Kazarinov, R. F. & Suris, R. A. Possibility of amplication of electromagnetic waves in a semiconductor with a superlattice. Sov. Phys. Semiconductors 5 , 707–709 (1971)..

Faist, J. et al. Quantum cascade laser. Science 264 , 553–556 (1994)..

Hall, R. N. et al. Coherent light emission from GaAs junctions. Phys. Rev. Lett. 9 , 366–368 (1962)..

Razeghi M. The MOCVD challenge: a survey of GaInAsP-InP and GaInAsP-GaAs for photonic and electronic device applications . 2nd edition. CRC Press: Boca Raton, 2011.

Wang, F. et al. Room temperature quantum cascade lasers with 22% wall plug efficiency in continuous-wave operation. Opt. Express 28 , 17532–17538 (2020)..

Li, L. H. et al. Multi-Watt high-power THz frequency quantum cascade lasers. Electron. Lett. 53 , 799–800 (2017)..

Wang, F. et al. Continuous wave quantum cascade lasers with 5.6 W output power at room temperature and 41% wall-plug efficiency in cryogenic operation. AIP Adv. 10 , 055120 (2020)..

Bai, Y. et al. Room temperature quantum cascade lasers with 27% wall plug efficiency. Appl. Phys. Lett. 98 , 181102 (2011)..

Khalatpour, A. et al. High-power portable terahertz laser systems. Nat. Photonics 15 , 16–20 (2021)..

Khalatpour, A. et al. Enhanced operating temperature in terahertz quantum cascade lasers based on direct phonon depopulation. Appl. Phys. Lett. 122 , 161101 (2023)..

Bosco, L. et al. Thermoelectrically cooled THz quantum cascade laser operating up to 210. K. Appl. Phys. Lett. 115 , 010601 (2019)..

Razeghi, M. High-Performance InP-Based Mid-IR Quantum Cascade Lasers. IEEE J. Sel. Top. Quantum Electron. 15 , 941–951 (2009)..

Köhler, R. et al. Terahertz Semiconductor-Heterostructure Laser. Nature 417 , 156–159 (2002)..

Walther, C. et al. Quantum cascade lasers operating from 1.2 to 1.6 THz. Appl. Phys. Lett. 91 , 131122 (2007)..

Williams, B. S. et al. Operation of terahertz quantum-cascade lasers at 164 K in pulsed mode and at 117 K in continuous-wave mode. Opt. Express 13 , 3331–3339 (2005)..

Landau, L. D.&Lifshitz, E. M. Quantum Mechanics: Non-Relativistic Theory, Volume 3. (Amsterdam: Elsevier, 2013).

Bastard, G. Wave Mechanics Applied to Semiconductor Heterostructures (New York: John Wiley and Sons Inc, 357, 1991).

Jirauschek, C. & Kubis, T. Modeling techniques for quantum cascade lasers. Appl. Phys. Rev. 1 , 011307 (2014)..

Cardona, M. Electron effective masses of InAs and GaAs as a function of temperature and doping. Phys. Rev. 121 , 752–758 (1961)..

Chuang, S. L. Physics of Photonic Devices. (Hoboken: Wiley, 2009).

Sirtori, C. et al. Nonparabolicity and a sum rule associated with bound-to-bound and bound-to-continuum intersubband transitions in quantum wells. Phys. Rev. B. 50 , 8663–8674 (1994)..

Miyoshi, T. & Ban, D. Y. Investigation of coulomb scattering in terahertz quantum cascade lasers. J. Appl. Phys. 129 , 153102 (2021)..

Nelson, D. F., Miller, R. C. & Kleinman, D. A. Band nonparabolicity effects in semiconductor quantum wells. Phys. Rev. B. 35 , 7770–7773 (1987)..

Hendorfer, G. et al. Enhancement of the in-plane effective mass of electrons in modulation-doped In x Ga 1 − x As quantum wells due to confinement effects. Phys. Rev. B. 48 , 2328–2334 (1993)..

