We examined the effective utilization of Particle Swarm Optimization (PSO) to enhance the light absorption performance in thin CuIn
_{1x}
Ga
_{x}
Se
_{2}
(CIGS) solar cells with dual (top and bottom) gratings. The PSO tuned structure was demonstrated to be capable of achieving high and ultra broadband absorption spectra due to wellspaced and welldefined absorption peaks, which were SPPs and photonic modes induced by the metal and dielectric gratings. For only TM polarization and both polarizations, the fully optimized net absorptions exhibit 85.6% and 78.1%, which correspond to ~35.4% and ~23.5% improvement compared to optimized flat structures, respectively.
I. INTRODUCTION
Among various thin film solar cells progressing more rapidly, the polycrystalline chalcopyrite CuIn
_{1x}
Ga
_{x}
Se
_{2}
(CIGS) is a one of the most attractive candidates due to low material cost, longterm thermal stability, largescale and flexible manufacturing, and superior conversion efficiency
[1

4]
. In particular, compared to other thin film solar cells based on aSi or polymers, the highest conversion efficiency of up to 20.3%
[5]
is a result of the unique material properties of CIGS as the IIIIVI
_{2}
semiconducting compounds, in which the bandgap energy range can be controlled by ratio x≡Ga/(In+Ga)
[4
,
6

7]
.
Recently, one of most attractive methods for enhancing light trapping in thin film solar cells is to utilize the effect of surface plasmon polaritons (SPPs) by applying metallic nanostructures
[8

12]
. The excited SPPs cause strong nearfield enhancement and increase optical path length via broadangle light scattering. Because the dissipation energy is proportional to the electric field intensity, the strongly confined energy at the interface between metallic element and active medium boost solar cell efficiency. For example, metallic nanogratings placed at the bottom of the active layer
[11

12]
enable the broadband and polarization insensitive absorption by taking advantage of effective coupling to SPP (plasmonic) or photonic modes. Meanwhile, metallic nanogratings or nanoparticles placed at the top surface of the active layer also increase absorption performance although the effect can occur only at certain wavelengths and a considerable amount of incident light is blocked by metallic elements. To further improve absorption performance compared to these single nanogratings based structures, recently, there has been growing interest in using dual grating structures to enhance absorption thin film solar cells
[13

16]
. These structures will enhance the coupling between incoming light and different types of resonance modes, by changing geometry parameters of dual grating structures (top and bottom), resulting in an overall broadband absorption enhancement.
Most previous works on optimization of grating structures used a parametric scan method to enhance absorption
[10
,
12

13]
. It becomes a much more difficult and timeconsuming process when the number of relevant geometric parameters increases. In this study, we demonstrate the maximal utilization of the ‘particle swarm optimization (PSO)’ method for a thin film CIGS solar cell with the dual nanogratings composed of a top CIGS grating and a bottom Ag grating, which have 7 degrees of freedom (active layer thickness, period, each thickness and width of top and bottom gratings, and the lateral shift between both gratings). The PSO method, developed by Kennedy and Eberthart
[17]
, was a robust stochastic optimization technique. Inspired by social behavior of bird flocks or fish schools, PSO has been used for various design optimization problems, including diffractive grating structures
[18

20]
. PSO is particularly well suited to the optimization of solar cell designs due to the fact that PSO is a metaheuristic algorithm  no assumption is necessary on the topology of the search space  which can be quite complicated when complex structures are employed to maximize performance. The flowchart of the PSO algorithm employed in this paper is shown in
Fig. 1
.
The flowchart of the PSO algorithm.
Due to wellspaced and welldefined absorption peaks obtained from PSO, the optimized shifted dual grating structure exhibits the broadband absorption at the wide range wavelengths from 300 nm to 1200 nm, with the net absorption of
A
= 85.6% and
A
= 78.1% for TM only and for both TM and TE polarizations, respectively. In this paper, wellspaced absorption peaks mean the peaks’s wavelengths are positioned in such a way that the entire absorption spectrum is divided into equally (or roughly equally) spaced regions. Welldefined absorption peaks means the absorption peaks achieve maximum total absorption of 100%.
