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The spatial dependence of absorption in a structured thin film solar cell is investigated through the rigorous coupled-wave analysis method. The investigated structure allows strong localized surface plasmon and surface plasmon polaritons, simultaneously. The absorptance of silver and amorphous silicon can be separately accounted for by calculating the time-averaged energy dissipation although only the absorption of amorphous silicon contributes to the photocurrent. In our studied case, the metallic material absorbs around 15%-20% of the total impinging sunlight while the active layer absorbs only ~50%.
©2010 Optical Society of America
1. Introduction
Solar energy helps provide a reasonable solution to a future without fossil fuels. However a large energy amount is needed to fabricate crystalline silicon solar cells (the first generation solar cells). The development of thin film solar cells would make for much less energy wasted during the fabrication process. The light absorption of a solar cell, which is the first step to converting sunlight into electricity, is always a challenge in thin film solar cells. In simulations of light absorption in the past, it has been assumed that light behaves as a ray. The simple exponential decay rule has been applied to determine the spatial distribution of the absorption [1]. However, the ray treatment method does not work well for the case of a thin-film solar cell and/or a material with a small absorption coefficient. In this case, the standing wave generated by the multi-beam interference cannot be simply described by an exponential function. Full wave treatment of the light to determine the spatial distribution of the absorption is required. The rigorous coupled-wave analysis (RCWA) method is a very powerful tool for wave simulation in a periodic structure [2–4]. In recent years, there has even been a series of studies seeking to expand the RCWA method to cover non-period situations [5]. Various methods have been utilized to enhance the light absorption of solar cells, such as surface roughness [1], photonic crystals [6,7] and surface plasmon (SP) resonances [8,9]. However, the light absorption of solar cells is usually overestimated [10]. This can occur because some materials have the ability to absorb light but do not contribute to the photocurrent. This is especially true for enhancement using surface plasmon resonance, which is related to the interaction between electromagnetic waves and a metallic material [11]. Surface plasmon resonance can be interpreted as the collection of electrons at the metallic surface, induced by the electromagnetic wave, and these collected electrons vibrate together in a particular way. Two basic properties of a surface plasmon on a metal surface are: (1) the surface plasmon propagates alone the metal surface; (2) the energy of the surface plasmon is concentrated at the metal surface. Therefore, surface plasmon resonance can be applied to enhance the absorption of thin film solar cells constructed using nano metallic particles [12–14], metallic gratings [15], or a multilayer metallic structure [16]. The metallic materials are usually placed as near as possible to the active layer in a solar cell, since the surface plasmon (SP) only influences the adjacent region. The active layer, which separates the excited electron-hole pairs, can be made of semiconductors, organic molecules, or dye materials. Only light absorption in the active layer can contribute to the external electricity; the light absorption of the other materials is useless. Thus one cannot simply just measure the absorptance spectrum of a thin film solar cell and assume that all light absorption happens in the active layer. The absorption of each component in the solar cell should be separated. Furthermore, to determine the external quantum efficiency (EQE) [17] precisely it is necessary to find the spatial distribution of absorptance in a solar cell. The EQE is the product of light absorptance and the internal quantum efficiency (IQE). The IQE is a spatial function describing the probability that an electron-hole pair will separate and reach the electrodes without recombination. Another application of obtained absorption distribution would to design the intermediate reflector for a tandem type solar cell [18]. There is a serious problem with the tandem type solar cell, which is the low current limitation. This problem must be solved by light management to maximize the output current. When the spatial distribution of the light absorption is known, the output current can be optimized through simulation.
In this study, we discuss the issue of absorption. We derive the equation for spatial distribution of absorptance based on the classical electromagnetic wave theorem. The rigorous coupled-wave analysis (RCWA) method is used to calculate the angle-resolved absorptance spectrum and electromagnetic field distribution of a plasmonic multilayer structure (PMS), which is considered as a simplified thin film solar cell structure. Finally, we will show the spatial distribution of absorptance of the PMS and the separation of absorptance from the active layer and metallic material.
