## Abstract

We propose a strain-modified effective two-band model to calculate the conduction band (CB) structure of strain-compensated quantum cascade lasers (QCLs). The proposed model can consider the effect of strain and remote band (RB) on the band-edge energy, electron effective mass, and nonparabolicity parameter although the currently used empirical two-band model can be applicable to only the unstrained QCLs. Based on the three-band second-order k·p Hamiltonian along with the Pikus-Bir Hamiltonian, analytical formula for the electron effective mass and nonparabolicity parameter are derived at the zone center, where the effects of strain and RB interaction are included. Then, the three-band first-order k·p Hamiltonian is reduced to the strain-modified effective two-band Hamiltonian, where the effective Kane energy, determined by the electron effective mass and nonparabolicity parameter, is used to include the nonparabolicity of the CB. By numerically solving the proposed strain-modified effective two-band model based on the finite difference method, we calculate the CB structure of several strain-compensated or unstrained QCLs in the mid-IR and terahertz range and predict their lasing wavelengths, which are well matched with the measured values in the literatures.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Since the first demonstration of the quantum cascade laser (QCL) in 1994, it has attracted much attention as one of the promising candidates for a mid-IR light source that has many chemical and biological applications [1]. The QCL has been intensively investigated in the III-V semiconductor compounds. Among them, there have been many reports of their designs based on the In_{0.53}Ga_{0.47}As/In_{0.52}Al_{0.48}As multiple quantum well (QW) active region lattice-matched to the InP substrate, from which high-performance QCLs with large output power and high operation temperature can be made in the lasing wavelength of larger than 7 μm [2–5]. Because QCLs make use of the intersubband transitions in the conduction band (CB), a precise calculation of the electron subband band-edge energies in InGaAs/InAlAs multiple QWs is very important to perform the accurate design of the emission wavelength. In the band structure calculation of QCLs, the effect of nonparabolicity on the electron effective mass has to be carefully considered because QCLs utilize high-order electron states whose confined energies are much higher than the band-edge energy of the CB [6,7].

According to Kane’s eight-band k·p formulation in unstrained bulk semiconductor, the nonparabolicity of the CB (Γ_{6c}) can be described by the coupling effect caused by the valence bands (VBs) (Γ_{7v}+Γ_{8v}), comprising the heavy hole (HH), light hole (LH) and spin-orbit (SO) split-off bands, as well as the remote bands (RBs) such as Γ_{7c}+Γ_{8c} [8]. In the k·p Hamiltonian matrix, the first-order perturbation term describes the interaction of the CB with the VB and the second-order term deals with the interaction of the CB with the RB. The nonparabolicity of the CB in QW layers was first described by Bastard’s two-band model, where only the CB and LH bands are considered and the diagonal second-order k·p perturbation terms were ignored at the zone center [9,10]. Bastard’s two-band model could be transformed into a one-band nonlinear Schrödinger equation for the CB, where the energy-dependent electron effective mass was determined by the effective bandgap energy. Bastard’s two-band model could be extended into Bastard’s three-band model, where the SO band was added to Bastard’s two-band model [11]. However, two Bastard’s models could not completely consider the nonparabolicity of the CB because they did not include the contribution of the second-order k·p perturbation terms that describe the interaction of the CB with the RBs [12,13].

To overcome the limitation of two Bastard’s models, an empirical two-band model was suggested, where an adjustable effective bandgap energy or effective Kane energy was determined by experimentally-measured or theoretically-calculated two inputs of electron effective mass ${m^\ast }$ and nonparabolicity parameter γ_{e}, which include the effects of the interaction of the CB with the RBs in the second-order k·p perturbation terms [12]. According to an extended Bastard model that theoretically calculated the electron effective mass and nonparabolicity parameter based on the second-order k·p perturbation terms, the calculated two inputs of ${m^\ast }$ and γ_{e} were very close to the two inputs used in the empirical two-band model. In addition, the CB subband band-edge energies of GaAs/AlGaAs QWs calculated by both models were in a very good agreement [13]. Although the empirical two-band or extended Bastard model has been widely used in the band structure calculation of QCLs with unstrained In_{0.53}Ga_{0.47}As/In_{0.52}Al_{0.48}As active regions, they cannot be applied to any QCL with strained QW layers because they did not consider the effect of strain on the subband energies of the CB.

QCLs, having the short operation wavelength of 3-5 μm, requires a large CB discontinuity of more than ΔE_{c} ≈ 0.52 eV provided by the unstrained In_{0.53}Ga_{0.47}As/In_{0.52}Al_{0.48}As material. A strain-compensated In_{x}Ga_{1-x}As/In_{y}Al_{1-y}As/InP material system has been used in the implementation of short wavelength QCLs, where QW well and barrier layers consisted of compressively strained In_{x}Ga_{1-x}As and tensile-strained In_{y}Al_{1-y}As, respectively [14–18]. The strain applied to QW layers results in the band-edge energy shifts of both the CB and the VB as well as the band couplings between the LH and SO bands at the zone center [19]. Thus, the applied strain significantly affects the bandgap energy, electron effective mass, and nonparabolicity parameter, which will change the subband band-edge energies of the CB.

In the case of the electronic band structure of a strain-compensated QCL based on the In_{0.7}Ga_{0.3}As/In_{0.4}Al_{0.6}As multiple QWs, the value of the nonparabolicity parameter γ_{e} for In_{0.7}Ga_{0.3}As was determined by scaling its value for unstrained In_{0.53}Ga_{0.47}As through the inverse ratio of their respective bandgap energy while strain effect was not included in the used value of the electron effective mass for In_{0.7}Ga_{0.3}As [15]. Because of the inappropriately-determined input parameters of ${m^\ast }$ and γ_{e}, the calculated lasing wavelength of 3.16 μm was greatly different from the measured luminescence peak wavelength of 3.58 μm. Sugawara *et al.* theoretically investigated the strain-induced electron effective mass including the effect of biaxial strain on the k·p coupling although the second-order k·p perturbation terms to describe the interaction of the CB with the RBs were not considered [20]. When the strain-induced electron effective mass theoretically predicted by Sugawara *et al.* was used, the calculated lasing wavelength of 3.4 μm became closer to the measured value of 3.58 μm, but there was still a disagreement [21]. Thus, there has not yet been any complete theoretical model to calculate the electronic band structure of strain-compensated QCLs considering the effect of strain and RB on the bandgap energy, electron effective mass, and nonparabolicity parameter.