II. DEVICE ARCHITECTURE AND NUMERICAL METHOD
A thin film solar cell device using shifted dual grating structure was designed with the silver grating in the bottom and the CISG grating on the top. The W
_{Ag}
and G
_{Ag}
are the width and thickness of the silver grating. The W
_{CIGS}
and G
_{CIGS}
are the width and thickness of the CIGS grating. In all calculations, the silver reflector thickness is 100 nm and the incident light is normal to the surface. The optical constants of CuIn
_{1x}
Ga
_{x}
Se
_{2}
(x = 0.31) and silver are extracted from Ref.
[7]
and
[21]
, respectively. The absorption efficiency (
a(λ )
) as a function of wavelength in an active layer is investigated using rigorous coupled wave analysis (RCWA) based commercial software, DiffractMOD. To evaluate solar cell performance, the net absorption efficiency
(A)
is also calculated by taking into account the standard AM 1.5G solar illumination (
S(λ )
), as follows:
III. OPTIMIZATION OF SINGLE GRATING STRUCTURES
In this section, we examine absorption performances of two single (top and bottom) grating based structures by using ‘parametric scan’ as the optimization method under both polarizations. We fixed the CIGS layer thickness to H
_{CIGS}
= 40 nm, at which net absorption has a highest value of
A
= 63.2% in flat structure (
Fig. 2
). Due to the effect of a FabryPerotlike resonance caused by the reflector and the surface of a flat solar cell, increasing the thickness of the CIGS layer does not always increase its performance.
Net absorption efficiency as the function of thickness of flat structure.
 3.1. Bottom Grating
The schematic of bottom grating structure is shown in
Fig. 3(a)
. For the purpose of maximizing the net absorption efficiency, we employed the parametric scan for period range of P
_{Ag}
from 100 to 900 nm with 100 nm intervals, fill factor range of F
_{Ag}
from 0.1 to 0.7 with 0.1 intervals, and grating thickness range of G
_{Ag}
from 5 to 35 nm with 5 nm intervals. This means that calculations of 9×7×7 times are conducted. The optimized averaged
A
of 66.8% is obtained in the case of P
_{Ag}
= 400 nm, F
_{Ag}
= 0.3 (W
_{Ag}
= 120 nm), G
_{Ag}
= 20 nm.
Figure 3(b)
shows the absorption efficiency spectra of optimized bottom grating structureand optimized flat structure with H
_{CIGS}
= 40 nm for both polarizations. There are two absorption peaks for TM wave at wavelength range of 500 to 800 nm. The magnetic field profiles (
Fig. 3(c)
and
3(d)
) reveal that the absorption peaks at λ= 560 nm and λ= 740 nm correspond to SPPs modes characterized by strongly confined fields at interface between silver and CIGS layer. For TE polarization, absorption peak at λ= 640 nm originates from waveguide mode coupling, as estimated from strong electric field confined in only the CIGS layer in
Fig. 3(e)
.
(a) Schematic diagram of bottom grating structure. (b) Absorption efficiency spectra of optimized bottom grating structure for normal incidence light (black dashed line for TM, green dashed line for TE, red solid line for average of TM and TE polarizations) and optimized flat structure (blue triangle lines). (cd) Magnetic field distributions for TM polarization at λ = 560 nm and λ = 740 nm, respectively. (e) Electric field distribution for TE polarization at λ = 640 nm. The optimized structure parameters are H_{CIGS} = 40 nm, P_{Ag} = 400 nm, W_{Ag} = 120 nm and G_{Ag} = 20 nm. The average net absorption is A = 66.8%.