2. Absorption issue and simulation setup
The spatial distribution of the absorption of a solar cell can usually be understood based on the concepts of geometrical optics. The spatial distribution follows an exponential decay of intensity and the Fresnel reflection at each interface. This method can give a good solution if the solar cell is thick enough or the absorption coefficient of the material is high. However, given the interference phenomenon that occurs in some much thinner solar cells or when the absorption coefficient is small, for instance when there is small absorption in the band tail range, the spatial distribution of the absorption must be considered as a full wave result. Some researchers have used the divergence of the Poynting vector to determine the absorption at an arbitrary position in a solar cell [18]. The divergence of the Poynting vector is shown in the second part of Eq. (1). The Poynting theorem in the differential form is as follows [19]:
where E and H are the electric field and magnetic field, respectively. The divergence of the Poynting vector denotes the net energy flow into/out of some spatial position. However, we would like to use the power density for the data processing of absorption, while the divergence operation is inconvenient within a grid system. If the materials are assumed to be linear and non-magnetic, the time-average power density can be derived to the form:where ε is the complex permittivity of a dispersive media. Equation (2) is applied to calculate the spatial distribution of the absorption. Although the light interaction with Ag and amorphous silicon (a-Si) are different, the optical properties of both can be defined by the complex permittivity and permeability. We can assume that the energy just dissipates, regardless of the physical mechanism, which may create electron-hole pairs in a semiconductor or become heat in a metal. This formula for energy dissipation has been applied to calculate the absorptance distribution in multilayer structures [12,15,20,21].In order to study the more general case characteristic of real solar cells, we expand the concept to a two-dimensional structure. The field distribution is calculated by the rigorous coupled-wave analysis (RCWA) method. The RCWA method splits the simulated structure into several parts that each of them has different optical properties. The splitting is according the rule that the optical properties of each part are independent in the main propagation direction, the z direction in this study. Fourier wave expansion is then applied in a homogeneous layer and Block wave expansion in a periodic structured layer, which has variation of permittivity in the lateral direction. The wave expansion has a truncation called the harmonics number, which indicates the number of expanded waves needed to reach the simulation solution. The RCWA method directly obtains a steady state solution, unlike the finite-difference time-domain (FDTD) method, which involves time revolution and is time-consuming. The studied plasmonic multilayer structure (PMS) is shown schematically in Fig. 1 . The PMS consists of an Ag grating-like front electrode, a thin a-Si active layer, and a backside Ag reflector/electrode. A TM-polarized wave, in which the magnetic field is parallel to the grating grooves, is launched from the bottom at an incident angle of θi with respect to the normal direction of the surface. The TM-polarized wave is required to excite the SP resonances. This structure is a simplified thin film solar cell structure similar to that studied before [22,23].
Fig. 1 Schematic diagram of the plasmonic multilayer structure (PMS), showing simplified structure of a thin film solar cell. The PMS consists of a silver (Ag) grating-like front electrode, a thin amorphous silicon (a-Si) active layer, and a backside Ag reflector/electrode (from bottom to top). The TM-polarized wave is launched from the bottom with an incident angle of θi.
The a-Si layer is assumed to be a monolayer, i.e., each layer of the original structure although having different doping is assumed to have the same refractive index. The thickness of the a-Si layer (ta) is only 30nm, which is advantageous for the collection of the generated electron-hole pairs, due to the short distance between the front and back electrode. The period of the Ag grating (Λ) is 350nm and the thickness (tg) is 100nm. The backside of the Ag reflector/electrode is 500nm thick. This is to make sure that there is no transmission through it. We discuss two widths of Ag grating line (W), to see the shifting of the localized SP resonance in the spectrum. The complex dispersive permittivity for Ag (εm) and a-Si (εa) are referred from the reference [24].
3. Angle-resolved absorptance spectrum
The simulated angle-resolved absorptance spectra of the PMS for W = 50nm and 120nm are shown in Figs. 2(a) and (b) , respectively. The horizontal axis is the parallel wave vector of the launch light (kx), i.e., kx = k0sinθi, where the k0 is the wave vector of light in a vacuum and the launch angle (θi) varies from 0° to 60°. The spectrum scans the wavelength from 300nm to 900nm, corresponding to photon energies of 4.132eV and 1.377eV on the vertical axis. The red and blue colors indicate higher and lower absorptance, respectively. The structure is interesting due to the mixing state characteristics of the localized SP and the surface plasmon polaritons (SPPs) on the Air/Ag and a-Si/Ag interfaces. These surface modes provide channels to confine the sunlight in the PMS, but they also cause difficulty for precisely analyzing the contribution of each surface mode. These SP modes are usually couple to each other.
