## Abstract

The optics of microcrystalline thin-film silicon solar cells with textured interfaces was investigated. The surface textures lead to scattering and diffraction of the incident light, which increases the effective thickness of the solar cell and results in a higher short circuit current. The aim of this study was to investigate the influence of the frontside and the backside texture on the short circuit current of microcrystalline thin-film silicon solar cells. The interaction of the front and back textures plays a major role in optimizing the overall short circuit current of the solar cell. In this study the front and back textures were approximated by line gratings to simplify the analysis of the wave propagation in the textured solar cell. The influence of the grating period and height on the quantum efficiency and the short circuit current was investigated and optimal grating dimensions were derived. The height of the front and back grating can be used to control the propagation of different diffraction orders in the solar cell. The short circuit current for shorter wavelengths (300-500 nm) is almost independent of the grating dimensions. For intermediate wavelengths (500 nm – 700 nm) the short circuit current is mainly determined by the front grating. For longer wavelength (700 nm to 1100 nm) the short circuit current is a function of the interaction of the front and back grating. An independent adjustment of the grating height of the front and the back grating allows for an increased short circuit current.

©2011 Optical Society of America

## 1. Introduction

Reducing the cost and increasing the conversion efficiency is a major objective of research and development on solar cells. An approach that simultaneously achieves these two objectives is to use light-trapping or photon management. Light trapping facilitates the absorption of sunlight by a thin-film solar cell that is much thinner than the absorption length of the material. We will focus in this study on the analysis of thin-film solar cells based on hydrogenated microcrystalline silicon. Microcrystalline silicon thin-film silicon solar cells have a typical thickness of 0.8–1.5 µm, which is significantly less than the thickness of conventional wafer based silicon solar cells (200 µm – 300 µm) [1].

Different light trapping or photon management methods have been proposed for increasing the effective optical thickness of the solar cell, while preserving the actual absorber layer thickness in the range of 1 μm. One approach for increasing the effective thickness of the solar cell is introducing randomly textured surfaces in the solar cell structure. When light is diffracted and scattered upon entering the solar cell, conditions for multiple light reflections inside the cell are created. It has been shown that introduction of randomly textured surfaces in microcrystalline thin-film solar cells leads to a strong increase in the short circuit current and the conversion efficiency [2–6]. The random textured surface can be realized by wet etching of sputtered zinc oxide films, direct deposition of textured zinc oxide or tin oxide films by Low Pressure Vapor Chemical Vapor Deposition (LPCVD) or using nano particle based films [7–9].

Two-dimensional simulations of the wave propagation in such solar cells show that the short circuit current of the solar cell with absorber layer thickness of 1 µm can be enhanced from 16.3 to 23.4 mA/cm^{2} [10]. Simulations were carried out for solar cells with integrated periodic line grating or pyramid textures [11–17]. In the case of periodic line gratings, the highest short circuit current was determined for grating period of 600 nm and a grating height of 300 nm [17]. Similar results were observed by different research groups [11–13]. Haase et al. calcucated the highest short circuit current for a grating period of 700 nm and a grating height of 350 nm [11]. Catchpole et al observed the highest current for a grating period of 650 nm and a grating height of 300 nm. Haase et al. determined an optimal grating period of 850 nm and a pyramid height of 400 nm [19]. Zanotto et al. studied the simultaneous application of a patterned surface and an antireflection coating. Improvements of the short circuit current of 12.4% and up to 36.5% were observed, respectively, for amorphous silicon and crystalline silicon as an absorbing material [20]. Even though similar results were observed by different research groups it remains unclear why the highest short circuit current is observed for such grating dimensions. The aim of the investigation is to identify the underlying operation principle and to derive optimization criteria for other thin-film solar cells with textured interfaces. The optical wave propagation was investigated by Finite Difference Time Domain (FDTD) method and Rigorous Coupled Wave Analysis (RCWA) [21,22].

In section 2 the basic optical simulation model will be introduced and the simulation results will be described. In section 2.1 the device structure and the optical tools are introduced. In section 2.2 the short circuit current as a function of the grating period will be presented. The influence of the grating shape on the short circuit current will be described in section 2.3. The dependence of the short circuit current on the grating height will be presented in section 2.4. A separate analysis of the diffraction profiles of the front and the back grating is described in section 2.5. Finally, the behavior of solar cells with combined front and back grating will be described in section 2.6. In the discussion section (section 3) design rules for the grating dimensions are derived for improving the light trapping properties of thin-film solar cells.

