Download Optical properties - Outline

Optical properties
and
Interaction of radiation with matter
S.Nannarone
TASC INFM-CNR & University of Modena
Outline
•Elements of Classical description of E.M. field propagation in
absorbing/ polarizable media  Dielectric function
•Quantum mechanics microscopic treatment of absorption and
emission and connection with dielectric function
Physics related to a wide class of Photon-in
Photon-out experiments including Absorption,
Reflectivity, Diffuse scattering, Luminescence and
Fluorescence or radiation-matter interaction
[some experimental arrangements and results, mainly in
connection with the BEAR beamline at Elettra
http://new.tasc.infm.it/research/bear/]
Systems
•bulk materials  the whole space is occupied by matter
•Surfaces  matter occupies a semi-space, properties of the
vacuum matter interface on top of a semi-infinite bulk
•Interfaces  transition region between two different semiinfinite materials
Information [see mainly following lectures]
•Electronic properties  full and empty states, valence
and core states, localized and delocalized states
•Local atomic geometry /Morphology  electronic
states – atomic geometry different faces of the same
coin
Energy range Visible, Vacuum Ultraviolet, Soft X-rays)
Synchrotron and laboratory sources/LAB
Conceptually Shining light
on a system, detecting the
products and measuring
effects of this interaction
This can be done by Laboratory sources
They cover in principle the whole energy range nowadays covered by
synchrotrons (J.A.R.Samson Techniques of vacuum ultraviolet spectroscopy)
•Incandescent sources
•Gas discharge
•X-ray e- bombardment line emission
•Bremsstrahlung continuous emission sources
•Higher harmonic source
Synchrotron and laboratory sources / Synchrotron
Some well known features
•Collimation
•Intrinsic linear and circular polarization
•Time structure (typically 01-1 ns length, 1 MHz-05GHz repetition rate)
•Continuous spectrum, high energy  access to core levels
• Reliable calculability of absolute intensity
•Emission in clean vacuum, no gas or sputtered materials
•High brilliance unprecedented energy resolution
• High brilliance  small spot  Spectromicroscopy
“The one important complication of synchrotron source is, however, that
while laboratory sources are small appendices to the monochromators, in a
synchrotron radiation set-up the measuring devices becomes a small appendices
to the light source. It is therefore recommendable to make use of synchrotron
radiation only when its advantages are really needed.”
C.Kunz, In Optical properties of solids New developments, Ed.B.O.Seraphin, North Holland, 1976
Radiation-Matter Interaction  Polarization and current induction in
E.M. field
Matter polarizes in presence of an electric field  Result is the establishment in
the medium of an electric field function of both external and polarization
charges
Matter polarizes in presence of a magnetic field  Result is the establishment
in the medium of a magnetic field function of both external and polarization
currents
The presence of fields induce currents
•Mechanisms and peculiarities of polarization and currents induction in
presence of an E.M. field
•Scheme to calculate the E.M. field established and propagating in the
material
•Basis to understand how this knowledge can be exploited to get information
on the microscopic properties of matter
Basic expressions - Charge polarization
and induced currents
P
p
i
i
Polarization vectors
V
M
m
i
i
V
 pol    P
J pol

 P
t
J mag  c M
J cond   transport Escalar potential   optical Eem wave  J transport  J optical
Charge and Magnetic/current polarization – closer look
P
p
i
M
i
V
m
i
i
V
m
E
-Ze-
B
+Ze
pi  e
P0
J mag
Induced currents
J cond   transport Escalar potential   optical Eem wave  J transport  J optical
E ( )
V
Ze+
e+
_
J transport  qNv   transport E
  transport (V )
 transport
Ne 2

2m
B ( )
 optical ( )  E ( )
Motion of charge under the
effect of the electric field of the
E.M. field but in an
environment where it is
present an E.M. field
Expansion of polarisation
Pi    ij E j    ij E j Ek  higher order terms
j
ij
Linear and isotropic media
P
e E
D  E  4 P
Physical meaning Elastic limit
 the potential is not deformed
by the field
M
m B
E  4e E   E
H  B  4 M B  4m B  B(1  4 m )
  1  4 e
Dielectric function

1
1  4 m
Permeability function
Linear versus non linear optics
Formally linear optics implies neglecting terms corresponding
to powers of the electric field
Physically it means E.M. forces negligible with respect to electronnuclei coulomb attraction
E 109 V / cm  breakdown fields 108 V / cm
Nuclear atomic potential is deformed  not harmonic (out of the
elastic limit) response  distortion  higher harmonic generation
Dielectric function and response
In very general way
D( r , t ) 

All space
External stimulus
t
dr '  dt ' (r , r ' , t , t ' ) E (r ' , t ' )   (r , t ) E (r , t )

 D  4ext
Responce function
Note  is defined as a real quantity

-1
Summary material properties within linear approximation
  1  4 e
1

1  4 m
And
 optical
Conduction in an e.m. field
J   EE.M .
 transport
Conduction under a scalar
potential – Usual ohmic
conduction
J   Estatic
Maxwell equations in matter for the linear case
  E  4ext
 H  0
1 
1 
 E  
H  H 
 E  4 optical E  4 J ext
c t
c t
Corresponding equations for vacuum case
 E  4ext

