**3. Results**

A direct comparison of the population residual at individual desired subbands under different injection methods (RT and SA injections) is presented in **Figure 2**. For simplicity, both the designs are based on the two-well quantum structure with a lasing *High-Lying Confined Subbands in Terahertz Quantum Cascade Lasers DOI: http://dx.doi.org/10.5772/intechopen.105479*

### **Figure 2.**

*Design structures with RT injection (a) and SA injection (b). The designs are based on simple quantum structures containing only two wells (three desired subbands). The population shares of the injector subband and the upper laser subband at both low/high temperatures ((c) 50 K and (d) 300 K) are shown to illustrate the advantage of SA injection under the different applied biases (the black dash lines represent the operational bias condition).*

frequency of 3.8 THz. The RT-QCL design in **Figure 2a** precisely follows the previously used scheme for *T*max > 200 K [7, 25]. The design in **Figure 2b**, i.e. the scheme designed in this study, employs SA injection. In the former design, electrons are pumped from the injector subband *i* into the upper laser subband *u* via RT process. Subsequently, diagonal radiation transition occurs between the laser subbands *u* and *i*. Electrons are then depopulated via intrawell LO phonon (direct-phonon) resonance and moved into the next injector subband to repeat the previous steps. In the latter design, electrons are injected from subband *i* following direct-phonon resonance (phonon emission in a vertical manner) and then perform diagonal transition

radiation, after which the depopulation of subband *l* follows an RT process. It is clear from **Figure 2c** and **d** that, for RT injection, because the bias is applied until it reaches the operational bias (dashed line labeled in **Figure 2c** and **d**), most of the electrons are residual at injector subband *i*. Under the operational bias, the injector subband *i* population ratios are 56%/60% at 50 K/300 K. By contrast, the upper laser subband *u* only maintains 33%/22% at 50 K/300 K. For SA injection, most of the population is injected into the upper laser subband *u* (72%/58% at 50 K/300 K under the operational bias). Here, the total electrons population in each period is normalized as 100%.

The two-well SA-QCL designs are shown in **Figure 3** with the different lasing frequencies of 2.2 THz (a, b) and 3.8 THz (c, d). To study the effect of high-lying subbands, the number of confined subbands in each period is controlled by tuning the axial cut-off energy range, that is, the narrow range in **Figure 3a** and **c** which only contains three desired subbands (*i*, *u*, and *l*), and the large range in **Figure 3b** and **d** which contains one more high-lying subband together with the desired subbands (*i*, *u*, *l*, and *h*). To further enhance injection selectivity, the lasing barrier positioned between the upper and lower subbands is relatively thick. As a result, the designs in this study feature a reduced oscillator strength, that is, the oscillator strength for 2.2 THz, 3 THz, and 3.8 THz lasers are 0.15, 0.2, and 0.24, respectively. To compensate

**Figure 3.**

*Conduction band profile with the modulus square of the Wannier-Stark subbands under the operational bias, for the 2.2 THz design (a, b) and 3.8 THz design (c, d) including/excluding the high-lying subband* h*.*

for such low oscillator strength, the doping level is correspondingly increased. Meanwhile, to avoid too strong Coulomb scattering effects, the periodic doping levels are balanced at 5.5 <sup>10</sup><sup>10</sup> cm<sup>2</sup> , 5 <sup>10</sup><sup>10</sup> cm<sup>2</sup> , and 4.7 <sup>10</sup><sup>10</sup> cm<sup>2</sup> (sheet doping density) for the 2.2 THz, 3 THz, and 3.8 THz designs, respectively.

**Figure 4** shows the changes in population inversion (Δ*n* = *nunl*) and the optical gain peak as functions of the lattice temperature, the plots are shown with the inclusion or exclusion of the high-lying subband *h* during modeling. 1) *Population inversion*. Regardless of the frequencies and lattice temperatures, the population inversion Δ*n* increases when the subband *h* is included (**Figure 4-a1, b1** and **c1**). This differs from RT-QCLs in that the high-lying subbands are always treated as thermally activated electron leakage channels to reduce population inversion. In order to quantitatively estimate the changes, **Table 1** enumerates the magnitude of the changes in population inversion Δ*n.* The table shows a 0.7% and 2.5% increasing share in Δ*n* after including the subband *h* for the 2.2 THz and 3.8 THz designs at 300 K, respectively. The more increasing trend at 3.8THz can be explained by the quantum structures in **Figure 3**, that is, the upper well is positioned to encourage LO-phonon emission. Hence, the thickness of this well should be the largest. Likewise, the lower well should also be sufficiently wide to move down the lower laser subband *l* and satisfy the required THz radiation frequencies. Therefore, the injector subband *i* (the second excited subband in the upper well) and high-lying subband *h* (the second excited subband in the lower well)

