A transmission line model for the optical simulation of

JOURNAL OF APPLIED PHYSICS 110, 114506 (2011)

A transmission line model for the optical simulation of multilayer structures and its application for oblique illumination of an organic solar cell with anisotropic extinction coefficient N. A. Stathopoulos,1,a) L. C. Palilis,2,b) S. R. Yesayan,3 S. P. Savaidis,1 M. Vasilopoulou,3 and P. Argitis3 1

Department of Electronics, Technological Educational Institute (TEI) of Piraeus, Aigaleo 12244, Greece Department of Physics, University of Patras, Patras 26500, Greece 3 Institute of Microelectronics, National Center of Scientific Research (NCSR) “Demokritos,” Athens 15310, Greece 2

(Received 9 March 2011; accepted 22 October 2011; published online 2 December 2011) A transmission line model for the calculation of optical interference phenomena in dielectric multilayered structures is adopted as an alternative option to the transfer matrix model (TMM). The method is based on the transmission line theory and is exact, easy to implement and uses closed iterative forms instead of the TMMs matrix formalism. The proposed model has been appropriately modified and then applied for performance evaluation of a typical organic photovoltaic device under inclined illumination. Optical field distribution, short-circuit photocurrent and reflectivity have been calculated under different angles of light incidence. The theoretical simulations have been discussed and compared with experimental photocurrent measurements, while the influence of the photoactive layer thickness on the device efficiency has been evaluated for different angles of light incidence, C 2011 American Institute of Physics. taking into account its extinction coefficient anisotropy. V [doi:10.1063/1.3662952] I. INTRODUCTION

Polymer-based optoelectronic device structures like organic light-emitting diodes (OLEDs) and organic photovoltaics (OPVs) and components of them such as distributed Bragg reflectors (DBRs) are typically comprised of successive dielectric layers deposited on glass or plastic substrates. Light wave interference through a multilayer structure could cause various interesting optical phenomena, which would actually determine device efficiency, like OLEDs’ wide angle or multiple beam microcavity effects,1 OPVs’ variation in internal reflection/absorption,2 or DBRs’ enhanced reflectivity characteristics.3 Recently, the analysis of light interference throughout the dielectric multilayer structure of an OPV has attracted an increased interest because of its significant influence on the OPVs external quantum efficiency (EQE).4,5 Although there is an extended literature for the analysis of EQE for normal incidence of light on an OPV, only recently oblique incidence has been examined and modelled.6–8 A detailed investigation of the angular response of OPVs would enable a better understanding of the device physics under real-time conditions, and its findings could be beneficial in the development of novel device architectures (i.e., the “folded” solar cell).9 The aim of this paper is to present a newly developed interference calculation method based on the transmission line theory and then apply it in the analysis of inclined light incidence onto a typical OPV structure. Recently, we have successfully applied the transmission line method (TLM) for the EQE evaluation of a hybrid polymer solar cell under normal light incidence.10 Here, we modify the aforementioned a)

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b)

0021-8979/2011/110(11)/114506/10/$30.00

model in order to be suitable for the EQE calculation under oblique light incidence, while it would also be useful for the inclined reflectivity and transmittance calculation of any type of multilayer structure (i.e., DBRs). Although the transfer matrix method (TMM) has been, almost exclusively, used for optical simulation of onedimensional multilayer structures, we propose that TLM could be an appealing, alternative, method for optical simulations as it has an increased flexibility to tackle more complicated problems. More specifically, TLM is based on the closed-form equations of the elementary transmission line theory and avoids the use of matrices. These closed-form iterative analytical formulas are easy to implement while for more complicated structures, where inhomogeneous or nonlinear media are involved, TLM could be advantageous. By decomposing each transmission line into a chain of successive equivalent two-ports, the proposed model11 may tackle similar problems effectively. It is also worth mentioning that TLM embodies the physics of stationary waves, giving a better perception of the involved interference phenomena. Apart from the aforementioned characteristics, TLM can efficiently simulate the effect of optical anisotropy, since it handles independently the refractive indices and the extinction coefficients in any direction. This effect appears to be of significant importance in the most efficient, to-date, bulkheterojunction OPVs as their photoactive layer is comprised of a blend of two compounds (poly-hexylthiophene, P3HT, and a methyl ester-fullerene derivative, [6,6]-phenyl-C70-butyric acid methyl ester C70 (C70-PCBM)) that, upon thermal annealing, exhibits an increasingly anisotropic extinction coefficient as a result of the preferential polymer chain orientation that is aligned parallel to the substrate.12–14 Using TLM, a theoretical simulation of the role of optical

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anisotropy of the blend extinction coefficient for various angles of incidence on the device external quantum efficient will also be conducted and discussed herein. This paper is organized as follows: In Sec. II, a detailed presentation of TLM and its modification for non-normal illumination are provided, and results are given for the short-circuit photocurrent calculation, based on the optical field distribution across a P3HT:C70-PCBM (1:0.8 wt. %) blend photoactive layer. In Sec. III, we examine the influence of the photoactive layer thickness in the produced photocurrent by using TLM, under different angles of incidence. The calculations are made for monochromatic illumination at 532 nm due to the strong absorption of the active material at this wavelength. Results from the optical simulations are compared with experimental results of the short-circuit photocurrent on the optically simulated device structure under inclined light incidence.

