1. Introduction

Due to their low materials cost, wide choice of possible substrates, and solution processability, organic photovoltaics (OPVs) are a topic of considerable interest for solar energy conversion technology [1]. Despite these desirable attributes, the exciton and carrier diffusion lengths in organic materials are typically much shorter than the optical absorption length; thus, it is challenging to simultaneously absorb photons and collect carriers with high efficiency [2]. Nanophotonic light-trapping architectures [3–21] offer a potential solution to this inherent tradeoff for OPVs, enabling large optical absorption over a broad spectral range in ultra-thin active layers.

An important aspect of organic materials is their propensity to self-assemble into highly ordered morphologies with strong optical anisotropies. For instance, small-molecule [22–25], polymer [26–28], and polymer:fullerene blend [29–32] thin-films have all been shown to exhibit significant anisotropy in their optical coefficients. Regardless of these morphology-induced anisotropies, previous studies of OPV light-trapping schemes have generally only considered optically isotropic organic layers [3–14]. In this paper, we explore the interplay between optical anisotropies and nanophotonic light-trapping in OPV thin-films. We find that 2D light-trapping architectures (highly confined in the out-of-plane z-direction) based on surface-plasmon modes (SP) and low-index-inclusion gap waveguide modes (GM) predominantly redistribute electromagnetic field energy into the out-of-plane electric field component, E_z [Fig. 1(a)]. Thus, material and light-trapping anisotropies must be properly aligned to maximize absorption. To demonstrate this effect, we compare light-trapping in the OPV material P3HT with a model morphology, termed zP3HT, which comprises the same molecular constituents and intermolecular ordering but with the polymer chains aligned perpendicular to the substrate [Figs. 1(b)]. The deposition of films with the zP3HT molecular orientation has recently been demonstrated using directional solvent evaporation [33]. We show that zP3HT-based nanophotonic light-trapping architectures exhibit superior absorptive properties across the solar spectrum, irrespective of film thickness, due to the alignment of the strongly absorbing axis with the dominant field component, E_z .

 

Fig. 1 (a) Surface plasmon (SP) and gap mode (GM) OPV architectures and |E_z|² mode profiles, (b) molecular orientation of P3HT and a model material, zP3HT, which consists of P3HT molecules oriented such that the polymer chains are aligned out-of-plane, (c) anisotropic real and imaginary parts of the P3HT and zP3HT dielectric functions versus wavelength. The P3HT dielectric function is measured by spectroscopic ellipsometry, while zP3HT is modeled by switching the in-plane (||) and out-of-plane (⊥) optical coefficients of P3HT.

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2. Effects of anisotropy on absorption

Under typical processing conditions, P3HT molecules align with the polymer chains parallel to the substrate surface [34,35] as shown in Fig. 1(b). At optical length scales there is no preferred in-plane orientation and the optical properties are homogeneous in the x-y plane. The chains are always perpendicular to the out-of-plane (z) direction, however, defining an optical axis with distinct optical properties. Thus, P3HT is a uniaxial material with a complex dielectric function which can be described by a 2^nd rank tensor,

The effect of these anisotropies can be clearly seen by considering the steady-state power absorbed (A) within a volume (V ) of anisotropic material:

Untextured planar OPVs benefit from the large in-plane absorption coefficient of P3HT thin-films, which couple efficiently to normally incident sunlight. Nanophotonic light trapping architectures, however, predominantly redistribute electromagnetic energy into the out-of-plane field component, E_z. To demonstrate the effect of this redistribution, we compare light trapping in P3HT thin-films to our model morphology, zP3HT, where the polymer chains are assumed to align perpendicular to the substrate [Fig. 1(b)]. As a conservative estimate, zP3HT is modeled as a uniaxial material where the in-plane and out-of-plane optical coefficients of P3HT have been switched as shown in Fig. 1(c). This assumption likely underestimates the absorptive properties of a material comprising vertically aligned P3HT molecules (see Appendix A for further discussion). However, E_z becomes the dominant field component contributing to optical absorption, and comparisons between P3HT and zP3HT light-trapping reveal significant differences which are entirely attributable to molecular orientation. As we will show subsequently, zP3HT exhibits superior absorption to P3HT in both SP and GM light-trapping architectures.

3. Surface plasmon light trapping with anisotropy

SP modes [(Fig. 1(a)] are highly confined waveguide modes that propagate along the surface of a metal/dielectric interface, and have been studied extensively as a means to enhance light absorption in thin-film photovoltaics [10, 14, 16, 19, 20]. For a uniaxial dielectric cladding with a z-oriented (out-of-plane) extraordinary axis, the propagation constant of a SP mode can be written as [38, 39]

 

Fig. 2 (a) Real part of the SP effective index for P3HT (black), zP3HT (red), and air (blue, dashed), and the real part of the out-of-plane top dielectric material index for P3HT (black) and zP3HT (red). (b) Imaginary part of the SP effective index and out-of-plane dielectric index for the same materials, showing that out-of-plane absorption is the dominant contributor to SP modal loss. The dashed line indicates λ_p, the wavelength at which the plasma resonance ${{\epsilon }^{\prime }}_{d,\left|\right|}{{\epsilon }^{\prime }}_{d,\perp }={{\epsilon }^{\prime }}_m^2$ occurs.

