1. Light Scattering Spectroscopy
References :
H. Ibach and H. Luth, “Solid-State Physics” (Springer-Verlag, 1990)
P.Y. Yu and M. Cardona, “Fundamentals of Semiconductors” (Springer-Verlag, 1996)
J.S. Blakemore, “Solid State Physics” (Cambridge University Press, 1985)
J.I. Pankove, “Optical Processes in Semiconductors” (Dover Publications, Inc., 1971)

2. Light Scattering Spectroscopy
Inelastic scattering of light can result in vibrational excitations of the atoms
Longitudinal waves in 1-D monatomic crystal (linear chain of atoms) : a
r-2
r-1 r
r+1
r+2 +x

3. Light Scattering Spectroscopy
Hooke’s law for atom r :
Fr = -m(xr – xr+1) – m(xr – xr-1)
= m (xr+1 + xr-1 – 2xr) (Eq. 1)
Longitudinal displacements :
xr = A exp i(kra – wt)
Fr = m d2xr/dt2 = - m w2xr (Eq. 2) a
r-2
r-1 r
r+1
r+2 +x

4. Light Scattering Spectroscopy
Equating Eq. 1 & 2 :
w2 = (m/m) (2 – xr+1/xr – xr-1/xr)
Substituting in Eq. 2 :
w2 = (m/m) (2 – eika – e-ika)
= (4m/m) sin2(ka/2)
w = ± 2(m/m)½ sin (ka/2)
(dispersion relationship)

5. Light Scattering Spectroscopy
w = ± 2(m/m)½ sin (ka/2)
(dispersion relationship)
From Blakemore, Fig. 2-3, p. 94

6. Light Scattering Spectroscopy
w = ± 2(m/m)½ sin (ka/2)
(dispersion relationship)
Smallest wavelength is 2a
Largest k = ± p/a a
r-2
r-1 r
r+1
r+2 +x

7. Light Scattering Spectroscopy
Can describe dispersion curve entirely within k = 0 to k = G = p/a
From Blakemore, Fig. 2-3, p. 94

8. Light Scattering Spectroscopy
For small k, w = [ (m/m)½a ] k
Speed of sound in the material
From Blakemore, Fig. 2-3, p. 94

9. Light Scattering Spectroscopy
Must also include transverse vibrations
Atomic spacing, a, will vary for different directions in the scrystal; dispersion curves are plotted for different directions
Deviations from simple model occur due to forces from remote neighbors
from
Blakemore,
Fig. 204,
p. 96

10. Light Scattering Spectroscopy
Longitudinal waves in 1-D diatomic crystal : a
r-2
r-1 r
r+1
r+2 m M

11. Light Scattering Spectroscopy
Longitudinal displacements :
xr = A exp i(kra – wt)
xr+1 = B exp i[k(r+1)a – wt ]
-m w2 xr = m (xr+1 + xr-1 – 2xr)
-M w2 xr+1 = m (xr+2 + xr – 2xr+1) a
r-2
r-1 r
r+1
r+2 m M

12. Light Scattering Spectroscopy
Substituting and rearranging as before :
A(2m-mw2)=2mBcoska
B(2m-Mw2)=2mAcoska
Combining terms to eliminate A and B gives:
w2 = m(m-1 + M-1) ± m [ (m-1 + M-1)
– (4sin2ka)/mM ]½

13. Light Scattering Spectroscopy
w2 = m(m-1 + M-1) ± m [ (m-1 + M-1)
– (4sin2ka)/mM ]½
From Blakemore, Fig. 2-10, p. 107

14. Light Scattering Spectroscopy
2 curves separated by gap
Lower curve
Acoustic branch
Similar to previous monatomic chain
Upper curve
Optical branch
From Blakemore, Fig. 2-10, p. 107

15. Light Scattering Spectroscopy
Optical branch
Near k = 0 atoms move in opposite directions
Ionic bonding has a dipole moment
Optical phonons can be excited optically
From Blakemore, Fig. 2-11, p. 109

16. Light Scattering Spectroscopy
3-D Polyatomic Crystals :
Any 3-D crystal can be described by a unit cell and a basis
The basis are the atoms and their orientation with respect to each lattice point
There are 14 possible 3-D unit cells (Bravais lattices)

