Photon recycling in perovskite solar cells and its impact on device design

3.1 Radiative emission yield

The primary necessity for efficient photon recycling is a high radiative emission yield. This implies that photogenerated carriers recombine radiatively rather than via competing SRH or Auger recombination processes. A high radiative emission implies that noncollected electron-hole pairs predominantly recombine radiatively, potentially setting off reabsorption/re-emission cycles. The total charge-carrier recombination in a perovskite layer under open-circuit conditions is quantified by the general rate equation [39]:

In this, G _ph is the carrier generation rate by absorbed photons, k ₁, k ₂ and k ₃ are rate constants associated with SRH recombination, band-to-band radiative recombination, and Auger recombination, respectively, and R _T is the total charge recombination rate; n c is the charge carrier density.

SRH recombination refers to nonradiative recombination caused by defect states which can be present in both the bulk and at the surfaces. Here, we consider the k ₁ factor to be associated with bulk SRH recombination [Figure 2(a)]. Bresnes et al. [32] showed from device simulations that perovskite films need to have a k ₁ value lower than 1 × 10⁴ s⁻¹ to truly benefit from photon recycling to increase the V _OC and thus PCE on the device level, as shown in Figure 2(b). If k ₁ increases, the presence of photon recycling starts to diminish and becomes insignificant upon reaching the value of 3 × 10⁶ s⁻¹, as depicted in Figure 2(b)–(d). Furthermore, Bresnes et al. [32] also demonstrated the sensitivity of k ₂ on photon recycling, indicating that a value of k ₂ = 9.2 × 10⁻¹⁰ cm³ s⁻¹, as measured in state-of-the-art PSCs devices, may be sufficient [42]. Auger recombination is also associated with nonradiative recombination, however it does not play a significant role in PSCs since it is only significant at extremely high carrier injection levels (>10¹⁸ cm⁻³) while being negligible at 1-sun equivalent carrier densities [43]. We note that k ₁ can be engineered by defect passivation strategies [44] whereas k ₂ and k ₃ are intrinsic material properties. Simulation results revealed that photon recycling only becomes significant if the SRH lifetime, τ _SRH, is more than 300 ns which is the typical transient photoluminescence lifetime value of perovskite films on glass [40]. Here, the model was assuming perfectly passivated interfaces (expressed by the surface recombination velocity, S = 0) and used the bulk lifetime as the main parameter (Figure 2(e) and (f)). For lower values of τ _SRH, photon recycling is hardly present and does not influence the V _OC of the device.

Figure 2:

(a) Schematic diagram of nonradiative recombination (bulk and interfaces) in PSCs. Simulated current density-voltage (J–V) curve of PSCs of 1.6 eV bandgap, including and excluding photon recycling (PR) for SRH recombination constants (b) k ₁ = 1 × 10⁴ s⁻¹, (c) k ₁ = 2 × 10⁵ s⁻¹ and (d) k ₁ = 3 × 10⁶ s⁻¹. (e) Efficiency (η) and (b) V _OC as function of SRH carrier lifetime τ _SRH (s) with and without photon recycling for perfectly passivated PSCs. (g) and (h) panels show the same results as (e) and (f) but include the effect of surface recombination (S > 10 cm s⁻¹) with and without photon recycling in PSCs. Here, the authors in their calculation assumed that S ₁ = S ₂ at each interface. (i) Schematic depiction of defects at the surface and bulk of perovskite and the various passivation strategies using functional groups. Figure (b)–(d) are adapted with permission from Ref. [32], Figure (e)–(h) are taken with permission from Ref. [40] and Figure (i) is adapted from Ref. [41].

