# QCL Fundamentals

The quantum cascade laser differs significantly from conventional semiconductor lasers, posing a number of potential advantages. In such bipolar lasers, emission takes place as a result of the recombination of carriers across the bandgap. The earliest double heterostructure devices consisting of adjacent p- and n-type layers, had their operation wavelength exclusively set by the bandgap energy, Figure 1(a). Due to the existence of few compatible narrow-gap semiconductors and the problem of Auger processes in such materials, wavelengths were limited to around 1 μm. Introducing a thin layer of a smaller bandgap semiconductor to form a Type I quantum well can extend the wavelength and improve efficiency by narrowing the depletion region. With a quantum well, or indeed multiple quantum wells for further enhancing recombination, carriers can be confined to energy levels within these wells enabling the formation of a lower energy transition, Figure 1(b). Using this scheme, lasing up to 5.2 μm has been demonstrated in the III-Vs* and up to 30 μm in the IV-VIs (lead-salts)*, albeit at cryogenic temperatures. Although most commercial lasers are of the quantum well variety, the design inhibits terahertz operation as the emission wavelength is still ultimately limited by the material bandgap. An alternative is provided by the QCL.

## Unipolar Operation

The quantum cascade laser is a unipolar device that suffers from no such “bandgap slavery”. Transitions wholly take place within the conduction (or valence) band between subbands of a quantum well, Figure 1(c). With a knowledge of quantum mechanics, structures can be engineered with transitions unconstrained by the material bandgap enabling long wavelength operation. Moreover, since carriers do not undergo a one-time recombination process, it is possible for them to cascade through a device to take part in multiple radiative transitions, hence the term “quantum cascade”. This boosts efficiency and increases output power.

## Quantum Wells

Quantum wells are the building blocks for cascade structures, so a
brief overview of their formation will be given here. A more detailed
understanding can be gained by consulting appropriate
texts*. Quantum wells with quantised
states are formed by a layer of semiconductor thinner than the
deBroglie wavelength of the carrier (approximately 10 nm)
sandwiched between semiconductors of a larger bandgap. This acts as a
one dimensional potential well of finite depth V_{0},
confining carriers to discrete levels (subbands) of energy E in the
direction of growth, x. Since carriers are unbound parallel to the
growth direction, an in-plane wavevector k_{//} can exist to
give an energy dispersion of each subband. This becomes particularly
significant in scattering processes discussed in Section
“Scattering Mechanisms”. See Figure 2.

The solutions of the time-independent Schrödinger equation for
the system (below) provide the subband wavefunction ψ(x) for any
energy E for a carrier of mass m. Applying the necessary continuous
boundary conditions to the wavefunction and its derivative yields the
quantisation conditions. These conditions can then be solved via a
graphical method involving the well parameters to give the discrete
energies E_{1}, E_{2}, etc. From this argument it is
evident the primary variable in determining the subband energies is
the well width, or layer thickness, w. With an increase in well width
the subbands become more closely packed in energy in addition to an
overall shift to lower energies.

## Coupled Wells

The penetration of wavefunctions into the walls of the potential well enables them to be coupled across multiple wells. Taking the simple case of two identical wells separated by a thick barrier, the two wells are uncoupled and share the same subband structure, Figure 3(a). Reducing the barrier thickness, the exponential tails of each wavefunction can reach the adjacent well and experience its potential confinement. The delocalised wavefunctions are now no longer degenerate in energy and have an energy separation ΔE, Figure 3(b). The thinner the barrier, the stronger the coupling, giving a larger ΔE, with an exponential relation between the two. From this simple case, in the first instance it is evident a variation of well and barrier thickness enables subband energies to be finely tuned.

The aforementioned argument can be extended to multiple wells. Taking the case of five identical wells separated by sufficiently thin barriers, each wavefunction will be delocalised across the entire structure with each slightly displaced in energy from the degenerate case. These closely spaced levels form a broad energy continuum known as a miniband. This applies to excited states as well as the ground state, thus multiple minibands are formed, separated by a minigap, depicted in Figure 4. This periodic layer structure is a so-called “superlattice” and will prove important in the design and operation of the QCL.

## Quantum Mechanical Tunnelling

Quantum mechanical tunnelling is the actual mechanism that enables the coupling of wells and the transfer of a carrier between them. According to classical mechanics the carrier would require an energy greater than the height of the potential barrier in order to surmount it. In quantum systems however, there is a probability the carrier will tunnel through the barrier to the adjacent well. Figure 5 shows an explicit example of the exponential decay of the wavefunction into a potential barrier, as already mentioned in the coupled well case. The probability of tunnelling is expressed in terms of the transmission coefficient T, below, and decays rapidly with energy. Hence, for good tunnelling probability, a short tunnel distance, low potential barrier and small effective mass are required. These parameters will be revisited in the design of the cascade in Chapter "Silicon Heterostructure".

Normally the quantum barrier does not exist in isolation. Flanking a potential well with two such barriers forms a useful resonant tunnelling structure. Depending on the incident energy of the carrier, the bound states within the well can either enhance or suppress the wavefunction by a change in the transmission coefficient. A carrier coincident in energy with that of a subband (on resonance) will have a transmission coefficient of near certainty, whilst the further away from this case (off resonance), the wavefunction is increasingly suppressed, Figure 6.

The transmission coefficient for the complete structure can be
calculated from the coefficients for each barrier, T_{L} and
T_{R}, and the subband energies.
Equations~\ref{eqn:trans_coeff_double_off} and
\ref{eqn:trans_coeff_double_on} below offer a good approximation of T
depending upon the resonance condition.

In practice however, the transmission coefficient in Equation~\ref{eqn:trans_coeff_single} is approximated by the Wentzel Kramers Brillouin (WKB) integral approximation for an arbitrary shaped well, such as a triangular well formed under an electric field. Furthermore, a transfer matrix method is used in all but the simplest cases to calculate the coefficient across multiple barrier, superlattice structures*. Figure 7 shows the outcome of such a calculation for the double barrier structure.

## Application of Electric Field

Doping layers either side of the double barrier structure provides a carrier accumulation region that can be increased in potential with the application of an electric field. Figure 8 depicts how band-alignment and its characteristic Negative Differential Resistance (NDR) occurs in a simple Resonant Tunnelling Device (RTD).

- At low bias the band edge is near flat and the carrier density of states lies near the bottom of the well. Conduction is low and minimal current will flow.
- The field increases and the carrier energy begins to align to a bound state, enhancing the transmission coefficient and increasing current flow.
- The resonant voltage is achieved. The band edge aligns to a bound state, giving a local maximum for the transmission coefficient and a peak in current flow.
- A further increase in field and alignment moves away from the resonant condition, resulting in a sharp decrease in current flow.

In the case of the RTD, the transmission coefficient is modified due to the deformation of the square well by an applied field. It therefore lies away from unity on resonance and in fact decreases at each resonance with increasing energy. Moreover, the actual current flow is different to that predicted. Particularly in the off-resonance case, a number of scattering mechanisms (Section “Scattering Mechanisms”) can divert carriers to other states. Electronic transport is also effected by thermal effects such as hot carriers and level broadening.

## QCL Design

The design of a quantum cascade laser requires the understanding of a number of principles to fulfill the criteria for lasing, as laid down in Section “Laser Fundamentals”. The band structure and scattering mechanisms must be tailored for forming population inversion with appropriate radiative transitions as well as suitable waveguiding. QCL designs are presented by example in the following section, whilst Chapter “Silicon Heterostructure” contains a more detailed discussion of these factors in the specific case of a silicon heterostructure.