 B   E
t
 B  0
1 
4
 B 
E
J ext
c t
c
Wave equation - Vacuum
2
1

2
 E 2 2E
c t
E  E0 exp i(q  r  t )
2


2
q 
c
2
q
c
Vacuum supports the propagation of plane E.M. waves with dispersion /
wave vector energy dependence
Wave equation - Matter
4 optical  
 E  2 2 E
E
2
c t
c
t
2
  2
q2  
2
c
2
(  i
4

)
2
c
2
n2
Matter supports the propagation of E.M. waves with this dispersion
Formally q is a complex wavevector
Wave vector eigenvalue/dispersion depends on the properties of matter
through    (all real quantities)
Complex refraction index
q2  
2
c2
(  i
4

)
2
c2
n2




E  E0 exp  ksˆ  r   exp i( nsˆ  r  t )
c
c


Absorption
Phase velocity
  1  i 2  n 2
1   n 2  k 2  / 
Real and imaginary parts
not independent
 2  2nk /   4 / 
Absorption coefficient
IE
2
c
v
n
dI   Idr
Lambert’s law

2 k
c
I (d ,  )  I 0 ( )e

4 k

 ( ) d
Complex dielectric constant – Complex wave vector
q 
2
2
c2
(  i
4

)
n  n  ik
1   n 2  k 2  / 
  1  i 2  n 2 / 
 2  2nk /   4 / 




E (r , t )  E0 exp ( kqˆ  r )  exp i ( nqˆ  r   t )
c
c


I  E  dI   Idr    2 k  4 k
c

2
Supported/propagating E.M. modes depend on the properties of
matter through   
The study of modes of the e.m. field supported/propagating in a
medium and the related spectroscopical information is the essence of
the optical properties of matter
Relation between (r,t), (r,t) (r,t) or (q,) (q, ) (r ) and
the properties of matter
1st part  Classical scheme / macroscopic picture
2nd part  Quantum mechanics / microscopic picture
Spatial dispersion
q
2


qˆ
extension on which the average is made
  a
 (q ,  )   (q
q
2

qˆ
0
0,  )   (0,  )
Note  0 wavevector does not mean lost of dependence on
direction  anisotropic materials excited close to origin
q
2

qˆ
0
Unknowns and equations
   (real quantities) are the unknowns related with the material properties
(r,t) is close to unity at optical frequencies  magnetic effects are small
J mag  c M
(not to be confused with magneto-optic effects: i.e. optics in
presence of an external magnetic field)
Generally a single spectrum – f.i. absorption – is available from experiment
(An ellipsometric measurement provides real and imaginary parts at the same time.
It is based on the use of polarizers not easily available in an extended energy range)
Real and imaginary parts are related through Kramers – Kronig relations
Sum rules
Kramers – Kronig dispersion relations
Under very general
hypothesis including
causality and linearity

 ' 2 ( ' )
'
1  1  P  ' 2
d

 0 ( )   2
2
'


(

)  1 '
2
1

 2 ( )  
P
d
' 2
2
 0 ( )  

Models for the dielectric constant / Lorentz oscillator
Mechanical dumped oscillator forced by a local e.m. field
ev  BLocal / c
Neglecting the magnetic term
e-
ELocal ( )
d2
d
m 2 r  m r  m02 r  eELoc
dt
dt
2
Induced dipole
e ELoc
1
p
  ( ) ELoc
2
2
m (0   )  i
Out of phase – complex/dissipation – polarizability (Lorentzian line shape)
e2 ELoc
1
 ( ) 
m (02   2 )  i
Complex dielectric function
From
P  N E
D  E  4 P
E  4e E   E
 ( )  1  4 ( )  1 ( )  i 2 ( )
4 Ne2
1
  1
m (o2   2 )  i
(o2   2 )
4 Ne2
1  1 
m (o2   2 )2   2 2
4 Ne2

2 
m (o2   2 )2   2 2
Lorentz oscillator Dielectric function

Lorentz oscillator Refraction index
Lorentz oscillator Absorption Reflectivity Loss function
Physics
Difference between transverse and longitudinal excitation
EEL spectroscopy
Optical
spectroscopy
Non linear Lorentz oscillator
Anarmonic potential
d2
d
e
2
2
r  2 r   0 r   r   ELoc
2
dt
dt
m
r  r1  r2  ....
r1  1E r2  2 E 2
d2
d
e
2
r  2 r1  0 r1   E
2 1
dt
dt
m
 i n t
E
(

)
e
 n
e
r1  
m ( 0   n ) 2  2i n
d2
d
2
2
r

2

r


r


r
2
2
0 2
1
dt 2
dt
• induced dipole at frequency  and 2 
• the system is excited by a frequency 
oscillates also at frequency 2 
• re-emitting both  and 2 
 2 ( n )  
e
1
m  2   2  2i  2  2   2 2  2i  2  
n
n
n
n
 0
 0

but
Lorentz oscillator in a magnetic field 1/2
d2
d
dr
2
m 2 r  m r  m0 r  e( ELoc   BExt ) BExt  (0, 0, B)
dt
dt
dt
m 2 x  im x  m02 x  eEx  eBy (i )
m 2 y  im y  m02 y  eE y  eBx(i )
x and y motions are
coupled
m 2 z  im z  m02 z  0
Px   Nex ; Py   Ney
Solving for x and y


