Absorption of Light in Solids

Abstract

In this chapter we present the absorption of light in solids.

Problems

18.1

Consider the Burstein shift phenomena for heavily doped semiconductors. Determine the energy at which appreciable interband (band-to-band) absorption begins at 300 K for a p-type GaAs sample $E_g=1.42$ eV with a hole concentration of $1\times 10^{20}$ cm$^{-3}$. Take into account both the light- and heavy-hole bands, which are degenerate at the $\varGamma$ point in the Brillouin zone ($m_{lh}=0.074 m_0$, $m_{hh}=0.62 m_0$).

18.2

Suppose that you prepare a quantum well structure by molecular beam epitaxy (MBE) from GaAs ($E_g = 1.42$ eV) and $\mathrm Al_xGa_{1-x}As$ $(E_g = \mathrm 1.80\,eV)$, where $x=0.3$ so that $\mathrm Al_xGa_{1-x}As$ is a direct gap semiconductor. Assume that the band off–set of the conduction band is three times greater than in the valence band ($m_{e}=0.067 m_0$).

• (a) Assuming a width of the quantum well of $L_z = 15$ nm, find the photon energies at which optical absorption can take place due to optical transitions between the highest heavy hole and light hole bound states to the lowest conduction band bound state? Use the approximation of an infinite rectangular well in obtaining the energy levels in the bound state.

• (b) Are there selection rules that suppress the transitions between selected valence and conduction band bound states?

• (c) What is the dependence of the threshold photon energy for optical transitions on $L_z$ for a lightly n-doped system? Use the notation $n=n_0$ and $\tau = \tau _0$.

• (d) What is the free carrier contribution to the dielectric function at room temperature?

• (e) What is the free carrier contribution to the optical absorption coefficient?

• (f) What is the difference in the optical spectrum between the superlattice where $\mathrm Al_xGa_{1-x}As$ ($x=0.3$) is used as the wide bandgap semiconductor [part (a)] and the case of AlAs ($x=1.0$), where we note that AlAs is an indirect bandgap semiconductor ($E_g = 2.2$ eV), for which the X point conduction band minimum is 0.23 eV below the lowest $\varGamma$ point ($k=0$) conduction band.

18.3

In many physical cases, the momentum matrix element coupling the highest lying valence band and the lowest lying conduction band vanishes by symmetry at the extremal point $\mathbf {k}_0$. Thus, optical transitions at $\mathbf {k}_0$ are “forbidden”. However, in these cases the momentum matrix element is non–zero as we move away from $\mathbf {k}_0$ by an arbitrary amount. This gives rise to “forbidden direct interband transitions” which have a different frequency dependence for the optical absorption coefficient than their “allowed counterparts”.

• (a) By making a Taylor expansion of the wave function $\psi _{n\mathbf {k}}(\mathbf {r})$ about the band extremum $\mathbf {k}_0$, where $\mathbf {k}= \mathbf {k}_0+\mathbf {\kappa }$, find the dependence of the matrix element $\langle v | \mathbf {p}| c\rangle$ on the magnitude of $\mathbf {\kappa }$, assuming that the matrix element vanishes by symmetry at $\mathbf {k}_0$.

• (b) Using the result from part (a), find the frequency dependence of the optical absorption coefficient for the case of a forbidden direct interband transition around an $M_0$ type critical point.

• (c) Compare the frequency dependence of the optical absorption coefficient in (b) to that for direct allowed interband transitions and for indirect optical transitions.

• (d) What is the frequency dependence of the optical absorption coefficient for a two-dimensional electron gas for allowed and forbidden interband transitions?

18.4

Temperature dependence and isotopic shift of the bandgaps:

• (a) Show that the temperature (T) dependence of an interband gap energy $E_g$ can be written as

  $$ E_g(T) - E_g(0) = A \left( \frac{2}{\exp [\hbar \varOmega /(k_BT)] -1} + 1 \right) , $$

  where A is a temperature-independent constant, $k_B$ is the Boltzmann constant, and $\hbar \varOmega$ represents an average phonon energy. Hint: the term inside the parenthesis in the above equation represents the ensemble-averaged square of the phonon displacement.

• (b) Show that $\varDelta E_g(T) =E_g(T) - E_g(0)$ becomes linear in T in the limit of $k_B T \gg \hbar \varOmega$.

• (c) For small T, $\varDelta E_g(T)$ can also be written as

  $$ \varDelta E_g(T) = \left( \frac{\partial E_g}{\partial V}\right) _T \left( \frac{dV}{dT} \right) _P \varDelta T + \left( \frac{\partial E_g}{\partial T}\right) _V \varDelta T $$

  where the first term describes the change in $E_g$ caused by thermal expansion. Its sign can be positive or negative. The second term is the result of electron-phonon interaction. Its sign is usually negative. Estimate the contribution of these effects to $E_g(0)$ by extrapolating $E_g(T)$ to $T=0$ using its linear dependence at large T. The resultant energy is known as the renormalization of the bandgap at $T=0$ by electron-phonon interaction. Determine this energy for the $E_0$ gap of Ge from Fig. 18.14.

• (d) The result in part (c) can be used to estimate the dependence of the bandgap on isotopic mass. Since the bonding between atoms is not affected by the isotopic mass, the average phonon energy $\hbar \varOmega$ in solids with two identical atoms per unit cell, like Ge, can be assumed to depend on atomic mass M as $M^{-1/2}$. Calculate the difference in the $E_0$ bandgap energies between the following isotopes: $^{70}$Ge, $^{74}$Ge, and $^{76}$Ge.

Fig. 18.14

Ge energy bandgap versus temperature

18.5

For a two level system with energy levels $E_1$ and $E_2$ (where $E_1 < E_2$), assume that before time $t=0$ when a light wave of frequency $\omega$ and intensity $I_0$ is applied, the system is in the ground state $E_1$.

• (a) Find the transition probability for transitions to the state $E_2$ as a function of time t. Consider the response of the system as a signal of frequency $\omega$ is tuned over the resonant frequency $\omega _R$ allowing measurement of the linewidth in energy at $\hbar \omega _R = E_2 - E_1$.

• (b) Suppose that the system is in state $E_2$ at time $t_0$ when the light wave is switched off, find an expression for the probability that state $E_2$ is still occupied after a time ($t_f - t_0$).

• (c) Sketch the occupation of states $E_1$ and $E_2$ over the time interval $0 \le t \le t_f$ and indicate the change in behavior occurring at time $t_0$.

18.6

For GaAs nanowires, it is very difficult to determine the free carrier concentration using traditional Hall effect measurements. As an alternative, photoluminescence measurements can be used to estimate the carrier density based on the Burstein shift.

• (a) For an n-type GaAs nanowire, a blueshift of 20 meV is observed in the photoluminescence emission relative to an undoped nanowire. Based on this information, estimate the Fermi level and carrier density of these nanowires.

• (b) How would your answer change if this shift was observed in p-doped GaAs nanowires? Explain your answer.