## Importance of the microscopic effects on the linewidth enhancement factor of quantum cascade lasers |

Optics Express, Vol. 21, Issue 23, pp. 27804-27815 (2013)

http://dx.doi.org/10.1364/OE.21.027804

Acrobat PDF (1581 KB)

### Abstract

Microscopic density matrix analysis on the linewidth enhancement factor (LEF) of both mid-infrared (mid-IR) and Terahertz (THz) quantum cascade lasers (QCLs) is reported, taking into account of the many body Coulomb interactions, coherence of resonant-tunneling transport and non-parabolicity. A non-zero LEF at the gain peak is obtained due to these combined microscopic effects. The results show that, for mid-IR QCLs, the many body Coulomb interaction and non-parabolicity contribute greatly to the non-zero LEF. In contrast, for THz QCLs, the many body Coulomb interactions and the resonant-tunneling effects greatly influence the LEF resulting in a non-zero value at the gain peak. This microscopic model not only partially explains the non-zero LEF of QCLs at the gain peak, which observed in the experiments for a while but cannot be explicitly explained, but also can be employed to improve the active region designs so as to reduce the LEF by optimizing the corresponding parameters.

© 2013 Optical Society of America

## 1. Introduction

*μ*m) and Terahertz (THz, from 1.2 to 5 THz, or 60 to 250

*μ*m) ranges of the electromagnetic spectrum. As compact and coherent radiation sources, they have received considerable attentions since their first demonstrations [1

1. R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature **417**(6885), 156–159 (2002). [CrossRef] [PubMed]

2. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science **264**(5158), 553–556 (1994). [CrossRef] [PubMed]

*α*) plays an important role in determining the optical emission linewidth of the laser systems [3

3. C. H. Henry, “Theory of the linewidth of semiconductor-lasers,” IEEE J. Quantum Electron. **18**(2), 259–264 (1982). [CrossRef]

4. T. Chattopadhyay and P. Bhattacharyya, “Role of linewidth enhancement factor on the frequency response of the synchronized quantum cascade laser,” Opt. Commun. **309**, 349–354 (2013). [CrossRef]

3. C. H. Henry, “Theory of the linewidth of semiconductor-lasers,” IEEE J. Quantum Electron. **18**(2), 259–264 (1982). [CrossRef]

5. M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor-lasers - an overview,” IEEE J. Quantum Electron. **23**(1), 9–29 (1987). [CrossRef]

*k*-space. In contrast, due to the intersubband transition characteristics, QCLs are expected to have a narrow and symmetric gain spectrum, and hence, according to the Kramers-Kronig relation, resulting in a zero LEF at the peak gain wavelength predicted by Faist

*et al*[2

2. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science **264**(5158), 553–556 (1994). [CrossRef] [PubMed]

6. M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Linewidth enhancement factor of a type-I quantum-cascade laser,” J. Appl. Phys. **94**(8), 5426–5428 (2003). [CrossRef]

11. J. Kim, M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Theoretical and experimental study of optical gain and linewidth enhancement factor of type-I quantum-cascade lasers,” IEEE J. Quantum Electron. **40**(12), 1663–1674 (2004). [CrossRef]

*μ*m [6

6. M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Linewidth enhancement factor of a type-I quantum-cascade laser,” J. Appl. Phys. **94**(8), 5426–5428 (2003). [CrossRef]

*μ*m (~2.55 THz) [7

7. R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. **92**(7), 071106 (2008). [CrossRef]

10. N. Kumazaki, Y. Takagi, M. Ishihara, K. Kasahara, A. Sugiyama, N. Akikusa, and T. Edamura, “Detuning characteristics of the linewidth enhancement factor of a midinfrared quantum cascade laser,” Appl. Phys. Lett. **92**(12), 121104 (2008). [CrossRef]

11. J. Kim, M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Theoretical and experimental study of optical gain and linewidth enhancement factor of type-I quantum-cascade lasers,” IEEE J. Quantum Electron. **40**(12), 1663–1674 (2004). [CrossRef]

11. J. Kim, M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Theoretical and experimental study of optical gain and linewidth enhancement factor of type-I quantum-cascade lasers,” IEEE J. Quantum Electron. **40**(12), 1663–1674 (2004). [CrossRef]

*macroscopic*picture. Furthermore, a theoretical analysis on the LEF of THz QCLs is lacked. Since the active region structures of THz QCLs are different from those of mid-IR QCLs, they shall show different characteristic of non-zero LEF. To disclose more physical underlining mechanisms of non-zero LEF value in both mid-IR and THz QCLs, a microscopic model is required.

