#### Abstract

A tandem solar cell consisting of a bottom c-Si high-efficiency subcell and a top low-cost Cu_{2}O subcell in parallel configuration is evaluated for the first time by a use of an electrical model. A numerical simulation based on the single-diode model of the solar cell is performed. The numerical method determines both the model parameters and the parameters of the subcells and tandem from the maximization of output power. The simulations indicate a theoretical limit value of the tandem power conversion efficiency of 31.23% at 298 K. The influence of temperature on the maximum output power is analyzed. This tandem configuration allows a great potential for the development of a new generation of low-cost high-efficiency solar cells.

#### 1. Introduction

Recently, tandem solar cells made up of various materials such as silicon (monocrystalline, polycrystalline, and amorphous) [1–3], perovskite [4–6], polymer [7, 8], dye-sensitized solar cells [9], and quantum dot solar cells [10] have been theoretically or experimentally studied. However, tandems with a high efficiency at low cost have still not been realized. The silicon-based tandem heterojunction solar cells made up of a high-efficiency crystalline silicon (c-Si) bottom subcell and a low-cost upper subcell are a promising candidate for both reducing the fabrication costs and increasing the efficiency above the silicon single-junction record of 25.6% [11]. The band gap of the top subcell is required to be higher than the band gap of silicon (1.1 eV) in order to absorb the photons of higher energy, thus yielding two complementary absorbing subcells. Generally, the bottom subcell generates a higher current and has a lower open-circuit voltage than the top subcell.

The cuprous oxide (Cu_{2}O) solar cell represents one of the best choices as top subcell. This semiconducting metal oxide has a band gap of 2.1 eV, a high optical absorption, is nontoxic, and has the potential of a low manufacturing cost [12]. The theoretical limit of the power conversion efficiency for a Cu_{2}O solar cell is approximately 20% [13, 14] under the solar radiation spectrum AM 1.5G. However, the highest conversion efficiency of 81% achieved experimentally for a solar cell made up of zinc oxide (ZnO) and Cu_{2}O based on thermally oxidized copper sheets [15] suggests the potential of further increase of the conversion efficiency.

In this paper, a comprehensive numerical simulation of the performance parameters of a tandem solar cell based on c-Si and Cu_{2}O subcells in parallel configuration is presented. The ideal working conditions for tandem are defined. The single-diode model for solar cells is used [16]. The parameters of the subcells and tandem are calculated from the maximum condition of the output power. The values of the main parameters, that is, the short-circuit current density (*J*_{sc}), the open-circuit voltage (*V*_{oc}), the fill factor (FF), and the power conversion efficiency (*η*), are calculated and discussed. The value of the maximum power conversion efficiency of this configuration type is estimated. The influence of temperature (*T*) on the performance of the tandem is analyzed.

All the programs used for the simulations are written in Mathcad environment.

#### 2. Theoretical Model

A tandem solar cell based on a bottom c-Si subcell and a top Cu_{2}O subcell as active layers of p-type materials is investigated. For the Cu_{2}O subcell, a wide range of possible n-type materials, including ZnO, ZnS, TiO_{2}, CdS, and Ga_{2}O_{3} [13–15] can be used. There are two possibilities of electrical connection for the two subcells, namely, series and parallel. The parallel connection is more advantageous [17] since it does not require current matching between the subcells and allows independent optimization of the subcells. A four-terminal device has the subcells electrically decoupled and independently controllable that can be connected in parallel, ensuring a maximum output power at all times [18]. The tandem efficiency is estimated to reach values above 30% [1, 2, 6, 19]. A possible four-terminal configuration is presented in Figure 1 [20].

The following ideal conditions of the solar cell operation—including illumination conditions, subcell materials, radiation absorption processes, and carrier generation processes in subcells—are considered: (1)the solar radiation spectrum AM 1.5G [21];(2)no optical losses due to nonactive layers (glass, n-type materials, encapsulation layer, etc.);(3)the carrier-injection efficiency for each subcell;(4)no resistive and recombination losses, , for each subcell;(5)the standard temperature .

