Light trapping in thin-film solar cells: the role of guided modes

Light trapping in thin-film solar cells: the role of guided modes T. Søndergaard *, Y.-C. Tsao, T. G. Pedersen, and K. Pedersen Department of Physics and Nanotechnology, Aalborg University, Skjernvej 4A, DK-9220 Aalborg East, Denmark ABSTRACT This paper studies theoretically light trapping in a solar cell configuration consisting of a 50-500 nanometer-thin planar silicon (asi:h) film with a planar silver back-reflector, and scatterer(s) placed directly on the silicon surface. The usual picture for thicker films is that part of the light incident on the scatterer(s) can be coupled into the silicon film at a continuum of angles above the critical angle for the silicon-air interface, in which case light will be trapped and subsequently absorbed. However, for thin films a more appropriate picture is that of light being coupled into the guided modes of the air-silicon-silver geometry corresponding to discrete angles. The scattering of light into each guided mode, and out-of-plane scattering, will be quantified by the related scattering cross section. It will be shown that scatteringcross-section spectra have sharp resonances near cut-off wavelengths of guided modes, with more closely spaced resonances for thicker films. Total resonant cross sections can easily exceed physical cross sections by a factor 10. This study also includes light trapping due to coupling into the Surface-Plasmon-Polariton mode that exists due to the silver surface. It will be shown that peaks in scattering cross sections can be tuned via the geometry to the appropriate wavelength range where light trapping is advantageous due to weak absorption in the silicon, resulting in an optimum thickness around 250 nanometers. We consider both theoretical calculations with and without material losses, and both dielectric and metal scatterers are considered. The calculations were carried out with Green s function integral equation methods. Keywords: Scattering theory, extinction cross section, scattering cross section, absorption cross section, guided mode, light-trapping, thin-film solar-cell, mode-index, guided-mode cut-off wavelength 1. INTRODUCTION Light-trapping is a strategy for enhancing the absorption of light in thin-film solar cells [1-3]. For wavelengths of light where the active material of the solar cell is weakly absorbing the incident light will either pass straight through the thinfilm active material, or it will be reflected back out of the solar cell, depending on the geometry. This is because the path length of light-propagation in the active material will be short compared to the absorption length. The absorption can be enhanced by modifying the geometry such that light will be scatterered into directions where it will propagate longer inside the active material. Scattering of light into directions or angles above the critical angle for interfaces can lead to light-trapping via total-internal reflection. Light trapped in this way will continue to propagate in the active material until it is either absorbed or scattered again by another scatterer. A thin-film geometry will usually also be a waveguide supporting several guided modes, where each mode corresponds to specific discrete angles of light propagation in the active material. Again, light coupled into these modes will be trapped and continue to propagate in the active material until it is either absorbed or re-scattered by another scatterer. Previously light trapping has been studied for the case of using periodic arrays of scatterers on the front-side [1-4], and for a textured back contact [5,6], and both periodic and random scattering arrangements are being investigated [7,8]. The objective of this paper is that, instead of considering an array of scatterers as in previous work, we will here study the light-trapping due to a single scatterer. This will provide insight into how a single scatterer couples to guided modes without the complication arising from multiple scattering effects, and we will demonstrate that scattering is dominated by strong scattering peaks located around cut-offwavelengths of the guided modes of the waveguide geometry. The geometry considered in this paper (see Fig. 1) consists of an optically thick silver substrate with a thin silicon-film (asi:h) on top of thickness d. From an optical point of view this can represent a simple model of a thin-film solar cell. For wavelengths where the silicon is weakly absorbing most of the incident light that passes through the first air-silicon interface will be reflected back out of the geometry at the silicon-silver interface, and there will only be little absorption since the path length of light propagation back and forth in the silicon will be short (because d is small). We will study Thin Films for Solar and Energy Technology VI, edited by Louay A. Eldada, Michael J. Heben, Proc. of SPIE Vol. 9177, 91770M 2014 SPIE CCC code: 0277-786X/14/$18 doi: 10.1117/12.2061660 Proc. of SPIE Vol. 9177 91770M-1

