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The expectation of perfectly geometric shapes of subwavelength grating (SWG) structures such as smoothness of sidewalls and sharp corners and nonexistence of grating defects is not realistic due to micro/nanofabrication processes. This work numerically investigates optical properties of an optimal solar absorber comprising a single-layered silicon (Si) SWG deposited on a finite Si substrate, with a careful consideration given to effects of various types of its imperfect geometry. The absorptance spectra of the solar absorber with different geometric shapes, namely, the grating with attached nanometer-sized features at the top and bottom of sidewalls and periodic defects within four and ten grating periods are investigated comprehensively. It is found that the grating with attached features at the bottom absorbs more energy than both the one at the top and the perfect grating. In addition, it is shown that the grating with defects in each fourth period exhibits the highest average absorptance (91%) compared with that of the grating having defects in each tenth period (89%), the grating with attached features (89%), and the perfect one (86%). Moreover, the results indicate that the absorptance spectrum of the imperfect structures is insensitive to angles of incidence. Furthermore, the absorptance enhancement is clearly demonstrated by computing magnetic field, energy density, and Poynting vector distributions. The results presented in this study prove that imperfect geometries of the nanograting structure display a higher absorptance than the perfect one, and provide such a practical guideline for nanofabrication capabilities necessary to be considered by structure designers.
© 2014 Optical Society of America
1. Introduction
Solar cells are commonly considered as promising candidates for alternative green energy sources replacing fossils fuels. With the emergence of nanotechnologies over the past few decades, photonic nanostructures including one, two, and three dimensional (1, 2, and 3D) features have emerged as important devices for photovoltaic collectors [1–13]. Besides various theoretical investigations on nanostructures available, e.g., extraordinary optical transmission or absorption, micro/nanofabrication techniques have attracted significant attention due to their reproducibility, precision, cost, and so on [14–20]. Although many excellent fabrication techniques have been proposed to achieve the perfect geometry of nanograting structures, it has not always been successful during the actual micro/nanofabrication [19, 21, 22]. For example, photoresist residues or intrinsic limits of a fabrication process may leave nanometer-sized features attached to profiles of nanograting structures or damage gratings. As a result, the practical structures possess more degrees of freedom from the actual simulation and design point of view. This problem has not been completely solved yet and is thus considered here.
Usually in simulations, structured profiles of simple or complex gratings, e.g., rectangular, circular, sinusoidal, or triangular cross sections, are assumed to be smooth to simplify geometric shapes and boundary conditions. The assumption is a good enough agreement between theoretical and experimental studies. However, free standing metal gratings possess tiny features attached to sidewalls that significantly affect their optical responses (transmittance, reflectance, and absorptance) at wavelengths in the infrared (IR) region [23]. Additionally, grating slits with different attached features and materials may cause different physical effects (surface plasmon resonance or cavity resonance) that may enhance optical transmittance or absorptance at different wavelength regions [12, 21, 23, 24]. Although optical responses of the free standing gratings in the IR range can be tuned, the structures are not widely used because of the challenges of their fabrication. Therefore, a question is posed whether similar characteristics for layered gratings on a substrate might occur for solar cells in the visible and near IR regions.
Accordingly, this present study performs a careful investigation into the effects of tiny features and grating defects on transverse magnetic spectra of an optimized solar absorber based on a single-layered Si grating on an Si substrate using numerical analyses. The optical performance of the solar absorber at wavelengths in the range of 330 to 1100 nm is first optimized by varying the geometry parameters using a combined numerical method, namely, the rigorous coupled-wave analysis (RCWA) and a genetic algorithm (GA) [25–29]. In calculations, the RCWA method is used to determine the absorptance spectra of the grating, while the GA is used to optimize the grating parameters (i.e., the grating thickness d, the grating period Λ and the filling factor f) in such a way as to maximize the absorptance at wavelengths of interest. Then, the absorptance spectrum of the solar absorber is further analyzed with added tiny features at the top and the bottom of grating sidewalls, and with periodic defects located within fourth and tenth periods. In addition, the absorptance enhancement of all SWG structures is examined by computing magnetic field patterns, energy density, and Poynting vector distributions using the finite element method-based COMSOL Multiphysics [30]. In general, the results show that SWG structures featuring imperfect geometries exhibit higher absorptance than the perfect one due to more energy being trapped in the structure via nanometer-sized features and grating defects positions.