o o @ Hy jky Hz ¼ jxe0 n2xy Ex ; @z o

(2d)

o o o @ Hx jkx Hz ¼ jxe0 n2xy Ey ; @z o

o

(2e)

o

jkx Hy jky Hx ¼ jxe0 n2z Ez :

(2f)

For the above Eqs. (2), we define the following variables:10 Magnetic voltage: o

o

VM ¼ kx Hx þ ky Hy :

(3a)

Magnetic current: o

o

o

IM ¼ ky Ex kx Ey ¼ xl0 Hz :

(3b)

II. THE TRANSMISSION LINE MODEL

We consider a plane optical wave with inclined incidence on a multilayer structure as shown in Figure 1. For each layer, the dielectric permittivity ei (¼ ex ¼ ey, ez) could be considered anisotropic, whereas the equivalent refractive index is a wavelength-dependent complex number, i.e., ni ¼ n0i jn00i , apart from the incidence medium (air or glass), which is considered lossless, i.e., a real number. For a monochromatic wave, each field component Wðx; y; z; tÞ has the form (1a) Wðx; y; z; tÞ ¼ Re Wðx; y; zÞejxt ;

Electric voltage: o

o

o

(3d)

Substituting the above variables in Eq. (2), we deduce the following forms that correlate the above mentioned electric and magnetic voltages with the relevant currents:

1

o o @ Ey ¼ jxl0 Hx ; @z

o

IE ¼ kx Hy ky Hx ¼ xe0 n2z Ez :

By transforming Maxwell equations in Fourier space, the following system of six equations is obtained: o jky Ez

(3c)

Electric current:

and its spatial Fourier transformation along the x and y axes will be ð þ1 ð þ1 o Wðx; y; zÞejkx x ejky y dxdy: (1b) W kx ; ky ; z ¼ 1

o

VE ¼ kx Ex þ ky Ey :

(2a)

c2xy @VM ¼ IM ; @z jxl0

(4a)

@IM ¼ jxl0 VM ; @z

(4b)

@VE c2z ¼ IE ; @z jxe0 n2z

(5a)

@IE ¼ jxe0 n2xy VE ; @z

(5b)

where

o o o @ Ex jkx Ez ¼ jxl0 Hy ; @z

(2b)

c2xy ¼ kx2 þ ky2 x2 l0 e0 n2xy ;

(6a)

o jkx Ey

(2c)

c2z ¼ kx2 þ ky2 x2 l0 e0 n2z :

(6b)

o jky Ex

o

¼ jxl0 Hz ;

FIG. 1. Schematic representation of TE and TM optical waves impinging on a stack of successive dielectric layers.


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FIG. 2. Equivalent magnetic and electric transmission lines for TE and TM waves, respectively.

Equations (4) represent magnetic transmission lines for TE waves, while Eqs. (5) represent electric transmission lines for TM waves with characteristic impedances ZM and ZE given by qffiffiffiffiffiffi c2xy ; (7a) ZM ¼ jxl0 pffiffiffiffiffi c2z ZE ¼ ; (7b) jxe0 nz nxy n

whereas cxy and c0z ¼ nxyz cz are their transmission constants, respectively. Furthermore, for isotropic materials nxy ¼ nz ¼ n, the transmission constants will be common for both lines, c2 ¼ c2xy ¼ c2z ¼ kx2 þ ky2 x2 l0 e0 n2 :

(8)

Introducing, now in Eq. (8), the wavenumber ku for the horizontal plane x-y, we can replace the horizontal wavevector components kx and ky. If the angle of incidence is h0 and k0 ¼ 2p/k, the free space wavenumber, then we define the wavenumber ku as follows: ku2 ¼ kx2 þ ky2 ¼ k02 n20 sin2 h0 :

(9)

Therefore Eq. (6) or (8) will be converted into the following form: c2xy ¼ ku2 x2 l0 e0 n2xy ;

(10a)

c2z ¼ ku2 x2 l0 e0 n2z ;

(10b)

c2 ¼ ku2 x2 l0 e0 n2 :

(10c)

For lossy dielectric media, with complex index of refraction ni ¼ n0i jn00i , Snell’s law (n0 sin h0 ¼ n1 sin h1 ¼ ) is valid for complex propagation angles hj.2,8,15,16 By applying Snell’s law on Eq. (9), the horizontal wavenumber ku will be common for all layers, while the transmission constant of their respective transmission line will be different, due to the different refractive index of each layer. From the above analysis, it is clear that for each angle of light wave incidence, we simulate the light propagation and its interference inside a multilayer structure, by a single chain of transmission lines connected in tandem as shown in

