Comparison of THz-QCL Designs Supporting Clean N-Level Systems

Abstract

Three different Terahertz quantum-cascade-laser designs supporting clean n-level systems were analyzed using nonequilibrium Green’s functions. In clean n-level systems, most of the electrons occupy the active laser levels, with thermally activated leakage channels being suppressed almost entirely up to room temperature. Simulations of the three designs, namely a resonant phonon design, a two-well design, and a split-well direct-phonon design were investigated. The results from the simulations indicated that the two-well design would perform best overall, in terms of variations in current density, interface roughness, and ionized impurity scattering. We conclude that future research aiming to improve the temperature performance of such laser designs should be based on a two-well design.

1. Introduction

The discovery of Terahertz (THz) quantum cascade lasers (QCLs) in 2002 [1] was a major event in the scientific world. The “THz gap” frequencies have much potential as light sources for use in processes such as the nondestructive detection of materials, imaging, and gas spectroscopy [2,3,4,5]. Fields that could be advantaged by utilizing THz-QCLs include astrophysics [6], medicine [7,8], security [9], and chemistry [10,11]. However, THz frequencies remain the least used across the electromagnetic spectrum. The main limitation has been the need to confine THz-based technologies to the laboratory environment, primarily because of the requirement for substantial cooling equipment. The highest maximum operating temperature ($T_{max}$) reported for pulsed operation of a GaAs/AlGaAs THz-QCL until recently was 210 K [12]. The first high-power portable THz-QCL was developed only in 2021, reaching a new $T_{max}$ of 250 K [13]. This achievement involved producing a clean three-level system, where no leakage channels were thermally activated, even at room temperature, using a two-well (TW) design. A similar TW design was previously researched and led to a $T_{max}$ of only 192 K [14,15].

In achieving a clean n-level system, all mechanisms that could possibly limit the laser’s performance have to be considered. Diagonal structures significantly reduce nonradiative, thermally activated, longitudinal optical (LO)-phonon scattering from the upper laser level (ULL) to the lower laser level (LLL) [16,17,18], which is the main mechanism limiting the temperature performance of THz-QCLs. Other mechanisms include thermally activated carrier leakage into the continuum [17] and into excited bound states [19]. These mechanisms have been addressed in previous work [13,17,19], as indicated by the negative differential resistance (NDR) behavior in their current—voltage curves even at room temperature. The NDR signature indicates that a clean n-level system has been achieved, with no parallel leakage channels having been activated.

Previous research on clean n-level systems has focused on three different designs, all of which have demonstrated the NDR signature up to room temperature. The first is a highly diagonal, high-barrier, resonant phonon (RP) design [19], the second is a highly diagonal, high-barrier, direct-phonon TW design [15], and the third is a highly diagonal, split-well direct-phonon (SWDP) design [20,21,22]. In this paper, we compare these three designs by simulation, aiming to identify the type of design with the greatest potential for room-temperature applications.

All calculations in this paper were performed using the nextnano.NEGF simulation package [23,24,25]. Previous work has been performed using full nonlinear simulations of quantum wells (QW) based on Bloch equations [26]. Here, the analysis of carrier transport and gain properties is based on the nonequilibrium Green function (NEGF) method [27,28,29,30,31,32,33,34] and includes all elements that must be considered in achieving accurate results for tunneling through the injection barrier. This includes correlation between different quantum states and nondiagonal elements in self-energy components, which involves density matrices. The scattering and the quantum evolution are treated through consistent perturbations [27]. All Green’s functions are based on the Dyson equation, discussed in previous works [27,30,31,32]. Two steps are taken in order to perform the calculations: first a self-consistent stationary solution of the quantum kinetic equations for a given applied bias is evaluated in order to solve the transport problem [31]. A numerical study of transport including systems with no periodic boundary conditions is presented in Ref. [35]. A broad quantum field theoretical approach for lasers in stationary state can be found in Ref. [36]. We can derive this way the Green’s functions describing a state far from equilibrium. Then, an additional weak radiation field is considered. The formulation used in the simulations employs the full two-time structure of the quantum transport theory [31]. A complete discussion on dynamical mean field theory in the non-equilibrium can be found in Ref. [37]. Non-equilibrium correlations are discussed in several papers; an insight on non-equilibrium correlations in open quantum system and their derived equations can be found in Ref. [38].

