Supplementary Information Supplementary Figures Supplementary Figure S1. Schematic defining the relevant physical parameters in our grating coupler model. The schematic shows a normally-incident light beam that couples to left and right propagating modes of a planar waveguide via a grating consisting of an array of metal stripes. The distance between two neighboring grating elements labeled and is given by. The amplitude of the incident wave is C and the effectiveness of the coupling of normally-incident light to Si waveguide modes is expressed in terms of coupling coefficients and appropriate for coupling to the left and right respectively. The effectiveness with which guided light is decoupled from the waveguide into freespace modes is expressed in terms of the coupling coefficient. Whereas the schematic only depicts the coupling to a single waveguide mode, in real solar cells featuring thick semiconductor layers, the coupling to multiple modes can assist in increasing the absorption of sunlight. 1
Supplementary Figure S2. The total power coupled into the waveguide due to a coupling resonance. Plot of the normalized coupled power into a 50 nm Si waveguide via a periodic array of 50 nm wide and 70 thick Ag stripes. Calculation for TE and TM polarizations are shown. Good agreement is obtained between the direct field calculation in Eq. 6 and the Lorentzian approximation in Eq.14. 2
Supplementary Figure S3. The absorbed power in the semiconductor layer due to a coupling resonance. Fraction of the absorbed power in a 50 nm thick Si waveguide decorated with an array of Ag stripes (blue and green). The same, 50 nm wide and 70 thick, Ag stripes are considered in these calculations as in Supplementary Figure S2. The spacing of the stripes was chosen to be 500 nm. The optical response function in red shows how much each resonance contributes to total absorption. The results are shown for both TE (panel a) and TM (panel b) illumination. 3
Supplementary Figure S4. The shape of the spectral response function. Plots of spectral dependence of the optical response function (blue) which governs the fraction of the absorbed power for a coupling resonance and the guided power in the waveguide (green). These plots are shown for both TE (a) and TM (b) illumination. The spectral location of coupling resonances that effectively contribute to the light absorption are substantially shifted from the spectral regions in which the most power is coupled to guided modes. 4
Supplementary Figure S5. Maps of absorption enhancement comparing performance of periodic and non-periodic arrays. (a) Map of the absorption enhancement in a 50 nm thick Si film vs. the incident photon energy and inverse period of the periodic strip-array for TM illumination. The green curve separates out the frequencies that have a large coupling contribution to the absorption (b) Similar map as in (a), but now for TE polarization. Similar maps as (a,b) plotted for (c,d) Fibonacci (e-f) quasiperiodic and (g-h) random arrays. All of the color bars are on logarithmic scale to the base 10. 5
Supplementary Figure S6. Performance Evaluation of the Waveguide Coupling Model. Increase in the spectral short-circuit current density for our model cell shown in Fig 1(a) of the main text with a quasi-random array of Ag stripes exhibiting an average spacing of 300 nm. The plots compare calculations using the proposed analytical model (red) and the full-field simulations (blue) under (a) TE and (b) TM illumination. The difference between the two methods for TM polarization is due to bare Si absorption (green) and contributions from the surface plasmon resonance of the stripes that are properly taken into account by the full-field simulations. 6
Supplementary Figure S7. Schematic and Fourier Spectra of the non-periodic arrays. Schematic of (a) periodic, (b) Fibonacci, (c) quasi-periodic and (d) Rudin-Shapiro patterns and their corresponding Fourier Spectra (e-h). All structure patterns have same filling ratio, i.e. an equal average spacing between the stripes of 290 nm. 7
Supplementary Note 1 A Semi-Analytical Model for Optimization of Nonperiodic Structures Analytical Modeling of Resonant Waveguide Coupling In this section we describe a model that treats the grating-induced coupling of a normally incident free space beam of light to the optical modes of a planar semiconductor waveguide. There is excellent work in the literature on the topic of grating coupling to waveguides and our model specifically builds on reference 16. Below, we describe the model in detail and list the assumptions and simplifications made to ultimately enable rapid optimization of non-periodic grating structures. Supplementary Figure S1 shows a schematic of the considered geometry and identifies the relevant physical parameters in the model. In the model we consider two types of events: First, wave propagation between two stripes and second, power exchange between the radiative and bound modes via the metallic strips that act as scatter sites. For