Photovoltaic light trapping theory and experiment do not always clearly demonstrate how much useful optical absorption is enhanced, as opposed to parasitic absorption that cannot improve efficiencies. In this work, we develop a flexible flux plane method for capturing these parasitic losses within finite-difference time-domain simulations, which was applied to three classical types of light trapping cells (e.g., periodic, random and plasmonic). Then, a 2 µm-thick c-Si cell with a correlated random front texturing and a plasmonic back reflector is optimized. In the best case, 36.60 mA/cm2 Jsc is achieved after subtracting 3.74 mA/cm2 of parasitic loss in a 2-µm-thick c-Si cell slightly above the Lambertian limit.
© 2015 Optical Society of America
There has been a strong trend in the solar industry toward thinner photovoltaic cells, particularly in the last decade. This has been driven not only by cost considerations, but also enabled by enhanced light trapping strategies [1–3 ] as well as the performance benefits associated with photon recycling [4, 5]. However, additional improvements in light trapping are needed for this approach to continue further, particularly in important solar materials with indirect bandgaps such as crystalline silicon (c-Si). The limits of current approaches such as random texturing  and plasmonic structures  should be considered, to see where particular opportunities for improvement may lie.
Figure 1 illustrates how three particular experimental light trapping schemes can enhance path length: periodic texturing on the front and back surfaces shown in Fig. 1(a) [7, 8], random texturing on the front and back surfaces shown in Fig. 1(b) [6, 9] and random texturing on the front surface combined with plasmonic nanoparticles (NPs) near the back surface shown in Fig. 1(c) [10, 11]. To date, incorporating these structures into 2 µm c-Si solar cells has resulted in experimentally measured short circuit currents between 22 and 29 mA/cm2. [6–11 ]. However, it is believed that substantially better performance should be possible. The upper limit of light trapping for purely random texturing is known as the Lambertian limit, which represents a maximum enhancement of path length of 4n 2 in weakly absorptive materials ; for 2 µm thick c-Si solar cells with perfect carrier collection, it would yield a short circuit current around 35 mA/cm2 .
This gap between experiment and theory strongly implies that parasitic losses are occurring in these designs, which may be associated with metallic structures in the back [6, 9, 11]. Nevertheless, in most experiments, parasitic absorption was not considered  or was only indirectly measured by comparing the external quantum efficiency (EQE) with the absorption spectrum [6, 8–11 ]. However, EQE measurements cannot distinguish between optical and electronic losses (e.g., from recombination). On the other hand, simulation based approaches have demonstrated great potential to help quantify optical parasitic losses [14–16 ]. Specifically, finite difference time domain (FDTD) methods can employ the integral of the power loss over the volume of the photovoltaic materials to quantify the fraction of useful and parasitic optical absorption at any layer of the solar cells . Unfortunately, this volume integration (VI) method increases the time complexity of the FDTD algorithm to a higher order. This makes optimization based on the FDTD method extremely slow and computationally expensive. [18–21 ]. Due to this critical limitation of the VI method, most previous simulations were conducted only within very thin layers (100—300 nm thick) [14, 16, 17] and limited wavelength ranges (λ = 600—800 nm) . A promising approach to sidestep this great computational expense would use flat flux planes  surrounding the volumes of interest, such as sets of flux planes forming a cube , or flux planes in the middle of a cell . However, the geometries allowed by this method are still limited to boxlike shapes.
In this work, we propose a novel simulation technique that can quantify the absorption in complicated (or generalized) 3-D cell structures both within a reasonable amount of computational time and with a negligible amount of memory consumption. Figure 2 indicates the computational cost of the main FDTD algorithm (without absorption), the VI method and the proposed method. The number of multiplication operations is nearly proportional to the computational time of the algorithm. As shown in Fig. 2(a), the VI method significantly delays the original FDTD method when the enclosed photovoltaic material is a few hundred nanometers thick. Also, the memory consumption of the VI method exceeds 4 GB, a common limit for RAM per computational core, when the thickness of the material is larger than 500 nm. This calculation is based on the grid size of 10 nm; a smaller grid size will increase the computational burden of the volume integration method. However, in the proposed method, both the speed of algorithm and the memory consumption are well below the original FDTD algorithm, so it does not significantly delay the main FDTD computation.
In this work, the proposed simulation technique will be discussed in detail, and then verified through a detailed comparison to experimental thin-film c-Si light trapping structures from Fig. 1. Next, an improved light trapping scheme for 2 µm c-Si cells with lower parasitic losses is proposed, which combines a recently reported correlated random front surface  with a plasmonic back reflector (BR). Finally, the optimized result is compared with both the ideal Lambertian limit and a totally random structure with limited roughness feature heights.