Figure 2. For TE polarization (s-waves), the chain of lines will be magnetic, whereas for TM polarization (p-waves), they will be electric. In both cases, the individual lines will have different lengths, different characteristic impedances, and different transmission constants. The semi-infinite substrate will be simulated by its characteristic impedance, while the ambient medium with a semi-infinite transmission line (see Figure 2). For a photovoltaic cell, the number of photogenerated excitons at position z inside the active layer should be proportional to the optical field intensity at the same position. Particularly, the photocurrent should be proportional to the internal light absorption across the photoactive layer which can be, simply, calculated by means of the light intensity integration over the photoactive layer’s thickness.16 Alternatively, the light absorption could be calculated by computing the energy flux through the photoactive layer.16 Although the latter way is more elegant, the calculation of Poynting vector is necessary in order to compute the energy flux throughout the active layer. Since the calculation of Poynting vector throughout lossy materials is very demanding, we are adopting the former calculation scheme involving the computation of optical intensity, or equivalently, the optical-electric field modulus square distribution across the photoactive layer (see the Appendix). By using the model of transmission lines, the electrical field distribution along any layer of a multilayer structure corresponds to the equivalent current or voltage transmission lines’ distribution. In particular, taking into account the magnetic variables defined by Eq. (3), the electric field component of a TE wave is proportional to the magnetic current, o Ey ¼ IM =kx , and since, according to Figure 1 ky ¼ 0, the field intensity at any point inside a layer should be written as o 2 I ðzÞ2 M 2 : (11a) jEs ðzÞj ¼ Ey ðzÞ ¼ ku In Eq. (11a), the equivalence between the field intensity in Fourier space and the one in the Cartesian space can be derived from the inverse Fourier transformation as follows: Eðu; zÞ ¼

ð þ1

o

Eðku ; zÞdðku k0 n0 sin h0 Þejku u dku

1 o

) jEðu; zÞj ¼ j Eðk0 n0 sin h0 ; zÞejk0 n0 sin h0 u j;


(11b)

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where d is the Kronecker’s delta function and the wavenumber ku ¼ k0n0sinh0 is considered a real number, since we assume a lossless incidence medium. This assumption is necessary to be made, otherwise the angle of incidence should be treated as a complex one and the under study problem will not be a one-dimensional one as the proposed method impose. In particular, since the incidence medium is lossless, the horizontal component of the wavenumber is real and, due to the satisfaction of Snell’s law for complex refractive indices, is also continuous along the structure. According to the former conditions, the constant field amplitude planes are parallel to Cartesian z-plane, and as a result, the calculation of the field amplitude turns to be a spatially one-dimensional problem. In this context, the proposed method employs a single transmission line and treats the problem in Fourier space as a onedimensional one. Next, by applying the above mentioned inverse Fourier transform, the one-dimensional spatial field distribution can be calculated without any inconsistencies. Using a similar approach for TM waves, the electric field intensity inside the ith layer 2 can be expressed through its Ex and Ez components as Ep ¼ jEx j2 þ jEz j2 . Using Eqs. (3c) and (3d), the field intensity at any point inside the ith layer should be written in terms of the electric transmission line parameters as 2 2 Ep ðzÞ2 ¼ VE ðzÞ þ IE ðzÞ : k xe n2 u 0 z

(12)

From Eqs. (11) and (12), we conclude that the optical field intensity at any point inside the active layer of a photovoltaic cell can be evaluated by calculating the distribution of the current on the magnetic line that corresponds to the active layer and the distribution of voltage and current across the relevant electric one. Next, we analyze the calculation procedure of the above mentioned current and voltage distributions for both lines. Using the elementary transmission lines’ input impedance formulation, we derive the following iterative forms for the calculation of input impedance for the ith layer (see also Figure 2): ZM þ ZiM tanh cxy;i di M M iþ1;in ; (13a) Zi;in ¼ Zi M M tanh cxy;i di Zi þ Ziþ1;in E Ziþ1;in þ ZiE tanh c0z;i di E : ¼ ZiE (13b) Zi;in E ZiE þ Ziþ1;in tanh c0z;i di ZiM and

ZiE

are the magnetic and electric line In Eq. (13), characteristic impedances, respectively, of the ith layer calM E and Ziþ1;in are the input impeculated through Eq. (7); Ziþ1;in dances of the next layer, and ci are the transmission constants that can be calculated from Eq. (10). Starting now M E ,Zmþ1 ) from the substrate’s characteristic impedances (Zmþ1 and using the recursive equations (13), the input impedances M E and Z1;in at the left-most side of both transmission lines Z1;in can be easily calculated. Next, the current and voltage reflection coefficients are given by

qM I ¼

M Z0M Z1;in M M ; Z0 þ Z1;in

(14a)

qEV ¼

E Z1;in Z0E ; E þ ZE Z1;in 0

(14b)

qEI ¼

E Z0E Z1;in : E Z0E þ Z1;in

(14c)

