1. Introduction

Thin ($\lesssim$1 $\mu$m) and ultra-thin ($\lesssim$100 nm) solar cells are emerging device concepts for the photovoltaics community, with a single junction, 205 nm (active region) GaAs solar cell having demonstrated an efficiency of 19.9$\%$ in 2019 [1]. Advantages of these devices are the cost reductions corresponding to reduced materials usage and an increased fabrication throughput [2], the increased efficiency of carrier extraction [3] and the increased open circuit voltage as a result of a reduced bulk recombination [4,5]. Additionally, the low weight, radiation tolerance [6] and flexibility [7] of ultra-thin solar cells make them attractive for extraterrestrial applications.

Despite these enabling characteristics, a well-known challenge in ever-thinner photovoltaics is a low light absorption, particularly near the band edge, as a result of reduced optical path length within the active layer [8]. This effect demands the introduction of light management techniques [9] in the device architecture to enhance the absorption of incident illumination, whether by the design of an anti-reflection coating (ARC) to suppress reflection losses, the introduction of a rear surface mirror to double the optical path length, or the introduction of scattering structures to couple light to supported optical modes. Ultimately, a synergy of all these strategies is required for efficient light management in devices with reduced length-scales, turning their optimisation into a design step of paramount importance, particularly in terms of the ARC and the scattering structures. Whilst the efficiency of the former depends mainly on its thickness and refractive index [10] (and multilayered ARCs can also be implemented), the scattering structures commonly form a periodic arrangement and so their efficiency is highly dependent on the dimensions of the unit cell that defines this periodicity (i.e. the pitch). Other relevant parameters for these structures are their geometry [11], thickness and component materials.

Considering that the efficiency of light management depends on the interplay of multiple design parameters which are also wavelength dependent, their optimisation commonly focuses on maximising a figure of merit that encompasses all relevant phenomena in terms of absorption enhancement. This is usually the ideal short circuit current (${Jsc}$) of the cell (or a related figure such as integrated absorption or ultimate efficiency [12]), assuming that all absorbed photons generate charge carriers that are then extracted at the device terminals. To search the broad parameter space for the optimal device design, the optimisation process commonly employed in other studies is to select ranges for some key parameters and perform extensive full-field simulations to scan for the highest figure of merit [1,11,13–23].

While this optimisation approach can be effective, it does not fully explore the underlying optical phenomena that make certain device parameters optimal and it can overlook the characteristics of the optical modes responsible for the absorption enhancement. The phenomena behind absorption enhancement can be analysed by examining the field distribution throughout the stack at certain wavelengths [14,23–25], but this analysis provides limited insight. Another option is matching the observed absorption peaks to calculated optical modes [1,5,26], but these are sometimes found by considering simplified solar cell models [20,27], neglecting layers which can have an impact on the availability of the optical modes. Ultimately, even in the cases where the optical modes are analysed, this tends to justify the design of a device rather than guide it based on knowledge of the underlying parametric trends.

In contrast, this work proposes a framework for light management optimisation, which harnesses the optical modes and is developed after gaining fundamental knowledge of the impact that different device parameters have on light absorption enhancement in realistic devices. We limit our study in this paper to waveguide modes and focus on 80 nm GaAs solar cells, given the interest in ultra-thin photovoltaic devices and their potential for space applications, which has been previously demonstrated for a device with this absorber thickness [6]. The developed approach, however, can be applied to different material systems and device geometries. The parametric trends of light absorption are studied by implementing a transfer matrix method [28] (TMM) for waveguide analysis and interfacing it with rigorous coupled-wave analysis (RCWA) full-field simulations [29]. The light management optimisation framework we build with these tools yields an equivalent device design to the one obtained by exhaustively simulating different parameter combinations looking for the best figure of merit. However, being guided by the underlying parametric trends of light absorption, our framework reduces the parameter space and inherently explains the phenomena behind the absorption enhancement in the optimal design. Additionally, our framework is applicable to multiple device architectures and has the potential to further our understanding of light management in reduced length-scales. We show this potential by using our framework to compare the light management performance of photonic crystal and quasi-random gratings in the ultra-thin devices of interest in this work, finding a superior performance for the former.

2. Results and discussion

2.1 General simulated structure

The simulated structure throughout this paper, based on a realistic architecture, is shown in Fig. 1. The active layer consists of two 40 nm GaAs layers, n and p doped, which are considered as a single 80 nm GaAs layer. InGaP and InAlP layers, both 20 nm thick, are introduced for passivation. The top of the structure consists of a single layer SiO$_2$ ARC. The scattering layer, introduced below the epitaxial layers, is formed by SiO$_2$ and Al with a square unit cell. Finally, the substrate is an Al mirror. It should be noted that the simulated structure is only the optical component of the photovoltaic device. The electrical contacts would be on a different length-scale and are not considered, since they would form a grid and leave areas of several square millimetres for the optical component.

 

Fig. 1. General solar cell structure studied in this work. The thicknesses of the ARC and the scattering layer, as well as the pitch and geometry of the unit cell in the latter, were chosen arbitrarily for illustrative purposes.

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The device parameters that are varied and studied in this work are the thicknesses of the ARC and the scattering layer, as well as the spatial distribution, pitch and volume ratio of the materials within the scattering layer’s unit cell. In this device of interest, SiO$_2$ and Al were chosen as dielectric and metal, respectively, due to their appropriate index characteristics, well-characterised optical properties, low cost and availability.