2
2
e 2  0    i Ex  2i L E y
Px   n
2
m
 02   2  i  4 L 2




2
2
e 2  0    i E y  2i L E x
Py  n
2
m
 02   2  i  4 L 2


L 
eB
2m
Larmor frequency
Lorentz oscillator in a magnetic field 2/2
  xx  1  xy
 xz   Ex 
 Px 

 
 
P





1

yy
yz   E y 
 y  0  yx
P 
  zx
 0 



1
zy
zz
 z

 
P   0 (  1) E
Px  ( xx  1) Ex   xy E y   xz Ez


    i
e
( xx  1)  ( yy  1)  n
m 0  2   2  i 2  4  2
0
L
2

2
0
2

2i L
e2
 xy  n
m 0  2   2  i 2  4  2
0
L


2i L
e2
 xy   yx  n
m 0  2   2  i 2  4  2
0
L


 zz  1
 ij ( B)
Lorentz oscillator in a magnetic field 1/3
P   0 ( 1) E
02   2  i
ne2
 xx  1   yy  1 
m 0  2   2  i 2  4 2 2
0
L


2i L
ne2
 xy 
m 0  2   2  i 2  4 2 2
0
L


2i L
ne2
 yx   xy 
m 0  2   2  i 2  4 2 2
0
L

The dielectric
function is a tensor
[ Physically lost of symmetry for
time reversal ]

  xx
    xy
 0

 xy
 yy
0
0

0 ;
 zz 
B  (0, 0, B)
Propagation in a magnetised medium 1/2
Note ≠ 0 in anisotropic media
2
(  E )   E    0 2 D
t
2
2
2
D



E





Ej
i
0  ij
j
0  ij
2
2
t
t
j
j
Wave
equation
2
ki  k j E j  k 2 Ei   0  0 2   ij E j
Eigenvalue
equation
j
 n 2  n 2  k x2 / k 2    xx

 n 2  k k / k 2   
y x
yx

 n 2  k k / k 2   
z x
zx

with
i 1, 2 ,3
n 2  k y k x / k 2    xy
n 2  n 2  k y2 / k 2    yy
n 2  k z k y / k 2    zy
k
n
k0
j
n 2  k z k x / k 2    xz 

n 2  k y k z / k 2    yz   0

n 2  n 2  k z2 / k 2    zz 
Propagation in a magnetised medium 2/2
Considering the medium with B||z
 n 2   xx

  yx
 0

 xy
n 2   yy
0
0 

 yz   0
 zz 
n2   xx  i zy
Magneto-optics effects
Dichroism
N+  Right circular polarized wave
N-  Left circular polarized wave
Two waves propagating with two different velocities and different absorption
Magneto-optic effects e.g. Faraday and Kerr effects/geometries
Longitudinal
geometry
M
M
Linear polarized
Elliptically polarized
 Rotation according to n+-n-
Dielectric tensor
 (q,  ),  (q,  ),  (q,  ),  e (q,  ),  m (q,  )
are in general tensorial quantities
D  E  4 P
E  4e E   E
  xx
 Dx   Ex 

   
 Dy    E y   4   yx
D  E 
  zx
 z  z

 xy
 yy
 zy
 xz   Px 
 
 yz   Py 
 zz   Pz 
Dielectric tensor
  xx

    yx
  zx

  xx  xy  xz 
 xy  xz    xx  xy  xz 
 
 4 

 yy  yz     yx  yy  yz  



yy
yz 
 yx

  zx  zy  zz 
 zy  zz    zx  zy  zz 


Scalar medium
 xx   yy   zz  
Magnetized medium
  xx

  yx
  zx

 xy
 yy
 zy
 xz   

 yz    0
 zz   0
  xx
    xy
 0

0

0
0

0
 
 xy 0 
 yy  yz 
0  zz 
Longitudinal and transverse dielectric constant 1/2
Any vector field F can be decomposed into two vector fields one of
which is irrotational and the other divergenceless
  FL  0
  FL    F
If a field is expanded in plane
waves FT is perpendicular to
the direction of propagation.
  FT  0
  FT    F
D
E
D  0  D  k
BE  0
k
B
Longitudinal and transverse dielectric constant 2/2
J T (q ,  ) 
i
(1  T ) ET (q ,  )
4
i
J L (q ,  ) 
(1   L ) EL
4
 2

1 2
4
q
(1

)