As part of the nextnano.NEGF simulation package, both elastic and inelastic scattering processes, including scattering caused by charged impurities, interface roughness (IFR) scattering, optical and acoustic phonons, and alloy disorders, were considered [23,24,25]. A full quantum-kinetic description of the gain within linear response theory [39] was used to perform the gain calculations self-consistently. Our careful study of these processes was aimed at identifying those that affect the performance of the various designs and thereby determining which of the designs would be optimal with respect to room-temperature performance.

The results of our NEGF calculations are presented in Figure 1 in terms of the conduction-band diagrams and the envelope functions for the three simulated designs. The three modules of each laser’s structure are displayed in all three diagrams. All designs have been shown experimentally to have an efficient isolation of laser levels from the excited bound and the continuum states [15,19,20]. The band structures in Figure 1 show that the laser levels are well separated from the excited states. This fact, together with the NDR signature previously experimentally observed up to room temperature [15,19,20], let us assume there is no carrier leakage, neither LO-phonon thermally activated, nor IFR-assisted [40]. More information about NDR and negative differential conductivity in QCLs can be found in Ref. [41].

The IFR amplitude used for the calculations of all the devices was of 0.12 nm. The correlation length is of 8 nm and its type is exponential. All IFR parameters are in agreement with previous experimental results [22,42]. For simplicity, the analyses have been performed assuming that the well/barrier interfaces are sharp. In reality, the well/barrier interfaces for any QCL are graded [43], which may reduce the temperature performance results, as will be later discussed.

2. Results

Figure 1a shows the structure of the simulated RP design. The RP design used can be described [19] as a highly diagonal resonant phonon THz-QCL, with an oscillator strength of f~0.2, and based on GaAs/Al_0.3Ga_0.7As materials (using barriers with a higher Al composition than usual). The doping density was ~6.0 × 10¹⁰ cm⁻² at the center of the injection well (more details are given in

Table 1. Main nominal design parameters and design data.

Design	Wafer Number	E21 [meV]	Lasing Energy (meV)	Oscillator Strength	Layer Sequence [#ML *] Barrier Composition and Doping Level
RP	VB0676	25	25	0.2	10.3/37.2/6.4/38.6/8.2/65.9 213 periods GaAs/Al0.3Ga0.7As 1.24 × 1017 cm−3 in the center 17 ML of the 65.9 ML well. (6.0 × 1010 cm−2)
TW	VB0747	56	13	0.18	13.5/25.5/11.0/50.2 354 periods GaAs/Al0.3Ga0.7As 1.26 × 1017 cm−3 in the centered 17 ML of the 50.2 ML well (6.0 × 1010 cm−2)
SWDP	VB0837	26	11	0.26	9.0/24.8/3.5/24.8/17.3/24.8 353 periods total thickness 10 μm GaAs/mixed barriers Al0.55Ga0.45As (Inj.) and Al0.15Ga0.85As (Rad., Intraw.) 2.13 × 1016 cm−3 in the 24.8 ML wells (2.98 × 1010 cm−2)

* #ML is the number of monolayers. The doped layers in the sequence are underscored, with the AlGaAs barriers in bold and the GaAs wells in plain text. The barriers’ composition and doping details are described in the following lines.

). There were four active levels, with all other levels being considered parasitic, namely the injector (level 1), an LLL doublet (levels 2 and 3), and the ULL (level 4). The ULL was aligned with the injector of the next module, where resonant tunneling occurs. The energy passage from the ULL to the LLL is radiative and the LO-phonon scattering occurs from the LLL doublet into the injector of the next module. The design supports a clean four-level system, as indicated by its NDR signature [19].