From Eq. (14), we can easily derive optical-electric field the 2 (for TE polarization) intensity coefficients qM I E 2 reflection E 2 and qV , qI (for TM polarization), which are the figures of particular interest in periodic multilayer structures like DBRs. Nevertheless, for organic photovoltaic devices, the optical field distribution inside the active layer is more significant as it is a direct measure of photocurrent generation. For the calculation of the electric field distribution inside the active layer, the transmission lines’ input current and voltage are required. Considering I0M and I0E ,V0E as the currents and voltage that represent the TE and the TM incident waves, respectively, from the ambient on the interface with the first dielectric layer 1, we can calculate the currents I1M , I1E and voltage V1E that are transmitted inside layer 1 as follows: I1M ¼ I0M 1 þ qM (15a) I ; (15b) I1E ¼ I0E 1 þ qEI ; (15c) V1E ¼ V0E 1 þ qEV : Using Eq. (15) and starting from I1M , I1E , and V1E , the currents and voltage on any interface between two successive layers will be given by the following iterative equations: M ¼" Iiþ1

IiM

#;

(16a)

#;

(16b)

#: ZiE 0 0 cosh cz;i di þ E sinh cz;i di Ziþ1;in

(16c)

M Ziþ1;in cosh cxy;i di þ M sinh cxy;i di Zi

E ¼" Iiþ1

cosh

E ¼" Viþ1

c0z;i di

IiE

E Ziþ1;in þ E sinh c0z;i di Zi

ViE

Furthermore, the expressions of current and voltage at any point z inside the ith layer of the structure are as follows: " # M Z iþ1;in M cosh cxy;i z þ M sinh cxy;i z ; (17a) IiM ðzÞ ¼ Iiþ1 Zi " E IiE ðzÞ ¼ Iiþ1

# E Z iþ1;in cosh c0z;i z þ E sinh c0z;i z ; Zi

" ViE ðzÞ

¼

E Viþ1

cosh

c0z;i z

þ

ZiE E Ziþ1;in

sinh

c0z;i z

(17b)

# :


(17c)

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For the OPV application, the number of photogenerated excitons inside the photoactive layer (i.e., the ith layer) at a position z under normal or non-normal incidence should be directly related to the modulus square of the current and voltage at the same position.4 Considering that the unpolarized incident light is split equally into TE and TM waves, the time average of the energy dissipated in the photoactive layer in position z is estimated as follows (see also the Appendix for the complete proof): 2 2p ce0 0 00 ~ ~y ðz;kÞ2 nxy;i nxy;i Ex ðz; kÞ þ n0xy;i n00xy;i E k ~z ðz;kÞ2 þ n0z;i n00z;i E

2 2 p ce0 0 00 M ¼ 2 nxy;i nxy;i Ii ðz; kÞ þ n0xy;i n00xy;i ViE ðz;kÞ ku k E 2 0 00 2 Ii ðz; kÞ ; (18) þ nz;i nz;i ku xe0 n2z

Qi ðz; kÞ ¼

where Ii ðz; kÞ and Vi ðz; kÞ are the currents and voltage across the active layer. In particular, IiM ðz; kÞ is the current of the magnetic line and can be obtained from Eq. (17a), whereas IiE ðz; kÞ and ViE ðz; kÞ are the current and the voltage of the electric line, which can be obtained from Eqs. (17b) and (17c). If we assume a bulk-heterojunction OPV device, then both the magnitude and the spatial distribution of the optical field inside the blend photoactive layer should contribute to exciton dissociation and charge photogeneration. Consequently, a device efficiency metric should take into account the total absorbed energy in the photoactive layer that is proportional to the optical-electric field modulus square. Thus, the following efficiency metric can be adopted for a discrete wavelength k: ð di 0 Qdi i ðkÞ ¼ ð 1

Qi ðz; kÞdz :

(19)

Qi ðz; kÞdz

0

This parameter gives the total absorbed light energy in the active layer, normalized by the total absorbed energy in an infinitely thick active layer. It could, therefore, be used for thickness optimization purposes, when the light absorption of the active blend material is dominated by a narrow spectrum around that wavelength. In this case, an estimation of short-circuit photocurrent density could be derived assuming internal quantum efficiency ¼ 1 (i.e., each absorbed photon is converted to an extracted electron): Jsc ¼

qe Pin hc

ð di 0

Qdi i dz;

(20)

where Pin is the monochromatic source power and Qdi i denotes Qi divided by the incident light intensity, 2 2 Qdi i ¼ 2k0 n0xy;i n00xy;i IiM ðz; kÞ þ n0xy;i n00xy;i ViE ðz;kÞ 2 þ n0z;i n00z;i IiE ðz; kÞ : (21) 2 In Eq. (21), IiM ðz; kÞ expresses the magnetic current modulus squared, normalized by the incident TE optical-electric

2 2 field modulus square, while ViE ðz;kÞ and IiE ðz; kÞ express the electric voltage and current modulus squared, normalized by the incident TM optical-electric field modulus square, respectively. Extending now the calculation of the short-circuit photocurrent density for a non-monochromatic light source, Eq. (20) could be written as follows: Jsc ¼

qe hc

ð kH kL

P0in ðkÞ

ð di 0

Qdi i dzkdk:

(22)

In Eq. (22), P0in ðkÞ is the source power density spectrum, i.e., the one provided by the solar light. III. NUMERICAL AND EXPERIMENTAL RESULTS