2. Design model and numerical algorithms
2.1 Grating geometry and material
Figure 1(a) illustrates a schematic diagram of the grating structure considered in this study. As shown, the structure comprises a rectangular Si grating on top of an opaque Si substrate with a thickness of 500 nm. This selected structure for optimization possesses a simple profile and a single material for the grating and the substrate, and thus results in fabrication easiness and reduced cost compared to that of 2D and 3D structures. As shown in Fig. 1(a), the geometry of the grating structure is characterized by the grating period (Λ), the grating thickness (d), and the filling factor (f). In simulations, only the transverse magnetic (TM) wave is considered for simplicity. It is incident on the grating from free space at an angle θ between the wavevector k and the surface normal z; H and E indicate oscillation directions of magnetic and electric fields, respectively.
Fig. 1 (a) Schematic diagram of the grating structure of which geometry is defined by the grating period Λ, the grating thickness d, and the filling factor f. The transverse magnetic wave (H) (parallel to the grating grooves or y-axis) is incident on the grating with a wavevector k and an incident angle θ; (b) Optical constants of the Si material used in this study include the extinction index κ and the refractive index n [31]
In computing absorptance spectra of the grating structure, the wavelength-dependent dielectric function as shown in Fig. 1(b), ελ = (n + i*κ)2, of Si is taken from Ref [31]. after interpolation between neighboring data of the extinction index κ and the refractive index n. The optical constants of free space at all wavelengths are known as n = 1 and κ = 0. Figure 1(b) also shows the dependence of κ and n within the wavelength range from 300 to 1100 nm. This range is usually used for solar cells because the solar energy irradiating the ground lies in the range between 295 and 2500 nm [32]. As shown, κ decreases significantly and reaches approximately 0 between 400 nm and 1100 nm. Accordingly, Si is selected for simulations here due to its high refractive index and low losses properties that are suitable for absorbers; Si is also easy to be integrated with other optoelectronic devices.
2.2 Optimization of the grating structure using numerical algorithms
In the present study, the simulation is performed using the RCWA for calculating absorptance spectra of SWG structures, and the GA algorithm in combination with the RCWA for optimizing a solar absorber. Meanwhile, the FEM-based COMSOL Multiphysics is used to plot magnetic field, energy density, and Poynting vector distributions to confirm absorptance enhancement. It is noted that results obtained from the RCWA and FEM were also validated before demonstrations of the enhancement of absorptance.
In optimizing the grating structure shown in Fig. 1, the parameters of interest include the grating thickness (d), the filling factor (f) and the grating period (Λ). These parameters of the SWG were specified under consideration of some theoretical fundamentals which reveal physical origins of enhancing absorptance. The theory is briefly described as follows. First, surface plasmon polaritons (SPPs) propagate along the interface between dielectric and metal media because real parts of optical constants of two media are of opposite signs. The relation of the SPP resonance of the dielectric and metal interface is given by, where j represents the diffraction order; εm and εd are optical constants of metal and dielectric media, respectively [33]. Next, Rayleigh-Wood’s (RW) anomaly also creates some peaks or dips in narrow spectra because there is a diffraction order (j) emerging at the grazing angle. This effect can be predicted by [7, 34]. From the RW equation, the resonance wavelength can be found as λ ≤ 2Λ due to , and thus RW occurs when Λ ≥ 150 nm with λmin = 300 nm. The last physical mechanism is called Fabry-Perot (FP) resonance which is excited to enhance absorptance within structures via coupling of incident light and gratings [12, 35].
