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9
Phonons
9.1 Infrared active phonons
9.2 Infrared reflectivity and absorption
in polar solids
9.3 Polaritons
9.4 Polarons
9.5 Inelastic light scattering
9.6 Phonon lifetimes
10.1 Infrared active phonons
The resonant frequencies of the phonons occur
in the infrared spectral region, and the modes
that interact directly with light are called
infrared active (IR active).
The phonon modes of a crystal are
subdivided into two general categories:
• acoustic or optical;
• transverse or longitudinal.
It is the “optical” rather than the acoustic
modes that are directly IR active. These
optically active phonons are able to absorb light
at their resonant frequency. Conservation laws
require that the photon and the phonon must
have the same energy and momentum.
Dispersion curves for the acoustic and optical phonon branches
in a typical crystal with a lattice constant of a. The dispersion
of the photon modes is shown by the dotted line. At small wave
vectors the slope of the acoustic branch is equal to vs, the
velocity of sound, while the optical modes are essentially
dispersionless near q=1. At the intersection , which corresponds
to the resonance, the frequency is equal to that of the optical
mode at q=0.
Photons couple to TO phonons through the driving force
exerted on the atoms by the AC electric field of the light wave.
This can only happen if the atoms are charged, or the crystal
must have some ionic character.
The crystals with ionicity contain the ionic crystals and the
crystals with partly covalent bonds (polar bonds)
10.2 Infrared reflectivity and absorption in polar solids
d2
K
q
( x  x )   ( x  x )  E (t ),
2
dt


10.2.1 The classical oscillator model
where
1
1
1


.
 m m
By putting x = x+ -- x- for the relative displacement of positive
and negative ions with their unit cell, the equation of motion in
the simpler form:
d 2x
q
2
 TO
x  E (t ),
2
dt

Interaction of a TO phonon mode propagating in the z
direction with an electromagnetric wave of the same
wave vector. The black circles represent positive ions,
while the grey circles represent the negative ions. The
solid line represents the spatial dependence of the
electric field of the electromagnetric wave.
When the wave vectors and frequencies match, the
strong interaction at resonance between a TO phonon
mode and an infrared light occurs at q  0 (~10 m).
There will be thousands of unit cells (usually less than
10-9m) within one period of the wave.
The equation of motion for the positive and negative
displaced ions:
d 2 x
m
  K ( x  x )  qE (t )
dt 2
d 2 x
m
  K ( x  x )  qE (t )
dt 2
where 2TO for K/. represents the natural vibrational
frequency of the TO mode at q = 0 in the absence of the extend
light field. The equation represents the equation of motion for
undamped oscillations of the lattice driven by the forces
exerted by AC electric field of the light wave. In reality, a
damping term to account for the finite lifetime of the phonon
mode should be incorporated into the eqn by introducing a
damping rate :
d 2x
dx
q
2




x

E (t ).
TO
dt 2
dt

The response of the medium to a light field of angular frequency
 with E (t) = E0 e it:
Nq 2
1
 r ()  1   
,
2
 0 (TO
 2  i)
where  represents the non-resonant susceptibility of the medium,
and N is the number of unit cells per unit volume.
10.2.1 The classical oscillator model
In the limits of low frequency:
Nq 2
 st   r (0)  1   
,
2
 0TO
In the limits of high frequency:
  r ()  1  .
Thus we can write:
2
TO
 r ()     ( st    ) 2
(TO  2  i).
where "   " in relative sense here.
The waves are transverse. If r = 0, the equation can be
 
satisfied with waves in which k  E  0, that is, with
longitudinal waves.
The dielectric can support longitudinal electric field
waves at frequencies which satisfy r(’) = 0.
The waves at  = ’ correspond to LO phonon wave,
namely LO, thus
 2LO  st
 .
2
TO

This is the Lyddane-Sachs-Teller (LST) relationship.
10.2.2 The Lyddane-Sachs-Teller relationship
For a lightly damped system with  =0, assume r = 0 at
’:
2
 r ( ' )  0     ( st    )
TO
.
(   ' 2 )
2
TO
This can be solved to obtain:
1
2
 
'   st  TO .
  
