*3.1. Design of Hybrid Hetero-Homojucntion Cell*

Firstly, we will discuss the main design rule that governs the operation of the p-n homojunction part of the hybrid hetero-homojunction cell. The doping densities of the two regions of a p–n junction must fulfill a condition to ensure that the depletion width is considerably lesser than the absorber layer thickness (*t*). Therefore, the doping densities must meet the following condition [53],

$$N\_{A/D} > \frac{2\varepsilon\_o \varepsilon\_r V\_{bi}}{qt^2} \tag{7}$$

where *Vbi* is the built-in voltage. Figure 3 shows the variation of the depletion width vs donor or acceptor doping density for two different values of *Vbi*. The horizontal line indicates the perovskite absorber thickness, which is 350 nm in our design. Therefore, the threshold (minimum) doping density depends on the value of *Vbi* and it is in the order of 1 × <sup>10</sup><sup>16</sup> cm−3, given the reported experimental thickness of *<sup>d</sup>* = 350 nm and *<sup>ε</sup><sup>r</sup>* = 8.2 for a typical value of *Vbi* ~ 1 V. Our initial design starts by *ND* (n-CH3NH3SnI3) = *NA* (p-CH3NH3SnI3)=1 × 1016 cm<sup>−</sup>3.

**Figure 3.** The variation of the depletion width vs donor or acceptor doping density.

#### 3.1.1. Influence of n- and p-Region Thickness

Regarding the hybrid hetero-homojunction lead-free PSC shown in Figure 1b, the whole n-CH3NH3SnI3/p-CH3NH3SnI3 homojunction functions as the perovskite absorber region. When the cell is illuminated, the photoinduced charge carriers are produced in both the p-type and n-type layers. The variations of the photovoltaic parameters are examined with p-type CH3NH3SnI3 layer thickness varying from 0 nm to 350 nm, taking into account that the sum of the n-type and p-type layers is fixed at 350 nm, which is the original, experimentally reported, overall thickness of the absorber layer [47].

The *Jsc*, *Voc*, *FF*, and PCE vs. p-layer thickness are given in Figure 4. Referring to the figure, it can be observed that all performance parameters are gradually increasing with the increase of p-CH3NH3SnI3 thickness. Beyond a thickness of 300 nm, *Jsc* saturates at a value of about 20 mA/cm2. The thickness of p-CH3NH3SnI3 affects *Voc* as it increases and then decreases slightly beyond a thickness of 300 nm. *FF* is increased because the series resistance declines with the decreasing thickness of the n-CH3NH3SnI3 layer (as the p-CH3NH3SnI3 layer thickness increases). The combination of the *Jsc*, *Voc* and *FF* results in the variation trend of PCE shown in the figure. The PCE is first enhanced with increasing the thickness of p-CH3NH3SnI3 and then slowly decreases beyond a p-type layer thickness of 300 nm. Accordingly, in the following simulations, we set thicknesses of 300 and 50 nm as optimized values of the p-CH3NH3SnI3 and n-CH3NH3SnI3 thickness, respectively.

**Figure 4.** Impact of p-layer thickness on the photovoltaic parameters.

Figure 5 shows the *J*–*V* characteristics under illumination (Figure 5a) and the quantum efficiency (Figure 5b) for three selected cases. The first case is the conventional pin structure for which the absorber is intrinsic and the other two cases are for *xp* > *xn* (taking *xp* = 300 μm) and *xn* > *xp* (taking *xp* = 50 μm) to demonstrate the difference between the impact of ntype and p-type thicknesses on the terminal characteristics compared to the conventional case. The results are confirmed in Figure 4, as both *Voc* and *Jsc* are degraded when the n-type thickness is higher than the p-type thickness. It was observed that when the p-type thickness is higher, the performance is enhanced over the conventional pin structure.

**Figure 5.** Difference between hetero-homo p-n junction (for *xn* > *xp* and *xp* > *xn*) vs conventional pin solar cell (**a**) illuminated *J*–*V* under AM1.5 and (**b**) quantum efficiency.

To give a physical insight about the dependence of the performance parameters on the thickness of the p- (or n-) layer, we drew the generation and the recombination rates across the device distance at a voltage of 0.5 V, as illustrated in Figure 6. The generation rate is the same for the different cases, as is clear from Figure 6a. It can be seen in Figure 6b that it is better to choose a wider p-layer to suppress the recombination losses. When using a wider n-layer, the recombination increases in the n-layer. Although it is suppressed in the p-layer when compared to the case of intrinsic absorber, the overall recombination losses are higher when the n-layer thickness exceeds the p-layer thickness.

