17.2.1 Monolithic Two‐Terminal Solar Cells

In monolithic two‐terminal cells the two subcells are electronically connected in series. To understand, how such a solar cell works, we look at Figures 17.4 and 17.5, where we follow Ref. [6]. Figure 17.4a shows current density (J) – voltage (V) characteristics for single‐junction solar cells under AM1.5 illumination for cells with a large bandgap “t” and a small bandgap “b” as well as the J–V characteristics of a monolithic series‐connected tandem solar cell consisting of these two subcells “b+t.”

To understand why the combined J–V curve looks as plotted in the graph, we take a look at Figure 17.4b, which shows an equivalent circuit of the tandem cell: in series connection, the voltages add up and the same current flows through both cells – it is determined by the subcell generating the lower current. This important property of monolithic tandem cells is called current matching and negatively affects the overall performance if the currents generated in the two subcells differ significantly. We will discuss this topic in more detail in Section 17.5.

Figure 17.5 sketches a band diagram of a tandem solar cell (a) in the dark and (b) under illumination. In (a) the situation is drawn under thermal equilibrium, meaning that the Fermi energy E_F is constant across the whole device. Here we assume that the two solar cells are homojunctions, hence their respective bandgaps are constant across the whole junction. In real devices, often different materials with different bandgaps will be used, hence the band diagram looks more complicated.

Under illumination, high‐energy (short‐wavelength) photons are absorbed by the top cell while low‐energy (long wavelength) photons can pass through the top cell and are absorbed in the bottom cell, as illustrated in Figure 17.5b. In both subcells the Fermi level separates into quasi‐Fermi levels for electrons and holes, respectively, as shown in Figure 17.5b. In the ideal case the two voltages add up, which is also depicted. Therefore, it is necessary, that electrons and holes can combine efficiently at the interface between the top and bottom cells. This can be either done with tunnel junctions or with a (metallic) recombination layer. Both concepts are discussed in Section 17.5.

17.2.2 Four‐Terminal Tandem Solar Cells

Figures 17.3b,c sketch four‐terminal (4T) tandem solar cells – the other common tandem‐cell concept. As in monolithic 2T solar cells, light passes first through the high‐bandgap top cell and then through the low‐bandgap bottom cell. In contrast to 2T cells, here each subcell has its own electric connections and can be operated independently from the other subcell. Therefore, both subcells can be operated at their respective maximum power points (MPPs) – no matter how the light is distributed between the two subcells.

This makes 4T cells much less sensitive to changes in the spectrum of the light, which illuminates the cell. Hence, the overall energy yield in principle can be higher than with 2T solar cells. However, for the bottom contact of the top cell and for the top contact of the bottom cell two additional contacting layers need to be established for an efficient lateral transport of the charge carriers. Such contacts can be transparent conducting oxide (TCO) layers such as indium tin oxide, zinc oxide or tin oxide. For wafer‐based silicon solar cells, also a metal grating system can be used. In any case, optical losses because of parasitic absorption and/or shading by metal fingers will be larger than for 2T devices, where only vertical transport of charge carriers is required between the two subcells. If the bottom cell is an interdigitated back‐contact cell, as sketched in Figure 17.3c, these losses can be slightly reduced.

Further, every module has four connections, therefore the number of cables in a photovoltaic (PV) system consisting of 4T modules roughly doubles, which increases the balance of system (BOS) costs of the PV system.

Several numerical studies compared the energy yield of photovoltaic systems based on two‐terminal and four‐terminal tandem solar cells with a perovskite top cell – we will review a few of them in Section 17.6.

17.2.3 Other Concepts

As discussed above, the major drawback of two‐terminal tandem solar cells is current mismatch, which means that the overall performance goes down when the current densities generated in the top and bottom cells differ. On the other hand, four‐terminal cells suffer from higher optical losses because of the contacting layers on bottom of the top cell and on top of the bottom cell.

Figure 17.3d–f shows sketches of three‐terminal (3T) tandem solar cells, which aim to combine the advantages of the two concepts by adding a third terminal to a 2T cell [5]. The third terminal matches the currents of the two subcells and therefore overcomes the performance losses due to current mismatch. The two subcells are monolithically connected. Therefore, optical losses are less a problem than in 4T cells.

Figure 17.3d illustrates a mid‐contacted three‐terminal solar cell, where the third terminal connects to the back of the top cell and the front of the bottom cell. However, also this concept needs a contacting layer which ensures sufficient lateral transport. If a TCO is used, lateral transport can be controlled by the layer thickness as well as by the conductivity – both measures increase parasitic absorption.

If the bottom cell is designed as interdigitated‐back‐contacted cell with both the electron‐ and hole‐selective contacts at the back, the optical losses are not larger than in 2T cells. Such 3T cells can be designed as two anti‐series connected diodes with a common ground (Figure 17.3e) or as series‐connected diodes with a coupling layer (Figure 17.3f). The power output of such 3T solar cells can be almost as high as for 4T solar cells [5].

Another concept is the series–parallel connected tandem solar cell, which is sketched in Figure 17.3g [7]. When two cells are connected in parallel, their voltages have to be matched for proper operation. To achieve this in a tandem configuration, where two types of cells with different bandgaps are used, n₁ top cells and n₂ bottom cells are connected in series, respectively, such that their string voltages roughly match, n₁ × V₁ ≈ n₂ × V₂. These two strings are then connected in parallel. The cell voltage only depends weakly (logarithmically) on the illumination, hence the voltages of the two strings are roughly independent from the illumination. Therefore, series–parallel connected tandem solar cells are much less susceptible to the illumination conditions than monolithic two‐terminal cells. As a consequence, the bandgaps of the top and bottom cell materials are much less constraint than for monolithic 2T cells. Further, PV modules based on this concept would only have two connections which is an advantage with respect to 3T and 4T concepts. However, they would still be sensitive toward spectral changes, which naturally occur for outdoor illumination as discussed in Section 17.6.

17.2.4 Bifacial Solar Cells

At the end of this section we briefly discuss the concept of bifacial solar cells and solar modules, which not only can utilize light that is impinging on the front side of the solar module, but also light impinging at the solar module back. Depending on the reflectivity of the ground, this can significantly increase the energy yield over the year. For example, a study by Tillmann et al. showed that more than 10% of all the photon power incident on the solar module arrives at the back side for standard PV‐field configurations, when the reflectivity of the ground is assumed to be 30% [8]. Because of the rise of new technologies for silicon solar cells, such as the passivated emitter rear contact concept discussed in Section 17.5.1, bifacial solar cells are expected to have a market share of around 50% by 2029 [9].

Bifacial tandem solar cells in a four‐terminal configuration can be used just as standard monofacial cells, because the two subcells operate separately from each other. Hence, the additional light on the back side will only increase the current density of the bottom cell, shifting the MPP of the bottom cell with respect to a monofacial cell. In contrast, for monolithic tandem solar cells the additional current density at the bottom requires a change of design to achieve current matching, such that the current density in the top cell increases. We can achieve this by increasing the thickness of the absorber layer in the top cell, by using top cell materials with lower bandgaps or by using more advanced light management techniques, as discussed in Section 17.5. Finally, because of changing illumination conditions (position of the sun, clouds, etc.) the ratio between front and back‐side illumination can vary throughout the day and the year.