 3.2. Top Grating
The optimization process using the parametric scan was also conducted for the top grating, while keeping the similar parametric ranges to the bottom grating; period range of P
_{CIGS}
from 100 to 900 nm with 100 nm intervals, fill factor range of F
_{CIGS}
from 0.1 to 0.7 with 0.1 intervals, and grating thickness range of G
_{Ag}
from 10 to 70 nm with 5 nm intervals. The schematic of top grating structure is shown in
Fig. 4(a)
. At P
_{CIGS}
= 600 nm, F
_{CIGS}
= 0.3 (W
_{CIGS}
= 180 nm), and G
_{CIGS}
= 70 nm, the optimized averaged
A
of 73.35% is obtained.
Fig. 4(b)
shows the absorption efficiency spectra of optimized bottom grating structure and optimized flat structure with H
_{CIGS}
= 40 nm for both polarizations. The optimized top grating presents two SPPs absorption peaks in the longer wavelength range under TM polarization, as shown in
Fig. 4(b)
. The absorption peaks at λ= 770 nm and λ= 1110 nm come from SPP mode coupling, as estimated from strong field confinement at the interface between silver and CIGS layer in
Fig. 4(c)
and
3(d)
respectively, similar to bottom grating case. For TE polarization, coupling of waveguide mode into CICS grating element occurs at λ= 985 nm (seen in
Fig. 4(e)
).
(a) Schematic diagram of top grating structure. (b) Absorption efficiency spectra of optimized top grating structure for normal incidence light (black dashed line for TM, green dashed line for TE, red solid line for average of TM and TE polarizations) and optimized flat structure (blue triangle line). (cd) Magnetic field distributions for TM polarization at λ = 770 nm and λ = 1110 nm, respectively. (e) Electric field distribution for TE polarization at λ = 985 nm. The optimized structure parameters are H_{CIGS} = 40 nm, P_{CIGS} = 600 nm, W_{CIGS} = 180 nm and G_{CIGS} = 70 nm. The average net absorption is A = 73.35%.
IV. DUAL GRATING STRUCTURES
Now, we address the dual grating structures. Under the limitation of the design tools, we restricted the period of the top and bottom gratings to be the same. The double grating structure has 7 degrees of freedom; active layer thickness (H
_{CIGS}
), period (P), each thickness (G
_{CIGS}
, G
_{Ag}
) and width (W
_{CIGS}
, W
_{Ag}
) of gratings, and the lateral shift (S) between both gratings. To reliably, albeit roughly, estimate the global optimization, the parametric scan can be attempted, similar to the single grating case of previous section. Assuming that each parameter has 10 steps, total number of iterations would be 10
^{7}
(10 million); it is a prohibitive value which is highly timeconsuming.
Also, separated tuning of each parameter, as done in
[13]
, is not the answer; changing one design parameter would almost certainly shifts the local optimal value of all of the remaining parameters, rendering the search for the global optimum value impossible. Indeed, in
[22]
, virtually any absorption in the active layer was not obtained from 700 nm to near infrared, where strong plasmonic modes can be potentially excited.
 4.1. PSO Optimization of TM Polarized Performance
We employ the PSO method to solve the problem previously mentioned, and we consider the net absorption under TM polarization as the fitness function for the PSO algorithm. In PSO algorithm, tuning of different modes would involve tuning of the resonance wavelengths for maximizing broadband absorption in active layer, while minimizing resistive losses by metallic layer and reflection. Due to the highly absorbing nature of CIGS in the broadband wavelength ranges, a determining factor of absorption performance is not the number of resonance modes, but the spacing of prevailing resonance modes so that it is possible to populate the broader solar spectrum in contrast with Sibased solar cells. For example, resonance width at 90% of peak absorption is roughly 200 nm and remains relatively constant in
Fig. 3(b)
and
Fig. 4(b)
, which mean the full absorption spectrum from 300 nm to 1200 nm range can be divided into 4 resonance peaks with negligible reflective loss when we assume any optimal situation. Each welldefined and wellspaced peak requires at least two degrees of freedom  one to tune its maximum absorption, and one to tune its peak position (wavelength). As such, for our proposed device with 7 degrees of freedom, one can expect an absorption spectrum with at least three wellspaced and welldefined absorption peaks. Indeed, PSO result showed four resonance peaks with spacing from 190 nm to 295 nm, fully populated the 300 nm to 1200 nm spectrum, as discussed later.