(1


)
E
(
q
,

)

i
JT (q,  )
T  T

2
2
 c
c


1
2

2
q (1  ) 
2
c
(T   L )
Optics  EELS/e- scattering
The description in terms of longitudinal and transverse dielectric function is equivalent to the description in
terms of the usual (longitudinal) dielectric function and magnetic permeability. They are both/all real
quantities together with conductivity. They combine together to forming the complex dielectric constant
defined here.
Transverse and longitudinal modes 1/3
Propagating waves and excitation modes of matter are two different
manifestation of the same physical situation
Plasmon is a charge oscillation at a frequency defined
by the normal modes oscillation produces a field 
only a field of this kind is able to excite this mode
+
E ( ) _
q
Modes can be transverse or longitudinal in the same meaning of
transverse and longitudinal E.M. field
 searching for transverse waves is equivalent to searching for
transverse modes
q
Transverse and longitudinal modes 2/3
Searching for modes  eigenvectors of
 ij
Transverse modes  Polaritons
sˆ  D(q ,  )  0
E 
sˆ  E (q ,  )  0
N
k
c
The quantum particles are coupled modes of radiation field and of the elementary excitations of the system, called
Polaritons including transverse (opical) phonons, excitons,….
Longitudinal modes
E  Esˆ E  0
Di ( , k )   ij ( , k ) E j ( , k )  0
 ij ( , k )  0
the quantum particles are coupled modes of radiation field and of the elementary excitations of
the system: Plasmons, longitudinal opical phonons, longitudinal excitons,….
Transverse and longitudinal modes 3/3
Polarization waves
D0
D
P
4
E ( , k )  0
0  Ei ( , k )   ij Dij ( , k )


0
 ( , k )   1
  ij ( , k ) 


1
ij
1
Sum rules for the dielectric constant
Examples of sum rules

1 2
0  Im  ( )d  2 p
'

'
'
 
  '  1d '  2 2 (0)
Re
 

0
Of use in experimental spectra interpretation
Quantum theory of the optical constants
Macroscopic optical response
Microscopic structure
Transition probability
Ground state HRADIATION + HMATTER perturbed by radiation-matter interaction
Two approaches
• fully quantum mechanics
• semi classical
Three processes
• Absorption
• Stimulated emission
• Spontaneous emission
Microscopic description of the absorption and emission process
H  HR  HM  HI
System
O °
O
°
O
O
° ° O
Radiation
mi,ei
1
H R   ( Pk2   k2Qk2 )
k 2
E   (nk ,
k
1
 ) k
2
....nk ....
Matter
HM
 1
 
i  2mi


ei
p

 i
c



Aj (ri )   ei   j (ri )   H spin

j i
j i


2
Term neglected for
non relativistic
particles
mj,ej
•Interaction
Hamiltonian HI
•Effect of the
interaction on the
states of the
unperturbed HR + HI
Hamiltonian of a charged particle in E.M. field
H   A
1 
E
A  
c t
 A  0
1 
e  
e  
e 
HR 
p

A

p

A

p

Az 
 x
x
y
 y
 z
2m 
c  
c  
c 
2
px  i
 1

F  e  E  v  H  
 c

2
2

  e


x
1  2 2
e
e
e2 2 
HR 
    i   A  2i A    2 A   e
2m 
c
c
c

1  2 2
e
e2 2 
HR
    2i A   2 A 
2m 
c
c

2


e
 
2 
A p
mc
 2m

1  2 2
e




2
i
A



2m 
c

Particle radiation interaction
2

ei
e 2
2
H   
i  2i
AR (ri )   2 AR    ei   j (ri )  H spin
c
c
i  2mi
 i j i
2


ei
2
 j (ri )  H spin   2i
AR (ri ) 
i  2m i  i ei 
c
j i
i


Matter Hamiltonian
Problem to be solved
+
perturbation Hamiltonian
H  E 
Eigenstate and eigenvector of the matter radiation system in interaction
Important notes
The solution is found by a perturbative method
• it is assumed here – formally - that the problem in absence of
interactions has been solved.
• In practice this can be done with more or less severe
approximations.
• The calculation of the electronic properties of the ground state is
a special and important topic of the physics of matter
n  n (ri , si ...nk ...
 n (ri , si )
n
 H M  H R  n
 En n
Many particles state
Generally obtained by approximate methods
Transition between states of ground state due to the perturbation
term
The effect of perturbation HI on the eigenstates of H0
n  .....nk .... n n (ri , si )
Obtained by time dependent perturbation theory
t 0 
n
 (t )   cn (t ) n (t )
'
'
n'
dcm (t )
i
  m H I n
dt
m
Matrix elements 1/3
cn (0)  1  m (0)  n
The evolution of the state m is obtained calculating the matrix element
m (t )  cn (0) m (0) H I n (0)
System states under perturbation due to
A  A0 exp i(k  r  ik t )
n  ...nk .... n (ri , si )
Changes of photon occupation and matter (f.i. electronic) state
Matrix elements 2/3
It is found that for photon mode k, only
nk  nk  1
'
contribute linear terms to matrix elements
+1  photon emission
-1 photon absorption
Probability of transition of the system from state
n

 n'
Matrix element 3/3
1

sin 2   n'n   k t 
2
2
8 (nk )
2
  (E  E   )
cn' (t ) 
M n' n ( k ,  )
n
2
n'
V 
 n' n   k




Absorption
1

sin 2   n'n   k t 
2
2
8 (nk  1)
2
  ( E  E   ) Emission
cn' (t ) 
M n' n (  k ,  )
n
2
n'
V 
 n' n   k


