Enhanced light trapping in solar cells with a meta-mirror following generalized Snell ’ s law

As the performance of photovoltaic cells approaches the Shockley-Queisser limit, appropriate schemes are needed to minimize the losses without compromising the current performance. In this paper we propose a planar absorber-mirror light trapping structure where a conventional mirror is replaced by a meta-mirror with asymmetric light scattering properties. The meta-mirror is tailored to have reflection in asymmetric modes that stay outside the escape cone of the dielectric, hence trapping light with unit probability. Ideally, the meta-mirror can be designed to have such light trapping for any angle of incidence onto the absorber-mirror structure. We illustrate the concept by using a simple gapplasmon meta-mirror. Even though the response of the mirror is non-ideal with the unwanted scattering modes reducing the light absorption, we observe an order of magnitude enhancement compared to single pass absorption in the absorber. 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Introduction and background
The photovoltaic (PV) devices converts sunlight to electricity in two steps: sunlight is absorbed to create electron-hole pairs within the active layer, and the photo-generated carriers are extracted by the built-in electric field of the device.As solar cell technologies move closer toward the ideal thermodynamic (Shockley-Queisser, S-Q) limit of PV performance [1], the new designs need to be more effective in reducing the remaining loss components without introducing new ones [2].Unfortunately, non-ideal material characteristics couple absorptiontransport processes in a fundamental way: the finite optical absorption sets the minimum (optical) thickness of the absorber (optical thickness), while finite mobility and radiative/nonradiative recombination set the maximum (electrical) thickness limit of the cell.
The strategies that address the issue of absorption or carrier extraction all rely on decoupling optical and electrical thicknesses of solar cells.For example, solar cell configurations such as radial junction nanowire (NW) PV [3,4], and projected electrodes [5,6] focus on decoupling the light absorption from the carrier extraction length by absorbing light vertically, but separating the carrier laterally.These structures provide a shorter and more efficient path for carrier collection.Light trapping (LT) schemes instead focus on collecting the carrier vertically so that electrical path-length is reduced, while reducing reflection and increasing lateral scattering of the light so that optical absorption is enhanced.We can subdivide the LT schemes in two broad categories: (i) nano-structuring the active layer itself, and (ii) redesigning the dielectric environment around it.In the first category, random texturing, NW array, nanocone (NC) structures [7][8][9] reduce reflection of the incoming light and induce scattering for overall improved absorption.Other photonic crystal structures can scatter light inside the dielectric and yield longer light passage in the absorber [10][11][12].Also, embedded plasmonic nano-particles [13,14] can create localized high fields which in turn greatly enhance localized absorption.Unfortunately, these strategies also increase the surface area for recombination and the corresponding degradation in charge collection may ultimately negate any gain from light trapping [15].
In the second category, a separate dielectric or metallic scatterer layer is used to enhance absorption of the incident light inside the un-patterned, planar absorber layer.These approaches include scattering of light from metal particles at the top surface, or generation of surface plasmon polaritons (SPPs) at the back metal grating [16,17].However, at optical frequencies, typical plasmonic materials absorb a fraction of the incident photons, introducing an additional loss mechanism in the device.The patterning could also be done in a separate lossless dielectric to scatter the light going into the absorber.These designs aim at scattering light into higher angles to enhance the light path yielding good absorption [18,19].The key idea here is to couple the normally incident light into guided modes.However, the lossless patterned dielectric scatterers usually have complex designs and could be challenging to implement in practice.
In this paper, we wish to explore the potential of a new approach to dielectric light scattering into lateral guided modes by meta-mirrors.Remarkably, even a nominal design improves light absorption by an order of magnitude.Note that, absorption enhancement limits have been discussed for both geometric [20,21] and wave optics regime [22].Yablonovitch et al. [20,21] showed that absorption in a weak absorber can be enhanced up to a factor of 2 4n .Our proposed concept suggests that these limits may be exceeded (and the light may be trapped with unit probability) by using a meta-mirror as a back reflector, although the specific mirror design that achieves this perfect absorption is yet to be determined.
Briefly, recent developments in the concepts of meta-surface [23][24][25][26][27][28][29] have shown the possibility of rearranging light propagation in planar structures.In this work, we propose inserting a reflecting meta-mirror at the back of the solar cell to asymmetrically scatter the rays.The asymmetric scattering follows the generalized Snell's law of reflection [23].In principle, the meta-mirror can be designed to scatter light primarily into a single mode that stays away from the critical angle of the front surface; and thereby ensuring total internal reflection (TIR).As we show in Sec. 2, an idealized meta-mirror that scatters the light into such single mode has the potential to perfectly trap the entire incident light irrespective of the dielectric thickness (i.e., both in geometric or wave optics regime) for any angle of incidence.In practice, exclusive scattering into a single mode may not be possible.Also, a solar cell is a broadband absorber (although absorption close to the band-edge is most critical), and, therefore, wavelength dependence of asymmetric scattering associated the practical design must be explored.Despite these issues, a nominal design discussed in Sec. 3 anticipates an order of magnitude improvement over single pass absorption.We conclude by suggesting that an optimized structure has the potential for considerably higher enhancement of optical light trapping and the bandwidth is always sufficiently large for the scheme to be a practical light trapping scheme for high efficiency solar cells working close to the S-Q limit.