## 2. Optical model of thin-film silicon solar cell

#### 2.1 Device structure and simulation tools

The schematic cross section of a microcrystalline silicon solar cell on a smooth substrate is shown in Fig. (1a)
. The microcrystalline solar cell structure, investigated in the study, consists of a 500 nm thick aluminum doped zinc oxide (ZnO:Al) front contact, followed by a hydrogenated microcrystalline silicon p-i-n diode (*µc*-Si:H) with a total thickness of 1000 nm and a back reflector consisting of an 80 nm thick ZnO:Al layer and a perfect metal reflector. The thickness of the p- layer was assumed to be 30 nm. A detailed investigation of the influence of the p-layer on the quantum efficiency of textured solar cells is given in Ref. 23. Investigations of the light absorption in the n-layer show that only 1-2% of the longer wavelength light is absorbed in the layer [23].

A schematic cross section of a microcrystalline thin-film silicon solar cell with randomly textured interfaces is presented in Fig. 1(b). When incoming light reaches the ZnO/Si interface, it is diffracted by the textured surface and propagates at different angles within the silicon absorber layer. The incoupled light is further diffracted and reflected by the textured back contact. Thus, conditions for multiple reflections in the thin-film solar cell structure are created and the optical path within the silicon layer is increased. As a consequence, the optical path length of the device is enhanced. The cross sections in Fig. 1(a) and 1(b) represent solar cells in superstrate configuration. The light enters the solar cells through a glass substrate. The texturing of the solar cell is realized by texturing the transparent conductive oxide (TCO) based front contact prior to the deposition of the microcrystalline silicon thin film solar cell and the back contact. The microcrystalline silicon thin film solar cells are prepared by Plasma Enhanced Chemical Vapor Deposition (PECVD), which leads to the conformal deposition of the layers of the solar cell. Therefore, the surface texture of the front contact propagates all the way through the layer stack. No additional texturing of the back contact is required. Further details on the fabrication of the solar cells are given in Ref. 7-10.

To investigate the optics in nanotextured thin-film solar cells, it is imperative to use numerical models for the analysis of the optical losses in all layers of the thin-film silicon solar cells [11–18]. However, the investigation of the wave propagation within a randomly nanotextured solar cell is complex; hence we investigated a solar cell with a periodic surface texture. We assumed that the randomly textured surface can be approxiamted by a periodic surface structure. The invistigations of Haase and associates show that this assumption is valid [12]. They studied the optical wave propagation of microcrystalline silicon thin film solar cells on Quasi-random surfaces. Such surfaces consist of a large number of randomly arranged pyramids. The short circuit current of the solar cell on the Quasi-random surfaces was calculated and compared to simulations of periodic surface textures. The short circuit current could be determined by an area weighted superposition of the short circuit current for periodic surface textures. The simulations show that for relatively thin solar cells like amorphous or microcrystalline silicon solar cells the optics can be approximated by the area weighted superposition of optical simulations of periodic stuctures.

The cross sections of the investigated periodic structures are shown in Fig. 1(c) and Fig. 1(d). The periodic surface texture can be described by the grating period (P) and grating or triangular height (Hg). In this study the grating period and heigth of the texture were varied from 50 nm to 3000 nm and 0 nm to 500 nm, respectively. Since the period of the surface texture is in the range of the incident wavelength, Maxwell’s equations have to be rigorously solved [21]. The finite difference time domain (FDTD) method was used for numerically solving Maxwell’s equations. The unit cell was illuminated under normal incidence for the entire spectrum of wavelength 300 – 1100 nm. The complex optical constants of the layers were determined by optical measurements of the individual layers [25]. As a first step the values of the electric field at every point within the specified simulation grid were calculated. Afterwards, the time average power loss was determined. The quantum efficiency was calculated as the ratio of the power absorbed by the solar cell to the total power incident on the unit cell. In the last step the short circuit current was calculated for a spectral irradiance Air Mass 1.5 (AM 1.5). Further details on how to calculate the quantum efficiency and the short circuit current are given in Ref. 19.