In this paper, we propose a so-called strain-modified effective two-band model to calculate the electronic band structure of strain-modified QCLs. Using the Kane’s three-band second-order k·p Hamiltonian combined with the Pikus-Bir Hamiltonian, which includes the effects of both applied strain and the RB interaction on the CB, we derive analytical formula for the anisotropic electron effective mass and nonparabolicity parameter at the zone center. Based on the three-band first-order k·p Hamiltonian, we derive a strain-modified effective two-band Hamiltonian for the CB, where the effective bandgap or effective Kane energy, determined by the electron effective mass and nonparabolicity parameter, is used to include the nonparabolicity of the CB. By numerically solving the proposed strain-modified effective two-band model based on the finite difference method, we calculate the electronic band structure of five strain-compensated or unstrained QCLs in the mid-IR and terahertz (THz) range. The calculated lasing wavelengths of five QCLs are in a good agreement with the measured values in the literatures.

## 2. Theory

#### 2.1 Strain-modified effective mass and nonparabolicity parameter for the CB

Figure 1 represents a schematic diagram of the energy dispersion of a direct-gap bulk semiconductor, where the electron effective mass and nonparabolicity of the CB originate from the interactions of the CB with the VB and RB. The VB consists of the HH, LH, and SO bands. In Fig. 1, *E _{g} *=

*E*–

_{c}*E*indicates the bandgap energy of the unstrained bulk semiconductor, where

_{v}*E*is the band-edge energy of the CB and

_{c}*E*is that of both the HH and LH bands. The band-edge energy of the SO band is given by

_{v}*E*- Δ, where Δ is spin-orbit split-off energy. The biaxial strain makes the shifts of the band-edge energies and the coupling between the LH and SO bands, making the effective mass and nonparabolicity of the CB anisotropic.

_{v}When biaxial strain is applied, the CB dispersion to the order *k*^{4} in the electron wave vector becomes anisotropic. Let the QW growth direction be along the z-axis. Then, the anisotropic CB dispersion can be expressed as

According to the theory of Luttinger, Kane, and Bir-Pikus [8,22,23], the band structure of strained bulk semiconductors can be described by the 8 × 8 second-order k·p perturbation Hamiltonian, of which the basis functions including spin are composed of the CB, HH, LH, and SO bands. The details about the basis functions and the Hamiltonian matrix elements used in this derivation can be found in the Appendix A. The 8 × 8 second-order k·p Hamiltonian matrix in (24) can be reduced to a 6 × 6 Hamiltonian matrix due to the decoupling of the HH band at the zone center. In addition, owing to the spin degeneracy, the 6 × 6 Hamiltonian matrix can be further reduced to two equivalent 3 × 3 Hamiltonian matrixes, one of both is given by

*m*is the free electron mass. In the case that ${U_k}$ is defined a $({1 / {\sqrt 3 }}){P_{cv}}{k_z}, {P_{cv}}$ indicates the momentum matrix element moment and is equal to $i\sqrt {{\hbar ^2}{E_p}/(2{m_o})}$, where

_{o}*E*is the Kane energy. The strain-induced energy changes of the VB are defined as ${P_\varepsilon } ={-} {a_v}({\varepsilon _{xx}} + {\varepsilon _{yy}} + {\varepsilon _{zz}})$ and ${Q_\varepsilon } ={-} ({b / 2})({\varepsilon _{xx}} + {\varepsilon _{yy}} - {\varepsilon _{zz}})$ when

_{p}*a*and

_{v}*b*are the Bir-Pikus deformation potentials for the VB [22]. Finally, the term

*F*represents the interaction of the CB with the RB in Kane’s formulation [8] while ${\gamma _1} = \gamma _1^{LK} - {E_p}/(3{E_g})$ and ${\gamma _2} = \gamma _2^{LK} - {E_p}/(6{E_g})$ are the modified Luttinger parameters that represent the interaction of the VB with the RB [24]. Here, $\gamma _1^{LK}$ and $\gamma _2^{LK}$ are original Luttinger parameters. It is noticeable that three terms (

*F*, γ

_{1}, and γ

_{2}), characterizing the interaction with the RB, are the second-order terms in the k·p perturbation [8,23].

The energy eigenvalue of (2) is obtained by

where*E*represents the energy eigenvalue and ${I_{3 \times 3}}$ indicates the 3 × 3 identity matrix. Based on (2) and (3), the secular equation is expressed as

Based on (1), (4), and (33), we can obtain the analytic formulas of the strain-modified anisotropic electron effective masses by only taking into account the second-order *k*, where the details of the derivation steps can be found in the Appendix B. Here, the anisotropic electron effective masses are written as

*F*= 0 in (5) and (6), the calculation results obtained by (5) and (6) become the same as those calculated by the analytic formulas for the anisotropic electron effective mass derived by Sugawara

*et al.*in [20]. In the unstrained semiconductor, we have the relation of $P_\varepsilon ^c = {P_\varepsilon } = {Q_\varepsilon } = 0$, ${E_{C - HH}} = {E_{C - LH}} = {E_g}$, and ${E_{C - SO}} = {E_g} + \Delta .$ Then, both (5) and (6) become identical and the same as the analytical equation of the isotropic electron effective mass derived by Yoo

*et al.*in the unstrained semiconductor [13].

