1. Introduction

A detailed understanding of carrier dynamics is crucial for the design and improvement of modern semiconductor nanodevices such as quantum cascade lasers (QCLs). However, neither a classical or semiclassical nor a strictly coherent, ballistic quantum mechanical theory can capture the tight interplay between incoherent relaxation processes and quantum interference effects. A general and rigorous framework to capture all of these effects is the nonequilibrium Green’s function (NEGF) theory. Today, it is well established that this theory is among the most general schemes for the prediction of quantum transport properties [1–3]. Unfortunately, the basic NEGF equations are complex, mathematically demanding, and a quantitative implementation is still a highly challenging task, even with the recent advances of modern computer hardware. Furthermore, it is very difficult to develop approximations within the NEGF formalism which maintain charge and current conservation and obey the Pauli principle [4].

In this work, we present a novel method to calculate stationary quantum transport properties that we termed multi-scattering Büttiker probe (MSB) model [5]. It generalizes the so-called Büttiker probe model [1, 6, 7] and accounts for individual scattering mechanisms. If the semiconductor device equations are discretized on a spatial grid, then the standard Büttiker probe scattering model associates a single momentum and energy sink with each device grid point. Unfortunately, it does not capture effects of different scattering mechanisms. In contrast, our new MSB model accounts for individual scattering mechanisms by considering multiple scattering potentials at each grid position [Fig. 1]. In this article we first outline the concepts of semiconductor heterostructures most readers are familiar with. Then we briefly summarize the NEGF scheme and the original Büttiker probe model before we explain the details of our novel method. Finally, we apply our MSB model to a new quantum cascade laser structure and present results which is the main motivation for this work.

2. Method

2.1. The one-particle Hamiltonian

A quantum cascade laser is a heterostructure consisting of different layers that are laterally homogeneous. Therefore we can separate the growth coordinate (z axis) and the homogeneous layers (xy plane). We describe the electronic structure by a single-band effective mass Hamiltonian

2.2. The Poisson equation

The electrostatic potential ϕ is obtained by solving the nonlinear Poisson equation

The right hand side of this equation is the charge density which consists of the electron concentration n and the ionized donor concentration $N_{\mathrm{D}}^{+}$. Here, e is the positive elementary charge, ε₀ is the electric vacuum permittivity and ε_s is the static relative permittivity. The Poisson equation includes electron–electron scattering within the Hartree approximation. We solve this equation by a Newton–Raphson method. In order to solve it very efficiently, we apply a predictor–corrector approach to ‘predict’ the quantum density n(z) upon small changes dϕ(z) of the electrostatic potential [8]. We thus avoid the computationally expensive calculation of the Green’s functions [Eq. (5)–(8)] within each Poisson iteration cycle.

2.3. The NEGF scheme

The quantum transport problem of a QCL can be regarded as a carrier scattering problem between reservoirs. Hence, the system of interest must be treated as an open quantum system. A common approach is to divide the quantum system into a device region and two lead regions (source and drain). The leads are assumed to be in equilibrium and consequently, all relevant scattering processes occur inside the device. The effect of the coupling of the device to the source and drain leads is described via a so-called contact self-energy ${\mathbf{\Sigma }}_{\mathrm{C}}^{\mathrm{R}}$, which incorporates transitions between the device and the leads. A full implementation of the NEGF method requires the self-consistent solution of the retarded and lesser Green’s functions, G^R and G^<. The first one characterizes the width and energy of the scattering states and is related to the local density of states. The second quantity characterizes the state occupancy and determines the charge and current densities, and the optical gain. Under stationary conditions, the retarded and lesser Green’s functions are sufficient to completely describe a nonequilibrium system.

The self-consistent cycle involves the solution of four coupled integro-differential equations. In operator form, they read for a given energy E,

The term ‘self-consistent’ means that the Green’s functions determine the self-energies and at the same time the self-energies determine the Green’s functions. The self-energies [Eq. (6), Eq. (7)] are calculated in the self-consistent Born approximation. H is the electronic device Hamiltonian [Eq. (1)] that includes the electrostatic potential ϕ. Consequently, the Poisson equation [Eq. (4)] has to be solved self-consistently with the Green’s functions [Eq. (5)–(8)], which introduces another self-consistent cycle. Σ^R and Σ^< are the retarded and lesser total scattering self-energies. These self-energies are built-up from the contact self-energy and the sum of all environmental Green’s functions D^R and D^< that contain the scattering vertices.

