Parallel polymer tandem solar cells containing comb-shaped common electrodes

Solar Energy Materials & Solar Cells 132 (2015) 56–66

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Parallel polymer tandem solar cells containing comb-shaped common electrodes Hee Yoon Han, Hongkee Yoon, Choon Sup Yoon n Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Republic of Korea

art ic l e i nf o

a b s t r a c t

Article history: Received 23 April 2014 Received in revised form 7 August 2014 Accepted 18 August 2014

The poor optical transmittance of common electrodes based on thin metal films in three-terminal parallel tandem solar cells poses a major hindrance to attaining high power conversion efficiency (PCE). High optical transmittance and electrical conductivity of the common electrodes are crucial to achieving high PCE. We report a parallel polymer tandem solar cell that contains a comb-shaped electrode (CSE) based on gold (Au), which provides both high optical transmittance and low electrical resistance. We studied the performance of the tandem cell devices as a function of number of teeth N of the CSEs with tooth dimensions of 50 μm (width)
30 nm (thickness)
3 mm (length). A maximum PCE of 3.72%, which amounts to 85% of the PCE of an ideal tandem cell, was obtained with N ¼5 in the tandem structure ITO/ZnO/P3HT:PC71BM/PEDOT:PSS þTriton X-100/Au CSE/PEDOT:PSS/PTB7:PC71BM/LiF/Al, where ITO is indium tin oxide, P3HT is regio-regular poly(3-hexylthiophene), PC71BM is [6,6]-phenylC71-butyric acid methyl ester, PEDOT:PSS is poly(3,4-ethylenedioxythiophene):poly(styrene sulfonic acid), Triton X-100 is 4-(1,1,3,3-tetramethylbutyl)phenyl-polyethylene glycol, and PTB7 is thieno[3,4-b]thiophene/benzodithiophene. The combination of high optical transmittance and low electrical resistance of the Au CSE and the conductivity-enhanced hole transport layer of PEDOT:PSS þTriton X-100 resulted in a 20% higher PCE than that obtained using conventional common electrodes based on 12 nm thick Au films. We analyzed the resistance of the combined anode, which consists of the Au CSE and the PEDOT:PSSþ Triton X-100 film, theoretically. The theoretical results enabled us to predict the optimum number of teeth required for the Au CSEs to give the maximum PCE in three-terminal parallel tandem cells, and showed excellent agreement with the experimental results. & 2014 Elsevier B.V. All rights reserved.

Keywords: Parallel polymer tandem solar cells Polymer solar cells Comb-shaped electrodes Transparent common electrodes Three-terminal parallel tandem cells Combined anode

1. Introduction Since the invention of bulk heterojunction polymer solar cells [1], the power conversion efficiency (PCE) of polymer solar cells has increased rapidly and reached about 10% recently [2,3]. However the PCE of polymer solar cells is still lower than that of inorganic counterparts such as polycrystalline silicon, copper– indium–gallium–selenium, and cadmium-telluride, whose PCEs are more than 15% [4]. One of the fundamental explanations for the low PCEs of polymer solar cells is their narrow absorption band in the solar spectrum compared with the wide absorption spectrum of crystalline silicon solar cells, which ranges from 400 to 1100 nm [5]. A variety of strategies, including low band-gap semiconducting polymers [6–9] and tandem and multi-junction structures [10–13], have been studied to expand the absorption band of polymer solar cells. Because bulk heterojunctions can be easily integrated into tandem and multi-junction structures,

n

Corresponding author. Tel.: þ 82 42 350 2532; fax: þ82 42 350 8780. E-mail address: [email protected] (C.S. Yoon).

http://dx.doi.org/10.1016/j.solmat.2014.08.018 0927-0248/& 2014 Elsevier B.V. All rights reserved.

structures containing donor materials with complementary absorption spectra are considered to be strong candidates for high PCEs [14]. While series-connected tandem and multi-junction polymer solar cell structures have been studied the most and used to produce the highest PCEs [2,3], the series connection poses problems with respect to the photocurrent. Because the total photocurrent is limited by the smallest photocurrent of the subcells, photocurrent matching between these subcells is essential to achieve a high PCE. When more than two subcells are connected in series, the photocurrent of the rearmost subcell is bound to be small because most of the light will be absorbed by the front subcells. Therefore, photocurrent matching is difficult to achieve in a series connection [15]. As a result, efficiency enhancements in series-connected multi-junctions have thus far been limited to up to three junctions [16,17]. Parallel tandem and multi-junction structures have no such problems with photocurrent matching. Because the total photocurrent of tandem and multi-junction cells is simply the sum of the photocurrents of each subcell the short-circuit current density (JSC) can, in principle, be increased by increasing the number of

H.Y. Han et al. / Solar Energy Materials & Solar Cells 132 (2015) 56–66

subcells while maintaining the open-circuit voltage (VOC) in the region between those of the subcells. In parallel tandem and multi-junction structures, however, low optical transmittance of the metal interlayer electrodes presents serious problems with achieving a high PCE [15]. In these respects parallel tandem and multi-junction structures have the potential to improve the PCE, provided that their interlayer electrodes have high optical transmittance. To minimize the power losses, the resistance of the interlayer electrodes should be very small. Thin metal films [15,18] have been used for the interlayer electrodes to ensure low resistance, but their optical transmittance can be poor; the transmittance of a 12 nm-thick gold (Au) film was only approximately 50–60% in the visible range. To ensure high optical transmittance, graphene [19] and multiwalled carbon nanotubes (MWCNTs) [20] have been used but the resistances of both graphene and MWCNTs are about 20 times higher than that of the 12 nm-thick Au films. Therefore, a method that can be used to achieve high PCEs with parallel tandem and multi-junction solar cells by the creation of an interlayer electrode scheme that provides both high electrical conductivity and high optical transmittance is urgently required. We report a parallel polymer tandem solar cell that contains interlayer electrodes with high optical transmittance and low electrical resistance using comb-shaped electrodes (CSEs) based on Au and a conductivity-enhanced hole transport layer, leading to 20% higher PCEs than those obtained using conventional Au thinfilm electrodes.

