Understanding the nano-photonics absorption limit in both front-side and front/rear-side textured slabs

Surface texturing is one of the main techniques to enhance light absorption in solar cells. In thin film devices, periodic texturing can be used to excite the guided resonances supported by the structure. Therefore, total absorption is enhanced largely due to the excitation of these resonances. Although the maximum absorption enhancement limit in both bulk and photonic structures is known already, the weight of each resonance type in this limit is not yet clear. In this contribution, we extend the temporal couple-mode theory, deriving a closed formula to distinguish the contribution of Fabry-Perot and wave-guided modes within the absorption limit for 1-D grating structures. Secondly, using this analytical approach, we can clearly address cases of bulk and thin absorber thicknesses. Our results, supported by rigorous electromagnetic calculation, show that absorption enhancement in a 1-D grating structure can be much higher than the nano-photonic limit (2πn) reported by Yu et al. Thirdly, beyond the framework put forward by Yu et al., we extended our theory to describe the absorption enhancement in double side textured slabs. We have found that when the periods of top and bottom gratings are aliquant, absorption is enhanced in a wider frequency range. We provide rigorous numerical calculations to support our theoretical approach.

Understanding the nano-photonics absorption limit in both front-side and front/rear-side textured slabs 1

. Introduction
The performance of solar cells is strongly linked to their light absorption capabilities [1]. Therefore, a significant amount of research has been dedicated to different light management techniques, in order to improve light absorption in the device active layer [2,3]. Light management is mostly focused on (i) the effective use of solar spectrum [4][5][6][7][8][9][10], (ii) maximizing the number of photons reaching the absorber layer [11][12][13][14], (iii) keeping the solar radiation energy inside the absorber long enough, until it is absorbed by the material. The first aspect can be achieved by employing spectrum splitters and multi-junction devices [15][16][17][18]. The second aspect is accomplished by applying anti-reflection coatings and by minimizing light absorption in non-active layers [19][20][21][22][23][24]. The third aspect, usually referred to as light trapping or optical path enhancement, is attained by altering the propagation direction of light such that total internal reflection inside the absorber occurs [25]. This is possible only if one or both surfaces of the absorber are textured. After hitting an interface with random texturing, incident light is scattered into propagation directions with a wide angular range [26][27][28]. The greater the scattering angle, the larger the optical path length enhancement. Optical path length enhancement is defined as l d , with d representing the absorber physical thickness and cos l d θ = , where θ is the scattering angle [29]. In 1982, Yablonovitch elaborated that in a weakly absorbing slab with a 2-D Lambertian scattering interface, an optical path length enhancement of 2 4n can be achieved (where n is the refractive index of the absorber material) [30]. The angular intensity distribution (AID) of a Lambertian scatterer obeys Lambert's cosine law [31] and has an isotropic angular response, independently of the light incident angle. In defining the 2 4n limit, the material absorption is considered to be very small, because the angular intensity distribution (AID) follows Lambert's cosine law only if the slab is transparent (weakly absorbing). Furthermore, the thickness of slab is considered to be much larger than the incidence wavelength, and thus the light path can be defined in the framework of geometric optics. The 2 4n limit which is based on statistical ray optics, is nowadays commonly referred to as the Yablonovitch limit or Lambertian limit. The Lambertian limit holds for any angle of incidence from 2 π − to 2 π + (acceptance angle π ) with zero being the normal and 2 π ± grazing to the slab surface. By decreasing the acceptance angle of the cell from π to 2β, it is possible to increase the optical path enhancement from 2 4n to 2 4 sin2 n β [32,33]. The limit of 2 4 sin2 n β , commonly employed in concentrator photovoltaic, is valid for bulk structures and can easily be calculated using geometrical optics. On the other hand, when the texturing size or the absorber layer thickness is comparable to the wavelength, wave effects of light become prominent [34]. Thus, some of the basic assumptions leading to conventional limit are no longer applicable. In 2010, Yu et al. showed that -in the wave optics regime -the light absorption enhancement factor can go far beyond the conventional Lambertian limit, using wavelength-scale grating structures [35,36]. This absorption enhancement (F), which is called the nano-photonic limit, was derived using statistical temporal coupled-mode theory and can be calculated using: where M and N are the total number of resonances and reflection (leaking) orders supported by the structure in the frequency range ∆ω, respectively; d is the absorber thickness; i γ is the decay rate and it is linked to the material absorption coefficient α via i c n γ α = , where c is the speed of light in vacuo and n is the real part of material refractive index. At each frequency range, M and N are strongly dependent on grating symmetry, arrangement of unit cell and thickness of the absorber. At each frequency, a grating waveguide supports many modes. Each mode at each frequency in the waveguide propagates in different angle (different propagation angle). On the other hand, the grating scatters the incidence energy into many different angles (diffraction angles). If, at one particular frequency, a diffraction angle is equal to the propagation angle of a mode, then that mode can be excited via the grating. Both diffraction and propagation angles change with the light frequency. Therefore, in a perfectly designed grating structure, we can assume that each diffraction order can excite at least one resonance (mode) within the frequency range ∆ω. Therefore, the total number of resonances is the sum of resonances excited by different diffraction orders. Generally speaking, each diffraction order has different efficiency, meaning that the energy of incident light is not distributed uniformly among them. Consequently, each diffraction order has a different contribution to total absorption. For this reason, calculating the maximum absorption enhancement factor for each diffraction order is of great importance in grating design [37][38][39], especially for thin-film solar cells, where the absorber thickness is too small to efficiently absorb photons with energy close to the absorber band gap [40]. Among all different materials used in thin-film solar cells, (nano)-crystalline silicon (Si) is a good bench mark to study different light trapping schemes, due to its low absorption coefficient in the near infrared part of the spectrum [41].
