Population shares at individual subbands (n_{i}, n_{u}, n_{l}, n_{h}) in the 2.2 THz and 3.8 THz designs, with the corresponding population inversions (∆n_{ul}). The total electrons population in each period is normalized as 100%. The dipole matrix elements for radiation transition between upper and lower laser subbands z_{ul} is also shown. A “normal” peak gain G_{p}^{*} 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 G_{p} is the “real” value.

## Abstract

In designing the terahertz quantum cascade lasers, electron injection manner indeed plays a significant role to achieve the population inversion. The resonant tunneling process is commonly employed for this injection process but waste more than 50% fraction of populations out of the active region owing to resonance alignment, and the injection efficiency is obviously degraded due to thermal incoherence. An alternative approach is to consider the phonon-assisted injection process that basically contributes to most of the populations to the upper lasing level. However, this manner is still not realized in experiments if a short-period design only containing two quantum wells is used. In this work, it is found in this design that the population inversion is indeed well improved; however, the optical gain is inherently low even at a low temperature. Those two opposite trends are ascribed to a strong parasitic absorption overlapping the gain. The magnitude of this overlap is closely related to the lasing frequency, where frequencies below 3 THz suffer from fewer effects.

### Keywords

- intersubband transition
- terahertz
- quantum cascade lasers
- parasitic channels
- optical gain

## 1. Introduction

Thus far, the profusion of terahertz wave applications, including high-speed communications, industrial quality control, non-destructive cross-sectional imaging, gas and pollution sensing, biochemical label-free sensing, pharmacology, and security screening, has been demonstrated [1, 2, 3]. Moreover, the development of terahertz quantum cascade lasers (THz-QCLs) based on semiconductor quantum structures affords an attractive THz radiation source with coherent and compact wave features [4]. The basic radiation mechanism in this type of laser is intersubband transitions relying on quantum transport between discrete subbands. This method prevents the semiconductor bandgap limit at significantly low THz photon energies. The subbands can be freely tailored via engineering the thickness of quantum layers; therefore, the THz radiation frequency coverage is broad. However, THz-QCLs always suffer from temperature-triggered lasing quenching; consequently, the maximum operating temperature (_{max}) is still limited to below room temperature and thus requires additional cooling. Notably, despite stable progress since the first reported THz-QCL operating at 50 K (4.4 THz, pulsed mode) [5], _{max} has been stalled since 2012 (199.5 K) [6]. A recent breakthrough, achieving 250 K operation [7], indeed soothes the uncertainties regarding whether a 300-K-operation is prevented by any physical limit. The result has since spurred further efforts to achieve room-temperature operation.

With regard to the high-temperature THz-QCLs designs, different theoretical models, including density matrix formalism [8, 9, 10], non-equilibrium Green’s function (NEGF) [11, 12, 13], and Monte Carlo techniques [14, 15], have been proposed to understand the effects of temperature on quantum transport, that is, the loss of coherence, parasitic tunneling channels, and non-radiative processes with an increase in temperature. Numerous designs have been proposed, for example, by using diagonal radiative transitions [3] to suppress the thermally activated non-radiation channels (that triggers longitudinal optical (LO)-phonon emission instead of photon emission) between the upper and lower laser subbands, by using phonon resonance to depopulate the lower laser subband [6] yielding higher population inversion and partially relaxing the thermal backfilling, or by using clean subband systems [7] to avoid perturbation from high-lying subbands. Most of these designs use the resonant tunneling (RT) injection mechanism to populate the upper laser subband. In fact, the core feature of QCL design is electrons cascading across hundreds of stacked radiation periods. Therefore, a critical innovation in QCL design is the development of an injector region to maintain stable electrical bias in operations, enabling the first successfully operation of QCLs in the mid-infrared range [4].

