Numerically efficient density-matrix technique for modeling electronic transport in mid-infrared quantum cascade lasers

Abstract

We present a numerically efficient density-matrix model applicable to mid-infrared quantum cascade lasers. The model is based on a Markovian master equation for the density matrix that includes in-plane dynamics, preserves positivity of the density matrix and does not rely on phenomenologically introduced dephasing times. Nonparabolicity in the band structure is accounted for with a three-band ${\bf k}\cdot {\bf p}$ model, which includes the conduction, light-hole, and spin-orbit split-off bands. We compare the model to experimental results for QCLs based on lattice-matched as well as strain-balanced InGaAs/InAlAs heterostructures grown on InP. We find that our density-matrix model is in quantitative agreement with experiment up to threshold and is capable of reproducing results obtained using the more computationally expensive nonequilibrium Green’s function formalism. We compare our density-matrix model to a semiclassical model where off-diagonal elements of the density matrix are ignored. We find that the semiclassical model overestimates the threshold current density by 29% for a 8.5-$\upmu$m-QCL based on a lattice-matched heterostructure and 40% for a 4.6-$\upmu$m-QCL based on a strain-balanced heterostructure, demonstrating the need to include off-diagonal density-matrix elements for accurate description of mid-infrared QCLs.

Appendices

Appendix 1: Electronic states

Within the envelope function approximation, the electron eigenfunctions near a band extremum ${\bf k}_0$ ($\varGamma$ valley in this work) can be written as a combination of envelope functions from different bands [61, 62]

$$\begin{aligned} \varPsi _{n{\bf k}}({\bf r},z)= \sum _b \phi _{n,{\bf k}}^{(b)}({\bf r},z)u_{{\bf k}={\bf k}_0}^{(b)}({\bf r},z), \end{aligned}$$

(53)

where the b sum is over the considered bands, $u_{{\bf k}_0}^{(b)}({\bf k},z)$ is a rapidly varying Bloch function that has the period of the lattice, and $\phi _{n,{\bf k}}^{(b)}({\bf r},z)$ are slowly varying envelope functions belonging to band b. We make the approximation that the envelope functions factor into cross-plane and in-plane direction according to

$$\begin{aligned} \phi _{n,{\bf k}}^{(b)}({\bf r},z)= N_{\rm norm} \psi _n^{(b)}(z){\rm e}^{i{\bf k}\cdot {\bf r}}, \end{aligned}$$

(54)

where $\psi _n^{(b)}(z)=\phi _{n,{\bf k}=0}^{(b)}({\bf r},z)$ is the envelope function for ${\bf k}=0$ and $N_{\rm norm}$ is a normalization constant. This separation of in- and cross-plane motion is a standard approximation when describing QCLs and other superlattices. [9] Without it, a separate eigenvalue problem for each ${\bf k}$ would need to be calculated, which is very computationally expensive.

In order to model QCLs based on III-V semiconductors, accurate calculation the $\varGamma$-valley eigenstates is crucial. The simplest approach is the Ben Daniel–Duke model, in which only the conduction band is considered [61]. The Schrödinger equation for the conduction-band envelope function $\psi _n^{(c)}(z)$ in Eq. (54) reduces to the 1D eigenvalue problem [9]

$$\begin{aligned}&\left[ -\frac{\hbar ^2}{2}\frac{{\rm d}}{{\rm d}z}\frac{1}{m^*(z)}\frac{{\rm d}}{{\rm d}z}+V^{(c)}(z) \right] \psi _n^{(c)}(z) =E_n\psi _n^{(c)}(z) \end{aligned}$$

(55a)

$$\begin{aligned}&E_{n,{\bf k}}=E_n + E_k, \end{aligned}$$

(55b)

where $V^{(c)}(z)$ is the conduction-band profile, and $m^*(z)$ is the spatially dependent cross-plane effective mass of the conduction band. Typically, a parabolic dispersion relation $E_k=\hbar ^2k^2/(2m_\parallel ^*)$ is assumed. A single-band approximation such as Eq. (55b) can be justified when electrons have energy close to the conduction-band edge (compared with the bandgap $E_{\rm gap}$) and quantum well widths are larger than $\sim$10 nm [61, 63]. This is the case for THz QCLs, where single-band approximations can give satisfactory results [42, 64]. However, in mid-IR QCLs, the well thicknesses can be below 2 nm [52] and the energy of the lasing transition can exceed 150 meV, which is a significant fraction of typical bandgaps of the semiconductor materials, especially for low-bandgap materials containing InAs. In this case, including nonparabolicity in the electron Hamiltonian is important.

Nonparabolicity has previously been included in semiclassical [9, 53] density-matrix [34, 65], and NEGF-based [32, 34, 66] approaches using single-band models, where only the conduction band is modeled, but the influence of the valence bands is included via an energy-dependent effective mass. One of the most important effects of nonparabolicity is the energetic lowering of the high-energy states, such as the upper lasing level. This effect results from higher effective mass at high energies, reducing the kinetic energy term in the Schrödinger equation [67]. However, an energy-dependent effective mass has an undesirable trait in density-matrix and semiclassical models, the resulting eigenfunctions are not orthogonal [9]. In this work, we will calculate the band structure using a 3-band ${\bf k}\cdot {\bf p}$ model, which includes the conduction (c), light hole (lh) and spin-orbit split-off (so) bands [68]. The three-band ${\bf k}\cdot {\bf p}$ model has previously been used in semiclassical work on QCLs [16, 17]; present work is a quantum-mechanical QCL simulation relying this model.

The three-band Hamiltonian can be derived from an eight-band Hamiltonian for bulk zinc-blende crystals [69], which, at in-plane wave vector ${\bf k}=0$, reduces to three-bands due to spin degeneracy and the heavy-hole band factoring out. The $3\times 3$ Hamiltonian for a strained bulk crystal can be written as a sum of three terms

$$\begin{aligned} H_{\rm B}=H_{\rm B,edge}+H_{\rm B,kin}+H_{\rm B,strain}. \end{aligned}$$

(56)

The first term in Eq. (56) is diagonal and contains the band edges

$$\begin{aligned} H_{\rm B,edge}= \begin{pmatrix} E_{\rm c} &{} 0 &{} 0 \\ 0 &{} E_v &{} 0 \\ 0 &{} 0 &{} E_v-\varDelta _{\rm so} \end{pmatrix}, \end{aligned}$$

(57)

where $E_{\rm c}$ ($E_v$) is the conduction (valence) band edge and $\varDelta _{\rm so}$ is the spin-orbit split-off energy.

The second term in Eq. (56) contains the kinetic contributions as well as coupling between the different bands which involve the cross-plane wavenumber $k_z$

$$\begin{aligned}&H_{\rm B,kin} \nonumber \\&\quad = \begin{pmatrix} (1+2 F) \frac{\hbar ^2k_z^2}{2m_0} &{} \sqrt{\frac{\hbar ^2E_{\rm P}}{3m_0}} k_z &{} \sqrt{\frac{\hbar ^2E_{\rm P}}{6m_0}} k_z \\ \sqrt{\frac{\hbar ^2E_{\rm P}}{3m_0}} k_z &{} - (\gamma _1+2\gamma _2)\frac{\hbar ^2k_z^2}{2m_0} &{} -\sqrt{2}\gamma _2 \frac{\hbar ^2k_z^2}{m_0} \\ \sqrt{\frac{\hbar ^2E_{\rm P}}{6m_0}} k_z &{} -\sqrt{2}\gamma _2 \frac{\hbar ^2k_z^2}{m_0} &{} -\gamma _1 \frac{\hbar ^2k_z^2}{2m_0} \end{pmatrix}, \end{aligned}$$

(58)

with $E_{\rm P}$ the Kane-energy, [70] $m_0$ the free-electron mass, F is a correction to the conduction band effective mass due to higher energy $\varGamma$ valleys [70], $\gamma _1$ and $\gamma _2$ are the modified Luttinger parameters [71] given by [69]

$$\begin{aligned} \gamma _1&=\gamma _1^{L}-\frac{E_{\rm P}}{3E_{\rm gap}+\varDelta _{\rm so}} \\ \gamma _2&=\gamma _2^{L}-\frac{1}{2}\frac{E_{\rm P}}{3E_{\rm gap}+\varDelta _{\rm so}}, \end{aligned}$$

(59)

where $\gamma _1^L$ and $\gamma _2^{L}$ are the standard Luttinger parameters [72] and $E_{\rm gap}$ is the $\varGamma$-valley bandgap. For a brief overview of the different material parameters, we refer the reader to Ref. [73].

The third term in Eq. (56) contains the effects of strain for a layer grown on a (001)-oriented (z-direction) substrate [62],

$$\begin{aligned}&H_{\rm B,strain}&=\begin{pmatrix} A_\varepsilon &{} 0 &{} 0 \\ 0 &{} - P_\varepsilon +Q_\varepsilon &{} \sqrt{2} Q_\varepsilon \\ 0 &{} \sqrt{2}Q_\varepsilon &{} -P_\varepsilon \end{pmatrix}, \end{aligned}$$

(60)

with

$$\begin{aligned} A_\varepsilon&= 2 a_{\rm c} \left( 1- \frac{c_{12}}{c_{11}}\right) \frac{a_0-a}{a} \end{aligned}$$

(61a)

$$\begin{aligned} P_\varepsilon&= 2 a_v \left( 1- \frac{c_{12}}{c_{11}}\right) \frac{a_0-a}{a} \end{aligned}$$

(61b)

$$\begin{aligned} Q_\varepsilon&= - b \left( 1+2\frac{c_{12}}{c_{11}}\right) \frac{a_0-a}{a}, \end{aligned}$$

(61c)

where $a_{\rm c}$, $a_v$ and b are the Bir-Pikus deformation potentials [74] with the sign convention of all being negative, $a_0$ and a are the lattice constant of the substrate and the grown material (in the absence of strain), respectively, and $c_{11}$ and $c_{12}$ are the elastic stiffness constants of the strained material. Numerical values for all material parameters used in this work are given in “Appendix 6”.

Equation (56) is valid for a bulk III-V semiconductors, where $k_z$ refers to crystal momentum in the [001] direction. However, for a heterostructure under bias, $k_z$ is no longer a good quantum number and must be replaced by a differential operator $k_z\rightarrow -i \frac{\partial }{\partial z}$. In addition, all the material properties become functions of position. Care must be taken when applying Eq. (56) to heterostructures because the proper operator symmetrization must be performed to preserve the hermiticity of the Hamiltonian. In this work, we follow the standard operator symmetrization procedure in ${\bf k}\cdot {\bf p}$ theory for heterostructures [75]

$$\begin{aligned} f(z) k_z&\rightarrow \frac{1}{2} \left( f(z) k_z + k_z f(z) \right) , \\ g(z) k_z^2&\rightarrow k_z g(z) k_z, \end{aligned}$$

(62)

where f(z) and g(z) are arbitrary functions of position (e.g., Kane energy, or Luttinger parameter). The electron Hamiltonian for a heterostructure can then be written as

$$\begin{aligned} H_{\rm het}= \begin{pmatrix} H_{\rm c} &{} H_{c,lh} &{} H_{c,so} \\ H_{c,lh}^\dagger &{} H_{lh} &{} H_{lh,so} \\ H_{c,so}^\dagger &{} H_{lh,so}^\dagger &{} H_{so} \end{pmatrix}, \end{aligned}$$

(63)

with

$$\begin{aligned} H_{\rm c}&= E_{\rm c}(z) -\frac{\hbar ^2}{2m_0} \frac{\partial }{\partial z}(1+2 F(z)) \frac{\partial }{\partial z}+ A_\varepsilon (z) , \\ H_{lh}&= E_v(z) + \frac{\hbar ^2}{2m_0} \frac{\partial }{\partial z}(\gamma _1(z)+2\gamma _2(z)) \frac{\partial }{\partial z}\\&- P_\varepsilon (z) + Q_\varepsilon (z) , \\ H_{so}&= E_v(z)-\varDelta _{\rm so}(z)+ \frac{\hbar ^2}{2m_0} \frac{\partial }{\partial z}\gamma _1(z) \frac{\partial }{\partial z}- P_\varepsilon (z) , \\ H_{c,lh}&= -\frac{i}{2} \sqrt{\frac{\hbar ^2E_{\rm P}(z)}{3m_0}} \frac{\partial }{\partial z}- \frac{i}{2} \frac{\partial }{\partial z}\sqrt{\frac{\hbar ^2E_{\rm P}(z)}{3m_0}} , \\ H_{c,so}&= -\frac{i}{2} \sqrt{\frac{\hbar ^2E_{\rm P}(z)}{6m_0}} \frac{\partial }{\partial z}-\frac{i}{2} \frac{\partial }{\partial z}\sqrt{\frac{\hbar ^2E_{\rm P}(z)}{6m_0}} , \\ H_{lh,so}&= \sqrt{2} \frac{\hbar ^2}{m_0} \frac{\partial }{\partial z}\gamma _2(z) \frac{\partial }{\partial z}+ \sqrt{2}Q_\varepsilon (z) . \end{aligned}$$

(64)

From Eq. (64), we see that the electron Hamiltonian now depends on spatial derivatives of material functions such as the Kane energy $E_{\rm P}(z)$ and modified Luttinger parameters $\gamma _1(z)$, and results will be sensitive to how an interface between two different materials is treated. One choice is to treat the material functions as piecewise-constant. However, with that choice, spatial derivatives become delta functions (and derivatives of delta functions) that are difficult to treat numerically. In this work, we will assume that material parameters are smooth functions that have slow variations on length scales shorter than a single monolayer $L_{\rm ML}$ (in heterostructures grown on InP, $L_{\rm ML}\simeq 0.29$ nm at room temperature). The material parameters that we use are obtained using

$$\begin{aligned} f(z)=\frac{1}{\sqrt{\pi } \sigma } \int f_{{\rm pc}}(z') {\rm e}^{-(z-z')^2/\sigma ^2} {\rm d}z' , \end{aligned}$$

(65)

where we use $\sigma =\tfrac{1}{2} L_{\rm ML}$ and $f_{{\rm pc}}(z)$ is a piecewise constant material function that is discontinuous at material interfaces. As an example, Fig. 11 shows the Kane energy as well as the piecewise-constant variant for a single period of the QCL studied in Sect. 4.1, with $\sigma =0.29$ nm.