In realistic devices surface recombination is likely to occur, which – as stated – is predominately caused by SRH processes [40]. Figure 2(g) and (h) shows the effect on V _OC and PCE as function of the bulk lifetime (τ _bulk, denoted as τ _SRH in the figure taken from Ref. [40]), and the impact of S on photon recycling. We can see that above S = 10² cm s⁻¹, the V _OC and PCE considerably decrease, even with high values for τ _bulk, illustrating that both bulk and surface passivation of metal halide perovskite defects play a crucial role for enhancing photon recycling and thus, considerably impacts the efficiency potential of PSCs. The surface recombination velocity S influences the effective lifetime τ _eff which is also influenced by τ _bulk (combining the effects of SRH, radiative, and Auger recombination), the diffusion coefficient D and the film thickness W, as:

It should be noted that, under low excitation intensity, τ _bulk for lead halide perovskites is primarily determined by τ _SRH, due to bulk and grain boundary defects [45]. To increase τ _eff, the surface recombination velocity should be decreased and the τ _SRH of perovskite films should be increased. To achieve this, several passivation strategies have been reported [44], as summarized in the sketch in Figure 2(i). After implementing these passivation strategies, contemporary state-of-the-art PSCs exhibit experimental values of k ₁ between 1 × 10⁵–1 × 10⁷ s⁻¹ with a trap density of 10¹⁵–10¹⁷ cm⁻³ [12, 39, 46]. Comparing these values with those mentioned earlier suggests that a further reduction of nonradiative recombination losses is required to reduce k ₁ in order to observe a substantive impact of photon recycling in PSCs. In the context of surface passivation, Yang et al. [45] reported a value of S ∼ 10² cm s⁻¹ for polycrystalline halide perovskite films coated with polymethylmethacrylate (PMMA). Moreover, Staub et al. [33] demonstrated the utility of combining time-resolved photoluminescence (TRPL) measurements with analytical models to determine values for S. For materials with high purity and good surface passivation, the estimated S values are in the order of ∼10 cm s⁻¹ for typically observed perovskite TRPL lifetimes (few hundred nanoseconds). Meanwhile, Krogmeier et al. [47] obtained S ≈ 200 cm s⁻¹ for the interface between the halide perovskite film and PCBM. Recently, Wang et al. [48] determined S values in the range of 10⁻¹–10⁴ cm s⁻¹ using TRPL for various electron/hole transport and polycrystalline halide perovskite interfaces, values that may be suitable to enable photon recycling, if combined with sufficiently high τ _bulk values.

In the context of devices, the requisite high radiative yield for effective photon recycling can be expressed by the need for high internal (Q _i ) and external (Q _e ) photoluminescence quantum efficiencies [35, 36, 49], determined at open-circuit conditions. Q _e is the ratio of the emitted photons per absorbed photons in the material, whereas Q _i is the ratio of the radiative recombination rate to the device’s total (radiative and nonradiative) recombination rate [50]. Q _e is an important parameter affecting the V _OC and thus PCE. If Q _e  = 1, the V _OC is equal to the radiative open-circuit voltage (V _{OC, rad}). The total recombination current (expressing the loss in charge carriers per unit time via radiative and nonradiative processes) increases in case of any additional nonradiative recombination, reducing the V _OC and Q _e [40]. Similarly, failure to achieve a high Q _i is an indicator for high nonradiative recombination losses [36]. To estimate the influence of radiative and nonradiative recombination on device performance, Q _e and Q _i values should be known. Simulations by Kirchartz et al. [51], revealed that both a high Q _e and Q _i are critical parameters in the context of photon recycling and therefore the V _OC of the device. The relationship between Q _i and Q _e can be expressed using the following equation [51]:

Here, p _e stands for the probability of emission or outcoupling of photons, p _r  = 1 − p _e  − p _a is the reabsorption probability in the perovskite, and p _a is the probability of parasitic absorption, most prominently via free-carrier absorption within the contact layers of the device. Given that in the case of photon recycling, photons may undergo multiple reabsorption and re-emission events within the absorber and may bounce back and forth may times within a thin-film solar cell, the optical design must play a vital role in curtailing parasitic absorption [52]. Figure 3 clearly shows how Q _e depends on Q _i and p _a , assuming p _e  = 12.7% for flat PSCs [55]. From Eq. (5), it is clear that Q _e increases with the higher p _r Q _i and thus increased photon recycling. Conversely, an increase in p _a results in a substantial decrease in Q _e , as shown in Figure 3(a). A reduced Q _i directly lowers Q _e and hence the Δ V o c P R as depicted in Figure 3(b). Q _i can be calculated with the help of the experimentally given Q _e , through rearranging Eq. (5) and assuming that there is no parasitic absorption (p _a  = 0) [56]:

Figure 3:

Simulation results illustrating the importance of Q _i and parasitic absorption probability (p _a ) for (a) Q _e and (b) Δ V o c P R of PSCs. Calculations are based on Eq. (5) with p _e  = 12.7%, associated with the flat PSCs. (c) Relationship between Q _i and Q _e for various photon escape probabilities (p _e ) and assuming no parasitic absorption p r = 1 − p e − p a → p a = 0 = 1 − p e ) losses. Dotted line with Q _e  ∼ 50% (highest reported value for perovskite films [53], [54]) and Q _i of 80% corresponds to p _e  ∼ 24%. This analysis shows the importance of p _e to achieve the Q _i value that is derived from the experimentally determined Q _e value. (d) Relationship between Q _i and Δ V o c P R for various photon reabsorption probabilities p _r .

As introduced by Abebe et al. [31], a significant enhancement in photon recycling only occurs for very high Q _i and p _r as per the following equation:

Hence, based on Eqs. (6) and (7), specific knowledge of p _e is required to define Q _i and thus minimize the nonradiative recombination in PSCs and evaluate the potential for ( Δ V o c P R ) in PSCs. Moreover, Figure 3(d) displays the increase in photon recycling ( Δ V o c P R ) over Q _i for various p _r . For low Q _i   < ∼  40%, Δ V o c P R is low and similar for all p _r , while for higher Q _i , the value of Δ V o c P R more sensitively depends on p _r . Hence, the above calculation demonstrates the dependence of Q _i on Q _e and also highlights the importance of accurately determining p _e . In recent reports, p _e values are reported between ∼5 and 13% for planar perovskites films [31, 36, 49, 52, 56]. Employing such values in Eq. (7), the noticeable value of Δ V o c P R  = 30 mV can be achieved for Q _i  ∼ 90% (Q _e  ∼ 40–50%) in PSCs as shown in Figure 3(c) and (d). The recent advancements in passivation strategies for perovskite absorbers enable lower nonradiative recombination and defect densities, allowing for internal Q _i values above 90% [57]. These results underline the promise for realizing photon recycling in practical applications.

3.2 High reabsorption: minimum Stokes shift

Improving only the radiative emission yield is insufficient to observe photon recycling. The increase in Δ V o c P R is only obtained by reabsorbing internally emitted photons, resulting in an increased carrier concentration within the absorber. If the emitted photon energy is smaller than the perovskite bandgap, or, more specifically, is emitted at a photon energy that corresponds to a negligible absorption coefficient, the emitted photon is unlikely to be reabsorbed. Hence, to enable photon recycling, understanding the reabsorption mechanism is as essential as increasing the radiative emission yield.

The Stokes shift is a useful figure of merit to estimate the influence of reabsorption on photon recycling, and is defined as the energy difference between the maximum positions of the emission and absorption spectra as sketched in Figure 4(a) [12]. To effectively reabsorb the re-emitted photons (a necessary condition for photon recycling), the Stokes shift must be minimal. In other words, a significant overlap between the absorption/emission spectra is essential for the reabsorption mechanism to occur [59]. Lee et al. [12] demonstrated that Δ V o c P R decreases exponentially with increasing Stokes shift, as shown in Figure 4(b). In Table 1, we summarize the reported Stokes shift in GaAs and perovskites. For GaAs, the Stokes shift is quite small (4 meV) compared to perovskites. A small Stokes shift implies that the bandgap of the absorber is direct. In addition, Wong et al. [60] demonstrated that the smaller the Urbach energy (γ) of a semiconductor, the smaller the Stokes shift, where the Urbach energy is extracted from the logarithmic slope of the absorption coefficient below the band edge and is indicative of the material’s static and dynamic disorder. In the case of γ < kT, the absorption and emission spectra are overlapping and this leads to a minimal Stokes shift. However, in the case of γ > kT, the emission spectra are notably broadened and move away from the absorption band edge, resulting in a larger Stokes shift as shown in Figure 4(c). With the presence of deeper, yet nonlocalized bandedge states, radiative recombination of electrons and holes populating these deeper states results in a redshifted emission of photons with energies below the absorption band edge [Figure 4(d)].