For a multi-quantum well structure such as a QCL grown along the z axis, the lateral momentum conservation simplifies the arguments to the following functional form, G = G (z, z′, k_║, E). All quantities are therefore functions of space, momentum and energy. The key simplifications in our new MSB approach consist in (i) sidestepping the self-consistent solution of Σ^< and G^< and (ii) performing the k_║ integration analytically so that the Green’s functions become functions of space and energy only, G = G(z, z′, E_z). The total energy E is the sum of the energy E_z along the direction of propagation and energy E_║ within a layer

As we will later show, we reduce the mathematical complexity within the MSB model, so that the self-consistent cycle only involves two coupled equations, namely G^R [Eq. (5)] and ${\mathbf{\Sigma }}^{\mathrm{R}}={\mathbf{\Sigma }}_{\mathrm{C}}^{\mathrm{R}}+{\mathbf{G}}^{\mathrm{R}}{\mathrm{D}}^{\mathrm{R}}$ [Eq. (6)]. The iterative solution of only two coupled equations is far more robust and faster than the iteration over four coupled equations within the full NEGF formalism. Moreover, current conservation and the Pauli principle are inherently guaranteed during every step of the iteration within the MSB model, since we explicitly calculate the virtual chemical potentials μ of the Büttiker probes as we will show further below. Therefore, even in a nonself-consistent Born approximation, current conservation is not violated.

The retarded self-energy

The wave vector k(E_z) is calculated for each energy E_z of the discretized energy grid from the lead dispersion which is given for a discrete lattice by

 

Fig. 1 Left: The standard Büttiker probe model associates a single momentum and energy sink with each device grid point i in position space using a phenomenological scattering parameter η, which is fixed during the calculation. Right: The proposed multi-scattering Büttiker probe model accounts for individual scattering mechanisms such as longitudinal acoustic (LA) or longitudinal optical (LO) phonon scattering by including multiple scattering potentials for each node that are calculated self-consistently.

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2.4. The multi-scattering Büttiker probe (MSB) model

One of the difficulties of the NEGF formalism is the fact that current conservation and the Pauli principle are not automatically guaranteed. Particularly, the calculation of the self-energies in any nonself-consistent approximation violates current conservation. To remedy such a result, the iteration over the four coupled Green’s functions and self-energies has to be carried out to infinite order [4]. In contrast, the main advantage of the Büttiker probe model is that it bypasses any self-consistent calculation between the Green’s functions and their corresponding self-energies. Moreover, it inherently guarantees current conservation, and the Pauli principle is always obeyed since a quasi-equilibrium expression for G^< is assumed and thus, the fluctuation–dissipation theorem holds. However, the Büttiker probe model introduces a phenomenological scattering parameter η,

In general, the derivation of the scattering kernels B_LA and B_LO for the MSB model follows the derivation of the scattering self-energies within the full NEGF scheme. In this scheme B_LA depends on the retarded Green’s function G^R, and B_LO on both the retarded and the lesser Green’s functions, G^R and G^<. The MSB model, however, makes the following two approximations. First, we neglect the contribution of G^< on B_LO which essentially decouples the retarded and the lesser Green’s functions making the overall calculation faster and more robust. Secondly, we take only the diagonal part of G^R into account by assuming that the scattering kernels only depend on the local density of states. The local density of states ρ(z,E_z) is the diagonal part (z = z′) of the spectral function A divided by 2π

The spectral function

The scattering potential B(z,E_z) is a functional of the local density of states ρ(z,E_z), i.e. it has to be calculated self-consistently. The local density of states can be directly calculated from the retarded Green’s function G^R [Eq. (19)–(20)]. Consequently, Eq. (21) and Eq. (22) constitute a self-consistent relation between Σ^R and G^R that, in principle, has to be solved in an iterative scheme. Fortunately, the density of states ρ(z,E_z) and thus also the self-energy Σ^R depend on the diagonal part of G^R only. Therefore, only the diagonal part of G^R is required during the iteration which can be calculated very efficiently, e.g. with the FIND (Fast Inverse Using Nested Dissection) algorithm [9]. In order to decouple the lesser and the retarded Green’s functions, we neglected in Eq. (22) the contribution of the charge carrier density to enhance the convergence while the Pauli principle is still inherently guaranteed since the probes are assumed to be in a local equilibrium. This approximation essentially corresponds to dropping the term G^<D^< in Eq. (6). In principle, the scattering self-energy for LO phonon coupling depends on the momentum k_║. However, since we assume a Fröhlich coupling, which favors small momentum transfers, we can neglect the k_║ dependence of the self-energies. Thus, they depend solely on the energy E_z along the growth axis [Eq. (9)]. We can therefore analytically perform integrations with respect to k_║ [5] that appear in the calculation of observables like density, current and optical gain.

