We study dielectric diffraction gratings for light-trapping in quantum well solar cells and compare their performance with plasmonic and Lambertian light-trapping structures. The optimum structural parameters are identified for symmetric uni-periodic, symmetric bi-periodic and asymmetric bi-periodic gratings. The enhancement in short-circuit current density from the quantum well region with respect to a reference cell with no diffraction grating is calculated. The ratio of this enhancement to the maximum achievable enhancement (i.e. no transmission losses) is 33%, 75% and 74%, respectively for these structures. The optimum asymmetric and symmetric bi-periodic structures perform closest to Lambertian light-trapping, while all three optimum grating structures outperform optimum plasmonic light-trapping. We show that the short-circuit current density from the quantum well region is further enhanced by incorporating a rear reflector.
© 2013 OSA
Incorporating quantum wells is a novel approach to enhance the overall efficiency of a standard p-i-n junction solar cell and extends the spectral response of the solar cell beyond the bandedge of the bulk active material [1–3]. Quantum well solar cells (QWSCs) have been successfully fabricated using the GaAs-AlGaAs and InGaAs-GaAs material systems [1, 2, 4–6]. GaAs-AlGaAs QWSCs have demonstrated an extended spectral response compared with an AlGaAs reference cell . Single junction GaAs cells achieve high efficiencies, but are limited by its bandgap of 1.42 eV which is smaller than the optimum of 1.34 eV (AM 1.5G). InGaAs quantum wells (QWs) integrated with a GaAs solar cell offer the possibility of extending its photo-response below 1.42 eV. However, the absorption fraction of a single QW is on the order of 1%. To increase the absorption of longer wavelength light, many QWs are required. Unlike GaAs-AlGaAs QWs, the epitaxial growth of successive InGaAs-GaAs QWs results in an accumulation of strain due to lattice mismatch that leads to the formation of defects and a reduction in the material quality of the QWSC . Introducing strain-balancing layers into an InGaAs-GaAs quantum well stack may allow the number of quantum wells to be increased without significant strain accumulation, but the epitaxial growth of these structures is substantially more challenging. Moreover, stacking many InGaAs-GaAs QWs reduces the carrier extraction efficiency from the intrinsic region and can lead to a reduction of the open circuit voltage. A limited number of strained InGaAs-GaAs QWs can be grown without a severe reduction in material quality and so, given the low absorption fraction of a single InGaAs-GaAs QW, the overall absorption fraction of an InGaAs-GaAs QW stack is restricted. Consequently, wavelengths that lie beyond the bandedge of GaAs are not completely absorbed in a single pass through the QW stack. Therefore, increasing the absorption in the QWs without having to significantly increase the number of QWs becomes necessary to enhance the overall efficiency of an InGaAs-GaAs QWSC.
Light-trapping in a solar cell can increase its photocurrent by enhancing the path-length of weakly absorbed light. To increase path-length, light is coupled into modes that lie outside the loss cone of the solar cell – the loss cone is defined by the critical angle for total internal reflection at the cell-air interface. These modes then undergo total internal reflection at the cell-air interface thereby increasing their path-length and hence the absorption and photocurrent of the solar cell . Light-trapping structures at the rear of a solar cell increase the path-length of light that is not completely absorbed in the first pass. Light-trapping structures can also reduce material costs by maintaining the external quantum efficiency (EQE) of a solar cell with an active material thinner than its absorption length. There are a variety of structures that facilitate light-trapping, achieving their respective enhancements in path-length through different physical mechanisms. Dielectric diffraction gratings and metallic nanoparticles (plasmonics) are two examples that have recently received significant attention [8, 9]. The focus of this paper is on dielectric diffraction gratings for light-trapping in InGaAs-GaAs QWSCs.
A common benchmark that is often employed to gauge the performance of any light-trapping structure is that of the isotropic limit. Defined as 4n2, where n is the real refractive index of the solar cell’s active material, this limit sets the maximum achievable path-length enhancement for a wafer-based solar cell . The isotropic limit is commonly referred to asthe Lambertian light-trapping limit wherein incident light is isotropically scattered irrespective of the angle of incidence. In this work, the performance of dielectric diffraction gratings is compared with that of both Lambertian and plasmonic light-trapping.
A diffraction grating can be either a uni-periodic or bi-periodic structure whose periodicities are larger than, comparable to, or smaller than the particular wavelengths of interest. Dielectric diffraction gratings that lie at the interface of two materials with different refractive indices couple light into both materials as a series of higher order modes centered about a zero order principal mode. Equation (1) is the two-dimensional grating equation and applies to uni-periodic diffraction gratings, describing the angle at which the orders in each material are diffracted.
