1 Introduction

In order to ensure a large part of the worldwide energy production, photovoltaics has to increase its watt produced over price ratio and match, if not exceed, the one of nuclear and fossil energies. In recent years, two strategies have emerged from this objective. The first path consists in increasing significantly the solar-cell efficiencies. In that sense, one solution is to develop group III–V multi-junction solar cells (MJSC), which combine absorbers with different bandgap energies allowing to harvest a wider range of solar spectrum while lowering intrinsic losses such as thermalisation losses (Chen et al. 2015; Hirst and Ekins-Daukes 2011), and can therefore surpass the well-known Schockley–Queisser limit of an ideal single-junction solar cell (Shockley and Queisser 1961). Such multigap systems have achieved conversion efficiencies exceeding 40% (Dimroth et al. 2014). However, these MJSC are currently developed on expensive GaAs and Ge substrates and required high-quality materials, which incurs a substantial associated cost. Targeting terrestrial applications, less expensive alternatives have been proposed with two junction terminal cells containing a Si bottom sub-cell, and developed on a low-cost Si substrate (Kurtz et al. 1990; Liu et al. 2013; Loper et al. 2014, 2015; Durand et al. 2014; Filipič et al. 2015; Bailie et al. 2015; Green et al. 2016; Bush et al. 2017). For a Si bottom sub-cell with a bandgap of 1.12 eV, it has been shown that the maximum conversion efficiency are achieved with a top sub-cell bandgap lying between 1.6 and 1.8 eV (Coutts et al. 2002; Geisz and Friedman 2002; Almosni et al. 2013).

Following a different strategy, hybrid organic–inorganic perovskite (HOP) materials, such as lead methyl ammonium tri-iodide CH3NH3PbI3, in so-called ‘perovskite cells’ have recently attracted huge attention (Kojima et al. 2009; Lee et al. 2012; Liu et al. 2013; Burschka et al. 2013; Jeon et al. 2015; Minemoto and Murata 2015; Tsai et al. 2016; Blancon et al. 2017). Not only have these solar cells demonstrated remarkably high efficiencies (around 20%) and high open-circuit voltages (Voc) (Snaith 2013), but also they offer great potential for low costs of production. Taking advantage of both these strategies, perovskites are ideal candidates as a top sub-cell absorber for tandem cells built on CIGS or Si due to their high radiative recombination efficiencies, excellent absorption and charge transport characteristics and their broad ability to tune the bandgap (Bailie et al. 2015; Mailoa et al. 2015; Albrecht et al. 2015).The current record for a two terminal perovskite tandem cell is obtained with a perovskite–silicon tandem cell at 23.6% (Green et al. 2016; Bush et al. 2017).

As already widely developed in the III–V multigap systems (Geisz et al. 2008; Baudrit and Algora 2008; Friedman 2010; Rolland et al. 2014), our approach to connect electrically both subcells is to use a tunnel junctions (TJ), or Esaki diode (Esaki 1958), placed between them. An all-Si made interband TJ offers a low resistance behavior due to the high doping values reachable in Si (Bailie et al. 2015; Mailoa et al. 2015).

Simulations of perovskite-based solar cells are a key point in the optimization of the device architecture (Minemoto and Murata 2014, 2015; Hörantner and Snaith 2017). As a major feature, the behavior of photo-carriers in HOP materials can be dealt with in the same way than in inorganic materials (Minemoto and Murata 2015). This similarity enables the use of existing drift–diffusion based device simulator widely used in the context of conventional semiconductors.

In this work, we conduct numerical simulations of two-terminal tandem cells based on HOP materials as top sub-cell absorber monolithically integrated on a bottom Si sub-cell. The sub-cells are electrically connected through a low-resistance TJ. As it has recently been shown that introducing a mixture of bromide and iodide in the halide composition of lead methyl ammonium perovskites allows for tuning the band gap (Sadhanala et al. 2014), we chose CH3NH3PbI3(1−x)Br3x as top sub-cell absorber material. Besides halide concentration, the thickness of the absorber will also influence the overall performances of the total cell. Our computational study allows us to design an optimal HOP based tandem cell. The best efficiency is obtained for a bromide ratio x = 0.2 and a thickness of the HOP layer between 300 and 400 nm. In these conditions the tandem cell exhibits a conversion efficiency of 27% higher than the efficiencies of the Top and Bottom sub-cell which are respectively 17.9 and 17.3%. The Hoke effect under light soaking is also mentioned and alternative materials designed to limit or avoid such bromide/iodide segregation are briefly discussed.