M n' n   n'


ei
i m exp(ik  r )(i i  n
i
Transition probabilities
Integrating in time from 0 to infinity for the transition probabilities
per unit time
Stimulated emission
Spontaneous emission
2
4 2 (nk  1)
M n'n (k , )  ( En  En'   ) Emission
V k
2
4 2 (nk )
M n'n (k , )  ( En'  En   )
V k
Absorption
Spontaneous emission present only in quantum mechanics treatment
Dielectric function and microscopic properties
Dissipated
power
T


 ( )  1  4 ( )  i
4

4 2 nk
 2 8 nk 
V
 
4
V
V
M nn '   n '
f ( n)
E 2 nk


8
V
1
P( )   J  E dt   E 2
T0
M
nn'
 2
 
4
 ( En  En   )  f (n)  f (n ')
2
nn'
'
ei
 i m exp(ik  ri ) pia  n
i
probability of finding the state in a state n, at thermodynamic equilibrium
f (n)  f (n ')
for
En  En '
Microscopic expression of the dielectric function
4 2 1
2
Im  ( )  2  M n ' n  ( En '  En   k )  f (n)  f (n ') 
 V nn'
Physical meaning  Sum of all the absorbing channels at that photon energy
Note dissipation originates from non radiative de-excitation channels
Intuitive meaning of the expression for absorption coefficient
4 2 1
2
Im  ( )  2  M n ' n  ( En '  En   k )  f (n)  f (n ') 
 V nn'
En’
N ( En ' )
N(E) density of states
(Number of states/eV)
En
Im  ( )
N ( En )
Joint Density Of States
N(E) N(E’)
N ( En' ) N ( En ) M nn'  ( En '  En   k )  f (n)  f (n ') 
Dipole approximation
2
M n' n ( k ,  ) 
8 (nk )
 n'
V k
ei
i m exp(ik  ri ) pi  n
i
2
exp ik  r  1
ei
M n'n ( )  
 n' pi  n 
i mi
 i n'n  n'
 e r
i i
i
n
Matrix elements
of position
operator
Semi classical approach
Note that the same result can be obtained by considering the transition
probability between quantized states of the matter system under the
effect of classical external perturbation of the E.M. field with given by
the same expression of
AR
ei
 A0 exp i (k  r  i k t )  2i
AR (ri ) 
c
This semi classical approach gives identical results for absorption and
stimulated emission probabilities, but does not account for
spontaneous emission
Selection rules for Hydrogen atom
 n'
 e r
i i
ˆ  (sin  cos  ,sin  sin  , cos )
n
i
Generic light polarization
 n 'l ' m '
 n'l 'm' er  nlm 
ˆ  ' ' ' er sin  cos   nlm
ie
nl m
ˆje  ' ' ' er sin  sin   nlm
nl m
 nlm
ˆ  ' ' ' er cos  
ke
nlm
nl m
nlm  R(r ) Pl (cos )eim
m
Selection rules/2
m'
i ( m  m )
drd

d

cos

sin

sin

r
R
(

)
P
(

)
e

' Rn P '
l

n
l
2
m'
'
m
  drr Rn' Rn  d cos ( ) Pl ' ( ) Pl ( )  dei ( m m )
2
'
m
 n' l ' l 1m'
For radiation polarized along z
 n'l 'm' er  nlm 

  kˆ  n'l 'm' er cos  nlm
m  m'
l'  l 1

 nlm
[linear polarized light ]
Expressions valid in any central field
Hydrogen - Selection rules Circular polarization
r sin   ( x  iy )
E

or

l  1
2
r sin   ( x  iy )
e
i

2
 i
m  1
Calculation of matrix elements - Optical properties of matter
ei
 n' pi  n 
mi
M n'n ( )  
The basic step in calculation involves
i
many particles wavefunctions
 i n'n  n'
 e r
i i
n
i
Born - Oppenheimer approximation
Nuclear motions separated
from electronic motions
One electron description
n (r , R)   n (r , R)  n ( R)
M m'k ,mk ( ) 
One electron WF Solution of motion in an
average potential generated by all other
electrons
e

*
dr

(
k
,
r
)(

i
) m (k , r )

m
r
Dielectric function in one electron approximation
Case of crystals
K reduced vector within the
Brillouin zone
Crystal states  E(k)
Im  ( ) 
4

2

m' m
2
dk M

 2 
3
2
m' k , mk
( ) ( Em' ( k )  Em ( k )
BZ
  f F ( Em (k ))  f F ( Em' (k )) 
M m'k ,mk ( )
2
 dk ( E
m'
2
(k )  Em (k )  M m'k ,mk ( ) JDOS
BZ
e