Since the FDTD method has been considered a promising tool for optical simulations, a great deal of work has been presented on improving the accuracy of this method [26–32 ]. The FDTD method generally represents curved material boundaries as stepped edges, resulting in discretization errors [27, 28]. One study used a contour FDTD method, which utilizes subpixel modeling to reduce this error. In this work, arbitrary surfaces of both metal and dielectric materials are considered. Then, we use a general tri-linear interpolation to reduce this discretization error.
2.1. Flexible flux planes
The surface integration of the Poynting vector (SI) over a flat plane successfully calculates the total absorption of solar cells within a moderate computational time and memory (compared to the propagation of Maxwell’s equation) by this equation for the transmitted power :1]. We can define N s as the number of grid points in each spatial direction and Nω as the total number of distinct frequencies. In this case, the flat flux plane method has time complexity, which is of the same order as the original FDTD calculations. Of course, FDTD is still with respect to the total number of voxels M in 3D, but as defined here. However, typically the flux plane calculation is generally slightly faster than the FDTD propagation, in the common case where only a subset of all possible frequencies below the Nyquist limit are of interest.
For problems concerning irregular shapes that cannot be solved with the flat flux plane approach, let us now consider the more general energy loss calculation by the VI method:Fig. 2(a), the higher order of computational time of VI method now significantly delays the original FDTD algorithm . Also, the memory storage is increased from a three dimensional data structure (x, y, ω) to a four dimensional data structure (x, y, z, ω), making volume integration of losses within FDTD an extremely expensive approach.
To overcome the computational burden of the VI method while preserving its flexibility, we now introduce a flexible flux plane (FFP) method, whose computational time is just for arbitrary surfaces, with a speed similar to that of the flat flux planes, but with generality comparable to that of volume integration. As shown in Fig. 3(c), the FFP method assumes that there is a pre-defined material boundary, which is discretized into triangles within the Yee grid. Then, as shown in Fig. 3(d), it calculates the centroid points of the triangles, their normal vectors, and their cross-sectional areas. Note that these calculations are performed only once (before FDTD starts its time iteration), so they do not affect the computational time of the main FDTD loop. Using predefined information, FFP now tracks the net transmitted electromagnetic flux along a material boundary in time iteration of a FDTD simulation with a given calculation of the transmitted power:Eq. (3). As shown in Fig. 2(b), the FFP calculation does not significantly burden the computational time, because it is only performed at the material boundary, which results in complexity. As explained in this work, the FFP approach is relatively straightforward to implement, so it is applicable to all FDTD tools (in principle).
2.2. Parasitic absorption modeling
Traditional approaches to enhancing light trapping in thin-film solar cells are illustrated in Fig. 1. In these approaches, incident rays are scattered at the front texturing surface into the active Si material, or toward the Ag BR while potentially exciting the surface propagating polariton (SPP) mode. A large fraction of the SPP and transmitted rays are eventually absorbed by the Ag BR, causing parasitic absorption. In this modeling section, Jsc and Jpar are calculated by
In order to accurately model the behavior of the materials used in the experiments (such as c-Si and silver), a recently developed time-domain dispersion model, called quadratic complex rational function (QCRF), is used [25, 33]. Other experimental materials of interest (e.g., ZnO and ITO) were assumed to be non-dispersive and non-absorptive, since thin ITO and ZnO layers generally appear transparent in the visible wavelengths and their single path absorption of 70 nm ITO does not exceed 2.42% for near-infrared wavelengths below 1000 nm. The overall impact of realistic dispersion on the short-circuit current is relatively modest for sample cells, not exceeding 0.6 mA/cm2 for the designs tested in the next section, and are not highly sensitive to the cell design. Thus, ITO dispersion is not included in our subsequent analysis. Sources are placed above the cell, perfectly matched layers are imposed at the vertical ends, and because the photovoltaic cells are very large in the transverse direction, periodic boundary conditions are applied in the transverse directions with 1 µm periodicity. The spectrum of incident waves considered ranges from 400 nm to 1100 nm (close to the band edge of silicon).
3. Result : modeling of experimental light trapping approaches
3.1. Periodically textured texturing cell
As illustrated in Fig. 1(a), periodically textured cells diffract incoming light into highly localized modes inside the active Si material. Commonly, periodic (grating like) cells are deposited conformally, thus the front and back surfaces have similar morphologies [7, 8, 34, 35]. The experimental periodic texturing cells typically have a simple pattern on the back surface, which may not excite strong SPP modes, but they still have a few edges on their BRs, which may introduce localized fields near these edges.
