*2.1.1. Including the effect of strain in the nanostructure of QWSC*

As mentioned previously, the best host material for a QWSC should be GaAs. However is very difficult to grow high quality quantum wells in a GaAs p-i-n solar cell because there are no high quality, lattice matched materials with a lower band-gap than GaAS. Therefore is required to use strained materials for that goal, but, once a critical thickness of strained material is deposited, it relaxes and causes the formation of misfit dislocations, which serve as centres for non-radiative recombination [20].

The increase in average strain can be limited by including tensile and compressively strained layers alternatively, choosing appropriately alloy composition and layer thickness, and taking into account each elastic constant. This way is possible to obtain structures which are locally strained, but exert no net force on the substrate or neighbouring repeat units. Such strain-balanced structures have demonstrated great photovoltaic performance, and the strain-balanced quantum well solar cell (SB-QWSC) is currently the most efficient QWSC [21].

One of the methods to reach the strain balance condition is the average lattice method [22]. If we considered *LB* as the barrier thickness, *LW* as the well thickness, and we denote *aB* and *aW* as the respective well and barrier lattice constants; its defined as:

$$a\_{GaAs} \equiv \langle a \rangle = \frac{L\_B a\_B + L\_W a\_W}{L\_B + L\_W} \tag{8}$$

The total strain in the layer may be separated into a hydrostatic component and an axial component. In the case of unstrained bulk material, the heavy hole (hh) and light hole (lh) bands are degenerate at the Brillouin zone centre. The hydrostatic component of strain acts on the band edges changing the band gaps. On the other hand, the axial strain component acts on degeneration of bands. In the valence band, the axial component broken the degeneracy that exists at the band edge (*Γ* point).

Under compressive strain, the bottom energy of the conduction band is shifted to higher energies and the valence band splits, with the light hole band moving further away from the conduction band than the heavy hole band, suppressing the lh transition [23]. In contrast, under tensile strain the material band gap is reduced and the higher energy valence band is the light hole band.

where <sup>∑</sup> *<sup>α</sup>en*−*hhm* (*E*) and <sup>∑</sup> *<sup>α</sup>en*−*lhm* (*E*) are sums over well states *<sup>n</sup>* and *<sup>m</sup>*, which numbers depend on the quantum wells width and depth, *<sup>α</sup>en*−*hhm* (*E*) and *<sup>α</sup>en*−*lhm* (*E*) are the absorption coefficients due to electron-heavy hole and electron-light hole transitions to conduction band, respectively; *α<sup>W</sup>* is the well layer absorption coefficient and Λ is called "quantum thickness

The exciton absorption is taken into account in the theoretical calculation and exciton binding energies are analytically evaluated in the framework of fractional-dimensional space developed by Mathieu et al [19]. Once the total *QE* is calculated, by using the AM1.5 incident solar spectrum represented by *F*(*λ*), the photocurrent is then determined by integration

where *λ*<sup>1</sup> and *λ*<sup>2</sup> are limits of the taken solar spectrum. Then, equation (1) is completely

As mentioned previously, the best host material for a QWSC should be GaAs. However is very difficult to grow high quality quantum wells in a GaAs p-i-n solar cell because there are no high quality, lattice matched materials with a lower band-gap than GaAS. Therefore is required to use strained materials for that goal, but, once a critical thickness of strained material is deposited, it relaxes and causes the formation of misfit dislocations, which serve

The increase in average strain can be limited by including tensile and compressively strained layers alternatively, choosing appropriately alloy composition and layer thickness, and taking into account each elastic constant. This way is possible to obtain structures which are locally strained, but exert no net force on the substrate or neighbouring repeat units. Such strain-balanced structures have demonstrated great photovoltaic performance, and the strain-balanced quantum well solar cell (SB-QWSC) is currently the most efficient QWSC [21]. One of the methods to reach the strain balance condition is the average lattice method [22]. If we considered *LB* as the barrier thickness, *LW* as the well thickness, and we denote *aB* and

*aGaAs* <sup>≡</sup>*<sup>a</sup>* <sup>=</sup> *LBaB* <sup>+</sup> *LW aW*

The total strain in the layer may be separated into a hydrostatic component and an axial component. In the case of unstrained bulk material, the heavy hole (hh) and light hole (lh) bands are degenerate at the Brillouin zone centre. The hydrostatic component of strain acts on the band edges changing the band gaps. On the other hand, the axial strain component acts on degeneration of bands. In the valence band, the axial component broken

*LB* + *LW*

(8)

*F*(*λ*)*QETOTAL*(*λ*)*dλ* (7)

*JPH* = *q*

*aW* as the respective well and barrier lattice constants; its defined as:

the degeneracy that exists at the band edge (*Γ* point).

determined and conversion efficiency *η* can be evaluated.

as centres for non-radiative recombination [20].

