Design guides for artificial photosynthetic devices consisting of voltage-matched perovskite/silicon tandem solar-cell modules and electrochemical reactor modules

The current density of a single-junction SC (j^(SC)) is described by the following formulation derived from an equivalent circuit model consisting of a constant current source, a diode, and a series resistance, ^28,35,36)

$$\begin{eqnarray}&&{j}^{({\rm{SC}})}\left[v\right]={j}_{{\rm{ph}}}-{j}_{0}\,\left(\exp \left[q\,\left(v+{r}_{{\rm{s}}}\,{j}^{({\rm{SC}})}\left[v\right]\right)/{k}_{{\rm{B}}}\,{T}_{{\rm{cell}}}\right]\right),\end{eqnarray} \tag{ 1 }$$

where j_ph, j₀, and r_s are the current density, reverse saturation current density of the diode, and series resistance, respectively, and q, k_B, and T_cell denote the elementary charge, Boltzmann constant, and cell temperature, respectively. The external quantum efficiency (η_EQE) of the photovoltaic conversion can be approximated to be a constant independent of the photon energies, and then j_ph is determined from the bandgap (E_g) of the light-absorbing material used in the SC and the spectral photon flux of solar illumination (${n}_{\mathrm{sun}}\left[\hslash \omega \right]$),

$$\begin{eqnarray}&&{j}_{\mathrm{ph}}=q\,{\eta }_{\mathrm{EQE}}{\int }_{{E}_{{\rm{g}}}}^{\infty }d(\hslash \omega )\,{n}_{\mathrm{sun}}\left[\hslash \omega \right].\end{eqnarray} \tag{ 2 }$$

For the Si bottom SC, the upper bound of the integration range in Eq. (2) is E_g of the top PVK SC (${E}_{{\rm{g}}}^{(\mathrm{PVK})}$). This means no overlap between the PVK and Si absorption ranges, and hence will lead to slight overestimations of η_SC and the impact of spectrum variation on η_SC. The component originating from the radiative recombination involved in the recombination current density in Eq. (1) is determined by the generalized Plank law. ³⁷⁾ In addition, the external radiative efficiency (η_ERE) is introduced for the expression of the nonradiative component. ^36,38) Thus, by replacing the Fermi–Dirac distribution function with the Boltzmann distribution function,

$$\begin{eqnarray}&&{j}_{0}=\frac{2\,\pi \,{k}_{{\rm{B}}}\,{T}_{\mathrm{cell}}}{{h}^{3}\,{c}^{2}\,{\eta }_{\mathrm{ERE}}}\,\left({E}_{{\rm{g}}}^{2}+2{E}_{{\rm{g}}}\,{T}_{\mathrm{cell}}+2\,{T}_{\mathrm{cell}}^{2}\right)\,\exp \left[-{E}_{{\rm{g}}}/\left({k}_{{\rm{B}}}\,{T}_{\mathrm{cell}}\right)\right]\,\end{eqnarray} \tag{ 3 }$$

is derived, with h and c being the Plank constant and light velocity in vacuum, respectively.

The VM tandem SC module consists of the parallel-connected top module and bottom module, in which n_PVK PVK SCs and n_Si Si SCs are connected in series, respectively. The J–V relation per a single PVK SC is derived as follows,

$$\begin{eqnarray}&&J={j}_{{\rm{PVK}}}\left[V\right]+{j}_{{\rm{Si}}}\left[V/\left({n}_{{\rm{Si}}}/{n}_{{\rm{PVK}}}\right)\right]/\left({n}_{{\rm{Si}}}/{n}_{{\rm{PVK}}}\right).\end{eqnarray} \tag{ 4 }$$

It should be noted that the ohmic losses in the transparent electrodes and connecting wires and the photon losses arising from reflection and parasitic absorption that do not contribute to photovoltaic conversion are neglected.

