## Abstract

Recent work has shown that using a high-index cladding atop a lower-index photovoltaic absorber enables absorption of light beyond the ergodic (4n^{2}) limit. In this paper, we propose a generalized optimization method for deriving optimal geometries that allow for such enhancement. Specifically, we adapted the direct-binary-search algorithm to optimize a complex 2-D multi-layer structure with the explicit goal of increasing photocurrent. We show that such an optimization results in enhancing the local density of optical states in an ultra-thin absorber, which forms a slot-waveguide geometry in the presence of a higher-index overcladding. Numerical simulations confirmed optical absorption approaching 100% and absorption-enhancement beyond the ergodic (4n^{2}) limit for specific spectral bands of interest. Our method provides a direct, intuitive and computationally scalable approach for designing light-trapping nanostructures.

© 2013 Optical Society of America

## 1. Introduction

Photovoltaic devices with ultra-thin absorbers allow for high charge-transport and carrier-collection efficiencies [1,2]. Furthermore, such devices could be manufactured with inexpensive scalable technologies [3]. Recent improvements in material quality have resulted in world-record device efficiencies of 20.4%, 18.7% and 12% for CIGS, CdTe and organic absorbers, respectively [4,5]. However, ultra-thin layers are intrinsically poor absorbers of incident sunlight. Previously, random or simple geometries of micro- and nanostructures were utilized to scatter normally incident light at large angles into the absorber layer [6–9]. The resulting increased optical path lengths lead to higher absorption. It was pointed out that for thick absorbers, light absorption may be enhanced compared to an unpatterned absorber by a factor of up to 4n^{2}, where n is the refractive index of the absorber [10]. This is the so-called ergodic limit [10–12].

Recent theoretical work suggested that light absorption might be enhanced beyond this limit in
the case of absorbers with deep subwavelength thicknesses [13–16]. Specifically, this may be
achieved in the case of an ultra-thin absorber that is sandwiched between two cladding layers
with a higher refractive index [Fig. 1(a)].This slot-waveguide configuration results in an increased local density of optical states
(LDOS) into which incident light may couple [15,16]. In this paper, we prescribe a specific methodology for
designing nanophotonic structures to efficiently couple incident light into guided modes within
such a slot-waveguide absorber. As a result, light absorption may be increased beyond the
4n^{2} limit and thereby, the efficiency of ultra-thin photovoltaic devices can be
increased significantly.

## 2. Simulation model

The schematic of the nanostructured photovoltaic device in 2 dimensions is illustrated in Fig. 1(a). The variation of the designed dielectric nanostructure is along X axis while it is uniform along Z axis. As opposed to previous work, here, we generalize the geometries based upon four nanostructured interfaces, namely those between air and the top cladding, top cladding and the absorber, absorber and the bottom cladding, and finally, bottom cladding and the back reflector. Two adjacent rectangular scatterers are placed at each interface. We have full freedom in controlling their heights, widths and positions. As a result, we identify 22 variables that define the device geometry. Specifically, these structures scatter light-waves such that incident sunlight can be efficiently coupled into waveguide modes inside the low-index absorbing layer. The device is assumed to be periodic in X direction with period of *Λ*. The average thicknesses of the top cladding, absorber and bottom cladding layers are *H _{c1}*,

*H*and

_{a}*H*, respectively. For each design, the average thickness of the absorber,

_{c2}*H*is kept constant, while the remaining 22 geometric parameters are treated as optimization variables as described in the following section.

_{a}We assume that the absorber is P3HT:PCBM [17,18] and that the cladding material is GaP, which has a higher refractive index than the absorber [15,16]. Perfect metal is assumed for the back reflector for simplicity. The numerical simulation was implemented via the finite-difference time-domain (FDTD) method provided by an open-source software package [19]. The illumination is assumed to be spatially collimated, temporally continuous light propagating in the + Y direction from the top of the computational domain. AM1.5G standard solar spectrum is also assumed [20]. Bloch periodic boundary condition is imposed in the X direction since guided mode resonances (GMRs) are sustained by periodically repeated structures. And perfectly matched layers (PMLs) are utilized at both the top and bottom terminals along Y axis in the simulation unit cell. Although only normal incidence is considered during optimization, the responses of the optimized designs to oblique illumination are carefully analyzed as discussed later. The angle of incidence *θ* and two orthogonal polarization states are defined in Fig. 1(b).