Energy yield analyses, explained in Section 17.6, can help to find the best band gap/thickness combination for the top cell (under the assumption that the bottom cell parameters are fixed, as e.g. for silicon solar cells). Further, these analyses can be used to assess estimate the differences in energy yield for 4T and 2T tandem cells both in monolithic and bifacial configurations.

17.3.1 Efficiency Limit for Four‐Terminal Tandem Solar Cells

For four‐terminal cells we can follow the derivation by De Vos from 1980 Ref. [11]. The overall efficiency η_4T is given by

(17.1)

where E_t and E_b denote the bandgap energies of the top and bottom cells, respectively, and p_t and p_b denote the power densities at the respective MPPs. Finally, p_in denotes the incident power density, which is 1000 W/m² for the standardized AM1.5g solar spectrum, which we use for this derivation [2]. For the whole section we assume E_t > E_b.

The maximum power density for the top cell is given by

(17.2)

where the maximum can be found numerically. The current density – voltage characteristic is based on the Shockley equation, which describes an ideal diode,

(17.3)

with the dark current density , the photocurrent density the elementary charge q, the Boltzmann constant k and the temperature of the environment of the solar cell T_e, which is set to 25 °C for solar cell standard testing conditions (STCs) [12]. To calculate the dark current density, we use the detailed balance assumption, which assumes that the solar cell behaves as a black body for wavelengths up to the bandgap and is fully transparent for longer wavelengths and that the black body radiation is driven by radiative recombination in the solar cell,

(17.4)

with the Planck constant h, the speed of light c and the solid angle element dΩ = sin θdθ dφ. The factor 2 arises from the fact that the solar cell is assumed to emit from both front and back sides. The factor cosθ arises from Lambert's cosine law of emission. The wavelength λ_t is given by λ_t = hc/E_t. More details can be found in Ref. [6], chapter 10.

For the bottom cell Eqs. (17.2)–(17.4) hold as well, the subscript t needs just to be only replaced by b.

The photocurrent density for the top cell is given by

(17.5)

where the first term is the photocurrent density due to solar illumination with the spectral photon flux Φ_{ph, λ} and the second term arises from the blackbody emission from the bottom cell, cut off at the bandgap of the top cell (the bottom cell also emits blackbody radiation between λ_t and λ_b, but this cannot be utilized by the top cell).

For the bottom cell, we find

(17.6)

where the first term is the photocurrent density due to solar illumination, which passes through the top cell with the spectral photon flux Φ_{ph, λ} and the last term arises from the blackbody emission from the top cell.

The second term J_LC in Eq. 17.6 accounts for luminescence coupling (LC, also called luminecent coupling): in a high‐quality top cell the part of the generated current , which is not utilized electrically, can lead to radiative recombination and hence to emission of light by the top cell. A fraction of this light reaches the bottom cell, where it can be absorbed. A simple expression for the additional current density generated in the bottom cell is given by [13, 14]

(17.7)

with the luminescent‐coupling efficiency η_LC. No further conversion factors are needed, because one electron–hole pair in the top cell, which recombines radiatively, leads to the emission of one photon, which may generate one electron–hole pair in the bottom cell. η_LC might also depend on the bias voltage V_t of the top cell, which, however, is out of the scope of this book [13]. In principle also LC from the bottom cell into the top cell should be considered. However, as we may safely assume that the light is emitted with energies close to the bandgap, light emitted by the bottom cell has energies below the top‐cell bandgap and cannot be utilized.

Figures 17.6a,b shows the theoretical efficiency limit for four‐terminal cells as a function of the top‐ and bottom‐cell bandgaps. For 4T cells the efficiencies of the top and bottom subcells can be maximized independently of each other for each combination of bandgaps. Without luminescence coupling, shown in Figure 17.6a, the maximum PCE is 45.2% with top and bottom bandgap energies of 1.73 and 0.94 eV, respectively. This value is significantly higher than the detailed balance limit for single junction solar cells, which is slightly above 30%. For 4T cells, maximal luminescence coupling (η_LC = 100%) only would change the picture slightly, as shown in Figure 17.6b, where the maximum efficiency is 45.4% at 1.73 eV top‐ and 0.95 eV bottom‐cell bandgaps. The little effect of LC can be understood as follows: in a 4T cell, the top cell operates at its MPP with a current density J_t close to . Hence, and subsequently J_LC are very small.

17.3.2 Efficiency Limit for Two‐Terminal Tandem Solar Cells

For two‐terminal cells, the two subcells cannot be treated independently because of their electrical series connection. As a consequence, the same current density flows through both subcells. Hence, we write

(17.8)

where the maximum power density is given by

(17.9)

In contrast to Eq. 17.2, here the voltages V have to be treated as dependent variables – hence the Shockley equation 17.3 must be inverted. Therefore, identifying the maximum power is numerically more involved than for 4T solar cells. The dark current density and the photocurrent densities can be calculated as in the 4T case using Eqs. (17.4)–(17.6).

Figure 17.6c shows the theoretical efficiency limit for two‐terminal tandem solar cells with no and 100% luminescence coupling. Without LC, the maximum PCE is 44.8% with top and bottom bandgap energies of 1.63 and 0.96 eV, respectively. While this limit is only slightly below that of 4T cells, the sweet spot is much smaller for this architecture. Especially, the top‐cell bandgap strongly affects the overall efficiency: at higher top‐cell bandgaps, the top photocurrent density decreases, increasing the potential photocurrent density of the bottom cell. However, this increase cannot be utilized because of current matching. On the other hand, 1.63 eV is closer to the bandgap of high‐performance perovskites than 1.73 eV, which is the optimum for 4T cells. If 100% LC was occurring, as shown in Figure 17.6d, the maximum would be 45.2% at 1.58 eV top‐ and 0.94 eV bottom‐cell bandgaps. For top‐cell bandgaps below the maximum, Figure 17.6d looks almost like for 4T cells, meaning that luminescence coupling can (partially) compensate for the disadvantages of 2T cells.

Figure 17.7 shows the efficiency limit for 2T and 4T tandem solar cells as functions of the top‐cell bandgap for a bottom‐cell bandgap of 1.1 eV, which corresponds to the bandgap of silicon – the semiconductor on which most solar cells are based on. While the top‐cell bandgap hardly affects the overall efficiency for 4T cells, 2T cells without luminescence coupling are very susceptible to changes of the top‐cell bandgap with a very narrow maximum. This also makes them more sensitive to spectral changes in the illumination, which naturally occur throughout the course of time. We will discuss this topic in Section 17.6. For ideal luminescence coupling (η_LC = 100%), the maximum efficiency is almost as high as for 4T cells, when the top‐cell bandgap is below the maximum: in this regime the 2T‐cells are bottom‐cell limited, which can be partially compensated by the additional current from luminescence coupling. However, for larger wavelength, the 2T cells are top‐cell limited and luminescence coupling has no added value. As discussed above, luminescence coupling can only improve 4T cells slightly.

Experimentally, luminescence coupling is not a relevant effect in perovskite‐based tandem solar cells yet. For perovskite solar cells, external luminescence quantum efficiencies of up to 15% were reported as of summer 2020 [15–20]. For III–V multi‐junction solar cells, where the strongest LC is observed, efficiencies of 33% were reported [21].