Figure 4:

(a) Schematic diagram representing emission, absorption, reabsorption spectra, and Stokes shift ( Δ E stokes ). The overlap is essential between emission and absorption spectra as it increases the reabsorption and consequently enhances the Δ V o c P R . (b) Calculated Δ V o c P R as a function of ( Δ E stokes ) in perovskite films. The table shows the Stokes shift in GaAs and perovskites, as reported in the literature. (c) Effects of Urbach energy (γ) on absorption and emission (photoluminescence) spectrum; Normalized photoluminescence (PL, dashed line), absorptance (Abs, solid line) spectrum and distribution of band gaps (dotted line) of solar cell with increasing γ. (d) Schematic diagram of density of states (DOS) with excitation and recombination of carrier. Figure (a)–(b) are derived from Ref. [12] and Figure (c)–(d) are taken with permission from Ref. [60].

Practically, at room temperature, γ is about 15 meV for lead-halide perovskites, compared with GaAs = 7 meV [61], which likely explains also the difference in Stokes shift for these materials. It is of note that the Urbach energy scales with temperature [62]. Therefore, with increasing temperature the Stokes shift will also increase, likely impairing photon recycling. This further underlines the importance of addressing parasitic heating of solar cells; the hotter the devices, the less photon recycling will be present, due to the increased Urbach energy [63]. Overall, γ can be used as be metric for the material’s high structural quality, with small values leading to a smaller Stokes shift [61]. We note that different perovskite compositions and the use of various additives may be explored to improve the material’s quality, expressed through γ and ΔE _stokes, which may aid in enhancing photon recycling in these materials.

3.3 Efficient light management

Apart from radiative emission and reabsorption, light management also plays an essential role in thin-film solar cells for effective photon recycling. Figure 5(a), shows the optically ideal structure for photon recycling which minimizes the escape-cone angle from the front side of the cell and increases the photon concentration by perfectly internally reflecting all the emitted photons that reach the rear side of the device. In particular, following Snell’s law, the large refractive index contrast between the absorber and free space leads to a narrow escape cone angle [64]. To analyze the impact of optical design on photon recycling in PSCs, Abebe et al. [31] discussed three types of light interaction with semiconductors [Figure 5]. The first possible light interaction is perfect incoupling of light (ideal case of a perfect antireflective coating, ARC), where the absorption response is based on the Beer–Lambert law but without considering light-trapping effects [Figure 5(c)]. A second interaction is light propagation in a multilayer medium where constructive and destructive interference effects occur. The interference effect is produced only if each layer’s (optical) thickness in the thin-film stack is on the order of the wavelength of light. In general, the transfer matrix method (TMM) is used to model such structures. The TMM relies on wave optics, in which a matrix represents the propagation of the electromagnetic field (light) through each layer of the device. The matrices associated with each layer are multiplied together to achieve a global matrix and is used to calculate the electromagnetic field on each side of the layer that allows to calculate reflection and transmission coefficients [65]. Essentially, with the TMM, the absorptance of a planar experimental solar cell device can be calculated [Figure 5(b) and (d)].

Figure 5:

(a) Schematic diagram of optically ideal solar cell structure for efficient photon recycling with no optical loss (No reflection from the front side and perfectly reflected from the back side). Photons are trapped due to the narrower escape-cone angle and reabsorbed and re-emitted within the active layer. Moreover, when the escape-cone angle is narrower than the emission angle, the emitted photons will be reflected from the cell’s front surface. (b) Schematic representation of layer stack [ITO(150 nm)/TiO₂(20 nm)/CH₃NH₃PbI₃(300 nm)/Spiro-OmeTAD(250 nm)/Au(60 nm)] of PSCs considered in the photon recycling calculation by Abebe et al. [31]. θ _in is the incident angle of incoming light and θ _out is the angle relative to the surface normal i.e., defined as the escape-cone angle. (c) Schematic illustration of perfect incoupling of incoming light and a perfect back reflector with no light-trapping. The absorptance is calculated through Beer–Lambert law which assumes zero front reflection. (d) Schematic depiction of a planar stack of PSC for which the absorption is calculated using the transfer matrix method (TMM). (e) Schematic design of front side Lambertian scatterer to consider optimum light-trapping in solar cell. (f) Absorption spectrum of PSCs with a Lambertian front texturing, perfect ARC (Beer–Lambert Law) and planar stack of PSCs. (g) Calculated V _OC and (h) Δ V o c P R as function of Q _i for Beer–Lambert, or wave optics, or Lambertian light-trapping angle-independent absorption response of PSC for θ _out = 90°. All plots are plotted for cases with and without parasitic absorption in other layers of PSCs. (i) Calculated Δ V o c P R as function of parasitic absorption probability (p _a ) in supporting layers of PSCs at Q _i  = 1.0 and for various escape cone angles ( θ _out). Figure (f)–(i) are taken with permission from Ref. [31].