Fig. 11

Kane energy as a function of position for a single period of the InGaAs/InAlAs-based mid-IR QCL described in Sect. 4.1. The dashed curve shows the piecewise constant version and the solid curve is the smoothed version given by Eq. (65), with $\sigma =0.29$ nm. This is double the value of $\sigma$ we typically use and is chosen for visualization purposes. The region of higher Kane energy corresponds to InGaAs

The eigenfunctions of the Hamiltonian $H_{\rm het}$ given in Eqs. (63)-(64) satisfy

$$\begin{aligned} H_{\rm het}\overline{\psi _n}(z)=E_n\overline{\psi _n}(z), \end{aligned}$$

(66)

where $\overline{\psi _n}$ are three-dimensional vectors of envelope functions $\overline{\psi _n}(z)=\left( \psi ^c_n(z),\psi ^{\rm lh}_n(z),\psi ^{\rm so}_n(z) \right) ^T$. We will use Dirac notation $| n\rangle$, with $\langle z | n \rangle =\bar{\psi _n}(z)$ and $| n,{\bf k}\rangle$ with $\langle {\bf r}, z | n,{\bf k} \rangle =N_{\rm norm} \bar{\psi _n}(z) {\rm e}^{i{\bf k}\cdot {\bf r}}$ and normalization $\langle n,{\bf k} | n',{\bf k}' \rangle =\delta _{n,n'}\delta ({\bf k}-{\bf k}')$. The electron probability density is obtained by summing over the different bands

$$\begin{aligned} |{\overline{\psi }}_n(z)|^2=|\langle z | n \rangle |^2=\sum _b |\psi _n^{b}(z)|^2 \end{aligned}$$

(67)

and matrix elements of an operator B such as $\langle n | B| n' \rangle$ are understood as

$$\begin{aligned} \langle n | B| n' \rangle = \sum _b \int {\rm d}z \left[ \psi _n^{(b)}(z) \right] ^*B^{(b)}\psi _{n'}^{(b)}(z) \ , \end{aligned}$$

(68)

where the sum is over the considered bands, with

$$\begin{aligned} B = \begin{pmatrix} B^{\rm (c)} &{} 0 &{} 0 \\ 0 &{} B^{\rm (lh)} &{} 0 \\ 0 &{} 0 &{} B^{\rm (so)} \end{pmatrix} . \end{aligned}$$

(69)

Note that $B^{(b)}$ is an operation working on component b of the envelope function ${\overline{\psi }}_n(z)$. In this work, all operators are assumed to be identical for all three bands ($B^{(c)}=B^{(lh)}=B^{(so)}$) except for the interaction potential due to interface roughness. The interaction potential due to interface roughness is not the same for all bands because the band discontinuities at material interfaces are not the same for different bands [see Appendix C.1].

Fig. 12

(a) Conduction band $E_{\rm c}(z)$ and valence band $E_v(z)$ used for the calculation of eigenfunctions. Dashed rectangle shows a single period of the conduction band. Potential barriers are imposed to the far left and far right for both conduction and valence bands. Also show, are two of the Hermite functions ($n=0$ and $n=10$) used as basis functions. (b) After discarding the high- and low-energy states, about 80 states are left, most of which have an energy high above the conduction band or are greatly affected by the boundary conditions on the right. (c) After choosing 8 states with the lowest ${{\tilde{E}}}_n$ and discarding copies from adjacent stages (see main text), the states in the central stage (bold curves) are left. States belonging to adjacent periods (thin curves) are calculated by translating states from the central stage in position and energy

The eigenfunctions $| n\rangle$ are calculated in a similar manner as in previous semiclassical work [9, 16, 17]. A typical computational domain is shown in Fig. 12(a), where the conduction band of a single period of the QCL from Ref. [52] is marked by a dashed rectangle. The computational domain contains 3–4 periods, padded with tall potential barriers, placed far away from the central region in order for the boundary conditions not to effect the eigenfunctions near the center of the computational domain. We solve the eigenvalue problem in (66) using a basis of Hermite functions (eigenfunctions of the harmonic oscillator), with two example basis functions shown in Fig. 12(a). We used Hermite functions because they are easy to compute and required less computational resources than finite-difference methods. We typically use a basis of $\sim 400$ Hermite functions for each band, resulting in a matrix with dimension $1200\times 1200$. The diagonalization procedure results in 1200 eigenstates, most of which are far above the conduction band or far below the valence bands. For transport calculations, we only include the states close to the conduction-band edge. This bound-state approach is known to produce spurious solutions, which are states with high amplitude far above the conduction band edge. In order to systematically single out the relevant states, we first “throw away” the states that have energies far ($\sim 1$ eV) below or far above the conduction-band edge. This step typically reduces the number of states to $\sim 100$. Figure 12(b) shows the conduction-band edge along with the remaining eigenstates. Next, we remove the states that are obtained by translation of a state from the previous/next period (e.g., remove duplicates). We do accomplish this by calculating each eigenstate’s “center of mass”,

$$\begin{aligned} z_n =\int z |\psi _n(z)|^2 {\rm d}z, \end{aligned}$$

(70)

and only keep states within a range $z\in [z_{\rm c},z_{\rm c}+L_{\rm P}]$, where $L_{\rm P}$ is the period length. We will refer to this range as the center period. The choice of $z_{\rm c}$ is arbitrary and we typically use $z_{\rm c}=-L_{\rm P}/4$. Lastly, we follow the procedure in Ref. [9] and, for the remaining states, calculate the energy with respect to the conduction-band edge using

$$\begin{aligned} {{\tilde{E}}}_n=E_n-\int E_{\rm c}(z)|\psi _n(z)|^2 {\rm d}z . \end{aligned}$$

(71)

States with ${{\tilde{E}}}_n<0$ are valence-band states and are discarded. We keep $N_{\rm s}$ states, with the lowest positive values of ${{\tilde{E}}}_n$. The value of $N_{\rm s}$ is chosen such that all eigenstates below the conduction band top are included. After this elimination step, $N_{\rm s}$ is the number of states used in transport calculations. The remaining states for $N_{\rm s}=8$ are shown in bold in Fig. 12(c). States belonging to other periods (needed to calculate coupling between different periods) are finally obtained by translation in space and energy of the $N_{\rm s}$ states belonging to the center period. These states are identified by thin curves in Fig. 12(c).

Appendix 2: Dissipator for electron–phonon interaction

Expanding the dissipator (commutators in Eq. (6)) gives 4 terms, which can be split into two Hermitian conjugate pairs

$$\begin{aligned}&{\mathcal {D}}[\rho _{\rm e}(t)]=\int _0^{\infty } ds {\rm Tr}_{\rm ph} \{ \\&\quad +H_{\rm i} \rho _{\rm e}(t)\otimes \rho _{\rm ph} {\rm e}^{-i(H_0+H_{\rm ph})s}H_{\rm i} {\rm e}^{i(H_0+H_{\rm ph})s} \\&\quad -H_{\rm i}{\rm e}^{-i(H_0+H_{\rm ph})s}H_{\rm i}{\rm e}^{i(H_0+H_{\rm ph})s}\rho _{\rm e}(t)\otimes \rho _{\rm ph} \} \\&\quad + {\rm h.c..} \end{aligned}$$

(72)

In order to proceed, we assume a Frölich-type Hamiltonian for the description of an electron interacting with a phonon bath in the single-electron approximation [76]

$$\begin{aligned} H_{\rm i}= \frac{1}{\sqrt{V}} \sum _{{{\bf Q}}} {\mathcal {M}}({{\bf Q}})(b_{{\bf Q}}{\rm e}^{i{{\bf Q}}\cdot {{\bf R}}} -b^\dagger _{{\bf Q}}{\rm e}^{-i{{\bf Q}}\cdot {{\bf R}}}). \end{aligned}$$

(73)

Here, $b_{{\bf Q}}^\dagger$ ($b_{{\bf Q}}$) is the creation (annihilation) operator for a phonon with wave vector ${{\bf Q}}$ and the associated matrix element ${\mathcal {M}}({{\bf Q}})$ and V is the quantization volume. Note that the matrix elements are anti-Hermitian ${\mathcal {M}}({{\bf Q}})^*=-{\mathcal {M}}({{\bf Q}})$, making the interaction Hamiltonian Hermitian.

Inserting Eq. (73) into Eq. (72) gives

$$\begin{aligned} {\mathcal {D}}&=\frac{1}{V}\int _0^{\infty } ds \sum _{{\bf Q},{{\bf Q}}'} {\mathcal {M}}({{\bf Q}})^* {\mathcal {M}}({{\bf Q}}') {\rm Tr}_{\rm ph} \Big \{ \nonumber \\ -&\left( b_{{\bf Q}}{\rm e}^{i{\bf Q}\cdot {\bf R}}-b^\dagger _{{\bf Q}} {\rm e}^{+i{\bf Q}\cdot {\bf R}} \right) \rho _{\rm e}(t)\otimes \rho _{\rm ph} {\rm e}^{-i(H_0+H_{\rm ph})s} \times \nonumber \\&\left( b_{\bf Q'}{\rm e}^{i{\bf Q'}\cdot {\bf R}}-b^\dagger _{\bf Q'} {\rm e}^{-i{\bf Q'}\cdot {\bf R}} \right) {\rm e}^{i(H_0+H_{\rm ph})s} \nonumber \\ +&\left( b_{{\bf Q}}{\rm e}^{i{\bf Q}\cdot {\bf R}}-b^\dagger _{{\bf Q}} {\rm e}^{-i{\bf Q}\cdot {\bf R}} \right) {\rm e}^{-i(H_0+H_{\rm ph})s}\times \nonumber \\&\left( b_{\bf Q'}{\rm e}^{i{\bf Q'}\cdot {\bf R}}-b^\dagger _{\bf Q'} {\rm e}^{-i{\bf Q'}\cdot {\bf R}} \right) {\rm e}^{i(H_0+H_{\rm ph})s} \rho _{\rm e}(t)\otimes \rho _{\rm ph} \Big \} \nonumber \\ +&{\rm h.c..} \end{aligned}$$

(74)