For the numbering of the spatial grid points we assume that i = 1 and i = N refer to the source and drain grid points, respectively. At all grid points i = 1,…,N we locate exactly one Büttiker probe p which has the scattering strength B(z,E_z) at this position z_p. We can map the Büttiker scattering potential onto the retarded Büttiker probe self-energy of each probe as follows

The δ function indicates that the self-energy ${\mathbf{\Sigma }}_p^{\mathrm{R}}$ for probe p is nonzero only at the position coordinate z_p where the probe is located. All other matrix elements apart from this one are zero. The retarded Büttiker probe self-energy ${\mathbf{\Sigma }}_{\mathrm{B}}^{\mathrm{R}}$ is the sum of the self-energies of each probe p,

It has nonzero matrix elements only on the diagonal (z = z′). This self-energy corresponds to the term G^RD^R in Eq. (6).

For a system in equilibrium with chemical potential μ, the lesser Green’s function G^< is calculated using

This equation is known as the fluctuation–dissipation theorem. Hereby, the spectral function is filled up according to the Fermi distribution function. Note that we use F₀ which is the appropriate zero-order Fermi–Dirac integral for homogeneous semiconductor heterostructures defined in Eq. (33). From the diagonal part (z = z′) of G^<, we can directly extract the energy resolved electron density

A device is driven out of equilibrium if the source and drain contacts have different chemical potentials, μ_S ≠ μ_D. In the special case of a ballistic situation where scattering is absent, G^< can be calculated also in a nonequilibrium situation. However, we have to separate the total spectral function A into the spectral functions due to the source and drain contacts, A = A_S + A_D. We then fill up each spectral function by the corresponding Fermi distribution function of the source or drain contact, respectively,

Within the MSB model, the total spectral function A is altered by adding a further term A_B. This is the spectral function due to the Büttiker probe contacts. In fact, A_B is the sum of the spectral functions of each individual Büttiker probe,

The spectral function of each probe A_p (z,z′,E_z) is given by

It is determined by its self-energy ${\mathbf{\Sigma }}_p^{\mathrm{R}}$ through the broadening function Γ_p (z,z′,E_z),

In our implementation the broadening function Γ_p of each probe is only nonzero at the position coordinate of the probe (z_p = z = z′). By replacing the index p with S or D in the last two equations, the corresponding quantities for the source and drain are obtained. The spectral function A_p of each probe is in general nonzero everywhere, i.e. each probe contributes to all other grid points and their correlations (z,z′) a specific spectral function, similar as the source and drain contacts. Therefore, we need a (local) Fermi distribution for each probe, in addition to the Fermi distributions of the source and drain contacts.

Now we introduce the approximation to calculate G^< in nonequilibrium using the expression for an equilibrium system. By doing so, we use a local Fermi distribution F₀(z,E_z) ≡ F_0,p(E_z, μ_p) for the occupation of the spectral function A_p of each probe p, which therefore depends on the spatial coordinate z, i.e. on a local virtual chemical potential μ(z). Now G^< is given by

The physical motivation for this is straightforward. Scattering of carriers leads to a redistribution of their energy and momentum while charge and current conservation laws must remain valid. This leads to a constraint for determining the local Fermi level μ(z) which means that the virtual chemical potentials μ are not determined a priori. They can therefore be calculated in such a way that current conservation is fulfilled. Hence, scattering events can be viewed as absorption of charge carriers into the probes and a reinjection of equilibrated charge carriers back into the device because the virtual probe contacts are assumed to be in a thermodynamic equilibrium, analogously to the real contacts. This is physically motivated by the fact that scattering turns a distorted system back to equilibrium. However, in our implementation of two-contact devices we do not determine μ(z). Instead, current conservation can also be fulfilled by solving a linear system of equations for the coefficients c(z) ∈ [0,1] that are defined as follows

