**2.1. Modelling the Quantum Well Solar Cell**

The derived current-voltage relationship of a QWSC with *NW* wells each of length *LW* into the intrinsic region of length W, with barrier band gap *EgB* and well band gap *EgW* is given by equation 1, after [15, 16];

$$f(V) = f\_0(1 + r\_R \beta) \left[ \exp(\frac{qV}{\mathbf{k\_B}T}) - 1 \right] + (ar\_{NR} + f\_S) \left[ \exp(\frac{qV}{2\mathbf{k\_B}T}) - 1 \right] - f\_{PH} \tag{1}$$

where *q* is the electron charge, *V* is the terminal voltage of the device, **kB***T* the thermal energy, *<sup>α</sup>* <sup>=</sup> *qWABniB* and *<sup>β</sup>* <sup>=</sup> *qWBBn*<sup>2</sup> *iB <sup>J</sup>*<sup>0</sup> are parameters defined following Anderson [17]. *J*<sup>0</sup> is the reverse saturation current density; *AB* is the nonradiative coefficient for barriers in the depletion region, which is related to barrier non-radiative lifetime *τ<sup>B</sup>* by *AB* = <sup>1</sup> *<sup>τ</sup><sup>B</sup>* ; *BB* is the radiative recombination coefficient of the host material; *niB* is the equilibrium intrinsic carrier concentration for the host material; *rR* and *rNR* are the radiative enhancement ratio and non-radiative enhancement ratio respectively given by the equations 2 and 3.

$$r\_R = 1 + f\_W \left[ \gamma\_B \gamma\_{DOS}^2 \exp\left(\frac{\Delta E - qFz}{\mathbf{k\_B}T}\right) - 1\right] \tag{2}$$

$$r\_{\rm NR} = 1 + f\_W \left[ \gamma\_A \gamma\_{\rm DOS} \exp\left(\frac{\Delta E - qFz}{2\mathbf{k\_B}T}\right) - 1\right] \tag{3}$$

Those enhancement ratios represent the fractional increase in radiative and non-radiative recombination in the intrinsic region, due to the influence of quantum wells. In the equations 2 and 3, ∆*E* = *EgB* − *EgW*, *fW* is the fraction of the intrinsic region volume substituted by the quantum well material, *<sup>γ</sup>DOS* <sup>=</sup> *gW gB* is the density of states enhancement factor, *gW* and *gB* are the effective density of states for the wells and barriers, and *γ<sup>B</sup>* and *γ<sup>A</sup>* are "oscillator enhancement factor" and "lifetime reduction factor", respectively [17]. The built-in field is denoted by *F* and *z* is the position in the wells, so *rR* and *rNR* are position dependent. The photocurrent *JPH* is calculated from the quantum efficiency of the cell. The p-region and n-region contribution to QE was classically evaluated solving the carrier transport equations at room temperature within the minority carrier and depletion approximations. The quantum efficiency (*QE*) is calculated by the expression:

for this devices is obtained. The effect of the superlattice characteristics on the conversion efficiency is discussed. The SLSC conversion efficiency is compared with the maximum conversion efficiency obtained for the QWSC. Finally, we present GaAs/ GaInNAs SLSC conversion efficiency as a function of solar concentration, showing an clear increment in its

In this section, we will develop the model, starting with the QWSC, whose structure is shown in figure 1. The make use of the common assumptions of homogeneous composition in the doped and intrinsic layers, the depletion approximation in the space-charge region, and total photogenerated carrier collection, assuming an equal carrier temperature in all regions. Transport and Poisson equations were used to compute the quantum efficiency in the charge-neutral layers while the quantum efficiency of the intrinsic region is determined taking into account the absorption coefficient of the nanostructure involved. The overall photocurrent is simply expressed in terms of superposition, adding photocurrent, obtained form the calculated quantum efficiency, to the dark current in order to find out the