12. T. Liu, K. E. Lee, and Q. J. Wang, “Microscopic density matrix model for optical gain of terahertz quantum cascade lasers: Many-body, nonparabolicity, and resonant tunneling effects,” Phys. Rev. B **86**(23), 235306 (2012). [CrossRef]

## 2. LEF in mid-IR QCLs

### 2.1. The microscopic model

6. M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Linewidth enhancement factor of a type-I quantum-cascade laser,” J. Appl. Phys. **94**(8), 5426–5428 (2003). [CrossRef]

13. C. Gmachl, F. Capasso, J. Faist, A. L. Hutchinson, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, S. N. G. Chu, and A. Y. Cho, “Continuous-wave and high-power pulsed operation of index-coupled distributed feedback quantum cascade laser at λ≈8.5 μm,” Appl. Phys. Lett. **72**(12), 1430–1432 (1998). [CrossRef]

9. J. von Staden, T. Gensty, W. Elsässer, G. Giuliani, and C. Mann, “Measurements of the *α* factor of a distributed-feedback quantum cascade laser by an optical feedback self-mixing technique,” Opt. Lett. **31**(17), 2574–2576 (2006). [CrossRef] [PubMed]

_{51}is the coupling strength), is shown through the injector barriers. The energy states within one period are coupled through scattering processes, but at the injector barrier, the transport is modeled by tunneling. In order to conveniently treat the many body effects, we derive the dynamic equations of motion in the second quantized representation. The Hamiltonian of the system of mid-IR QCL in Fig. 1, which characterizes the electron-light coupling, the tunneling effects, free electrons and electron-electron Coulomb interactions, can be written as

*ξ*is the slowly varying complex electric field amplitude,

*μ*is the electron charge times the dipole matrix element of laser transition between energy level

_{j5}*j*and 5, Δ

_{51’}is the injection coupling strengths.

*j*th subband energy,

**k**is the in-plane wave vector.

12. T. Liu, K. E. Lee, and Q. J. Wang, “Microscopic density matrix model for optical gain of terahertz quantum cascade lasers: Many-body, nonparabolicity, and resonant tunneling effects,” Phys. Rev. B **86**(23), 235306 (2012). [CrossRef]

15. S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B **80**(24), 245316 (2009). [CrossRef]

*G*and carrier-induced refractive index change

*δn*are given by where

*ε*

_{0}is the vacuum permittivity,

*n*is the refractive index,

*c*is the light speed in vacuum, and

*V*is the volume of one period of active region.

_{m}*α*can be obtained by the ratio of the change in the real part of the refractive index change

*δn*to the change in the gain

*G*with respect to the carrier density

*N*[14]

_{0}### 2.2. Results and discussions

*ω*. The experimental α-value of this active region design at gain peak is around −0.5 [6

_{λ}**94**(8), 5426–5428 (2003). [CrossRef]

**40**(12), 1663–1674 (2004). [CrossRef]

16. C. Gmachl, F. Capasso, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, and A. Y. Cho, “High-power, continuous-wave, current-tunable, single-mode quantum-cascade distributed-feedback lasers at lambda ~5.2 and lambda ~7.95 μm,” Opt. Lett. **25**(4), 230–232 (2000). [CrossRef] [PubMed]

9. J. von Staden, T. Gensty, W. Elsässer, G. Giuliani, and C. Mann, “Measurements of the *α* factor of a distributed-feedback quantum cascade laser by an optical feedback self-mixing technique,” Opt. Lett. **31**(17), 2574–2576 (2006). [CrossRef] [PubMed]

9. J. von Staden, T. Gensty, W. Elsässer, G. Giuliani, and C. Mann, “Measurements of the *α* factor of a distributed-feedback quantum cascade laser by an optical feedback self-mixing technique,” Opt. Lett. **31**(17), 2574–2576 (2006). [CrossRef] [PubMed]

