## Introduction

Organic–inorganic metal hybrid perovskites have been consistently arousing extraordinary research interest in the photovoltaic community owing to their exceptional semiconductor properties such as facile fabrication process, long diffusion length1, long carrier lifetime2, panchromatic absorption of light3, etc. To date, the maximum power conversion efficiency (PCE) achieved in single-junction perovskite solar cells (PSCs) has been as high as 25.5%1. So as to further enhance the PCE constrained by the Shockley–Queisser (SQ) limit, some different strategies were pursued, namely, the carrier multiplication effect to harvest the additional energy (hυ-Eg) of photons with energy larger than bandgap (Eg)4 and multijunction absorbers to harvest photons with energy smaller than Eg5. Whereas it still is impractical and elusive to gain the PCE via carrier multiplication phenomena, multijunction (tandem) PSCs have successfully achieved the PCE as large as 29.15%6. However, inspired by the achievements of counterparts of tandem PSCs, GaAs and GaInP-based multijunction solar cells which have reached a maximum PCE of 38.8%7, there is still a burgeoning interest in the further improvement of the multijunction PSC performance. This has spurred the search for new materials and architectures for multijunction PSCs.

Semiconducting transition metal dichalcogenides (TMDs), including MoS2, MoSe2, MoTe2, WS2, and WSe2, are emerging as highly impressive absorbers for solar cells owing to their ultrahigh absorption coefficients8, mechanical flexibility9, high carrier mobility10, together with an ideal bandgap for photovoltaic applications8. Notably, a TMD layer thinner than 20 nm is able to absorb light even ten times larger than well-known direct bandgap semiconductors8. While the TMDs, especially MoS2, have been widely employed as carrier transport layers (HTLs) in the PSCs11,12, there is no report of deriving a benefit from the TMDs absorption capacity in order to improve the light absorption efficiency in PSCs. Although most TMDs have almost the same bandgap magnitude as perovskites, bulk MoTe2 with a bandgap of around 1 eV would be a complementary absorbing material for perovskite to harvest the near-infrared (NIR) range of sunlight. The strong NIR absorption capability of MoTe2, along with the absence of dangling bonds at its surface, a property of TMDs which originate from their weak van der Waals (vdW) interlayer interaction, underpin MoTe2 aa a suitable candidate to be heterostructured with perovskite materials for tandem solar cells13,14. Experimentally, the cost-effective chemical and mechanical exfoliation methods available allow for uniform and homogeneous thin MoTe2 film preparation15,16. Thus, it would be more valuable to explore the exploitation of MoTe2 materials as a supportive absorbing layer, to benefit from the MoTe2 absorption.

Herein, we numerically present and propose a Planar type of parallel multijunction PSCs with an absorbing region made of a thin MoTe2 and CH3NH3PbI3. The main device is composed of ITO/TiO2/CH3NH3PbI3/MoTe2/Spiro-OMeTAD/Ag layers, a configuration that was likewise fabricated with MoS216. The excellently desirable band alignment of MoTe2 with other layers, along with its high NIR absorption capacity, remarkably paves the way for achieving higher photovoltaic efficiency. By comparing to single-junction PSCs, the proposed device yields an increase in PCE from 14.01 to 18.52%. By performing an accurate numerical analysis of the MoTe2 thickness-dependent device performance, an optimum thickness of 25 nm was obtained, which is several orders of amplitude thinner than the previous supportive absorbing layers so far reported in multijunction PSCs17.

Nonetheless, it is a well-established fact that the utilization of a low band gap absorber is detrimental to the open-circuit voltage (VOC) of solar cells owing to the limited electron and hole quasi fermi level separation. Likewise, we have observed a reduction in Voc after turning the structure into a multijunction device. In order to compensate these photovoltage losses, we replace a reduced graphene oxide (rGO) sheet with Spiro-OMeTAD as an HTL to improve the hole extraction and transportation. Outstandingly, the rGO sheet enhances the device VOC and PCE up to 0.928 and 20.32%, respectively. It is noteworthy that the efficacious performance of the rGO layer as both interlayer and charge transport layer has been well proved in PSCs18,19,20,21,22,23.