Note that the theoretical limit for silicon solar cells (29.4%) is a few per cent below the detailed balance limit for 1.1 eV [22]. Silicon is an indirect semiconductor, therefore recombination happens mainly via Auger processes, while radiative recombination, which is assumed in the Shockley–Queisser limit, hardly occurs.

17.3.3 Efficiency Limit for Cells with More Junctions

As we have seen, tandem solar cells can yield higher PCEs than single‐junction solar cells, because the top cell with the higher bandgap can utilize the energy of high‐energy photons more effectively. The efficiency can be increased even more by building solar cells with more than two junctions.

Figure 17.8 shows the bandgap utilization and the efficiency limits for cells with one, two, three, four, and infinitely many junctions according to data from Martí and Araújo [23]. If the sunlight is not concentrated, a cell consisting of one, two, three, or four junctions would yield a PCE of 32.5%, 44.3%, 50.1%, and 54.0%, respectively. In the limit of infinitely many junctions 65.4% of the incident sunlight would be converted into electricity. The additional gain decreases with every additional junction the optimal number of junctions therefore finally depends on the economics: is the cost of additional junctions outweighed by the gain in performance. Note that the efficiency limit for tandem cells (44.3%) slightly differs from the limits presented in Figure 17.7. The reason for this is probably different spectral data for the AM1.5 spectrum as well as other numerical effects.

17.5.1 Perovskite/Silicon Tandem Solar Cells

Crystalline silicon photovoltaics is the leading technology in the global PV market, with 95% market share from total module production in 2018 [9]. Therefore, out of the different possible bottom cell technologies for perovskite‐based tandem solar cells, silicon receives by far the highest attention resembling in the highest tandem efficiency realized so far and the highest number of published reports, as shown in Table 17.1 and Figure 17.10. Most silicon solar cells consist of a diffused homojunction and an aluminium back surface field (Al BSF) for enhanced charge extraction. In the improved passivated emitter rear contact (PERC) design, most of the back‐side contact is passivated with a dielectric, such as silicon oxide and or silicon nitride, to reduce surface recombination losses. Only local electrical contacts are formed. For PERC cells, efficiencies up to 24% in commercial wafer size are reported [73]. For passivated emitter and rear locally diffused (PERL) cells with passivated emitter and rear locally diffused design, the long‐time efficiency record has been 25.0% [74]. Single‐junction silicon solar cells with even higher PCE values can be fabricated by implementing full‐area contacts, which are both passivating and selective. One example for such contacts are ultrathin silicon oxides, which are combined with doped polycrystalline layers. Devices with such layers are called polycrystalline silicon on oxide (POLO) or TOPCon and they have demonstrated efficiencies up to 26.1% (with interdigitated back contact [IBC] contacts) [75] and 25.8% (with double sided contacts) [76], respectively. Another concept are silicon heterojunction (SHJ) solar cells, where two ultrathin intrinsic and doped hydrogenated amorphous silicon (a‐Si;H) layers are deposited onto both sides of the solar cell. SHJ cells reach the highest efficiencies for silicon single cells of 25.1% and 26.7% for double sided [77] and IBC [78] contacts schemes, respectively. For further reading on high efficiency silicon single junction solar cells, we refer to reviews by Battaglia et al. [79] and Hermle et al. [80].

In this section, we present the development of perovskite/silicon tandem solar cells until mid 2020; first for 2T monolithic and then for 4T architectures. The 2T tandem section includes a summary of the efficiency improvement, the development of certain device structures such as the bottom and top cell design and electrode polarities as well as implementation of textured surfaces and light management strategies. The main focus here is directed towards the majority of reported results for perovskite/silicon tandem cells that are shown in purple color in Figure 17.10 with rather similar device design being described in Figure 17.11 and the corresponding performance data also found in Table 17.1. In addition, we highlight results on textured wafers and other (non‐silicon‐heterojunction) bottom cells, both categories are also listed in Table 17.1. For more information on the efficiency evolution of perovskite/silicon tandem solar cells, we refer the reader to several review papers by Werner et al. [81], Tai et al. [82], Torabi et al. [83] or, most recently, by Song et al. [84], Hu et al. [85], Jost et al. [31], and the progress report by Shen et al. [86].

The first perovskite/silicon 2T tandem solar cell was realized on a silicon homojunction bottom cell processed with classical high temperature diffusion process [39]. The tandem cell reached a PCE of 13.7%, strongly limited by its low J_SC of 11.5 mA/cm² which was limited by the highly absorptive, doped Spiro‐OMeTAD film present in the front contact stack. Thus, the overall parasitic absorption equivalent to a loss in photogeneration current density was over 8 mA/cm² strongly limiting the PCE.

Since then, many researchers focused on SHJ solar cells as bottom cells. The advantage of these cells is their high efficiency, which is reached because of very high V_oc values with records up to 750 mV [87]. In addition, the optical loss arising from the front side selective contact made from a‐Si:H does not limit tandem cell operation, because this fraction of the incident light is already absorbed by the perovskite top cell. Besides the high efficiency, typical SHJ cells already feature a TCO, which fully covers the front side. This TCO layer can serve as the bottom contact for the perovskite top cell, just as in single‐junction perovskite solar cells fabricated on glass. This allows relatively simple process integration. Another aspect is the flexibility of the polarity that is offered by the use of SHJ: the bottom cells can be utilized with both polarities and such also the polarity of the perovskite top‐cell can be selected as well which was a key step toward >25% for perovskite/silicon tandem cells, as described below.

The first 2T perovskite/SHJ tandem solar cell was demonstrated in 2015 by Albrecht et al. [88] with 19.9% JV‐scan efficiency and – due to hysteresis – 18.1% efficiency from stabilized MPP‐tracks efficiency, as introduced in Chapter 16. The SHJ utilized as bottom cell was made from an n‐type, double‐side polished float zone (FZ) c‐Si wafer with a front emitter fabricated from thin layers of intrinsic and p‐doped hydrogenated amorphous silicon (a‐Si:H). The p/n recombination junction interconnecting the two sub‐cells was formed between the p‐doped (p)a‐Si:H film and a layer of ITO. ITO is a well‐established electrode material for perovskite single‐junction cells. The perovskite cell was then processed at temperatures below 120 °C in order to avoid degradation of the SHJ cell: The ITO was covered by an atomic layer deposition (ALD)‐processed SnO_x film, to form the electronic contact of the perovskite sub‐cell. The FAMAPbI₃ perovskite absorber (E_g ∼ 1.57 eV) was 600 nm thick being processed by spin‐coating. The hole selective contact and top contact stack was Spiro‐OMeTAD and thermally evaporated MoO_x buffer combined with sputtered ITO. Finally, a LiF film was evaporated on top of the front ITO layer as an antireflective coating. Later that year, Werner et al. showed a strongly improved efficiency in a similar device [89]: They used an almost identical SHJ cell, but with slightly different TCOs and another electron selective (fullerene C₆₀ instead of ALD SnO₂) contact for the perovskite top cell to reach a PCE from MPP tracks of 21.2%.