#### 2.2 Influence of grating period on short circuit

The calculated short circuit current of microcrystalline silicon thin film solar cells with an integrated triangular and line grating is shown in Fig. 2 (a) and (b)
. The dashed line in both graphs show the short circuit current of a solar cell prepared on a smooth substrate [Fig. 1(a)]. The short circuit current calculated for such a solar cell structure is 12.4 mA/cm^{2}. Irrespective of the grating height and grating period, a distinct increase of the short circuit current is observed for the solar cell with integrated triangular and line grating. For both surface textures an optimal short circuit current is observed for grating periods in the range from 500 nm to 800 nm. For smaller and larger grating periods a decrease of the short circuit current is observed. The trend for the short circuit current as a function of the grating period can be explained by the grating equation

*P*is the grating period,

*n*denotes the refractive index of the propagating media after diffraction,

*m*specifies the diffraction order, and

*θ*designates the diffraction angle. For large grating periods the diffraction angle is reduced. As a result, the effective thickness of the solar cell is decreased. With further increasing period size the effective thickness of the solar cell converges toward its real thickness. Subsequently, the short circuit current for such structures converges toward the short circuit current for a microcrystalline thin-film silicon solar cell on a smooth substrate. For small grating periods the diffraction angle is increased. Light being diffracted at higher diffraction orders leads in turn to an increased optical path length in the structure. However, for small grating periods (P<450nm) higher diffraction orders cannot propagate anymore and as a result a drop of the short circuit current is observed. The maximal short circuit current is observed for grating periods ranging from 500 nm to 800 nm. In the case of the triangular grating the the highest short circuit current of 21 mA/cm

_{m}^{2}was determined for a grating period of 600 nm and grating height of 500 nm. Compared to the short circuit current of 12.4 mA/cm

^{2}for a solar cell on a smooth substrate, the short circuit current is enhanced by 70%. The solar cell with the integrated line grating exhibits a highest short circuit current of 19.7 mA/cm

^{2}for a grating period of 600 nm and a grating height of 300 nm

#### 2.3 Influence of grating shape on the short circuit current

A comparison of the solar cell with an integrated triangular and line grating in Fig. 2 (a) and (b) indicates that the shape of the grating has a relatively small effect on the short circuit current of a microcrystalline silicon thin film solar cell. Both structures exhibit maximum short cicuit currents in the range of 20 mA/cm^{2}. The maximum short circuit current is observed for structures with grating periods ranging from 500 nm to 800 nm and grating heights of of 300 nm to 500 nm. A difference between solar cells with triangular and line gratings is observed for larger grating periods. The short circuit current drops for solar cells with a line grating with periods larger than 900 nm, while the short circuit current for the triangular grating remains high until the grating period exceeds 1500 nm.

In the following chapters simple design rules will be derived for the optimization of silicon thin film solar cells with line gratings. A direct transfer of the derived grating dimensions from a line to triangular grating is not possible. Neverthelss, the analysis of a line grating provides insights in the optics of a microcystalline solar cell with a nanotexture surface.

#### 2.4 Influence of grating height on short circuit current

Together with the grating period, the grating height is the other important grating parameter that has an influence on the short circuit current. To gain insights in the optics of textured solar cells the influence of the front and the back grating on the wave propagation was investigated separately. We will start with the front grating. The front grating acts as a phase grating; hence, the phase difference can be calculated using the equation

where n_{si}and n

_{ZnO}are the real parts of the complex refractive indices of silicon and zinc oxide, and h

_{g}being the grating height of the line grating. By varying the grating height the phase difference of the interfering waves is changed. For simplicity we will concentrate in the following on two cases, a phase change of π and 2π. In the case of farfield diffraction (Fraunhofer diffraction), a phase difference of π results in the propagation of even diffraction orders [24]. As a result, the zero order is canceled out and light is diffracted and transmitted only at orders m = ± 1, ± 3, etc. as shown in the sketch in Fig. 3(a) . Hence, all light propagates at large diffraction angles, which increases the optical path length of the light in the solar cell. For a grating structure causing a phase change of π and an incident wavelength of 700 nm the corresponding grating height of the front grating was calculated to be 180 nm.