## Basic equations and models

In this work, we employ a hybrid optical and electrical model to calculate and evaluate the presented structures. We present their traditional formulation (i.e., in the frequency domain) and then discuss the extension to the time domain. A finite element method (FEM) is used to solve the partial differential equations (PDEs).

### Optical model

Figure 1A depicts the schematic diagram of our basic planar PSC scheme. From top to bottom, the structure is stacked by a transparent indium tin oxide (ITO) electrode, a compact titanium dioxide (TiO2) layer, a methylammonium lead iodide perovskite (CH3NH3PBI3) film, an N,N-di(4-methoxyphenyl)amino]-9,9′-spirobifluorene (spiro-OMeTAD) layer, and a silver (Ag) rear electrode. The incident light enters the cell from the ITO layer and is absorbed by the perovskite film to some extent. Also, the incoming light experiences a multireflection because of the rear Ag reflector which gives rise to an absorption enhancement. To quantity the interaction between electromagnetic waves and the layers, as well as the electric field (E) distribution, the Helmholtz equation (represented as follows) was solved:

$$\nabla \times \left(\nabla \times \mathrm{E}\right)-{k}_{0}^{2}{\varepsilon }_{r}E=0$$

where k0 is the free space wave number and εr is the dielectric constant. Clearly, to solve the above equation, one needs all the complex refractive index ($$N=n=ik$$) of layers as a function of wavelength. Subsequently, the E distribution obtained from solving the above Helmholtz equation enables us to compute the light absorption and carrier generation rate (Gopt). The transfer-matrix method (TMM) is applied to estimate Gopt in each layer of the structure. The Gopt formula is as follows,

$${G}_{OPT}=\frac{{\varepsilon }^{"}{E}^{2}}{2\hslash }$$

where is the reduced Planck constant, and ε" is the imaginary part of the relative permittivity. As the formula obviously indicates, Gopt is proportional to the square of the E intensity in a certain wavelength. The total generation rate (Gtot) can be calculated by integrating Gopt over an incident light wavelength bandwidth.

$${G}_{tot}={\int }_{{\lambda }_{min}}^{{\lambda }_{max}}{G}_{opt}\left(\lambda \right)d(\lambda )$$

The resulting Gtot is used for the input of electric model.

### Electrical model

The following well-known J–V relation is used to describe electrical characteristics of the present PSCs:

$$J\left(V\right)={J}_{dark}+{J}_{sc}={J}_{0}\left(\mathrm{exp}\left(\frac{eV}{nKT}\right)-1\right)-q{G}_{opt}({L}_{n}+{L}_{p})$$

where Jdark depicts the electric current of the PSCs in the absence of light illumination, Jsc is photocurrent, e is the electron charge, n is an ideality factor, K is the Boltzmann’s constant, and T is the temperature in kelvin. In order to calculate the currents, the following Poisson and continuity equations should be solved across the device:

$$\nabla .\left({\varepsilon }_{0}.{\varepsilon }_{r}.{\nabla }_{\varphi }\right)=-\rho$$
$$\frac{\partial n}{\partial t}=\frac{1}{q}{\nabla }_{{j}_{n}}+{G}_{n}-{U}_{N}$$
$$\frac{\partial p}{\partial t}=\frac{1}{q}{\nabla }_{{j}_{p}}+{G}_{p}-{U}_{P}$$

where ε0 is the permittivity of free space, ϕ is the electrostatic potential, ρ is the charge density, and q is the electron charge. Also, the Jn and Jp show the current densities arising from electrons and holes, respectively, the UN and UP illustrate the electron and hole recombination rates, respectively, and Gn and Gp are the electron and hole generation rates, respectively. By assuming that every absorbed photon creates one electron–hole pair, Gn and Gp are considered same to the Gtot obtained from the optical part.

In this study, the influence of grain boundaries and the carrier recombination at the interfaces between semiconductors are neglected. Additionally, we assume that trap-assisted recombination (SRH) inside bulk materials is the fastest and most dominant recombination mechanism in our devices.

