1. Introduction

Organic materials are promising candidates for photovoltaics due to their high quantum yield, tunable bandgaps, potential of low-cost production and mechanical flexibility [1–4]. Decades of studies have been focused on developing new materials and tailoring junction morphology of organic absorbers [5–8]. However, organic solar cells suffer from low carrier-diffusion lengths, which severely limit their overall optical-to-electrical power-conversion efficiency [1,5,9,10]. Organic films with thicknesses less than 100nm are desirable for efficient carrier extraction. However, this severely limits their optical absorption [1,10–12]. Recently, a number of approaches to enhance absorption in such ultra-thin devices have been proposed using light scattering via dielectric nanostructures [12], cylindrical substrates [13] and plasmonic nanostructures [14–17]. Periodic dielectric nanostructures placed on top of an ultra-thin silicon absorber have been shown to generate guided-mode resonances and thereby, provide broadband enhancement of optical absorption in the silicon layer [18]. Previous designs of these geometries were based on simple parametric-search methods. Here, we apply a modified version of the direct-binary-search (DBS) algorithm [19,20] to generate geometric designs that maximize optical-to-electrical power conversion with ultra-thin organic solar cells. We also introduce a novel architecture where the organic absorber is sandwiched between two conductive (but transparent) patterned layers, where the pattern on the top and bottom layers could be different from one another. Optical fields were computed using the finite-difference-time-domain (FDTD) method [21] and the electrical properties were characterized using the conventional diode model. The optimized designs show improvement in total power output by factors up to 16 times compared to flat reference devices for an absorber thickness of 10nm.

The simulated organic solar cell sandwiched between two nanostructured light-trapping layers is illustrated in Fig. 1(a) . The device is comprised of (from top to bottom) a top-cladding layer, an active layer, a bottom-cladding layer, and finally, a back-contact layer. The simulation region is defined in one dimension and it is assumed to be periodic with a spacing of Λ. Note that the two cladding layers are patterned with rectangular profiles as indicated. The cladding layers are made of indium tin oxide (ITO), which offers excellent charge-carrier collection properties [22]. The active layer is composed of two thin-film materials: a 20nm-thick PEDOT:PSS layer, which serves as the top contact to enhance charge-collection efficiency [23–25], and the absorber, P3HT:PCBM as the carrier-generation medium [1–7,12–17,26]. Two reference devices were used as controls in our simulations. The first device in Fig. 1(b) was comprised of a 3-layer stack of PEDOT:PSS, P3HT:PCBM and a back-reflector. Although this device lacks a top electrode and hence, would not be functional, we used it as a simplified reference for optimization and parametric analysis to gain insight into the mechanisms of nanophotonic light trapping. The second reference device shown in Fig. 1(c) represents a more realistic architecture as it included an optimized anti-reflection coating as the top layer. This layer also serves as the top-electrical contact and is composed of ITO. Our optimized devices were compared to this reference geometry in the final discussions. The back-contact was assumed to be a perfect-electric conductor.

 

Fig. 1 Schematics of the simulation models. (a) Ultra-thin absorber layer sandwiched between two nanostructured light-trapping layers. (b) First reference cell with bare, unpatterned surface. The incident angle is defined as $\theta$. (c) Second reference cell with an optimal anti-reflection coating.

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The variables for optimization were the parameters that describe the 1D geometry of the device in Fig. 1(a). These are: the period of the simulated region (Λ), the fill-factor of the top-cladding layer (ff₁), the fill-factor of the bottom-cladding layer (ff₂), the displacement factor, s between the patterned areas of the top and bottom cladding layers, the etch-depth of the top-cladding layer (t_s), the thickness of the remaining top-cladding layer (t_c1), the thickness of the unetched bottom-cladding layer (t_c2) and the etch-depth into the bottom electrode (t_b). The two reference devices both have fixed geometric parameters and have the same film thicknesses for the PEDOT:PSS (t_PED = 20nm) and P3HT:PCBM (t_a) layers as the optimized device.

