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)

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

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

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 r+2 +x

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 r+1 r+2 m M

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

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 1 longitudinal (LA) 2 transverse (TA) (p-1) longitudinal (LO) 2(p-1) transverse (TO)

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)

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 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

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 detect stress 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