Theory of perfect absorption
Consider an optical 'mirror' which adds a phase shift Φ( ) x at any point x on the mirror surface upon each reflection of the incident wave.In the special case when Φ( ) Here, ( ) k and ( ) x r k are the x -components of the wave vectors of the incident and reflected plane waves, respectively.The plane wave is assumed to be in xz -plane in a dielectric with refractive index n .Equation (1) states that a plane wave propagating in a dielectric will not be reflected with an angle equal to the incidence angle expected from the conventional Snell's Law of reflection.We can rewrite the previous relationship as follows, 0 0 sin sin ( ) Here, 0 i θ = and r θ are the incidence and reflected angles of the plane wave at the mirror surface.0 k is the free-space wave vector at wavelength of interest 0 λ .The mirror effectively steers the incident plane wave into higher angles breaking the conventional rule of specular reflection in planar mirrors.And, for sin (1 ) the scattered waves become surface waves confined close to the mirror dielectric interface.This causes asymmetric scattering, which can be referred to as the 'generalized Snell's law' of reflection.We will show in the following discussions how this asymmetric scattering property of the mirror can be ideally used to develop a scheme for perfect light trapping.
Figure 1(a) shows a light trapping scheme in a dielectric.Assume that the mirror follows generalized Snell's law of reflection.Light incident from air refracts into the dielectric with some angle 1 i θ inside the escape cone (see Fig. 1(b)).Thus we have 1 is the critical angle of refraction).Then the plane wave hits the back mirror and reflects back with a larger angle, 1 r θ θ θ > > .This cycle will continue with the light attaining a higher angle after each bounce off the back mirror.Rays with larger angle of propagation will always stay outside the escape cone and remain trapped inside the dielectric.Note that, after some N bounce cycles, the incident angle on the mirror will be ( ) In this case, instead of reflecting back into a propagating mode, the light wave will scatter into a surface wave along the dielectric-mirror interface.
The scheme provides means for trapping the incident light indefinitely-a dielectric with the smallest of absorption coefficient will have perfect absorption.Ideally, the structure absorbs 100% of the light, irrespective of how weak the absorption coefficient may be.Thus, the conceptual structure can have an absorption enhancement factor limited only by the inverse of the original optical absorption -for poor absorbers, this breaks the conventional absorption enhancement limit (e.g., the Yablonovitch 4n 2 limit [20,21]).This perfect absorption should not be misunderstood to be unphysical, because detailed balance ensures that at open circuit condition, absorption is counterbalanced by appropriately high radiation [30].Figure 1 shows an example of the path followed by light ray inside the dielectric ( 3.5 n = ) of the proposed structure.Here, for the ray tracing calculations, we have assumed sin C δ θ = .