In this paper, we apply the statistical temporal coupled-mode theoretical approach to calculate the maximum enhancement factor for each type of resonance in a grating structure. The maximum absorption enhancement for 2-D grating structure varies according to the dimensions (lattice constant) and arrangement of unit cell (hexagonal, square, triangle…). Therefore, for 2-D grating, it is not possible to write down a closed formula independent of grating arrangement and unit cell dimensions. For this reason, in this article we only deal with one dimensional (1-D) grating structures. For our analysis, we consider normal incidence and only one polarization. Total number of resonances is then multiplied by two in order to consider two polarizations. This work is divided in two parts: first, we investigate the case of texturing at one surface. Then, we extend our calculations for double side textured structures.

Distinguishing the contribution of Fabry-Perot modes and wave-guided modes
Consider a periodically corrugated dielectric slab with thickness d and periodicity L, made of Si. The slab is placed on a perfect mirror and it is illuminated from above under normal incidence ( Fig. 1(A)). The incident plane wave, after hitting the air-Si interface, scatters into many distinct propagating directions inside the slab. The scattered waves bounce along the z and x directions, exciting resonances. In such a structure, the value of the wave-number of resonances inside the film is not continuous, but is described by the following equation: where, ω is the angular frequency of light. Each p and q pair correspond to one resonance. The index p (q) refers to resonances excited in the z (x) direction, due to the thickness of the slab, d (period of the grating, L). Any frequency lower than ω ( ± 1,0) (cut off frequency) cannot propagate inside the structure. In a frequency range [ω, ω + ∆ω], the total number of resonances supported by a 1-D structure can be calculated as [42]: 2 2 area of circle with radius ... area of a resonance ...
where c is the speed of light in vacuo, P is the total number of polarizations that is taken into account ( 1 P = for polarized light, TE or TM, 2 P = for non-polarized light). In this work, we only consider one polarization, therefore, 1 P = throughout this article.ψ is grating symmetry coefficient and has only two values, 1 ψ = for asymmetric and 1 2 ψ = for symmetric grating. In this context, an asymmetric grating is characterized by a grating whose cross-sectional period does not have mirror symmetry. According to reference [36] (2)). In the frequency range between [ω, ω + ∆ω | ω > ω (1,0) ] the number of pure FP resonances can be calculated by: According to Eq. (2), any frequency lower than cut-off frequency cannot propagate inside the structure. Therefore, Eq. (5) is valid only for ω > ω (1,0) . Now that we can count all the resonances and distinguish their origin (guided or Fabry-Perot), the total enhancement factor for each type of resonance can be calculated, using Eq. (1). If we determine M q using Eq. (4) and then substitute the obtained value in Eq. (1), we can calculate the total enhancement factor for q th diffraction order as: where N = 2.ψ. s + 1 is the number of channels (reflection orders) for this structure, and x represents the largest integer that is smaller than x. Equation (6) shows the enhancement factor for a particular guided resonance, when ( ) ( ) factor for pure FP in an asymmetric grating can calculated by: The numerator indicates that a FP resonance enhances the light path 2 times. Equation (7) is only valid for ( ) s L d n > ⋅ , which is the cut-off frequency of the structure. Absorption enhancement due to FP resonance decreases as frequency increases. The symmetry of grating does not affect the total number of excited FP resonance (Eq. (5) has not dependency onψ ) but it affects absorption enhancement due to FP resonance. As an asymmetric grating could couple a FP resonance into resonances with odd or even modal amplitude profile, a FP resonance in such a grating structure has more chance to escape the system. This means that for s ≥ 1, absorption enhancement due to FP resonance is smaller in asymmetric grating than in symmetric one. For normalized frequency 0 ≤ s < 1, Eq. (7) shows identical result for symmetric and asymmetric gratings. The total (cumulative) enhancement factor is obtained by adding the enhancement factor for all diffraction orders and the pure FP resonances: Using Eq. (8), we propose the weight of Fabry-Perot contribution to the enhancement factor (W FP ) as: We shall use this aggregated metric in the remainder of this contribution to quantitatively distinguish the impact of Fabry-Perot resonances on total absorption enhancement factor.

