1. Introduction

Economically viable photovoltaic (PV) solar energy conversion systems have the potential to reform our energy system and reduce current problems associated with the use of fossil fuels.[1] One important route towards such a development, is to improve the light capturing ability of the PV material, so that thinner layers can be used without compromising light absorption. This would reduce the amount of expensive PV material, improve conditions for charge carrier collection and raise the efficiency by virtue of the concentrated energy density in the PV layer,[2] factors which together would increase the energy output per unit cost.

Noble metal nanoparticles support so-called localized surface plasmon resonances (or nanoparticle plasmon resonances).[3] The latter can readily be tuned into the ideal spectral range for PV applications by means of size, distribution, shape and local dielectric environment. The optical cross sections for light absorption and scattering resulting from these nanoparticle plasmons are extremely high and may extinct most of the incident light within layers, where the effective metal thickness (volume metal per unit area) is only a few nanometers. If the oscillator strength associated with these excitations could efficiently be coupled to useful electron-hole pair production in a solar cell layer, the thickness of present-day thin film PV layers (~1 to 10 µm) could be reduced by up to a factor of 10²[4] (based on the Thomas-Reiche-Kuhn sum rule[5]). This would pave way for a new class of solar cells, for which materials and concepts not applicable to thicker layers, could be viable. We have previously investigated systems, experimentally and theoretically, where either near-field (dominated by non-propagating field components)[4, 6] or far-field[7, 8] effects characterized the response.

In the present work, optimization of plasmon near-field enhanced absorption in a very thin photovoltaic layer is addressed, with particular emphasis on the role of the extinction coefficient of the PV material. It is demonstrated that the large thickness reduction indicated above is feasible. Several recent studies have discussed the optimization of plasmonic structures for photovoltaics,[9-12] but without analyzing and varying the properties of the photovoltaic material systematically, and with the main focus on far-field effects. In these studies and many of those preceding them,[6, 7, 13-15] the gain from the presence of plasmonic structures has often been quantified in terms of an enhancement factor, defined as the ratio between the photocurrent with and without the plasmonic structure. Although this number is of high interest for specific solar cell configurations, it leads to extremely thin, weakly absorbing PV layers in an optimization of the plasmon enhanced solar cell as a whole.[16] While in such case peak enhancement factors of 10²-10³ can be achieved, the useful absorption, contributing to the photovoltaic output of the PV layer, may still be very low and typically weak compared to the Joule heating losses in the metal nanoparticles. In other words it is possible to achieve a very high plasmonic enhancement factor which still results in a low absolute efficiency. To avoid this shortcoming one may instead consider a branching ratio, defined as the absorption leading to creation of charge carriers in the PV layer, divided by the total absorption in the system. However, this branching ratio is in itself not useful as a figure of merit, since it is maximized in the complete absence of plasmonic material and leads to conventional PV designs. As an alternative we therefore propose to use the plasmon induced quantum efficiency, defined as

as the figure of merit. Here, Q is the quantum efficiency (‘number of electrons out per photons in’) of the solar cell with a plasmonic structure, and Q_ref is a reference quantum efficiency for the same solar cell structure but with the plasmon active metal part removed (see Fig. 1). ΔQ can be thought of as the fraction of incident photons that, because of the plasmon resonances, contributes to (or are subtracted from, if negative) the photocurrent. It reaches unity, that is 100% efficiency, if all incident light is absorbed and creates one useful electron-hole pair in the PV layer per absorbed photon (Q=1), and if all absorption occurs via the plasmon resonance only (Q_ref=0). In order to develop an understanding and to simplify the separation of various contributions, we will here limit the optimization to the peak value of ΔQ at a particular wavelength. As will be shown, this approach is still useful, since the plasmon resonances attain a significant spectral width even under such an optimization. Since PV devices typically suffer most from a lack of absorption near the band gap threshold, which is ideally at 1.34 eV (≈ 920 nm wavelength) for a single band gap solar cell,[17] we target a wavelength band centered at 900 nm as our exploratory example. It is further assumed that the solar cell has an ideal response in the sense that the fraction A of incident light absorbed in the PV absorption layer, results in useful charge carriers with unit quantum yield (Q=A), so that the plasmon induced quantum efficiency ΔQ [Eq. (1)] equals the plasmon induced absorption,