The optical constants of all the epitaxial layers were measured with ellipsometry [17]. No significant difference in the optical constants was observed between the n and p doped GaAs. Those of Al were modelled following a Lorentz-Drude model [30], and SiO$_2$ was considered as a lossless medium with $n$ = 1.46 for all studied wavelengths. All reported full-field simulations consider normally-incident light from air, and their absorption results are reported as a fraction of the incident power, considering an average of s and p polarised light (for normal incidence, this corresponds to the electric field polarised along the y and x axes, respectively).

2.2 Applicability of the implemented transfer matrix method

Waveguide modes are described by propagation constants, which correspond to the wavevector components of the fields in the modes along the direction of propagation. To find these propagation constants for our ultra-thin solar cells of interest, we implement a TMM for the waveguide analysis of multi-layered planar structures (see Supplement 1). In our TMM implementation the scattering layer is modelled using effective medium theory, having a refractive index equal to the average of those of its component materials, weighted by their volume ratio within its unit cell.

To exemplify the applicability of the TMM implementation, we use it for a device with scattering layer and ARC thicknesses equal to 100 nm. For the scattering layer we arbitrarily select a 2D photonic crystal grating with a square unit cell, which consists of a square SiO$_2$ feature surrounded by Al (Fig. 2(a)). We label this general photonic crystal geometry as PC. For our particular example, the size of the square dielectric feature is such that the dielectric coverage (volume ratio of dielectric in the unit cell) is 0.25.

 

Fig. 2. Applicability of the TMM implementation. Results correspond to a wavelength of 850 nm, for a structure as shown in Fig. 1 having both scattering layer and ARC thicknesses = 100 nm. (a) Unit cell of the grating simulated in this study. (b) Transverse field distribution across the z axis for the modes in Table 1. (c) GaAs absorption as a function of the pitch of the grating, calculated with RCWA. Dotted lines correspond to pitches where Eq. (1) predicts different sets of optical states coupling to the bound modes in b. (d) Reciprocal space representation of the optical states in the simulated device, showing the labelling system followed in this work.

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Table 1. Bound waveguide modes for the example structure

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For a wavelength of 850 nm, two bound modes [31] (TE0 and TM0) are found for this device with the TMM implementation, one for transverse electric (TE) and one for transverse magnetic (TM) polarisation. Their propagation constants (of the form $\gamma_{guided} = \alpha + \mathrm {i}\beta$) are shown in Table 1, with the field profiles plotted in Fig. 2(b). In order to evaluate the potential of these modes for light absorption enhancement, we calculate their confinement factor, or the fraction of the power (along the direction of propagation) in each mode that is confined within the active layer (Table 1). The confinement factor is significantly higher for the TE0 mode than for the TM0 one. Similar results where TE modes show better absorption enhancement have been reported previously [3]. It is important to highlight that this confinement factor can only be an indication of the absorption enhancement potential of a bound mode, as not all incident illumination will couple to a mode and the scattering layer can also outcouple light from it.

We use an RCWA implementation to determine if the TMM results agree with those of full-field simulations. RCWA (sometimes referred to as Fourier modal method [32]) describes the interaction of incident light with the photovoltaic devices of interest by treating the electromagnetic fields in the modelled stack as Fourier expansions and solving them according to Maxwell’s equations. Knowledge of the fields then allows to calculate the Poynting vector and obtain the transmitted and reflected power, as well as that absorbed in different layers of the device. We follow this process to determine the absorption in the GaAs layer of the example structure as a function of the pitch of the grating (Fig. 2(c)). The relationship between the pitch and the allowed in-plane wavevectors for diffracted light is given by the following equation (assuming normal light incidence):

The full-field simulations in Fig. 2(c) show absorption peaks at pitches where Eq. (1) predicts different sets of optical states coupling to the bound modes shown in Table 1, by yielding a $k_ {xy}$ value equal to the real part ($\alpha$) of their propagation constants. These predicted resonant pitches are labelled in Fig. 2(c) according to the set of optical states responsible for the given resonance. Also, in agreement with the confinement factor calculations, the TE0 resonances show a stronger absorption enhancement, with only one weak resonance being distinguished for the TM0 mode. The relevance of the TE0 mode for the structure under consideration was confirmed by studying different wavelengths and finding a less significant absorption contribution from the TM0 mode, to the point where for some wavelengths no resonance was distinguishable. For a comparison between the RCWA resonant pitches in Fig. 2(c) and those predicted with the TMM implementation, see Supplement 1, section 2.

These results highlight the potential of the TMM implementation to predict resonant pitches for waveguide mode coupling in a nanostructured ultra-thin solar cell, and guide the selection of the pitch accordingly to enable a desired coupling event at a certain set of optical states. Complementing this technique with RCWA then allows to determine the absorption in the active layer when a certain coupling event occurs under incident illumination, by considering all the optical phenomena happening in parallel with this predicted coupling at other available optical states. In fact, the absorption in Fig. 2(c) for different pitches, regardless of whether or not a coupling event is observed, is a result of various processes happening at different optical states. These processes involve first pass and specular reflection (which correspond to the zeroth optical state, or $m_ {1}$ = $m_ {2}$ = 0 in Eq. (1)), as well as different diffraction events at other optical states, the nature of which depends on the pitch. In general, increasing this pitch results in more optical states being available for diffraction, which initially lie outside the escape cone and can contribute positively to the overall absorption (even when not coupled to a bound mode). Eventually, however, increasing the pitch leads to leakage losses when optical states are introduced within the escape cone, which for the example in Fig. 2(c) occurs at OS1 for $\Lambda >$ 850 nm. Ultimately, using the TMM implementation in tandem with full-field calculations enables a unique opportunity to gain complete and fundamental insight into how bound mode coupling and absorption enhancement are affected by different device parameters.