## Results and discussion

In this simulation, the refractive index of bulk MoTe2 was obtained from the Ref34. Also, in all calculations, the input light source is conformed to the AM1.5G spectrum. The wavelength bandwidth is chosen from 300 to 1200 nm in a resolution of as much as 20 nm. The periodic boundary condition (PBC) is used for each side of the insulating region in the structures and the Au layer sides are set to a perfect electric conductor (PEC). The bottom and top contacts are considered ideal ohmic and Schottky with a surface recombination velocity of 107 cm/s, respectively. Furthermore, a swept mesh is applied to more precisely resolve the fields around the thin layer. Table 1 includes all optical and electrical input values used in the simulations. Herein, εr is dielectric constant, NC and NV are effective density of states of conduction and valence bands, μn and μp are electron and hole mobilities, χ is electron affinity, Eg is bandgap energy, NA and ND are acceptor and donor densities, and τn and τp are electron and hole lifetimes, respectively. The MoTe2 materials are known to be naturally P-doped35. In addition, in the bulk limit, the semiconducting TMDs bear photogenerated carrier lifetimes up to a few nanoseconds36,37.

The current density–voltage (J–V) characteristics of our reference PSC under one sun condition are demonstrated in Fig. 2a. The PSC shows a PCE of 14.01%, with Jsc of 15.20 mA/cm2, Voc of 1.14 V, and FF of 0.81. Benefiting the NIR light absorbed in the MoTe2 layer, Jsc considerably increases by 26.2 mA/cm2 in the multijunction PSC with an optimized thickness of MoTe2. But, the Voc drops to 0.84 V due to the electron and hole quasi fermi level separation is now restricted by the MoTe2 bandgap. Altogether, notwithstanding the Voc is destroyed after inserting the MoTe2 layer, the enhancement of Jsc is highly predominated over the Voc reduction, leading to a noticeable increase in PCE from 14.01% to 18.52%. This PCE increase is also contributed by a suitable band alignment between MoTe2 and the perovskite layer and HTL, as indicated in Fig. 2b. Indeed, the desired band alignment between absorbing layers can effectively mitigate Voc loss in multijunction PSCs as a result of charge transport improvement and charge recombination reduction38. In order to provide a broader perspective on the TMDs capability for light absorption, we compare the absorption spectrum of the present structure with when the MoTe2 layer was replaced by three other TMDs, WSe2, MoSe2, and MoS2, as illustrated in Fig. 2c. The refractive index and band structure parameters of WSe2, MoSe2, and MoS2 are obtained from the literature34,39,40,41. While all TMDs show strong light–matter interaction under light illumination, their bandgaps cover a broad range from 1–2 eV42. Herein, WSe2 and MoSe2 with the bandgap around 1.3 eV can absorb a wider spectrum of light compared to MoS2 with a bandgap of 1.45 eV. Of these, MoTe2 clearly is more able to absorb NIR light, making it the best choice to be cascaded with the PSK. Figure 2d,e exhibit the interaction between the light electric fields and different layers at the wavelength of 600 and 1000 nm. One can see that the MoTe2 layer interacts with light when the wavelength is set to 1000 nm, whereas its contribution to light absorption in the visible wavelength of 600 nm is negligible. It is also worth knowing that the utilization of TMDs in PSCs has shown successful outcomes to enhance stability16,43. On the other side, TMDs in each thickness can be easily prepared through environment-insensitive and non-destructive approaches such as dry or liquid-phase exfoliation16, then transferred by dry or wet methods. Thus, a combination of PSK materials and TMDs can potentially improve PSC performance, not only photovoltaic operation but also stability.