The local distribution function F(z,E_z) of the probe at position z is assumed to be a linear combination of the Fermi distribution functions F_S and F_D of the source and drain contacts. If c = 0, the probe is filled up completely by the drain contact, and if c = 1, the probe is filled up completely by the source contact. For an equilibrium simulation, it obviously must hold c = 0.5. The assumption of a linear combination of the source and drain distributions for a probe at position z is reasonable because the carriers inside a two-contact device originate either from the source or drain and also leave through the source or drain. Since both leads are assumed to be in equilibrium, scattering mechanisms inside the device result in a mixture of the equilibrium distributions. Consequently, we do not have to calculate the virtual chemical potentials μ(z) that are typically used in the standard Büttiker probe model. Instead, it is sufficient to calculate the coefficients c(z) which define a local distribution function. This has numerical advantages and makes the overall iteration cycle more stable. In quantum cascade lasers electron–electron scattering is responsible for efficient energy redistribution and effective thermalization. This has been predicted theoretically [10] and observed experimentally in mid-infrared quantum cascade structures [11], resonant phonon THz QCLs [12] or bound-to-continuum THz QCLs [13–15]. Thus we believe that using a local Fermi–Dirac distribution is reasonable. Unfortunately, our simplified model does not give insight on individual subband temperatures. Therefore such an influence on the thermal performance of quantum cascade lasers cannot be answered.

Although the novel MSB method is a rather rigorous approximation that is orders of magnitude faster in terms of computational work compared to the full NEGF model, it reproduces the results of the latter and also experimental data very nicely as demonstrated in [5] for a few QCL structures that we investigated. Concretely, we tested the MSB method against experiments [16] of an AlGaAs/GaAs THz QCL (three quantum well resonant phonon design), and against experiments [17, 18] and NEGF simulations [19] of another AlGaAs/GaAs THz QCL (four quantum well resonant phonon design). Furthermore, we compared our MSB model with NEGF simulations [20] of an InGaAs/GaAsSb THz QCL proposal with four quantum wells and a diagonal optical transition. Here, the MSB method reproduces the major observable quantities very well, although rough interfaces and alloy disorder are neglected. Finally, we benchmarked our calculations with respect to experimental data of a strain-balanced short-injector AlInAs/InGaAs mid-infrared QCL (diagonal transition design) [21] and an injectorless AlInAs/InGaAs mid-infrared QCL [22]. The peculiarity of the latter design is that there are in fact two different lasing frequencies depending on the applied bias. This feature was excellently reproduced by our calculations. In all these cases, the agreement was very encouraging. However, we were not able to simulate experimental results of the AlInAs/InGaAs mid-infrared QCL design of [23]. It turned out that the reason for this disagreement was the neglect of the nonparabolicity of the effective mass within our MSB model. This significantly changed the alignment of the higher-lying electronic states and thus the operation principle could not be reconstructed at all. This was proved by a full NEGF calculation using the software presented in [3] which includes nonparabolic effects and thus reproduced the experimentally observed transition correctly. On the other hand, assuming a parabolic mass in this full NEGF model leads to the same results as our MSB model for this structure. Obviously, the simplifications within the MSB model retain the main physical features very well for both THz and mid-infrared QCLs although special care has to be taken in terms of designs where nonparabolicity effects play an important role. In these cases we recommend complementary simulations using other models that take this into account. The MSB algorithm has been implemented into the nextnano.MSB software (http://www.nextnano.com) and allows finding an optimal QCL layout very quickly, as variations in alloy concentration, barrier thickness, and further parameters can be calculated in parallel.

