

26

27

12

12 ,E J J s s

r D exchange integral

Sz=0 sector: 1 1

2s s

3 1

2s s

( ) | 0s h i s

Since determinants differ by 2 spin-orbitals, only the interaction contributes.

Configuration Interaction

1

does not depend on . Compute splitting ini

i i j ij

H h Sr

1 1 1

3 3 3

E H s H s s H s

E H s H s s H s

12

1 1[ ( (1) (1) (2) (2) (1) (1) (2) (2)) ( (1) (1) (2) (2) (1) (1) (2) (2))

2 a a a a J s s s s

r

27

28

0.88 eV for N

0.68 eV for O

triplet is lower (lower binding energy) by:

12

1( (1) (2) | | (1) (2)) J s s

r

12 12

1 1 1[( (1) (2) | | (1) (2)) ( (1) (2) | | (1) (2))], that is,

2J s s s s

r r

Taking the spin scalar products, two terms vanish, and writing the two—electron integrals

28

Ratio 3:1 (triplet versus singlet)

In a similar way, the 1s spectrum of O2 (binding energy ∼ 547 eV ) has two components separated by 1.1 eV with an intensity ratio 2:1 (quartet to doublet ratio).

29

Intial state effects

final state effects

electrostatic potential surrounding the atom

before ionization (several eV of either sign )

Polarization around hole (several eV, to lower BE)

One can tell valence and ionicity from shifts

Chemical shifts

BE eV 535 540

O1s

295 29o

C1s

Binding Energy eV

Acetone

30 The C bound to O is more electropositive and has larger Binding Energy

Pauling electronegativity scale

Binding energy

eV

Intial state effects, mainly

300

295

Pauling charge

0 10 20

CH4

CF4

CHF3

CO2

CO

CH3OH

CS2

33

the missing line 1

2

4 p Extreme initial state effects:

According to Dirac-Fock, a 4p1/2 line should exist between 4s and 4p3/2, but none is seen

4s 3

2

4 p

1

2

4 ?p

34

11.1 eV away from DSCF

virtual processes:

4p1/2 hole 2(4d) holes + electron

and Back

Xe+ Xe++ +e resonance

9.4 eV away from DSCF

A large self-energy merges 4p ½ with Auger continuum. Many body theory beyond HF is not a matter of refining the position of peaks!

Core level XPS spectra: large

relativistic effects for large Z

Core level XPS spectra- chemical shift

37

Hole Screening satellites Energy shifts to lower binding

The ion is left excited because of correlation, coupling to phonons, plasmons, etc.

Low –energy satellites arise from excited final states

Screening wins at threshold (final-state shift)

Useful approximate scheme: final-state Hamiltonian is different because of the potential of the hole.

Final-state effects in photoemission spectroscopy

By energy conservation: h = final ion energy + photoelectron energy

Postcollisional interactions seldom involved for fast photoelectrons

excited ion slower photoelectron, but hole screening faster photoelectron.

38

Shake-up satellites DHF approximation)

We can treat Hfin in Hartree-Fock approximation if we allow for a different final-state Hamiltonian while initial state |i> is the ground state without the hole. Then we can treat the initial state Hiniz in Hartree-Fock as well.

†

f

iniz i

in

niz without core hole,

with core hole potential.

Core Photoemission line shape: core GF

1Im core

, has core hole and frozen orbitals :

no eigenstate

O

of H

D S

c

c c c

fin

c

fin

G

G i a H a

H i E i

H

a i f

i

i

w

w w w

DHF approximation

Method to obtain the answer from the difference of eigenvalues of two HF calculations

40

2| |if

w w

ion eigenstates

ifFrozen determinant (N-1 spin-orbitals, obtained by removing core state spin-orbital from neutral HF determinant)

eigenstates of Hfin; in HF, they are determinants of N-1 relaxed spin-orbitals computed with core-hole.