Similarly, for a grating structure with a phase difference of 2π destructive interference of the odd diffraction orders is observed, so that only even diffraction orders can propagate (m = 0, ± 2, ± 4, etc.) as shown in Fig. 3(b). Hence, a large fraction of the transmitted light propagates in the 0th order. In this case the optical path length of the light is reduced in comparison to the grating generating a phase difference of π. For a phase difference of 2π and an incident wavelength of 700 nm the corresponding grating height of the front grating was calculated to be 360 nm.

We will continue with the back grating. The back grating acts as a reflection grating. It diffracts and reflects the incident light transmitted through the front grating. The phase difference introduced by the back grating can be calculated by:

where n_{si}is the real part of the complex refractive index of silicon, and h

_{g}being the grating height of the line grating. The phase difference of the back grating can be controlled by the grating height.

The influence of the front and the back grating on the wave propagation was studied by calculating the diffraction pattern of the front and the back grating separately. To analyze the diffraction behavior of the solar cells as a function of the wavelength we used the Rigorous Coupled Wave Analysis [21,22]. By using the Rigrorous Coupled Wave Analysis we can calculate the diffraction efficiency of the transmitted and reflected waves for the different orders as a function of the incident wavelength.

In Fig. 4 (a) and (b) , a contour plot of the specular and diffused transmission of the front grating for a grating period of 700 nm is shown. The transmission was calculated for a front grating fabricated on top of an infinite silicon slab to study the behavior of the front grating. The diffused transmission is defined as the transmitted light that is diffracted in diffraction orders other than the 0th order. If the phase change introduced by the front grating is equal to π, the light is diffracted in the odd orders m = ± 1, ± 3. Consequently, most of the light contributes to the diffused transmission, while the specular transmission is minimal. The red line in Fig. 4(b), overlaying the contour maps, shows the grating height, for which a phase difference of π is observed. In Fig. 4(a) the specular or 0th order transmission is plotted. It can be seen that for a phase difference of π the specular transmission is minimal, whereas the specular transmission for a phase change of 2π is maximal. For a phase change of 2π more light is coupled into the zero order and propagates straight into the structure in comparison to the light diffracted and transmitted through the higher diffraction orders. The opposite is observed for the diffused transmission in Fig. 4(b). Most of the energy is coupled in the odd diffraction orders. The diffused transmission is maximized for a phase difference of π and minimized for a phase difference of 2π. If the incident wavelengths is larger than the period of the unit cell (P = 700 nm) the specular transmission [Fig. 4(b)] for the 2π case (black line) is diffracted in the 2nd order and not in the 0th order. This behavior is of major importance for an efficient light trapping of longer wavelength in the solar cell.

Similar to the transmission of the front grating of the solar cell, the specular and diffused reflection were calculated for the back grating. The simulated reflection for a grating period of 700 nm is shown in Fig. 4 (c) and (d). The specular reflectance is shown in Fig. 4 (c). For a grating height corresponding to π, the specular reflectance has a minimal value. For such a structure, when the light reaches the back grating it is diffracted in the higher orders and almost no light is reflected straight into the zero mode. The specular reflectance is maximized, when the back grating creates a phase difference of 2π and its even multiples. Figure 4 (d) presents a contour map of the diffuse reflection of the back grating. The diffuse reflectance is maximized for a phase difference of π or odd multiples of π. As for the case of 2π and its even multiples, the diffuse reflectance is minimal, since a large portion of the light is reflected into the zero order. The observations in Fig. 4 (c) and (d) comply with the derivations and theoretical explanation in Eq. (3). For a wavelength of 700 nm phase difference of π is generated by a back grating of 46 nm or its odd multiples, while for a phase difference of 2π a back grating with a height of 92 nm or its even multiples is required. The RCWA analysis confirms that a small variation of the grating height has a more influential effect on the back grating than on the front grating.