The derived current-voltage relationship of a QWSC with *NW* wells each of length *LW* into the intrinsic region of length W, with barrier band gap *EgB* and well band gap *EgW* is given

where *q* is the electron charge, *V* is the terminal voltage of the device, **kB***T* the thermal

is the reverse saturation current density; *AB* is the nonradiative coefficient for barriers in the depletion region, which is related to barrier non-radiative lifetime *τ<sup>B</sup>* by *AB* = <sup>1</sup>

the radiative recombination coefficient of the host material; *niB* is the equilibrium intrinsic carrier concentration for the host material; *rR* and *rNR* are the radiative enhancement ratio

*DOS* exp

*γAγDOS* exp

Those enhancement ratios represent the fractional increase in radiative and non-radiative recombination in the intrinsic region, due to the influence of quantum wells. In the

+ (*αrNR* + *JS*)

 <sup>∆</sup>*<sup>E</sup>* <sup>−</sup> *qFz* **kB***T*

 <sup>∆</sup>*<sup>E</sup>* <sup>−</sup> *qFz* 2**kB***T*

exp( *qV*

*<sup>J</sup>*<sup>0</sup> are parameters defined following Anderson [17]. *J*<sup>0</sup>

 − 1 

> − 1

<sup>2</sup>**kB***<sup>T</sup>* ) <sup>−</sup> <sup>1</sup>

− *JPH* (1)

*<sup>τ</sup><sup>B</sup>* ; *BB* is

(2)

(3)

performance.

**2. Model details**

154 Solar Cells - New Approaches and Reviews

illuminated current-voltage characteristic.

by equation 1, after [15, 16];

*J*(*V*) = *J*0(1 + *rRβ*)

energy, *<sup>α</sup>* <sup>=</sup> *qWABniB* and *<sup>β</sup>* <sup>=</sup> *qWBBn*<sup>2</sup>

**2.1. Modelling the Quantum Well Solar Cell**

*rR* = 1 + *fW*

*rNR* = 1 + *fW*

exp( *qV*

**kB***<sup>T</sup>* ) <sup>−</sup> <sup>1</sup>

*iB*

and non-radiative enhancement ratio respectively given by the equations 2 and 3.

 *γBγ*<sup>2</sup>

$$QE(\lambda) = \left[1 - R(\lambda)\right] \exp\left(-\sum\_{i=1}^{3} a\_i z\_i\right) \left[1 - \exp\left(-a\_B W - N\_W a\_W^\*\right)\right] \tag{4}$$

where *R*(*λ*) is the surface reflectivity of the antireflection layer. The first exponential factor is due to the attenuation of light in the precedent layers to the depletion layer as showed in Figure 1. The layers numbered in Figure 1, are: (1) antireflection coating, (2) emitter and (3) space-charge region of the emitter; *α<sup>i</sup>* is the the absorption coefficient of each layer and *zi* its corresponding width, *α<sup>B</sup>* is the absorption coefficient of the bulk barrier material, and *α*<sup>∗</sup> *<sup>W</sup>* is the dimensionless quantum well absorption coefficient, used for energies below the barrier band gap.

**Figure 1.** Sketch of energy band diagram of a GaAs p-i-n solar cell with quantum wells inserted within the intrinsic region.

When mixing between light and heavy valence sub-bands is neglected, the absorption coefficient can be calculated as follows[18]

$$
\boldsymbol{\alpha}\_W^\* = \boldsymbol{\alpha}\_W \boldsymbol{\Lambda} \tag{5}
$$

$$\mathfrak{a}\_{\mathcal{W}}(E) = \sum \mathfrak{a}\_{\mathfrak{e}\_n - lh\_m}(E) + \sum \mathfrak{a}\_{\mathfrak{e}\_n - lh\_m}(E) \tag{6}$$

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 of the heterostructure" [18].

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 following equation 7:

$$J\_{PH} = q \int\_{\lambda\_1}^{\lambda\_2} F(\lambda) Q E\_{TOTAL}(\lambda) d\lambda \tag{7}$$

where *λ*<sup>1</sup> and *λ*<sup>2</sup> are limits of the taken solar spectrum. Then, equation (1) is completely determined and conversion efficiency *η* can be evaluated.
