A modeling framework to quantify the intermediate layer impact in III–V//Si multijunction solar cells

Multijunction solar cells (MJSCs) are capable of converting sunlight to electricity more efficiently than single-junction solar cells. The intermediate scattering layers between the individual junctions contribute to high efficiency by impacting the generated currents, photon recycling (PR), as well as luminescent coupling (LC) in the device. The MJSC efficiency can be simulated using expressions that involve a simplified and idealized intermediate layer structure but cannot accurately reflect its actual performance. This work, however, aims to establish a systematic optical model for MJSCs with complicated intermediate layers. It begins with incorporating the LC and PR effects into the developed model, emphasizing requirements for the cut-off wavelength and long-wavelength transmission of the intermediate layer. Furthermore, a three-dimensional metallic nanocylinder array is designed as the intermediate layer to improve device performance. With the model, high-performance MJSCs can be designed and optimised by quantifying the impact of PR and LC on device parameters.


Introduction
Si-based multijunction solar cells (MJSCs) provide a path toward high-performance and low-cost photovoltaic devices. 1,2) Since Si is the dominating solar cell technology and has suitable properties to act as the bottom cell, a series of top cells with a high bandgap have been considered as partners with Si for tandem cells, including perovskite, 3) CZTS, 4) CIGS, 5) and III-V. 6) The highest efficiency of 35.9% 7) has been demonstrated with both four-terminal (4 T) 6) and two-terminal (2 T) 8) Si-based III-V MJSC devices. 4 T tandem cells are usually fabricated using simple mechanical stacking, 9) while the 2 T tandem cells need more delicate fabricating procedures. 8,10) In a 4 T configuration, the current match between subcells is not required as the III-V and Si subcells are parallelly connected. Therefore, 4 T III-V//Si tandem cells are more tolerant to the variations in bandgap energy and solar spectrum. 11,12) Furthermore, the III-V and Si subcells can be fabricated separately before integration, making optimization of individual subcells and the intermediate layer simple. The commercial textured Si bottom cells can also be applied in the 4 T III-V//Si tandem solar cells. Thus, this work focuses on the optical optimization of 4 T III-V//Si tandem cells. The high-performance MJSC device with multiple interfaces requires optimized optical management at each interface. Various approaches have been attempted for optical management. A textured polydimethylsiloxane (PDMS) anti-reflection (AR) foil on the front surface of the wafer bonded III-V//Si tandem solar cells has been applied for broadband AR, 13) and a rear diffraction grating was reported in III-V/Si MJSCs enhancing the internal light trapping of the Si bottom cell. 14) The optical optimization of the intermediate layer between the III-V and Si subcells should consider its impact on the external and internal illumination. For the external illumination, the intermediate layer enables the photon redistribution, which may impact the absorption of each individual junction.
For the III-V cell, the intermediate layer works as the back reflector to extend the length of the optical path within the device and improve the absorption. While for the Si bottom cell, the intermediate layer acts as the filter and may bring a considerable transmission loss. For the internal illumination, the intermediate layer reflects re-emitted photons from radiative recombination to enhance photon recycling (PR) of the III-V top cells, while suppressing the luminescence coupling (LC) to the Si bottom cell will lead to lower photocurrent in the Si bottom cell. It has been shown that the main impact of PR on the III-V is the reduction of dark current, which is manifested in an improved open-circuit voltage. 15,16) The suppression of LC reduces the current in the Si bottom cell. Therefore, the trade-off between PR and LC should be considered for the design of an intermediate layer. Enhanced PR has been extensively reported in III-V solar cells, and a substantial improvement in open-circuit voltage has been achieved experimentally by rear metal coating. 17,18) However, there are fewer efforts on the investigation of PR in semitransparent III-V top cells for III-V//Si MJSCs, which is challenging but necessary for further improving the MJSC performance.
Experimentally, Palladium (Pd) nanoparticles are incorporated between III-V and Si subcells to construct the 2 T tandem cells and provide both conductivity and transparency, 19) indicating the potential of using metallic nanoparticles as the intermediate layer in the tandem cell. The absorbance peak of Pd nanoparticles is normally below 300 nm, 20,21) thus there is no obvious interaction with photons within the absorption band of solar cells for optical manipulation. Besides this, simple spacers such as a low-index epoxy 22) or an air gap 23,24) with angle-dependent reflection have demonstrated enhanced PR for semitransparent III-V top cells, which worked as angle restriction filters that block emission under certain angles. In terms of simulation, a few studies considered the use of sophisticated nanostructures as the intermediate layer to boost tandem device efficiency. 25,26) These research works emphasized the potential of boosting the tandem performance by optimal design of the intermediate layer, but they focused exclusively on harvesting the external sunlight without considering radiative recombination in the top cell. Radiative recombination is essential and worth investigating in a device composed of semiconductors with high radiative recombination efficiency, such as III-V and perovskite materials. A few studies examined the radiation recombination effect using numerical 16,27) and analytical techniques. [28][29][30] However, only simplified and idealized intermediate layer structures are investigated in these works due to the challenge of using analytical expressions to calculate complex structures. As a result, the performance of complex structure intermediate layer or fine optical optimization in MJSCs cannot be accurately simulated.
In order to tackle these issues, a comprehensive model is established in this work to quantify the effects of the intermediate layer on MJSCs performance. First, a model is established combining an analytical framework with threedimensional (3D) simulation calculation, allowing the quantification of the impact of the nanostructured intermediate layer on the device performance. The analytical model is built upon the multi-layer detailed balance equation, and the effects of PR and LC on the individual junction as well as the overall device performance are analyzed. With this model, the underlying mechanism for the enhanced overall tandem device performance from optical optimization is revealed, and the design principles of the intermediate layer in GaInP/GaAs//Si are derived. Moreover, the optical response and angle dependence of the optimized intermediate layer are investigated, which strongly affect the PR and LC effects in GaInP/GaAs//Si MJSCs. Additionally, the effects of parasitic absorption loss from the optimized nanostructure on both photocurrent and LC are quantified to assess their impact. Using this model, the design of the intermediate layer is optimized over a range of geometric parameters for obtaining the desired wavelength-selective reflection and transmission properties.