*
M m'k ,mk ( )   dr (k , r )(i
) m (k , r )
m
r
Joint Density of States - JDOS
Phenomenology of absorption
•Interband transitions
-direct/indirect
-Intraband absorption
-Phonon contribution
•Core/localized (e.g. molecular) level absorption
Local field effects - Local (Lorentz) field corrections
P
EL  E 
3 0
Decay and relaxation of excited states
Probability of relaxation/decay of excited state as integral on all the
spontaneous emission channels of field and matter states
1
R


n , k ,
'
4 2 (nk  1)
V k
2
M n'n (k , ) ( Em  En'  k )
As a consequence the dependence of Im () has to be modified
•Lorentzian broadening
• function substituted for by
Lorentzian curve
(e.g. see Lorentz oscillator)

(
1

1

1
 R  e ph  ee
)
Lorentzian broadening
 ( Em  En'   k )
1
(02   2 )  i

Exploitation of emission / radiative decay
Total/Partial yield  measurement of absorption through electron
(Secondary, Auger,..) and photon (fluorescence, luminescence,…)
yields
De-excitation spectroscopies
•Fluorescence
•Luminescence  XEOL
•Auger electron and photon induced – Selection rules and
surface sensitivity
3
2
I c ( )   3   dk  ( En (k )  Ec   )  dr n* (k , r ) er c (r ) f F ( En (k ))
n
 1
matrix element ~ constant I  Density of states and  3
Boundaries  reflectivity
From material filling the whole space to material with boundaries and
matter-vacuum interfaces
Reflectivity - Measure of the reflected intensity as a function of incident
intensity
Fresnels relations based on boundary conditions of fields link
reflected intensity with dielectric function
Reflectivity from a semi-infinite homogeneous material
Surface plane
Normal to surface
Modellisation of surfaces and interfaces
Multiple boundaries
S and p reflectivity
Diffuse scattering 1/2
Small and/or rough objects
Scattered wave
scattering medium
Incident field
2 E (r ,  )  k 2 (r ,  ) E (r ,  )     E (r ,  )  ln  (r ,  )   0
 2 E (r ,  )  k 2 n 2 (r ,  ) E (r ,  )  0
Term neglected if   dimensions
Inhomogeneous filling of space
Scalar theory of scattering
(single Cartesian component)
1 2 2
F (r ,  ) 
k  n (r ,  )  1
4
Defining: Scattering potential
 U (r ,  )  k U (r ,  )  4 F (r ,  )U (r ,  )
2
2
ŝ
Diffuse scattering 2/2
ŝ0
Born approximation
U (rsˆ)
f1 (sˆ, sˆ0 )   F (r )e
'
e
ik ( sˆ0 r )
ik ( sˆ  sˆ0 )r '
 f1 ( sˆ, sˆ0 )
d r   F (r )e
3 '
'
e
iksˆr
r
iqr '
d 3r '
The scattering amplitude is the Fourier transform of the
scattering potential
Inverting  F(r)  n(r)
q
q  ksˆ  ks0
Conclusions
Classical scheme Introduction of the dielectric
function
Microscopic (quantum mechanics) treatment of
emission and absorption
Relation between macroscopic dielectric
function (measured quantity) and microscopic
properties
http://www.gfms.unimore.it/
Calculation of ij elements
Source
Source: 3.3 m of arc, 3.1 m x 3.3 m vertical x horizontal
two fields – vertical and horizontal – out of
phase of ±/2 according to the sign of take off
angle  (J.Schwinger PR 75(1949)1912)
Electric fields
E0 z ( )  i
e
3 23
3 23
(
)  Ai [(
) (1   2 2 )]
4 c
2  0 cr 4 c
e
3 13 ' 3 23
E0 y ( ) 
(
) A i [(
) (1   2 2 )]
4 c
2  0 cr 4 c
≈ 103 photons/bunch - bunch duration ≈ 20 ps
4.5
320eV
r=1m
0.1% BW
3.5
2.5
3
2
Eoz N/C
Eoy N/C
4
3
x 1012
T
x 10125
//
2
1
0
320eV
-1
1.5
r=1m
0.1% BW
1
-2
0.5
0-2
-1.5
-1
-0.5

