Reflection and transmission calculations in a multilayer structure with coherent, incoherent, and partially coherent interference, using the transmission line method N. A. Stathopoulos,1,* S. P. Savaidis,1 A. Botsialas,1 Z. C. Ioannidis,2 D. G. Georgiadou,3 M. Vasilopoulou,3 and G. Pagiatakis4 1

Technological and Educational Institute of Piraeus, Department of Electronics Engineering, 12244 Aegaleo, Greece 2

National and Kapodistrian University of Athens, Physics Faculty, 15784 Athens, Greece

NCSR “Demokritos”, Institute of Advanced Materials Physical Chemistry Processes and Micro-Nano-Electronics, P.O. Box 60228, 15310 Agia Paraskevi Attiki, Greece

3

4

School of Pedagogical and Technological Education (ASPETE)–Marousi Athens, Greece *Corresponding author: [email protected] Received 11 November 2014; revised 17 January 2015; accepted 20 January 2015; posted 20 January 2015 (Doc. ID 226745); published 20 February 2015

A generalized transmission line method (TLM) that provides reflection and transmission calculations for a multilayer dielectric structure with coherent, partial coherent, and incoherent layers is presented. The method is deployed on two different application fields. The first application of the method concerns the thickness measurement of the individual layers of an organic light-emitting diode. By using a fitting approach between experimental spectral reflectance measurements and the corresponding TLM calculations, it is shown that the thickness of the films can be estimated. The second application of the TLM concerns the calculation of the external quantum efficiency of an organic photovoltaic with partially coherent rough interfaces between the layers. Numerical results regarding the short circuit photocurrent for different layer thicknesses and rough interfaces are provided and the performance impact of the rough interface is discussed in detail. © 2015 Optical Society of America OCIS codes: (310.6860) Thin films, optical properties; (240.5770) Roughness; (240.0310) Thin films; (230.4170) Multilayers; (250.2080) Polymer active devices; (250.3680) Light-emitting polymers. http://dx.doi.org/10.1364/AO.54.001492

1. Introduction

Thin-film multilayer structures with dielectric materials are used in numerous optical applications and photonic devices. Among them, semiconducting organic multilayer structures for light-emitting and photo-detecting devices have been attracting increased interest in the last decades. Their common characteristics are the transparency in the visible 1559-128X/15/061492-13$15.00/0 © 2015 Optical Society of America 1492

APPLIED OPTICS / Vol. 54, No. 6 / 20 February 2015

spectrum and their small vertical dimensions in comparison with their horizontal dimensions. Several modeling efforts of such multilayer structures consider the interfaces between their layers as ideally smooth and parallel to each other, while thick layers are considered semi-infinite. Consequently, the stack of the successive layers is a one-dimensional (1D) problem and the calculation of light reflection and transmission throughout these structures is a typical problem of wave propagation through coherent layers and interfaces. However, the roughness of the interface between two layers could be taken into

account by considering the propagation of optical waves through them as being partially coherent [1–3], whereas bulk layers of optical materials could also be taken into account by considering them as incoherent optical media [1–3]. The calculation of the light reflection and transmission throughout generic multilayer structures, with not only coherent but also partial coherent and incoherent layers, has been treated so far by using the transfer matrix method (TMM) [1–3]. An alternative to the TMM method, based on transmission line theory, has been recently proposed to treat coherent 1D calculations in organic light-emitting diodes (OLEDs) and organic photovoltaics (OPVs) devices [4–6]. In this context, the transmission line model (TLM) can be extended to consider partial coherency and incoherency and thus to introduce a global calculation scheme for both OLED and OPV devices. Furthermore, TLM is an electrical model, which is much closer to the Maxwell’s equations which are the common origin of all solutions in propagation problems. In this sense, the TLM provides a more straightforward physical insight and flexibility in handling refractive index anisotropy, inhomogeneity, and nonlinearity, when appropriate. These inherent virtues of the TLM are the motivation for the present work to generalize it, in order to include the partial coherent and incoherent layers and thus establish a global calculation framework equivalent to the TMM alternative presented in [1–3]. In addition, due to the numerous potential applications of OLED and OPV devices in the fields of solid-state lighting, modern display devices, and renewable energy [7–15], the present work aims also to demonstrate how this global transmission line calculation scheme may facilitate specific fabrication and design requirements that may appear in these devices. For example, the general method for fabricating OLEDs is vapor deposition under vacuum, which demonstrates critical drawbacks. The most notable of them is the pixilation, since the use of evaporation masks limits the fabrication scalability and resolution. Other limitations include the low use of the expensive OLED materials (∼20%) and the high manufacturing costs. On the contrary, solution processed OLEDs do have advantages over thermal evaporation processing because of their low-cost and large-area manufacturability using spin-coating, inkjet-printing, blade coating, and screen printing techniques [16,17]. However, with these solutionbased deposition techniques, it is rather difficult to achieve an accurate determination of each individual layer thickness even though the film’s thickness together with its spectral refractive index and spectral extinction coefficient are playing a key role in the determination of the interference effects and thus in the device performance. Actually, the optical interference effects, throughout the stack of the OLED’s dielectric layers, determine the outcoupling efficiency, the optical radiation pattern, and the losses due to the waveguiding phenomena. The interference