Optical model
A theoretical model under ideal assumptions is presented that allows performance estimation of GaInP/GaAs//Si MJSCs with an intermediate layer. The multiple-layer detailed balance condition 31) is applied by analyzing individual junctions and incorporating the optical properties of the intermediate layer. The investigated device structure is shown in Fig. 1. In this device, the III-V and silicon subcells are under independent electrical connections in a 4 T mechanically stacked configuration, excluding the limitation imposed by current matching between III-V and Si. A Si cell with a front texture and a rear mirror with perfect reflection is assumed as the bottom cell. 1-sun AM 1.5 G illumination with the incident intensity of 1000 W m −2 is used to optimize and analyze the structure.
To simplify the calculation, the carrier collection probability of the III-V subcell is set to unity. The radiative limit is assumed (η int = 1) for GaInP and GaAs subcells, neglecting nonradiative recombination in the III-V junctions against radiative recombination. Based on common device fabricate techniques, the thickness of GaInP and GaAs junctions are set as 0.4 μm and 3 μm, respectively, which leads to the GaAs subcell being the current-limiting junction to investigate the effect of the intermediate layer. The thickness of the Si junction is set as 200 μm according to standard Si wafer structure in the PV industry. The following sections give a description of the parameters related to photocurrent and radiative recombination. The superscripts of (1), (2), and (3) in the equations indicate the GaInP, GaAs, and Si junctions, respectively.

GaInP top junction
For the GaInP top junction, a perfect AR coating with no reflection under the normally incident of the external light is assumed. And the refractive indices of GaInP and GaAs are assumed identical to each other. Therefore, single-pass absorption is considered in the GaInP junction, and the absorptance of GaInP can be calculated by is the characteristic absorption coefficient of the GaInP from Ref. 32 and the bandgap of the GaInP junction E g (1) is set as 1.9 eV.
The saturation current density of the GaInP J 0 (1) consists of: the current density of radiative recombination in GaInP with photons escaping the top surface of the cell J out (1) , the current density of radiative recombination with photons escaping out of the rear side to the GaAs junction J LC (1) , and the current loss due to nonradiative recombination J nr (1) . In the radiative limit (η int = 1), non-radiative recombination is ignored (J nr (1) = 0). The perfect AR coating at the front surface of the GaInP junction leads to no reflection within the critical angle (θ crit,air ).
The radiative recombination loss of J out (1) with the integrations over the photon energies and angle can be calculated by is the background blackbody flux at ambient temperature 33) which depends on the refractive index of the emitting medium. A (1) (E, θ) is the angular dependence of the absorptance that stems from the angle-dependent path length of photons. The angular dependency needs to be included for radiative recombination since the photons are emitted isotropically in the junction and reach the interface with different angles. When θ > θ crit,air , the light emitted to the top surface is reflected the rear and contributes to the J LC (1) . Combining this, J LC (1) can be calculated by the integrations over the photon energies and angle.
The external luminescence yield η ext (1) is the ratio of J out (1) to the total recombination loss J loss (1) , which implies the fraction of electron-hole pairs recombining to yield a photon that ultimately escapes from the front surface of the cell, 34) which becomes

