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Abstract
Intersubband polar-optical-phonon (POP) scattering plays an important role in determining the population inversion and optical gain of mid-infrared (mid-IR) quantum cascade lasers (QCLs). In particular, the nonparabolicity of the conduction band (CB) significantly affects the energy dispersion relation and intersubband POP scattering time. However, the currently used parabolic-band (PB) and nonparabolic-band (NPB) energy dispersion models are not appropriate for mid-IR QCLs because they are unsuitable for high electron wave vectors and do not consider the effect of applied strain on the energy dispersion relation of the CB. The eight-band k·p method can provide a relatively accurate nonparabolic energy dispersion relation for high electron wave vectors but has the disadvantages of high computational complexity and spurious solutions to be discarded. Consequently, we propose a strain-modified improved nonparabolic-band (INPB) energy dispersion model that has no spurious solution and acceptable accuracy, compared to the eight-band k·p method. To demonstrate the accuracy and efficiency of our proposed INPB model compared with those of the PB, NPB, and eight-band k·p models, we calculate the energy dispersion relations and intersubband POP scattering times in a strain-compensated QCL with a lasing wavelength of 3.58 µm. Calculation results reveal that our proposed model is almost as accurate as the eight-band k·p model; however, it enables much faster calculations and is free from spurious solutions.
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1. Introduction
Quantum cascade lasers (QCLs) have been intensively investigated as compact and efficient light sources for chemical and biological applications in the mid-infrared (mid-IR) range of 3 and 20 µm [1–3]. Over the last decade, QCLs have attracted significant interest for the development of frequency comb sources in both the mid-IR and terahertz (THz) ranges, with applications in spectroscopy, free-space communication, and imaging fields [4,5]. In band structure calculations of QCLs based on intersubband optical transitions within the conduction band (CB), it is essential to consider the nonparabolicity effect of the CB [6,7]. This significantly affects the energy dispersion relation between the energy and wave vector of the conduction subband and influences the momentum and energy conservation conditions of the carrier scattering times, such as the intersubband polar-optical-phonon (POP) scattering times. In QCLs, the nonparabolicity of the CB significantly affects the degree of population inversion and the magnitude of the optical gain, which are determined by the ratio of the intersubband POP scattering times between the upper and lower electron subbands involved in the lasing action [8].
According to Kane’s eight-band k·p formulation for unstrained bulk semiconductors, the nonparabolicity of the CB (Γ6c) is due to the coupling effect of valence bands (VBs) (Γ7v + Γ8v), comprising the heavy hole (HH), light hole (LH), and spin-orbit (SO) split-off bands, as well as remote bands (RBs) such as Γ7c + Γ8c [9]. In the band structure calculation of QCLs, the nonparabolicity effect on the band-edge energies of the conduction subbands has been intensively investigated at the zone center. When the HH band is decoupled at the zone center (zero electron wave vector) and the spin is degenerated, Kane’s eight-band k·p model can be reduced to a three-band model [10,11]. An effective two-band model was proposed by replacing the LH and SO bands in the three-band model with one effective VB through a unitary transformation [10,11], where the two-band Schrödinger equation could be numerically solved using the finite difference method (FDM) [8,11,12]. Furthermore, the effective two-band model could be recast into a one-band nonlinear Schrödinger equation with an energy-dependent effective mass determined by the effective bandgap energy [6,11,13]. This nonlinear eigenvalue problem has been numerically solved using the transfer matrix method [13,14] or FDM [15]. The aforementioned models have contributed to a precise calculation of the lasing wavelength for mid-IR QCLs, where the nonparabolicity effect results in the lowering of the subband band-edge energies of higher quantum well (QW) states [8].