In order to proceed, we use the fact that the thermal equilibrium-phonon density matrix is diagonal, so only terms with ${{\bf Q}}={{\bf Q}}'$ survive the trace over the phonon degree of freedom. We also use that only terms containing both creation and annihilation operators are nonzero after performing the trace. Using the two simplifications, we get

$$\begin{aligned}&{\mathcal {D}}=\frac{1}{V}\int _0^{\infty } ds \sum _{{\bf Q}}|{\mathcal {M}}({{\bf Q}})|^2 {\rm Tr}_{\rm ph} \Big \{ \nonumber \\&\quad +b_{{\bf Q}}{\rm e}^{i{\bf Q}\cdot {\bf R}} \rho _{\rm e}(t)\otimes \rho _{\rm ph} {\rm e}^{-iH_0s}{\rm e}^{-iH_{\rm ph}s} b_{{\bf Q}}^\dagger {\rm e}^{iH_{\rm ph}s} {\rm e}^{-i{\bf Q}\cdot {\bf R}}{\rm e}^{iH_0s} \nonumber \\&\quad +b_{{\bf Q}}^\dagger {\rm e}^{-i{\bf Q}\cdot {\bf R}} \rho _{\rm e}(t)\otimes \rho _{\rm ph} {\rm e}^{-iH_0s}{\rm e}^{-iH_{\rm ph}s} b_{{\bf Q}}{\rm e}^{iH_{\rm ph}s} {\rm e}^{i{\bf Q}\cdot {\bf R}}{\rm e}^{iH_0s} \nonumber \\&\quad -b_{{\bf Q}}{\rm e}^{i{\bf Q}\cdot {\bf R}}{\rm e}^{-iH_0s}{\rm e}^{-iH_{\rm ph}s}b_{{\bf Q}}^\dagger {\rm e}^{iH_{\rm ph}s} {\rm e}^{-i{\bf Q}\cdot {\bf R}}{\rm e}^{iH_0 s} \rho _{\rm e}(t)\otimes \rho _{\rm ph} \nonumber \\&\quad -b_{{\bf Q}}^\dagger {\rm e}^{-i{\bf Q}\cdot {\bf R}}{\rm e}^{-iH_0s}{\rm e}^{-iH_{\rm ph}s}b_{{\bf Q}}{\rm e}^{iH_{\rm ph}s} {\rm e}^{i{\bf Q}\cdot {\bf R}}{\rm e}^{iH_0s}\rho _{\rm e}(t)\otimes \rho _{\rm ph} \Big \} \nonumber \\&\quad + {\rm h.c..} \end{aligned}$$

(75)

Performing the trace, and going from a sum over ${{\bf Q}}$ to an integral, we get

$$\begin{aligned}&{\mathcal {D}} = \frac{1}{(2\pi )^3} \sum _{\pm } \int _0^{\infty }ds \int d^3Q |{\mathcal {M}}({{\bf Q}})|^2\nonumber \\&\quad \times \big \{ \left( \tfrac{1}{2} \mp \tfrac{1}{2} +N_{{\bf Q}} \right) {\rm e}^{\mp iE_{{\bf Q}} s}{\rm e}^{i{\bf Q}\cdot {\bf R}}\rho _{\rm e}(t) {\rm e}^{-iH_0s}{\rm e}^{-i{\bf Q}\cdot {\bf R}}{\rm e}^{iH_0s} \nonumber \\&\quad -\left( \tfrac{1}{2} \mp \tfrac{1}{2} + N_{{\bf Q}} \right) {\rm e}^{\pm iE_{{\bf Q}} s}{\rm e}^{-i{\bf Q}\cdot {\bf R}}{\rm e}^{-iH_0s}{\rm e}^{i{\bf Q}\cdot {\bf R}}{\rm e}^{iH_0s} \rho _{\rm e}(t) \big \} \nonumber \\&\quad +{\rm h.c.,} \end{aligned}$$

(76)

where $E_{{\bf Q}}$ refers to the energy of a phonon with wave vector ${{\bf Q}}$ and we have defined the expectation value of the phonon number operator ${\rm Tr}_{\rm ph} \{\rho _{\rm ph} b_{{\bf Q}}^\dagger b_{{\bf Q}}\}=N_{{\bf Q}}$. Note that the upper (lower) sign refers to absorption (emission) terms. In the following, we will assume phonons are dispersionless, so $E_{\bf Q}=E_0$ and $N_{{\bf Q}}=N_0=({\rm e}^{E_0/k_{\rm B}T}-1)^{-1}$, T is the lattice temperature, and $k_{\rm B}$ is the Boltzmann constant. In order to keep expressions more compact, we define the scattering weight

$$\begin{aligned} {\mathcal {W}}^\pm ({{\bf Q}})=\left( \tfrac{1}{2} \mp \tfrac{1}{2} + N_0 \right) |{\mathcal {M}}({{\bf Q}})|^2. \end{aligned}$$

(77)

We can then write the dissipator due to a single dispersionless phonon type as (after reintroduction of $\hbar$),

$$\begin{aligned}&{\mathcal {D}} = \frac{1}{\hbar ^2(2\pi )^3} \sum _{\pm } \int _0^{\infty }ds \int d^3Q \big \{ \nonumber \\ +&\mathcal W^{\pm }({{\bf Q}}){\rm e}^{\mp iE_0 s/\hbar } {\rm e}^{i{\bf Q}\cdot {\bf R}}\rho _{\rm e}(t){\rm e}^{-iH_0s/\hbar }{\rm e}^{-i{\bf Q}\cdot {\bf R}}{\rm e}^{iH_0s/\hbar } \nonumber \\ -&{\mathcal {W}}^{\pm }({{\bf Q}}){\rm e}^{\pm iE_0 s/\hbar }{\rm e}^{-i{\bf Q}\cdot {\bf R}}{\rm e}^{-iH_0s/\hbar }{\rm e}^{i{\bf Q}\cdot {\bf R}}{\rm e}^{iH_0s/\hbar } \rho _{\rm e}(t) \big \} \nonumber \\ +&{\rm h.c.,} \end{aligned}$$

(78)

where the total dissipator will be obtained later by summing over the different relevant phonon branches. The dissipator can be written as ${\mathcal {D}}={\mathcal {D}}^{{\rm in}}+\mathcal D^{{\rm out}}$, corresponding to the positive and negative terms in (78), respectively. We will derive the form of the matrix elements of the in-scattering term. The out-scattering term is less complicated and can be obtained by inspection of the in-scattering term and only the final results will be given.

By using the completeness relation four times, we can write

$$\begin{aligned} {\mathcal {D}}^{{\rm in}}&=\frac{1}{\hbar ^2(2\pi )^3} \int d^3Q\int _0^{\infty }ds \sum _{n_{1234},\pm }\int d^2k_{1234} \nonumber \\&\times \langle n_1,{\bf k}_1 | {\rm e}^{i{\bf Q}\cdot {\bf R}}| n_2,{{\bf k}}_2 \rangle {\rm e}^{\mp iE_0 s/\hbar } {\mathcal {W}}^{\pm }({{\bf Q}}) \nonumber \\&\times \langle n_3,{\bf k}_3 | {\rm e}^{-iH_0s/\hbar }{\rm e}^{-i{\bf Q}\cdot {\bf R}}{\rm e}^{iH_0s/\hbar }| n_4,{\bf k}_4 \rangle \nonumber \\&\times \langle n_2,{\bf k}_2 | \rho _{\rm e}| n_3,{{\bf k}}_3 \rangle | n_1,{{\bf k}}_1 \rangle \langle n_4,{\bf k}_4 | + {\rm h.c.,} \end{aligned}$$

(79)

where $\sum {n_{1234}}$ and $\int d^2 k_{1234}$ refer to summation over $n_1$ through $n_4$ and integration of ${{\bf k}}_1$ through ${{\bf k}}_4$, respectively. Integration of ${{\bf k}}_2$ through ${{\bf k}}_4$ and shifting the integration variable ${\bf Q}\rightarrow -{{\bf Q}}+{{\bf k}}_1$ gives

$$\begin{aligned} {\mathcal {D}}^{{\rm in}} &=\frac{1}{\hbar ^2(2\pi )^3} \int d^3Q\int _0^{\infty }ds \sum _{n_{1234},\pm }\int d^2k_{1} \nonumber \\&\quad\times {\mathcal {W}}^\pm ({{\bf Q}}-{{\bf k}}_1) \left( n_1|n_2\right) _{Q_z} \left( n_3|n_4\right) _{Q_z} ^* \rho _{n_2,n_3}^{E_q} \nonumber \\&\quad\times {\rm e}^{-i\frac{s}{\hbar }(\varDelta _{n_3,n_4}-E_{k_1}\pm E_0+E_q ) } | n_1,{{\bf k}}_1 \rangle \langle n_4,{{\bf k}}_1 | + {\rm h.c.,} \end{aligned}$$

(80)

where $E_k$ is the in-plane dispersion relation, $q=\sqrt{Q_x^2+Q_y^2}$ is the magnitude of the in-plane component of ${\bf Q}$, $\varDelta _{nm}=E_n-E_m$ is the energy spacing of the cross-plane eigenfunctions and

$$\begin{aligned} \left( n|m\right) _{Q_z} = \sum _b \int _{-\infty }^{\infty } [ \psi ^{(b)}_n(z) ]^* {\rm e}^{iQ_zz}\psi ^{(b)}_m(z) {\rm d}z\ , \end{aligned}$$

(81)

where the sum is over the considered bands (conduction, light-hole and split-off bands in this work). In order to perform the s integration, we use

$$\begin{aligned} \int _0^\infty {\rm e}^{-i\varDelta s/\hbar }=\pi \hbar \delta (\varDelta )-i\hbar {\mathcal {P}}\frac{1}{\varDelta }, \end{aligned}$$

(82)

where ${\mathcal {P}}$ denotes the Cauchy principal value, which causes a small shift to energies called the Lamb shift. [44] By omitting the Lamb shift term and dropping the subscript on the ${{\bf k}}_1$ integration variable, we get

$$\begin{aligned} {\mathcal {D}}^{{\rm in}}&=\frac{\pi }{\hbar (2\pi )^3} \sum _{n_{1234},\pm }\int d^2k \int d^3Q {\mathcal {W}}^\pm ({{\bf Q}}-{{\bf k}}) \nonumber \\&\left( n_1|n_2\right) _{Q_z} \left( n_3|n_4\right) _{Q_z} ^* \rho _{n_2,n_3}^{E_k+\varDelta _{n_4n_3}\mp E_0} \nonumber \\&\delta [E_q-(E_k+\varDelta _{n_4n_3}\mp E_0)] | n_1,{\bf k} \rangle \langle n_4,{{\bf k}} | + {\rm h.c..} \end{aligned}$$

(83)

Multiplying by $\langle N |$ from the left and $| N+M\rangle$ from the right and renaming the sum variables $n_2\rightarrow n$ and $n_3\rightarrow m$ gives the matrix elements of the in-scattering contribution of the dissipator, giving

$$\begin{aligned} \left( \frac{\partial f_{N,M}^{E_k}}{\partial t} \right) ^{\rm in}&= \frac{\pi }{\hbar (2\pi )^3} \sum _{n,m,\pm }\int d^3Q \mathcal W^\pm ({{\bf Q}}-{{\bf k}}) \nonumber \\&\left( N|n\right) _{Q_z} \left( m|N+M\right) _{Q_z} ^* \times \rho _{n,m}^{E_k+\varDelta _{N+M,m}\mp E_0} \nonumber \\&\delta [E_q-(E_k+\varDelta _{N+M,m}\mp E_0)] + {\rm h.c..} \end{aligned}$$

(84)

The sums involving n and m are over all integers. However, it is convenient due to numerical reasons to shift the sum variables in such a way that terms with small |n| and |m| dominate, and terms with large |n| and |m| approach zero. This can be done by making the switch $n\rightarrow N+n$ and $m\rightarrow N+M+m$, giving

$$\begin{aligned}&\left( \frac{\partial f_{N,M}^{E_k}}{\partial t} \right) ^{\rm in} = \frac{\pi }{\hbar (2\pi )^3} \sum _{n,m,\pm }\int d^3Q \mathcal W^\pm ({{\bf Q}}-{{\bf k}}) \times \nonumber \\&\left( N|N+n\right) _{Q_z} \left( N+M+m|N+M\right) _{Q_z} ^* \times \nonumber \\&\rho _{N+n,N+M+m}^{E_k+\varDelta _{N+M,N+M+m}\mp E_0} \delta [E_q-(E_k+\varDelta _{N+M,N+M+m}\mp E_0)] \nonumber \\&\quad + {\rm h.c..} \end{aligned}$$

(85)

In order to write the RHS in Eq. (85) in terms of $f_{N,M}^{E_k}$, we use

$$\begin{aligned} \rho ^{E_k}_{N+n,N+M+m}=\rho ^{E_k}_{N+n,N+n+(M+m-n)}=f_{N+n,M+m-n}^{E_k} \end{aligned}$$

(86)

and get

$$\begin{aligned}&\left( \frac{\partial f_{N,M}^{E_k}}{\partial t} \right) ^{\rm in} = \frac{\pi }{\hbar (2\pi )^3} \sum _{n,m,\pm }\int d^3Q \mathcal W^\pm ({{\bf Q}}-{{\bf k}}) \nonumber \\&\left( N|N+n\right) _{Q_z} \left( N+M+m|N+M\right) _{Q_z} ^* \times \nonumber \\&f_{N+n,M+m-n}^{E_k+\varDelta _{N+M,N+M+m}\mp E_0} \delta [E_q-(E_k+\varDelta _{N+M,N+M+m}\mp E_0)] \nonumber \\&\quad + {\rm h.c..} \end{aligned}$$

(87)

or

$$\begin{aligned} \left( \frac{\partial f_{N,M}^{E_k}}{\partial t} \right) ^{\rm in}&= \sum _{n,m,\pm }\varGamma _{NMnmE_k}^{\pm ,{\rm in}} f_{N+n,M+m-n}^{E_k+\varDelta _{N+M,N+M+m}\mp E_0} \nonumber \\&\quad + {\rm h.c.}, \end{aligned}$$

(88)

where we have defined the in-scattering rates

$$\begin{aligned} \varGamma _{NMnmE_k}^{\pm ,{\rm in}}&= \frac{\pi }{\hbar (2\pi )^3} \int d^3Q {\mathcal {W}}^\pm ({{\bf Q}}-{\bf k}) \nonumber \\&\left( N|N+n\right) _{Q_z} \left( N+M+m|N+M\right) _{Q_z} ^* \times \nonumber \\&\delta [E_q-(E_k+\varDelta _{N+M,N+M+m}\mp E_0)]. \end{aligned}$$

(89)

Note that the RHS in Eq. (89) looks like it depends on the direction of ${{\bf k}}$. However, the scattering weight only depends on the magnitude $|{{\bf Q}}-{{\bf k}}|$, so the coordinate system for the ${{\bf Q}}$ integration can always been chosen relative to ${{\bf k}}$, which means the RHS only depends on the magnitude $|{{\bf k}}|$.