However, RT injection in THz-QCLs has several drawbacks, which are illustrated in Figure 1a. 1) * Resonance alignment between the injector and upper laser subbands.* Electrons always wait for resonance before being injected into the upper laser subband (

*→*i

*). Ideally, in the coherent transport regime, the upper laser subband*u

*holds as many carriers as the injector subband*u

*, which is half of the total available electrons (that means a maximum of 50% share in the upper laser subband). In reality, with a thick injector barrier and the presence of multiple scattering channels, the population inversion generally falls below 50% at low temperatures. 2)*i

*. Moreover, as the injector subband*Thermal backfilling

*is already mostly populated, thermal backfilling to the lower laser subband*i

*cannot be neglected at high temperatures [16, 17]. This process introduces twice the adverse impacts, reducing the population inversion by decreasing the injected population and refilling the lower laser subband*l

*simultaneously. Given that the population inversion is defined as the difference between the upper and lower laser subband populations, the inversion will undergo significant degradation. 3)*l

*. Owing to the small subband energy separation, THz-QCLs face difficulty in selectively injecting electrons into the upper laser subband, while avoiding incorrect injection into the lower laser subband. To enhance this selectivity, the injector barrier (Figure 1a) should be chosen meticulously, on one hand, the barrier needs to be sufficiently thick to suppress incorrect injection and reduce the negative differential resistance (NDR) during parasitic alignment bias (i.e. the injector subband*Selective injection issue

*and lower laser subband*i

*will first align (*l

*↔*i

*) before reaching the operational bias). However, on the other hand, this barrier must also be sufficiently thin to increase the dynamic range of the laser current density. It is important to note that, owing to the close energy spacing between subbands*l

*and*u

*, the subband alignments*l

*↔*i

*and*u

*↔*i

*occur with similar electric fields. The constraint on the injector barrier worsens when the device lasing frequency approaches small. All the aforementioned RT-QCL issues result in significant challenges in terms of maintaining the population inversion when the operating temperature approaches 300 K, thus impelling designers to develop novel approaches to overcome the bottlenecks associated with RT injection.*l

An indirect injection scheme (scattering-assited (SA) injection), by designing the injector subband * i* to lie one LO-phonon energy above the upper laser subband

*(Figure 1b), has also been reported [16, 18, 19, 20]. SA injection can be traced back as early as 2001 by Scamarcio*u

*. [21], where the method was successfully employed in mid-infrared QCLs [17] to inject electrons into the upper laser subband by resonantly emitting phonons from the injector subband (*et al

*→*i

*). In this case, by combining a diagonal radiative transition, the parasitic off-resonant tunneling from the injector subband*u

*to the lower laser subband*i

*also can be suppressed. Therefore, this scheme may offer a venue for realizing higher population inversion, even at 300 K. Previous reports [16, 17, 19, 21] provide comprehensive discussions on how SA-QCL designs address the shortcomings of RT-QCLs. However, only three-well [22], four-well [16], and five-well [19] SA-QCL designs have been studied. Moreover, in high-temperature THz-QCLs designs, it is found that a higher*l

T

_{max}also results from a shorter period length by decreasing the number of desired subbands (seven-well [23], four-well [24], three-well [6], and two-well [25]). In this work, we attempt to study the SA injection in THz-QCL designs based on a short period length that contains only two wells. The result demonstrates a strong deviation between the optical gain and population inversion. In other words, although high population inversion can remain in such designs, when the lasing frequency exceeds 3 THz, the peak gain is considerably small even at low temperatures. This phenomenon is ascribed to the emergence of specific parasitic absorption, which can closely overlap with the optical gain. However, this limitation can be avoided in low-frequency lasing. This implies that specific strategies are required to alleviate the shortcomings of different lasing frequencies when introducing short-period SA injection designs; here in the final part of this work, we also study the feasibility in using step well quantum structures to suppress the absorption effects on the gain.

