## Abstract

We investigate the accuracy of rigorous coupled-wave analysis (RCWA) for near-field computations within cylindrical GaAs nanowire solar cells and discover excellent accuracy with low computational cost at long incident wavelengths but poor accuracy at short incident wavelengths. These near fields give the carrier generation rate, and their accurate determination is essential for device modeling. We implement two techniques for increasing the accuracy of the near fields generated by RCWA and give some guidance on parameters required for convergence along with an estimate of their associated computation times. The first improvement removes Gibbs phenomenon artifacts from the RCWA fields, and the second uses the extremely well-converged far-field absorption to rescale the local fields. These improvements allow a computational speedup between 30 and 1000 times for spectrally integrated calculations, depending on the density of the near fields desired. Some spectrally resolved quantities, especially at short wavelengths, remain expensive, but RCWA is still an excellent method for performing those calculations. These improvements open up the possibility of using RCWA for low-cost optical modeling in a full optoelectronic device model of nanowire solar cells.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Nanowire solar cells (NWSC) are a new solar cell technology with the potential to improve upon existing solar cell devices. Their potential stems from their ability to effectively absorb incident light while using less semiconductor material than planar solar cells.

The optimal nanowire solar cell arrays consists of nanowires that are a few microns in height and with diameters and periodicities comparable to the wavelengths present in the solar spectrum [1, 2]. These small sizes require full wave optics simulations to accurately model their optical properties, unlike in standard planar devices [3, 4]. Both experimental measurements and modeling have shown high levels of absorption with low sensitivity to the incident angle of light [5]. Additionally, the finite in-plane dimensions of nanowires can accommodate strain due to growth on lattice-mismatched substrates without introducing dislocation faults in the crystal lattice [6]. This capability opens up the possibility for high-efficiency III-V tandem cells grown on silicon [7].

The larger design parameter space of NWSC relative to planar solar cells requires careful optimization of geometric parameters to maximize device performance [8]. There is a need for fast, accurate modeling tools to enable rapid exploration and optimization of nanowire designs. Conventionally, finite element [4, 9–11] and finite difference methods [1, 12, 13] have been used in optical models of NWSC. While these techniques are highly accurate, they are computationally expensive, limiting their usefulness in a closed-loop global device optimization. Rigorous coupled-wave analysis (RCWA) is another wave-optics modeling technique that lacks the memory and computational requirements of competing techniques [14]. RCWA is a Fourier domain technique ideally suited to periodic arrays. It is promising for its speed and is highly accurate when computing far-field quantities such as total absorptance, reflectance, and transmittance. RCWA simulations become more accurate as the number of plane waves *N _{G}* increases, and the computational cost scales as ${N}_{G}^{3}$. However, naive implementations lack accuracy at reasonable

*N*when computing near-fields internal to the device due to the well-known Gibbs phenomenon [15]. Such near-fields are required to compute carrier generation rates and are thus an essential component of a fully-coupled optoelectronic device model.

_{G}In this work, we show that RCWA can be used for accurate optical modeling of nanowire solar cells. We examine a test device (see Fig. 1 and Table 1), indicate where standard RCWA formulations lack sufficient accuracy, and provide two techniques for increasing accuracy of the near fields. The first is an implementation of an already published technique for introducing proper discontinuities in the near fields and mitigating the Gibbs phenomenon [15–18]. We extend the open source RCWA library S^{4} to include this technique, and show it greatly increases the accuracy of local field computations [19]. The second is a new rescaling technique that increases the accuracy of device simulations while keeping computational cost reasonable when computing spectrally integrated quantities. We show that even with our two improvements, some spectrally resolved quantities continue to require more expensive calculations. Using our improvements, RCWA shows promise as an effective technique for rapid optical modeling of nanowire solar cells.

