Light absorption and emission in nanowire array solar cells

Jan Kupec; Ralph L. Stoop; Bernd Witzigmann

doi:10.1364/OE.18.027589

1. Introduction

In the past few years several photovoltaic devices based on nano and microwires emerged [1–13]. A plethora of remarkable experimental as well as theoretical results was presented. In terms of theoretical analysis, researchers investigated the unique optical and electronic properties intrinsic to nanowire solar cells constituting a true advantage over thin-film devices. Promising results were obtained with regard to reduced semiconductor material use through absorption enhancement [14] and orthogonalization of the optical and transport paths within the device [15] improving carrier collection.

Concerning the optical properties of nanowire array solar cells, literature reports that an array of nanowires can exhibit superb absorptivity even at a fairly sparse arrangement [16, 17]. To explain these properties the rigorous solution of the Maxwell equations is imperative, as the characteristics cannot be described by ray optics or effective refractive index considerations such as Maxwell-Garnett [14].

There is a large variety of different concepts of nanowire and microwire solar cells with structures of diameter of a few nanometer [1, 18] to a few micrometer [19] being considered for photovoltaic power generation. Furthermore, different materials and arrangements (random, square array, triangular or hexagonal array) are considered. In this publication, we are focusing on InP nanowires standing upright arranged in a regular array of various diameters and distances, illuminated from the top. They are under investigation and promising results concerning their application in solar cell are reported in literature [5] and research is continuing to create and improve such devices [20].

In the first part of this paper, we highlight the absorption properties of those structures. By weighting the results from the optical absorptivity simulations with the solar spectrum we compute the short-circuit current. This calculation is only a generalized model to estimate the performance of a solar cell, however, it is suitable for comparing various device designs and illustrating their relative performance.

In the second part of this paper, we calculate the device efficiency within the radiative limit. For this task, we compute the local density of photon states within the structure to assess the emission enhancement of the radiative recombination. We include both micro-concentration and emission enhancement into the detailed balance framework and conclude with a comparison of the efficiency limits of planar thin-film and nanowire solar cells.

2. Absorption analysis

We define the geometric fill factor as the ratio between the footprint area of the nanowire and the area of the unit cell. We consider nanowires of circular cross-section (diameter d₀) to stand in a square array of unit cell size a₀. The geometric fill factor thus reads

f_{NW} = \frac{d_{0}^{2} π}{4 a_{0}^{2}} with d_{0} \leq a_{0} .

The highest fill factor occurs when the diameter d₀ equals the unit cell size a₀, i.e. the nanowires are in contact with their neighbors (geometric fill factor equal to $\frac{π}{4} \approx 0.785$ ). The pn-junctions within the wires are axial, i.e. a junction has the same area as the nanowire footprint.

We assume that the nanowires consist of indium phosphide (InP), however, as will be highlighted by our results the properties of the array are mainly determined by the geometry and the modal light dispersion within the array, thus, the findings about the governing physical effects are also applicable to other III–V semiconductor nanowires of same dimensions.

The space between the nanowires and above them is assumed to be void. We cover a range of diameters 20nm ≤ d₀ ≤ 400nm and a corresponding range of unit cell sizes a₀ to result in a geometric fill factor f_NW range from a dense packing to less than 0.1. The common length of the nanowires in all designs is equal to h₀ = 2000nm. The nanowires are assumed to stand on an infinitely thick InP substrate.

The diameter range of choice addresses many geometries currently subject of research. We intend to study the wave-optical properties expecting effects not rooted in the absorptivity of bulk material alone. From drift-diffusion transport considerations it is known that efficient carrier transport by diffusion of minority carriers towards the space charge region is critical [15]. As we are considering InP nanowires, we select the length h₀ attending their strong absorptivity as well its moderate minority carrier lifetime leading to a moderate diffusion length, which aggravates carrier extraction. Thus, we select the nanowire length h₀ = 2000nm for high absorptivity as well as acceptable carrier extraction properties.

As the starting point of our analysis, we solve the electromagnetic wave equation

\nabla \times ({\bar{\bar{μ}}}^{- 1} \nabla \times E) - k_{0}^{2} \bar{\bar{ɛ}} E = - j k_{0} Z_{0} J

where E(x, λ) the electric field and J a sheet current causing a normally incident plane wave. We use the three-dimensional Finite Element Method (FEM) in the frequency domain on a tetrahedral mesh with the electric field being expanded in terms of first order curl-conforming Nedelec elements [21], implemented in our custom computer program LUMI3 [22].

We incorporate the absorption in the materials using wavelength dependent complex refractive indices given in tabular form [23]. The doping is assumed to be sufficiently low to not alter the optical properties of the material. The simulations cover the range of 300nm ≤ λ ≤ 920nm, where the lower limit is the start of significant contributions of the AM1.5d solar spectrum (spectral power p_AM1.5d(λ)) and the upper limit corresponds to the bandgap energy of InP at which absorption decreases rapidly. By applying a proper set of boundary conditions, the simulation domain can be reduced to a quarter unit cell of the array yielding the result for a two-dimensional nanowire array of infinite extension.

