Basis of a Bipolar Semiconductor Laser

Abstract

We treat the basis of bipolar semiconductor lasers. We discuss: condition of gain; joint density of states; gain coefficient; laser equation; bipolar character of the active medium. And we derive, by use of Planck’s radiation law, the Einstein coefficients for an ensemble of two-level systems that is governed by Fermi’s statistics.

Problems

21.1

Wave vector of nonequilibrium electrons in GaAs.

• (a) Calculate the wave vector k of electrons in GaAs that have an energy of 100; 10; and 1 meV. Compare the values with the wave vector $q_\mathrm{p}$ of a photon with the energy $h \nu =E_\mathrm{g}$ (m$_\mathrm{e} =0.07$ m$_0$; m$_0 =0.9 \times 10^{-30}$ kg; $E_\mathrm{g} =1.42$ eV; $n =3.6$).

• (b) Determine the energies $\epsilon _\mathrm{c}$ and $\epsilon _\mathrm{v}$ if $q_\mathrm{p} =k$.

21.2

Wave vector of radiative pair levels. We supposed that the wave vector of a photon involved in a radiative transition is small compared to the wave vector of the electron and the hole that are involved in a radiative transition. Show that this is justified for electrons and holes of sufficient energies.

21.3

Electron and holes in an undoped GaAs quantum film in thermal equilibrium.

• (a) What is the condition with respect to the quasi-Fermi energies that the electron gas and the hole gas are in thermal equilibrium?

• (b) What is the corresponding condition with respect to $\epsilon _\mathrm{Fc}$ and $\epsilon _\mathrm{F v}$?

• (c) Estimate the electron density $N_\mathrm{thermal}^{2\mathrm{D}}$ of subband electrons ($=$density of subband holes) in a quantum film at temperature T. Show that $N_\mathrm{thermal}^{2\mathrm{D}}$ is by many orders of magnitude smaller than the transparency density $N_\mathrm{tr}^{2\mathrm{D}}$ of electrons in the quantum film.

21.4

Condition of gain. Show that the condition of gain, $E_\mathrm{F c} - E_\mathrm{F v} > E_{21} =E_2 - E_1$, follows from the condition $f_2 - f_1 > 0$.

21.5

Gain mediated by a quantum well. Given are the following quantities:

• $D_\mathrm{c}^{2\mathrm{D}}$ ${=}$ two-dimensional density of states of electrons in the conduction band.

• $D_\mathrm{v}^{2\mathrm{D}}$ ${=}$ two-dimensional density of states of electrons in the valence band ($=$two-dimensional density of states of holes).

• $N^{2\mathrm{D}}$ ${=}$ two-dimensional density of nonequilibrium electrons in the conduction band (assumed to be equal to the two-dimensional density of nonequilibrium holes in the valence band).

• $g (h \nu - E_{21})$ ${=}$ lineshape function.

• $a_2$ ${=}$ height of a photon mode that contains the quantum well; the plane of the quantum well is oriented parallel to the propagation direction of the radiation.

• $F \equiv f_2 - f_1 =d^{2\mathrm{D}} \times (N^{2\mathrm{D}} - N_\mathrm{tr}^{2\mathrm{D}})$; this expansion implies that the quantum well is operated near the transparency density.

Formulate equations, which are suited to determine the following quantities:

• (a) $E_\mathrm{F c}$ ${=}$ quasi-Fermi energy of electrons in the conduction band.

• (b) $E_\mathrm{F v}$ ${=}$ quasi-Fermi energy of electrons in the valence band.

• (c) $N_\mathrm{tr}^{2\mathrm{D}}$ ${=}$ two-dimensional transparency density.

• (d) $R_{\mathrm{s p, h} \nu }^{2\mathrm{D}}$ ${=}$ spontaneous emission rate per unit photon energy in the cases that the lineshape function is broad or narrow.

• (e) $R_\mathrm{s p}^{2\mathrm{D}}$ ${=}$ total spontaneous emission rate (at a broad or a narrow lineshape function).

• (f) $\tau _\mathrm{s p}$ ${=}$ lifetime of the nonequilibrium electrons with respect to spontaneous emission of radiation.

• (g) $H^{2\mathrm{D}}$ ${=}$ two-dimensional gain profile.

• (h) $\gamma$ ${=}$ modal growth coefficient.

• (i) $\alpha$ ${=}$ modal gain coefficient.

• (j) $b_\mathrm{eff}$ ${=}$ effective growth rate constant.

• (k) $\sigma _\mathrm{eff}$ ${=}$ effective gain cross section.

• (l) $G_1 - 1$ ${=}$ gain of light traversing a quantum well.

The answers are found in Sect. 21.8.

21.6

Quantum well laser. Given are the quantities:

• $H^{2\mathrm{D}}$ ${=}$ two-dimensional gain profile of a quantum well.

• $a_2$ ${=}$ extension of the resonator perpendicular to the quantum well.

• $a_1$ ${=}$ width of the resonator.

• $a_1 \times L$ ${=}$ area of the quantum well.

• L ${=}$ length of the resonator.

• $N^{2\mathrm{D}}$ ${=}$ two-dimensional density of nonequilibrium electrons.

• $r^{2\mathrm{D}}$ ${=}$ two-dimensional pump rate.

• $f_2 - f_1{\,=\,}d^{2\mathrm{D}} \times (N^{2\mathrm{D}} - N_\mathrm{tr}^{2\mathrm{D}})$; operation near the transparency density.

• $b_\mathrm{eff}$ ${=}$ growth rate constant.

• $\sigma _\mathrm{eff}{\,=\,}n b_\mathrm{eff}/c$ ${=}$ effective gain cross section.

• (a) Formulate the laser equations (rate equations).

• (b) Derive the threshold condition.

• (c) Determine the threshold current and the threshold current density.

• (d) Formulate the threshold condition of a quantum well laser operated at an electron density near the transparency density; neglect lineshape broadening. The answers can be found in Sect. 21.9.

21.7

Determine the de Broglie wavelength $\lambda _\mathrm{dB}$ ${=}$ h / p of electrons of an energy of 10 meV that are propagating (a) in free space and (b) as conduction electrons in a GaAs crystal.

21.8

A three-dimensional GaAs semiconductor at zero temperature contains nonequilibrium electrons of a density that corresponds to a quasi-Fermi energy $\epsilon _\mathrm{Fe} =25$ meV. Determine the following quantities.

• (a) Density of electrons in the conduction band.

• (b) Fermi momentum $k_\mathrm{F}$, i.e., the momentum of the electrons at the Fermi surface.

• (c) The de Broglie wavelength of the electrons that have Fermi momentum.

• (d) Quasi-Fermi energy $\epsilon _\mathrm{F h}$ of the nonequilibrium holes in the valence band assuming crystal neutrality.

• (e) Fermi momentum of the nonequilibrium holes.

• (f) The de Broglie wavelength of the holes that have Fermi momentum.

21.9

Answer the questions of the preceding problem with respect to a two-dimensional GaAs semiconductor at zero temperature containing nonequilibrium electrons of a density that corresponds to a quasi-Fermi energy $\epsilon _\mathrm{Fe}{\,=\,}25$ meV.

21.10

Answer the same questions with respect to a one-dimensional GaAs semiconductor at zero temperature containing nonequilibrium electrons of a density that corresponds to a quasi-Fermi energy $\epsilon _\mathrm{Fe}{\,=\,}25$ meV.