The J–V characteristics of the 2T tandem SC module are obtained by solving the simultaneous equation,

$$\begin{eqnarray}&&\left\{\begin{array}{l}J={j}_{{\rm{top}}}\left[{v}_{{\rm{top}}}\right]={j}_{{\rm{bot}}}\left[{v}_{{\rm{bot}}}\right]\\ V={v}_{{\rm{top}}}+{v}_{{\rm{bot}}}\end{array}\right..\end{eqnarray} \tag{ 5 }$$

When the SC module is coupled with a power conditioner, η_SC is the ratio of the maximal value of the output power density that is equal to the product of J and V as a function of V to the solar intensity (P_sun),

$$\begin{eqnarray}&&{\eta }_{\mathrm{SC}}={\left.J\left[V\right]\cdot V\right|}_{\max }/{P}_{\mathrm{sun}}.\end{eqnarray} \tag{ 6 }$$

For numerical evaluations, η_EQE, η_ERE, and r_s for the PVK and Si SCs were determined so that the photovoltaic properties calculated using Eq. (1) are approximately the same as the values of the world-record SCs measured under the standard condition for SC evaluation of the AM 1.5 G illumination (P_sun = 100 mW cm⁻²) ³⁹⁾ and T_cell = 25 °C: η_EQE = 0.90, η_ERE = 0.15, and r_s = 3.1 Ω cm² for the PVK SC (${E}_{{\rm{g}}}^{(\mathrm{PVK})}$ = 1.51 eV); and η_EQE = 0.96, η_ERE = 0.0062, and r_s = 0.28 Ω cm² for the heterojunction (HJ) Si SC. These parameter sets reproduce the experimental results well, as summarized in Table I. ^17,40) Although more complicated equivalent circuits including the diode ideal factor and two diodes have been proposed for more precise fitting, ^41,42) Eq. (1) well describes the J–V characteristics of a highly efficient SC at around its maximal power point, which fits the objective of the present study. For the PVK SCs with different ${E}_{{\rm{g}}}^{(\mathrm{PVK})}$ values, the same parameter set was used neglecting the ${E}_{{\rm{g}}}^{(\mathrm{PVK})}$-dependence. This approximation leads to a higher η_SC than those previously demonstrated at a wider ${E}_{{\rm{g}}}^{(\mathrm{PVK})}.$ For example, η_SC = 23.8% at ${E}_{{\rm{g}}}^{(\mathrm{PVK})}$ = 1.7 eV is higher than the record efficiency of 22% using metal electrodes ⁴³⁾ and 20% of semi-transparent SCs with ${E}_{{\rm{g}}}^{(\mathrm{PVK})}$ = 1.6 eV, ^34,44) and therefore would be a target value.

When an SC operates outdoors, T_cell changes. A higher T_cell narrows E_g and hence increases j_ph. However, the detriments of a smaller open-circuit voltage and a lower filling factor arising from more significant radiative and nonradiative recombination surpass the benefit of the larger j_ph. Consequently, η_SC lowers. Thus, the dependences of η_SC and η_STC on T_cell are essential for evaluation of annual production. ^31,45,46) Therefore, the explicit T_cell dependence and T_cell-dependent E_g ($d{E}_{{\rm{g}}}/d{T}_{\mathrm{cell}}$ = 0.31 meV K^–1 for PVK ⁴⁷⁾ and −0.27 meV K^–1 for Si ⁴⁸⁾) in Eqs. (1)−(3) are involved, whereas η_EQE, η_ERE, and r_s are approximated to be constant independent of T_cell. The resultant T_cell-dependences of η_SC, i.e. $d{\eta }_{{\rm{SC}}}/d{T}_{{\rm{cell}}}$ is −0.15% K^–1 for the PVK SC and −0.27% K^–1 for the Si SC, are close to the experimental values of −0.17% K^–1 (PVK) ⁴⁹⁾ and −0.26% K^–1 (HJ Si). ⁵⁰⁾

The operating point of the VM-nEC artificial photosynthetic device illustrated in Fig. 1(d) is determined from the J–V curve of the VM tandem SC module and the load curve of the EC module. When n_EC EC reactors are connected in series in the EC module powered by the VM SC module, the J–V curve per a single EC reactor is

$$\begin{eqnarray}\begin{array}{ccc}J & = & {j}_{\mathrm{PVK}}\left[V/\left({n}_{\mathrm{PVK}}/{n}_{\mathrm{EC}}\right)\right]/\left({n}_{\mathrm{PVK}}/{n}_{\mathrm{EC}}\right)\\ & & \,+{j}_{\mathrm{Si}}\left[V/\left({n}_{\mathrm{Si}}/{n}_{\mathrm{EC}}\right)\right]/\left({n}_{\mathrm{Si}}/{n}_{\mathrm{EC}}\right).\end{array}\end{eqnarray} \tag{ 7 }$$

The intersection of this J–V curve and the load curve of the EC reactor (${j}_{{\rm{EC}}}\left[V\right]$) determines the operating point (J_op and V_op).