0
0.5
(mrad)
1
1.5
2
-3
-2
-1.5
-1
-0.5

0
0.5
(mrad)
1
1.5
2
Polarimetry 100 eV ellipse
0
-0.4
-100
Vertical
I(A*e-12)
-50
Electric field at E= 100 eV
+0.4
E = 100 eV
Exit slits
900um * 30 um
-150
-200
-250
15
20
25
30
Zc
35
40
45
-0.4
-1
Horizontal
+1
Electric field at E = 100eV
Erepresentation
+0.4
0
Selector fully open:
Zc= 45 mm, Zg = 1 mm
S1=-0.9 S2=0.011 S3=0.068
Ey=0.95 Ez=0.04 =-1.4 
Ellipticity,  =0.04
Vertical
I , A*e-12
-50
-100
-150
-200
0
Zc = 34
Zg = 41
100 eV
-250
20
30
Zc, mm
40
-0.4
-1
0
+0.4
-50
Horizontal
+1
=-1.44 
Ellipticity,  =0.33
-100
-150
-200
Ey=0.98 Ez=0.04
Electric field at E = 100 eV
Vertical
I , A*e-12
Polarization selector position:
Zc = 34 mm, Zg = 41 mm
(aperture 4 mm)
S1=-0.97 S2=0.011 S3=0.082
Zc = 31
Zg = 31
100 eV
-250
20
30
Zc, mm
40
-0.4
-1
Horizontal
+1
Polarization selector position:
Zc = 31 mm, Zg = 31 mm
(aperture 14 mm)
S1=-0.77 S2=0.08 S3=-0.57
Ey=0.93 Ez=0.31 =1.43 
Transport and conditioning optics
Source  4 m HxV
Mirrors in sagittal focusing  reduction of slope errors effects in
the dispersion plane
GAS CELL
Helicity
selector
P2
BPM
Intensity
monitor
P1
EXIT SLITS
MONO
Light spot
plane-grating-plane mirror monochromator based on
the Naletto-Tondello configuration
Energy range 3- 1600 eV
Energy resolution E/E ≈ 3000 (peak 5000) at vertical slit (typically 30 μm) x 400 μm (variable) Variable
divergence (maximum, variable) 20 m vert x hor
ellipticity variable horizontal/vertical (typically in the range 1.5 – 3.5, Stokes parameters (normalized to
the beam intensity) S1 0.5 - 0.6, S2 0 - 0.1, S3 0.75 -0.85 )
helicity variable (typical value for rate of circular polarization P or S3 0.75 – 0.95)
Examples and experimental arrangements at
BEAR (Bending magnet for Absorption Emission and Reflectivity)
Bulk materials
Absorption
Surfaces
Reflectivity
Interfaces
Fluorescence
Luminescence – XEOL
Diffuse scattering
Experimental arrangements
BEAR (Bending magnet for Emission
Absorption Reflectivity) beamline at
Elettra
Experimental/scattering chamber
Detection
e- analyser / photodiodes
(2 solid angle)
VIS Luminescence
monochromator
Goniometers  , 0.001°
M
A
A
0.01°
(Positive S
0.05°
Differentially
0.1°
pumped joints) C
Sample manipulator
6 degree of freedom
Rotation around beam axis  any position
of E in the sample frame
Optical constants of rare hearths
See f.i. Mónica Fernández-Perea, Juan I. Larruquert, José A. Aznárez1, José A. Méndez Luca Poletto,
Denis Garoli, A. Marco Malvezzi, Angelo Giglia, Stefano Nannarone, JOSA to be published
2
10
1
10
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
k -6
10
-7
10
-8
10
-9
10
-10
10
-11
10
-12
10
-13
10
Endriz
Drude
O2,3
Larruquert
N4,5
Current
M4,5
L1,2,3
Henke
K
Chantler
Extrapolation
10
-2
10
-1
10
0
10
1
10
2
10
3
10
photon energy (eV)
4
10
5
10
6
Interfaces & surface physics in periodic structures
(multilayer optics) See also poster P III 26
ML : Artificial
periodic stack of
materials
(Optical technology  Band pass mirrors)
Z
At Bragg Z dependent standing e.m. field
establishes both inside the structure and
at the vacuum-surface interface
modulated in amplitude and position
•Devices of use in spectroscopy
BRAGG
ultra-thin deposited films
 buried interface spectroscopy
Standing waves & excitation
Scanning
through
Bragg peak
In energy or
angle
Si
Mo
Si
Local modulation
of excitation
Spectroscopy of
interfaces
Physics of
mirror/Reflection
Photoemission,
Auger, fluorescence,
luminescence etc..
Cr/Sc Cr-Oxide interface
573 eV
X 60
(6 Å)

Cr O
2 3
Cr
Sc
(15 Å)
(25 Å)
Qualitative analysis
-Opposite behavior of Cr and Sc
-Different chemical states of the buried Sc
-Two signals from oxygen: one bound to Cr at the
surface, the second coming from the interface
- Carbon segregation at the interface
(As received )
Ru (Clean) -Si buried interface
Angular scan through
the Bragg peak
st
1 component
nd
2
Ru
(15 Å)
Si
(41.2 Å)
Ru 3d
838 eV
Peak area (arb. units)
at
component
Mo
(39.6 Å)
X 40
Ru 3d
st
Intensity (arb. units)
1 component
nd
2 component
background
5.0
5.5
6.0
6.5
Grazing angle (°)
7.0
7.5
h = 838 eV
silicide
Intensity (arb. units)
Mo 3d
Ru 3d
Si 2p
278
280
282
284
286
Binding energy (eV)
288
290
560
580
600
620
Kinetic energy (eV)
728
732
736
740
Model – Ru-Si interface
• Interface morphology
• Calculation of e.m. field inside Ml
•Photoemission was calculated, (Ek= h - EB)
 Minimum position and lineshape depend critically on
Ru
Ru-Si
the morphology profile
Mo-Si ML & i.f. roughness
Motivation  role of ion kinetic energy and flux during ML growth
ML (P  8 nm,   0.44) Performance- R (10°)
Ion assistance
Mo/Si
61.2 %
58.4 %
Wavelength [nm]
Ions EK: 5 eV (1st nm), 74 eV > 1nm  Controlled activation of surface mobility
Performance & diffuse scattering
Performance differences are to be related to
interface quality
Diffuse scattering around the specular beam was
measured
KS= Ki + qZ + q//
diffuse scattering - MLs
I.f.roughnes produces diffuse scattering around the specular beam
I.f.roughnes can/can not be coherently correleted through the ML
Description on a statistical base, ….fractal properties
Single interface – Autocorrelation function