As illustrated by Eq. (1), deep sub-wavelength scale diffraction gratings only couple incident light into the principal diffraction orders and therefore do not facilitate light-trapping as the zero orders are not subject to total internal reflection at the cell-air interface. For periodicities significantly larger than the wavelengths of interest, the incident light is coupled to a continuum of diffraction orders, a large fraction of which lie within the loss cone. A diffraction grating on this scale is therefore not appropriate for light-trapping. Effective light-trapping requires wavelength-scale diffraction gratings that support only the first and second diffraction orders outside the loss cone. This has been demonstrated through statistical temporal coupled-mode theory by Fan. et al [9, 11]. Figure 1 depicts the cell-air interface and illustrates the zero, first and second diffraction orders of a wavelength-scale diffraction grating.
Maximum path-length enhancement requires not only that light is preferentially coupled into the first and second diffraction orders, but that these orders lie outside the loss cone and that the reflection at the cell-air interface is simultaneously maximized. For this reason, the design of wavelength-scale dielectric diffraction gratings is the focus of this study. However, irrespective of grating design, the transmission of light into the zero order cannot be completely eliminated. Therefore, this research will further consider the additional benefit of a reflector at the rear of InGaAs-GaAs QWSCs that have been coupled with an optimum grating.
2.1 Finite-difference time-domain modeling methodology
Finite-difference time-domain (FDTD) simulations are used to study the effect of dielectric diffraction gratings. An In0.21Ga0.79As-GaAs QWSC is modeled, comprising ten 7 nm In0.21Ga0.79As QWs inserted into the intrinsic region with 50 nm GaAs barriers between 300 nm layers of GaAs. The wavelength-scale dielectric diffraction grating designs are all composed of TiO2 because of its large real refractive index and low losses over the wavelengths of interest. Simulations are performed using Lumerical’s FDTD Solutions package  and short-circuit current densities (Jsc) are calculated under AM 1.5G conditions . These simulations incorporate periodic boundary conditions in the plane of the QWSC and perfectly matched layers (PML) normal to the surface of the solar cell. This allows a single period of the TiO2 diffraction gratings to be considered and prevents transmitted or reflected light from re-entering the simulation volume. Figure 2(a) illustrates the structure of the In0.21Ga0.79As-GaAs QWSC, the boundaries of the simulation volume, the plane-wave source and the positions of the power monitors used to measure the absorption in the QWSC. The Jsc that are evaluated throughout this paper only consider absorption by the In0.21Ga0.79As-GaAs QW stack over the QW region (875 – 1010 nm) and represent theoretical maxima achieved with TiO2 rear-side diffraction gratings.
Theory suggests that bi-periodic gratings are more efficient light-trapping structures than uni-periodic gratings and that the introduction of grating asymmetry further improves efficiency. In order to maximize the enhancement in Jsc by a dielectric diffraction grating the structural parameters must be optimized, including periodicity (L), height (h) and fill-factor (ff). The fill-factor is defined as the fraction of the single period’s volume occupied by the grating. Figures 2(b)-2(d) illustrate a single period of the three wavelength-scale TiO2 diffraction gratings that are studied for the In0.21Ga0.79As-GaAs QWSC. Figures 2(b) and 2(c) depict symmetric uni-periodic and bi-periodic gratings, respectively. Both of these gratings are optimized over their height, fill-factor and periodicity. Figure 2(d) depicts an asymmetric, bi-periodic grating that is optimized over its height and periodicity with a fill-factor equal to 0.33. The slope of the skewed pyramid grating is dependent on the period and height given the fill-factor is fixed at 0.33. For the purpose of this research, the gratings illustrated in Figs. 2(b)-2(d) are referred to as rectangular strip grating, square pillar grating and skewed pyramid grating, respectively. Both the square pillar and the skewed pyramid gratings are assumed to have equal periodicities in the x and y directions, i.e. a square single period.
A reference In0.21Ga0.79As-GaAs QWSC without a TiO2 diffraction grating at its rear is modeled to serve as a comparison to quantify the enhancement in photocurrent provided by the optimum gratings. Plasmonic light-trapping is also investigated for comparison with the TiO2 diffraction gratings. The optimum dimensions for a periodic array of silver nanoparticles have been identified by Mokkapati. et al . The nanoparticles are modeled with a square base of width 200 nm, pitch 400 nm and height 50 nm. Absorption losses by the silver nanoparticles are taken into account when evaluating the QWSC absorption. Lastly, for comparison, an isotropically scattering rear reflector (Lambertian light-trapping) is numerically modeled for the wavelengths beyond the bandedge of GaAs using the theoretical framework outlined by Goetzberger .