The Faradaic efficiency (η_FE) of the EC reaction of H₂O → H₂ + 1/2 O₂ is virtually unity. The thermodynamic threshold voltage of this reaction is 1.23 V. Thus, the conversion efficiency of solar energy to chemical energy of H₂ (η_H2) is derived as follows,

$$\begin{eqnarray}&&{\eta }_{{\rm{H2}}}=1.23\,{J}_{{\rm{op}}}/{P}_{{\rm{sun}}}.\end{eqnarray} \tag{ 8 }$$

On the other hand, η_FE of the CO₂ → CO + 1/2 O₂ reaction is lower than unity and dependent on V_op. Therefore, the solar-to-CO energy conversion efficiency (η_CO) is calculated by

$$\begin{eqnarray}&&{\eta }_{\mathrm{CO}}=1.34\,{J}_{\mathrm{op}}\,{\eta }_{{\rm{FE}}}\left[{V}_{\mathrm{op}}\right]/{P}_{\mathrm{sun}}\end{eqnarray} \tag{ 9 }$$

using the thermodynamic threshold voltage of 1.34 V.

For numerical evaluations, the load curve of the H₂-producing reactor was taken from Ref. 51, the load curve and η_FE for the CO production from Ref. 52, and these curves are depicted in Fig. 3. CO production requires a higher voltage by approximately 0.2 V than H₂ production. Another point is that the η_FE of CO production lowers with decreasing voltage from 1.7 V.

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The standard condition for SC evaluations is the combination of AM 1.5 G illumination (P_sun = 100 mW cm⁻²) ³⁹⁾ and T_cell = 25 °C. However, this is a rare condition for outdoor use; mostly, P_sun is smaller and T_cell is higher. Therefore, the average values of P_sun and T_cell ($\left\langle {P}_{{\rm{sun}}}\right\rangle$ and $\left\langle {T}_{{\rm{cell}}}\right\rangle ,$ respectively) and their distributions were obtained for a south-facing 32° slope using a database of P_sun, solar spectrum, and ambient temperature (T_a) measured in Tsukuba, Japan (N36° and E140°, in 2015). ⁵³⁾ The weighted values of $\left\langle {P}_{{\rm{sun}}}\right\rangle$ and $\left\langle {T}_{{\rm{cell}}}\right\rangle$ by P_sun were calculated as follows because the electricity of the SC modules and the H₂ and CO production rates of the artificial photosynthetic devices are approximately proportional to P_sun,

$$\begin{eqnarray}&&\left\langle {P}_{{\rm{sun}}}\right\rangle =\displaystyle \int dt\,{P}_{{\rm{sun}}}^{2}\left[t\right]/\displaystyle \int dt^{\prime} \,{P}_{{\rm{sun}}}\left[t^{\prime} \right],\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\left\langle {T}_{{\rm{cell}}}\right\rangle =\displaystyle \int dt\,{T}_{{\rm{cell}}}\left[t\right]\,{P}_{{\rm{sun}}}\left[t\right]/\displaystyle \int dt^{\prime} \,{P}_{{\rm{sun}}}\left[t^{\prime} \right].\end{eqnarray} \tag{ 11 }$$

An empirical relation of P_sun and T_a to T_cell was adopted for determining T_cell, ⁴⁹⁾

$$\begin{eqnarray}&&\begin{array}{l}{T}_{{\rm{cell}}}=\left(NOCT-20\left[^\circ {\rm{C}}\right]\right)\,{P}_{{\rm{sun}}}/80\,\left[{\rm{mW}}\,{{\rm{cm}}}^{-2}\right]+{T}_{{\rm{a}}},\\ NOCT=44\,^\circ C,\end{array}\end{eqnarray} \tag{ 12 }$$

where NOCT denotes the nominal operating cell temperature defined as T_cell at P_sun = 80 mW cm⁻², T_a = 20 °C, and a wind speed of 1 m s⁻¹.

The results were $\left\langle {P}_{{\rm{sun}}}\right\rangle$ = 63.9 mW cm⁻² and $\left\langle {T}_{{\rm{cell}}}\right\rangle$ = 37.7 °C, and their distributions are depicted in Figs. 4(a) and 4(b), respectively. Eighty percent of the relative frequency distribution of P_sun locates in a range of 20–95 mW cm⁻², and 80% of the T_cell distribution in 20 °C–55 °C.