To achieve the multijunction PSC peak performance, an analysis of the cell performance dependence on the MoTe2 thickness has been carried out, while other input parameters in Table 1 are left unchanged. According to Fig. 3, the absorption, carrier generation, and photovoltaic parameters of the cell change as the MoTe2 thickness increase from 5 to 100 nm. Figure 3a exhibits the absorption spectra of four different thicknesses of the MoTe2 layer inside the multijunction PSC. As expected, the thicker the MoTe2 layer, the more light absorption in the MoTe2 layer. However, the light absorption rate becomes slower as the MoTe2 thickness increases, until it reaches saturation at a certain thickness. Even though too much light is absorbed by the MoTe2 at the longer wavelengths around 1100 nm, the carrier generation is poor at such wavelengths, as illustrate in Fig. 3b. This can be ascribed to resonant cavity effect and interference that play a role in absorption spectra, but do not exert any influence on carrier generation. As shown in Fig. 3c,d, photovoltaic parameters of the cell, PCE, Jsc, Voc, and FF vary with the MoTe2 thickness. With increasing MoTe2 layer thickness, the Jsc gradually increases until it reaches a point of saturation. Conversely, the Voc reduces as the MoTe2 thickness increases. The Voc initially experiences a quick decrease and then the decrease rate becomes slower with the increase of MoTe2 thickness. The decreasing Voc value can be assigned to an increase in charge carrier recombination in the thicker absorbing layer and to the increased series resistance44. When the absorbing layer thickness is smaller than the carrier diffusion length, the carrier recombination rate significantly diminishes, resulting in a sharp increase in Voc. On the other hand, after a distance as much as carrier diffusion length, a Voc reduction occurs arising from the carrier recombination increase. Also, it is worth noting that the FF parameter has a negligible dependency on the MoTe2 thickness. Consequently, as indicated in the Fig. 3d, the PCE initially undergoes a relatively intense increase in the response to both Voc and JSC sharp changes in the thinner MoTe2 thicknesses and then reaches a maximum (~ 18.52%) at the MoTe2 thickness of 25 nm, and subsequently, it drops off as the Jsc increase is saturated.

In order to compensate the destructive effect of parallelly stacking low and high bandgap materials, we replace the spiro layer with a 60 nm rGO layer to improve the carrier transfer. Arguably, graphene oxide (GO) and rGO can provide multi-benefit to PSCs, namely, the improvement of stability, electrical, and thermal conductivity45. Hence, the materials were widely used for different functions in PSCs such as carrier transport layers, interlayers, and transparent conductive oxides. Here, the GO layer is selected to insert as a HTL due to its well-aligned bandstructure with the adjacent layers band edges. The electronic energy band parameters of rGO are obtained from the Ref46. As illustrated in the Fig. 4a, the utility of rGO as HTL notably improves both FF and Voc up to 0.89 and 0.928, respectively, in the comparison with the multijunction PSC without the rGO layer. Consequently, it yields a PCE as high as 20.32, around 1.77% larger than the multijunction PSC with spiro HTL. The significant improvement of photovoltaic performance in rGO-based multijunction PSC is devoted to more efficient charge transport and better energy band alignment, alongside a reduction in the increased series resistance owing to the expected charge recombination reduction at the interface.

Figure 4b compares the photovoltaic performance of multijunction PSC with different HTLs including Spiro, PTAA, rGO, and CuS materials. The input parameters of these materials are tabulated in Table 2. The rGO layer acts as HTL better than other materials due to its high hole mobility47, along with a nice band alignment with MoTe2. Conversely, CuS is not well energetically aligned with MoTe2, leading a VOC reduction. The band diagram of multijunction PSC with different HTLs is shown in the Fig. 4c.

## Conclusion

In summary, the significant successes achieved in multijunction (tandem) PSCs have intensified scientific efforts to address existing challenges and enhance their current performance. In this direction, we designed and proposed an n–i–p multijunction perovskite solar cell made of ITO/TiO2/CH3NH3PbI3/MoTe2/Spiro-OMeTAD/Ag layers, including two absorbers, the CH3NH3PbI3 and MoTe2, with cascaded bandgaps to absorb a wider solar spectrum. The MoTe2 layer with a bandgap around 1 eV enables us to harvest photons with energies smaller than the perovskite bandgap. The calculated results show an appreciable increase in the perovskite solar cell efficiency originating from the short circuit current, compared to the cell without MoTe2. Nevertheless, in a sharp contrast to the short circuit current, stacking the absorbers with different bandgaps has led to a fall in the open circuit voltage because of hole transport deterioration in the absorbing area. In order to alleviate the unavoidable issue, we inserted a graphene oxide layer with a thickness of 1.5 nm. Consequently, we observed that the open circuit voltage increases as much as 0.1 eV, leading to an efficiency improvement from 18.52% to 20.32%. Both MoTe2 and graphene oxide layers energy bandstructure are perfectly matched with their nearby layers band edges, allowing for achieving a high performance. It is also important to mention that the MoTe2 and graphene oxide layers chosen in this research have been experimentally utilized for various functions, such as stability improvement, transport layers, etc.