Song, Z. G. et al. Corrigendum: band structure of Ge1− x Sn x alloy: a full-zone 30-band k · p model (2019 New J . Phys . 21 073037). N. J. Phys. 22 , 019502 (2020). .

Ma, X. P. et al. Two-band finite difference method for the bandstructure calculation with nonparabolicity effects in quantum cascade lasers. J. Appl. Phys. 114 , 063101 (2013)..

Gao, X., Botez, D. & Knezevic, I. X -valley leakage in GaAs∕AlGaAs quantum cascade lasers. Appl. Phys. Lett. 89 , 191119 (2006)..

Gao, X., Botez, D. & Knezevic, I. X -valley leakage in GaAs-based midinfrared quantum cascade lasers: a Monte Carlo study. J. Appl. Phys. 101 , 063101 (2007)..

Ikonić, Z., Harrison, P. & Kelsall, R. W. Intersubband hole-phonon and alloy disorder scattering in SiGe quantum wells. Phys. Rev. B. 64 , 245311 (2001)..

Ikonić, Z., Kelsall, R. W. & Harrison, P. Monte Carlo simulations of hole dynamics in SiGe∕Si terahertz quantum-cascade structures. Phys. Rev. B. 69 , 235308 (2004)..

Ikonić, Z., Harrison, P. & Kelsall, R. W. Self-consistent energy balance simulations of hole dynamics in SiGe∕Si THz quantum cascade structures. J. Appl. Phys. 96 , 6803–6811 (2004)..

Tan, I. et al. A self-consistent solution of Schrödinger–Poisson equations using a nonuniform mesh. J. Appl. Phys. 68 , 4071–4076 (1990)..

Jirauschek, C., Matyas, A. & Lugli, P. Modeling bound-to-continuum terahertz quantum cascade lasers: the role of Coulomb interactions. J. Appl. Phys. 107 , 013104 (2010)..

Valavanis, A. n-type silicon-germanium based terahertz quantum cascade lasers. PhD thesis, University of Leeds, Leeds, 2009.

Valavanis, A. et al. Theory and design of quantum cascade lasers in (111) n -type Si/SiGe. Phys. Rev. B 78 , 035420 (2008)..

Lever, L. et al. Simulated [111 ] Si–SiGe terahertz quantum cascade laser. Appl. Phys. Lett. 92 , 021124 (2008)..

Donovan, K., Harrison, P. & Kelsall, R. W. Self-consistent solutions to the intersubband rate equations in quantum cascade lasers: analysis of a GaAs/Al x Ga 1−x As device. J. Appl. Phys. 89 , 3084–3090 (2001)..

Jacoboni, C. & Reggiani, L. The Monte Carlo method for the solution of charge transport in semiconductors with applications to covalent materials. Rev. Mod. Phys. 55 , 645–705 (1983)..

Shi, Y. B. & Knezevic, I. Nonequilibrium phonon effects in midinfrared quantum cascade lasers. J. Appl. Phys. 116 , 123105 (2014)..

Callebaut, H. et al. Importance of electron-impurity scattering for electron transport in terahertz quantum-cascade lasers. Appl. Phys. Lett. 84 , 645–647 (2004)..

Li, H. et al. Monte Carlo simulation of extraction barrier width effects on terahertz quantum cascade lasers. Appl. Phys. Lett. 92 , 221105 (2008)..

Lü, J. T. & Cao, J. C. Monte Carlo simulation of hot phonon effects in resonant-phonon-assisted terahertz quantum-cascade lasers. Appl. Phys. Lett. 88 , 061119 (2006)..

Demić, A. et al. Infinite-period density-matrix model for terahertz-frequency quantum cascade lasers. IEEE Trans. Terahertz Sci. Technol. 7 , 368–377 (2017)..

Pan, A. et al. Density matrix modelin g of quantum cascade lasers without an artificially localized basis: a generalized scattering approach. Phys. Rev. B. 96 , 085308 (2017)..