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The average of ${n}_{{\rm{sun}}}\left[\hslash \omega \right]$ ($\left\langle {n}_{{\rm{sun}}}\left[\hslash \omega \right]\right\rangle$) was determined in a similar manner,

$$\begin{eqnarray}&&\left\langle {n}_{{\rm{sun}}}\left[\hslash \omega \right]\right\rangle =\displaystyle \int dt\,{n}_{{\rm{sun}}}\left[\hslash \omega ,t\right]\,{P}_{{\rm{sun}}}\left[t\right]/\displaystyle \int dt^{\prime} \,{P}_{{\rm{sun}}}\left[t^{\prime} \right].\end{eqnarray} \tag{ 13 }$$

As is clear from Figs. 4(c) and 4(d), $\left\langle {n}_{{\rm{sun}}}\left[\hslash \omega \right]\right\rangle$ is close to the AM 1.5 G photon flux, except that the high photon-energy components are relatively only slightly more intense. The variation in the spectrum was quantified by the relative difference (${\rm{\Delta }}{\tilde{j}}_{{\rm{ph}}}$) between the photocurrent densities of the PVK with ${E}_{{\rm{g}}}^{(\mathrm{PVK})}$ = 1.7 eV and Si SCs (${j}_{{\rm{ph}}}^{({\rm{PVK}})}$ and ${j}_{{\rm{ph}}}^{({\rm{Si}})},$ respectively),

$$\begin{eqnarray}&&{\rm{\Delta }}{\tilde{j}}_{{\rm{ph}}}=2\,\left({j}_{{\rm{ph}}}^{({\rm{PVK}})}-{j}_{{\rm{ph}}}^{({\rm{Si}})}\right)/\left({j}_{{\rm{ph}}}^{({\rm{PVK}})}+{j}_{{\rm{ph}}}^{({\rm{Si}})}\right).\end{eqnarray} \tag{ 14 }$$

The frequency distribution of ${\rm{\Delta }}{\tilde{j}}_{{\rm{ph}}}$ is depicted in Fig. 4(e); 80% of the relative distribution ranges from −0.05 to 0.2 with ${\rm{\Delta }}{\tilde{j}}_{{\rm{ph}}}$ = 0.036 for $\left\langle {n}_{{\rm{sun}}}\left[\hslash \omega \right]\right\rangle .$

Although the characteristics of the EC reactors are also affected by their temperature, the general trends are not clear because the effects differ depending on the target reactions and catalyst materials. ^54,55) In addition, the temperature variation is smaller than that of the SC module that is directly solar-illuminated. Therefore, the temperature dependence was neglected, and j_EC and η_FE measured at RT, shown in Fig. 3, were always used.

Recently, high-performance PVK SCs with up to ${E}_{{\rm{g}}}^{(\mathrm{PVK})}$ = 1.7 eV have been realized. ^15,16) Therefore, we employed a combination of the Si SCs and PVK SCs with ${E}_{{\rm{g}}}^{(\mathrm{PVK})}$ = 1.7 eV to design the VM tandem SC modules and artificial photosynthetic VM-nEC devices. First, ${n}_{{\rm{Si}}}/{n}_{{\rm{PVK}}}$ of the VM module was optimized so that η_SC was maximized under the "average" condition of $\left\langle {P}_{{\rm{sun}}}\right\rangle ,$ $\left\langle {T}_{{\rm{cell}}}\right\rangle ,$ and $\left\langle {n}_{{\rm{sun}}}\left[\hslash \omega \right]\right\rangle$ derived above, and subsequently other practical conditions depicted in Fig. 4. Although n_PVK and n_Si are integers, ${n}_{{\rm{Si}}}/{n}_{{\rm{PVK}}}$ can be finely tuned for a large-sized module with large n_PVK and n_Si. Therefore, they were dealt with as continuous variables. Then, the annually averaged η_SC (${\eta }_{{\rm{SC}}}^{({\rm{annual}})}$) determined by the ratio of the accumulated electricity production to the accumulated solar intensity was calculated. Thus, the suitable optimization condition was sought by investigating the impacts of the optimization conditions on ${\eta }_{{\rm{SC}}}^{({\rm{annual}})}.$ The design parameters of the artificial photosynthetic devices, i.e. ${n}_{{\rm{PVK}}}/{n}_{{\rm{EC}}}$ and ${n}_{{\rm{Si}}}/{n}_{{\rm{EC}}}$, were similarly dealt with as continuous variables and optimized. The effects of the optimization conditions on the annually averaged η_H2 and η_CO (${\eta }_{{\rm{H2}}}^{({\rm{annual}})}$ and ${\eta }_{{\rm{CO}}}^{({\rm{annual}})},$ respectively) were investigated, and thus the design guide was derived. Finally, we discussed the feasibility of improvements in ${\eta }_{{\rm{SC}}}^{({\rm{annual}})},$ ${\eta }_{{\rm{H2}}}^{({\rm{annual}})},$ and ${\eta }_{{\rm{CO}}}^{({\rm{annual}})}$ assuming realization of PVK SCs with ${E}_{{\rm{g}}}^{(\mathrm{PVK})}$ wider than 1.7 eV, by calculating ${\eta }_{{\rm{SC}}}^{({\rm{annual}})},$ etc. as functions of ${E}_{{\rm{g}}}^{(\mathrm{PVK})}.$