Overlap of determinants=determinant of overlaps: all N-1 body states contribute

Ground-stateground state = threshold, other peaks = satellites

†

fin

1Im c c cG i a H a i w w w

41 Satellites perfectly balance the relaxation shift

Shake-up

satellites

Discrete excited states

Shake-off

satellites

Continuum excited states

Sum rules

w w w w

2 2

, , 1d d i f i f

ww w w w w

2 2

, ,

, , ,fin fin

d d i f i f

i f H if i f H i f

From Siegbahn’s lectures. In solids, vibrations but also plasmons

electron optics allow resolution 0.001:

sees rotovibrational structure

E

E

UPS

D

http://www.casaxps.com/help_manual/manual_updates/xps_spectra.pdf

http://www.fisica.unige.it/~rocca/Didattica/Fisica%20dello%20Stato%20Solido%20(Scienza%20ed%20Ingegneria%20dei%20Materiali)/7%20plasmons%20and%20surface%20plasmons.pdf

46

Transverse electromagnetic wave in conducting (J=E) homogeneous non magnetic medium

iSi

( ( ),0,0) (0, ( ),0) (0

mpl

,0, )

: no and the current ( ( ),0,0s )e t case

i t i t

i t

nd

E E z e B B z e S E B S

E J J z e

w w

w

Consider the plane wave going upwards i.e. Poynting vector along z

iMaxwell equations: div 4 0 and div 0ndE B

( , , ) (0

1rot ( ) (

, ( ),0)

)

y z z y z x x z x y y x z

t z

rotE E E E E E E E z

iE z B

cE B

cz

w

( , , ) ( ( ),0

1 4rot

4( ) ( ) ( )

,0)y z z

z

y z x x z x y y x z

tB E Jc c

iB z E z

rotB B B B B B B B

J zc c

z

w

47

rot ( ) ( )1

t zE Bc

iE z B z

c

w

1 4 4( ) ( )ot ( )r zt

iB E BJ

c ccz E z J z

c

w

22

2 2

4( ) ( )z

iE z E z J

c c

w w

Putting together the above inhomogeneus equations, namely,

One finds inhomogeneous wave equation

( ) ( ) ( )J z E z w

Assuming a transverse elecromagnetic wave in homogeneous local medium

2 22

2 2 2

4 4( ) ( ) 1 ( ) ( )z

i iE z E z J E z

c c c

w w w w

w

Maxwell equations in medium with constant dielectric constant:

yield waves with , refraction index, ,

and for non-magnetic media, assuming =1. Thus,

cc n n

n

22 thedielectric func

4, ( ) 1 ( )

( )tion

c ic

w w

w w

22

2

4( ) 1 ( ) ( )z

iE z E z

c

w w

w

50

Drude’s classical theory

v vrelaxation tim, e

d mm F

dt

( )F e E v B

In case of constant field, for times t>>

one finds : , mobilitye

v Em

0

0

In case of oscillating field :

v v

i t

i t

E E e

d mm eE e

dt

w

w

51

0

0 0

solved by v v

1( )v

i t

i t i t

is e

m i e eE e

w

w ww

0 00v .

1 1( )

eE Ee

m im i

ww

0 00v .

1 1( )

eE Ee

m im i

ww

00This produces a current ( ) v v

1

i t i tEeJ t ne ne e ne e

m i

w w

w

22 2 2

00

22

Hence we get a conductivity

( ) 1 1 4( ) ,

( ) 1 1 4 4

4where where plasma frequency.

p

p p

J t ne ne ne

E t m i i m m

ne

m

w w

w w

w w

51

22

2

22 the Drude dielectric functi

e.m wave in cond Recall:

4( ) 1 ( ) ( )

4, wh

ucti

ere

ng mediu

on( ) 1 )

m

(( )

z

iE z E z

c

c ic

w w

w

w w

w w

Drude dielectric function

2

01 2

2 2 2 2

1 22 2 2 2

( )( ) 1 ( ) ( ) 1 ( ) ( )

1( )

( ) 1 , ( )1 (1 )

p

p p

i iii

w w w w w w w

w w w w

w w w w

w w w

w

w

2 2

2 2

1 2

2 2

1

Low frequency region: 1, ( ) 1 , ( )

Typically 1, is large negative

p

p

p

w w w w w

w

w

2 2

1 22 3intermediate frequency region: 1 , ( ) 1 , ( )

large refractive index, metal reflects

p p

p

w ww w w w

w w

2 2

1 22 3high frequency region: , ( ) 1 1

is transpare

, ( ) 0

meta nl .t

p p

p

w ww w w w

w w

Maxwell equations imply 2

2

2k

c

w w

2 2

2

2

41 ,

p

p

ne

m

w w w

w

Simple metals have

propagating waves have

Summary on plasmons

3

12, 1 13.6

5.64 3.51

3.0 9.07

2.0 16.66

p

s

s p

s p

s p

Ry Ry eVr

Cs r eV

Au r eV

Al r eV

w

w

w

w

From Blaber et al., J.Chem.Phys (2009) experimental, with g=1/

http://www.phy.cuhk.edu.hk/course/surfacesci/mod3/m3_s2.pdf

surface plasmon

Surface plasmon polaritons

are plasmon-photon modes

localized at an interface.