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To further illustrate the impact of morphology on OPV light trapping, we evaluate the fundamental upper limit of SP modal absorption [10, 14, 19, 20] in P3HT and zP3HT thin-films. Following the procedures outlined in [40], the incident radiation is assumed to couple to SPs via a Lambertian surface which equally populates all available optical states of the structure. Under this assumption, the fraction of incident light absorbed by the SP mode is given by (see Appendix C for derivation):

To achieve unity absorption, the SP mode must have large n_SP and κ_SP. The local density of optical states (LDOS) is proportional to n_SP; an increase in SP mode index corresponds to a larger number of SP states into which incident solar radiation can couple. The rate of absorption is proportional to κ_SP; an increase in this decay constant improves the likelihood that an SP is absorbed before escaping (i.e. re-radiating) via the Lambertian surface. However, some losses arise from undesired Ohmic dissipation in the metal substrate. To account for these effects we calculate the fraction of SP losses attributable to exciton generation in the polymer layer ξ_a (see Appendix D for derivation). As seen in Fig. 3(a), the majority of the losses take place within the organic layer, and this fraction is larger in zP3HT at all wavelengths. Taking parasitic losses into account, we can derive an upper limit for light absorption within the polymer layer:

The results presented in Figs. 2 and 3(a) demonstrate that light trapping is significantly more effective in zP3HT than P3HT. By reorienting the polymer morphology we can increase the LDOS, absorption rate, and fraction of light absorbed within the polymer relative to the metal substrate.

 

Fig. 3 (a) Fraction of power absorbed in the polymer for an SP mode. The dashed line indicates λ_p. (b) Integrated absorption weighted by the AM1.5 solar spectrum [41] for increasing thickness, h, of the polymer thin film atop a Ag substrate (shown in inset). Single pass absorption through each material is plotted with dashed lines for comparison.

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So far, we have only discussed properties of SPs at an interface with an infinitely thick organic layer. Ultimately, however, we are concerned with light trapping in very thin films. Using a transfer matrix method (TMM) for lossy, anisotropic slab waveguide modes [42], we numerically evaluate n_SP, κ_SP, ξ_a, and A_SP, polymer for finite thickness films with the geometry illustrated in the inset of Fig. 3(b). The thickness, h, dependent integrated absorbance (defined as ${\int }_{300\hspace{0.17em}\text{nm}}^{650\hspace{0.17em}\text{nm}}{A}_{\text{SP},\text{polymer}}B\text{d}{\lambda }_0$, where B is the weighted AM1.5 solar spectrum [41] normalized such that ${\int }_{300\hspace{0.17em}\text{nm}}^{650\hspace{0.17em}\text{nm}}B\text{d}{\lambda }_0=1$) for SP-coupled P3HT and zP3HT thin-films is shown in Fig. 3(b). zP3HT absorbs more incident solar radiation for all film thicknesses, reaching a maximum increase of a factor of ≈ 2. For thick films, the absorption saturates for both morphologies, reaching a higher value in zP3HT since the polymer more efficiently absorbs energy contained in SP modes as compared to the metal substrate. Normal-incidence single-pass absorption through a P3HT (black, dashed) and a zP3HT (red, dashed) film is also plotted in Fig. 3(b). The in-plane oriented polymers in P3HT couple strongly to normal-incidence light, and coupling to SPs can actually lead to a reduction in integrated absorbance for certain film thicknesses. The opposite is true in zP3HT where coupling to SPs can enhance the integrated absorbance by more than a factor of 6. Clearly, molecular orientation strongly impacts light trapping processes in organic thin-films. Properly aligning polymer and small-molecule domains to capture energy contained in the large z-fields characteristic of SP modes is essential to maximizing the potential of SP-based light-trapping architectures. In the following section, we show that such light-trapping anisotropies are not unique to SP architectures.

4. Gap mode light trapping with anisotropy

GM architectures [7, 17] have recently been proposed as a method to enhance light trapping in low-index (n_L) thin-films beyond the Ray Optics limit [43] of $4n_L^2$. A representative GM architecture [7], comprising a thin polymer layer embedded between a perfect electrical conductor (PEC) and a thick high-index dielectric (n_H), is illustrated in Fig. 1(a). The structure supports a hybrid TM slab waveguide mode with mode profile |E_z|² superimposed on the figure in blue. The field intensity within the active layer exhibits large enhancement, arising entirely from the out-of-plane field component E_z. This enhancement results from distinct electromagnetic boundary conditions: the in-plane electric fields E_x and E_y and out-of-plane displacement field D_z = ε₀ ε̃_⊥ E_z are each continuous across an interface. As a result, the out-of-plane electric field amplitude is enhanced by a factor of $n_H^2/{\tilde{n}}_{L,\perp }^2$ and the field intensity by a factor of $n_H^4/{\left|{\tilde{n}}_{L,\perp }\right|}^4$, resulting in large absorption enhancements. We calculate absorption enhancements in P3HT and zP3HT GM architectures using three different methods: employing analytical electrostatic (ES) approximations based on the formalism of [7] and [13], numerically solving for the fundamental TM waveguide mode in a GM architecture using TMM (as done earlier with SPs), and simulating the coupling of normal-incident solar radiation into GMs with a broadband coupling grating via rigorous-coupled wave analysis (RCWA) [44].