First, we considered the VM tandem SC modules illustrated in Fig. 1(c). Figure 5(a) shows the dependence of η_SC on P_sun for the VM module (${n}_{{\rm{Si}}}/{n}_{{\rm{PVK}}}$ = 1.98) optimized under the average condition of $\left\langle {P}_{{\rm{sun}}}\right\rangle$ = 63.9 mW cm⁻², $\left\langle {T}_{{\rm{cell}}}\right\rangle$ = 37.7 °C, and $\left\langle {n}_{{\rm{sun}}}\left[\hslash \omega \right]\right\rangle$ (${\rm{\Delta }}{\tilde{j}}_{{\rm{ph}}}$ = 0.036). The V_MPP values of the VM module and top/bottom modules are also depicted. In the calculated P_sun range, V_MPP of the Si SC monotonically increases, proportionally to $\mathrm{log}\left[{P}_{{\rm{sun}}}\right].$ By contrast, the V_MPP of the PVK SC changes only slightly, because its r_s is an order of magnitude larger than that of the Si SC and hence the ohmic loss is more significant at a large P_sun. ^56,57) As a result, the change in η_SC is 0.9% at most in the P_sun range of 20–95 mW cm⁻² in which the relative frequency distribution accounts for 80%. Although the difference between the V_MPP values of the PVK and Si SCs, namely, the voltage mismatch, increases when P_sun shifts from $\left\langle {P}_{{\rm{sun}}}\right\rangle ,$ it is as small as 0.05 V at most in the relevant P_sun range. Thus, it causes only a slight lowering in η_SC, smaller than 0.2% at a given P_sun relative to that of the 4T tandem SC that is free from the voltage mismatching problem. The details are discussed in Appendix A.

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By contrast, the impact of T_cell is notable, as is clear from Fig. 5(b). When T_cell rises from the lower bound (20 °C) to the upper bound (55 °C) of 80% of the relative frequency distribution, η_SC lowers by 2.2% because the V_MPP of each submodule is a decreasing function of T_cell. In addition, the V_MPP of the Si SC decreases more rapidly than that of the PVK SC. However, the detrimental impact of the resultant voltage mismatch is marginal. The maximal mismatch is 0.04 V, which lowers η_SC relative to that of the 4T module by only 0.2%.

On the other hand, it is apparent that the impact of ${\rm{\Delta }}{\tilde{j}}_{{\rm{ph}}}$ is extremely weak, as shown in Fig. 5(c), even though the present model of no overlap between the PVK and Si absorption ranges would overestimate the impact. The voltage mismatch is 0.005 V at most, corresponding to a negligibly small change in η_SC of 0.0002%, in the ${\rm{\Delta }}{\tilde{j}}_{{\rm{ph}}}$ range of the 80% frequency distribution between −0.05 to 0.2.