According to the conventional nonparabolic-band (NPB) model, the conduction subband energy of unstrained bulk semiconductors is expanded up to the fourth-order of the electron wave vector near the zone center, where a nonparabolicity parameter γ is used to represent the analytic NPB energy dispersion relations [6,7,11,16]. The nonparabolicity parameter could be obtained by considering the coupling terms originating from the VBs and RBs used in the multi-band k·p model [6,7,11]. However, the conventional NPB dispersion relation was only valid close to the band-edge of the CB, and they broke down as the transverse wave vector increased [16]. There is another approach to obtain the NPB energy dispersion relations based on the numerical solutions of the stationary nonlinear Schrödinger equation that includes an energy-dependent electron effective mass [17]. In case of semiconductor QW structures, the energy-dependent electron effective masses along the longitudinal and transverse directions can be determined using the multi-band k·p model [18–20]. Because the nonlinear Schrödinger equation includes both longitudinal and transverse energy-dependent effective masses, obtaining numerical solutions is very complicated [20].
Because short-wavelength QCLs with a lasing wavelength of 3-5 µm, require a large discontinuity in the CB, they were implemented based on a strain-compensated InxGa1-xAs/InyAl1-yAs/InP material system, which consisted of compressively strained InxGa1-xAs well and tensile-strained InyAl1-yAs barrier layers [21,22]. The biaxial strain applied to a QW layer also alters the nonparabolicity effect of QCLs because it can modify the subband band-edge energies [11,23,24] and electron effective masses [11,25,26]. The effect of applied strain on the subband band-edge energies was investigated in In0.18Al0.82As/In0.8Ga0.2As/In0.18Al0.82As QWs [24]. The authors derived an analytical formula for the strain-modified anisotropic electron effective masses [11], the calculation results of which were the same as those obtained using the analytic formulas for the anisotropic electron effective mass derived by Sugawara et al. [25]. Nevertheless, the effect of applied strain on the nonparabolic energy dispersion relations in mid-IR QCLs has not yet been thoroughly investigated.
Although the parabolic-band (PB) energy dispersion relation is not as accurate even near the zone center, it has been widely used in the calculations of the intersubband POP scattering time in mid-IR QCLs because analytical expressions for the intersubband scattering time can be easily derived through integration over transverse wave vectors [27–29]. A few theoretical studies have calculated intersubband POP scattering times based on the NPB energy dispersion model in GaAs/Al0.3Ga0.7As QWs [16] and GaAs/AlxGa1-xAs-based mid-IR QCLs [30]. The effect of the applied strain on the electron-electron scattering times was investigated based on an In0.7Ga0.3 As/AlAsSb strain-compensated structure [24]. However, there is still a lack of research on the validity of the energy dispersion relations and intersubband POP scattering times calculated based on the PB and NPB models in mid-IR QCLs.
A relatively accurate nonparabolic energy dispersion curve in a high transverse wave vector region can be obtained by numerically calculating the eigenvalues of the eight-band k·p Hamiltonian matrix with respect to the transverse wave vector [31,32]. Although the eight-band k·p method has the advantage of having a relatively higher accuracy than the PB and NPB models, it has the disadvantage of a long computation time owing to its four-fold matrix dimension. In addition, because it provides many spurious solutions that need to be discarded, complicated numerical techniques must be applied to avoid them, even at the cost of reduced calculation accuracy [33,34]. A bulk-like NPB model approximates the energy dispersion of the QW subband as bulk material and uses the energy-dependent electron effective mass to represent the nonparabolic energy dispersion relation [29]. According to the energy dispersion relation calculated for a single unstrained QW layer, the bulk-like NPB model, which has no spurious solution, shows good agreement with the eight-band k·p method [29]. However, its validity in semiconductor superlattices comprising multiple QW layers has not yet been demonstrated. Moreover, the effect of the biaxial strain on the nonparabolic energy dispersion relation has not yet been considered in the bulk-like NPB model.
In this study, we propose a strain-modified improved nonparabolic-band (INPB) energy dispersion model that efficiently calculates the nonparabolic energy dispersion relation with acceptable accuracy. The proposed model can effectively calculate the nonparabolic energy dispersion relation based on the longitudinal and transverse energy-dependent electron effective masses, while also including the effect of the biaxial strain on the energy dispersion relation. The proposed INPB model is computationally fast and has no spurious solutions because it solves the roots in a third-order polynomial equation using numerical methods. To demonstrate the accuracy and efficiency of the proposed INPB model compared with the PB, NBP, and eight-band k·p models, we present the calculation results of the energy dispersion relations and intersubband POP scattering times in a strain-compensated QCL with a lasing wavelength of 3.58 µm. According to the calculated energy dispersion curves, in addition to the accuracy of the proposed model being almost as good as that of the eight-band k·p model, it does not have spurious solutions and has a much faster computation time. In addition, we demonstrate that the calculated intersubband POP scattering times based on the proposed INPB or eight-band k·p model can be smaller by a factor of 0.3 than that obtained by the widely used PB model.