The proposed method has been previously applied to optically simulate a hybrid polymer solar cell and optimize its structure for maximum efficiency with an optical spacer layer inserted between the photoactive layer and the metal cathode. Nevertheless, only normal light incidence had been examined under sunlight spectrum conditions. Herein, the method has been modified in order to evaluate a typical bulk-heterojunction OPV device under an oblique monochromatic light source. The device structure consists of a 100 nm indium tin oxide (ITO) layer on a glass substrate, a 50 nm poly(3,4-ethylenedioxythiophene):poly(styrenesulfonate) (PEDOT:PSS) layer, a blend of poly(3-hexylthiophene) (P3HT) and C70-PCBM (1:0.8 wt. %) with variable thickness as the photoactive material, and an aluminum layer of 150 nm thickness on the top of them. Although the model could be implemented for a non-monochromatic light source, in our calculations, we decided to use a monochromatic one for two reasons: First, the active material’s peak absorption is located at a wavelength of 530 nm with a relative narrow bandwidth, and second, it is more convenient to use a monochromatic light source in order to evaluate accurately the influence of the angle of incidence on the OPVs quantum efficiency. Next, we optically simulate the aforementioned structure (glass/ITO/PEDOT:PSS/P3HT:C70-PCBM/Al) using refractive indices of the individual materials from the literature6,14,17–19 at a wavelength of 532 nm and assuming that the active layer is optically isotropic (this generally corresponds to a non-annealed layer). In Figure 3, the opticalelectric field distribution inside the ITO, PEDOT:PSS, and P3HT:C70-PCBM (thickness of 210 nm) layers is illustrated for 0 , 30 , and 60 angle of incidence. A continuous, constructive and destructive, interference pattern is evident at all angles, irrespective of the polarization. The magnitude of the field modulus square is largest for normal incidence. As the angle of incidence increases, the field magnitude is reduced, thus resulting in a reduced photogeneration and a lower photocurrent. Also from Figure 3, we note that the peaks and valleys (local maxima and minima) of the field across the various layers shift away from the metal cathode, when the angle of incidence is increased. Additionally, for TM polarization, a field discontinuity is observed on both PEDOT:PSS and aluminum/active layer interfaces, and the


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field does not completely vanish at the metal interface. These discontinuities are well explained from the boundary conditions that hold only for the normal to the interface electrical field component. In our case, it is the Ez field component which is discontinuous due to the strong refractive index difference between the two dielectric materials. However, it should be mentioned that this discontinuity is revealed for higher angles of incidence, where the z component of electric field is larger. Therefore, field discontinuity on these interfaces, for high incident angles, needs additional attention when calculating the device’s internal quantum efficiency.8 Moreover, with increasing angle of incidence, the interference pattern is expanded and the distance between field maxima (i.e., antinodes) and minima (i.e., nodes) is increased, according to 1/cosu where u is the angle of light refraction (angle to the interface normal), consistent with Bragg’s law.8 Generally, we note that our findings, which are in excellent agreement, both qualitatively and quantitatively, with previously reported OPV simulations using TMM,8 can easily be verified using TLM.

Furthermore, we examine the influence of active layer thickness in the OPVs efficiency. In Figure 4, the external quantum efficiency (normalized with respect to an infinite active layer thickness) has been calculated versus the active layer thickness for different angles of incidence. Due to light wave interference inside the multilayer structure, local efficiency minima and maxima are found for discrete active layer thicknesses with a progressive increase of the maximum efficiency as the active layer thickness becomes larger (and up to 60 ).4 Now, examining the first local maxima, it is evident that for a thickness of approximately 80 nm, the efficiency appears to be relatively insensitive to the incident angle. However, it should generate less photocurrent when compared to a device with a 210 nm active layer thickness, which, on the contrary, is more sensitive to the angle of incidence. Furthermore, for angles greater than 30 , the interference pattern appears to be compressed, particularly for photoactive layer thicknesses above 150 nm. Actually, for higher inclination angles, the optical wave is expected to travel along a longer path inside each layer, and this causes compression of interference pattern.8 This behavior, however, is mainly determined by the enhanced optical reflectivity that the device exhibits in higher angles of incidence. Choosing an active layer thickness of 200 nm, we then calculate the optical field reflection coefficient on the glass/ ITO interface for angles from 0 to 60 . Due to the quite large refractive index difference between glass and ITO, this interface is expected to dominate the reflectance characteristics of the fabricated OPVs with the other internal interfaces playing a lesser role. Typically, the OPVs wavelengthdependent reflection is more informative, since it illustrates its behaviour as an optical mirror in the visible spectrum. In Figure 5, this behaviour is clearly revealed at the wavelength of 530 nm. The most striking feature is that, for angles greater than 30 , the reflectivity is much higher (for the aforementioned wavelength), in particular, for the TE polarization. This result clearly shows that as the angle increases both more TE and TM polarized light are reflected at the

FIG. 4. The normalized external quantum efficiency, for different angles of incidence, versus the active layer thickness (assuming an isotropic active layer).

FIG. 5. The optical field reflection coefficient on the glass/ITO interface, for TE and TM waves and for different angles of incidence, versus the optical wavelength.

