Optical Properties of Semiconductors

Abstract

In previous chapters, we introduced the reader to the fundamental concepts of quantum mechanics, band structure, and semiconductor physics. In this chapter we have the opportunity to apply this acquired knowledge of the electronic structure of solids to understand the optical properties. We do this by modeling the optical response properties, in particular the permittivity of the solid. We present the formalism which allows one to calculate the permittivity and then study how this permittivity affects the light penetrating the solid. We shall demonstrate how band structure and free electrons determine the permittivity, and therefore the way light propagates in a solid, and how much of this light gets absorbed. We shall investigate under what circumstances the lattice can couple to photons and how this coupling can affect the velocity of light in a medium. But we shall see in the next chapters that band structure depends on the dimensionality of the system, and we have already seen in Chaps. 8 and 9 that carriers can be added or neutralized in semiconductors. So it turns out that just in the same way that the energy bands can be engineered, so can the optical properties. Atom by atom growth and miniaturization are modern key engineering tools, but so is the application of external electric and magnetic fields. In the last sections of this chapter, we therefore investigate how an electric or a magnetic field modifies the band structure and how this reflects on the optical properties. The fundamental concepts developed in this chapter are a necessary prerequisite to understand the way optical methods can be used to characterize the electronic structure of semiconductors as is described in Chap. 15.

Problems

• 1. Calculate the real and imaginary part of the frequency-dependent admittance of a wire as a function of frequency, if the area is 1 cm², the length 0.1 cm, the charge density 10²¹ cm⁻³, and the relaxation time τ = 10⁻¹³s and effective mass 0.1m ₀. Write down the results as a function of frequency. What are the conductance and the capacitance?

• 2. Calculate the oscillator strength F₁₂ linking the ground state n = 1 and first excited state n = 2 of box eigenstates with box size L = 1 nm and effective mass m* = 0.023 m ₀.

• 3. Calculate the reflectivity of a metal as a function of frequency using the Drude permittivity formula with free carrier concentration n _c = 10²² cm⁻³, relaxation time τ = 10⁻¹² s, and m* = 0.045 m ₀. Plot the result and compare with Fig. 10.2.

• 4. Explain the difference between direct and indirect bandgap materials. Sketch the two situations. If phonons were not allowed to provide the necessary momentum in an indirect bandgap excitation, what other mechanisms can you think of which could make the absorption process happen in another way?

• 5. Calculate the density of states per unit volume of a three-dimensional nearly free electron gas with effective mass m* in a magnetic field B _z perpendicular to the x-y plane including spin. Remember that the number of allowed k _y states per Landau level is given byL _x L _y qB/h for an area of size L _x L _z and that there is another (free electron) z-degree of freedom in the z-direction.

• 6. What is meant by the permittivity of a solid? How is it calculated? How is it related to the refractive index? What does the real and imaginary part of the refractive index signify? How would you design a material which is a perfect reflector?

• 7. Using the definition of the complex refractive index given by Eq. (10.9), derive the pair of equations given by Eq. (10.14) which show that this leads to a quadratic equation from which the real and imaginary part of the complex refractive index $\overline{n}$ and κ can be computed.

• 8. What is a phonon-polariton? Write down the explicit algebraic solutions which give the two branches of the dispersion relation ω ²(k) for the phonon-polariton equation using Eq. (10.102). Explain how and why the group velocity of this new particle changes with wavenumber.

• 9. What is an exciton? In GaAs the effective mass of an electron is m _e = 0.067 m ₀, and the effective mass of the hole is m _h = 0.082m ₀. The relative static permittivity ε _r is 13.1. Using Eq. (10.85) and Eq. (10.86), calculate the exciton radius and binding energy. At what temperatures would you expect the excitons to be detectable by experiment?

• 10. With the help of Eq. (10.125), derive the magnetic field-dependent complex conductivity of an electron gas as given by Eq. (10.127): $\sigma \left(B,\omega \right)=\frac{nq^2\tau }{m^{\ast }}\left(\frac{1}{\tau}\right)\left\{\frac{1/\tau - i\omega}{{\left( i\omega -1/\tau \right)}^2+{\omega}_c^2}\right\}$. Discuss the behavior of the real part as a function of the magnetic field. What happens when the magnetic field becomes very large? Give a physical interpretation. How does a magnetic field affect the reflectivity of a free electron gas?