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Light Scattering Spectroscopy
References :
1. H. Ibach and H. Luth, “Solid-State
Physics” (Springer-Verlag, 1990)
2. P.Y. Yu and M. Cardona,
“Fundamentals of Semiconductors”
(Springer-Verlag, 1996)
3. J.S. Blakemore, “Solid State Physics”
(Cambridge University Press, 1985)
4. J.I. Pankove, “Optical Processes in
Semiconductors” (Dover Publications,
Inc., 1971)
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
+x
r+2
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
+x
r+2
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)
Light Scattering Spectroscopy
w = ± 2(m/m)½ sin (ka/2)
(dispersion relationship)
From Blakemore, Fig. 2-3, p. 94
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
+x
r+2
Light Scattering Spectroscopy
• Can describe dispersion curve
entirely within k = 0 to k = G = p/a
From Blakemore, Fig. 2-3, p. 94
Light Scattering Spectroscopy
• For small k, w = [ (m/m)½a ] k
Speed of
sound in
the material
From Blakemore, Fig. 2-3, p. 94
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
Light Scattering Spectroscopy
• Longitudinal waves in 1-D diatomic
crystal :
a
r-2
r-1
r
m
M
r+1
r+2
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
m
M
r+1
r+2
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
Light Scattering Spectroscopy
w2 = m(m-1 + M-1) ± m [ (m-1 + M-1)
– (4sin2ka)/mM ]½
From Blakemore, Fig. 2-10, p. 107
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
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
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)
Light Scattering Spectroscopy
From Blakemore, Fig. 1-21, p. 36
Light Scattering Spectroscopy
From Blakemore, Table 1-6, p. 37
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
2
1
longitudinal transverse
(TA)
(LA)
2(p-1)
(p-1)
longitudinal transverse
(TO)
(LO)
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
2
1
longitudinal transverse
(TA)
(LA)
2
1
longitudinal transverse
(TO)
(LO)
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
Light Scattering Spectroscopy
Incident
light
Ei = ħwi
pi = ħki
Reflected
light
(Rayleigh
scattering)
Ei = ħwi
|ps| = |pi|
Inelastic
scattering
of light
Es = ħws
ps = ħks
Phonon
Ep = ħw(k)
pp = ħk
• Frequency shifts of scattered light
are characteristic of the material
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
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|
Light Scattering Spectroscopy
|k| << |G| for visible light
 Can only excite phonons near k ~ 0
From Pankove, Fig. 12-15, p. 273
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
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
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
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)
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)
Light Scattering Spectroscopy
double
monochromator
PMT
sample
laser
Raman Spectroscopy
• Can
determine composition and crystal
structure
• Can detect stress
From Schroder, Fig. 9.33, p. 632
Raman Spectroscopy
• Surface Enhanced Raman Scattering
(SERS)
• Coherent anti-Stokes Raman
Scattering (CARS)
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. 12-18, p. 276
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
Light Scattering Spectroscopy
• Use Fabry-Perot interferometer rather than
grating spectrometer
From Yu & Cardona, Fig. 7.29, p. 388