the effect of placing a single silver or silicon nanostrip of width w and thickness t on the front surface as illustrated in Fig. 1 for scattering of light into the guided modes of the air-silicon-silver waveguide geometry. As illustrated in Fig. 1 there will not only be scattering of light into the guided modes but also into out-of-plane propagating modes. Furthermore, there will be absorption in the silicon and silver. Figure 1. Illustration light incident on a silver or silicon strip placed on a silicon-film-on-silver waveguide geometry. Scattering of light into out-of-plane propagating modes, and into various wave-guide modes, are illustrated. The width and height of the silver strip are denoted w and t, respectively, and the silicon film thickness is denoted d. The guided modes can be described as a function of y times an exponential function, exp( ik0n x), where the sign determines the direction of propagation, k 0 2 / is the free-space wave-number, and n guided is the mode-index which will be useful in the following chapters. We will consider a plane wave being incident on the geometry with power per unit area I i. Due to the nanostrip a certain amount of power P OUP will be scattered into out-of-plane propagating modes, and a certain amount of power P guided will be scattered into the guided modes of the waveguide geometry. It is now convenient to define scattering cross sections for scattering into out-of-plane propagating and guided modes as P / I and P I, respectively. The calculations of these cross sections presented OUP OUP i guided guided / i throughout the paper have been made using the Green s function surface integral equation method following a procedure similar to that used in Refs. [9,10]. We should note here that when the absorption losses are very large, as will be the case for some of the wavelengths considered in this paper, then it is not always perfectly meaningful to talk about the amount of power coupled into the guided modes. Firstly, the modes will no longer (necessarily) be power orthogonal, and summing up the power of each mode will not necessarily amount to the total amount of power coupled into guided modes, and secondly the power of a guided mode will also depend on where this power is measured. If it is a distance far from the scatterer most of the power of the mode may have been absorbed. When absorption losses are absent the concept of a guided-mode scattering cross section is, however, perfectly meaningful. When absorption is present it may be more useful to instead consider the extinction. The extinction power P EXT can be defined as the power of the reflected beam in the case where the nanostrip is absent minus the power of the (specularly) reflected beam in the case where the nanostrip is present. We can then define the extinction cross section as P I. Note that in the EXT guided EXT / i lossless case. This will not be the case when there are absorption losses. However, when these EXT OUP guided losses are small we can still estimate the scattering into guided modes as. The paper is organized in the following way. In Section 2 we will study light-trapping for the geometry in Fig. 1 for a case where absorption losses in materials are neglected. Here the concept of scattering into guided modes and are perfectly well-defined. We will then in Section 3 present a few preliminary results for the case where guided absorption losses are included. Finally, we will offer our conclusions in Section 4. A more detailed study of the situation with absorption losses included, and measurements of out-of-plane scattering and extinction for a structure similar to Fig. 1 will be presented elsewhere [11]. guided 2. THEORETICAL RESULTS NEGLECTING ABSORPTION LOSSES In this section we consider the scattering situation illustrated in Fig. 1 but neglect ohmic losses in the materials, which means that as refractive index of silver we keep only the imaginary part of the actual silver refractive index obtained from Ref. [12], and for silicon (asi:h) we keep only the real part of the refractive index. For the refractive index of asi:h we use the measured refractive index from Ref. [13]. In Fig. 2(a) we consider the case of a silver strip of width w = 60 nm and thickness t = 25 nm, placed on a d = 50 nm asi:h-film on silver, and s-polarized normally incident light. The EXT OUP Proc. of SPIE Vol. 9177 91770M-2

mode-index (top panel) as a function of wavelength shows that there are no guided modes in the air-asi:h-silver waveguide geometry for wavelengths larger than app. 1000 nm, while cut-off-wavelengths for guided modes are located at wavelengths of app. 450 nm and 1000 nm. The total scattering cross section, being the sum of out-of-plane scattering and scattering into guided modes, shows resonant peaks at the cut-off wavelengths of guided modes. For e.g. a wavelength of 800 nm we can notice that out-of-plane scattering is practically absent, and there the total scattering is dominated by the scattering of light into guided modes. The scattering cross sections are normalized with the width w of the nanostrip, and the figure thus shows how large is the scattering cross section compared with the physical width of the strip. Results for other film thicknesses d = 150 nm, 250 nm, and 500 nm, are shown in Figs. 2(b), (c) and (d), respectively. It can be seen that as d increases the waveguide will support an increasing number of guided modes, and there will be an increasing number of cut-off-wavelengths of guided modes in the studied wavelength range. Near each of these cut-off-wavelengths a peak is found in the total scattering, and as wavelengths decrease slightly below the cutoff-wavelength a sharp increase is seen in the scattering into guided modes. The cut-off-wavelengths depend sensitively on d and thus the position of scattering peaks can be controlled by changing d. Figure 2. Mode-index for the guided modes of a planar asi:h-film-on-silver waveguide for s-polarized light, and out-ofplane scattering cross section, guided-mode scattering cross section, and sum of cross sections, for s-polarized light being normally incident on a silver strip of width w = 60 nm and thickness t = 25 nm placed directly on the asi:h-film-on-silver waveguide. The asi:h film has thickness (a) 50 nm, (b) 150 nm, (c) 250 nm, and (d) 500 nm. Absorption losses in silver and asi:h have been ignored. Proc. of SPIE Vol. 9177 91770M-3