where A_ref is the absorption in the reference PV layer. Under this condition, the photocurrent enhancement factor becomes A/A_ref, and the branching ratio Γ=A/(A+A_met). In the latter expression A_met is the absorption in the metal part, which we assume to be lost as heat.

 

Fig. 1. Representations of the simulated metal (grey) plasmonic nanoparticle arrays, in contact with the PV absorbing material (blue), the latter in the form of shells around or as a layer underneath the plasmonic particles. A plane wave is incident along positive x, with its E-field in parallel with z. The boundary conditions simulate infinite periodicities Λ_y and Λ_z in the y-and z-coordinates, respectively. The particles are spheroidal or hemi-spheroidal, with major semi-axis a along z, b along y and minor semi-axis c along the propagation direction x. Unless otherwise stated the metal particle volume is 25600 nm³, corresponding to an effective thickness of 1 nm for 160 nm average particle spacing. In (a), each metal particle constitutes the core of a core-shell structure, where the shell is absorbing. The shell is defined by a spheroid having semi-axes lengths a+d, b+d and c+d, in the z, y and x-directions, respectively. The reference for this system is an identical array, but with the metal cores replaced by the same absorbing material as in the shells. In (b), hemi-spheroidal metal particles are placed on top of a planar absorbing layer of thickness d. The reference for this system is obtained by removing the metal.

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We examine the plasmon induced absorption for two-dimensional (2D) arrays of both core-shell structures and particles on top of planar PV layers (Fig. 1). The parameters investigated include the optical constants of the PV material and surrounding media, the PV layer thickness, the particle shape, size and 2D particle density and arrangement. In our first model system, we consider 2D arrays of spheroidal metal cores embedded in absorbing PV shells, as shown in Fig. 1(a). Such core-shell structures are of particular interest for dye sensitized solar cells[18] (where a dye sensitized metal oxide semiconductor shell would cover the metal core) and related solar cell concepts, where the metal cores may also be envisioned to serve as electron conduction paths.

2. Methods

In the calculations, either the optical constants of Ag obtained from [19] are employed, or a loss-free metal representing an ideal case. Ag is chosen because it has plasmon resonances of very high quality factors for visible light to near-infrared frequencies.[20] Since the particle sizes typically exceed the electronic mean free path (~50 nm), we do not correct for surface scattering effects.[15] All media are assumed nonmagnetic. For the core-shell system, the absorbing shell is chosen to have a complex refractive index N=n+iκ (in the following we refer to its real part simply as the refractive index), while the dielectric environment is described by a single real refractive index n_e. As a reference system, the same array is considered but with homogeneous particles having the optical constants of the shell throughout. The reference absorption is then obtained by integration over all absorbing material. The procedure for the layered systems (Fig. 1b) were similar and is described in further detail in connection to the results (section 3).

The extinction coefficient κ at the primarily targeted wavelength λ₀=900 nm is typically varied from a small value (10^-3, which is less than for crystalline Si[21]) up to at least unity (higher than for typical organic solar cell films[22-24] and CIS[25]). We choose a wavelength independent absorption coefficient (α=4πκ/λ) for the PV material itself, by taking κ(λ)=κ(λ ₀)λ/λ ₀ with λ ₀=900 nm. Although real PV materials typically have a wavelength dependendent absorption coefficient, this choice allows for a clear distinction of the influence of the plasmon resonance on the absorption, and may be used for a first estimate of quantities such as peak width even when α depends on λ.

