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

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results

_{0.3}Ga

_{0.7}As materials (using barriers with a higher Al composition than usual). The doping density was ~6.0 × 10

^{10}cm

^{−2}at the center of the injection well (more details are given in Table 1). 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].

^{10}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].

_{0.55}Ga

_{0.45}As) and with radiative and intrawell barriers containing 15% Al (i.e., Al

_{0.15}Ga

_{0.85}As). The doping density was ~3 × 10

^{10}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 × 10

^{10}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.

^{−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.

_{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

_{21}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.

_{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.

_{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.

## 4. Conclusions

_{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.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Kohler, R.; Tredicucci, A.; Beltram, F.; Beere, H.E.; Linfield, E.H.; Davies, A.G.; Ritchie, D.A.; Iotti, R.C.; Rossi, F. Terahertz semiconductor-heterostructure laser. Nature
**2002**, 417, 156–159. [Google Scholar] [CrossRef] - Darmo, J.; Tamosiunas, V.; Fasching, G.; Kroll, J.; Unterrainer, K.; Beck, M.; Giovannini, M.; Faist, J.; Kremser, C.; Debbage, P. Imaging with a Terahertz quantum cascade laser. Opt. Express
**2004**, 12, 1879–1884. [Google Scholar] [CrossRef] - Hubers, H.-W.; Pavlov, S.G.; Richter, H.; Semenov, A.D.; Mahler, L.; Tredicucci, A.; Beere, H.E.; Ritchie, D.A. High-resolution gas phase spectroscopy with a distributed feedback terahertz quantum cascade laser. Appl. Phys. Lett.
**2006**, 89, 061115. [Google Scholar] [CrossRef] - Williams, B.S. Terahertz quantum-cascade lasers. Nat. Photonics
**2007**, 1, 517–525. [Google Scholar] [CrossRef][Green Version] - Lee, M.; Wanke, M.C. Searching for a solid-state terahertz technology. Science
**2007**, 316, 64–65. [Google Scholar] [CrossRef] [PubMed] - Kidd Walker, C.; Kulesa, C.; Goldsmith, P.; Groppi, C.; Helmich, F.; Hollenbach, D.; Kawamura, J.; Langer, W.; Melnick, G.; Neufeld, D.; et al. GUSTO: Gal/Xgal U/LDB Spectroscopic-Stratospheric Terahertz Observatory; American Astronomical Society: Washington, DC, USA, 2018. [Google Scholar]
- Mittleman, D.M. Twenty years of terahertz imaging. Opt. Express
**2018**, 26, 9417–9431. [Google Scholar] [CrossRef] [PubMed] - Rahman, A.; Rahman, A.K.; Rao, B. Early detection of skin cancer via terahertz spectral profiling and 3D imaging. Biosens. Bioelectron.
**2016**, 82, 64–70. [Google Scholar] [CrossRef] [PubMed] - Federici, J.F.; Schulkin, B.; Huang, F.; Gary, D.; Barat, R.B.; Oliveira, F.; Zimdars, D.A. THz imaging and sensing for security applications—Explosives, weapons and drugs. Sci. Technol.
**2005**, 20, 266–280. [Google Scholar] [CrossRef] - Korter, T.; Plusquellic, D.F. Continuous-wave terahertz spectroscopy of biotin: Vibrational anharmonicity in the far-infrared. Chem. Phys. Lett.
**2004**, 385, 45–51. [Google Scholar] [CrossRef] - Ogawa, Y.; Hayashi, S.; Oikawa, M.; Otani, C.; Kawase, K. Interference terahertz label-free imaging for protein detection on a membrane. Opt. Express
**2008**, 16, 22083–22089. [Google Scholar] [CrossRef] - Bosco, L.; Franckie, M.; Scalari, G.; Beck, M.; Wacker, A.; Faist, J. Thermoelectrically cooled THz quantum cascade laser operating up to 210 K. Appl. Phys. Lett.
**2019**, 115, 010601. [Google Scholar] [CrossRef] - Khalatpour, A.; Paulsen, A.K.; Deimert, C.; Wasilewski, Z.R.; Hu, Q. High-power portable terahertz laser systems. Nat. Photonics
**2020**, 15, 16–20. [Google Scholar] [CrossRef] - Franckié, M.; Bosco, L.; Beck, M.; Bonzon, C.; Mavrona, E.; Scalari, G.; Wacker, A.; Faist, J. Two-well quantum cascade laser optimization by non-equilibrium Green’s function modelling. Appl. Phys. Lett.