## 2. RCWA

RCWA is a Fourier-space method for solving the frequency domain Maxwell’s equations with frequency *ω*:

**E**is the electric field, $\mathbf{D}=\u03f5\mathbf{E}$ is the displacement field,

*ϵ*is the electric permitivitty,

*μ*is the magnetic permeability, ${\mathbf{J}}_{f}$ is the free current density, and

*ρ*is the free charge density. RCWA relies on two critical assumptions about the geometry of the system. First, the device must be composed of discrete, axially-invariant layers such that at a given

_{f}*x*-

*y*point within a layer, the material parameters along the z-direction remain constant. Second, the device must be decomposable into fundamental unit cells that are 2D periodic in the plane. If these conditions are satisfied, then the longitudinal and transverse dimensions are separable and the fields in a single layer can be written as:

**G**is one of

*N*in-plane reciprocal lattice vectors,

_{G}**k**is the in-plane component of the excitation wave vector, and $\mathbf{r}=x\widehat{\mathbf{x}}+y\widehat{\mathbf{y}}$. Note the

**G**is generally chosen to be an array of reciprocal lattice points with a circular truncation, keeping all

**G**with $\left|\mathbf{G}\right|$ less than some constant, which maintains symmetry in Fourier space [19]. The in-plane dielectric profile $\u03f5\left(\mathbf{r}\right)$ may depend on the material, allowing it to have piecewise-constant dependence on the transverse spatial coordinates. Vertical nanowire arrays (see Fig. 1) satisfy these geometric constraints.

We consider only non-magnetic materials, so $\mu ={\mu}_{0}$, the vacuum permeability. At optical frequencies, which we consider, *ρ _{f}* is negligible. ${\mathbf{J}}_{f}$ can be used to describe emission from within structures; we will not make such considerations and will thus consider ${\mathbf{J}}_{f}=0$ in the examples below, but such currents could be added without difficulty.

The approximation of step changes in *ϵ*, while convenient mathematically, is known to produce singularities in local fields and their gradients at internal corners and edges [20]. These singularities are square integrable, as is required for conservation of energy, but are generally not fully resolvable with standard FDTD, FEM, and RCWA methods, though specialized methods have been proposed to treat some of the singularities in particular geometries [21–23]. The cylindrical geometry considered here contains no corners and the active region of GaAs is convex, so the field singularities inside the nanowire material are relatively benign, but they are not accurately modeled here. Instead, we effectively spread the associated optical generation over a region given by the Fourier resolution, which scales as the unit cell period divided by $\sqrt{{N}_{G}}$. Since this length scale is smaller than material diffusion lengths, we do not anticipate effects in device modeling.

The essential part of RCWA is determining ${\mathbf{H}}_{\mathbf{G}}\left(z\right)$ in Eq. (5) for a given set of reciprocal lattice vectors **G**. One can assume the coefficients in Eq. (5) take the form [19]

*ϕ*are expansion coefficients and the z-component has been chosen to satisfy the $\nabla \cdot \mathbf{H}=0$ condition. This form of the fields illustrates one of the key advantages of RCWA over competing techniques, namely the analytic dependence on the z coordinate. By inserting Eq. (6) into Eq. (1), one arrives at an eigenvalue equation for determining the set of eigenvalues

*q*and the components of the eigenvectors

*ϕ*for a single layer. Once the eigenmodes of each layer have been determined, multilayer structures are joined together by introducing propagation amplitudes for the eigenmodes and using the scattering matrix method to join solutions at layer interfaces [24–28]. Results increase in accuracy with

*N*. Our work is an extension to S

_{G}^{4}, an open-source implementation of RCWA built on the scattering matrix method [19]. In the remainder of the manuscript, we refer to S

^{4}as the standard RCWA method, but it has included a significant number of improvements from the original RCWA methods; for details, see [19].

