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Abstract
Non-equilibrium Green’s function (NEGF) formalism is used to optimize the gain region of a quantum cascade laser (QCL) tailored to emit radiation at ∼5 µm wavelength, originally designed by Evans et al. [Appl. Phys. Lett. , 88,051105(2006) [CrossRef] ]. The optimization strategy uses electron-photon selfenergies to find characteristics of devices under the "operating conditions," i.e., interacting with the laser field. These conditions can be quite different from the one when the device is in no-lasing state and the unsaturated gain is being optimized. The saturation caused by the optical field can push the structure from strong to weak coupling conditions, what changes laser parameters in a non-linear manner. Moreover, the NEGF method does not require any phenomenological parameters (such as, e.g., the phase relaxation times), so the quantities dependent on these parameters are determined solely on physical grounds. The use of the above procedure for the structure under investigation shows that the increase of the quantum efficiency by 24% and the output power by 83% in comparison to the original design can be achieved when the widths of injection and extraction barriers are changed to their optimal values.
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1. Introduction
Engineering of the active (gain) region of a quantum cascade laser (QCL) fundamentally determines its properties. Even if the target wavelength and the manufacturer’s technological capabilities dictate the material solutions and the choice of a specific emission scheme (two-phonon resonance, bound-to-continuum, excited-state injection, etc.), there is still room for an optimization of the designed structure. The appropriate sequence of layers (their number and widths) affects the laser properties through the value of the dipole element for the laser transition and the form factors determining electron lifetimes. It is also important to optimally couple the light-amplifying (gain) region to emitter and collector. It can be accomplished by adjusting the widths and/or heights of injection and extraction barriers. Their optimal values depend on the adapted criterion. For the injection barrier, the value of the optical gain $g$ was commonly used as a quality factor [1,2]. Recently, however, it has been shown that electromagnetic (e-m) field in the laser cavity can make the tunneling transition through the potential barrier crossover from the strong coupling regime, in which the transport through the barrier is coherent, to the weak coupling regime, in which this transport becomes incoherent [3]. This crossover is quantitatively described by the so-called gain recovery time, $\tau _{rec}$, which for mid-infrared QCLs operating at room temperature gets subpicosecond values [4,5]. It is a dynamic parameter, which is also important for the steady-state operation of the laser, because it determines its optical power $P$. Then, if $P$ is maximized the more adequate quality factor should involve not only the gain but also the time of its recovery.
For a device with a Fabry-Perot waveguide defined by its width $w$, and thickness $dN_{p}$, where $d$ is the thickness of a single module (period) and $N_{p}$ is the periods number, one has [3]
where $R$ is the reflection coefficient for the waveguide materials, $E_{\nu }=h \nu$ is the energy of emitted photons, and $\Phi _{cl}$ is the value of the photon flux $\Phi$, for which the gain $g$ is clamped to its threshold value, $g_{th}$. The last one is the value required to compensate all optical losses, i.e., where $\alpha _{w}$ and $\alpha _{m}$ denote losses in the waveguide and on the mirrors, and $\Gamma$ is the confinement factor [6]. The second equality in Eq. (1) was obtained using the relation which describes the non-linear gain saturation caused by an e-m field in a two-level atomic system [7]. In Eq. (3), $g_{c} \equiv \frac {\partial g}{\partial \Delta n}$ is the gain cross-section - a coefficient derived from the dependence of the gain on the population inversion $\Delta n$. Alternatively, in Eq. (3), one may use the saturation flux, $\Phi _{s}=1 / (\tau _{rec}g_{c}d)$ [8], or the saturation time, $\tau _{sat}$, defined in [9].2. Density matrix modeling
In the simplest model of the QCL, at least three energy levels must be considered with an additional pumping/extracting level (1, 1’) located in the injector [10] (Fig. 1.(a)). For such a case, Eq. (3) still holds, with the time, $\tau _{rec}$, determined by the interlevel (intersubband) relaxation times, $\tau _{32}$, $\tau _{21}$, and the tunneling time through the injection barrier, $T_{in}$ [3]. Namely,
The tunneling time is defined as [11] where $\tau _{in}$ is a dephasing time of the tunneling transition, $\Delta _{13} \equiv (E_{1}-E_{3}) / \hbar$, is the detuning from the resonance of states 1 and 3, and $\Omega _{13}$ is the Rabi angular frequency determined by the matrix element of these states. This element introduces a dependence of $T_{in}$ on the width and height of the injection barrier [12,13].