In the similar manner with the anisotropic electron effective mass, the anisotropic nonparabolicity parameter can be obtained by only considering the fourth-order *k*, where the derivation steps associated with the anisotropic nonparabolicity parameters can be found in the Appendix B. The anisotropic nonparabolicity parameters for the CB are expressed as

In the similar fashion to the strain-modified anisotropic electron effective masses in (5) and (6), the strain-modified anisotropic nonparabolicity parameters in (7) and (8) become the same and identical to that derived by Yoo *et al.* [13] if there is no strain applied to the semiconductor.

#### 2.2 Strain-modified effective two-band model for the CB

In principle, the electronic subband energies and their wave functions of the strain-compensated QCL can be obtained by numerically solving the 3 × 3 second-order k·p Hamiltonian in (2), which comprises the strain-induced terms $(P_\varepsilon ^c,\textrm{ }{P_\varepsilon },\textrm{ }{\textrm{Q}_\varepsilon })$, the first-order k·p perturbation term (${U_k} = i({1 / {\sqrt 6 }}){k_z}\sqrt {{\hbar ^2}{E_p}/{m_o}}$), and the second-order k·p perturbation terms (*F*, γ_{1}, γ_{2}). However, a direct numerical solution of (2) can provide many spurious solutions, which results from the second-order perturbation terms [25,26]. Although some numerical techniques are proposed to avoid the spurious solutions, they have the limitations to neglect the interaction of the CB with the RB [25] or to affect the nonparabolicity [26]. Thus, it will not be easy or be very complicated to obtain accurate energy eigenvalues for the CB by applying those methods suggested in [25,26] in the strain-compensated QCL structures.

According to the so-called extended Bastard’s model [13], we can obtain considerably correct subband energies of the CB by neglecting the second-order k·p perturbation terms of the 3 × 3 Hamiltonian in (2) and instead using the electron effective mass and nonparabolicity parameter determined by the second-order terms, as shown in (5) and (7). After some mathematical manipulations, the 3 × 3 first-order k·p Hamiltonian can be reduced to the effective 2 × 2 Hamiltonian, where the effective Kane or bandgap energy, determined by the electron effective mass and nonparabolicity parameter, is used to include the nonparabolicity of the CB [6]. Thus, we use the 3 × 3 first-order k·p perturbation Hamiltonian matrix elements to derive the strain-modified effective two-band model. To include the effect of the second-order k·p terms in the strain-modified effective two-band model, the electron effective mass and nonparabolicity parameter are taken from (5) and (7), which include the effect of applied strain along with the interaction of the CB with the VB and the RB [12,13].

In order to derive the strain-modified effective two-band model, we define the new VB basis set (${\phi _c}$, ${\phi _v}$, and ${\phi _s}$) minimizing the contribution of ${\phi _s}$ to ${\phi _c}$. It is given by [6]

The third row of (10) can be represented as $|{{\phi_s}} \rangle = \left( {{{\sqrt 2 } / 3}} \right)\Delta /(E - {E_v} + {{2\Delta } / 3} + {P_\varepsilon } + {Q_\varepsilon })|{{\phi_v}} \rangle .$ The maximum of the energy-dependent coefficient can be obtained at the band edge (${E_c} + P_\varepsilon ^c$) [6]. Correspondingly, the maximum contribution of the $|{{\phi_s}} \rangle $ to the $|{{\phi_v}} \rangle $ can be expressed as $C_{s - v}^{\max } = ({{2 / 9}} ){\Delta ^2}/({E_{C - SO}} + {Q_\varepsilon } - {\Delta / 3})({E_{C - LH}} - {Q_\varepsilon } + {\Delta / 3}) \approx ({{8 / 9}} ){\Delta ^2}/{({E_{C - LH}} + {E_{C - SO}})^2}.$

Figure 2 shows the calculated changes of $C_{s - v}^{\max }$ in strained In_{x}Ga_{1-x}As and In_{y}Al_{1-y}As on the InP substrate as a function of the InAs composition (x or y). The materials parameters of GaAs. AlAs, and InAs used in this calculation are shown in Table 1. To approximate the dependence on the alloy composition x or y, we use the bowing parameters shown in Table 2, where the zero value of the bowing parameter indicates a linear interpolation. Because the values of $C_{s - v}^{\max }$ are inversely proportional to the bandgap energy of the alloy materials, they increase when the InAs composition of In_{x}Ga_{1-x}As or In_{y}Al_{1-y}As increases. The overall values of $C_{s - v}^{\max }$ is less than 4.7%, which is comparable to the maximum contribution of $C_{s - v}^{\max } \simeq 0.04$ in unstrained In_{x}Ga_{1-x}As [6]. In Fig. 2, the blue and green dots on the solid lines represent the respective value of $C_{s - v}^{\max }$ in the In_{0.7}Ga_{0.3}As/In_{0.4}Al _{0.6}As [15] and In_{0.72}Ga_{0.28}As/In_{0.3}Al_{0.7}As [18] well/barrier materials of the strain-compensated QCL used in this calculation. The values of $C_{s - v}^{\max }$ are 3.1% for In_{0.7}Ga_{0.3}As and 3.18% for In_{0.72}Ga_{0.28}As, which are a still small faction and can be negligible in (10). Thus, by neglecting the wave function component of $|{{\phi_s}} \rangle ,$ we can obtain the reduced 2 × 2 k·p Hamiltonian acting on the two-dimensional envelope function $\phi = {({\phi _c},{\phi _v})^T}$ in the strained semiconductor at the zone center, where the subscript (*T*) represents the transpose of a vector.

The reduced 2 × 2 k·p Hamiltonian is expressed as [6,7]

The effective Kane energy depends on both the longitudinal electron effective mass in (5) and longitudinal nonparabolicity parameter in (7), which are determined by the applied strain and the second-order k·p perturbation terms that give rise to the nonparabolicity of the CB.

