**2.2. Modelling the Superlattice Solar Cell**

In contrast with a QWSC where the different quantum wells are considered independent and there is no coupling between neighboring quantum wells, in the superlattice solar cell an interaction exist between neighboring wells and the wave function becomes extended over the whole nanostructure. Therefore the discrete levels in isolated quantum wells spread into a miniband, as it can be seen in Figure 2.

**Figure 2.** Sketch of energy band diagram of a typical *Al*0.1*Ga*0.9*As* p-i-n solar cell with inserted quantum wells (a) and superlattices (b) in the intrinsic region

In order to achieve the quantum well coupling in the intrinsic region, which is inside an electric field, a variable spaced superlattice is proposed. In this case is necessary that each well width changes in the way that electron levels of the wells are resonant at the operating bias. Then the conditions are fullfilled for resonant tunneling of carriers in the whole nanostructure for a specific value of the electric field, which depends of the doping of the p- and n- regions.

From the theoretical point of view, the advantages of an SLSC over an QWSC are the following: (i) provides quantum levels for electrons and holes within specific eigen-energies (minibands), (ii) improves the miniband photon absorption, (iii) cancels deep-level recombination between single and double heterojunction, (iv) the carriers are able of tunneling along the growth direction through thin barriers while they are essentially free along the transverse direction, and (v) allows an efficient escape rate of carriers out of quantum wells, which are collected in the emitter and base regions [25].

In order to extend the model to the SLSC, the coefficients *rR* and *rNR* (Eqs. (2) and (3)) and the photocurrent *JPH* should be related to superlattice structure. Now, the *fW* factor is the intrinsic region fraction replaced by superlattices, with *gSL* and *gB* as the effective density of states for superlattices and barriers, *γ<sup>B</sup>* = *BSL*/*BB* and *γ<sup>A</sup>* = *ASL*/*AB* are the radiative and non radiative recombination coefficients referred to superlattices and barriers respectively. The photocurrent is evaluated using absorption coefficients of the transitions of the minibands. The effective density of states for electrons in the superlattice was found [3]:

$$\begin{split} g\_{SL\varepsilon} &= \frac{m\_{\varepsilon}}{\pi \hbar^{2} d\_{SL}} \int\_{0}^{\Gamma\_{\varepsilon}} \left( \frac{1}{2} + \frac{1}{2} \arcsin \left( \frac{E - \frac{\Gamma\_{\varepsilon}}{2}}{\frac{\Gamma\_{\varepsilon}}{2}} \right) \right) \exp \left( -\frac{E + E\_{\varepsilon}}{\mathbf{k}\_{\mathbf{B}} T} \right) dE \\ &+ \frac{m\_{\varepsilon}}{\pi \hbar^{2} d\_{SL}} \mathbf{k\_{B}T} \left[ \exp \left( -\frac{E\_{\varepsilon} + \Gamma\_{\varepsilon}}{\mathbf{k\_{B}} T} \right) - \exp \left( -\frac{\Delta E\_{\varepsilon}}{\mathbf{k\_{B}} T} \right) \right] \\ &+ 2 \left( \frac{2 \pi m\_{\varepsilon} \mathbf{k\_{B}} T}{\hbar^{2}} \right)^{\frac{3}{2}} \left[ 2 \sqrt{\frac{\Delta E\_{c}}{\pi \mathbf{k\_{B}} T}} \exp \left( -\frac{\Delta E\_{c}}{\mathbf{k\_{B}} T} \right) + \pi f c \sqrt{\frac{\Delta E\_{c}}{\mathbf{k\_{B}} T}} \right] \end{split} \tag{23}$$

where *Ee* is the electron miniband bottom, *er f c* is the complementary error function, *me* is the electron effective mass, ∆*Ec* = *Qc*(*EgB* − *EgW*) is the well depth in the conduction band, *Qc* is the band offset factor, *dSL* is the superlattice period and *Γ<sup>e</sup>* is the miniband width in the conduction band. Analogous expressions are obtained for heavy hole and light hole effective density of states (*gSLhh* , *gSLlh* ). Then the total superlattice effective density of states was calculated as:

$$\log\_{SL} = \sqrt{\mathcal{g}\_{SL\mathcal{L}}(\mathcal{g}\_{SL\_{\text{th}}} + \mathcal{g}\_{SL\_{\text{th}}})} \tag{24}$$

The absorption coefficient for the transitions between light hole and electron minibands was also determined as a function of their widths, *Γlh* and *Γ<sup>e</sup>* which we will use in our model as a parameter:

In order to achieve the quantum well coupling in the intrinsic region, which is inside an electric field, a variable spaced superlattice is proposed. In this case is necessary that each well width changes in the way that electron levels of the wells are resonant at the operating bias. Then the conditions are fullfilled for resonant tunneling of carriers in the whole nanostructure for a specific value of the electric field, which depends of the doping of

From the theoretical point of view, the advantages of an SLSC over an QWSC are the following: (i) provides quantum levels for electrons and holes within specific eigen-energies (minibands), (ii) improves the miniband photon absorption, (iii) cancels deep-level recombination between single and double heterojunction, (iv) the carriers are able of tunneling along the growth direction through thin barriers while they are essentially free along the transverse direction, and (v) allows an efficient escape rate of carriers out of