Fig. 1. (a) Three-level and (b) four-level models of QCL. The electron extraction from the active region (states $|2 \rangle, |3 \rangle$) to injector state $|1 \rangle$ may occur by: (a) scattering process characterized by the relaxation time $\tau _{21}$, (b) coherent tunneling through the extraction barrier characterized by the matrix element $\Omega _{24}$. In both models, the injection from the injector state of the previous cascade $|1' \rangle$ is by coherent tunneling through injection barrier characterized by the matrix element $\Omega _{13}$. In the active region, the transitions occur through the non-radiative process characterized by the relaxation time $\tau _{32}$ and the radiative process characterized by the stimulated-emission time $\tau _{st}$. More details on these models can be found e.g in [14].
When a wide tuning range is not a desired feature of the laser (broadband QCLs), then the emitted power and/or quantum efficiency becomes a quality factor. Its large value can be achieved for devices with narrow gain spectrum. In such a case, the population inversion results in a large value of gain peak at the expense of spectral width. For these purposes, structures with a wide extraction barrier are preferred. Wide barrier prevents the broadening of the lower laser state caused by its mixing with extractor states. In such designs, the extraction of carriers from the active region is described by the coherent tunneling through the extraction barrier (see Fig. 1.(b)) rather than by the scattering processes characterized by the relaxation time, $\tau _{21}$. For a system in Fig. 1.(b), the Hamiltonian $H$ and the dissipation matrix $S$ in the Lindblad equation
take the formsWhile the detailed results of Eqs. (7) and (8) are strongly influenced by the simplicity of the applied model, the most important conclusion of the above consideration is that the width of extraction barrier affects (through the matrix element $\Omega _{24}$) the laser performance not only through the value $g(0)$, but also through the value of gain recovery time. Both these parameters should be optimized with respect to the chosen quality factor using an approach that correctly handles not only scattering but also the quantum tunneling. An example of such optimization applied, however, only to the injection barrier was presented in [3].
3. Structure optimization
The models in Fig. 1 consider only the homogeneous broadening of the gain spectrum. In real structures, however, there are numerous phenomena, like, e.g., the non-parabolicity, which lead to the inhomogeneous broadening of the spectrum. In this case, an additional compression of the gain is observed [17]. Then, Eq. (3) may no longer be valid. However, the optimization can still be performed using first equality in Eq. (1) and the value of the clamped flux, $\Phi _{cl}$. For example, if the optimized quantity is the (external) quantum efficiency of a single cascade, then for a Fabry-Perot waveguide, the quality factor can be estimated as
It is the ratio between the photon current generated in one stage of the cascade and the particle current of the flowing electrons. Both unknown quantities, i.e., current density $J$ and $\Phi _{cl}$ should be estimated from a model that (i) best describes the designed structure and (ii) its physics. Of importance is (iii) their quantum-mechanical description, since - as shown - the tunneling process cannot be simply replaced by an equivalent scattering process. Moreover, (iv) the interaction with the e-m field should enable the estimation of the gain spectrum, not only its energy-averaged value. It is necessary because the clamped flux is reached, when the gain peak is clamped to $g_{th}$. Semi-classical models can fulfill these requirements only in part. An example is the analysis in [9], which accounts for gain saturation and compression, but not for the quantum coherence.Condition (iii) restricts acceptable approaches to quantum models based on density matrices or NEGFs [13,18,19]. They equally account for tunneling and scattering. The NEGF method is more complex and time-consuming, but does not require any arbitrary assumed values of pure dephasing times. The quantities affected by these times, such as the width and peak of the gain spectrum, are therefore realistically estimated. For the condition (iv), the light-matter interaction must be taken into account. In the NEGF method, applied to QCLs, this interaction was first implemented through the decomposition of the Green’s functions and the selfenergies into higher harmonics of the lasing frequency [4,5,20]. Such an approach generates huge numerical load and is less efficient than the approach presented in this submission, which implements light-matter interaction through electron-photon selfenergies. While ’selfenergy’ approach has already been used for interband devices [21–23], it is new for devices relying on intersubband transitions [24]
The rest of the section illustrates the application of the NEGF-based model to optimize the gain region of the QCL emitting $\cong 5.2 \, \mathrm {\mu }$m wavelength radiation. Such devices are used, e.g., in nitrogen dioxide $(\mathrm {NO}_{2})$ detection systems [25]. The structure selected for the optimization is that of Evans et al. [26], in which a two-phonon resonance emission scheme [27] was adapted. The gain region in this laser is separated from the injectors by the injection and extraction barriers, with the widths $w_{in}, w_{ex}$, which are to be optimized. The quantity defined in Eq. (10) is used as a quality factor.