The reduced 2 × 2 Hamiltonian matrix in (11) can be further simplified into a one-dimensional nonlinear energy-dependent Schrödinger equation, which is given by [6]

Here, the energy-dependent electron effective mass is given by

When there is no applied strain ($P_\varepsilon ^c = {P_\varepsilon } = {Q_\varepsilon } = 0$), the effective band-edge energy of the VB $E_v^{eff}$ approaches ${E_v} - \Delta /3.$ In this case, the energy-dependent longitudinal electron effective mass in (14) becomes the same formula as that derived in the unstrained semiconductor [6]. Finally, the energy-dependent electron effective mass can be also expressed as

## 3. Calculation results

#### 3.1 Strain-modified effective mass and nonparabolicity parameter for the CB

Figure 3 shows the longitudinal and transverse electron effective masses of In_{x}Ga_{1-x}As and In_{y}Al_{1-y}As grown on the InP substrate as a function of the InAs composition x or y. The unstrained alloy compositions of x = 0.53 and y = 0.52, lattice-matched to the InP substrate, are designated in the vertical dotted line. The right (left) arrow represents the compressive (tensile) strain applied to In_{x}Ga_{1-x}As and In_{y}Al_{1-y}As on the InP substrate. The material parameters together with bowing parameters are shown in Table 1 and 2. In Fig. 3, the black solid line represents the calculated electron effective mass in the unstrained material, which is obtained by the linear interpolation between InAs and GaAs (AlAs) together with the bowing parameters [27]. The red solid line indicates the strain-modified electron effective mass obtained by (5) and (6), where both the applied strain and contribution of the RB to the CB are considered.

While the electron effective mass in the unstrained alloy is isotropic, the strain-modified electron effective mass is anisotropic. The longitudinal electron effective mass in (5) is proportional to the strain-induced bandgap energies of ${E_{C - LH}} = {E_g} + P_\varepsilon ^c + {P_\varepsilon } - {Q_\varepsilon }$ and ${E_{C - SO}} = {E_g} + P_\varepsilon ^c + {P_\varepsilon } + \Delta $. As shown in Fig. 1, the compressive strain increases the values of ${E_{C - LH}}$ and ${E_{C - SO}}$ so that the longitudinal electron effective masses in the compressively strained In_{x}Ga_{1-x}As and In_{y}Al_{1-y}As materials become greater than those in the unstrained materials. In contrast, the tensile strain decreases the longitudinal electron effective mass because the tensile strain reduces the strain-induced bandgap energies of ${E_{C - LH}}$ and ${E_{C - SO}}.$ In addition, the transverse electron effective mass in (6) is only proportional to the strain-induced bandgap energy of ${E_{C - HH}} = {E_g} + P_\varepsilon ^c + {P_\varepsilon } + {Q_\varepsilon }$. As shown in Fig. 3(c) and (d), the transverse electron effective mass in the compressively strained (tensile-strained) alloy becomes greater (less) than that in the unstrained alloy, which is similar to the calculated longitudinal electron effective mass in Fig. 3(a) and (b). However, the amount of the change in the strain-modified transverse electron effective mass in reference to the electron effective mass in the unstrained material is less than that of the strain-modified longitudinal electron effective mass because the sign of the strain-induced variation in ${E_{C - HH}}$ is opposite to those in ${E_{C - LH}}$ and ${E_{C - SO}}$.

Figure 4 shows calculated longitudinal and transverse nonparabolicity parameters for the CB in In_{x}Ga_{1-x}As on the InP substrate as a function of the InAs composition x. The vertical black dotted lines represent the lattice-matched alloy composition of x = 0.53, where the lattice constant of In_{0.53}Ga _{0.47}As is equal to that of the InP substrate. The right (left) arrow represents the compressive (tensile) strain applied to In_{x}Ga_{1-x}As on the InP substrate. The black solid line represents the isotropic nonparabolicity parameter of unstrained *(ε _{ii}* = 0) In

_{x}Ga

_{1-x}As. The red solid line represents the strain-modified (

*ε*≠ 0) anisotropic nonparabolicity parameters with the contribution of the RB to the CB considered. The green dots represent the nonparabolicity parameters found in the literatures [6,12,15]. According to the analytical formula in (7), the longitudinal nonparabolicity parameter is inversely proportional to the bandgap energies $({E_{C - LH}},\textrm{ }{E_{C - SO}})$ and electron effective mass $({m_e^ \bot } )$ while it is proportional to the RB’s contribution (

_{ii}*F*) and modified Luttinger parameters (γ

_{1}, γ

_{2}). In Fig. 4, the nonparabolicity parameters of the unstrained In

_{x}Ga

_{1-x}As are isotropic and become larger as the alloy composition of InAs (x) increases. As shown in Table 1, InAs has the smaller bandgap energy, larger modified Luttinger parameters, and larger RB’s contribution than GaAs. Thus, as the alloy composition of InAs increases in the unstrained In

_{x}Ga

_{1-x}As, the isotropic nonparabolicity parameter increases. In Fig. 4(a), it is noticeable that the calculated isotropic nonparabolicity parameter shows a good agreement with the values found in the literature [6,12,15]. The calculation results in the longitudinal nonparabolicity parameters show that the data shown in the literatures did not consider the effect of the applied strain.