2.2 Modeling the optical constants of an In0.21Ga0.79As-GaAs quantum well
Bulk optical data are used for the 300 nm layers of GaAs  and the TiO2 diffraction gratings are assumed to be loss-less with a real refractive index of 2.3 . The FDTD simulations model the In0.21Ga0.79As-GaAs QW stack as a single block of material with homogeneous and isotropic optical properties. The real refractive index of this block is assumed to be equal to that of bulk GaAs. That is, the real refractive indices of In0.21Ga0.79As and GaAs are assumed to be equal, implying no reflections are present at any of the In0.21Ga0.79As-GaAs interfaces. This is a valid assumption for wavelengths beyond the bandedge of GaAs . For wavelengths below the bandedge of GaAs the differences in the real refractive indices are larger for In0.21Ga0.79As and GaAs . However, these wavelengths are strongly absorbed by the top 300 nm of GaAs, so reflections at any of the In0.21Ga0.79As-GaAs interfaces are assumed not to significantly contribute to the overall dynamics of the FDTD simulations.
The band structure and absorption spectrum of an In0.21Ga0.79As-GaAs QW are calculated by an eight band k•p model based on , with material parameters provided by . The QW valence and conduction band density of states is calculated and taken into consideration by the absorption computational routines.
The fraction of light absorbed due to a single In0.21Ga0.79As-GaAs QW is calculated for light incident in the normal direction to the QWSC surface (x-y polarized light). The absorption fraction for a single QW is linearly extrapolated to obtain the fraction of light absorbed by ten QWs. The overall absorption fraction is then converted to an absorption coefficient, by dividing by the total thickness of the QW stack, and so assuming homogeneous and isotropic absorption. In order to simplify the FDTD simulations, only the absorption fraction for x and y polarized light is considered. This aligns with the consideration of the In0.21Ga0.79As-GaAs QW stack as a block of material with homogeneous and isotropic optical properties. The absorption coefficient is subsequently converted to an extinction coefficient using Eq. (2) . This extinction coefficient and the real refractive index of bulk GaAs are used as the complex refractive index of the QW stack used in Lumerical’s FDTD Solutions package.
3. Results and discussion
The focus of section 3.1 is to identify the optimum structural parameters for the rectangular strip, square pillar and skewed pyramid gratings; subsequently drawing performance comparisons and discussing the role of grating asymmetry. Section 3.2 then compares the performance of these optimum gratings with the Jsc achieved by plasmonic and Lambertian light-trapping. Finally, section 3.3 investigates the effect of a rear reflector behind the optimum square pillar grating to determine whether the grating-reflector separation is critical, as postulated by Wang. et al .
3.1 Design of TiO2 diffraction gratings for In0.21Ga0.79As-GaAs quantum well solar cells
The Jsc of a QWSC with both the rectangular strip and square pillar gratings is simulated as a function of fill-factor and height for periodicities over the range of 400 – 1000 nm, while the skewed pyramid grating is simulated as a function of period and height with a constant fill-factor of 0.33. At each periodicity there is an optimum fill-factor and height that maximizes the Jsc. Figure 3 illustrates these maxima as a function of period. FDTD simulation results for the gratings are averaged over both polarizations of the plane-wave source. Figure 3 illustrates that all three grating structures have maxima in Jsc at a period of 850 nm. The first and second orders exist in the QWSC over the QW region for this periodicity. At this optimum period the skewed pyramid grating (Jsc = 3.1 mAcm−2) performs comparably to the square pillar grating (Jsc = 3.2 mAcm−2), while both structures significantly outperform the rectangular strip grating (Jsc = 2.0 mAcm−2). That is, both bi-periodic gratings achieve significantly larger enhancement than the uni-periodic grating, however the asymmetric bi-periodic grating provides only comparable enhancement to the symmetric bi-periodic grating.
Two theoretical guidelines have been proposed to govern the upper limits of absorption enhancement for wavelength-scale dielectric diffraction gratings . The upper limit in absorption enhancement that can be achieved by a bi-periodic grating is greater than that of a uni-periodic grating, while the upper limit in absorption enhancement that can be achieved by an asymmetric grating is greater than that of its symmetric counterpart. These guidelines are determined through statistical temporal coupled-mode theory employed by Fan. et al [9, 11] and consider the contribution of a single resonance to the overall absorption of a given medium coupled with a diffraction grating. The upper limits in absorption enhancement are evaluated by summing the contribution of each individual resonance over a broad spectral range. The theoretical analysis assumes the solar cell’s active material to be in the weakly absorbing limit and much thicker than the wavelengths considered. Moreover, it assumes that each individual resonance is in the over-coupling regime. The results previously discussed and illustrated in Fig. 3 are consistent with the theoretical framework of Fan. et al  insofar as the bi-periodic gratings outperform the uni-periodic grating, however the asymmetric grating does not outperform the symmetric grating.