 R 




   
2
H ( x  x' , y  y ' )  H ( R )  2 1  e   






2 2 1
 qz 

1 qz2 2

S (q // )  2 e
d
R
e

qz
 H ( R)
2

e
 
iq// R
In plane Fourier transform on q// of potential
 f.i. Stearns jAP
84,1003
MLs : S(q)  two terms
See f.i. Stearns JAP, 84,
1003, 1998
KS= Ki + qZ + q//
• incoherent scattering by single interfaces
• correlated/coherent scattering among i.f.
(interlayer replica of roughness)
Diffuse scattering
Detector
qdetector  0.003 nm-1
 - scan
At 0.48 nm (13.1 eV)
 Rocking scan
Incident beam
Divergence
qdiv  0.0005 nm-1/m
At 0.48 nm (13.1 eV)
Mo-Si ML diffuse scattering
See also poster Borgatti et al. P III 17
Mo/Si
Ion assistance
ξ= 300 Å
ξ=400 Å
ξ=120 Å
ξ=200 Å
In plane correlation function - absence of interface correlation
Correlation function


2h
2
C ( R )   exp  R /  

ξ , correlation length  h, fractal dimension/jaggedness
Pentacene on Ag(111)
See also poster Pedio et al. P II 33
Chemisorption morphology - tilt angle & electronic structure
( Concentrating on 1 Mono layer )
Premise about C22H14/substrates
•He scattering on pentacene deposited by hyperthermal beams  1ML planar
•Morphology and electronic properties ( delocalization of the  electrons) 
transport properties highly anisotropic;
•on Metals: nearly planar orientation  a condition hindering the formation of an
ordered overlayer;
on semiconductors/oxides: SiO2 standing
GeS lying.
Surface Cell
( By He scattering )
Danışman et al. Phys. Rev. B 72,
085404 (2005)
C3 symmetry
Oblique cell Periodicity (6 x 3) ,
1 monolayer
XAS
C K-edge
 resonances
(At magic angle 54.7°)
LUMO+1
Gas phase XAS
LUMO
2
(Alagia et al.JChemPhys 122(05)124305)
6
4
4
3
13 5
6
5
1
gas phase
2
Multilayer
3 ML
2 ML
10°
1 ML
27°
0.5 ML
25°
0.3 ML
284
286
288
290
292
Photon Energy (eV)
Redistribution of the oscillator strength in
the C1s – LUMO excitation region (1-3 of
gas phase)
294
296
298
Tilt angle
VB photoemission
LDA calculations C22H14/Al(100)
Simeoni et al. S.Science 562,43 (2004)
Photoemission
EV
h=30 eV
0

6.6 eV
7.4
8.3
3b3g
3b2g
2au
3b3g
EF
2au
3b2g
3 ML
2 ML
1 ML
0.5 ML
0.3 ML
clean
Redistribution of  states upon chemisorption
-4
-3
-2
-1
Kinetic Energy (eV)
0
1

HOMO-LUMO gap increasing
XAS 1 ML - precession scan
Dichroism/Bond directionality & Tilt angle of the molecule
 resonances
1 Ml C22H14/Ag(111)
ε= EV/EH = 0.29
Nex_185_int
Nex_186_6_int
Nex_187_6_int
Nex_188_6_int
Nex_189_int
absorption coefficient a. u.
C=90°
i= 10°
 resonances
beam
zM
yM
C=0°
xM
280
285
290
295
300
Photon Energy (eV)
305
310
XAS – 1 ML - deconvolution
0.5
1 Ml C22H14/Ag(111)
absorption coefficient a.u.
0.4
data
fit
0.3
i
iCi
Step
A
C=54.7°
C
E
B
D
0.2
F
ACA
BCB
0.1
CCC
0.0
284
286
288
Photon Energy (eV)
290
292
294
0.5
absorption coefficient a.u.
Precession scan - Formulae
1 Ml C22H14/Ag(111)
0.4
data
fit
i
iCi
Step
0.3
A
C=54.7°
C
E
B
D
0.2
F
ACA
BCB
0.1
CCC
0.0
Fit function for Single domain
284
286
288
Photon Energy (eV)
P
2
E  p  p 2 EH2 

   2 cos 2  C  sin 2  C  2 cos C sin  C cos    sin  M sin  cos   cos  M cos   
2
  cos 2  C   2 sin 2  C  2 cos C sin  C cos   sin 2  sin 2  
2 cos C sin  C   2  1   cos   cos 2  C  sin 2  C   
  sin  M sin  cos   cos  M cos   sin  sin  
Fit parameter: θ
(polar angle of
dinamic dipole )
290
292
294
Tilt angle - Fit
Coverage
Tilt angle
precession scan
Tilt angle
Polar scan
0.3
25° +/- 5°
0.6
27° +/- 5°
28° +/- 4°
1.0
10° +/- 4°
8° +/- 4°