In order to proceed, we need to specify the in-plane dispersion relation $E_k$, as well as the scattering mechanism, which determines the form of the scattering weight ${\mathcal {W}}^\pm ({\bf Q}-{{\bf k}})$. In this work, we will assume a parabolic dispersion relation $E_k=\hbar ^2k^2/2m_\parallel ^*$, with a single in-plane effective mass $m_\parallel ^*$, defined as the average effective mass of the $N_{\rm s}$ considered eigenstates

$$\begin{aligned} \frac{1}{m_\parallel ^*}=\frac{1}{N_{\rm s}} \sum _{n=1}^{N_{\rm s}}\sum _b \int \frac{|\psi _n^{(b)}(z)|^2}{m_b^*(z)}{\rm d}z , \end{aligned}$$

(90)

where $m^*_b(z)$ is the position-dependent effective mass of band b, calculated using the procedure given in “Appendix 6”. The approximation of a parabolic in-plane dispersion relation can fail in short-wavelength QCLs, where electrons can have in-plane energies comparable to the material bandgap [53]. It is possible to go beyond a parabolic dispersion relation by including a term proportional to $k^4$ (in-plane nonparabolicity). However, the inclusion of a $k^4$ term would significantly complicate the integration over q in Eq. (89) and is beyond the scope of this work.

Since we do not have an analytical expression for the eigenfunctions, the products $\left( n|m\right) _{Q_z}$ have to be calculated numerically. For a more compact notation, we define

$$\begin{aligned} I_{n,m,k,\ell }(Q_z)= \left( n|m\right) _{Q_z} \left( k|\ell \right) _{Q_z} ^* \ . \end{aligned}$$

(91)

Switching to cylindrical coordinates $d^3Q\rightarrow q dq d\theta dQ_z$, making the change of variables $E_q=\hbar ^2q^2/2m_\parallel ^*$ and integrating over $E_q$ gives

$$\begin{aligned}&\varGamma _{NMnmE_k}^{\pm ,{\rm in}} = \frac{m_\parallel ^*}{2\pi \hbar ^3}u[E_k+\varDelta _{N+M,N+M+m}\mp E_{\rm s}] \nonumber \\&\quad \times \int _{0}^{\infty } dQ_z I_{N,N+n,N+M+m,N+M}(Q_z) \nonumber \\&\quad \times \left[ \frac{1}{2\pi } \int _0^{2\pi } d\theta {\mathcal {W}}^\pm ({{\bf Q}}-{{\bf k}}) \right] _{E_q=E_k+\varDelta _{N+M,N+M+m}\mp E_0} , \end{aligned}$$

(92)

where the term in the square brackets is the scattering weight averaged over the angle between ${{\bf Q}}$ and ${{\bf k}}$, with the in-plane part of ${{\bf Q}}$ constrained by energy conservation, and u[x] is the Heaviside function, which is zero when no positive value of $E_q$ can satisfy energy conservation. Note that we have also taken advantage of the fact that the real part of the integrand is even in $Q_z$, allowing us to limit the range of integration to positive $Q_z$. The resulting expressions from the angle-averaged scattering weight can be quite cumbersome, so we define the angle-averaged scattering weight

$$\begin{aligned} \widetilde{\mathcal W}^\pm (E_k,Q_z,E_q)=\frac{1}{2\pi } \int _0^{2\pi } d\theta \mathcal W^\pm ({{\bf Q}}-{{\bf k}}), \end{aligned}$$

(93)

simplifying the in-scattering rate to

$$\begin{aligned} \varGamma _{NMnmE_k}^{\pm ,{\rm in}}&= \frac{m_\parallel ^*}{2\pi \hbar ^3}u[E_k+\varDelta _{N+M,N+M+m}\mp E_0] \nonumber \\&\quad \times \int _{0}^{\infty } dQ_z I_{N,N+n,N+M+m,N+M}(Q_z) \nonumber \\&\quad \times \widetilde{\mathcal W}^\pm (E_k,Q_z,E_k+\varDelta _{N+M,N+M+m}\mp E_0) . \end{aligned}$$

(94)

In order to proceed, we need to specify the scattering weight, which depends on the kind of phonon interaction being considered. Calculations of scattering rates for longitudinal acoustic phonons and polar optical phonons are performed in “Appendix 4”.

Appendix 3: Dissipator for elastic scattering mechanisms

In addition to electron–phonon scattering, there are numerous elastic scattering mechanisms that are relevant to QCLs. The most important ones are interface roughness, ionized impurities, and alloy scattering [9, 31, 33]. The derivation of the corresponding scattering rates is slightly different from the case for phonons and will be demonstrated for the case of interface roughness, which is a very important elastic scattering mechanism for both THz and mid-IR QCLs [50, 77, 78] and has been found to decrease the upper lasing state lifetime by a factor of 2 at room temperature in mid-IR QCLs [78]. For other elastic scattering mechanisms, we only briefly discuss the interaction potential and give final expressions for rates in “Appendix 4”.

1.1 Interface roughness

For an electron in band b, deviation from a perfect interface between two different semiconductor material can be modeled using an interaction Hamiltonian on the form [79]

$$\begin{aligned} H_{\rm IR}^{(b)}= \sum _{\rm i} \varDelta _{\rm i}({\bf r}) \left. \frac{\partial V^{(b)}}{\partial z}\right| _{z=z_{\rm i}} \simeq \sum _{\rm i} \varDelta V_{\rm i}^{(b)} \delta (z-z_{\rm i}) \varDelta _{\rm i}({\bf r}) \ , \end{aligned}$$

(95)

where i labels an interface located at position $z_{\rm i}$, $V^{(b)}(z)$ the band edge corresponding to band b and $\varDelta V_{\rm i}^{(b)}$ is the band discontinuity at interface i. The function $\varDelta _{\rm i}({\bf r})$ represents the deviation from a perfect interface at different in-plane positions ${\bf r}$. For the conduction band of an InGaAs/InAlAs heterostructure, the sign of $\varDelta V_{\rm i}^{(b)}$ is positive when going from well material to barrier material (from the left) and negative when going from barrier to well material. For the light-hole and split-off bands, the signs are opposite (with respect to the conduction band).

The introduction of the interaction Hamiltonian in Eq. (95) breaks translational invariance in the xy-plane, and the resulting density matrix will no longer be diagonal in the in-plane wave vector. In order to get around this limitation, we follow the treatment in Refs. [9, 31, 80] and only consider statistical properties of the interface roughness, which is contained in the correlation function

$$\begin{aligned} \langle \varDelta _{\rm i}({\bf r}) \varDelta _j({\bf r}')\rangle =\delta _{ij} C(|{\bf r} - {\bf r}'|) \end{aligned}$$

(96)

where $\langle ... \rangle$ denotes an average over the macroscopic interface area, $C(|{\bf r}|)$ is the spatial autocorrelation function and the Kronecker delta function means that we neglect correlations between different interfaces. In this work, we employ a Gaussian correlation function $C(|{\bf r}|)=\varDelta ^2{\rm e}^{-|{\bf r}|^2/\varLambda ^2}$, characterized by standard deviation $\varDelta$ and correlation length $\varLambda$. However, calculations will be taken as far as possible without specifying a correlation function. When calculating rates, we will mainly use its Fourier transform,

$$\begin{aligned} C(|{\bf q}|)= \int d^2 C(|{\bf r}|){\rm e}^{-i{\bf r}\cdot {\bf q}}. \end{aligned}$$

(97)

The derivation for the interface-roughness scattering rates proceeds the same way as for electron–phonon scattering until Eq. (72). For the case of interface roughness, there is no phonon degree of freedom to trace over and we get

$$\begin{aligned} {\mathcal {D}}_{\rm IR}&=\int _0^{\infty } ds \{ +H_{\rm IR}\rho _{\rm e}(t) {\rm e}^{-iH_0 s}H_{\rm IR} {\rm e}^{iH_0 s} \nonumber \\&\quad -H_{\rm IR} {\rm e}^{-iH_0 s} H_{\rm IR}{\rm e}^{iH_0s}\rho _{\rm e}(t) \} + {\rm h.c..} \end{aligned}$$

(98)

After using the completeness relation 4 times and performing the s integration, the in-scattering term in Eq. (98) becomes

$$\begin{aligned} {\mathcal {D}}_{\rm IR}^{\rm in}&=\frac{\pi }{\hbar }\sum _{n_{1234}} \int d^2 k_{1234} \delta [ E(n_4,{\bf k}_4)-E(n_3,{\bf k}_3)] \\&\langle n_2,{\bf k}_2 | \rho _{\rm e}| n_3,{\bf k}_3 \rangle \times \langle n_1,{\bf k}_1 | H_{\rm IR}| n_2,{\bf k}_2 \rangle \\&\langle n_3,{\bf k}_3 | H_{\rm IR}| n_4,{\bf k}_4 \rangle | n_1,{\bf k}_1 \rangle \langle n_4,{\bf k}_4 | + {\rm h.c.}. \end{aligned}$$

(99)

Integrating over ${\bf k}_3$ gives

$$\begin{aligned} \mathcal D_{\rm IR}^{\rm in}&=\frac{\pi }{\hbar }\sum _{n_{1234}} \int d^2 k_{124} \delta [ E(n_4,{\bf k}_4)-E(n_3,{\bf k}_2)] \rho _{n_2,n_3}^{E_{k_2}} \nonumber \\&\quad \times \langle n_1,{\bf k}_1 | H_{\rm IR}| n_2,{\bf k}_2 \rangle \langle n_3,{\bf k}_2 | H_{\rm IR}| n_4,{\bf k}_4 \rangle \nonumber \\&\quad \times | n_1,{\bf k}_1 \rangle \langle n_4,{\bf k}_4 | + {\rm h.c.} \end{aligned}$$

(100)

Now let us focus on the part of the integrand containing the interaction Hamiltonian

$$\begin{aligned}&\langle n_1,{\bf k}_1 | H_{\rm IR}| n_2,{\bf k}_2 \rangle \langle n_3,{\bf k}_2 | H_{\rm IR}| n_4,{\bf k}_4 \rangle \nonumber \\&\quad = \sum _{i,j} \left( n_1|n_2\right) _{z_{\rm i},{\rm IR}} \left( n_3|n_4\right) _{z_j,{\rm IR}} ^* \nonumber \\&\quad \times \int d^2 r d^2 r' \langle {\bf r} | {\bf k}_1 \rangle ^*\langle {\bf r} | {\bf k}_2 \rangle \langle {\bf r}' | {\bf k}_2 \rangle ^*\langle {\bf r}' | {\bf k}_4 \rangle \varDelta _{\rm i}({\bf r})\varDelta _j({\bf r}'), \end{aligned}$$

(101)

where we have defined

$$\begin{aligned} \left( n|m\right) _{z_{\rm i},{\rm IR}} = \sum _b \varDelta V_{\rm i}^{(b)} \psi _n^*(z_{\rm i})\psi _m(z_{\rm i}), \end{aligned}$$