The finite element method (FEM) was used to solve the vector wave equation for the scattered electric field in a symmetry reduced geometry, by means of software from COMSOL according to the procedures described in [7]. The results of the model were tested successfully against analytically solvable cases (multilayer reflectance) as well as experimental results.[7, 26] In a typical series of calculations, the Helmholtz wave equation was solved over a range of wavelengths for the scattered electric field. Data, such as absorption in different regions and energy fluxes across boundaries, were extracted by integration of the relevant field quantities. While keeping the particle volume fixed, the particle eccentricities were then adjusted iteratively to position the plasmon absorption peak for a small κ at a wavelength of 900±0.5 nm. Two values of the refractive indices in the film and the surroundings were primarily investigated, of 1.5 and 3. The former is of higher relevance for organic semiconductors and dye sensitizers, while the latter is more typical for solid semiconductors. For oblate spheroids, as depicted in Fig. 1(a), the major semi-axes ended up at a=42±3 nm for n=n_e=1.5, and at a=24±2 nm for n=n_e=3. The array constants were chosen to avoid the lowest order grating modes at 900 nm wavelength.[27] Subsequent variation of the extinction coefficient κ of the PV absorption layer was then performed without further adjustments of the particle shape. This procedure was justified, since variations of κ only caused minor shifts of the plasmon peak position (<10 nm) up to a point, typically on the order of κ=0.1 or higher, above which the plasmon peak smeared out completely. This is characteristic of a forced, damped harmonic oscillator, for which the resonance has a fixed peak position given by the natural, undamped resonance frequency. The peak disappears when the quality factor decreases below 2^-1/2.[28] For reference, critical damping occurs at a quality factor equal to 1/2, at which point no plasmon resonance exists. The position and amplitude of maxima in ΔA were finally estimated by cubic spline interpolation between data points obtained for discrete values of λ and κ. The associated error was estimated to be below 1% for the peak height.

3. Results

3.1 Ideal core-shell particles

It is instructive to first consider an ideal case, where the metal core is loss-free with a purely real dielectric constant taken to equal the real part of the Ag dielectric constant. Fig. 2 shows the results of calculations for square arrays of such core-shell particles, where the particles have a fixed core volume of 25600 nm³ and a shell thickness d=10 nm. The refractive indices of the environment and shell are chosen to be n_e=n=1.5, respectively.

 

Fig. 2. Absorption A in a PV shell, plasmon induced absorption ΔA, far-field transmittance T and reflectance R for arrays of core-shell particles. The array constants are Λ_y=Λ_z=480 nm in (a) and (b), Λ_y=Λ_z=160 nm in (c) and (d), and Λ_y=Λ_z=110 nm in (e) and (f). The dependence on the shell extinction coefficient κ at the plasmon resonance is shown in the left column [(a), (c) and (e)] for a fixed plasmon resonance wavelength of 900 nm. The dependence on wavelength is shown in the right column, for the extinction coefficients maximizing A in the left column. These values are κ=0.014 in (b), κ=0.14 in (d) and κ=0.27 in (f), respectively.

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From the left column of Fig. 2, it is clear that the shell extinction coefficient κ strongly affects the absorption in the PV material, and that there are pronounced maxima for particular values of κ. The optical extinction is at maximum for low κ, where the shell absorption A is low but the reflectance approaches 100%. This is in line with recent predictions for perfectly conducting cylinder arrays,[29] and also the case for perfectly conducting planar films. When increasing κ, both absorption in the shell and transmittance increases initially, at the expense of reflectance. The absorption A then reaches a maximum of close to 50% for an extinction coefficient at which the transmittance and reflectance are both close to 25%. The value of 50% for the maximal absorption is not a coincidence, but a general upper limit for the absorption of normal incident light in these type of systems, as was recently shown for ordered cylinder arrays[30] and 2D nanoparticle arrays.[31] A corresponding restriction can further be derived for homogeneous thin films in the zero thickness (2D) limit,[32] which therefore applies to any thin film system (including disordered) that can be described by effective medium theory. Thus, for normal incident light and systems characterized by small length scales compared to the wavelength, the plasmon near-field induced absorption ΔA=A-A_ref, and hence our figure of merit ΔQ [see Eq. (2)], are limited to 50%. However, it is important to note that additional light management in the far-field regime or 3D arrangements of the particles are viable options to increase the plasmon induced absorption towards 100%. One may for instance place a reflector at a distance (~1/4 wavelength) behind the layer, in order to achieve a constructive interference maximum in the layer position,[33] or use scattering/light trapping structures.[34] As discussed further for the planar PV layer system presented below, the use of a high refractive index medium (and an anti-reflective coating) in front of the PV structure will also improve the situation.