## 3. Figure-of-merit

In this paper, the enhancement of short-circuit current density *J _{sc}* compared to a reference device with bare flat absorber of P3HT:PCBM [Fig. 1(b)] is considered as the figure-of-merit (FOM) during optimization. The effective thickness of the absorber in the reference device,

*H*is chosen such that the volume of the absorber is the same as that in the optimized device. Once the electric field distributions are determined for all discrete wavelengths by 2D MEEP FDTD simulations [19], we can proceed to compute the absorbance and the short-circuit current density. The absorbance spectrum is expressed as [21]:

_{a}^{ref}*λ*is the wavelength,

*ω*is the optical frequency,

*ε”*(

*λ*) stands for the dispersion of the imaginary part of the permittivity of P3HT:PCBM absorber,

*P*(

_{inc}*λ*) is the incident solar power at wavelength of

*λ*, and

*Λ*is the period of light trapping structure. The integral of electric field

*E*(

*x*,

*y*,

*λ*) is over the active region. The attainable short-circuit current density

*J*from the designed solar cell is [22]:

_{sc}*IQE*(

*λ*) represents internal quantum efficiency of P3HT:PCBM [17],

*q*is the elementary charge,

*h*is the Plank’s constant and

*c*is the velocity of light. The absorbance and

*J*of the reference device is calculated likewise. The integral over the entire spectrum of interest from

_{sc}*λ*= 350nm to

_{min}*λ*= 850nm (limited by the bandgap of the absorber) is numerically approximated by sum of a finite set of discrete wavelengths.

_{max}The enhancement spectrum is plotted as the wavelength dependence of the ratio of the absorbance spectrum of the optimized design to that of the reference:

*A*(

_{ref}*λ*) stands for absorbance spectrum of the reference device and

*E*(

_{ref}*x*,

*y*,

*λ*) is the electric field distribution inside the active region of the reference cell at wavelength of

*λ*. The fraction defined in Eq. (3) is also equivalent to the ratio of the total electromagnetic energy in the optimized versus the reference devices. This is denoted by

*U*(

*λ*) and

*U*(

_{ref}*λ*), respectively, each of which is obtained by integrating the local electromagnetic energy density,

*u*(

*x*,

*y*,

*λ*) over the active layer.

The local electromagnetic energy density is a product of the LDOS and the modal occupation number [15]. By assuming maximum modal occupation number, *i.e.*, 100% coupling efficiency from external source into absorber, the enhancement spectrum can be simplified as the ratio of LDOS in the optimized and reference devices. Interestingly, the LDOS of a flat slab can potentially be modified through its surrounding medium [15]. Applying an outer environment with a higher index of refraction is able to greatly enhance the LDOS, since in such slot-waveguide configuration, larger propagation constant, smaller group velocity and greater confinement factor of the guided modes are obtained [16]. As a result, it is possible to achieve light trapping beyond the conventional ergodic limit, especially at vicinity of the semiconductor band edge [15]. The extent to which the LDOS is improved is constrained primarily by the refractive index of the surrounding medium, especially when the thickness of slab shrinks to subwavelength regime [16]. The optimization technique described in this study aims at realizing nanophotonic structures with maximized modal occupation numbers to take best advantage of the enhanced LDOS in the slot-waveguide illustrated in Fig. 1(a). As indicated by Eqs. (1)–(3), the increase of *J _{sc}* is directly proportional to the enhanced absorption, which itself is related to the elevated LDOS due to the high-index cladding.