The RCWA simulations provide a good understanding of the diffraction of the light in the microcrystalline silicon solar cell. The calculations performed with the Rigorous Coupled Wave Analysis approach comply with the derivations in Eqs. (2) and (3) and the schematic sketch in Fig. 3. However, the model does not take into account the optical near field diffraction in the solar cell. To consider the optical nearfield FDTD simulations of the front and the back grating were carried out. The power loss maps of the front grating with a grating height of 180 nm (π phase shift) and 360 nm (2π phase shift) are shown in Fig. 5(a) and (c) . The period of the grating was kept again constant at 700 nm. The wavelength of the incoming light was 700 nm. The power loss was simulated for a front and back grating which was prepared on top of an infinite silicon slab. The colored lines in Figs. 7(a) and 8(a) indicate the different diffraction orders. The power loss close to the groove is determined by the optical nearfield diffraction (Fresnel diffraction), whereas the power loss a couple hundred nanometers away from the interface is determined by the farfield diffraction (Fraunhofer diffraction). The maxima in the power loss are determined by the constructive interference of diffraction orders of two neighboring unit cells. The structure with a grating height of 180 nm (π phase shift) exhibits maxima in the absorption for interference of 1th order and 3rd order and 1st order of neighboring unit cells. For a structure with a grating height of 360 nm the maxima of the absorption are determined by the interference of the 2nd order diffractions. The power loss of the back grating for grating heights of 180 nm and 360 nm, respectively, are shown in Fig. 5(b) and (d). Again the different colored lines indicate the diffraction orders of the back grating. As observed for the front grating, the back grating power loss profile is determined by constructive and destructive interference of the propagating modes of the neighboring cells. The maximum of the power loss in both structures is obtained by constructive interference of the 1st and 2nd diffraction orders of the neighboring cells.

A systematic analysis of the back grating shows that the phase change introduced by the back grating cannot be approximated very well by Eq. (3). The coverage of the back grating by the ZnO layer has a small influence on the phase change. Simulations of the back grating reveal that the phase change of the back grating can be approximated by

_{si}is the real part of the complex refractive index of silicon, P is the period of the unit cell, n

_{ZnO}is the real part of the complex refractive index of ZnO, d

_{ZnO}is the thickness of the deposited layer, and h

_{g}being the grating height of the line grating.

If the back contact is exposed to light with a wavelength of 700 nm, a grating height of 180 nm corresponds to a phase difference of 3.7π (effectively 1.7π), whereas a grating height of 360 nm leads to a phase difference of 7.4π (effectively 1.4π). For a grating height of 180 nm light that reaches the back reflector under perpendicular incidence is mainly diffracted in even orders as shown in Fig. 3(a), whereas for a grating height of 360 nm light is diffracted in odd orders as shown in Fig. 3(b). The analysis of the power loss maps indicates that the back grating with grating height of 360 nm diffracts the incoming light more effectively.

#### 2.5 Front grating versus back grating

In the following the short circuit current was calcualted for solar cells with separate front and back grating, respectively. The short circuit current as function of the grating dimensions is shown in Fig. 6 (a) and (b)
. When studying solar cells with separate line gratings in the front and back the fabrication process of the cells has to be taken into account. A solar cell with a front grating but no back grating can only be realized by fabricating a solar cell in substrate configuration, as it is shown in the inset of Fig. 6 (a). A solar cell with a textured back contact can be realized by fabricating a solar cell in superstrate configuration. The back contact of the solar cell has to be textured, while the front contact remains not textured. In the optical simulations the thickness of the microcrystalline silicon solar cell was kept constant at 1 μm. The grating height was varied again from 100 nm to 500 nm. Compared to the solar cell on a smooth substrate the short circuit current is increased by 2.0-4.0 mA/cm^{2}, whereas for a solar cell with only a back grating the short circuit current increases by 2.0-2.5 mA/cm^{2}. The combination of a front and back grating leads to an increase of the short circuit current by 7.0 mA/cm^{2}. The current is increased mainly due to the increased absorption of light in the red and infrared part of the optical spectrum. Consequently, the interaction between the front and the back grating plays a significant role in increasing the short circuit current.

#### 2.6 Solar cells with front and back grating

The power loss maps for microcrystalline silicon thin film solar cells with integrated front and back grating are shown in Fig. 7 (a) and (b) for grating heights of 180 nm and 360 nm, respectively. The absorption profiles in the silicon absorber layer are determined by the superposition of the diffraction pattern of the front and the back grating. The diffraction orders from the front grating are overlaid with the reflected and diffracted beams from the back grating. Comparing the diffraction patterns of both structures in Fig. 7 (a) and (b) indicates that the grating structure with a grating height of 360 nm exhibits a higher power loss in the silicon absorber layer.