Consider vacuum for z>0 with =1 and metal with w for z<0, wave propagating along x.

X

z

1

i

nothing depen

Look for solutions with:

( ( , ), ds on y0, ( , )) (0, ( , ),0)

0 0

x

i t i t

x z y

nd

iq

E E x z E x z e B B x z e

J

w w

iMaxwell equations inside solid: div 4 0 and div 0ndE B

( , , ) ( , ,0)

1

y z y z z x x z

y z x x z

x y y x y z z x x zrotE E E E E E E E E

i B E E

E

BrotE

c tw

(

( , , ) ( ,0, )

1 )

( )

y z y z z x x z x

z

y y x z y x

y

x y y z

y

x

rotB B B B B B B B B

D ErotH rotB

c t

B i E

B iqB i Ec t

w w

w w

2 2

21 1

( )

p p

i

w w w

ww w

E

Derivation of the wave equation for (0, ( , ),0) inside the solid:

equations say:

( ) ( )y z x x z z y x x y y z

i t

y

i B E E B i E B iqB i E

B B x z e

Maxwell

w

w w w w w

2

2

2 2

Differentiating the second ( ) ( )

and substituting in the first

, that is, ( ) ( )( )

z y x z y z x

z y

y x z y z y x z

B i E B i E

Bi B E B B i E

i

w w w w

w w w w ww w

22 2

2

2

2

22 2

E is eliminated using the third,

( ) ( ) ( )

and substituting we get a wave equation for :

( ) ( )

(changes from free case: and c

inside

(

:

z

x y

y z y

x

y z y x z

y

B

B iqB i E i

q

q B i E

B

Bc

cq

w

w w w w

w

))w

22 2

2outside ( ) (same with 1 instead of ) : y z yB q B

c

w

0 localized e[ ( xcita) ( ) ] tionz z

yB B z e z eg g

2

2

0

0

, 0

, 0

z

x

z

ci B e z

Ec

i B e z

g

g

g

w

g

w

Next, we find the electric field, which is also localized:

evanescent wave solution: exponentially localized at surface

2 22 2 2

02 2

2 22 2 2

02 2

inside ( ) ( ) ( ) , ( )

outside ( ) ( ) ,

:

:

z

y z y y

z

y z y y

B q B B B z e qc c

B q B B B z e qc c

g

g

w w w g w

w w g

Substituting into ( )z y xB i Ew w

Continuity condition for electric field at z=0 dispersion law 2

2

0

0

2 22 2

, 0

continuity of

, 0

requires ( )( )

z

x

z

ci B e z

Ec

i B e z

c c

g

g

g

w

g

w

g gg g

w w w

This requires 0 that is, below .p w w

2 2

2 22 2

2 2

2 22 2 2

2 2

( ( )) establishes a link between q and ω through ( ) :

indeed, recall and ( ) .

( ) ( ) ( )

q qc c

q qc c

g w g w

w wg g w

w w w w

22

2

22

2

inside ( )

outside

:

:

qc

qc

wg w

wg

Polariton Dispersion law

2 2 2 2 2 2 2Expand ( ) ( ) ( ) 0c q c qw w w w

2 2 2 2 2 2 2So, ( ) ( 1)*something else. Indeed,c q c qw w

2 22 2 2

2 2Solve ( ) ( ) ( ) for .q q

c c

w w w w w

Remark: for =1 any must work (indeed, =cq). w w

2 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2 2

( ) 0 can be rewritten ( (c q - )+c q )=0,

(just multiply to see)

so thecondi

( -

tion is: ( ) (c q - )+c q

1)

0.

c q c q

really

w w w w

w w

2 2 2 2 4 4 41( ) [ 2 4 ]