Thus, it was found that the impact of T_cell on η_SC is notable, whereas the other two operating conditions of P_sun and ${n}_{{\rm{sun}}}\left[\hslash \omega \right]$ have only a slight effect. Therefore, ${n}_{{\rm{Si}}}/{n}_{{\rm{PVK}}}$ was optimized under three T_cell conditions: T_cell = 25 °C, $\left\langle {T}_{{\rm{cell}}}\right\rangle$ = 37.7 °C, and 50 °C with the average conditions for P_sun and ${n}_{{\rm{sun}}}\left[\hslash \omega \right].$ The results are compared in Fig. 6. Although the slope of the η_SC vs. T_cell relation depends on the T_cell value adopted for the optimization, the difference is minor. Consequently, all three ${\eta }_{{\rm{SC}}}^{({\rm{annual}})}$ values, which are the average of η_SC over various T_cell values, are almost the same: 34.6%–34.7%. In other words, ${\eta }_{{\rm{SC}}}^{({\rm{annual}})}$ is scarcely affected even when the design parameter of ${n}_{{\rm{Si}}}/{n}_{{\rm{PVK}}}$ changes between 1.9–2.0. Further, when ${n}_{{\rm{Si}}}/{n}_{{\rm{PVK}}}$ was optimized under the condition of the AM 1.5 G illumination (P_sun = 100 mW cm⁻²) and T_cell = 25 °C that is the standard for SC evaluation, ${\eta }_{{\rm{SC}}}^{({\rm{annual}})}$ was 34.5%, virtually the same as the abovementioned three values. The resultant ${\eta }_{{\rm{SC}}}^{({\rm{annual}})}$ values are more realistic than the results of over 40% in the radiative limit, ²⁹⁾ although there remains a slight discrepancy from the practically achievable values because the parasitic optical and ohmic losses are neglected in the present model as described in Sect. 3.1. In addition, improvements in semi-transparent PVK SCs are obviously essential.

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Consequently, ${n}_{{\rm{Si}}}/{n}_{{\rm{PVK}}}$ of the VM tandem SC module can be optimized under any practical conditions. This is because the lowering in η_SC caused by the voltage mismatch under varying operation conditions is smaller than 0.2% in most cases, and hence scarcely affects η_SC in the annual average. This is also the reason that ${\eta }_{{\rm{SC}}}^{({\rm{annual}})}$ = 34.7% of the 4T tandem SC module calculated in a similar manner is almost the same as that of the VM module. A detailed comparison between the VM module and 4T module is described in Appendix A.

Next, we calculated η_H2 of the H₂-producing artificial photosynthetic devices consisting of the VM tandem SC modules and EC modules shown in Fig. 1(d). The requisite for a high η_H2 is that the V_MPP of the SC module is a little larger than 1.4 eV, as stated in Sect. 2. Figure 7 depicts the results for the device optimized under the average condition (${n}_{{\rm{PVK}}}/{n}_{{\rm{EC}}}$ = 1.22 and ${n}_{{\rm{Si}}}/{n}_{{\rm{EC}}}$ = 2.43). The significant difference from η_SC of the VM SC module coupled with a power conditioner displayed in Fig. 5 is that η_H2 is a monotonically decreasing function of P_sun. The lowering in η_H2 is as large as 1.6% when P_sun increases from 20 mW cm⁻² to 95 mW cm⁻². Another difference is that η_H2 becomes only slightly higher at a lower T_cell than $\left\langle {T}_{{\rm{cell}}}\right\rangle$ = 37.7 °C although the lowering in the higher T_cell range is similar to that for η_SC. On the other hand, the effect of ${\rm{\Delta }}{\tilde{j}}_{{\rm{ph}}}$ is again marginal.

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These dependences of η_H2 on P_sun and T_cell arise from the mismatch between V_MPP of the SC module and V_op, as depicted in Fig. 2. A larger P_sun and a higher T_cell lead to V_MPP < V_op, and therefore η_H2 that is proportional to ${J}_{{\rm{op}}}/{P}_{{\rm{sun}}}$ rapidly lowers. On the other hand, when P_sun increases and T_cell falls, V_op decreases below V_MPP, J_op shifts from J_MPP toward J_ph, and consequently η_H2 increases slightly.

Thus, P_sun and T_cell strongly affect η_H2; a large P_sun is equivalent to a high T_cell. Therefore, ${n}_{{\rm{PVK}}}/{n}_{{\rm{EC}}}$ and ${n}_{{\rm{Si}}}/{n}_{{\rm{EC}}}$ were optimized under five conditions in which both P_sun and T_cell were simultaneously changed. The results are shown in Fig. 8. When a small P_sun and a low T_cell are adopted for the optimization, η_H2 lowers remarkably with increasing P_sun and T_cell although η_H2 is high at a small P_sun and a low T_cell. By contrast, η_H2 changes moderately when the design parameters of ${n}_{{\rm{PVK}}}/{n}_{{\rm{EC}}}$ and ${n}_{{\rm{Si}}}/{n}_{{\rm{EC}}}$ are optimized under a condition of a large P_sun and a high T_cell. Reflecting these trends, ${\eta }_{{\rm{H2}}}^{({\rm{annual}})}$ notably depends on the optimization conditions, in contrast to ${\eta }_{{\rm{SC}}}^{({\rm{annual}})}.$ The best optimization condition, i.e. P_sun = 80 mW cm⁻² and T_cell = 45 °C (Condition D in Fig. 8) are both slightly greater than the average values. The maximal ${\eta }_{{\rm{H2}}}^{({\rm{annual}})}$ is as high as 28.7%, although there is a slight discrepancy from the practically achievable values, as discussed in Sect. 4.1. However, it should be noted that too large a P_sun and too high a T_cell (Condition E) decrease ${\eta }_{{\rm{H2}}}^{({\rm{annual}})}$