(102)

where the sum is over the conduction (c), light hole (lh), and spin-orbit split-off bands (so). We use a convention where the band discontinuity $\varDelta V_{\rm i}^{(b)}$ is positive for the conduction band. In order to proceed, we replace the product $\varDelta _{\rm i}({\bf r})\varDelta _j({\bf r}')$ with the Fourier decomposition of its spatial average using Eqs. (96)-(97) and get

$$\begin{aligned}&\langle n_1,{\bf k}_1 | H_{\rm IR}| n_2,{\bf k}_2 \rangle \langle n_3,{\bf k}_2 | H_{\rm IR}| n_4,{\bf k}_4 \rangle \nonumber \\&\quad = \sum _{i} \left( n_1|n_2\right) _{z_{\rm i},{\rm IR}} \left( n_3|n_4\right) _{z_{\rm i},{\rm IR}} ^* \frac{1}{(2\pi )^2} \nonumber \\&\int d^2q \delta [{\bf k}_2-({\bf k}_1-{\bf q})]\delta [{\bf k}_4 -({\bf k}_2+{\bf q})] C(|{\bf q}|) \ . \end{aligned}$$

(103)

Inserting Eq. (103) into Eq. (100) and integrating over ${\bf k}_4$ and ${\bf k}_2$, dropping the subscript index of ${\bf k}_1$ and shifting the integration variable ${\bf q}\rightarrow -{\bf q}+ {\bf k}$ gives

$$\begin{aligned} \mathcal D_{\rm IR}^{\rm in}&=\frac{\pi }{\hbar (2\pi )^2} \sum _{n_{1234}} \int d^2k \int d^2q \delta [E_q-\varDelta _{n_4,n_3}-E_{k}] \nonumber \\&\sum _{\rm i} \left( n_1|n_2\right) _{z_{\rm i},{\rm IR}} \left( n_3|n_4\right) _{z_{\rm i},{\rm IR}} ^* C(|{\bf q}-{\bf k}|) \nonumber \\&\rho _{n_2,n_3}^{E_q} | n_1,{\bf k} \rangle \langle n_4,{\bf k}_1 | + {\rm h.c..} \end{aligned}$$

(104)

Switching to polar coordinates $d^2q\rightarrow q dqd\theta$ and performing the change of variables $E_q=\hbar ^2q^2/(2m_\parallel ^*)$ gives

$$\begin{aligned} {\mathcal {D}}_{\rm IR}^{\rm in}&=\frac{m_\parallel ^*}{2\hbar ^3} \sum _{n_{1234}} \int d^2k \sum _{\rm i} \left( n_1|n_2\right) _{z_{\rm i},{\rm IR}} \left( n_3|n_4\right) _{z_{\rm i},{\rm IR}} ^* \nonumber \\&\times \frac{1}{2\pi }\int _0^{2\pi } \int _0^{\infty } {\rm d}E_q \delta [E_q-\varDelta _{n_4,n_3}-E_{k}] \nonumber \\&\times C(|{\bf q}-{\bf k}|) \rho _{n_2,n_3}^{E_q} | n_1,{\bf k} \rangle \langle n_4,{\bf k}_1 | + {\rm h.c..} \end{aligned}$$

(105)

Performing the $E_q$ integration, renaming the dummy variables $n_3\rightarrow m$ and $n_2\rightarrow n$ and multiplying from the left by $\langle N |$ and from the right with $| N+M\rangle$ gives

$$\begin{aligned}&\left( \frac{\partial f_{N,M}^{E_k}}{\partial t} \right) ^{\rm in}_{\rm IR} =\frac{m_\parallel ^*}{2\hbar ^3} \sum _{n,m} u[\varDelta _{N+M,m}+E_k] \\&\quad \times \sum _{\rm i} \left( N|n\right) _{z_{\rm i},{\rm IR}} \left( m|N+M\right) _{z_{\rm i},{\rm IR}} ^* \\&\quad \times {{\tilde{C}}}(E_k,\varDelta _{N+M,m}+E_k) \rho _{nm}^{\varDelta _{N+M,m}+E_k} + {\rm h.c..} \end{aligned}$$

(106)

where we have defined the angle-averaged correlation function

$$\begin{aligned}&{{\tilde{C}}}(E_k,E_q) \nonumber \\&\quad = \frac{1}{2\pi }\int _0^{2\pi }d \theta C\left( \sqrt{\frac{2m_\parallel ^*}{\hbar ^2}} \sqrt{E_k+E_q-2\sqrt{E_kE_q}\cos (\theta )} \right) . \end{aligned}$$

(107)

Shifting the sum variables $n\rightarrow N+n$, $m\rightarrow N+M+m$ and using Eq. (86), we can write

$$\begin{aligned}&\left( \frac{\partial f_{N,M}^{E_k}}{\partial t} \right) ^{\rm in}_{\rm IR} =\frac{m_\parallel ^*}{2\hbar ^3} \sum _{n,m} u[\varDelta _{N+M,N+M+m}+E_k] \nonumber \\&\quad \times \sum _{\rm i} \left( N|N+n\right) _{z_{\rm i},{\rm IR}} \left( N+M+m|N+M\right) _{z_{\rm i},{\rm IR}} ^* \nonumber \\&\quad \times {{\tilde{C}}}(E_k,\varDelta _{N+M,N+M+m}+E_k) f_{N+n,M+m-n}^{\varDelta _{N+M,N+M+m}+E_k} + {\rm h.c.} \nonumber \\&\quad =2 \sum _{n,m} \varGamma ^{{\rm in},{\rm IR},\pm }_{NMnmE_k} f_{N+n,M+m-n}^{\varDelta _{N+M,N+M+m}+E_k} + {\rm h.c.}, \end{aligned}$$

(108)

where we have defined $\varGamma ^{{\rm in},{\rm IR},\pm }_{NMnmE_k}$, the in-scattering rate due to interface roughness. The purpose of the factor of two in the definition of the rate is so we can write

$$\begin{aligned} \left( \frac{\partial f_{N,M}^{E_k}}{\partial t} \right) ^{\rm in}_{\rm IR}&=\sum _{n,m,\pm } \varGamma ^{{\rm in},{\rm IR},\pm }_{NMnmE_k} f_{N+n,M+m-n}^{\varDelta _{N+M,N+M+m}+E_k} + {\rm h.c.}. \end{aligned}$$

(109)

In other words, we have split the interface roughness scattering into two identical absorption and emission terms to be consistent with the notation for phonon scattering in “Appendix 2”. In order to calculate the rates, we need to choose a correlation function. The rates are calculated in “Appendix 4” for a Gaussian correlation function, which is often used to model QCLs [9, 34]. The reason we choose a Gaussian correlation function is that it results in a simple analytical expression for the angle-averaged correlation function ${{\tilde{C}}}$, given by Eq. (107). Another popular choice is an exponential correlation function [9, 31, 33]. However, angle-averaged expressions for the correlation function are more complicated, involving elliptical integrals [9, 31].

1.2 Ionized impurities

The effects of the long-range Coulomb interaction from ionized impurities are already included to lowest order with a mean-field treatment (Poisson’s equation). The goal of this section is to include higher order effects by including scattering of electrons with ionized impurities. We assume a 3D doping density of the form

$$\begin{aligned} N(z)=N_{\rm 2D}\delta (z-z_\ell ) \ , \end{aligned}$$

(110)

which is a delta-doped sheet at cross-plane position $z_\ell$. The advantage of Eq. (110) is that an arbitrary doping density can be approximated by a linear combination of closely spaced delta-doped sheets. Typically, in QCLs, only one [64] or a few [52, 59] layers are doped and the doping is kept away from the active region to minimize nonradiative transitions from the upper to lower lasing level. Note that, in this context, a layer refers to a barrier or a well, not atomic layers. The corresponding scattering rates due to the ionized-impurity scattering distribution in Eq. (110) is calculated in Appendix E.4.

1.3 Random alloy scattering

In ternary semiconductor alloys $A_xB_{1-x}C$, the energy of band b can be calculated using the virtual crystal approximation, [79] where the alloy band energy is obtained by averaging over the band profiles of the two materials AC and BC

$$\begin{aligned} V_{b,{\rm VCA}}({\bf R})&= \sum _n \big [ x V_{b,{\rm AC}}({\bf R}-{\bf R}_n) \\&\quad + (1-x)V_{b,{\rm BC}}({\bf R}-{\bf R}_n) \big ], \end{aligned}$$

(111)

where the sum is over all unit cells n and $V_{b,{\rm AC}}({\bf R}-{\bf R}_n)$ and $V_{b,{\rm BC}}({\bf R}-{\bf R}_n)$ are the corresponding bands of the AC and BC material, respectively. The difference between an alloy and a binary compound is that, in alloys, periodicity of the crystal is broken by the random distribution of AC and BC unit cells. This deviation from a perfect lattice can be modeled as a scattering potential, which can be considered much weaker than Eq. (111). The scattering potential is given by the difference between the real band energy and the band energy obtained using the virtual crystal approximation. We will follow the treatment of alloy scattering in Refs. [9, 33, 81] and write the scattering potential (which we assume is the same for all bands) as

$$\begin{aligned} V_{\rm alloy}({\bf R})= \sum _n C_n^x\delta ({\bf R}-{\bf R}_n), \end{aligned}$$

(112)

where the sum is over all unit cells and $C_n^x$ is a random variable, analogous to the interface-roughness correlation function in Eq. (96), with

$$\begin{aligned} \langle C_n^x \rangle&=0 \\ \langle C_n^x C_m^x \rangle&= \delta _{nm} x_n(1-x_n) (V_n \varOmega )^2, \end{aligned}$$

(113)

where $\varOmega =a^3/4$ is equal to one-fourth of the volume of a unit cell and $V_n$ the alloy scattering potential in unit cell n, which can be approximated as the conduction band offset of the two binary materials, corresponding to $x=0$ and $x=1$. The presence of the Kronecker delta function $\delta _{nm}$ means that the location of A and B atoms are assumed to be completely uncorrelated. The derivation of corresponding scattering rates proceeds in the same way as for interface roughness scattering and results are given in “Appendix 4”. The final expressions for the rates are considerably simpler than the ones due to interface roughness due to the local nature of the interaction potential in Eq. (112).

Appendix 4: Is the dissipator really of the Lindblad form?

For each of the scattering mechanisms considered in this work, it can be shown that the corresponding dissipator is of the Lindblad form [43]. In fact, each dissipator was derived according to a procedure that guarantees a Lindbladian dissipator [42, 44, 82, 83]. However, one can readily convince oneself of this fact. In what follows, we prove the Lindblad form for the electron–phonon interaction dissipator as an example.

According to Eq. (78), we can write the dissipator due to a single dispersionless phonon type as

$$\begin{aligned}&{\mathcal {D}} = \frac{1}{\hbar ^2(2\pi )^3} \sum _{\pm } \int _0^{\infty }ds \int d^3Q \big \{ \nonumber \\&\quad +\mathcal W^{\pm }({{\bf Q}}){\rm e}^{\mp iE_0 s/\hbar } {\rm e}^{i{\bf Q}\cdot {\bf R}}\rho _{\rm e}(t){\rm e}^{-iH_0s/\hbar }{\rm e}^{-i{\bf Q}\cdot {\bf R}}{\rm e}^{iH_0s/\hbar } \nonumber \\&\quad -{\mathcal {W}}^{\pm }({{\bf Q}}){\rm e}^{\pm iE_0 s/\hbar }{\rm e}^{-i{\bf Q}\cdot {\bf R}}{\rm e}^{-iH_0s/\hbar }{\rm e}^{i{\bf Q}\cdot {\bf R}}{\rm e}^{iH_0s/\hbar } \rho _{\rm e}(t) \big \} \nonumber \\&\quad + {\rm h.c.,} \end{aligned}$$

(114)

The dissipator can be written as a summation of in- and out-scattering terms (${\mathcal {D}}={\mathcal {D}}^{{\rm in}}+\mathcal D^{{\rm out}}$), corresponding to the positive and negative terms in Eq. (78), respectively. By using the completeness relation four times, we can write

$$\begin{aligned} {\mathcal {D}}^{{\rm in}}&=\frac{1}{\hbar ^2(2\pi )^3} \int d^3Q\int _0^{\infty }ds \sum _{n_{1234},\pm }\int d^2k_{1234} \nonumber \\&\quad \times \langle n_1,{\bf k}_1 | {\rm e}^{i{\bf Q}\cdot {\bf R}}| n_2,{{\bf k}}_2 \rangle {\rm e}^{\mp iE_0 s/\hbar } {\mathcal {W}}^{\pm }({{\bf Q}}) \nonumber \\&\quad \times \langle n_2,{{\bf k}}_2 | \rho _{\rm e}| n_3,{{\bf k}}_3 \rangle \nonumber \\&\quad \times \langle n_3,{\bf k}_3 | {\rm e}^{-iH_0s/\hbar }{\rm e}^{-i{\bf Q}\cdot {\bf R}}{\rm e}^{iH_0s/\hbar }| n_4,{\bf k}_4 \rangle | n_1,{{\bf k}}_1 \rangle \langle n_4,{{\bf k}}_4 | + {\rm h.c.}, \end{aligned}$$