Beji, G. et al. Coherent transport description of the dual-wavelength ambipolar terahertz quantum cascade laser. J. Appl. Phys. 109 , 013111 (2011)..

Grange, T. et al. Room temperature operation of n -type Ge/SiGe terahertz quantum cascade lasers predicted by non-equilibrium Green’s functions. Appl. Phys. Lett. 114 , 111102 (2019)..

Kubis, T. et al. Theory of nonequilibrium quantum transport and energy dissipation in terahertz quantum cascade lasers. Phys. Rev. B 79 , 195323 (2009)..

Bugajski, M. et al. Mid-IR quantum cascade lasers: device technology and non-equilibrium Green’s function modelingof electro-optical characteristics. Phys. Status Solidi (B) 251 , 1144–1157 (2014)..

Faist, J. et al. Bound-to-continuum and two-phonon resonance, quantum-cascade lasers for high duty cycle, high-temperature operation. IEEE J. Quantum Electron. 38 , 533–546 (2002)..

Chen , G. et al. Self-consistent approach for quantum cascade laser characteristic simulation. IEEE J. Quantum Electron. 47 , 1086–1093 (2011)..

Harrison P.&Valavanis, A. Quantum Wells, Wires and Dots: Theoretical and Computational Physics of Semiconductor Nanostructures. 4th edn. (Chichester: Wiley, 2016).

Bufler, F. M., Schenk, A. & Fichtner, W. Efficient Monte Carlo device modeling. IEEE Trans. Electron Devices 47 , 1891–1897 (2000)..

Terazzi, R. Transport in quantum cascade lasers. PhD thesis, ETH Zurich, Zurich 2012. https://www.research-collection.ethz.ch/bitstream/handle/20.500.11850/153241/eth-5287-01.pdf https://www.research-collection.ethz.ch/bitstream/handle/20.500.11850/153241/eth-5287-01.pdf .

Callebaut, H. & Hu, Q. Importance of coherence for electron transport in terahertz quantum cascade lasers. J. Appl. Phys. 98 , 104505 (2005)..

Jirauschek, C. Density matrix Monte Carlo modeling of quantum cascade lasers. J. Appl. Phys. 122 , 133105 (2017)..

Razavipour, S. G. Design, analysis, and characterization of indirectly-pumped terahertz quantum cascade lasers. PhD thesis, University of Waterloo, Waterloo, 2013.

Dupont, E. et al. A phonon scattering assisted injection and extraction based terahertz quantum cascade laser. J. Appl. Phys. 111 , 073111 (2012)..

Lee, S. C. & Galbraith, I. Intersubband and intrasubband electronic scattering rates in semiconductor quantum wells. Phys. Rev. B. 59 , 15796–15805 (1999)..

Lugli, P. et al. Monte Carlo algorithm for hot phonons in polar semiconductors. Appl. Phys. Lett. 50 , 1251–1253 (1987)..

Bai, Y. High wall plug efficiency quantum cascade lasers. PhD thesis, Northwestern University, Evanston, 2011.

Oresick, K. M. et al. Highly efficient long-wavelength infrared, step-tapered quantum cascade lasers. Proceedings of SPIE 11705, Novel In-Plane Semiconductor Lasers XX. SPIE, 1170515 https://doi.org/10.1117/12.2582436 https://doi.org/10.1117/12.2582436 (2021).

Kirch, J. D. et al. 86% internal differential efficiency from 8 to 9 µm-emitting, step-taper active-region quantum cascade lasers. Opt. Express 24 , 24483–24494 (2016)..

Bai, Y. et al. Highly temperature insensitive quantum cascade lasers. Appl. Phys. Lett. 97 , 251104 (2010)..

Wang, F. et al. Room temperature quantum cascade laser with ~31% wall-plug efficiency. AIP Adv. 10 , 075012 (2020)..

Wang, F., Slivken, S. & Razeghi, M. High-brightness LWIR quantum cascade lasers. Opt. Lett. 46 , 5193–5196 (2021)..