The layers of the whole structure have the following widths (in nm) and order: 3.02 1.44 2.71 1.52 2.41 1.60 2.30 1.76 2.19 1.76 1.95 2.03 1.95 ${\boldsymbol{w}}_{\boldsymbol{in}}$ 1.36 1.14 4.75 1.14 4.18 1.23 4.02 ${\boldsymbol{w}}_{\boldsymbol{ex}}$. Barriers are marked in bold. The first fourteen layers form the injector and the next eight layers form the gain region. Doped layers (in the injector) are underlined. In the original design, shown in Fig. 2, $w_{in}=2.8$ nm and $w_{ex}=2.02$ nm [26].
Fig. 2. Conduction band edge of the structure subjected to optimization. Labels ’in’ and ’ex’ refer to injection and extraction barriers, respectively. Laser levels are marked with thick lines. States responsible for depopulation of the lower level and pumping the upper level are also shown.
In the first step, the optimization of the width $w_{in}$ of the injection barrier was carried out for six structures, which differ only in this width. The values of $w_{in}$ are collected in Table 1. The values of remaining parameters of the model were kept identical in all six structures. They are gathered in Table 2. The value of the threshold gain, $g_{th}=9.5 \; \mathrm {cm}^{-1}$, corresponds to the overall losses: $\alpha _{m}=2 \; \mathrm {cm}^{-1}, \alpha _{w}=3.8 \; \mathrm {cm}^{-1}$ in a typical HR coated waveguide, with InGaAs core and the confinement factor, $\Gamma =0.6$ [28].
Table 1. Structures With Various Injection Barrier Widths.
Table 2. Parameters of the QCL Model.
The calculations were performed with the NEGF-based simulator described in [29]. It uses a position basis with the non-uniform discretization grid applied to single-band effective-mass Hamiltonian in the conduction band [6,30]
Fig. 3. (a) Current-voltage (left axis) and light-current (right axis) characteristics, (b) gain peak-current characteristics calculated using the NEGF method for structures #0-#5 without $(\bullet )$ or with $(\times )$ the e-m field included in the calculations through the electron-photon selfenergies. Dashed lines depict theoretical estimation of the threshold current for the device analyzed in [26].
The results obtained for the original structure, #0, need to be compared with the experiments reported in [26]. However, direct comparison would be incorrect because of the differences in the threshold gain. The simulations were performed for the waveguide defined in [28], for which $g_{th}=9.5 \, \mathrm {cm}^{-1}$. For the waveguide defined in [26] $g_{th}$ drops to $7.1 \, \mathrm {cm}^{-1}$. This value was estimated with the use of the Drude model and material parameters evaluated from mathematical models provided in [35]. In our implementation, the confinement factor was not evaluated. Instead, the trial value of the gain in active region (AR) was assumed. It was then tuned until the waveguide losses $\alpha _{w}$ fully compensate the mirror losses $\alpha _{m}=2.18 \, \mathrm {cm}^{-1}$ (3 mm long cavity with HR-coated facet). Results of these calculations are shown in Fig. 4.(a).