Extending the temporal couple-mode theory to different thicknesses
According to Eq. (2), the wavenumber must reach the minimum value , in order to excite the lowest mode (p = 1) of the q th diffraction  (8) for an overly thick slab with d = 1 mm (Fig.  3(A)) is presented in Fig. 3(B). The black curve represents the cumulative absorption enhancement in a film with the physical thickness of 1 mm ( ( ) 4 1.5 10 L n d − ⋅ = × ) with n = 4, endowed with 1-D asymmetric texturing. At s = q / n, a new diffraction order appears. Therefore, a peak in the guided mode enhancement (grey curve) and in the total enhancement (black curve) is observed. Since the diffraction angle decrease as frequency increases, the enhancement drops dramatically until s = (q + 1) / n, where a new peak appears. At s = 1, the number of channels (reflection orders) increases from 1 to 3 and therefore the enhancement drops suddenly. In Fig. 3(C) we present W FP related to the structure shown in Fig. 3(A). For s < 0.25 (considering s = q / n, with n = 4 and q = 1), no diffraction order exists and all the absorption enhancement is due to FP resonance, therefore, F TOT = F FP and W FP = 100%. At s = 0.25 the first diffraction order arises. The absorption is significantly enhanced due to presence of such first diffraction order (see the black curve at s = 0.25 in Fig. 3(B)), whereas F FP does not change (see the blue curve in Fig. 3(B), for 0 < s < 1, F FP does not vary). As F TOT = F FP + F q = ± 1 , where F q = ± 1 >> F FP , W FP drops suddenly. F q = ± 1 decreases as the normalized frequency increases and, therefore, W FP grows rapidly until reaches its maximum (around 30%) at s → 0.5. For s = 0.5 another diffraction order arises and thus W FP drops again. This sequence repeats: for s approaching q / n, W FP is peaks and for s = q / n the W FP is minimum.  So far, only asymmetric grating has been considered. The number of channels are also influenced by the symmetry of the grating and becomes N sym = s + 1. Therefore, the maximum enhancement achieved by symmetric grating would be half of asymmetric one. The graph in Fig. 4(B) shows the absorption enhancement in a symmetric grating structure with the unrealistic thickness of 1 mm (identical thickness to the structure in Fig. 3(A)). One can clearly see that the maximum absorption enhancement is half of the one for asymmetric grating (Fig. 3(B)). Thus, for a film with d → ∞, we observe that (a) the maximum theoretical enhancement tends to infinite for both symmetric and asymmetric gratings and (b) the symmetry of the grating profile does not play any role in defining maximum absorption enhancement. In Fig. 4(C) the W FP related to the structure shown in Fig. 4(A) is presented. Comparing Fig. 4(C) and Fig. 3(C), one can observe that in a structure with symmetric grating (Fig. 4(A)) the W FP is larger than in a structure with asymmetric grating (Fig. 3(A)). This is due to the fact that a structure with symmetric grating has smaller F TOT than a structure with asymmetric grating, while both structures have the same F FP . Therefore, W FP is larger in structure endowed with symmetric grating.

Maximum enhancement frequency in bulk regime
The maximum enhancement frequency for bulk structures (i.e. the frequency at which the enhancement is maximum) occurs when the denominator in Eq. (6) tends to zero, i.e. s → q/n. On the other hand, for s ≥ 1 the number of channels increases, which results in a drop of enhancement. From these two conditions, we can conclude that s max can be calculated using equation below: The refractive index n can be any positive real number, whereas q can only be an integer. Therefore, depending on whether n is an integer or not we get two cases: (i) if n is an integer the maximum frequency occurs for q = n -1; (ii) if n is not an integer, s max occurs for q = n.
