**3. Transmission coefficient of multiple quantum well structure**

For analyzing electrical property of MQW structure, first transmission coefficient is calculated by considering a three-well four-barrier structure with rectangular potential profile configuration having GaN as well layer and Al*<sup>x</sup>* Ga1−*<sup>x</sup>* N as barrier layer. It is observed that by increasing well width, transmission probability increases. This is due to the fact that by increasing tunneling dimension of the well, quantum confinement decreases, so transmission can be achieved at lower energy values. By changing the material composition of barrier material, it is observed that transmission coefficient decreases with increase of Al mole fraction. It is plotted in **Figure 1**. This is due to the fact that by increasing Al percentage, potential height increases and effective mass mismatch at the junction also increases. This increases quantum confinement, which increases the eigenenergy. By increasing the thickness of the contact barrier, transmission probability reduces, whereas it increases if contact barrier thickness is reduced compared to that of the internal barriers. This is shown in **Figure 2** for both parabolic and nonparabolic band structures. By increasing number of wells, it is observed the eigenvalue reduces, and ultimately becomes constant for higher numbers, as evident from **Figure 3**. Eigenvalue increases with increase of material composition, as suggested from previous results, plotted in **Figure 4**. Obviously, for nonparabolic band structure, eigenenergy is less compared to the ideal parabolic concept. The variation is almost linear, and the gap reduces with increase of al mole fraction.

**Figure 1.** Transmission coefficient with energy for different barrier material composition for both parabolic and nonparabolic structure.

**Figure 2.** Transmission coefficient with energy for asymmetric barrier width for both parabolic and nonparabolic structure.

**Figure 3.** Eigen energy with dimension of well for both parabolic and nonparabolic structure.

**Figure 1.** Transmission coefficient with energy for different barrier material composition for both parabolic and nonparabolic

**Figure 2.** Transmission coefficient with energy for asymmetric barrier width for both parabolic and nonparabolic structure.

structure.

34 Quantum Cascade Lasers

**Figure 4.** Eigen energy with barrier material composition for both parabolic and nonparabolic structure.

## **4. Density of states of MQW structure**

In **Figure 5**, density of states (DOS) is plotted for lowest two eigenstates with different well widths. It is observed from the plot that by increasing the well width, eigenenergy appears at lower energy values due to the reduction of quantum confinement. It may also be seen that higher bandgap system provides eigenstate at lower energy range. Hence, optical device based on GaN/AlGaN system can be tuned at lower bias. Also due to closeness of first two energy levels in this composition, intersubband transition energy is higher for InAs/GaInAs system, which makes it efficient candidate for high frequency laser.

**Figure 5.** Density of states for the lowest two eigenstates for different well widths and different material compositions with nonparabolic dispersion relation.

**Figure 6** shows the comparative study of DOS in presence and absence of electric field for InAs/GaInAs material composition. From the graph, it may be noted that application of transverse field lowers the magnitude of eigenstates irrespective of dispersion relation considered for simulation. Results are also compared with overestimated parabolic assumption. It may be observed from the comparison that incorporation of band nonparabolicity reduces the eigenvalue. Hence, for fine wavelength tuning purpose to design photonic transmitter/detector, realistic band structure consideration plays a vital role.

**4. Density of states of MQW structure**

36 Quantum Cascade Lasers

system, which makes it efficient candidate for high frequency laser.

In **Figure 5**, density of states (DOS) is plotted for lowest two eigenstates with different well widths. It is observed from the plot that by increasing the well width, eigenenergy appears at lower energy values due to the reduction of quantum confinement. It may also be seen that higher bandgap system provides eigenstate at lower energy range. Hence, optical device based on GaN/AlGaN system can be tuned at lower bias. Also due to closeness of first two energy levels in this composition, intersubband transition energy is higher for InAs/GaInAs

**Figure 6** shows the comparative study of DOS in presence and absence of electric field for InAs/GaInAs material composition. From the graph, it may be noted that application of transverse field lowers the magnitude of eigenstates irrespective of dispersion relation considered for simulation. Results are also compared with overestimated parabolic assumption. It may be observed from the comparison that incorporation of band nonparabolicity reduces the eigenvalue. Hence, for fine wavelength tuning purpose to design photonic transmitter/detec-

**Figure 5.** Density of states for the lowest two eigenstates for different well widths and different material compositions

tor, realistic band structure consideration plays a vital role.

with nonparabolic dispersion relation.

**Figure 6.** Density of States for the lowest two eigenstates in presence of absence of electric field for nonparabolic and parabolic structures.

By varying the material composition of barrier layers, it is observed that quantum states appear in higher energy values with increase of GaN mole fraction. The result is shown in **Figure 7**. This is because with increase of *x*, mismatch of effective mass increases as well as the conduction band discontinuity. This enhances the quantum confinement. Hence, eigenenergy increases. This is reflected via density of states plot.

**Figure 7.** DOS of superlattice with different material compositions in absence of field considering band nonparabolicity.
