Quantum Cascade Lasers

Abstract

The quantum cascade laser represents an important accomplishment in quantum photonics, combining quantum excitation of the electromagnetic field, electron transport and controllable quantum mechanical tunneling all working cooperatively in a single device. In cascade lasers, the carriers are “recycled” from one stage to the next, so that the evacuation region of stage 1 is connected to the injection region of stage 2. A single carrier traveling through the structure may emit a photon at each stage. The successful operation of the quantum cascade laser is a balancing act between population inversion maintained by resonant phonon evacuation of carriers from the lower level and non-radiative depletion of the upper level population caused by optical phonon scattering. The interband cascade laser eliminates this major difficulty of the quantum cascade laser, by using a conduction band to valence band optical transition to generate the laser emission. As a result, the threshold power density in the interband cascade laser is lower by an order of magnitude compared to the quantum cascade laser.

Exercises

• 8.1 Evaluate the intersubband optical dipole matrix element between 2 levels i and j, and show, following West and Eglash, that the general expression is:

$$ {\boldsymbol {\mathcal{M}}}_{ij} = qL_{w} \frac{8}{{\pi^{2} }}\frac{ij}{{\left( {i^{2} - j^{2} } \right)^{2} }} $$

• 8.2 Quantum cascade lasers have been successfully designed using a wide variety of structures. This direction-transition device, emitting at 4.6 µm was designed by Carlo Sirtori, and co-workers (1996) (Fig. 8.18).

  Fig. 8.18

  

  Quantum cascade laser structure. Epitaxial layers are formed using Ga_0.47In_0.53As and Al_0.52In_0.48As. Layer dimensions are given in Table 8.2

Table 8.2 Layer geometry for a direct transition quantum cascade laser operating at 4.6 µm

• (a) Using the data given, calculate the position of the quantized levels in each quantum well. Determine the energy difference between the n = 1 and n = 2 levels in the emission layer.

• (b) Calculate the miniband width of a series of 5 quantum wells having a width of 1.8 nm confined by 6 layers having width of 2.0 nm.

• (c) At zero applied bias voltage, what is the energy difference between the n = 1 quantum well in the emission layer and the n = 1 quantum well in the photon transfer layer.

• (d) What range of applied bias voltage would align the quantum levels in the evacuation and injection regions (within the miniband width that you calculated in 8.2b)?

• 8.3 Design a direct tunneling structure for the injection section of the ICL shown in Fig. 8.16: that is, 5 barriers and 4 quantum wells. Assume that the materials used are AlSb for the barrier and InAs for the well regions.

  Assume that the operating electric field is $E = 10^{4} \,\text{V}\, \text{cm}^{ - 1}$, and that the emission energy is 0.37 eV.

• (a) Determine appropriate thicknesses for each of the barriers and wells. (Hint treat each of the quantum wells independently to determine the lowest energy bound state in each of the 4 quantum wells. Then apply the electric field and adjust the well parameters so the ground-state energies are aligned energetically at this value of the electric field. Show that this sequence will also give the desired emission energy.

• (b) Estimate the energy width of the lowest energy tunneling miniband for the integrated 5-barrier/4-well structure? (Hint: use the average of the well widths determined in 8.3a and compute the transmission coefficient for the complete structure, ignoring the electric field).

• 8.4 The pseudopotential method introduced in Chap. 4 can be used to calculate the band structures of many III-V semiconductors.

• (a) Using the pseudopotential method, estimate the valence band offset between InAs and GaSb by calculating each bandstructure and comparing the position of the valence bands.

• (b) The accepted value for the valence band offset is −0.51 eV. The parameters of the pseudopotential calculation can be varied to yield this value of the valence band offset. Which parameters should be changed, and by how much?