Design rules for meta-mirror
Phases are only defined per modulo 2π , i.e., the added phase of ( ) Φ x has the same effect as the added phase of ( ) , where m is an integer.Therefore, we can wrap the linear profile of ( ) Φ x into a periodic triangular form, where the added phase varies linearly from 0 to 2π inside a period.This profile of Φ( ) x for the mirror can be created by an extra graded index dielectric on top of a conventional mirror [31] or by a meta-surface [23,27] [29].
The period Γ x of the mirror structure depends on the required slope of Φ( ) x .Hence Γ x will be determined by Δ Φ( ) / ).As explained earlier, for the light trapping scheme to work, the value of δ is constrained by C θ at a given Thus the mirror structure period should be limited by constraints set by C θ , or equivalently by the refractive index n of the active layer dielectric and the wavelength 0 λ .However, as we will show in the following analysis, design of Γ x only depends on the wavelength of interest.
As explained earlier, the light rays initially reach the mirror at an angle of ( ) .Thus the period for the mirror design is found as follows: The above mentioned condition is sufficient to trap light inside the planar dielectric (or absorber) for any incidence angle from air.If we focus on normal incidence, i.e., 1 0 i θ = then the design condition is relaxed to 0 Γ x λ ≤ .We will consider only normal incidence for the design studies presented in this paper.Note that, wavelengths away from 0 λ may be scattered into multiple modes.However, for a range of wavelengths ( ) λ close to 0 λ , most of the scattered wave will couple into the intended mode following the generalized Snell's law.Ofcourse, for short-wavelengths and thus there will be no light trapping for such wavelengths.The bandwidth for this response depends on the specifics of the design.The concept of a periodic meta-mirror is most simply illustrated by a graded refractive index layer on top of a conventional mirror (see Appendix A).
While the periodic meta-mirror based on graded index layer explains the basic concepts of asymmetric scattering intuitively, in practice it is easier to design equivalent meta-mirror based on the concept of 'gap-plasmon' meta-surface [29], discussed as follows.

Properties of designed meta-mirror
x ranging from 0 to 2π .The details of the meta-mirror design are given in Appendix B. The numerical calculations are done using 4  S [32], which is based on Rigorous Coupled Wave Analysis (RCWA).Metal components have been assumed to be loss-less having refractive index of 7 metal n i = .It is also taken as non-dispersive for clear demonstration of the concept.The stability of the results has been checked by testing for higher number of Fourier components.Since RCWA does not automatically yield reliable results for metal inclusions in the structure, we validated the results by using Finite Difference Time Domain (FDTD) simulations in Meep [33] as well.We choose period Γ 550 x = nm for the design wavelength 0 600 λ = nm.The structure is uniform along y-direction.And, we only consider y Hpolarization for the incident light.Our design will only focus on one polarization to provide a clear illustration and simple explanation of the design idea.However, it is possible to design a meta-mirror to simultaneously address both polarizations, as explained in Ref [29].
The transfer characteristics ( r θ vs. i θ ) of the mirror is presented in Fig. 2(c) as a spectrum of scattered y H field components.The ( + 1)-diffraction mode (also marked in the figure ) corresponds to the generalized Snell's law of reflection, Eq. ( 3).The angle of reflection for normal incidence 0 i θ = can be calculated for the ( + 1)-diffraction mode as follows: ( ) ( ) For 600 λ = nm, Γ 550 x = nm, and n = 3.5, we find ( ) relationship of the 'generalized Snell's law of reflection', Fig. 2(c) shows that there are other scattered modes as well.This set of parasitic couplings can only contribute to some specific modes as a consequence of the periodicity of the mirror.For example, for 0 i θ = , the scattered angle is expected to assume a value for any structure with period Γ x , where the order of the modes are given by 0, 1, 2, m = ± ± …, etc.However, a linear phase profile Φ( ) x created by the sub-cells can ensure exclusive coupling into 1 m = + mode.Although only 1 m = + is desired here, some amplitude contribution in other modes cannot be avoided because of the non-linear phase profile of 'sub-cells' in one period (details about subcells are in Appendix B), and finite mutual coupling between adjacent sub-cells.This is especially prominent at higher angles of incidence i θ on the meta-mirror.For example, as presented in Fig. 2(b), the scattered wave for normal incidence (top profile) at 600nm show minimum distortion in the plane wave.However, as the angle of incidence increases (e.g., 15 i θ =  ), we start to see perturbation in the scattered field profile (middle profile) due to contribution of the unwanted modes.The existence of these parasitic modes degrades light trapping, as they allow the light to exit through the escape cone.θ <  ).However, larger i θ reverts to Snell's Law, i.e., specular reflection (with small contribution from other diffraction modes).For this specific structure, no surface wave is observed for the high incidence angles-this is a deviation from the ideal characteristics.A meta-mirror using a 'xylophone'-like gapplasmon configuration, as in [27], is likely to show much improved characteristics.Also Eq. ( 5) predicts that ( ) 1 r θ + would increase with λ .This is seen in Fig. 2(d nm.This means that the meta-mirror is deviating from the generalized Snell's law and showing more specular reflection outside the aforementioned wavelength range.This indicates that light trapping may be degraded for 800 λ > nm for the structure built using this meta-mirror.We will address the issue further in the following discussions.