The three papers discussed above are plotted as the first three unlabeled dark purple datapoints in Figure 17.10. In 2017, a key achievement toward higher efficiencies followed by Bush et al. [41]: they switched the tandem polarity from n‐i‐p top cells with front‐emitter SHJ bottom cells to rear‐junction SHJ and p‐i‐n top cells. With that, the parasitic absorption in the n‐type front contact layer stack could be reduced significantly by thinner and less absorbing films. This inverted design has been used ever since and resulted in most of the results summarized in Table 17.1 exceeding 25% PCE.

The above‐mentioned tandem design with rear‐junction SHJ and p‐i‐n perovskite top cells utilized for most tandem reports with efficiency above 25% from Table 17.1 is schematically represented in Figure 17.11. Figure 17.11c displays the general monolithic 2T tandem design with a zoom into the back side and the top cell. Typically, the SHJ back side is fabricated on textured wafers by passivating the wafer with intrinsic (i)a‐Si:H, utilizing (p)a‐Si:H as selective contact and implementing a low absorptive TCO such as ITO or aluminium‐doped zinc oxide (AZO) between the selective contact and the reflecting silver (Ag) back electrode contact. Some tandem designs use silica nanoparticles between the TCO and silver layers to enhance the NIR absorption in the bottom cell and leave approximately 5% of the area uncovered for direct electrical contact between the TCO and the Ag [41, 54]. On the planar (or polished) front side, (i)a‐Si:H forms the passivated, selective contact together with n‐doped (n)a‐Si:H [42] or (n)nc‐SiO_x:H [46, 47] and the subcells are interconnected by TCOs such as ITO or tunnel junctions formed by doped silicon layers [47, 52]. The hole‐selective contact of the perovskite top cells is typically formed by organic molecules such as Spiro‐TTB or polymers such as poly(triaryl amine) (PTAA) [42, 49] and the perovskite is processed on top. Spin coating is still used for most of the reported perovskite absorbers in the tandems with compositions such as the “triple cation” FAMACsPb(I_xBr_1−x)₃ or double cations such FACs(I_xBr_1−x)₃. Band gaps of these compositions utilized so far range from 1.63 to 1.68 eV and a constant increase in band gap toward 1.68 eV was found in the development of 2T perovskite/SHJ tandems as shown in Table 17.1. The semitransparent n‐type top cell stack is typically formed by fullerenes or derivatives such as C₆₀ or phenyl‐C₆₁‐butyric acid methyl ester (PCBM), ALD SnO₂ as buffer layer, TCOs sputtered at low temperature such as ITO or indium zinc oxide (IZO) and anti‐reflection coating such as LiF or MgF₂. Metal fingers or grids are typically made from evaporated Ag. This tandem layer design and material stack enabled reported PCEs up to 29.15% [50]. Future improvements might be possible – next to the optical optimization – by reduction of non‐radiative recombination for these perovskite top‐cells (see also Figure 17.9) in order to improve the top cell V_oc and fill factor (FF) and with that the tandem PCE. In addition, Figure 17.11b also visualizes the yet reported strategies to cover textured silicon wafers by either complete conformal coverage [42, 52] or utilization of thick absorber layers in combination with mild textures to fully cover the texture height. Both being explained in more detail below. Figure 17.11 also includes in (a) a scale bar to highlight the size difference between the bottom and top cells.

Usually, single‐junction silicon solar cells are made from textured silicon wafers to effectively reduce reflection and to extend the light path within the absorber layer (light trapping). If light trapping happens via the rear side of the silicon bottom cell only, the perovskite top cell can be processed on the flat front side of the Si wafer without constraints. Therefore, standard texturing processes can be used on the silicon rear side, and indeed, most of today's high efficiency perovskite/SHJ tandem cells are processed on rear side textured wafers with beneficial light trapping in the NIR – data are presented in Table 17.1. Optical simulations, which are briefly discussed in Section 17.6.2, can help to estimate the effect of different light‐trapping measures. The optical simulation studies by Jošt et al. [90] and Santbergen et al. [91] of 2T perovskite/SHJ tandems with such a random pyramid textured rear side showed a gain around ∼0.7 mA/cm², if a rear side texture is present, which was also demonstrated experimentally by Werner et al. [92]. For reducing reflective losses, the front side has to be textured as well. Jost et al. carried out sophisticated optical simulations of perovskite/SHJ tandems on flat, rear side and both side textured silicon wafers, also considering an optimization of the perovskite band gap [45]. They find a difference of ∼0.5 mA/cm² in photocurrent density between optimized rear side texture and a double side textured cells, which can result in 1% absolute efficiency gain. For monocrystalline silicon, random pyramid textures with a height and base length of a few micrometers are typically used [42]. However, such feature sizes are incompatible with spin‐coating, which is typically applied for high efficiency lab‐based solar cells. Here, typical thicknesses of around 500 nm and well below 100 nm for the absorber and charge selective layers are utilized, respectively. Therefore, usually additional chemical and/or mechanical polishing is required to planarize the Si surface and enable perovskite spin coating atop [42].

Despite the potential advantages not only in optical properties, but also in terms of process compatibility to standard silicon cell technology, only few results on implementing perovskite cells on bottom cells with random pyramid textures on both wafer sides are reported. As illustrated in Figure 17.11b, 2T perovskite/silicon tandem solar cells with front side textures can be implemented in two ways: (i) a fully conformal deposition of the top cell layers or (ii) utilizing a mild texture and a thick perovskite absorber to fully cover the texture height with the perovskite absorber layer. The first strategy of fully conformal coverage was realized by Sahli et al. in 2018 [42]: To achieve a conformal coating of the perovskite film, they used a hybrid process consisting of first co‐evaporating the inorganic precursor species PbI₂ and CsBr to form a porous but conformal film on the texture. Then, this film was infiltrated by a spin‐coated solution of FAI and FABr perovskite precursors and annealed at 150 °C to form the Cs_xFA_1−xPb(I,Br)₃ perovskite film. Interestingly, spin coating the organic components on top of the textured and conformally covered (with PbI₂ and CsBr) wafer yielded a conformal perovskite film. The final tandem cell has a certified MPP‐tracked PCE of 25.2%. Indeed, the reflection losses are minimal between 400 and 1100 nm and the external quantum efficiency (EQE) of the fully textured tandem reported by Nogay et al. shows low parasitic losses as well as a good current matching between the sub‐cells [52]. The later strategy of utilizing a thick absorber to fully cover the texture height one is reported by Chen et al. [54] and Hou et al. [55]: utilizing a mild sub‐micron to micron texture and a thick perovskite absorber from blade‐ or spin‐coating enabled the complete coverage of perovskite films that showed a rather planarized perovskite top surface on top of the textured silicon wafer. Remarkable monolithic tandem cell efficiencies of 25.7% and 26.1% were realized by the teams of Chen and Hou/Aydin, respectively. Another interesting approach is mechanical stacking into 2T monolithically tandem designs as shown by Ramirez Quiroz et al. [78] andLamanna et al. [53]. Ramirez Quiroz et al. used a planar bottom cell and a conductive adhesive to mechanically and electrical connect both sub cells realizing remarkable fill factors above 80% [78]. Lamanna et al. utilized a fully textured bottom cell by contacting the back electrode of the perovskite top cell with the texturized and metalized front contact of the silicon bottom cell [53]. As of July 2020, the utilization of front side texturing has not enabled highest photocurrents compared to front side flat wafers (compare e.g. the results by Köhnen et al. [47] and Sahli et al. [42]), but it was shown that double side textured wafers, i.e. the standard in silicon PV manufacturing, can be utilized for highly efficient tandems as well and that these could have a better optical efficiency limit.