simplicity we will ignore the inter-coupling between waveguide modes of different order. We start our analysis by considering a wave propagating to the right over a distance D m from a grating element labeled to a neighboring element labeled. Whereas in a non-periodic grating the distances D m can vary, in a periodic grating this distance reduces to 1 single distance that we will define as D. For simplicity, the periodic case will be considered first. If the complex propagation constant for this wave is k, the field for a wave propagating to right will then become:, (S1) where is the amplitude of the wave. As the guided wave reaches the st grating element, part of the energy will be transmitted, part will be reflected, and the rest will be scattered into free-space radiation modes (we ignore coupling from radiation modes back into guided modes in this model). The transmitted part of the guided field will interfere with fields that propagate in the opposite direction reflected from the nd grating and the incoming field with an amplitude C that couples into the guided mode with a coupling efficiency of :, (S2). (S3) Here, t and r are transmission and reflection coefficients respectively and is the phase pick up in coupling free space photons into the waveguide via a stripe. In steady-state the field amplitudes for a uniform periodic grating are given by:, (S4), (S5) 8
where for modes with a transverse electric (TE) polarization and for modes with a transverse magnetic (TM) polarization. We also define to be the magnitude of and. By using Eqs. S1 through S5, the field amplitude at steady state can be written as:, (S6) where and are the real and imaginary parts of the complex propagation constant. The attenuation coefficient has contributions from both scattering and absorption:. Once the coupling and scattering coefficients (,, and ) are known, it is straightforward to calculate the field of the guided wave using Eqs. S1 and S6. By taking the square of the field amplitude given above, it is clear how coupling resonances result with a Lorentzian lineshape. A resonance occurs at an optical frequency at which reflected and transmitted waves propagating in the waveguide add up constructively, i.e. when the phase of the denominator goes to zero. The Lorentzian nature of the resonance can be brought out by first rewriting the expression in Eq. S6 in terms of its modulus and phase: (S7) Next, we express the modulus in terms of the attenuated field amplitude: (S8) where is the scattering loss per period, D, and s is an attenuation constant that captures the loss per unit length due to scattering. With this knowledge, we see that Eq. S6 predicts a simple condition for a coupling resonance condition to occur: (S9) Here, n labels the various coupling resonances that can occur. We can also find a simple expression for the frequency dependence of the coupling resonance. Using Eq.S8, the denominator of the right-hand side of Eq. S6 can be expanded around the resonance frequency ( ) in a simple form: ( ) ( ) In the low loss limit, we find: (S10) ( ) ( ) ( ). (S11) where, is the group velocity. Using this final expression, the square of the field amplitude can be simplified to a Lorentzian form with a full-width at half-maximum of ( ) : 9
( ) ( ) (S12) The last part of this equation defines a linear cross section that captures the effectiveness by which incident light builds intensity in the semiconductor layer. From this equation it is clear that this coupling model produces a collection of Lorentzian resonances with each resonance frequency determined using the condition:. In other words, the propagation constant needs to equal specific spatial frequencies present in the array (apart from a small phase shift) for a resonance to occur:. (S13) The magnitudes of the resonant frequencies dispersion relationship of the planar waveguide. are linked to the propagation constants β n through the Determination of the total power coupled into the waveguide Using Eq. S12, we can find simple expressions for the total coupled power and the power absorbed in the semiconductor waveguide. The coupled power per unit frequency and unit length around each resonance related to a spatial frequency K n in the array can be found using, where is the incoming power per unit frequency per period (D is for normalization of coupling) and the factor of 2 accounts for coupling into both right and left propagating waves: (( )) ( ) (S14) The ratio of the optical cross-section of the strip and the inter-stripe spacing directly gives the fraction of the coupled incident power to the left or right. With expression S14, we can now determine the total coupled power for a specific resonance by integrating over the resonance : (S15) In order to evaluate the validity of this model for the types of structures and waveguides used in the main text, we numerically calculated the amount of power absorbed in a representative 50 nm thick Si waveguide on a silica substrate. Using full-field electromagnetic simulations, we numerically determined the coupling coefficients for a 50 nm wide and 70 thick Ag stripe placed on the Si waveguide with a 20- nm-thick silica spacer layer between the Si and the stripe. The coupling coefficients are determined for a single stripe by calculating the amount of power coupled into the guided modes of the structure for a normally incident plane wave coming from the top. Similarly the scattering coefficient (transmission and 10
