PH475/575 Spring 2005 May 29, 2019 PH575 Spring 2019 Lecture #17/18 Free electron metals: Optical response, Kittel Ch. 14 pp. 152-156; 395-403 http://nuclphys.sinp.msu.ru/solidst/images/cu.jpg http://perso.wanadoo.fr/michel.hubin/physique/couche/fermi1.jpg L19 Monday, May 9 2005

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 9 2005

PH475/575 Spring 2005 May 29, 2019 https://physics.stackexchange.com/questions/72368/why-are-most-metals-gray-silver L19 Monday, May 9 2005

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 http://www.pueschner.com/engl/images/phys_basics/diagram2.gif L19 Monday, May 9 2005

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 ≈ 10-1000 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 9 2005

Dielectric function of the (undamped) electron gas: PH475/575 Spring 2005 May 29, 2019 Dielectric function of the (undamped) electron gas: L19 Monday, May 9 2005

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 9 2005

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 2005

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 9 2005

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 9 2005

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 9 2005

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 9 2005