Transcript PowerPoint Sunusu

Light Scattering Spectroscopies
Raman Scattering
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Macroscopic Theory of Phonon Scattering
System : infinite, isotropic medium
 

 
F(r, t)  Fi (ki ,i ) cos(ki  r - i t)
 

 
P(r, t)  Pi (ki ,i ) cos(ki  r - i t)
: Plane electromagnetic wave
: Induced sinusoidal polarization

 
 
Pi (ki ,i )   (ki ,i )Fi (ki ,i )
: Electric susceptibility
(fluctuates at T >0)
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Macroscopic Theory of Phonon Scattering
Atomicdisplacements
  associated

 with
 a phonon
Q(r, t)  Q(q,0 ) cos(q  r - 0 t)
Adiabatic approximation:Characteristic electronic frequencies are>> ω0
(ki,wi,Qi)
Taylor expansion in Qi(r,t)



  
 (ki , i , Q)   0 (ki , i )   Q(r , t )  ...
Q0
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Macroscopic Theory of Phonon Scattering
Thus, the polarization of the medium in the presence
of atomic vibrations is obtained as
   
  
P(r, t,Q)  P0 (r, t)  Pind (r, t,Q)
where

 
 

P0 (r , t )  0 (ki ,i )Fi (ki ,i ) cos(ki  r  it )
(Polarization vibrating in phase with F)
&

 
 
  
Pind (r , t , Q)   Q(r , t ) Fi (ki , i ) cos(ki  r  i t )
Q0
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(Polarization wave induced by the phonon)
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Macroscopic Theory of Phonon Scattering
Now rewriting the equation as
 
 
 
 
  
Pind (r , t , Q)   Q(q, 0 ) cos(q  r  0t )  Fi (ki , i ) cos(ki  r  i t )
Q0
 
  
  Q(q ,  0 ) Fi (ki ,  i ) 
Q0
  
  
cos (ki  q )  r  ( i   0 )t  cos (ki  q )  r  ( i   0 )t
 



The induced polarization consists of two sinusoidal waves!
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Macroscopic Theory of Phonon Scattering
Pind
Stokes shifted sinusoidal
ks=(ki-q) and s=(i- o)
Anti-Stokes shifted wave
kAS=(ki+q) and AS=(i+o)
The light produced by these polarization waves
Stokes scattered
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anti-Stokes scattered
light
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Macroscopic Theory of Phonon Scattering
Terminology
phonon
incident photon
frequency =
frequency
ph
i




scattered photon
frequency
s
Energy and momentum are conserved
One phonon scattering probes zone center phonons
Higher order terms lead to ± ωa ±ωb (ωa>ωb)
Hence two phonon scattering. Combination and difference
modes. If two phonons are identical, the two phonon peak is
called an overtone.
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Raman Tensor and Selection Rules
Intensity of scattered rad can be calculated from time
averaged induced polarizations Pind into solid angle.
Intensity of the scattered
radiation depends on the
polarization of the scattered
as well as of the incident
radiation.
   

I s  ei   Q(q ,  0 )  es
Q0
2
Define a complex second rank tensor with q=Q/|Q| as
   ˆ
R     Q ( 0 )
 Q  0
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Macroscopic Theory of Phonon Scattering


I s  ei  R  es
2
(R : the Raman tensor)
Raman scattering can be used to determine
frequency and symmetry of a zone center phonon mode
Raman tensors for zincblende structure
0 0
R ( X )  0 0
0 d
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0
d 
0 
0 0 d 
R (Y )   0 0 0 
d 0 0 
0
R ( Z )  d
 0
0
0 0
0 0
d
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Macroscopic Theory of Phonon Scattering
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Nonclassical View of Raman Scattering
The quantity measured in experiments =  (scattering efficiency)

EM energyscatteredintoa solid angle / (unit time unit frequencyinterval)
Energyof theincident EM modescrossing thescatteringarea / unit time
   
 s 
    VL  eˆi     Q( 0 )  eˆi
 c 
 Q 0
4
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L:scattering length
V=AL
s:frequency of
the scattered light
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Nonclassical View of Raman Scattering

EM energyscatteredintoa solid angle / (unit time unit frequencyinterval)
Energyof theincident EM modescrossing thescatteringarea / unit time
Raman scattering inelastic scattering of photons by quantized excitations in the medium
Ns

Ni
: total scattering cross section
Ni:flux of incident photon beam per unit area
Ns:flux of scattered photon beam per unit area
  i   d 2 
  

  
  s A   dd s 
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(non classical result)
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Nonclassical View of Raman Scattering
 : total cross section corresponding to |ei.R(i, s). es|
 Ns=  (i, s)Ni
Assuming that the incident photon flux is such that only
one scattered phonon is produced
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 Ni= 1/ (i, s)
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Nonclassical View of Raman Scattering
• Raman scattering possesses time-reversal symmetry
Ns= (Ni-1) + Ni A (s, i)
From the time reversal symmetry we have the requirement
Ns= Ni
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 1= NiA (s, i)
(i, s) = A(s, i)
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Experimental Setup
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Selected Results
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Frequency Shift
 (T) = 0 + 1(T )+ 2(T ),







where 0 + 2(0) is the Raman frequency as T approaches 0 K, 1(T )
represents the volume dependence of the frequency due to the thermal
expansion of the crystals and 2(T ) specifies the contribution of
anharmonic coupling to phonons of other branches.
1(T ) can be written as
1(T) = 0 exp( 3   (T )dt )  1

where (T ) is the coefficient of linear thermal expansion.
In general, the purely anharmonic contribution to the frequency shift can
be modeled as
2(T ) = A 1  x 1  x 1 




e 1  1 e 2  1
which represents the optical phonon coupling to two different phonons
(three-phonon processes). Here, x1 = hc1 / kBT and x2 = hc2 / kBT .
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
1
1 

1  x1

 e  1 e x 2  1
Linewidth


The temperature dependence of the phonon
linewidth can be described as
follows
1
1 

 = C 1  x  x 

e 1  1 e 2  1
where C is the broadening of the phonon line due
to the cubic anharmonicity at absolute zero (the
decrease in phonon lifetime,  , due to the decay of
the optical phonon into two different phonons).
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TlInS2
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TlInS2:
6
C 2h



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At BZCenter Group
Theory gives:
10 Ag +14 Bg + 10
Au + 14 Bu ,
where Au + 2 Bu
modes are acoustic.
There should be 10
Ag + 14 Bg Ramanactive modes
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Modes in TlInS2


Most modes shifts to smaller frequencies as
temperature increases.
Modes at 280.9 ans 292.3 cm-1 sohws
hardening with increasing temperature
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Translational Mode Frequency at 57.1 cm-1
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Intramolecular Mode at 292.3 cm-1
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57.1 and 292.3 cm-1



Curve 1:Exp freq shift
Curve 2: Purely
thermal expansion
1(T)
Curve 3:Purely
anharmonic
contribution, 2(T)
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Frequency and Lineshift Analysis
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ν0
ν1
ν2
A
C
37.6
18.8
18.8
0.40
0.32
61.8
30.9
30.9
0.67
0.29
83.8
41.9
41.9
-0.57 1.06
139.1
69.5
69.6
-0.79
0.44
278.3
104.6
173.7
1.34
0.79
284.4
78.9
205.5
4.86
0.70
303.8
48.2
255.6
2.46
0.98
347.5
48.9
298.6
0.59
1.54
348.3
139.3
209.0
2.11
0.93
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