1. PH475/575 Spring 2005
May 29, 2019
PH575 Spring 2019
Lecture #17/18
Free electron metals: Optical response, Kittel Ch. 14 pp ;
L19 Monday, May

2. PLASMA frequency separates reflection (low frequency)
PH475/575 Spring 2005
No propagation;
reflection
Vacuum
Metal
May 29, 2019
AIR
METAL
Transparent
PLASMA frequency separates reflection (low frequency)
from transparency (high frequency)
L19 Monday, May

3. PH475/575 Spring 2005
May 29, 2019
L19 Monday, May

4. Dielectric function e of an electron gas:
PH475/575 Spring 2005
May 29, 2019
Dielectric function e of an electron gas:
In intro physics, e is the (real) factor measuring field enhancement in a dc capacitor. Related to polarizability of a material.
In general, e is frequency-dependent, because charges respond differently to different frequency electric fields. It is also complex because there is an in-phase and an out-of-phase response.
Frequency dependence of the dielectric properties of water
L19 Monday, May

5. Visible light l ≈ 10-1000 nm; mfp Cu RT ≈ 80 nm.
PH475/575 Spring 2005
May 29, 2019
Dielectric function of the free electron gas (i.e. electrons in a metal)
Material's response to electromagnetic radiation with frequency w and wave vector k = 2π/ l.
Long wavelength response (spatial electric field of light wave changes little over the mfp of an electron).
Visible light l ≈ nm; mfp Cu RT ≈ 80 nm.
Free mobile electrons against fixed background of positive charge (total system is neutral).
Permittivity of free space 8.8x10-12 F/m
c: electric susceptibility
Permittivity relative to free space
E: electric field; P: dipole moment /volume
L19 Monday, May

6. Dielectric function of the (undamped) electron gas:
PH475/575 Spring 2005
May 29, 2019
Dielectric function of the (undamped) electron gas:
L19 Monday, May

7. Dielectric function of the electron gas:
PH475/575 Spring 2005
May 29, 2019
Dielectric function of the electron gas:
wp is called “plasma frequency”
Magnitude of wp?
Fp=10^15Hz
Ep=7eV
For n=3e28/m^3
Eps0=8.854e-12 F/m
L19 Monday, May

8. r - relative permittivity
PH475/575 Spring 2005
May 29, 2019
Maxwell, free space:
velocity:
Maxwell, medium:
velocity:
r - relative permittivity
n – refractive index (can be complex, real or imag)
L19 Monday, May

9. Optical constants e = dielectric constant n = index of refraction
PH475/575 Spring 2005
May 29, 2019
Optical constants
E, electric field
k: wave vector
v=/k=c/nc
e = dielectric constant
n = index of refraction
a = absorption coefficient
= extinction coefficient
R = reflection coefficient
Unfortunate double use of k as wave vector and as extinction coeff
Sign of ik in n_complex depends on sign of wt-kx
Optical properties depend on 2 quantities (n, k) so generally need to measure 2 things (R,T for us, phase & amp for ellipsometry)
But can get from one to the other with Kramer’s-Kronig analysis, but need full spectrum.
L19 Monday, May

10. w  wp, rel > 0; n real - propagating wave
PH475/575 Spring 2005
May 29, 2019
w  wp, rel > 0; n real - propagating wave
w  wp, rel < 0; n imag. - damped wave
"Transparent conductor" In2O3. Arrows show plasma wavelengths for two different carrier concentrations. (Which one is higher?)
L19 Monday, May

11. No propagation; reflection Vacuum Metal AIR METAL Transparent
PH475/575 Spring 2005
No propagation;
reflection
Vacuum
Metal
May 29, 2019
AIR
METAL
Transparent
L19 Monday, May

12. Dielectric function of the electron gas: doped semiconductors
PH475/575 Spring 2005
May 29, 2019
Dielectric function of the electron gas: doped semiconductors
Till now only the free electron part, but in semiconductors, dielectrics, molecuar crystals, also other bound charge
The “background” dielectric constant is the one associated with the positive cores.
L19 Monday, May