2. Experimental The zinc oxide (ZnO) precursor solution was prepared by dissolving 1.0 g of zinc acetate dihydrate (C4H6O4Zn  2(H2O), 99.5%, Sigma-Aldrich, St. Louis, MO, USA) and 0.27 g of ethylamine (CH3CH2NH2, 99.0%, Sigma-Aldrich) in 10 mL of 2-methoxyethanol (CH3OCH2CH2OH, 99.9%, Sigma-Aldrich) with continuous stirring for 24 h [21]. Indium–tin oxide (ITO) glass substrates with a sheet resistance of 15 Ω/& were sequentially cleaned in deionized water with detergent, acetone, and isopropyl alcohol. The ZnO precursor solution was spin coated at 3000 rpm for 40 s onto the ITO and then the film was annealed at 200 1C for 1 h. Regio-regular poly(3hexylthiophene) (P3HT, Rieke Metals, Lincoln, NE, USA) and [6,6]phenyl C71 butyric acid methyl ester (PC71BM, Nano-C, Westwood, MA, USA) were blended with a weight ratio of 1.5:1 in 1,2dichlorobenzene (ODCB) to form the mixture denoted as A1, which was spin coated onto the ZnO layer at 1000 rpm for 40 s. We used poly(3,4-ethylenedioxythiophene):poly(styrene sulfonic acid) (PEDOT:PSS, Clevios P VP AI 4083, Heraeus, Hanau, Germany) as a hole transport layer. To improve the wettability of the PEDOT:PSS on the surface of the A1 active layer, 1 wt% of 4-(1,1,3,3-tetramethylbutyl)phenyl-polyethylene glycol (Triton X-100, Sigma-Aldrich) was added to the PEDOT:PSS as a surfactant. PEDOT:PSSþ Triton X-100 was spin coated at 2000 rpm for 40 s onto the A1 active layer, followed by annealing at 150 1C for 5 min to remove the residual solvent. Au CSEs, 30 nm thick, with different numbers of teeth (N¼1, 2, 3, 5, 10, and 20), and Au films were thermally evaporated individually on top of the PEDOT:PSSþ Triton X-100 layer at a pressure of 10  7 Torr. The dimensions of the teeth were 50 μm (width, w)
30 nm (thickness, t)
3 mm (length, L) and the thickness of the Au film was 12 nm (Fig. 1). The current density–voltage (J–V) curves of the front subcell were measured without encapsulation before fabricating the back subcell. PEDOT:PSS was spin coated on top of the Au CSE at 2000 rpm for 40 s, followed by annealing at 120 1C for 5 min. Thieno[3,4-b]thiophene/benzodithiophene (PTB7, One Material, Dorval, Quebec, Canada) and PC71BM were blended with a weight

57

Fig. 1. Schematic diagrams of a parallel polymer tandem solar cell structure. The Au CSE was used as an anode.

ratio of 1:1.5 in ODCB to form the mixture denoted as A2, which was spin coated onto the PEDOT:PSS layer at 1000 rpm for 40 s and dried for 15 min. Both A1 and A2 were used as active materials. A 0.5 nm thick LiF layer and 100 nm thick aluminum (Al) electrode were thermally evaporated on top of the A2 active layer. Fig. 1 shows a schematic diagram of a parallel tandem cell device with three-terminal structure. Al and ITO were used as a cathode and the Au CSE was used as an anode. We fabricated seven types of parallel tandem cells; six types contained Au CSEs with six different numbers of teeth, and one type contained an Au film electrode. To ensure reliable device performance, four devices were fabricated for each type of parallel tandem cell. The photovoltaic parameters of all the seven types of the parallel tandem cells with four devices for each type are listed in Tables A.1 and A.2 in Appendix A along with those of the front and back subcells. Also, 14 devices for A1 single cells based on the A1 active layer, nine devices for A2 single cells based on the A2 active layer, and four devices for the A2 single cell masked by the A1 layer are fabricated and their photovoltaic parameters are listed in Table B.1 in Appendix B. We measured J–V curves under illumination using a source meter (Keithley 2400, Cleveland, Ohio, USA) and a solar simulator (ABET 10500, Baltimore, MD, USA) with AM 1.5 G illumination at 100 mW/cm2. A metal aperture 0.3
0.3 cm2 in area was used to define the illumination area exactly [22]. Absorption spectra of the two donor materials and the transmittance of a combination of the PEDOT:PSS þTriton X-100 film and the Au CSE were measured using a spectrophotometer (JASCO V-530, Easton, MD, USA). The thickness of the spin-coated films was measured by a Dektak-8 surface profilometer (Veeco, Plainview, NY, USA). The electrical conductivity of the PEDOT:PSS and PEDOT:PSS þTriton X-100 films and the contact resistivity between the Au CSE and PEDOT: PSSþ Triton X-100 layer were measured using the transmission line method [23]. The area of the Au electrode used in the conductivity and contact resistivity measurements was 50 μm (width, w)
2 mm (length, L), and the thicknesses (t) of the PEDOT:PSS and PEDOT:PSS þTriton X-100 films were 50 nm and 30 nm, respectively.

3. Results and discussion The PEDOT:PSS hole transport layer had no wettability at all to the A1 active layer. Triton X-100 is widely used as a non-ionic surfactant to improve the wettability of PEDOT:PSS to the A1 active layer [24,25]. The resistances of the PEDOT:PSS and PEDOT: PSSþ Triton X-100 films are shown in Fig. 2 as a function of the distance between two contact points in the plane of the films, and the sheet and contact resistances (Table 1) are calculated from the

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H.Y. Han et al. / Solar Energy Materials & Solar Cells 132 (2015) 56–66

Fig. 2. Resistances of the PEDOT:PSS and PEDOT:PSS þ Triton X-100 films as a function of the distance between two contact points in the plane of the films.

Table 1 Electrical properties of PEDOT:PSS and PEDOT:PSS þTriton X-100 thin films.