We applied the FDTD method for rigorous numerical simulations. A broadband plane-wave light source was positioned above the cell. Its spectrum was weighted according to the standard AM1.5 solar spectrum ranging from 350nm to 800nm [27]. For computational simplicity, only normal incidence is assumed during optimization. However, the optimized devices were analyzed under broad-angle illumination and their performance was compared to the reference devices. Perfectly-matched layers (PML) were included in the model to avoid numerical reflection in the direction of wave propagation (Y-direction as indicated in Fig. 1(a)) [28]. The PMLs were located at the top and bottom boundaries of the simulation region and placed parallel to X-axis. Bloch-periodic boundary conditions were imposed in the X-direction [29,30]. The execution time of FDTD simulation is sufficiently long to guarantee numerical convergence to steady state.

2. Optimization algorithm and figure-of-merit

The direct-binary-search (DBS) algorithm has been successfully adapted and implemented in our previous work on unconventional diffractive optics [19,20]. Here, we apply the same search algorithm to a set of geometric parameters with the objective of maximizing the electric power extracted from a solar cell. DBS operates in an iterative fashion. During each iteration, each geometric parameter is perturbed by a small pre-determined value. Then, a figure-of-merit (FOM) or objective function is calculated for the resulting device. If the FOM is improved, the perturbation is kept and the next parameter is perturbed, and the FOM evaluated. If the FOM is not improved, the perturbation is discarded. At this time, an alternate perturbation (of the opposite sign) may be applied and the FOM re-evaluated. This perturbation cycle continues until all the parameters have been addressed. This completes one iteration of the DBS algorithm. Such iterations are continued until the FOM converges to a stable value. An upper bound on the total number of iterations and a minimum change in FOM are defined to enforce numerical convergence [20]. It is well known that the DBS and associated search algorithms are sensitive to the starting point and tend to suffer from premature convergence in local optima [20]. Therefore, we repeated this process with several random starting designs. Each independent parameter of an initial solution is randomly generated by functions in Matlab. Since different initial cases will lead to different optimal designs, only the ones with the highest enhancements are considered for discussions in this paper. Note that there is a trade-off between the granularity of the search (unit of perturbation of each geometric parameter) and the number of iterations for convergence. Here, we empirically determined this granularity based on preliminary simulations. The perturbation and range of the optimization variables are listed in Table 1 .

Table 1. Ranges and unit perturbations of geometric parameters for optimization

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The FOM for optimization was the output electrical power-density of the solar cell. We first computed the light intensity distribution, I_λ(x,y,λ) within the absorber layer of the device for a given geometry. Then, the short-circuit current density was computed as

We also calculated F, the enhancement of the integrated light intensity within the active layer as well as J, the corresponding enhancement of the short-circuit current density. These metrics were defined in our previous works [18,32].

3. Optimization results

The thickness of the absorber plays a critical role in determining the ultimate enhancement [18]. Here, we applied the optimization algorithm on devices with three different absorber thicknesses,10nm, 30nm and 50nm.

The simulation and optimization results of the first design with a 10-nm thick P3HT:PCBM layer are summarized in Fig. 2 . A power-enhancement factor, E = 31 is reached after 19 iterations (see Fig. 2(a)). A random parameter set was used as the initial condition. The geometry of the optimized design is shown in Fig. 2(a). In the optimized design, the period, Λ = 320nm, and the fill factor of the top-cladding layer, ff₁ = 0.19, which gives a width of the scattering line of 60nm. The bottom-cladding layer is offset by a half period from the center of the top-cladding layer and it has a fill factor, ff₂ = 0.43, which gives a width of 138nm for the scattering line. Interestingly, the heights of scattering lines for the top and bottom cladding layers are 75nm and 25nm, respectively. This suggests that strong optical scattering and hence, light trapping is achievable even with relatively shallow structures. Note that such low-aspect ratio nanostructures can be reliably replicated using nanoimprint lithography [33]. As plotted in Fig. 2(a), both E and J follow F in the process of optimization. This further demonstrates that both the increased short-circuit current density and the increased power output are due to the enhanced light absorption enabled by the periodic dielectric nanostructures. As we show later, such nanostructures generate guided-mode resonances, which dramatically increase the light intensities within the active layer. The electrical characteristics of the optimized and the reference devices were simulated and plotted in Fig. 2(b). The short-circuit current density and the peak power-density are both increased dramatically in the nanostructured device.

