1 8 Chapter 11. “Continuum Theory”“Atomic Structure of Solids”“Quantum Mechanics”

2 Summary: 1.In the case of no conductivity: 2.When there is a conductivity: From Maxwell equation of EM wave. Hagen-Rubens equation

3 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 Some electrons in a metal can be considered to be free and can be accelerated by an external electric field. Moving electrons colliding with certain metal atoms (Friction force): Can’t explain fluctuation of reflectivity (absorption band ) Electric dipole: Presented each atom as an 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 Reflectivity of (a) metals and (b) dielectrics

4 8.2 Free Electrons Without Damping, 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) Momentary value of the field strength of a plane-polarized light wave:

5 The reflectivity is 100% (2) <1. Then is real and imaginary part is zero (for large frequency) The material is essentially transparent for these wavelengths (1) >1. Then is imaginary and real part is zero, (for large frequency),

6 : the condition for a plasma oscillation Plasma frequency

7 N eff can be obtained by measuring n and k in the red or IR spectrum and by applying (in a frequency range without absorption bands)

8 8.3 Free Electrons With Damping (Classical Free Electron Theory of Metals) Equation of motion forced oscillation With Damping At drift velocity, V’=constant (saturation drift velocity) 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.

9 Put this solution to the above equation

10 = Damping frequency

11 Where plasma frequencydamping frequency

Special Cases Thus,, When For the UV, visible, and near IR regions, the frequency varies between and s -1, while 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. (Table 11-1)

Reflectivity The reflectivity of metals is calculated using in conjunction with and

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

(Supplement)

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

17 where,

18 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.

Discussion of the Lorentz Equations for Special Cases High Frequencies 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 From eq’n So what?

Absorption Near which shows that the absorption becomes large for small damping 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.

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