3. Results

The emission wavelength of quantum cascade lasers ranges from the mid-infrared to the farinfrared or THz domain. QCLs are designed to enable transitions from an upper energy state to a lower energy state within the conduction band. The emitted photons have exactly the energy difference of the aforementioned states. This enables the engineering of the exact energy of the emitted photons. This photon energy is determined by the alignment of the states within the QCL and it can be tailored over a wide range through the variation of the well and barrier thicknesses, barrier heights, doping concentrations, as well as the employed materials. To optimize and design quantum cascade laser devices an efficient yet accurate calculation of dissipative transport properties is essential. Supported by recent calculations with our new MSB method, we propose a novel two-well injectorless THz quantum cascade laser design with a photon emission frequency of 2.85 THz which corresponds to a photon energy of E_γ = 11.8 meV. The design is based on the following principles. First, the device is based on the GaAs/AlGaAs material system. Although, devices based on InAs or AlSb intrinsically achieve better gain performances due to their lower effective masses [24], GaAs-based technology offers several advantages, e.g. the GaAs material system is well understood and mature processing technologies are available [25,26]. Secondly, we use an injectorless design which has the advantage of high gain [27] since the device is shorter and the intermediate injector–collector states are neither needed to fill the upper laser states nor to empty the lower laser states. Thus, in an injectorless two-well QCL the active region extends over almost the whole period and consequently the ratio of gain to nongain regions is much higher [28]. Thirdly, the design is based on narrow wells and barriers. This pushes the bound states to higher energies and therefore, so-called parasitic leakage currents from one period to the next are suppressed. Additionally, we do not impose the restriction of uniform barrier heights, i.e. we use different aluminum contents for the barriers. This additional degree of freedom for the QCL design can be used to improve temperature performance [29] and to find a design that is robust against fluctuations of material compositions. Finally, the design is based on the ‘complete thermalization’ of carriers [20]. Typical THz QCLs are designed such that the potential drop per period ϕ_th at threshold bias equals the sum of the emitted photon energy E_γ and the energy E_LO of one longitudinal optical (LO) phonon, eϕ_th = E_γ + E_LO. As most electrons pass the period without emitting a photon, electrons might get heated up unless thermalized by electron–electron scattering. Hot electrons can be avoided by designing the threshold bias to be an integer multiple of the LO phonon energy, for instance eϕ_th = 2E_LO [20, 30], thereby maximizing the carrier thermalization within each period. Our design uses eϕ_th = E_LO and hence, for the energy balance of one period, the photon energy is negligible since only a minority of carriers emits a photon. Consequently, the majority of carriers, i.e. those that are not emitting light, are efficiently thermalized and nonperiodic effects are remedied [19,31].

Our concrete suggestion of a novel THz quantum cascade laser consists of GaAs wells and alternating Al_0.1Ga_0.9As and Al_0.27Ga_0.73As barriers. The barrier heights used in the calculation are 107 meV and 237 meV, respectively. The layer sequence for one period is 2.1/7.35/2.1/7.35 nm (Al_0.1Ga_0.9As/GaAs/Al_0.27Ga_0.73As/GaAs). The AlGaAs barriers are shown in bold and the GaAs wells in normal font. 7.35 nm correspond to 26 molecular layers of GaAs, and 2.1 nm to 7 molecular layers of Al_xGa_1−xAs, respectively. Only the underlined GaAs layer of each period is doped with a concentration of 3 × 10¹⁶ cm⁻³ where we assume the donors to be fully ionized. The widths of the layers as well as the heights of the barriers, which are defined by the alloy compositions, were optimized with a genetic algorithm based on the MSB method. Concretely, in terms of barrier widths w_b and well widths w_w, we were searching for structures that fit on a grid space resolution of a = 0.15 nm using the constraint w_b = na and w_w = ma with n and m being integers. Since the MSB method is very efficient in terms of computational memory and time, various alloy concentrations, barrier thicknesses, and further parameters can be calculated in parallel and an optimal QCL layout can be found quickly.

The operation scheme of this QCL is shown in Fig. 2 and Fig. 3(a). In principle, this QCL design consists of alternating upper and lower laser states only. The upper laser states are labeled ‘1a’ and ‘2a’ in Fig. 2 and are located in the wells labeled ‘A’ in Fig. 3(a). From this upper laser level a diagonal photon transition into the states ‘1b’ and ‘2b’, respectively, is possible. The laser transition between these states is diagonal and thus suppresses nonradiative transitions. Rather than emitting a photon, however, the majority of carriers scatters efficiently from state ‘1a’ into the upper laser state ‘2a’ of the next period because this energy difference is designed to be the energy of a longitudinal optical phonon. For the GaAs material system, the energy of a LO phonon is assumed to be E_LO = 34 meV and thus, the threshold voltage of our novel QCL design is expected to be slightly below 34 mV where the states are aligned correctly for the LO phonon cascade. The threshold voltage is the applied bias voltage for which the optical gain compensates all optical losses, and thus the QCL starts to emit photons.

 

Fig. 2 (a) Calculated conduction band profile (white line) and contour plot of the energy and position resolved spectral function A(z = z′,E_z) at vanishing in-plane momentum k^║ = 0 for the suggested QCL at the threshold bias voltage of 36 mV per period and a lattice temperature of 100 K. (b) Corresponding contour plot of the energy and position resolved current density.