*2.1.1. Including the effect of strain in the nanostructure of QWSC*

 *λ*<sup>2</sup> *λ*1

of the heterostructure" [18].

156 Solar Cells - New Approaches and Reviews

following equation 7:

During strained growth, the lattice constant of the epitaxial layer is forced to be equal to the lattice constant of the substrate. Then, there is an biaxial in-plane strain *εi*,*j*, where i, j = x, y, z. In the case of biaxial strain in (001) plane, the values along [001] direction are *εxx* = *εyy* = *εzz*. The strain causes changes of the band edges as explained above at Γ point, which are given by [24]:

$$E\_{h\text{h}}^{\varepsilon} = a\_{\nu} (2e\_{\text{xx}} + e\_{\text{zz}}) - b(e\_{\text{xx}} - e\_{\text{zz}}) \tag{9}$$

$$E\_{\rm ll}^{\varepsilon} = a\_{\nu} (2e\_{\rm xx} + e\_{\rm zz}) + b(e\_{\rm xx} - e\_{\rm zz}) - b^2 \frac{(e\_{\rm xx} - e\_{\rm zz})^2}{\Delta\_{\rm SO}} \tag{10}$$

$$E\_c^\varepsilon = E\_\mathcal{S} + a\_\mathcal{c} (\varepsilon\_{xx} + \varepsilon\_{yy} + \varepsilon\_{zz}) \tag{11}$$

where *E<sup>ε</sup> hh*, *<sup>E</sup><sup>ε</sup> lh* and *<sup>E</sup><sup>ε</sup> <sup>c</sup>* are the energy level values under strain for heavy holes, light holes and electrons, respectively, *Eg* is the band gap, *av* and *ac* are the hydrostatic deformation potentials, *b* is the shear deformation potential and ∆*SO* is the spin-orbit splitting of the valence band at Γ point. The separation of the total hydrostatic deformation potential in conduction (*ac*) and valence band (*av*) contributions is important at heterointerfaces.

The band structure dispersion relations for strained InGaAs and GaAsP, are shown in Figure 8. Note that due to strains, the *In*0.2*Ga*0.8*As* layer experiences a band increment of 121 meV, while for *GaAs*0.7*P*0.3 layer a decrease up to 176 meV is obtained. When the In and P compositions and layer widths are changed, such that the condition given by the definition (8) is satisfied, the strain is changed in the well and barrier layers, causing in both layers a variation in the absorption threshold.

The envelope function approximation is here assumed to determine QW energy levels in the conduction band. The electron energy *Ee* and wave function *ψ<sup>e</sup>* can be then calculated within effective mass approximation. Then, the shift of the conduction band electron in the QW is described by Schrödinger equation. In order to obtain the QW energy levels in the hh and lh bands under varying compressive strain, a 4 × 4 *k* · *p* Kohn-Luttinger Hamiltonian was used:

$$H\_{\rm KL}^{\varepsilon} = H\_{\rm KL} + H^{\varepsilon} \tag{12}$$

where *HKL* is the Kohn-Luttinger Hamiltonian without strain and *H<sup>ε</sup>* is the strain Hamiltonian for epilayers grown in [001] direction, which is given by:

$$H^{\varepsilon} = \begin{bmatrix} E\_{lh}^{\varepsilon} & 0 & 0 & 0 \\ 0 & E\_{lh}^{\varepsilon} & 0 & 0 \\ 0 & 0 & E\_{lh}^{\varepsilon} & 0 \\ 0 & 0 & 0 & E\_{hh}^{\varepsilon} \end{bmatrix} \tag{13}$$

In a QW system, some splitting of the confined valence band levels takes place due to the differences in effective mass, which can be greatly enhanced if the QW is strained. When the well material is strained we have the confinement effects, but in addition we have the consequences of the non-degenerate bulk band edges. For example under (001) biaxial compressive strain the lh band is shifted further energetically from the conduction band.

