Wave-optics analysis is performed to investigate the benefits of utilizing Bragg-reflectors and inverted ZnO opals as intermediate reflectors in micromorph cells. The Bragg-reflector and the inverted ZnO opal intermediate reflector increase the current generated in a 100nm thick upper a-Si:H cell within a micromorph cell by as much as 20% and 13%, respectively. The current generated in the bottom μc-Si:H cell within the micromorph is also greater when the Bragg-reflector is used as the intermediate reflector. The Bragg-reflector outperforms the ZnO inverted opal because it has a larger stop-gap, is optically thin, and due to greater absorption losses that occur in the opaline intermediate reflectors.
© 2010 OSA
The micromorph , is a two-terminal, thin-film tandem solar cell comprising an upper (top) a-Si:H cell deposited onto an underlying (bottom) mc-Si:H cell. The respective band-gaps of a-Si:H and mc-Si:H are approximately 1.7 eV and 1eV, which is nearly optimal for a dual-junction tandem cell operating under AM 1.5 solar irradiance . Moreover, in an optimal two-terminal tandem cell the current generated by each component cell must be equal. However, the thickness of a-Si:H cells must be kept to a minimum in order to mitigate the effects of light induced degradation by means of the Staebler-Wronski effect , thereby ensuring a sufficiently large internal electric field . Consequently, micromorph cells are top-limited, meaning their output is limited by the relatively lower current produced in the top a-Si:H cell as compared to the higher current generated by the bottom mc-Si:H based cell. One solution to alleviate this limitation is to deposit a ZnO layer, typically less than 100nm thick, between the a-Si:H and mc-Si:H junctions . In this configuration the ZnO layer acts as an intermediate reflector (IR), ideally returning incident photons of energy greater than ~1.7 eV to the a-Si:H cell in order to boost the current generated in the upper cell while transmitting photons of lesser energy to the underlying mc-Si:H cell.
Recently, the benefits of fabricating IRs for micromorph cells from alternate materials or structures have been investigated. For example, 12.2% efficient micromorph cells featuring doped silicon oxide IRs (SOIRs) have been fabricated, the primary advantage being that the SOIR is deposited from the same forming gases as the thin-film silicon-based cells and can thus be produced in the same reactor . Additionally, research directed towards utilizing three-dimensional photonic crystals (PCs), in the form of inverted ZnO opals, as IRs has been initiated . This latter idea wherein PCs [8,9] are utilized as IRs in micromorph cells is particularly interesting because PCs are a unique class of optical materials that interact with electromagnetic waves causing them to diffract and form interference patterns within the PC, giving rise to many interesting optical phenomena . For example, PCs can be engineered to possess a photonic band-gap over a specified spectral range within which light cannot propagate; in this case incident light will undergo 100% reflection, suggesting the use of PCs as back-reflectors for thin-film PV. Herein we perform theoretical calculations in order to evaluate and compare the benefits of utilising Bragg-reflectors, commonly referred to as one-dimensional PCs, and opaline PCs as IRs in micromorph tandem cells.
2. Background – inverted ZnO opals
Opals, the most commonly known example of a PC found in nature, are iridescent gems comprising closely packed silica spheres. The bright colours of these gems are determined by the size of their silica spheres and the nature of their packing . Furthermore, the large-scale synthesis of opaline PCs possessing a one-dimensional photonic band-gap, also referred to as a stop-gap, can be achieved through sedimentation techniques wherein colloidal dispersions of monodisperse silica spheres are deposited into a close-packed face-centered-cubic (FCC) crystal structure . Although, these opaline PCs do not possess a full photonic band-gap they do exhibit a narrow stop-gap (∆ω/ω = 5.5%) in the  direction. Fortuitously, the  direction in opaline PCs is aligned with the normal of the surface on which the PC is grown, and thus the opaline PC makes an excellent reflector for normally incident light over its stop-gap frequencies. It can be noted that as the angle of incidence moves away from the normal of the opal surface the stop-gap shifts to higher energies . Moreover, it has recently been demonstrated that two attributes of opaline PC back-reflectors optically coupled to thin semiconductor films contribute to enhanced photoconductivity in the semiconductor [14,15]. Namely, (i) the opaline PC back-reflector behaves as a perfect mirror, exhibiting complete reflection over stop-gap frequencies; and (ii) the PC-semiconductor film interface couples incident light into resonant states that propagate along the plane of the film, thereby further enhancing absorption.