In [7] the authors derive an ES approximation for the absorption enhancement in a thin low-index inclusion embedded between two thick high-index dielectric slabs:

 

Fig. 4 (a) Effective indices of a gap mode with P3HT and zP3HT in the gap as a function of gap height and wavelength, (b) integrated absorbance of the gap-mode structure for both P3HT (black) and zP3HT (red) as a function of gap height using the following methods: transfer matrix method (TMM, left panel, dashed-dotted lines), electrostatic limit (ES, left panel, solid lines), rigorous coupled wave analysis (RCWA, right panel, solid lines), and single pass absorption (right panel, dashed lines). (inset) Out-of-plane field profile at 550 nm in the active layer beneath the numerically optimized coupling grating from [7]. (c) Side view of the grating structure used for RCWA calculations and indices used for TMM calculations where the scattering layer has been replaced with an average index of n_s = 1.87.

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The analysis described above assumes a sufficiently thin active layer, such that the fields within can be described by ES approximations (e.g. E_z is constant across the gap), and that the high index layer is optically thick (see Appendix E for further discussion). These assumptions overestimate the total fraction of absorbed light and do not accurately capture what happens as active layer thickness is increased. Using the TMM method [42] discussed earlier in the paper, we calculate n_GM and κ_GM for the lowest order GM waveguide mode supported by the structure in Fig. 4(c). Results are shown in Fig. 4(a) and reveal a significantly larger for zP3HT especially above 450 nm where κ_GM is up to 18 times larger than in P3HT for h < 10 nm. Plugging these values into Eq. (4) (Fig. 4(b), left panel), we see that the integrated absorbance for a zP3HT GM is ≈ a factor of 2 larger than that for a P3HT GM (dashed-dotted lines). We note that a similar treatment for an isotropic organic layer is performed in [10] and [14].

Lastly, we move beyond the assumption of Lambertian coupling and use RCWA to numerically model how a broadband grating couples to anisotropic GMs. The grating considered here (identical to the design in [7]) is shown in Fig. 4(c) and the inset of Fig. 4(b). The RCWA results are shown in the right panel of Fig. 4(b) (solid lines). Similar to TMM the results are less than those predicted by ES approximations; however, zP3HT still exhibits superior integrated absorbance properties. For all thicknesses and methods of analysis considered here, zP3HT provides significantly larger integrated absorption than P3HT–reaching a maximum of 113% larger absorption in the ES limit, 150% larger absorption in the single mode TMM calculated limit, and ≈ 63% in the RCWA grating structure. All cases far exceed single-pass absorption for both orientations (dashed lines, Fig. 4(b), right panel).

5. Discussion

The results presented in Sections 3 and 4 illustrate the need to align the strongly absorbing axis of P3HT with the anisotropic field enhancement of these light trapping architectures. Similar anisotropies can be expected in other light trapping architectures that share traits common to SPs and GMs. For instance, SPs and GMs are both TM waveguide modes characterized by in-plane E_x and out-of-plane E_z components. Because of the distinct continuity conditions for these fields, the out-of-plane E_z fields can exhibit much larger field enhancements and also need not go to zero in active layers that are adjacent to a metal substrate with large optical conductivity. Additionally, both light-trapping architectures couple to modes which are evanescent within the organic active layer (n_eff > n_⊥). Such modes have purely out-of-plane oriented electric fields when n_eff = n_⊥, and only exhibit an appreciable in-plane E_x component if n_eff ≫ n_⊥.

In general, these results illustrate the necessity for the explicit modeling of anisotropies in the active layer materials used in nanophotonic OPV architectures. It can be seen from Eq. (2) that the absorption for a uniaxial material is maximized when the energy contained in the field component pointing in the direction of strongly absorbing axis is maximum. For P3HT, where the strongly absorbing axis is in-plane, architectures with strong in-plane field enhancements such as hyperbolic metamaterials [45] could prove to be very beneficial. This concept can be extended to architectures where the localized field enhancements are not as anisotropic as the cases explored here. For example, Bloch modes, dielectric resonances and localized SP resonances have all been demonstrated for OPV nanophotonic light trapping. In these architectures the field enhancements must be designed with the morphology of the active layer in mind. Alternatively, through materials engineering, the morphology of the active layer can be tailored to maximize the benefit of the light trapping strategy being used. This may be beneficial for carrier collection considerations as well.

6. Conclusion

In summary, we have shown that light-trapping in SP and GM OPV architectures can be strongly influenced by morphology-dependent optical anisotropies of the organic active layer. Using P3HT as a representative system, we have shown that orienting organic domains such that the strongly absorbing axes are aligned perpendicular to the substrate leads to significant absorption enhancements. This orientation-dependent coupling arises from the significant anisotropic field enhancements intrinsic to these light trapping architectures. More generally, these results highlight the importance of considering optical anisotropies when modeling light absorption in any OPV architecture comprising well-ordered morphologies.