The first silicon/perovskite tandem solar cell based on an n‐type Si homojunction bottom cell with a standard diffused junction was built by Mailoa and coworkers in 2015 [39]. Soon after, Werner et al. demonstrated 16.3% efficient cells using ZnSnO as recombination layer, a stack of sputtered, compact TiO₂ and mesoporous TiO₂ as electron transport layer (ETL), and MAPbI₃ as top cell absorber [93]. Just as in the cell by Mailoa, they used Spiro‐OMeTAD as hole‐selective layer, which suffered from parasitic absorption at wavelengths shorter than 400 nm limiting the photocurrent density. In 2017, We and coworkers manufactured a tandem cell with a silicon homojunction bottom cell featuring passivating Al₂O₃ and SiN_x films with local contact openings and Cr/Pd/Ag point contacts, covered on the front with an ITO electrode; hence their passivation/point contact schemes was very similar to those in industrial PERC/passivated emitter rear totally diffused (PERT) Si cells. The top cell was similar to Werner's device; the PCE was 22.5% [56]. In 2018, Zheng et al. built tandem solar cells with 16 cm² device area with an interconnecting layer of SnO₂ processed by spin coating from a colloidal precursor and annealing at 150 °C. For this device, a reverse scan PCE of 17.6% and an MPP‐tracked PCE of 17.1% were demonstrated, whereas on 4 cm² a PCE of 21.0% under reverse scanning and a stabilized PCE of 20.5% was shown [94]. Later that year, the same group demonstrated a 16 cm² cell with 21.8% PCE; here (FAPbI₃)_0.83(MAPbBr₃)_0.17 was used as perovskite [57]. Shen and coworkers applied a different approach by implementing an interconnecting layer stack consisting of ∼54 nm ALD‐grown compact TiO₂, followed by a mesoporous TiO₂ layer, on two different high‐temperature processed front p/n junctions. Their n‐type c‐Si cells had either a diffused front (p+/n)c‐Si homojunction, or a (p+)polysilicon/silicon oxide/(n)c‐Si passivated front junction (being called POLO or TOPCon contact). The perovskite composition in the top cell was CsRbFAMAPbI_3−xBr_x with 1.63 eV bandgap and the front contact was a Spiro‐OMeTAD/MoO_x buffer/sputtered ITO stack. With an additional antireflection foil, the 1 cm² cell with passivated polysilicon contact reached an MPP‐tracked efficiency of 24.1% whereas the homojunction‐based tandem had a PCE of 22.9% [58]. In 2019, Zheng et al. presented a 2T perovskite/silicon‐homojunction tandem solar cell with 23.0% MPP efficiency on a 4 cm² cell. It incorporated a down‐shifting phosphor in the polydimethylsiloxan (PDMS) antireflective film, which absorbs UV photons and re‐emitting green light leading to a small improvement of the integrated photocurrent improvement in external quantum efficiency spectra of around 0.3 mA/cm² [59].

In 4T perovskite/silicon tandem cells both subcells are processed individually and mechanically stacked on top of each other. A selection of relevant results is listed at the end of Table 17.1. We summarize here the latest developments in efficiency improvements only. 4T tandem cells suffer from less restrictions on the layer integration and enjoy more freedom toward the perovskite top cell design than 2T cells. Free choice of polarity and layer arrangement (substrate vs. superstrate) enabled a continuous efficiency improvement but also more spread and diverse device designs. This can also be seen by the various perovskite top cell band gaps utilized for recent high efficiency designs ranging from 1.57 to 1.72 eV. Typically, the active layer sizes are not matched for the bottom and top cells; Table 17.1 therefore lists the areas of both cells. For unmatched areas the small semitransparent top cell is measured first and then a larger top cell‐like device with similar optical properties (smaller devices have reduced losses from serial losses) is used as a filter for the measurement of the larger bottom cell (due to the rather large diffusion length in crystalline silicon, the bottom cell suffers from edge effects at smaller active areas).

The first report on 4T perovskite/silicon tandem cells with efficiency close to the best silicon single junction was presented by Duong et al. in 2017 [35]. Using a “quadruple” cation composition with MA, FA, Cs, and Rb stabilized the top cell absorber at 1.73 eV bandgap and therefore enhanced efficiency, and reduced both hysteresis and degradation under operation. The mesoporous top cell with PTAA instead of Spiro‐OMeTAD as hole‐transporting layer had an enhanced NIR transmission and allowed to realize a 26.4% tandem solar cell. In early 2018, Ramirez Quiroz et al. achieved a tandem efficiency of 26.7% using a planar p‐i‐n top cell with a CH₃NH₃PbI₃ absorber (1.57 eV bandgap) and silver nanowires as transparent top contact [60]. In late 2018, Jaysankar et al. improved the 4T tandem further by using a CsFAPbIBr perovskite composition with 1.72 eV band gap [61]. Further, they passivated the perovskite surface by a sub‐nm layer of atomic‐layer‐deposited Al₂O₃ to further enhance the open‐circuit voltage of the top cell. Combining this planar n‐i‐p top cell with an IBC silicon bottom cell, a PCE of 27.1% was reached. In early 2020, Duong and coworkers achieved a new efficiency record of 27.7% tandem efficiency with a size‐unmatched IBC bottom cell and 26.2% for 1 cm² size‐matched cell using a PERL bottom cell [62]. They improved their mesoporous n‐i‐p cell by passivating the surface of the quadruple cation perovskite with a large cation n‐butylammonium bromide to enhance both the V_oc and also slightly the FF. As of July 2020, the highest 4T tandem efficiency combining perovskite and silicon is 28.2% and was reached using a triple cation perovskite composition with 1.63 eV band gap [63]. They used urea as additive to the perovskite, which enhanced the carrier diffusion length and therefore allowed to increase the perovskite thickness, leading to increased absorption and an unprecedented photocurrent density exceeding 22 mA/cm² in the semitransparent top cell and 39.5 mA/cm² in the 4T tandem.

After discussing the efficiency and device‐development of 2T and 4T perovskite/silicon tandem solar cells, we now will briefly look at more practical properties such as operational stability and upscaling toward larger. To evaluate the operational stability, long‐term MPP tracking is used and the most stable perovskite single cells reach T80 stabilities (the time when they are degraded to 80% of the initial performance) of >1000 hours [95, 96]. The stability under MPP tracking depends on the measurement conditions, such as device temperature, spectral illumination condition such as with or without UV irradiation, and the atmospheric conditions such as oxygen and moisture levels. As good practice, writing these conditions directly into the MPP tracking figures has evolved. However, long‐term MPP tracking of tandem solar cells is only rarely reported in the literature and yet non‐standardized. Tandem devices without encapsulation were tested in ambient conditions and retained 90% after 61 hours [42] and 92% after 100 hours [54], as reported by Sahli and Chen, respectively. Encapsulated tandem devices on the other hand by Hou et al. were light‐soaked under 400 hours MPP load at 40 °C but showed no degradation [55].