(115)

where $\sum {n_{1234}}$ and $\int d^2 k_{1234}$ refer to summation over $n_1$ through $n_4$ and integration of ${{\bf k}}_1$ through ${{\bf k}}_4$, respectively. Since, $\rho$ and ${\mathcal {D}}$ are diagonal in ${{\bf k}}$, this can be further simplified to

$$\begin{aligned} {\mathcal {D}}^{{\rm in}}&=\frac{1}{\hbar ^2(2\pi )^3} \int d^3Q\int _0^{\infty }ds \sum _{n_{1234},\pm }\int d^2k_{1,2} \nonumber \\&\quad \times \langle n_1,{\bf k}_1 | {\rm e}^{i{\bf Q}\cdot {\bf R}}| n_2,{{\bf k}}_2 \rangle {\rm e}^{-i(\varDelta _{n3,n4}\pm E_0 +E_{{{\bf k}}_2} - E_{{{\bf k}}_1}) s/\hbar } {\mathcal {W}}^{\pm }({{\bf Q}}) \nonumber \\&\quad \times \langle n_2,{{\bf k}}_2 | \rho _{\rm e}| n_3,{{\bf k}}_2 \rangle \nonumber \\&\quad \times \langle n_3,{{\bf k}}_2 | {\rm e}^{-i{\bf Q}\cdot {\bf R}}| n_4,{{\bf k}}_1 \rangle | n_1,{{\bf k}}_1 \rangle \langle n_4,{{\bf k}}_1 | + {\rm h.c.}. \end{aligned}$$

(116)

By defining

$$\begin{aligned}&{\mathcal {A}}_Q = \langle n_1,{{\bf k}}_1 | {\rm e}^{+i{\bf Q}\cdot {\bf R}}| n_2,{\bf k}_2 \rangle | n_1,{{\bf k}}_1 \rangle \langle n_2,{{\bf k}}_2 |\, , \end{aligned}$$

(117)

$$\begin{aligned}&\varGamma _{n_{1} n_{2}}^{{{\bf k}}_{1}{{\bf k}}_2}= \frac{{\mathcal {W}}^{\pm }({\bf Q})}{\hbar ^2(2\pi )^3}\int _0^{\infty }ds {\rm e}^{-i(\varDelta _{n1,n2}\pm E_0 +E_{{{\bf k}}_2} - E_{{{\bf k}}_1}) s/\hbar }\, , \end{aligned}$$

(118)

Equation (116) can be written compactly as

$$\begin{aligned} {\mathcal {D}}^{{\rm in}} = 2\sum _{n_{1,2,3,4}}\int d^3Q\int d^2{{\bf k}}_{1,2} \;\varGamma {\mathcal {A}}\rho {\mathcal {A}}^{\dagger }. \end{aligned}$$

(119)

Note that $\varGamma$, as defined in Eq. (118), is Hermitian (transposition of the top two or bottom two indices results in the complex conjugate of the original). Similarly, the out-scattering terms can be written as

$$\begin{aligned} {\mathcal {D}}^{{\rm out}}&=-\frac{1}{\hbar ^2(2\pi )^3} \int d^3Q\int _0^{\infty }ds \sum _{n_{1234},\pm }\int d^2k_{1234} \nonumber \\&\quad \times \langle n_1,{\bf k}_1 | {\rm e}^{-i{\bf Q}\cdot {\bf R}}| n_2,{{\bf k}}_2 \rangle {\rm e}^{ i(\varDelta _{n4,n3}\pm E_0+ E_{{{\bf k}}_1} - E_{{{\bf k}}_2}) s/\hbar } {\mathcal {W}}^{\pm }({{\bf Q}}) \nonumber \\&\quad \times \langle n_2,{{\bf k}}_2 | {\rm e}^{+i{\bf Q}\cdot {\bf R}}| n_3,{{\bf k}}_1 \rangle \nonumber \\&\quad \times \langle n_3,{{\bf k}}_2 | \rho _{\rm e}| n_4,{{\bf k}}_2 \rangle \ | n_1,{{\bf k}}_1 \rangle \langle n_4,{{\bf k}}_1 | + {\rm h.c.}\, . \end{aligned}$$

(120)

Using Eqs. (117) and (118), ${\mathcal {D}}^{out}$ is written as

$$\begin{aligned} {\mathcal {D}}^{{\rm out}} = -\sum _{n_{1,2,3,4}}\int d^3Q\int d^2{{\bf k}}_{1,2} \;\varGamma \left( {\mathcal {A}}^{\dagger }{\mathcal {A}}\rho + \rho {\mathcal {A}}^{\dagger }{\mathcal {A}}\right) ], . \end{aligned}$$

(121)

Summing in- and out-scattering terms would give us the total dissipator for electron-phonon interaction of the form

$$\begin{aligned} {\mathcal {D}} = \sum _{n_{1,2,3,4}}\int d^3Q\int d^2{{\bf k}}_{1,2} \;\varGamma \left( 2{\mathcal {A}}\rho {\mathcal {A}}^{\dagger } - \left\{ {\mathcal {A}}^{\dagger }{\mathcal {A}},\rho \right\} \right) , \end{aligned}$$

(122)

which is of the Lindblad form.

Appendix 5: Calculation of scattering rates

1.1 Longitudinal acoustic phonons

We will start with longitudinal acoustic (LA) phonons, which have the following scattering weight for absorption [79]

$$\begin{aligned} {\mathcal {W}}^+_{\rm LA}({{\bf Q}})=N_Q \frac{\hbar ^2 D_{\rm ac}^2Q^2}{2\rho E_Q} = \left( {\rm e}^{E_Q/kT}-1 \right) ^{-1} \frac{D_{\rm ac}^2E_Q}{2\rho v_{\rm s}^2}, \end{aligned}$$

(123)

where we have used the LA phonon dispersion relation $E_Q=\hbar v_{\rm s} Q$, where $v_{\rm s}$ is the speed of sound in the medium, $\rho$ the mass density and $D_{\rm ac}$ is the acoustic deformation potential. We cannot directly use the expression in Eq. (123) because the phonon energy is not constant, which was an approximation we used to derive Eq. (94). It is possible to discretize the relevant range of Q and treat LA phonons with different magnitude Q as independent scattering mechanism. However, we would need to calculate and store the corresponding rates for each value of Q, which is prohibitive in terms of CPU time as well as memory use. Another option is to consider the limit $E_Q\ll kT$ and treat the LA phonons as an elastic scattering mechanism. However, because the LA phonons can exchange an arbitrarily small energy with the lattice, they play the important role of low energy thermalization at energies much smaller than the polar optical phonon. A second reason to include a nonzero energy for the LA phonons is that they tend to smooth the in-plane energy distribution of the density matrix, greatly facilitating numerical convergence. For this reason, we follow the treatment in Ref. [31] and approximate acoustic phonons as having a single energy $E_{\rm LA}\ll kT$. Typically, we will assume the LA phonon energy is equal to the energy spacing in the simulation (0.5-2 meV). Using this approximation, we get

$$\begin{aligned} {\mathcal {W}}^\pm _{\rm LA}({{\bf Q}})=\left( \tfrac{1}{2}\mp \tfrac{1}{2} + N_{\rm LA} \right) \frac{D_{\rm ac}^2E_{\rm LA}}{2\rho v_{\rm s}^2}, \end{aligned}$$

(124)

with $N_{\rm LA}=( {\text {exp}}(E_{\rm LA}/kT)-1 )^{-1}$. The expression in Eq. (124) is isotropic (does not depend on ${{\bf Q}}$), so the angular integration in Eq. (94) is trivial and we get

$$\begin{aligned} \varGamma _{NMnmE_k}^{\pm ,{\rm in},{\rm LA}}&= \left( \tfrac{1}{2}\mp \tfrac{1}{2} +N_{\rm LA} \right) \frac{D_{\rm ac}^2E_{\rm LA}m_\parallel ^*}{4\pi \hbar ^3\rho v_{\rm s}^2} \\&u[E_k+\varDelta _{N+M,N+M+m}\mp E_{\rm LA}] \\&\int _{0}^{\infty } dQ_z I_{N,N+n,N+M+m,N+M}(Q_z) . \end{aligned}$$

(125)

The corresponding out scattering rates are

$$\begin{aligned}&\varGamma _{NMnmE_k}^{\pm ,{\rm out},{\rm LA}} = \left( \tfrac{1}{2}\mp \tfrac{1}{2} +N_{\rm LA} \right) \frac{D_{\rm ac}^2E_{\rm LA}m_\parallel ^*}{4\pi \hbar ^3\rho v_{\rm s}^2} \\&\quad \times u[E_k+\varDelta _{N+n,N+M+m}\pm E_{\rm LA}] \\&\quad \times \int _{0}^{\infty } dQ_z I_{N+M,N+M+m,N+M+m,N+n}(Q_z) . \end{aligned}$$

(126)

1.2 Polar longitudinal-optical phonons

The scattering weight for polar longitudinal-optical (LO) phonons is [79]

$$\begin{aligned} {\mathcal {W}}^\pm _{\rm LO}({{\bf Q}})= \left( \tfrac{1}{2}\mp \tfrac{1}{2} + N_{\rm LO} \right) \beta _{\rm LO} \frac{Q^2}{(Q^2+Q_{\rm D}^2)^2} \ , \end{aligned}$$

(127)

with

$$\begin{aligned} \beta _{\rm LO}= \frac{{\rm e}^2 E_{\rm LO}}{2\varepsilon _0} \left( \frac{1}{\varepsilon _{\rm s}^\infty }-\frac{1}{\varepsilon _{\rm s}^0} \right) , \end{aligned}$$

(128)

where $\varepsilon _0$ is the permittivity of vacuum, $\varepsilon _{\rm s}^\infty$ and $\varepsilon _{\rm s}^0$ are the high-frequency and low-frequency relative bulk permittivities, respectively, $E_{\rm LO}$ is the polar-optical phonon energy and $Q_{\rm D}$ is the Debye screening wave vector defined by $Q_{\rm D}^2=n_{3D}{\rm e}^2/(\varepsilon _{\rm s}^\infty k_{\rm B}T)$, with $n_{3D}$ the average 3D electron density in the device. We note that QCLs are typically low-doped, so the effect of screening is minimal. However, the screening facilitates numerical calculations by avoiding the singularity when $|{{\bf Q}}|=0$ in Eq. (127). Integration over the angle between ${{\bf Q}}$ and ${{\bf k}}$ gives

$$\begin{aligned}&\frac{\widetilde{{\mathcal {W}}}^\pm _{\rm LO}(E_k,Q_z,E_q) }{\left( \tfrac{1}{2} \mp \tfrac{1}{2}+N_{\rm LO} \right) \beta _{\rm LO}} = \frac{1}{2\pi }\int _0^{2\pi } \frac{|{\bf Q}-{\bf k}|^2}{( |{\bf Q}-{\bf k}|^2+Q_{\rm D}^2)^2} \\&\quad = \frac{1}{2\pi }\int _0^{2\pi } d\theta \frac{Q_z^2+q^2+k^2-2qk\cos (\theta )}{( q_z^2+q^2+k^2-2kq\cos (\theta ) +Q_{\rm D}^2 )^2} \\&\quad = \frac{1}{Q_z^2+q^2+k^2} \\&\quad \times \left[ 1+\frac{Q_{\rm D}^2}{Q_z^2+q^2+k^2} - \frac{4k^2q^2}{(Q_z^2+q^2+k^2)^2 } \right] \\&\quad \times \left[ \left( 1+\frac{Q_{\rm D}^2}{Q_z^2+q^2+k^2}\right) ^2 -\frac{4k^2q^2}{(Q_z^2+q^2+k^2)^2} \right] ^{-\frac{3}{2}}, \end{aligned}$$

(129)

which can be written as

$$\begin{aligned}&\widetilde{{\mathcal {W}}}^\pm _{\rm LO}(E_k,Q_z,E_q) = \frac{\hbar ^2}{2m^*}\frac{\left( \tfrac{1}{2} \mp \tfrac{1}{2}+N_{\rm LO} \right) \beta _{\rm LO}}{E_z+E_k+E_q} \\ \quad&\times \left[ 1+\frac{E_{\rm D}}{E_z+E_q+E_k} - \frac{4E_kE_q}{(E_z+E_q+E_k)^2 } \right] \\&\quad \times \left[ \left( 1+\frac{E_{\rm D}}{E_z+E_q+E_k}\right) ^2 -\frac{4E_kE_q}{(E_z+E_q+E_k)^2} \right] ^{-\frac{3}{2}} \ , \end{aligned}$$

(130)

with $E_z=\hbar ^2Q_z^2/(2m_\parallel ^*)$. Note that this definition of $E_z$ is only for convenience, it has nothing to do with energy quantization in the z-direction. The in- and out-scattering rates can now be written as

$$\begin{aligned} \varGamma _{NMnmE_k}^{\pm ,{\rm in},{\rm LO}}&= \frac{m_\parallel ^*}{2\pi \hbar ^3}u[E_k+\varDelta _{N+M,N+M+m}\mp E_{\rm LO}] \nonumber \\&\quad \times \int _{0}^{\infty } dQ_z I_{N,N+n,N+M+m,N+M}(Q_z) \nonumber \\&\quad \times \widetilde{\mathcal W}^\pm _{\rm LO}(E_k,Q_z,E_k+\varDelta _{N+M,N+M+m}\mp E_{\rm LO}) \end{aligned}$$

(131)

$$\begin{aligned} \varGamma _{NMnmE_k}^{\pm ,{\rm out},{\rm LO}}&= \frac{m_\parallel ^*}{2\pi \hbar ^3} u[E_k+\varDelta _{N+n,N+M+m}\pm E_{\rm LO}] \nonumber \\&\quad \times \int _{0}^{\infty } dQ_z I_{N+M,N+M+m,N+M+m,N+n}(Q_z) \nonumber \\&\quad \times \widetilde{\mathcal W}^\pm _{\rm LO}(E_k,Q_z,E_k+\varDelta _{N+n,N+M+m}\pm E_{\rm LO}), \end{aligned}$$

(132)

with the angle-averaged scattering weights $\widetilde{\mathcal W}^\pm _{\rm LO}$ given by Eq. (130).