Finally, it should be mentioned that effective medium approximations can have limitations related to the length-scale of the studied features in comparison to the wavelengths under consideration [33], as well as the presence of certain resonances [34]. The approximation that we follow in our TMM implementation to describe the scattering layer, however, proved to be applicable to obtain the modal structure of our devices of interest. Such applicability is evidenced by the good match (within a relative difference of $\pm 3\%$) between our predicted resonant pitches and those found with full-field calculations (where no approximations are involved), shown in this and the following section for different wavelengths and dielectric coverages.

2.3 Parametric dependence of bound mode coupling

We now study the parametric dependence of bound mode coupling and absorption enhancement. For simplicity, we keep the PC grating used previously. Given its improved absorption enhancement contribution, we only focus on the TE0 mode for any given structure. We note that some wavelengths, particularly those closer to the blue end of the spectrum, can have other bound modes (e.g. TE1). However, the confinement factor of these higher-order modes is lower than that of TE0 [35] and these wavelengths have less significant absorption contributions from bound mode coupling, which justifies limiting our study to the TE0 mode.

The device parameters we aim to study are the dielectric coverage of the scattering layer, the thickness of the grating and that of the ARC. Studying their impact on bound mode coupling sets an important requirement for the pitch, which must guarantee this coupling. Considering that the TE0 dispersion depends on all the device parameters of interest, in this section we calculate the TE0 propagation constant for each different combination of parameters with our TMM implementation, and then determine the pitch that will couple OS1 to this bound mode. This predicted resonant pitch is then introduced to the RCWA simulations to calculate the absorption in the full device. To confirm the expected coupling, in the RCWA simulations we look for a GaAs absorption peak within $\pm 3\%$ of the predicted resonant pitch and report the data at the peak. Note that we choose to study the coupling of OS1 to the bound mode of interest given that for this condition all diffraction occurs at this set of optical states, since higher sets would correspond to wavevectors beyond the maximum in-plane wavevector for light in the device ($k_ {xy} > n_ {max}k_ {0}$, with $k_ {0}$ being the free-space wavevector and $n_ {max}$ the maximum refractive index in the structure for a given wavelength). Meeting this condition then allows us to directly study the parametric dependence of bound mode coupling by looking at the diffracted absorption component, without the influence of other effects that could take place simultaneously at other optical states.

The first parameter we analyse is the dielectric coverage of the scattering layer, which is changed by varying the size of the dielectric square feature in the unit cell of the PC grating whilst keeping its aspect ratio constant (Fig. 3(a)). We fix the grating and ARC thicknesses to 100 nm, and simulate the absorption in the active layer for different dielectric coverages.

 

Fig. 3. Effect of the grating’s dielectric coverage on GaAs absorption and bound mode (TE0) coupling, for a device as in Fig. 1 with grating and ARC thicknesses = 100 nm. (a) Representative images (not to scale) of the simulated 2D photonic crystal grating (PC). (b) Total absorption in the GaAs layer as a function of the dielectric coverage of the grating for wavelengths between 600 and 850 nm. All measurements were individually checked for convergence and correspond to pitches that ensure coupling OS1 to the TE0 mode at the corresponding wavelengths given a certain dielectric coverage. (c) Contribution to the absorption in b from first pass and specularly reflected light. (d) Contribution to the absorption in b from OS1. In b-d, no data is shown for 0.05 dielectric coverage at 600 nm given that no evidence of coupling to the TE0 mode was found with full-field calculations (i.e. no GaAs absorption peak was found within $\pm 3\%$ of the predicted resonant pitch).

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Figure 3(b-d) show the results of this analysis for six different wavelengths: 600, 650, 700, 750, 800 and 850 nm. The pitches of all the simulated devices are included in Supplement 1, section 3 (these pitches are in the range from 175 to 315 nm, and have a variation below 20 nm in the full range of dielectric coverage at a given wavelength). The limits of 0 and 1 dielectric coverage correspond, respectively, to cases where the scattering layer becomes a planar Al or SiO$_2$ slab. The total absorption results in Fig. 3(b) show that the introduction of a grating has a significant and positive impact for wavelengths near the GaAs band edge ($\sim$870 nm). For shorter wavelengths ($<$700 nm), however, this positive effect is reduced, to the point where for 600 nm having just a rear Al mirror (i.e. a planar structure) is better than introducing a scattering layer.

These trends can be better understood by independently looking at the absorption contribution from two different phenomena: first pass and specular reflection in Fig. 3(c) (which we will call the specular component and occurs at the zeroth optical state), and TE0 coupling in Fig. 3(d) (which we will call the TE0 component). The TE0 absorption component is clearly higher for longer wavelengths ($>$700 nm) which are less readily absorbed in the device, and is what drives the trends of the total absorption for near band edge wavelengths. At 700 nm the absorption contributions from the specular and TE0 components are comparable, and shorter wavelengths have a more significant contribution from specular absorption. These wavelengths are more readily absorbed and have reduced absorption contributions from bound mode coupling.