Figure 1b shows the structure of the simulated TW design. The TW design used (also called “HB2” [15]) was a high-barrier, highly diagonal, TW structure with thin wells and with all barriers having a 30% aluminum composition. Thin wells have demonstrated to improve gain active regions [44]. Due to the thin wells in the TW design, the excited states are pushed to higher energies. The first experimental demonstration of the concept of introducing tall barriers in the active region in order to push up the excited states and reduce leakage was published more than a decade ago for mid-IR QCLs [45]. Resonant leakage channels to the next module states are also minimized. The doping of the design was high (~6.0 × 10¹⁰ cm⁻²) to compensate for reduced gain caused by the highly diagonal (f~0.18) optical transition [46]. The simulated design reached the lasing state at the lowest oscillator strength reported so far for THz-QCLs [15] (further information can be found in Table 1). There were three active laser levels in this design, with the other levels being considered parasitic. The ULL (level 3) and the injector of the next module (level 4) were aligned to give DP characteristics (DP designs have shown advantages over RP designs in previous research, one being this exact TW design [15]). This design approach leads to a clean three-level system and has demonstrated THz-QCL performance up to a $T_{max}$ of 250 K [13].

Figure 1c shows the third design investigated. The simulated structure is of a highly diagonal (f~0.25) SWDP, with an injector barrier containing 55% Al (i.e., Al_0.55Ga_0.45As) and with radiative and intrawell barriers containing 15% Al (i.e., Al_0.15Ga_0.85As). The doping density was ~3 × 10¹⁰ cm⁻² (more data and specifications can be found in Table 1). This design has been presented previously [20], with experiments demonstrating a $T_{max}$ of 170 K. Because the same design with a doubled doping of ~6 × 10¹⁰ cm⁻² proved to have poor performance [47] in comparison to the original design [20], we used the original design in our calculations. As was the case for the TW design, which included three ground levels, the original design supported a clean three-level system, and the levels were all aligned to form a DP configuration. However, the difference in this novel design was that a thin intrawell barrier was introduced. Such a barrier helps tune the energy splitting between the LLL (level 2) and the injector (level 1) to provide the optimal LO-phonon energy. This enables the fastest depopulation rate for the LLL.

The optical gain obtained for the three designs at various temperatures is shown in Figure 2 and

Table 2. Gain data for the three designs.

Design	Peak Gain at 10 K (1/cm)	Photon Energy at Peak Gain (meV)	Full Width at Half Maximum (meV)
RP	27.28	25	6.64
TW	24.67	14	6.43
SWDP	22.42	11	6.13

. Multiple studies on gain mechanisms in different kinds of lasers, such as random fiber lasers, have been performed in the past [48]. Previous works have shown the importance of the gain in quantum wells [49]. For the RP design (see Figure 2a), the peak gain at all temperatures is at a photon energy of around 25 meV. Assuming that the losses are around 15 ${\mathrm{cm}}^{-1}$, this design has its gain exceeding its losses up to its $T_{max}$ of 170 K (at temperatures above this, the design will not lase because the losses exceed the gain). Note also that, after reaching its maximum, the curve drops very sharply.

A similar tendency is seen with the SWDP design (see Figure 2c), although the point where the gain exceeds losses of 15 ${\mathrm{cm}}^{-1}$ is lower (110 K). Lasing was again achieved up to a $T_{max}$ of 170 K because of very low losses at around 13.5 cm⁻¹, attributable to the high-quality fabrication process used for the original design [20]. The photon energy at the peak optical gain for this design was lower than for the RP design, at 11 meV.