Conductivity (S/cm) Sheet resistance (kΩ/&) Contact resistance with Au electrode (Ω) Contact resistivity with Au electrode (10  7 Ω m2) a

PEDOT:PSSa (50 nm thick)

PEDOT:PSS þTriton X-100b (30 nm thick)

3.48 57.4 63.1

32.05 10.4 46.0

2.77

8.13

PEDOT:PSS represents poly(3,4-ethylenedioxythiophene):poly(styrene sulfonic

acid). b

Triton X-100 represents 4-(1,1,3,3-tetramethylbutyl)phenyl-polyethylene glycol. The concentration of Triton X-100 in PEDOT:PSS was 1 wt%.

gradient and the y-intercept of the resistance line shown in Fig. 2 [23]. It is remarkable that the addition of 1 wt% Triton X-100 not only solved the wettability problem but also increased the conductivity of PEDOT:PSS by a factor of 9.2, which may be explained by the fact that poly(ethylene glycol) chains in Triton X-100 improve the connectivity between the grains of PEDOT:PSS particles [26,27]. Triton X-100 also reduced the contact resistance of the PEDOT:PSS layer with the Au CSE by 27%. 3.1. Optical and electrical properties of PEDOT:PSSþ Triton X-100 plus Au CSE Because the conductivity of PEDOT:PSS combined with Triton X-100 is very high, the combination of both the Au CSE and the PEDOT:PSS þTriton X-100 layer can play the role of an anode and is called a combined anode. In Fig. 3(a), the optical transmittance of the combined anode with different numbers of Au CSE teeth (N) is shown in the spectral range of 350–850 nm. Also the absorption spectra of the A1 and A2 active materials are shown in the same figure. The transmittance of the combined anode with different N at a wavelength of 600 nm is summarized in Table 2. The transmittance of the combined anode was higher than 80% over the spectral range 350–850 nm, when N was less than or equal to 5. However, the transmittance reduced to 76.8% and 69.3% for N¼10 and 20, respectively. On the other hand the transmittance of a 12 nm thick Au film was less than 57.5% over the same spectral range. Fig. 3(b) shows the resistance and optical transmittance of the combined anode as a function of number of teeth, N, of the Au CSE. It is worth noting that the resistance of the combined anode with N ¼2 decreased drastically to less than 28% of that of N ¼1. The resistance of the combined anode with N Z5 is similar to that of

Fig. 3. (a) Transmittance of the combined anode with different numbers of teeth N of the CSEs and the normalized absorption spectra of the films of two donors, P3HT blended with PC71BM (▲) and PTB7 blended with PC71BM (●), which were used in the active layers A1 and A2, respectively. (b) Resistance (●) and optical transmittance (▲) of the combined anode as a function of the number of teeth of the CSEs; “film” on the x-axis indicates a 12 nm thick Au film.

the 12 nm thick Au film. The transmittance of the combined anode showed a gradual decrease with increasing number of CSE teeth until N ¼5. This result clearly indicates the advantage of the Au CSE over the Au film electrodes; the Au CSE with N ¼5 has a resistance similar to that of the Au film but shows far higher optical transmittance. The resistances of the PEDOT:PSS þTriton X-100 layer and the Au CSEs, the contact resistances between the PEDOT: PSSþTriton X-100 layer and the Au CSEs, and the resistances of the combined anode are also summarized in Table 2. 3.2. J–V characteristics of front and back subcells and parallel tandem cells The group of J–V curves in the upper part of Fig. 4 shows the J–V characteristics of the front subcell in the parallel tandem devices, which is measured before building the back subcell and has the structure ITO cathode/ZnO/A1/PEDOT:PSS þTriton X-100/ Au CSE or Au film. The more the number of CSE teeth, the larger the JSC of the front subcell because more light was reflected back to the front subcell to be absorbed with increasing N. The resistance and transmittance of the Au CSE with large N approach those of the Au film whose J–V curve is indicated by dash–dotted line in Fig. 4. The FF of the front subcell also increased with increasing N because the resistance of the CSE decreased as N increased. On the other hand, VOC of the front subcell remained nearly the same with increasing N and was distribited over a range of 0.60–0.63 V.

H.Y. Han et al. / Solar Energy Materials & Solar Cells 132 (2015) 56–66

59

Table 2 Resistance and optical transmittance of a combined anode. Number of Au CSE teeth

Resistance of PEDOT:PSS þTriton Resistance of Au CSE Contact resistance between R1 X-100 (R1) (Ω) (R2) (Ω) and R2 (RC) (Ω)

Resistance of combined anode (R1 þ R2 þ RC) (Ω)

Optical transmittance (600 nm) (%)

1 2 3 5 10 20 Filma

325.0 81.2 36.1 13.0 3.2 0.8 –

356.8 97.1 46.7 19.3 6.4 2.4 1.3

87.9 87.2 85.9 82.8 76.8 69.3 57.5

a

22.4 11.2 7.4 4.4 2.2 1.1 1.2

9.4 4.7 3.1 1.8 0.9 0.4 0.05

Thickness: 12 nm.

Table 3 Photovoltaic parameters and efficiencies of the front and back subcells and the parallel tandem cells. Standard deviations are shown in parentheses. Cell type

Number of CSE teeth

VOC (V)

JSC (mA/cm2)

FF (%)

PCE (%)

Front subcella

1 2 3 5 10 20 Filmc 1 2 3 5 10 20 Filmc 1 2 3 5 10 20 Filmc

0.60 0.60 0.62 0.61 0.62 0.62 0.62 0.65 0.65 0.66 0.67 0.62 0.60 0.62 0.63 0.63 0.63 0.63 0.62 0.60 0.62

5.0 5.3 5.4 5.3 5.4 5.7 5.8 6.3 6.1 6.2 6.2 5.6 5.1 3.4 11.4 11.4 11.6 11.5 11.1 10.8 9.2

47.4 49.1 55.6 58.0 56.4 55.8 57.0 33.9 46.4 44.2 46.2 53.2 55.8 53.1 37.2 47.2 49.8 51.4 53.5 56.0 54.4

1.42 1.56 1.86 1.87 1.89 1.97 2.05 1.39 1.84 1.81 1.92 1.85 1.71 1.12 2.67 3.39 3.64 3.72 3.68 3.65 3.10

Back subcellb

Parallel tandem cell

a b c

Fig. 4. The upper group of curves with short-circuit current densities distributed between –5 and –6 mA/cm2 represents the J–V curves of the front subcells with different numbers of Au CSE teeth, N, while the lower group of curves with the short-circuit current densities distributed between –9 and –12 mA/cm2 represents the J–V curves of the tandem cells with different numbers of Au CSEs teeth, N; “film” in the legend indicates a 12 nm thick Au film.