### **Figure 4.**

*Changes in population inversion (a1, b1, and c1) and real peak gain (a2, b2, and c2) in three different lasing frequencies SA-QCL designs (2.2 THz, 3 THz, and 3.8 THz). The x-axis corresponds to the lattice temperature, ranging from 50 K to 300 K.*


### **Table 1.**

*Population shares at individual subbands (ni, nu, nl, nh) in the 2.2 THz and 3.8 THz designs, with the corresponding population inversions (Δnul). The total electrons population in each period is normalized as 100%. The dipole matrix elements for radiation transition between upper and lower laser subbands zul is also shown. A "normal" peak gain Gp \* is estimated based on the change in population inversion compared with the cases of (i, u, and l) and (i, u, l, and h), setting the value of (i, u, and l) as a standard. The peak gain Gp is the "real" value.*

are close to each other. Especially for high lasing frequencies, the energy separation between those two subbands decreases more (29.5 meV for 2.2 THz and 18.5 meV for 3.8 THz in **Figure 2-b**, **d**). Meanwhile, the barrier between them cannot be excessively thick, as this barrier plays a role in tuning the oscillator strength between the laser subbands *u* and *l*. As a result, parasitic coupling is formed between *i* and *l*. The magnitude of this coupling can be quantified by the energy splitting between them (2*ħ*Ω*ih*). The splitting energy is 3 meV and 10.5 meV in the 2.2 THz and 3.8 THz designs, respectively. Therefore, the high-lying subband *h* can act as an additional depopulation channel for the lower laser subband *l* in upstream period (noted that subbands *i* and *l* in neighboring periods are with full resonance alignment), this further depopulation can increase the population inversion. For higher lasing frequencies, this channel will be stronger owing to the enhanced coupling, leading to more increase in inversion share (i.e. 2.5% for the 3.8 THz designs). In addition, from **Table 1**, it can be observed that the non-equilibrium occupation of the high-lying subband *h* at 300 K is considerably low, that is, *nh* = 1.5%/2.6% at 2.2 THz/3.8 THz. This demonstrates that the role of subband *h* is to act as a channel to partly redistribute the populations among the desired subbands, rather than itself storing a high share of population. 2) *Optical gain.* As shown in **Figure 4-a2, b2** and **c2**, after the inclusion of the high-lying subband *h*, the changes in the peaks of optical gain are inconsistent with the population inversion. In the 2.2 THz design, the peak gain is almost the same in the cases of (*i*, *u*, and *l*) and (*i*, *u*, *l*, and *h*), regardless of the temperature. However, when the lasing frequency is higher, the peak gain is strongly reduced and surprisingly even negative in the 3.8 THz designs, despite temperatures even as low as 115 K (**Figure 4-c2**). It should be noted that the inclusion or exclusion of the high-lying subband *h*, the dipole matrix elements *zul*, and the radiation transition linewidth Γ*ul* remain almost unchanged. Therefore, following the semiclassical manner to predict optical gain, G*<sup>p</sup>* Δ*n*\**zul* 2 /Γ*ul*, after the inclusion of high-lying subband *h*, an increased population inversion should improve the peak gain in the 3.8 THz design. **Table 1** presents the "nominal" gain peak G*p*\* for (*i*, *u*, *l*, and *h*) case at 300 K, which is estimated based on the changes in population inversion. In the 3.8 THz design, G*p*\* is 26.6 cm<sup>1</sup> , representing a 1.6 cm<sup>1</sup> increase over the (*i*, *u*, and *l*) case, whereas the "real" peak gain G*<sup>p</sup>* is only 38 cm<sup>1</sup> , showing a sharp decrease of 63 cm<sup>1</sup> when compared with G*p*\*.

