Limits on Broadband Absorption Enhancement in the Presence of Multiple Lossy Materials
Aaswath Raman Zongfu Yu Shanhui Fan



Enhancing the absorption and emission of electromagnetic waves over a broad range of wavelengths is a topic of fundamental and applied interest in photonics and energy research. In the context of light trapping in solar cells, for example, significant interest in the past decade has focused on overcoming limits in the ray-optics regime with nanophotonic structures. However, many such structures, in particular plasmonic structures, or PT-symmetric systems can posses multiple materials with varying values of intrinsic loss. Here, we rigorously determine the effect of parasitic loss on the achievable absorption enhancement in arbitrary electromagnetic structures. We show that the fundamental limit of broadband absorption enhancement, even in the presence of large parasitic loss in an alternate material, can exceed conventional ray-optics limits on light trapping and absorption enhancement. We numerically verify this behavior by determining the absorption enhancement factor of a canonical system, a metal-insulator-metal waveguide, whose core is a low-index organic semiconductor, in the presence of varying intrinsic loss values in the metal.

Spin coherence as a function of depth for high-density ensembles of silicon vacancies in proton-irradiated 4H-SiC
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§ INTRODUCTION

Enhancing broadband light absorption in thin, sub-wavelength layers of materials is a topic of fundamental interest in contemporary materials, photonics and photovoltaics research. Following early work in light trapping for solar cells<cit.>, a recent wave of work has sought to apply nanophotonic techniques to enhancing light absorption in thin active layers for next-generation solar cells<cit.>. Related to the wide array of systems and devices considered, the theoretical framework required to understand the fundamental limit of absorption enhancement was concurrently extended beyond the ray optics approximation, to account for wavelength- and subwavelength-scale effects present in nanoscale active layers <cit.>. This nanophotonic light trapping theory<cit.> established mechanisms by which one can exceed the conventional limit on the absorption enhancement factor F of 4n^2, where n is the bulk refractive index of the bulk absorber for which conventional limits do apply. 

One particular category of nanophotonic absorption enhancement schemes that has been extensively investigated is the use of metallic nanostructures, which support plasmonic modes at deep subwavelength scales, to enhance light absorption in a thin absorber <cit.>. It has been shown that in the case of ultra-thin absorbers, that this approach offers the possibility of exceeding the conventional broadband 4n^2 limit, where n is the refractive index of the absorbing material <cit.>. However, a key challenge with the use of metallic nanostructures for light trapping has been parasitic absorption of light in the metal. Many numerical and analytical analyses have been undertaken on specific plasmonic light trapping structures where the loss in the metal has been taken into account<cit.>. Theoretical works on limits to light trapping<cit.> on the other hand have primarily focused on assuming that the active semiconductor layer to be the only absorbing layer present, with analyses of parasitic loss considered for specific scenarios and approximations. A rigorous, universal analysis of the impact of parasitic loss on the fundamental limits of absorption enhancement remains to be developed. 

Beyond metallic nanostructures, a better understanding of light absorption or emission in subwavelength volumes in the presence of multiple intrinsic loss mechanisms is a topic of broad relevance. All solar cells in practice have lossy regions, such as electrodes or heavily doped regions in semiconductor cells, which are essential for the operation of the cell, but where absorption of light does not contribute to photocurrent. By Kirchhoff's law, understanding how broadband absorption enhancement behaves in the presence of multiple lossy materials also reveals how thermal emission might be enhanced in such complex photonic structures. From thermal radiation applications such as radiative cooling<cit.> and thermophotovoltaics, to the use of high-index nanostructures in photodetectors and solar cells, there is thus a general need to understand how the ability to enhance broadband absorption and emission in subwavelength volumes is affected by the presence of non-zero absorption in non-active materials in, or near, the active volume.

In this paper we develop and evaluate a formalism that describes how the introduction of multiple lossy materials influences the broadband absorption enhancement limit in nanophotonic structures and metamaterials. This formalism is in general applicable for any system with multiple lossy materials, and explicitly accounts for the impact of parasitic absorption in a non-active material, and for any active layer volume. We focus in particular on how parasitic absorption from metal influences the capability of plasmonic light trapping structure to exceed the conventional limit of broadband absorption enhancement. We rigorously show that in the weak active-layer absorption regime, parasitic absorption reduces the achievable absorption enhancement factor F_p in a desired material or region of a nanophotonic structure. However, with appropriate design choices, this lowered enhancement factor F_p can still exceed the conventional limit of F = 4n^2 across a broad range of wavelengths. We then numerically examine the effect of parasitic absorption across all absorption regimes by considering a plasmonic system with high local density of state: a metal-insulator-metal (MIM) waveguide. We show how the metal's material loss reduces the absorption enhancement in a thin core layer of a high-efficiency organic semiconductor, but still holds the potential to exceed conventional absorption enhancement limits.



