Simple quantum models of
optical response of condensed matter
v. 3
11.4.2021
1
FK020_Electrodynamics of solids
Josef Humlíček, humlicek@physics.muni.cz
The simplest quantum models based on perturbation theory.
Probability of transitions to excited states and dissipation of energy.
Response functions related to one-electron picture of direct interband transitions of valence electrons in
crystals.
Examples: optical functions of doped GaAs.
2
Quantum transitions in perturbation theory
3
The light wave is composed by a train of photons, carrying quantized energy. They can be absorbed during
the interaction with matter.
In the manybody system of condensed matter, elementary excitations in the form of quasiparticles can be
identified (electrons, holes, excitons, phonons etc.).
The photon and quasiparticle fields influence each other via their interactions (“scattering”, “collisions”).
One of the basic processes is the absorption of a photon, with the transfer of its energy to the quasiparticle
system.
Other processes are possible, such as elastic or inelastic scattering of photons, when a photon survives the
collision in a modified form (direction of propagation and/or energy). Important processes involve
spontaneous or stimulated emission of photons, carrying energy taken from quasiparticles.
In the case of small changes of the studied system, caused by a weak optical field, the prediction of
response functions can be based on the standard perturbation theory of quantum mechanics.
4
A convenient quantity to be calculated is the energy taken from a harmonic electromagnetic wave in a unit
volume per unit of time, linked to the imaginary part of the dielectric function,
2
2
1
,
2
oQ E 
which is proportional to the time average of the power of the wave. Its macroscopic form is usually the
(Joule) heat.
The procedure involves a calculation of the increase of the mean energy of the condensed system, and use
the above equation for the evaluation of the absorptive part of the dielectric function. The real part can be
obtained via Kramers-Kronig transform of the imaginary part calculated for all frequencies.
Matter will be divided into small areas (of the volume V, with their dimensions much smaller than the
wavelength of the optical field). In these areas, the electric field intensity of the wave is independent of the
position; we retain solely the harmonic time dependence
  i t
oE t E e 

from the time to=0 of switching the perturbation on. Magnetic component will be neglected.
The perturbative part of hamiltonian can be expressed via the operator of dipole moment (charge times its
displacement) as
 
   ( ) Re , where /
2
oi t i t i t
o o o
d n E
H t d E e e e n E E   

     
is a unit vector in the direction of force. The force performs work due to the displacement of the charge,
equal to the scalar product of the vectors of force and displacement.
(4.1)
(4.2)
(4.3)
5
2
2 0
2
2
2
2
2
1
( , ) ( ) d
( , ) ( , ) , where
sin
2
, ( , ) .
2
f iE E
T i t
if
i f i f o
f i
i f
p T f H t e t i
f d n i
T F T F T E
T
x
E E
F T x
T
x

   




     
 
    

Assume the system in a stationary state i (with the energy Ei) at the initial time to. The probability of a
transition to a stationary final state f (with the energy Ef) at the time T (which is the squared modolus of
the probability amplitude) is
.if  
For large T, the probability is negligible except for the fulfillment of the “resonance condition”
(4.4)
(4.5)
(4.6)
6
(
, )d for any 0 ; lim ( , ) ( ) .
T
F T x x T F T x x 

 
  
For T → ∞ the function F can be replaced by the Dirac :
Owing to the transition i→f , the light field performs work, which (per unit volume and time) reads
2
2
( , )
( ) ( ) ( ) .if
if o i f i f
p T
Q f d n i E
TV V
  
             
The work vanishes whenever the resonance condition is not fulfilled, and diverges otherwise. This is a
consequence of the stationary initial and final states. Quasistationary states have finite lifetimes; for the
exponential temporal dependence of the probability Ps of the decay of the state of mean energy Eo during
the time t (Pn means the probability preserving the state during the time t),
2 2
( ) , ( ) 1 ,
t t
n sP t e P t e
 
 
  
the probability density of finding the energy E is
2 2
1
( ) .
( )
o
o
E E
E E



 
  
This is so called Breit-Wigner, or Lorentz, or Cauchy distribution.
(4.7)
(4.8)
(4.9)
(4.10)
7
The positive parameter  has the dimension of energy; it is inversely proportional to the lifetime of
quasistationary ( <<Eo) state,
For
.
2
 

Thus, the Fermi golden rule of the perturbation theory can be complemented by the assumption concerning
the random values of the energy
2 2 2 2
2 2
/ /
( ) d
( ) ( )
( ) /
.
( ) ( )
f i
if o o
f f f i i i
i f
o
if if i f
E E
E E E E E E
E E
 