As analyzed above, the parameters of the grating structure were selected as follows: the minimum grating period was set as 30 nm according to present nanofabrication technology constraints while the maximum period was defined as Λ < 150 nm in accordance with subwavelength gratings without RW resonance excitation; the filling factor was specified in the range of 0 < f < 1; and the grating thickness (d) was set by the aspect ratio of the thickness-to-linewidth (d/w, w = fΛ), 0 < d < 150 nm, in consideration of fabrication easiness. In optimizing the solar absorber, the GA first generates an initial solution for the grating thickness, the grating period, and the filling factor. The RCWA method is then applied to calculate the TM absorptance spectrum for each solution at wavelengths in the range of 330 ~1100 nm. The absorptance values are then substituted into a fitness function in order to evaluate how close the solution is. If the difference between the calculated absorptance and the ideal one is large, the GA performs its operators to extract or recalculate results [29]. Note that the fitness function as the ideal absorptance spectrum is set to unity at wavelengths of interest and the calculated spectrum (A) is obtained indirectly from the reflectance R, i.e., A = 1-R. It is also noted that the objective of this study is to optimize the SWG structure with the highest absorptance in the whole wavelength region, and thus FP resonance excitation is one of the effects which may provide a good solution.
In addition, the FEM-based COMSOL Multiphysics was used in this study because it has advantages in modeling arbitrary grating shapes and inputting various materials manually or from an available library. This graphical user interface software provides an efficient means of computation because tiny features attached to grating sidewalls make the structure more complex. The FEM method was implemented using a 2D model for one period, and a mesh size based on the relation Δ = λ/10 where λ is the wavelength of incident light. Additionally, the boundary condition in the x-direction was selected as the periodic boundary condition with Floquet periodicity while the port boundary condition was applied in the z-direction (at the top and bottom of the structure). From the ports the reflection and transmission coefficients are directly extracted based on S parameters, e.g., R0th order = and T0th order = [30], respectively.
3. Results and discussion
3.1 Absorptance spectrum of the optimized solar absorber
Figure 2 shows the absorptance spectra of the optimized SWG structure with Λ = 130 nm, d = 90 nm, and f = 0.5 at the wavelength range of 330 ~1100 nm for TM wave with normal incidence and different angles of incidence. Note that the inset in Fig. 2(a) illustrates the TE absorptance spectrum of the optimized absorber, and both TM and TE spectra of plain Si are also plotted by dash lines for comparison with the SWG structure. Note also that the aspect ratio d/fΛ = 1.38 is feasible to fabricate using available techniques [36–38]. From Fig. 2(a), it is seen that the average absorptance of the SWG structure is about 0.86 while that of the plain Si is only 0.62. A similar finding is observed at the TE spectra as shown in the inset figure, but the absorptance is not high enough within the whole wavelength range. At normal incidence, two absorptance peaks appear at wavelengths of 405 nm and 530 nm while a dip occurs at the wavelength of 440 nm. At angles between 0° and 30° as shown in Fig. 2(b), the absorptance is 100% and over 80% at the wavelength range of 500 ~700 nm and at outside of this range, respectively. However, the absorptance decreases when increasing angles of incidence of the incoming light. In general, the results presented in Fig. 2 show that the optimized grating structure exhibits higher absorptance of about 25% than that of the plain Si. Further, its spectrum is insensitive to incident with angles of 0° ~30° at wavelengths of 500 ~700 nm.
Fig. 2 a) Absorptance spectra of SWG Si grating and plain Si for TM waves at normal incidence and b) Contour plot of the absorptance as a function of the wavelength and angle of incidence for TM waves.
3.2 Investigation of absorptance spectra of imperfectly geometric shapes of the solar absorber
3.2.1 Nanometer-sized features attached to grating sidewalls
In this section, the absorptance of the optimized grating structure is further investigated in consideration of nanometer-size features attached to grating sidewalls, which may be occur as a result of nanofabrication. For example, Fig. 3 shows four cases of imperfectly geometric shapes of the optimized absorber whose geometry includes the grating period Λ = 130 nm, the grating thickness d = 90 nm, the filling factor f = 0.5, and the square feature size h = 10 nm. Note that the coordinate system used to analyze is the same as shown in Fig. 1 in which the TM wave is incident from free space on the grating structure with an angle θ. As shown in Fig. 3, SWG structures possess different profiles: symmetric rectangular features attached at the bottom of grating sidewalls (GI), symmetric rectangular features attached at the top and the bottom of grating sidewalls (GII), asymmetric features at the bottom of grating sidewalls (GIII), and rectangular features with their heights h lengthened to 2h vertically at the bottom (GIV).
