**3.2. Strain-balanced GaAsP/InGaAs/GaAs SB-QWSC**

The SB-QWSC is a GaAs p-i-n solar cell with quantum well layers incorporated into the i-region with InGaAs as well material and GaAsP as barrier material. Figure 7 shows the band-structure of the GaAsP/InGaAs/GaAs SB-QWSC modeled by C. I Cabrera et al. [27].

**Figure 7.** The band-structure of the SB-QWSC. The QW stack is embedded within the depletion zone of the GaAs cell and extends the absorption edge of the cell beyond that of a classical GaAs solar cell.

The compressive strain in the InGaAs QW is matched by tensile strain in GaAsP barriers, overcoming the lattice-mismatch limitation. The GaAsP and InGaAs layer widths were chosen to ensure the average lattice parameter across the i-region was equal to that of GaAs following equation 8, where the barrier material is GaAsP and well material InGaAs.

Elastic constants were considered to evaluate the tensile and compressive strain in GaAsP and InGaAs layers and examine the effect on the band structure of both materials under strain. Consequently, when the In an P compositions are varied, the strains in the barrier and well layers modify absorption threshold in both layers.

in particular, when 10 clusters of superlattices are inserted in the intrinsic region. Therefore,

We researched the AlGaAs/GaAs SLSC efficiency, which was compared with the AlGaAs/GaAs QWSC efficiency. Figure 6 illustrates the normalized efficiency versus Aluminium composition and cluster number in the QWSC. The normalized efficiency in this figure is defined as the ratio between the efficiencies of SLSC and its corresponding QWSC for the graph versus Aluminium concentration. In the case of the graph in function of the cluster number the QWSC efficiency in the ratio is the highest. The SLSC efficiency is better than the highest QWSC efficiency for five or more clusters of superlattices in the intrinsic region which meas that under these conditions the photocarrier generation in the SLSC overcome the recombination. However, the best SLSC efficiency is just 4% better than the QWSC efficiency because the increase of SLSC photocurrent does not increase enough. This suggest that the miniband absorption and the absorption of wide quantum wells are comparable. On the other hand, normalized efficiency was plotted versus Al concentration in the QWSC, for 10 clusters in the SLSC intrinsic region, 15 nm well width, and 24 wells in the QWSC intrinsic region and in this case Figure 6 exhibits that SLSC efficiency is always higher than the QWSC efficiency, and become larger as the well barrier height increases. Because of the results that our model predictions are neither compared nor confirmed experimentally, it

would be interesting to see if future experiments will corroborate our findings.

The SB-QWSC is a GaAs p-i-n solar cell with quantum well layers incorporated into the i-region with InGaAs as well material and GaAsP as barrier material. Figure 7 shows the band-structure of the GaAsP/InGaAs/GaAs SB-QWSC modeled by C. I Cabrera et al. [27].

**Figure 7.** The band-structure of the SB-QWSC. The QW stack is embedded within the depletion zone of the GaAs cell

The compressive strain in the InGaAs QW is matched by tensile strain in GaAsP barriers, overcoming the lattice-mismatch limitation. The GaAsP and InGaAs layer widths were chosen to ensure the average lattice parameter across the i-region was equal to that of GaAs following equation 8, where the barrier material is GaAsP and well material InGaAs.

and extends the absorption edge of the cell beyond that of a classical GaAs solar cell.

**3.2. Strain-balanced GaAsP/InGaAs/GaAs SB-QWSC**

this mechanism becomes the most relevant for SLSC.

168 Solar Cells - New Approaches and Reviews

**Figure 8.** Representation of the effect of strain on band structure of *In*0.2*Ga*0.8*As* and *GaAs*0.7*P*0.3 around of the first Brillouin zone center compared with unstrained bulk material (dashed line) which is degenerate degenerate at the zone centre for valence band. *c*(*s*) is the strained conduction bands, *lh*(*s*) and *hh*(*s*) are the strained light and heavy hole bands, respectively (straight lines). The vertical axes give the energy in eV, the lateral axes give the wave number *kx* and *kz* , respectively. (a) Shift and deformation of *In*0.2*Ga*0.8*As* energy bands for compressive strain, *εxx* = − 0.014, *εzz* = 0.013, (b) of *GaAs*0.7*P*0.3 energy bands for tensile strain, *εxx* = 0.019; *εzz* = −0.010.