Consequently the PCE of the front subcell increased with increasing N because PCE is proportional to the product of VOC, JSC, and FF. J–V curves of the back subcell with different N could not be shown in Fig. 4 because most of them overlapped with the curves of the front subcell. Therefore we summarize the device performance of the back subcell in Table 3 instead, along with the performances of the front subcell and the tandem cell. Unlike the front subcell, JSC of the back subcell decreased with increasing N because the optical transmittance through the CSE decreased as N increased. The FF of the back subcell increased with increasing N because the resistance of the CSE decreased as N increased. The combination of decreasing JSC and increasing FF with increasing N in the back subcell results in a maximum PCE of 1.92% at N ¼5. When using an Au film for the common anode the PCE of the front subcell is greater than that of any of the front subcells containing the CSE, but the PCE of the back subcell is smaller than that of any of the back subcells containing the CSE. The group of J–V curves in the lower part of Fig. 4 shows the J–V characteristics of the parallel tandem cells containing Au CSEs or

(0.02) (0.02) (0.01) (0.01) (0.02) (0.02) (0.02) (0.01) (0.01) (0.01) (0.01) (0.02) (0.03) (0.02) (0.03) (0.01) (0.01) (0.02) (0.02) (0.02) (0.02)

(0.14) (0.17) (0.10) (0.10) (0.15) (0.26) (0.08) (0.13) (0.36) (0.30) (0.33) (0.39) (0.26) (0.24) (0.15) (0.39) (0.19) (0.39) (0.39) (0.50) (0.25)

(1.35) (2.09) (2.55) (3.81) (1.33) (1.70) (1.50) (1.68) (3.30) (2.60) (2.26) (2.73) (3.57) (2.13) (0.93) (2.68) (2.05) (1.47) (4.08) (3.32) (1.89)

(0.06) (0.10) (0.11) (0.13) (0.03) (0.12) (0.07) (0.05) (0.05) (0.06) (0.04) (0.09) (0.06) (0.12) (0.12) (0.15) (0.14) (0.07) (0.12) (0.14) (0.02)

ITO/ZnO/ P3HT:PC71BM/PEDOT:PSS þ Triton X-100/Au CSE or film. Au CSE or film/PEDOT:PSS/PTB7:PC71BM/aluminum film (100 nm). Thickness: 12 nm.

an Au film as a common anode. The device performance of the parallel tandem cells is summarized in Table 3. VOC of the parallel tandem cells varied only minimally with increasing N (Table 3 and Fig. 4). Because JSC of the parallel tandem cells is most affected by the transmittance of the common anode, JSC also varied minimally until N ¼ 5 and decreased with increasing N [Table 3; Figs. 4 and 5(a)]. On the other hand, the FF increased with increasing N [Fig. 5(a)] because the resistance of the Au CSE decreased as N increased. As a result, the PCE of the parallel tandem cell showed a maximum of 3.72% at N ¼5 [Table 3 and Fig. 5(b)] because PCE is proportional to the product of VOC, JSC, and FF. The maximum PCE of 3.72% corresponds to 98% of the sum of the PCEs of the front and back subcells (Table 3) and 85% of the PCE of an ideal tandem cell, which is composed of the sum of the PCEs of the A1 single cell (2.23) and the A2 single cell masked by the A1 layer (2.17) [28–30], as listed in Table 4. It should be noted that the JSC of any parallel tandem cells containing Au CSEs is larger than that of the parallel tandem cell containing an Au film electrode.

4. Theoretical analysis of parallel tandem cells containing Au CSEs We analyzed the resistance of the combined anode using the results that were reported for inorganic solar cells [31]. The purpose

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of the analysis was to determine the optimum number of CSE teeth that gives the maximum efficiency of three-terminal parallel tandem cells, when the photovoltaic parameters of the two single cells based on the front and back subcells were given. An equivalent circuit to the parallel tandem cell containing an Au CSE is shown in Fig. 6(a). The resistance and optical transmittance of the CSE depend on the number of teeth N and the dimensions of the CSE, but in this analysis we fixed the thickness and length of a tooth for simplicity. Thus, the resistance of the CSE and the contact resistance between the Au CSE and the PEDOT:PSSþ Triton X-100 layer depend only on the number of teeth N and width w of a tooth. The resistance of the PEDOT:PSSþTriton X-100 layer also depends on N and w because the horizontal and vertical components of the hole currents from the front subcell are determined by the space between CSE teeth. Therefore the resistance of the combined anode, RCA, is represented by the sum of the resistances of the Au CSE, the PEDOT:PSSþ Triton X-100 layer, and the contact resistance between them [31,32]: RCA ðN; wÞ ¼ RCSE þ RPEDOT=Triton þ RC ;

where RCSE and RPEDOT/Triton represent the resistances of the CSE and the PEDOT:PSSþTriton X-100 layer, respectively, and RC represents the contact resistance between the Au CSE and the PEDOT: PSSþTriton X-100 layer. Because the shunt current between the anode and cathode is negligible compared with the current in the tandem cell, it is reasonable to assume that the shunt resistance of the combined anode is infinite. The optical transmittance of the combined anode depends not only on N and w but also on the wavelength of light λ and is denoted by TCA(N,w,λ). The dashed and solid lines in Fig. 6(b) represent an ideal J–V curve for which RCA ¼0 and

ð1Þ

Fig. 5. (a) The short-circuit current density (●) and fill factor (▲) of the parallel polymer tandem solar cells as a function of number of teeth of the Au CSEs. (b) The PCE of the parallel polymer tandem solar cells as a function of number of teeth of the Au CSEs; “film” on the x-axis indicates a 12 nm thick Au film.

Fig. 6. (a) An equivalent circuit to a parallel tandem solar cell containing the Au CSE. RFs , RFsh , and J F are the series resistance, shunt resistance, and current density of the front subcell, respectively. RBs , RBsh and J B are the series resistance, shunt resistance, and current density of the back subcell, respectively. J Fi is the current density that defines the FF of the ideal J–V curve for the front subcell and J Bi is the current density that defines the FF of the ideal J–V curve for the back subcell. RCA and TCA are the resistance and optical transmittance of the combined anode, respectively. (b) The dashed line represents an ideal J–V curve for which RCA ¼ 0 and TCA ¼ 100%, and the solid line represents a practical J–V curve for which RCA 40 and TCA o 100%. J i and V i are the current density and voltage that define the FF of the ideal J–V curve, respectively, and J p and V p are the current density and voltage that define the FF of the practical J–V curve, respectively.