Fig. 3 Schematic illustration of imperfect gratings with the grating period Λ = 130 nm, the grating thickness d = 90 nm, the filling factor f = 0.5, and the square feature size h = 10 nm. The coordinate system used is the same as that in Fig. 1.
The absorptance spectra of the SWG structures with attached nanoscale features and the absorptance peaks and dips of the optimized grating marked by A, B, and C are shown in Fig. 4(a).The validated results of peaks B and C calculated with various angles of incidence using the RCWA and FEM as examples are plotted in Fig. 4(b). Note that subscripts A, B, and C represent the name of grating types, e.g., AI symbolizes the peak A of the grating I (GI). It is seen from Fig. 4(a) that the absorptance of all imperfect gratings is higher than that of the perfect one, and their spectra almost remain the same with respect to the position of peaks and dips. However, peak AII of GII (the orange solid line with filled triangulars) goes higher and shifts to a longer wavelength, which is due to attaching asymmetric features at the bottom of grating sidewalls. In order to understand underlying physical mechanisms of peaks A and C and the dip B, the magnetic field, energy density, and Poynting vector distributions were calculated and plotted next. Prior to plotting these patterns, the absorptance dip B and the peak C were computed at angles of incidence using the RCWA and the FEM-based COMSOL Multiphysics. The obtained results were in a very good agreement, as shown in Fig. 4(b).
Fig. 4 a) Absorptance spectra of SWG structures at normal incidence and b) Absorptance peaks of the perfect absorber with different angles of incidence calculated using the RCWA and the FEM-based COMSOL Multiphysics.
Following Figs. 5, 6 , and 7 present magnetic fields (top panels), and the time-average Poynting vector and the energy density (bottom panels) at peaks A, AII, and AIV, dips B, BII, and BIV, and peaks C, CII, and CIV, within one period of the perfect, GII, and GIV structures, respectively, obtained by the FEM method. The background contour of the top panels is the y-component intensity of magnetic fields while that of the bottom ones is the time-average energy density. The arrows indicate the Poynting vectors representing the energy flow rate with their directions and magnitudes. Note that GII and GIV are selected to compare with the perfect grating here due to their unique spectrum while the others’ spectrum is not much different, as shown in Fig. 4.
Fig. 5 Near-field patterns of the SWG structures at normal incidence including the optimized grating, GII, and GIV at peaks A (at λ = 405 nm), AII (at λ = 410 nm), and AIV (at λ = 380 nm). Note that the top of Fig. 5 represents magnetic field distributions, and the bottom of Fig. 5 shows energy density and the Poynting vector in one period.
Fig. 6 Near-field patterns of the SWG structures at normal incidence including the optimized grating, GII, and GIV at dips B (at λ = 440 nm), BII (at λ = 450 nm), and BIV (at λ = 430 nm). Note that the top of Fig. 6 represents magnetic field distributions, and the bottom of Fig. 6 shows energy density and the Poynting vector in one period.
Fig. 7 Near-field patterns of the SWG structures at normal incidence including the optimized grating, GII, and GIV at peaks C, CII, and CIV at the same wavelength of λ = 530 nm. Note that the top of Fig. 7 represents magnetic field distributions, and the bottom of Fig. 7 shows energy density and the Poynting vector in one period.
It can be seen from Fig. 5 that the magnetic field at peaks A (located at λ = 405 nm), AII (located at λ = 410 nm), and AIV (located at λ = 380 nm) is confined in the lamella width much more than in the groove grating regions as illustrated by red and yellow. However, the magnetic field at the peak AIV is trapped much more in the attached tiny features and less in the grating lamella widths. Hence, it can be seen that the incident light strongly couples with the grating; this is known as an FP resonance excitation [35]. Additionally, the absorptance enhancement is confirmed by an inspection of the energy density and the Poynting vector plots in the bottom panels. As shown, energy is trapped inside the lamella widths, the top edge of grating sidewalls, and the substrate via grooves of the perfect grating and GII while much energy passes through edges of the bottom tiny features of GIV at corners of the substrate and the grating. More clearly, the Poynting vectors flow through the top nanoscale features, lamella widths and the substrate of GII much more than that of the perfect and GIV ones; as a result, the absorptance peak AII displays the highest value of the absorptance spectra.