The p- and n- regions were designed with 200 and 500 nm width respectively, with a 40 nm *Al*0.8*Ga*0.2*As* window layer before the p-region to reduce front surface recombination, with a MgF: SiN layerfor anti-reflective coating (ARC) of 70 nm width. The electron and hole concentrations are *n* = 1018*cm*−<sup>3</sup> and *p* = 1018*cm*−3, respectively. Finally a passivation layer in the solar cell rear was assumed with 200 cm/s surface recombination velocity.

Figure 9 depicts TE and TM spontaneous emission rates and the ratio TM/TE as a function of In fraction. The discontinuous steps at approximately 6% In are due to the emergence of QW energy levels, *e*<sup>2</sup> − *hh*<sup>2</sup> and *e*<sup>2</sup> − *lh*<sup>2</sup> transitions as the well depth increases. It is evidenced that as the In fraction increases, the QWs influence a higher compressive strain. As a result of this, radiative transitions from the conduction band to *hh* band (TE) are favoured over those to *lh* band (TM). However, both polarized emissions increase with well depth, indicating that biaxial compressive strain does not suppresses a mode of radiative recombination in the plane of the QWs, although certainly the TM/TE ratio is reduced. We, consequently, observe that increasing the In composition leads to larger radiative recombination that increases the total recombination dark current.

In Figure 10, spontaneous emission rates and gain are presented as a function of In fraction, where is important emphasize the great difference between the values of the spontaneous

**Figure 9.** Modeled spontaneous emission rate for TE and TM modes and TM/TE ratio versus In composition for 10 well embedded within i-region of the SB-QWSC. Well width, *LW* =15 nm and P composition, y = 0.05. The TE mode is favored over TM mode, but both polarized emissions increase with well depth.

**Figure 10.** Spontaneous emission rate and gain versus In composition for 10 well embedded within i-region of the SB-QWSC. Well width, *LW* =15 nm and P composition, y =0.05. Gain is several orders greater than radiative recombination

emission rates evidenced in the ratio TM/TE. The results of this study indicate that the generation of electron-hole pairs in the QWs is much higher than the radiative recombination, and if we add to this the influence of transverse electric field in the depletion region, which favors thermally assisted tunneling, then the carriers escape from the QWs with unity efficiency.

A distributed Bragg reflector (DBR) is a region consisting of layers of alternating refractive indices optimized for a specific wave-length such that each layer is a quarter wavelength thick. As a result, partial reflections from each interface interfere constructively and the reflectivity is high over a narrow wavelength band.

Photon-recycling is the generation of an electron-hole pair via the absorption of a photon emitted elsewhere in the cell. The increased absorption is due to the reflection of incident solar radiation which has not been absorbed on its first pass through the cell and may then be reabsorbed on its second pass. It is equivalent to say that a DBR doubles the optical path length of a SB-QWSC without altering the length over which minority carriers must travel. The photons emitted by recombination into quantum wells were also considered.

The net solar incident radiation flow on front surface of a solar cell was modeled as a Fabry-Perot cavity. We derived an expression to calculate the contribution of the multiple internal reflections inside the device and then the net spectrum, *Fnet*(*λ*), which takes into account all contributions and the incident AM1.5 solar spectrum *F*(*λ*) is given by:

**Figure 9.** Modeled spontaneous emission rate for TE and TM modes and TM/TE ratio versus In composition for 10 well embedded within i-region of the SB-QWSC. Well width, *LW* =15 nm and P composition, y = 0.05. The TE mode is

**Figure 10.** Spontaneous emission rate and gain versus In composition for 10 well embedded within i-region of the SB-QWSC. Well width, *LW* =15 nm and P composition, y =0.05. Gain is several orders greater than radiative

emission rates evidenced in the ratio TM/TE. The results of this study indicate that the generation of electron-hole pairs in the QWs is much higher than the radiative recombination, and if we add to this the influence of transverse electric field in the depletion region, which favors thermally assisted tunneling, then the carriers escape from the QWs with unity

A distributed Bragg reflector (DBR) is a region consisting of layers of alternating refractive indices optimized for a specific wave-length such that each layer is a quarter wavelength

favored over TM mode, but both polarized emissions increase with well depth.