Table 4 Photovoltaic parameters and efficiencies of the A1 single cell, the A2 single cell, and the A2 single cell masked by the A1 layer. Standard deviations are shown in parentheses. Cell type a

A1 single cell A2 single cellb A2 single cell masked by the A1 layerc a b c

VOC (V)

JSC (mA/cm2)

FF (%)

PCE (%)

0.62 (0.00) 0.73 (0.00) 0.73 (0.00)

6.5 (0.24) 8.6 (0.59) 6.6 (0.47)

55.3 (2.01) 49.5 (3.66) 45.0 (3.44)

2.23 (0.08) 3.11 (0.15) 2.17 (0.14)

ITO/ZnO/P3HT:PC71BM/PEDOT:PSS þ TritonX-100/Au film (100 nm). ITO/PEDOT:PSS/PTB7:PC71BM/aluminum film (100 nm). A1/ITO/PEDOT:PSS/PTB7:PC71BM/aluminum film (100 nm).

H.Y. Han et al. / Solar Energy Materials & Solar Cells 132 (2015) 56–66

TCA ¼100% and a practical J–V curve for which RCA 4 0 and TCA o100%, respectively. The PCE η of practical parallel tandem solar cells can be expressed as ηP 0 ¼ J p V p ¼ J p V i  J p ðV i  V p Þ ¼ J p V i  J p ½J p RCA ðN; wÞ ¼ J p V i  J 2p RCA ðN; wÞ;

ð2Þ

where P 0 is the incident light power density, J p and V p are the current density and voltage that define the FF of the practical J–V curve for the tandem cell, respectively, and V i is the voltage that defines the FF of the ideal J–V curve for the tandem cell. Eq. (2) shows that the PCE of practical parallel tandem cells is equal to the PCE for RCA ¼0 minus the power loss due to non-zero RCA. As shown in Fig. 3(a), the variation of transmittance over wavelength λ for different N is relatively small for λ 4500 nm. If the average transmittance of the combined anode over wavelength is denoted by T CA ðN; wÞ;J p can be expressed as [33] J p ¼ J Fi þJ Bi T CA ðN; wÞ;

ð3Þ

where J Fi is the current density that defines the FF of the ideal J–V curve for the front subcell and J Bi is the current density that defines the FF of the ideal J–V curve for the back subcell. Substituting J p from Eq. (3) into Eq. (2) yields ηP 0 ¼ ½J Fi þJ Bi T CA ðN; wÞfV i ½J Fi þJ Bi T CA ðN; wÞRCA ðN; wÞg:

ð4Þ

Because the conductivity of the PEDOT:PSSþTriton X-100 layer is 9.2 times greater than that of the PEDOT:PSS layer (Table 1), the hole currents from the front subcell and also most of the hole currents from the back subcell can flow via the PEDOT:PSSþTriton X-100 layer to reach the Au CSE. Considering that the space between the teeth of the Au CSE for N¼ 10 (0.23 mm) is much wider than the thicknesses of the PEDOT:PSS layer (60 nm) and the PEDOT:PSSþTriton X-100 layer (80 nm) and the width of a tooth of the Au CSE (50 μm), a large part of the hole currents in the PEDOT:PSSþTritonX100 layer will flow in the direction parallel to the layer plane. A small fraction of the hole currents in the PEDOT:PSSþTriton X-100 layer will flow perpendicular to the layer plane, which are the currents directly below the teeth of the Au CSE flowing from the front subcell [Fig. 7(a)]. Thus the current pattern in the PEDOT:PSSþTritonX-100 layer can be modeled as a combination of dominant currents flowing parallel to the layer plane and minor currents perpendicular to the layer plane, as shown in Fig. 7(b). If we denote the hole conductivity of the PEDOT:PSS layer and the contact resistivity between the PEDOT:PSS layer and the Au CSE by σ 0 and ρ0C , respectively, the resistance to the hole currents in the PEDOT:PSS film and the contact resistance to the currents

Fig. 7. (a) Patterns of hole currents in the PEDOT:PSS and PEDOT:PSS þTriton X-100 layers. (b) Simplified current patterns of (a) for the analysis of the combined anode.

61

between the PEDOT:PSS layer and the Au CSE can be expressed by t=σ 0 L2  10  5 Ω and Nwρ0C =L3  10  2 Ω, respectively. Because both resistances are negligible compared with the resistance to the currents flowing parallel to the layer plane in the combined anode, which are 0.5 Ω for N ¼ 20 and 300 Ω for N ¼1, it is sufficient to consider only the resistance of the combined anode RCA in our analysis. RCSE and RPEDOT=Triton were derived using Eqs. (24) and (18) from Ref. [31], respectively, and RC was derived using Eq. (19) from Ref. [31] and Eq. (26) from Ref. [34]. The detailed derivation of RCSE , RPEDOT=Triton , and RC can be found in Appendix C. The resistance of the combined anode in Eq. (1) can be expressed as sffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρC Rsheet ρ L R Rsheet coth w RCA ðN; wÞ ¼ Au þ sheet2 þ ; ð5Þ 3Nwt 12N 2NL ρC where Rsheet is the sheet resistance of the PEDOT:PSSþTriton X-100 layer, ρC is the contact resistivity between the PEDOT:PSSþTriton X-100 layer and the Au CSE, and ρAu is the resistivity of Au [35]. T CA ðN; wÞ is equal to the average transmittance of the PEDOT: PSSþTriton X-100 layer minus the reflectance by the Au CSE: T CA ðN; wÞ ¼ T PEDOT=Triton ½1  ð1  T Au ÞNw=L;

