2.1 A generic conduction band profile in two stages of a QCL under an applied
bias. The arrow displays the transition 3 → 2, responsible for the laser action,
and the moduli squared of the corresponding wave functions are denoted by
solid curves.

2.2 Timeline of significant QCL developments.

3.1 Flowchart of the selfconsistent Schrödinger-Poisson solver. The initial value for the electrostatic potential is set to zero. In the second step the envelope functions and the eigenenergies are calculated according to the Schrödinger equation. The Poisson equation is solved after the determination of the Fermi level. A check of the electrostatic potential update decides whether the iteration terminates.

4.1 Anitcrossing between two single well states.

4.2 Difference between the semiclassical and coherent picture of coupled quantum wells.

4.3 Schematic of three adjacent stages of a QCL under an applied field. Each time an electron in state |k_{∥ } , νηλ⟩ undergoes an interstage scattering to a state |k_{∥}^{′},ν^{′}η^{′}(λ±1)⟩,
the electron is reinjected into the central region and the corresponding electron
charge contributes to the current.

4.4 Flow chart of the ensemble Monte Carlo algorithm. The input is provided by a selfconsistent Schrödinger-Poisson solver.

6.1 Band structure of the considered FSL at an applied field of 50 kV/cm.

6.2 Current-density voltage characteristics of a GaAs/Al_{0.3}Ga_{0.7}As
Fibonacci superlattice at T = 200K.

6.3 A schematic diagram of the conduction band profile for one and a half periods of the GaAs/Al_{0.33}Ga_{0.67}As QCL for an
electric field of 48 kV/cm.

6.4 Comparison of the current-voltage characteristics of a GaAs/ Al_{0.33}Ga_{0.67}As QCL calculated using the Robin boundary condition
approach with a nonequlibrium Green’s functions simulation [1].

6.5 The spatial dependences of the real parts of the envelope functions are plotted for different values of the electric field. The solid lines correspond to the solution of the Dirichlet problem and the dashed lines represent the wavefunctions of the Robin problem.

6.6 The spatial dependences of the imaginary parts of the envelope functions belonging to the solution of the Robin Problem at F = 10^{3}kV/cm
(solid line) and F = 10^{-6}kV/cm (dashed line).

6.7 Optical gain of a THz GaAs/Al_{0.15 } Ga_{0.85}As QCL driven at 160 A cm^{-2}. The solid line represents the
result calculated using the Robin boundary condition approach and the dashed
line corresponds to measured values.

6.8 Calculated values of the dipole matrix elements for F = 10^{3} kV/cm and F = 10^{-6} kV/cm. The solid lines represent
the solution of the Dirichlet problem and the dashed lines correspond to the
solution of the Robin problem.

6.9 Calculated conduction band diagram and squared wave functions for a GaAs/Al_{0.15}Ga_{0.85}As QCL under an applied field
of 10 kV/cm.

6.10 Calculated LO-phonon scattering rates 4 → 4 and 4 → 3 at an applied bias of 10 kV/cm and an operating temperature of 70 K.

6.11 Time evolution of the current density at an electric field of 10 kV/cm, with and without Γ-X intervalley scattering, illustrating the good convergence behavior. The current densities averaged over the time are 3.45⋅10^{6} A/m^{2} by including the X valley, and
3.08⋅10^{6 } A/m^{2 } for the Γ valley only.

6.12 Population of the individual subbands as a function of the kinetic energy.

6.13 Current density as a function of the applied electric field with and without X valley transport.

6.14 Comparison of the energy distributions of subband 4 for different temperatures. The dashed lines show the results without interface roughness scattering and the solid ones illustrate the results including scattering on rough interfaces.

6.15 Comparison of the energy distributions of subband 3 for different temperatures. The dashed lines show the results without interface roughness scattering and the solid ones illustrate the results including scattering on rough interfaces.

6.16 Current density as a function of the applied electric field with and without interface roughness scattering at 70 K.

6.17 Current density as a function of the temperature with and without interface roughness scattering.

6.18 Conduction band diagram and squared wave functions for a GaAs/Al_{0.45}Ga_{0.55} As QCL in the MIR region under an applied field of 40
kV/cm [2]. The conduction band discontinuity between GaAs and Al_{x}Ga_{1-x}As
layers is taken to be ΔE_{c} = 1.1x eV for 0 ≤ x ≤ 0.45, and ΔE_{c} = 0.43+0.14x eV
for 0.45 < x ≤ 1.

6.19 Overlap between the lower Γ-state of the central stage and the upper X-state of the next stage at an electric field of 40 kV/cm in dependence on the Al content.

6.20 Γ-X intervalley scattering between the lower state 2 and the upper state 9^{′} of the next stage at F = 40 kV/cm.