FIG. 3. Modulus square of the optical-electric field and its spatial distribution across a 210 nm thick P3HT:C70-PCBM based OPV for both TE and TM incident monochromatic light (k ¼ 532 nm) polarizations and three (0 , 30 , and 60 ) angles of incidence.


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ITO/glass interface. At 60 , the reflection coefficient of the TE polarized light is a factor of 2 higher than that of the TM polarized light. This subsequently reduces the actual light power that is transmitted and then absorbed in the active layer, thus obviously resulting in a lower photocurrent at larger angles. It also explains the lower degree of interference effects and the compression of the familiar interference pattern inside the device for higher angles of incidence that was observed in Figure 4. If we now consider that the active layer becomes strongly anisotropic after thermal annealing,12 the simulation results of Figure 4 (isotropic case) change drastically. If we define n00xy and n00z as the parallel and perpendicular component of the extinction coefficient, respectively, and n0 xy and n0 z the corresponding parallel and perpendicular component of the refractive index, then by using the anisotropy of the active layer’s complex refractive index (ni ¼ n0i jn00i ),12 we calculate the device external quantum efficiency (normalized with respect to an infinite active layer thickness) versus the active layer thickness for different angles of incidence. For n0xy ¼ 1.81, n0z ¼ 1.828, and n00xy ¼ 0.409, n00z ¼ 0.164 (Ref. 14) (whereas for the isotropic case of Figure 4, n ¼ 1:853 j0:15), the results are illustrated in Figure 6 and show that the quantum efficiency is significantly reduced when the angle of incidence is increased for all active layer thicknesses. Moreover, the interference pattern and, in particular, the ripple of local minima and maxima have been severely compressed in comparison with the respective isotropic ones, illustrated in Figure 4. However, the first local maximum at approximately 80 nm thickness shows an improved EQE in comparison with the respective isotropic one, for angles of incidence up to 60 . In Figure 7, a similar simulation plot is shown, considering now a smaller anisotropy of the extinction coefficient, i.e., n00xy ¼ 0.254 and n00z ¼ 0.127. In this case, the interference pattern demonstrates a state between the fully expanded and the highly compressed pattern of the fully isotropic and the highly anisotropic extinction coeffi-

FIG. 6. The normalized external quantum efficiency, for different angles of incidence, versus the active layer thickness. An anisotropic active material has been considered with parallel and perpendicular extinction coefficient values of 0.409 and 0.164, respectively.

J. Appl. Phys. 110, 114506 (2011)

FIG. 7. The normalized external quantum efficiency, for different angles of incidence, versus the active layer thickness. Parallel and perpendicular extinction coefficient values of 0.254 and 0.127, respectively, have been used for the active layer.

cient of the active material, respectively, with local maxima and minima in approximately the same active layer thicknesses, as the ones shown in Figs. 5 and 6. These simulation results are potentially very important as they clearly illustrate that, in a strongly anisotropic P3HT:PCBM layer, there appears to be almost no reward in making the absorbing layer thicker as the first local maximum is also the highest maximum, in terms of quantum efficiency and photocurrent generation. As the optical anisotropy of this layer decreases, the effect of the absorbing layer thickness becomes more critical in photocurrent generation as shown in Figs. 4, 6, and 7, respectively, with the limit being the isotropic case where a continuous increase of the absorber thickness would create local absolute quantum efficiency maxima. On the other hand, it is worthwhile to discuss the underlying physics that interprets the external quantum efficiency dependence on the angle of light incidence. In general, the horizontal field components have an increased contribution to the total photocurrent, if the index anisotropy is higher for the horizontal direction. Note that anisotropy will be enhanced under standard thermal annealing processing used for high-performance P3HT:PCBM solar cells as it preferentially and increasingly (as a function of temperature/time) orients the P3HT backbone chains (and the corresponding transition dipole moments) parallel to the substrate, thus resulting in an increased light absorption for TE polarization and partially for TM polarization, i.e., only for the horizontal field components. We note that as the angle of incidence is increased, the direction of the optical-electric field component of the TM-polarized light will become increasingly perpendicular to the substrate, thus reducing the significance of extinction coefficient n00xy on the value of Q (and consequently on the light absorption), while the direction of the TE-field component will not change as a function of angle as it will always be parallel to the substrate, thus the significance of n00xy will remain the same. Conclusively, TLM simulations clearly elucidate the important role that the optical anisotropy, particularly in the