In a sense, compressive strain acts in the same way as the effect of confinement. Then, biaxial strain, in well and barriers layers, lifts the degeneracy in the valence band such that it is possible to consider independently the hh and lh bands. Under above approximations, the QW energy levels for hh and lh bands where found. The Schrödinger equation corresponding to Hamiltonian is not separated so it is assumed that the "off-diagonal" terms, which lead to valence band mixing, are small enough that they can be neglected. With this assumption the Schrödinger equation becomes separable:

$$\left[-\frac{\hbar^2}{2m\_0}\left(\gamma\_1 - 2\gamma\_2\right)\frac{d^2}{dz^2} + \mathcal{U}(z) + E\_{h\hbar}^\varepsilon - E\_{h\hbar}\right] \Psi\_{h\hbar}^\uparrow(z) = 0\tag{14}$$

$$\left[-\frac{h^2}{2m\_0}\left(\gamma\_1 + 2\gamma\_2\right)\frac{d^2}{dz^2} + \mathcal{U}(z) + E\_{\text{Ih}}^{\varepsilon} - E\_{\text{Ih}}\right] \Psi\_{\text{Ih}}^{\uparrow}(z) = 0\tag{15}$$

where Ψ↑ *hh* and <sup>Ψ</sup><sup>↑</sup> *lh* are the envelope functions with spin direction. The equations (14) and (15) are solved in barrier and well regions with the corresponding *U*(*z*) potential and Konh-Luttinger parameter values (*γ*<sup>1</sup> and *γ*2) in each layer. Computed the *Ee*, *Ehh* and *Elh* values the optical transitions are calculated by Fermi's golden rule and the equation (7) is evaluated to determine the absorption in the quantum wells. Then, the procedure to calculate the photocurrent is analogous to that described in 2.1.

Compressive strain results in lower thermal occupancy of the *lh* band relative to the *hh* band, and radiative transitions from the conduction to the *hh* band are favored over those to the *lh* band. If the splitting becomes greater than a few **kB***T*, *lh* transitions could be suppressed almost entirely[23]. In order to examine this behavior, the anisotropic radiative recombination and gain, as consequence of the strain in the quantum wells, is investigated in order to determinate their influence in the SB-QWSC performance.

As a result of the dislocation-free material, the radiative recombination dominates in SB-QWSCs at high current levels bringing the structure close to the radiative limit. This allows the exploitation of radiative photon recycling by means of the growth of distributed Bragg reflectors (DBRs) between the active region and the substrate of the cell, increasing the efficiency of the SB-QWSC.

This observed increase in solar cell efficiency due to DBRs is due to first to the reduction of the dark current via radiative photon recycling, and second to an increase in photocurrent. The first is due to the reflection of radiative emission back into the cell, reducing the net radiative emission and there-fore, the radiative dark current which, as we have seen, dominates in these structures. The second is the product of the reflection of photons back into the cell of photons which would otherwise have been absorbed in the substrate, leading to decreased photon loss to the substrate, and increasing the photocurrent, equivalent to a net increase in the quantum well absorption.

In a QW system, some splitting of the confined valence band levels takes place due to the differences in effective mass, which can be greatly enhanced if the QW is strained. When the well material is strained we have the confinement effects, but in addition we have the consequences of the non-degenerate bulk band edges. For example under (001) biaxial compressive strain the lh band is shifted further energetically from the conduction band. In a sense, compressive strain acts in the same way as the effect of confinement. Then, biaxial strain, in well and barriers layers, lifts the degeneracy in the valence band such that it is possible to consider independently the hh and lh bands. Under above approximations, the QW energy levels for hh and lh bands where found. The Schrödinger equation corresponding to Hamiltonian is not separated so it is assumed that the "off-diagonal" terms, which lead to valence band mixing, are small enough that they can be neglected. With this assumption the

*dz*<sup>2</sup> <sup>+</sup> *<sup>U</sup>*(*z*) + *<sup>E</sup><sup>ε</sup>*

*dz*<sup>2</sup> <sup>+</sup> *<sup>U</sup>*(*z*) + *<sup>E</sup><sup>ε</sup>*

and (15) are solved in barrier and well regions with the corresponding *U*(*z*) potential and Konh-Luttinger parameter values (*γ*<sup>1</sup> and *γ*2) in each layer. Computed the *Ee*, *Ehh* and *Elh* values the optical transitions are calculated by Fermi's golden rule and the equation (7) is evaluated to determine the absorption in the quantum wells. Then, the procedure to calculate