GaAs middle junction
External radiation current of GaAs junction, J abs (2) , is determined by the transmittance from the GaInP junction T (1) (E) and the absorbance of GaAs A (2) When a back mirror with the reflectance of R inter is located underneath the GaAs junction, the average length of the optical path is extended. Thus, the enhanced absorptance of GaAs can be expressed as a a Since the optical absorption edge is a key property determining the optical emission spectrum in GaAs, an analytical fit 35) to experimental data is applied in this work to obtain the absorption coefficient α (2) for GaAs with E g = 1.4 eV. J out (2) and J LC (2) stem from photons emitting out of the device through the front surface and rear side of the device, respectively. In analogy to Eq. (4), and considering that the GaInP junction is transparent to luminescence from the GaAs junction, J out (2) for the GaAs junction is defined as sin cos 11 When applying the nanostructure as the intermediate layer, the internal emission from the GaAs junction impinges from the front surface of GaInP with θ > θ crit,air is reflected and contributes to J LC (2) , leading to the absorptance above the critical angle of The dark current due to radiative recombination from the rear side of the GaAs junction can be defined as where R inter (E,θ) is the energy and angle-dependent reflectance from the nanostructured intermediate layer.
The presence of the intermediate layer between GaAs and Si with enhanced reflection leads to less light escaping from the rear side of the cell, thus boosting η ext (2) . 36) Furthermore, the luminescence coupled into the GaAs junction (LC (1) ) is determined by the voltage of the GaInP junction.
=J e LC 14 The J-V characteristic of the GaAs junction can be rewritten as

Si bottom junction
The external irradiation and internal luminescence LC (2) resulting from radiative recombination in the GaAs junction contribute to the current density in the Si bottom cell.
gives the reported performance under standard illumination. 28) The photocurrent of the Si bottom cell is determined by the absorption of incident light and carrier collection probability f c , where f c of 0.978 is the effective collection probability of Si derived by comparing the measured photocurrent with the calculated result in Ref. 28.
The spectral absorption of the Si cell with textures at the front surface and a perfect mirror at the rear is given in Refs. 28 In this work, LC (2) is obtained when the III-V top cell is operated at its maximum power point.
The change in the overall efficiency of GaInP/GaAs//Si solar cells is presented as a function of λ R and R in Fig. 2(a). An increase in overall efficiency is observed when λ R is between 825 nm and 925 nm, as indicated by the dashed area. This range is defined as the acceptable window of the intermediate layer. Within this window, higher R can lead to more enhancement. The most significant improvement of the overall performance is 0.44% absolute efficiency, achieved when R = 100% and λ R = 895 nm. This result indicates that more short-wavelength reflection will improve overall efficiency if the cut-off wavelength is set appropriately. A detailed analysis of the ideal reflector (R = 1) is required to reveal the mechanism of the overall efficiency change. As shown in Figs. 2(b) and 2(c), the overall efficiency improvement peaks at λ R = 895 nm since the GaAs external luminescence yield η ext (2) improves while the Si short circuit current J sc (3) drops with increasing λ R . The improvement of III-V top cell efficiency is mainly attributed to the rise of V oc , which is 4.38% when λ R = 895 nm. The V oc increase with the increase in η ext (2) reveals the enhanced PR effect in GaAs, which suppresses the saturation current J loss (2) from 5.36 × 10 -16 to 1.03 × 10 −16 mA cm −2 . A 1.48% improvement in J sc of III-V top cells stems from the enhanced absorption of the band-edge photons owing to the extended optical path. The PR has a more substantial impact on the voltage than on the current of the III-V top cell, which is consistent with the results from Ref. 34  of J sc is mainly from a reduced transmission of the normally incident sunlight, which is 0.72 mA cm −2 , and the rest of the loss is from a reduction of LC (2) to the silicon accounting for 0.11 mA cm −2 . When λ R increases from 850 to 870 nm, the J sc of the Si bottom cell is almost constant since the external luminescence within this wavelength range will not reach the bottom cell and the majority of internal emission from the GaAs junction is at the band edge of GaAs.
The analysis reveals a trade-off between the impact of the intermediate reflector on the top and bottom cells. The improvement of the top cell requires a cut-off wavelength of the intermediate layer equal to or larger than the band edge of GaAs. But the current of the bottom cell reduces as the cutoff wavelength increases due to the transmission losses of both external luminescence and LC (2) from the III-V top cell. Therefore, the cut-off wavelength needs to be within an acceptable window (825-925 nm in this case) to gain overall performance improvement of III-V//Si MJSCs.