The result of the optical simulation is the spatially and spectrally resolved optical power dissipation density p(x, λ). It is defined as the divergence of the real part of the complex Poynting vector, or equivalently as the power dissipated in the lossy dielectric InP

p (x, λ) = \frac{1}{2} \nabla \cdot ℜ {E (x, λ) \times H {(x, λ)}^{*})}

where λ is the free space wavelength of the incident light and H(x, λ)^* the conjugate-complex magnetic field.

We integrate the spatially and spectrally resolved optical power dissipation density p(x, λ) as evaluated by a rigorous solution of the Maxwell equations over the volume V_NW of the nanowire yielding the spectrally resolved power dissipated in the nanowires. Normalizing this figure with the incident power p_inc, the absorptivity a(λ) of the nanowires is obtained.

a (λ) = \frac{1}{p_{inc}} \int_{NW} p (x, λ) d x

As the optical problem is linear, the optical simulation can be performed with arbitrary incident power p_inc canceling out in the normalization step. This circumstance allows the reuse of the optical absorptivity data for different spectra and concentrations. Note that the integration in Eq. (4) removes all information concerning the spatial distribution of the optical power dissipation within the nanowire, a(λ) is not a spatially resolved quantity. Power dissipation in the substrate is present but not included in the absorptivity of the nanowire, carriers generated there are lost.

In order to condense the absorptivity curves of all the geometries considered to a single figure of merit, we calculate the photo-generated current by weighting the incident solar spectrum with the absorptivity [17]. We assume that all generated carriers can be extracted out of the device by connecting it to a short-circuit. This is identical with the initial step in the detailed balance analysis presented by Shockley and Queisser in 1961 [24]. The photo current which is identical to the short-circuit current under the aforementioned assumption reads

I_{ph} = I_{sc} = A_{SC} \frac{q_{e}}{h c_{0}} \int_{0}^{λ_{g} = \frac{c_{0} h}{E_{g}}} λ a (λ) p_{AM 1.5 d} (λ) d λ

where A_SC is the macroscopic area of a nanowire array solar cell constituting a large integer number of unit cells.

We normalize the short-circuit current with respect to the current generated in a perfectly absorbing solar cell with unity absorptivity for photons exceeding the bandgap energy.

a (λ) = {\begin{matrix} 0 & λ > \frac{h c_{0}}{E_{g}} \\ 1 & λ \leq \frac{h c_{0}}{E_{g}} \end{matrix}

For the bandgap of InP and the unconcentrated AM1.5d solar spectrum the respective value is 330 Am⁻². We shall use this figure to normalize the short-circuit current in the results section. The normalized short-circuit current is then defined as

η_{sc} = \frac{\int_{0}^{λ_{g} = \frac{c_{0} h}{E_{g}}} λ a (λ) p_{AM 1.5 d} (λ) d λ}{\int_{0}^{λ_{g} = \frac{c_{0} h}{E_{g}}} λ p_{AM 1.5 d} (λ) d λ}

2.1. Results

For the optimum design within our parameter space the dissipation in the substrate is smaller than 2% of the incident power for most wavelengths λ₀ < 900nm, except in the vicinity of λ₀ = 550nm where it mounts to 7.2%.

The spectrally resolved absorptivity a(λ) of the nanowire array for various diameters at constant geometric fill factor of 0.196 (and thus equal material use) is illustrated in Fig.1. For comparison, the figure also illustrates the absorptivity of an InP thin-film with thickness of equal material use, i.e. 2000nm × 0.196 = 392nm. We assume that the thin-film has a perfect anti-reflective coating and the active region is located on a substrate of same refractive index so its absorptivity is not calculated by solving the Maxwell equations but rather by a direct application of the Beer-Lambert law, which is sufficient under these assumptions. For a real planar thin-film device of tangible dielectic composition, it is possible to use 1D FEM [21], or any other suitable method. Assuming a perfect anti-reflective coating favors the thin-film device. Our justification for this step is twofold: First, excellent broadband anti-reflective coatings do exist. Second, we want to compare the nanowire to the theoretically best case thin-film device. The reference thin-film solar cell is therefore a design idealized for high efficiency, in order to serve as benchmark for evaluating the intrinsic properties of the nanowire achitecture.