Fig. 4. (a) Refractive index (black/blue dashes) and $\mathrm {TM}_{0}$ mode intensity (red) calculated for the waveguide of [26]. Layers starting from the left are as follows: InP $n$-doped to $2 \times 10^{19}$ (1 $\mathrm {\mu }$m), InP $n$-doped to $2 \times 10^{17}$ (2.5 $\mathrm {\mu }$m), InGaAs $n$-doped to $8 \times 10^{16}$ (0.1 $\mathrm {\mu }$m), AR with the gain $g=g_{th}=7.1 \, \mathrm {cm}^{-1}$, InP $n$-doped to $2 \times 10^{17}$ - substrate. (b) Gain versus flux of photons of energy $\hbar \nu _{max}=235$ meV for the structure biased with the voltage $U=352$ mV/period, for which the flux gets maximum. Symbols are the simulation points. Line is the plot of Eq. (3). Dotted line depicts the value of the threshold gain $g_{th}=9.5 \, \mathrm {cm}^{-1}$ used in the optimization process. Solid lines depict the value of threshold gain $(7.1 \, \mathrm {cm}^{-1})$ and respective clamped flux $(3.6 \times 10^{12} \mathrm {photons/nm}^{2}/\mathrm {s})$ for the device from [26]. (c) Unsaturated gain spectrum calculated for the bias $U=352$ mV/period.
The value of the threshold current can be deduced from the gain vs. current characteristic in Fig. 3.(b): for $g_{th}=7.1 \, \mathrm {cm}^{-1}, J_{th} \cong 1.18 \, \mathrm {kA}$. In the model, the self-heating of the structures is not included, so the calculated characteristics should be compared to the characteristics measured in a pulsed mode. At 288 K the experimental threshold current $J_{th}=1.1 \, \mathrm {kA/cm}^{2}$ is very close to the value from the simulations. The L-I-V characteristics were measured only in cw mode, so neither the threshold voltage nor the peak current can be compared with the simulated values. The maximum average optical power of $\cong 85$ mW was measured for the lowest, 5%, duty cycle. Theoretical estimate can be obtained using the first part of Eq. (1) and the value of photon flux $\Phi _{cl}=3.6 \times 10^{12} \, \mathrm {photon/nm}^{2}$s, which clamps the gain to $g_{th}=7.1 \, \mathrm {cm}^{-1}$. This value was inferred from Fig. 4.(b), where the clamping process is illustrated. The corresponding optical power equals $P=1.66$ W. For 5% duty cycle, the simulated average power $\cong 83$ mW excellently agrees with the experimental observation. Eventually in Fig. 4.(c), the unsaturated gain spectrum is shown. The gain peaks at $\hbar \nu _{max}=0.235$ eV ($\lambda =5.28 \, \mathrm {\mu }$m), which well correspond to experimental lasing wavelength $\lambda _{lase}=5.25 \, \mathrm {\mu }$m.
In summary, the achieved capability of simulated and experimental characteristics validates the model, the choice of its parameters and the method used in the simulations. Then, their use as an optimization tool would provide referential results. They are presented in the following.
The maximum values of $\Phi _{cl}$ and the corresponding current densities $J$, found for structures #0-#5, were used to calculate the values of the quality factor. Results are gathered in Table 1 and also are plotted as a function of the barrier thickness in Fig. 5. The optimal thickness $w_{in}=2.4$ nm of the injection barrier is obtained for structure $\#3$. This value was used in the second stage of the optimization, in which the optimal width of the extraction barrier was sought out. The calculations utilized the same optimization procedure. The examined structures labeled #3__x and the respective results of calculations are summarized in Table 3.
Fig. 5. Quantum efficiency $QE / stage$ calculated by means of the NEGF method for structures #0 - #5 with different injection barrier thicknesses. Values are referenced to the maximum.
Table 3. Structures With Various Extraction Barrier Widths.