**2018**, 112, 021104. [Google Scholar] [CrossRef][Green Version] - Albo, A.; Flores, Y.V.; Hu, Q.; Reno, J.L. Two-well terahertz quantum cascade lasers with suppressed carrier leakage. Appl. Phys. Lett.
**2017**, 111, 111107. [Google Scholar] [CrossRef] - Albo, A.; Hu, Q. Investigating temperature degradation in THz quantum cascade lasers by examination of temperature dependence of output power. Appl. Phys. Lett.
**2015**, 106, 131108. [Google Scholar] [CrossRef] - Albo, A.; Hu, Q. Carrier leakage into the continuum in diagonal GaAs/Al
_{0.15}GaAs terahertz quantum cascade lasers. Appl. Phys. Lett.**2015**, 107, 241101. [Google Scholar] [CrossRef] - Kumar, S.; Hu, Q.; Reno, J.L. 186 K operation of terahertz quantum-cascade lasers based on a diagonal design. Appl. Phys. Lett.
**2009**, 94, 131105. [Google Scholar] [CrossRef] - Albo, A.; Hu, Q.; Reno, J.L. Room temperature negative differential resistance in terahertz quantum cascade laser structures. Appl. Phys. Lett.
**2016**, 109, 081102. [Google Scholar] [CrossRef] - Albo, A.; Flores, Y.V.; Hu, Q.; Reno, J.L. Split-well direct-phonon terahertz quantum cascade lasers. Appl. Phys. Lett.
**2019**, 114, 191102. [Google Scholar] [CrossRef][Green Version] - Lander Gower, N.; Piperno, S.; Albo, A. The significance of carrier leakage for stable lasing in split-well direct phonon terahertz quantum cascade lasers. Photonics
**2020**, 7, 59. [Google Scholar] [CrossRef] - Lander Gower, N.; Piperno, S.; Albo, A. Self-consistent gain calculations and carrier transport analysis for split-well direct-phonon terahertz quantum cascade lasers. AIP Adv.
**2020**, 10, 115319. [Google Scholar] [CrossRef] - Grange, T. Electron transport in quantum wire superlattices. Phys. Rev. B
**2014**, 89, 165310. [Google Scholar] [CrossRef][Green Version] - Grange, T. Nanowire terahertz quantum cascade lasers. Appl. Phys. Lett.
**2014**, 105, 141105. [Google Scholar] [CrossRef][Green Version] - Grange, T. Contrasting influence of charged impurities on transport and gain in terahertz quantum cascade lasers. Phys. Rev. B
**2015**, 92, 241306(R). [Google Scholar] [CrossRef][Green Version] - Majer, N.; Lüdge, K.; Schöll, E. Cascading enables ultrafast gain recovery dynamics of quantum dot semiconductor optical amplifiers. Phys. Rev. B
**2010**, 82, 235301. [Google Scholar] [CrossRef] - Wacker, A.; Lindskog, M.; Winge, D.O. Nonequilibrium Green’s function formulation of intersubband absorption for nonparabolic single-band effective mass Hamiltonian. IEEE J. Sel. Top. Quantum Electron.
**2013**, 19, 1200611. [Google Scholar] [CrossRef] - Wang, K.; Grange, T.; Lin, T.-T.; Wang, L.; Jéhn, Z.; Birner, S.; Yun, J.; Terashima, W.; Hirayama, H. Broadening mechanisms and self-consistent gain calculations for GaN quantum cascade laser structures. Appl. Phys. Lett.
**2018**, 113, 061109. [Google Scholar] [CrossRef] - Yasuda, H.; Kubis, T.; Hosako, I.; Hirakawa, K. Non-equilibrium Green’s function calculation for GaN-based terahertz-quantum cascade laser structures. J. Appl. Phys.
**2012**, 111, 083105. [Google Scholar] [CrossRef] - Lee, S.-C.; Wacker, A. Nonequilibrium Green’s function theory for transport and gain properties of quantum cascade structures. Phys. Rev. B
**2012**, 66, 245314. [Google Scholar] [CrossRef][Green Version] - Wacker, A. Gain in quantum cascade lasers and superlattices: A quantum transport theory. Phys. Rev. B
**2002**, 66, 085326. [Google Scholar] [CrossRef][Green Version] - Kubis, T.; Yeh, C.; Vogl, P.; Benz, A.; Fasching, G.; Deutsch, C. Theory of nonequilibrium quantum transport and energy dissipation in terahertz quantum cascade lasers. Phys. Rev. B
**2009**, 79, 195323. [Google Scholar] [CrossRef][Green Version] - Schmielau, T.; Pereira, M.F., Jr. Nonequilibrium many body theory for quantum transport in terahertz quantum cascade lasers. Appl. Phys. Lett.
**2009**, 95, 231111. [Google Scholar] [CrossRef] - Yasuda, H.; Kubis, T.; Vogl, P.; Sekine, N.; Hosako, I.; Hirakawa, K. A phonon scattering assisted injection and extraction based terahertz quantum cascade laser. Appl. Phys. Lett.