For optoelectronic device modeling, we are most concerned with determining the local carrier generation rate, which is determined from the local electric field strength in each material. RCWA expresses the fields using the Fourier series in Eq. (5). Any finite Fourier series representation is always continuous, even across in-plane material interfaces, as between the core and shell of a nanowire. In an exact solution, the normal components of **E** should be discontinuous across material boundaries, but a Fourier reconstruction requires an intractable number of terms to accurately model such a discontinuity, even though far-field quantities such as the total absorptance may be well converged. For any finite *N _{G}*, standard RCWA-produced fields have spurious oscillations, especially near material interfaces.

To assess the convergence of RCWA with *N _{G}*, we define two methods for computing the absorptance of a layer of the device. The first method relies only on the power passing through the top and bottom of the layer. These powers can be computed entirely in Fourier space [19], and do not suffer from convergence issues in the reconstruction of the near fields. The net emitted powers of layer

*i*are defined as:

*S*is the z-component of the Poynting vector and the integration is over the top or bottom surface of the unit cell, with the appropriate sign for emitted power. Considering the top (layer 1) and bottom (layer

_{z}*n*) together, the total reflectance and transmittance are where

*P*is the input power of the incident plane wave. Then total absorptance is:

_{in}The contribution of a single layer to the device absorptance can be calculated similarly. We consider the test structure detailed in Table 1 and use S^{4} [19] to perform RCWA calculations with normally-incident circularly polarized light. Due to the circular symmetry of the nanowires, total absorption is independent of the handedness of the incident light, so we can consider just one polarization and determine the local field strengths even for incoherent illumination [11]. We consider 60 equally spaced frequencies corresponding to wavelengths from 300 nm to 900 nm, just beyond the GaAs absorption edge of 871 nm. Figure 2 shows that the far-field absorptance spectrum of the full device converges rapidly with basis terms, and is self-converged within 0.5% with *N _{G}* = 75.

Equation (11) expresses the power absorbed in a layer in terms of the fluxes into and out of the layer. The divergence theorem and Maxwell’s equations allow rewriting that power in terms of the local fields, instead. The absorbed power can then be written,

The complex dielectric at each frequency is constructed from tabulated real *n* and imaginary *k* parts of the index of refraction in each material [9, 29].

We calculate ${A}_{\text{near}\phantom{\rule{0.2em}{0ex}}\text{field}}$ by extracting $\mathbf{E}\left(\mathbf{r}\right)$ on a cubic mesh with 1 nm spacing in the plane for all layers. We use 3 nm spacing along the z-direction in the ITO layer and 3.5 nm spacing in the nanowire layer. A sparser mesh of 16 nm spacing is used in the substrate due to the weak absorption there. This choice of mesh is sufficiently dense to converge the result better than 1% using a simple trapezoidal rule integration. Figure 3(a) shows the convergence of ${A}_{\text{near}\phantom{\rule{0.2em}{0ex}}\text{field}}$ with *N _{G}*. Though the near fields are well converged for $\lambda >450$ nm, they are not converged at short wavelengths even for

*N*= 997.

_{G}Figure 4 shows some results of Figs. 2 and 3 as a function of *N _{G}* at three wavelengths and spectrally integrated with the AM1.5G solar spectrum [32], to more clearly show that the far-field method (green line) converges rapidly at all wavelengths, while the standard local-field method (blue line) only converges to the far-field result for long wavelength.

In the following sections, we provide two techniques for improving the accuracy of the near fields in RCWA. The first is an implementation of an existing technique, which we call the continuous variable formulation (CVF), which mitigates the Gibbs phenomenon and ensures proper discontinuities at interfaces by modifying the field computations such that only quantities that are continuous in real space are reconstructed from their Fourier components [15]. The discontinuities across in-plane material boundaries are then handled in real space. The second technique uses the well-converged, highly accurate far-field computation of each layer’s absorption to rescale the near fields, ensuring correct total generation within a device layer.