2sp p pq c q c qw w w

the other sign is not acceptable because it gives

2

2that implies 1 0.

p

P

ww w w

w

2 2

2

2

2 2 2 2 2 4 2 2 2 2 2 2 2

2

1 1 wecan solve for ( ) :

( )

(

w

1 )(c q - )+c q 0

ith Dr de

2

u

0.

p p

p

p p

qi

c q c q

w w w w

ww w

ww w w w w

w

2 2 2 2 2Polariton dispersion: ( ) (c q - )+c q 0; w w

Explicit Polariton Dispersion law-Drude dielectric function

2 2 2 2 4 4 41However ( ) [ 2 4 ]

2

gives a photon-like non-localized mode

sp p pq c q c qw w w

0.5 1.0 1.5 2.0

0.5

1.0

1.5

2.0

2.5

2 2 2 4 4 41[ 2 4 ]

2 2

p

p pc q c qw

w w

2 2 2 4 4 41[ 2 4 ]

2p pc q c qw w

p

cq

w

p

cqw

w

6 1 8 1/ 1 10 10pcq for q cm cma

w

( )

p

qw

w

There are also localized phonon modes at interfaces.

surface plasmon branch

photon branch

62

h

New equilibrium position for Harmonic oscillator

Plasmon oscillator shift by uniform field E

E

-3 -2 -1 1 2 3

2

4

6

8

63

h

New radial equilibrium position for Harmonic oscillator

Plasmon oscillator shift by inner charge

The potential is still harmonic, but minimum is shifted (x=radial coordinate)

22

0( ) ( ) ( )V x V x m x Ow

†1H g d d

ddgddH 0wFinal-state:

64

Phonon and Plasmon Satellites in Core Spectra- Lundqvist model

Boson mode with frequency 0w before photoemission

†

0 0H d dw† , 1d d

Initial state: i vac

excited states:

†

!

n

d

dn vac

n

2 2

0

1( )

2V x M xwHarmonic oscillator

Sudden change of H due to photoionization

65

2 2

0

1( )

2V x M xwPhonon oscillators:

x=0 equilibrium bond length. After ionization, bond length changes

-3 -2 -1 1 2 3

2

4

6

8

Sudden change of H due to photoionization

66

ddgddH 0wFinal-state:

† † *,

Parameter to be determined

s d s dg g

g

shifted bosons

† †, , 1s s d d

2

0

| | n

n

L i H i i n Ew w w

Spectrum =Density of states

0 unshifted vacuum (prevails before core-hole)di

To compute L we use a canonical transformation to new boson coordinates.

67

)())((~ *

0 ggggw ssgssH

Canonical transformation is time-independent

New H is just the old one in terms of s operators:

ddgddH 0wFinal-state:

† † *,

Parameter to be determined

s d s dg g

g

shifted bosons

68

† * † *

0( )( ) ( )H s s g s sw g g g g

Canonical transformation is time-independent

New H is just the old one in terms of s operators:

Hamiltonian in diagonal form

2†

0

0

.g

H s sww

ddgddH 0wFinal-state:

0if we choose 0gw g

†,no linear terms in s s

0

gg

w

† * † † * *

0 0 0 0( ) ( )H s s s s g s s gw w g w g g g w g g

69

Relaxation shift to higher KE 2

0

.g

Ew

D

Remark: 2° order approximati

2

on

0

0 .g

H d d g d d Eww

D

Ground state of

2†

0

0

gH s sw

w s vacuum 0s

†

0 0H d dw† , 1d d

Initial state: 0di

† † *,s d s dg g shift

instead, the shifted boson vacuum is su

0 0 0 0 co

ch tha

herent state for

t

s s ss d dg

Hamiltonian in diagonal form 2

†

0

0

.g

H s sww

0

gg

w

70

†

0

10 n shifted bosons

!

n

s

s n s n

n sn

H n E n E E nw

D

Solution of Lundqvist model

Franck-Condon factors 2

si n

n shifted plasmons d vacuum

0 unshifted vacuumdi

2

0

| |s n

n

L i H i i n Ew w w

Density of states

71

2

si n

0 0d sLet C† †

0 0

0 1 0 0 0 0d s d s d s

g gs d C

w w

† †

0

0

1 10 0 0 0 ( ) 0

! !