In contrast to the absorption A, which peaks close to 50% for all particle spacings considered in Fig. 2, the ideal plasmon induced absorption ΔA is maximized for the lowest particle density. The reason is that when the particle distances are reduced below the typical near-field length scale (approximately twice the particle diameters[35] or λ/2π, depending on which is larger), the interparticle interactions result in stronger coupling to radiation.[27, 35] Not surprisingly, increasing the particle size has the same effect; light scattering scales with the square of the particle volume for small particles[36]. To compete effectively with such an enhanced radiative damping, a higher shell extinction coefficient is required and the optimal extinction coefficient κ=κ_o at which ΔA is maximized thus increases, as seen in the left column of Fig. 2. Conversely, the optimal extinction coefficient is roughly proportional to the particle core volume for otherwise unchanged conditions (not plotted). However, when higher values of κ are required to maximize A, the absorption A_ref in the reference system also increases at the optimum. This reduces the peak value of ΔA slightly and leads to the observed advantage of low particle densities and small particle sizes. On the other hand, an important effect of more radiative damping and more damping in the shell at the increased κ_o is an increase of the peak width (Δλ, a measure of the total damping of the plasmon resonance[36]), as seen with increasing particle density in the right column of Fig. 2. This effect is useful for covering a broader range of the solar spectrum.

3.2 Realistic core-shell particles

Having discussed the ideal case, we now turn to more realistic systems where the full dielectric function of Ag[19] is employed for the particle core. This complicates the situation in the sense that the metal dissipation opens an additional relaxation channel so that the plasmon resonance may decay via Joule heating in addition to useful absorption and (inevitable) emission. The effect of the Joule heating is minimized if conditions are such that the two other damping channels dominate, which means that coupling to radiation must be strong. This is achieved for a higher particle density (or larger particle size), but the increase of κo associated with this situation again has a negative impact on ΔA through the increased reference absorption. Therefore, an optimal particle density results, as can be seen from the calculations presented in Fig. 3(a) and (b). The optimal particle spacing is found close to the characteristic near-field extent (~4a), but it should be noted that the dependence is quite flat around this optimum, especially on the high particle density side. The maximum ΔA exceeds 40% and corresponds to an absorption enhancement factor of about 100.

In Fig. 3, results for particle distributions deviating from square arrays are also included. Individual variation of either array constant Λ_y or Λ_z, leads to modifications of the interparticle couplings (more constructive or destructive in nature), and to different shifts of the plasmon resonance wavelength.[37] However, after compensating the wavelength shift by adjustments of the particle eccentricity, the effect of asymmetry on the plasmon induced absorption is seen to be very weak (plots of common color have the same particle density).

The metal filling fraction is thus the main characteristic of the particle distribution for determining its response, as expected for a system well described as an effective medium.[36]

 

Fig. 3. (a) Plasmon induced absorption for arrays of core-shell particles, in the metal cores (A_met, +-symbols) and absorbing shells with d=10 nm (ΔA, no symbols), as a function of κ. The refractive indices of the dielectric environment and the shell are n_e=n=1.5, respectively, and the array constants (Λ_y, Λ_z) are varied as shown in nm units. In (b), the peak values ∇A(κ _o) and the optimal extinction coefficients κ _o, are shown as functions of the average particle spacing (Λ_yΛ_z)1/2. The data for (b) were obtained by using the spline interpolated curves shown in (a). The lines in (b) are drawn between data points from the square arrays (Λ_y=Λ_z), but data points for the asymmetric arrays are also included (see symbols).

