## Abstract

We analyze the electromagnetic properties of InP/InAs nanowire solar cells for different geometries. We address both eigenvalue calculations to determine the wave propagation as well as source problems to simulate direct perpendicular illumination by three-dimensional finite element calculations. We demonstrate the validity of a 2D waveguide modal analysis as a method of estimating the results of the computationally far more demanding 3D analysis. The resulting data is employed in a detailed balance analysis in order to determine the optimum set of bandgap energies for a single-junction and dual-junction cell as well as the corresponding efficiency limit. The efficiency of the nanowire design can approach the efficiency of conventional thin-film designs despite the low volume fill-factor.

© 2009 Optical Society of America

## 1. Introduction

Photovoltaic devices play a significant role as a future source of renewable energy. We believe that efficiency is paramount for photovoltaic devices used in connection with concentrators. In order to achieve this goal, thin-film multi-junction heterostructure devices are fabricated to harvest the solar spectrum in a more efficient way. However, the selection of the bandgap energies in a pseudomorphic setup is constrained by aspects of interface strain due to varying lattice constants. A promising method of relaxing this constraint is the use of semiconductor nanowires placed in an array covering an area of macroscopic dimensions. Such devices belonging to the so-called third generation are — among different other devices types — currently under intense research. For a detailed review about contemporary directions of research the reader is referred to [1].

There remains the question of the ultimate efficiency that can be achieved in nanowire array devices. The well-known Shockley-Queisser detailed balance calculation is commonly used to determine the optimal bandgap energies and efficiency limit of a multi-junction device. This approach is based on simplifications in the electronic (semiconductor transport) and optical domain. In its original form, the detailed balance calculation assumes perfect light absorption and no recombination other than radiative, so that each exciton contributes to the short-circuit current [2].

In contrast to thick-film or thin-film solar cells, light propagation in devices nanostructured in the device plane is not straightforward to calculate. The assumption of perfect absorption within an arbitrarily thin layer of semiconducting material as used in the detailed balance approach in its simplest form is not justified by any means. Nanowire arrays with features of size in the range of the solar spectrum wavelengths form metamaterials, i.e. their absorption characteristics are mainly influenced by their structure rather than material composition. Therefore, it is necessary to evaluate the maximum efficiency in a way encompassing the geometry of the nanostructure array.

In this publication we calculate the light propagation and absorption in a nanowire array using the full-wave vectorial finite element method (FEM). We relate the obtained results to a waveguide calculation to demonstrate that the incident light is propagating along the nanowires mainly in discrete modes. By doing so, we illustrate that the absorption characteristics of the nanowire solar cell are determined by the dispersion relation of the structure. As a post-processing step we perform a detailed balance calculation to determine the optimum bandgap energies for a dual-junction nanowire solar cell. As the finite element method is appropriate for the analysis of arbitrarily-shaped structures, the method herein is applicable to any nanostructured solar cell designs, including single-nanowire photovoltaics [3].

## 2. Geometry

We investigate a 2D-periodic nanowire array in the *xy*-plane with a unit cell *U* of square footprint. It is embedded in an infinite homogeneous medium of refractive index *n*
_{0}. We shall refer to the distance between the centers two adjacent nanowires as the array period *a*
_{0}, the cylindrical nanowires have a diameter of *d*
_{0}, optionally they are embedded in a coaxial cylindrical waveguide of the diameter *d*
_{WG}.

The height of the nanowires is denoted as *h*
_{0}. Each unit cell is assumed to be illuminated by a plane wave through its top boundary *∂U*
_{top}. The interface between free space and the plane of all top boundaries is located by definition at *z*=0. These conventions are illustrated in Fig. 1.

The exemplary numerical results discussed in this paper were calculated assuming *h*
_{0}=2 µm and *n*
_{0}=1. The passive waveguide consists of silicon dioxide with a diameter of *d*
_{WG}=200 nm (referred to as “with oxide”) or *d*
_{WG}=0 nm (referred to as “without oxide”). Furthermore we conduct two series of simulations assuming the nanowires consist of InP and InAs, respectively. We use the dielectric parameters measured for thin films and bulk provided in [4].