2. Theory
2.1 Energy dispersion relation models for the CB
A QW layer, which is pseudomorphically grown on a (001)-oriented substrate, is considered such that only the diagonal components of the strain tensors remain [35]. We assume the QW layer to be grown along the z-axis. Let a three-dimensional electron wave vector be $k = {\hat{a}_x}{k_x} + {\hat{a}_y}{k_y} + {\hat{a}_z}{k_z}.$Then, ${k_z}$ and ${k_t} = {\hat{a}_x}{k_x} + {\hat{a}_y}{k_y}$ represent longitudinal and transverse wave vectors, respectively. Four theoretical models for the anisotropic energy dispersion relations of the CB are derived when the biaxial strain is applied to the QW layer.
2.1.1 Strain-modified PB model
When the biaxial strain is applied, the PB energy dispersion relation can be expressed as
In the QW well/barrier structure for the mid-IR QCL, the quantization effect occurs along the z-axis such that the longitudinal wave vector kz in (1) can be replaced by $- i\partial /\partial z$. At the zone center, a one-band linear stationary Schrödinger equation for the strain-modified PB model can be written as
In Section 1 of Supplement 1, we present the detailed derivation of (4) and (5). The energy dispersion relation of $E_n^{(PB)}({k_t})$ in (4) can be also calculated through the direct numerical solution of the stationary Schrödinger equation outside the zone center (kt ≠ 0). By replacing kz with $- i\partial /\partial z$ in (2), we obtain:
2.1.2 Strain-modified NPB model
Similar to [11], the NPB energy dispersion relation is expanded up to the fourth order of the electron wave vector and is expressed as
The one-band nonlinear Schrödinger Eq. (9), which is identical to the strain-modified effective two-band model proposed in [11,12], can be expressed as
Then, the one-band nonlinear stationary Schrödinger equation in (9) can be rewritten as
where $\psi _n^{({NPB} )}(z )= {({\psi_{n,c}^{({NPB} )}(z ),\psi_{n,v}^{({NPB} )}(z )} )^T}$ comprises the CB and effective VB parts of the wave functions [11,12]. Here, the superscript (T) represents the transpose of a vector. The wave function in the strain-modified NPB model is given by:In a manner similar to (4), the analytical expression of the NPB energy dispersion relation at subband n is given by
2.1.3 Proposed strain-modified INPB model
To consider the energy-dependent effective mass effects outside the zone center, we assume that the contribution of the barrier material in the QW structure to the energy dispersion relations for the CB is negligible [29]. In addition, when a biaxial strain is applied to the QW layer, the energy dispersion relation becomes anisotropic [11]. In the proposed strain-modified INPB model, the energy dispersion relation at subband n can be represented in terms of the anisotropic energy-dependent electron effective mass, and is expressed as
2.1.4 Eight-band k·p method
The 8 × 8 second-order k·p perturbation Hamiltonian of the eight-band k·p model can be reduced to two equivalent 4 × 4 Hamiltonian matrices through block diagonalization under the axial approximation [31]. Because of the axial approximation, the electronic band structure of the CB is isotropic in the transverse direction, which is normal to the QW growth direction. The details of the block diagonalization basis functions and two equivalent 4 × 4 Hamiltonian matrix elements can be found in Section 5 of Supplement 1. When only the upper 4 × 4 Hamiltonian matrix is considered, the wave function at subband n is expressed as
2.1.5 Computation methods
First, the stationary Schrödinger equations are numerically solved using the FDM to obtain subband energies and wave functions in one period of the QCL [12]. The z-axis is discretized into N + 1 points uniformly with a step size Δz = 0.1 nm when the total z-directional calculation point is N = 631. We apply the FDM to the Schrödinger equations such that we can construct an N × N matrix for (2) in the PB model, a 2N × 2N matrix for (10) in the NPB and INPB models, and a 4N × 4N matrix for (26) in the k·p method. A built-in MATLAB function called eig is used to extract the eigenvalues and eigenvectors of the two-dimensional matrices.