*μ*m blue-shift and red-shift positions relative to the peak under different biases computed from the microscopic model “many body + non-parabolicity” (considering both many body and non-parabolicity effects), the microscopic free-carrier model (considering both free carriers and non-parabolicity but neglecting the renormalization of band structure and Rabi frequency), the macroscopic density-matrix model with and without including the resonant-tunneling effect, respectively. As shown in Fig. 3(a), the many body Coulomb interaction, coherence of resonant tunneling and non-parabolicity all induce a finite value of LEF at the gain peak. Furthermore, by the comparison of these contributions to LEF, the many body Coulomb interaction and non-parabolicity parameter have more important effects on the LEF at gain peak. This is because the many body Coulomb interaction and non-parabolicity all tend to distort the shape of gain spectrum, as shown in Fig. 3(b), where the non-parabolicity can more greatly modify the gain spectrum as compared with Coulomb interaction.

_{0.53}Ga

_{0.47}As quantum wells is higher than that in GaAs quantum wells. Therefore, the gain spectrum in mid-IR QCLs can be more greatly influenced by the non-parabolicity. If the non-parabolicity (proportional to the ratio of the effective mass at the upper laser level and the lower laser level) slightly increases, the LEF will increase according to Fig. 4(a).The increase is attributed to the influence of non-parabolicity on symmetry of gain spectrum, as shown in Fig. 4(b). Therefore, according to the above simulations, it is expected that, for similar structures, mid-IR QCL emitting at a shorter wavelength and with a higher non-parabolicity has a larger α-value.

## 3. LEF in THz QCLs

17. S. Fathololoumi, E. Dupont, C. W. I. Chan, Z. R. Wasilewski, S. R. Laframboise, D. Ban, A. Mátyás, C. Jirauschek, Q. Hu, and H. C. Liu, “Terahertz quantum cascade lasers operating up to ~200 K with optimized oscillator strength and improved injection tunneling,” Opt. Express **20**(4), 3866–3876 (2012). [CrossRef] [PubMed]

12. T. Liu, K. E. Lee, and Q. J. Wang, “Microscopic density matrix model for optical gain of terahertz quantum cascade lasers: Many-body, nonparabolicity, and resonant tunneling effects,” Phys. Rev. B **86**(23), 235306 (2012). [CrossRef]

*ω*. It is noted that, when the bias goes above the designed value and becomes 13.3 kV/cm, the LEF becomes −0.3. This is mainly attributed to the symmetric changes of gain spectrum with the extraction detuning due to tunneling effects i.e. the energy splitting due to the coupling between the lower laser level and the extraction level, as shown in Fig. 5 in [12

_{λ}**86**(23), 235306 (2012). [CrossRef]

*ω*. Furthermore, the non-parabolicity, in contrast to mid-IR QCLs, can only induce a slight influence on the LEF according to the comparisons between free-carrier model and macroscopic one with resonant tunneling, but the many body Coulomb interaction causes a large variation of LEF at gain peak with the comparison of the model “many-body + non-parabolicity” and free-carrier one due to its strong modifications to gain spectrum, as shown in Fig. 8. (The Coulomb interaction include the Hartree-Fock, dephasing and scattering contributions, more details can be seen in [12

_{λ}**86**(23), 235306 (2012). [CrossRef]

## 4. Conclusion

*e.g.*the increase of LEF with injection current for DFB QCLs. The discrepancy between the experimental value and the proposed theoretical models can be attributed to the refractive index change due to device self-heating, which should also be considered in the future explorations.

## Appendix A

*O*is operator,

*H*is the Hamiltonian. Due to the Coulomb interaction, the result is an infinite hierarchy of coupled differential equations. The hierarchy describes the correlation effect in the Coulomb potential. The first order correlation is induced by the Hartree-Fock contributions, which results in band structure and Rabi frequency renormalizations. Scattering and dephasing contributions cause the second order correlation in the Coulomb potential, and so on. In this paper, we only include the Hartree-Fock contributions, and dephasing and scattering contributions at the level of a relaxation-rate approximation. Then one can get the following equations of motion for the slowing varying polarization

*i*and

*j.*

*j*,

*i*and

*j*.