### *High-Lying Confined Subbands in Terahertz Quantum Cascade Lasers DOI: http://dx.doi.org/10.5772/intechopen.105479*

To study the inconsistency of the changes in population inversion Δ*n* and the "real" optical gain G*p*, additional data are extracted from the optical gain mappings and spectra. **Figure 5** shows the optical gain spectra for the 2.2 and 3.8 THz designs at both low/high temperatures (50 K/300 K). The black and colored solid curves represent the gain spectra for (*i*, *u*, and *l*) and (*i*, *u*, *l*, and *h*) cases, respectively. It is clear that the appearance of high-lying subbands *h* introduces strong parasitic absorption, as labeled by arrows a\*/b\* in both the 2.2 THz and 3.8 THz designs. The arrows a/b indicate the peak gain at the designed lasing frequencies. At both 50 K and 300 K, for the 2.2 THz design, the peak gain area is separated from this absorption, as a result, the net gain peak is not significantly different when comparing (*i*, *u*, and *l*) and (*i*, *u*, *l*, and *h*). By contrast, for the 3.8 THz design, this absorption can overlap with the peak gain area and induce a dramatic reduction in the net gain, regardless of the temperature. Considering the pairs of subbands, this absorption originates from the coupling between the injector subband *i* and high-lying subband *h*, where the energy separation is 29.5 meV in the 2.2 THz design and 18.5 meV in the 3.8 THz design. In particular, at high temperatures, electrons from the upper laser subband *u* (which shares most of the population in SA-QCL) will be thermally back to the injector subband *i*, thus enhancing this parasitic absorption. For the 3.8 THz design, as shown in **Figure 4-c2** labeled by the double-sided black arrows, the deviation of the peak gain between (*i*, *u*, and *l*) and (*i*, *u*, *l*, and *h*) is 45 cm<sup>1</sup> at 50 K, and 63 cm<sup>1</sup> at 300 K, respectively.

**Figure 6** shows the gain mappings resolved based on the spatial position and lasing frequencies. Clearly, the emergence of parasitic absorption between the subbands *i* and *h* (**Figure 6b** and **d**) overlaps the gain in the 3.8 THz design (**Figure 6-d1** and **d2**).

### **Figure 5.**

*Optical gain spectra of the 2.2 THz and 3.8 THz designs at 50 (a) and 300 K (b) under operational bias. Arrows a/b denote the peak gain, and arrows a\* /b\* represent the parasitic absorption peak.*

**Figure 6.**

*Spatial and energy resolved gain mappings of the 2.2 THz design (a1, b1 at 50 K; a2, b2 at 300 K) and 3.8 THz design (c1, d1 at 50 K; c2, d2 at 300 K) under operational bias.*

In general, 3–4 THz is the frequency band desired to achieve high-temperature operation [7, 25]. Therefore, this significant reduction in the optical gain reinforces the need for specific strategies to suppress this parasitic absorption, for example, by engineering the high-lying subband *h*.

Here, we study the feasibility of using step well to engineer the subband *h*. As shown in **Figure 7**, the use of AlGaAs in upper well (instead of GaAs) is proposed, and the Al composition in this ternary alloy can be controlled to tune the energy of parasitic absorption between subbands *i* and *h*. The upper well is set to make both the injector and upper laser subbands high in energy; meanwhile, the depopulation efficiency between them remains by keeping an energy separation same as the design in **Figure 3**. Consequently, the lower well can be narrowed correspondingly to satisfy the THz radiation frequencies. By doing this, the high-lying subband in the lower well is

*High-Lying Confined Subbands in Terahertz Quantum Cascade Lasers DOI: http://dx.doi.org/10.5772/intechopen.105479*

### **Figure 7.**

*Parasitic absorption energy between injector and high-lying subbands (i, h) as functions of the Al composition in upper well AlGaAs (a). Design structure (band profile and subbands) with Al% of 3% in upper well AlGaAs (b).*

significantly upward. The parasitic absorption energy can be enlarged from 18.5 meV with AlGaAs (Al% = 0) to 170 meV with AlGaAs (Al% = 7%) (**Figure 7a**). A reprehensive design structure with Al% of 3% is shown in **Figure 7b**. Those strategies show possible pathway to guide the experiments, and the initial results [33] are convinced this strategy.