Fig. 2 Angle-resolved absorptance spectra for (a) W = 50nm and (b) W = 120nm. The other geometric parameters are identical. The parallel wave vector of the launch light (kx) is obtained from kx = k0sinθi. The launch angle (θi) varies from 0° to 60° and the spectrum scans wavelength from 300nm to 900nm. The absorptance is obtained from the far field results (1-R-T) through the RCWA method.
There is an absorptance peak around 3.5eV in Fig. 2(a), which is induced by localized SP of the Ag grating. The localized SP is less affected by the launch angle of the light; the resonance wavelength of the localized SP is dependent on the geometric properties of a unit cell. This localized SP resonance red shifts to around 2.75eV when the W becomes 120 nm. The localized SP resonance forms the horizontal absorption peak shown in Fig. 2(b). In contrast to the localized SP, the surface plasmon polaritons (SPPs) must preserve the moment conservation equation, ksp = kx + m(1/Λ), where ksp is the wave vector of the SPPs on a flat surface; 1/Λ is the reciprocal lattice vector of the Ag grating; and m is an integer. The ksp can be calculated by the well-known momentum conservation equation ksp = k0((εm × εd)/(εm + εd))1/2, where εd is the permittivity of the adjacent dielectric. Therefore, a mixing state of SPPs from the Air/Ag and a-Si/Ag interfaces shows clear angle-dependent behavior around 2.25eV. If only the SPPs of Air/Ag is considered, this angle-dependent band is related to its m = −1 momentum conservation condition. In Figs. 2(a) and (b), it can be seen that there is relatively high absorptance in the lower photon energy region from ~1.5eV to ~2.0eV. This additional absorptance comes from the mixing state of the intrinsic Fabry Perot resonance in the thin a-Si layer (30nm in thickness) and SPPs on the a-Si/Ag interface, which is excited by the evanescent field of the Ag grating. Such excitation of the SPPs on the a-Si/Ag interface can also be stimulated by a dielectric grating instead of an Ag grating, provided the gap between the grating and the a-Si/Ag interface is small enough. Thus we can see that, when the W becomes 120nm from 50nm, there is only a little influence on this region. We should note that the absorptance in Fig. 2 is calculated by (1-R-T) instead of Eq. (2), where R and T are the total reflectance and transmittance in the far field. It has to mention that the solar light is unpolarized. The TE-polarized light is shuttered by the subwavelength metallic grating. However, replacing the grating by an array of isolated metallic islands in a practical case, there is no significant cut-off of TE-polarized light. Even the metallic grating form is reserved for some reason; the TE light might be trapped by using the guided-mode resonance (GMR) effect [25] when the dielectric grating in GMR structure is replaced by the metallic grating. However, the design of structure will be much complex for both enhancing light absorption in TM and TE polarization through the SP and GMR, respectively.
4. Spatial distribution of the absorptance
Figure 3 shows the spatial distribution of the absorptance and the Hy amplitude distribution in a unit cell at the resonant wavelengths of 350nm and 760nm when W is 50nm and the launch angle is 0°. The absorptance values in Figs. 3(a) and (c) were calculated directly from Eq. (2) then normalized by the input power. The white arrows in Figs. 3(b) and (d) indicate the corresponding time-averaged Poynting vector. The localized SP dominates at the wavelength of 350nm; however Ag always has high absorption at this wavelength due to the nature of the d-band absorption. Figures 3(a) and (b) respectively show the spatial concentration of the absorptance and the Hy field for the Ag grating part at this wavelength. Figure 3(a) shows that there are two small regions inside the a-Si material, which are adjacent to two top corners of the Ag grating, have high absorptance. This is due to the evanescent properties of the localized SP. As mentioned above, the energy of SPPs and localized SPs is always confined to the corresponding interface. Therefore, only the a-Si region near the interface benefits from the SP resonances leading to enhanced absorptance in an a-Si material. The a-Si material region, when not directly blocked by the Ag grating, absorbs less than that predicted by the ray treatment method. At a wavelength of 350nm, the optical response of Ag is more like a lossy dielectric media and then the electromagnetic wave can penetrate deeply into the Ag grating; the spatial distribution of the absorptance also spreads into the Ag grating. As assumed, no absorption presents in the air region. There is no significant absorption on the backside Ag reflector, since the SPPs on the a-Si/Ag interface is not excited at this wavelength. The Hy amplitude distribution is shown in Fig. 3(b). The property of the localized SP on the Ag grating has an asymmetric resonant mode, i.e., resonance between the two top corners and bottom surface. In addition, the direction of the Poynting vector in the area surrounding this Ag grating varies greatly due to the localized SP.