In the next step the quantum efficiency was calculated for the two solar cell configurations with different grating heights. The quantum efficiency for a grating height of 180 nm (π front change) is 0.40, while the structure with grating height of 360 nm (2π front change) exhibits a quantum efficiency of 0.50. Furthermore, the short circuit current was calculated for the two solar cell structures with grating heights of 180 nm and 360 nm. The short circuit current for the structure with grating height of 360 nm is equal to 18.5 mA/cm^{2}, while for the structure with a grating height of 180 nm it is 16.3 mA/cm^{2}. Similar to the quantum efficiency data, the higher short circuit current was determined for the structure with a front phase difference of 2π. An improvement of 2.2 mA/cm^{2} of the short circuit current is observed.

## 3. Discussion

In this study the random texturing of microcrystalline silicon thin film solar cells was approximated by triangular and line gratings [Fig. 1(c), 1(d)]. The analysis of different device structures with integrated grating structures reveals that the solar cell with a front and back grating is clearly superior to a structure with a front or a back grating alone. Furthermore, the optical simultions show that the wave propagation in a solar cell with triangular surface texture is not fundamentally different from a solar cell with a line grating. Therefore, a solar cell with a line grating can be used as a model system to study the influence of the front and back grating on the short circuit current for such solar cells.

Experimental results show that surface texturing of a 1 μm thick microcrystalline thin-film solar cells leads to an increase of the short circuit current from 16.3mA/cm^{2} to 23.4mA/cm^{2} [10,11]. Therefore, the experimentally measured short circuit currents are higher than the simulated short circuit currents. This is caused by the fact that we used 2D rather than 3D simulations to determine the short circuit current of the microcrystalline solar cells. In our simulations the light is scattered or diffracted in 2 rather than 3 dimensions. In reality the light is scattered and diffracted in 3 dimensions, so that the total short circuit current is higher. Simulations of 3D structures exhibit distinctly higher short circuit currents. We have carried out optical simulations in 3D for some of the structures in this study. Since the optical simulations in 3D is computationally complex we did not carry out 3D optical simulations for all possible combinations of grating height and grating period. In the 3D simulations we assumed a pyramidal texture with a square shaped base of the pyramid. In general, the 3D simulations confirm the trends observed for the 2D calculations. The short circuit current is maximized for pyramid periods of 500 nm - 900 nm and pyramid heights of 300 nm - 500 nm. A maximum short circuit current of 24.5 mA/cm^{2} was determined for a pyramid period of 600 nm and a pyramid height of 400 nm. At this stage it remains unclear if the simulated short circuit current of 24.5 mA/cm^{2} represents an upper limit of the short circuit current. The optical simulations depend on the shape of the base of the pyramidal texture. Changes of the base will have an influence on the short circuit current. Further investigations are needed to study the influence of the base of the pyramids on the short circuit current.

The optical simulations of solar cells with integrated line grating indicate that the short circuit current is maximized for a grating height of 2π. In this case the front grating transmits a large fraction of the incident light, which is reflected and effectively diffracted by the back grating at larger diffraction angles. The efficiency of the solar cell is maximized if the incident light is efficiently incoupled and transmitted straight into the structure by the front grating and then strongly diffracted by the back grating. The light is transmitted straight into the structure, when the front grating causes a phase difference of 2π. The back grating has to correspond to a phase difference of π in order to strongly diffract the incoming light into the higher diffraction orders. In our simulation the highest short circuit current was achieved for a grating height of 340 nm. Thus, the phase difference of the front grating corresponds to 1.9π (approximately 2π) and the back grating height phase shift is 7π (effectively π). The quantum efficiency of the solar cell with a grating height of 180 nm, 340 nm and 360nm for an incident light spectrum from 300 nm to 1100 nm is shown in Fig. 8. The three structures exhibit similar quantum efficiency for shorter wavelengths (< 550 nm). Short wavelengths are absorbed within the first few nanometers of the solar cell due to the strong wavelength dependent absorption coefficient of microcrystalline silicon. Only a fraction of the light reaches the back contact, where it is diffracted. Beyond wavelength of 550 nm the absorption profile of the solar cell is determined by interference and superposition of the diffraction modes of the front and the back grating. The structures with a grating height of 360 nm and 340 nm exhibit higher quantum efficiencies than the structure with a grating height of 180 nm for the wavelengths from 650 nm to 750 nm. For a grating height of 180 nm the corresponding phase differences of the front grating is equal to π and the back grating causes a phase shift of 1.7π. As a consequence a large fraction of the incident light reaches the back contact where it is diffracted in the 0th order resulting in an increased reflection of the solar cell. Subsequently the quantum efficiency is low for this wavelength region. A grating height of 360 nm results in a phase difference of 2π for the front grating and 1.4π for the back grating. The phase difference of the front grating matches the optimization rule, however the phase difference for the back grating is still relatively far away from the optimal phase change of π. A grating with a height of 340 nm causes a phase change of 1.9π in the front and the phase change of π in the back. As a result, the solar cell with 340 nm grating height exhibits the highest QE. The optimized structure exhibits an advantage of the QE in the region 625 nm – 750 nm, since the structure was adjusted for optimal performance for wavelength of 700 nm.