Design of nanostructured intermediate layer
The analytical model showed that a proper cut-off wavelength of the intermediate layer is critical for achieving performance improvement for III-V//Si solar cells. This section proposes employing 3D nanostructures as the intermediate layer and evaluates their effects on cell performance when applied between III-V and Si subcells.
The optical properties of the nanostructures are controllable via their geometrical parameters, making them an outstanding candidate in the optical manipulation of a wide wavelength range. It is well known that nanocylinder or nanowire arrays can be used as an anti-reflectance layer when placed on a high optical index substrate and illuminated from the superstrate direction. 38) In this work, the nanocylinders are located below the high-index III-V cell and embedded in a low-index matrix to form the intermediate layer between III-V and Si. The detailed structure is shown in Fig. 3(a).
The RF module of COMSOL Multiphysics is employed to calculate this proposed nanostructured intermediate layer.
The electric and magnetic fields are calculated in the frequency domain using a finite element method to solve Maxwell's equations in 3D. The optical constants for metals vary considerably 39) and have a significant impact on the design of the nanophotonic light trapping schemes. 40) In any practical device, the optical constants should be measured, so here the required optical constants are derived from Refs. 41, 42 to calculate the optical response. The extinction coefficient of GaAs is set to 0 to extract the optical properties of the intermediate layer. Periodic conditions and ports with tunable incident angle θ are applied. A range of geometrical parameters of the periodic nanostructures is used in the calculation to investigate their impacts on optical response.
The calculated reflection spectra of nanocylinders under normal incidence with fixed height (h = 100 nm), diameter to lattice constant (d/l = 0.6), and varying diameters (d) are shown in Fig. 3(b). It reveals how the geometries of the nanocylinder tune the location of resonances. By varying d from 30 nm to 125 nm, the resonance peak blue-shifts from 1100 nm to 840 nm, while the intensity of reflectance only slightly declines. The inset in Fig. 3(b) shows the simulated distribution of equivalent charges and electric potential produced on the nanocylinder (d = 30 nm, h = 100 nm) near the plasmon resonance frequency (excited by 840 nm external light). The red and blue colors represent the produced positive and negative charges, respectively, while the direction and length of the black arrows indicate the direction and magnitude of the electric field. These results illustrate the excitation of the electric dipole mode resonance at the interface of the semiconductor and the nanocylinder by the external light. The resonance wavelength shifts to a longer wavelength with increasing diameter. The relationship between the nanocylinder diameter and the resonance peak is presented in Fig. 3(c) with an exponential fitting. The resonance peak shift reflects the strength of the inter-particle near-field coupling and this is directly determined by the volume of nanoparticles and the gap between nanoparticles. As reported in Refs. 43-45, the exponential trend is used to describe the size scaling of the plasmon coupling in complex assemblies. In this system, an empirical fit gives the exponential trend between the plasmon shift and the nanocylinder diameter. The calculated reflection spectra of nanocylinders surrounded by materials with different refractive indices are presented in Fig. 3(d). The increasing refractive index n of the surrounding material causes a slightly red-shift in the wavelength of the resonance. This shift might be attributed to the increased effective refractive index of the whole intermediate layer with increasing n. 46) This phenomenon has been observed and reported previously 47,48) to effectively manipulate the resonance peak.
In addition to the reflection, the impact of the nanocylinder height h on transmittance is also investigated, as shown in Fig. 3(e). For the long-wavelength transmission from 950 nm to 1300 nm, when h = 140 nm, the maximum value of averaged long-wavelength transmittance of 99% can be achieved. The calculated distribution of electric charges and electric current is presented in the inset of Fig. 3(e) for nanocylinders with d = 30 nm, h = 120 nm, excited by 1200 nm light. Higher simulated charge density distributes at the tip of the nanocylinder away from the III-V material, indicating the location of the electric dipole mode resonance. This resonance will allow the photons to pass through the intermediate layer reaching the bottom cell. The simulation results indicate that metallic nanocylinder arrays can be designed to exhibit tunable cut-off wavelength for reflection and high long-wavelength transparency when placed below the III-V semiconductor. Therefore, the proposed metal nanostructures can serve as an effective intermediate layer for performance improvement.