reflection coefficients) is determined by calculating the transmitted and reflected amount of field for a guided wave incident on a stripe. Supplementary Figure S2 shows a comparison of the guided power in the Si slab normalized to incoming power per period,, calculated using the direct field calculation in Eq. S6 and the Lorentzian approximation in Eq. S14. The two plots are in very good agreement (lie on top of each other) for both TE and TM polarizations throughout the spectrum. Note that the coupled power at some of the resonances can be over an order of magnitude larger than the incident power. Although the amount of guided power gives some information about the amount of absorption increase due to waveguide coupling, the spectral behavior of these two quantities can be very different due to the strong spectral variation in the materials absorption of a semiconductor, such as Si. In the next section we derive a quantitative expression for the actual absorbed power based using the developed coupling model. Determination of the absorbed power in the semiconductor layer due to a coupling resonance With knowledge of the coupled power, we can also calculate the actual absorbed power in the semiconductor layer. In a similar way to the previous scheme we can calculate the contribution to the total absorbed power of resonance n from knowledge of the frequency-dependence of the absorbed power near that resonance: ( ) ( ) ( ) ( ) (S16) where is the coupled power per unit frequency and per period and determines the amount of power lost per period due to absorption. When we use our previous assumption of low attenuation, such that ( ), then and we find the total contribution of a resonance by performing an integration over frequency: (S17) We can now define an optical response function one resonance by the incident light intensity: by normalizing the absorbed power produced by (S18) This quantity provides the contribution to the absorbed power that is related to the presence of one coupling resonance or spatial frequency K n in the array. The total absorbed power can then be obtained by a simple summation over all resonances: 11
{ } (S19) The summation is required as a periodic array contains several spatial frequencies K n as per Eq. S13. Equation S18 deserves special attention as it determines the total contribution of any coupling resonance to the overall absorption in the semiconductor layer. As all of the coefficients as well as the group velocity in this expression are strongly dependent on the angular frequency, a spatial frequency K n that is present in the array can only induce effective coupling of sunlight over a band of angular frequencies. Typically, this frequency band will be narrow compared to the solar spectrum. For this reason we define an absorption band (AB) in the main text as the spectral region in which waveguide coupling can provide a significant benefit to the overall light absorption in the semiconductor layer. Knowledge of the response function and thus the spectral location of the AB allows for a rational choice for the period of our periodic array (and later the spatial distribution of stripes in a non-periodic array). In order to maximize the light absorption from the sun it is essential to maximize the optical coupling of sunlight in the AB. To illustrate this point, we plot the absorbed power in the 50-nm-thick Si waveguide considered in the main text (normalized to the incoming power) as a function of the excitation wavelength for the previously discussed stripe array under TE and TM illumination (green curves in Supplementary Figure S3a S3b). Note that there are only a few coupling resonances ( ) contributing to the overall absorption and these show up as absorption peaks in the spectral plots. We also plot the response function (Eq. S18) for each polarization assuming a frequency-independent illumination power equal to unity, I( ) = 1 (red curves in Supplementary Figure S3a S3b). It is clear that the response function effectively defines a spectral region in which coupling resonances can make a significant contribution to the overall absorption. We will (somewhat arbitrarily) define the AB as that spectral region within which. For each coupling resonance the area under the Lorentzian-lineshape resonance is given by the function at the corresponding resonance frequency. Comparing Supplementary Figure S3 to Supplementary Figure S2, we confirm that even the strongest peaks in the guided power plots have no contribution to the absorption if they do not exhibit good overlap with the response function. In order to verify our semi-analytical model we performed Finite-Difference Frequency-Domain (FDFD) simulations on the same structures. Comparing the plots of absorbed power in Supplementary Figure S3a, a reasonable match between the analytical and numerical plots is observed for the location, width, and magnitude of coupling resonance peaks. The broad background in the numerical plot is due to the absorption in the Si thin film that is not related to waveguide coupling/light trapping. The small peak at 2.9eV is a second order waveguide mode that does not show up in the analytical calculation as in this example we only accounted for the fundamental mode. 12
