Within the diversified class of sub-wavelength photonic crystals, planar gratings with high index contrasts can exhibit unexpected behaviours for light management [1]. Wide band mirror properties were in particular recently discovered, finding a relevant application in ultra-compact high-Q cavity resonators [2,3]. Such gratings were thought to work within a surrounding medium of low optical index (free standing membranes, for instance) so that no diffraction occurs anywhere in the spectral range of interest. Other general configurations, however, were not explored, albeit very useful for some opto-electronic applications. In this paper, on the basis of rigorous electromagnetic simulations, I consider highly non-symmetrical grating structures with rather moderate index contrasts. I will not present an analytical study complementary to already known theoretical works [4–6], but essentially highlight parametric possibilities that can functionalize the optical system differently depending on the direction of the incident light. A potential application for thin silicon solar cells is especially discussed.

Let us take a sub-wavelength dielectric grating designed to behave as a broad-band mirror under some excitation conditions. Such gratings are usually cladded with a low optical index medium to prevent diffraction orders of high number which could open radiation channels and then break the mirror property. However, a key remark is that only the transmission region of the reflecting photonic crystal must be of sufficiently low index compared to that of the grating bars (or grating membrane). The reflector behaviour can work even if the incidence medium is of high index, and even partly absorbing, as we will see. Such a non-symmetrical configuration has not been really treated earlier. Consider the system sketched in Fig. 1(a), where a one-dimensional photonic grating (whose profile is not necessarily rectangular) forms the interface between two (non-metallic) half-spaces, with n₁ > n₂ ≥ n₃. Light comes from the high index medium. The period is such that P < λ_c/n₃, where n₃ is the index of the transmission region and λ_c is the lowest incidence wavelength over which we want the grating to be highly reflecting.The electromagnetic behaviour can be classically simulated with a modal method or a FDTD one. Taking the silicon index for n₁ (> 3.45), n₂ = 2, n₃ = n_air, and a period P = 650nm, we can progressively find a set of geometrical parameters for which broad-band and high reflectivity ranges are obtained for wavelengths λ > λ_c (> 650nm), and that for both fundamental polarizations of light. Figure 1(b) shows typical spectra of the internal reflectivity. To calculate the real power reflected by the interface, the imaginary part of n₁ has been willingly omitted (taking into account this imaginary part within the grating does not change the optical response). Because of significant variations of the silicon index in the visible, the reflectivity lines do not exhibit flat maxima (plateau) but rather perturbed profiles. Optimal responses are not the same with TM or TE polarization although a trade-off is here found to have a close highly reflecting behaviour in a given spectral window. Such a polarization trade-off may be more difficult to obtain with symmetrical structures (air/grating/air) [1]. Better global performances could be achieved by choosing one working polarization and a bi-periodical structuration. Let us remark that the structure behaves less as a diffraction grating than as a specular mirror. Although diffraction orders exists in the high index half-space (the grating is not subwavelength with respect to the incidence medium), they do not transport so much real power and the energy is totally reflected in the zero-order, as shown on Fig. 1(c).

 

Fig. 1 Sketch of the non-symmetrical grating structure and parameters. Red arrows are opposite directions of incident light (direction 1 by default). (b) Reflectivity of the system taking a rectangular profile and high contrast indexes: n₁ = n_Si(λ) (the imaginary part is omitted), n2 = 2, n3 = 1, P = 650 nm, h₁ = 325nm, h₂ = 0, w/P = 0.55. (c) Efficiencies of the first reflected orders in the TM case.

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As analysed previously [6], conditions for total reflection of high contrast gratings are met when at least two excited modes, propagating vertically in the grating, destructively interferes, i.e. have nearly the same amplitude and exhibit a π-phase difference between them at the output interface, hence no coupling to the zero-order transmitted diffraction ray. It joins a similar interpretation based on the Mach-Zehnder interferometer analogy [4]: the effective indexes of the grating modes i are such that ${\left(n_{\mathit{eff}}^{(i)}\right)}^2<\mathit{max}\left(n_1^2,n_2^2\right)$ and decreases as their order increases (if $n_{\mathit{eff}}^2<0$, the mode is said to be evanescent). Then a reflecting condition would obey the relation:

 

Fig. 2 (a) Case of a low contrast index grating, with fixed n1 = 3.5 and n₂ = 3.0, needing to take a sufficiently thick grating to retrieve a high and braadband internal reflectivity. (b) Reflectivity diagram in function of the filling factor for optical indexes globally low: n₁ = 2.5, n₂ = 1.5, n₃ = 1, P = 700nm, h₁ = 400nm (h₂ = 0), in TE polarization.