As for the dielectric coverage dependence, this is wavelength dependent and different for the TE0 and the specular components. For the TE0 component, near band edge wavelengths are better absorbed in the active layer with higher dielectric coverage. We envision this dielectric coverage dependence to encompass a complex interplay of different effects such as index contrast for different wavelengths, the Fourier spectrum of the grating, the boundaries available for scattering or changes in the relative absorption of the scattering layer compared to that of the active layer. As for the specular absorption component, this is in all cases reduced when introducing a scattering layer, with the maximum absorption lying at either ends of the plot. The preferred dielectric coverage (0 or 1) is defined by the index contrast in the structure. Wavelengths beyond 700 nm show the highest absorption of the specular component with a dielectric coverage of 1, whereas from 700 nm and below a dielectric coverage of 0 gives the best absorption. The strong dependence on the specular component for shorter wavelengths explains the shift in the total absorption maximum towards a planar structure.

These results highlight the existence of two different wavelength regimes, one below 700 nm where the specular component is most significant, and one starting at 700 nm and beyond, driven by the absorption in the bound mode. We expect the presence and extent of these regimes to depend on the thickness and absorption coefficient of the active layer. A thicker GaAs layer would be more absorptive on a single pass and thus the wavelength where specular contributions become less significant should extend to longer wavelengths. Equally, materials with lower absorption coefficients [36] would see the relevance of diffracted contributions extending to shorter wavelengths.

Similar studies to the one in Fig. 3 were carried out to determine the impact that the thicknesses of the grating and the ARC have on bound mode coupling and absorption enhancement (Supplement 1, section 4). It was found that increasing the thickness of the grating had a detrimental consequence on the specular absorption component in the active layer. The TE0 absorption component was initially enhanced upon increasing the thickness of the grating as a result of a more efficient scattering. Eventually, this component reached a plateau and decreased given that the bound mode propagates throughout the whole structure, and at high grating thicknesses the field enhancement within the grating itself becomes significant and preferentially increases its parasitic absorption. Ultimately, a beneficial effect of increasing the grating’s thickness on the total absorption in the active layer requires the TE0 absorption enhancement exceeding the reduction in specular absorption. This demands a dielectric coverage that ensures good diffraction and TE0 coupling, but a good field confinement within the active layer and a low absorption in the grating compared to the one in the active layer are also important. Finally, the studies on the thickness of the ARC confirmed that the effect of this parameter is a variation in the amount of reflected and, consequently, incoupled power. In this case both the specular and TE0 absorption components followed the same trends, increasing when less power was reflected.

2.4 Harnessing the parametric dependence for multiple wavelength absorption enhancement

We now build on the previous results to study the enhancement of light absorption in a broad spectral range, so that our framework for light management optimisation can be built. In this and the following section, we use the TMM implementation to calculate the pitch required for the desired bound mode coupling, and directly use it as input in the RCWA absorption calculations.

To begin, we study the full spectrum (300 - 900 nm) absorption enhancement consequences of varying the dielectric coverage of the PC grating when its pitch couples OS1 to TE0 at two different wavelengths: 850 nm (OS1@850TE0, Fig. 4(a)) and 700 nm (OS1@700TE0, Fig. 4(b)). We study dielectric coverages in the range from 0.3 to 0.8, and fix the thicknesses of the grating and the ARC at 100 nm. In all cases, the curves below 500 nm are very similar due to a strong absorption in the specular component. Varying the dielectric coverage causes the peak around 500 nm to shift slightly, given that this is a result of thin-film effects which depend on the index contrast in the structure. On average, however, there is no pronounced difference in absorption at these short wavelengths in the studied devices. For both OS1@700TE0 and OS1@850TE0, intermediate wavelengths (between 500 and 700 nm) show an increased absorption as the dielectric coverage is lowered. This spectral region has a significant absorption contribution from the specular component, which at these wavelengths was shown to be higher at low dielectric coverages. However, the presence of absorption peaks at intermediate wavelengths evidences that diffraction contributions are still in play, as these correspond to higher optical states than OS1 coupling to the bound mode at the corresponding wavelengths.

 

Fig. 4. Full spectrum (300 - 900 nm) performance of the PC grating as a function of its dielectric coverage and the wavelength coupled to TE0 at OS1, considering a device as the one in Fig. 1 with grating and ARC thicknesses = 100 nm. (a) GaAs absorption when OS1 is coupled to the TE0 mode at 850 nm (OS1@850TE0), for gratings with dielectric coverages ranging from 0.3 to 0.8. (b) GaAs absorption when OS1 is coupled to the TE0 mode at 700 nm (OS1@700TE0), for gratings with dielectric coverages ranging from 0.3 to 0.8. (c) Ideal short-circuit current (${Jsc}$) as a function of the dielectric coverage of the PC grating, for four different wavelengths (700, 750, 800 and 850 nm) coupled to TE0 at OS1.

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For OS1@850TE0 (Fig. 4(a)), there is a clear shift in the absorption trends beyond 700 nm for different dielectric coverages. In this spectral region, the significant contribution from diffraction and TE0 coupling (Fig. 3(d)) results in higher dielectric coverages becoming more efficient at enhancing light absorption. For OS1@700TE0 (Fig. 4(b)), however, the absorption trends beyond 700 nm are similar regardless of the dielectric coverage. In this case no TE0 coupling is allowed beyond 700 nm, and even when most wavelengths in such spectral range do have absorption contributions from diffraction at OS1 outside the escape cone, the weak absorption enhancement compared to a planar structure evidences that these are not very significant.