Slivken, S. et al. Sampled grating, distributed feedback quantum cascade lasers with broad tunability and continuous operation at room temperature. Appl. Phys. Lett. 100 , 261112 (2012)..

Bai, Y. B. et al. Quantum cascade lasers that emit more light than heat. Nat. Photonics 4 , 99–102 (2010)..

Slivken, S. et al. Gas-source molecular beam epitaxy growth of an 8.5 μm quantum cascade laser. Appl. Phys. Lett. 71 , 2593–2595 (1997)..

Slivken, S. et al. High-power, continuous-operation inters ubband laser for wavelengths greater than 10 μ m. Appl. Phys. Lett. 90 , 151115 (2007)..

Wang, F., Slivken, S. & Razeghi, M. Harmonic injection locking of high-power mid-infrared quantum cascade lasers. Photonics Res. 9 , 1078 (2021)..

Zhou, W. J. et al. High performance monolithic, broadly tunable mid-infrared quantum cascade lasers. Optica 4 , 1228–1231 (2017)..

Beck, M. et al. Continuous Wave Operation of a Mid-Infrared Semiconductor Laser at Room Temperature. Science 295 , 301–305 (2002)..

Knigge, A. et al. Passively cooled 940 nm laser bars with 73% wall-plug efficiency and high reliability at 98 W quasi-cw output power. CLEO/Europe. 2005 Conference on Lasers and Electro-Optics Europe, 2005. Munich, Germany: IEEE, 2005, 108

Lyakh, A. et al. 5.6 μ m quantum cascade lasers based on a two-material active region composition with a room temperature wall-plug efficiency exceeding 28. Appl. Phys. Lett. 109 , 121109 (2016)..

Troccoli, M. et al. Long-wave IR quantum cascade lasers for emission in the λ = 8-12μm spectral region. Optical Mater. Express 3 , 1546–1560 (2013)..

Spitz, O. et al. Free-space communication with directly modulated mid-infrared quantum cascade devices. IEEE J. Sel. Top. Quantum Electron. 28 , 1200109 (2022)..

Bismuto, A. et al. High performance, low dissipation quantum cascade lasers across the mid-IR range. Opt. Express 23 , 5477–5484 (2015)..

Becker, C. & Sirtori, C. Lateral current spreading in unipolar semiconductor lasers. J. Appl. Phys. 90 , 1688–1691 (2001)..

Hatakoshi, G. I. Analysis of beam quality factor for semiconductor lasers. Optical Rev. 10 , 307–314 (2003)..

Heydari, D. et al. High brightness angled cavity quantum cascade lasers. Appl. Phys. Lett. 106 , 091105 (2015)..

Lyakh, A. et al. Multiwatt long wavelength quantum cascade lasers based on high strain composition with 70% injection efficiency. Opt. Express 20 , 24272–24279 (2012)..

Xie, F. et al. Watt-level room temperature continuous-wave operation of quantum cascade lasers with λ > 10 μm. IEEE J. Sel. Top. Quantum Electron. 19 , 1200407 (2013)..

Fathololoumi, S. et al. Terahertz quantum cascade lasers operating up to ~ 200 K with optimized oscillator strength and improved injection tunneling. Opt. Express 20 , 3866–3876 (2012)..

Lu, Q. Y. et al. Widely tuned room temperature terahertz quantum cascade laser sources based on difference-frequency generation. Appl. Phys. Lett. 101 , 251121 (2012)..

Lu, Q. Y. et al. Room temperature single-mode terahertz sources based on intracavity difference-frequency generation in quantum cascade lasers. Appl. Phys. Lett. 99 , 131106 (2011)..

Belkin, M. A. et al. Terahertz quantum-cascade-laser source based on intracavity difference-frequency generation. Nat. Photonics 1 , 288–292 (2007)..

Wang, F. H. et al. Generating ultrafast pulses of light from quantum cascade lasers. Optica 2 , 944–949 (2015)..