In this work, since we assume n = 4 for both symmetric and asymmetric structures, the maximum enhancement occurs at s = 0.75, where the third diffraction order has the largest propagation angle and there is only one channel (reflection order).

Maximum enhancement in thin film regime
So far, we have discussed the case of a film with a very large thickness. As mentioned earlier, when L/d increases, the lowest frequency to excite the first resonance for a diffraction order is increased. Therefore, the peaks in Eq. (6) do not occur at s = q/n, but rather at . This situation is illustrated in Fig. 5(B) where the absorption enhancement for an asymmetric grating structure with thickness of 1 μm is shown (Fig. 5(A)). The maximum absorption enhancement occurs at around s = 0.76 rather than 0.75. Additionally, since the structure is thin, its cut off frequency occurs at higher frequency. This can be observed in Fig. 5(B) where F FP ≠ 0 for s ≥ 0.15 rather than s = 0. Although, changing the thickness influences the cut-off frequency, it does not affect the W FP much . In Fig. 5(C) the W FP is only slightly shifted to the right with respect to Fig. 3(C), but its maximum and minimum values remains unchanged. The absorption enhancement is lower for a thin film with a symmetric grating. The graph in Fig. 6(B) presents the absorption enhancement achieved in a 1000-nm thick film with a symmetrical grating (Fig. 6(A)). Although symmetric gratings have much lower maxima than asymmetric gratings, it still higher than what is predicted by Yu et al., but for a small frequency range. The graph in Fig. 6(C) illustrates the W FP related to the structure shown in Fig. 6(A). As it has been elaborated earlier, changing the thickness does not influence the maximum or minimum values for W FP , but the W FP is shifted to left or right for decreasing of increasing the thickness, respectively. To test our theoretical analysis, we rigorously calculate the absorption enhancement using a symmetric and an asymmetric grating structure shown in Fig. 7(A) and Fig. 8(A) respectively. Both gratings have a duty cycle of 50%, height h = 20 nm and period of L = 600 nm. The structures have a thickness of d = 1000 nm. We use TE polarized light under normal incidence within the wavelength range [750 nm, 1100 nm] (0.54 < s < 0.8). We use COMSOL Multiphysics, a finite element analysis (FEA) solver, as a modelling tool to rigorously calculate the electromagnetic field inside the absorber. Periodic boundary condition is used to calculate the electric field in the periodic structure. The electric field is then decomposed into its spatial components, using Fourier expansion to calculate the absorption enhancement for each diffraction order [43]. Since Si is a dispersive material, in this simulation we use wavelength-dependent ( ) n λ and ( ) k λ data of nc-Si:H [44]. The wavelength-dependent absorption coefficient, α , can be calculated using ( ) k λ via ( ) 4 ( ) k α λ π λ λ = .
The wavelength-dependency of α implies that i γ is also a function of wavelength. The electric field inside the structure is calculated for each wavelength, using the wavelength-dependent n and k data. The result for symmetric grating is illustrated in Fig.  7(B). Each color represents the absorption enhancement for one diffraction order. There are peaks well beyond both the asymmetric and symmetric 1-D nano-photonic limit (2πn and πn, respectively), and one even exceeding the 2-D Lambertian bulk limit (4n 2 ). This example shows that for a limited frequency range, the enhancement can significantly exceed the nanophotonic limit. Comparing Fig. 6(B) and Fig. 7(B) one can realize that the absorption enhancement is larger than what is predicted by our theoretical model. One of the reasons is that the analytical approach does not consider the multiple scattering in the structure. It counts each resonance only for the first scattering event (T i ). Whereas, in the rigorous calculation, multiple scattering events are taken into account (T i and T i '). Therefore, the simulation shows a higher absorption enhancement. ] at corresponding ch grating order is n asymmetric g ymmetry of the n enhancement to Fig. 7 thin film processes. ed with a p (T i ) and interface with different periodicity allows the light to be re-scattered with different angular distribution, thus exciting more resonances. The scale of the structure in Fig. 9 is only for presentation purposes and it is not related to our calculation. Attempting to find the maximum absorption enhancement achievable in such an optical system entails to extend the temporal coupledmode theory as put forward by Yu et al. [35,36], since it handles only one side texturing.