ð6Þ

where T Au and T PEDOT=Triton are the average transmittance of a 30 nm thick Au film and the PEDOT:PSSþ Triton X-100 layer over wavelength, respectively. By substituting Eqs. (5) and (6) into Eq. (4), the practical PCE of parallel tandem solar cells can be obtained as a function of N and w of the Au CSE for given V i , J Fi , and J Bi . However, the three parameters could not be obtained directly from the parallel tandem cells because they were defined for the ideal J–V curve with RCA ¼0 and TCA ¼100% [Fig. 6(b)]. As an approximation, V i , J Fi , and J Bi were obtained experimentally using a single cell based on the A1 active layer and a single cell based on the A2 active layer masked by the A1 active layer. These experimental setups are termed the A1 single cell and the A2 single cell masked by the A1 layer, correspondingly. The device structures and photovoltaic parameters of the two single cells are listed in Table 4. Because the open circuit voltage of parallel tandem solar cells is determined by the smaller value of the open circuit voltages of each of the front and back subcells, V i is taken as the smaller value of the voltages that define each of the FFs of the J–V curves for the A1 single cell and the A2 single cell masked by the A1 layer. J Fi was taken as the current density that defined the FF of the J–V curve for the A1 single cell and J Bi was taken as the current density that defined the FF of the J–V curve for the A2 single cell masked by the A1 layer. The PCEs calculated using Eq. (4) are shown in Fig. 8(a) as a function of tooth width of the Au CSE for different numbers of teeth. It is worth noting that the more the number of teeth, the larger the maximum PCE value at a narrower width of CSE teeth. The decrease of the PCE on the left side of the PCE curves is attributed to the increase in the contact resistance RC with decreasing tooth width and the decrease of the PCE on the right side of the PCE curves is ascribed to the decrease of optical transmittance with increasing electrode area covered by the Au CSE. Considering that the tooth width of our tandem cell device is 50 μm, the vertical line at 50 μm in Fig. 8(a) indicates that the optimum number of CSE teeth that maximizes the PCE lies somewhere between 5 and 10. To evaluate our analysis, the distribution of tooth number that gives the maximum PCE is plotted in Fig. 8(b) as a function of V i and J Bi for J Fi ¼ 4:5 mA=cm2 because the front cell is fixed in our analysis. The tooth width was fixed at 50 μm. As J Bi increases the number of teeth that maximizes the PCE also increases, which implies that the resistance of the combined anode is more important than the optical transmittance when the photocurrent

62

H.Y. Han et al. / Solar Energy Materials & Solar Cells 132 (2015) 56–66

cell that contained the Au CSE was studied as a function of tooth number; the tandem cells had the structure ITO/ZnO/ P3HT:PC71BM/PEDOT:PSS þTriton X-100/Au CSE/PEDOT:PSS/PTB7: PC71BM/LiF/Al. A combination of the high optical transmittance and low electrical resistance of the Au CSE and the conductivityenhanced hole transport layer of PEDOT:PSS þTriton X-100 results in a maximum PCE of 3.72% for N ¼5, which amounts to 85% of the PCE of an ideal tandem cell and is 20% higher than that obtained using conventional common electrodes based on 12 nm thick Au films. The resistance of the combined anode was also analyzed theoretically. The theoretical results enabled us to predict the optimum number of teeth required for the Au CSEs to give the maximum PCE in three-terminal parallel tandem cells, and showed excellent agreement with the experimental results.

Acknowledgments This work was supported by a Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Science, ICT & Future Planning (2013R1A1A2012950).

Appendix A. Performance data of the front and back subcells and the tandem cells Table A.1 lists the photovoltaic parameters of the front and back subcells and the tandem cells for various numbers of CSE teeth, N, and Table A.2 lists the same parameters for the front and back subcells and the tandem cells with Au film electrodes.

Appendix B. Performance data of the A1 and A2 single junction devices Table B.1 lists the photovoltaic parameters of the A1 and A2 single junction devices. Fig. 8. (a) The PCE of parallel tandem solar cells as a function of tooth width of the Au CSEs for different numbers of teeth (N ¼1, 2, 3, 5, 10, and 20). The vertical line at 50 μm indicates the tooth width used in our tandem cell device. (b) The distribution of the optimum tooth number that gives the maximum PCE is plotted as a function of J Bi and V i , where J Bi is the current density that defines the FF of the ideal J–V curve for the back subcell and V i is the voltage that defines the FF of the ideal J–V curve for the tandem cell. The tooth width was fixed at 50 μm. The optimum number of teeth for our tandem cell device, which has J Bi ¼ 4:6 mA=cm2 and V i ¼ 445 mV, is marked by a white square.

density of the back subcell is high. On the other hand, as V i increases, the number of teeth that maximizes the PCE decreases. This result implies that the transmittance of the combined anode is more important than the resistance when the photo-induced voltage of the tandem cell is high. The number of teeth that maximizes the PCE for our tandem cell device ðJ Bi ¼ 4:6 mA=cm2 and V i ¼ 445 mVÞ is marked in Fig. 8(b) by a white square (N¼7). In Table 4, N¼5 gives a maximum PCE of 3.72% and N¼10 gives a second maximum PCE of 3.68%, suggesting that the maximum PCE can be obtained for a tooth number between 5 and 10, which is confirmed by the analysis.

Appendix C. Derivation of Eq. (5) Fig. C.1(a, b) shows schematic diagrams of the solar cell electrodes that are used in Ref. [31] and those that are used in this study, respectively. The geometries of both electrodes are practically identical, apart from the notations used to represent the dimensions of each electrode. Here the number of grid fingers, n, in Ref. [31] is equivalent to the number of CSE teeth, N, in this study. RCSE and RPEDOT=Triton were derived from Eqs. (24) and (18) of Ref. [31]. The power loss associated with the current flow along the grid fingers is represented by Eq. (24) in Ref. [31]: 2

P finger ¼

4J 2L na3 b ρf ; 3tw

ðC:1Þ

where t is the electrode thickness. Because n in Ref. [31] is equivalent to N in this study, a is equivalent to L, 2nb is equivalent to L, and ρf is equivalent to ρAu ; thus the power loss associated with the current flow along the CSE is 2

5. Conclusion In order to solve the problem of poor optical transmittance of common electrodes based on thin metal films in parallel polymer tandem solar cells we proposed a CSE based on Au, which provided both high optical transmittance and low electrical resistance. The performance of a parallel polymer tandem solar

P CSE ¼

J 2L a3 ð4n2 b Þρf J 2L L5 ρAu I 2total ρAu L ; ¼ ¼ 3Nwt 3ntw 3Nwt

ðC:2Þ

where I total ¼ J L L2 . According to Ohm's law the power loss, PCSE, is equal to I 2total RCSE . Comparison of this with Eq. (C.2) gives RCSE ¼

ρAu L : 3Nwt

ðC:3Þ

Table A1 Photovoltaic parameters of the front and back subcells and the tandem cells for various numbers of CSE teeth, N. N¼1