## 3. Methods

#### 3.1. Optical Field Calculation

The time-harmonic vectorial wave equation for anisotropic media reads as

In order to keep consistency with the notation used in the solar cell community, we choose to represent the temporal frequency ω_{0} of a harmonic wave in terms of its free-space wave-vector k0 or free space wavelength λ_{0}. In our calculation we are employing the finite element method and the underlying variational formulation is equal to [5, 7]

In this publication, this variational formulation is solved in two distinct ways using the finite element method.

### 3.1.1. 3D Finite Element Simulation

In the 3D simulation the electric field is expanded in terms of first order curl-conforming Nedelec basis functions on a tetrahedral mesh [6]. By the proper application of the Dirichlet and Neumann boundary condition, we simulate an infinite periodic array of nanowires of finite height illuminated by a normally incident plane wave. The incident wave is induced by an infinite sheet current with the *z*-oriented Poynting vector S_{inc}. We assume that reflection can occur at the interface between free space and the nanowire array, therefore the total power flux is assumed to be the superposition of power fluxes of the incident and scattered fields.

Note that the power flux of the scattered field S_{sc}(λ_{0}) has components in -*z*-direction. The simulation domain in -*z*-direction is terminated by perfectly matched layers (PML) sufficiently far away from the array structure. The materials composing the structure are represented by their respective wavelength-dependent complex refractive index. We perform simulations over a wavelength range of 300 nm≤λ_{0}≤2000 nm in order to cover the relevant parts of the solar spectrum. The resulting local spectral power dissipation *p*(x,λ_{0}) in a lossy medium serves as the basis for the subsequent detailed balance calculation.

We refer to the ratio of the power absorption within a volume *V*⊆*U* with respect to the power incident upon the unit cell *U* as relative power dissipation *η _{p}*(

**x**,λ

_{0}). In this context, note that the unit cell does not contain any sources.

with **z**̂ being the unit direction vector along the *z*-axis. Note that the denominator represents the integral over the entire top boundary of the unit cell while the volume *V* may be any volume within *U*. In the scope of this publication, *V* comprises the entire semiconductor nanowire. Furthermore we refer to the relative power dissipation of the entire unit cell as total relative power absorption *η*
_{abs}(*λ*
_{0}). The transmittivity of the surrounding medium-to-nanowire-interface *T* is defined as the ratio of the power flux into the array compared to the power flux in an infinite volume of the surrounding medium.

Note that due to the orientation of **S**
_{sc}(*λ*
_{0}), the value of *T*(*λ*
_{0}) is within 0≤*T*(*λ*
_{0})≤1. Finally we want to introduce the convention that the power *P*(*z*,*λ*
_{0}) penetrating a surface within the unit cell is calculated as

and the incident power *P*
_{inc} defined in analogy as an integral over *S*
_{inc}(**x**,*λ*
_{0}).

### 3.1.2. 2D Waveguide Simulation

In the waveguide simulation we assume homogeneity of the array in the *z*-direction. We are thus modeling the nanowire array as an infinite array of infinitely extended nanowires. The validity of this assumption is not evident in any way because the nanowires have a length of *h*
_{0} which is in the order of a few wavelengths. For the geometries we investigated, however, direct comparisons between the 2D and the 3D calculation revealed good matching. We employ a separation of variables ansatz by inserting

into Eq. (2) eliminating the dependence of the electric field on the *z*-coordinate other than a scaling with exp(-*jβ z*) leading to the reduction of the problem dimensionality. Hereby, we require the permittivity and the permeability to be isotropic with respect to the *z*-axis. This limitation is intrinsic to the ansatz in Eq. (7) and does not limit the generality of the method in case of isotropic materials or at negligible anisotropy along the *z*-direction.