Appendix A: Estimation of absorptive properties for out-of-plane aligned P3HT molecules

We compare typical P3HT morphologies, wherein polymer chains lie parallel to a substrate, with a model morphology, zP3HT, in which the polymer chains are oriented perpendicular to the substrate (see reference [33] of the manuscript for a recent experimental demonstration). A typical P3HT film, exhibits a uniaxial dielectric tensor with the optical axis oriented out-of-plane along the z-direction:

(8)$${\overleftrightarrow{\mathit{\epsilon }}}_{\text{P}3\text{HT}}=\left[\begin{array}{ccc}{\tilde{\epsilon }}_{\left|\right|}& 0& 0\\ 0& {\tilde{\epsilon }}_{\left|\right|}& 0\\ 0& 0& {\tilde{\epsilon }}_{\perp }\end{array}\right]=\left[\begin{array}{ccc}{\tilde{\epsilon }}_{\text{SA}}& 0& 0\\ 0& {\tilde{\epsilon }}_{\text{SA}}& 0\\ 0& 0& {\tilde{\epsilon }}_{\text{WA}}\end{array}\right]$$

Here, ε̃_SA and ε̃_WA refer to the relative permittivities in the strongly absorbing and weakly absorbing directions, respectively. The simplest model for zP3HT is to rotate the P3HT dielectric tensor:

(9)$${\overleftrightarrow{\mathit{\epsilon }}}_{\text{P}3\text{HT}(\text{rotated})}=\left[\begin{array}{ccc}{\tilde{\epsilon }}_{\text{WA}}& 0& 0\\ 0& {\tilde{\epsilon }}_{\text{SA}}& 0\\ 0& 0& {\tilde{\epsilon }}_{\text{SA}}\end{array}\right].$$

Physically, this dielectric tensor describes a morphology in which polymer backbones are randomly oriented within the y-z plane, defining an optical axis along the x-direction. Although such a morphology could be realized using an ultramicrotome, for example, typical fabrication techniques exhibit in-plane (x-y plane) symmetry at optical length scales; the substrate has no preferred in-plane direction. In the manuscript we model this more physically realistic morphology with a conservative assumption: that both in-plane permittivities are equal to ε̃_WA,

(10)$${\overleftrightarrow{\mathit{\epsilon }}}_{z\text{P}3\text{HT}}=\left[\begin{array}{ccc}{\tilde{\epsilon }}_{\text{WA}}& 0& 0\\ 0& {\tilde{\epsilon }}_{\text{WA}}& 0\\ 0& 0& {\tilde{\epsilon }}_{\text{SA}}\end{array}\right].$$

It is instructive to compare the results of our calculations for these two optical models of a zP3HT morphology. In the manuscript, we assume the dielectric function in Eq. (10). The SP effective indices are identical along all in-plane directions and are plotted in Figs. 5(a) and 5(b) (red curves). In the rotated model Eq. (9), SPs propagating along the x-axis exhibit identical dispersion as the red curves in Fig. 5. SPs propagating along the y-axis experience an identical out-of-plane dielectric function but a different in-plane dielectric function and consequently have different properties (blue curves). Despite these physical differences, the SP properties are very similar, confirming that SPs are largely insensitive to in-plane permittivities. However, both of the optical models described above underestimate the dielectric function expected of a P3HT morphology in which the polymer chains are predominantly aligned out-of-plane.

 

Fig. 5 (a) Real part of the SP effective index for P3HT, zP3HT, P3HT(rotated, y-direction), and zP3HT(max). (b) Imaginary part of the SP effective index for the same set of materials showing the theoretical bounds of losses with reorientation.

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In a typical P3HT morphology, polymer chains align parallel to the substrate, but are randomly oriented within this plane. The in-plane permittivities thus arise from a mixture of transitions along the polymer backbone and between polymer chains. Although spectroscopic signatures of inter-molecular excitations in P3HT have been identified [46], these transitions are relatively weak and the dielectric function is dominated by intra-molecular excitations aligned along the polymer backbone [29]. Assuming purely intra-chain dipole transitions with susceptibility χ, the effective in-plane susceptibility (χ_eff) for P3HT is:

(11)$${\chi }_{\mathit{eff}}=\chi \frac{{\int }_0^{\pi }\text{cos}\left(\theta \right)\cdot \text{cos}\left(\theta \right)\text{d}\theta }{{\int }_0^{\pi }\text{d}\theta }=\frac{\chi }{2}.$$

The strongly absorbing dielectric function is then:

(12)$${\tilde{\epsilon }}_{\text{SA}}=1+{\chi }_{\mathit{eff}}=1+\frac{\chi }{2}.$$

Assuming instead that all polymer chains are aligned out-of-plane, the z-oriented dielectric function equals:

(13)$${\tilde{\epsilon }}_{z\text{P}3\text{HT}(\text{max}),\perp }=1+\chi =2{\tilde{\epsilon }}_{SA}-1.$$

Unlike the typical P3HT morphology, all chains are aligned along a single axis and the dielectric tensor for this assumption equals:

(14)$${\overleftrightarrow{\mathit{\epsilon }}}_{z\text{P}3\text{HT}(\text{max})}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 2{\tilde{\epsilon }}_{\mathit{SA}}-1\end{array}\right].$$

The resulting real and imaginary (SP) effective indices are also plotted in Fig. 5 as zP3HT(max) (orange curve). The real (imaginary) SP effective index is 20% (50%) larger assuming the dielectric function in Eq. (14) compared to the values of Eq. (10), used in the manuscript, when averaged over wavelengths from 450 nm to 650 nm. Thus, the calculations used in the manuscript likely underestimate the light-trapping benefits arising from a reorientation of polymer constituents. Regardless, we chose the most conservative assumption possible in order to highlight differences in light-trapping which arise solely from reorienting the dielectric functions of a material. Further studies are needed to accurately quantify the dielectric functions of morphologies similar to zP3HT.

