**5. Absorption coefficient of MQW structure**

Absorption coefficient is calculated as a function of operating wavelength for different well width, as shown in **Figure 8**. For lower well dimension, it is found that the absorption coefficient value is less compared to the higher well dimension. Again, when the energy difference between the bands increases, interband transition energy increases. This reduces the peak of absorbance amplitude, provided half-width at half-maximum is kept constant throughout the simulation.

**Figure 8.** Absorption coefficient for different well dimension with wavelength variation for parabolic, nonparabolic dispersion.

With increase of well dimension, redshift is observed for all the profiles corresponding to the transition energies; also peak value is shifted rightwards and the wavelength value increases. When the parabolic and nonparabolic dispersion relation is considered for same well dimension the peak value is greater for lower well dimension nonparabolic case. But with change in eigenenergy, for higher well dimension the parabolic relation gives a higher absorption peak value. Thus the nonparabolic dispersion relation takes the priority for simulation.

With parabolic and nonparabolic relations into consideration the material composition is varied in accordance with the wavelength. The absorption coefficient profiles are depicted in **Figure 9**. For parabolic dispersion, the transition energies are low, so the absorption peak values are less compared to that of nonparabolic case. Also for higher material composition the shift in absorption value is considerable. The peak absorption values are greater for higher material composition and also for nonparabolic dispersion.

**Figure 9.** Absorption coefficient for different material composition with wavelength variation for nonparabolic and parabolic dispersion.

## **6. Band structure of quantum cascade laser**

**5. Absorption coefficient of MQW structure**

the simulation.

38 Quantum Cascade Lasers

simulation.

dispersion.

Absorption coefficient is calculated as a function of operating wavelength for different well width, as shown in **Figure 8**. For lower well dimension, it is found that the absorption coefficient value is less compared to the higher well dimension. Again, when the energy difference between the bands increases, interband transition energy increases. This reduces the peak of absorbance amplitude, provided half-width at half-maximum is kept constant throughout

With increase of well dimension, redshift is observed for all the profiles corresponding to the transition energies; also peak value is shifted rightwards and the wavelength value increases. When the parabolic and nonparabolic dispersion relation is considered for same well dimension the peak value is greater for lower well dimension nonparabolic case. But with change in eigenenergy, for higher well dimension the parabolic relation gives a higher absorption peak value. Thus the nonparabolic dispersion relation takes the priority for

**Figure 8.** Absorption coefficient for different well dimension with wavelength variation for parabolic, nonparabolic

With parabolic and nonparabolic relations into consideration the material composition is varied in accordance with the wavelength. The absorption coefficient profiles are depicted in **Figure 9**. For parabolic dispersion, the transition energies are low, so the absorption peak values are less compared to that of nonparabolic case. Also for higher material composition the shift in absorption value is considerable. The peak absorption values are greater for higher

material composition and also for nonparabolic dispersion.

Electronic band structure of quantum cascade laser is analytically computed in presence of electric field applied along the direction of quantum confinement. At first, electronic and optoelectronic properties of the MQW structure is calculated. Next, miniband formation is observed for quantum cascade laser for a few precise magnitudes of electric fields. Separation of the miniband w.r.t. lowest energy band is calculated [68]. Al*<sup>x</sup>* Ga1−*<sup>x</sup>* As/GaAs composition is taken into account with the incorporation of Ben-Daniel Duke boundary conditions at heterointerfaces. Eigenstates are calculated in presence and absence of Kane-type first-order conduction band nonparabolicity. Its variation as a function of applied field is determined, and is compared with the findings of zero bias conditions.

**Figures 10**–**12** represent the electronic band structure of quantum cascade laser. For biasing, electric field is applied along the direction of quantum confinement, and eigenstates are computed. Results are also obtained when field is absent. While designing the structure, the most important aspect is that a distinguishable separation between injector and active region should be kept along with the miniband formation [68]. **Figure 10** shows the band diagram when external field is totally absent, and **Figure 11** demonstrated it for very high electric field (56.8 × 10<sup>5</sup> V/m). Here it may be noted that all the high magnitudes of electric field will not provide desirable QCL operation, as discrete miniband formation is necessary for that purpose. From the simulated observations, it may be indicated that the eigenenergy states are totally discrete in absence of external excitation, which are nothing but the eigenstates of a simple multiple quantum-well (MQW) structure.

**Figure 10.** Electronic band structure of QCL without external bias.

**Figure 11.** Electronic band structure of QCL with 56.8 × 10<sup>5</sup> V/m.

From **Figure 11**, it is observed that after application of very high electric field, miniband is formed over the ground state energy band. In this context, it may be noted down that after the injector region, active region starts, and the wavefunction starts to produce minibands. The energy gap between miniband and ground state energy band is distinguishable, which ensures the QCL operation. In this context, it may be pointed out that the magnitude of external field is chosen based on the structural dimensions and the material composition taken for the simulation. The structure is titled compared to **Figure 10**, along the direction of electric bias. Thus, the periodic growth of wavefunction in the miniband appears outside the confinement region, that is, in the quasi-continuous region. This band structure modulation is absent if the field is moderate (35.8 × 10<sup>5</sup> V/m).

Further reduction of electric field makes position of the miniband almost inside the confinement region, that is, miniband position is below the quasi-continuous region. This is shown in **Figure 12**. Eigenenergy variation with the applied bias for the laser is shown in **Figure 13**. Simulation result suggests that with increase of applied field, eigenenergy increases monotonically, and the rate reduces once filed crosses the value 35 × 10<sup>5</sup> V/m. Thus, stimulated emission between the miniband and the ground state energy band is effectively controlled by external bias.

**Figure 12.** Electronic band structure of QCL with 35.8 × 10<sup>5</sup> V/m.

**Figure 10.** Electronic band structure of QCL without external bias.

40 Quantum Cascade Lasers

**Figure 11.** Electronic band structure of QCL with 56.8 × 10<sup>5</sup> V/m.

**Figure 13.** Variation of eigenenergy with applied field.

## **7. Conclusion**

In this article, a detailed investigation on electrical and optoelectronic properties of multiple quantum well structure is carried out. Based on that structure, wavefunction in a quantum cascade laser at various biased conditions is analytically calculated and also for unbiased condition. It is proven and shown that miniband formation is only possible at some precise electric field. The method showed can also be applicable for the other structures. Key factor in this calculation is that position-dependent effective mass is considered for the simulation.

## **Author details**

## Arpan Deyasi

Address all correspondence to: deyasi\_arpan@yahoo.co.in

Department of Electronics and Communication Engineering, RCC Institute of Information Technology, Kolkata, India