Bacon, D. R. et al. Gain reco very time in a terahertz quantum cascade laser. Appl. Phys. Lett. 108 , 081104 (2016)..

Choi, H. et al. Gain recovery dynamics and photon-driven transport in quantum cascade lasers. Phys. Rev. Lett. 100 , 167401 (2008)..

Green, R. P. et al. Gain recovery dynamics of a terahertz quantum cascade laser. Phys. Rev. B 80 , 075303 (2009)..

Wang, C. Y. et al. Coherent instabilities in a semiconductor laser with fast gain recovery. Phys. Rev. A 75 , 031802(R) (2007)..

Barbieri, S. et al. Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis. Nat. Photonics 5 , 306–313 (2011)..

Maysonnave, J. et al. Mode-locking of a terahertz laser by direct phase synchronization. Opt. Express 20 , 20855–20862 (2012)..

Freeman, J. R. et al. Direct intensity sampling of a modelocked terahertz quantum cascade laser. Appl. Phys. Lett. 101 , 181115 (2012)..

Oustinov, D. et al. Phase seeding of a terahertz quantum cascade laser. Nat. Commun. 1 , 69 (2010)..

Wang, F. H. et al. Short terahertz pulse generation from a dispersion compensated modelocked semiconductor laser. Laser Photonics Rev. 11 , 1700013 (2017)..

Wang, F. H. et al. Ultrafast buildup dynamics of terahertz pulse generation in mode-locked quantum cascade lasers. Phys. Rev. Appl. 18 , 064054 (2022)..

Wang, F. H. et al. Ultrafast response of harmonic modelocked THz lasers. Light Sci. Appl. 9 , 51 (2020)..

Kundu, I. et al. Ultrafast switch-on dynamics of frequency-tuneable semiconductor lasers. Nat. Commun. 9 , 3076 (2018)..

Riccardi, E. et al. Short pulse generation from a graphene-coupled passively mode-locked terahertz laser. Nat. Photonics 17 , 607–614 (2023)..

Seitner, L. et al. Theoretical model of passive mode-locking in terahertz quantum cascade lasers with distributed saturable absorbers. Nanophotonics 13 , 1823–1834 (2024)..

Jirauschek, C. Theory of hybrid microwave–photonic quantum devices. Laser Photonics Rev. 17 , 2300461 (2023)..

Hillbrand, J. et al. Mode-locked short pulses from an 8 μm wavelength semiconductor laser. Nat. Commun. 11 , 5788 (2020)..

Täschler, P. et al. Femtosecond pulses from a mid-infrared quantum cascade laser. Nat. Photonics 15 , 919–924 (2021)..

Shahili, M. et al. Continuous-wave GaAs/AlGaAs quantum cascade laser at 5.7. THz. Nanophotonics 13 , 1735–1743 (2024)..

Razeghi, M. et al. Recent progress of quantum cascade laser research from 3 to 12 μm at the Center for Quantum Devices [Invited ] . Appl. Opt. 56 , H30–H44 (2017)..

Vitiello, M. S. et al. Quantum cascade lasers: 20 years of challenges. Opt. Express 23 , 5167–5182 (2015)..

Kumar, S. et al. A 1.8-THz quantum cascade laser operating significantly above the temperature of ℏω/ k B . Nat. Phys. 7 , 166–171 (2011)..

Chassagneux, Y. et al. Limiting factors to the temperature performance of THz quantum cascade lasers based on the resonant-phonon depopulation scheme. IEEE Trans. Terahertz Sci. Technol. 2 , 83–92 (2012)..

Kumar, S. et al. Two-well terahertz quantum-cascade laser with direct intrawell-phonon depopulation. Appl. Phys. Lett. 95 , 141110 (2009)..

Wittmann, A. et al. Distributed-feedback quantum-cascade lasers at 9 μ m operating in continuous wave up to 423 K. IEEE Photonics Technol. Lett. 21 , 814–816 (2009)..