Appendix B: Derivation of the anisotropic surface plasmon dispersion

Given the topology shown in Fig. 6, we derive the dispersion relation for an anisotropic SP mode. The SP is a TM mode with a complex magnetic field amplitude that decays evanescently into both media,

(15)$$H_y=H_0\{\begin{array}{cc}\text{exp}\hspace{0.17em}\left(-j\tilde{\beta }x+j{\tilde{k}}_{z\text{I}}z\right)& \text{Region}\hspace{0.17em}\text{I}\\ \text{exp}\hspace{0.17em}\left(-j\tilde{\beta }x-j{\tilde{k}}_{z\text{I}}z\right)& \text{Region}\hspace{0.17em}\text{II}\end{array}.$$

Region I is described by a complex dielectric tensor,

(16)$${\overleftrightarrow{\mathit{\epsilon }}}_d=\left(\begin{array}{ccc}{\tilde{\epsilon }}_{d,\left|\right|}& 0& 0\\ 0& {\tilde{\epsilon }}_{d,\left|\right|}& 0\\ 0& 0& {\tilde{\epsilon }}_{d,\perp }\end{array}\right),$$

and region II is described by the complex dielectric function of Ag, ε̃_m. Applying Ampère’s law ∇ × H⃗ = − jωε₀ ε⃡ · E⃗, the electric field components of the mode are obtained,

(17)$$E_x=\{\begin{array}{cc}\frac{{\tilde{k}}_{z\text{I}}}{\omega {\epsilon }_0{\tilde{\epsilon }}_{d,\left|\right|}}{H}_y& \text{Region}\hspace{0.17em}\text{I}\\ -\frac{{\tilde{k}}_{z\text{II}}}{\omega {\epsilon }_0{\tilde{\epsilon }}_m}{H}_y& \text{Region}\hspace{0.17em}\text{II}\end{array}$$

(18)$$E_z=\{\begin{array}{cc}\frac{\tilde{\beta }}{\omega {\epsilon }_0{\tilde{\epsilon }}_{d,\perp }}{H}_y& \text{Region}\hspace{0.17em}\text{I}\\ \frac{\tilde{\beta }}{\omega {\epsilon }_0{\tilde{\epsilon }}_m}{H}_y& \text{Region}\hspace{0.17em}\text{II}\end{array}.$$

Applying Gausss law in region I,

(19)$$\nabla \cdot {\overleftrightarrow{\mathit{\epsilon }}}_d(\omega )\overrightarrow{\mathbf{E}}(\omega )=\frac{\partial }{\partial x}{\tilde{\epsilon }}_{d,\left|\right|}{E}_x+\frac{\partial }{\partial z}{\tilde{\epsilon }}_{d,\perp }{E}_{z\text{I}}=0,$$

as well as for region II, we can now relate the z-field components in both media to the x-field component:

(20)$$E_x=\frac{{\tilde{k}}_{z\text{I}}{\tilde{\epsilon }}_{d,\perp }}{\tilde{\beta }{\tilde{\epsilon }}_{d,\left|\right|}}{E}_{z\text{I}}=-\frac{{\tilde{k}}_{z\text{II}}}{\tilde{\beta }}{E}_{z\text{II}}$$

Applying the continuity of the displacement fields at the interface along with Gausss law provides a relation between the out-of-plane k-vectors in each medium,

(21)$${\tilde{k}}_{z\text{II}}=-\frac{{\tilde{\epsilon }}_m}{{\tilde{\epsilon }}_{d,\left|\right|}}{\tilde{k}}_{z\text{I}},$$

while Faradays law ∇ × E⃗ = − jμ₀ωH⃗(ω) can be combined with the above substitutions:

(22)$$\frac{{\tilde{k}}_{z\text{I}}^2}{{\tilde{\epsilon }}_{d,\left|\right|}}+\frac{{\tilde{\beta }}^2}{{\tilde{\epsilon }}_{d,\perp }}=k_0^2$$

(23)$$\frac{{\epsilon }_m{\tilde{k}}_{z\text{I}}^2}{{\tilde{\epsilon }}_{d,\left|\right|}}+\frac{{\tilde{\beta }}^2}{{\tilde{\epsilon }}_m}=k_0^2.$$

Solving Eqs. (22) and (23) simultaneously results in the dispersion relation given in Eq. (3) of the main text, and is consistent with [38, 39].

 

Fig. 6 Geometry and propagation vector definitions for anisotropic SP dispersion derivation.