## 4. Optimization algorithm and results

A modified version of the direct-binary-search algorithm was previously implemented to design broadband diffractive optics [23,24], diffractive phase masks for three-dimensional microlithography [25] and nanophotonics for light trapping [26]. Here, we further adapt this algorithm to optimize the geometric variables that define the 2D photovoltaic device as illustrated in Fig. 1(a). It operates in an iterative manner with the explicit goal of increasing the short-circuit current density of the device. The search algorithm is detailed in Fig. 2 and the basic version was also described previously in [26]. The parameters are traversed in a random order within each iteration. Proper termination conditions such as a minimum improvement in FOM are imposed to guarantee numerical convergence. Because of the algorithm’s tendency of premature convergence to local maxima, we repeated the same optimization process with several randomly generated initial candidates, among which the best optimized solution was chosen. Note that our proposed optimization approach can be readily adopted to design other categories of nanophotonic devices of both dielectrics [13] and plasmonics [6] by considering the geometric parameters as variables to be perturbed.

Due to the complexity of the geometry, additional constraints are required to preserve realistic designs. For instance, the organic absorber must be continuous along the X direction in order to prevent any electrical shorts. An additional check to avoid physical contact of the top and bottom cladding layers is employed after each parameter perturbation. After optimization, the short-circuit current density enhancement factor (compared to the reference device shown in Fig. 1(b)) as a function of incident wavelength, which we call the enhancement spectrum, is computed.

In this paper, we describe four design examples. The ranges and the unit perturbations of the
geometric parameters for two cases of *H _{a}* = 10nm and

*H*= 50nm are listed in Table 1.As shown in Fig. 1(b), 22 parametric variables are required to define the complex multi-layer nanostructures. These geometric parameters fall into five categories: period (

_{a}*Λ*), fill-factor of scatterer (

*ff*), shift-factor of scatterer (

*s*), thickness of cladding layer (

*H*) and thickness of scatterer (

*h*). The period is selected to be comparable to the incident wavelength. Fill-factor is the ratio of the scatterer width and

*Λ*. The shift-factor is the relative distance to the center of one period divided by

*Λ*. The height of scatterer covers negative (concave) to positive (convex) values, further expanding the space of design freedom.

The evolution of optimization in terms of iterations for four different designs are
plotted in Fig. 3, with the schematics of the optimized
designs illustrated as insets. Major geometric dimensions are labeled, while the details of the
absorber region are depicted by the small figure under each plot. The reference absorber
thicknesses (*H _{a}^{ref}*) for these designs are 33nm, 42nm,
68nm and 88nm. The devices in Figs. 3(a) and 3(b) have

*H*= 10nm, while the devices Figs. 3(c) and 3(d) have

_{a}*H*= 50nm. They are selected out of 12 different initial solutions with randomly generated parameters (6 for

_{a}*H*= 10nm and 6 for

_{a}*H*= 50nm). The optimized values of period,

_{a}*Λ*, for the devices in Figs. 3(a)–3(d) are 450nm, 500nm, 500nm and 300nm, respectively. They are close to the wavelength of peak power density in AM1.5G solar spectrum [20]. The calculated short-circuit current densities of these designs at normal incidence are 8.9mA/cm

^{2}, 10.1mA/cm

^{2}, 11.0mA/cm

^{2}and 11.9mA/cm

^{2}. These numbers correspond to enhancement factors (compared to the reference device) of 2.71, 2.88, 2.49 and 2.13 under the condition of normal incidence. As demonstrated in previous work [26], with thicker effective thickness

*H*, the optimized cell in general tends to generate higher short-circuit current density and at the same time reduced enhancement in

_{a}^{ref}*J*[14,26]. Although these structures are challenging to pattern, recent breakthroughs in a variety of nanolithography techniques [27], especially roll-to-roll nanoimprint lithography, could pave a path towards low-cost mass production of such nanophotonic structures with reasonably high fidelity and repeatability [28,29].

_{sc}## 5. Absorbance and enhancement spectrum analysis

Absorbance spectra of the optimized (in blue) and of the reference (in red) devices,
calculated by Eq. (1), at normal (solid lines) and
oblique (*θ* = 40°, dashed lines) incidence are shown in Figs. 4(a), 4(d), 4(g) and 4(j). More
than 60% of the incident photons between 450nm and 650nm are absorbed for carrier generation.
Interestingly, this absorbance approaches 100% at ~600nm in Fig.
4(d) and at ~550nm in Fig. 4(g). Unfortunately,
the parasitic absorption of light by GaP at shorter wavelengths (shown by the yellow curves in
Figs. 4(a), 4(d),
4(g) and 4(j))
constraints the achievable *J _{sc}*. These results are consistent with
previous work, where randomly textured GaP was used as the cladding layer [15].