The maximum quantum efficiency per stage was obtained for structure #3__1 with the extraction barrier width, $w_{ex}=1.22$ nm. In Fig. 6, the characteristics of the original structure (#0) described in [26] and the best structure (#3__1) with barrier widths, $w_{in}=2.4$ nm, $w_{ex}=1.22$ nm, were compared. As can be seen, the increase in the maximum value of the clamped flux $\Phi _{cl}$, and so the maximum optical power, is approximately 83%. The improvement of $\approx 24\%$ in the $QE/stage$ is lower. One can ascertain from Fig. 6.(b) that this increase was achieved not due to the increase in the gain peak, which gets almost the same value $\approx 30 \, \mathrm {cm}^{-1}$ in both structures, but due to the significant reduction of the gain recovery time, $\tau _{rec}$. As stems from Eq. (3), this reduction results in the increase of the photon flux necessary to clamp the gain to its threshold value and accordingly the increase of electrical current that could be converted into this flux. The data in Fig. 6.(a) confirm this reasoning. The amount of electrical current available for the conversion can be estimated as the difference between the I-V characteristics measured with and without the e-m field. For the original structure, this difference is $\approx 1.4 \, \mathrm {kA / cm}^{2}$, while for the optimized structure it increases to $\approx 2.8 \, \mathrm {kA / cm}^{2}$. This increase is due to the smaller width of both barriers, so that the loss of electrons in the upper laser state, caused by the e-m field stimulated emission, can be quickly replenished by their injection from the injector states. The speed of this process is described by the gain recovery time, $\tau _{rec}$. Due to the periodicity of the QCL structure, it depends on the width of both injection and extraction barriers.
Fig. 6. Comparison of the characteristics of the original structure from [26] (black squares) and the optimized structure (red circles). Filled/empty symbols refer to the structures which interact/do not interact with e-m field. Horizontal dashed-line-bars in (a) estimate the amount of the current that can be converted into photon flux.
In QCLs emitting at $5.2 \, \mathrm {\mu }$m wavelength, the band offset is artificially increased by adding strain to the layers [36]. It needs to be carefully balanced to prevent the structure relax. This is achieved by choosing material compositions that yield a lattice constant (parallel to the layers) $a_{||}$ of the superlattice equal to the one of the substrate material [6,37,38]. For the devices analyzed in this paper, the one gets $a_{||}=0.586\underline {69}$ nm for the original design of [26] and $a_{||}=0.586\underline {89}$ nm for the optimized design. These numbers have been obtained using model solid theory [39] and material parameters taken from [39] and [40]. The respective value for the InP - the substrate material - is $a_{||}=0.586\underline {94}$ nm [40]. Comparison of these numbers show that the strain-balance condition is even better fulfilled in the optimized structure than in the original one. Thus, there is no need to adjust material composition to restore strain balance (therefore simulations for both structures were performed for the same set of parameters). Moreover, as the optimized structure better balances the strain, it presents an improvement of the original design also in this aspect.
4. Summary and conclusions
The theoretical analysis carried out with the use of advanced simulation tools based on the non-equilibrium Green’s function formalism shows that the QCL structure devoted to emit the radiation at $\sim 5.2 \, \mathrm {\mu }$m wavelength proposed by Evans et al. [26] can be further optimized. Namely, the optimal injection and extraction barrier widths are respectively: 2.4 nm instead of 2.8 nm and 1.22 nm instead of 2.02 nm. For these values, the quantum efficiency larger by 24% and the output power larger by 83% than those for the original structure can be achieved. The quality factor used in the optimization process (i.e., quantum efficiency) is known, but the method used for its evaluation is new. It uses the electron-photon selfenergy, recently adapted for intraband transitions in QCLs [24], to find the characteristics of devices interacting with the laser field. The optimal value is then achieved for the device under the "operating conditions". These conditions can be quite different from those in no-lasing state because the saturation caused by the laser field can push the structure from strong to weak coupling regime. Also important is that the method is devoid of the burden associated with the phenomenological determination of the values of certain model parameters, such as, e.g., the phase relaxation times for tunneling transitions. In the applied NEGF method, all relevant times are self-determined by the processes included in the model. This applies to both incoherent scattering and coherent tunneling processes. Therefore, the quantities dependent on these times, such as the tunneling time, the absorption line width, and the optical power, are determined solely on physical grounds.
Funding
Narodowe Centrum Nauki (UMO-2020/37/B/ST7/01830).
Acknowledgments
Critical reading of the manuscript and much valuable advice from Professor Maciej Bugajski (Łukasiewicz Research Network - Institute of Microelectronics and Photonics, Warsaw, Poland) are gratefully acknowledged.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. T. V. Dinh, A. Valavanis, L. J. M. Lever, Z. Ikonić, and R. W. Kelsall, “Extended density-matrix model applied to silicon-based terahertz quantum cascade lasers,” Phys. Rev. B 85(23), 235427 (2012). [CrossRef]
2. J. B. Khurgin and G. Sun, “Enhancement of light absorption in a quantum well by surface plasmon polariton,” Appl. Phys. Lett. 94(19), 191106 (2009). [CrossRef]
3. A. Kolek, “Light-enhanced incoherence of electronic transport in quantum cascade lasers,” Sci. Rep. 10(1), 9318 (2020). [CrossRef]
4. D. O. Winge, “Quantitative Modeling of Gain in Quantum Cascade Lasers under Operational Intensities,” Ph.D. thesis, Lund University, Faculty of Science, Department of Physics, Division of Mathematical Physics, Lund (2016).