§ STATISTICAL COUPLED-MODE THEORY FOR MULTIPLE LOSSY CHANNELS


Consider a resonance in a optical structure excited by an external plane wave. The incident plane wave comes from a particular channel, as specified, for example, by a particular angle of incidence. In addition, the resonance can couple to a total of N different channels in free space, including and in addition to the channel where the incident wave is coming from. The resonance amplitude a is then described by the temporal coupled mode formalism:

    d/dta =  ( i ω_0 - Nγ + γ_i + γ_p/2 ) a + i √(γ)S

 

Here a is the resonance amplitude, ω_0 the resonance frequency, γ the external coupling rate, and γ_i the intrinsic absorption rate in the active material. We assume that the resonance has an isotropic response, i.e. its coupling rate to all external channels is the same at γ. Unlike previous formulations of light trapping and new to this work, we also introduce a parasitic absorption rate γ_p which corresponds to modal loss in a non-active material. S is the amplitude of the incident plane wave, with |S|^2 corresponding to its intensity.

We consider an incident wave at a particular frequency ω, 

    S(t)    = S(ω)exp(iω t)

we then have

    a(t)    = a(ω)exp(iω t).

Substituting into Eq. (<ref>) we find the following expression for the resonance amplitude a:

    a(ω) = i√(γ) S(ω)/i (ω - ω_0 ) + (γ_i + γ_p + Nγ)/2.

This leads to separate expressions for absorption in the active layer A_act and parasitic absorption A_p due to the resonance:

    A_act   = γ_i γ/(ω - ω_0 )^2 + (γ_i + γ_p + Nγ)^2 /4
    
    A_p    = γ_p γ/(ω - ω_0 )^2 + (γ_i + γ_p + Nγ)^2 /4


To characterize the contribution of the resonance to the broadband absorption enhancement, we then compute the corresponding spectral absorption cross section σ_act<cit.> 

    σ_act = ∫_-∞^∞ A_act(ω) dω = 2πγ_i 1/N + γ_i/γ + γ_p/γ


A larger σ_act corresponds to the stronger contribution of the resonance to the overall broad band absorption. To reach its maximum value of σ_max = 2πγ_i / N for σ_act the structure must operate in the overcoupling regime where γ≫γ_i and γ≫γ_p. The above analysis is for a single resonance in the absorbing structure. To get the total absorption coefficient A one must sum over all m resonances within a frequency range Δω:

    A = 2πγ_i/Δω∑_mγ_m/Nγ_m + γ_i + γ_p

Ref. <cit.> provides the following upper bound on the coupling rates γ_m, assuming M resonances in the relevant bandwidth [ω, ω + Δω] 

    γ_m ≤N/MΔω/2π.

Plugging this into Eq. (<ref>) we find our first main result, which bounds the absorption in its most general way, in the presence of parasitic absorption:

    A ≤1/1 + γ_p/γ_i + N/MΔω/2π

To render this expression more readable we introduce two new terms, the single-pass absorption of the active material α_0 d and the raw enhancement factor F_0 for the structure in question in the weak-absorption limit and assuming no parasitic absorption. We can then rewrite Eq. (<ref>) as

    A ≤α_0 dα_0 d  (1 + γ_pγ_i ) + 1F_0

where

    F_0 = M/N2πγ_i/α_0 d Δω.

In general, we emphasize that F_0 and γ_p/γ_i are functions of wavelength in most realistic photovoltaics materials and light trapping schemes. This wavelength-dependence is important to consider, and thus the effect of parasitic absorption across all absorption regimes will be illustrated in the numerical example section with real material systems. 

Furthermore, previous work has indicated that if the parasitic material is a metal, γ_p is subject an upper bound defined by the metal's material parameters<cit.>. In general, we further observed that if the mode in the absorber has sub-wavelength volume, γ_p is typically very near this upper bound. Specifically, if the metal is well described by the Drude model, then for strongly confined modes, such as surface modes near the surface plasmon frequency, γ_p ∼Γ/2 where Γ is the Drude damping rate of the metal.

For an optical mode in an arbitrary nanostructure we can express γ_i as γ_i = α^wg_i· c / n_wg, where α^wg_i is the the modal absorption coefficient in the active layer and n_wg the group index of the mode. If we then define the modal overlap factor of the electric field intensity with the active material V_act = α^wg_i / α_0.