 

       
  

    

This is again the Breit-Wigner distribution; it is centered about the difference of the individual centers.
0 ( ) , ( ) ( ) .o oE E E E       
if f i ifE E E   
taken from the wave via the i→f transition. The width parameters of the Breit-Wigner distribution of the
initial and final energies can differ; assuming independent occurrence of both energies, the probability
density of the energy difference is the convolution (“Faltung”, “svjortka”)
(4.11)
(4.12)
(4.13)
(4.14)
8
The width parameters and the corresponding relaxation times are
With the interpretation of inverse relaxation time as the frequency of collisions, the result for independent
events concerning the initial and final states is the sum of frequencies.
The spectral dependence of the energy absorbed via transitions i→f between quasistationary states is a
broadened version of the delta-function singularity for stationary states. Multiplication by the probability
density and summation over all possibilities leads to
1 1 1
, .
2
if i f
if if i f  
       

 
o 2 2 o 2 2
( ) ( ) ( )d( )
( / ) ( / ) ( )d( )
( ) ( )
1 1
.
( ) ( )
i f i f i f i f
i f
if if if if
x x x x
        
    
   
    




    
   
  
 
  
       


.if  The resonances are of finite magnitude and width, they are centered close to
(4.15)
(4.16)
9
The former result leads to the following approximate expression for the contribution of the i→f transitions
to the imaginary part of the dielectric function,
2 22
2
o 2 2 o 2 2
2 4
( )
4
.
( ) ( )
i f i f
if
o oo
i f i f
o if if if if
Q Q
EE
f d n i
V


    
  
   
  
       
This formula can be further simplified, and, the need for the Kramers-Kronig transform (to obtain the real
part) avoided.
(4.17)
10
the contribution to the absorptive part of the dielectric function can be expressed in the form
Since
2 2 2 2
2 2 2
( ) ( )
21 1
Im Im ,
2
o o
o
o o oi i i
 
     

         

   
   
     
         
2
o
2 2
o 2 2
2
24 1
( ) Im .
2
( )
if
if
if ifo
if
f d n i
V
i



  
 
   
   
    
  
2
2 2
1 .o
o
S
i



 

 
 
This is to be compared with the result of classical Lorentz model for the complex dielectric function,
(4.18)
(4.19)
(4.20)
11
the eigenfrequency and damping time are
The dimensionless “strength” of the i→f transitions is
2
o 2
, 2
( ) , .
2
if
o if if if
if
  

  

2
,
2 2,
,
( ) 1 .if o if
i f
o if
if
S
i

 

 

 
 

The identical spectral dependencies of the imaginary parts imply identical spectra of the real parts, since
they are related by Kramers-Kronig integral transform. Thus, in summing the contributions of (possibly
many) independent contributions of transitions from different initial states to different final states, we may
include the dispersive (real) part of the dielectric function in the sum:
2
o
8
,if
o if
f d n i
S
V 


Each of the terms has poles in the lower plane of complex frequencies.
(4.21)
(4.22)
(4.23)
12
can be expressed in terms of the momentum operator, using the commutator of the unperturbed
hamiltonian Ho with the position:
The matrix element involving the charge and position,
,ifD f d n i e f r n i   
Since
 
 
, ,
, ( ) ,
,
.
o f o i
o f i
o
f i
f H r i E f r i f rH i E f r i
f H r i E E f r i
f H r i
f r i
E E
 
 


 , ,o
dr p
H r i i
dt m
   
we arrive at
.
( )
if
f i
e
D i f p n i
m E E
  

(4.24)
(4.25)
(4.26)
(4.27)
13
This form is usually used for the one-electron transitions in crystals, which do not involve other excitations
(such as phonons). As the momentum is proportional to the k–vector of the Bloch states, the matrix
element is zero for states of different k (the state vectors are orthogonal). The allowed transitions are called
“direct” (in k-space). This selection rule can also be interpreted as the requirement of momentum
conservation, since the momentum of the involved photon is negligible.
Adding independent contributions from all available pairs of initial and final states provides the total
response. In the picture of transitions between different electronic bands, occupied l and unoccupied l’, the
Fermi golden rule and conservation of momentum implies the dielectric function (4.23) in the form
The transition probability is proportional to
2
22
.
( )
if
f i
e
D f p n i
m E E
 

(4.28)
2
' '
2
2, '
'
( ) ( ) ( )
( ) 1 .
( ) ( )
( )
ll l l
l lk BZ
l l
if
S k k k
k k i
k
 
 

  


   
    