However, the gain plots for the TW design (see Figure 2b) show that, using this design, the gain exceeds the losses across all temperatures. This means that, by using this design, lasing can be achieved even at room temperature. Despite this, the experimental results showed lasing up to a temperature of 250 K and not up to room temperature as in the simulations. This is, as mentioned before, probably due to the sharp interfaces in the simulation, as opposed to graded interfaces in reality [43]. Graded interfaces may likely result in carrier leakage through excited states since the energy differences with respect to the ULL decrease and the wavefunctions of the excited states and the ULL start to overlap. The temperature performance may be specially affected by thermally-activated IFR assisted carrier leakage [40], which, as pointed out by Boyle, is 4–10 times higher, in mid-IR QCLs, than LO-phonon carrier leakage. The photon energy at peak optical gain for this design was around 13 meV.

A comparison of the data for the three designs of peak gain versus temperature is shown in Figure 3. Although the gain values start higher for the RP design, it is evident from the graphs that the gain for the TW design is the most consistent across the range of temperatures. The SWDP and the RP plots both have very sharp declines when the temperature increases, whereas the TW plot declines less steeply.

The I-V curves for all three designs are presented in Figure 4. The maximum current density (J_max) values for the RP, TW, and SWDP designs at 10 K, as taken from the simulations, are 1032 $\raisebox{1ex}{$A$}\!\left/ \!\raisebox{-1ex}{${\mathrm{cm}}^2$}\right.$, 1393 $\raisebox{1ex}{$A$}\!\left/ \!\raisebox{-1ex}{${\mathrm{cm}}^2$}\right.$ and 642.2 $\raisebox{1ex}{$A$}\!\left/ \!\raisebox{-1ex}{${\mathrm{cm}}^2$}\right.$, respectively. The high J_max value for the TW design indicates how effective it is for current transport. No NDR signature can be seen in the NEGF simulation graphs in Figure 4, which differs from the graphs derived by experiment [15,19,20].

3. Discussion

The graph of Jmax versus temperature carries extremely important information. This data was extracted from the simulations and is presented in the graphs shown in Figure 5, for all three designs. In designs that support clean n-level systems, the transport through the injector barrier can be described by the Kazarinov-Suris formula [20,50] for tunneling involving dephasing such as:

$$J=eN\times \frac{2{\mathsf{\Omega }}^2{\tau }_{\parallel }}{4{\mathsf{\Omega }}^2\tau {\tau }_{\parallel }+{\omega }_{21}^2{\tau }_{\parallel }^2+1}$$

(1)

where τ is the ULL lifetime, τ‖ is the dephasing time between the injector and the ULL subbands, Ω is the coupling between the injector and the ULL subbands across the barrier, and ω₂₁ is the energy misalignment between the two. Equation (1) describes a Lorentzian centered at ${\omega }_{21}=0$, whose width is proportional to $\sqrt{4{\mathsf{\Omega }}^2\tau {\tau }_{\parallel }+1}$. The maximum current ($J_{max}$) is then given by $J\left({\omega }_{21}=0\right)=J_{max}$. The upper-state-lifetime-limited transport regime corresponds to the case where $\tau \gg \frac{1}{4{\mathsf{\Omega }}^2{\tau }_{\parallel }}$ [51,52] and $J_{max}~\frac{1}{\mathsf{\tau }}\approx \frac{1}{{\mathsf{\tau }}_{\mathrm{n}r}}$. Transport limited by resonant tunneling corresponds to the case where $\tau \ll \frac{1}{4{\mathsf{\Omega }}^2{\tau }_{\parallel }}$, and $J_{max}=2eN{\mathsf{\Omega }}^2{\tau }_{\parallel }$. Within the upper-state-lifetime-limited transport regime, another subregime describing the state of lasing can be derived from the Kazarinov-Suris formula [20], but it is irrelevant with respect to our NEGF simulations.