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When the device is optimized under the standard condition for SC evaluation of AM 1.5 G illumination (P_sun = 100 mW cm⁻²) and T_cell = 25 °C, the resultant ${\eta }_{{\rm{H2}}}^{({\rm{annual}})}$ = 28.4% is approximately the same as the maximal value determined under the best optimization condition. However, this is attributed to the combination of a P_sun close to the upper bound of the measured frequency distribution and a T_cell close to the lower bound shown in Fig. 4 having an effect similar to that of the best optimization condition, because a large P_sun is equivalent to a high T_cell. It should be noted again that the standard condition are far from typical conditions or the average condition.

Figure 9 compares the J–V curves of the VM tandem SC module used in the device optimized under the best optimization condition (Condition D in Fig. 8) with the load curve of the EC reactor. When the device operates under the optimization condition (the solid line), V_op determined by the intersection of the two curves, coincides with V_MPP, leading to a high η_H2. However, V_op shifts away from V_MPP and hence η_H2 lowers when P_sun and T_cell change, although these detriments are minimized. This drawback of the direct connection of the SC module and EC module is solved by using a DC–DC converter for connecting these two modules. However, it is found that the energy loss originating from the voltage mismatch for the optimized device is smaller than the internal loss of a commonly used DC–DC converter with an efficiency of 90%–95%. ^58,59) In short, the direct connection yields a higher ${\eta }_{{\rm{H2}}}^{({\rm{annual}})}.$ The details are discussed in Appendix B.

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We also considered the CO-producing artificial photosynthetic devices. Figure 10 shows η_CO, V_op, and V_MPP of the device optimized under the average condition (${n}_{{\rm{PVK}}}/{n}_{{\rm{EC}}}$ = 1.34 and ${n}_{{\rm{Si}}}/{n}_{{\rm{EC}}}$ = 2.78). As is clear from Fig. 10(a), η_CO rapidly lowers with decreasing P_sun. This qualitative difference from the P_sun dependence of η_H2 depicted in Fig. 7(a) arises from the η_FE of the EC reactor that lowers at a low V_op and a small J_op; see Fig. 3. The hypothetical η_CO supposing η_FE = 1 (dotted line) depends on P_sun similarly to η_H2.

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The impacts of the optimization conditions on the P_sun- and T_cell-dependent η_CO depicted in Figs. 11(a) and 11(b), respectively, are also similar to those for the H₂-producing devices, except for the rapid lowering in the small P_sun range arising from the low η_FE; see Fig. 8. Thus, ${\eta }_{{\rm{CO}}}^{({\rm{annual}})}$ is also maximized when the design parameters of ${n}_{{\rm{PVK}}}/{n}_{{\rm{EC}}}$ and ${n}_{{\rm{Si}}}/{n}_{{\rm{EC}}}$ are optimized under the condition of P_sun = 80 mW cm⁻² and T_cell = 45 °C (Condition D in Fig. 11) that are both slightly greater than the average values. However, the lowering in ${\eta }_{{\rm{CO}}}^{({\rm{annual}})}$ when optimized with the larger P_sun and higher T_cell (Condition E) is moderate, reflecting that a smaller P_sun and a resultant smaller J_op lead to a lower η_FE. Therefore, the more precisely selected best P_sun for the optimization should be larger than that for the H₂ production. It should be again noted that the present model of the SC modules will slightly overestimate ${\eta }_{{\rm{CO}}}^{({\rm{annual}})}.$

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The J–V curves of the VM tandem SC module used in the device optimized under the best optimization condition (Condition D in Fig. 11) are plotted in Fig. 12, with the load curve and η_FE of the EC reactor. Although η_FE is close to unity under the intense illumination of P_sun = 90 and 80 mW cm⁻² (Conditions E and D, respectively), it lowers to 0.91 at P_sun = 40 mW cm⁻² (Condition A). However, it should be noted that how much η_FE lowers depends on the EC reactor structure, catalysts used in the reactor, etc. It was confirmed again that the direct connection of the optimized VM SC module and EC module yields a higher ${\eta }_{{\rm{CO}}}^{({\rm{annual}})}$ than the use of a DC–DC converter; the numerical results are shown in Appendix B.