Thirdly, light may interact at the interface in which a Lambertian light-trapping structure is implemented, as shown in Figure 5(e). Here, the frontside-textured surface is considered to allow complete randomization of the direction of transmitted light and results in an absorptance enhancement by a factor up to the Yablonovitch limit, assuming the absorption in the active layer is weak. The absorption spectra were calculated with thickness h = 300 nm of the perovskite absorber at normal incidence.

To analyze the performance of these designs, Abebe et al. [31] derived an explicit expression to predict the Δ V o c P R . They also derived an expression to describe the dependence of Δ V o c P R on Q _i , the probability of parasitic absorption of radiatively emitted photons (p _a ), and the escape probability of emitted photons (p _e ), respectively, as defined earlier. In particular, they calculated p _e without any angular restriction ( θ _out = 90°, angle-independent) while comparing the impact of the various absorption responses on photon recycling in a solar cell.

The Lambertian response leads to a higher and ideal absorption response but results in a lower V _OC  = 1290 mV at Q _i  = 1 [Figure 5(e)], and Δ V o c P R [green plots in Figure 5(g) and (h)] compared to the Beer–Lambert and wave-optics response [red and blue plots in Figure 5(g) and (h)]. This relates to the high escape probability of the Lambertian scheme (p _e  = 17.8%) as compared to the Beer–Lambert (p _e  = 5.32%) and wave optics (p _e  = 3.32%) responses. The Lambertian response shows the highest p _e due to the lack of total internal reflection inherent for such an idealized front texture. Figure 5(g) and (h) show the V _OC of the PSC as a function of Q _i and the obtained maximum V _OC of 1317 and 1305 mV for the PSCs including charge-transport layers (wave optics) and Beer–Lambert cases at Q _i  = 1, respectively. The Beer–Lambert response leads to a higher absorption compared to the wave-optics analysis but provides less V _OC and Δ V o c P R . Following this, the authors argued that the perfect incoupling of light through the Beer–Lambert response results in a larger escape probability of re-emitted photons compared to the wave-optics analysis and hence to a less Δ V o c P R as shown in Figure 5(h). Indeed, the wave-optics case results in a larger V _OC when Q _i  > 0.95, where the photon recycling impact is sufficiently large. Moreover, they have calculated Δ V o c P R based on different escape-cone angles ( θ _out) and found that at θ _out = 2.5°, one could reach Δ V o c P R  = 240 mV, as shown in Figure 5(i). The increase in Δ V o c P R is mainly due to re-emitted photons having a lower escape probability due to a smaller escape cone. Note that upon introducing parasitic absorption p _a , even though Q _i approaches unity, the increase in Δ V o c P R substantially reduces [Figure 5(h)]. This is because the parasitic absorption probability p _a profoundly affects the probability of reabsorption of photons within the perovskite. Furthermore, parasitic absorption’s unfavorable influence on photon recycling is more obvious in the case of varying θ _out [Figure 5(i)]. Therefore, dedicated attention must be given to light management to minimize p _a to exploit photon recycling in PSCs.

In practical terms, rugate structures could establish a narrow θ _out angle over a broader wavelength range [66], [67]. This structure relies on dielectric coatings, such as a solar cell with an ARC (which simultaneously may act also as transparent electrode, depending on its electrical conductivity) with a refractive index equal to the geometric mean of that of its adjacent materials, i.e. glass (n = 1.5) and the absorber material (the optical thickness of the ARC then dictates the minimum of reflection via quarter-wavelength conditions). The refractive index of the ARC could also be made gradual, e.g. by SiO₂/TiO₂ codeposition [68], [69]. However, light-trapping strategies that involve narrowing θ _out, compared to Lambertian light-trapping concepts, come at the cost of reducing J _sc of the device due to the optical reciprocity principle.