It is expected that ZnO inverse opals can function as intermediate reflectors in micromorph tandem cells because, unlike other opaline PCs, they are fabricated from a material that is both conductive and transparent. Inverted ZnO opals can be fabricated by infiltrating opals comprised of polystyrene spheres, rather than silica spheres, via atomic layer deposition and subsequently removing the polystyrene template by firing at elevated temperatures [16,17]. Similarly to opaline PCs, ZnO inverse opals also possess a  directional stop-gap aligned normal to the surface of the substrate on which the PC is deposited. This  directional stop-gap, occurring in the Γ-L direction, has a width of ∆ω/ω ~14.3%. Furthermore, the spectral position of this stop-gap can be tuned by setting the diameter of the polystyrene spheres in the initial opal template. In light of the results shown in references fourteen and fifteen, in functioning as an IR in the micromorph cell the inverted ZnO opal is expected to enhance absorption in the upper a-Si:H cell by reflecting photons of energies within its stop-gap back into the top cell and also by diffracting incident photons such that they propagate in the lateral direction thereby extending their path-length through the cell.
3. Simulations and results
In this Section we perform calculations to compare the performance of Bragg-reflectors and inverted ZnO opal IRs to that of the more common IR consisting of just a homogeneous ZnO film. The calculations determine the current generated in the upper a-Si:H cell and also evaluate the ability of these IRs to transmit light by providing an upper limit on the amount of current that can be generated in the bottom µc-Si:H cell. The methods and assumptions used to perform the calculations are described in Section 3.1 while the performance of micromorph tandem cells featuring a single ZnO film, a Bragg-reflector, or an opaline PC functioning as an IR are compared to the case in which the IR is absent in Sections 3.2, 3.3 and 3.4, respectively. Also, the performance of the aforementioned IRs in micromorph cells subjected to direct solar irradiance at off-normal angles is reported in Section 3.5. Finally, the performance of the IRs as a function of the thickness of the upper a-Si:H cell is also presented in Section 3.6.
3.1 Model assumptions
Each of the aforementioned micromorph cell configurations is first represented as a two- dimensional structure. The scattering matrix method [18,19] is then used to calculate the amount of light absorbed in both the top a-Si:H and bottom μc-Si:H cells upon exposure to the ASTM AM1.5 (Global tilt) solar spectrum . In the scattering matrix method the structure is divided into a number of vertically uniform layers that can be periodic in the lateral direction. The electromagnetic field in each of these layers is represented by an infinite set of plane waves. The method rigorously solves the Maxwell equations by imposing matching conditions for the tangential field components at each boundary. The amount of light absorbed in any layer(s) within the structure is determined from the difference between the Poynting vectors calculated at the upper and lower boundaries of the layer(s) of interest. Also, the optical absorption is calculated in the wavelength range 300-1100nm and the number of calculated sampling (wavelength) points is 500.