The two regimes applicable to our simulations from this analysis are the upper-state-lifetime-limited transport regime and the resonant-tunneling-limited regime. If the transport is limited by the upper state lifetime, ${\tau }_{nr}$ decreases with an increase of temperature, causing $J_{max}$ to rise and the slope in the $J_{max}$ versus temperature plot to be positive. However, if the transport is limited by resonant tunneling, the dephasing time decreases with temperature, $J_{max}$ falls, and the slope is negative. Considering these two regimes, the $J_{max}$ versus temperature graphs will tend to have a bell-like shape, rising at low temperatures when the upper-state-lifetime-limited transport is dominant, and decreasing at higher temperatures when resonant tunneling is the dominant regime.

For the RP and TW designs, the graphs of the maximum current J_max versus temperature given in Figure 5 demonstrate the two regimes described above, with the rising of the J_max versus temperature curve being best explained in terms of the upper-state-lifetime-limited transport regime. The curves enter the resonant-tunneling-limited regime only at higher temperatures when J_max starts to drop. These temperatures are approximately 250 K and 200 K for the RP and TW designs, respectively. The drop in both cases is very weak even at room temperature. Moreover, comparing all three plots in Figure 5 indicates that the J_max value for the TW design is the highest of the three at all temperatures.

The behavior described by the curves in Figure 5 for the SWDP design matches the behavior previously described [20] for a resonant-tunneling-limited regime. In this case, the maximum current density (J_max) drops as a function of temperature from the beginning. We can, therefore, deduce that the SWDP design is controlled by the resonant-tunneling-limited regime alone, without showing signs of being subject to the upper-state-lifetime-limited transport regime. Because this decreasing J_max is a sign of the transport being limited by resonant tunneling (where the dephasing time decreases and the line broadens as the temperature rises), it means that the design is controlled by a regime that has very strong dephasing and decoherence, even at low temperatures.

To further understand the designs’ behavior, we calculated the contribution of the various scattering mechanisms to the optical gain by a process of elimination. The resulting plots are presented in Figure 6. From these plots, we could identify that IFR scattering is the mechanism that most limits SWDP performance, as has previously been observed [22], but it seems not to be the dominant mechanism for the TW and RP designs. Furthermore, for the TW design, the gain appears almost independent of any IFR scattering. We could also identify the significant contribution of the ionized impurities scattering (IS) mechanism, particularly for the TW design. This mechanism is the main problematic factor to be addressed for this design and it can be particularly problematic if the doping is increased. For the RP design, the main limiting mechanism turned out to be electron-electron scattering.

As previously noted [22], designs with wider wells would result in lower IFR scattering and designs with a decreased overlap of the doped region with active laser states would ensure lower IS scattering, which would increase the active region gain by minimizing gain-broadening effects.

4. Conclusions

In conclusion, we compared three THz-QCL structures presented in previous work, all of which had shown NDR up to room temperature. From our results, we found the TW design to be the most promising of the three as a focus for future research. According to the data obtained by our simulations, the gain for this design exceeds the losses up to room temperature, whereas the gain drops below the losses at high temperatures for the RP and SWDP designs. The J_max for the TW design proved to be relatively constant and high, and its transport is less resonant-tunneling-limited than for the SWDP design. Of the three designs, the J_max for the TW design was the highest, making it the most effective for current transport. We also found that the DP design has several advantages over the RP design. It has lower sensitivity to misalignment of the laser levels caused by the Poisson effect, and it has very fast depopulation of the LLL, caused solely by LO-phonon scattering.

Regarding the mechanisms that limit THz-QCL performance, IFR scattering effects have been minimized in the TW design because of its optimized structure and broader wells. It is, therefore, no longer the dominant mechanism limiting temperature performance, unlike in the SWDP design. However, the IS remains very strong and could be problematic when trying to increase the doping and testing the design experimentally. Optimization of the TW design should include careful placement of the doping within its structure. We aim to continue the investigation and optimization of THz-QCLs using clean-n-level designs based on the TW design.