As a next step the short circuit current was calculated for the three structures. The short circuit current values are summarized in Table 1
. The short circuit current of the solar cell with a grating height of 340 nm and 360 nm is significantly higher (2.2-2.8 mA/cm^{2}) than the current of a structure with a grating height of 180 nm. The solar cell with a grating height of 360 nm exhibits a short circuit current of 18.5 mA/cm^{2}, which is 2.2 mA/cm^{2} higher than the structure with 180 nm grating height (16.3 mA/cm^{2}). The optimal grating structure with a grating height of 340 nm leads to another increase of the short circuit current by 0.6 mA/cm^{2}, resulting in a short circuit current of 19.1 mA/cm^{2}.

The derived design rules are not limited to microcrystalline silicon thin film solar cells. The concept can be easily transferred to other thin film solar cells with integrated line grating. We have tested the design rules for amorphous silicon thin film solar cells. In our study we assumed a solar cell thickness of 350 nm, which is common for amorphous silicon thin film solar cells. For such solar cells we found optimal grating periods of 300 nm and optimal grating heights of 200 nm. Due to the large optical bandgap of 1.75 eV longer wavelength cannot be effeciently absorbed by the solar cell. Therefore, the optimal short circuit current is observed for smaller grating periods. Smaller grating periods allow for the efficient scattering of short wavelengths, which can be efficiently absorbed by the solar cell. The highest short circuit current is again observed for a front grating phase difference of close to 2π and a back grating phase shift of close to π.

The current work shows that the phase difference can be controlled by the grating height. Design rules for optimizing the quantum efficiency and short circuit current were derived and tested. It was shown the highest quantum efficiency and short circuit current was achieved for a front grating phase difference of close to 2π and a back grating phase shift of close to π.

A topic for further investigation is the separate control of the front and the back grating height in an attempt to achieve the optimal phase change. For example the adjustment of the thickness of the zinc oxide layer between the microcrystalline diode and the back contact allows for tuning the phase change of the back contact. As an alternative different materials can be used to form a back contact in the solar cell structure.

## 4. Summary

The influence of the front and back grating on light trapping in microcrystalline silicon thin-film solar cells was studied using a Finite Difference Time Domain and Rigorous Coupled Wave Analysis approach. The influence of the grating period and height on the quantum efficiency and short circuit current was investigated. An optimal short circuit current is observed for grating periods in the range of 500 – 800 nm. The phase change of the front and the back grating has a significant effect on the quantum efficiency and the short circuit current of the microcrystalline thin-film silicon solar cell. Furthermore the interaction of the front and the back grating plays a major role in optimizing the short circuit current of the solar cells. The short circuit is maximized if the phase change of the front grating is close to 2π, whereas the phase change of the back grating is close to π. A grating height of 340 nm leads to a 1.9π front grating phase shift and a π back grating phase shift. This results in a maximal short circuit current of 19.1 mA/cm^{2}. The short circuit current can be further increased by separately controlling the phase change of the front and the back grating.

## Acknowledgements

The authors acknowledge the Institute of Energy Research (IEF-5) Photovoltaics, Research Center Jülich. Furthermore, the authors acknowledge financial support from Embedded Microsystems Bremen and Jacobs University Bremen's graduate program.

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