First, we used the PSO algorithm to optimize the dual grating structure (
Fig. 5 (a)
) under TM polarization only. The optimized geometry parameters are as follows; H
_{CIGS}
= 52 nm, P = 300 nm, G
_{CIGS}
= 70 nm, W
_{CIGS}
= 145 nm, G
_{Ag}
= 27 nm, W
_{Ag}
= 220 nm, and S = 74 nm. The corresponding absorption efficiency in
Fig. 5(b)
shows ultra broadband enhancement. An important fact is that the absorption capability of CIGS, as the most active medium, is comparatively much lower at longer wavelengths
[7]
, and thus effective light trapping mechanism is required to achieve high absorption performance, which plasmonic modes are capable of. Indeed, in our optimized structure, three of four absorption peaks pertaining to broadband absorption enhancement are attributed to SPP modes excited at longer wavelength ranges. We present here the total absorption spectrum, including metal absorption (black solid line in the
Fig. 5(b)
). These SPP modes cause strong field confinement at the interface between Ag and CIGS, as shown in
Fig. 5(d)
,
5(e)
, and
5(f)
. The other corresponds to waveguide mode (
Fig. 5(c)
). In the
Table 1
, the net absorption efficiency is
A
= 85.6% for TM polarization, which indicates improvement of 35.4% compared to an optimized flat structure.
PSO optimization of TM polarized performance. (a) Schematic diagram of optimized dual grating structure. (b) Absorption efficiency spectra of optimized dual grating structure for normal incidence light (black dashed line for TM, green dashed line for TE, black solid line for total absorption of TM, red solid line for average of TM and TE polarizations) and optimized flat structure (blue triangle lines). (cf) Magnetic field profiles under TM illumination at absorption peak wavelengths, (c) λ = 365 nm, (d) λ = 650 nm, (e) λ = 900 nm, and (f) λ = 1090 nm, respectively. The optimized structure parameters are H_{CIGS} = 52 nm, P = 300 nm, G_{CIGS} = 70 nm, W_{CIGS} = 145 nm, G_{Ag} = 27 nm, W_{Ag} = 220 nm, and S = 74 nm. The average net absorption is A = 85.6%.
Net absorption efficiency of PSO optimization dual grating structure optimized for TM polarization
Net absorption efficiency of PSO optimization dual grating structure optimized for TM polarization
 4.2. PSO Optimization of Simultaneous TE and TM Polarized Performance
Next, we applied the PSO algorithm to optimize the dual grating structure under nonpolarization, in other words, we consider the average of the net absorption under TE and TM polarization as the fitness function for the PSO algorithm. The optimized geometry has H
_{CIGS}
= 39 nm, P = 570 nm, G
_{CIGS}
= 70 nm, W
_{CIGS}
= 131 nm, G
_{Ag}
= 18 nm, W
_{Ag}
= 129 nm, and S = 187 nm. The optimized net absorption efficiency of 78.1% corresponds to improvement of 23.5% compared to flat structure as seen in
Table 2
. The absorption performance is somewhat smaller than the only TM optimized case with
A
= 85.6%. This means that all of the design goals cannot reach optimal values simultaneously due to introducing of additional optimization goals to a system with the same limited number of degrees of freedom. The PSO algorithm is designed to find a tradeoff between TE and TM performance, and thus the result is no longer simple to analyze due to the complex coupling of the different modes supported, unlike the TM optimized structure. Therefore, absorption peaks are no longer welldefined and wellspaced for both TM and TE polarizations.
Fig. 6(a)
indicates absorption spectra for the optimized dual grating and optimized flat structures.