Second, the energy dispersion relations are calculated with respect to the transverse wave vector. Regarding the PB, NPB, and INPB models, the stationary Schrödinger equations are solved only once at the zone center. The energy dispersion relations for the PB and NPB models are calculated based on the analytical formulas expressed in (4) and (13), respectively. The energy dispersion relation for the INPB model is obtained through a numerical solution for the third-order polynomial expressed in (22). We used a built-in MATLAB function named roots, which can calculate the roots of a single-variable polynomial represented by a vector of coefficients. On the other hand, the eigenvalues and eigenvectors of the 4N × 4N matrix are extracted at each transverse wave vector to obtain the nonlinear dispersion relation based on the k·p method.
2.2 Intersubband POP-emission scattering times
The intersubband POP scattering time plays an important role in determining the degree of population inversion [8,27,29]. The analytical formula associated with the intersubband POP-emission scattering time from the initial to final subbands is given by [27,37]
3. Calculation results
In this section, using the PB, NPB, INPB and eight-band k·p models, we calculate the energy dispersion relations and intersubband POP scattering times of a strain-compensated QCL structure, which shows a lasing wavelength of 3.58 µm with the common three-QW active region. The energy dispersion relations calculated using the PB and NPB models are determined using the analytical formulas in (4) and (13), respectively. Conversely, those calculated by the INPB and eight-band k·p models are obtained by numerically solving (22) and (26) at a transverse-wave-vector interval of Δkt = 0.001 nm−1. We then compare the intersubband POP-emission scattering times of the QCL based on the four calculated energy dispersion relations. The material parameters are described in Section 8 of Supplement 1. The parameters used for the eight-band k·p model are interpolated using bowing parameters based on the material compositions of the well and barrier regions. The POP emission scattering parameters are linearly interpolated based on the material compositions of the well and barrier regions.
3.1 Comparison of energy dispersion relations for the CB
Figure 1(a) shows the CB band-edge energy diagram and the moduli squared-wave functions at the zone center of the strain-compensated short-wavelength QCL [38], which are calculated using the PB, NPB, INPB, and eight-band k·p models. The CB band-edge energy and wave functions obtained by NPB and INPB are identical and are plotted as dotted lines. In the strain-compensated short-wavelength QCL, only three wave functions are designated by the red (state 1 or subband 1), green (state 2 or subband 2), and blue (state 3 or subband 3) colors, where the lasing action occurs at the optical transition from states 3 to 2, while carrier depopulation is achieved through a single POP emission between states 1 and 2. Approximately 10 spurious solutions obtained by the eight-band k·p model, whose subband energies range between states 1 and 3, are shown in Fig. 1(b). The barrier and well, having a CB discontinuity of 0.74 eV, comprise In0.4Al0.6As and In0.7Ga0.3As on the InP substrate. The layer sequence of one period of the active/injector region, starting from the injection barrier, is 4.5/0.5/1.2/3.5/2.3/3.0/2.8 /2.0/1.8/1.8/1.8/1.9/1.8/1.5/2.0/1.5/2.3/1.4/2.5/1.3/3.0/1.3/3.4/1.2/3.6/1.1, where thickness is in nanometers. Barriers and wells are denoted in bold and normal, respectively. The QCL has a measured emission wavelength of 3.58 µm at 300 K under an electric field of −108 kV/cm [38]. For a simple comparison of the CB energy dispersion relations calculated by the four models, a self-consistent calculation that considers the effect of n-doped layers is not considered. In Fig. 1(a), all wave functions and subband band-edge energies calculated using the proposed INPB model are similar to those obtained using the eight-band k·p method. However, the band-edge energy of subband 3 calculated by the PB model is more than 100 meV greater than those obtained by both the INPB and eight-band k·p models. This indicates that the band-edge energy lowering of the higher subbands caused by the longitudinal nonparabolicity effect is significant in short-wavelength QCLs.