It must be mentioned that, in Fig. 3(b), the introduction of a grating showed a reduction in the GaAs absorption of 600 nm light compared to a planar device. Those results corresponded to OS1@600TE0. In contrast, for OS1@850TE0 and OS1@700TE0 (Fig. 4(a) and (b)), the absorption at 600 nm is enhanced over the planar case for some of the lowest dielectric coverages. This difference is a consequence of variations in the available optical states for diffracted light. For OS1@600TE0, the only available optical states for diffracted waves at a wavelength of 600 nm are the zeroth one and OS1. However, for OS1@700TE0 and OS1@850TE0, all dielectric coverages have OS2 available for diffraction at 600 nm together with OS1. Furthermore, for OS1@850TE0, the spectral region where OS4 becomes available lies around 600 nm for all dielectric coverages. Certainly, none of the devices in Fig. 4(a) or (b) have a TE0 resonance at 600 nm, but nonetheless all the available optical states at this wavelength lie outside the escape cone and, together, they contribute positively to the overall absorption.

We now evaluate how varying the dielectric coverage of the PC grating, as well as the wavelength coupled to TE0 at OS1, impact the performance of the solar cell. We use the ${Jsc}$ for this purpose, which can be calculated according to the following expression:

So far, in terms of optimal light management we find the importance of introducing a bound mode resonance near the band edge, and balancing the effects of specular and diffracted components present in different wavelength regimes by selecting an appropriate dielectric coverage. However, the absorption in the device with the best figure of merit (Fig. 4(a), 0.5 dielectric coverage) is reduced in the spectral region around 750 nm. In this device, the next TE0 resonance after the one at 850 nm appears slightly below 700 nm and corresponds to OS2. Thus, a strategy towards improving the performance of the solar cell would be to introduce more TE0 resonances in this region of low absorption. However, these resonances cannot be arbitrarily introduced, given that the dispersion of the TE0 mode is fixed, as is the relative separation in reciprocal space between different sets of optical states. As a result, to keep a resonance near the band edge whilst having more resonances around 750 nm, a different set of optical states should be coupled to TE0 at 850 nm. This strategy will shift the spectral position of other available resonances at higher optical states.

We calculate the full spectrum absorption of devices with the PC grating, having a dielectric coverage of 0.5 and different sets of optical states coupled to TE0 at 850 nm: OS2, OS4, OS5, OS8 and OS9 (Fig. 5(a-e)). All the studied devices have grating and ARC thicknesses of 100 nm, and different pitches (included in Supplement 1, section 6) that provide the desired TE0 coupling. These pitches are in the range from 436 to 925 nm, with larger pitches being required to couple higher sets of optical states to the bound mode near the band edge. The ${Jsc}$ of the studied devices is included in Fig. 5(f). In this case the shapes of the absorption curves differ strongly amongst themselves due to differences in the available optical states and the corresponding resonances. The results show that going to higher sets of optical states coupled to the bound mode near the band edge results in an enhanced device performance up to OS4, which achieves a ${Jsc}$ of 16.07 mA/cm$^2$. The performance enhancement is brought about by an increased number of available TE0 resonances and an increased number of optical states available for diffraction outside the escape cone, which even when not coupled to TE0 result in an overall absorption enhancement.

 

Fig. 5. Performance of ultra-thin solar cells as shown in Fig. 1 (ARC and grating thicknesses = 100 nm, dielectric coverage = 0.5) with a PC grating having different sets of optical states coupled to the TE0 mode at 850 nm. (a-e) Absorption as a function of wavelength for devices with (a) OS2, (b) OS4, (c) OS5, (d) OS8 and (e) OS9 coupled to TE0 at 850 nm. Areas labelled as “higher states within the escape cone” correspond to spectral regions where at least OS1, OS2 and OS4 lie within the escape cone, but higher states may lie within it as well for certain wavelengths. (f) Ideal short circuit current for the simulated devices in a-e. The short circuit current for OS1 coupled to TE0 at 850 nm corresponds to the absorption spectrum in Fig. 4(a) (0.5 dielectric coverage).

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For OS4@850TE0 (Fig. 5(b)), the best device within the studied range, there is an abrupt drop in absorption below 616 nm. Below this wavelength, OS1 falls within the escape cone with air. However, the resultant drop in absorption is not very pronounced as it occurs at a wavelength where there is a significant absorption contribution from the specular component. In contrast, for OS5@850TE0 (Fig. 5(c)), the point where OS1 enters the escape cone appears at 689 nm. At this wavelength the specular contribution is not as significant as for 616 nm and thus the drop in absorption is more pronounced. Consequently, the ${Jsc}$ of the device decreases compared to the previous case. For OS8@850TE0 and OS9@850TE0 (Fig. 5(d) and (e)), all wavelengths below the band edge have OS1 within the escape cone, and so their absorption is significantly reduced. For these devices, the wavelength where OS2 also falls within the escape cone starts approaching the regime where the specular component is less significant ($\lambda \geq$ 700 nm). Having more optical states within the escape cone, these devices then have a further reduced ${Jsc}$. It must be noted that the optimal set of optical states to couple to the bound mode at 850 nm is the one which maximises the ${Jsc}$, but this does not imply that such condition is optimal for the absorption of all individual wavelengths. Also, varying the dielectric coverage could impact the optimal set of optical states to couple to the bound mode near the band edge, given that changing this parameter comes with associated variations in specular and diffracted contributions to absorption.