In 2019, Kamino et al. reported on a perovskite/SHJ tandem cell with 57.4 cm² aperture area and 22.6% PCE [97]. However, for commercial applications, at least the full wafer formats M2 or better M6 need to be developed, with areas of around 244 and 274 cm², respectively. The continuous trend toward larger wafers might also force the required area for top cells to M12 format, which is 21 cm × 21 cm = 441 cm² area. It is very important to demonstrate homogeneous top‐cell layer coverage on this area, which can be potentially achieved by blade coating as used by Chen et al. [54] or vacuum‐based methods [98]. In addition, also low‐temperature curing grids and grip pastes with minimal contact resistance to the TCO need to be developed and implemented. First results on this aspect were reported by Kamino et al. [99]. For further reading, we highlight reviews on large area processing for perovskite single junctions that were e.g. published by Chen et al. [100].

17.5.2 Perovskite‐Chalcogenide Tandem Solar Cells

Besides crystalline silicon, also CIGS [Cu(In,Ga)Se₂] or copper indium selenide (CIS) [CuInSe₂] are suitable materials for the bottom cells of tandem cells with a perovskite top cell. In 2019, 1.6 GWp of PV modules based on CIGS or CIS were produced, which is around 1.2% of the totally produced PV modules [101]. Because of the direct bandgap of CIGS, the absorbers can be made much thinner than in Si solar cells. Hence, this technology enables low material and energy consumption. In other words, CIGS solar cells are a thin‐film technology and therefore would allow true thin‐film tandem cells, which can be fabricated as light‐weight modules with a high power‐to‐weight ratio, flexibility and radiation hardness for space applications [102]. The bandgap of CIGS can be tuned by varying the In:Ga ratio between ∼1.7 eV for CuGaSe₂ and ∼1.0 eV for CuInSe₂. Comparing the best CIGS single‐junction cell with 23.25% PCE [103] to the best front‐side‐planar silicon solar cell with 22.1% PCE [104] shows that CIGS might have an obvious advantage.

In this section we discuss the efficiency evolution of 2T and 4T perovskite/CIGS tandems solar cells until mid 2020 and conclude with a short paragraph on stability and upscaling. The number of published reports on perovskite/CIGS tandem solar cells is significantly lower than for perovskite/silicon tandem solar cells, which can be seen by the number of data points in Figure 17.10 and when comparing the numbers of entries in Tables 17.1 and Table 17.2. Only few reports on 2T perovskite/CIGS solar cells have been published. This is likely because of the roughness of the CIGS bottom cell surface, as we will discuss below.

Figure 17.12a shows a thickness comparison of a 2–3 μm thick CIGS bottom cell and a 500 nm thick perovskite top cell. In contrast to wafer‐based silicon tandem solar cells, here a true thin film tandem with an overall thickness below 5 μm can be built. Figure 17.12c shows the typical device structure of CIGS bottom cells: it consists of a substrate, a molybdenum (Mo) back contact, the p‐type CIGS absorber, an n‐type CdS absorber and a zinc oxide (ZnO) front contact. Here, the CIGS and CdS layers form a p–n junction. Due to this defined polarity in the typical substrate configuration, only the p‐i‐n perovskite top cell configuration has been used in 2T tandem devices so far. As a consequence, the top cell designs used for CIGS or SHJ bottom cells were recently merged and p‐i‐n perovskite top cells were implemented in both designs (compare Figures 17.11 and 17.12). Luckily, this restriction is not crucial, because optical losses in a p‐i‐n top cell configuration are much smaller than in n‐i‐p designs, as explained in Section 17.5.1.

Figure 17.12b illustrates two examples of monolithic integration for 2T perovskite/CIGS tandem solar cells by either using polishing of the intermediate TCO (left) or conformal coverage of the charge selective layers (right). Both will be described below. Overall, only a few reports on monolithic perovskite/CIGS tandem solar cells can be found in the literature, with best ones summarized in Table Table 17.2. Reasons for this low number are – amongst others – polarity restrictions and processing restrictions. Most importantly, the deposition of CIGS absorber results in a relatively rough surface, in the range of σ_RMS = 50–200 nm, with surface features of high aspect ratio [65]. While the typical perovskite absorber thickness of around 500 nm is sufficient to fully cover the rough CIGS surface, high‐performance hole selective layer for the perovskite top cell, commonly deposited via solution processing, cannot cover the rough CIGS surface. This very likely leads to a shunted top cell and makes fabrication of a monolithic perovskite/CIGS tandem very challenging. Consequently, only three reports on 2T perovskite/CIGS cells with efficiencies higher than 20% were published by July 2020; they are summarized in Table Table 17.2.

Todorov et al. reported the first 2T perovskite/CIGS tandem device in 2015 with 11% PCE [40]. This tandem cell was based on a bottom cell being processed from solution instead of typical sputtering or co‐evaporation processes, thus reducing the surface roughness. As a hole selective layer between top and bottom cells a thick poly(3,4‐ethylenedioxythiophene):polystyrene sulfonate (PEDOT:PSS) layer was used, which smoothened the bottom‐cell surface even further. While this layer protected the top cell from shunting, the overall tandem cell performance might have suffered from reduced V_oc because of high recombination losses at the PEDOT:PSS/perovskite interface [105] and an ultrathin semitransparent metal front contact.

In 2018, Han and coworkers presented a top‐cell integration, where the front surface is polished to obtain a flat surface [64], similar to what is done for perovskite/silicon tandems. They sputtered a thick ITO recombination layers on top of the CIGS bottom cell and polished it to reduce roughness as shown in Figure 17.12b. With that, the spin‐coating of perovskite and very thin hole selective layers could be implemented. Consequently, a PCE of 20.8% on 0.52 cm² was obtained and a certified PCE of 22.4% on 0.042 cm² (see also Table Table 17.2). This tandem shows a high V_oc of 1.78 V, which is the highest value reported for this tandem technology, as of July 2020. Because of the thick ITO recombination layer, the free‐carrier absorption increases in the NIR, reducing the generated current density in the bottom CIGS sub‐cell. Thus, the overall tandem J_SC of only 17.3 mA/cm² is rather moderate.

Another approach was reported by Jošt et al. in early 2019: instead of polishing the CIGS cell to obtain a flat surface, they conformally deposited a hole transport layer directly on top of the rough as‐grown CIGS surface [65]. 10 nm of ALD grown NiO_x were sufficient to form a conformal, closed layer. In addition, a spin‐coated PTAA polymer layer reduced interface recombination between the conformal NiO_x and the perovskite, even though, the polymer itself might not have been covered fully conformal in a dense film, leaving some of the NiO_x uncovered. This enabled 21.6% PCE on an area of 0.78 cm². The current density of this cell was 18.0 mA/cm² and therefore higher than by Han et al., despite a strong current mismatch between the sub‐cells. This high current density could be achieved by low parasitic absorption in the recombination layer and the rough internal interfaces, which reduced internal reflection. However, this tandem device has a rather low V_oc below 1.6 V.