Fig. 1. (a) Schematic diagram of the CB band-edge energy and moduli squared-wave functions of the strain-compensated short-wavelength QCL [38], which are calculated by the PB, NPB, INPB, and eight-band k·p models. The CB band-edge energy and wave functions obtained by the NPB and INPB are identical and plotted in the dotted lines. Only three wave functions participating in lasing action are designated by the red (state 1 or subband 1), green (state 2 or subband 2), and blue (state 3 or subband 3) colors. The layer sequence on one period of the active/injector region, starting from the injection barrier, is given by 4.5/0.5/1.2/3.5/2.3/3.0/2.8 /2.0/1.8/1.8/1.8 /1.9/1.8/ 1.5/ 2.0/1.5/2.3/1.4/2.5/1.3/3.0/1.3/3.4/1.2 /3.6/1.1, where the thickness is in nanometers. The barrier and well are designated in bold and normal. The barrier and well are composed of In0.4Al0.6As and In0.7Ga0.3As on the InP substrate, of which the CB discontinuity is 0.74 eV. The QCL has a measured emission wavelength of 3.58 µm at 300 K under the electric field of −108 kV/cm [38]. For simple comparison of the energy dispersion relations calculated by the PB, NPB, INPB, and eight-band k·p models, the self-consistent calculation that considers the effect of the n-doped layers is not considered. (b) Moduli squared-wave functions of approximately 10 spurious solutions obtained by the eight-band k·p model, whose subband energies range between states 1 and 3 of the strain-compensated short-wavelength QCL.
Figure 2 shows the energy dispersion relations for the three subbands of the strain-compensated 3.58-micrometer-lasing QCL, which are calculated by the PB, NPB, INPB, and eight-band k·p models. Only three energy dispersion relations involved in the lasing action are designated as state 1 (or subband 1), state 2 (or subband 2), and state 3 (or subband 3). Dotted (green), dash-dotted (purple), dashed (blue), and solid (red) lines indicate the energy dispersion relations obtained by the PB, NPB, INPB, and eight-band k·p models, respectively. In the eight-band k·p model, the effects of the CB, HH, LH, SO, and remote bands are included in the energy dispersion relations for the CB. In addition, the eight-band k·p model considers the influence of both the well and barrier materials in the QCL structure. Hence, the energy dispersion curves calculated by the eight-band k·p model are more accurate than those obtained by the other three models. However, approximately 10 spurious solutions, which were manually discarded, are obtained through the eight-band k·p model at each transverse wave vector with an interval of Δkt = 0.001 nm−1. Thus, about 10,000 spurious solutions obtained by the eight-band k·p model are discarded to obtain the energy dispersion relations in Fig. 2.
Fig. 2. Energy dispersion relations of the strain-compensated 3.58-micrometer-lasing QCL obtained by the PB, NPB, INPB, and eight-band k·p models. Only three energy dispersion relations involved in lasing action are designated for the state 1 (or subband 1), state 2 (or subband 2), and state 3 (or subband 3). Dotted (green), dash-dotted (purple), dashed (blue), and solid (red) lines indicate the energy dispersion relations obtained by the PB, NPB, INPB, and eight-band k·p models, respectively. Approximately 10,000 spurious solutions obtained by the eight-band k·p model are discarded to obtain the energy dispersion relations.
The PB model does not consider both longitudinal and transverse nonparaboliciy effects, which give rise to the reduction of the subband band-edge energies as well as the curvature lowering of the subbands. Correspondingly, the energy dispersion curves calculated using the PB model, as shown in Fig. 2, are significantly different from those obtained using the eight-band k·p model. The energy-dependent longitudinal electron effective mass, determined by the longitudinal electron effective mass and nonparabolicity parameter, only modifies the subband band-edge energies at the zone center. However, the subband energies outside the zone center are described by the second- and fourth-order polynomials of kt, which are insufficient for correctly calculating the nonparabolic energy dispersion relation. Hence, the energy dispersion relation obtained by the NPB model is broken down at a high transverse-wave vector and begins to have a negative curvature as kt becomes greater than 0.6 nm−1.