*j*,

*j*. The chemical potentials and temperatures are determined by electron number conservation and energy conservation, which are described in details in [12

**86**(23), 235306 (2012). [CrossRef]

_{.}For subband

*j*, we have

18. U. Ekenberg, “Nonparabolicity effects in a quantum well: sublevel shift, parallel mass, and Landau levels,” Phys. Rev. B Condens. Matter **40**(11), 7714–7726 (1989). [CrossRef] [PubMed]

19. S. Panda, B. K. Panda, and S. Fung, “Effect of conduction band nonparabolicity on the dark current in a quantum well infrared detector,” J. Appl. Phys. **101**(4), 043705 (2007). [CrossRef]

## Appendix B

20. E. Dupont, S. Fathololoumi, and H. C. Liu, “Simplified density-matrix model applied to three-well terahertz quantum cascade lasers,” Phys. Rev. B **81**(20), 205311 (2010). [CrossRef]

## Acknowledgments

## References and links

1. | R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature |

2. | J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science |

3. | C. H. Henry, “Theory of the linewidth of semiconductor-lasers,” IEEE J. Quantum Electron. |

4. | T. Chattopadhyay and P. Bhattacharyya, “Role of linewidth enhancement factor on the frequency response of the synchronized quantum cascade laser,” Opt. Commun. |

5. | M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor-lasers - an overview,” IEEE J. Quantum Electron. |

6. | M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Linewidth enhancement factor of a type-I quantum-cascade laser,” J. Appl. Phys. |

7. | R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. |

8. | T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. |

9. | J. von Staden, T. Gensty, W. Elsässer, G. Giuliani, and C. Mann, “Measurements of the |

10. | N. Kumazaki, Y. Takagi, M. Ishihara, K. Kasahara, A. Sugiyama, N. Akikusa, and T. Edamura, “Detuning characteristics of the linewidth enhancement factor of a midinfrared quantum cascade laser,” Appl. Phys. Lett. |

11. | J. Kim, M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Theoretical and experimental study of optical gain and linewidth enhancement factor of type-I quantum-cascade lasers,” IEEE J. Quantum Electron. |

12. | T. Liu, K. E. Lee, and Q. J. Wang, “Microscopic density matrix model for optical gain of terahertz quantum cascade lasers: Many-body, nonparabolicity, and resonant tunneling effects,” Phys. Rev. B |

13. | C. Gmachl, F. Capasso, J. Faist, A. L. Hutchinson, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, S. N. G. Chu, and A. Y. Cho, “Continuous-wave and high-power pulsed operation of index-coupled distributed feedback quantum cascade laser at λ≈8.5 μm,” Appl. Phys. Lett. |

14. | W. W. Chow, S. W. Koch, and M. S. I. I. I. Semiconductor-Laser Physics, (Springer-Verlag, Berlin, 1994). |

15. | S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B |

16. | C. Gmachl, F. Capasso, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, and A. Y. Cho, “High-power, continuous-wave, current-tunable, single-mode quantum-cascade distributed-feedback lasers at lambda ~5.2 and lambda ~7.95 μm,” Opt. Lett. |

17. | S. Fathololoumi, E. Dupont, C. W. I. Chan, Z. R. Wasilewski, S. R. Laframboise, D. Ban, A. Mátyás, C. Jirauschek, Q. Hu, and H. C. Liu, “Terahertz quantum cascade lasers operating up to ~200 K with optimized oscillator strength and improved injection tunneling,” Opt. Express |

18. | U. Ekenberg, “Nonparabolicity effects in a quantum well: sublevel shift, parallel mass, and Landau levels,” Phys. Rev. B Condens. Matter |

19. | S. Panda, B. K. Panda, and S. Fung, “Effect of conduction band nonparabolicity on the dark current in a quantum well infrared detector,” J. Appl. Phys. |

20. | E. Dupont, S. Fathololoumi, and H. C. Liu, “Simplified density-matrix model applied to three-well terahertz quantum cascade lasers,” Phys. Rev. B |

**OCIS Codes**

(140.0140) Lasers and laser optics : Lasers and laser optics

(140.3070) Lasers and laser optics : Infrared and far-infrared lasers

(140.5965) Lasers and laser optics : Semiconductor lasers, quantum cascade

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: July 5, 2013

Revised Manuscript: October 29, 2013

Manuscript Accepted: October 29, 2013

Published: November 6, 2013

**Citation**

Tao Liu, Kenneth E. Lee, and Qi Jie Wang, "Importance of the microscopic effects on the linewidth enhancement factor of quantum cascade lasers," Opt. Express **21**, 27804-27815 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-27804