Light-trapping in planar dielectric with back meta-mirror
Next let us calculate the light trapping using the proposed scheme.As we have discussed, the gap-plasmon periodic meta-mirror design proposed here is not ideal: (i) there is scattering into modes other than the desired mode, (ii) surface waves are not generated, and (iii) there is considerable angle and wavelength dependence of the mirror response.
For the study of absorption, let us choose the absorber thickness 500 L = nm (which would be typical for thin film solar cell) on top of the mirror.To reduce reflections off the top surface, a quarter wave matched (for 0 600 ) for better comparison of the light trapping properties.Note that the dielectric and metal in the metasurface structure are also assumed to be non-dispersive.The blue line in Fig. 3(b) represents the absorptance in the structure constructed using the meta-mirror configuration at the back (as shown in Fig. 3(a)).The black line shows the absorption in the same cell structure only with a conventional mirror at the back.As expected, the absorption in the planar cell is 2 0.1 L α ≈ = (black line).For the meta-mirror configuration in Fig. 3(b), there is minimal change in absorptance for Γ 550 x λ < = nm as the design constraint is not satisfied in this wavelength range.Also, note that, the absorption starts to go down for 800 λ > nm as predicted from the previously studied mirror properties.It is clear that there is an average absorption enhancement of a factor of ~5.75, compared to the conventional back mirror structure, for 550nm 800nm λ < < .Obviously, the band of wavelengths where the absorption enhancement occurs can be shifted to arbitrary wavelengths by tailoring the meta-mirrors at appropriate periodicities.The study considered here only focuses on normally incident light.This reflects the operating conditions for solar cells with dual-axis tracking that could experience direct normal incidence throughout the day.Of course, a meta-mirror designed using 0 Γ / 2 x λ ≤ can ideally trap light for all angles of incidence (see Sec. 2.2 for details).
Although the solar cell is a broadband absorber, the finite bandwidth over which a metamirror sustains significant gain is not a concern.For a practical solar cell (e.g., Si), the absorption is strong for high energy photons (lower wavelengths) away from the band-edge.Thus, almost all the high energy photons are absorbed within practical cell thickness-the light trapping schemes are irrelevant.However, the absorption coefficients close to the bandedge are considerably lower and it is within this narrow bandwidth where light-trapping is relevant and important.So long as the meta-surface mirrors are designed to overlap this weak absorption regime, the improvement in light trapping can be significant.Also, the proposed scheme is expected to be fairly insensitive to small changes in the optical real refractive index for practical PV materials.
Finally, we realize that the meta-mirror structure proposed here consists of subwavelength components that may be difficult to fabricate.Despite the difficulty, however, recent works presented in Refs [25,27,29].show that such structures are feasible and can be demonstrated experimentally.

Conclusions
In this paper, we have proposed a planar structure for light trapping which utilizes the asymmetric scattering of the meta-mirror at the back.The meta-mirror only reflects light into a single mode following generalized Snell's law of reflection.The ideal configuration anticipates perfect absorption for all angles of incidence onto the planar absorber.Practical implementation based on gap-plasmon meta-mirrors may inherently allow scattering light into multiple diffraction modes.The undesired modes cause the light to leak out of the structures.Although perfect absorption is no longer expected, an optimized design can lower coupling to unwanted modes and improve overall absorption considerably.Indeed, a nominal design presented here shows an absorption gain of a factor of ~11.5 compared to single pass absorption in the dielectric (i.e., factor of ~5.75 compared to a back mirrored planar structure).Obviously, the gap-plasmon meta-mirror is just one example of a meta-surface being capable of scattering light asymmetrically.It remains an important open question to see if other implementations might provide improved absorption approaching the ideal limit.