Finally, the proposed INPB model simultaneously considers the longitudinal and transverse nonparabolicity effects by integrating the energy-dependent longitudinal and transverse electron effective masses, which can consider the dependence of kt on the energy dispersion relation. Correspondingly, the calculation accuracy of the dispersion curve obtained by the INPB model is significantly enhanced, compared to that calculated by the NPB model. In Fig. 2, the energy dispersion curves of subbands 1 and 2 based on the INPB model match well with those calculated using the eight-band k·p model. The energy dispersion curve of subband 3 calculated by the INPB model deviates slightly from that obtained by the eight-band k·p model.
To quantify the computation accuracy of the four energy dispersion models, we use the mean absolute percentage difference (MAPDn) of subband n, which is defined as
Table 1. Comparison of MAPDs of the energy dispersion relations between the eight-band k·p model and the other three models in Fig. 2.
Table 2 lists the computation times required to calculate the energy dispersion relations depicted in Fig. 2. All calculations are performed using MATLAB on a personal computer with an Intel Core i7-8700 K CPU (3.70 GHz). Besides the 1,000 calculation points for transverse wave vectors between 0 and 1.0 nm−1, numerical calculations are also performed with calculation points of 250, 500, and 750 to investigate how computation speed is affected by the number of computation points. As shown in Table 2, the PB, NPB, and INPB models, which use analytical formulas for the energy dispersion relation, have similarly short computation durations and are nearly insensitive to the number of calculation points for the transverse wave vectors. In contrast, the eight-band k.p model, which uses full-blown numerical solutions, has a relatively long computation duration, which is proportional to the number of calculation points.
Table 2. Comparing the computation time required to calculate the energy dispersion relation depicted in Fig. 2 with respective to the number of calculation points for the transverse electron wave vectors between 0 to 1.0 nm−1.
3.2 Comparison of intersubband POP-emission scattering times
Figure 3 shows the effect of different energy dispersion relations on the intersubband POP-emission scattering of the strain-compensated short-wavelength QCL. The black arrows of $s_{i \to f}^{(PB)}, s_{i \to f}^{(NPB)}, s_{i \to f}^{(INPB)},$ and $s_{i \to f}^{(k\cdot p)}$ indicate the intersubband POP-emission scattering processes occurring in the energy dispersion curves calculated by the PB, NPB, INPB, and eight-band k·p models when the initial electron state at the zone center of the i-th subband relaxes to the final electron state of the f-th subband. Owing to the nonparabolicity effect considered in the NPB, INPB, and eight-band k·p models, the curvatures of the energy dispersion relations calculated using the three nonparabolic dispersion models are lower than those obtained using the PB model. Compared with the PB model, this curvature lowering of the energy dispersion relation results in an increase in the final transverse wave vector (ktf) and a decrease in the electron group velocity at the final transverse wave vector $({\partial {E_f}({k_{tf}})/\partial {k_{tf}}} ),$ both of which contribute to increasing the electronic DOS in (28). In Fig. 3(b), the intersubband POP-emission processes of $s_{3 \to 2}^{(NPB)}$ and $s_{3 \to 1}^{(NPB)}$ do not occur because the energy and momentum conservation between the initial and final electron states cannot be satisfied in the NPB energy dispersion relations.