Fig. 1 Top: Modal electric field intensities of the two most relevant modes used to compute absorptivity using the 2D modal approach in [17]. Center: Relative incoupling into the modes plotted on the top, for definition, refer to Eq. (8). Bottom: Spectral absorptivity of an InP nanowire array and an equivalent plane perfectly AR-coated thin-film with same material use. The geometric fill factor is constant for all plots, the footprint area of the nanowires covers a fraction of 0.196 of the unit cell area. For thin nanowires up to 180nm diameter, the absorptivity of low energy photons can be improved by increasing the diameter while the absorptivity of high energy photons is high even for small diameters. Green dashed curve illustrates absorptivity of the (a₀ = 360nm, d₀ = 180nm)-geometry obtained from 2D waveguide modal analysis as discussed in [17].

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The plot (Fig.1) illustrates that the absorptivity of the nanowire array can approach unity and confirms the presence [14, 17, 25] of both an intrinsic anti-reflective property of the structure as well as power collection exceeding the nanowire footprint area. This observation can be interpreted as an absorption efficiency exceeding unity.

The data also shows that nanowire arrays featuring thin wires exhibit low absorptivity at long wavelength. This aspect is caused by their geometry and not primarily by the material properties as all absorptivity curves in Fig.1 show nanowires of an identical material and even geometric fill factor. The material properties of InP are merely responsible for the sharp decrease of the absorptivity common to all geometries for wavelengths above 900nm where the photon energy decreases towards the bandgap of InP and absorption of the material ceases.

As shown in Fig.1, for the design of d₀ = 180nm we observe absorption enhancement compared to the thin-film, meaning that an equal amount of material shaped as a nanowire is capable of absorbing more light within the same active volume.

The spectral absorptivity of the various geometries can be understood by calculating the eigenmodes of the nanowire array [17, 26]. In a preceding publication [17] we have shown that most of the incident light is propagating within the nanowire array in discrete modes. This statement alludes to the light propagation along the nanowires in the direction perpendicular to the plane of the array rather than to light propagating along the plane of the array, which is in any event irrelevant for the given geometry at perpendicular light incidence.

Designing a proper dispersion of the light (i.e. modal attenuation and field pattern correlation) can lead to reduced material use, especially for high wavelengths where the penetration depth of the light in a plane thin-film is increasing.

Performing the 2D modal analysis of the optimum (a₀ = 360nm, d₀ = 180nm)-geometry along the lines of [17] reveals the near-field optical phenomena causing the enhancement. In the analysis we observe that for the given geometry only two modes are of significance: Over a wide spectral range, one mode is confined to the free space between the nanowires, we shall refer to it as mode A. The second mode is mainly confined to the nanowire over a wide spectral range, we shall refer to it as mode B. The modal electric field intensities of mode A and mode B at various frequencies (in terms of free space wavelength) are plotted in Fig.1. The expressions

c_{A} (λ) = \frac{t_{A} (λ) η_{A} (λ)}{η_{A} (λ) + η_{B} (λ)} and c_{B} (λ) = \frac{t_{B} (λ) η_{B} (λ)}{η_{A} (λ) + η_{B} (λ)}

describe the relative incoupling of the incident TEM light into the respective modes. In Eq. (8) the quantity λ denotes the frequency in terms of free space wavelength and the quantities η_A,B and t_A,B are the modal coupling efficiencies and modal transmittivities, respectively, with their definitions presented in [17] denoted as Eq. (9) and Eq. (10).

The quantities described by Eq. (8) are illustrated in Fig.1. Both coupling and absorption of mode A are high for wavelengths up to λ = 600nm, for larger wavelengths, however, the correlation of the modal field with the incident TEM light deteriorates as the mode starts to feature an electric field pointing in both positive and negative x-direction (the incident TEM light is x-polarized). Therefore the relative incoupling of mode A decreases for λ > 600nm in favor of mode B. For large wavelengths, mode A does not contribute to absorptivity even though its modal confinement to the wire is high and the imaginary part of its modal effective refractive index is negative and high in magnitude.

Mode B in turn expands its energy confinement to the outside of the nanowire and features a well correlating electric field. At λ = 800nm, we observe absorption enhancement compared to a planar film. The absorptivity is high because both mode A and B feature sufficient confinement to the nanowire. At λ = 800nm, the absorptivity decreases as mode B loses its confinement to the nanowire (see the darkening cyan color depicted in Fig.1 at the core of the nanowire for mode B between λ = 800nm and λ = 900nm). The vanishing absorptivity is further supported by the decreasing material absorption as confinement alone does not yield high absorptivity in the case of near-lossless nanowire material. However, we want to emphasize that the absorptivity of the optimum geometry exceeds the absorptivity of the equivalent thin-film by far in the wavelength range 800nm < λ < 900nm. While the absorptivity of the bulk material is monotonously decreasing for wavelengths exceeding 600nm, the absorptivity of the nanowire array can be maintained at a high level by a proper design of the light dispersion.

In our optimal design, mode A is dominant and coupling of incident light into this mode is very efficient over a large wavelength range meaning that there is little reflection away from the top of the nanowire array and the modal field correlates well with the incident TEM wave. As long as the decay of the mode A along the length of the nanowire h₀ is sufficiently strong, high absorptivity occurs.