Front subcell

Back subcell

Sample number

VOC

JSC

FF

PCE

Sample number

VOC

JSC

FF

PCE

#1 #2 #3 #4 Average Standard deviation #1 #2 #3 #4 Average Standard deviation #1 #2 #3 #4 Average Standard deviation

0.60 0.63 0.62 0.58 0.61 0.02 0.65 0.66 0.67 0.66 0.66 0.01 0.63 0.61 0.63 0.56 0.61 0.03

5.00 5.30 5.20 5.30 5.20 0.14 6.30 6.50 6.50 6.60 6.48 0.13 11.40 11.70 11.50 11.70 11.58 0.15

47.40 46.00 44.10 45.90 45.85 1.35 33.90 30.00 31.70 30.80 31.60 1.68 37.20 37.10 35.20 36.80 36.58 0.93

1.42 1.54 1.42 1.41 1.45 0.06 1.39 1.29 1.38 1.34 1.35 0.05 2.67 2.65 2.55 2.41 2.57 0.12

#1 #2 #3 #4 Average Standard deviation #1 #2 #3 #4 Average Standard deviation #1 #2 #3 #4 Average Standard deviation

0.56 0.60 0.61 0.61 0.60 0.02 0.66 0.65 0.65 0.66 0.66 0.01 0.61 0.63 0.63 0.63 0.63 0.01

5.10 5.30 5.50 5.20 5.28 0.17 6.80 6.10 6.90 6.50 6.58 0.36 11.80 11.40 12.30 11.60 11.78 0.39

50.40 49.10 48.80 45.50 48.45 2.09 40.10 46.40 39.10 40.50 41.53 3.30 42.90 47.20 41.30 41.80 43.30 2.68

1.44 1.56 1.64 1.44 1.52 0.10 1.80 1.84 1.75 1.74 1.78 0.05 3.09 3.39 3.20 3.05 3.18 0.15

N¼ 3

Front subcell

Back subcell

Tandem cell

N¼ 5

Sample number

VOC

JSC

FF

PCE

Sample number

VOC

JSC

FF

PCE

#1 #2 #3 #4 Average Standard deviation #1 #2 #3 #4 Average Standard deviation #1 #2 #3 #4 Average Standard deviation

0.60 0.60 0.62 0.60 0.61 0.01 0.67 0.67 0.66 0.67 0.67 0.01 0.61 0.61 0.63 0.63 0.62 0.01

5.30 5.30 5.40 5.50 5.38 0.10 6.80 6.80 6.20 6.40 6.55 0.30 12.00 12.00 11.60 11.80 11.85 0.19

52.30 50.70 55.60 49.80 52.10 2.55 40.00 40.70 44.20 45.30 42.55 2.60 45.30 45.70 49.80 46.50 46.83 2.05

1.66 1.61 1.86 1.64 1.70 0.11 1.82 1.85 1.81 1.94 1.86 0.06 3.32 3.35 3.64 3.46 3.44 0.14

#1 #2 #3 #4 Average Standard deviation #1 #2 #3 #4 Average Standard deviation #1 #2 #3 #4 Average Standard deviation

0.61 0.60 0.61 0.60 0.61 0.01 0.67 0.66 0.66 0.66 0.66 0.01 0.63 0.60 0.63 0.63 0.62 0.02

5.30 5.50 5.50 5.40 5.43 0.10 6.20 6.80 6.60 6.10 6.43 0.33 11.50 12.20 12.00 11.40 11.78 0.39

58.00 49.70 56.50 52.30 54.13 3.81 46.20 43.90 46.20 49.40 46.43 2.26 51.40 48.50 49.10 51.20 50.05 1.47

1.87 1.64 1.90 1.69 1.78 0.13 1.92 1.97 2.01 1.99 1.97 0.04 3.72 3.56 3.71 3.68 3.67 0.07

JSC

FF

PCE

55.80 55.90 57.60 59.40 57.18 1.70

1.97 1.80 1.87 2.07 1.93 0.12

N ¼10

Front subcell

H.Y. Han et al. / Solar Energy Materials & Solar Cells 132 (2015) 56–66

Tandem cell

N ¼2

N ¼ 20

Sample number

VOC

#1 #2 #3 #4 Average Standard deviation

0.58 0.60 0.61 0.62 0.60 0.02

JSC 5.60 5.70 5.40 5.40 5.53 0.15

FF

PCE

Sample number

VOC

58.10 55.20 55.40 56.40 56.28 1.33

1.89 1.89 1.82 1.89 1.87 0.03

#1 #2 #3 #4 Average Standard deviation

0.62 0.62 0.58 0.60 0.61 0.02

5.70 5.20 5.60 5.80 5.58 0.26

63

Tandem cell

The tandem cell device with the highest PCE value among the samples of each type is selected and its parameters are listed in Table 3; the sample numbers and the corresponding VOC, JSC, FF, and PCE values of the tandem cells and their front and back cells are highlighted in italic.

55.80 52.90 51.90 47.20 51.95 3.57 56.00 53.70 52.80 48.10 52.65 3.32 5.10 5.00 5.60 5.30 5.25 0.26 10.80 10.00 11.10 11.00 10.73 0.50 0.60 0.66 0.62 0.66 0.64 0.03 0.60 0.63 0.58 0.63 0.61 0.02 #1 #2 #3 #4 Average Standard deviation #1 #2 #3 #4 Average Standard deviation Back subcell

#1 #2 #3 #4 Average Standard deviation #1 #2 #3 #4 Average Standard deviation

0.65 0.64 0.66 0.62 0.64 0.02 0.58 0.61 0.61 0.62 0.61 0.02

5.80 6.40 6.30 5.60 6.03 0.39 11.30 12.00 11.60 11.10 11.50 0.39

50.70 46.60 49.70 53.20 50.05 2.73 55.60 46.70 48.94 53.50 51.19 4.08

1.91 1.91 2.07 1.85 1.93 0.09 3.65 3.43 3.48 3.68 3.56 0.12

FF JSC VOC Sample number Sample number

VOC

JSC

FF

PCE

N ¼ 20 N ¼10 Table A1 (continued )

1.71 1.75 1.80 1.65 1.73 0.06 3.65 3.38 3.41 3.33 3.44 0.14

H.Y. Han et al. / Solar Energy Materials & Solar Cells 132 (2015) 56–66

PCE

64

Table A.2 Photovoltaic parameters of the front and back subcells and the tandem cells with Au film electrodes. Au film