Figures 4(b), 4(e), 4(h) and 4(k) give the enhancement spectra of the four optimal designs by using Eq. (3), demonstrating significant enhancement at wavelengths close to the bandgap of the organic absorber (~750nm). This is achieved by the excitation of multiple guided-mode resonances inside the dielectric slot waveguide formed by the GaP-organic-GaP structure [13]. The optimized multi-layer nanostructures at the interfaces allow incident light to couple energy efficiently into these GMRs. An alternative but equivalent view is that the high-index cladding (GaP) surrounding the low-index absorber significantly increases the available LDOS [15,16]. The increased LDOS, in turn, allows for a more efficient coupling of energy between the incident light field and the GMRs, with significant field intensities within the absorber layer. As plotted in Figs. 4(b), 4(e), 4(h) and 4(k), such resonances are polarization-dependent. As expected for a lossy slot waveguide, the resonances with TE polarization are weaker than those with TM polarization [30]. The intensity distributions at the wavelengths of maximum enhancement are plotted in Figs. 4(c), 4(f), 4(i) and 4(l), for both polarizations. This strong mode confinement inside the thin film absorber with sub-wavelength thickness directly contributes to the increased LDOS, as claimed in [16]. Note that by replacing the perfect electric conductor (PEC) by a real metal silver as back reflector, the short-circuit current density of design 1 is slightly reduced to 8.5mA/cm^{2}, representing an enhancement factor of 2.58. This is primarily ascribed to parasitic loss due to the finite skin depth of silver and coupling to plasmonic modes. Design 2 generates 2.36 times the short-circuit current density when compared to a reference device with *H _{a}^{ref}* = 42nm with a PEC back reflector.

In order to avoid tracking the sun, it is important that the light trapping mechanism operates even when the angles of incidence are oblique [31]. We analyze the performance of our optimized designs under oblique illumination with incidence angle *θ* as large as 60°. The resulting absorbance spectra are shown as dashed lines in Figs. 4(a), 4(d), 4(g) and 4(j). As expected for GMRs, the resonances are red-shifted and the resonance peaks are reduced [26,31]. The result is that the enhancement of *J _{sc}* is decreased (see Fig. 4(e), for example). We have previously shown for simple geometries that a nonlinear optimization algorithm that takes into account oblique incidence can alleviate this problem [26,31]. Nevertheless, the

*J*enhancement factors when averaged over all angles from 0 to 60° are only slightly reduced to 2.39, 2.22, 2.12, and 2.29 for the four designs. Even though only the normal incidence was assumed during optimization, the performance of these devices is excellent over a large range of incident angles.

_{sc}## 6. Enhancement beyond the ergodic limit

In order to explicitly study the impact of the increased LDOS in our optimized designs, we
compared the design in Fig. 3(b) (design 2) to a
previously optimized design that uses a lower-index cladding (indium-tin oxide or ITO) [26]. This latter design shown in Fig. 5(c) has an absorber thickness of 50nm and a slightly simpler geometry
than the design in Fig. 3(b). Nevertheless, the
comparison is instructive. The geometry and absorbance spectra of design 2
(*H _{a}^{ref}* = 42nm) presented in Figs. 4(d)–4(f) are reproduced
in Figs. 5(a) and 5(b). For comparison, the geometry of the design with ITO cladding and its absorbance
spectra are shown in Figs. 5(c) and 5(d). Despite a large fraction (~26.7%) of the incident solar spectrum being
absorbed by the GaP layer [Fig. 5(b)], the benefit of
utilizing GaP nanostructures instead of ITO is exemplified by the prominent absorption in
P3HT:PCBM near its bandgap (600nm<λ<750nm). On the other hand, the design in Fig. 5(c) suffers from broadband parasitic absorption by the
ITO layer.