In a recent publication [27] the propagation properties of the HE₁₁ mode of a single nanowire were identified to be responsible for the significant drop of absorptivity beyond a critical wavelength. The analysis presented in [27] address the properties of the mode B presented in this paper. As can be seen in Fig. 1, the modal field of mode B only exhibits qualitative similarity to the HE₁₁ of a solitary nanowire. Most importantly, the apparent distortions of the field due to the proximity of the wires cannot be encompassed by considering a single nanowire alone. Therefore, investigating solely the HE₁₁ is suitable for obtaining a qualitative explanation of the absorptivity.

Incorporating all modes of relevant incoupling, their modal dispersion and correlation with the incident field yields a quantitative understanding of the absorption process in the nanowire. The good agreement of the 3D-FEM and 2D waveguide modal analysis [17] for the optimum geometry is illustrated in Fig. 1 (solid and dashed green curve).

We want to emphasize that even thin nanowires can absorb superbly given their length is increased because there is little reflection at the array-to-free-space-interface. The length of the nanowires is, however, limited by fabricational feasibility and making use of the carriers generated along a very long nanowire is yet another difficult issue as noted previously when we commented on our selection of the nanowire length.

The result of the absorptivity analysis of a wide range of geometries is illustrated in Fig.2. Every marker in the plot symbolizes an optical absorptivity FEM calculation such as illustrated in Fig.1 subjected to the post-processing described by Eq. (5).

Fig. 2 Normalized short-circuit current of InP nanowire solar cells for various diameters at various area fill factors and reference thin-film normalized short-circuit currents (markers on the right). The nanowires are arranged in square array, common height of nanowires in all designs is 2000nm. For constant geometric fill factor the usage of InP is constant. The device constitutes a photonic device, the absorptivity cannot be predicted by effective medium considerations based on geometric aspects alone. Absorption enhancement leads to decreased material use compared to thin-film. Refer to Fig.1 for the plot of a few absorptivity curves serving as the input for the analysis depicted herein.

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In Fig.2 the short-circuit current is normalized to the photo current generated assuming perfect absorption described by Eq. (6). The normalization is applied across the entire unit cell to account for a nanowire solar cell of macroscopic size to allow direct comparison to a planar thin-film solar cell, i.e. the normalized photo current cannot exceed unity. For our device, increasing the nanowire diameter beyond d₀ = 180nm corrupts the photo-generated current (as well as the efficiency as calculated in the second part) as it either causes more reflection at the top of the array and/or changes the dispersion characteristics reducing the favorable absorption enhancement at high wavelengths.

Increasing the density of an array of nanowires featuring an optimal diameter, leads to increased reflection at the top interface of the nanowire array. The normalized short-circuit currents of a few perfectly AR-coated InP thin-film cells are plotted on the right at f_NW = f_TF ≡ 1. The figure highlights a few remarkable observations:

As the nanowire height is constant for all simulations, the material usage per macroscopic area of such a solar cell is constant at every point of the abscissa. A design featuring nanowires of 20nm diameter exhibits only approximately one third of the normalized short-circuit current of a design based on 180nm diameter at the same material usage (geometric fill factor 0.196). The normalized short-circuit current of the 20nm diameter design is 0.76 at the densest packing of $\frac{π}{4}$ , the 180nm design equalizes this value at a significantly sparser array (geometric fill factor of 0.06). It is remarkable that for a wide range of geometries the normalized short-circuit current at the densest packing is virtually invariant with regard to the diameter. Also it is worth noting that for diameters exceeding 100nm increasing the spacing can lead to an increased efficiency limit up to a certain sparsity when efficiency decreases again due to the insufficient absorptivity of the array.

Within our design space, we identify the (a₀ = 360nm, d₀ = 180nm)-geometry as the design featuring highest normalized short-circuit current (η_sc = 0.92). To equalize this figure, a perfectly AR-coated layer of 760nm thickness, i.e. 1.94 times more InP material is required. For equal material use compared to f = 0.196 the thin-film solar cell merely offers a normalized short-circuit current of 0.82. This efficiency limit can be achieved by a (a₀ = 400nm, d₀ = 140nm) nanowire array with only a fraction $\frac{0.096}{0.192} = 0.50$ of the 392nm thin-film device InP consumption.

This data highlights the merit of nanowire devices even under the bias that the thin-film reference assumes perfect AR-coating while the optical properties of the nanowires are modeled rigorously including reflection.

3. Efficiency analysis

In contrast to photo diodes which are operated at passive operational points and their pn-junctions are highly reverse biased to extract the carriers generated by the incident light in a fast way, solar cells are operated in active operational regimes. This difference is significant as radiative recombination is an inevitable effect that occurs at the optimum operational point of maximum terminal power output. As the optimum operational point normally features a current which deviates only by a few percent from short-circuit operation only a fraction of the carriers generated do recombine in a radiative way, however, this process determines the terminal voltage and ultimately the efficiency.