Fig. 3 Spatial distribution of absorptance and Hy amplitude field distribution of the PMS with W = 50nm at wavelengths of 350nm and 760nm. Localized SPs happen at λ = 350nm and SPPs happen at λ = 760nm.
At a resonant wavelength of 760nm, the SPPs on the a-Si/Ag interface are excited by the evanescent field of the adjacent Ag grating nano-structure. Figure 3(d) shows a strong Hy field located along the interface of the a-Si/Ag. The simultaneous absorptance distribution is shown in Fig. 3(c). It spreads widely into the a-Si layer. However, we have difficulty seeing the clear SPPs behavior due to the mixing state of the Fabry Perot and SPPs in this situation. An indirect characteristic of the existence of the SPPs is the variation in the Poynting vector near the backside of the Ag reflector, which does not appear at a wavelength of 350nm. The concentration of absorptance near/inside the Ag grating also disappears. However, at this wavelength, the blocking effect of the Ag grating becomes significant, due to the metal-insulator-metal (MIM) structure. This structure modifies the effective SPPs mode inside [26] so that the originally excited SPPs at the a-Si/Ag interface are not suitable. In general, this kind of absorptance distribution does not lead to a high EQE due to the greater distance between the location of generation and electrodes. This absorption distribution consideration is important to a real thin film solar cell because of the spatial dependence of the IQE. We have to consider both the properties of the light and the electrical properties of the thin film solar cell when designing higher efficiency structures in future. However, in this study we only sketch out the spatial distribution of the absorptance in the active layer. This is first step.
Figure 4 shows the spatial-integrated absorptance spectrum of each material in the PMS in different situations. The spatial-integrated absorptance means the absorptance, which is a function of space as shown Fig. 3(a) and (c), is summed spatially based on the different materials in this structure. The green and blue lines, labeled “a-Si” and “Ag”, indicate the spatial-integrated absorptance inside the a-Si and Ag material, respectively. The spatial distribution of absorptance is calculated from Eq. (2). The red line indicates the summation of “a-Si” and “Ag”. The results are in agreement with the far field results; see the black line labeled “far field” (1-R-T) in Fig. 4. The upper and lower two diagrams in Fig. 4 show the simulated results using launch angles of 0° and 30°, respectively. Comparing Fig. 4(a) with (b), the most significant difference is that the localized SP absorptance peak red shifts when the W becomes longer. This localized SP resonance becomes significant at 450nm as shown in Fig. 4(d). The mixing state of SPPs on Air/Ag and a-Si/Ag is presents around 550nm as seen in Figs. 4(c) and (d), when the launch angle rises to 30°. The separation of absorptance between a-Si and Ag also makes it easier to distinguish the dominating mode in a mixing state. We can see that the absorptance of Ag becomes larger with increasing wavelength in the spectral range from 600nm to 900nm. This phenomenon can attribute to the excitation of a-Si/Ag SPPs on the Ag reflector. Originally Ag should behave more like a perfect conductor, i.e., no absorption, with increasing wavelength. The coupling between evanescent fields and a-Si/Ag SPPs is more efficient at longer wavelengths, then the absorptance of Ag increases. Unfortunately, the absorptance spectrum of the a-Si material, which acts as an active layer, is relatively low in this spectral range. An additional simulation shows that the a-Si layer has, on average, ~50% of the absorptance in a similar case without an Ag grating. The Ag material, including the Ag grating and Ag backside reflector, provides around 15%-20% of the absorptance on average. Obviously, this additional part of the absorption from metal cannot contribute to the photocurrent of the solar cell. The SP resonances might enhance the light absorption. However, in our case, the SP-enhanced light absorptance cannot be totally taken account as enhancing the external quantum efficiency of a solar cell. A similarly negative result has been reported for absorption enhancement through SP resonances [27]; additionally, the optimization and limits are proposed in the configuration without the backside Ag reflector [28]. We should perhaps focus on using the scattering properties of metal material to trap light instead of exciting these surface modes to cause large absorption [8,13,16].