**Figure 6.** (**a**) The generation rate and (**b**) recombination rate across the device distance at a voltage of 0.5 V for three distinctive cases of absorber.

In addition, the electric field distribution along the device structure (from ETL to HTL direction) at a voltage of 0.5 V supports the idea of recombination behavior, and this distribution is shown in Figure 7. The field distribution illustrates that the field direction of the case when *xn* > *xp* is reversed at the absorber/ETL interface, so it is in opposite direction to the two other cases. This field reversal affects the carrier collection and results in an increase in recombination rates and thus a reduction in the current. On the other hand, the field direction of the two other cases is in the proper route at the absorber/ETL interface in such a way as to enhance the electron collection by pushing the electrons from the absorber to the ETL. In addition, it is noted that when *xp* > *xn*, the field has the highest peak value amongst the other cases. This electric field behavior explains the reduction of the recombination rate inside the absorber near the ETL for the case when *xp* > *xn*. It also explains the higher rate near the HTL, as the field for *xp* > *xn* is reversed which limits the carrier collection which, in turn, slightly increases the recombination rate. However, this reduction in the carrier collection has a minor impact. Further, the electron and hole concentration distributions, which strongly depend on the electric field behavior, are shown in Figure S1 (see Supplementary Material).

**Figure 7.** Electric field distribution across the device distance for the three cases of absorbers. The field is calculated at a voltage of 0.5 V. The electric field direction is also indicated in the figure by arrows.

#### 3.1.2. Influence of Perovskite Doping Concentration

In this subsection, the impact of the doping concentrations of the p-type and n-type CH3NH3SnI3 films on the performance parameters of the proposed structure is investigated. Figure 8 demonstrates the changes of performance parameters vs both donor concentration *ND* of n-CH3NH3SnI3 and acceptor concentration *NA* of the p-CH3NH3SnI3 from 10<sup>16</sup> to 1018 cm<sup>−</sup>3, with maintaining other material parameters as they are recorded in Table 1.

**Figure 8.** Variations of solar cell performance parameters with the donor concentration *ND* of n-CH3NH3SnI3 and acceptor concentration *NA* of the p-CH3NH3SnI3 (**a**) open circuit voltage, (**b**) shortcircuit current, (**c**) fill factor and (**d**) efficiency.

It can be seen from Figure 8a that for higher values of *ND* (>1017 cm<sup>−</sup>3) in n-CH3NH3SnI3, *Voc* rises with rising *NA* in p-CH3NH3SnI3, which can be clarified by drawing the energy band diagram, as shown in Figure 9a, at the dark condition, in which *NA* in p-CH3NH3SnI3 varies (1016 and 1018 cm−3) and *ND* in the n-side is maintained at 3 × <sup>10</sup><sup>17</sup> cm−3. We can observe that by increasing *NA*, the degree of band bending increases on the n-side, which causes *Vbi* to increase, leading to the rise of *Voc* [54]. The built-in voltage is calculated from the conduction band (or valence band) energy difference between the n-side and p-side. Meanwhile, for lower values of *ND* (<1017 cm−3) in n-CH3NH3SnI3, the open circuit voltage increases slightly with increasing *NA* and then decreases, also marginally, further increasing *NA* beyond about 1017 cm−3. Therefore, the effect of *NA* is minor for lower values of *ND*. This can also be attributed to the behavior of *Vbi*, as illustrated in Figure 9b, which shows the energy band diagram (at dark) for two values of *NA* (3 × <sup>10</sup><sup>16</sup> and 1018 cm<sup>−</sup>3) at a fixed value of *ND* = 3 × <sup>10</sup><sup>16</sup> cm<sup>−</sup>3. As can be inferred from the figure, the variation of *Vbi* is insignificant, which reflects on the *Voc* trivial change.

**Figure 9.** The energy band diagrams of the proposed structure at the dark condition for (**a**) *ND* in the n-side is maintained at <sup>3</sup> <sup>×</sup> <sup>10</sup><sup>17</sup> cm<sup>−</sup>3, (**b**) fixed value of *ND* = 3 <sup>×</sup> 1016 cm<sup>−</sup>3.

Moreover, in Figure 8b, *Jsc* declines with the rise in *NA* in p-CH3NH3SnI3, as the higher doping concentration results in thinner depletion region width and a broader neutral region, in which the greater bulk recombination takes place and thus reduces the collection chance of photoexcited electrons and holes. In addition, *ND* of n-CH3NH3SnI3 has a minor influence on the *Jsc*. This is because the n-region has the smallest thickness. Although light is illuminated from the TCO side, the number of photogenerated carriers of the n-CH3NH3SnI3 region is considered low when compared to that at the wider p-CH3NH3SnI3 region.