In the previous section, we discussed the absorption enhancement due to microscopic dielectric features of the nanowire array. As a consequence of reciprocity, a complete study of the device also needs to include the emission enhancement of the radiative recombination caused by the photonic properties of the array. Due to our observation of an effective nanowire aperture exceeding its physical footprint in our absorption analysis we assume that the dual process of emission is also enhanced for the same direction, i.e. in the direction perpendicular to the array plane.

For geometric optical concentrators such as a Fresnel lens, which also increases the aperture of a conventional solar cell, the discussion of emission enhancement is trivial: The Fresnel lens enlarges the solid angle from which solar irradiation is incident upon the solar cell while it does not change the rate of radiative recombination as it does neither suppress nor enhance emission. Light emitted by a solar cell with a lens at the front is also subjected to the refraction of the lens. For reasons of reciprocity not only the solar cell receives more light from the sun but also the sun receives more light emitted by the solar cell. These circumstances are leading to increased efficiency [24,28–31].

In case a nano-structured device, enhancement of the radiative recombination, i.e. the Purcell effect, can lead to decreased efficiency. Therefore its discussion is imperative and device performance cannot be assessed by its absorptivity alone. These considerations apply to all devices featuring absorption enhancement induced by near-field concentration [16,32,33] or plasmonic scattering [34–36].

The number of photons emitted by an ideal photovoltaic cell at chemical potential Δμ for photons exceeding the bandgap energy at a cell temperature T₀ per solid angle dΩ per area per energy dE at a location on the surface reads [37]

N (E, x) = \frac{c}{4 π} \frac{ρ (E, x)}{exp (\frac{E - Δ μ}{k_{B} T_{0}}) - 1} d E d Ω

where ρ(E,x) is the local photon density of states (LDOS) at the location x and Δμ = q_ev is the chemical potential [38]. With zero chemical potential Δμ = 0 Eq. (9) yields Planck’s law of radiation [37, 39].

With non-zero chemical potential at the ideal operational point, we conclude from Eq. (9) that most of the photon emission occurs near the wavelength of the InP electronic bandgap, i.e. near λ = 920nm free-space wavelength.

In material with unity relative magnetic permeability, the LDOS can be deducted from the trace of the imaginary part of the dyadic Green’s function of the electric type, which relates the vectorial current of a Hertzian dipole at x to the resulting vectorial electric field at x′ [40, 41].

ρ (E, x) dE = - \frac{2 π E}{h^{2} c^{2}} ℑ (Tr (G^{E} (x, x, E))) d E

For free space the LDOS yields [40]

ρ_{vac} (E) d E = \frac{8 π E^{2}}{c^{3} h^{3}} d E .

ρ (E, x) d E = β (E, x) \frac{8 π E^{2}}{c^{3} h^{3}} d E .

In a photonic device governed by near field optics the spectrally and spatially resolved LDOS can deviate from the behavior of bulk dielectric. It can lead to inhibited emission at certain locations due to the unavailability of photon states as formulated by Yablonovich in 1987 as well as to emission enhancement reported by Purcell in 1946 [42]. The Purcell effect is described by the ratio β (E, x) of the LDOS observed at point x in the specific dielectric structure and the constant free-space LDOS.

β = \frac{ρ (E, x)}{ρ_{vac} (E, x)}

Thus, the Purcell factor can be interpreted as a normalized LDOS. In an analogous way, a normalized projected LDOS (PLDOS) can be obtained for each xyz dipole orientation by contracting the dyadic Green’s function with the corresponding dipole momentum [40].

\begin{array}{l} β_{i} (E, x) = - \frac{2 π}{k} ℑ (e_{i}^{T} \cdot G^{E} (x, x, E) \cdot e_{i}) with i \in x, y, z \\ β (E, x) = β_{x} (E, x) + β_{y} (E, x) + β_{z} (E, x) \end{array}

The spatially and spectrally resolved emission enhancement β (x, E) scales the rate of radiative recombination [43, 44] and can be incorporated into a microscopic analysis of the carrier transport, which includes spatially resolved radiative and non-radiative recombination.

Both micro-concentration and emission enhancement can be incorporated into the detailed balance analysis as micro-concentration scales the current and emission enhancement scales the photo-generated current of radiative recombination in the diode equation-representation of the Shockley-Queisser analysis [29, 30].

i (v) = I_{ph} (v) - I_{dark} (v) = I_{ph} (v) - I_{0} \cdot (exp (\frac{q_{e} v}{k_{B} T}) - 1)

Just as in the case of the carrier generation, the spatially resolved emission enhancement has to be reduced to as single scalar to fit into the zero-dimensional detailed balance analysis. We define a quantity β′(E) equal to the average of β (E, x) across a cross section of the nanowire near the top of the array. Other approaches such as to calculate an average weighted with the local power dissipation can be envisioned.

