The QCL is an unipolar device made of a sequence of alternating wide band gap and narrow band gap semiconductor layers with typical thicknesses of a few nanometers. Semiconductor layers of narrow bandgap thinner than the De Broglie wavelength of the carriers form quantum wells with quantized states. This one dimensional potential well of finite depth confines carriers to discrete levels in the direction of growth. Orthogonal to the growth direction the carriers are unbound resulting in an energy dispersion of each subband.
These multi quantum well structures form repeated stages where each stage contains an active region and a carrier injector region. Figure 2.1 illustrates the generic conduction band profile in two adjacent stages of a QCL under an applied electric field. Depending on the barrier thicknesses, the exponential tails of the delocalized wave functions can reach the adjacent wells.
Due to the quantum confinement multiple minibands are formed in the injector region, whereas in the active region discrete energy levels arise. Coupling of the wells and carrier transfer is provided by quantum mechanical tunneling. The upper lasing level 3 in the active region is filled by electrons from the injector region, that tunnel through the barriers. Radiative transitions from the upper level to the lower lasing level 2 occur if the population inversion condition is satisfied, i.e. the occupation of the upper state exceeds the number of electrons in the lower state significantly. In other words, the relaxation time τ32 from the upper state into the lower state has to be greater than the lifetime of the lower state τ2 , i.e. τ32 > τ2. Due to interaction of electrons with longitudinal optical (LO) phonons level 2 gets depopulated fastly to level 1. Subsequently, the electrons escape by means of tunneling into the injector region of the adjacent stage. While an electric current flows through a quantum cascade structure, electrons cascade down an energy staircase emitting a photon at each step in the ideal case. Thus an injected electron can theoretically generate as many photons as stages are present. The cascade of light generated in this way makes the optical power proportional to the number of stages, which points up the capability of QCLs.
So far, we have described the principle of population inversion between the upper and lower state, which can be achieved by designing properly the layer thicknesses and electric field in the active region, assuming the ideal carrier transport path where sufficiently many electrons are injected into level 3 from the preceding injector region after relaxation from level 2 to 1 via LO phonons. In reality also other scattering paths for the electrons are possible. To minimize these effects of leakage currents is a task of QCL design optimization. To achieve lasing, it is necessary to suppress unwanted escape routes by tunneling from the upper level [15]. In order to ensure highly selective injection, electrons are injected into the upper laser level by a resonant tunneling process. Typical tunneling times are of the order of sub-picoseconds and can be approximated as [16]
 | (2.1) |
where ΔE denotes the energy separation of the delocalized wave functions in the coupled well system. At too low bias, conduction is low and only minimal current will flow. When the field increases, the transmission coefficient gets enhanced and the current flow increases. At resonance, where the band edge aligns to a bound state, a local maximum for the transmission coefficent is achieved, leading to a peak in the current flow. Further increase in field results in a sharp decrease in the current flow. Especially in the off-resonance case, scattering mechanisms are capable to route carriers.
Carrier transport can also be affected by thermal effects, like in short-wavelength lasers, where the energetically high upper laser level is located close to the quasi continuum above the barriers. There the electrons may undergo thermal excitation, and thermionic emission from the confined states constitutes. In the continuum, the electric field accelerates the electron freely. Thus they do not partake in the laser action any more.
In general, laser action for a wide range of electric fields can be achieved if a large gain coefficient and a low waveguide loss can be established. The current has to be large enough that the gain compensates the loss. The gain itself mainly depends on the dipole matrix element between the corresponding laser levels and the phenomenological broadening of the transition. It has been shown that a long lifetime for the upper laser level and a short one for the lower level is essential to obtain a high peak gain in QCLs [17], the same criterion as for an efficient population inversion.
Evidently, the understanding of numerous physical processes is essential for the design and optimization of QCLs. Predominantly, suitable waveguiding and population inversion with appropriate radiative transitions must be provided by tailoring the band structure and the lifetimes.