When compressive or tensile strain is applied to In_{x}Ga_{1-x}As, the strain-modified nonparabolicity parameters become anisotropic and are either larger or smaller than the isotropic nonparabolicity parameter of unstrained In_{x}Ga_{1-x}As depending on the type of strain. The compressive strain applied to In_{x}Ga_{1-x}As increases the strain-induced bandgap energies of ${E_{C - LH}}$ and ${E_{C - SO}}$ as well as the electron effective mass in reference to the unstrained In_{x}Ga_{1-x}As. In contrast, the tensile strain decreases both the strain-induced bandgap energies $({E_{C - LH}},\textrm{ }{E_{C - SO}})$ and the electron effective mass. In Fig. 4(a), the longitudinal nonparabolicity parameter calculated in the compressively strained In_{x}Ga_{1-x}As is less than that in unstrained In_{x}Ga_{1-x}As. On the other hand, the longitudinal nonparabolicity parameter calculated in the tensile-strained In_{x}Ga_{1-x}As is greater than that in unstrained In_{x}Ga_{1-x}As. The transverse nonparabolicity parameter in (8) is inversely proportional to the bandgap energies (${E_{C - HH}}$, ${E_{C - LH}}$, and ${E_{C - SO}}$) and transvers effective mass ($m_e^\parallel $). As the biaxial strain is applied to In_{x}Ga_{1-x}As, the sign of the variation in ${E_{C - HH}}$ is opposite to those in ${E_{C - HH}}, {E_{C - SO}},$ and $m_e^\parallel .$ In a similar fashion to the longitudinal nonparabolicity parameter, the transverse nonparabolicity parameter also decreases (increases) in the compressive (tensile)-strained In_{x}Ga_{1-x}As. However, the amount of the change in the strain-modified transverse nonparabolicity parameter in reference to the isotropic nonparabolicity parameter of unstrained In_{x}Ga_{1-x}As is smaller than that of the strain-modified longitudinal nonparabolicity parameter.

#### 3.2 Comparison of calculated wave functions and subband energies of short-wavelength QCLs between the strain-modified effective two-band and second-order three-band models

The validity of the strain-modified effective two-band model is demonstrated by comparing the wave functions and their subband energies located at the active region of short-wavelength and THz QCL structures between the strain-modified effective two-band model in (11) and the second-order three-band model in (2). Two short-wavelength mid-IR and one THz QCL structures have so-called three-QW or four-QW active region design [15,18,28]. The self-consistent calculation, which considers the modification of the electric potential caused by the presence of ionized donors in the injector region, is considered [29]. The wave functions and subband energies of a few electron states located at the active region are calculated based on the finite difference method. In the numerical calculation of the second-order three-band model, we do not apply any numerical techniques to avoid the spurious solutions suggested in [25,26]. Instead, we manually identify and discard spurious solutions of the second-order three-band model by comparing those calculation results with numerical solutions obtained by the strain-modified effective two-band model, which provides no spurious solution.

Figure 5 shows the schematic conduction band diagrams and moduli-squared wave functions of a strain-compensated mid-IR QCL structure calculated by the strain-modified effective two-band and second-order three-band models. Only four wave functions and their subband energies located at the active region of the short-wavelength QCL are designated. It is noticeable that total 16 spurious solutions obtained by the second-order three-band model, whose subband energies range between the state 1 and state 4, are not shown in Fig. 5. The QCL, having the measured emission wavelength of 3.58 μm at 270 K, includes the three-QW active design under the electric fields of -96 kV/cm [15]. The lasing action takes place at the optical transition between the state 3 and state 2 while the carrier depopulation is achieved through the single optical phonon resonance between the state 2 and state 1. The barrier and well regions are made of In_{0.4}Al_{0.6}As and In_{0.7}Ga_{0.3}As on the InP substrate. The layer sequence of one period of the active/injector region, in nanometers, 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 barrier and well layers are designated in bold and normal. The Si-doped layers (*N _{d}*=2.5 × 10

^{17}cm

^{-3}) in the injector region are underlined. In the self-consistent calculation, two periods of the active/injector regions are considered to accurately model the effect of the modified electric potential caused by the ionized donors on the wave functions and their subband energies of the QCLs. The CB discontinuity of 0.74 eV is determined by the application of the model solid theory. The input parameters used for the strain-modified effective two band model are 0.045 ($m_w^ \bot /{m_o}$), 0.079 ($m_b^ \bot /{m_o}$), and 1.117 × 10

^{−18}($\gamma _w^ \bot $ in m

^{2}), where w(b) represents the well(barrier) region.

Table 3 shows the comparison of the calculated well occupancy probabilities, subband energies of four wave functions in Fig. 5 between the strain-modified effective two-band and second-order three-band models. To compare the degree of the similarity of the wave functions calculated by the two models, the well occupancy probabilities (O_{w}), defined as the ratio of the moduli-squared wave function distributed at the In_{0.7}Ga_{0.3}As well regions, are calculated by [12]

Here, the index *i* represents the i-th component of the wave function and the numerical integration is taken over all the well regions in Fig. 5. In Table 3, the difference of the well occupancy probabilities and subband energies between the two models are within 8% and 6 meV, respectively. The emission wavelengths, calculated by *λ _{emit}* = 1241(meV×μm)/(

*E*–

_{3}*E*), are 3.56 μm (348.06 meV) for the three-band model and 3.55 μm (349.13 meV) for the two-band model. These calculated emission wavelengths are in a good agreement with the measured luminance peak wavelength of 3.58 μm (346.64 meV) in the literature [15].

_{2}Figure 6 shows the schematic conduction band diagrams and moduli-squared wave functions of a strain-compensated mid-IR QCL structure calculated by the strain-modified effective two-band and second-order three-band models. Only five wave functions and their subband energies located at the active region are represented. Total 18 spurious solutions obtained by the second-order three-band model, whose subband energies range between the state 1 and state 5, are not shown in Fig. 6. The QCL having the measured emission wavelength of 4.06 μm at 298 K comprises a four-QW active design under the electric fields of -85 kV/cm [18]. The lasing action takes place at the optical transition between the state 4 and state 3. The carrier depopulation takes place through the double optical phonon resonance among the state 3, state 2, and state 1. The barrier and well regions are made of In_{0.3}Al_{0.7}As and In_{0.72}Ga_{0.28}As on the InP substrate. The layer sequence of one period of the active/injector region, starting from the injection barrier, is given by **3.88/**1.01/**1.36**/3.88/**1.36**/3.49/**1.46**/3.1/**2.21**/2.75/**1.66**/ 2.36/**1.75**/2.13/**1.84**/1.87/**1.94**/1.79/**2.12**/1.71/**2.12**/1.6/**2.77**/1.6 in nanometers. The barrier and well layers are designated in bold and normal, respectively. The Si-doped layers (*N _{d}*=3 × 10

^{16}cm

^{-3}) in the injector region are underlined. In the self-consistent calculation, two periods of the active/injector regions are considered. The CB discontinuity is set to be 0.87 eV. The input parameters used for the strain-modified effective two-band model are 0.046 ($m_w^ \bot /{m_o}$), 0.085 ($m_b^ \bot /{m_o}$), and 1.119 × 10

^{−18}($\gamma _w^ \bot $ in m

^{2}), where w(b) represents the well(barrier) region.