 

Fig. 2 Results of the optimized device with 10-nm thick absorber. (a) Evolution of the enhancement factors as a function of the iteration number. The optimized design is shown in the inset. (b) Calculated J-V and P-V curves for both the optimized device and the reference device (inset). (c) Enhancement spectrum analysis (inset: electric-field-intensity distributions inside the active layer at the peaks of the TE and TM polarizations). (d) Angular response analysis of the optimal design (inset: spectral-angular analysis).

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We also computed the enhancement spectrum (see Fig. 2(c)), which is defined as the ratio of the total light intensity integrated over the active region for each wavelength for the optimized design to that for the reference device. The spectrum is plotted for both TM and TE polarizations as indicated. There is broadband enhancement over the entire spectrum, which is likely due to the anti-reflection properties of the top-cladding layer. However, there are multiple resonance peaks that make large contributions to the overall enhancement. For the TE mode with electric field polarized parallel to the nanostructures (Z-direction), the enhancement-spectrum peak appears at λ = 414nm. The strongest resonance for the TM mode with electric field polarized perpendicular to the nanostructure (X-direction) is located at λ = 652nm. Intriguingly, the peaks at the two polarizations approach the peak of the solar spectrum (~500nm for AM1.5) and the bandgap of the absorber (~750nm for P3HT:PCBM). The intensity distributions of electric fields of both polarizations inside the active layer at their corresponding resonance peaks are illustrated in the insets in Fig. 2(c). The enhancement factors were calculated as a function of the incident angle, θ for the optimized design and plotted in Fig. 2(d). This plot is a good measure of the performance of the device when not tracking the sun or under diffuse illumination. The broadband enhancement of output power remains greater than 5 × for incidence angles as large as 60°. Furthermore, a closer look at the enhancement spectrum as a function of incident angle (inset of Fig. 2(d)) shows that guided-mode resonances exist even at large angles (>40°) although a shift in resonance wavelength with incident angle is also seen. This shift could be advantageous when both broadband and broad-angle enhancement is desired. Note that the effect of an anti-reflection coating is typically characterized by enhancement with very little wavelength dependence. Therefore, we can also conclude that the light scattering effects of both cladding layers predominate over the anti-reflection effects of the top-cladding layer for angles less than about 60°.

In the second design, an absorber thickness of 30nm was assumed and the results are summarized in Fig. 3 . A power-enhancement, E = 11 is obtained after 23 iterations. Interestingly, in the optimized design the top-cladding layer is absent, and light trapping arises from the relatively thin patterned bottom-cladding layer. Although this optimized design represents an unrealistic device (since there is no top electrode), we can evaluate the performance of nanophotonic light-trapping by numerically studying the properties of such a device. Note that this situation may be readily avoided in the future by including a minimum thickness constraint on the top-cladding layer (recall that this also serves as the top electrode) during optimization. The J-V and P-V curves are plotted in Fig. 3(b), and the spectral and the angular responses are shown in Figs. 3(c) and 3(d), respectively. The electric field intensities at the peaks of the enhancement spectrum for TE and TM polarizations are shown as insets in Fig. 3(c). The first peak at λ = 430nm for TE polarization is close to the maximum of the AM1.5 solar spectrum, offering strong light enhancement at the maximal power input. The second peak at 690nm for TM polarization approaches the bandgap of the P3HT:PCBM absorber, which attempts to boost the low intrinsic absorption there, by enhanced light-trapping. The performance of the optimized device under oblique incidence is relatively good since the enhancement has a slower drop-off with incident angle compared to the first design (see Fig. 2(d)). The enhancement spectrum as a function of incident angle is shown in the inset of Fig. 3(d). The presence of sharp peaks in the spectra even at large incidence angles (~40° and greater) confirms that the nanophotonic light trapping dominates the total contribution to power enhancement compared to the anti-reflection effect.