In the waveguide approach the electric field is expanded in the simulation plane (*xy*-plane) in terms of first order curl-conforming Nedelec basis functions, the electric field in the *z*-direction is approximated by linear node-based shape functions [8]. In the waveguide simulation no excitation is applied, we are solving the resulting eigenvalue problem. The eigenvalues and eigenvectors can be interpreted as a complex effective refractive index *n*
_{eff,m}(*λ*
_{0})=*λ*
_{0}
*β _{m}*(

*λ*

_{0})/2

*π*of a mode

*m*and as the corresponding field profile

**E**

*(*

_{m}*x,y*), respectively. The relation

*β*(

_{m}*λ*

_{0}) is the well-known dispersion relation of the waveguide structure with the frequency being represented by

*λ*

_{0}. For reasons of clarity we shall omit to mention the dependence of

*n*

_{eff,m}(

*λ*

_{0}) on the wavelength

*λ*

_{0}.

In case of a nanowire solar cell with an area of several hundred mm^{2}, the size and symmetry of the arrangement leads to a virtually infinite number of modes with high degeneracy of each mode. We can safely reduce the total number of modes by analyzing the modal behavior of a unit cell of the structure, thereby imposing spatial periodicity of the electromagnetic mode. This reduction leads to four distinct symmetry expansions of the field. Assuming an excitation of the cell by perpendicularly incident plane waves, only a field expansion with one Dirichlet and one Neumann boundary condition on two opposing edges of the unit cell remains to be considered.

The dispersion relation of a specific setup including the field distributions of the modes is illustrated in Fig. 2. Note that the structure exhibits modes that are mostly confined to either the semiconductor nanowire or the space between them. This fact also reflects in the modal effective refractive indices. Modes that are strongly confined in air have effective refractive indices close to unity with small imaginary part while modes mostly confined to the semiconductor material feature high absorption and refractive indices.

The attenuation of optical power *P _{m}* propagating in the mode

*m*can be calculated using the Beer-Lambert law

We assume the waveguide is free of any perturbation and therefore there is no cross-talk between two arbitrary modes. Consequentially, the coupling between free space and the respective modes remains to be analyzed.

The dispersion relation alone does not indicate the distribution of power into the modes. Impedance matching has to be applied to determine the in-coupling of radiation power into the respective modes yielding the total absorption of the structure. The incident radiation is assumed to be a TEM wave outside of the nanowire forest, at the interface between the nanowire array and free space the light is coupled into the waveguide modes. The coupling efficiency into mode *m* is defined as [9]:

being the overlap integral between the electric field of the incident plane wave and the electric field of the mode *m*. This integral represents the correlation of the incident wave and the modal fields. The modal transmittivity *t _{m}* is equal to

It is thus necessary to calculate the dispersion relation of all modes with significant overlap integral. The value of the overlap integral depends on geometry as well as the wavelength. The dependence of the overlap integral on the nanowire diameter *d*
_{0} at *a*
_{0}=600 nm and *λ*
_{0}=300 nm is provided in Fig. 3. The figure illustrates two characteristic values of the modes: the modal penetration depth defined as the distance in *z*-direction leading to a decrease of intensity by 1/*e* and the modal transmittivity *t _{m}* as introduced in Eq. (10). The graphs indicate that for thin diameters the coupling into a weakly absorbing mode with most energy confined in the free space between the wires (mode no. 3 in Fig. 2) is dominant. This observation can be explained by the fact that the modal electric field distribution of mode no. 3 is fairly uniform throughout the entire unit cell and correlates well with the incident plane wave.

The coupling into the highly absorbing mode with most energy confined to the nanowire (no. 4 in Fig. 2) increases with increasing nanowire diameter as its field profile correlates stronger with the incident TEM wave. In Fig. 2 a mode with a strong variation of the direction of the electric field is illustrated (mode no.1). Due to the small correlation with the uniform electric field direction of the incident plane wave, the coupling at *d*
_{0}=180 nm is minuscule. The variation of the direction of the modal electric field distribution also reduces the coupling efficiency of mode no. 2 which exhibits — in contrast to the well-coupling mode no. 3 — a change of the direction of the electric field within the unit cell. The associated zero-crossing of mode no. 2 is indicated by the blue strip through its field profile in Fig. 2. Note, the coupling into mode no. 3 decreases with increasing diameter as the free space between the nanowires (to which the mode is confined) is reduced.