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We come now to the focal point of the paper. The non-symmetrical character of the structure induces a more subtle behaviour: whereas light coming from the high index medium is well totally reflected, a wave coming from the low index medium can be efficiently transmitted over the same working spectral window. In other words, there is one passing direction for normal radiation, which makes the system a kind of photonic valve exhibiting some diode-type behaviour between two electromagnetic tanks, as sketched in Fig. 3(a). The high index, i.e. the medium presenting a high density of states for the photons, would correspond to a cathode (for instance, in TE polarization, the photon eigenmodes within the grating exactly respect the same eigenvalue equation than the wave functions of electrons in a periodic potential (Kroenig-Penney model); the high optical index would thus correspond to a low potential medium). In the passing (forward) direction 2, light can couple to transmitted diffraction orders and energy is mainly transported through a diffraction order ±1 not propagating when light comes with the opposite direction. In this case, we actually retrieve the classical problem of anti-reflection surfaces based on sub-wavelength patterns [5,7]. To benefit optimally from this dual-behaviour, a layer h₂ thick of low index may be added under the grating (see Fig. 1(a)) so as to increase the transmission power in the direction 2 without deteriorating the reflection plateau in the blocking direction 1, as clearly illustrated in Fig. 3(b). It is worth noting that the angular acceptance of the mirror is limited since for great incidence angles, the zero-order diffraction condition is not fulfilled anymore in the region of index n₃, except if the period is very small. In our example (Fig. 3(c)), the mirror behaviour is robust for incidence angles (internal to the substrate) lower than ±10 degrees (reflectivity remains above 60% for greater angles). Transmission leaks could be returned towards the system as the angular acceptance is very large for light arriving with direction 2. For sake of completness, let us mention that different structuration profiles can be used, that we do not encounter in symmetrical high contrast grating membranes. Figure 4(a) shows examples of similar reflection spectra obtained with trapezoidal and hemi-cylindrical profiles. Aspect ratio w/h₁ are often close to 1. These possibilities are particularly interesting for applicative purposes as they allow technological tolerance and degrees of freedom in the fabrication process. Let us thus tackle the relevance of such a structure for photo-detecting devices, in particular solar cells.

 

Fig. 3 (a) Sketch showing a parallel (not an analogy) between the structure and a diode. The system acts as a passive photonic valve. (b) Reflection/Transmission spectra demonstrating this dual optical behaviour, with an additional sub-layer h₂ thick (see Fig. 1(a)) optimizing transmission in the passing direction. (c) Angular responses in the blocking directon (for an incidence angle taken within the high index medium).

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Fig. 4 Reflectivity spectra for different structuration profiles, in TE polarization (numerical examples). The period is P = 700nm.

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Nowadays, the typical thickness of standard crystalline silicon cells is between 150 and 200μm, and tends to decrease progressively. However, as the silicon substrate becomes thinner (under 100μm), the optical absorption in the near-infrared region is incomplete (see Fig. 5(a)). An other important aspect is that a full metallization of the rear side (back electrode) ought to be avoided to prevent strong mechanical strenghs (wafer bow) and then breaking of the thin device during firing processes. Instead, a localized rear metallization is prefered, but passivated areas between contacts are poorly reflecting. This is where the nano-texture illustrated here could be applied, confering mirror properties without metal. It is an alternative to many photonic/diffractive solutions previously reported in this technical context [8–12]. However, it will be important to analyse and control the electrical properties of such a structure (surface recombination velocity, interface states, crystal damage due to etching for instance) [13, 14], knowing that the medium of index n₂ can play the advantageous role of a thin passivation layer (SiN, SiO2, transparent conductive oxide), possibly conforming to the geometrical profile. The advanced texture is also compatible with module integration: whereas light normally coming from the sun is internally reflected in the semi-conductor, surrounding light coming from the lateral and back sides of the module can be fully transmitted in the (bifacial) cell, as shown in Fig. 5(b). Here again, the chosen period is P < λ_c/n₃ where n₃ is the index of the encapsulation medium (≈ 1.5 typically) and λ_c is the wavelength over which the optical aborption of the thin substrate significantly decreases. At end, it is worth noting that the texture let infrared radiation (heat) escape, which might favour the conversion efficiency. Advanced devices where a thermo-PV (lower energy gap) cell is integrated just under the Si one are then possible.

 

Fig. 5 (left) Internal optical absorption of monocrystalline silicon for different substrate thicknesses (one passage of light). (right) Sketch of a PV module architecture integrating a bifacial c-Si cell with a photonic valve at the rear side and a conformal passivation layer. Arrows indicate various directions of light with respect to the texture.

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In conclusion, I highlighted optical properties of a non-symmetrical photonic grating directly realized at the rear side of a high index substrate. The structure exhibits some photonic valve type behaviour, i.e. both high and broad-band internal reflectivity, and efficient anti-reflection properties for light coming from the opposite side. Examples in the visible/near infrared regions were given. High contrast indexes are not systematically required. Depending on wanted specifications, the configuration may apply to different kinds of opto-electronic devices as thin crystalline Si or SiGe cells, imaging systems like CMOS devices, laterally structured SOI waveguides [15]. One can finally imagine novel directionnal and/or spectrally selective light emitters if the semi-conducting region is photo-active.