2 Device structure and simulation method

Simulations are performed using the Silvaco Atlas device simulator (Silvaco Inc. 2012), which allows to numerically solve Poisson’s equation coupled with the continuity equations for both electrons and holes under steady state conditions. It is possible to account for quantum effects, which are mandatory to simulate a TJ. Different levels of mesh are required for tandem cell devices in order to consider different layer thicknesses that vary from tens of nanometers to hundreds of micrometers in realistic devices. Figure 1 shows a scheme of a perovskite tandem solar cell monolithically integrated on a n-doped silicon substrate. The structure consists in a 850 nm thick hole transport material (HTM) layer, a CH3NH3PbI3(1−x)Br3x absorbing layer of various thickness, and a 300 nm thick TiO2 electron transport layer. The 2 × 20 nm thick Si(n ++)/Si(p ++) TJ insures the electrical connection between the two subcells. Finally, a 280 μm thick n-type Si substrate (ND = 1016 cm−3) includes deep trap levels, with an energy level located at 0.5 eV above the valence band edge and a density corresponding to a diffusion length of 600 µm, and a 100 nm thick n ++ Si layer (ND = 1019 cm−3) for the back contact.

Fig. 1
figure 1

Overview of the structure showing the different layers taken into account in the simulation

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It is not obvious to define relevant values of the physical parameters needed to accurately simulate the total cell, particularly for the HOP materials since their growth procedures are not yet standardized. Table 1 summarizes the physical key parameters from the literature used in this study for HTM, HOP, TiO2 and Si materials (Noh et al. 2013; Minemoto and Murata 2014, 2015). Following the work by Minemoto and Murata (2015), HOP electron (μn) and hole (μp) mobilities are set to 2 cm2 V−1 s−1. HOP materials are assumed to be n-type with a low doping level of 1013 cm−3. Deep traps are also considered with an energy level located 0.5 eV below the conduction band edge and a density NT = 1015 cm−3 (Stranks et al. 2013; Yang et al. 2015), which corresponds to a carrier diffusion length of about 700 nm in good agreement with available experimental data (Stranks et al. 2013; Minemoto and Murata 2015).

Table 1 Input parameters for device simulation
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The imaginary part of the refractive index k for HTM, TiO2 and CH3NH3PbI3(1−x)Br3x layers was calculated from the absorption coefficient given by \(\alpha = A\sqrt {(E - E_{g} )}\) where A is a prefactor, Eg the material bandgap energy, and \(k = \frac{1.24\alpha }{4\pi E}\) (with α in cm−1 and E = hν in eV). A is kept constant for all the subsequently considered materials. In the work by Minemoto and Murata (2015), A amounts to 105 cm−1 eV−1/2. Here, we chose 2.5 × 105 cm−1 eV−1/2, which leads to a better agreement with the experimental absorption coefficient reported in ref. (Ball et al. 2015). In this work, the real part of the refractive index is kept constant for the whole wavelength range of the solar spectra: n = 2.54: average value of n published in ref. (Ball et al. 2015).

In addition to the thickness of the absorber, one essential parameter of a tandem cell is the bandgap of the light absorbing material. Here, the choice of CH3NH3PbI3(1−x)Br3x mixed halide allows to introduce a wide range of bandgaps from 1.55 eV (x = 0) to 2.3 eV (x = 1). Bandgap energies and electron affinities of mixed compositions were computed from linear interpolations of the values reported for the pure tri-iodide and tri-bromide compounds (Table 1). In the following we monitor the tandem cell efficiencies for various bromide concentrations and various absorbing layer thicknesses.

3 Effect of absorber layer bandgap energy and thickness

3.1 The current matching issue in the HOP/Si tandem cell

In this section, we show that a good current matching between the Top and Bottom sub-cells is required to obtain a maximum efficiency of the tandem cell. Figure 2 shows the simulated current–voltage characteristics (J–V) of the top HOP sub-cell (with x = 0.2), the bottom silicon sub-cell and the HOP-based tandem cell under AM1.5 illumination. The doping level of the silicon TJ is fixed at 1020 cm−3.

Fig. 2
figure 2

J–V characteristics of the bottom cell (silicon absorber, black line), top cell (HOP absorber, green line) and tandem cell (silicon and HOP absorbers, red line) under AM1.5 illumination. The HOP absorber material is CH3NH3PbI3(1−x)Br3x with x = 0.2 and a thickness of 300 nm. (Color figure online)

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The results are in good agreement with the literature (Kojima et al. 2009; Lee et al. 2012; Liu et al. 2013; Burschka et al. 2013; Masuko et al. 2014; Jeon et al. 2015; Minemoto and Murata 2015). The efficiencies of the silicon bottom cell, and of the HOP cell for x = 0.2, are 17.3 and 17.9% respectively. For the tandem cell, the open-circuit voltage is VOC = 1.56 V, which corresponds to the sum of the open-circuit voltages of each sub-cell. On the other hand, the same short-circuit current value than for the top cell is obtained, which is the maximum of the total current that can be reached in the tandem cell. It shows that in this device the current limitation is mainly due to the HOP top sub-cell which leads to a non-ideal current matching between top and bottom sub-cells.