Table 4 shows the comparison of the calculated well occupancy probabilities, subband energies of five wave functions in Fig. 6 between the strain-modified effective two-band and second-order three-band models. As shown in Table 3, the difference of the well occupancy probabilities and subband energies between the two models are within 2% and 11 meV, respectively. The biaxial strains $({{\varepsilon_{xx}} = {\varepsilon_{yy}}} )$ applied to In_{0.72}Ga_{0.28}As well and In_{0.3}Al_{0.7}As barrier are -1.26% and 1.55%, respectively. These applied strains of the short-wavelength QCL shown in Fig. 6 are greater than those shown in Fig. 5, where the biaxial strains applied to In_{0.7}Ga_{0.3}As well and In_{0.4}Al_{0.6}As barrier are -1.13% and 0.85%, respectively. Because of the lasing action between the state 4 and state 3, the emission wavelength is calculated by *λ _{emit}* = 1241(meV×μm)/(

*E*–

_{4}*E*). The calculated emission wavelengths are 4.08 μm (303.81 meV) for the three-band model and 4.11 μm (302.17 meV) for the two-band model, which is close to the measured luminance peak wavelength of 4.06 μm (305.66 meV) in the literature [18].

_{3}Besides short-wavelength mid-IR QCLs based on the InGaAs/InAlAs system, the proposed strain-modified effective two-band model can be also applied to the electronic band calculation of THz QCLs based on the InGaAs/InAlAs system. Figure 7 shows the schematic conduction band diagrams and moduli-squared wave functions of a THz QCL structure calculated by the strain-modified effective two-band and second-order three-band models. Only four wave functions and their subband energies located at the active region are represented. The spurious solutions obtained by the second-order three-band model, whose subband energies range between the state 1 and state 4, do not exist. This will be explained later in the discussion section. The QCL, having the measured emission wavelength of 84 μm at 10 K, comprises a three-QW active design under the electric fields of -3.3 kV/cm [28]. The lasing action takes place at the optical transition between the state 3 and state 2. The carrier depopulation takes place through the optical phonon resonance between the state 2 and state 1. The barrier and well regions are made of In_{0.52}Al_{0.48}As and In_{0.53}Ga_{0.47}As, respectively. Both are latticed-matched with the InP substrate so that the well and barrier layers of the THz QCL are unstrained. The layer sequence of one period of the active/injector region, in nanometers, starting from the injection barrier, is given by **1.9/**15.5/**0.3**/26.0/**0.4**/21.0/**0.6**/ 17.5/**0.8**/16.0/**1.0**/28.0, where barrier and well layers are designated in bold and normal, respectively. The Si-doped layers (*N _{d}*=2.2 × 10

^{16}cm

^{-3}) in the injector region are underlined. In the self-consistent calculation, two periods of the active/injector regions are considered. The CB discontinuity is set to be 0.51 eV. The input parameters used for the strain-modified effective two-band model are 0.043 ($m_w^ \bot /{m_o}$), 0.073 ($m_b^ \bot /{m_o}$), and 1.11 × 10

^{−18}($\gamma _w^ \bot $ in m

^{2}), where w(b) represents the well(barrier) region.

Table 5 shows the comparison of the calculated well occupancy probabilities, subband energies of four wave functions in Fig. 7 between the strain-modified effective two-band and second-order three-band models. As shown in Table 5, the difference of the well occupancy probabilities and subband energies between the two models are within 0.1% and 0.5 meV, respectively. Because of the lasing action between the state 3 and state 2, the emission wavelength is calculated by *λ _{emit}* = 1241(meV×μm)/(

*E*–

_{3}*E*). The calculated emission wavelengths are 84.69 μm (14.65 meV) for the three-band model and 86.43 μm (14.36 meV) for the two-band model, which is close to the measured lasing peak wavelength of 84 μm (14.77 meV) in the literature [28].

_{2}## 4. Discussion

Since Sirtori *et al.* had fabricated the QCL based on GaAs/AlGaAs superlattice lattice-matched with the GaAs substrate, many researchers have investigated the GaAs/AlGaAs material system, making the mid-IR and THz QCLs [30–36]. Compared with the QCL based on the InGaAs/InAlAs material system, the QCL based on the GaAs/AlGaAs material system has advantages of low fabrication cost and good interface quality. However, there are several limitations in the GaAs-based QCL. The optical gain of the GaAs-based QCL is lower than that of the InGaAs-based QCL because the electron effective mass of GaAs is greater than that of InGaAs. Moreover, the optical transition energy of the GaAs-based QCL cannot be greater than 150 meV due to the relatively shallow CB discontinuity of the GaAs/AlGaAs as well as the electron leakage from the Γ- to X-valley or Γ- to L-valley [31–33]. Thus, the optical emission wavelength of the mid-IR QCL based on the GaAs/AlGaAs material system has been designed longer than 8 μm. In addition to the QCL based on the InGaAs/InAlAs material system, our proposed strain-modified effective two-band model can be also applied to the band structure calculation of the QCL based on the GaAs/AlGaAs material system. Its validity is demonstrated by comparing the wave functions and their subband energies of GaAs-based mid-IR and THz QCLs between the strain-modified effective two-band and the second-order three-band models.