 

Fig. 3 Results of the optimized device with 30-nm thick absorber. (a) Evolution of the enhancement factors as a function of the iteration number. The optimized design is shown in the inset. (b) Calculated J-V and P-V curves for both the optimized device and the reference device (inset). (c) Enhancement spectrum analysis (inset: electric-field-intensity distributions inside the active layer at the peaks of the TE and TM polarizations). (d) Angular response analysis of the optimal design (inset: spectral-angular analysis).

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For the third design, an absorber thickness of 50nm was assumed and the results are summarized in Fig. 4 . The power-enhancement factor (E = 3.6 in Fig. 4(a)) of the optimized design is lower than that of the devices with thinner active regions. This is consistent with previous work [34] and is related to the near-field effect of nanophotonic light-trapping. As the absorber layer increases, the reference device has increased absorption and the improvement one can gain from nanophotonic light trapping decreases. The geometric parameters, together with the illustration of the optimized design, are depicted in the inset of Fig. 4(a). Using Eq. (2), we plotted the electrical characteristics of the optimized and reference devices in Fig. 4(b). Note that both the short-circuit-current density and the peak-power density are increased due to the enhanced absorption of photons within the active region. The two resonance peaks are clearly visible in the enhancement spectrum (see Fig. 4(c)). The locations of these peaks are consistent with the previous designs and can be explained in a similar manner. The angular response of the device shown in Fig. 4(d) shows a less dramatic drop-off in enhancement compared to the devices with thinner active layers. It is noteworthy that the power-enhancement remains above 2 × for incident angles over 70°. The enhancement spectrum as a function of incident angle shown in the inset in Fig. 4(d) confirms the presence of resonant peaks even at angles as large as 60°. A relatively lower short-circuit current density (see Fig. 4(b)) was achieved compared to the designs with thinner absorbers as plotted in Fig. 2(b) and 3(b). This is likely due to parasitic light absorption in the top ITO layer, which will be discussed later.

 

Fig. 4 Results of the optimized device with 50-nm thick absorber. (a) Evolution of the enhancement factors as a function of the iteration number. The optimized design is shown in the inset. (b) Calculated J-V and P-V curves for both the optimized device and the reference device. (c) Enhancement spectrum analysis (inset: electric-field-intensity distributions inside the active layer at the peaks of the TE and TM polarizations). (d) Angular response analysis of the optimal design (inset: spectral-angular analysis).

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4. Parametric analysis

The sensitivity of power-enhancement to variations in the geometric parameters of the optimized design is important, since fabrication of these devices will inevitably introduce errors. Furthermore, such a parametric analysis can provide additional insight into the mechanisms of light trapping involved. We conducted such an analysis and these results are summarized in Fig. 5 . Each parameter was scanned over a predetermined range of values while the other parameters were kept constant at the optimal design. For each parameter, we plot the power-enhancement factor as well as the enhancement spectrum as a function of the parameter value. Only the first design with the 10nm-thick active layer is analysed for brevity.

 

Fig. 5 Parametric analysis of the optimized device with an active-layer thickness of 10nm. (a) (E) vs. Λ; (b) Enhancement spectrum vs. Λ; (c) (E) vs. s; (d) Enhancement spectrum vs. s; (e) (E) vs. ff₁; (f) Enhancement spectrum vs. ff₁; (g) (E) vs. t_s; (h) Enhancement spectrum vs. t_s; (i) (E) vs. ff₂; (j) Enhancement spectrum vs. ff₂; (l) (E) vs. t_b; (l) Enhancement spectrum vs. t_b; (m) (E) vs. t_c1; (n) Enhancement spectrum vs. t_c1; (o) (E) vs. t_c2; (p) Enhancement spectrum vs. t_c2.