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extinction coefficient of the active material, plays in determining the external quantum efficiency and the short-circuit photocurrent angular pattern characteristics of OPVs. In this sense, it is also revealed that the various post-fabrication treatments (e.g., thermal or solvent annealing) may have a significant influence on the device efficiency. Next, we evaluate the simulation results when optical anisotropy is included by measuring the short-circuit photocurrent of three representative OPV devices with active layer thicknesses of 250, 180, and 110 nm, respectively, as a function of the incident angle. In brief, OPVs were fabricated on oxygen plasma-cleaned ITO coated glass substrates (ITO layer thickness 100 nm). PEDOT:PSS, obtained from Aldrich, was used as an anode interfacial layer to planarize the ITO and improve hole extraction. The 40 nm thick PEDOT:PSS layer was spin-cast from an aqueous solution and then annealed at 130 C for 15 min. Next, a layer of the blend of P3HT, obtained from Rieke Metals, and C70-PCBM (obtained from Solenne) with a P3HT:C70-PCBM mass ratio of 1:0.8 was spin-cast on PEDOT-PSS from a 2 wt. % chlorobenzene solution. Note that the polymer solution had been stirred overnight to allow complete dissolution and, prior to spin coating, was filtered using a 0.20 lm polytetrafluoroethylene (PTFE) filter. After spin coating, the polymer blend layer was allowed to slowly dry in a Petri dish and then annealed at 150 C for 5 min. Finally, an 150 nm thick Al layer was vapour deposited through a shadow mask, onto all samples, to define an active cell area of 12.56 mm2. The thicknesses of the polymer layers were measured with an Ambios XP-2 profilometer, whereas the device short-circuit photocurrent was measured with a Keithley 2400 sourcemeasure unit. The light source was a 5 mW green laser (k ¼ 532 nm) that was attenuated by an appropriate neutral density filter. Oblique illumination measurements were performed by rotating the light source in steps of 10 and then focusing the laser beam to cover the entire device area at each angle. All measurements were carried out in air at room temperature.

FIG. 8. Calculated (solid lines) and measured (symbols) short-circuit photocurrent for three different active layer thicknesses versus the angle of light incidence. A best-fit to the experimental measurements was carried out by varying the active layer’s extinction coefficient anisotropy.

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In Figure 8, the normalized experimental short-circuit photocurrent (to its value at normal incidence) versus the angle of incidence is illustrated for the aforementioned devices with variable active layer thickness. Experimental curves for the thicker (250 nm) film show a maximum photocurrent at normal incidence with a continuous decrease at larger angles. Overall, the thicker active layer exhibits the larger photocurrent, regardless of the incident angle. However, the photocurrent of the other two samples (having 110 and 180 nm thick active layers) shows a different bahaviour with distinct, local maxima at 40 and 30 , respectively. These local maxima as well as the overall angular dependence can be explained from the general constructive/destructive form of the stationary wave and the spatial distribution and magnitude of the modulus-square of the optical-electric field that is present, in particular, inside a very thin photoactive layer. For the particular combination of the angle of incidence, the wavelength of the incident light, and the layer’s thickness, it appears that constructive effects may be enhanced, producing these photocurrent local maxima. Simulation curves have been fitted to the experimental photocurrent values by changing the optical anin00 sotropy ratio A ¼ nxy00 . As a result, the best-fit of the optical z anisotropy ratio is found to be A 2 with n00xy ffi 0:25. This ratio appears to reproduce very well the experimental data curves for the thicker (250 nm) and the thinner (110 nm) active layers, while a slight discrepancy is noted for the 180 nm thick active layer. It also appears to be slightly lower than the ones reported in literature (typical value of A is 2.5)12,14 but differences on materials properties, processing methods, as well as environmental conditions could reasonably explain this small deviation. In particular, the anisotropic refractive index values are influenced by several material properties and processing parameters. For example, it has been reported that both the type of annealing, i.e., thermal vs. solvent annealing as well as various annealing processing parameters, temperature and annealing time play a significant role in determining, primarily, the extinction coefficient anisotropy. Active materials characteristics such as the mass fraction of PCBM in the blend as well as the degree of regioregularity and the molecular weight of P3HT affects also the anisotropy of the film optical properties. In addition, fabrication process methods, namely, the solvent, the spin casting speed and time, and the environmental conditions, have also been proposed to influence optical anisotropy. Furthermore, taking into account the previously mentioned remarks, we extend the photocurrent simulations to P3HT:PCBM solar cells, under inclined incidence of the solar light. A typical relative experiment should use a solar simulation lamp, while the simulated results could be calculated using the TLM. Here, we focus on device simulations and in Figures 9 and 10, we calculate the normalized short circuit photocurrent density Jsc under the AM1.5G solar spectrum versus the photoactive layer thickness for various incidence angles. In this calculation, Jsc is normalized with respect to the maximum photocurrent that can be produced by an infinitely thick active layer. The results illustrated in Figure 9 concern the isotropic active material of Figure 4,


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IV. CONCLUSIONS

FIG. 9. Normalized short circuit photocurrent, for different angles of incidence, versus the active layer thickness (assuming an isotropic active layer and AM1.5G solar light incidence).

while those in Figure 10 concern the anisotropic one of Figure 6. A general remark for both sets of results is that the interference ripple pattern is smoother for the solar light compared to the green monochromatic one, especially for the isotropic case. This is reasonable due to co-propagation of almost equally strong optical waves with different wavelengths, which are absorbed in the photoactive layer and which appear to counteract and reduce interference phenomena. However, the active layer thicknesses that correspond to the local minima and maxima have also been observed around the same thicknesses as in Figures 4 and 6, because the active layer’s material presents a strong absorption spectral selectivity in a narrow band around 530 nm (not shown here). This latter characteristic is more intense for the anisotropic active material, and as a result, higher ripples in the interference pattern are illustrated in Figure 10, in comparison to Figure 9, while a higher percentage of the maximum produced short-circuit photocurrent could also be achieved for a smaller active layer thickness.