Figures 4(a) and 4(b) illustrate the contour plots of the Jsc for the QW region (875 – 1010 nm) for a period of 850 nm for the rectangular strip and square pillar gratings, respectively. Figure 4(c) illustrates the contour plot of the Jsc for the QW region (875 – 1010 nm) for the skewed pyramid grating as a function of period and height with a fixed fill-factor of 0.33. Figure 4(a) indicates that the optimum rectangular strip grating is a low aspect ratio structure with a specific optimum fill-factor and height. Figure 4(b) illustrates that although the optimum square pillar grating is also a low aspect ratio structure, there is a broader range of optimal fill-factors. The optimum skewed pyramid grating illustrated by Fig. 4(c) is a high aspect ratio structure that provides comparable enhancement to that of the square pillar grating at the expense of practicality with respect to fabrication. The optimum structural parameters corresponding to the maximum Jsc are outlined in Table 1 for each of the three TiO2 diffraction gratings.
To gauge the efficiency of the TiO2 diffraction gratings as light-trapping structures, the relative enhancements in Jsc (ΔJ/ΔJMax) are calculated as the ratio of the enhancement in Jsc (ΔJ = Jsc - Jsc:Ref) to the maximum possible enhancement (ΔJMax = Jsc:Max – Jsc:Ref). Enhancement in Jsc is calculated with respect to the reference In0.21Ga0.79As-GaAs QWSC. The maximum possible Jsc is defined as that of an In0.21Ga0.79As-GaAs QWSC with zero transmission losses and no anti-reflection coating. For the rest of the manuscript we choose a fill-factor of 0.6 and height of 500 nm for the optimum square pillar structure and a height of 1000 nm for the optimum skewed pyramid structure, for comparison with Lambertian and plasmonic light-trapping. Table 2 lists the relative enhancements for the optimum rectangular strip, square pillar and skewed pyramid gratings as 33%, 75% and 74%, respectively.
These enhancements illustrate the effectiveness of each grating as a light-trapping structure, reiterating the point that both bi-periodic gratings significantly outperform the uni-periodic grating. The optimum skewed pyramid grating does not outperform the square pillar grating; it only provides Jsc enhancement comparable to the optimum square pillar grating. Moreover, as discussed earlier the high aspect ratio of the optimum skewed pyramid structure is challenging with regard to fabrication. The lack of enhancement of the skewed pyramid grating over the square pillar grating may be the result of the theoretical upper limit for absorption enhancement not being reached by the skewed pyramid grating. Fan. et al  also observed smaller than expected absorption enhancement for numerically simulated asymmetric gratings and attributed the short-fall below the theoretical upper limit to the assumption that all individual resonances are in the over-coupling regime not being satisfied.
3.2 Comparison with plasmonic and Lambertian light-trapping structures for In0.21Ga0.79As-GaAs quantum well solar cells
An optimum silver nanoparticle array (nanoparticles with a square base, width 200 nm, pitch 400 nm and height 50 nm ) and a Lambertian rear reflector with zero transmission losses are investigated for comparison with the optimized TiO2 diffraction gratings. Figure 5 illustrates the In0.21Ga0.79As-GaAs QWSC absorption spectra for the optimum square pillar grating, plasmonic nanoparticles, Lambertian rear reflector and reference. Although the optimum skewed pyramid grating provides comparable enhancement to that of the optimum square pillar grating, its fabrication would prove challenging because of the high aspect ratio and asymmetric shape, and so only the absorption spectrum of the optimum square pillar grating is considered and plotted in Fig. 5 for comparison. Figure 5 illustrates these three light-trapping structures all provide enhancement over that of the reference QWSC for the entire QW region. Lambertian light-trapping demonstrates uniform enhancement over the QW region while the optimum square pillar grating and plasmonic nanoparticles both exhibit strong resonances. The resonances in absorption for the optimum square pillar grating are stronger than that of the plasmonic nanoparticles and in parts of the QW region, larger than the uniform enhancement of Lambertian light-trapping.
Table 2 lists the relative enhancements of the optimum gratings and their associated structural parameters, showing all three outperform plasmonic nanoparticles as light-trapping structures. The bi-periodic square pillar and skewed pyramid gratings however, provide significantly greater enhancement nearest to that of Lambertian light-trapping.