It is known that the bandgap energy of the top sub-cell absorber material and the thickness of the absorbing layer are key parameters to obtain a good current matching between the two sub-cells (Geisz et al. 2008). Figure 3 shows the band diagram of the tandem cell at room temperature and at thermal equilibrium for various halide compositions. Due to the low residual doping level of the HOP material, the absorbing layer acts almost as an intrinsic layer and the top sub-cell can be considered as a PIN junction. Moreover, whatever the halide composition, when an electron–hole pair is created in the perovskite layer, the electrons will easily flow to the TiO2 layer without encountering any potential barrier. The behavior of the holes is similar from the perovskite layer to HTM layer.

Fig. 3
figure 3

Energy Band diagrams of the tandem cell. The valence and conduction band edges (resp. EV and EC) are depicted in green and blue, respectively. The position of the top cell is highlighted between red dash lines. Effect of halide composition in the HOP absorber CH3NH3PbI3(1−x)Br3x is illustrated for x = 0.0 (solid line), 0.47 (dashed line) and 1.0 (dashed dotted line). The thickness is kept constant at 300 nm. (Color figure online)

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3.2 Optimization of the HOP single cell

In order to validate our simulation, the influence of the absorbing layer bandgap and of its thickness on the HOP cell efficiency is examined, and the theoretical results are compared to published experimental data.

Figure 4 shows the mapping of JSC, VOC and Eff as a function (1) of the bromide concentration in the absorbing layer, and (2) of its thickness in a HOP solar cell. For a given thickness, the short circuit current JSC decreases when the bandgap energy increases because the amount of photo-generated carriers decreases. Concurrently, the open circuit voltage VOC increases. These contradictory trends lead to an optimum for the top sub-cell efficiency which is currently obtained for an absorber bandgap energy of about 1.4 eV (Dimroth 2006). From our simulations, we find a maximum efficiency around 18.5% obtained for CH3NH3PbI3 with a layer thickness between 300 and 500 nm. With a similar thickness we find an efficiency of 11.3% for the CH3NH3PbBr3 compound. These values compare well with the corresponding experimentally measured efficiencies of 17% (CH3NH3PbI3) (Im et al. 2014) and 9% (CH3NH3PbBr3) (Chen et al. 2014; Sheng et al. 2015) and show that our simulation is well suitable for further study of the HOP tandem cell.

Fig. 4
figure 4

HOP top cell a short-circuit current Jsc, b open-circuit voltage Voc, and c efficiency Eff

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3.3 Silicon tunnel junction

The silicon tunnel junction used to interconnect both top and bottom sub-cell plays an important role and behaves as a short circuit when the tandem-cell operates (Baudrit and Algora 2008, 2010; Rolland et al. 2014). Therefore, in order to optimize the whole device, the knowledge of the minimum doping level of the n ++ and p ++ region of this junction for efficient band to band tunneling effect is mandatory. To approximately calculate the tunneling current, an elastic model for interband tunneling is used and the resullting current is given by:

$$J_{T} = \frac{qkT}{{2\pi^{2} h^{3} }}\sqrt {m_{e} m_{h} } \int_{E} {T(E)\ln \left( {\frac{{1 + \exp \left( {\frac{{E_{F,r} - E}}{kT}} \right)}}{{1 + \exp \left( {\frac{{E_{F,l} - E}}{kT}} \right)}}} \right)} dE$$

where JT is the tunneling current, h is Planck constant, and me (mh) is the electron (hole) effective mass. JT is evaluated for all energies at which tunneling is possible, up to the maximum of the band energy at the interface. T(E) is the tunneling probability which is given by the WKB approximation (Silvaco Inc. 2012). EF,r and EF,l are the quasi-Fermi levels on the right and left side of the tunnel junction respectively. JT is converted into a recombination rate and inserted into the current continuity equation at each end of the tunneling paths.

Figure 5 shows the variation of the tunnel junction peak current and the differential negative resistance, which is representative of the peak-valley current ratio, as a function of the doping level of each region. The doping level is assumed to be the same on both side of the junction. A variation of more than six orders of magnitude is observed for an increase of the doping level from 3 × 1019 to 1020 cm−3. For the latter value, the reverse tunnel junction current is greater than 105 mA cm−2 under a reverse bias of 0.2 V.