Overall, the results in Fig. 5 evidence another requirement for optimal light management: having as many optical states available for diffraction whilst ensuring that no diffraction within the escape cone is allowed for wavelengths which rely on the diffracted absorption component. Together with the premise of keeping a resonance near the band edge, these two simple requirements form guidelines for the optimal distribution of optical states and, in turn, the selection of the appropriate pitch. Meeting these guidelines depends on the dielectric coverage and the thickness of the ARC and the grating, as these all have an impact on the dispersion of the bound mode. However, these three parameters must also be tailored individually so that i) the absorption contributions from specular and diffracted components are balanced to preserve a good performance in the long and intermediate wavelength regimes (varying dielectric coverage), ii) the amount of incoupled photons is maximised (varying ARC thickness) and iii) incoupled light is efficiently diffracted to the active layer without excessive enhancement of parasitic absorption in the grating (varying scattering layer thickness). The light management framework we propose is to optimise each of these three phenomena individually and sequentially, but guided by the dispersion of the bound mode to ensure that the pitch constantly provides the optimal distribution of optical states.

To apply this framework to the device of interest, we build on our previous results and start from a PC grating with a dielectric coverage of 0.5. Then, fixing the grating thickness at 100 nm, we calculate the GaAs absorption and corresponding ${Jsc}$ in devices with ARC thicknesses ranging from 20 to 200 nm (Fig. 6(a)). The pitch of every simulated device provides OS4@850TE0, as this yields the best distribution of optical states. The optimal ARC thickness is found to be 80 nm. We note that this optimal value is also found when simulating the absorption in the active layer of a planar structure for different ARC thicknesses. Next, having selected the thickness of the ARC as 80 nm and keeping the dielectric coverage at 0.5, we calculate both the absorption and ${Jsc}$ in the active layer of devices with grating thicknesses ranging from 80 to 260 nm (Fig. 6(b)). Once again, in all cases the pitch is selected to provide OS4@850TE0. The results of this step indicate that the optimal grating thickness is 140 nm.

 

Fig. 6. Optimisation of the grating and ARC thicknesses in the ultra-thin GaAs solar cell of interest (Fig. 1) according to our framework, considering a PC grating with a dielectric coverage of 0.5. Every calculated figure of merit corresponds to a device with a pitch that allows OS4 to couple to TE0 at 850 nm (OS4@850TE0). (a) ${Jsc}$ as a function of ARC thickness. In all cases the thickness of the grating is 100 nm. (b) ${Jsc}$ as a function of grating thickness. In all cases the thickness of the ARC is 80 nm. (c) Reflected, transmitted and absorbed power in all the layers of the optimised device (ARC thickness = 80 nm, grating thickness = 140 nm, pitch = 618 nm and ${Jsc}$ = 16.56 mA/cm$^2$).

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At this point the optimisation process is finished considering the parameter space established. The optimal device with a PC grating has a ${Jsc}$ of 16.56 mA/cm$^2$, with a dielectric coverage of 0.5, a grating thickness of 140 nm, an ARC thickness of 80 nm and a pitch of 618 nm. Figure 6(c) shows the power flow (reflected, transmitted and absorbed power in all layers of the simulated device) through the optimised structure. Compared to the absorption of the device in Fig. 5(b), which also had OS4@850TE0 but in which the thicknesses of the grating and the ARC were not optimised, in this case the GaAs absorption curve shows clearer peaks in the long wavelength regime where diffraction is essential. The more pronounced modulation indicates a better coupling to available resonances at OS4 and higher states. From the power flow shown in Fig. 6(c), it can also be seen that a main loss mechanism in the optimised device is the parasitic absorption in the grating, particularly at longer wavelengths. At short wavelengths there is also significant absorption in the InGaP passivation layer, but, in practice, some of the photogenerated charge carriers in this layer might also be extracted. It should also be mentioned that, in our guided optimisation process, the changes in the ARC and grating thicknesses did not significantly change the TE0 dispersion. As a result, having OS4@850TE0 still achieved the best distribution of optical states. Had the TE0 dispersion changed significantly, it is possible that the pitch which resulted in OS4 coupling to the bound mode near the band edge would have resulted in the introduction of diffraction within the escape cone at wavelengths with significant absorption contributions from diffraction. The opposite effect is also possible.

To conclude, we compare the performance of the device optimised with our guided design framework to the one obtained by performing full-field simulations across the parameter space and selecting the combination that yields the best figure of merit. To carry out this optimisation, we begin by fixing the ARC thickness at 80 nm, as this optimal value is found by just maximising the GaAs absorption in a planar device. We then simulate, using RCWA, different combinations of the pitch (400 - 750 nm in 50 nm intervals), grating thickness (100 - 200 nm in 20 nm intervals) and dielectric coverage (0.3 - 0.7 in intervals of 0.1) of the PC grating. The highest ${Jsc}$, 16.60 mA/cm$^2$, is obtained for a device with a pitch of 600 nm, a grating thickness of 140 nm and a dielectric coverage of 0.5. The correlation between the optimal parameters found with both optimisation procedures supports the applicability of our framework and the validity of the design guidelines it is based on (note that the difference between the optimal pitches of both optimisation procedures is comparable to the one between the resonant pitches found with RCWA and with our TMM implementation). However, our framework proves advantageous by being applicable to different device architectures, extending our knowledge on the parametric trends of light management, having a reduced parameter space and inherently explaining the phenomena behind the absorption enhancement in the optimal design.