5. D. O. Winge, M. Lindskog, and A. Wacker, “Nonlinear response of quantum cascade structures,” Appl. Phys. Lett. 101(21), 211113 (2012). [CrossRef]
6. J. Faist, Quantum Cascade Lasers (Oxford University, 2013).
7. A. Yariv, Quantum electronics (Wiley, 1989), 3rd ed.
8. C. Jirauschek, M. Riesch, and P. Tzenov, “Optoelectronic Device Simulations Based on Macroscopic Maxwell–Bloch Equations,” Adv. Theory Simul. 2(8), 1900018 (2019). [CrossRef]
9. H. D. Tholl, Q. Yang, and J. Wagner, “Modeling the Electro-Optical Performance of High Power Mid-Infrared Quantum Cascade Lasers,” Photonics 3(2), 30 (2016). [CrossRef]
10. S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B 80(24), 245316 (2009). [CrossRef]
11. G. Scalari, R. Terazzi, M. Giovannini, N. Hoyler, and J. Faist, “Population inversion by resonant tunneling in quantum wells,” Appl. Phys. Lett. 91(3), 032103 (2007). [CrossRef]
12. H. Callebaut and Q. Hu, “Importance of coherence for electron transport in terahertz quantum cascade lasers,” J. Appl. Phys. 98(10), 104505 (2005). [CrossRef]
13. C. Jirauschek and T. Kubis, “Modeling techniques for quantum cascade lasers,” Appl. Phys. Rev. 1(1), 011307 (2014). [CrossRef]
14. R. Terazzi, Transport in Quantum Cascade Lasers: Investigation of numerical transport models in quantum cascade lasers (LAP LAMBERT Academic Publishing, 2012).
15. 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]
16. H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. B. Norris, “Gain Recovery Dynamics and Photon-Driven Transport in Quantum Cascade Lasers,” Phys. Rev. Lett. 100(16), 167401 (2008). [CrossRef]
17. J. Huang and L. W. Casperson, “Gain and saturation in semiconductor lasers,” Opt. Quantum Electron. 25(6), 369–390 (1993). [CrossRef]
18. O. Jonasson, S. Mei, F. Karimi, J. Kirch, D. Botez, L. Mawst, and I. Knezevic, “Quantum Transport Simulation of High-Power 4.6-µm Quantum Cascade Lasers,” Photonics 3(2), 38 (2016). [CrossRef]
19. A. Pan, B. A. Burnett, C. O. Chui, and B. S. Williams, “Density matrix modeling of quantum cascade lasers without an artificially localized basis: A generalized scattering approach,” Phys. Rev. B 96(8), 085308 (2017). [CrossRef]
20. A. Wacker, M. Lindskog, and D. O. Winge, “Nonequilibrium Green’s Function Model for Simulation of Quantum Cascade Laser Devices Under Operating Conditions,” IEEE J. Sel. Top. Quantum Electron. 19(5), 1–11 (2013). [CrossRef]