Supplementary Figure S3b also shows good agreement between the numerical and analytical calculations for TM polarization. Besides the waveguide peaks, numerical plots have two extra features. First, there is a broad feature near the surface plasmon resonance at 2.1 ev nm, which is caused by enhanced absorption due to increased near-fields close to the metal stripes. Second, there are two relatively weak and sharp peaks at 2.45 ev and 1.65 ev that are caused by light coupling at the air light line and glass light line (i.e. related to diffracted modes). Even though there is a strong absorption in the Si at short wavelengths and we are at the limit of our low loss regime approximation (( ) ), the analytical model captures the contributions to the absorption from all of the coupling resonances sufficiently well to allow for rapid optimization of spatial distributions. In this section we showed that there can be a striking difference between the spectral distribution of the coupled sunlight (as determined in section S1c) and the light absorbed in the semiconductor waveguide. This shows that obtaining large absorption enhancements at some frequencies is not sufficient to realize a good light trapping structure (despite the fact that large absorption enhancements may sound impressive at first). It is clear that a careful optimization of the light trapping array is required to maximize the optical coupling in the AB. Only then robust contributions to absorbed sunlight can be made by coupling to waveguide modes. In the next section we discuss the parameters that determine the shape of the spectral response function. The shape of the spectral response function In the previous section we showed that the integrated absorption contributed by each resonance is directly related to the guided power (Eq.S15) where the difference between these two quantities is dependent on the materials absorption and modal properties of the semiconductor waveguide. Supplementary Figure S4 compares the two quantities for different illumination polarizations. Clearly the spectral dependence is quite different for TE and TM polarizations. Looking at Eq.S18 there are several factors that determine the spectral shape of : the coupling coefficient, the scattering coefficient, the group velocity, and the grating period. The coupling and scattering coefficients and v g depend on the waveguide mode and the polarization state of the excitation. Under TE illumination (E-field directed along the length of the strips, surface plasmons are not excited) the coupling coefficient is solely determined by the overlap of the field distribution of the relevant waveguide mode with the metallic stripes. As a result, the coupled power is maximized at low energies (1.1-1.5 ev) where the Si absorption and refractive index are smallest and the overlap is largest (Supplementary Figure S4a). The spectral dependence of the absorbed power then results from the balance between a decreasing amount of guided power and an increasing amount of materials absorption with increasing energy. The interplay between these quantities causes the coupling 13
resonances that mostly contribute to the overall absorption to be located 0.3 ev away ( max ~ 1.75 ev) from the wavelength at which optimum coupling occurs. Under TM polarization on the other hand (Supplementary Figure S4b), one can excite surface plasmons on the stripes, which enhance the scattering cross-sections near the surface plasmon resonance frequency. This results in very large coupling coefficients. The coupling coefficient for TM polarization has a maximum around the stripe resonance (~ 2.4 ev) with a quite narrow-band spectral response. For this reason an efficient coupling occurs only over a narrow band of the spectrum. The coupling resonance also has a linear dependence on the group velocity. For the SOI structure considered in the main text, the TM waveguide mode is cut off near 1.5 ev. The group velocity strongly increases from 3 ev to 2eV which extends the region of strong coupling (guided power) to lower energies. Despite this fact, the strongest contributions to the absorbed power still occur at relatively high energies (2.4 ev) due to the sharp increase in the materials absorption of Si at higher photon energies. The shape and bandwidth of the response function