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So far, the PV shell thickness has been held fixed at d=10 nm. In Fig. 4, the influence of this parameter and the particle shape are explored. The shell thickness primarily affects the optimal extinction coefficient κ_o, which we find is to good approximation given by an expression of the form

where d′ and κ _o,inf are constants, see Fig. 4(b). There is no marked deviation from this relation in the range of d values investigated, although it is expected to fail when different shells overlap or when thin film interference effects begin to have an influence (for d~λ/4n). Eq. (3) demonstrates a useful possibility to tune κ_o without significantly affecting the balance between the different damping channels (dissipation in the metal, emission and useful absorption).

 

Fig. 4. (a) Plasmon induced absorption in the metal cores (A_met, with +-symbols) and in the PV shells (ΔA, no symbols) for arrays of core-shell particles with Λ_y=Λ_z=250 nm. The refractive indices of the dielectric environment and the shell are n_e=n=1.5, respectively, and the shell thickness is varied. The solid lines are for oblate particles (semi-axes a=b>c), and the dashed lines are for a prolate particle (semi-axes a>b=c) of the same core volume, oriented with their longest axis in the E-field direction. (b) The optimal value κo of the extinction coefficient is shown as a function of the inverse PV layer thickness d, interpolated from the data shown in (a) for the oblate core-shell structures, and for the data of planar PV layers presented in Fig. 7. The inset equations are used for the fitted lines.

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Thus, by adjusting the shell thickness, close to optimal conditions can be reached for a given photovoltaic material, within certain limits. For the core-shell particle array of Fig. 4(b), d'=0.44 nm and κ_o,inf=0.026. Thus for d=a (≈ 42 nm), κ _o is within a few percent of the asymptotic value of 0.026 for large d. In the other extreme, we note that a 1 nm thick shell would require κ=0.47 for optimal damping, which may lie within reach for some molecular absorbers;[18] a 1 nm thick monolayer of dye, absorbing 1% of the incident light at 900 nm, corresponds to κ=0.72. However, the validity of this classical calculation would clearly have to be evaluated for the latter case. For example, a reduced screening of the interaction between the metal and an external absorber on this length scale has been suggested to lead to stronger coupling and higher enhancement factors.[38]

A comparison with a prolate spheroid of the same core volume (a≈51 nm, b=c, d=10 nm) is also included in Fig. 4(a) and shows that particle shape has a very weak influence on the maximal value of ΔA, and a relatively small effect on κ _o. The slight decrease of κ _o observed is attributed to a more efficient coupling to the prolate shell, which actually contains less material than the oblate. Because the local absorption rate is to good approximation proportional to the E-field energy (proportional to |E|²), a PV shell shape bounded by an E-field energy isosurface maximizes the coupling to the material. This general feature becomes clear if considering a deviating shape; it is then possible to move material into regions of higher |E|² and hence increase the absorption rate locally. A lower value of the optimal extinction coefficient then results. For plasmon resonances dominated by electric dipole contributions, as is the case here, the |E|²-isosurfaces have bulbs close to the poles, see Fig. 5. The shell surrounding the prolate spheroid therefore comes closer to the ideal than the shell in the oblate case, since the mass of the former is more distributed towards the poles. Thus, our previous conclusion[7] that plasmon particle shape is important mainly to the extent that it affects the coupling to the PV layer, is here confirmed and accentuated. It is also quite clear that a distribution of the PV material bounded by an |E|²-isosurface around the plasmonic structure, is ideal for minimizing the amount of PV material required to reach an optimal damping, and for maximizing the enhancement factors for these thin PV layer thicknesses.