1.3 Interface roughness

In order to calculate the scattering rates due to interface roughness in Eq. (108), we need to calculate the angle-averaged correlation function ${{\tilde{C}}}(E_k,E_q)$. In this work, we employ a Gaussian correlation function $C(|{\bf r}|)=\varDelta ^2{\rm e}^{-|{\bf r}|^2/\varLambda ^2}$, with a corresponding Fourier transform $C(|{\bf q}|)=\pi \varDelta ^2\varLambda ^2{\rm e}^{-|{\bf q}|^2\varLambda ^2/4}$. The angle-averaged correlation function can be calculated using

$$\begin{aligned}&{{\tilde{C}}}(E_k,E_q)=\frac{1}{2\pi }\int _0^{2\pi }C(|{\bf q}-{\bf k}|) d\theta \nonumber \\&\quad =\frac{1}{2\pi }\int _0^{2\pi } d\theta C\left( \sqrt{\frac{m_\parallel ^*}{2\hbar ^2}} \sqrt{E_k+E_q-2\sqrt{E_kE_q}\cos (\theta )} \right) \nonumber \\&\quad =\varDelta ^2\varLambda ^2\int _0^{\pi } d\theta {\rm e}^{ -\frac{m_\parallel ^*\varLambda ^2}{2\hbar ^2}(E_k+E_q-2\sqrt{E_kE_q}\cos (\theta )) } \nonumber \\&\quad =\pi \varDelta ^2\varLambda ^2 {\rm e}^{ -\frac{m_\parallel ^*\varLambda ^2}{2\hbar ^2}(E_k+E_q)} I_0(\sqrt{E_kE_q}m_\parallel ^*\varLambda ^2/\hbar ^2), \end{aligned}$$

(133)

where we have used Eq. (3.339) from Ref. [84]

$$\begin{aligned} \int _0^{\pi } {\rm e}^{-x\cos (\theta )} d\theta = \pi I_0(x), \end{aligned}$$

(134)

with $I_0(x)$, the modified Bessel function of the first kind of order zero. The in-scattering rates can now be written as

$$\begin{aligned} \varGamma ^{{\rm in},{\rm IR},\pm }_{NMnmE_k}&= \frac{m_\parallel ^*}{4\hbar ^3} u[\varDelta _{N+M,N+M+m}+E_k] \\&\quad \times \sum _{\rm i} \left( N|N+n\right) _{z_{\rm i},{\rm IR}} \left( N+M+m|N+M\right) _{z_{\rm i},{\rm IR}} ^* \\&\quad \times {{\tilde{C}}}(E_k,\varDelta _{N+M,N+M+m}+E_k), \end{aligned}$$

with the corresponding out-scattering rates

$$\begin{aligned}&\varGamma ^{{\rm out},{\rm IR},\pm }_{NMnmE_k} = \frac{m_\parallel ^*}{4\hbar ^3} u[\varDelta _{N+n,N+M+m}+E_k] \nonumber \\&\quad \times \sum _{\rm i} \left( N+M|N+M+m\right) _{z_{\rm i},{\rm IR}} \left( N+M+m|N+n\right) _{z_{\rm i},{\rm IR}} ^* \nonumber \\&\quad \times {{\tilde{C}}}(E_k,\varDelta _{N+n,N+M+m}+E_k) , \end{aligned}$$

(135)

where we have split the rates into identical absorption ($+$) and emission (−) terms.

1.4 Ionized impurities

The potential due to a single ionized impurity situated at ${\bf R}_{\rm i}$, can be written as

$$\begin{aligned} V({\bf R})=-\frac{{\rm e}^2}{4\pi \varepsilon } \frac{{\rm e}^{-\beta |{\bf R}-{\bf R}_{\rm i}|}}{|{\bf R}-{\bf R}_{\rm i}|}, \end{aligned}$$

(136)

where $\beta$ is the inverse screening length. Now suppose there are many impurities, situated on a sheet ${\bf R}_{\rm i}=({\bf r}_{\rm i},z_\ell )$. The resulting interaction Hamiltonian can be written as

$$\begin{aligned} H_{\rm II}&=-\frac{{\rm e}^2}{4\pi \varepsilon } \sum _{i} \frac{{\rm e}^{-\beta \sqrt{ ({\bf r}-{\bf r}_{\rm i})^2+(z-z_\ell )^2 }}}{\sqrt{({\bf r}-{\bf r}_{\rm i})^2+(z-z_\ell )^2}} \nonumber \\&=\frac{-{\rm e}^2}{8\pi ^2\varepsilon }\sum _{\rm i} \int d^2q {\rm e}^{i{\bf q}\cdot {\bf (r-r_{\rm i})}} \frac{{\rm e}^{-\sqrt{\beta ^2+q^2}|z-z_\ell |}}{\sqrt{\beta ^2+q^2}} , \end{aligned}$$

(137)

where the second line in Eq. (137) is the 2D Fourier decomposition of the first line. We now proceed as we did with interface roughness scattering up until Eq. (100) and get the in-scattering part of the dissipator due to ionized-impurities

$$\begin{aligned}&{\mathcal {D}}_{\rm II}^{\rm in}=\frac{\pi }{\hbar }\sum _{n_{1234}} \int d^2 k_{124} \delta [ E(n_4,{\bf k}_4)-E(n_3,{\bf k}_2)] \rho _{n_2,n_3}^{E_{k_2}} \\&\quad \times \langle n_1,{\bf k}_1 | H_{\rm II}| n_2,{\bf k}_2 \rangle \langle n_3,{\bf k}_2 | H_{\rm II}| n_4,{\bf k}_4 \rangle | n_1,{\bf k}_1 \rangle \langle n_4,{\bf k}_4 | \\&\quad + {\rm h.c..} \end{aligned}$$

(138)

Inserting Eq. 137 into Eq. (138), we can simplify the integrand using

$$\begin{aligned}&\langle n_1,{\bf k}_1 | H_{\rm II}| n_2,{\bf k}_2 \rangle \langle n_3,{\bf k}_2 | H_{\rm II}| n_4,{\bf k}_4 \rangle \nonumber \\&\quad = \frac{{\rm e}^4}{64\pi ^4\varepsilon ^2} \sum _{r_{\rm i},r_j}\int d^2q d^2q' \frac{1}{\sqrt{\beta ^2+q^2}}\frac{1}{\sqrt{\beta ^2+q'^2}} \nonumber \\&\quad \times {\rm e}^{-i{\bf q}\cdot {\bf r}_{\rm i}}{\rm e}^{-i{\bf q'}\cdot {\bf r}_j} \left( n_1|n_2\right) ^{{\rm II}}_{q,\ell } \left( n_2|n_3\right) ^{{\rm II}}_{q',\ell } \nonumber \\&\quad \times \langle {\bf k}_1 | {\bf k}_2+{\bf q} \rangle \langle {\bf k}_2 | {\bf k}_4-{\bf q}' \rangle \ . \end{aligned}$$

(139)

with

$$\begin{aligned} \left( n|m\right) ^{{\rm II}}_{q,\ell } = \sum _b \int {\rm d}z [\psi _n^{(b)}(z)]^*\psi _m^{(b)}(z) \frac{{\rm e}^{-\sqrt{\beta ^2+q^2}|z-z_\ell | }}{\sqrt{\beta ^2+q^2}}, \end{aligned}$$

(140)

where the sum is over the considered bands. Equation (139) depends the detailed distribution of impurities, just as Eq. (101) depends on the detailed in-plane interface roughness $\varDelta _{\rm i}({\bf r})$. By averaging over all possible distribution of impurities ${\bf r}_{\rm i}$ and ${\bf r}_j$, we can see that terms with ${\bf r}_{\rm i}={\bf r}_j$ and ${\bf q}={\bf q}'$ dominate. Limiting the sum to ${\bf r}_{\rm i}={\bf r}_j$ and going from sum over ${\bf r}_{\rm i}$ to integral gives

$$\begin{aligned} \sum _{i}{\rm e}^{-i{\bf r}_{\rm i}\cdot ({\bf q}-{\bf q}')}&\simeq \int d^2 r_{\rm i} N_{\rm 2D}{\rm e}^{-i{\bf r}_{\rm i}\cdot ({\bf q}-{\bf q}')} \\&=(2\pi )^2 N_{\rm 2D}\delta [{\bf q}-{\bf q}']. \end{aligned}$$

(141)

Inserting Eq. (141) into Eq. (139) gives

$$\begin{aligned}&\langle n_1,{\bf k}_1 | H_{\rm II}| n_2,{\bf k}_2 \rangle \langle n_3,{\bf k}_2 | H_{\rm II}| n_4,{\bf k}_4 \rangle \nonumber \\&\quad = \frac{{\rm e}^4}{16\pi ^2\varepsilon ^2} N_{\rm 2D} \int d^2q \frac{1}{\beta ^2+q^2} \left( n_1|n_2\right) ^{{\rm II}}_{q,\ell } \left( n_2|n_3\right) ^{{\rm II}}_{q,\ell } \nonumber \\&\quad \times \delta [{\bf k}_2-({\bf k}_1-{\bf q})]\delta [{\bf k}_4-({\bf k}_2+{\bf q})]. \end{aligned}$$

(142)

Inserting Eq. (142) into Eq. (138) and integrating over ${\bf k}_4$ and ${\bf k}_2$, dropping the subscript on ${\bf k}_1$ and shifting the integration variable ${\bf q}\rightarrow -{\bf q}+{\bf k}$ gives

$$\begin{aligned} {\mathcal {D}}_{\rm II}^{\rm in}&= -\frac{{\rm e}^4 N_{\rm 2D}}{16\pi \varepsilon ^2\hbar } \sum _{n_{1234}} \int d^2 k \int d^2q \delta (E_q-\varDelta _{n_4,n_3}-E_k) \nonumber \\&\quad \times \left( n_1|n_2\right) ^{{\rm II}}_{|{\bf q}-{\bf k}|,\ell } \left( n_3|n_4\right) ^{{\rm II}}_{|{\bf q}-{\bf k}|,\ell } \rho _{n_2,n_3}^{E_q} . \end{aligned}$$

(143)

Calculating the scattering rates from Eq. (142) proceeds the same way as for interface roughness in Eqs. (104)-(108). The total rate due to ionized impurities is obtained by summing over all delta-doped sheets located at positions $z=z_\ell$ with sheet densities $N_{{\rm 2D},\ell }$. The result for the in-scattering rates is

$$\begin{aligned}&\varGamma _{NMnmE_k}^{\pm ,{\rm in},{\rm II}}= \frac{{\rm e}^2m_\parallel ^*}{16\varepsilon ^2\hbar ^3} u[E_k+\varDelta _{N+M,N+M+m}] \sum _{\ell } \frac{N_{{\rm 2D},\ell }}{2\pi } \\&\quad \times \Big [ \int _0^{2\pi } d\theta \left( N|N+n\right) ^{{\rm II}}_{|{\bf q}-{\bf k}|,\ell } \\&\quad \times \left( N+M+m|N+M\right) ^{{\rm II*}}_{|{\bf q}-{\bf k}|,\ell } \Big ]_{E_q=\varDelta _{N+M,N+M+m}+E_k}\\ \end{aligned}$$