Impact of the 3D intermediate layer
The numerical simulation and the analytical model were combined to identify the optimum geometric parameters of the nanocylinder, which has the dimensions of d = 30 nm, h = 140 nm, and l = 50 nm. Epoxy (n epoxy = 1.5, k = 0) is chosen as the surrounding material since its integration with the nanocylinder gives a suitable optical resonance peak and is easy to process.
The optical response of the nanostructure under the normal incident (θ = 0°) is studied firstly to estimate its impact on the photocurrents of each subcell. The reflectance and transmittance spectra of the designed nanostructures under the normal incidence are shown in Fig. 4. The most striking feature of the designed structure is its high and flat  Table I. Using the optimized longwavelength transmission of the nanostructure, the current loss of the bottom cell from the nanostructured intermediate layer ΔJ abs (3) is 1.36 mA cm −2 , which is lower than the current loss of 2.15 mA cm −2 due to the presence of an epoxy layer. The cut-off wavelength (the wavelength of the half maximum reflection) is designed to be 890 nm within the acceptable window identified in Sect. 2. The peak reflectance reaches 71%, thus leading to an increase in top cell photocurrent density J abs (2) by 0.11 mA cm −2 compared to the device without any intermediate layer. This nanostructure with wavelength-selectivity is competitive since it shows high transmission at the long wavelength combined with high reflection at the band edge of GaAs, resulting in improved photocurrent for the bottom and top cells at the same time, compared with the pure epoxy intermediate layer.
The well-optimized optical response of the nanostructured intermediate layer under normal incidence improving the photocurrent of subcells is one of the factors determining the tandem cell's overall performance. The other critical factor is related to PR and LC effects which influence both voltage and current density of subcells. As mentioned in Sect. 1.2, the internal emission from the GaAs junction impinges on the intermediate layer at all angles in contrast to the external illumination. Therefore, the optical response of the nanocylinder at oblique angles needs to be calculated. The corresponding angular-dependent transmittance and reflectance spectra are calculated by considering s-and p-polarization, as shown in Fig. 5. Within the small-angle range (0°to 20°), the line shapes of the reflection and transmission spectra are similar to those under normal incidence, while an abrupt change in reflection spectra appears, and no transmission can be observed when the incident angle becomes larger than 30°. This indicates that the nanostructured intermediate layer obtains the combined properties of plasmon resonance from   metal nanoparticles and the total internal reflection at large angles at the semiconductor/epoxy interface. This angulardependent reflection R inter (E, θ) within the wavelength range of GaAs absorptance will affect the dark current due to the radiative recombination from the rear side of GaAs junction J LC (2) , as demonstrated in Eq. (13). The achieved T inter (E, θ) determines the internal emission reaching Si LC (2) from GaAs, as expressed in Eq. (18).
The results above of the angular-dependent reflection and transmission are then applied to the analytical model established in Sect. 2, to quantify the impact of the intermediate layer on the PR effect in the GaAs junction and the LC effect in the Si junction. The calculated results are listed in Table II (2) × e qV/kT . Less radiative current escapes from the rear surface of the device, and more escapes through the front side of the GaAs junction resulting in a higher η ext (2) . 55 mV and 30 mV improvements in V oc (2) are achieved by this enhanced PR effect for epoxy and nanostructure intermediate layers, respectively. However, a huge difference between J LC (2) × e qV/kT and LC (2) is observed for the nanostructured intermediate layer. This indicates that most of the radiative current escaping from the rear surface of the GaAs junction does not reach the Si bottom cell, which is instead consumed by parasitic absorption in the metal nanocylinders and converted to heat. Even though high transmission from the designed nanostructured intermediate layer is achieved under normal irradiation, the absorption loss in the structure under oblique incidence significantly weakens the LC effect. This suggests that it is necessary to consider the absorption loss of the structure under non-normal incidence to maximize the benefits for the Si bottom cell from the luminescence of the top cell when designing the intermediate layer structure in the future.
The above discussion describes the effect of the selective reflector on the light absorption of the subcell and the effect on PR and LC.   from the top cell to the Si cell. Lower absorption loss is possible by using alternative nanomaterials such as highindex dielectric or semiconductor nanostructures to produce light resonance 49) which is driven by displacement currents rather than actual currents. 50,51) According to the calculation in this work, 0.51% overall device performance improvement can be achieved despite the parasitic loss in the metallic nanostructure.

Conclusion
In