Al‐Ashouri et al. developed the idea of conformally deposited layers even further and used a self‐assembled monolayer (SAM) acting as the hole selective layer [66]. This ultrathin SAM layer binds to oxide surfaces (e.g. ITO, ZnO) and thus connects the tandem recombination contact with the perovskite top‐cell absorber. Despite being only a single molecular layer, the strong binding affinity and self‐assembly ensures closed layers even on rough surfaces. The almost lossless SAM/perovskite interface additionally resulted in an improved V_oc, which, combined with improved light management, yielded a certified current record PCE for perovskite/CIGS tandems of 23.3% on more than 1 cm² area (see also Figure 17.10).

In January 2020, a team from Helmholtz‐Zentrum Berlin (HZB) manufactured a perovskite/CIGS tandem solar cell with a certified efficiency of 24.16% [106]. This result was realized by improving the V_oc to the value by Han et al. Based on these results, the future efficiency development is sketched in Figure 17.10. As the efficiency potential for perovskite/CIGS tandem is highly comparable to perovskite/silicon tandems [31], it is assumed that the efficiency will quickly increase to well above 26% for perovskite/CIGS tandem and could ultimately catch up with perovskite/silicon tandems above 30%.

Now we discuss developments on four‐terminal perovskite/CIGS solar cells starting with the first cell that was significantly outperforming the best CIGS single junction: Shen et al. reported in early 2018 a combined tandem efficiency of 23.9% by using a quadruple‐cation perovskite composition with 1.62 eV band gap as also utilized for their 2T perovskite/silicon tandems [58, 68]. In addition, they also report on a MA‐free perovskite compoisiton with 1.75 eV band gap that is characterized by higher stability toward oxygen atmosphere. It is supposed that MA cations and oxygen molecules interact and enhance the surface recombination, which is identified as main driver for oxygen‐induced degradation [68]. A further enhancement toward efficiency was reported by IMEC in September the same year: They improved the NIR transmittance of the perovskite top cell by adding optical coupling layers and by optimizing the transparent electrodes. In addition, the perovskite itself was optimized toward a 1.72 eV band gap which is close to the fundamental optimum for higher tandem efficiency [69]. The latest report with the highest 4T tandem efficiency of 25.9% was announced by Hoe Kim and coworkers: using additives such as the large organic cation phenethylammonium iodide (PEAI) and lead thiocyanate (Pb(SCN)₂) improved the perovskite film quality with enhanced crystallinity, reduced formation of excess PbI₂, a lower defect density and also higher charge carrier mobility as well as lifetime [70]. This enabled a highly performing perovskite top cell with around 20% and a combined tandem efficiency of 25.9%, the yet highest reported value for any kind of perovskite/CIGS tandem solar cells.

The results in this section highlight the versatile potential of perovskite/CIGS tandem solar cells. Although less results and lower performance was reported in comparison to perovskite/silicon tandems, Jošt et al. recently compared the efficiency potential between silicon and CIGS and found that comparable efficiency metrics above 33% could be achieved [31] with realistic material properties and device designs. Therefore, we forecast a strong efficiency improvement for perovskite/CIGS tandem solar cells toward 30% within the next two to three years.

At the end of this section, we briefly summarize the more practical properties such as operational stability and upscaling toward larger areas for perovskite/CIGS tandems although the number of reports is yet limited. Larger area perovskite mini modules and their integration into tandem devices were reported by Jaysankar et al. [107]. When scaling the semi‐transparent from 0.13 toward 4 cm² (seven series connected sub cells) and 16 cm² the efficiency drops from 16.7% to 14.8% and to 12.0%, respectively. Combining the 4 cm² mini module with a CIGS bottom cell, an area matched tandem efficiency of 21.3% was achieved [107]. For perovskite/CIGS tandems, module integration by series connection of sub‐cells and the deposition on the m² area are two important challenges toward commercialization. Another important aspect for outdoor tandem modules is operational stability. Long term MPP tracking was reported by Han et al.: non‐encapsulated tandems retained 88% of the initial performance after 500 hours under continuous one‐sun illumination at 30 °C ambient conditions. After storing the tandems in dark for 12 hours, it recovered to 93% of its initial performance [64]. The perovskite‐specific dark recovering might induce higher operational stability under outdoor conditions [108], ensuring high energy yields over a long tandem module operation time scale.

17.6.1 Illumination Model

First, the irradiance onto the module needs to be calculated. It depends, amongst others, on the position of the Sun in the Sky, the direct normal irradiance (DNI) of the sunlight, the diffuse horizontal irradiance (DHI) irradiance of the light from the sky and the orientation of the module. These data are provided by many institutions, such as national weather services or the National Renewable Energy Laboratory (NREL) in the USA. Some sources provide data for many years, here the annual energy yield would be calculated by averaging over several years. Others, such as NREL, provide data for a typical meteorological year (TMY). For tandem solar cells, it is especially relevant to have spectral data, because this determines how the incoming light is distributed between the two subcells. Currently, most databases do not provide these data, however. If no spectral data is available the spectral distribution of the light can also be described by the average photon energy (APE), which for example was used by Futscher and Ehrler [109]. A high APE value corresponds to a blue‐rich spectrum, while a red‐rich spectrum leads to a low APE. While some models consider a singular module with no shadowing effects by other modules [110], other models consider the PV module to be in a large PV field, therefore taking shadowing effects into account [8]. The most complex models also consider the surrounding, including detailed shadowing by buildings at a specific location [111].

17.6.2 Optical Model

When the irradiance is determined, it can be used to calculate the spectral intensity profile throughout the absorber layer I_λ(z), where z is the vertical position within the absorber. A detailed optical model should take all layers of the solar cell into account; it is relevant to know the thickness and the wavelength‐dependent complex refractive index data of every layer. Especially in the quickly evolving field of perovskite solar cells, where material compositions of the different layers change often, this task is very relevant and often requires an individual characterization of the layers in the respective labs. Further, textures of the different surfaces and interfaces in the solar cell strongly affect the optics. Taking them into account in an appropriate manner always increases the simulation time and is not always possible. While large pyramids, which e.g. are commonly present in silicon solar cells can be simulated with approaches such as ray tracing [4, 112], periodic structures with dimensions in the range of the wavelength need rigorous approaches for reliable results, for example the finite elements method [113, 114].

From the intensity profile the absorption in the absorber layer is given by

(17.10)

where d is the thickness of the absorber. If light traverses the absorption layer only once, the intensity decays according to the Lambert–Beer law,

(17.11)

with the absorption coefficient α. However, this relation does not hold if light is reflected from the back into the absorber, if interference is relevant or if the device contains scattering structures.

For a rough estimate, simple optical approaches can be used. For example, every solar cell absorber layer can be assumed to absorb all the light with energies larger or equal then the bandgap: then, all the light reaching the solar cell with energies larger than the top‐cell bandgap is considered to be absorbed there while longer wavelength can reach the bottom cell. In a more realistic but still idealistic approach, the absorption in the weakly absorbing region close to the bandgap can be estimated by the Tiedje–Yablonovitch limit [115] or by an approach by Green [116]. However, these approaches only will yield a layer absorptivity A but no depths‐dependent information.