PSO optimization of dual grating structure for both TM and TE polarizations. (a) Absorption efficiency spectra of optimal dual grating structure for normal incidence light (black dashed line for TM, green dashed line for TE, red solid line for average of TM and TE polarizations) and optimized flat structure (blue triangle lines).
Net absorption efficiency of PSO optimization dual grating structure optimized for both TM and TE polarizations
Net absorption efficiency of PSO optimization dual grating structure optimized for both TM and TE polarizations
A design with larger number of degrees of freedom would be more capable of fully optimizing such a system. Specifically, in order to realize a structure with four welldefined and wellspaced absorption peaks under both TE and TM polarizations, a system with more degrees of freedom would be required. One might note that it is possible to realize such a system in 2D asymmetrical dual grating
[24]
, meanwhile, in such large numbers of optimization parameters, any kind of brute force parametric scanning would not be practical at all.
In practical applications, the short circuit current density is one of the popular indicators of solar cell performance in terms of optical absorption, assuming that each absorbed photon creates an electronhole pair that gets collected. In this work, we expressed the short circuit current density (
J
_{sc}
) as below:
where,
q
is the electron charge,
λ
is the wavelength of incident light,
h
is the Planck’s constant and
C
is the velocity of light in vacuum.
The short circuit current density of the PSO optimization dual grating structure and the flat structure with the equal volume of CIGS is shown in the
Table 3
. In the case of TE polarization, the
J
_{sc}
of PSO optimization dual grating structure is improved over 20% in comparison to the flat structure and for TM polarization the
J
_{sc}
is improved over 28%. The overall (average) improvement is 24% compared to the flat structure.
Short circuit current densityJscof of PSO optimization dual grating structure optimized for both TM and TE polarizations
Short circuit current density J_{sc} of of PSO optimization dual grating structure optimized for both TM and TE polarizations
 4.3. Sensitivity
In practical device fabrication, some geometry parameters may not match with optimized parameters. In order to investigate the effective tolerance of geometry parameters, we examined the sensitivity of absorption performance to varying geometry parameters around the optimal condition (
Fig. 7
). The optimized structure is rather insensitive to deviations of its design parameters. For P, W
_{CIGS}
, W
_{Ag}
and S, an error of 20 nm results in a reduction in the averaged net absorption of less than 0.3% compared with optimal value. For G
_{CIGS}
, G
_{Ag}
and H
_{CIGS}
, an error of 5 nm results in a reduction in the average net absorption of 0.8%. In fabrication, these tolerances of our optimized design structure are suitable with nanofabrication equipment
[25]
.
Examining the sensitivity of PSO optimization of dual grating structure optimized for both TM and TE polarizations. Examined parameters: Period (yellow line), CIGS grating thickness (blue line), CIGS grating width (green line), CIGS layer thickness (pink), Silver grating thickness (magenta line), silver grating thickness (black line) and shift between two gratings (red line).
V. CONCLUSION
In conclusion, we investigated the optimization capability of PSO in thin film CIGS solar cell with dual gratings (composed of top dielectric and bottom metal gratings). The optimized structure produces strong and ultra broadband absorption in the range of 300~1200 nm. PSO algorithm allows the formation of wellspaced and welldefined absorption peaks, as the results of a combination of SPPs and photonic modes induced by metal and dielectric gratings. The potential of PSO is strikingly useful in structure with the larger number of degree of freedom, meanwhile conventional optimization method such as parametric scan requires a prohibitive calculation time. Under only TM and both polarization, the PSO optimized net absorption of
A
= 85.6% and 78.1%, correspond to ~35.4% and ~23.5% improvement compared to flat structure, respectively. The average improvement of both polarization of the optimized structure in term of the short circuit current density
J
_{sc}
with respect to the flat structure is 24%.
Acknowledgements
This work was supported by National Research Foundation of Korea Grant (NRF2014R1A2A2A01006720, NRF20080061906 and NRF20090094046).
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