Figure 8 shows the schematic conduction band diagrams and moduli-squared wave functions of an unstrained GaAs-based mid-IR QCL structure calculated by the strain-modified effective two-band and second-order three-band models. The barrier and well regions are made of Al_{0.45}Ga_{0.55}As and GaAs, of which the CB discontinuity is 0.39 eV. The material parameters of Al_{x}Ga_{1-x}As are linearly interpolated between those of GaAs and AlAs shown in Table 1 except that the energy band gap has a bowing parameter of -0.127 + 1.31x [27]. To include the effect of the modified electric potential caused by the ionized donors, the self-consistent calculation is performed in two periods of the active/injector regions. The layer sequence of one period of the active/injector region, in nanometers, starting from the injection barrier, is given by **4.6/**1.9/**1.1**/5.4/**1.1**/4.8/**2.8**/3.4/**1.7**/3.0/**1.8**/2.8/**2.0**/3.0/**2.6**/3.0. The barrier and well layers are designated in bold and normal. The Si-doped layers (*N _{d}*=3.8 × 10

^{11}cm

^{-2}) in the injector region are underlined. Only four wave functions and their subband energies located at the active region are represented. Total eight spurious solutions obtained by the second-order three-band model, whose subband energies range between the state 1 and state 4, are not shown in Fig. 8. The QCL, having the measured emission wavelength of 9.4 μm at 300 K, comprises a three-QW active design under the electric fields of -48 kV/cm [31]. The lasing action takes place at the optical transition between the state 3 and state 2. The carrier depopulation takes place through the single optical phonon resonance between the state 2 and state 1. The input parameters used for the strain-modified effective two-band model are 0.067 ($m_w^ \bot /{m_o}$), 0.104 ($m_b^ \bot /{m_o}$), and 4.164 × 10

^{−19}($\gamma _w^ \bot $ in m

^{2}), where w(b) represents the well(barrier) region.

Table 6 shows the comparison of the calculated well occupancy probabilities, subband energies of four wave functions in Fig. 8 between the strain-modified effective two-band and second-order three-band models. As shown in Table 6, the difference of the well occupancy probabilities and subband energies between the two models are within 3% and 6 meV, respectively. Owing to the lasing action between the state 3 and state 2, the emission wavelength is calculated by *λ _{emit}* = 1241(meV×μm)/(

*E*–

_{3}*E*). The calculated emission wavelengths are 9.25 μm (134.07 meV) for the three-band model and 9.17 μm (135.25 meV) for the two-band model, which is close to the measured luminance peak wavelength of 9.4 μm (132.02 meV) in the literature [31].

_{2}Figure 9 shows the schematic conduction band diagrams and moduli-squared wave functions of a GaAs-based THz QCL structure calculated by the strain-modified effective two-band and second-order three-band models. The barrier and well regions are made of Al_{0.15}Ga_{0.85}As and GaAs, of which the CB discontinuity is 0.13 eV. To include the effect of the modified electric potential caused by the ionized donors, the self-consistent calculation is performed in two periods of the active/injector regions. The layer sequence of one period of the active/injector region, in nanometers, starting from the injection barrier, is given by **4.4/**7.7/**2.8** /6.9/**3.6**/15.7/**1.7**/10.2/**2.5**/8.3, where barrier and well layers are designated in bold and normal, respectively. The Si-doped layers (*N _{d}*=2.9 × 10

^{16}cm

^{-3}) in the injector region are underlined. Only four wave functions and their subband energies located at the active region are represented. Total two spurious solutions obtained by the three-band model, whose subband energies range between the state 1 and state 4, are not shown in Fig. 9. The barrier electron effective mass used for the two-band model are 0.080 ($m_b^ \bot /{m_o}$), where b represents the barrier region.

Table 7 shows the comparison of the calculated well occupancy probabilities, subband energies of five wave functions in Fig. 9 between the two-band and three-band models. As shown in Table 7, the difference of the well occupancy probabilities and subband energies between the two models are within 0.6% and 0.2 meV. Because of the lasing action between the state 3 and state 2, the emission wavelength is calculated by *λ _{emit}* = 1241(meV×μm)/(

*E*–

_{3}*E*). The calculated emission wavelengths are 105.51 μm (11.76 meV) for the three-band model and 106.35 μm (11.66 meV) for the two-band model. This is close to the measured luminance peak wavelength of 100.8 μm (12.4 meV) in the literature [34].

_{2}To investigate the calculation accuracy of the proposed strain-modified effective two-band model, Table 8 shows the absolute errors of the transition energies calculated by the three- and two-band models compared with the measured transition energies in the literatures. In the case of three mid-IR QCLs, the absolute errors of the calculated transition energies are less than 2.05 meV for the three-band model and less than 3.49 meV for the two-band model. Regarding two THz QCLs, the absolute errors are less than 0.64 meV for the three-band model and less than 0.74 meV for the two-band model. Although the absolute error is reduced to 0.74 meV, the difference of the transition wavelength between the calculation (106.35 μm) and measurement (100.8 μm) is about 6 μm in the THz QCL in Fig. 9. This fact can limit the applicability of the proposed two-band model to the conduction band structure calculation of THz QCLs.

In Fig. 7, it is noticeable that there is no spurious solution of the wave functions between the state 1 and state 4, which are calculated by the second-order three-band model. On the other hand, the other four QCL structures have spurious solutions when they are calculated through the three-band model. To explain the reason why only the THz QCL in Fig. 7 has no spurious solution, it is necessary to analyze the theoretical condition that the second-order three-band model produces spurious solutions [26]. When we expand the polynomial (4) for k_{z}, we have