The photo-generated current is given by Eq. (5). The current density of radiative recombination can be calculated by [30]

j_{0} = \frac{4 π q_{e}}{h^{3} c_{0}^{2}} \int_{E = 0}^{E = \infty} β^{'} (E) \frac{E^{2}}{exp (\frac{E - qV}{k_{B} T} - 1)} d E

On the basis of the following assumptions: (a) Boltzmann approximation valid, (b) the emissivity of the structure for energies above the materials electronic bandgap energy E_g is unity and (c) the quantity β′(E) is constant within the spectrum of the radiative recombination, the expression can be simplified to [30]

j_{0} = \frac{4 π q_{e}}{h^{3} c_{0}^{2}} β^{'} k_{B} T (2 k_{B}^{2} T^{2} + 2 k_{B} T E_{g} + E_{g}^{2}) exp (- \frac{E_{g}}{k_{B} T})

For comparison with the thin-film cell the respective LDOS has to be computed. Incorporating the LDOS enhancement of bulk dielectric and the Snell’s law of internal reflection, which limits the solid angles within the semiconductor material from which photons can be emitted, the factor of emission enhancement from the planar thin-film cell reads [30]

{β^{'}}_{TF} = \frac{n_{t}^{2} + n_{b}^{2}}{2}

where n_t and n_b denote the refractive indices at the top and at the bottom of the absorbing layer, respectively. For our reference thin-film cell we assume n_t = 1.0 and n_b = 1.0. This assumption again constitutes a bias in favor of the thin-film reference: When calculating absorption, we assumed a perfect AR-coating, i.e. impedance matching. Yet when considering emission, we assume the top and bottom layer to be air, thus having strong total internal reflection.

In the case of the nanowire solar cell, micro-concentration increases the density of the photo-generated current by a factor of 1/f_NW so that for identical absorptivity of a nanowire and thin-film, the current density in the nanowire is increased.

At the same time, emission enhancement due to the Purcell effect, which manifests itself in the increased LDOS, leads to higher radiative current density. It is advantageous if the effect of non-unity geometric fill-factor is not diminished by the effect of emission enhancement. The former leads to smaller, the latter to higher recombination currents. In a direct comparison with a planar thin-film cell, if the condition

β^{'} f_{NW} < {β^{'}}_{TF}

holds, the efficiency limit of the nanowire solar cell can exceed the efficiency limit of the thin-film cell. This observation is analogous to the geometric optical case to either increase the solid angle from which the solar cell receives solar radiation or to decrease the solid angle into which the solar cell radiates the photons generated by radiative recombination.

Introducing the relation g = β′_NW/β′_TF provides a comparison between the thin-film and the nanowire device.

v_{TF} (i) = \frac{k_{B} T}{q_{e}} ln (\frac{I_{ph} - i}{I_{0}} + 1)

v_{NW} (i) = \frac{k_{B} T}{q_{e}} ln (\frac{I_{ph} - i}{g f_{NW} I_{0}} + 1) \approx \frac{k_{B} T}{q_{e}} ln (\frac{I_{ph} - i}{g f_{NW} I_{0}}) \approx v_{TF} (i) - \frac{k_{B} T}{q_{e}} ln (g f_{NW})

As implied by Eq. (17) and Eq. (18), the applied formalism is identical for both nano-structured as well as thin-film devices. In the case of planar thin film devices such as analyzed by [29, 30], the optical density of states (or in other words the spatially invariant LDOS) can be computed by integrating the per-solid-angle density of states of the active layer over a restricted solid angle as given by Snell’s law [29, 30] without the need of solving the source problem Eq. (2) for obtaining the dyadic Green’s function.

This aspect is rooted on the fact that the ray-optical Snell’s law and the ray-optical concept of total internal reflection are the ray-optical interpretation of the photon density of states and its wave-optical definition via the dyadic Green’s function. For more details concerning the fact that Snell’s law is a manifestation of the interaction of the interplay between optical states in two media, the reader is referred to literature (see [45], pp. 12).

Let us discuss the thin-film solar cell from the wave-optical picture to understand this analogy:

In the ray-optical picture radiative recombination can occur through rays of light, however, only photon states that result in rays of light that are not subjected to total internal reflection can be occupied in the radiative recombination process.
In the wave-optical picture, radiative recombination can occur through x-, y- or z-oriented Hertzian dipoles (or linear combinations thereof), however, only available photon states permitted by the PLDOS of the respective dipole orientation can be occupied in the radiative recombination process.

The analogy between the ray-optical and the wave-optical approach becomes apparent when investigating how the dipole orientation is related to the propagation direction of light rays: A Hertzian dipole emits electromagnetic power according to the well-known sinθ-shaped lobe. Most importantly, it does not emit any light in the direction of its orientation. In a ray-optical picture, a Hertzian dipole emits rays of light radially, weighted with the sinθ-shaped lobe.