Fig. 3. Energy dispersion curves and intersubband POP-emission scattering processes of the strain-compensated QCL obtained by (a) the PB, (b) NPB, (c) INPB, and (d) eight-band k·p models. Black arrows represent the intersubband POP-emission scattering from the initial electron state at the zone center of the i-th subband to the final electron state of the f-th subband. The terms $s_{i \to f}^{(PB)}, s_{i \to f}^{(NPB)}, s_{i \to f}^{(INPB)},$ and $s_{i \to f}^{(k\cdot p)}$ indicate the intersubband POP-emission scattering processes occurring at the energy dispersion relations calculated by the PB, NPB, INPB, and eight-band k·p models, respectively. The parenthesis next to $s_{i \to f}^{(P)},s_{i \to f}^{(NP)}, s_{i \to f}^{(INPB)},$ or $s_{i \to f}^{(k\cdot p)}$ is denoted by (${k_{tf}}(nm^{-1}), \rho _f^{i \to f}$ (eV−1nm−2)), where ktf is the magnitude of the transverse wave vector of the final subband and $\rho _f^{i \to f}$ is the DOS of the final electron state when the initial electron state is located at the zone center $({k_{ti}} = 0).$
Table 3 shows the calculated intersubband POP-emission scattering times from the initial subband at the zone center to the final subband based on the energy dispersion relations of the PB, NPB, INPB, and eight-band k·p models in the strain-compensated short-wavelength QCL. The parentheses next to the intersubband POP-emission scattering times indicate the ratio of the calculated intersubband POP emission scattering time with reference to the PB model. The values of τ21 obtained by the NPB, INPB, and eight-band k·p models only decrease by 14%, 14%, and 21% with reference to that calculated by the PB model. Because the curvature lowering of the energy dispersion curves is more pronounced at high transverse wave vectors, as shown in Fig. 2, the values of τ32 (τ31) calculated by the INPB and eight-band k·p models are significantly reduced by 60% (72%) and 48% (71%) with respect to those obtained by the PB model. In addition, the calculation results of τ32, τ31, and τ21 based on the INPB model are quite close to those obtained using the eight-band k·p model. By contrast, the value of τ32 (τ31) calculated using the NPB model are unavailable, as shown in Fig. 3(b).
Table 3. Calculated intersubband POP-emission scattering times from the initial subband at the zone center to the final subband based on the energy dispersion relations of PB, NPB, INPB, and eight-band k·p models in the stain-compensated QCL.a
Table 4 presents the calculated population inversion values τ3(1 - τ21/τ32) of the four energy dispersion models. Here, the population inversion defined in Section 9 of Supplement 1 is proportional to the peak optical gain of the mid-IR QCL [27]. The population inversion based on the NPB model is unavailable because τ32 and τ31 can not be calculated. In contrast, the calculated population inversions are reduced by a factor of 0.31 for the INPB model and 0.39 for the eight-band k·p model compared with that of the PB model. This indicates that a precise calculation of the energy dispersion relation is essential to accurately calculate and design POP scattering times and optical gains of the short-wavelength QCL, where a high transverse electron wave vector is involved in the intersubband POP scattering processes.
Table 4. Population inversions obtained using the PB, NPB, INPB, and eight-band k·p models in the strain-compensated short-wavelength QCL.a
4. Discussion
According to Panda et al., the NPB energy dispersion relation can be obtained through numerically solving the stationary nonlinear Schrödinger equation, which includes energy-dependent longitudinal and transverse effective masses [17]. Because it is very complicated to obtain these numerical solutions, they proposed using an approximated analytical equation that modifies the analytical expression of the conventional NPB model (13) as
We also discuss whether the proposed INPB model can be applied to THz QCLs. Because the INPB model is universal, it can be applied to any QCL emission wavelength, including the THz QCLs. For instance, in Section 11 of Supplement 1, the electronic band structures (Fig. S3) and energy dispersion relations (Fig. S4) are calculated by applying the PB, NPB, INPB, and eight-band k·p models in the lattice-matched InGaAs/InAlAs 84-microleter-lasing QCL [39]. Because the intersubband transition energies between the upper and lower electron subbands are close to one POP energy, the final electron states in the lower subbands are close to the zone center. Therefore, the energy dispersion relations shown in Fig. S4 are sufficient for calculating up to kt = 0.3 nm−1. In Figs. S3 and S4, the moduli-squared wave functions and energy dispersion relations calculated by the four models for the THz QCL are comparable, which differs from the calculation results of the short-wavelength QCL depicted in Figs. 1 and 2. According to Table S7, the average MAPDs of the PB, NPB, and INPB models with reference to the eight-band k·p model are 1.6%, 0.3%, and 1.0%, respectively. Observe that these average MAPDs of the PB and NPB models are markedly lower than those of the strain-compensated short-wavelength QCL listed in Table 1. This discrepancy is attributed to the small nonparabolicity effect near the zone center in the THz QCL.