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The first parameter we analyzed was the period of the computational domain of the device, Λ, which was varied from 50nm to 800nm. The power-enhancement factor peaks at about Λ = 350nm (see Fig. 5(a)). This arises from multiple resonances over the incident spectrum. These resonances are particularly strong at wavelengths shorter than 500nm as observed in the enhancement spectra shown in Fig. 5(b).

The second parameter we analyzed was s, the relative displacement factor between the patterned regions of the top- and bottom-cladding layers. Although, the power-enhancement factor shown in Fig. 5(c) has low sensitivity to changes in s, the enhancement spectra shown in Fig. 5(d) illustrate very interesting behavior. For all values of s, there is a strong resonance at λ~650nm. However, when s is close to 0.5, i.e., when the patterned regions of the top- and bottom-cladding layers are shifted relative to each other by Λ/2, a second strong resonance starts to appear at λ~400nm. We can relate this to the field distribution plots shown in the insets of Fig. 2(c). It looks like the top-cladding layer concentrates TE polarized light in the center of the active region, while the bottom-cladding layer concentrates TM polarized light along the edges of the active region. The net result of this design is that for unpolarized light, the active region is uniformly illuminated with enhanced light intensities. Note that uniform illumination is important to avoid hot-spots within the absorber material.

The analysis of the top-cladding structure is shown in Figs. 5(e)-5(h). When the fill-factor, ff₁ or the thickness of the layer, t_s becomes 0, a single resonance at λ~600nm is seen in the enhancement spectra (Figs. 5(f) and 5(h)). This resonance clearly arises due to light scattering from the patterned bottom-cladding layer. As ff₁ or t_s increases, multiple resonances emerge until the power-enhancement factor peaks at ff₁ = 0.2 or t_s = 70nm. We can surmise that the bottom-cladding layer primarily affects the resonance at long wavelength, while the short-wavelength resonance (where the AM1.5 spectrum peaks) is a result of the interaction between the top- and bottom-cladding layers. As ff₁ increases further, two distinct regions of resonance arise. The top-cladding layer modulates the position and strength of the second resonance. This resonance shifts to longer wavelength with increasing fill factor and thickness of scattering layer (see Figs. 5(f) and 5(h)). The long-wavelength resonance lies slightly above the bandgap and hence, doesn’t contribute significantly to device performance and the overall power-enhancement drops. In Fig. 5(h), the resonance is weakened gradually at t_s>50nm and the secondary peak in Fig. 5(g) is likely due to the third resonance occurring at λ~500nm when t_s>100nm.

The analysis of the impact of the bottom-cladding structure is shown in Figs. 5(i)-5(l). As hinted earlier, the bottom-cladding layer disproportionally impacts the second resonance peak close to the bandgap of P3HT:PCBM absorber. At lower values of ff₂, a broad resonance exists from 400nm to 600nm. However, as ff₂ increases, the resonance shifts to wavelengths above 600nm. The biggest impact on the power-enhancement factor arises from absorption near the peak of the AM1.5 spectrum (~500nm) and this seems to occur when ff₂~0.4. On the other hand, as the thickness, t_b increases beyond 30nm, the second resonance splits into three peaks, and the power-enhancement E is reduced. This is likely due to the sharing of energy between these three distinct resonances.

Intriguingly, the parametric analyses of t_c1 and t_c2, the thicknesses of the unpatterned regions of the top- and bottom-cladding layers, respectively, indicate that these are essential to optimal light-trapping performance (see Figs. 5(m)-5(p)). A minimum thickness is required to support guided-mode resonances. In addition, these layers act as optimal electrodes with high transparency and low carrier-recombination. Despite the peak of the power-enhancement factor, E at t_c1 = 120nm via our optimization algorithm, the parametric analysis reveals another peak when t_c1 = 0nm (see Fig. 5(m)). From the enhancement spectrum at t_c1 = 0nm, we can observe that there is a relatively flat broadband enhancement from 400nm to 650nm, and very high enhancements beyond 650nm (see Fig. 5(n)). Although this peak has a higher value for E, it clearly will not be a practical device since no top electrode would be present. Yet a design with cladding layer thickness close to the second data point in Fig. 5(m) may serve as an optimal design with higher enhancement factor compared to that with t_c1 = 120nm. This particular example also demonstrates that our DBS algorithm is apt to converge to local maxima and care must be taken to interpret and analyze the optimized designs. As illustrated in Figs. 5(o) and 5(p), the unpatterned region of the bottom-cladding layer sustains strong broadband resonance at t_c2 = 60nm. It is also clear that the power-enhancement factor is strongly influenced by t_c2 and care must be taken to ensure its precision during fabrication.