 

Fig. 5. Cuts through core-shell particles (x-z plane) showing the distribution of absorption rates in the Ag cores and in (a) a weakly absorbing PV shell and (b) a PV shell with a close to optimal extinction coefficient (giving ΔA>40%). The conditions are otherwise as for Fig. 3, with symmetric particle spacings of 160 nm. The streamlines show the associated Poynting vector energy flow.

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We now turn to the influence of refractive indices in the core-shell system, see Fig. 6. Matched indices (n=n_e) are seen to produce the highest plasmon induced absorption in the shell, which can loosely be attributed to minimized scattering imposed by the external shell boundary. Further, the lower matched refractive indices investigated leads to the highest ΔA and also to the highest ratio between the peak ΔA (at κ _o) and the maximal absorption in the metal (at low κ). The influence of refractive indices are discussed in further detail in connection to the results for planar PV layer systems presented next.

 

Fig. 6. Plasmon induced absorption in the metal core (+-symbols) and PV shell for a shell thickness fixed to d=10 nm, and varying refractive indices according to the (n_e, n) pairs indicated. Conditions are otherwise as in Fig. 4.

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3.3 Ag-hemispheroids on a planar PV layer

We now turn to the case of plasmonic particles placed on top of a planar absorbing PV layer, see Fig. 1(b). This is a system of high practical interest as it represents a straightforward implementation of nanoparticle plasmon enhanced photovoltaics, which may readily be further optimized with respect to the refractive indices immediately in front of and behind the PV layer. We assume hemi-spheroidal particles of fixed volume in contact with the absorbing PV layer at x=0. The PV layer extends to x=d and has a complex refractive index N=n+iκ. A substrate of real refractive index n_s is assumed for x>d, and a medium with a real refractive index n_i for x < 0. The reference system is taken as the layered structure without particles. When n_i=n=n_s=1.5, a resonance wavelength of 900 nm is again achieved for a semi-axis length of a≈42 nm.

 

Fig. 7. Plasmon induced absorption in an array of metal hemi-spheroids (A_met, with +-symbols) and in the planar PV layer underneath (ΔA, no symbols), as a function of the PV layer extinction coefficient. The array constants are fixed at Λ_y=Λ_z=250 nm. In (a), the refractive indices are n_i=n=n_s=1.5, and the layer thickness is varied according to the inset. In (b), the layer thickness is kept constant at d=10 nm, and the refractive indices are varied according to the triplets (n_i, n, n_s) as indicated.

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Many of the trends observed here are strongly reminiscent of the core-shell case. Fig. 7(a) illustrates that there is again a clear optimal extinction coefficient κ_o, which can readily be tuned by the layer thickness as shown in Fig. 4(b). However, the maximal ΔA now increases somewhat with decreasing layer thickness d, and appears to only asymptotically approach the same level (about 37%) as that found for all core-shell structures considered in Fig. 4(a). This means that materials with a high κ have an advantage over less absorbing materials in this situation. Further, the optimal extinction coefficient is generally higher than for the corresponding thickness in the core-shell cases, due to a less efficient coupling between the nanoparticles and the PV material. Yet, even with a PV layer as thin as 2.5 nm, the value of κ_o (≈0.27) is within reach of existing semiconductors (κ=0.44 for CIS at 900 nm wavelength[25]) in the example of Fig. 7(a). For a close to optimal particle spacing of Λ_y=Λ_z=160 nm and a layer thickness of d=10 nm, ΔA peaks above 40% for κ_o=0.22 (not plotted here).