Unlike the behavior of peaks A, AII, and AIV, the magnetic field at dips B (located at λ = 440 nm), BII (located at λ = 450 nm), and BIV (located at λ = 430 nm) for TM normal incidence are confined almost entirely in the lamella widths regions of the SWG structures as shown in Fig. 6. Moreover, it is observed that the magnetic field gets trapped more inside the substrate of GIV via grating grooves, marked by yellow. This implies that the energy concentrates in those regions due to the light-grating coupling. It is clearly seen from the bottom panels of Fig. 6 that most energy density and Poynting vector distributions are in the lamella widths of three structures; however, the energy of GIV also passes into the substrate more. Accordingly, the dip BIV exhibits high absorptance compared with dips B and BII.
Similar to Fig. 6, magnetic field, energy density, and Poynting vector distributions at peaks C, CII, and CIV at the same wavelength of λ = 530 nm for TM normal incidence are plotted in Fig. 7. Similar characteristic as in Fig. 6 are seen in which all patterns are located at the same position, i.e., the lamella widths of the grating. As can also be seen, the energy absorbed by GIV having tiny features with a height of 2h is larger than that of the perfect and GII gratings. That is due to tiny features at the bottom of the grating corners. It is observed that the Poynting vectors are guided into GIV via the edges of grating sidewalls and the substrate with attached features more than those of the other gratings. Thus, the absorptance of GIV with higher height features is the highest (see the peak CIV in Fig. 4.) It is noted that the height of the tiny features 2h is an optimal value for practitioners because when increasing this value the peak absorptance will decrease (data are not shown here for simplicity.)
3.2.2 Periodic grating defects
In this section, other imperfectly periodic grating structures are discussed, including the grating GVI lacking one lamella width (marked by the red dash line), the grating GVII with one air groove filled with Si (marked by the red dash rectangular), and the grating GVIII lacking two lamella widths (marked by the red dash line), within the fourth and tenth periods, as shown in Fig. 8.Note that the perfect grating GV is also plotted to compare with other gratings and the numerical calculation as well.
Fig. 8 Schematic illustration of imperfect gratings with periodic defects within a*Λ periods. Note that geometry parameters are the same as the ones for the optimized grating including the grating period Λ = 130 nm, the grating thickness d = 90 nm, and the filling factor f = 0.5; the coordinate system is the same as that in Fig. 1; and the constant a is the number of the grating periods considered in the simulation defined as a = 4 and 10.
Figure 9 illustrates the absorptance spectra of imperfectly periodic structures within the fourth and the tenth periods compared with the perfect spectra calculated for one, four, and ten periods. The grating geometry used to compute the absorptance includes Λ = 130 nm, d = 90 nm, and f = 0.5 at the wavelength range of 330 ~1100 nm for TM wave with normal incidence. Note that in the case of the fourth and the tenth periods the value of the grating period is larger than that of the incident wavelength, and thus their absorptance spectra are calculated for zeroth order diffraction such that they can be compared with that of the subwavelength structures as discussed above.
Fig. 9 Absorptance spectra of imperfectly periodic structures at TM normal incidence: a) in the fourth periods and b) in the tenth periods. Absorptance spectra at zero order diffraction are also calculated for comparison between the fourth and the tenth periods with one period using the RCWA.
It is also seen from Fig. 9(a) that the agreement result of absorptance spectra of the perfect grating for one period (red solid line) and for four period (black dash line) is obtained from the RCWA calculation. It is also observed that GVII has the highest average absorptance of 91% (the light blue dot line) amongst the others, and its spectrum has the same feature as that of the gratings discussed above, e.g., peaks A4VI and C4VII, and the dip B4VII. Figure 9(b) also shows the agreement of results obtained from the RCWA for one period and ten periods. The absorptance spectra of imperfect structures do not fluctuate much as in Fig. 9(a) while GVII calculated within ten periods possesses the highest absorptance (89%). In general, periodic defects within the fourth period of GVII provide highest absorptance for solar cells among many defected gratings in this study.