The INPB model can be used along with theoretical models associated with frequency comb sources [40–42]. They require the electron transition lifetime as an input parameter. To accurately reproduce the spectral and temporal properties of the frequency comb, such as power spikes or linear chirps, the thermal dependence and nonparabolicity effects should be considered. The average scattering lifetime over two-dimensional carrier distributions is necessary to consider the thermal distribution of the electrons [28]. To calculate the average scattering lifetime, it is necessary to know the scattering lifetimes and energy dispersion relations from the subband band-edge energy up to ten times the thermal energy. The proposed INPB model not only provides the nonparabolic energy dispersion relations comparable to the eight-band k·p model, but also easily incorporates the nonparabolicity effects into average electron transition lifetimes. Hence, developing theoretical models for frequency comb sources can be facilitated if the INPB model is incorporated into calculating the average electron transition lifetimes by considering the thermal dependence and nonparabolicity effects. However, our INPB model can not be applied to the calculation of optical characteristics such as optical confinement factor, which requires numerical solutions of Maxwell equations.
5. Conclusion
We proposed a strain-modified INPB energy dispersion model that efficiently calculated the nonparabolic energy dispersion relation of mid-IR QCLs. The currently used PB and NPB energy dispersion models were not valid for high electron wave vectors and did not include the effect of applied strain on the energy dispersion relation of the CB. On the other hand, the eight-band k·p method could provide a relatively accurate nonparabolic energy dispersion relation, even for high electron wave vectors. However, it had the disadvantages of many detrimental spurious solutions that must be discarded and a relatively long computation time.
The proposed strain-modified INPB model could effectively calculate the nonparabolic energy dispersion relation based on longitudinal and transverse energy-dependent electron effective masses. To demonstrate the accuracy and efficiency of our proposed model, we presented the calculation results of the energy dispersion relations and intersubband POP-emission scattering times among the PB, NBP, INPB, and eight-band k·p models in the strain-compensated 3.58-micrometer-lasing QCL. With reference to the calculated energy dispersion curve of the eight-band k·p model, the MAPDs of the INPB model were less than 3.3%, which was much lower than the MAPDs of conventional models, PB (47.1%) and NPB (31.7%). Regarding the computation time of the energy dispersion curve, the proposed model was 4,000 times faster than the eight-band k·p model when the calculation points for the transverse electron wave vectors are 1,000. Thus, we demonstrated that the accuracy of the INPB model was nearly as good as that of the eight-band k·p model. Additionally, it had the advantage of a much faster computation time.
According to the calculated intersubband POP scattering times, the intersubband POP-emission scattering times based on the proposed INPB model, which were comparable to those calculated by the eight-band k·p model, could be reduced by a factor of 0.3, compared to those obtained by the widely used PB model. However, certain intersubband POP-emission scattering times based on the currently used NPB model were unavailable because of the breakdown of the energy dispersion relation for high electron wave vectors. We found that a precise calculation of the nonparabolic energy dispersion curve based on our proposed model is essential for accurately calculating and predicting the intersubband POP scattering times, and the population inversion of the strain-compensated short-wavelength QCL, where a high transverse electron wave vector is involved in the intersubband POP scattering process.
The computation time for our INPB model can be further reduced without sacrificing a degree of accuracy if we use an analytical formula rather than currently used numerical approach to obtain roots for the third-order polynomial Eq. (22). In addition, it can be applied to calculate the energy dispersion relation of any emission wavelength of QCLs grown on well/barrier materials with any material composition and strain characteristics. Besides the intersubband POP scattering times, it can be incorporated with other carrier scattering mechanisms, such as electron-electron scattering, so that the carrier scattering times at high transverse wave vectors can be efficiently calculated with an acceptable accuracy. This can help improve the calculation of the average electron transition lifetimes by considering the thermal dependence and nonparabolicity effects, which can be used for frequency comb sources.
Funding
National Research Foundation of Korea (2020M3H4A3081665, 2021R1F1A1062591).
Acknowledgments
This research was in part supported by the Basic Science Research Program (NRF-2021R1F1A1062591) and in part by the Materials Innovation Project (NRF-2020M3H4A3081665) funded by the National Research Foundation of Korea.
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.
Supplemental document
See Supplement 1 for supporting content.
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