Appendices Appendix A. Graded index mirror
For a more qualitative understanding, let us consider a hypothetical mirror which can demonstrate the properties discussed in Sec.2.1.The simplest of examples for this purpose would be a graded refractive index layer on top of a conventional mirror (see Fig. 4(a)).Each segment has the same thickness, but different refractive index, and therefore, provides varying optical paths for the incoming wave.The phase added to the light wave by the segments will follow the index gradient; therefore, as the wave reflects from the graded index layer, it will accumulate a spatially gradient phase Φ( ) x .The graded index layer has periodic and linearly varying indices ranging from L n to H n .For a large number of sub-segments, if the phase added by the dielectric with index L n is L Φ then the phase added by segment H n would be L Φ 2π + .Let us assume that every period consists of S N segments and the refractive index changes by Δn between consecutive segments.The phase added at the end of each period is 2π , or equivalently, the extra optical path length would be 0 λ (design wavelength).Thus, for the graded index layer of thickness GI t , ( ) Here, Δ S n N corresponds to the refractive index variation over one period.The factor of two is due to the fact that the light travels through the layer twice as there is a mirror at the back.Finally, we can write: Δ .
The refractive indices of the graded index layer sub-segments are given by GI n in the above equation; where, ( ) Figure 4(a) shows the graded index mirror embedded in a dielectric ( 3.5 n = ).We introduce a matching layer on top of the graded index layer to minimize specular reflection.We choose period Γ 550 x = nm for the design wavelength 0 600 λ = nm (as mentioned in Sec.2.2, the light trapping scheme requires that 0 Γ x λ < ).The structure is uniform along ydirection.The design parameters are summarized in Table 1.We only consider θ <  ).However, larger i θ reverts to Snell's Law, i.e., specular reflection (with small contribution from other diffraction modes).No surface wave is observed for the high incidence angles.These characteristics are similar to those of the meta-mirror discussed in Sec.3.1.Note that we have ( )  ).We see that ( ) which does not support the light trapping constraint for our proposed scheme.Also, observe that a more prominent mode emerges at 0 r θ = for 800 λ > nm-similar to the meta-mirror structure.The absorption enhancement using the graded index mirror is presented here for absorber thickness of 500 L = nm, and absorption coefficient of

Appendix B. Design of gap-plasmon periodic meta-mirror
The meta-mirror has a periodic structure.Each period, i.e., the supercell, is constructed from 10 sub-cells, see Fig. 6(a).Each sub-cell is intended to add increasing phases to get Φ( ) x of 0 to 2π within each period.We design the meta-mirror as a 'gap-plasmon' meta-surface structure [27,29].A dielectric material (refractive index 1.38) is placed between a metal strip of width W and a metal plate.The thicknesses of these layers are marked in Fig. 6(a).Varying W will add different phases to the scattered (reflected) wave.Of-course this response has to be investigated inside the dielectric (active layer of the device) with 3.5 n = .The metal for the numerical studies is assumed to be loss-less and non-dispersive with refractive index 7 that, as the change in phase added Φ spans from 0  to ~300  for W range of 0 to 55nm , the sub-cells do not contribute to the overall phase equally, see Fig. 6(b).This non-uniform contribution to the overall phase, however, has only a minor effect on the meta-mirror performance for normally incident light.These simulations were done using 4 S [32] as explained in Sec.3.1.
The supercell constructed from the selected sub-cells (as tabulated in bottom of Fig. 6) is expected to show a spatially linear phase profile.It is presented in Fig. 6(c) as a variation in the relative phases.It is clear that, for normal incidence ( ) onto the meta-mirror, a linear phase profile is preserved (the red circles).However, the linear phase profile is not observed for oblique incidence (blue crosses, 40

Fig. 1 .
Fig. 1.(a) The concept of light trapping is shown.A dielectric with refractive index n is placed on top of a meta-mirror which follows the generalized Snell's law of reflection.The meta-mirror is shown here as a combination of a perfect mirror and a gradient phase adding interface layer.The ideal path for the light ray is shown.(b) The red line shows the law of reflection at the back meta-mirror.The black line represents specular reflection for total internal reflection at the top dielectric-air interface.The shaded region represents the escape cone angles.The blue circles are the incident angles at the meta-mirror after multiple bounces for the example illustrated in (a).

