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Light Scattering Spectroscopy

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)

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Light Scattering Spectroscopy

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

  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 optical acoustic 2 transverse (TA) 2 transverse (TO) 1 longitudinal (LO) 1 longitudinal (LA)

  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 10-3Å-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. 12-15, 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 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. 12-18, 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

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