  (4.29)
14
The energy differences between different bands (“interband energies”) resonate with the energy
of incoming photons.
Joint density of states (JDOS) is usually very instrumental in dealing with most of the
crystalline matter. In fact, neglecting the k-dependence of the matrix elements in (4.29), the
frequency dependence of the probability of a photon being absorbed is proportional to the
number of available energy differences between the occupied initial and free final electron
states.
JDOS changes strongly with frequency in the neighborhood of the critical points (minima,
saddle points, maxima) of the interband energy.
lattice dynamics, electronic bandstructure: quasiparticles are
bosons fermions
15
The transitions produced by the light waves can be illustrated conveniently in the case of GaAs
(a prototype polar material):
Electron density in the ground state
(Cohen and Chellikowsky, Electronic Structure and Optical Properties of Semiconductors, nonlocal pseudopotential)
suggests a difference in the IR response between Si and GaAs due to lattice vibrations
Si
GaAs
16
Spectral structure due to critical points (CP) of JDOS
Differentiation of high-quality optical spectra, enhancement of critical points of JDOS
30
M0
M0
M0
M1
M1
M1
M2
M2
M3
3D
2D
1D
PrehledCP_eps

PHOTON ENERGY
10
M0
M0
M0
M1
M1
M1
M2
M2 M3
3D
2D
1D
PrehledCP_D1
'(a.u.)
PHOTON ENERGY
3
M0
M0
M0
M1
M1
M1
M2
M2
M3
3D
2D
1D
PrehledCP_D2
''(a.u.) PHOTON ENERGY
removes any constant
background removes any linear
background
30
M0
M0
M0
M1
M1
M1
M2
M2
M3
3D
2D
1D
PrehledCP_eps

PHOTON ENERGY
removes any
quadratic background,
spans a narrow ()
spectral range centered at the
critical-point energy

M0
M0
M0
M1
M1
M1
M2
M2 M3
3D
2D
1D
PrehledCP_D3
'''(a.u.)
PHOTON ENERGY
Low—noise data: even triple differentiation might be possible (not very often, though)
An instructive case study: doped crystalline GaAs
19
Doped GaAs – dielectric function (Drude, Lorentz, ∞, interband transitions)
0.00 0.02 0.04 0.06 0.08 0.10
-200
-100
0
100
200
2.7x10
18
cm
-3
Im
Re
n-GaAs
DIELECTRICFUNCTION
PHOTON ENERGY (eV)
5 10 15 20
-10
0
10
20
Im
Re
GaAs
DIELECTRICFUNCTION
PHOTON ENERGY (eV)
20
Doped GaAs - conductivity
0.00 0.02 0.04 0.06 0.08 0.10
-200
0
200
400
600
800
2.7x10
18
cm
-3
Im
Re
n-GaAs
CONDUCTIVITY(
-1
cm
-1
)
PHOTON ENERGY (eV)
5 10 15 20
-10000
-5000
0
5000
10000
15000
Im
Re GaAs
CONDUCTIVITY(
-1
cm
-1
)
PHOTON ENERGY (eV)
21
Doped GaAs – refractive index
0.00 0.02 0.04 0.06 0.08 0.10
0
5
10
15
2.7x10
18
cm
-3Im
Re
n-GaAs
REFRACTIVEINDEX
PHOTON ENERGY (eV)
5 10 15 20
0
1
2
3
4
5
Im
Re
GaAs
REFRACTIVEINDEX
PHOTON ENERGY (eV)
22
Doped GaAs – inverse dielectric function
0.00 0.02 0.04 0.06 0.08 0.10
-0.5
0.0
0.5
2.7x10
18
cm
-3
Im
Re
n-GaAs
-1/
PHOTON ENERGY (eV)
5 10 15 20
-0.5
0.0
0.5
Im
Re
GaAs
-1/
PHOTON ENERGY (eV)
23
Doped GaAs – penetration depth
0.00 0.02 0.04 0.06 0.08 0.10
1
10
2.7x10
18
cm
-3
n-GaAs
PENETRATIONDEPTH(m)
PHOTON ENERGY (eV)
5 10 15 20
0.01
0.1
1
10
GaAs
PENETRATIONDEPTH(m)
PHOTON ENERGY (eV)
24
Doped GaAs – normal-incidence reflectivity
0.00 0.02 0.04 0.06 0.08 0.10
0.0
0.2
0.4
0.6
0.8
1.0
2.7x10
18
cm
-3
n-GaAs
REFLECTIVITY
PHOTON ENERGY (eV)
5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
GaAs
REFLECTIVITY
PHOTON ENERGY (eV)
25