## 2. Non-equilibrium Green’s function method

The most fundamental tool in the design of THz-QCLs structures and analysis is a numerical package to calculate subband wavefunctions and energies. Because the subband energy position is critical for the discussion of the parasitic absorption in this work, it needs to estimate the high-lying energy position more precisely. Here, two factors effecting the energy separation between subbands are considered, ** a.** In THz-QCLs, the quantum structure contains the layers with thickness of only several nanometers; the non-parabolicity can largely effect on the energy of confined subbands, especially on HCS as it is lifted further away from the bottom of the conduction band; and the high-electric filed operation of THz-QCLs will worse this issue. Here, the band structures are based on three-band Hamiltonian that accounts for the conduction (

*), light-hole (*c

*), and split-off (*lh

*) bands, as follows:*so

where * P* is the interband momentum matrix element related to the Kane energy

*through:*E

_{p}

**the alignment of subbands and also the energy position is also very sensitive to the conduction band offset (CBO) values, especially in short period with tall barriers (here, AlAs% of AlGaAs barrier is 0.3). We follow the latest calibration of CBO based on a machine learning method reported in Ref. [7]. The exact numbers of subbands participating in the transports were controlled by the axial cut-off energy. These subbands were transformed into localized basis modes (reduced real space basis) and used in the NEGF algorithm [26, 27]. The subband energy broadening can play significant roles for estimating the tunneling current (increase the dephasing) and the optical gain (widen the radiation linewidth), especially for THz-QCL studied in this work, as this subband energy broadening (∼10 meV) is similar as the photon energy (15 meV). In THz-QCLs, this broadening originates from multiple scattering couplings. Here, the self-energy terms are calculated for all scatterings, including the optical phonons, acoustic phonons, charged impurities, interface roughness, alloy disorder, and electron-electron interactions [28, 29, 30, 31, 32]. The critical part of the model is a self-consistent NEGF solver that starts from an initial guess of the Green’s functions, the self-energies are then presented roughly, and the Green’s functions are again calculated iteratively. Simultaneously, the mean field electrostatic potential is calculated self-consistently (Poisson’s equation). Such iterations are performed until convergence is reached. The current density as well as the carrier density distribution is finally displayed. The optical gain or absorption in pairs of intersubband transitions follows linear response theory.**b.

## 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 frequency of 3.8 THz. The RT-QCL design in Figure 2a precisely follows the previously used scheme for _{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

*via RT process. Subsequently, diagonal radiation transition occurs between the laser subbands*u

*and*u

*. 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*i

*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*l

*. Under the operational bias, the injector subband*i

*population ratios are 56%/60% at 50 K/300 K. By contrast, the upper laser subband*i

*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%.*u

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*,

*, and*u

*), and the large range in Figure 3b and d which contains one more high-lying subband together with the desired subbands (*l

*,*i

*,*u

*, and*l

*). 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 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 × 10*h

^{10}cm

^{−2}, 5 × 10

^{10}cm

^{−2}, and 4.7 × 10

^{10}cm

^{−2}(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* =

*−*n

_{u}

*) 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*n

_{l}

*during modeling. 1)*h

*. Regardless of the frequencies and lattice temperatures, the population inversion Δ*Population inversion

*increases when the subband*n

*is included (Figure 4*h

**and**-a1, b1

**). 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**c1

*The table shows a 0.7% and 2.5% increasing share in Δ*n.

*after including the subband*n

*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*h

*and satisfy the required THz radiation frequencies. Therefore, the injector subband*l

*(the second excited subband in the upper well) and high-lying subband*i

*(the second excited subband in the lower well) 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*h

*and*u

*. As a result, parasitic coupling is formed between*l

*and*i

*. The magnitude of this coupling can be quantified by the energy splitting between them (2*l

*Ω*ħ

*). 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*

_{ih}

*can act as an additional depopulation channel for the lower laser subband*h

*in upstream period (noted that subbands*l

*and*i

*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*l

*at 300 K is considerably low, that is,*h

*= 1.5%/2.6% at 2.2 THz/3.8 THz. This demonstrates that the role of subband*n

_{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)*h

*As shown in Figure 4*Optical gain.