A similar calculation but for p-polarized incident light is shown in Fig. 3. In this case we find for d = 50 nm [Fig. 3(a)] that there is a guided mode for long wavelengths ( > 700 nm) and for short wavelengths ( < 450 nm), while there is a wavelength range in between with no guided modes. The long-wavelength mode is actually the surface-plasmonpolariton (SPP), which is bound to and propagating along the silicon-silver interface. Figure 3. Mode-index for the guided modes of a planar asi:h-film-on-silver waveguide for p-polarized light, and out-ofplane scattering cross section, guided-mode scattering cross section, and sum of cross sections, for p-polarized light being normally incident on a silver strip of width w = 60 nm and thickness t = 25 nm placed directly on the asi:h-film-on-silver waveguide. The asi:h film has thickness (a) 50 nm, (b) 150 nm, (c) 250 nm, and (d) 500 nm. Absorption losses in silver and asi:h have been ignored. We notice for most of the wavelength range from 700 nm to 1400 nm that the total scattering is dominated by scattering into the SPP mode. For the case of film thickness d = 150 nm [Fig. 3(b)] guided modes are available at all considered wavelengths. A total scattering peak is located near the cut-off-wavelength of each of the three ordinary guided modes (a double peak in one of the cases). No scattering peak is found to be related to the SPP. This mode now has higher modeindex and will be well-confined to the silicon-silver interface, and will not extend much into the air-region above the waveguide, which can explain that this mode is not excited (no scattering into guided modes for > 1050 nm). As the Proc. of SPIE Vol. 9177 91770M-4

film thickness is increased further then similar to s-polarization there will be more guided modes available and with more cut-off-wavelengths of guided modes in the studied wavelength range, and near each cut-off-wavelength a scattering peak is observed. In order to study the dependence of the wavelengths of scattering peaks on the scattering situation we have calculated the scattering for a silver strip on 150-nm-aSi:H-on-silver for different angles of light incidence i = 20, 40, and 60. Results for s- and p-polarized light are shown in Fig. 4(a) and Fig. 4(b), respectively. The main point is that the wavelengths of scattering peaks are not sensitive to the angle of light incidence, which is in good agreement with the cut-offwavelengths of guided modes being an intrinsic property of the waveguide geometry. Figure 4. Out-of-plane scattering cross section, guided-mode scattering cross section, and sum of cross sections, for (a) s- polarized light, and (b) p-polarized light, being incident under an angle of incidence of 20, 40 and 60 on a silver strip of width w = 60 nm and thickness t = 25 nm placed directly on a 150-nm-aSi:H-film-on-silver waveguide. Absorption losses in silver and asi:h have been ignored. Proc. of SPIE Vol. 9177 91770M-5

As an alternative to using a silver strip we consider a silicon (asi:h) strip with the same dimensions (w = 60 nm, t = 25 nm) for normally incident s-polarized light in Fig. 5. For a 50-nm asi:h-film [Fig. 5(a)] and long wavelengths ( > 700 nm) the shape of the scattering peaks for out-of-plane scattering and guided modes are very similar to the case of a silver strip [see Fig. 2(a)] with the main difference that the peaks are slightly smaller. Just before the onset of the higher-order guided mode at a wavelength of app. 450 nm another scattering peak is found for scattering into the fundamental guided mode. Thus, the enhanced scattering near the cut-off wavelength of the higher-order guided mode does not only increase scattering into out-of-plane modes but also into the already available guided mode. Apart from this the scattering peaks are located at the same wavelengths as for the case of a silver strip. We should note that the refractive index of asi:h increases for decreasing wavelengths [13] which can be part of the explanation of the increased height of scattering peaks found for short wavelengths. Figure 5. Out-of-plane scattering cross section, guided-mode scattering cross section, and sum of cross sections, for s- polarized light being normally incident on a silver strip (asi:h) of width w = 60 nm and thickness t = 25 nm placed directly on an asi:h-film-on-silver waveguide. The asi:h film has thickness (a) 50 nm, (b) 150 nm, (c) 250 nm, and (d) 500 nm. Absorption losses in silver and asi:h have been ignored. A similar calculation but for p-polarized light is shown in Fig. 6. Comparing with the case of a silver strip (Fig. 3) we notice for long wavelengths ( > 700 nm) that the scattering is significantly reduced. However, for short wavelengths we see the same trend of increasing height of scattering peaks as found for s-polarization. Again, the location of scattering peaks does not depend on if it is a silver or silicon strip. Proc. of SPIE Vol. 9177 91770M-6