A schematic of the cell modeled in this work, without an IR, is shown in Fig. 1(a) . The top a-Si:H cell is coated with a 60nm thick transparent conducting oxide, specifically indium tin oxide (ITO), whose thickness was chosen to minimize reflection losses. The ITO is assumed to be completely non-absorbing and its index of refraction is set to 1.9 for all wavelengths of incident light. Moreover, in order to investigate the potential of the aforementioned IRs to enhance the current generated in a top-limited micromorph cell it is assumed that the thickness of the top a-Si:H cell is 100nm. As previously mentioned, it is desirable to keep the a-Si:H cell as thin as possible to avoid degradation of its electrical performance and also to reduce the duration of its deposition. However, since the absorption in the a-Si:H cell is highly reliant on its thickness this dependence is investigated in Section 3.6 where the a-Si:H thickness is varied from 100nm to 500nm. The underlying µc-Si:H cell is assumed to be infinitely thick for all cases and thus the rear metal contact shown in Fig. 1(a) is not included in the calculations. One may be concerned that if the µc-Si:H cell had a finite thickness, then reflections from the metal/µc-Si:H interface would increase the absorption in the upper a-Si:H cell. To address this concern we have calculated the current generated in the a-Si:H cell within the micromorph cell shown in Fig. 1(a) for the case in which the μc-Si cell is 2 µm thick and has a perfect mirror on its rear side. The calculation reveals that the current generated in the a-Si:H cell for the case in which the μc-Si cell is 2 µm thick is increased by just 1% as compared to the case for which the μc-Si cell is infinitely thick. Furthermore, in choosing its thickness to be infinite the current generated in the underlying µc-Si:H cell becomes an upper limit since each photon that is transmitted into this cell is accounted for. Here we point out that, in designing an optimal micromorph cell, once the current generated in the upper a-Si:H cell is maximized the absorption in a bottom μc-Si cell of finite thickness can be enhanced by either making it thicker or incorporating enhanced light trapping schemes, such as a diffraction grating, into its rear side. Also, for simplicity, the optical constants of the p- and n- regions are assumed to be identical to those of the i- region for both the upper a-Si:H and underlying μc-Si:H cells, respectively. As a final note, it is common practice to purposely introduce rough interfaces into thin-film solar cells in order to scatter incident light into the planar direction thereby increasing the path length of the light through the absorbing medium. However, our calculations have been simplified by assuming the interfaces within the micromorph cells are planar.
Presenting the micromorph cells investigated in this work in two dimensions is straight forward with the exception of the case in which the IR is an inverted ZnO opal. In order to model the inverted ZnO opal IR in two dimensions we have projected the filling fraction of this structure onto its (110) crystallographic plane [Fig. 1(b)]. To maintain consistency with the opal fabrication process the  crystallographic direction on this map has been aligned in the vertical direction. A discretization process is then carried out in order to transfer the map of the projected filling fraction onto a grid that has 34 points in the lateral direction and 16 points per sphere layer in the vertical direction. The wavelength dependant index of refraction of ZnO is then multiplied by each point on this grid to represent the opal in two dimensions. Also, to further simplify the calculations performed in this work, the incident light is assumed to be E-polarized where, as shown in Fig. 1(b), E is perpendicular to the (110) crystallographic plane. For normally incident light the results are polarization independent since the E- and H- polarized wave modes are degenerate along the  direction in the inverted opal. However, it should be noted that the results presented in Fig. 5 , where light is incident from off-normal angles, may differ for H- polarized light.
The optical constants of ZnO and a-Si:H are taken from the literature [21–23] while the optical constants for the μc-Si:H is determined using the effective medium approximation assuming a two-phase mixture of a-Si:H and crystalline silicon (c-Si) both having a volume fraction of 0.5 [24,25]. The indices of refraction and extinction coefficients for ZnO, a-Si:H and μc-Si:H used to perform the calculations herein are plotted in Section 7. It should also be noted that the distribution of current generated in the a-Si:H and µc-Si:H cells within the micromorph depends on the band-gap energy of the a-Si:H cell, however, this parameter is not varied in our calculations. Based on the definition that the band-gap of a-Si:H is the energy at which its absorption coefficient is equal to 104·cm−1, the band-gap energy of the a-Si:H used for the calculations in this work is ~1.75 eV which, as cited in the introduction, is nearly optimal for a dual-junction tandem cell operating under AM 1.5 solar irradiance.
Once the scattering matrix method has been used to calculate the absorption in the a-Si:H and µc-Si:H cells the current generated in these cells are then determined using the assumption that each absorbed photon generates and contributes one electron to the output current. Under the aforementioned assumptions the short circuit current generated in the upper a-Si:H and underlying μc-Si:H cells can be calculated using Eqs. (1) and (2), respectively:Eqs. (1) and (2) is has been set to 4.1 eV because the number of photons of energy greater than 4.1 eV, which corresponds to λ ≈300nm, in the AM 1.5 spectrum is negligible.
We have also used the scattering matrix method to calculate the amount of light lost due to reflection from the front surface of the micromorph cells as well as the amount of light lost in the IRs. Light absorbed in the IRs is considered an absorption loss and is referred to as parasitic absorption since this light does not contribute to the current generated in either the a-Si:H or the µc-Si:H cells. These two loss mechanisms can be equated to a current loss through Eqs. (3) and (4).