## 3. Continuous variable formulation

The CVF is a modification to RCWA that only Fourier reconstructs quantities that are continuous across material interfaces in real space. Near material interfaces, these quantities are the components of the displacement field **D** that are normal to, and the components of the electric field **E** that are tangential to, the interface. Using these real-space continuous quantities, one can determine the full electric field everywhere by using the constitutive relationship

*ϵ*. S

^{4}already uses a related technique for calculating the Fourier modes, but it does not use this method when extracting real-space quantities.

To perform this decomposition with arbitrarily shaped 2D regions within a layer, one must construct a locally-defined vector field that is tangent to all material interfaces and periodic in the plane, which can be generated automatically, as is done by S^{4} [19, 30]. This vector field induces an associated projection operator *T* that can be used to project the Cartesian components of the electromagnetic fields onto this local coordinate system such that:

**E**along the tangential vector field and ${\mathbf{D}}_{N}$ is the component of

**D**perpendicular to the tangential vector field. The total field satisfies

By taking the Fourier transform of $T\left(\mathbf{r}\right)$, the projection onto the tangential vector field can also be done in Fourier space. Mirroring the notation of [15], we denote discrete real-space quantities with upper case letters (as in *E _{x}* to represent the vector ${E}_{x}\left({\mathbf{r}}_{i}\right)$ for many points ${\mathbf{r}}_{i}$), vectors of Fourier coefficients with lower case letters surrounded by single brackets (as in $\left[{e}_{x}\right]$), and Fourier space matrix operators with double brackets (as in $\left[\text{}\left[T\right]\text{}\right]$). Using this notation, the Fourier transform of Eq. (15) is given by [15]:

**G**, ${\mathbf{G}}^{\prime}$) element of $\left[\text{}\left[T\right]\text{}\right]$ is $\tilde{T}\left(\mathbf{G}-{\mathbf{G}}^{\prime}\right)$.

We extract $\left[{e}_{x}\right]$ and $\left[{e}_{y}\right]$ from a standard RCWA implementation and then construct $\left[{d}_{N,x}\right]$ and $\left[{d}_{N,y}\right]$. Since $\u03f5\left(\mathbf{r}\right)$ and ${\mathbf{E}}_{N}\left(\mathbf{r}\right)$ are both discontinuous, the proper Fourier factorization takes [31]

After finding ${d}_{N,x}$ and ${d}_{N,y}$ we reconstruct the real space electric field

*ϵ*jumps. Finally, the real space electric fields in Cartesian coordinates can be recovered using:

_{r}Figure 5 shows the norm-squared components of the electric fields computed using unmodified RCWA and the CVF on a line cut along the x-direction through the center of the nanowire. In this cut, *E _{x}* is normal to the interface and should therefore be discontinuous, while

*E*should be continuous. Note the Gibbs oscillations in the standard result for

_{y}*E*, while the CVF result has introduced discontinuities at the boundaries and significantly reduced the amplitude of the Gibbs oscillations.

_{x}Figure 3(b) shows the improved agreement between the CVF absorptance and the well-converged far field absorptance. Figure 6 shows the relative difference between the far and near field absorptances calculated with and without the CVF. It is clear that the CVF significantly improves the agreement at all wavelengths, but the disagreement is still significant for wavelengths shorter than 450 nm. Figure 6 indicates the AM1.5G spectrum, which shows that the CVF-based ${A}_{\text{near}\phantom{\rule{0.2em}{0ex}}\text{field}}$ agrees well with the far field results through the most important parts of the solar spectrum. In the next section, we introduce a simple rescaling technique to increase accuracy of the near fields at all incident wavelengths.