Compressive strain results in lower thermal occupancy of the *lh* band relative to the *hh* band, and radiative transitions from the conduction to the *hh* band are favored over those to the *lh* band. If the splitting becomes greater than a few **kB***T*, *lh* transitions could be suppressed almost entirely[23]. In order to examine this behavior, the anisotropic radiative recombination and gain, as consequence of the strain in the quantum wells, is investigated

As a result of the dislocation-free material, the radiative recombination dominates in SB-QWSCs at high current levels bringing the structure close to the radiative limit. This allows the exploitation of radiative photon recycling by means of the growth of distributed Bragg reflectors (DBRs) between the active region and the substrate of the cell, increasing the

This observed increase in solar cell efficiency due to DBRs is due to first to the reduction of the dark current via radiative photon recycling, and second to an increase in photocurrent. The first is due to the reflection of radiative emission back into the cell, reducing the net radiative emission and there-fore, the radiative dark current which, as we have seen, dominates in these structures. The second is the product of the reflection of photons back into the cell of photons which would otherwise have been absorbed in the substrate, leading

*hh* − *Ehh*

*lh* − *Elh*

*lh* are the envelope functions with spin direction. The equations (14)

 Ψ↑

 Ψ↑

*hh*(*z*) = 0 (14)

*lh*(*z*) = 0 (15)

Schrödinger equation becomes separable:

 <sup>−</sup> *<sup>h</sup>*<sup>2</sup> 2*m*<sup>0</sup>

 <sup>−</sup> *<sup>h</sup>*<sup>2</sup> 2*m*<sup>0</sup>

*hh* and <sup>Ψ</sup><sup>↑</sup>

158 Solar Cells - New Approaches and Reviews

efficiency of the SB-QWSC.

where Ψ↑

(*γ*<sup>1</sup> <sup>−</sup> <sup>2</sup>*γ*2) *<sup>d</sup>*<sup>2</sup>

(*γ*<sup>1</sup> <sup>+</sup> <sup>2</sup>*γ*2) *<sup>d</sup>*<sup>2</sup>

in order to determinate their influence in the SB-QWSC performance.

the photocurrent is analogous to that described in 2.1.

We will discuss the theoretical background to radiative recombination, gain and photon recycling. The electron and hole quasi-Fermi separation was calculated in order to determine the spontaneous TE and TM emission rate from QW and gain as a function of In composition. Similarly, the optical transitions in quantum well are evaluated to calculate the QW absorption coefficient. Then, we present the results of simulations of the SB-QWSC that it takes account of DBR and anisotropic effects. We calculate quantum and conversion efficiencies and observe an increment in the SB-QWSC performance, particularly under solar concentration.

The radiative recombination current could be suppressed in SB-QWSC devices with deep QWs relative to the prediction of the generalized Planck formula assuming isotropic emission. Following Adams et al [23], emission can be defined as TE, which is polarized in the plane of the QWs, and TM, which is polarized perpendicular to the plane of the QWs. It is therefore possible for TE-polarized light to be emitted either out of the face or the edge of the QWs, whereas TM polarized light can only be emitted out of the edge of the QWs. The *hh* transition couples solely to TE-polarized emission, and the lh band couples predominantly to TM-polarized emission with a minor TE-polarized contribution.

The spontaneous emission rates, *Rspon*, were calculated by ab-initio methods, where the transition from bulk to quantum wells structures was carried out by converting the 3D density of states to the 2D density states. Calculations of TE and TM emission out of the faces and edges of a quantum well include the strain modifications to the spontaneous emission rate resulting from varying the In and P compositions and their layer widths such that the condition given by equation (8) is satisfied.