Figure 2 (
Figure 2(a) shows a gap-plasmon meta-surface mirror embedded in a dielectric ( 3.5 n =).The various metal strip-sizes in gap-plasmon configuration add gradient phase Φ( )x ranging from 0 to 2π .The details of the meta-mirror design are given in Appendix B. The numerical calculations are done using4  S[32], which is based on Rigorous Coupled Wave Analysis (RCWA).Metal components have been assumed to be loss-less having refractive index of 7

Fig. 2 .
Fig. 2. (a) The meta-surface mirror configuration embedded in the dielectric.(b) Scattered y H explained earlier in Sec.2.1, the situation mentioned above breaks the condition for our light trapping scheme.The higher wavelengths gradually allow more contributions from modes other than 1 m = + -this is reflected in the perturbed scattered field shown in Fig.2(b) (bottom profile).Also, note that the larger contribution for the Both n and α are assumed to be non-dispersive in the wavelength range of interest ( 200nm 1000nm λ < <

Fig. 3 .
Fig. 3. (a) The light trapping scheme for thin film solar cell is shown using the meta-surface mirror.(b) The black line represents the absorption in the structure with a conventional mirror at the back.The blue line in (b) shows the absorption in the absorber layer for structures in (a).
incident light.The transfer characteristics ( r θ vs. i θ ) of the mirror is presented in Fig. 4(b).The i r θ θ − relationship predominantly follows the generalized Snell's law of reflection for low i θ (i.e., 20 i  for this structure as well.

Fig. 4 .
Fig. 4. (a) The graded index mirror configuration embedded in the dielectric.(b) The transfer characteristics ( r θ vs. i θ ) are shown as a spectrum of scattered y H field amplitudes at The study is similar to the #206689 -$15.00USD Received 18 Feb 2014; revised 28 Mar 2014; accepted 31 Mar 2014; published 23 Apr 2014 (C) 2014 OSA one done for the meta-mirror configuration (in Sec.3.1) for 200nm 1000nm λ < <.The absorption in the structure shown in Fig.5(a) is represented by the blue line in Fig.5(b).The black line is the absorption in a structure having a conventional mirror at the back.As expected, the enhancement of absorptance is minimal for Γ 550x λ < = nm (also see Sec.2.2).However, absorption peaks are observed in this λ -range.This can be associated with the resonances created by the series of dielectric layers.

Fig. 5 .
Fig. 5. (a) The light trapping scheme for thin film solar cell is shown using the graded index mirror.The ARC layer on top of the dielectric is used to minimize reflection of the incident light.(b) The black line shows the absorption in the structure with a conventional mirror at the back.The blue line show the absorption in the absorber layer for structures in (a).

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e., relative permittivity of −49).The phases added to the reflected waves under such constraints are presented in Fig. 6(b) as a function of W .The design is done under normal incidence condition (blue solid line in Fig. 6(b)).Now, 10 sub-cells that add 36 β ×  phases (here 0,1, ,9 β = … ) need to be used to construct the supercell.The period of the sub-cell is Γ 55nm x sub − = ; thus period of the meta-mirror is The widths ( ) W of individual stripes are tabulated in Fig. 6.Note

Fig. 6 .
Fig. 6.(a) Sub-cell of the meta-mirror periodic structure.The various lengths have been defined here.Varying metal strip widths W are used to form the supercell (single period) of the meta-mirror.(b) The phase added by super-cells as a function of W at normal incidence (blue solid line).10 components/sub-cells are used in a supercell (red circles).These components have W values as listed in the table (bottom).(c) The 10 sub-cells create a linearly varying (relative) phase profile for normal incidence ( 0 i = θ varies ( To ensure the design works for all incident angles, we start with the worst- #206689 -$15.00USD Received 18 Feb 2014; revised 28 Mar 2014; accepted 31 Mar 2014; published 23 Apr 2014 (C) 2014 OSA