The traditional light-trapping limit suitable for a bulk absorber, also referred to as the ergodic limit, is expressed by [10–12]:

in which*n*is the real part of the refractive index of the active layer of P3HT:PCBM [32]. In [16], an upper bound by using enhanced local density of optical states was presented, which is defined by:where

_{L}*n*is the real part of the refractive index of the cladding material GaP [33]. The high-index GaP is deployed to potentially achieve enhancement surpassing the ergodic limit since

_{H}*n*>

_{H}*n*[15].

_{L}Enhancement spectra plotted in logarithmic scale clearly illustrate that a nanostructured high-index cladding provides significantly higher enhancement factors compared to that with a low-index cladding [13–16]. The solid blue line in Fig. 5(e) indicates the ergodic (4n^{2}) limit [10]. The optimized device with a high-index cladding (black solid line) can provide enhancement beyond this limit for wavelengths close to the bandgap of the absorber. In fact, the TM resonant mode at 672nm (dark-green dashed line) allows for an enhancement factor that reaches the LDOS limit (red solid line). We can therefore, conclude that the increased LDOS enabled by the high-index cladding is essential to couple incident light efficiently into GMRs and hence, enable high light absorption in the ultra-thin absorber layer.

## 7. Triangular scattering elements

It is also interesting to know how geometric profiles of the scattering elements affect the light
trapping performances [9,31]. The simulation results of the nanophotonic light trapping designs when the
originally optimized rectangular scatterers are simply replaced by triangular elements with the
same heights and widths are summarized in Fig. 6.Note that due to the transformation of geometries, the effective thicknesses
*H _{a}^{ref}* of the active layer are shrunken as stated under
the schematic of each structure. The curves in Figs.
6(a), 6(c), 6(e) and 6(g) represent the enhancement spectra
for different angles of incidence (0, 20°, 40° and 60°) for comparison.
They are plotted in waterfall manner offset in Y axis. As anticipated, the triangular structures
offer completely different guided mode resonances compared to the rectangular cases [31], especially at longer wavelengths, which is critical for
efficient light trapping. The new resonances inside the slot waveguide of low-index organic
absorber inserted between high-index claddings, induced by triangular elements, are likely to
give rise to wide-angle responses, which may potentially lead to high-performance angularly
insensitive solar cells. The employment of low aspect-ratio structures is generally beneficial
for releasing the acute angular responses induced by the original high aspect-ratio rectangular
geometries (see Figs. 6(b), 6(d), 6(f) and 6(h)). Other exotic geometries such as sinusoid, trapezoid and inverted
trapezoid, may serve as scattering elements in the similar way [9]. This, however, is realized at the expense of reduced enhancements at small incident
angles (e.g.

*θ*= 0). The angle-averaged enhancements over the range from 0 to 60° are 2.27, 2.20, 2.14 and 2.28 for the four designs. Note that with proper optimization higher enhancement factors are likely feasible.

## 8. Conclusions

In this paper, we described a new technique for designing light-trapping nanostructures within photovoltaic devices comprised of an ultra-thin absorber sandwiched between two high-index cladding layers. Our optimized designs show short-circuit current-density enhancement of up to 2.88 times compared to a bare reference device under normal incidence. When considering oblique illumination (up to 60°), the enhancement is still up to 2.39 times greater than that of the reference device. We further analysed the details of the optical resonances generated within the optimized devices, and clearly showed that the enhancement is primarily attributed to the increased local density of optical states enabled by the high-index cladding layers. The basic framework of utilizing LDOS was previously proposed however with the assumption of ideal coupling efficiency [16]. Our technique renders practical geometries that maximize the coupling efficiencies.

Enhancement spectrum of the optimized device was demonstrated to exceed the traditional ergodic limit within a finite bandwidth. Scatterers of alternative geometries are supposed to mitigate angular sensitivity, further promoting the applicability of the photonic design principles. Extension to 3D nanostructures will further increase LDOS and corresponding light absorption [13]. Additional research into charge-separation and collection schemes within the nanostructured stack would be beneficial as well. Furthermore, a fully integrated opto-electronic model should be introduced to simulate nanophotonic solar cells with better accuracy [34,35]. Although the optimized designs have complex geometries, advances in aligned nanoimprint lithography [28,29,36] should enable the cost-effective manufacture of such devices in a layer-by-layer fashion.

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