Figure 6. Out-of-plane scattering cross section, guided-mode scattering cross section, and sum of cross sections, for p- polarized light being normally incident on a silver strip (asi:h) of width w = 60 nm and thickness t = 25 nm placed directly on an asi:h-film-on-silver waveguide. The asi:h film has thickness (a) 50 nm, (b) 150 nm, (c) 250 nm, and (d) 500 nm. Absorption losses in silver and asi:h have been ignored. 3. THEORETICAL RESULTS INCLUDING ABSORPTION LOSSES In this section we will now turn to the case where absorption losses in the silver and silicon are included. For silver and asi:h we will still use the refractive index from Refs. [12] and [13] but now we will keep both real and imaginary parts. In that case the out-of-plane scattering cross section is still well-defined. However, the cross section for scattering into guided modes is no longer strictly well-defined as explained in the introduction. Instead the extinction cross section is well-defined. In the previous section where absorption losses were absent the sum of cross sections for scattering into out-of-plane propagating and guided modes (denoted total) was exactly equal to the extinction cross section. In that case the cross section for scattering into guided modes could also straightforwardly have been obtained as the difference between the extinction cross section and the out-of-plane scattering cross section. This approach to estimate the scattering into guided modes can still be used in an approximate sense when there are absorption losses, as long as those losses are not too large. In any case the extinction minus out-of-plane scattering is a direct measure of the increase of the absorbed power in the geometry due to the presence of the nanostrip, where most of this power is related to coupling of light into guided modes when the losses are small. We have calculated the out-of-plane scattering cross section, the extinction cross section, and the difference, for s- and p- polarized light being normally incident on a silver strip (w = 60 nm, t = 25 nm) placed directly on an asi:h film of thickness d = 290 nm, 310 nm, and 330 nm (see Fig. 7) for the case with all absorption losses included. Proc. of SPIE Vol. 9177 91770M-7

Figure 7. Out-of-plane scattering cross section, extinction cross section, and extinction minus scattering cross section, for (a) s-polarized, and (b) p-polarized light, being normally incident on a silver strip (asi:h) of width w = 60 nm and thickness t = 25 nm placed directly on an asi:h-film-on-silver waveguide for asi:h film-thicknesses 290 nm, 310 nm, and 330 nm. Absorption losses of silver and asi:h are included. When comparing with the case where absorption losses were ignored the main difference is that scattering peaks for wavelengths smaller than app. 700 nm are either weak or absent compared with the case without losses. This is related to the fact that the guided modes for these wavelengths are now highly lossy due to absorption in asi:h, and resonant scattering is practically eliminated. For the longer wavelengths ( > 800 nm) there is no absorption in asi:h and at the same time absorption in the silver is weak. Thus a calculation with and without losses are practically identical. We can see from the mode-index calculations in Figs. 2 and 3 that the cut-off-wavelengths of guided modes red-shift with increasing film thickness. Here we consider small increases in film thickness from 290 nm to 310 nm and 330 nm. We clearly see (Fig. 7) that peaks in the scattering and extinction cross sections red-shift as well with increasing film thickness, which gives a possibility for fine-tuning the wavelength of a peak to match with the wavelengths where asi:h is weakly absorbing. We also notice that for short wavelengths, and especially for s-polarization, it can occur that the extinction cross section is negative. This does not happen when absorption losses are ignored. Here a negative extinction means that because the reflected beam in the case without the strip has less power than the incident due to absorption in the asi:h film it is possible to actually increase the power of the reflected beam. This can be done by placing a structure Proc. of SPIE Vol. 9177 91770M-8