plotted in Supplementary Figure S4 (Blue curves) determines the spectral region in which coupling resonances significantly contribute to the total absorption. By engineering the geometry of the stripes and their position with respect to the Si waveguide layer, it is possible to shift the surface plasmon resonance of the stripe and to maximize the interaction of the guided mode with the stripes. In this way, we can control the coupling and scattering coefficients in a real solar cell. In addition, by changing the thickness of the semiconductor layer we can selectively determine the number and properties of the waveguide modes and further optimize the coupling and light absorption. Based on the discussion above, it is clear that the optimization of a solar cell structure can proceed in two consecutive steps. If a planar cell structure is known, one can tailor the response function based on a choice of scattering particle and a choice of its placement with respect to the active semiconductor layer of the cell structure. These properties set the spectral response function of that defines the AB. When the response function is known, we can subsequently optimize the spatial arrangement of the scattering particles in periodic and non-periodic arrays. This is the topic of the next section. Spatial Engineering of Coupling Resonances in Non-Periodic Gratings The previous sections describe the definition and frequency-dependent nature of the response function. Knowledge of the response function enables one to calculate the contributions that different spatial frequencies in a light trapping structure can make to the total amount of absorbed sunlight. Thus far, we have only considered the relatively simple case of a periodic array of stripes. However, we could have gone through the mathematical derivation to derive the frequency-dependent nature of coupling resonances by letting go of the requirement of an equal spacing between stripes. By looking at Eq.12, it is 14
clear that the magnitude of the coupled optical field is solely determined by the scattering properties of a single stripe and the waveguide propagation loss per unit length, = s + a, the group velocity of light in a waveguide and the stripe density of stripes D. The magnitude of is a property of a single stripe and thus independent of the spatial distribution of the stripes. The other physical quantities are all the same when the density of stripes is equal. As such, the only difference between the periodic and non-periodic arrays in the low-loss regime is the location and relative contribution of different spatial frequencies to the waveguide coupling. This conclusion that Eq.12 holds for non-periodic arrays could also be argued by applying the Array theorem in Fourier optics. This theorem states that we can find the scattering properties of any array (periodic or non-periodic) of identical stripes by convolving the scattering function of the stripe with an appropriate array of delta functions sitting at the origin of the stripes. Here, the scattering function describes the complex amplitude of the field near the stripe and the delta function describes their spatial organization in the array. This theorem (and the previous argument) holds as long as the scattering properties of the stripes are unaffected by interaction between the stripes. Intuitively this will hold when the stripes are not too close (empirically less than 30 nm from our simulations) together and in practical situations the validity can be checked by full field simulations. The above discussion essentially argues that we can use the same response function for stripes in periodic and non-periodic arrays. As such, the difference between periodic and non-periodic arrays boils down to the difference in the spatial distribution of the stripes. Similar to our procedure for a periodic array, we now determine the total amount of absorbed sunlight for any non-periodic array by summing the contributions from all of the spatial frequencies, K present in the array. These contributions should (as before) be weighted by the optical response function. For a non-periodic array, we now also need to weigh every contribution by the amplitude squared of the spatial Fourier transform for the relevant spatial frequencies. Moreover, if the solar cell supports multiple optical modes i, we will need to perform this summation for each modes. The total amount of absorbed sunlight thus becomes: ( ) ( ) (S20) This is equation is Eq. 1 in the main text and plays a central role in our paper. 15