The coupling efficiencies at a given wavelength do not necessarily sum up to 1, after normalizing the coupling efficiencies of m modes, we calculate the power *P _{m}*(

*z*=0) of the mode

*m*at the top interface of the solar cell to be equal to

By applying the Beer-Lambert law the total relative spectral absorption of the waveguide approach can be calculated allowing direct comparisons between the 3D and waveguide ansatz.

The comparison of Eq. (12) to the 3D calculation is illustrated in Fig. 4.

### 3.2. Detailed Balance Calculation

Based on the assumptions of the detailed balance approach published by Shockley and Queisser [2], we assume that only photons with energy *hν*>*E _{g}* generate electron-hole pairs with an internal quantum efficiency

*η*≤1. The local spectral generation rate reads as

_{eqe}and the photogenerated short circuit-current within a volume *V* of constant *E _{g}*

with *λ _{g}* being the wavelength with photon energy of

*E*. Assuming radiative recombination only the corresponding IV-characteristic can be derived to be [10]

_{g}with

Note that in a multi-junction solar cell with the junctions connected in series, the total current across all junctions is constant by definition and limited by the smallest short-circuit current of any cell. Furthermore, the local generation rate depends on the local spectral power dissipation which is in turn dependent on the illumination, dielectric properties of the materials and the structure geometry.

Thus, we calculate the efficiency limit of a n-junction solar cell by relating the electrical output power to the integral over the incident concentrated solar spectral power density.

In contrast to the classical Shockley-Queisser approach, our calculation treats the absorption of light in a more refined way. We do not assume total light absorption up to the bandgap energy in a lossy region [2]. Consequentially, in a multi-junction cell design, photons of energy higher than the bandgap of a certain region may traverse it. We merely assume that excitons — once generated — do not recombine unless they are not extracted out of the solar cell according to the cell’s electrical operating point. Furthermore, in our model the generation of carriers is local allowing a spatially resolved detailed balance calculation.

## 4. Discussion

We performed 3D calculations of a nanowire array for various diameters and material compositions and compared the resulting total relative power absorption *η*
_{abs}(*λ*
_{0}) to the value calculated using the waveguide approach *η*
_{abs,2D}(*λ*
_{0}). The results are in good agreement as illustrated in Fig. 4. Two designs of conventional thin-film InAs and InP solar cells were added for comparison illustrating the implicit anti-reflective effect of the nanowire array. The hypothetical 2 layer anti-reflective coating with *n*
_{1}=1.6 and *n*
_{2}=2.3 was obtained using the genetic algorithm discussed in [11].

The consequences of our observations are threefold: First, we can assume that light is propagating in the solar cell in the discrete modes of the infinitely extended waveguide although the nanowire structure is only a few wavelengths in dimension. The dispersion relation of and the field distribution in the nanowire array has therefore to be matched to the incident plane wave. Second, the waveguide method is particularly appealing for the evaluation of designs over a wide parameter space as the computational demand of the waveguide method is smaller by a factor of more than 1,000. Third, the effect of the variation of the nanowire length can be readily established using the Beer-Lambert law making repeated 3D calculations superfluous for structures homogeneous in *z*-direction.

As illustrated in Fig. 5 as well as in Fig. 4 the absorption of thin nanowires is significantly improved by the presence of a dielectric waveguide cylinder. This effect decreases for thicker nanowires with a diameter *d*
_{0} approaching the diameter of the waveguide *d*
_{WG}. The structure also exhibits local maxima of the total relative power absorption that are caused by the dispersion characteristic of the structure. In contrast, the dielectric properties of bulk do not exhibit this phenomenon.