According to the Descartes’s rule of signs for the polynomial equation, the number of positive roots of a polynomial is equal to the number of sign changes of the coefficient sequence of the polynomial [26]. When the number of sign changes of the coefficient sequence is one, the polynomial has one positive root and no spurious solution. If the number of sign changes of the coefficient sequence is zero or two, the polynomial has spurious solutions [26]. In the case of the THz QCL in Fig. 7 [28], the sign of the coefficient sequence (A, B, C, D) in (18) is (+, +, +, -) for both the In_{0.47}Ga_{0.53}As well and the In_{0.52}Al_{0.48}As barrier. Here, the energy eigenvalues of $E > E_c^\varepsilon$, which indicates the subband energies of the CB, are considered in the calculation of the sign of the coefficient sequence. Because the number of sign changes of the coefficient sequence is one for both the well and barrier, the THz QCL in Fig. 7 has no spurious solution calculated by the three-band model. In contrast, the sign of the coefficient sequence of the other four QCL structures is (+, +, +, -) for the well (In_{0.7}Ga_{0.3}As [15], In_{0.72}Ga_{0.28}As [18], GaAs [31,34]) and (-, +, +, -) for the barrier (In_{0.4}Al_{0.6}As [15], In_{0.3}Al_{0.7}As [18], Al_{0.45}Ga_{0.55}As [31], Al_{0.15}Ga_{0.85}As [34]), respectively. Because the number of sign changes of the coefficient sequence is two for the barrier regions, the other four QCL structures have spurious solution.

## 5. Conclusion

We proposed the strain-modified effective two-band model that could calculate the wave functions and subband energies of strain-compensated QCLs by considering both applied strain and the interaction of the RB on the CB. Based on the second-order 3 × 3 k·p Hamiltonian along with the Luttinger, Kane, and Pikus-Bir Hamiltonians, we derived the analytical formulas for the anisotropic electron effective masses and nonparabolicity parameters, where the effects of strain and RB interaction were included. The calculated anisotropic electron effective masses in the compressive-stained In_{x}Ga_{1-x}As and In_{y}Al_{1-y}As materials became greater than those in the unstrained In_{x}Ga_{1-x}As and In_{y}Al_{1-y}As materials. In the case of tensile-strained In_{x}Ga_{1-x}As and In_{y}Al_{1-y}As materials, the calculated anisotropic electron effective masses were less than those in the unstrained materials. Then, the three-band first-order k·p Hamiltonian was reduced to the strain-modified effective two-band Hamiltonian, where the effective Kane energy, determined by the longitudinal electron effective mass and nonparabolicity parameter, was used to include the nonparabolicity of the CB.

By numerically solving the proposed strain-modified effective two-band and second-order three-band models based on the finite difference method, we calculated the electronic band structures of three mid-IR and two THz QCLs based on either InGaAs/InAlAs or GaAs/AlGaAs material systems and compared their well probabilities and subband energies for the electron states involved in lasing action. The calculation results obtained by the strain-modified effective two-band model were very close to those calculated by the three-band model, which required distinction and deletion of many spurious solutions. In addition, the calculated emission wavelengths of five QCLs were in a good agreement with the measured values in the literatures.

In conclusion, our proposed strain-modified effective two-band model could include the effect of applied strain and RB interaction on the CB structure of strain-compensated QCLs although the application of the currently used empirical two-band model was limited to unstrained QCLs. The proposed strain-modified effective two-band model showed considerable calculation accuracy compared with the second-order three-band model. Furthermore, unlike the three-band model, the proposed strain-modified effective two-band model provided no spurious solution to be discarded. In addition, it had faster computation speed due to the reduced matrix dimension compared with the second-order three-band model.

## Appendix

## A. 8 × 8 Hamiltonian for the second-order k·p perturbation

Based on the theories of Luttinger, Kane, Bir, and Pikus [8,22,23], we use the following set of basis functions, which are given by

*S*has the atomic s-orbital symmetry while X, Y, and Z have the atomic p-orbital symmetry. The symbols $\uparrow $ and $\downarrow $ represent the spin of an electron. Here, ($|1 \rangle $, $|5 \rangle $), ($|2 \rangle $, $|6 \rangle $), ($|3 \rangle $, $|7 \rangle $), and ($|4 \rangle $, $|8 \rangle $) indicate the Bloch functions for the CB, HH, LH, and SO, respectively. The CB and VB structures in the strained zinc-blend structure semiconductor are described by the 8 × 8 Hamiltonian in envelope function space, taking into account the Hamiltonian to the second-order. The second-order 8 × 8 Hamiltonian matrix is given by

For a strained-layer semiconductor pseudo-morphically grown on a (001)-oriented substrate, only the diagonal parts of strain tensor components remain. They are given by

_{11}and C

_{12}indicate the elastic stiffness constants.

## B. Derivation of the anisotropic electron effective mass and nonparabolicity parameter

In this section, the details of the derivation step associated with the anisotropic electron effective mass and nonparabolicity parameters are shown. The longitudinal electron effective mass and nonparabolicity parameter can be derived from the secular equation in (4). When the bulk energy dispersion relation for the CB at the *k _{t}* = 0, shown in (1), is inserted into (4), we have

The longitudinal electron effective mass can be obtained by considering only the second-order terms for k_{z} in (27), which is given by

Arrangement of the appropriate coefficients on both sides leads to

After some mathematical manipulations, the longitudinal electron effective mass is given by

The longitudinal nonparabolicity parameter can be also obtained by taking into account only the fourth-order terms for k_{z} in (27), which is expressed as

After some mathematical manipulations, we obtain the analytical expression of the longitudinal nonparabolicity parameter, which is written as

We can derive the transverse electron effective mass and nonparabolicity parameter using the secular equation shown in (24) at the k_{z}=0. This secular equation is represented by

In the similar manner with the longitudinal electron effective mass and nonparabolicity parameter, we use the bulk energy dispersion relation for the CB at the *k _{t}* = 0. To obtain the transverse electron effective mass, we consider only the second-order terms for k

_{t}, which is given by

When we arrangement the appropriate coefficients on both sides, we have

Finally, the transverse nonparabolicity parameter can be derived by considering only the fourth-order terms for *k _{t}*, which is expressed as

Some mathematical manipulations lead to

## Funding

National Research Foundation of Korea (2020M3H4A3081665).

## Acknowledgments

This research was supported by the Materials Innovation Project funded by National Research Foundation of Korea (2020M3H4A3081665).

## 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.

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