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### References

- R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002). [CrossRef] [PubMed]
- J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science264(5158), 553–556 (1994). [CrossRef] [PubMed]
- C. H. Henry, “Theory of the linewidth of semiconductor-lasers,” IEEE J. Quantum Electron.18(2), 259–264 (1982). [CrossRef]
- T. Chattopadhyay and P. Bhattacharyya, “Role of linewidth enhancement factor on the frequency response of the synchronized quantum cascade laser,” Opt. Commun.309, 349–354 (2013). [CrossRef]
- M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor-lasers - an overview,” IEEE J. Quantum Electron.23(1), 9–29 (1987). [CrossRef]
- M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Linewidth enhancement factor of a type-I quantum-cascade laser,” J. Appl. Phys.94(8), 5426–5428 (2003). [CrossRef]
- R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett.92(7), 071106 (2008). [CrossRef]
- T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett.89(9), 091121 (2006). [CrossRef]
- J. von Staden, T. Gensty, W. Elsässer, G. Giuliani, and C. Mann, “Measurements of the α factor of a distributed-feedback quantum cascade laser by an optical feedback self-mixing technique,” Opt. Lett.31(17), 2574–2576 (2006). [CrossRef] [PubMed]
- N. Kumazaki, Y. Takagi, M. Ishihara, K. Kasahara, A. Sugiyama, N. Akikusa, and T. Edamura, “Detuning characteristics of the linewidth enhancement factor of a midinfrared quantum cascade laser,” Appl. Phys. Lett.92(12), 121104 (2008). [CrossRef]
- J. Kim, M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Theoretical and experimental study of optical gain and linewidth enhancement factor of type-I quantum-cascade lasers,” IEEE J. Quantum Electron.40(12), 1663–1674 (2004). [CrossRef]
- T. Liu, K. E. Lee, and Q. J. Wang, “Microscopic density matrix model for optical gain of terahertz quantum cascade lasers: Many-body, nonparabolicity, and resonant tunneling effects,” Phys. Rev. B86(23), 235306 (2012). [CrossRef]
- C. Gmachl, F. Capasso, J. Faist, A. L. Hutchinson, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, S. N. G. Chu, and A. Y. Cho, “Continuous-wave and high-power pulsed operation of index-coupled distributed feedback quantum cascade laser at λ≈8.5 μm,” Appl. Phys. Lett.72(12), 1430–1432 (1998). [CrossRef]
- W. W. Chow, S. W. Koch, and M. S. I. I. I. Semiconductor-Laser Physics, (Springer-Verlag, Berlin, 1994).
- S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B80(24), 245316 (2009). [CrossRef]
- C. Gmachl, F. Capasso, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, and A. Y. Cho, “High-power, continuous-wave, current-tunable, single-mode quantum-cascade distributed-feedback lasers at lambda ~5.2 and lambda ~7.95 μm,” Opt. Lett.25(4), 230–232 (2000). [CrossRef] [PubMed]
- S. Fathololoumi, E. Dupont, C. W. I. Chan, Z. R. Wasilewski, S. R. Laframboise, D. Ban, A. Mátyás, C. Jirauschek, Q. Hu, and H. C. Liu, “Terahertz quantum cascade lasers operating up to ~200 K with optimized oscillator strength and improved injection tunneling,” Opt. Express20(4), 3866–3876 (2012). [CrossRef] [PubMed]
- U. Ekenberg, “Nonparabolicity effects in a quantum well: sublevel shift, parallel mass, and Landau levels,” Phys. Rev. B Condens. Matter40(11), 7714–7726 (1989). [CrossRef] [PubMed]
- S. Panda, B. K. Panda, and S. Fung, “Effect of conduction band nonparabolicity on the dark current in a quantum well infrared detector,” J. Appl. Phys.101(4), 043705 (2007). [CrossRef]
- E. Dupont, S. Fathololoumi, and H. C. Liu, “Simplified density-matrix model applied to three-well terahertz quantum cascade lasers,” Phys. Rev. B81(20), 205311 (2010). [CrossRef]

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