21. S. Steiger, “Modelling nano-LEDs,” Ph.D. thesis, ETH Zurich, Zürich (2009).
22. U. Aeberhard and R. Morf, “Microscopic nonequilibrium theory of quantum well solar cells,” Phys. Rev. B 77(12), 125343 (2008). [CrossRef]
23. A. Shedbalkar and B. Witzigmann, “Non equilibrium Green’s function quantum transport for green multi-quantum well nitride light emitting diodes,” Opt. Quantum Electron. 50(2), 67 (2018). [CrossRef]
24. A. Kolek, “Implementation of light–matter interaction in NEGF simulations of QCL,” Opt. Quantum Electron. 51(6), 171 (2019). [CrossRef]
25. P. Kluczynski, S. Lundqvist, J. Westberg, and O. Axner, “Faraday rotation spectrometer with sub-second response time for detection of nitric oxide using a cw DFB quantum cascade laser at 5.33 µm,” Appl. Phys. B 103(2), 451–459 (2011). [CrossRef]
26. A. Evans, J. Nguyen, S. Slivken, J. S. Yu, S. R. Darvish, and M. Razeghi, “Quantum-cascade lasers operating in continuous-wave mode above 90 °C at λ ∼ 5.25 µm,” Appl. Phys. Lett. 88(5), 051105 (2006). [CrossRef]
27. J. Faist, D. Hofstetter, M. Beck, T. Aellen, M. Rochat, and S. Blaser, “Bound-to-continuum and two-phonon resonance, quantum-cascade lasers for high duty cycle, high-temperature operation,” IEEE J. Quantum Electron. 38(6), 533–546 (2002). [CrossRef]
28. L. Diehl, D. Bour, S. Corzine, J. Zhu, G. Höfler, M. Loncar, M. Troccoli, and F. Capasso, “High-temperature continuous wave operation of strain-balanced quantum cascade lasers grown by metal organic vapor-phase epitaxy,” Appl. Phys. Lett. 89(8), 081101 (2006). [CrossRef]
29. G. Hałdaś, “Implementation of non-uniform mesh in non-equilibrium Green’s function simulations of quantum cascade lasers,” J. Comput. Electron. 18(4), 1400–1406 (2019). [CrossRef]
30. T. Kubis, C. Yeh, P. Vogl, A. Benz, G. Fasching, and C. Deutsch, “Theory of nonequilibrium quantum transport and energy dissipation in terahertz quantum cascade lasers,” Phys. Rev. B 79(19), 195323 (2009). [CrossRef]
31. C. Sirtori, F. Capasso, J. Faist, and S. Scandolo, “Nonparabolicity and a sum rule associated with bound-to-bound and bound-to-continuum intersubband transitions in quantum wells,” Phys. Rev. B 50(12), 8663–8674 (1994). [CrossRef]
32. T. Kato and S. Souma, “Study of an application of non-parabolic complex band structures to the design for mid-infrared quantum cascade lasers,” J. Appl. Phys. 125(7), 073101 (2019). [CrossRef]
33. R. Lake, G. Klimeck, R. C. Bowen, and D. Jovanovic, “Single and multiband modeling of quantum electron transport through layered semiconductor devices,” J. Appl. Phys. 81(12), 7845–7869 (1997). [CrossRef]
34. G. Hałdaś, A. Kolek, and I. Tralle, “Modeling of Mid-Infrared Quantum Cascade Laser by Means of Nonequilibrium Green’s Functions,” IEEE J. Quantum Electron. 47(6), 878–885 (2011). [CrossRef]
35. M. Sotoodeh, A. H. Khalid, and A. A. Rezazadeh, “Empirical low-field mobility model for III–V compounds applicable in device simulation codes,” J. Appl. Phys. 87(6), 2890–2900 (2000). [CrossRef]
36. J. Faist, F. Capasso, D. L. Sivco, A. L. Hutchinson, S.-N. G. Chu, and A. Y. Cho, “Short wavelength (λ ∼ 3.4 µm) quantum cascade laser based on strained compensated InGaAs/AlInAs,” Appl. Phys. Lett. 72(6), 680–682 (1998). [CrossRef]
37. T. Gresch, “Gain and waveguide engineering in mid-infrared quantum cascade lasers,” Ph.D. thesis, ETH Zurich, Zürich (2009). Diss., Eidgenössische Technische Hochschule ETH Zürich, Nr. 18732, 2009.
38. A. Evans, J. S. Yu, S. Slivken, and M. Razeghi, “Continuous-wave operation of λ ∼ 4.8 µm quantum-cascade lasers at room temperature,” Appl. Phys. Lett. 85(12), 2166–2168 (2004). [CrossRef]
39. C. G. Van de Walle, “Band lineups and deformation potentials in the model-solid theory,” Phys. Rev. B 39(3), 1871–1883 (1989). [CrossRef]
40. I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for III–V compound semiconductors and their alloys,” J. Appl. Phys. 89(11), 5815–5875 (2001). [CrossRef]
