Chapter Questions
Identify the various semiconductors (including alloys) that can be used for light emission at $1.55 \mu \mathrm{~m}$. Remember that light emission occurs at an energy near the bandgap.
Calculate and plot the optical absorption coefficients in GaAs, InAs, and $\operatorname{In} \mathrm{P}$ as a function of photon energy assuming a parabolic density of states.
Consider a $100 \AA \mathrm{GaAs} / \mathrm{Al}_{0.3} \mathrm{Ga}_{0.7} \mathrm{As}$ quantum well structure. Assuming that the problem can be treated as that with an infinite barrier, calculate the absorption spectra for in-plane and out-of-plane polarized light for $E \leq 100 \mathrm{meV}$ from the effective bandedge. Assume the simple uncoupled model for the HH and LH states.
Calculate and plot the overlap of an electron and heavy hole ground state envelope function in a $\mathrm{GaAs} / \mathrm{Al}_{0.3} \mathrm{Ga}_{0.7} \mathrm{~A}$ s quantum well as a function of well size from $20 \AA$ to $200 \AA$. Assume a $60: 40$ value for $\Delta E_c: \Delta E_v$. At what well size does the overlap significantly differ from unity?
Calculate the gain in a GaAs region as a function of injected carrier density at room temperature. Plot your results in the form of gain vs. energy.
Estimate the strength of the intraband transitions in a $100 \AA \mathrm{GaAs} /$ $\mathrm{Al}_{0.3} \mathrm{Ga}_{0.7} \mathrm{As}$ quantum well structure at an electron carrier concentration of $10^{12} \mathrm{~cm}^{-2}$. Assume an infinite barrier model and a linewidth (full width at half maximum) of 1 meV for the transition.
Calculate the electron-hole recombination time $\tau_0$ for an HgCdTe alloy which has a bandgap of 0.1 eV . The momentum matrix element is the same as for GaAs .
In a GaAs sample at 300 K , equal concentrations of electrons and holes are injected. If the carrier density is $n=p=10^{17} \mathrm{~cm}^{-3}$, calculate the electron and hole Fermi levels using the Boltzmann and Joyce-Dixon approximations.
In a p-type GaAs doped at $N_a=10^{18} \mathrm{~cm}^{-3}$, electrons are injected to produce a minority carrier concentration of $10^{15} \mathrm{~cm}^{-3}$. What is the rate of photon emission assuming that all $e-h$ recombination is due to photon emission? What is the optical output power? The photon energy is $\hbar \omega=1.41 \mathrm{eV}$.
Calculate the electron carrier density needed to push the electron Fermi level to the conduction bandedge in GaAs. Also calculate the hole density needed to push the hole Fermi level to the valence bandedge. Calculate the results for 300 K and 77 K .
Calculate and plot the 300 K recombination rate in GaAs as a function of electron (hole) carrier density. Cover the electron (hole) density range from $10^{14} \mathrm{~cm}^{-3}$ to $10^{18} \mathrm{~cm}^{-3}$ in your calculations. Estimate the carrier density dependence of the recombination rate in the low carrier density and high carrier density regime.
The radiative lifetime of a GaAs sample is 1.0 ns . The sample has a defect at the midgap with a capture cross-section of $10^{-15} \mathrm{~cm}^2$. At what defect concentration does the non-radiative lifetime become equal to the radiative lifetime at i) 77 K and ii) 300 K ?
Electrons are injected into a p-type silicon sample at 300 K . The electron-hole radiative lifetime is $1 \mu \mathrm{~s}$. The sample also has midgap traps with a cross-section of $10^{-15} \mathrm{~cm}^2$ and a density of $10^{16} \mathrm{~cm}^{-3}$. Calculate the diffusion length for the electrons if the diffusion coefficient is $30 \mathrm{~cm}^2 / \mathrm{s}$.