Supplementary Note 2 Determining the Ideal Non-Periodic Pattern Engineering the Right Degree of Randomness to Maximize the Absorption of Sunlight In the main text we demonstrated that moving away from a perfectly periodic grating by introducing some randomness can lead to a desirable increase in the absorption efficiency from 40% to 55% (Fig. 1b). In this section we investigate the changes in the overall absorption that occur when an increasing number of spatial frequencies are introduced into a grating structure. To this end, we specifically chose to analyze the light trapping ability of arrays that feature very different spatial distributions: a periodic, a Fibonacci, a quasi-periodic array realized by randomly displacing the metallic stripes, and a virtually random Rudin- Shapiro pattern. A detailed procedure for generating these patterns is given in Supplementary Note 3. A convenient way to qualitatively and quantitatively show the impact of introducing randomness on the light absorption is by generating the absorption enhancement maps described in the main text (Fig. 3a-d) and in references 19 and 22. Supplementary Figures S5a-h show such maps generated using fullfield electromagnetic simulations for the four grating patterns mentioned above. For each pattern, we show what happens for both the TM and TE polarization. By investigating the trends in these maps an intuition for the optimum spacing and degree of disorder can rapidly be acquired visually. The maps exhibit two distinct features that can be correlated with large enhancements. The relatively dispersionless (i.e. flat), broad yellow bands in the TM maps result from an effective excitation of the surface resonance of an individual stripe. Their occurrence is not associated with a specific array configuration and is primarily related to the stripe geometry and dielectric environment. A slight redshift is observed with increasing G = 2 /D (i.e. decreasing particle spacing, D) because of the increased interparticle coupling. The sharper, dispersive features that are seen in both TM and TE maps are related to effective coupling to waveguide modes. To analyze the possible contributions of the various coupling resonances to the overall absorption, we also outline the AB by a set of green curves. From the plot it can be seen that regions exhibiting large absorption enhancements are not necessarily located in the AB. It shows that the often followed approach of generating structures that show large absorption enhancements at specific wavelength most likely will not give rise to the best broadband absorption across the solar spectrum. To accomplish this goal, it is essential to first identify the AB and then to make sure that strong coupling is obtained in the AB. The metallic stripe size and cross sectional shape primarily determine the spectral location of the broad yellow bands in the TM maps related to the excitation of the surface resonance of the individual stripes. It is by now well-established that wider and thinner stripes produce resonances at lower energies 30. In this paper, we focus on the unresolved challenge of optimizing the spatial arrangement after 16
the stripe geometry (or other scattering particle) has been chosen. To this end, we will explore which spatial arrangements produce the largest absorption enhancements in the AB. Supplementary Figure S5 shows that by going from a periodic to a non-periodic Fibonacci grating we introduce additional spatial frequencies for each average stripe separation D or G = 2 /D. From the maps it is clear that the Fibonacci grating provides more strong coupling resonances that lie within the AB for both TE and TM polarizations (Supplementary Figure S5c-d). Next we will look at a quasi-periodic grating pattern which is realized by randomly moving the stripes of the periodic grating to new positions with the displacements spread according to a Gaussian distribution. Here, the degree of disorder is controlled by changing the variance of the distribution. Unlike the periodic and Fibonacci patterns which exhibit a number of well-defined coupling resonances, this non-periodic grating exhibits many spatial frequencies at each photon energy. This results in a broadband increase in absorption (Supplementary Figure S5e-f) throughout the optical spectrum, albeit with smaller coupling efficiencies at each resonance (due to smaller Fourier amplitudes). As we integrate the absorbed flux over all spectral frequencies, we find that Fibonacci and quasi-periodic structures provide overall increases of 20% and 15% respectively over the periodic array with optimized average stripe spacing. This is expected as it is clearly seen in Supplementary Figure S5a-b that the periodic structure exhibit resonances that lie right outside of the AB, and a subtle change in the spatial frequencies leads to an increased overlap with AB region. For case of the virtually random Rudin-Shapiro gratings, the spatial frequency spectrum consists of a more-or-less continuous spectrum of spatial frequencies. As a result the coupling resonances occur well beyond the AB and the coupling resonances within the AB exhibit significantly decreased amplitudes. From this analysis it is clear why a too high degree of randomness is undesirable and in fact leads to an increase of only 35% in the overall absorption and about 5% smaller than the periodic grating. A final note is that the optimum absorption for periodic, Fibonacci and quasi-periodic arrays happen