What does r=0 mean physically? When  = 0,


  D    (r 0 E)  0,
For r  0 and wave solutions with the form:
 
 

E (r , t )  E0 ei ( k r t ) ,
we have
 
k  E  0.
The data in the table imply that the LO phonon and TO
phonon modes of non-polar crystals are degenerate.
There is no infrared resonance, and therefore r = .
This is indeed the case for the purely covalent crystals
of the group IV element namely diamond (c), silicon
and germanium.
10.2.3 Restrahlen
Set  = 0, the dielectric constant has the following
frequency dependence:
2
vTO
 r (v)     ( st    ) 2
.
(vTO  v 2 )
A wave number of 1 cm-1 is equivalent to a
frequency of 2.9981010 Hz.
1THz = 1012 Hz.
1ps = 10-12 s
III-V semiconductor
At low frequency the dielectric constant is just equal to r.
As v increases from 0, r (v) gradually increases until it
diverges when the resonance at vTO is reached. Between
vTO and vLO , r is negative. Precisely at v = vLO, r = 0.
thereafter, r is positive, and gradually increases towards
the value of .
The most important optical property of a polar solid in the
infrared spectral region is the reflectivity.
2
n~  1
R ~

n 1
r 1
r  1
2
.
The reflectivity is equal to 100 % in the frequency region
between vTO and vLO. This frequency region is called the
restralen band. Light cannot propagate in the restrahlen band.
Infrared reflectivity of InAs and GaAs at 4.2
K. there is a sharp dip in the reflectivity just
above the LO phonon resonance. This
reduction in reflectivity from 100 % is
caused by ignoring the damping term. The
damping also broadens the edge so that there
is only a minimum in R just above vLO rather
than a zero. The magnitude of  obtained in
this way are around 1011-1012 s-1, which
implies that the optical phonons have a
lifetime of about 1-10 ps.
10.2.4 Lattice absorption
The dispersion of the polaritons as q  0 and q   :
The absorption coefficients expected at the resonance with
the TO phonon can be calculated from the imaginary part of
the dielectric constant. At  = TO we have
 r (TO )     i( st    )
TO
.

The extinction coefficient  and the absorption coefficient
 (typically in the range of 106 – 107 m-1) can be work out
from r. The absorption is very high at the TO phonon
resonance frequency.
as q  0,
  vq 
c
q,
 st
as q 
  vq 
c
q,

 
 st
TO  LO

  TO
LO >  > TO, r < 0, restrahlen band;
 > LO, r > 0, v  c / ( )1/2..
10.3 Polariton
 
1
{(  st 02  c 2 q 2 )  ( st 02  c 2 q 2 ) 2  402 c 2 q 2   }
2 
Polariton dispersion predicted for a crystal with vTO = 10 THz,
st = 12.1 and  = 10. the asymptotic velocities are vst and v.
Dispersion of the TO and LO phonons in GaP
measured by Raman scattering. The solid lines are
the predictions of the polariton model with hvTO =
45.5 meV,  = 9.1 and st 11.0.
10.4 Polarons
The polaron theory cab be applied to holes or electrons in
the formula. In a non-polar crystal such as silicon,  =  st ,
and ep = 0, there is no polaron effect.
In the table the coupling constant increases from the III-V,
which is predominantly covalent, to the I-VII, which is
highly ionic. The effective mass can be measured using
cyclotron resonance.
1
The electron energy: En   n  C ,
2

Where
This is a schematic representation of a polaron. A free
electron moving through an ionic lattice attracts the
positive ions, and repel the negative ones. This
produces a local distortion of lattice within the polaron
radius that is a new elementary excitation of the
crystal. The optical properties of the whole host can
depend on the polaron effect of the electron-phonon
coupling.
The strength of the electron-phonon interaction in a
polar solid can be quantified by the dimensionless
coupling constant ep, which is given by:
 ep 
1
2
1  mc  1
1

   ,
137  2 LO      st 
* 2
The effective mass:
1
 d 2E 
m    2  .
 dk 

2
C 
eB
.
m*
Optical transition (absorption) with n =  1 can
take place at a wavelength  ( in the far-IR) given by:
hc eB
 

m
The effective mass can be deduced from  and B at
resonance, however, it is usually not the value
determined by the curvature of the bands given by m*.
For the small ep:
m
1
1

 1   ep .

m
1   ep / 6
6
For III-V material like GaAs with ep<0.1, m** only
differs from m* by about 1%. The polaron effect is thus
only a small correction. The correction become more
significant for II-VI (e.g. ~7% for ZnSe). With highly ionic
crystals like AgCl, this is 0.43 (about 50% larger than the
rigid lattice value). The small ep approximation is not
valid.
The additional energy correction:
E g   ep  LO ,
This means a reduction in the band gap by the amount.
The radius, rp, which specifies how far the lattice distortion
extends. If ep is small:
10.5.1 General principles of inelastic light scattering
Inelastic light scattering can be subdivided into two types:
• Stokes scattering: emitting a phonon or other excitation;
• Anti-Stokes scattering: absorbing a phonon.
1