These results show the importance of photon management in PSCs toward efficient photon recycling. In addition, the authors’ theoretical analysis of photon management for photon recycling has assessed only planar PSCs while the optical treatment of nanostructure-based PSCs is yet to be reported. In particular, the analysis could be extended to perform a precise photon-recycling treatment in PSC designs that feature nanoparticles, nanowires, grating structures, or nanopatterned active layers.

In recent times, similar models were also demonstrated by Kirchartz et al. [49], who studied systems in the framework of ray optics. Furthermore, several groups have also investigated both randomly-textured and planar structures to quantify the impact of photon recycling for PSCs by using both simulations (ray-optics and statistical models) and experimental analyses [70], [71], [72]. Nevertheless, Abebe et al. [31] reported that conventional light-trapping structures that are based on Lambertian light scattering will not be applicable for efficient photon recycling in PSCs. For thin-film planar stacks and nanostructured solar cells, interference and light–matter interactions at the nanoscale need to be considered. Therefore, the PSC community should explore other light-trapping strategies for excellent light-trapping in PSCs. Some potential light-trapping structures are discussed in Section 5.

5.1 Lambertian light-trapping

Lambertian light-trapping is based on the complete randomization of incoming light directions after entering the absorber. By definition, an ideal Lambertian scatterer works independently of the angle of incidence of incoming light [71] resulting in an upper limit of light path enhancement to be α _ope = 4n ² for weakly absorbed light [37], [85] [Figure 5(e)], as defined above. For example, for silicon and lead halide perovskites, α _ope values of ≈50 and ≈21 are theoretically possible respectively.

However, the theoretical photon recycling models of Abebe et al. [31] and Kirchartz et al. [49] explained that Lambertian light trapping would undesirably improve light out-coupling (through optical reciprocity) and thereby actually reduce the amount of reabsorbed photons, and hence relatively reduce the carrier concentration and thus the V _OC of the device. Nevertheless, the optical benefits of Lambertian light-trapping via the increase in absorption and so J _sc outweigh these shortcomings. Consequently, a solar cell with a Lambertian light trapping [Figure 5(e)] structure can be of interest in those cases where the photon recycling effect is anyway relatively weak (e.g. when using an absorber with indirect bandgap or relatively large Stokes shift, possibly caused by high environmental temperatures).

5.2 Resonant light-trapping

Light-trapping structures that can achieve a larger redirection of path lengths are diffractive elements such as diffraction grating structures. A possible light path of emitted photons involving a diffraction grating in a solar cell is shown in Figure 10(a). In principle, the grating structure, placed on the front side, diffracts the light into at least one diffraction order m (…−2,−1,1,2…) other than the 0th order in the case where the grating period (d) is larger than the effective wavelength of light in the absorber (at normal incidence) [85]. The diffraction efficiency (the ratio between the diffracted and incident light intensity) is responsible for the amount of light energy going into a particular diffraction order m, which essentially relies on the grating structure’s geometry [22], [85]. In particular, for gratings with a periodicity d ≤ λ _r (λ _r is the wavelength of re-emitted photons), internally emitted photons can couple into a particular diffraction mode in air i.e. the 0th or the principal diffraction order. As the refractive index of the absorbing medium is higher than air, re-emitted photons can couple to the available propagating diffraction modes in the absorber [86]. Hence, the re-emitted photons can couple to a higher number of propagating diffraction modes inside the absorbing medium, compared to only one diffraction mode in air. Therefore, such gratings can act as an efficient light-trapping scheme for photon recycling. However, as the number of allowed propagating diffraction orders in air increases with an increased grating period, so does the light out-coupling efficiency. Therefore, the grating design is critical for maximizing the V _OC improvement by photon recycling.

Figure 10:

Schematic design of proposed light-trapping structure for photon recycling in PSCs.