The emission spectrum from a solar cell depends on the absorption coefficient and the carrier density through the quasi-Fermi-level separation, ∆*Ef* . To model the emission from either sample at a given generated strain, we first calculated the absorption coefficient using a quantum-mechanical model above mentioned. We assumed that the number of photogenerated carrier pairs is equal to the total emitted photo flux. In the absence of any photon density, the emission rate is the spontaneous emission rate, provided a state −→*<sup>k</sup>* is occupied by an electron and a hole is present in the same state −→*<sup>k</sup>* in the valence band. The rate depends on the occupation probability functions for electrons, *fe*, and holes, *fh*, with the same **kB**-value. The occupation probability function for electrons and holes depends on the corresponding quasi-Fermi level. The spontaneous emission rate expression for quantum well structures is obtained be integration over all possible electronic states

$$R\_{\rm spon} = \int d(\hbar\omega) A\hbar\omega \sum\_{n,m} \left[ \frac{d^2\mathbf{k}}{(2\pi)^2} |\overleftrightarrow{a}\cdot\overrightarrow{\mathbf{p}}\rangle\_f|^2 \delta\left(E\_n^{\varepsilon}(\overrightarrow{\mathbf{k}}\cdot) - E\_m^{\hbar}(\overrightarrow{\mathbf{k}}\cdot)\right) \times f\_\varepsilon(E\_n^{\varepsilon}(\overrightarrow{\mathbf{k}}\cdot)) f\_\hbar(E\_m^{\hbar}(\overrightarrow{\mathbf{k}}\cdot)) \right] \tag{16}$$

The integral over *d*(*h*¯ *ω*) is to find the rate for all photon emitted and the integration over *d*2*k* is to get the rate for all the occupied electron and hole subband state. Equation (16) summarizes the discrete energy states of the electrons (index n) and the heavy holes (index m) in the well. *E<sup>e</sup> n*( −→*<sup>k</sup>* ) and *<sup>E</sup><sup>h</sup> m*( −→*<sup>k</sup>* ) denote the QW subbands of the electrons and heavy holes and *<sup>δ</sup>* denotes the Dirac delta function. The factor *<sup>A</sup>* <sup>=</sup> <sup>2</sup>*q*2*nr m*2 0*c*3*h*<sup>2</sup> is a material dependent constant, where *nr* is the refractive index of the well material. The first term inside the element <sup>|</sup>*<sup>a</sup>* · −→*<sup>p</sup> i f* <sup>|</sup> represents the polarization unit vector, *<sup>a</sup>*, while the second term represents the momentum matrix element, −→*p i f* . The spontaneous emission rate of the QWs was calculated using the above formula.

In a semiconductor in nonequilibrium condition, the total electron concentration n and the total hole concentration p are described to be the, respectively, electron and hole quasi-Fermi levels. If detailed balance is applied when each photon produces one electron-hole pair and all recombination events produce one photon, the electrons and hole quasi-Fermi levels in the quantum well structure were calculated by solving the following system of equations [28]:

$$
\Delta n(E\_{\rm F\_c}) = p(E\_{\rm F\_h}); \qquad \Delta E\_f = E\_{\rm F\_c} - E\_{\rm F\_h} \tag{17}
$$

Determining ∆*Ef* is essentially a matter of normalizing the emission spectrum to the generation rate. If detailed balance applies, the number of photogenerated carrier pairs is equal to the total emitted photon flux, and the gain (G) is defined is defined as the number of photogenerated carrier pairs per unit area and time:

$$\mathcal{G} = \iint\limits\_{0}^{\infty} \mathcal{G}(\lambda, z) dz d\lambda = \int\limits\_{0}^{\infty} L(\hbar \omega) d(\hbar \omega) \tag{18}$$

where *G*(*λ*, *z*) is the electron-hole pair generation rate at *z* depth from the surface in the growth direction and is given by the expression:

$$G(\lambda, z) = \left[1 - R(\lambda)\right] a(\lambda, z) F(\lambda) \exp\left[-\int\_0^z a(\hbar \omega, z') dz'\right] \tag{19}$$

The exponential factor is due to the attenuation of light in the layers between the surface of the cell and the depletion layer. The layers considered in our calculus are antireflection layer, emitter layer, and space-charge region from to the emitter layer (see Figure 7). The emitted flux density *L*(*h*¯ *ω*), of photons of energy ¯*hω*, is given by:

$$L(\hbar\omega) = \frac{2n\_r^2}{h^3c^2} \frac{\mathfrak{a}(\hbar\omega)(\hbar\omega)}{e^{\frac{\hbar\omega - \Delta E\_f}{\hbar\_\mathbf{B}T}} - 1} \tag{20}$$

At low enough carrier density, where ∆*Ef* is much smaller than the effective band gap, the Boltzmann approximation is used, and Eq. (20) is simplified, then the dependence on ∆*Ef* is an explicit function. From Eqs. 18 and 20, we found:

$$\Delta E\_f = -\mathbf{k}\_\mathbf{B} T \ln \left[ \frac{1}{G} \int\_0^\infty \frac{2n\_r^2}{(2\pi\hbar)^3 c^2} a(\hbar\omega)(\hbar\omega)^2 \exp-\frac{\hbar\omega}{\mathbf{k}\_\mathbf{B} T} d(\hbar\omega) \right] \tag{21}$$

The total electron concentration is determined by:

m) in the well. *E<sup>e</sup>*

160 Solar Cells - New Approaches and Reviews

*n*(

calculated using the above formula.