Modeling the 3-D experimental periodically textured solar cells in FDTD simulations is challenging, since true experimental cells have roughness and randomness that departs from ideal periodicity. Our modeled cells have a structure consisting of 1-µm-thick c-Si textured with a 1.5 µm period and a nearly parabolic shape shown in the inset of Fig. 4(a), which is very close to the experiment structure in . In our model, the quantified useful Jsc was 24.21 mA/cm2 and Jpar was 4.05 mA/cm2. This compares reasonably well to the reported Jsc of 25 mA/cm2 in , with an error of 3.6 % relative. As shown in Fig. 4(b), high field intensities are observed near the surface of the Ag BR in the form of a standing wave. Our calculations using the FFP method indicate that the strong overlap of these standing waves with Ag causes substantial parasitic losses.
To quantify the role of material thickness in the periodically textured design, and for ease of comparison with other experiments, we modeled the effect of increasing the average thickness of crystalline silicon from the experimental value of 1 µm to a value of 2 µm, which was also used in the random texturing and plasmonic cells. In this case, we found that the useful Jsc increases from 24.21 mA/cm2 to a value of 26.04 mA/cm2.
The degree of light trapping observed may be sensitive to the maximum height of the surface features. We investigated this sensitivity by choosing a 2 µm-thick cell having a constant period of 2.5 µm, the best value observed in experiment . We then varied the periodic texture height in a range from 0 to 625 nm, and found that the best height value was 500 nm, much like in the experiment. In this case, the useful Jsc improved to 26.04 mA/cm2, while Jpar dropped to 3.77 mA/cm2. Since the 2-µm-thick periodic texturing cell has the same thickness with the other cells which will be studied in the next sections, it could be directly compared to other cells.
3.2. Random texturing cell
Random texturing on the front and back surfaces provides another pathway toward enhancing light trapping. As shown in Fig. 1(b), the typical random texture is deposited conformally . As mentioned earlier, the random surface at the front can enhance light trapping up to the Yablonovitch limit; however, the back side texturing may give rise to both light trapping enhancement and increased parasitic loss. Thus, in this section, we apply the FFP method to quantify the amount of parasitic loss occurring in randomly textured cells.
We modeled a conformal random texturing cell with 2 µm thickness (sample D in ). Since random textures will have random variations over small spatial regions, resulting in random spectral features, we must generate an ensemble of many random surfaces and average the results to converge on the large-area limit for light trapping. After 50 identical trials, we calculate the average useful Jsc to be 22.77 mA/cm2 in 2-µm-thick c-Si solar cells, while the average Jpar to be 5.16 mA/cm2, with a standard error below 1 mA/cm2. This predicted useful Jsc matches well with the spectrally-integrated experimental EQE data for sample D in , which corresponds to a J of 22 mA/cm2 sc for a relative error of 3.5 %. We avoid directly using the measured Jsc from the I-V curve because the additional complications associated with masking, the choice of solar simulator spectrum, and the overall solar simulator power go beyond the scope of this study. The total ensemble-averaged absorption predicted by our model matches the experiment very closely, as shown in Fig. 5(a). The amount of parasitic absorption was very substantial in the presence of a highly textured Ag BR. As shown in Fig. 5(b), strong localized fields are observed near the Ag BR, which may introduce strong SPP modes. Based on these observations, it appears that rough surfaces in the BR generally cause high parasitic losses. The mechanisms for rough surfaces enhancing losses may include SPP mode coupling and slow-light propagation near the metal surface; based on the large value of our ensemble-averaged random texture Jpar, it may be hard to avoid both mechanisms simultaneously.
On a related note, Sivec et al. recently demonstrated a promising random texturing method using a ZnO pre-coating to independently control ZnO and Ag surface morphologies . To explore the light trapping potential of a random texturing approach, we optimized the ZnO and Ag surfaces separately and then combined them with the front surface random texturing including the internal quantum efficiency from sample D in . We found that the best random textured 2-µm-thick cell could achieve 27.41 mA/cm2 Jsc with 3.11 mA/cm2 parasitic loss. A general improvement in the amount of useful absorption is consistent with the experimental observations of improved maximum power in . The optimized structure we calculated has relatively high texturing features on the front, moderate texturing features on the ZnO layer and small texturing features on the Ag layer. The important observation here is that although the metal back reflectors are slightly textured (feature heights of approximately 50 nm), the parasitic loss quantified in this type of cell was not negligible. This implies that in future studies of nano-structured BRs, it will be essential to determine the fraction of useful absorption and parasitic absorption to prove that the benefits of a new approach outweigh its disadvantages.
