Boundary conditions for Quantum Cascade Laser (QCL)

Question

The Quantum Cascade Laser (QCL) is a semiconductor device for the generation of radiation in the MIR region of the electromagnetic spectrum. One period of the device consists of two regions, the active region and the collector region and a typical device might consist of approximately 20-75 periods. A region may consist of several wells, with typical well depths $\approx 500$ meV and well widths $\approx 10$ nm.

In the simplest approach all that is modelled is the central region of  the structure, where the laser action takes place and the first step is to solve the 1D Schrodinger equation for three periods to determine the wave functions and the eigenvalues associated with each region of the device (namely the injector, active and the collector regions, where the injector region is identical to the collector region and the term comes from how the device is used: electrons are injected from the injector region to the active region where laser action takes place and then collected in the collector region .....).

To solve the Schrodinger equation the following approach is often taken (for example see R. M. Kolbas and N. Holonyak Jr.  Am. J. Phys. 52, 431 (1984), Paul Harrison Quantum Wells, Wires and Dots): The three period section is extended by some arbitrary amount and for some chosen value of the energy and some step size, $\delta z$, the wave function is numerically iterated from $z = 0$ to the extreme of the (extended) three period structure, $z = z_f$, with the following boundary condition applied: $\psi(0) = 0, \psi(\delta z) = 1$ (equivalent to fixing the wave function and its slope at the origin).

The energy is then changed until the boundary condition $\psi(z_f) = 0$ is satisfied.

However, for a (strictly) periodic lattice (of period a and length $L = Na$ where $N$ is a large integer), the wave function satisfies Bloch's Theorem, and you can label the wave function by an index $n$ such that the wave function in any one period must satisfy

$$\psi(x+a) = e^{iKa}\psi(x), K = (2 \pi n)/(N a), n = 0,\pm 1, \pm 2, ....$$

The question is, when you solve for the wave function you only seem to assume $n=0$. What about all the other states?

Answer 1

The principle of QCL is that the geometry is (quasi)periodic, but the choice of semiconductors differs slightly from unit to unit in order to obtain the necessary cascade of lasing levels.

That is, the design of energy levels in successive units of three geometric periods is such that once an electron goes through lasing within one unit, it necessarily transitions to the upper level in the next unit where it lases again, and then repeats the cycle for 25-30 units.

If the structure would be entirely periodic, the electrons would end up on the common ground level after only one lasing cycle and would have to be re-execited some other way. The cascade removes this step and makes things much more efficient.