Figure 10 plots the magnetic field (top panels), and the time-average Poynting vector and the energy density (bottom panels) at peaks A4VI (λ = 400 nm) and C4VII (λ = 600 nm) computed in the fourth periods and at peaks A10VI (λ = 400 nm) and C10VII (λ = 600 nm) computed in the tenth periods of GVII, using the FEM method. It can be seen that the magnetic field at these peaks is much concentrated in the lamella width and the substrate as illustrated by red and yellow. Accordingly, the energy is trapped in those regions and much higher than that in the others, as marked by red in the bottom panels. Note that the Poynting vector also enters in these areas. It is clearly seen from the bottom of Fig. 10 that the obtained energy in the grating with defects within four periods is much more than that one with ten periods. This is also consistent with the absorptance spectra shown in Fig. 9.
Fig. 10 Near-field patterns of GVII with defects at normal incidence in the fourth periods at peaks A4VI and C4VII and the tenth periods at peaks A10VI and C10VII at wavelengths of 400 nm and 600 nm, respectively. Note that the top of Fig. 10 represents magnetic field distributions, and the bottom of Fig. 10 shows energy density and Poynting vectors.
3.2.3 Dependence of absorptance spectra of imperfect structures upon angles of incidence
Figure 11 shows the absorptance contour plots for wavelengths of 300 ~1100 nm at TM incidence for imperfect structures GIV and GVII with defects in the fourth periods as it depends on incident angles of 0 ~80°. It is observed that the absorptance of GIV is maximum at wavelengths ranging from 450 ~750 nm corresponding to incident angles ranging from 0 ~45°. Meanwhile, GVII with periodic defects in the fourth periods exhibits very high absorptance of about 100% in a large range of wavelengths and angles of incidence, marked by a large area of red. Moreover, it is also seen that the energy absorbed (91%) by this defect grating working in a wide range of incident angles is much more than that of GIV with absorptance of 89%. Overall, the results presented in Fig. 11 confirm that the imperfect grating structures with the fourth periodic defects perform more efficiently than the perfect and the imperfect structures with attached tiny features.
Fig. 11 Contour plots of the absorptance as a function of the wavelength and angle of incidence for TM waves of imperfect structures exhibiting efficient performance: a) GIV and b) GVII with periodic defects in the fourth periods.
4. Conclusions
This study has optimized the geometry parameters of a single-layered SWG structure in order to maximize the absorptance in the visible and near IR wavelength regions (300 nm ≤ λ ≤ 1100 nm) using the rigorous coupled-wave analysis (RCWA) and the genetic algorithm (GA). The absorptance of the optimized solar absorber is then investigated with different imperfect geometric shapes that could exist as a result of the nanofabrication process, e.g., the grating with nanoscale features attached to its sidewalls and periodic grating defects within the fourth and the tenth periods. The absorptance enhancement with the SWG structures has been demonstrated with plotting magnetic field, energy density, and Poynting vector distributions using the finite element method (FEM) based on COMSOL Multiphysics. The results have shown that the grating with defects in each fourth period exhibits the highest average absorptance (91%) compared with that of the grating having defects in each tenth period (89%), the grating with attached features (89%), and the optimized one (86%). The FEM analysis results illustrated that the absorptance enhancement offered by the imperfect gratings is due to much light trapping via tiny features attached at the bottom of the grating sidewalls and via grating defect positions. Moreover, the results indicated that the absorptance spectrum of the imperfect structures is practically insensitive to angles of incidence. In general, this analysis provides practical guidelines for researchers working in both theory and experiment in order to manipulate optical responses of grating structures for the best possible performance of future real practical devices.
Acknowledgments
The authors gratefully acknowledge the financial support provided to this study by the Operational Programme Education for Competiveness, the European Social Fund, and the state budget of the Czech Republic with the project named Opportunity for young researchers reg No. #CZ.1.07/2.3.00/30.0016, the Regional Centre of Excellence RMTVC project No. #CZ.1.05/2.1.00/01.0040, and the Grant Agency of the Czech Republic (#P205/11/2137); and by ASPIRE (Applied Science in Photonics and Innovative Research in Engineering), NSERC (Natural Sciences and Engineering Council,) and CREATE (Collaborative Research and Training Experience program) program of Canada. Dr. Nghia also wishes to thank Profs. Y.-L. Lo and Y.-B. Chen (National Cheng Kung University, Taiwan) for useful discussions on the RCWA method.
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