##### 2.1. Optical Model

In the conditions above, the photoelectron generation rate determined by the solar radiation incident on the surface of the active layer is
where *L* is the thickness of the active layer (Cu_{2}O or c-Si), *λ* is the wavelength of incident radiation, the integration limits *λ*_{1} and *λ*_{2} are imposed by the absorption characteristics of the active layer through the absorption coefficient *α*(*λ*), and *ϕ*(*λ*) is the spectral incident photon flux density. The photoelectrons are then collected without any losses.

Note that there is a range of wavelengths in which both Cu_{2}O and c-Si absorb radiation. The incident radiation on the top surface of the Cu_{2}O layer is the entire AM 1.5G solar spectrum; at the top of the c-Si layer, the spectral irradiance of the AM 1.5G solar spectrum is diminished due to the partial absorption of Cu_{2}O layer. These lead to the possibility of optimization in choosing the thickness of the two layers so that the output power of tandem cell is at maximum.

##### 2.2. Electrical Model

The current density *J* through each subcell, in the single-diode model [16], is
where *J*_{L} is the light-generated current density, *J*_{0} is the reverse saturation current density of the diode, *V* is the output voltage, *γ* is the ideality factor, is the series resistance, is the shunt resistance, and is the thermal voltage, where *k*_{B} and *q* denote the Boltzmann constant and elementary electric charge, respectively. In the conditions above, (2) becomes

In the approximation of a negligible diffusion of carriers and assuming a uniform field across the active layer (subcell materials are without any defects and impurities), the light-generated current density is [22]
where *V*_{bi} is the built-in voltage [23] and the product *μτ* is given by the relation
where *μ*_{p}, *μ*_{n} and *τ*_{p}, *τ*_{n} are the hole and electron mobilities and lifetimes, respectively.

By use of (3) and (4), the output power density of the solar cell is

One can observe that, at a given temperature, *P* depends on seven parameters: *L*, *J*_{0}, *V, R*_{sh}, *γ, μτ*, and *V*_{bi}.

In the following, the two subcells will be considered electrically connected in parallel. In the steady state, the current densities and voltages obey the relations and

##### 2.3. The Influence of Temperature on Tandem Cell Performance

The solar cells for usual applications are used at temperatures ranging from −15°C to 100°C. Many studies have pointed out that the performance of solar cells degrades as temperature increases [24–28]. The variation of *R*_{sh} with temperature slightly affects the efficiency of a solar cell [27, 28]. We analyze the variation with temperature of parameters *V*_{th} and *J*_{0}.

The thermal voltage dependence on temperature is where is the thermal voltage for the standard temperature.

The dependence on temperature of the reverse bias saturation current density *J*_{0} is [25]
where and is the band gap energy of the semiconductor (the active layer) which depends on temperature by the relationship [26]
where *E*_{g}(0) is the band gap value at *T* ≈ 0 K, and *α* and *β* are constants.

#### 3. Results and Discussion

The operation of a tandem solar cell with a Cu_{2}O top subcell and a c-Si bottom subcell is simulated using the presented theoretical model. The performance of the tandem is obtained by numerical simulations in the conditions stated in Section 2. The solar radiation spectrum AM 1.5G has values between the wavelength limits 280 nm and 4000 nm and a flux power density of 1000 W/m^{2}. The active layers of c-Si and Cu_{2}O have the absorption wavelength range ([280 nm, 1450 nm] [29] and [280 nm, 640 nm] [30], resp.).

The parameters *R*_{sh}, *γ*, *J*_{0}, *μτ*, and *V*_{bi} of the single-diode model are material constants that depend on materials and the manufacturing of cells. Therefore, the calculated parameters of the c-Si and Cu_{2}O cells remain unchanged in the tandem configuration. These parameters can be calculated by different methods, such as fitting methods [31, 32], Lambert-*W* function [33], and asymptotic approximation [34] by use of *J–V* experimental data.