**2009**, 94, 151109. [Google Scholar] [CrossRef] - Kostadinova, E.G.; Padgett, J.L.; Liaw, C.D.; Matthews, L.S.; Hyde, T.W. Numerical study of anomalous diffusion of light in semicrystalline polymer structures. Phys. Rev. Res.
**2020**, 2, 043375. [Google Scholar] [CrossRef] - Lubatsch, A.; Frank, R. A self-consistent quantum field theory for random lasing. Appl. Sci.
**2019**, 9, 2477. [Google Scholar] [CrossRef][Green Version] - Lubatsch, A.; Frank, R. Evolution of Floquet topological quantum states in driven semiconductors. Eur. Phys. J. B
**2019**, 92, 215. [Google Scholar] [CrossRef][Green Version] - Morozov, V.; Ignatyuk, V. Energy conservation and the correlation quasi-temperature in open quantum dynamics. Particles
**2018**, 1, 285–295. [Google Scholar] [CrossRef][Green Version] - Banit, F.; Lee, S.-C.; Knorr, A.; Wacker, A. Self-consistent theory of the gain linewidth for quantum-cascade lasers. Appl. Phys. Lett.
**2005**, 86, 041108. [Google Scholar] [CrossRef][Green Version] - Boyle, C.; Oresick, K.M.; Kirch, J.D.; Flores, Y.V.; Mawst, L.J.; Botez, D. Carrier leakage via interface-roughness scattering bridges gap between theoretical and experimental internal efficiencies of quantum cascade lasers. Appl. Phys. Lett.
**2020**, 117, 051101. [Google Scholar] [CrossRef] - Almqvist, T.; Winge, D.O.; Dupont, E.; Wacker, A. Domain formation and self-sustained oscillations in quantum cascade lasers. Eur. Phys. J. B
**2019**, 92, 72. [Google Scholar] [CrossRef] - Flores, Y.V.; Albo, A. Impact of interface roughness scattering on the performance of GaAs/Al
_{x}Ga_{1–x}as terahertz quantum cascade lasers. IEEE J. Quantum Electron.**2017**, 53, 1–8. [Google Scholar] [CrossRef] - Grange, T.; Mukherjee, S.; Capellini, G.; Montanari, M.; Persichetti, L.; Di Gaspare, L.; Birner, S.; Attiaoui, A.; Moutanabbir, O.; Virgilio, M.; et al. Atomic-scale insights into semiconductor heterostructures: From experimental three-dimensional analysis of the interface to a generalized theory of interfacial roughness scattering. Phys. Rev. Appl.
**2020**, 13, 044062. [Google Scholar] [CrossRef] - Witzigmann, B.; Römer, F.; Martens, M.; Kuhn, C.; Wernicke, T.; Kneissl, M. Calculation of optical gain in AlGaN quantum wells for ultraviolet emission. AIP Adv.
**2020**, 10, 095307. [Google Scholar] [CrossRef] - Shin, J.C.; D’Souza, M.; Liu, Z.; Kirch, J.; Mawst, L.J.; Botez, D.; Vurgaftman, I.; Meyer, J.R. Highly temperature insensitive, deep-well 4.8 μm emitting quantum cascade semiconductor lasers. Appl. Phys. Lett.
**2009**, 94, 201103. [Google Scholar] [CrossRef] - Chan, C.W.I.; Albo, A.; Hu, Q.; Reno, J.L. Tradeoffs between oscillator strength and lifetime in terahertz quantum cascade lasers. Appl. Phys. Lett.
**2016**, 109, 201104. [Google Scholar] [CrossRef] - Lander Gower, N.; Piperno, S.; Albo, A. The effect of doping in split-well direct phonon terahertz quantum cascade laser structures. Photonics
**2021**, 8, 195. [Google Scholar] [CrossRef] - Chen, H.; Gao, S.; Zhang, M.; Zhang, J.; Qiao, L.; Wang, T.; Gao, F.; Hu, X.; Li, S.; Zhu, Y. Advances in random fiber lasers and their sensing application. Sensors
**2020**, 20, 6122. [Google Scholar] [CrossRef] - Meinzer, N.; König, M.; Ruther, M.; Linden, S.; Khitrova, G.; Gibbs, H.M.; Busch, K.; Wegener, M. Distance-dependence of the coupling between split-ring resonators and single-quantum-well gain. Appl. Phys. Lett.
**2011**, 99, 111104. [Google Scholar] [CrossRef] - Kazarinov, R.F.; Suris, R.A. Possible amplification of electromagnetic waves in a semiconductor with a superlattice. Sov. Phys. Semicond.
**1971**, 5, 707. [Google Scholar] - Bhattacharya, I.; Chan, C.W.I.; Hu, Q. Effects of stimulated emission on transport in terahertz quantum cascade lasers based on diagonal designs. Appl. Phys. Lett.
**2012**, 100, 011108. [Google Scholar] [CrossRef] - Sirtori, C.; Capasso, F.; Faist, J.; Hutchinson, A.L.; Sivco, D.L.; Cho, A.Y. Resonant tunneling in quantum cascade lasers. IEEE J. Quantum Electron.
**1998**, 34, 1722. [Google Scholar] [CrossRef]