The influence of the refractive indices, represented as the triplets (n_i, n, n_s), are explored in Fig. 7(b). The main effect on the peak value of ΔA may be summarized by considering three categories. The first is a symmetric category with n_i=n_s, which produces the four intermediate peak values of ΔA. Among these four cases, the lowest, matched values (1.5,1.5,1.5) yields the best results, as in the core-shell system. A second category is the asymmetric cases with n_i<n_s, which produces the two lowest peak values of ΔA. Finally, the two asymmetric cases with n_i>n_s results in the highest values of ΔA, in particular for the triplet (3,1.5,1.5). Considering the constructive interference between the internally reflected field and the incident field at the back of a high refractive index medium, and the opposite situation when light is incident upon a higher refractive index medium, these results are not too surprising. They are also in line with what is found when replacing the plasmonic particles and PV layer by a very thin, effective medium layer. The maximum absorption in the layer is then given by n_i/(n_i+n_s), and may thus exceed 50%.[32] On the other hand, to take full advantage of this for photovoltaic applications, one must typically (that is, when n_i significantly exceeds unity) minimize reflection losses at the front interface of the solar cell by, for instance, anti-reflective coatings.

4. Summary and discussion

We have proposed to use the plasmon induced quantum efficiency as the figure of merit for optimizing plasmon enhanced absorption in photovoltaic systems. We investigated the influence on this quantity of all relevant parameters in extremely thin, 2D metal-core/PV-shell arrays and in metal hemi-spheroid arrays on top of planar PV layers. In systems where the near-field dominates the response, an optimal useful damping by the absorbing PV material is established from a balance between absorption and radiation/emission. In the most simple case when the thin PV layer is surrounded by equal dielectric media, we confirm a 50% limit for the total absorption, which ultimately limits the useful plasmon induced absorption. When the plasmon resonance is “under-damped” by the PV layer, reflection losses are most important, while transmission losses are higher in the opposite situation. The PV layer extinction coefficient is the most direct way to control the useful damping (at least in theoretical studies), and it results in a pronounced maximum for the PV layer absorption when transmission and reflection losses are close to equal. In a more general asymmetric situation, the maximum absorption is given by n_i/(n_i+n_s).[32] When this is exploited together with anti-reflective layers at the front surface, and/or appropriately positioned reflective layers behind the structure, it is possible to approach 100% absorption without increasing the thickness of the PV layer.

We have demonstrated how the optimal value of the extinction coefficient can be tuned to the PV material at hand by means of the geometry, in particular by the PV layer thickness. In addition, although particle shape only has a weak influence on the maximum plasmon induced absorption as such, it does affect the coupling to the PV absorption layer. Distributions of PV material bounded by an E-field energy isosurface are argued to be ideal in the sense that they maximize the coupling between the plasmon near-fields and the PV material. A core-shell structure can come close to this ideal in the case of dipole like resonances, and may be employed for dye-sensitized solar cells and related concepts.

There is also a weak influence of particle arrangement on the near-field coupling strength to the PV material. Meanwhile, the branching ratio between metal absorption and other damping channels (useful absorption and emission) stays virtually unaffected when comparing the responses for different shapes and arrangements at the optimal extinction coefficients and a fixed plasmon resonance wavelength. The latter behaviour is to be expected in the quasistatic limit,[20] but less evident for the present case. It suggests that the response can be well understood in terms of effective medium theory, where the amount of metal on the surface (the filling factor) is the parameter of primary importance.[36] Previous experimental observation[7] of a lack of angular dependence other than that expected for a thin anisotropic (but homogeneous) film gives additional support to the effective medium representation. The details of this connection is further investigated in a forthcoming publication.[32]

