Quantum cascade lasers, for the first time fabricated in the 90-ies (Faist et al.
1994), are one of the most promising compact sources of mid-infrared and terahertz radiation. Since then, many different structure designs have been proposed with the aim of improving performance, achievement of room temperature operation in the mid-infrared range (Page et al.
2001), or to extend the emission spectra to the range of terahertz wavelengths (Köhler et al.
2002).
Design and fabrication of QCL is, without doubt, one of the principal successes of wavefunction engineering. Typical structure of QCL consists of a number of elementary periods, each of which is built of layers of different semiconductors forming quantum wells and barriers, grown with precision up to a single atomic layer. The electric field in QCL device is applied in the direction perpendicular to the semiconductor layers interfaces. The operation of the device mainly depends on the perpendicular electron transport, which is ruled by electron tunneling between subsequent quantum wells and electron scatterings between states belonging to discrete, quantized subbands. Moreover, the radiative emission is achieved between superlattice subband states, hence the emission wavelength of this type of laser is not limited by the materials bandgap, but can be intentionally tuned in a wide range by precise design of constituting layers thicknesses. Precise matching of energy subbands positions, scattering rates, and optical dipole matrix elements are required for this task. To assist with such a challenge, various theoretical models have been developed such as rate equations, Monte Carlo simulations, density matrix models or non-equilibrium Green functions (NEGF) approach.
1.1 MC versus other methods used for QCL modeling
It may be interesting to briefly review various methods that are commonly used in theoretical studies of QCL.
The rate equation method (Donovan et al.
2001; Indjin et al.
2002a,
b,
2003,
2004; Chen et al.
2011; Saha and Kumar
2016), often applied to the description of the properties of QCL structures is based on the semiclassical electron transport model (transport dominated by scattering) described by Boltzmann equation, which in fact, is the same physical model as in the MC method. The important point is that the algorithm is easier to implement and less demanding of computational time, at the price of an additional hypothesis about the shape of the electron distribution function, which is usually taken as a Fermi–Dirac distribution, or in even simpler models by using phenomenological values of electron scattering times between subbands.
Models entirely based on density matrix approach have been successfully used by various authors (Willenberg et al.
2003; Kumar and Hu
2009; Weber et al.
2009; Dupont et al.
2010; Terazzi and Faist
2010; Lindskog et al.
2014). Their main advantage, when compared to MC simulations, is proper accounting of electron resonant tunneling through quantum barriers and dephasing processes. However, density matrix models still rely on a number of approximations, such as scattering mechanisms based on Fermi’s golden rule and thermalized subbands.
Iotti and Rossi (
2001a) employed density matrix formalism to describe electron transport in QCL and compared with results obtained by MC simulations. They have found that for the typical structures, on a subpicosecond time scale, energy-relaxation and dephasing processes are strong enough to destroy any phase-coherence effect. Consequently, the incoherent (semiclassical) description of stationary charge transport can be justified. For studies of ultrafast phenomena, as for example, time evolution of population inversion, full-quantum description should be used (Iotti and Rossi
2001a,
2003,
2005).
The NEGF method (Wacker
2002; Lee and Wacker
2002; Lee et al.
2006; Banit et al.
2005; Kubis et al.
2009; Schmielau and Pereira
2009; Hałdaś et al.
2011,
2017; Wacker et al.
2013) can provide a fully quantum mechanical treatment of electron transport, where quantum mechanical effects such as coherent tunneling and incoherent scattering effects play an important role. An advantage unique to NEGF method is the ability to capture effects such as dispersive gain and gain linewidth reductions due to correlations. Unfortunately, this approach requires very demanding computations and provides results, the interpretation of which requires additional effort.
Mátyás et al. (
2009,
2010b) performed comparative studies of semiclassical ensemble MC and fully quantum mechanical NEGF methods for stationary transport in THz structure. They have found that when all contributing to transport electron states are clearly nondegenerate, the current density and spectral gain profile obtained by the MC method quantitatively agree with NEGF. In the case of bias conditions significantly below threshold, coherent multi-barrier tunneling dominates transport and MC underestimates the current density in this regime. Since, for the purpose of structures design, optimization conditions close or above threshold are studied, the MC modeling is suitable owing to less demanding computations.