Parasitic absorption in ITO can partially explain the effects of t_c1 and t_c2. Despite relatively good transparency of ITO, thicker top cladding layer may still potentially consume a significant amount of solar energy, compromising the power output, as plotted in Fig. 5(m) where E at t_c1 = 0 is higher than E at t_c1 = 120nm. The calculated fractions of light absorbed by ITO layers are 29%, 9% and 16% for the three designs above with 10nm, 30nm and 50nm thick absorbers, respectively.

In short, it is the relative strength and location of the two prominent guided-mode resonances that determinate the overall power-enhancement of the optimized device. Adjusting the geometric parameters such that the resonance peaks with balanced strengths approach the peak of the AM1.5 spectrum and the bandgap wavelength of the absorber, respectively, can increase the enhancement. In addition to the analyses above, the impact of the active-layer thickness, t_a on the enhancement factors (F, J and E) is illustrated in Fig. 6 . The enhancement factors decrease exponentially as t_a increases. This is expected from previous work [34].

 

Fig. 6 Power-enhancement factor, (E) of the optimized devices vs. thickness of the active layer, compared to both bare reference and reference device with ARC. The comparison is made for both normal incidence and averaged over a range of incident angles from 0° to 60°. The enhancement factor, (E) is plotted in log scale.

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5. Conclusions

We presented the application of a modified version of the direct-binary search (DBS) algorithm to design nanophotonic light trapping for ultra-thin organic solar cells. When compared to the bare reference device (see Fig. 1(b)), the optimized device produces 31 × , 11 × and 3.6 × more power for absorber thickness of 10nm, 30nm and 50nm, respectively (blue-solid line in Fig. 6). When compared to reference devices with anti-reflection coatings (see Fig. 1(c), ARC thickness of 25nm for the first design, 20nm for the second and the third designs), the optimized device produces 16.4 × , 4.6 × and 1.57 × more power for absorber thicknesses of 10nm, 30nm and 50nm, respectively (red-solid line in Fig. 6). The enhancement drops when the incident angle deviates from zero. Hence it is instructive to compute the enhancement when averaged over a range of incident angles [32]. For our simulations, we averaged the enhancement over incident angles from 0 to 60°. In this case, the optimized devices with absorber thicknesses of 10nm, 30nm and 50nm produced 14.6 × , 7.2 × and 2.85 × more power compared to the bare reference device (blue-dashed line in Fig. 6), and produced 7.6 × , 3.3 × and 1.2 × more power compared to the reference device with ARC (red-dashed line in Fig. 6), respectively.

The nanostructured geometries in the optimized devices are challenging to fabricate at low-cost using conventional patterning technologies such as scanning-electron-beam lithography or photolithography [35]. This can be mitigated somewhat by the recent advances in nanopatterning such as absorbance modulation [36], optically-saturable transitions [37], two–photon absorption [38] and interference lithography [39]. Furthermore, developments in roll-to-roll nanoimprint lithography [33,40] can enable the mass replication of such nanostructures.

Although, we used the example of organic absorbers and dielectric scatterers here, the DBS algorithm can be readily applied to other photovoltaic materials as well as plasmonic scatterers. In addition, the algorithm can be extended to include oblique angles of incidence during optimization, a step analogous to our previous work [32]. Finally, we can also extend this approach to two-dimensional nanostructures as well as alternative cross-sections such as sinusoid, triangle [18,32], cone [41,42], or pyramid [43].