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Appendix C: Upper limit of absorption for finite thickness surface plasmons and gap modes: a waveguide perspective

Equation 4 of the manuscript represents the upper limit of absorption in a SP (used later in the text for a GM), assuming that incident solar radiation couples equally into all optical states. This expression can be used to represent any waveguide mode that can be characterized by a complex effective index. We derive Eq. (4) using two different formalisms, beginning with [40], where the fraction of light absorbed in a single mode waveguide under the assumption of Lambertian in-coupling [7, 13, 14, 20, 40, 43]:

(24)$$A_{wg}=\frac{{\alpha }_{\mathit{eff}}h}{{\alpha }_{\mathit{eff}}h+{\left[4\left(\frac{{\rho }_{tot}{\nu }_g}{{\rho }_{0}c}\right)\right]}^{-1}}.$$

Here, α_eff is the modal absorption coefficient, v_g is the group velocity, h is the thickness of the absorbing layer, ρ₀ is the vacuum local density of optical states (LDOS), and c is the vacuum speed of light. The total LDOS within the structure, ρ_tot = ρ_rad + ρ_wg consists of the LDOS associated with a continuum of radiation modes (ρ_rad), and a set of discrete guided modes (ρ_wg). The bracketed quantity in the denominator of Eq. (24) can be interpreted as the path length enhancement of a waveguide mode as compared to a single pass through the material. It can be shown that as h becomes large, this path length enhancement approaches the 4n² ray optics limit [40].

For a single mode waveguide [14], ρ_rad ≪ ρ_wg and, the LDOS can be related to the waveguide effective index [14,40], ρ_wg = β_wg/(2πv_gh) = n_eff/(v_gλ₀h), and the free space LDOS is given by ρ₀ = ω²/(π²c³). Applying these substitutions to (24) results in Eq. (4) of the main text,

(25)$$A_{wg}=\frac{{\alpha }_{\mathit{eff}}h}{{\alpha }_{\mathit{eff}}h+{\left[n_{\mathit{eff}}\frac{{\lambda }_0}{h}\right]}^{-1}}=\frac{1}{1+{\left[4\pi n_{\mathit{eff}}{\kappa }_{\mathit{eff}}\right]}^{-1}},$$

which holds for both SPs and GMs. An additional step is required to account for parasitic metallic losses in SPs by introducing the fraction of light absorbed in the polymer ξ_a, given by Eq. (35), to find that A_SP, polymer = ξ_a A_SP,total. We note that (25) is only valid in the limiting case of a single mode waveguide.

We can reproduce Eq. (25) by solving for the maximum enhancement using the method of [7] while including material dispersion. The general method of this solution is similar; however we repeat it here for consistency. We start by defining the absorption enhancement factor of a waveguide mode in the limit of weak absorption:

(26)$$F_{wg}=\frac{2\pi {\gamma }_{\mathit{eff}}}{{\alpha }_{0}h\text{d}\omega }\frac{M_{wg}}{N},$$

where α₀ is the material absorption coefficient and the loss rate of the mode (γ_eff) is equal to the modal absorption coefficient times the group velocity, γ_eff = α_effv_g. The number of in-coupled channels, N, is characterized by the 2D density of optical states in free space,

(27)$$N=2\pi k_0^2{\left(\frac{L}{2\pi }\right)}^2,$$

whereas the number of optical states in the structure is

(28)$$m_{wg}=\pi {{\beta }^{\prime }}_{wg}^2{\left(\frac{L}{2\pi }\right)}^2.$$

Taking the derivative of Eq. (28) yields the number of available states per unit frequency inside of the structure,

(29)$$M_{wg}=\text{d}{m}_{wg}=\frac{2\pi {{\beta }^{\prime }}_{wg}^2}{{\nu }_g}{\left(\frac{L}{2\pi }\right)}^2\text{d}\omega$$

Substituting Eqs. (27) and (29) into Eq. (26) results in the path length enhancement in the limit of weak absorption,

(30)$$F_{wg}=\frac{{\alpha }_{\mathit{eff}}{n}_{\mathit{eff}}{\lambda }_0}{{\alpha }_{0}h},$$

and applying the second law of thermodynamics, as was done in [13], results in

(31)$$A_{wg}=\frac{{\alpha }_{0}h}{{\alpha }_{0}h+F_{wg}^{-1}}=\frac{{\alpha }_{\mathit{eff}}h}{{\alpha }_{\mathit{eff}}h+{\left[n_{\mathit{eff}}\frac{{\lambda }_0}{h}\right]}^{-1}}=\frac{1}{1+{\left[4\pi n_{\mathit{eff}}{\kappa }_{\mathit{eff}}\right]}^{-1}}.$$

Equation (31) exactly reproduces Eq. (25), and holds in general for any waveguide mode. Typically, the complex effective indices in Eq. (25) must be evaluated numerically. We have performed these calculations with a general TMM solver for lossy, anisotropic materials using the formalism of [42] for both finite thickness SPs and GMs.