Therefore, spectral absorption characteristics of the structures discussed herein cannot be described Maxwell-Garnett effective medium approximation

where the effective refractive index neff only depends on the refractive index of the nanowire material *n*
_{NW}, the surrounding refractive index *n*
_{0} and the volume fill-factor *δ*. The 2D waveguide approximation discussed in this paper, however, can be interpreted as more sophisticated effective medium approach where the modes which depend on the exact geometry constitute spatially superimposed effective media. This aspect also manifests itself in the validity of the Beer-Lambert law on a mode-by-mode basis.

The transmittivity of the structure is very high due to the fact that the structures exhibit modes with high overlap integral and an effective refractive index of ℜ{*n*
_{eff}}≈1. This mode is referred to as mode no. 3 in Fig. 2 and Fig. 3. The nanowire structure therefore acts as an anti-reflective coating by itself.

Based on the spectral total relative power dissipation determined using the 3D method, we calculated the Shockley-Queisser efficiency of various single-junction and dual-junction nanowire solar cell designs under 500×AM1.5d illumination with the junctions connected in series. The efficiency limit as defined by Eq. (17) is illustrated for the single-junction and dual-junction device in Fig. 5 and Fig. 6, respectively. The simulations assume the nanowires to have the dielectric properties of InAs with zero absorption for photon energies smaller then the corresponding bandgap energy. This simplification is based on the observation, that the total relative power absorption of the wires consisting of InAs or InP is quite similar apart from the vanishing absorption at the bandgap energy of InP. This assumption is further justified by the quantitative analysis of these materials at various mole-fractions described in [12]. Therefore, the calculation illustrated in Fig. 5 and in Fig. 6 is not self-consistent. The results for ideally absorbing (no reflection, complete absorption) thin-film devices as described in [10] are provided for the purpose of comparison. It is remarkable that a relative area *πd*
^{2}
_{0}/4*a*
^{2}
_{0} ≈0.1 is sufficient to yield seventy percent of the conventional thin-film solar cell efficiency. The resulting efficiency limits indicate that micro-concentration is occurring as the efficiencies the nanowire cells exceed the efficiency of the thin-film cell weighted with the relative nanowire area. Due to the long penetration depth at high wavelengths, we observe an increase of the optimum bandgap energies for a dual-junction design exceeding the bounds of the InAs/InP material system. In order to achieve high efficiency within the given material system, it is imperative to increase absorption as the efficiency drop is more pronounced in case the optimum bandgap energy lies outside of the material system.

## 5. Conclusion and Outlook

We demonstrated that in a nanowire solar cell large fractions of light are traveling in discrete modes of the corresponding waveguide structure. The absorption characteristics of the nanowire array are therefore not primarily influenced by the material composition but by the field distribution of the various modes, ergo the structure geometry. Also, the reduction of the problem size leads to decreased computational demand.

We believe that the present analysis of the coupling into various modes gives rise to ideas to shape the incident light to better correlate with well absorbing modes. This principle is commonly used in microwave engineering, where horn antennas are employed to couple the modes of a waveguide to the electromagnetic field in free space.

The nanowire array exhibits highly absorbing modes at low volume fill factors, in case a good coupling into these modes is accomplished, this circumstance can lead to decreased usage of expensive III-IV semiconductor material. Such a cost advantage is of particular importance for large-scale on-grid power generation.

Based on the field distribution within the nanowire, we calculated the spatially and spectrally resolved power dissipation leading to a detailed balance analysis that is based on a spatially resolved optical model. We demonstrated that the resulting efficiency limits of the nanowire solar cells are comparable to conventional thin-film single-junction and dual-junction designs. Micro-concentration of the incident light by the nanowire array was observed. Self-consistency of the calculation can be achieved by iterating within a material system until the optimum bandgap energies correspond to the dielectric properties used in the electromagnetic simulation. The described method allows further investigation on the spatial arrangement of the absorbing pn-junctions, this aspect — however — remains to be investigated further.

## Acknowledgements

The research outlined in this publication was conducted within the AMON-RA project (further information can be found online at http://www.amonra.eu) and was funded by grant agreement FP7-214814-1.

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