3.2 A single ZnO film functioning as the IR
The current generated in the upper a-Si:H and lower µc-Si:H cells of the micromorph without an IR, as shown in Fig. 1(a), is 11.5 mA/cm2 and 26.5mA/cm2, respectively. The addition of a ZnO film functioning as an IR and having an optimal thickness of 59nm increases the current generated in the upper a-Si:H cell to 12.5 mA/cm2. The current generated in the underlying µc-Si:H cell at this thickness is 21.4 mA/cm2. The absorption in both the upper a-Si:H and lower μc-Si:H cells in these micromorph tandem cells both with and without a 59nm thick ZnO IR are plotted in Fig. 2 . The parasitic absorption in the ZnO IR is small and accounts for a loss of the total current generated in the upper a-Si:H and underlying µc-Si:H cells of just 0.2 mA/cm2.
3.3 A Bragg-reflector functioning as the IR
A schematic diagram of a Bragg-reflector comprising alternating layers of μc-Si:H and ZnO functioning as an IR in a micromorph tandem cell is shown in Fig. 3(a) . The benefit of fabricating the Bragg-reflector from μc-Si:H and ZnO is the large contrast in their indices of refraction which will provide a broad and intense stop-gap. However, one drawback of using μc-Si:H is that it will increase the amount of parasitic absorption in the IR. It is assume that the μc-Si:H films within the IR can be doped and that its conductivity can be made sufficiently large such that ohmic losses occurring in the IR are negligible; this situation is comparable to that of reference six wherein a doped silicon oxide film serves as the IR.
The results indicate that the current generated in the upper a-Si:H cell of the micromorph featuring the ZnO/μc-Si:H Bragg-reflector functioning as an IR can be increased to 13.8 mA/cm2 and the corresponding current generated in the bottom μc-Si:H cell is 14.9 mA/cm2. This optimal case occurs when the IR is three and a half bi-layers thick with ZnO films serving as both the top and bottom layers. The thicknesses of the ZnO and μc-Si:H films are 65nm and 38nm, respectively. The absorption in the upper a-Si:H and lower μc-Si:H cells are plotted in Fig. 3(b). The absorption enhancements in the upper a-Si:H are the greatest over the stop-gap frequencies of this Bragg-reflector, where its reflection is at a maximum. The integrated parasitic absorption losses, also plotted in Fig. 3(b), equate to JIR = 2.2 mA/cm2. As the number of bi-layers in the Bragg-reflector is increased beyond three and a half the parasitic absorption in the IR increases and the current generated in the upper a-Si:H drops.
As a final point, given the additional work involved in fabricating a three and a half layer Bragg-reflector, it is noteworthy that utilising a ZnO/μc-Si:H IR comprising just one and a half layers, wherein a μc-Si:H film is sandwiched between two ZnO films, increases the current generated in the top cell of the micromorph to 13.5 mA/cm2; the corresponding current in the lower cell is 17.3 mA/cm2.
3.4 An inverted ZnO opal PC functioning as the IR
A schematic diagram of an inverted ZnO opal functioning as an IR in a micromorph tandem cell is shown in Fig. 4(a) . The maximum amount of current generated in the upper a-Si:H cell is 13.0 mA/cm2 and this occurs when the inverted ZnO opal IR is 13 layers thick and the diameter of its air-holes are 264nm. The corresponding current generated in the underlying μc-Si:H cell is 14.6 mA/cm2. The absorption in the upper a-Si:H and lower μc-Si:H cells for this optimal case is plotted in Fig. 4(b). The reflection from a 13-layered ZnO inverted opal, similar to that used as the IR, is also plotted in Fig. 4(b). The absorption enhancements in the upper a-Si:H are the greatest over the stop-gap frequencies of the inverted ZnO opal, where its reflection is at a maximum. The absorption occurring in the inverted ZnO opal IR is also plotted in Fig. 4(b). Despite the fact that μc-Si:H is a much stronger absorber than ZnO, the parasitic absorption is greater for the case in which the ZnO inverted opal functions as the IR compared to the case in which the ZnO/μc-Si:H Bragg-reflector functions as the IR. In fact, the integrated parasitic absorption losses occurring in the ZnO opal IR equate to JIR = 4.3 mA/cm2. The current generated in the top a-Si:H and bottom µc-Si:H cells within the various micromorph cell constructs investigated herein are summarized in Table 1 for convenience. The reflection and parasitic losses occurring in these cells are also listed in Table 1.