3.3. Plasmonic cell
The last type of experimental thin-film c-Si cell considered in this manuscript, shown in Fig. 1(c) is a plasmonic enhanced cell with a hemispherical rear Ag NP . Recently, it has been demonstrated that this specific structure can greatly improve light trapping in thin silicon solar cells [10,11,38]. In this approach, light trapping can be achieved both by the front surface and the back NP, but it also has two sources of parasitic absorption; the Ag NP and the Ag BR. The parasitic losses of the plasmonic NPs are a particular problem for this type of light trapping approach [39,40]. Thus, we quantify the total amount of parasitic losses in these plasmonic type cells by placing our proposed FFP right at the Si/ZnO boundary.
We have modeled the performance of a hemispherical Ag plasmonic NP using FFP. The modeled experimental structure, whose performance was described in Fig. 4(a) in , has a single 600 nm diameter NP in 1 µm periodicity and 200 nm thick ZnO layer encapsulating the NP. As shown in Fig. 6(a), and in contrast with other light-trapping techniques, the hemispherical NP embedded in ZnO does not create large parasitic losses. As shown in Fig. 6(b), the hemispherical NP does introduce a localized field around its surface, but it does not propagate along the surface of the NP as SPP modes – possibly because these NPs are physically separated from one another. Note that the absorption enhancement observed in excess of the experimental data in Fig. 6(a) is due to two differences between the simulation and experiment, in terms of periodicity and texturing. First, the simulation incorporates very small front surface texturing features not found in experiment that increase the short wavelength absorption (i.e., around 400 nm). Second only the simulated structure has a periodic arrangement of NPs in the ZnO layer, which strongly enhances absorption near the diffraction threshold (around 1100 nm). Nonetheless, 50 identical simulation trials of these cells yield Jsc =24.01 mA/cm2, which is only a 0.01 % error from the reported Jsc with 2.04 mA/cm2 of parasitic loss; this result demonstrates that these two errors effectively cancel. This in turn strongly indicates that the plasmonic rear NP does not introduce much parasitic loss, while successfully enhancing the absorption of the cell.
4. Result : Optimization toward the Lambertian limit
Further improvement of useful absorption for experimentally relevant solar cells may be achieved through optimization of three factors; the front scattering surface, the overall periodicity, and the back reflector. As mentioned previously, new deposition techniques have made it possible to control the surface of each layer separately  and deposit NPs between the layers. This capability could be used to improve the front scattering surface further with correlated random texturing for thin-film micromorph tandem cells, which was demonstrated by the current authors to outperform a pure random texturing over a broad range of wavelengths . This front surface design introduces short-range correlations to capture the smoothness of real structures, as well as periodicity to enhance diffraction into waveguided modes that are strongly absorbed [41, 42]. These results provide a concrete basis for detailed structural design and optimization. Note that in this section, we obtain a partial fraction of light absorption of silicon layer by FFP method, which is then multiplied by both an ideal IQE (without charge collection losses), and the AM1.5G solar spectrum to calculate Jsc,ideal and Jpar,ideal:
In our first investigation, we combine the experimental structure from Section 3.3 with a correlated random surface in the front, and compare it with a control, consisting of an identical front random surface and a flat BR. Starting with a 2-µm-thick c-Si cell, two 1 µm-periodic features are included: a 300 nm-high textured front surface created by the correlated random algorithm  and a rear NP with a 600 nm diameter. In Fig. 7(a), at longer wavelengths (over 700 to 1100 nm), nearly 20 % relative absorption enhancement is observed on average from combining the correlated random and rear NP features while, at shorter wavelengths, no meaningful enhancement is observed. Note that the solid red and black curves only include useful absorption of light in the silicon. In Figs. 7(b) and (c), the electric field profiles are shown for the nanoparticle and control (flat BR) cases, respectively, at a wavelength of 1058 nm. Even though they have the same front texturing, strong light trapping modes are observed only in the first case (with NP). Compared to the flat BR case, the rear NP does not introduce additional parasitic modes, but still strongly enhances absorption in this wavelength region.
