Electron Waves in Semiconductor Heterostructures

Abstract

We study electron waves in one-dimensional potentials and in semiconductor heterostructures.

Problems

30.1

Quantum well.

Estimate the eigenvalues $E_1$ and $E_2$ of an electron in an AlAs/GaAs/AlAs quantum well (barrier height 2.2 eV; $m_\mathrm{GaAs} = 0.07\ m_0$; $m_\mathrm{AlAs}$ $\sim$ 3 $m_\mathrm{GaAs}$) if the well consists of films of different thickness.

• (a) Film thickness = 14 GaAs monolayers

• (b) Film thickness = 2 GaAs monolayers.

30.2

Double-quantum well.

• (a) Determine the eigenvalues of a one-dimensional double well, which correspond to the two lowest energy levels (s $=$ 1, 2) of a single one-dimensional double well. [Hint: make use of the symmetry.]

• (b) Determine the energy level splitting $E_1^- - E_1^+$ for the two lowest levels.

• (c) Sketch the wave functions that correspond to the four lowest levels.

• (d) Calculate the level splitting occurring in an AlAs/GaAs/AlAs/GaAs/AlAs double quantum well (Fig. 30.6) for $a_1=10$ nm and $a_2=2$ nm.

30.3

Dispersion of electrons in a periodic potential.

Derive the dispersion relation of electrons in a periodic potential by the use of the matrix method.

30.4

Interface.

• (a) Electrons (energy $\epsilon$) propagate toward a GaAs/AlAs interface and are reflected. Determine the average penetration depth of electrons. [Hint: take into account the difference between the penetration depth of the wave function and of the electrons.]

• (b) Determine the penetration depth for $\epsilon = 10\,\mathrm {meV}$ and 100 meV.

• (c) Show that the reflectivity is $R = |k_1-I\kappa |/|k_1-I\kappa _1|$.

• (d) Explain the electron total reflector used in a GaN quantum well laser (Sect. 24.3, Fig. 24.3a).

30.5

Tunneling.

• (a) Determine the transmissivity of an AlAs barrier in a GaAs/AlAs/GaAs heterostructure for electrons of energy $\epsilon$.

• (b) Determine the transmissivity for electrons of energy $\epsilon = 10\,\mathrm {meV}$ and 100 meV at a barrier width of 2 monolayers of AlAs and for a barrier of 10 monolayers of AlAs.

30.6

Resonance state.

• (a) Given is a GaAs/AlAs/GaAs/AlAs/GaAs heterostructure. Determine the energy dependence of the transmissivity for electron waves of different energies.

• (b) Design a heterostructure that is transparent for electrons of $\epsilon = 10$ meV.

• (c) Design a heterostructure that is transparent for electrons of $\epsilon$ 100 meV.

30.7

Injector of a quantum cascade laser.

• (a) Design a quantum cascade laser of AlAs/GaAs/AlAs/GaAs/AlAs heterostructures embedded in chirped GaAs/AlAs superlattices for a quantum cascade laser that may be able to generate radiation at a frequency of 4 THz.

• (b) Estimate the thicknesses of the different layers.

• (c) Discuss the role of the superlattice, especially in view of the result of the preceding problem.

30.8

Semiconductor superlattice.

• (a) Determine the effective mass $m^*$ of an electron in a superlattice (for propagation along the superlattice axis) for an electron with k $\sim$ 0.

• (b) Determine $m^*$ of a GaAs/AlAs superlattice with 14 monolayers GaAs and 2 monolayers AlAs ($\epsilon _\mathrm{m}$ $\sim$ 140 meV).

• (c) Determine $m^*$ of a GaAs/AlAs superlattice with 4 monolayers GaAs and 2 monolayers AlAs ($\epsilon _\mathrm{m}$ $\sim$ 40 meV).

• (d) Determine the effective mass $m^*$ of an electron in a superlattice (for propagation along the superlattice axis) for arbitrary k and discuss the slope $m^* (k)$ and $m^* (\epsilon )$.

• (e) Determine the group velocity $\mathrm v_\mathrm{g}(k)$ and the peak group velocity.

• (f) Sketch the wave functions of the lowest miniband for $k \sim 0$ and $k = \pi /a$.

• (g) Sketch the wave functions of the second miniband for $k \sim 0$ and $k = \pi /a$.