Fig. 4 Absorptance spectra with W = 50nm and W = 120nm at (a) (b) normal incident or (c) (d) oblique incident conditions. The “far field” line indicates the absorptance calculated from the far field result (1-R-T). The “a-Si” line and “Ag” line respectively indicate the a-Si absorptance and Ag absorptance, which are calculated from Eq. (2) and are spatially integrated. The “a-Si+Ag” line indicates the summation of the “a-Si” line and “Ag” line.
Figure 5 shows the spatial distribution of the ideal electron-hole pair generation rate. They are calculated by summing the absorptance spectra (with θi = 0°) from 300nm to 900nm with additional weighting by the AM1.5d solar radiation spectrum, which is direct radiation of the sun located at 37°. It is assumed at the ideal electron-hole pair generation rate that the quantum efficiency is 1 for absorption occurred in the a-Si layer and 0 in the Ag material. The generation rate is the product of the photon absorption rate and quantum efficiency. The scale bar is in the unit of generation number/(second.nm3). The blocking effect of the Ag grating means that there is a relatively low generation rate in the region behind the Ag grating. The blocking effect is more serious when W = 120nm. A comparison of Fig. 5(a) and (b) shows that the blocking effect leads to enhancement in the adjacent area over that without blocking (with normal incidence). The generation rate reaches a maximum in the region in the center of the a-Si layer, which indicates the wave behavior of the thin film. This is good for separation of excited electron-hole pairs because the center region usually has lower recombination rate. Figures 5(c) and (d) show the oblique blocking effect of the Ag grating with a launch angle of 30°. It can be seen that the absorptance is concentrated on the left side. There are two small high absorptance spots adjacent to the two top corners of Ag grating. These formed due to the excited SP resonances in these four conditions.
Fig. 5 Electron-hole pair generation rate distribution with (a) (c) W = 50nm, and (b) (d) 120nm at launch angles of 0° and 30°, respectively. The generation rate is calculated by summing the spatial distribution of the absorptance in spectra from 300nm to 900nm with a weighting of AM1.5d solar radiation spectrum. We assume the quantum efficiency is 1 in a-Si and 0 in Ag.
5. Conclusion
In this study, the spatial distribution of absorptance in a PMS solar cell is evaluated using the power density calculation. The angle-resolved absorptance spectrum of the PMS, as in a simplified thin film solar cell, is studied to point out the several areas of the dominating SP resonance. The spatial distribution of the absorptance is demonstrated and the blocking effect from the Ag grating is shown. We can separate the contribution of absorptance from the a-Si and the Ag. The usually overestimated absorptance of a-Si is shown if we consider the absorption of Ag happens in a-Si layer. The Ag material, including the Ag grating and Ag backside reflector, absorbs around 15%-20% of the total sunlight (spectrum-average) while the a-Si material absorbs only ~50%. This separation is also helpful allowing us to distinguish several SP modes in a complex structure. Due to the large absorption of the Ag material, the application of SP resonances on a thin film solar cell to enhance the EQE might fail. The concepts in this study can be followed to complete similar analyses of other structures, such as nano particles and photonic crystals, or the design for intermediate reflectors. The EQE of a solar cell can be more precisely predicted through the obtained spatial distribution of absorptance. The spatial distribution of the absorptance can be also calculated using the FDTD, finite element method (FEM), and so on, to study the electromagnetic field distribution.
Acknowledgements
The authors are grateful for the financial support of this research received from the National Science Council of Taiwan, R.O.C. under the grant number NSC98-2221-E-259-004-.
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