For detailed electrical simulations, we need the generation rate G(z) (in cm⁻³), which tells us, how many electron‐hole pairs are generated at depths z. Starting from the spectral intensity I_λ(z), we can obtain the spectral generation rate by

(17.12)

where hc/λ is the photon energy at wavelength λ and η_G is the generation quantum efficiency, usually assumed equal to unity. The generation rate is then obtained via integration

(17.13)

Simpler electrical models or estimates do not require generation profiles, but only the short‐circuit current density. When we assume that every absorbed photon leads to the generation of an electron–hole pair, we obtain the maximum achievable current density,

(17.14)

where Φ_{ph, λ} denotes the spectral photon flux impinging on the solar modes. For standard optical modeling it is obtained from the AM1.5g spectrum at normal incidence [2]; for detailed energy yield analyses we obtain it from an illumination model as discussed in Section 17.6.1.

Note that the generation rate and the current density need to be calculated for the absorbers in the top cell and the bottom cell.

17.6.3 Electrical Model

The final step in the chain of models sketched in Figure 17.13 is the electrical model. A very rigorous method for electrical modeling is to calculate the depth profiles of the electron concentration n(z) and hole concentration p(z), from which the electrical potentials and hence the band diagram of the device can be calculated, which is the base for the current density – voltage characteristic. All this is explained in detail for example in Reference [6], Chapters 6–8. The calculations are based on three sets of equations:

• The continuity equation ensures the conservation of charge, not only when electrons and holes flow through the solar cell, but also when electron‐hole pairs are generated as described in Section 17.6.2, or when they recombine via recombination mechanisms such as radiative recombination, defect‐driven Shockley‐Read‐Hall recombination or Auger recombination, which is relevant in indirect semiconductors. More information on generation and recombination is for example given in Ref. [6], chapter 7.
• The transport equation relates the current density to gradients in potential and charge density. Here the current density J(z) is driven by diffusion, which is caused by concentration differences of the charge carriers at different positions z in the semiconductor, and by drift due to electric fields.
• The Poisson equation relates the electrical potential and hence the electrical field to the sum of all charge densities.

Using these equations, the electron and hole concentrations across a solar cell can be calculated and the current density – voltage characteristic can be extracted. Usually, these equations cannot be solved analytically but require a numerical approach. Next to commercial programs, several academic groups have developed solvers to study the device physics of solar cells, e.g. [117, 118]. While these programs are well‐suited to simulate single‐junction solar cells, they often have great difficulties in covering the physics happening at the recombination contact of monolithic tandem solar cells, where the top and bottom cells are electrically connected to each other. Further, this kind of simulations is relatively expensive in terms of computational capacity and takes relatively long, which makes them less suited for energy yield analyses, where the simulations must be conducted for many time steps.

A much simpler analytical approach is to electrically describe the solar cell via a one‐diode model, where we consider the solar cell to consist of a photocurrent source I_ph, a diode, a parallel (shunt) resistance R_p and a serial resistance R_s, as illustrated in Figure 17.14. Mathematically, the current density – voltage characteristic of this circuit is given by

(17.15)

with the dark current I₀, the diode ideality factor m, the unit charge q, the Boltzmann constant k, and the temperature T. For R_s = 0, R_p → ∞, and m = 1 the equation reduces to the Shockley equation 17.3, which describes an ideal diode. We used it in our derivation of tandem solar cell efficiency limits in Section 17.3. For some types of solar cells, such as silicon solar cells, models consisting of a single diode cannot describe the solar‐cell behavior accurately. Then, two‐diode models can be used where two diodes are assumed to be in parallel: one is an ideal diode with ideality factor m = 1, the other one has a different ideality factor (see e.g. Ref. [6], chapter 9): With such models, monolithic two‐terminal solar cells can be modeled as sketched in Figure 17.4b. One‐diode models have been used by several authors for their energy yield analysis of tandem solar cells [110, 119].

Each of the approaches described above leads to a current density – voltage characteristic, for which the MPP needs to be determined. The maximum power can then be used for the energy yield calculation.

In an even simpler approach, the external solar cell parameters of experimentally measured single‐junction solar cells are taken as starting points: the open‐circuit voltage V_{oc, ref}, the short‐circuit current density J_sc, and the fill factor FF. Using optical simulations, a maximum achievable current density J_ph is determined for each subcell and taken as new value for J_sc – for monolithic tandem solar cells the lower one of these two photocurrents is also used for the other subcell in order to mimic current matching. Then, the open‐circuit voltage is determined by accounting for the logarithmic relationship between current density and voltage,

(17.16)

Finally, the FF is assumed to stay constant. Assuming this approach, a power density is determined for the two subcells and the two power densities are added. This approach was for example used for layer thickness optimizations of monolithic tandem solar cells [45, 113, 120].

17.6.4 Temperature Model

The electrical performance is not only determined by the irradiance which reaches the solar cell, as described in Section 17.6.1, but also by the solar cell temperature. The temperature affects many electronic processes in the solar cell, such as recombination rates and – as a consequence – the voltage. For detailed energy‐yield analyses and electrical modeling, knowledge of the solar‐cell temperature, which might be significantly hotter than the surface of the solar module, is therefore very important.

The temperature of a solar cell within the solar module is determined by several factors: temperature of the surrounding, irradiance on the module, wind at the module – which is also dependent on the installation height of the module. Some models that can be used to estimate the module temperature are described in Ref. [6], section 20.3.

The effect of temperature on the performance of silicon PV modules is often approximated linearly, where the change per degree Kelvin of a parameter such as the module's PCE, is given as a temperature coefficient. Temperature coefficients are published in datasheets of silicon PV modules. For perovskite solar cells a temperature coefficient of −0.17% per K between 25 °C and 85 °C was published in 2020 [121], which is comparable with silicon solar cells. For perovskite‐based tandem solar cells no temperature coefficients have been determined yet. However, as both perovskite and silicon solar cells show similar temperature coefficients, it is likely that perovskite‐silicon tandem solar cells behave analogously.

17.6.5 Energy Yield Calculation

Above we described the models that are required for an energy yield analysis:

• An illumination model to determine the amount of radiation – and hence the wavelength‐dependent photon flux – hitting the solar module. For tandem modules, where the two cells are susceptible for different wavelength ranges, the illumination model should provide spectral information.
• An optical model that calculates, how much light is absorbed in the absorber layers of the subcells, with the output of the illumination model as input.
• An electrical model, which calculates the current density – voltage characteristic and subsequently the maximum power output at the given illumination and temperature.
• Finally, a temperature model that determines the temperature of the solar module, which is relevant for accurate electrical modeling.

All these models must be performed sequentially for every time step in the period, for which the energy yield should be calculated. Some models described above are very time consuming, such as the finite element method e.g. for optical modeling or numerical electrical models based on the continuity, transport, and Poisson equations. It is not practical to perform them for every time step. Therefore, for energy yield calculations we need to fall back on simpler and faster models. Alternatively, we can use surrogate modeling methods [122], where we perform the rigorous calculations only for a few points. An example for such methods is the reduced basis method [123], which can be applied to the finite elements method [124, 125].

Some authors use energy yield calculations for optimizing different aspects of tandem solar cells, e.g. to study the effect of bandgap and layer stack changes on the performance of 2T‐ and 4T solar cells [119, 126].