3.5 Performance of the IRs as a function of the incident angle of the solar irradiance
We have investigated the performance of IRs in micromorph cells subjected to direct solar irradiance at off-normal angles. Specifically, we have calculated the current generated in the upper a-Si:H and underlying μc-Si:H cells for the optimal cases of the single ZnO film, the ZnO/μc-Si:H Bragg-reflector made of either 1.5 or 3.5 bilayers, and the inverted ZnO opal functioning as the IR. Here we state once again that the optimal ZnO film IR is 59nm thick, and the optimal inverted ZnO opal IR is 13 layers thick. The current generated in these micromorph cells are plotted as a function of the incident angle of the solar irradiance in Fig. 5. At normal incidence the micromorph cells are top-limited for each of the IRs considered. However, for micromorph cells featuring either a ZnO/μc-Si:H Bragg-reflector or an inverted ZnO opal IR, the amount of current generated in the bottom μc-Si:H cell decreases below that generated in the upper a-Si:H cell at large incident angles. More specifically, micromorph cells with the ZnO/μc-Si:H Bragg-reflector that has either 1.5 or 3.5 bilayers become bottom-limited for incident angles exceeding θ ~75° and θ ~50°, respectively. When an inverted ZnO opal functions as the IR the micromorph cell becomes bottom-limited for incident angles exceeding θ ~45°. The micromorph with the inverted ZnO opal IR becomes bottom-limited for incident angles in excess of just 45° for two main reasons. Firstly, as previously mentioned, parasitic absorption occuring in the inverted ZnO opal prevents a significant portion of the solar irradiance from entering the bottom μc-Si:H cell. Secondly, the thickness of the inverted ZnO opal is much greater than that of the other IRs. For example, the optimal 13-layered inverted ZnO opal is 2.8μm thick while the thickness of the optimal IRs constructed in the form of a ZnO/μc-Si:H Bragg-reflector with 3.5 layers or a ZnO film are just 374nm and 59nm, respectively. For less energetic photons with longer wavelengths the optical behaviour of the inverted ZnO opal can be compared to a homogeneous film with an effective refractive index. As can be seen from the dashed black line in Fig. 4(b), Fabry-Perot resonances occurring in the optically thick inverted ZnO opal gives rise to many troughs in the absorption spectra of the bottom μc-Si:H cell. Furthermore, the intensity and spectral position of these troughs strongly depend on the incident angle of the solar irradiance. The angles at which micromorph cells with a 100nm thick upper a-Si:H cell becomes bottom-limited for the different cases of the IRs investigated herein are summarized in Table 2 .
3.6 Performance of the IRs as a function of the upper a-Si:H cell thickness
Here we investigate the performance of IRs in micromorph cells as a function of the thickness of their upper a-Si:H cell. Specifically, we have calculated the current generated in the top a-Si:H and bottom μc-Si:H cells in micromorph cells for the optimal cases of the different IRs considered herein as the a-Si:H cell thickness is increased from 100nm to 400nm. These results are plotted in Fig. 6 where it can be seen that all of the micromorph cells are top-limited when their upper a-Si:H cell is 100nm thick. As this thickness is increased from 100nm the current generated in the top a-Si:H increases, however, the current generated in the underlying μc-Si:H cell decreases since less light is transmitted through the top cell. At a certain critical thickness of the a-Si:H cell the current generated it the a-Si:H and μc-Si:H cells are equal. As the thickness of the a-Si:H cell is increased beyond this critical point the current generated in the a-Si:H cell surpasses that generated in the μc-Si:H cell and thus the micromorph cell becomes bottom-limited and its efficiency decreases. The critical thicknesses at which the micromorph cells with the various IRs investigated herein become bottom-limited are enclosed in circles in Fig. 6. The maximum amount of current generated in any of the micromorph cell configurations considered in this work is 19.0 mA/cm2 and this occurs for the case in which there is no IR and when the a-Si:H cell is 500nm thick (not shown). The critical thicknesses of the upper a-Si:H cells and the corresponding currents generated in micromorph cells with the optimal IR configurations considered in this work are summarized in Table 3 .