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In the previous subsections, the PVK SC with ${E}_{{\rm{g}}}^{(\mathrm{PVK})}$ = 1.7 eV was adopted for the design of the VM tandem SC modules because the performance of the PVK SCs with a wider ${E}_{{\rm{g}}}^{(\mathrm{PVK})}$ than 1.7 eV is not sufficiently high at present. ^15,16) Finally, we discuss the feasibility of improvements in ${\eta }_{{\rm{SC}}}^{({\rm{annual}})},$ ${\eta }_{{\rm{H2}}}^{({\rm{annual}})},$ and ${\eta }_{{\rm{CO}}}^{({\rm{annual}})}$ when highly efficient PVK SCs with a wider ${E}_{{\rm{g}}}^{(\mathrm{PVK})}$ are realized. The fitting parameters of η_EQE, η_ERE, and r_s obtained in Sect. 3.1 were used, neglecting their ${E}_{{\rm{g}}}^{(\mathrm{PVK})}$ dependences.

Figure 13(a) compares the ${\eta }_{{\rm{SC}}}^{({\rm{annual}})}$ values of the VM and 2T tandem SC modules as a function of ${E}_{{\rm{g}}}^{(\mathrm{PVK})}.$ The advantage of the VM module is the weak dependence on ${E}_{{\rm{g}}}^{(\mathrm{PVK})},$ which is in contrast to the ${\eta }_{{\rm{SC}}}^{({\rm{annual}})}$ of the 2T module being strongly affected by ${E}_{{\rm{g}}}^{(\mathrm{PVK})}$ because of the current matching problem. ^28,29,31,32) Although ${\eta }_{{\rm{SC}}}^{({\rm{annual}})}$ of the VM module is maximized at ${E}_{{\rm{g}}}^{(\mathrm{PVK})}$ = 1.84 eV, the gain compared with the value at ${E}_{{\rm{g}}}^{(\mathrm{PVK})}$ = 1.7 eV is as small as 0.4%.

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The results for the H₂- and CO-producing artificial photosynthetic devices using the VM-nEC configuration exhibit similar trends, as shown in Figs. 13(b) and 13(c), respectively. The maximal ${\eta }_{{\rm{H2}}}^{({\rm{annual}})}$ and ${\eta }_{{\rm{CO}}}^{({\rm{annual}})}$ at ${E}_{{\rm{g}}}^{(\mathrm{PVK})}$ = 1.84–1.85 eV are higher than those at ${E}_{{\rm{g}}}^{(\mathrm{PVK})}$ = 1.7 eV by only 0.3%–0.4%. Therefore, it is concluded that the room for improvements in ${\eta }_{{\rm{SC}}}^{({\rm{annual}})},$ ${\eta }_{{\rm{H2}}}^{({\rm{annual}})},$ and ${\eta }_{{\rm{CO}}}^{({\rm{annual}})}$ using a wider ${E}_{{\rm{g}}}^{(\mathrm{PVK})}$ PVK SC is extremely narrow, and hence the focus should be further improvements in η_SC and durability with ${E}_{{\rm{g}}}^{(\mathrm{PVK})}$ = 1.7 eV.

The maximal ${\eta }_{{\rm{SC}}}^{({\rm{annual}})}$ for the 2T SC module that satisfies the current matching under the optimization condition is close to that for the VM SC module. ^28,29,31,32) By contrast, the maximal ${\eta }_{{\rm{H2}}}^{({\rm{annual}})}$ and ${\eta }_{{\rm{CO}}}^{({\rm{annual}})}$ for the 2T-EC devices are remarkably lower than those for the VM-nEC devices. This is because the optimal ${E}_{{\rm{g}}}^{(\mathrm{PVK})}$ of the 2T modules are determined from the current matching condition, resulting in too high a V_MPP to power the present EC reactors. ²⁸⁾