2

 .
rp   
 2m  LO 
The gives rp = 4.0 nm for GaAs and 3.1 nm for ZnSe.
Large polaron: rp >> a (unit cell size, ~ 0.5 nm) and ep is
small;
Small polaron: rp ~ a and ep is not small.
Small polar existing in highly ionic crystals leads to selftrapping of the charge carriers. The local lattice distortion
is so strong that the carrier can get completely trapped in
its own lattice distortion. The only way they move is by
hopping to a new site. The electrical conductivity of most
alkali halide crystals is limited by this thermally activated
hopping process at RT. The effects are also improtant in
the conduction processes in organic semiconductors .
10.6 Inelastic light scattering
Inelastic light scattering can be medicated by many
different types of elementary excitations in a crystal such
as phonons, magnons or plasmons.
Inelastic light scattering from phonons is subdivided as:
• Raman scattering: scattered from optical phonons;
• Brillouin scattering: scattered from acoustic phonons
Conservation of energy:
Conservation of energy:
1  2  ,
  
k1  k 2  q.
The light is shifted down in frequency – Stokes;
The light is shifted up in frequency – anti-stokes.
The ratio of anti-stokes to stokes scattering is given by:
I anti Stokes
 exp(  / k BT ).
I stokes
The probability for anti-Stokes scattering is very low at
cryogenic temperatures.
The maximum phonon frequency in a typical crystal is
about 1012 – 1013 Hz. This is almost two orders of
magnitude smaller than the frequency of a photon in the
visible spectral region. Thus the maximum frequency
shift for photon will be around 1%.

 n
k 2  k1 
,
c
and
  
q  k1  k2 .
The maximum possible value of q thus occurs for the
back-scattering geometry:


n
q  k  (k )  2
.
c
The maximum value of q is of order 107 m-1. This is
very small compared to the size of the Brillouin zone in
a typical crystal (~ 1010 m-1). Inelastic scattering is thus
only able to probe small wave vector phonons.
10.5.2 Raman scattering
Raman scattering is mainly used to determine the
frequency of LO and TO modes near the Brillouin zone
centre, and gives little information about the dispersion
of the phonons (optical ponons are essentially
dispersionless near q = 0).
IR reflectivity and light scattering is of complementarity.
The full selection rule requires the use of group theory.
However, a simple rule can be given for crystals that
posses inversion symmetry:
• The odd parity modes are IR active;
• The even parity modes are Raman active.
This is called the rule of mutual exclusion.
In non-centrosymmetric crystals, some modes may be
simultaneously IR and Raman active.
Experimental apparatus used to record Raman spectra.
Raman spectra obtained from four III-V crystals at 300 K
using a Nd:YAG laser at 1.064 m. the spectra are plotted
against the wave number shift: 1 cm-1 is equivalent to an
energy shift of 0.124 meV. These correspond to Stokesshifted signals from the TO phonons and LO phonons. The
LO mode is the one at higher frequency.
10.5.3 Brillouin scattering
It refers to inelastic light scattering from acoustic
phonons. Its main purpose is to determine the dispersion
of these acoustic modes. The frequency shift of the
phonon in the scattering is given by
   s
One phonon is annihilated
and two new phonons are
created.
2n

sin ,
c
2
where s is the velocity of the acoustic waves, and  is
the angle through which is light is scattered. This can be
used to determined the velocity of the sound waves.
10.6 Phonon lifetimes
Analysis of the experimental data on the reflectivity in the
restrahlen band show that damping constant  is typically
in the range 1011—10 12 s-1. Since  is equal to lifetime -1,
the data implies that  is in the range 1 – 10 ps. The very
short lifetime of the optical phonons is caused by
anharmonicity in the crystal. In general, the atoms sit in a
potential well of the form:
Decay of an optical phonon
into two acoustic phonons by
a three -phonon interaction .
U ( x)  C2 x 2  C3 x 3  C4 x 4  .
The term in x2 is the harmonic term. The term in x3 and
higher are the anharmonic terms. These anharmonic terms
allow phonon – phonon scattering processes. For example,
the term in x3 allows interactions involving three phonons.
The lifetime of the optical phonons can be deduced from Raman data in three different ways. Firstly, the spectral width of
the Raman line is affected by lifetime broadening, which in frequency unit is expected to be (2)-1; Secondly,  can be
measured directly by time-resolved Raman spectroscopy using short pulse lasers; thirdly, the lifetime of the optical
phonons can be deduced from reflectivity measurements.