In order to extend the model to the SLSC, the coefficients *rR* and *rNR* (Eqs. (2) and (3)) and the photocurrent *JPH* should be related to superlattice structure. Now, the *fW* factor is the intrinsic region fraction replaced by superlattices, with *gSL* and *gB* as the effective density of states for superlattices and barriers, *γ<sup>B</sup>* = *BSL*/*BB* and *γ<sup>A</sup>* = *ASL*/*AB* are the radiative and non radiative recombination coefficients referred to superlattices and barriers respectively. The photocurrent is evaluated using absorption coefficients of the transitions of the minibands. The effective density of states for electrons in the superlattice was found [3]:

> *<sup>E</sup>* <sup>−</sup> *<sup>Γ</sup><sup>e</sup>* 2 *Γe* 2

− exp

 <sup>−</sup> <sup>∆</sup>*Ec* **kB***T*

<sup>−</sup> *Ee* <sup>+</sup> *<sup>Γ</sup><sup>e</sup>* **kB***T*

where *Ee* is the electron miniband bottom, *er f c* is the complementary error function, *me* is the electron effective mass, ∆*Ec* = *Qc*(*EgB* − *EgW*) is the well depth in the conduction band, *Qc* is the band offset factor, *dSL* is the superlattice period and *Γ<sup>e</sup>* is the miniband width in the conduction band. Analogous expressions are obtained for heavy hole and light hole effective density of states (*gSLhh* , *gSLlh* ). Then the total superlattice effective density of states

The absorption coefficient for the transitions between light hole and electron minibands was also determined as a function of their widths, *Γlh* and *Γ<sup>e</sup>* which we will use in our model as

 <sup>−</sup> <sup>∆</sup>*Ec* **kB***T*

exp 

<sup>−</sup> *<sup>E</sup>* <sup>+</sup> *Ee* **kB***T*

> ∆*Ec* **kB***T*

+ *er f c*

*gSLe*(*gSLhh* + *gSLlh* ) (24)

 *dE*

(23)

quantum wells, which are collected in the emitter and base regions [25].

the p- and n- regions.

162 Solar Cells - New Approaches and Reviews

*gSLe* <sup>=</sup> *me*

+

+ 2

was calculated as:

*<sup>π</sup>h*¯ <sup>2</sup>*dSL*

*me <sup>π</sup>h*¯ <sup>2</sup>*dSL*  *Γ<sup>e</sup>* 0

**kB***T* exp 

2*πme***kB***T h*¯ 2

 1 2 + 1 <sup>2</sup> arcsin

3 2 2 ∆*Ec <sup>π</sup>***kB***<sup>T</sup>* exp

*gSL* =

$$\begin{array}{rcl} \text{a } a\_{\text{lh}-\text{c}}(E) &=& \frac{q^2}{cm\_\varepsilon^2 \varepsilon\_0 n\_r d\_{SL} \hbar^2 \omega} \left| \hat{a} \cdot \overrightarrow{p} \right\rangle \left| \hat{a} \times \frac{m\_{\text{lh}} m\_\varepsilon}{m\_{\text{lh}} + m\_\varepsilon} \left\{ \frac{1}{2} + \frac{1}{\pi} \arcsin \left[ \frac{E - E\_{\text{S}0} - \frac{\Gamma\_\varepsilon + \Gamma\_\text{IR}}{2}}{\frac{\Gamma\_\varepsilon + \Gamma\_\text{IR}}{2}} \right] \right\} \right| \end{array} \tag{25}$$

where <sup>|</sup>*<sup>a</sup>* · −→*<sup>p</sup> i f* <sup>|</sup> is the optical matrix element between the initial *<sup>i</sup>* and the final *<sup>f</sup>* transition states, *a* is an unit vector in the direction of propagation, *p* is the momentum, *nr* is refraction index of the heterostructure, *ε*<sup>0</sup> is the vacuum dielectric constant, *ω* is the angular frequency of radiation, *mlh* is the light hole effective mass, *Eg*<sup>0</sup> = *EgW* + *Ee* + *Elh* and *Elh* is the light hole miniband top energy. An analogous expression for the absorption coefficient of the transitions between heavy hole and electron minibands (*αhh*−*e*(*E*)) is also found. The total superlattice absorption coefficient can be expressed by:

$$
\mathfrak{a}(E) = \mathfrak{a}\_{\text{Ih}-\mathfrak{e}}(E) + \mathfrak{a}\_{\text{Ih}-\mathfrak{e}}(E) \tag{26}
$$

According to the detailed balance theory, the radiative recombination coefficient is expressed by:

$$B = \frac{8\pi n\_r^2}{c^2 \hbar^3 n\_0 p\_0} \int\_{E\_1}^{E\_2} \frac{\varkappa\_{SL} E^2 dE}{\exp\left(\frac{E}{\mathbf{k}\_\mathbf{B} T}\right) - 1} \tag{27}$$

Now the quantum efficiency the intrinsic region is calculated by the expression 28, considering the absorption of photons through a miniband and not by quantum well levels:

$$QE(\lambda) = \left[1 - R(\lambda)\right] \exp\left(-\sum\_{i=1}^{3} a\_i z\_i\right) \left[1 - \exp\left(-a\_B W - N\_{SL} L\_{SL} a\_{SL}^\*\right)\right] \tag{28}$$

where *LSL* is the superlattice width. This superlattice could be considered an structural unit and we can repeat several of this superlattice units as much as they fit inside the intrinsic region. So, let's define this superlattice units as a cluster. Then, *NSL* is the number of clusters or superlattice units that we insert in the intrinsic region of the p-i-n solar cell.

Once the expressions for the effective density of states, the absorption coefficient, the radiative recombination coefficient, and photocurrent were found for SLSC, we are able to determine the J-V characteristic from equation (1) and then it is possible to calculate the conversion efficiency.