Front subcell

Back subcell

Tandem cell

Sample number

VOC

JSC

FF

PCE

#1 #2 #3 #4 Average Standard deviation #1 #2 #3 #4 Average Standard deviation #1 #2 #3 #4 Average Standard deviation

0.62 0.63 0.59 0.60 0.61 0.02 0.62 0.65 0.62 0.66 0.64 0.02 0.62 0.62 0.59 0.62 0.61 0.02

5.80 5.80 5.70 5.90 5.80 0.08 3.40 3.30 3.70 3.80 3.55 0.24 9.20 9.00 9.30 9.60 9.28 0.25

57.00 60.00 60.20 58.40 58.90 1.50 53.10 50.10 55.30 52.70 52.80 2.13 54.40 55.20 55.90 51.60 54.28 1.89

2.05 2.19 2.02 2.07 2.08 0.07 1.12 1.07 1.27 1.32 1.20 0.12 3.10 3.08 3.07 3.07 3.08 0.02

The tandem cell device with the highest PCE value among the samples is selected and its parameters are listed in Table 3; the sample numbers and the corresponding VOC, JSC, FF, and PCE values of the tandem cell and its front and back cells are highlighted in italic.

The power loss associated with the current through the semiconductor sheet is represented by Eq. (18) in Ref. [31]: 3

P sheet ¼

2J 2L nab Rsheet : 3

ðC:4Þ

The power loss associated with the current flow through the PEDOT:PSS þTriton X-100 film is therefore P PEDOT=Triton ¼

2J 2L NLðL=2NÞ3 Rsheet J 2L L4 Rsheet I 2total Rsheet ¼ ¼ : 3 12N 2 12N 2

ðC:5Þ

According to Ohm's law the power loss, PPEDOT/Triton is equal to I 2total RPEDOT=Triton . Comparison of this with Eq. (C.5) yields RPEDOT=Triton ¼

Rsheet 12N 2

:

ðC:6Þ

RC was derived from Eq. (19) in Ref. [31] and Eq. (26) in Ref. [34]. The power loss associated with the contact resistance between grid fingers and the semiconductor film is represented by Eq. (19) with I ¼ J L ab in Ref. [31]. 2

P contact ¼ 2J 2L na2 b Rcontact ;

ðC:7Þ

where Rcontact is the contact resistance between part of the grid finger with an area of aLT and the semiconductor film. Here, LT is the effective current transfer length defined in Ref. [31]. Therefore, the power loss associated with the contact resistance between the Au CSE and the PEDOT:PSS þTriton X-100 film is  P C ¼ 2J 2L NL2

L 2N

2 Rcontact ¼

I 2total Rcontact : 2N

ðC:8Þ

Using Eq. (26) from Ref. [34], Rcontact can be expressed as Rcontact ¼

sffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρc Rsheet Rsheet coth w ; L ρc

ðC:9Þ

where ρc is the contact resistivity between the grid finger and the semiconductor film.

H.Y. Han et al. / Solar Energy Materials & Solar Cells 132 (2015) 56–66

65

Table B.1 Photovoltaic parameters of the A1 and A2 single junction devices.

A1 single cell

A2 single cell

A2 single cell masked by A1 layer

Sample number

VOC

JSC

FF

PCE

#1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13 #14 Average Standard deviation #1 #2 #3 #4 #5 #6 #7 #8 #9 Average Standard deviation #1 #7 #8 #9 Average Standard deviation

0.62 0.63 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.63 0.62 0.62 0.00 0.73 0.73 0.73 0.73 0.73 0.73 0.73 0.73 0.73 0.73 0.00 0.73 0.73 0.73 0.73 0.73 0.00

6.67 6.52 6.49 6.61 6.39 6.08 6.29 6.66 6.97 6.82 6.82 6.64 6.35 6.52 6.56 0.24 9.11 8.39 8.85 9.14 9.29 8.02 8.73 7.47 8.62 8.62 0.59 6.82 6.43 5.73 6.59 6.39 0.47

56.26 58.77 56.21 58.81 58.33 61.11 60.73 57.08 56.56 56.55 56.55 59.31 54.27 55.25 57.56 2.01 52.98 54.16 49.49 48.22 48.05 57.39 54.69 56.90 49.46 52.32 3.66 49.51 50.80 53.24 45.04 49.65 3.44

2.33 2.41 2.26 2.41 2.31 2.30 2.37 2.36 2.44 2.39 2.39 2.44 2.17 2.23 2.34 0.08 3.52 3.32 3.20 3.22 3.26 3.36 3.49 3.10 3.11 3.29 0.15 2.46 2.38 2.23 2.17 2.31 0.14

The samples that are fabricated using the same polymer solution as that used to fabricate the tandem cell device with the highest PCE (3.72) shown in Table 3 and in Table A.1 are highlighted in italic. The single cell device with the highest PCE value among the italic-highlighted samples is selected for the A1 and A2 single junction devices, and the device parameters are listed in Table 4. For the A2 single cell masked by the A1 layer, the photovoltaic data of sample #9 are listed in Table 4 because the cell is fabricated using sample #9 of the A2 single junction devices.

Fig. C.1. Schematic diagrams of (a) the solar cell electrodes that were used in Ref. [31] and (b) those that were used in this study. The geometries of both electrodes are practically identical, apart from the notations used to represent the dimensions of each electrode. The number of grid fingers, n, in (a) is equivalent to the number of CSE teeth, N, in (b).

Using Eq. (C.9), Eq. (C.8) then becomes sffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρc Rsheet Rsheet 2 P C ¼ I total coth w 2NL ρc

PEDOT:PSSþ Triton X-100 film. Comparison of this with Eq. (C.10) yields ðC:10Þ I 2total RC .

Here, According to Ohm's law the power loss, P C , is equal to RC represents the contact resistance between the Au CSE and the

sffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρc Rsheet Rsheet coth w RC ¼ 2NL ρc

ðC:11Þ

66

H.Y. Han et al. / Solar Energy Materials & Solar Cells 132 (2015) 56–66

Therefore, the resistance of the combined anode is RCA ðN; wÞ ¼ RCSE þ RPEDOT=Triton þRC sffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρC Rsheet ρAu L Rsheet Rsheet þ coth w ¼ þ : 3Nwt 12N 2 ρC 2NL

ðC:12Þ

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