1. Lecture 21 Optical properties

2. Incoming lightReflected light Transmitted light Absorbed light Heat Light impinging onto an object (material) can be absorbed, reflected, or transmitted. If the medium is transparent (no absorption): Reflectivity (R) = Reflected fraction Incident fraction Transmitivity (T) = Transmitted fraction Incident fraction

3. In more detail: Reflection: the incident and exit angle with respect to the normal to the surface are identical. Transmission: the refractive index change from one material to the next determines the change in direction from outside to inside the medium. Scattering: on a rough surface, locally the surface normal varies, resulting in a broad macroscopic distribution of “reflected” light called scattering. Absorption: the incoming light partially penetrates the material, transfers energy to electron and/or lattice excitations. These in turn may relax back to the ground state by emitting light and or phonons. Refractive index n is the ratio of the vacuum speed of light c to the speed of light in the medium v. Frequency is the constant. Transmission

4. The amplitude of the reflected light depends on the polarization of the light and the dielectric properties of the material. The components of E parallel and perpendicular to the plane of incidence. Reflection

5. A finite fraction of the light intensity dI is absorbed over a small distance dx into the material. The absorption coefficient is a material property. In a semiconductor is proportional to the frequency of visible light and a material property. means that there is no absorption. Absorption

6. WindowsMirrors X-rays for detectors

7. The parameters T, R, and A (for absorption) along with n and are optical material constants. When there is no absorption A=0 is zero. Transmission Reflection Absorption Depth profiling of shallow water

8. Reflection data for Al, Ag, Au, and Cu Band diagram for Al

9. SiliconAluminumSilverCopper Density of states per energy Energy Fermi level Au 2.3 eV Ag 4.0 eV Cu 2.0 eV Energy from 3d levels 2 eV 4 eV Al more absorption

10. Background: Light moving in z-direction with electrical polarization in x-direction From Maxwell’s equations “trial” solution (note: complex) Complex index of refraction dampingplane wave From

11. Reflectivity In a metal with low frequencies and dielectric values less than 10: Hagan-Rubens relation

12. Theoretical Classical Drude model Electrons are free within a band and can be accelerated by external fields and can loose energy by scattering. They transition between energy states within a band. Typically phonons are involved. Transitions are caused by phonon-electron scattering or by impurity-electron scattering (in insulators). The phenomenological model by Drude includes these contributions results in And from there to the complex current density The real and imaginary parts are complex

13. Estimate of scale At =1THz=10 12 /s and and for the non-complex versions: ConductivityDamping Au Ag Al 5um 2um 1um 500nm 200nm 60 150 300 600 1500 THz

14. Theoretical background Lorentz theory of bound electrons electrons are bound to nuclei by “springs”, which determine the natural frequency. Recall the harmonic oscillator An external electric field displaces charges and creates a dipole. This is assumed to be the oscillator. Vibrations are forced by an external AC field. At the resonance frequency the maximum amount of energy is absorbed. The combination explains free electrons with high absorption (R near zero) for low frequencies ( ) for the IR region of the light spectrum. The bound electrons, oscillator explain the absorption bands. Insulators and semiconductors are explained by the harmonic oscillator of bound electrons. Failures Why should electrons be free at low frequencies and bound at higher frequencies? Quantum Mechanics Solves the dilemma and explains the absorption (or not) of light with intra-band and inter-band transitions as well as direct and indirect energy gaps.

15. This is mostly observed in metals, where small energy transitions are possible due to partially filled bands. In semiconductors and insulators this does not occur. Exceptions may be highly doped semiconductors where the Fermi Energy is right at the conduction (or valence) band. At low frequencies this effect dominates (not quantized). Window coatings with materials such as ITO are used to transmit visible light but reflect IR light. Loss of heat is minimized in the winter or the room temperature remains cooler in the summer. Conclusions Inter-band transitions correspond to the bound-state version of the Lorentz model, while Intra-band transitions correspond to free electron effects of the Drude model. The sharp absorption lines from atoms (i.e. single resonance frequencies) give way to broad bands and hence absorption bands and not absorption frequencies. The plasma frequency corresponds to the edge where the reflectance of a metal turns up.

16. MaterialsWork function (eV) threshold frequency (10 14 /s Ceasium1.914.62 Rubidium2.175.25 Potassium2.245.42 Lithium2.285.51 Sodium2.465.95 Zinc3.578.63 Copper4.1610.06 Tungsten4.5410.98 Silver4.7411.46 Platinum6.3015.23

17. Inter-band transitions Electrons transition from one band to another (usually from the valence to the conduction band) Direct gap GaN vsIndirect gap Si Direct: Electrons transition “vertically” without the “assistance” of phonons. The momentum vector k remains constant (the momentum of the photon is very much smaller and insignificant here). There is a vast number of near continuous transitions possible. The band gap merely represents the lower minimum. Indirect: Phonons are created or absorbed to accommodate the required change on momentum vector k. In metals these transitions play a miniscule role (100 to 1000 times smaller in intensity). However, in semiconductors they play a big role. Keep in mind that jumps to higher bands (not shown) are also possible.