In this context, z-oriented Hertzian dipoles can be identified as critical: For the thin-film cell, when light is incident from a small solid angle from the z-direction it is beneficial for efficiency to reduce recombination along z-oriented Hertzian dipoles as they exclusively emit power (or in other words, rays of light) to direction where no light is incident from.

Recombination along x- and y-oriented Hertzian dipoles is inevitable in the sense that due to reciprocity and Kirchhoff’s law of radiation, emission in the direction of the incident light does occur in case absorptivity at near-bandgap energies is present. The direction of the z-incident light corresponds with the maximum of the sinθ-shaped lobe of x- or y-oriented dipoles. Radiation of x- or y-oriented dipoles into solid angles from which no light is incident should also be reduced. Alternatively, concentrator optics can be used to redirect more emitted light towards the light source (including light radiated by z-oriented dipoles).

In the case of the thin-film cell the reduction of the photon density of states and the confinement of the solid angle of emission is achieved by total internal reflection. In the case of a nano-structured device forming a photonic crystal, the properties of the photon dispersion can be used to modify the LDOS: Dipoles oriented along the x- or y-axis (or linear combinations thereof) radiate power towards the top and bottom end of the array, furthermore, they radiate a TE-like wave along the nanowire array plane. The nanowires are of finite height, so there cannot be a perfect photonic bandgap and the dipoles — no matter how they are oriented — always can radiate some power into the free space. Even if the nanowires were of infinite length, the given structure (circular rods in square array) intrinsically does not exhibit a TE bandgap (see [46], p. 243).

In contrast to x- or y-oriented dipoles, the z-dipoles excite TM-like waves within the nanowire array. An array of infinitely long rods of our geometry does exhibit a TM-type bandgap (see [46], p. 243)), however as the nanowires are of finite length, light can always couple to free space modes. This circumstance is even more underlined by the fact that most generation occurs at the top.

3.1. Results

We computed the PLDOS along the lines of recent work [47] based on the solution of the source problem. We modeled the infinitely extended nanowire array by considering an array 5 × 5 unit cells where the unit cells next to the boundary of the computation domain are immersed into the PML. The effects of this artificial break of infinite periodicity, which we require to make our numeric computation feasible, can be estimated from literature [40] and we consider the calculation of the spectrally and 3D-spatially resolved PLDOS as sufficiently accurate.

More specifically, we derive this verdict from the published analysis of the cluster parameter N_C as defined in [40]. In a nutshell, N_C describes the number of unit cells in the quasi-periodic setup (see [40]). The data suggests that the LDOS converges fast and that when convergence is not sufficiently reached a slightly increased LDOS within the nanowire [40] is observed, which — applied to our case — would decrease the nanowire solar cell efficiency. The computational domain of the 3D FEM simulation for obtaining the spatially and spectrally resolved PLDOS and consequentially the LDOS as well as the electric field for various excitations is illustrated in Fig.3.

Fig. 3 Computational domain for the (a₀ = 360nm, d₀ = 180nm) geometry for the calculation of the PLDOS. The solutions for the source problem with dipoles located at the axis of the center nanowire array 100nm below the top. The space above the nanowires is not illustrated, the space between the nanowires is made transparent. The color is representing the absolute value of the real part of the electric field, the color bar is omitted due to the qualitative statement of the illustration. The length of the nanowires was shortened by 1000nm to decrease computational effort. The cluster parameter as defined in [40] is N_C > 9. A real refractive index of n = 3.5 was assumed.

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As we intend to compute the efficiency limit of a representative design within our parameter space, we choose the (a₀ = 360nm, d₀ = 180nm) geometry. For this design, we compute the PLDOS within the unit cell at various heights. Fig.4 depicts the result for a height 100nm below the top of the nanowire array which corresponds to a region where significant absorption occurs. Due to the C₄ symmetry of the unit cell, the x-PLDOS and y-PLDOS are identical with respect to a rotation of 90 degrees, therefore, Fig.4 shows their sum.

Fig. 4 Spectral PLDOS (normalized to the LDOS of free space) for the (a₀ = 360nm, d₀ = 180nm) geometry for various wavelength at 100nm below the top of the array separated for xy-dipole momentum giving rise to quasi-TE waves within the photonic crystal and z-dipole momentum agitating quasi-TM waves. The enhancement of spontaneous emission with respect to free space is given by the sum of all dipole momentum orientations. We observe an enhancement of approx. 2.5, partially degrading the ameliorating effect of micro-concentration. The length of the nanowires was shortened by 1000nm to decrease computational effort. The cluster parameter as defined in [40] is N_C > 9. A real refractive index of n = 3.5 was assumed.

