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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