1

!

snn

d s d s d s

n

gn s d

n n

gC

n

w

w

2

2 2

0 0 0

1C determined by | 0 | 1

!s s

n

d s

n n

gn C

n w

Calculation of

72

2

0

2 ,

w

gaeC

aNormalization fixes C

2

0

gE

wD Shift of the threshold, like

in second-order

0

0 !

na

n

aL e E n

nw w w

D

1

1 1 0

Poisson distribution!

( 1)! !

na

n

n na a

n

n n n

aP e

n

a an nP e e a

n n

73

0

0

0

0

2 2

0 0 020 0 0

( ) 1 Normalization OK!

( )!

0!

The center-of-mass of spectrum is the same as for g=0

Satellit

na

n

na

n

na

n

ad L e d E n

n

ad L e d E n

n

a g ge E n E n

n

w w w w w

ww w ww w w

w w ww w

D

D

D D

es balance relaxation shift exactly.

0

0

Properties of !

na

n

aDOS L e E n

nw w w

D

74

-4 -2 2 4

0.01

0.02

0.03

0.04

0.05

0.06

0.07

.3a

-2 -1 1 2 3 4 5

0.05

0.1

0.15

0.2 3.0a

-2 -1 1 2 3 4 5

0.02

0.04

0.06

0.08

0.1

0.12

.1a

-10 -5 5 10

0.01

0.02

0.03

0.04

0.05

.6a

10 w( ) 0.05,L with broadeningw

75

w

w w w

ww

w ww

w w

D

D

D

0 0 0

2

0

0

(

00

)

0 0

( ) ( )2

using ,

(

, and

)

!

! !

i t iHt

n ni Et in t ia t in ta a

na

t

n n

n

C

dL t L e i e i

gE a

a aL t e

aL e E

n

n

e e

n

e en

w w is the Fourier transform of ( ) iHtL i H i L t i e i

w

w

w

w

0

00

( )

0

( 1)

( ) ( 1)

exponential at exponent

( )

L(t)=e

where

i t

i t

ia

iHt C t

t a e

C t ia t a e

L t i e i e

76

Strongly coupled slow modes (e.g. phonons) a>>1

w w 2 2

0 00, finiteg a

22 2

2221

( ) limit2

g tgiHte e L e gaussian

g

w

w

The relaxation energy diverges in the Gaussian limit, which is a serious overestimate;

however the Gaussian line shape is often a good approximation

for phonon broadened core levels in solids and the width provides a sensible

measure of the electron-phonon interaction, although the Gaussian should be convolved with a Lorentzian (producing a so called Voigt profile) to account for the core hole lifetime.

2

22

0 0

0

1lim

! 2

na g

an

ae a n e

n g

w

w w w

From angular resolved spectra one can find the band structure E(k)

The photon momentum is small The photoelectron keeps its momentum

Time- and Angle-Resolved Two-Photon Photoemission (TR&AR 2PPE).

Review article: T. Fauster et al., Progr. Surf. Sci. 82, 224 (2007)

A pump photon h1 excites an electron

After a delay of the order of hundreds of femtoseconds (1 femtosecond = 10-15 second) a probe photon h2 takes the electron out.

The pump pulse creates a non-equilibrium distribution of electrons

in the sample and the resulting relaxation dynamics in the

intermediate state (e.g. population decay or carrier localization) are

monitored by the time-delayed probe pulse.

The kinetic energy of the photoelectrons is detected as a function of

their emission angle.

http://www.fhi-berlin.mpg.de/pc/PCres_methods.html

http://www.physik.fu-berlin.de/einrichtungen/sfb/sfb658/tutorials/dokumente/Tutorial_Wolf.pdf

83

BREMSSTRAHLUNG ISOCROMAT SPECTROSCOPY (bis)

Vacuum level

Fermi level

Free

electrons

Ene

rgy

kk

,

band

Core levels

monochromatic

electron beam

Photons

Inverse Photoemission

83

Photomultiplier

Electron gun

84

empty states

monochromatic

electron beam

k

J

h

photon detector

BREMSSTRAHLUNG ISOCROMAT SPECTROSCOPY (bis)

84

When the incoming electron energy corresponds to a resonant state , there is an enhanced pobability that the electron is captured end emits a photon

85 85

86 Image resonance: electron (almost) bound to metal by image potential

86