Finally, as an indication of the potential performance of the optimised device, an efficiency of 13.4% could be reached assuming an open circuit voltage of 1.022 V and a fill factor of 79.2% (both of which have been achieved in a device with a 205 nm GaAs absorber [1]). This efficiency could be increased further given the potential collection of photogenerated charge carriers from the InGaP layer, although some shading losses from the electrical contacts are also expected.

2.5 Photonic crystal vs quasi-random design

We now attempt to control the distribution of scattered light to further improve light management in our device of interest. We aim to scatter it preferentially at the spatial frequency range that contains the TE0 propagation constants of wavelengths which rely the most on bound mode coupling ($\lambda \geq$ 700 nm). We call this spatial frequency range the “diffraction regime”. Building on our previous results, to find the diffraction regime we define the dielectric coverage as 0.5, the ARC thickness as 80 nm and the grating thickness as 140 nm. The lower and upper boundaries of the diffraction regime are then found to be 20.32 $\mu$m$^{-1}$ and 27.66 $\mu$m$^{-1}$, as these correspond, respectively, to the real part ($\alpha$) of the propagation constant of the TE0 mode at 850 and 700 nm.

The distribution of scattered light is related to the Fourier spectrum of the scattering layer [19,21,37]. Consequently, performing the desired localisation of diffracted light is a matter of fitting optical states within the diffraction regime and maximising the Fourier amplitude of the grating at these states. Since different sets of optical states can be introduced within this region by varying the pitch, we study this systematically by keeping the premise we followed in our optimisation framework: always introducing a TE0 resonance near the band edge. As a result, we choose to design six different gratings having OS4, OS5, OS8, OS10, OS17 and OS20 coupled to the bound mode at 850 nm. This selection fixes the required pitch and the distribution of optical states for all gratings, and the only step left to complete the design process is to ensure that the unit cells of these gratings have their Fourier amplitudes localised at the states within the diffraction regime, as well as the required dielectric coverage.

Based on existing literature [21], we follow a stochastic unit cell design algorithm to obtain quasi-random (QR) gratings that satisfy these conditions (see Supplement 1, section 7). The unit cells of the designed quasi-random gratings are included in Fig. 7(a-f), labelled according to the set of optical states that couples to TE0 at 850 nm (QR4, QR5, QR8, QR10, QR17 and QR20). Also included in this figure are reciprocal space diagrams of each unit cell, where the radius of each optical state is proportional to the absolute value of the Fourier amplitude of the unit cell at that state. In all cases, the radii of the optical states within the boundaries of the diffraction regime highlight the effectiveness of the Fourier amplitude localisation algorithm. As higher optical states couple to TE0 at 850 nm, the Fourier amplitude of the gratings is localised at a higher number of optical states within the diffraction regime, which correlates with the gratings becoming more random.

 

Fig. 7. Quasi-random scattering structures and their absorption enhancement potential. (a-f) Unit cells of the designed quasi-random gratings, which have OS4 (QR4), OS5 (QR5), OS8 (QR8), OS10 (QR10), OS17 (QR17) and OS20 (QR20) coupled to TE0 at 850 nm. The thickness of the gratings is not to scale and in all cases is equal to 140 nm. Grey areas correspond to Al, and the voids to SiO$_2$. The dielectric coverage is 0.5 in all cases. Reciprocal space representations of the unit cells are also included. The radius of the zeroth optical state is in all cases reduced to a dot for clarity. (g) GaAs absorption in devices as the one in Fig. 1, with ARC thickness = 80 nm and the gratings QR4, QR8 and QR20. The active layer absorption in a planar structure (no grating) and in the optimised device with a PC grating are also included. (h) Ideal short circuit current of devices as shown in Fig. 1, with ARC thickness = 80 nm and all the quasi-random gratings designed. (i) Integrated absorption (averaged absorption over the spectral range from 300 to 900 nm) in the grating and the GaAs layer of devices as shown in Fig. 1, with ARC thickness = 80 nm and all the quasi-random gratings designed.

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We then simulate the full spectrum absorption of ultra-thin devices with these scattering layers. Example results are included in Fig. 7(g) for QR4, QR8 and QR20. There is a weaker modulation of the absorption curves as the gratings become more random, which implies a lower coupling efficiency to the bound modes. We expect this effect to be a consequence of the distribution of diffracted power among increasing numbers of optical states in more random structures, which compromises the amount of power that can be coupled to a guided mode. In contrast, a photonic crystal structure will diffract light at highly localised spatial frequencies and will offer a higher coupling efficiency [38]. This can be appreciated by comparing the absorption of QR4 to that of the optimised device with a PC grating (also included in Fig. 7(g) for comparison); both devices have all the same parameters except for the spatial distribution of materials within the unit cell. Whereas the absorption of a device with the photonic crystal structure shows clear peaks at near band edge wavelengths, this is not the case for the QR structure.

