1. 1 8 Chapter 11.

2. 2 8.1 Survey Hagen- Rubens Model Continuum theory: limited to frequencies for which the atomistic structure of solids does not play a major role Drude Model Moving electrons colliding with certain metal atoms (Friction force): Can’t explain fluctuation of reflectivity (absorption band ) Electric dipole: Alternating electric field to the atoms cause forced vibrations. This vibration thought to harmonic oscillator. An oscillator is known to absorb a maximal amount of energy when excited near its resonance frequency. Lorentz Model

3. 3 8.2 Free Electrons Without Damping Plane-polarized light wave where, We consider the simplest case at first and assume that the free electrons are excited to perform forced but undamped vibrations under the influence of an external alternating field. (forced harmonic vibration) (9.14 and 9.15)

4. 4 (1) >1. Then n has imaginary part, (for small frequency) The reflectivity is 100% (2) <1. Then n is real, (for large frequency) The reflectivity is 0% Plasma frequency

5. 5 : the condition for a plasma oscillation

6. 6 N eff can be obtained by measuring n and k in the red or IR spectrum and by applying

7. 7 8.3 Free Electrons With Damping (Classical Free Electron Theory of Metals) Equation of motion forced oscillation With Damping At drift velocity, V’=constant We postulate that the velocity is reduced by collisions of the electrons with atoms of a nonideal lattice. The damping is depicted to be a friction force which counteracts the electron motion.

8. 8 Put this solution to the above equation

9. 9 = Damping frequency

10. 10 Where plasma frequencydamping frequency

11. 11 8.4 Special Cases Thus,, When For the UV, visible, and near IR regions, the frequency varies between 10 14 and 10 15 s -1. The average damping frequency,    is 5x10 12 s -1. with   Thus, in the far IR the a.c. conductivity and the d.c. conductivity may be considered to be identical.

12. 12 8.5 Reflectivity The reflectivity of metals is calculated using 10.29 in conjunction with 11.26 and 11.27.

13. 13 8.6 Bound Electrons (Lorentz) (Classical Electron Theory of Dielectric Materials) Where Stationary solution for the above equation (Resonance frequency of the oscillator)

14. (Supplement)

15. 15 Since, (N a is the number of all dipoles)

16. 16 where,

17. 17 Depicts the absorption product in the vicinity of the resonance frequency. Resembles the dispersion curve for the index of refraction. Both of these equations reduce to the Drude equation when   is zero.

18. 18 8.7 Discussion of the Lorentz Equations for Special Cases 8.7.1 High Frequencies 8.7.2 Small Damping: When radiation-induced energy loss is small, namely,  ’ is small, We observe that for small damping,   (and thus essentially n 2 =(c/v) 2 ) approaches infinity near the resonance frequency. From the figures of    and    2 approaches 0 at high frequencies. And,  1 =n 2 -k 2 =1. Thus, n assumes a constant value 1.- No refraction

19. 19 8.7.3 Absorption Near which shows that the absorption becomes large for small damping 8.7.4 More Than One Oscillator Each atom has to be associated with a number of i oscillators, each having an oscillator strength, resonance frequency damping constant Electrons absorb most energy from light at the resonance frequency.

20. 20 8.8 Contributions of Free Electrons and Harmonic Oscillators to the Optical Constants free electronsbound electrons