Appendix D: Fraction of loss in organic layer for surface plasmon modes

For the slab geometry given Fig. 6 we can define the power absorption per unit area in x and y from Eq. (2):

(32)$$A=\frac{\omega {\epsilon }_0}{2}\left[{\int }_0^{\infty }{{\epsilon }^{″}}_{d,\left|\right|}{\left|E_x\right|}^2+{{\epsilon }^{″}}_{d,\perp }{\left|E_z\right|}^2\text{d}z+{{\epsilon }^{″}}_m{\int }_{-\infty }^0{\left|E_x\right|}^2+{\left|E_z\right|}^2\text{d}z\right].$$

Integrating Eq. (32) and using the E-field relations from Eqs. (17) and (18), the absorption per unit length in regions I and II can be expressed as,

(33)$$A_{\text{I}}=\frac{1}{4\omega {\epsilon }_0}\left[\frac{{{\epsilon }^{″}}_{d,\left|\right|}{\left|{\tilde{k}}_{z\text{I}}\right|}^2}{{\left|{{\epsilon }^{″}}_{d,\left|\right|}\right|}^2{\tilde{k}}_{z\text{I}}}+\frac{{{\epsilon }^{″}}_{d,\perp }{\left|\tilde{\beta }\right|}^2}{{\left|{{\epsilon }^{″}}_{d,\perp }\right|}^2{\tilde{k}}_{z\text{I}}}\right]H_0^2\text{exp}\hspace{0.17em}\left(-2{\beta }^{″}x\right),$$

(34)$$A_{\text{II}}=\frac{1}{4\omega {\epsilon }_0}\left[\frac{{{\epsilon }^{″}}_m{\left|{\tilde{k}}_{z\text{II}}\right|}^2}{{\left|{{\epsilon }^{″}}_m\right|}^2{\tilde{k}}_{z\text{II}}}+\frac{{{\epsilon }^{″}}_m{\left|\tilde{\beta }\right|}^2}{{\left|{{\epsilon }^{″}}_m\right|}^2{\tilde{k}}_{z\text{II}}}\right]H_0^2\text{exp}\hspace{0.17em}\left(-2{\beta }^{″}x\right).$$

Equations (33) and (34) can be combined to define the fraction of power absorbed in the polymer region,

(35)$${\xi }_a=\frac{A_{\text{I}}}{A_{\text{I}}=A_{\text{II}}}=\frac{\frac{{{\epsilon }^{″}}_{d,\left|\right|}{\left|{k^{″}}_{z\text{I}}\right|}^2}{{\left|{\tilde{\epsilon }}_{d,\left|\right|}\right|}^2{k^{″}}_{z\text{I}}}+\frac{{{\epsilon }^{″}}_{d,\perp }{\left|\tilde{\beta }\right|}^2}{{\left|{\tilde{\epsilon }}_{d,\perp }\right|}^2{k^{″}}_{z\text{I}}}}{\frac{{{\epsilon }^{″}}_{d,\left|\right|}{\left|{k^{″}}_{z\text{I}}\right|}^2}{{\left|{\tilde{\epsilon }}_{d,\left|\right|}\right|}^2{k^{″}}_{z\text{I}}}+\frac{{{\epsilon }^{″}}_{d,\perp }{\left|\tilde{\beta }\right|}^2}{{\left|{\tilde{\epsilon }}_{d,\perp }\right|}^2{k^{″}}_{z\text{I}}}+\frac{{{\epsilon }^{″}}_m{\left|{k^{″}}_{z\text{II}}\right|}^2}{{\left|{\tilde{\epsilon }}_m\right|}^2{k^{″}}_{z\text{II}}}+\frac{{{\epsilon }^{″}}_m{\left|\tilde{\beta }\right|}^2}{{\left|{\tilde{\epsilon }}_m\right|}^2{k^{″}}_{z\text{II}}}},$$

which is consistent with the definition provided in [20].

Appendix E: Upper limit of absorption for a gap mode in the electrostatic limit

Following the formalism of [7], we derive the upper limit of absorption for a GM in the electro-static (ES) limit. We begin with the definition of the loss rate for an optical mode m in terms the steady-state power absorbed by the mode, A_m, defined in Eq. (2), and W_m is steady-state energy contained in the mode:

(36)$${\gamma }_m=\frac{A_m}{W_m}=\frac{\omega {\int }_{\text{gap}}{{\epsilon }^{″}}_{L,\left|\right|}{\left|E_{x,m}\right|}^2+{{\epsilon }^{″}}_{L,\left|\right|}{\left|E_{y,m}\right|}^2+{{\epsilon }^{″}}_{L,\perp }{\left|E_{z,m}\right|}^2\text{d}V}{\int n^2(\overrightarrow{r}){\left|\overrightarrow{\mathbf{E}}\right|}^2\text{d}V}.$$

Assuming that the most of the energy in each mode is contained in the high index region, we can make the approximation that:

(37)$$\int n^2(\overrightarrow{r}){\left|\overrightarrow{\mathbf{E}}\right|}^2\text{d}V\approx n_H^2{\left|E_0\right|}^2{L}^{2}D.$$

Additionally, if h is sufficiently small such that the E⃗-field remain constant within the gap, the loss rate can now be written in terms of the fields in the high index region with continuity conditions applied:

(38)$${\gamma }_m=\frac{\omega \left({{\epsilon }^{″}}_{L,\left|\right|}{\left|E_{x,m}^0\right|}^2+{{\epsilon }^{″}}_{L,\left|\right|}{\left|E_{y,m}^0\right|}^2+{{\epsilon }^{″}}_{L,\perp }\frac{n_H^4}{{\left|{\tilde{n}}_{L,\perp }\right|}^4}{\left|E_{z,m}^0\right|}^2\right)L^{2}h}{n_H^2{\left|E_0\right|}^2{L}^{2}D}$$

Noting that under a Lambertian in-coupling assumption:

(39)$$\sum _m{\left|E_{x,m}^0\right|}^2=\sum _m{\left|E_{y,m}^0\right|}^2=\sum _m{\left|E_{z,m}^0\right|}^2=\frac{1}{3}M{\left|E_0\right|}^2,$$

the maximum absorption in the limit of small α_L,|| and α_L,⊥ can be found by using the same substitutions as [7]:

(40)$$a_{\text{ES}}=\sum _m\frac{2\pi {\gamma }_m}{N\text{d}\omega }=4n_H{n}_{L,\left|\right|}\left(\frac{2}{3}+\frac{1}{3}\frac{{{\epsilon }^{″}}_{L,\perp }}{{{\epsilon }^{″}}_{L,\left|\right|}}\frac{n_H^4}{{\left|{\tilde{n}}_{L,\perp }\right|}^4}\right){\alpha }_{L,\left|\right|}h.$$

We define the enhancement factor for a weak absorber in terms of α_L,|| for comparison to a single pass of normal-incidence light through a slab of anisotropic media in a similar manner to Eq. (26):

(41)$$F_{\text{ES}}\equiv \frac{a_{\text{ES}}}{{\alpha }_{L,\left|\right|}h}=4n_H{n}_{L,\left|\right|}\left(\frac{2}{3}+\frac{1}{3}\frac{{{\epsilon }^{″}}_{L,\perp }}{{{\epsilon }^{″}}_{L,\left|\right|}}\frac{n_H^4}{{\left|{\tilde{n}}_{L,\perp }\right|}^4}\right)$$

For large values of n_H the last term dominates Eq. (41), and the maximal absorption enhancement depends primarily on ε″_L,⊥. In the case where α₀h is not small, a thermodynamic upper limit on the fraction of absorbed light [13] can be related to the weak absorber limit, yielding Eq. (7) from the main text:

(42)$$A_{ES}=\frac{{\alpha }_{L,\left|\right|}h}{{\alpha }_{L,\left|\right|}h+F_{\text{ES}}^{-1}}=\frac{{\alpha }_{L,\left|\right|}h}{{\alpha }_{L,\left|\right|}h+{\left[4n_H{n}_{L,\left|\right|}\left(\frac{2}{3}+\frac{1}{3}\frac{{{\epsilon }^{″}}_{L,\perp }}{{{\epsilon }^{″}}_{L,\left|\right|}}\frac{n_H^4}{{\left|{\tilde{n}}_{L,\perp }\right|}^4}\right)\right]}^{-1}}.$$

We note that Eq. (37) assumes an optically thick high-index layer which supports a large number of planewave modes, while Eq. (38) assumes that the low-index inclusion is infinitesimally thin such that E_z remains constant across the inclusion. Neither of these assumptions hold for the grating structure explored in the main text, which is why the ES formalism overestimates the upper limit of absorption as shown in Fig. 4(b), left panel.

Appendix F: RCWA simulation details

The RCWA simulations were run with 29 harmonics in both the x and y directions, and the calculated absorption values were found to converge within < 3.5% when integrated over the solar spectrum. The absorption, A, as a function of λ₀ and h are plotted in Fig. 7(a) for P3HT and Fig. 7(c) for zP3HT, where it can be seen that zP3HT has a very large absorption band ranging from 450 nm to 650 nm. This large absorption band corresponds to the large κ_eff values in this wavelength range, shown in Fig. 4(a) of the main text. Field components and absorption density U_a [defined as the integrand of Eq. (2)] are shown at λ₀ = 550 nm and h = 5 nm [indicated as white circles in Figs 7(a) and 7(c)] in Fig. 7(b) for P3HT and Fig. 7(d) for zP3HT. All quantities are normalized relative to the peak of the E_x component for a P3HT active medium. These field plots clearly illustrate that the out-of-plane E_z component is the dominant contributor to U_a regardless of active material, and that U_a is significantly larger for a zP3HT active medium as compared to P3HT.

 

Fig. 7 (a),(c) RCWA calculated absorption versus wavelength and active layer thickness, (b),(d) field intensities and absorption density inside the active area at λ₀ = 550 nm and h = 5 nm for P3HT and zP3HT, indicated by white circles in (a), (c).

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Acknowledgments

The authors acknowledge L. Richter for P3HT ellipsometry data, A. Raman and S. Fan for discussions on numerical modeling of gap mode structures and article preprints. Research on plasmonic light-trapping was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, as part of the Center for Energy Efficient Materials Energy Frontier Research Center (award # DE-SC0001009). Research on dielectric gap-mode light-trapping was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, as part of the Center for Re-Defining Photovoltaic Efficiency through Molecule Scale Control Energy Frontier Research Center (award # DE-SC0001085).

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