(a) Grating coupler used as a light-trapping structure for trapping re-emitted photons through diffraction orders and waveguide modes. (b) Spherical nanoparticle array placed at the front side of the cell so that the re-emitted photons could be trapped through back scattering and waveguide modes. (c) Schematic design of angular-selective light-trapping concept. Incident light passes through the angularly selective optical filter and enters into the solar cell and randomized. In the schematic design, randomization is depicted through surface texturing (Lambertian scatterer). Re-emitted photons lying in an angle greater than the acceptance angle ( θ _a ) will be reflected back. (d) Double-array cone-like dielectric nanostructure, separated by an ideal reflector that facilities re-emitted photon trapping and reabsorption within the active layer.

Peters et al. [87] derived the maximum optical path enhancement for a diffraction-based light-trapping structure to be α _ope = 893n [85]. In addition, Eq. (2), describing Δ V o c P R can be rewritten with the optical path enhancement of diffraction-based light structure and described as follows:

Assuming n = 2.6 near the perovskite band edge [38] implies α _ope ≈ 2053 for lead halide perovskites and that Δ V o c P R of 200 mV may be obtained, with perfect absorption of re-emitted photons. Thus, this approach constitutes a diffraction-based theoretical upper limit and emphasizes the higher potential to trap the re-emitted photons via diffraction gratings, compared to Lambertian trapping concepts, albeit at the cost of restricting the acceptance angle to normal incidence, which will come at the expense of the J _sc of the device.

In addition to diffraction gratings, in literature various other concepts are suggested based on resonant structures including dielectric [88], [89], [90], [91], [92], [93], [94] as well as metallic nanoparticle arrays for light-trapping, as shown in Figure 10(b). Utilizing metallic nanoparticles allows efficient light scattering via localized surface plasmon resonances that originate from collective oscillations of free electrons in a metal [95]. The surface plasmon resonance condition can be tuned by varying the particles’ size, shape, and distribution. Although solar cell results based on plasmonic light-trapping have been reported by several research groups [96], [97], [98], [99], [100], the absolute PCE values are usually limited by dissipative losses, such as parasitic absorption within the metallic nanoparticle array [101]. Furthermore, parasitic absorption, i.e., photons that are absorbed within the material without resulting in collected charge carriers, can be detrimental for photon recycling. Recently, to overcome parasitic absorption and bring novel functionalities, dielectric Mie resonators can be used for photon recycling, where light can be confined via geometrical resonances [102]. However, the challenge when incorporating Mie resonators is to make their relatively narrowband resonances more broadband for covering the entire wavelength spectrum to overlap with the emission within the perovskite material’s bandgap. Moreover, extensive optimization of the particle size, shape, and distribution is required to adjust the desired range of wavelength spectrum. 3D photonic crystal structures are also excellent candidates for photon recycling as they combine the properties of a grating and a mirror [67]. However, apart from the recent work of Chen et al. [103], little effort has yet been made toward using 3D photonic crystals combined PSCs, where these challenge consists of implementing 3D photonic crystals with a photonic bandgap around the perovskite electronic bandgap.

5.3 Angular selective light-trapping

Another potentially useful light-trapping scheme consists of merging light scattering with an angularly selective optical filter [85, 104, 105]. The working principle of this concept is shown in Figure 10(c). Incoming light is allowed within the angular acceptance angle θ _a of the system. Once light enters the device, it scatters due to the surface’s roughness, where isotropically scattered light is confined by total internal reflection. In particular, the scattered light is trapped in case of light incident on the optical filter with an angle larger than θ _a . This mechanism achieves α _ope = 4n ² /sin θ _a with a Lambertian-scatterer and an ideal mirror at the rear side of the cell. Inserting θ _a  ∼ 1°, a higher optical path enhancement can be realized; for silicon α _ope ≈ 20,000 and lead halide perovskites α _ope ≈ 15,000 is theoretically possible. In addition, Eq. (2) of Δ V o c P R can be rewritten with optical path enhancement of angularly limiting light-trapping structure as follows:

Assuming a refractive index of 2.6 near the perovskite band edge [38], this implies that a Δ V o c P R of 240 mV may be obtained, with perfect absorption of re-emitted photons.