We now extend our analysis of Eq. (<ref>) by first considering the case of weak active layer absorption which has been a focus of all light trapping designs, before considering regimes of arbitrary α_0 d. An example of such a case is when plasmonic nanoparticles are used to enhancem absorption in the 800 - 1100 nm range for thin crystalline Silicon<cit.>. To facilitate study of this scenario we define a modified enhancement factor F_p in the weak-absorption regime (in the active layer) which does not assume γ_p is negligible, as one does when deriving F_0. Beginning with Eq. (<ref>) and assuming γ≫γ_i we find that

    F_p    = M/N2πγ_i/α d Δω ( 1 + MN2πγ_pΔω )^-1.
 
This then allows us to re-express Eq. (<ref>) in a manner analogous to the original upper bounds on absorption

    A ≤α dα d + 1F_p.

One can then rewrite Eq. (<ref>) in terms of the original weak-absorption, non-parasitic enhancement factor F_0, α_0 d and γ_p / γ_i:

    F_p = F_01+ α_0 d F_0·γ_pγ_i

Eq. (<ref>) shows that the raw enhancement factor is suppressed by the ratio of the modal absorption rate in the parasitic material, γ_p to the active material's intrinsic absorption rate. Furthermore it reveals that the greater the raw enhancement factor F_0 the more it will be suppressed by the presence of parasitic absorption. 




§ IMPACT ON ENHANCEMENT FACTOR


To develop physical intuition into how the enhancement factor F is reduced by the presence of parasitic absorption, we consider the simplest possible model: a parasitic absorber homogeneously distributed inside the active material.  In such a situation, where n is the bulk refractive index of the composite material, γ_p = α_p · c/n and γ_i = α· c/n. Eq. (<ref>) then reduces to

    F_p/F_0 = 1/1 + α_p d F_0.

We see immediately that the reduction of the raw enhancement factor F_0 is dependent on both the strength of the parasitic absorption α_p and the raw enhancement factor F_0 itself. What this immediately suggests is that a light trapping design which provides a nominally high enhancement factor F_0 due to mechanisms such as field confinement or scattering is in fact more sensitive to the presence of parasitic absorption than a design which provides a smaller enhancement factor. Thus, as we show in Fig. 1, design with higher raw enhancement factor but significant parasitic absorption can be worse than one with a significantly lower enhancement factor but lower parasitic absorption. 

The presence of the thickness d is indicative of the fact that the amount the incident field interacts with the active and parasitic materials plays an important role in suppressing the enhancement factor, an issue we next consider in detail.

In most nanostructures of interest the parasitic component will not be evenly distributed in the manner previously considered. In such cases the real-space profile of the modes in question becomes essential to understand the effect of parasitic absorption. For an optical mode in an arbitrary nanostructure we can express γ_i as γ_i = α^wg_i· c / n_wg, where α^wg_i is the the modal absorption coefficient in the active layer and n_wg the group index of the mode. If we then define the modal overlap factor of the electric field intensity with the active material V_act = α^wg_i / α_0 we can re-write Eq. (<ref>) as

    F_p/F_0 = 11+ γ_p F_0  ( n_wgdcV_act ).

This expression is useful for complex photonic nanostructures since we can directly calculate γ_p, the imaginary frequency of the mode in the limit of γ_p ≫γ_i, using analytical and numerical techniques. 

Alternatively, for non-plasmonic parasitic absorbers, it is useful to express Eq. (<ref>) in terms of the parasitic material's absorption coefficient. To do so we first define γ_p = α^wg_p· c / n_wg where α^wg_p is the modal absorption coefficient of the parasitic material. We can now also define V_par = α^wg_p / α_p, the modal overlap factor of the electric field with the parasitic material. Eq. (<ref>) then becomes 

    F_p/F_0 = 11+ α_p d F_0  ( V_parV_act ).

In the case of localized plasmon resonances, for example, where the field has equal intensity in the parasitic and active materials, V_par = V_act, and Eq. (<ref>) reduces to Eq. (<ref>).

In Fig. 1 we examine how an increased modal overlap ratio reduces F_p for different values of F_0 and α_p. We observe that a larger absorption coefficient in the parasitic material, α_p, has a greater effect on higher enhancement-factor (F_0) achieving designs. Thus, a nanophotonic design that nominally achieves a greater F_0 but relies on a lossy parasitic material with strong modal overlap may perform only slightly better than a simpler design with smaller F_0 but low-loss non-active materials. We also note that, even in the weak absorption regime, parasitic loss does indeed suppress a nanophotonic structure's achievable absorption enhancement limit.



