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Optical Processes in Crystals and Excitons: Theory and Applications

Explore the dielectric function, reflectivity, and excitons in crystals through Kramers-Kronig relations and Raman spectroscopy. Learn about response functions and the behavior of charged oscillators in an electromagnetic field, along with sum rules and applications in semiconductor energy gaps.

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Optical Processes in Crystals and Excitons: Theory and Applications

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  1. Dept of Phys M.C. Chang Optical processes and excitons • Dielectric functionand reflectance • Kramers-Kronig relations • Excitons • Frenkel exciton • Mott-Wannier exciton • Raman spectroscopy

  2. Dielectric function, reflectivity r, and reflectance R Response of a crystal to an EM field is characterized by (k,), (k0 compared to G/2) Experimentalists prefer to measure reflectivity r (normal incidence) Prob.3 It is easier to measure R than to measure   measure R() for   () (with the help of KK relations)  n()  ()

  3. Kramers-Kronig relations (1926) KK relation connects real part of the response function with the imaginary part examples of response function: • does not depend on any dynamic detail of the interaction • the necessary and sufficient condition for its validity is causality Ohm’s law polarization reflective EM wave Example: Response of charged (independent) oscillators For the j-th oscillator (atom or molecule with Z bound charges), steady state electric dipole

  4. Some properties of (): (1) () has no pole above (including) x-axis. (2) along (upper) infinite semi-circle (3) ’() is even in , ’’() is odd in . Collection of oscillators for identical oscillators Figs from http://www.physics.uc.edu/~sitko/LightColor/10-Optics1/optics1.htm

  5. ω and many more…, e.g., (4)+(5) (next page): By Ferrel and Glover (1958) From (1), (2), we have ( can be , or , or... ) property (3) used Also, A few sum rules: An example of the “fluctuation-dissipation” relation From (1) and (2), >>1 (Prob. 2): Thomas-Reiche-Kuhn sum rule From (3), >>1 (Prob. 4):

  6. KK relation for reflection Why take log? See Yu and Cardona for related discussion • constant R(s) doesn’t contribute • s>>, s<< don’t contribute ref: Cardona and Yu

  7. Conductivity of free electron gas (j=0, with little damping) (KK relations can be checked easily in this case) Recall that (chap 10) Real part of DC conductivity diverges (compare with ne2/m) Drude weight D Other sum rules (Mahan, p358): Related to the energy loss of a passing charged particle

  8. Dielectric function and the semiconductor energy gap (prob.5) band gap absorption (due to nonzero ’’) can be very roughly approximated by Si Ge GaAs InP GaP ’ 12.0 16.0 10.9 9.6 9.1 Eg (theo) 4.8 4.3 5.2 5.2 5.75 Ex (exp’t) 4.44 4.49 5.11 5.05 5.21 (ref: Cardona and Yu) Figs from Ibach and Luth

  9. Interband absorption for GaAs at 21 K Excitons (e-h pairs) in insulators • Tight bound Frenkel exciton: • (excited state of a single atom) • alkali halide crystals • crystalline inert gases

  10. Weakly bound Mott-Wannier excitons Similar to the impurity states in doped semiconductor: CuO2

  11. 1930 Raman effect in crystals

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