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Fig. 3 (a) Operation scheme of the novel QCL design. The red arrows indicate LO phonon emissions and the blue arrows indicate photon emissions. The two wells are labeled with ‘A’ and ‘B’ and only well ‘A’ is doped. (b) Calculated contour plot of the position resolved optical intensity gain as a function of the photon energy. The black solid contour line encloses the area of positive optical gain. The white solid line indicates the conduction band profile and is only meant to guide the eye. It is not related to the photon energy axis. The applied bias voltage is 36 mV and the temperature is 100 K.

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The effects of the narrow wells and high barriers are also shown in Fig. 2. The upper states labeled ‘1u’ and ‘2u’ are pushed more than 100 meV away from the conduction band edge. Thus they do not interact and interfere with other states of previous periods. Consequently, they do not carry any significant current, as shown in Fig. 2(b). The simulated structure extends from 0 nm to 105.6 nm. The left lead is positioned at 0 nm, followed by a 4.5 nm GaAs layer. Then we include 5 periods to minimize boundary effects, followed by a 2.1 nm Al_0.1Ga_0.9As barrier and an additional GaAs layer of 4.5 nm in front of the right lead. The labels ‘1a’ and ‘2a’ of Fig. 2 are located in the third period, the labels ‘2a’ and ‘2b’ in the fourth period.

We calculated the optical intensity gain as a function of the photon energy using the dipole approximation. We follow the approach of [32] where the gain is calculated for current-driven semiconductor heterostructures within the NEGF framework. The calculated position resolved optical intensity gain as a function of the photon energy is shown in Fig. 3(b). This plot gives insight into laser operation, i.e. one can easily see which parts of the structure are responsible for absorption or emission. Here, the black solid line encloses the area where the optical gain is positive. We find that almost the complete period contributes to the optical gain. Our calculated gain does not include optical losses which are typically at least of the order of 15 cm⁻¹ in other QCLs [16,21,33].

The calculated optical gain as a function of the lattice temperature is shown in Fig. 4(a). The corresponding calculated optical gain as a function of the photon energy is plotted for various temperatures in Fig. 4(b). We find significantly higher optical gain than for the four-well QCL published in [20]. Furthermore, we calculate gain even up to temperatures of 250 K which would be a new temperature record for THz quantum cascade lasers. This high temperature operation is confirmed by a complementary, independent simulation based on a different NEGF code [19]. Our previous simulations of a few other common THz QCL designs do not show such a high temperature operation, in agreement to experiment. In experiments, efficient cooling down to 230 K can be achieved easily with simple Peltier coolers in contrast to more sophisticated and more expensive cooling mechanisms which are required for lower temperatures [33,34]. The present temperature performance record of 199.5 K is achieved by a resonant phonon based three-well design [35].

 

Fig. 4 (a) Calculated optical intensity gain as a function of the lattice temperature. The solid black line was calculated for the QCL proposed in [20] whereas the solid blue line is calculated for the novel QCL design. (b) Calculated optical intensity gain as a function of the photon energy for the novel QCL design for various temperatures.

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Finally, we also calculated the DC current as a function of the applied bias voltage for our proposed QCL design. The current–voltage characteristics are plotted in Fig. 5. We calculated I–V curves for various temperatures from 100 K to 250 K. Our calculations do not show any negative differential resistance below the threshold voltage of 36 mV. The simulation indicates an electrically stable operation of our novel QCL design. Note that usually a negative differential resistance can be observed above the threshold voltage in NEGF as well as in our MSB calculations (not shown). This is caused by state misalignment and originates mainly from the fact that the coupling of the photon field is not included within the calculations. Furthermore, we found that the gain is not very sensitive with respect to layer thickness fluctuations.

 

Fig. 5 Calculated DC current as a function of the applied bias voltage for the novel QCL design for various temperatures. The current–voltage characteristics do not show any negative differential resistance and thus indicate an electrically stable design.

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4. Conclusion

We developed a novel method for the calculation of inelastic electron transport in quantum cascade lasers that takes into account individual scattering mechanisms. It is based on the nonequilibrium Green’s function formalism and treats the quantum cascade laser as an open quantum device. It offers the numerical efficiency of the simple Büttiker probe model but more importantly, it extends it by using realistic self-energies derived from the NEGF method for the probes. The assumption of local equilibrium simplifies the solution of the coupled equations. We presented a novel concrete THz QCL scheme based on a two-well design with alternating barrier heights where the threshold voltage is designed to match the energy of a LO phonon. The widths of the layers as well as the heights of the barriers were optimized with a genetic algorithm based on this newly developed multi-scattering Büttiker probe (MSB) model. Our calculations predict THz laser operation for temperatures up to 250 K.