In our second study, we consider the effects of varying the various diameters of NPs (100— 800 nm) placed within the rear ZnO layer. If we combine this with a correlated random surface and a periodicity of 1 micron, we find that a 600 nm diameter yields the best value of Jsc,ideal. Next, we optimize the correlated random front surface with respect to the feature height, when combining it with the best Ag NP geometry. The results are compared with cell structures having identical front texturing but a flat BR. In both cases, multiple simulations are performed for each texturing height, and the results are averaged to reduce the random fluctuation of Jsc,ideal. As shown in Fig. 8, the rear NP improves Jsc,ideal by nearly 6 mA/cm2 in the flat front case. Note that the parasitic loss is already subtracted off our result, and does not exceed 4 mA/cm2 due to its flat BR. The standard errors in each case are calculated as the quotient of the standard deviation and the square root of the number of identical trials n for each parameter value (here, n = 5) . This result is then graphed as error bars for each curve and parameter value, as can be seen in Fig. 8. As the silicon thickness and thus absorption increases, the importance of the rear NP for light trapping decreases. Eventually, both Jsc,ideal curves plateau for texturing heights greater than 1200 nm (not shown). This implies that in the presence of an ideal front scatterer, an additional back scatterer is unnecessary to enhance light trapping. However, since most experimental light trapping cells still suffer incomplete light absorption at wavelengths ranging from 700 to 1100 nm, which is driven by the limited availability of texturing heights and periods, it is still worth considering the plasmonic type BR with experimentally available texturing methods.
Finally, the performance of the optimal cell combining correlated random front texturing with a rear NP structure is presented in Fig. 9(a) as a solid red curve, and compared to competing light trapping strategies. This result is unique, because many previous researchers in light trapping have focused on total absorption, but do not subtract parasitic losses. In contrast, our result only includes useful absorption. The quantified Jsc,ideal of the optimum cell is 36.60 mA/cm2 with 3.74 mA/cm2 parasitic loss and 4.97 mA/cm2 reflected loss; this compares favorably with the Lambertian limit of an ideal random light-trapping structure in a 2 µm thick c-Si cell of around 35 mA/cm2 . Now let us consider the physical structure of this optimal absorber after accounting for parasitic losses. It is smooth over short distances, yet offers the large features required to support strong light-trapping modes, as shown in Fig. 9(b). By comparison, the purely random structure, shown in Fig. 9(c), has very narrow features densely distributed on the surface. Overall, our optimization results in this section have demonstrated that a front random scatterer can enhance broad-band absorption while the additional periodicity and a rear NP can enhance absorption at targeted wavelengths, particularly near the band edge. These principles are general, and can be applied to a broad range of PV materials, particularly indirect bandgap materials, with appropriate adjustments to the NP geometry.
In conclusion, a new simulation technique, called flexible flux plane (FFP) finite-difference time-domain (FDTD) is introduced in this manuscript. FFP FDTD allows one to accurately calculate parasitic losses in thin-film c-Si solar cells, while reducing time and memory consumption by two orders of magnitude compared to volume integration-based approaches. Applying this technique to experiment, three particular types of well-characterized light-trapping photovoltaic cells have been accurately modeled. First, conformally deposited periodic cells were shown to have moderate parasitic losses (4.05 mA/cm2) with good total absorption. Second, conformally deposited random cells were demonstrated to have the highest parasitic losses (5.16 mA/cm2) due to their rough metal back surfaces. Third, the flat Ag BR with a rear NP structure was shown to have the lowest parasitic losses (2.04 mA/cm2), with a strong absorption peak around higher wavelengths. This last structure was then combined with a correlated random front texture and optimized for the case of ideal charge collection to achieve the highest possible Jsc. It was found that the 600 nm diameter NP strongly enhances light absorption at wavelengths ranging from 700 nm to 1100 nm, while a correlated random surface has much broader absorption enhancement. In our optimized result, 36.60 mA/cm2 Jsc,ideal is achieved only with 3.74 mA/cm2 of parasitic loss. This result is slightly above the Lambertian limit of 2 µm thick silicon cells, even after including parasitic losses. However, this structure requires nanoscale structural control and high-performance thin-film electronic materials to fully realize this prediction.
The authors thank Mohammad Ryyan Khan for valuable discussions. Support was provided by the Department of Energy, under DOE Cooperative Agreement No. DE-EE0004946 (PVMI Bay Area PV Consortium), the Semiconductor Research Corporation, under Research Task No. 2110.006 (Network for Photovoltaic Technologies), the National Science Foundation, under Award EEC1454315-CAREER: Thermophotonics for Efficient Harvesting of Waste Heat as Electricity, and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2014R1A1A2054440).
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