**and**-a2, b2

**, after the inclusion of the high-lying subband**c2

*, 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 (*h

*,*i

*, and*u

*) and (*l

*,*i

*,*u

*, and*l

*), 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*h

**). It should be noted that the inclusion or exclusion of the high-lying subband**-c2

*, the dipole matrix elements*h

*, and the radiation transition linewidth Γ*z

_{ul}

*remain almost unchanged. Therefore, following the semiclassical manner to predict optical gain, G*

_{ul}

*∼ Δ*

_{p}

n

_{*}

z

_{ul}

^{2}/Γ

*, after the inclusion of high-lying subband*

_{ul}

*, an increased population inversion should improve the peak gain in the 3.8 THz design. Table 1 presents the “nominal” gain peak G*h

** for (*

_{p}

*,*i

*,*u

*, and*l

*) case at 300 K, which is estimated based on the changes in population inversion. In the 3.8 THz design, G*h

** is 26.6 cm*

_{p}

^{−1}, representing a 1.6 cm

^{−1}increase over the (

*,*i

*, and*u

*) case, whereas the “real” peak gain G*l

*is only −38 cm*

_{p}

^{−1}, showing a sharp decrease of 63 cm

^{−1}when compared with G

**.*

_{p}

Frequency (temperature) | Subbands number | _{h} | _{i} | _{u} | _{l} | Δ_{ul} | (nm)_{ul} | _{p}^{*} (cm^{−1}) | (cm_{p}^{−1}) |
---|---|---|---|---|---|---|---|---|---|

2.2 THz (300 K) | (, , ) | — | 14.5% | 52% | 33.6% | 18.4% | 2.4 | 21.5 | 21.5 |

(, , , ) | 1.5% | 13.6% | 52.3% | 32.6% | 19.7% (0.7% ↑) | 2.38 | 22.6 (1.1↑) | 20.5 (1↓) | |

3.8 THz (300 K) | (, , ) | — | 14.6% | 54% | 31.4% | 22.6% | 2.95 | 25 | 25 |

(, , , ) | 2.6% | 11.7% | 55.4% | 30.3% | 25.1% (2.5%↑) | 2.89 | 26.6 (1.6↑) | −38 (63↓) |

To study the inconsistency of the changes in population inversion Δ* n* and the “real” optical gain G

*, 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 (*

_{p}

*,*i

*, and*u

*) and (*l

*,*i

*,*u

*, and*l

*) 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 (*h

*,*i

*, and*u

*) and (*l

*,*i

*,*u

*, and*l

*). 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*h

*and high-lying subband*i

*, 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*h

*(which shares most of the population in SA-QCL) will be thermally back to the injector subband*u

*, thus enhancing this parasitic absorption. For the 3.8 THz design, as shown in Figure 4*i

**labeled by the double-sided black arrows, the deviation of the peak gain between (**-c2

*,*i

*, and*u

*) and (*l

*,*i

*,*u

*, and*l

*) is 45 cm*h

^{−1}at 50 K, and 63 cm

^{−1}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

*(Figure 6b and d) overlaps the gain in the 3.8 THz design (Figure 6*h

**and**-d1

**). 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**d2

*.*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

*and*i

*. 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 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.*h

## 4. Conclusions

In summary, to estimate the optical gain in the SA-QCL design, the high-lying subbands need to be included, and the optical gain and absorption arising from any coupled pairs of subbands should be calculated, especially when the SA-QCL design is based on a simple quantum structure. Notably, as shown in this work, for SA-QCLs with two wells, population inversion increases when additional high-lying subbands are included, owing to the activation of more depopulation channels. However, an increase in the optical gain does not correspondingly occur if the lasing frequency exceeds 3 THz; instead, the peak gain undergoes a significant decrease, even below zero. The strong decoupling between population inversion and optical gain is ascribed to the emergence of parasitic absorption, which is caused by transitions between the desired subband and high-lying subband. Owing to the engineering limit permitted in the simple quantum structure, this parasitic absorption unavoidably overlaps with the optical gain, resulting in a reduction in the peak gain intensity. This overlap is more severe when the lasing frequency exceeds 3 THz. This finding reinforces the need for engineering the specific high-lying subbands to suppress the overlaps, thus realizing the two-well SA-QCL design experimentally. Here in the final part of this work, the feasibility by employing AlGaAs ternary alloys instead of GaAs for upper well is shown, and the small Al composition can dramatically enlarge the energy of parasitic absorption, thus almost removing the overlaps.

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