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From Eq. (21) it is evident that for every power of e of the factor gf_NW the voltage is changed by one thermo-voltage. So for one order of the aforementioned product, the influence on the voltage and thus on the efficiency is negligible.

From our calculations depicted in Fig.4 we conclude that for the optimum geometry in terms of short-circuit current, the enhancement of the LDOS due to the photonic properties of the array is on the same scale as the micro-concentration 1/f_NW. We emphasize that the enhancement of the LDOS in the free space between the nanowires is of no relevance for device performance as carriers are confined to the nanowire and therefore no radiative recombination can occur in the void space between the nanowires.

For the optimum geometry, (a₀ = 360nm, d₀ = 180nm), we integrate the xyz-PLDOS over the cross section of the NW and obtain β′_NW = 2.48. With the geometric fill factor being fNW = 0.196, we do obtain gf_NW = 0.49 < 1. As a result, we observe a higher terminal voltage at the nanowire solar cell within our model, i.e. −25.9mV*ln(0.49) at same operational point (I_ph – i(v)).

We shall assume that absorption and emission enhancement neutralize for all geometries with respect to the radiative recombination, i.e. we assume gf_NW = 1. Note that this assumption does not affect the increased absorption per material. Instead, alludes to the aspect that the radiative recombination current is not decreased because the recombination current density is assumed to scale with 1/f_NW. For the optimum geometry, this is a cautious assumption.

The efficiency limit along the lines of the detailed balance analysis can be computed and illustrated in a similar manner as Fig.2. The result is shown in Fig.5.

Fig. 5 Detailed balance efficiency limit of InP nanowire solar cells for various diameters at various area fill factors, thermodynamic limits (dashed lines) and reference thin-film efficiencies (markers on the right). Nanowires (20nm ≤ d₀ ≤ 200nm) are arranged in square array, common height of nanowires in all designs is 2000nm. The dashed lines indicating the performance of planar, perfectly absorbing thin-film cells are calculated for concentrations of 1/f_NW.

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Under unconcentrated AM1.5d illumination, the optimum design of (a₀ = 360nm, d₀ = 180nm) exhibits an efficiency limit of 0.294, the same efficiency limit is achieved in a perfectly AR-coated thin-film cell of 770nm thickness, i.e. almost the double material use. A perfectly AR-coated thin-film solar cell of identical material use only attains to an efficiency limit of 0.258, this value is equalized by the (a₀ = 520nm, d₀ = 160nm) geometry and thus needs $\frac{0.744}{0.196} = 0.38$ of the thin-film material use.

The detailed balance limit assuming a perfectly absorbing thin-film for photon energies down to the InP bandgap predicts an efficiency limit of 0.316. At the same time, the tangible (a₀ = 360nm, d₀ = 180nm)-geometry (η = 0.294) virtually attains the same limit.

4. Conclusions

In summary, we report the absorption analysis of InP nanowire array solar cells for a wide range of geometries. Our simulations show two major results: a nanowire array solar cell can reach the efficiency of a nearly perfectly absorbing thin-film solar cell, using only 38% of active material. This results from a combination of near-field optics enhanced absorption and anti-reflection. Second, this result can only be achieved with a careful choice of wire diameter and distance, which needs to be optimized depending on the material, especially with regard to its bandgap energy where absorption enhancement has to be tailored to absorb well in the relevant spectrum.

Very promising experimental results showing improved absorption by nanowire array solar cell of similar geometry in contrast to a thin-film device were shown by Lu and Lal [7].

Our analysis shows that employing nanowires thinner than 100nm leads to suboptimal performance and cannot even be compensated by a dense packing and thus high material use. On the other hand the optimum absorptivity is achieved in a relatively sparse array with a geometric fill factor of approx. 0.2.

For the case of an InP single junction nanowire array cell, a nanowire diameter of 180nm and a spacing of 360nm yields highest short-circuit current. In this structures, intrinsic anti-reflective properties and absorption enhancement co-act, yielding high absorptivity with reduced material use. This observation cannot be explained by effective medium considerations based on the material composition alone, a rigorous solution of the electromagnetic wave equation is the sole way to determine the device performance limit.

The presented analysis can be performed for more complex geometries and material compositions as well. It is applicable at a very early design stage to evaluate the potential of a geometry regime.

Our numerical experiments suggest that the effect of emission enhancement due to the presence of nano-structures is of small importance for the optimally absorbing device in our design space. Given the same material usage, the absorption enhancement increases the efficiency significantly while the modification of the available photon states of the device does not deteriorate such efficiency enhancement effect. Nonetheless the computation of the LDOS is important to allow the calculation of an efficiency limit accounting for the optical properties with regard to absorption and emission with the same amount of rigor.

Acknowledgments

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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Light absorption and emission in nanowire array solar cells

Abstract

1. Introduction

2. Absorption analysis

2.1. Results

3. Efficiency analysis

3.1. Results

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Equations (21)

Optics Express