An optimal particle density is identified, which corresponds to an average spacing similar to the plasmon near-field length scale. In the case of a symmetric dielectric environment, the best performance is further achieved when the photovoltaic material refractive index is matched to the external medium. For favorable parameter values, a plasmon near-field induced absorption of close to 40% is predicted for both the core-shell and the planar geometries in this situation. The plasmon induced absorption stretches over a wavelength range (full width at half height) of about 200 nm under these conditions. Because the peak plasmon induced absorption at the targeted wavelength is a quite weak function of particle density close to the optimum, the spectral width can readily be increased without much loss in peak absorption. This is done by increasing the amount of plasmonic material in the system, and by compensating for the associated stronger coupling to radiation by increasing the useful damping of the plasmon resonances. For a particular PV material, the latter can be achieved by increasing the thickness of the PV layer or by improving the coupling between the PV layer and the plasmonic structure. A quite significant part (perhaps all) of the spectrum targeted for solar cell applications appears possible to cover in this way, especially for highly absorbing PV layers based on materials such as molecular absorbers, organic semiconductors, CIS/CIGS materials or III-V semiconductors.

To demonstrate the concept investigated in this study for an existing PV material, a calculation for a planar, 10 nm thick CIS layer[25] is presented in Fig. 8. Because some refractive index difference between the incidence and supporting medium is feasible (for instance resulting from the use of a transparent conducting layer on one side), we here consider a favorable situation where n_i=3 and n_s=1.5. After optimization of the particle spacings, we find a plasmon induced absorption peaking at 49%, and a total absorption (A +A_met) close to the thin film limit of 67% for this system. The peak width is then about 240 nm. A broader peak could be achieved by increasing the amount of Ag (here effectively 1.8 nm thick) and the useful damping by means of a thicker CIS layer and/or by embedding the particles in the CIS layer.

There are several important challenges to an efficient realization of plasmon near-field enhanced photovoltaics. One is to avoid the introduction of recombination channels via charge carrier trapping at the metal nanoparticles. Part of the solution to this may be to deploy the plasmonic nanostructures for the additional purpose of charge collection (by having them in contact with a conducting layer), so that efficient charge carrier separation can be achieved. Appropriate energy level engineering is then required. Encapsulation of the metal particles in thin insulating material is another option,[15] although the near-field coupling strength will then be reduced. A second challenge is the typically irrecoverable heat loss associated with direct absorption in the metal particles. We have chosen to study a favorable situation with the optical constants of bulk Ag representing the plasmonic material. Higher dissipation losses may be expected with more dissipative materials and in real systems, where grain boundaries, surfaces, defects and impurities will contribute to additional, unfavorable, damping. A too weak coupling between the plasmonic structure and the photovoltaic material would also lower the attainable branching ratio.

 

Fig. 8. Absorption (A in PV layer, A_met in particles) and other quantities (far-field transmittance T and reflectance R) for a 10 nm thick planar CIS layer with a square array of Ag hemispheroids placed on top. The hemi-spheroid volume was taken to the value used elsewhere in this study (25600 nm³), and the particle eccentricity was adjusted to position the resonance at about 900 nm wavelength. Refractive indices of n_i=3 and n_s=1.5 were assumed in front of and behind the film, respectively. The array constants were chosen to Λ_y=Λ_z=120 nm, which roughly maximize the plasmon induced absorption ΔA for this system. The reference absorption A_ref is included for a clear comparison.

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5. Conclusions

We have demonstrated a high potential for inducing absorption in extremely thin photovoltaic layers by means of plasmonic nanoparticles. The results show that the useful absorption can come close to a general upper limit applying to very thin effective medium layers. For instance, a plasmon induced absorption of 40% is possible to realize in the case of a symmetric environment where the theoretical limit is 50%. Higher absorption can be achieved in an asymmetric situation, for instance close to 50% in a CIS layer placed between a front and a back medium of refractive index equal to 3 and 1.5, respectively, provided that appropriate measures are taken to eliminate reflections at the front surface. The solar cell layer thicknesses investigated are on the nanoscale, on the order of 100 times thinner than conventional thin film solar cells. This means that this concept has a high potential to reduce material amounts and production costs, to increase the theoretical photovoltage (which to some extent compensates for lost absorption) and to enable the use of new materials and concepts not applicable to thicker films. These types of ‘two dimensional’ solar cells are thus of high interest for developing cheap, large scale solar energy harvesting.