An alternative, to the transfer matrix method, method for the calculation of optical interference phenomena in dielectric multilayered structures has been developed. The method is based on the transmission line theory and is exact, easy to implement and uses closed iterative forms instead of matrix formalism. The proposed method has been applied for the evaluation of a typical organic photovoltaic device under an inclined light incidence. Optical field distribution, short-circuit photocurrent, and reflectivity have been calculated under different angles of incidence for an isotropic active material. By introducing the photoactive material’s extinction coefficient anisotropy that is significantly enhanced after its thermal annealing, theoretical simulations show a balance between the local maxima of the EQE curves, versus the active layer’s thickness, at any angle of incidence. Using the proposed model, photocurrent experimental data have been examined for the best-fitted active layer’s anisotropy. Under the particular experimental conditions, the derived extinction coefficient anisotropy is found to be somewhat lower than the values reported in the literature and reasons for this deviation are discussed. APPENDIX: CALCULATION OF TIME AVERAGE OF THE ENERGY DISSIPATED IN THE PHOTOACTIVE LAYER

By applying the divergence theorem on the average ~ H ~ , we have Poynting’s vector S~r ¼ 12 Re E ð ð S~r d S~ ¼ rS~r dV; (A1) S

V

where S is the surface that encloses volume V. From Maxwell’s equations the divergence of the average Poynting’s vector will be as follows: 1 ~ ~ rS~r ¼ Re r E H 2 1 ~ ~ E ~ r H ~ rE ¼ Re H 2 1 ~ ~ E ~ jxe E ~ jxlH ¼ Re H 2

jxl ~ ~ jxe0 ~ ~ Ee E : HH þ ¼ Re 2 2

(A2)

For anisotropic and lossy media, e* is the conjugate of the complex relative dielectric diagonal tensor, 2 x 3 er þ jexi 0 0 6 7 0 5 eyr þ jeyi e ¼ 4 0 2 6 ¼4 FIG. 10. Normalized short circuit photocurrent, for different angles of incidence, versus the active layer thickness (assuming an anisotropic active layer with the complex refractive index characteristics used in the calculations of Fig. 6 and AM1.5G solar light incidence).

0

ezr þ jezi

0

ðnxr þ jnxi Þ2

0

0

3

0

ðnyr þ jnyi Þ2

0

7 5:

0

0

ðnzr þ

(A3)

jnzi Þ2

Substituting Eq. (A3) in Eq. (A2), we obtain the well known Poynting theorem for lossy anisotropic media,


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ð 2 xeo x ~ ~ Sr d S ¼ ei jEx j2 þ eyi Ey þ ezi jEz j2 dV: (A4) 2 V S

Introducing the anisotropic complex refractive index, Eq. (A4) becomes as follows: ð ð 2 S~r d S~ ¼ xeo nxr nxi jEx j2 þ nyr nyi Ey þ nzr nzi jEz j2 dV: S

V

xe0 DP ¼ DxDy 2

ð z2 z1

2 exi jEx ðzÞj2 þ eyi Ey ðzÞ þ ezi jEz ðzÞj2 dz: (A8)

Equation (A8) represents the optical power absorption from an elementary photocell area (DxDy). By introducing the relative complex refractive index, we deduce the form of Eq. (18) for the power absorption density inside the photoactive layer,

(A5) Furthermore, the negative sign in Eq. (A5) corresponds to power absorption inside volume V. Assuming now, without loss of generality, an orthogonal volume ranging from z ¼ z1 to z2, x ¼ 0 to Dx, and y ¼ 0 to Dy, both surface and volume integral can be calculated. Actually, the surface integral value represents the difference between the total incoming Pin T and the total outgoing power, through surface S. This difference is proportional Pout T to the optical power absorption inside the photoactive material’s volume V, ð in S~r dS~ ¼ Pout (A6) T PT ¼ DP: S

The amplitude of each component of the optical field inside E ~ ¼ EðzÞ ~ eko ay ek0 ax , the photoactive layer is of the form ~ represents the stationary wave along the z axis where EðzÞ due to interference phenomena from the multilayer structure perpendicular to z axis. On the other hand, the exponential decay along the horizontal plane is due to the assumption of lossy materials and the absence of standing waves, whereas a is the attenuation coefficient along x and y axis. Integrating now over dx and dy, we obtain a reduced version of the volume integral, ð z2 ð Dy ð Dx

~ 2 dxdydz ei e2ko ax e2ko ay EðzÞ z1 y¼0 x¼0 ð z2 2k aDy 1 ~ 2 dz: ei EðzÞ ¼ e o 1 e2ko aDx 1 2 2 4k0 a z1 (A7)

I¼

For small replacements Dx and Dy, from Eqs. (A5)–(A7), DP is calculated as follows:

DP=ðDxDyÞ ð z2 2 nxr nxi jEx ðzÞj2 þ nyr nyi Ey ðzÞ þ nzr nzi jEz ðzÞj2 dz: ¼ xeo z1

(A9) 1

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