Fig. 5
figure 5

Characteristic of the tunnel junction as a function of the doping level. The peak current density Jmax (mA cm−2) and the differential resistance Rmax (Ω cm2) are marked by red circles and blue squares, respectively. (Color figure online)

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We conduct the simulations of the tandem cell with different values of the tunnel junction doping levels. The optimum values of HOP bandgap energy and layer thickness mentioned above are used. The minimum doping level which leads to efficient tunneling effect and a good interconnection between top and bottom sub-cells is around 5 × 1019 cm−3. This value corresponds to the degeneracy limit of the material: below this value, the silicon material is no longer degenerated on the conduction band side, leading to a weak overlap of conduction and valence bands and a poor band to band tunneling effect.

3.4 Optimization of the HOP/silicon tandem-cell

In this section we show that an optimum of the efficiency can be obtained for the HOP tandem cell by varying both the HOP absorbing layer bandgap and thickness. Indeed, starting from the optimized doping level of the TJ, we can optimize the complete tandem solar cell. Figure 6 shows the mapping of JSC, VOC and Eff in a HOP/Si tandem cell as a function (1) of the bromide concentration in the absorbing layer, and (2) of its thickness.

Fig. 6
figure 6

Tandem cell a short-circuit current Jsc, b open-circuit voltage Voc, and c efficiency Eff. The black star in c marks the maximum efficiency 27% obtained for a bromide ratio of 20% and a thickness of 350 nm

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For a fixed bromide concentration value x, i.e. a given HOP bandgap energy, Jsc increases as a function of the thickness up to 1 µm. However, a small decrease in Jsc is noted for thicker absorbers likely due to carrier recombination effects. The trend of Voc variation as a function of the absorber thickness is a monotonous decrease whatever the value of the absorber bandgap energy. This is again due to the decrease of the electric field in the absorbing layer. This behavior is related to the balance between generation current in the absorber and the junction forward current. The cell efficiency presents an optimum value for each value of the absorber bandgap energy and with the parameters used in the present simulation, an optimum as high as 27%, i.e. higher than the best silicon solar cell record (optimized HIT structure) (Masuko et al. 2014), is obtained for CH3NH3PbI3(1−x)Br3x with a bromide concentration of x = 0.2, corresponding to a HOP bandgap energy of 1.7 eV, and a HOP absorber layer thickness between 300 and 400 nm. Table 2 shows the computed values of JSC and Voc, the filling factor (FF) and the cell efficiency for different halide compositions for top sub-cell absorbing layers with thickness of 300 and 400 nm.

Table 2 Tandem cell performances for different compositions of the absorber CH3NH3PbI3(1−x)Br3x for thickness of 300 and 400 nm including traps effect
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The maximum value obtained for Jsc is about 26.2 mA cm−2 for a HOP bandgap energy of 1.6 eV and a Voc rising from 1.6 V to 2.3 V is observed for absorber bandgap energy from 1.55 to 2.3 eV. Moreover, we find a maximum plateau of the efficiency for an absorber thickness from 300 to 400 nm. Thus, the alloy perovskite absorber with a bromide ratio of 0.2 and a thickness in the 300–400 nm range insures the best efficiency conditions of the whole tandem cell. Noteworthy, the ideal maximum conversion efficiency for a tandem cell with optimized absorbers band gaps for top and bottom cell, and without any recombination losses is 42.2%. It gives an hint about the importance of recombination losses in a realistic situation by comparison to the maximum efficiency reported here (27%).

4 Conclusion

In summary, the numerical simulation of a hybrid organic–inorganic perovskite on silicon tandem solar-cell using drift–diffusion modelling with traps recombination effect and including a tunnel junction has been performed. It is first shown that 5 × 1019 cm−3 p and n doping levels of Si are high enough to guarantee good enough tunneling properties in this device. From the simulations, the efficiency of the whole tandem device under AM1.5 illumination is derived and optimized with respect to the halide composition of the HOP materials and the thickness of the absorber layer. The optimum configuration is obtained for a perovskite thickness between 300 and 400 nm and a bromide concentration of 20%, which corresponds to a bandgap energy of 1.7 eV.

The corresponding optimal photovoltaic efficiency of the tandem cell is equal to 27%, showing that the perovskite methyl ammonium lead tri-iodide tri-bromide alloy CH3NH3PbI3(1−x)Br3x or related alloys could be suitable candidates as absorbing materials to realize high efficiency and low cost silicon based tandem solar cells, provided that the problem of light-induced halide segregation is circumvented.