(144)

and the corresponding out-scattering rates are given by

$$\begin{aligned}&\varGamma _{NMnmE_k}^{\pm ,{\rm out},{\rm II}} = \frac{{\rm e}^2m_\parallel ^*}{16\varepsilon ^2\hbar ^3} u[E_k+\varDelta _{N+n,N+M+m}] \sum _{\ell } \frac{N_{{\rm 2D},\ell }}{2\pi } \\&\quad \times \Big [ \int _0^{2\pi } d \theta \left( N+M|N+M+m\right) ^{{\rm II}}_{|{\bf q}-{\bf k}|,\ell } \\&\quad \times \left( N+M+m|N+n\right) ^{{\rm II*}}_{|{\bf q}-{\bf k}|,\ell } \Big ]_{E_q=\varDelta _{N+n,N+M+m}+E_k} . \end{aligned}$$

(145)

The products involving the wavefunctions $\left( n|m\right) ^{{\rm II}}_{|{\bf q}-{\bf k}|,\ell }$ are more complicated than in the case for both the phonon and interface roughness interaction. This is because the interaction potential for ionized impurities in Eq. (137) does not factor into the cross-plane and in-plane directions and the products involving eigenfunctions $\psi _n(z)$ are coupled with the in-plane motion. To overcome this issue, we numerically calculate and store the quantity

$$\begin{aligned} F_{nm\ell }(b)=\frac{1}{b}\int \psi ^*_n(z)\psi _m(z){\rm e}^{-b|z-z_\ell |} {\rm d}z \end{aligned}$$

(146)

using an evenly spaced mesh $[b_{\rm min},b_{\rm max}]$ for the relevant range of b. Typical values are $b_{\rm min}=\beta$ and $b_{\rm max}=10$ nm$^{-1}$. We then perform the $\theta$ integration numerically with the stored values of $F_{nm\ell }(b)$ and convert between b and $\theta$ using

$$\begin{aligned} b(\theta )=\sqrt{ \frac{2m_\parallel ^*}{\hbar ^2} }\sqrt{E_{\rm D}+E_k+E_q-2\sqrt{E_kE_q}\cos (\theta )} . \end{aligned}$$

(147)

with $E_{\rm D}=\hbar ^2 \beta ^2/(2m_\parallel ^*)$ the Debye energy.

1.5 Appendix 6: Alloy scattering

Inserting the alloy interaction potential given by Eq. (101) into Eq. (100) gives the in-scattering part of the dissipator due to random alloy scattering

$$\begin{aligned} \mathcal D_{\rm alloy}^{\rm in}&=\frac{\pi }{\hbar }\sum _{n_{1234}} \int d^2 k_{124} \delta [ E(n_4,{\bf k}_4)-E(n_3,{\bf k}_2)] \rho _{n_2,n_3}^{E_{k_2}} \\&\quad \times \langle n_1,{\bf k}_1 | V_{\rm alloy}| n_2,{\bf k}_2 \rangle \langle n_3,{\bf k}_2 | V_{\rm alloy}| n_4,{\bf k}_4 \rangle \\&\quad \times | n_1,{\bf k}_1 \rangle \langle n_4,{\bf k}_4 | + {\rm h.c..} \end{aligned}$$

(148)

The matrix elements involving the interaction potential can be simplified using

$$\begin{aligned}&\langle n_1,{\bf k}_1 | V_{\rm alloy}| n_2,{\bf k}_2 \rangle \langle n_3,{\bf k}_2 | V_{\rm alloy}| n_4,{\bf k}_4 \rangle \\&\quad = \sum _{i,j} \left( n_1 | n_2\right) _{\rm alloy}(z_{\rm i}) \left( n_3 | n_4\right) _{\rm alloy}(z_j) C_{\rm i}^x C_j^x \\&\quad \times \langle {\bf k}_1 | \delta ({\bf r}-{\bf r}_{\rm i})| {\bf k}_2 \rangle \langle {\bf k}_2 | \delta ({\bf r}-{\bf r}_j)| {\bf k}_4 \rangle \\&\quad = \varOmega _0 \sum _{\rm i} \left( n_1 | n_2\right) _{\rm alloy}(z_{\rm i}) \left( n_3 | n_4\right) _{\rm alloy}(z_{\rm i}) x_{\rm i}(1-x_{\rm i}) V_{\rm i}^2 \\&\quad \times \langle {\bf k}_1 | \delta ({\bf r}-{\bf r}_{\rm i})| {\bf k}_2 \rangle \langle {\bf k}_2 | \delta ({\bf r}-{\bf r}_{\rm i})| {\bf k}_4 \rangle \ , \end{aligned}$$

(149)

where we have replaced the product $C_{\rm i}^xC_j^x$ with the correlation function in Eq. (113) and defined the alloy matrix element

$$\begin{aligned} \left( n | m\right) _{\rm alloy}(z_{\rm i}) = \sum _b [\psi ^{(b)}_n(z_{\rm i})]^* \psi ^{(b)}_m(z_{\rm i}), \end{aligned}$$

(150)

where the sum is over the considered bands. Going from sum to integral using $\sum _{\rm i}\rightarrow \frac{1}{\varOmega _0}\int d^3 R_{\rm i}$ and following the same procedure as for interface roughness scattering in Eqs. (100)-(108) gives the scattering rates

$$\begin{aligned}&\varGamma _{NMnmE_k}^{\pm ,{\rm in},{\rm alloy}}= \frac{m_\parallel ^*\varOmega _0}{4\hbar ^3} u[E_k+\varDelta _{N+M,N+M+m}] \\&\quad \times \int {\rm d}z \left( N | N+n\right) _{\rm alloy}(z) \left( N+M+m | N+M\right) _{\rm alloy}(z) \\&\quad \times x(z)(1-x(z))V(z)^2 \end{aligned}$$

and

$$\begin{aligned}&\varGamma _{NMnmE_k}^{\pm ,{\rm out},{\rm alloy}}= \frac{m_\parallel ^*\varOmega _0}{4\hbar ^3} u[E_k+\varDelta _{N+n,N+M+m}] \\&\times \int {\rm d}z \left( N+M | N+M+m\right) _{\rm alloy}(z) \\&\left( N+M+m | N+n\right) _{\rm alloy}(z) \times x(z)(1-x(z))V(z)^2, \end{aligned}$$

where x(z) is the alloy fraction at position z and V(z) is the alloy strength parameter at position z, which we assume is the same for all bands.

Material parameters

Table 1 lists all material specific parameters relating to the III-V binaries used in this work, as well as other parameters such as polar-optical-phonon energy and density. The temperature-dependent lattice constant is calculated using

$$\begin{aligned} a(T)=a(T=300\ {\rm K}) + \alpha _L(T-300\ {\rm K}), \end{aligned}$$

(151)

with $a(T=300{\rm K})$ the lattice constant at 300 K and the thermal expansion coefficient $\alpha _L$ given in table 1. In order to calculate the temperature-dependent $\varGamma$-valley bandgap, we use the empirical Varshni functional form [85]

$$\begin{aligned} E_{\rm gap}(T)=E_{\rm gap}(T=0)-\frac{\alpha ^\varGamma T^2}{T+\beta ^\varGamma }, \end{aligned}$$

(152)

where $E_{\rm gap}(T=0)$ is the zero-temperature $\varGamma$-valley bandgap and $\alpha ^\varGamma$ and $\beta ^\varGamma$ are the Varshni parameters given in table 1. All parameters relating to band structure in table 1 are taken from Ref. [73] with the exception of Kane energy for GaAs and InAs, which were chosen to give room-temperature effective electron masses of $0.063 m_0$ and $0.023 m_0$, respectively ($m_0$ is the free electron mass). Note that the conduction-band effective mass can be written in terms of the parameters in table 1

$$\begin{aligned} \frac{m_0}{m^*_{\rm c}} = (1+2F) + \frac{E_{\rm P}(E_{\rm gap}+2\varDelta _{\rm so}/3)}{E_{\rm gap}(E_{\rm gap}+\varDelta _{\rm so})} . \end{aligned}$$

(153)

Table 1 Various material parameters for GaAs, AlAs and InAs

Material parameters such as Kane energy for ternary alloys ($A_{1-x}B_xC$) as a function of alloy composition x are obtained using a second-order interpolation formula on the form [86]

$$\begin{aligned} E_{\rm P}(A_{1-x}B_xC)&= (1-x)E_{\rm P}(A) + x E_{\rm P}(B) \\&\quad -x(1-x)C_b, \end{aligned}$$

(154)

where $E_{\rm P}(A)$ and $E_{\rm P}(B)$ are the Kane energies of binary AC and BC, respectively, and the bowing parameter $C_b$ accounts for deviation from linear interpolation. Generalization to other material parameters is straightforward. Exceptions to Eq. (154) are interpolations for the Kane energy and F parameters of the $\hbox {In}_{1-x}\hbox {Ga}_x\hbox {As}$ alloy, which contain a third-order term due to the sensitivity of energy spacing to the effective mass of the well material. This third-order term is described in more detail later in this appendix.

Nonzero bowing parameters for the $\hbox {In}_{1-x}\hbox {Ga}_x\hbox {As}$ and $\hbox {In}_{1-x}\hbox {Al}_x\hbox {As}$ alloys are given in table 2. All parameters are obtained from Ref. [73], except for the Kane energy and F parameter in $\hbox {In}_{1-x}\hbox {Ga}_x\hbox {As}$, which in this work were chosen to give a room-temperature electron effective mass of $0.041 m_0$ for $\hbox {In}_{0.53}\hbox {Ga}_{0.47}\hbox {As}$ (lattice-matched to InP). Note that when interpolating Luttinger parameters, the light holes ($m_{\rm lh}^*$) and heavy hole ($m_{\rm hh}^*$) masses should be interpolated using the bowing parameters in table 2 and corresponding Luttinger parameters calculated afterwards using

$$\begin{aligned} \frac{m_0}{m_{\rm lh}^*}&=\gamma _1^L+2\gamma _2^L \\ \frac{m_0}{m_{\rm hh}^*}&=\gamma _1^L-2\gamma _2^L . \\ \end{aligned}$$

(155)

Table 2 Nonzero bowing parameters for the $\hbox {In}_{1-x}\hbox {Ga}_x\hbox {As}$ and $\hbox {In}_{1-x}\hbox {Al}_x\hbox {As}$ alloys

As shown in Sect. 4.1, the material parameters presented in tables 1 and 2 give results in very good agreement with experiment as well as theory based on NEGF for lattice-matched designs. However, for the strain-balanced design in Sect. 4.2, the peak gain wavelength was overestimated by 0.2 $\upmu$m, with an associated shift to lower electric fields in the current density vs electric field curve. We attribute this disagreement to underestimation of the electron mass in the well material (lower mass means smaller energy spacing) and addressed the issue by adding a cubic term

$$\begin{aligned} -x(1-x)(x-0.47)D \end{aligned}$$

(156)

to the alloy interpolation formula [Eq. (154)] for Kane energy and F parameter in In$_{1-x}$Ga$_x$As. This functional form was chosen to not change the lattice-matched results for $x=0.47$, or the binary results ($x=0$ and $x=1$). Values of $D=32$ eV for Kane energy and $D=15$ for the F parameter were found to reproduce experimental results with an energy spacing of 270 meV (corresponding to $\lambda =4.6$ $\upmu$m) between the upper and lower lasing states. These values of D might seem excessively high. However, D is multiplied by 3 numbers that have magnitudes smaller than 1.

We will end this Appendix with a discussion of the parameters related to interface-roughness scattering. We used the same interface-roughness correlation length $\varLambda =9$ nm as in Refs. [32, 34]. We estimate the RMS value for interface roughness ($\varDelta$) by using the fact that interface-roughness scattering is the main broadening mechanism of the gain spectrum. [1] We followed the procedure in Ref. [32] and fix the correlation length at 9 nm and vary $\varDelta$ such that the gain spectra produces a peak with a FWHM of 25 meV, matching experimental results for electroluminescence spectra at a field strength of 53 kV/cm. [52] We note that the resulting value of $\varDelta =0.08$ nm is smaller than the value of $\varDelta =0.1$ nm used in Ref. [34] to simulate the same device. We attribute the difference to the different electron Hamiltonian used in this work (three-band ${\bf k}\cdot {\bf p}$), resulting in higher amplitude of eigenfunctions at material interfaces.