170 Solar Cells - New Approaches and Reviews

recombination

efficiency.

$$F\_{net}(\lambda) = F(\lambda) \left[ 1 + \frac{r\_B \left( r\_A + e^{a\_T^\*} \right)}{e^{2a\_T^\*} - r\_B r\_A} \right] \tag{29}$$

$$\mathfrak{a}\_T^\* = \sum\_j \mathfrak{a}\_j(\bar{\lambda}) l\_j \tag{30}$$

where *αj*(*λ*) are the absorption coefficients of each layer of Fabry-Perot cavity structure, where the exciton effect in the quantum well was taken into account, *lj* are layer widths, and *rA*, *rB* are the internal reflectivity from the front and back surface of the cell, respectively.

The photocurrent *JPH* is calculated from the total quantum efficiency (*QETOTAL*) of the cell:

$$J\_{PH} = q \int\_{\lambda\_1}^{\lambda\_2} F(\lambda) \left[ 1 + \frac{r\_B \left( r\_A + \varepsilon^{a\_\top^\*} \right)}{\varepsilon^{2a\_\top^\*} - r\_B r\_A} \right] Q E\_{TOTAL}(\lambda) d\lambda \tag{31}$$

where *λ*<sup>1</sup> = 400*nm* and *λ*<sup>2</sup> is the effective absorption threshold determined by the fundamental electron and hole confinement states.

Figure 11 shows modeled and experimental quantum efficiency (QE) versus wavelength for qt1897b sample from the Quantum photovoltaics Group of Imperial College. The cell is a p-i-n diode with an i-region containing five QWs that are 9.6 nm wide of compressively strained *In*0.16*Ga*0.84*As* inserted into tensile-strained *GaAs*0.91*P*0.09 barriers at strain-balance condition. The extra absorption is displayed in the inset of Figure 11, at wavelengths in excess of the GaAs band gap where the increase in quantum efficiency in the 880 nm is readily apparent. Figure 11 also shows the computed QE spectrum with DBR, using *rA* = 0.1 and *rB* = 0.95. The main feature of this plot is that for a highly reflecting mirror, nearly all photons absorbed contribute to the QE. This is clearly a desirable feature as it implies that carrier collection from the QWs is very efficient allowing the increase in short circuit current. It is a good indicator that the QE of an QWSC could well approach that of a bulk cell with

**Figure 11.** Modelled and experimental quantum efficiency versus wavelength for 5 well qt1897b sample from the Quantum photovoltaics Group of Imperial College. The inset shows the wavelength range dominated by the QW absorption with and without influence of DBR.

**Figure 12.** Contour plot for conversion efficiency as function of Bragg reflectivity and quantum well number. P composition y = 0.09, In composition x = 0.17 and *LW* = 9.6 nm

a similar band-gap if the light could be confined inside the cell until it was completely absorbed.

For qt1897b solar cell, the dependence of conversion efficiency on back mirror reflectivity and quantum well number (*NW*) is examined in Figure 12. This plot suggests that with the addition of a DBR in the device, fewer quantum wells are required to grow in the i-region in order to achieve high performance. In fact, low energy photons from the radiative recombination in the QWs can be reflected back into i-region and reabsorbed, lowering the radiative recombination current and improving the open-circuit voltage

It can be expected that SB-QWSC under concentration will be operating in a regime where recombination is dominated by radiative processes. Therefore, photon recycling effect is favoured under solar concentration when photons emitted from radiative recombination are subsequently reabsorbed by the solar cell. This can be explained as an increase in minority carrier lifetime or reduction in dark-current.This behavior is shown in Figure 13, where we have examined the conversion efficiency as a function of solar concentration for optimized *GaAs*0.96*P*0.04/*In*0.03*Ga*0.97*As*/*GaAs* solar cell with 20 nm quantum well width. We used *rA* = 0.1, *rB* = 0.95 and resistive effects were neglected. It can be observed how the conversion efficiency should increase with solar concentration up to 1000 suns. In any concentration range, the DBR cell efficiency improvement over the non-DBR cell which is explained by the fact of the lower dark current in the DBR cell. This effect also causes that the net increase in conversion efficiency is lower with increasing *NW*, as it can be noted in Figure 13.

**Figure 13.** Conversion efficiency as a function of solar concentration for several *GaAs*0.96*P*0.04/*In*0.03*Ga*0.97*As*/*GaAs* SB-QWSC which differ in the number of QW embedded within the i-region with and without influence of DBR.