Regarding the *FF* in Figure 8c, its variation with doping concentration of n- and p-CH3NH3SnI3 is the opposite of that of *Jsc*. The general tendency of rising *FF* with increasing *NA* can be noticed, which is nearly independent of the value of *ND*. As *NA* increases, the resistivity of the layer decreases and hence the series resistance declines, which improves *FF*. This phenomenon can be explained based on the dark characteristics of the cell. As shown in Figure 10a, the dark *J*–*V* is drawn for three different values of p-side acceptor concentration. Using the *J*–*V* dark characteristics, the local ideality factor can be extracted, as seen in Figure 10b. The fill factor is directly correlated to the value of the local ideality factor at the maximum power point (MPP) [55,56]. As can be inferred from Figure 10b, the local ideality factor decreases as *NA* increases, which proves the enhancement in the fill factor. Further, the corresponding values of the reverse saturation current and ideality factor of the equivalent single diode model are presented in Table S2 (see Supplementary Material).

Finally, in Figure 8d, the PCE is enhanced with moderate *NA* values of p-side which are in the order 4 × 1016 to 6 × 1016 cm<sup>−</sup>3. Higher *ND* of n-side values is required to obtain high efficiencies. Values of *ND* are in the range of 9 × 1016 to 2 × 1017 cm<sup>−</sup>3. Therefore, the n-side is suggested to be slightly at a higher doping level than the p-side doping. Hence, in the simulation, the optimized doping concentration of the n-side and the p-side is 1017 and 5 × 1016 cm<sup>−</sup>3, respectively. In this case, the photovoltaic parameters are: *Voc* = 0.7513, *Jsc* = 19.62, *FF* = 58.72 and PCE = 8.66%. Based on these reported values, the efficiency of the hybrid cell after a proper choice of the thickness and doping is higher than that of the conventional pin structure by more than 3.5%.

**Figure 10.** Impact of p-side acceptor concentration. (**a**) The cell dark *J–V* characteristics, (**b**) the cell local ideality factor.

#### 3.1.3. Comparison between Absorber Doping Types

In this subsection, we compare four different devices: namely, the reference intrinsic absorber (having a p-type doping concentration of 1 × <sup>10</sup><sup>15</sup> cm−3), a single p-layer (concentration of 1 × <sup>10</sup><sup>16</sup> cm−3) a single n-layer (concentration of 1 × <sup>10</sup><sup>16</sup> cm−3) and a pn-absorber, including the n- and p-layers. The impact of the doping concentration on the efficiency when using a single n-layer and p-layer absorbers is shown in Figure S2 (see Supplementary Material) in which the optimum efficiency occurs near the selected value of 1 × <sup>10</sup><sup>16</sup> cm−<sup>3</sup> for the single p-layer, while the impact of doping on the efficiency of the single n-layer is very weak. The *J*–*V* characteristics under illumination and EQE are presented in Figure 11a,b, respectively, for a series of devices with the differently tuned doping. The performance of the p-absorber cell is higher than that of the intrinsic case. However, the optimized hybrid cell gives a higher performance, indicated by the cell performance presented in Table 4. The situation is different if the n-absorber perovskite is used. Due to the strongly reduced carrier collection, a low *Jsc* of 7.91 mA/cm2 is obtained and the *FF* is as low as 32.98%. Moreover, the spectral response of the hybrid and single p-layer cells are close (see Figure 11b), while the intrinsic absorber cell is lower. The single n-layer has the lowest EQE, as expected from its low *Jsc*.

**Figure 11.** Comparison between different types of absorbers. (**a**) The cell *J*–*V* characteristics under illumination, (**b**) quantum efficiency.


**Table 4.** Cell performance parameters of the four studied cases of the absorber layer.

This can be explained by plotting the electric field distribution along the device distance, as previously discussed in Section 3.1.1. The distribution is shown in Figure S3 (see Supplementary Materials) for two cases, namely the short-circuit condition and at a voltage of 0.5 V. The results indicate that the field direction of the n-absorber is reversed, contrary to the other three cases, which results in higher recombination rates due to the poor electrons and hole extraction. The results also indicate that the pn absorber has the highest electric field peak compared to the other cases. To conclude this comparison, the cell containing the perovskite homojunction has a remarkable performance, especially its *Voc* (0.751 V) and *FF* (58.72%), which improved significantly compared to those of the other structures, even if *Jsc* is slightly less than the single p-layer case. The overall efficiency indicates the superiority of the hybrid homojunction lead-free cell design which gives 8.66%.