The thin-film limit is a case of extreme interest for solar cells as it presents numerous opportunities to exceed the conventional 4n^2 limit<cit.> and use less material to generate power equivalent to thicker cells. Moreover current state-of-the-art nanophotonic approaches are targeted and hold the greatest promise for thin, sub-wavelength active layer thicknesses. Previously, it was shown that the raw enhancement factor F_0 for each mode in the thin-film case is F^mode_0 = λ/d n_wg V_act<cit.>. We can substitute this into Eq. (<ref>) to find a modified modal enhancement factor 

    F^mode_p = λd n_wg V_act1 + γ_p V_act n^2_wgλc.

As discussed earlier, if the parasitic material is assumed to be a Drude metal with damping rate Γ, and when there is strong field confinement, one can substitute γ_p = Γ/2 in Eq. (<ref>). Similarly one can substitute into Eq. (<ref>) to find a modified modal enhancement factor written in terms of the parasitic material's absorption coefficient

    F^mode_p = λd n_wg V_act1 + α_pλ n_wg V_par.


To determine the overall value for F_p, one would then count the equivalent adjusted enhancement from each mode present, i.e. F_p = ∑_modes F_p^mode. The important feature of these equations is that it rigorously calculates the enhancement factor limit for potential beyond-4n^2 systems, even in the presence of parasitic loss. With it, we also now have a rigorous connection between F_p and the absorption coefficient of the parasitic material α_p and the modal overlap factor V_par. 



§ NUMERICAL STUDY OF A CANONICAL PLASMONIC SYSTEM




To capture the effect of parasitic loss on a light trapping scheme across all values of the active layer's absorption coefficient, we consider a canonical plasmonic system: a metal-insulator-metal (MIM) waveguide where the Drude metal has silver's plasma frequency and a variable damping rate Γ, and the dielectric core/ active layer is a high-efficiency organic bulk-heterojunction semiconductor PCDTBT:PC_70BM (n∼ 2)<cit.>. Since such organic materials need to be deposited in thin layers for efficient carrier extraction, we consider a very thin active layer thickness of 10 nm. The entire structure then supports the fundamental gap-plasmon mode across the entire relevant wavelength range, where we consider the resulting absorption enhancement.

To determine the absorption limit we first use Eq. (<ref>) to find F_p (λ). To do so, we directly calculate<cit.> the parasitic modal loss rate γ_p, modal overlap factor V_act and mode index n_wg of the fundamental gap plasmon mode at all relevant wavelengths with actual material parameters at those wavelengths. The calculated F_p (λ), γ_p (λ) and γ_i (λ) are then substituted into Eq. (<ref>) to calculate the modal absorption limit. In Fig. 2 we plot the absorption limit A(λ) for this mode as calculated for varying values of the Drude damping rate in the metal.  We emphasize that this limit is a modal calculation, and assumes ideal coupling into the fundamental gap plasmon mode, which in practice can be challenging in the geometry shown. We observe that realistic values of Γ∼ 0.01ω_p suppress the absorption limit below the idealized lossless case, but it remains above the conventional 4n^2 limit. Stronger Γ values however can cause the enhancement to go below the conventional limit, indicating the need for careful material choice and nanostructure design in using plasmonics for light trapping.

Finally, to illustrate the how parasitic absorption affects limits for varying thicknesses of the active layer, we examine the modal absorption limit for the MIM waveguide scenario in Fig. 3. We fix the parasitic loss rate γ_p at its worst value, Γ/2<cit.>, and consider two limits of the active layer thickness, 5 nm and 100 nm. Even with maximal parasitic loss in the metal, the modal confinement for the 5 nm thick active layer is sufficient to far exceed the conventional 4n^2 limit. However, with a thicker active layer the parasitic absorption actually brings the modal absorption limit below the conventional limit. This indicates that light trapping beyond the conventional limit is possible for very thin active layers even in the presence of very high parasitic losses. But, as one goes to thicker active layers, the effect of parasitic losses can outweigh the added benefit from modal confinement in terms of enhancement factors.



We have thus rigorously derived a theory on the effect of multiple lossy materials in any broadband photonic absorption enhancement scheme across all absorption regimes. These results indicate that, while parasitic loss in a non-active material can suppress the achievable absorption enhancement, conventional plasmonic schemes can still exceed conventional limits on light trapping with appropriate design and material selection. While we have focused on broadband light trapping for solar applications as a motivating scenario, these results point to a wide range of opportunities that may lie in studying the interaction of electromagnetic modes with multiple lossy materials in the same nanophotonic structure.


S. F. acknowledges the support of Department of Energy Grant No. DE-FG02-07ER46426






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