When we focus on stationary electron transport in QCL, theoretical description without accounting for the coherent carrier dynamics effects is demonstrated to be sufficient (Iotti and Rossi
2001a) for many cases. That means, that the semiclassical description in the framework of the Boltzmann Transport Equation can be used and the MC technique applied to solve such problems is capable to provide qualitatively and quantitatively reliable description. However, one should be aware (Callebaut et al.
2004) that especially in the case of THz structures, in which relatively thick barriers are used, semiclassical calculations, not taking into account the effect of dephasing, would always result in overestimation of the current density and gain values.
The great advantage of MC modeling is that it relies on well-established material parameters and structure specification and the need to include phenomenological parameters can be reduced or even completely avoided. MC simulation method is relatively intuitive and thus preparation of simulation codes is straightforward. In addition, when MC simulator is developed it can be easily adapted to study different structures. MC methods need higher computational resources than for example rate equation methods, but still they are far less demanding than the need of full quantum transport models. In addition, the results of the modeling can be easily interpreted and they give microscopic insight to elucidate the QCL operation, which often is not easy to be obtained by experiments.
1.2 Examples of MC studies of QCL
Various MC studies aimed to explain measured characteristics of experimentally prepared structures or to give microscopic description of their operation have been reported. Moreover, suggestions for the optimization of structures design or proposal of new structures have been provided. Furthermore, multiple MC studies have been conducted to estimate the importance of electron–electron (e–e) interactions, various screening models, nonequilibrium (hot) phonons, photon emission and other mechanisms.
The QCL structure prepared by Sirtori et al. (
1998) has been studied by means of MC modeling by Iotti and Rossi (
2000,
2001b) and that research allowed quantitative investigation of the importance of various mechanisms of electron relaxation.
The prototypical design for THz QCL structure proposed by Rochat et al. (
1998) was investigated by Köhler et al. (
2001), and starting from such studies, two other structures were designed. The first one is based on chirped-superlattice design concept (Tredicucci et al.
1998). Another designed structure using electron–phonon scattering as the depleting mechanism was based on double-quantum well superlattice (Wanke et al.
2001). Subsequent measurements (Köhler et al.
2002) have shown good agreement between predicted and observed results.
A 3.4 THz intersubband laser device (Williams et al.
2003a) has been investigated in several MC studies. Computed by Callebaut et al. (
2003) gain is in reasonable agreement with measurements but the current density is underestimated. The observed discrepancies were later explained (Callebaut et al.
2004) by adding impurities scattering mechanism omitted in their first studies. However, on the other hand, this scattering mechanism when included, leads to current and gain values overestimated when compared to experimental data. This discrepancy has been explained as resulting from lack in the semiclassical MC model of dephasing effects. Bonno et al. (
2005,
2006) used this structure as a testbed for the MC research of e–e scattering and multi-subband screening models. In other research (Williams et al.
2003b) reasonable agreement of maximal operating temperature has been reported: given by MC simulation
\(\sim\)85 K for surface-plasmon waveguide and
\(\sim\)160 K for metal-metal waveguide versus experimentally observed values of 92 and 137 K, respectively. In addition, calculations of the gain profile of this structure have been conducted in other research (Jirauschek and Lugli
2009; Jirauschek et al.
2009,
2010).
Jirauschek et al. (
2007a) theoretically compared three similar THz QCL structures based on longitudinal-optical phonon scattering depopulation, for which experimental comparison has been performed by Vitiello et al. (
2005). The MC simulation demonstrated that it allows to identify the parasitic processes affecting the operation and provided a meaningful explanation of the experimental findings. Calculated inversion peak is higher than experimental estimates, also calculated gain overestimates measurement by about 40%. Authors ascribe this deviation to various effects: absence of broadening mechanisms in modeling, uncertainty of free carrier density in the device, uncertainty in determination of device effective area that is used in calculations. In addition, the shifts of the simulated inversion peak and of the current peak density are ascribed to parasitic resistance in experimental setup or deviations in sample growth processes.
Li and coworkers, in a series of papers (Li et al.
2008a,
b,
c,
d,
e,
2009a,
b; Li and Cao
2011; Cao et al.
2008), extensively studied various aspects of the performance of THz structures based on resonant phonon depopulation. Agreement between experimental and theoretical results, as well as the influence of various parameters, has been demonstrated. Obtained by simulations (Li et al.