The calculation of the relative enhancement for plasmonic light-trapping takes into account the absorption losses of the nanoparticles. The light absorbed by the plasmonic grating does not generate any current and only results in losses. Our calculations indicate wavelength dependent resonances in the absorption spectrum of the plasmonic nanoparticles that are smaller than 5%. TiO2 diffraction gratings are loss-less over the QW region of interest. Figure 6 illustrates the fraction of light incident on the optimum TiO2 square pillar grating and the plasmonic grating that is reflected back into the solar cell. The fraction of reflected light that is coupled into the solar cell outside the loss cone is also plotted for both structures. The TiO2 grating has consistently higher reflection (approximately 15% or more) than the plasmonic grating over the QW region. Also, a higher fraction of reflected light is coupled outside of the loss cone for the TiO2 square pillar grating compared to the plasmonic grating. At the wavelength 930 nm the optimum square pillar grating couples 96% of reflected light outside the loss cone compared with 62% by the plasmonic grating. And therefore, although parasitic absorption by the plasmonic structures is present, their lower reflection and lower fraction of reflected light outside the loss cone over the QW region is the major contributing factor to the low Jsc when compared with the optimum TiO2 square pillar grating.
3.3 Investigation of a rear reflector for In0.21Ga0.79As-GaAs quantum well solar cells with TiO2 diffraction gratings
The optimum structural parameters for the TiO2 diffraction gratings reported in Table 1 maximize the Jsc of the In0.21Ga0.79As-GaAs QWSC for the QW region. These designs simultaneously maximize the reflection at the rear cell-air interface and the coupling efficiency to higher order diffraction modes. However, transmission losses into the zero order at the rear of the cell are still present. In order to minimize these losses, a silver reflector is incorporated at the rear of the In0.21Ga0.79As-GaAs QWSC to ensure that transmitted light can re-enter the QWSC upon reflection and subsequently allow for an improvement in its absorption. The rear reflector is separated from the base of the grating by free space. A fraction of the reflected light that is incident on the free space-grating interface can be diffracted into the QWSC as a series of higher order modes that are subject to total internal reflection, facilitating an improvement in QWSC absorption.
Wang. et al  demonstrated that a rear reflector integrated with both silver and TiO2 wavelength-scale diffraction gratings improved absorption by silicon solar cells. It was shown that the separation between the diffraction gratings and the rear reflector can be treated as a Fabry-Perot cavity and that the grating-reflector separation is critical in maximizing the absorption and hence Jsc of the silicon solar cells.
As discussed earlier, the structure of the optimum skewed pyramid grating would be challenging with regard to fabrication and so only the optimum square pillar grating is considered with a rear reflector. The Jsc contribution of the QWs is determined as a function of the grating-reflector separation for the optimum square pillar grating and is illustrated in Fig. 7 . The separation spans the range of 700 – 1700 nm and the reference Jsc for the optimum square pillar grating without a rear reflector is plotted for comparison. An inset is included in Fig. 7 depicting the simulation setup. Figure 7 illustrates that irrespective of the grating-mirror separation, incorporating a reflector at the rear of an In0.21Ga0.79As-GaAs QWSC with the optimum square pillar grating enhances the Jsc. However, specific grating-reflector separations maximise the enhancement. The Jsc averaged over the grating-reflector separation for the optimum square pillar grating is 3.74 mAcm−2.
The Jsc of the QWs as a function of grating-reflector separation is the unique superposition of periodic contributions provided by the transmitted wavelengths experiencing constructive and destructive interference, a Fabry-Perot cavity effect. The most strongly transmitted wavelengths over the QW region and the weighting introduced by the solar spectrum influence the overall amplitude of variation in Jsc as a function of grating-reflector separation.
Wavelength-scale TiO2 diffraction gratings are demonstrated as efficient light-trapping structures that significantly improve the absorption of light in the QW region, enhancing the Jsc of In0.21Ga0.79As-GaAs QWSCs. Symmetric uni-periodic, symmetric bi-periodic and asymmetric bi-periodic diffraction gratings provided relative Jsc enhancements of 33%, 75% and 74%, respectively. All three grating structures outperformed optimum plasmonic nanoparticles which provided a relative Jsc enhancement of only 27%. The optimum skewed pyramid and square pillar gratings performed comparably and provided enhancement nearest to that of Lambertian light-trapping. For the optimum square pillar grating, the Jsc over the QW region was further enhanced by a silver rear reflector.
We acknowledge the Australian Research Council (ARC) for financial support and the National Computational Infrastructure (NCI) for providing computational resources used for this work.
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