where s t = L top /λ and s b = L bot /λ are the normalized frequency of the top and bottom interfaces, respectively; and q t and q b are the q th diffraction order of the top and bottom gratings, respectively. If the solar cell is placed on a mirror, then the number of leakage channels (N) only depends on the periodicity of the top interface. If the mirror is not present, N is the sum of channels from top and bottom interfaces. Combining Eqs. (1) and (12), we can obtain the enhancement factor per diffraction order: This equation is a linear sum of the enhancement achieved by the top and bottom surfaces. If we plot the enhancement from top or bottom gratings individually, a graph very similar to Fig. 3 or Fig. 4 can be obtained, depending on the symmetry of the grating. Note that for double side texturing top and bottom gratings could have different symmetry. However, since normalized frequency is directly linked to the grating period, the plot for the bottom grating air Si Reflection ′ Perfect metal might be shifted to the right or left with respect to the plot for the top grating. The amount of shift depends on the ratio between L top and L bot . If L top /L bot > 1, there is a shift toward high frequencies. On the other hand, for L top /L bot < 1, a shift toward low frequencies occurs. For L top /L bot = 1, no shift is observed. The L top /L bot ratio can be defined according to the applications. In thin film solar cell, it is desirable to increase the absorption in a wide range of frequencies close to the band gap of the material. Therefore, the L top /L bot ratio should be smaller than one. L top should be smaller than the band gap wavelength. On the other hand, L bot should be larger than L top , to ensure maximum excitation of modes. Figure 10(A) schematically shows a slab with double side texturing, the top and bottom gratings are symmetric but they have different periods. The structure in Fig. 10(A) is only used for presentation purposes; the scale of top and bottom grating is exaggerated and it is not used for our calculation. In Fig. 10(B) one can see the cumulative enhancement factor calculated for a double side textured film with a thickness of 1000 nm. The period of top and bottom symmetric gratings are 600 nm and 660 nm respectively. Since L top /L bot < 1 the red curve is shifted to the left with respect to the blue curve. As it can be seen in Fig. 10-B, the solid black line has more peaks than either blue or red curves. The structure in Fig. 11(A) shows a 1-μm thick slab endowed with asymmetric grating on top and bottom interfaces. The top and bottom gratings have different periods, the scale of top and bottom grating is exaggerated and it is not used for our calculation. Figure 11(B) represents the absorption enhancement in a structure with double side texturing with asymmetric grating. Apart from the grating symmetry, the structure is identical to the one we used for Fig. 10(A).  In conformal texturing (L top /L bot = 1), peaks resulting from the excitation of modes by top and bottom grating overlap, creating larger peaks (F t,,b = 2F q ). However, the absorption is not enhanced spectrally. In solar cell application, it is desirable to design top and bottom gratings such that none of the peaks from top and bottom gratings overlap. Therefore, L top should be an aliquant part of L bot , or vice versa [45,46]. In this way, no multiple of the top grating vector (q top · k top ) is equal to a multiple of the bottom grating vector (q bot · k bot ). In other words, none of the diffraction angles from top surface matches with bottom diffraction angles.

Conclusions
Leveraging the temporal coupled-mode theory, we have used an alternative method to calculate total number of resonances with same x k in the frequency range [ω, ω + ∆ω] supported by a grating structure. Our results allow for calculating the maximum absorption enhancement for each diffraction order in 1-D grating structure. We discussed the influence of grating symmetry as well as the film thickness in absorption enhancement. We have shown that in a 1-D grating structure, the absorption can be enhanced much more than 2πn in multiple frequencies (Fig. 7). The maxima in cumulative absorption enhancement depends on the grating period, slab thickness and absorber refractive index (see Eq. (6)). For n = 4, the absolute maximum occurs when the third diffraction order is active. Furthermore, we have also shown that weight of Fabry-Perot resonances on the total enhancement factor can be computed. In all treated cases, such weight stays below 50% for symmetric and below 30% for asymmetric grating. Moreover, it has been shown that weight of Fabry-Perot resonances on the total enhancement factor does not change with the thickness of the slab. We have also shown that for achieving even higher absorption enhancement within a larger frequency range a double-side grating-textured absorption slab should be considered, in which L top should be an aliquant of L bot , or vice versa (see Eq. (13)). This last result further extends the temporal coupled-mode theory as proposed by Yu et al.
Finally, we did not consider 2-D gratings in this work, because it is not possible to derive a closed formula independent of grating arrangement and unit cell dimensions. Nevertheless, we expect similar trends in 2-D case. Meaning that, in higher normalized-frequency, the value for absorption enhancement in 2-D structure, would reach the 2-D bulk limit of 4n 2 , but for low frequencies we expect higher absorption enhancement.