−→*<sup>k</sup>* ) and *<sup>E</sup><sup>h</sup>*

of photogenerated carrier pairs per unit area and time:

*G* = ∞

flux density *L*(*h*¯ *ω*), of photons of energy ¯*hω*, is given by:

an explicit function. From Eqs. 18 and 20, we found:

growth direction and is given by the expression:

0

*G*(*λ*, *z*) = [1 − *R*(*λ*)] *α*(*λ*, *z*)*F*(*λ*) exp

*<sup>L</sup>*(*h*¯ *<sup>ω</sup>*) = <sup>2</sup>*n*<sup>2</sup>

*m*(

holes and *<sup>δ</sup>* denotes the Dirac delta function. The factor *<sup>A</sup>* <sup>=</sup> <sup>2</sup>*q*2*nr*

constant, where *nr* is the refractive index of the well material. The first term inside the element <sup>|</sup>*<sup>a</sup>* · −→*<sup>p</sup> i f* <sup>|</sup> represents the polarization unit vector, *<sup>a</sup>*, while the second term represents the momentum matrix element, −→*p i f* . The spontaneous emission rate of the QWs was

In a semiconductor in nonequilibrium condition, the total electron concentration n and the total hole concentration p are described to be the, respectively, electron and hole quasi-Fermi levels. If detailed balance is applied when each photon produces one electron-hole pair and all recombination events produce one photon, the electrons and hole quasi-Fermi levels in the quantum well structure were calculated by solving the following system of equations [28]:

Determining ∆*Ef* is essentially a matter of normalizing the emission spectrum to the generation rate. If detailed balance applies, the number of photogenerated carrier pairs is equal to the total emitted photon flux, and the gain (G) is defined is defined as the number

∞

0

 − *z*

0

*α*(*h*¯ *ω*, *z*

)*dz* 

(19)

(20)

*G*(*λ*, *z*)*dzdλ* =

where *G*(*λ*, *z*) is the electron-hole pair generation rate at *z* depth from the surface in the

The exponential factor is due to the attenuation of light in the layers between the surface of the cell and the depletion layer. The layers considered in our calculus are antireflection layer, emitter layer, and space-charge region from to the emitter layer (see Figure 7). The emitted

> *r h*3*c*<sup>2</sup>

At low enough carrier density, where ∆*Ef* is much smaller than the effective band gap, the Boltzmann approximation is used, and Eq. (20) is simplified, then the dependence on ∆*Ef* is

*e h*¯ *ω*−∆*Ef* **kB***<sup>T</sup>* − 1

*α*(*h*¯ *ω*)(*h*¯ *ω*)

−→*<sup>k</sup>* ) denote the QW subbands of the electrons and heavy

*m*2

*n*(*EFc* ) = *p*(*EFh* ); ∆*Ef* = *EFc* − *EFh* (17)

0*c*3*h*<sup>2</sup> is a material dependent

*L*(*h*¯ *ω*)*d*(*h*¯ *ω*) (18)

$$m = \int\_{E\_{\rm W\_\varepsilon}}^{E\_{\rm B\_\varepsilon}} g\_\varepsilon^{\rm QW}(E) f\_\varepsilon(E) dE + \int\_{E\_{\rm B\_\varepsilon}}^{\infty} g\_\varepsilon^{\rm Bulk}(E) f\_\varepsilon(E) dE \tag{22}$$

where *EWe* and *EBe* are the conduction valence band edge energy for quantum well and barrier material respectively, *gQW <sup>e</sup>* (*E*) is the electron quantum well density of state and *gBulk <sup>e</sup>* (*E*) is the electron bulk density of state in the quantum well material. The calculation of the total hole concentration is analogous. Then the equation system (17) may be solved and the quasi-Fermi level is determined. The spontaneous emission rate from QW region was then calculated according to Eq. (16).
