Plasmonics And Nanophotonics

MATL 6970 Homework #1

Spring 2022; Due by the end of March 11 1. Color Centers

Color center are responsible for beautiful colors of some ancient artwork. Such color centers are observed in many other materials as well, table salt for example. The figure below shows an absorption coefficient, α, of table salt, also known as NaCl. The absorption feature related to the color centers is strongly peaked in the visible and gives rise to the slightly yellow color of table salt. If you would take a salt crystal and irradiate it with energetic ions or X-rays, you would find that the crystal turns dark yellow. Color centers are formed by the creation of so- called Frenkel defects, an ion displaced from its usual site in the crystal. The vacancies that remain after the ions are displaced (by e.g. an energetic particle) can trap charges. Some configurations in which charge can be trapped are shown and named in Figure 2.

λ (nm)

α

Energy(eV)

Figure 1 Absorption Spectrum NaCl Figure 2 NaCl crystal with trapped charges Questions:

1) When light is incident on a NaCl crystal the bound electrons in an F site will experience a time varying electric field. The interaction of this field with the bound electron can be modeled using the Lorentz model described in class. Assuming that there are only F sites that trap charge, write down the equation of motion derived from the Lorentz model for this system and explain the terms. (Describe the properties of the trap using a damping factor, γ, and a binding strength determined by a spring constant, C.)

2) Derive an expression for the contribution this trap makes to the electric susceptibility, χ, of this system.

3) Treat the electron bound in the F center as an electron with a mass me that is trapped in the field of a point charge e in a medium of dielectric constant εNaCl = 1.6. What is the

E-field

x

εp y

εh

typical binding energy of the electron? Compare this to the absorption energy and draw your conclusions.

4) Explain why an annealing (heating) treatment of the crystal removes the color.

2. Metallic nanoparticles Some of the beautiful colors of 17th century church windows and ancient glass objects derive their color from the presence of nanometer size noble metal particles. We would like to study the optical properties of a piece of SiO2 glass with a refractive index of nSiO2 = 1.5 that has a low concentration of Ag nanoparticles (radius R) embedded in it. We illuminate the glass on one side with a white light source and notice a beautiful yellow color in transmission.

To understand the transmission properties let us first investigate the propagation in the glass of a monochromatic light wave at an angular frequency ω of the form:

When we choose a coordinate system that has its origin in the center of a nanoparticle, the potential inside that nanoparticle is given by:

1) Explain the role of α and β in the expression for a monochromatic wave. 2) Explain why the excitation of the nanoparticle is dipolar (and not for example

quadrupolar). 3) Using the Maxwell’s equations and the well-known boundary conditions stating

that the normal components of the electric flux, D, and the tangential components of the electric field, E, are continuous across an interface, show that the field inside the nanoparticle, Ei, is given by:

4) Show that the induced dipole moment is given by:

5) Figure 3 shows the real and imaginary parts of the dielectric constant of bulk Ag and for 10 nm diameter Ag particles. Estimate at what frequency the light is absorbed most efficiently in the sample with Ag nanoparticles. Does this explain the color of the glass observed in transmission?

7) Explain why the imaginary part of the dielectric constant of a nanoparticle is higher than the imaginary part of the dielectric constant for bulk Ag.

8) Make a sketch of the absorption in the glass as a function of frequency, using the dielectric constants appropriate for bulk Ag and those for the Ag nanoparticles.

9) Discus the phase relationship between the driving field E(x,t) and the induced dipole field P(x,t) inside the particle as a function of frequency.

10) Explain why a sample containing Ag nanoparticles mainly absorbs light and does not scatter much light. (Hint: Compare nanoparticle to microparticles…)

Figure 3. Dielectric constants appropriate for bulk Ag and Ag nanoparticles