The method used in this paper was previously applied to the study of solar cells containing a heterojunction with Cu_{2}O [14]. The method consists in the maximization of the output power density (7) which leads to the parameter values of the theoretical model. The obtained values, separately for the Cu_{2}O and c-Si cells, are presented in Table 1. The cell voltages are 1.61 V and 0.88 V, respectively, and the thickness of the active layers is 11.94 *μ*m and 115.00 *μ*m, respectively. The cell parameters calculated based on the values of Table 1 are shown in Table 2. The results indicate a maximum power conversion efficiency of 19.29% for the Cu_{2}O cell of 11.94 *μ*m active layer thickness and 31.29% for the c-Si cell of 115.00 *μ*m active layer thickness. The maximum values of the power conversion efficiency are in good agreement with the data reported in literature, ~20% for the Cu_{2}O cell [13, 14] and 32.9% [35], 29.8% [36], or 33.5% [5] for the c-Si cell.

The dependence of the tandem output power density, (6) where is , on the voltage and the total thickness of the active layers (Cu_{2}O and c-Si layers) is analyzed. The results for two ratios of the thicknesses of the active layers, and , are shown in Figure 2. They show a maximum of the tandem output power density. The output power density of tandem as a function of voltage, and the thickness of the two active layers and has been maximized. The calculated parameters for the two subcells and tandem are presented in Table 3. The obtained maximum output power density is 312.3 W/m^{2} for 0.88 V tandem voltage, 0.64 *μ*m Cu_{2}O layer thickness, and 60.62 *μ*m c-Si thickness. Therefore, the maximum power conversion efficiency is 31.23%. It can be noticed that in case of tandem, the thickness of the subcells is significantly reduced compared to the case of separate cells (see Table 2). In case of the c-Si and Cu_{2}O layer thicknesses obtained above, the *J–V* curves for the two subcells, as well as for the tandem cell, are presented in Figure 3. The open-circuit voltage of the tandem cell (see Table 3) is located between the *V*_{oc} values of separate subcells (see Table 2), practically being equal to that of the separate c-Si subcell. There is a limitation in the voltage for tandem operation, but this drawback is compensated by the advantages of parallel connection [18]. These results are consistent with theoretical or experimental values reported in literature for other types of tandem cells [5, 7, 8, 37].

**(a)**

**(b)**

Figure 4 shows the dependence of the maximum output power density on temperature for the thickness of active layers from Table 3 and for the following parameter values of (11): , , [38], , , and [26]. A decrease of the maximum output power density with increasing temperature is observed. This decrease is due to the temperature dependence of the reverse bias saturation current density (10). The drop of the output power density is 20.90% for a temperature increase from 298 K to 348 K. This result indicates that the temperature is a major factor in decreasing the tandem solar cell performance.

#### 4. Summary and Concluding Remarks

The paper presents a comprehensive numerical simulation of performance optimization of the tandem solar cells using c-Si and Cu_{2}O subcells. A numerical simulation based on the single-diode model of the solar cell is performed. The numerical method determines both the model parameters and the parameters of the subcells and tandem.

The thickness of active layers c-Si and Cu_{2}O in tandem cell is obtained from the maximization of output power. The tandem power conversion efficiency has a maximum of 31.23% for a 0.64 *μ*m Cu_{2}O layer thickness and 60.62 *μ*m c-Si layer thickness. These thicknesses are significantly reduced compared to the case of separate cells, and the efficiency represents the theoretical limit at 298 K. A realistic model of the tandem will include optical losses for each layer of the tandem as well as a smaller than unity injection efficiency in the active layers. Therefore, a lower value of the tandem efficiency is expected. The numerical simulation shows that temperature is a factor in decreasing tandem cell performance; therefore, temperature should be taken into account in tandem design.

The results of this paper give indications for the design and performance optimization of real tandem solar cells using Cu_{2}O and c-Si subcells.

An approach of the solar cell tandem using experimental data will be the subject of future research.

#### Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by (1) the project MultiscaleSolar MP1406, 2015–2019, supported by the European Commission through COST program (the article processing charges), and (2) the project SOLHET, 2016–2019, M-ERA.Net program, supported by the Research Council of Norway (RCN) and the Romanian Executive Agency for Higher Education, Research, Development and Innovation Funding (UEFISCDI).