to be around the same average separation (280 nm - 290nm). This is expected as the same response function determines the contribution of the resonances for all patterns. Non-periodic structures tend to feature a slightly larger optimum spacing due to relatively higher reflection and absorption losses seen for metallic structures that are spaced more closely than the average spacing. In summary, the absorption enhancement maps visualize that it is essential to choose a stripe distribution with optical resonances in the AB to maximally benefit from coupling to waveguide modes. To obtain the optimum stripe distribution one could first look at patterns that have promising Fourier Spectra that produce coupling resonances in the absorption band and then make a quantitative numerical comparison by generating absorption maps and evaluating the integrated absorption as a function of G. For periodic grating structures these maps can quite quickly be produced by taking advantage of periodic 17
boundary conditions in the full-field simulations. For non-periodic structures, it is extremely time consuming to generate absorption enhancement maps with full-field simulations as one cannot exploit periodic boundary conditions and large structures need to be simulated for each point in the map. For this reason, optimization of non-periodic structures becomes unfeasible using full-field simulations. Instead, we can use the presented semi-analytical model to realize the same absorption maps (with the approximations/limitations discussed above). As a result, we can perform a rapid, first-round optimization of different types of array structures using our semi-analytical model and ultimately verify the performance of the optimized structures using only one full-field simulation. In the next section we analyze the limitations of our coupling model. Performance Evaluation the Waveguide Coupling Model To show how well our model works and also to illustrate its limitations, we compare the predicted spectral short circuit current (SCC) density obtained with the model and full-field simulations for the quasi-periodic pattern discussed in the main text and generated by randomly displacing the metallic stripes in a periodic array. In order to compute the SCC we integrate the product of the absorbed power per unit wavelength and the solar irradiance assuming unity internal quantum efficiency (i.e. an electrically ideal cell). Supplementary Fig S6 shows the comparison for TM and TE polarizations respectively. For this polarization, we observe strong coupling to waveguide modes and sunlight absorption at short wavelengths. The difference between the spectra obtained with the semi-analytic model and the full-field simulation is due to the fact that the semi-analytical model only considers absorption enhancements due to waveguide coupling (i.e. light trapping). It does not capture the regular absorption in the Si slab without waveguide coupling and local field enhancement effects due to the excitation of a surface plasmon resonance of the stripe. The regular absorption of a Si slab (without the presence of the grating) is also shown (red curve). It is clear that in the spectral region where the bare Si slab is most strongly absorbing and near the surface plasmon resonance at 2.3 ev the difference between the numerical and model calculations is the largest. It is worth pointing out that there is a sufficiently good match between the model and the numerical results to allow for quick quantitative estimates of the SCC and comparisons between different structures. Whereas the magnitude of the calculated SCC may not be exact, making performance comparisons between different gratings featuring different spatial distributions of stripes is quite accurate as the contributions to the total absorption from non-coupled absorption and local plasmonic resonances of the stripes are largely independent on the spatial distribution of the stripes. As such we can focus on comparing the absorption enhancements related to coupling to waveguide modes. This is a commonly used procedure to estimate the effectiveness of light trapping layers 6. It is important 18
to also note that for thicker solar cells (micron and thicker), these plasmonic enhancement effects only play a very minor role in enhancing light absorption and optimum arrays can be determined even more accurately. The full-field simulations and coupling model predictions under TE illumination also match quite well with each other. Both the locations and amplitudes of the peaks in the short circuit current density are very well matched. Based on the good agreement between the full-field simulation and the quasianalytical coupling model, one can also generate absorption enhancement maps using this model. With these maps, one can again follow the optimization of the light trapping structures as for the maps generated by full-field simulations. As shown in the main text, this approach is very effective in rapidly identify promising non-periodic