## 4. Rescaling technique

In device simulations, the total optical generation rate must be determined accurately. The exact position where generation occurs is somewhat less important, as the carriers drift and diffuse, and deviations on the scale of a few nanometers are rarely significant. We can ensure that the total generation in each layer is calculated correctly, even with inexpensive RCWA calculations that have not fully converged the local fields. To achieve this goal, we use the well-converged far field results (as shown in Fig. 2) to rescale the components of the near fields in each layer such that ${A}_{\text{far}\phantom{\rule{0.2em}{0ex}}\text{field}}$ and ${A}_{\text{near}\phantom{\rule{0.2em}{0ex}}\text{field}}$ agree exactly. We define a rescaling factor *F* for each layer *i* and frequency *ω*:

Then, the components of **E** are rescaled such that:

This rescaling technique allows accurate determination of spectrally-integrated generation rates with small numbers of basis terms. Figures 7 and 8 show line cuts through the test structure at three representative wavelengths and spectrally integrated under AM1.5G illumination, calculated with 60 equally spaced frequencies. At 487 nm, the fields are quantitatively converged at small *N _{G}*, while the Gibbs oscillations are not entirely removed either at shorter or longer wavelengths. Calculations dependent on spectrally resolved local fields, such as external quantum efficiency (EQE), thus require relatively large

*N*

_{G}at some wavelengths. When the fields are spectrally integrated, however, the essentially random phases of the oscillations average away, and the spectrally-integrated fields are quantitatively converged by *N _{G}* = 197. Figure 9 compares the rescaled spectrally-integrated generation rates along a plane through the center of the nanowire at

*N*= 997 and 197, showing the excellent agreement that rescaling permits, even at low

_{G}*N*. This spectrally integrated generation rate is sufficient for optoelectronic modeling while reducing the requirements for

_{G}*N*by a factor of 5.

_{G}The computational cost of the RCWA method scales as ${N}_{G}^{3}$, so reducing *N _{G}* by a factor of 5 (from 1000 to 200) theoretically reduces the runtime by a factor of 125, and reducing to

*N*= 100 can reduce the runtime by a factor of 1000. Extracting the electric fields on a dense mesh of points, however, also has a computational cost, and for sufficiently small

_{G}*N*, this electric-field extraction limits the runtime. Figure 10 shows an estimate of the simulation run times for a single incident wavelength. Each desired wavelength must be calculated separately, and they all take approximately the same amount of processor time. Simulations were run on a single core of an Intel Xeon E5-2640 v4 CPU with a 2.40GHz clock speed. The figure shows both the simulation time for RCWA to determine the field amplitudes, in Fourier space, and for the extraction of those fields on the dense real-space mesh described above. The CVF method does not significantly change the run times. Efforts were made to minimize data input/output time and resource contention in all benchmarks. These results show that running at

_{G}*N*= 197 has a cost 25 times less than at

_{G}*N*= 997, and that cost is dominated by the field calculation and export. The field calculation and export time can possibly be optimized further and would certainly be reduced if a coarser mesh were requested. Decreasing that cost could allow computation times to be reduced by an additional factor of 5. Simulations at each frequency are completely independent, so computation time for a full spectral sweep can be easily reduced by running all frequencies in parallel [33]. With a sufficient number of CPU cores, a full spectral sweep can be run in the same clock time as a single simulation.

_{G}## 5. Conclusion

In this work, we investigate the accuracy of RCWA for optical modeling of nanowire solar cells. We find excellent accuracy with low computational cost at long incident wavelengths, but poor accuracy at short incident wavelengths. To increase the accuracy of RCWA we extend the open-source library S^{4} [19] to include an already published technique for improving near field computations in RCWA [15]. Our implementation mitigates the Gibbs phenomenon and introduces physically expected discontinuities in the fields at material interfaces, improving convergence of the near fields at all incident wavelengths. To bring convergence within a 1% tolerance, we introduce a simple rescaling technique that uses the well converged far field quantities to rescale the near fields on a per layer basis. These improvements open up the possibility of using RCWA as a low cost optical modeling technique in a full optoelectronic device model of nanowire solar cells.

## Funding

Natural Sciences and Engineering Research Council of Canada (STPGP 478768).

## Acknowledgments

We thank Anna Trojnar for helpful conversations.

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