In the devices with QR structures, it can also be observed that the absorption in both the intermediate and long wavelength regimes is reduced as the grating becomes more random, indicating lower absorption contributions from the diffracted components. The reduction in the absorption that results from diffraction still allows the total absorption near the band edge to exceed that of a planar structure. However, this is not the case in the intermediate wavelength region given the reduced contribution from diffraction and the detrimental effect of a grating on the absorption in the specular component. It should be mentioned that for all gratings but QR4 there are optical states within the escape cone at wavelengths with significant absorption contributions from diffraction, but the detrimental effect of diffraction into these states is expected to be reduced as a result of a low Fourier amplitude of the gratings at such states.

The ${Jsc}$ of devices with the QR gratings is included in Fig. 7(h). Even when they all have a higher figure of merit than a planar structure, they all show a considerably reduced performance compared to the optimised photonic crystal structure, with a difference of at least 2 mA/cm$^2$ in their ${Jsc}$. It is also clear that the more random the grating becomes, the more its performance is reduced. These results highlight the superior performance of a photonic crystal grating for the device architecture of interest in this work.

To better understand the observed trends, for each device with the designed QR structures we calculate the integrated absorption in both the grating and the GaAs absorber, defined as the average absorption in the layer under consideration over the spectral range from 300 to 900 nm (Fig. 7(i)). Whereas the integrated absorption in the GaAs follows the observed trend for the ${Jsc}$, the integrated absorption in the grating follows the opposite trend, increasing the more random this grating becomes. This increased absorption in the grating is observed in the intermediate and long wavelength regimes (see Supplement 1, section 8), so that the reduced diffraction contributions to the GaAs absorption in Fig. 7(g) are related to a loss of power in the scattering layer. We expect this phenomenon to be related to the lower coupling efficiency to the bound modes in more random structures. Since the constructive interference events in a guided mode can lead to an increased localisation of the field within the active layer, an improved coupling efficiency to these propagating modes would in turn reduce the parasitic absorption in the grating. In more random structures, the power that is diffracted at the states that are not coupled to guided modes would not have this localisation, and would then be more likely to be absorbed by the grating.

We note that similar results have been previously reported. An absorptive photonic crystal scattering structure containing Au has shown an improved performance over a QR design in an ultra-thin GaAs solar cell [19]. Another study [39], however, found increasingly random QR scattering structures to be beneficial for an ultra-thin GaAs solar cell. In this case the scattering layer was transparent in a broad spectral range, and the studied device also had a dielectric spacer to reduce parasitic absorption in the rear mirror. Reduced benefits from QR structures are expected in GaAs solar cells as a result of the direct band gap of this absorber material, which (depending on the thickness of the active layer) can limit the spectral range where diffraction is essential and so a more broadband absorption enhancement as the one offered by QR structures may not be needed for optimal absorption. Also, considering that ultra-thin GaAs solar cells have a reduced number of bound modes (e.g. our devices support a single mode for each polarisation of light at wavelengths near the band edge), accessing a broad range of spatial frequencies with QR structures can be less beneficial since this will not lead to an increased number of bound mode coupling events. However, as shown in Fig. 7(i), parasitic absorption in the device architecture can also pose limitations on the benefits of employing a QR structure in an ultra-thin GaAs solar cell.

Finally, we acknowledge that the optimal distribution of diffracted power among the optical states in the diffraction regime was not considered in our algorithm to design QR structures. Refinements to this algorithm may result in QR structures with better performance. We also highlight that the introduction of QR structures to our device architecture was not found to impact the applicability of our TMM implementation (see Supplement 1, section 9).

3. Conclusions

We report a TMM implementation capable of finding the bound waveguide modes in nanostructured, ultra-thin solar cells. This implementation can be used to guide the design of a device for optimal light management, in particular for the selection of an appropriate unit cell pitch and the corresponding distribution of optical states. Applied with an RCWA method to study light management in an ultra-thin GaAs solar cell, this TMM implementation provided insight into the parametric dependence of bound mode coupling and absorption enhancement in the active layer, which served as guidelines for the construction of a guided light management optimisation framework.

Following this framework, we optimised an ultra-thin GaAs solar cell with a 2D photonic crystal grating. The parameters and performance of this device were equivalent to the ones obtained by exhaustively scanning multiple combinations of device parameters in search for the highest ${Jsc}$, but our framework allows obtaining this optimal design with a reduced parameter space whilst furthering our fundamental understanding of light management in reduced length-scales. We also applied this framework to systematically compare the performance of quasi-random scattering structures against photonic crystal gratings. We found the latter to be significantly better for light management in our device architecture of interest, with increasingly random QR structures suffering more from parasitic absorption in the scattering layer.

The photonic crystal gratings that were studied, however, had features which were chosen arbitrarily and further improvements to device performance could potentially be achieved by altering their geometry. Changing the materials in the ARC and the scattering layer could also improve the absorption enhancement, for example by further reducing reflection or parasitic absorption (which may also improve the performance of QR designs). It should also be acknowledged that the present work only studied the performance of these devices considering normal light incidence, and future work should analyse their performance in the context of angular dependence. Future studies should also address the potential presence of different optical modes and corresponding resonances in other material systems (e.g. possible plasmonic features when introducing Ag, a commonly employed metal in light management designs [1,5,25,40,41]), so that our framework can be developed further. Additionally, the effect of varying the thickness of the absorber material should also be studied in detail, to determine the impact that this parameter has on the modal structure of the device, the extent of the spectral regime where diffraction is essential and the interplay of different parameters that determine the optimal design for light absorption. Overall, the presented design guidelines and framework remain applicable to many different device architectures and material systems, and are powerful tools for achieving high efficiency in ever-thinner photovoltaics.