2008b) threshold bias of
\({\sim }60\, \hbox {mV/module}\) and threshold current density
\({\sim }541\, \hbox {A/cm}^2\) can be compared with measured values of 63.6 mV/module and
\(738\, \hbox {A/cm}^2\), respectively. Calculated by Li et al. (
2008c) gain peak is slightly higher than measured value, which also leads to wider range of lasing domain: calculated 10.7–13.6 kV/cm versus measured 11.5–13.9 kV/cm. In addition, a blueshift due to the Stark effect in the emission spectra with increased drive current is observed, both in experiment and in simulations.
Multiple MC studies of mid-infrared QCL structure designed by Page et al. (
2001), operating at room temperature have been performed. Gao et al. (
2006,
2007a) investigated influence of electron leakage to
X valleys and compared the performance with other structures (Sirtori et al.
1998,
1999). We would like to notice that this particular structure design has been a basis for fabrication of QCL by Prof. M. Bugajski research group at Institute of Electron Technology, Warsaw, Poland (Kosiel et al.
2009,
2011), and extensively investigated experimentally (Pierściński et al.
2010,
2012; Bugajski et al.
2013,
2014). In collaboration with this project microscopic description of electron behavior (Borowik et al.
2010) and investigation of space-charge effect (Konupek et al.
2011; Borowik et al.
2012) have been performed by MC technique.
Temperature performance and limiting factors for high temperature operation of various designs of THz QCL have been addressed in multiple reports (Jirauschek and Lugli
2008a,
b; Li et al.
2009b; Mátyás et al.
2010a; Han and Cao
2010,
2012).
One of proposals in the quest for high temperature operating THz QCL suggested application of novel semiconductor heterostructures with larger LO-phonon frequencies. Structures based on wide bandgap materials GaN/AlGaN, ZnO/MgZnO have been examined by MC simulations (Bellotti et al.
2008,
2009; Bellotti and Paiella
2010).
Fathololoumi et al. (
2012) designed a THz structure, for which lasing up to 200 K has been observed. The tunneling barrier thicknesses were determined with the help of simplified density matrix model. In this research, the experimental lasing frequencies and their dependence on applied voltage are shown to be consistent with those deduced from the location of the maxima of gain spectra, calculated by MC simulation. Moreover, the calculated value of peak current is about 10–15% smaller than the experimental one and this discrepancy is explained by uncertainty of interface roughness and possible current leakage through not included in simulations high-energy states.
Further MC studies of similar structures have been conducted (Matyas et al.
2012; Jiang et al.
2014) and modified designs with variable barriers heights have been proposed with aim to increase the maximal operation temperature. Experimental studies (Jiang et al.
2014) have not confirmed operating temperatures superior to that of the reference record. To explain discrepancies between these MC results and experimental measurements several limitations of simulation model have been indicated. First, uncertainty of certain parameters as for example interface roughness, percentage of ionized donors. In addition, effects such as electron leakage to continuum states, tunneling, and aperiodicity of biased QCL structure, which are not included in MC model.
Shi et al. (
2012) extracted the heat generation rate from the electron–optical phonon scattering events recorded during the MC simulation and coupled it to the nonlinear heat diffusion equation in a self-consistent manner. The model has been used to investigate the cross-plane temperature distribution throughout a mid-infrared device.
1.3 Goal and organization of the paper
The cited above examples of MC studies confirm that this method is very useful in explaining various physical mechanism governing operation of QCL. Based on such deep understanding of device physics, limiting factors of operation have been demonstrated. In addition, optimizations of existing structures design or new structures designs could be proposed with the aim of increasing maximal operating temperature or extending the emission spectral range.
Recently Jirauschek and Kubis (
2014) published review article focused on technical details of various modelling techniques applied to the studies of QCL. The goal of the present paper is to provide more compact review of the research field and applications of the MC technique in this domain.
The paper is organized as follow: in the next section, we briefly describe the MC technique specific to the QCL studies. In the subsequent Sects.
3 and
4, main components of this method such as procedures to find electron states and electron scattering rates due to various scattering mechanisms are described. In addition, we review the influence of these effects on the functioning of QCL. In the Sect.
5, we briefly review types of MC simulations results that are commonly reported. Subsequently, in the Sect.
6 we mention the extension of standard MC simulations, in which also photon generation is considered. Finally, in the Sect.
7 we briefly review available extensions of the MC method focused to include important aspects of these more advanced models. In the summary Sect.
8 we also give our opinion about possible future extension of MC method used for QCL modeling.