6.21 Electron population of the Γ and X valleys in dependence on the electric field.

6.22 Current density as a function of electric field for structure A and structure B, with and without X-valley transport.

6.23 Conduction band diagram for an In_{0.53}Ga_{0.47}As/GaAs_{0.51}Sb_{0.49}
QCL under an applied field of 30 kV/cm [3]. The upper and lower laser levels
are labelled with 3 and 2, respectively.

6.24 Voltage-current characteristics at T = 78 K.

6.25 Current density versus temperature at 5 V and 13 V with and without interface roughness scattering.

6.26 Scattering rate due to LO phonon emission as a function of temperature at several bias points.

6.27 Energy distributions of the upper and lower laser level with and without taking into account polar optical phonon scattering.

6.28 Voltage-current characteristics at 78 K. Comparison between the measured values and the simulation results with and without alloy scattering.

6.29 Current density versus temperature at 5 V and 13 V with and without alloy scattering.

6.30 Current enhancement due to inclusion of alloy scattering for several temperatures.

2.2 Timeline of significant QCL developments.

3.1 Flowchart of the selfconsistent Schrödinger-Poisson solver. The initial value for the electrostatic potential is set to zero. In the second step the envelope functions and the eigenenergies are calculated according to the Schrödinger equation. The Poisson equation is solved after the determination of the Fermi level. A check of the electrostatic potential update decides whether the iteration terminates.

4.1 Anitcrossing between two single well states.

4.2 Difference between the semiclassical and coherent picture of coupled quantum wells.

4.3 Schematic of three adjacent stages of a QCL under an applied field. Each time an electron in state |k

4.4 Flow chart of the ensemble Monte Carlo algorithm. The input is provided by a selfconsistent Schrödinger-Poisson solver.

6.1 Band structure of the considered FSL at an applied field of 50 kV/cm.

6.2 Current-density voltage characteristics of a GaAs/Al

6.3 A schematic diagram of the conduction band profile for one and a half periods of the GaAs/Al

6.4 Comparison of the current-voltage characteristics of a GaAs/ Al

6.5 The spatial dependences of the real parts of the envelope functions are plotted for different values of the electric field. The solid lines correspond to the solution of the Dirichlet problem and the dashed lines represent the wavefunctions of the Robin problem.

6.6 The spatial dependences of the imaginary parts of the envelope functions belonging to the solution of the Robin Problem at F = 10

6.7 Optical gain of a THz GaAs/Al

6.8 Calculated values of the dipole matrix elements for F = 10

6.9 Calculated conduction band diagram and squared wave functions for a GaAs/Al

6.10 Calculated LO-phonon scattering rates 4 → 4 and 4 → 3 at an applied bias of 10 kV/cm and an operating temperature of 70 K.

6.11 Time evolution of the current density at an electric field of 10 kV/cm, with and without Γ-X intervalley scattering, illustrating the good convergence behavior. The current densities averaged over the time are 3.45⋅10

6.12 Population of the individual subbands as a function of the kinetic energy.

6.13 Current density as a function of the applied electric field with and without X valley transport.

6.14 Comparison of the energy distributions of subband 4 for different temperatures. The dashed lines show the results without interface roughness scattering and the solid ones illustrate the results including scattering on rough interfaces.

6.15 Comparison of the energy distributions of subband 3 for different temperatures. The dashed lines show the results without interface roughness scattering and the solid ones illustrate the results including scattering on rough interfaces.

6.16 Current density as a function of the applied electric field with and without interface roughness scattering at 70 K.

6.17 Current density as a function of the temperature with and without interface roughness scattering.

6.18 Conduction band diagram and squared wave functions for a GaAs/Al

6.19 Overlap between the lower Γ-state of the central stage and the upper X-state of the next stage at an electric field of 40 kV/cm in dependence on the Al content.

6.20 Γ-X intervalley scattering between the lower state 2 and the upper state 9

6.21 Electron population of the Γ and X valleys in dependence on the electric field.

6.22 Current density as a function of electric field for structure A and structure B, with and without X-valley transport.

6.23 Conduction band diagram for an In

6.24 Voltage-current characteristics at T = 78 K.

6.25 Current density versus temperature at 5 V and 13 V with and without interface roughness scattering.

6.26 Scattering rate due to LO phonon emission as a function of temperature at several bias points.

6.27 Energy distributions of the upper and lower laser level with and without taking into account polar optical phonon scattering.

6.28 Voltage-current characteristics at 78 K. Comparison between the measured values and the simulation results with and without alloy scattering.

6.29 Current density versus temperature at 5 V and 13 V with and without alloy scattering.

6.30 Current enhancement due to inclusion of alloy scattering for several temperatures.