on the surface that prevents light from being absorbed in the asi:h beneath that structure, which can lead to increased specular reflection, with a resulting negative extinction according to our definition. For wavelengths with a negative extinction, or actually extinction minus out-of-plane scattering, less light is absorbed in the solar cell due to the presence of the nanostrip. 4. CONCLUSION In conclusion, we have considered scattering of light by a silver or silicon nanostrip placed directly on a silicon (asi:h)- film-on-silver waveguide geometry and calculated scattering cross sections that govern scattering into guided modes and into out-of-plane propagating modes, respectively. It was found that the total scattering cross section spectra have resonant peaks at wavelengths that match the cut-off-wavelength of guided modes in the air-asi:h-film-silver waveguide geometry. The wavelengths of those peaks were shown to be insensitive to the direction of light incidence and the material of the nanostrip (silver or silicon). When comparing results obtained by assuming that material losses are absent (no absorption) with results where all material losses are included, it was found that for wavelengths with significant material losses the scattering peaks practically disappear. It was demonstrated that the wavelengths of scattering peaks can be tuned by changing the thickness of the asi:h film, and in particular such peaks can be tuned to wavelengths where asi:h is weakly absorbing, in which case the scattering of light into guided modes (light trapping) will lead to enhanced absorption. ACKNOWLEDGMENTS This work was funded by the Danish Council for Strategic Research as part of the project Thin-film solar cell based on nanocrystalline silicon and structured backside reflectors - THINC. Corresponding author*: ts@nano.aau.dk REFERENCES [1] Atwater, H. A., and Polman, A., "Plasmonics for improved photovoltaic devices," Nature Materials 9, 205-13 (2010). [2] Yu, Z., and Fan, S., "Nanophotonic light-trapping theory for solar cells," Appl. Phys. A 105, 329-339 (2011). [3] Spinelli, P., Ferry, V. E., van de Groep, H., van Lare, M., Verschuuren, M. A., Schropp, R. E. I., Atwater, H. A., and Polman, A., "Plasmonics light trapping in thin-film Si Solar cells," J. Opt. 14, 024002 (2012). [4] Stuart, H. R., and Hall, D. G., "Absorption enhancement in silicon-on-insulator waveguides using metal island films," Appl. Phys. Lett. 69, 2327-2329 (1996). [5] Springer, J., Rech, B., Reetz, W., Müller, J., and Vanecek, M., "Light trapping and optical losses in microcrystalline silicon pin solar cells deposited on surface-textured glass/zno substrates," Sol. Energy Mater. Sol. Cells 85, 1-11 (2005). [6] Paetzold, U. W., Meier, M., Moulin, E., Smirnov, V., Pieters, B. E., Rau, U., and Carius, R., "Plasmonic back contacts with non-ordered Ag nanostructures for light trapping in thin-film silicon solar cells," Materials Science and Engineering B 178, 630-634 (2013). [7] Battaglia, C., Hsu, C.-M., Søderstrøm, K., Escarre, J., Haug, F.-J., Charriére, M., Boccard, M., Despeisse, M., Alexander, D. T. L., Cantoni, M., Cui, Y., and Ballif, C., "Light Trapping in Solar Cells: Can Periodic Beat Random?," ACS Nano 6, 2790-2797 (2013). [8] Pala, R. A., Liu, J. S. Q., Barnard, E. S., Askarov, D., Garnett, E. C., Fan, S., and Brongersma, M. L., "Optimization of non-periodic plasmonic light-trapping layers for thin-film solar cells," Nat. Commun. 4:2095 doi: 10.1038/ncomms3095 (2013). [9] Jung, J., and Søndergaard, T., "Green s function surface integral equation method for theoretical analysis of scatterers close to a metal interface," Phys. Rev. B 77, 245310 (2008). [10] Søndergaard, T., Siahpoush, V., and Jung, J., "Coupling light into and out from the surface plasmon polaritons of a nanometer-thin metal film with a metal nanostrip," Phys. Rev. B 86, 085455 (2008). Proc. of SPIE Vol. 9177 91770M-9

[11] Søndergaard, T., Tsao, Y.-C., Kristensen, P. K., Pedersen, T. G., and Pedersen, K., "Light-trapping in guided modes of thin-film-silicon-on-silver waveguides by scattering from a nanostrip," J. Opt. Soc. Am. B, in press (2014). (at the time of writing the paper is online at the JOSA B website in the early postings section.) [12] Johnson, P. B., and Christy, R. W., "Optical Constants of the Noble Metals," Phys. Rev. B 6, 4370-4379 (1972). [13] Fisker, C., and Pedersen, T. G., "Optimization of imprintable nanostructured a-si solar cells: FDTD study," Opt. Express 21, A208-A220 (2013). Proc. of SPIE Vol. 9177 91770M-10