**Figure 1.**Band diagrams for the three THz_OCL designs: (

**a**) RP, (

**b**) TW, and (

**c**) SWDP. In each diagram, three sequential periods are shown (Table 1 describes the designs and design parameters in more detail).

**Figure 2.**Optical gain as a function of photon energy for various temperatures: (

**a**) for the RP design at a bias voltage of 86 mV/module, (

**b**) for the TW design at a bias voltage of 70 mV/module, and (

**c**) for the SWDP design at a bias voltage of 46 mV/module.

**Figure 3.**Peak gain as a function of temperature for a bias voltage of 86 mV/module, 70 mV/module, and 46 mV/module for the RP, TW, and SWDP designs, respectively.

**Figure 6.**The contributions of the various scattering mechanisms to the optical gain for (

**a**) RP, (

**b**) TW, and (

**c**) SWDP designs.

Design | Wafer Number | E_{21} [meV] | Lasing Energy (meV) | Oscillator Strength | Layer Sequence [#ML *] Barrier Composition and Doping Level |
---|---|---|---|---|---|

RP | VB0676 | 25 | 25 | 0.2 | 10.3/37.2/6.4/38.6/8.2/65.9213 periods GaAs/ Al 1.24 × 10_{0.3}Ga_{0.7}As^{17} cm^{−3} in the center 17 ML of the 65.9 ML well.(6.0 × 10 ^{10} cm^{−2}) |

TW | VB0747 | 56 | 13 | 0.18 | 13.5/25.5/11.0/50.2354 periods GaAs/ Al 1.26 × 10_{0.3}Ga_{0.7}As^{17} cm^{−3} in the centered 17 ML of the 50.2 ML well (6.0 × 10^{10} cm^{−2}) |

SWDP | VB0837 | 26 | 11 | 0.26 | 9.0/24.8/3.5/24.8/17.3/24.8353 periods total thickness 10 μm GaAs/mixed barriers Al and _{0.55}Ga_{0.45}As (Inj.)Al_{0.15}Ga_{0.85}As (Rad., Intraw.)2.13 × 10 ^{16} cm^{−3} in the 24.8 ML wells (2.98 × 10^{10} cm^{−2}) |

**AlGaAs**barriers in

**bold**and the GaAs wells in plain text. The barriers’ composition and doping details are described in the following lines.

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 |

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**MDPI and ACS Style**

Lander Gower, N.; Piperno, S.; Albo, A.
Comparison of THz-QCL Designs Supporting Clean N-Level Systems. *Photonics* **2021**, *8*, 248.
https://doi.org/10.3390/photonics8070248

**AMA Style**

Lander Gower N, Piperno S, Albo A.
Comparison of THz-QCL Designs Supporting Clean N-Level Systems. *Photonics*. 2021; 8(7):248.
https://doi.org/10.3390/photonics8070248

**Chicago/Turabian Style**

Lander Gower, Nathalie, Silvia Piperno, and Asaf Albo.
2021. "Comparison of THz-QCL Designs Supporting Clean N-Level Systems" *Photonics* 8, no. 7: 248.
https://doi.org/10.3390/photonics8070248