Angular selective light-trapping concepts restrict the escape cone angle, creating thereby a situation where the emitted photons are recycled back into the cell. Two components are required to accomplish such a system: a surface to induce a scattering mechanism and an appropriate angularly selective optical filter. Well-known possibilities to realize the first component includes the introduction of substrate roughness in the case of thin-film solar cells [106], [107] or front/backside texturing of a silicon solar cell [108], [109]. However, texturing the surface of perovskite films is not trivial [110], [111]. Since most perovskite films are deposited on glass superstrates, the best approach is to texture either the glass surface or the transparent conducting oxide (TCO) electrode or have a detached rear Lambertian reflector (like spectralon [112], or a textured Si wafer with a reflector for the relevant wavelengths on top). Recently, several works successfully explored the possibility to deposit solution-processed perovskite films on a textured TCO [113], [114]. Furthermore, vacuum-deposition approaches can also be utilized to deposit perovskite films on textured glass substrates [113]. In addition, solution-processed perovskite films on textured silicon tandems have been successfully demonstrated, paving the way for applications on textured glass [110, 115, 116]. The challenges in realizing an angularly selective light-trapping structure lie in the second component, which is finding the appropriate optical element. Peters et al. [104] suggested that a Bragg-like filter or opal (3D photonic crystal) could be used as a suitable optical filter to restrict the angular confinement of light. However, as also noted above, reducing the acceptance angle comes at the cost of reducing the J _sc and hence reducing the PCE of the device. When the acceptance angle is restricted, the angles at which light can go out of the cell are also limited due to optical reciprocity. As such, this approach may be compatible with concentrated PV designs which inherently limit the incoming light to direct radiation. In these cases, combining moderate-angle restriction with moderate concentration can offer more flexibility in system design, enabling higher PCEs. Besides, this method needs a solar tracking system on to which the solar cell is installed so that light always comes within the same acceptance angle. However, the concentration or tracking systems increases balance of the system (BOS) costs. In terms of angular restriction light-trapping structures, a low parasitic absorption probability p _a is required, i.e., at least in the range of 0–2% for efficient photon recycling [Figure 5(i)]. Achieving such low values of p _a is certainly challenging; however, devices with thicker absorber layers can usually support longer optical path lengths and offer low values of p _a [16], [31].

Based on a similar concept, Figure 10(d) illustrates the schematic design of dielectric double-array cone-like structures. In particular, the design comprises of a closely packed double array of cones set apart by a broadband band reflector shown in Figure 10(d). The radiatively emitted photons are reflected by the ideal broadband reflector. This kind of design has already been successfully implemented in GaAs solar cells to demonstrate photon recycling, inspiring its utilization in PSCs [26]. Another possible configuration to induce light-trapping is to nanopattern the active layer itself, such as by the use of cylindrical nanopatterned perovskite thin-films [28]. In this design, the light-trapping of re-emitted photons is achieved through the excitation of waveguide modes. Here, a significant portion of re-emitted photons are reabsorbed through the nanopatterned perovskite layer.

To date, there are no reports yet published that demonstrate the use of either a grating structure, a nanoparticle array or an angular-selective filter in the context of photon recycling for PSCs. However, in terms of optical path enhancement, the angular-selective light-trapping structures outclass both Lambertian and resonant light-trapping structures. Generally, these approaches aim at elevating the device-level J _SC by curtailing front-side reflection losses while concurrently increasing the optical pathlengths of the weakly absorbing in-coupled photons. Analogous to c-Si devices, numerous light management strategies have also been explored for improving the performance of perovskite-based single-junction and tandem solar cells. Typically, for light management, surface corrugations or periodic features are introduced in either the device stack’s supporting layers or within the photoactive layer itself. Examples of the former case include the use of fiber-array based anti-reflection electrodes which provide a lensing effect for realizing multiple reflection minima in the 360–750 nm wavelength interval [88], incorporating nanostructured frontside electrodes which address both anti-reflection and light-trapping aspects by allowing an increased number of higher order diffraction modes in the active layer while limiting those in the air [89] and the application of micro-pyramidal textured polydimethylsiloxane (PDMS) anti-reflection coating on an incident glass medium for improving the transmission to reflection ratio at the air/PDMS interface by means of multiple reflections at pyramid facets and refracting light at larger angles [90].