It must be noted that the generated currents reported in Table 3 are calculated assuming lossless carrier collection. In actuality the carrier collection efficiency in a p-i-n a-Si:H cell depends on a number of parameters including the mobility-lifetime product, diffusion and drift lengths as well as the thickness of the a-Si:H. Thus, it is difficult to predict the a-Si:H cell efficiency as a function of its thickness, however, in the literature it is generally suggested that this thickness should be kept below 300nm and that the Staebler-Wronski effect is detrimental to a-Si:H cells of thickness greater than 400nm [26,27]. The information presented in Table 3 also suggests that the intermediate Bragg-reflector with just 1.5 bilyers is a very attractive candidate for the IR because the thickness of the a-Si:H cell for this case is well within the acceptable range for optimal mitigation of the Staebler-Wronski effect reported in the literature and also because its fabrication process is the least challenging of the PC IRs investigated in this work.
With the AM1.5 solar spectrum at normal incidence the current generated in the upper a-Si:H cell of the micromorph cell shown in Fig. 1 increases from 11.5 mA/cm2 to 12.5 mA/cm2 when a single ZnO film is used as the intermediate reflector (IR) and to 13.0 mA/cm2 when a 13-layered ZnO inverted opal is used as the IR. However, the current generated can be increased to as much as 13.8 mA/cm2 if a ZnO/μc-Si:H Bragg-reflector comprised of 3.5 bilayers is used as the IR. The Bragg-reflector outperforms the ZnO inverted opal given its larger stop-gap. For example, using the optical constants of μc-Si:H and ZnO at λ = 600nm, the width of the stop-gap of the ZnO inverted opal is ∆ω/ω ~14.3% while that of the Bragg-reflector is ∆ω/ω ~45.2%.
Inverted ZnO opals functioning as IRs in a micromorph tandem cells are expected to enhance absorption in the upper a-Si:H cell by reflecting photons of energies within its stop-gap back into the top cell and also by diffracting incident photons such that they propagate in the lateral direction along the plane of the upper a-Si:H cell. However, diffraction and refraction into the planar direction also enhances parasitic absorption in the IR. Thus, absorption losses in opaline IRs are considerable and must be taken into account when modeling the performance of micromorph cells. Furthermore, this large amount of parasitic absorption hinders the design of better performing opaline IRs. For example, research directed towards improving the charge-transport properties in inverted silicon opals has recently been performed , and in this investigation we also considered using an inverted opal fabricated from μc-Si:H as the IR rather than a ZnO inverted opal. Using a μc-Si:H inverted opal would increases the width of the spectral stop-gap of the IR from ∆ω/ω ~ 14.3% to ∆ω/ω ~24.5%. However, the current generated in the upper a-Si:H cell for the case of the inverted μc-Si:H opal IR is lower than that of the inverted ZnO opal on account of the large amounts of parasitic absorption occurring in the IR.
As a final comment, our results show that the Bragg-reflector outperforms the opaline PC in functioning as an IR in micromorph tandem cells subjected to the AM1.5 solar irradiance. However, one may argue that these results depend on the choice of materials used in fabricating these different IR constructs. Indeed, it has been shown that inverted opaline PC IRs can be made to outperform Bragg-reflector IRs assuming a different choice of materials . Nevertheless, the work performed herein does show that, in functioning as an IR in the micromorph cell, the advantages of the Bragg-reflector are that it can be made to provide a larger stop-gap and less parasitic absorption than the inverted opaline PC.
The indices of refraction and extinction coefficients for the ZnO, a-Si:H and μc-Si:H used to perform the calculations herein are plotted in Figs. 7(a) and 7(b), respectively. Extrapolation was used to extend these values to the lower limit of λ = 300nm and the upper limit of λ = 1100nm. The extinction coefficients of the ZnO, a-Si:H and μc-Si:H in the spectral region within the stop-gap of the optimal intermediate Bragg-reflector and inverted ZnO, namely at λ =600nm, are ~8x10−4, ~2x10−2, and 1x10−2, respectively.
This work was supported through grants from the Natural Sciences and Engineering Research Council of Canada, Ontario Research Fund – Research Excellence program, and Arise Technologies Corporation.
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