17. Light Scattering Spectroscopy
From Blakemore, Fig. 1-21, p. 36

18. Light Scattering Spectroscopy
From Blakemore, Table 1-6, p. 37

19. Light Scattering Spectroscopy
3-D Polyatomic Crystals :
Twice as many transverse compared to optical branches
A basis of p atoms has 3p branches (3p vibrational modes)
basis = p atoms
acoustic
optical 1
longitudinal
(LA) 2
transverse
(TA)
(p-1)
longitudinal
(LO)
2(p-1)
transverse
(TO)

20. Light Scattering Spectroscopy
3-D Polyatomic Crystals :
e.g., C, Si
diamond structure = fcc unit cell + basis of p = 2 atoms
basis = 2 atoms
acoustic
optical 1
longitudinal
(LA) 2
transverse
(TA) 1
longitudinal
(LO) 2
transverse
(TO)

21. Light Scattering Spectroscopy
3-D Polyatomic Crystals :
In Si and C, both transverse branches are degenerate along [111] and [100] directions due to symmetry
From Blakemore,
Fig. 2-13,
p. 112

22. Light Scattering Spectroscopy
Reflected
light
(Rayleigh
scattering)
Ei = ħwi
|ps| = |pi|
Incident
light
Ei = ħwi
pi = ħki
Inelastic
scattering
of light
Es = ħws
ps = ħks
Phonon
Ep = ħw(k)
pp = ħk
Frequency shifts of scattered light are characteristic of the material

23. Inelastic Light Scattering
Conservation of energy and momentum:
Phonon absorption:
Es – Ei = + ħw(k)
ks – ki = + k
Phonon emission:
Es – Ei = - ħw(k)
ks – ki = - k

24. Light Scattering Spectroscopy
Maximum phonon wavevector excited using visible light is:
|k| = |ki – ks| = 2(2p)/l ~ 2 x 10-3 Å-1
|G| = p/a ~ few Å-1
|k| << |G|

25. Light Scattering Spectroscopy
|k| << |G| for visible light
Can only excite phonons near k ~ 0
From Pankove, Fig , p. 273

26. Light Scattering Spectroscopy
Must use neutrons with Ei ~ 0.1 – 1 eV for phonon spectroscopy
Brockhouse and Shull, Nobel prize in 1994 for inelastic neutron scattering work performed in 1950’s

27. Light Scattering Spectroscopy
Raman Scattering
Nobel prize in 1930
Inelastic scattering of light from optical phonons
E(k~0) ~ constant (LO, TO)
From Blakemore, Fig. 2-13, p. 112

28. Light Scattering Spectroscopy
Stokes shift: phonon is created; light loses energy
Anti-Stokes shift: phonon is destroyed; light gains energy
From Yu & Cardona, Fig. 7.21, p. 375

29. Light Scattering Spectroscopy
Raman scattered light intensity ~ 104 – 108 times weaker than elastically scattered light
Need high intensity light source (laser)
Sensitive detector with low background noise (cooled photodiode, PMT)

30. Light Scattering Spectroscopy
Frequency difference between Raman signal and laser light ~ 1% of laser frequency
Need good spectral resolving power, R = l / Dl > 104
Need high stray light rejection ratio (notch filter to block laser light, 2 or more spectrometers in series = double monochromator)

31. Light Scattering Spectroscopy
double
monochromator
PMT
sample
laser

32. Raman Spectroscopy Can detect stress
Can determine composition and crystal structure
Can detect stress
From Schroder, Fig. 9.33, p. 632

33. Raman Spectroscopy Surface Enhanced Raman Scattering (SERS)
Coherent anti-Stokes Raman Scattering (CARS)

34. Light Scattering Spectroscopy
Brillouin Scattering
Inelastic scattering of light with acoustical phonons
A continuum of k values exist → a continuum of Stokes and anti-Stokes components
From Pankove, Fig , p. 276

35. Light Scattering Spectroscopy
Frequency shifts are a few cm-1
Much smaller than Raman shifts (few 100 cm-1)
From
Yu & Cardona,
Fig. 7.30,
p. 389

36. Light Scattering Spectroscopy
Use Fabry-Perot interferometer rather than grating spectrometer
From Yu & Cardona, Fig. 7.29, p. 388