structures that can ultimately be verified using full-field numerical simulations. Supplementary Note 3 Generating Non-Periodic Arrays by Means of Inflation Rules. The Fibonacci Pattern A Fibonacci pattern/grating is a non-periodic structure generated using Fibonacci binary sequence 29. In mathematics, the sequence S n for a Fibonacci series starts with S 1 = A and S 2 = AB and defined by the famous concatenation rule S j = S j-1*s j-2 for each generation j > 2, where * denotes concatenation. Here, A and B are the two building blocks of the sequence. For example, if we start with a seed A, the generations up to five are thus going to be: A, AB, ABBA ABBABAAB, ABBABAABBAABABBA, etc. A Fibonacci structure can be formed by combining two different optical materials or building blocks according to the recurrence relation defined above. In our case, we form a grating structure in which A and B correspond to blocks of air and air/metal respectively, each having the same block width (Supplementary Figure S7b). The metal width is always kept constant so changing the block width changes the filling ratio of the metal stripe in the metal/air block. In calculations we used the 9 th generation of the sequence where the total number of elements is N(S 9) = 64 with 34 blocks of A and 30 blocks of B. In order to compare with different patterns we also calculated the average stripe separation. For instance a block width of 110 nm has an average particle separation of 290 nm. In numerical simulations we used periodic boundary conditions on both ends of the larger non-periodic structures. Supplementary Figure S7f shows the Fourier Spectra of the structure with periodic boundary applied for a block width of 110 nm. We find that using a longer series (i.e. a larger generation number) 19
does not significantly change the Fourier amplitudes, as expected for quasicrystals. Note that we calculate the Fourier transform of delta functions located at the center of each metal particle. In the calculations the volume and dielectric properties are already included in the coupling and scattering coefficients, therefore the array properties are only determined by the delta functions placed at each metal particle location. The Quasi-periodic pattern We form the quasi-periodic grating 31 by randomly moving the metallic stripes of a periodic structure to a new position within a Gaussian distribution. A 48 element structure was used in simulations with periodic boundary condition at the edges. Supplementary Figure S7c shows a small portion of such a pattern with a Gaussian distribution exhibiting a variance of 100 nm. For this pattern the degree of randomness can be controlled by changing the variance of the distribution. The spatial frequency components are shown in Supplementary Figure S7g. Although the displacements are random in nature, there are relatively sharp features in the Fourier transform that are due to the finite number of random elements used in the simulations. The Rudin-Shapiro pattern The Rudin-Shapiro pattern has a deterministic nonperiodic sequence generated using a four letter substitution rule to move from one generation of the sequence to the next 32, 33 : a ab, b ac, c db, d dc Starting with letter a (zeroth generation) generations up to four are: a, ab, abac, abacabdb, abacabdbabacdcac etc. Although the sequence has four unit cells, we only consider a binary form here by substituting every a and b with an A and every c and d with a B, which gives the final sequence: A, AA, AAAB, AAABAABA, AAABAABAAAABBBAB etc. Supplementary Fig S7d shows a small section of a structure consisting of 34 elements (5 th generation) where A is air and B is a combination of air and metal. The Fourier transform of the structure is shown in Supplementary Fig S7h for a unit cell width of 145 nm and average separation 290 nm. It is practically flat. For this reason, the Rudin-Shapiro mimics a purely random grating structure with a white Fourier spectrum. There is one spatial frequency component that stands out at K = 0.044nm -1. This spatial frequency corresponds to the inverse period of the sequence unit cell. The amplitude of the peak gets smaller and frequencies further spread out for bigger generation numbers (longer sequences) making it closer and closer to a purely random structure. We kept the structure size limited to 64 elements in order to manage our simulation times. 20
Supplementary References 29. Janot, C., Quasicrystals: a primer. 2nd ed.; Oxford University Press: Oxford, U.K., 1997. 30. Barnard, E. S., White, J. S., Chandran, A., Brongersma, M. L. Spectral properties of plasmonic resonator antennas. Optics Express 16, 16529-37 (2008). 31. Xing, B., Liu, H. C. Simulation of one dimensional quasi random gratings for quantum well infrared photodetectors. Journal of Applied Physics 80, 1214-1218 (1996). 32. Shapiro, R. S. Master, Massachusetts Institute of Technology, 1951. 33. Rudin, W. Some theorems on Fourier coefficients. Proc. Amer. Math. Soc 10, 855-859 (1959). 21