phenomena can be precisely determined, if the layers’ thicknesses and dielectric characteristics are known and the interfaces between them are considered coherent [4]. Well-known methods are available for the measurement of the refractive index (ellipsometry) and the thickness (profilometry and spectral reflectivity). Ellipsometry gives accurate results for the refractive index and the extinction coefficient of a material, whereas spectral reflectivity gives accurate results for the thickness of a thin film, when it is deposited on a reference (e.g., silicon) substrate. Nevertheless, in the case of the OLEDs, it is important to measure the thickness of each film during the coating process of layers, which typically starts with a coated glass substrate and continues with the deposition of each layer on top of a multilayer structure. In this case, the thickness measurement of individual layers is difficult, due to the small amount of light reflection from the surface of the multilayer structure. In addition, the correct interpretation of the reflected light measurement should take into account that the glass substrate acts as an incoherent bulk film. The herein presented TLM version can overcome these obstacles since it treats the glass substrate as an incoherent layer as well the rough interface as partial coherent. Moreover, TLM estimations are mainly based on constructive/destructive interference effects, which are rather evident even in the low-level measurement regime. In the previously mentioned context, as a first application of the TLM, an in-process film thickness measurement technique is proposed, considering that the successive layers are deposited on a bulk incoherent glass substrate. The reflected light is measured by a spectrometer and the reflectance spectrum is calculated. Next, by using a fitting approach between the experimental spectral reflectance measurements and the corresponding TLM calculations the film thickness is estimated. Furthermore, polymeric PV cells consist also of thin-film multilayer structures. They are based on a photoactive layer composed of a polymeric donor and a fullerene acceptor blend, which forms a phaseseparated bulk heterojunction (BHJ) nanostructured network that provides a large interfacial area for exciton dissociation. Currently, polymeric PV cells have reached certified power conversion efficiencies above 10% [18–20]. These successful device architectures present a stack of different organic and inorganic thin films, where their dielectric characteristics (thickness, spectral refractive index, and spectral extinction coefficient) play a dominant role in the determination of the optical interference effects and thus in their efficiency. In particular, since the distribution of the optical field intensity inside the photoactive layer determines the total photocurrent to be produced, the interference effects inside this layer have been studied extensively in recent years [5,6,21]. So far, the interference phenomena can be precisely determined if the layers’ thicknesses and dielectric characteristics are known and the 20 February 2015 / Vol. 54, No. 6 / APPLIED OPTICS

1493

interfaces between them are considered coherent. Nevertheless, the interfaces between polymer layers that are spin coated one on top of each other from orthogonal solvents, usually demonstrate a roughness, mainly due to the inter-diffusion between two soluble materials. Moreover, metal oxides that are frequently used as electron or hole transport layers, demonstrate typically a rough interface with the succeeding layer. Consequently, there is a strong motivation to analyze the influence of a potential interface roughness in the external quantum efficiency (EQE) of an OPV device. Thus, the TLM will be used in order to simulate an OPV structure with a partial coherent interface and numerical results will be provided. The paper is organized as follows: Section 2 describes the deployment of the TLM in order to simulate a general coherent, incoherent, and partial coherent planar dielectric layer structure. In Section 3, numerical results from the application of the method in film thickness measurement, as well as from its application on the calculation of the EQE of an OPV with rough interfaces, is discussed. In the last section, the conclusions of this work are presented. 2. Theory of the Transmission Line Model

The deployment of the TLM in order to calculate the reflection and the transmission coefficient of a multilayer dielectric structure is presented in this section. The presentation is organized in four subsections, which include an introduction to the equivalent TLM, the simulation of multilayer structures with coherent or incoherent layers as well as with rough partial coherent interfaces. Although each case is analyzed independently, the proposed calculation methods could be applied in structures with combined characteristics. A.

Equivalent Transmission Lines in Dielectric Media

We consider a monochromatic optical plane wave propagating along the z axis and impinging normally to a multilayer dielectric structure, as shown in Fig. 1. Each layer is characterized by its thickness di and complex refractive index ni , while the incidence medium #0 is lossless with refractive index n0 and the substrate is a semi-infinite medium with refractive index nN
1 . Using Maxwell’s equations,

Fig. 1. Normal incidence of an optical plane wave on a coherent multilayer dielectric structure. 1494

APPLIED OPTICS / Vol. 54, No. 6 / 20 February 2015

the electromagnetic field components Ey and H x fulfill the expressions [5] ∂Ey  jωμ0 H x ; ∂z

∂H x  jωε0 n2 Ey : ∂z

(1)

By defining the variables V M  H x and I M  Ey , Eqs. (1) represent magnetic transmission lines with characteristic impedance Zc and transmission constant γ as follows [5]: s














jωε0 n2 n ;  Zc  jωμ0 120π γ

q






































 jωε0 n2  ×  jωμ0   jk0 n;

(2)

(3)

where n is the wavelength-dependent complex refractive index of each layer at the normal direction and k0 is the wavenumber. The case of inclined incidence is rather complicated and can be simulated using the TLM, but in this case the field components are treated in Fourier space. In particular, by applying spatial Fourier transformation with respect to x and y, and by defining magnetic and electric current and voltage as a linear combination of electric and magnetic field components, a pair of transmission lines—one magnetic and one electric—is deduced, representing TE and TM waves, respectively [6]. For brevity reasons the hereinafter analysis of the TLM deployment considers only the normal incidence case, where the two polarizations coincide and the two transmission lines degenerate into one. Nevertheless, the methodology for treating the inclined incidence is fairly similar to the normal incidence with the above-mentioned difference that TLM is using a pair of transmission lines to take into account both TE and TM polarization. It is worth mentioning at this point that inclined incidence as well as index anisotropy has also been treated by TMM in [22,23]. B. Reflection and Transmission Calculation from a Coherent Multilayer Structure

We consider a stack of N thin films having a semiinfinite substrate and a semi-infinite incident medium [see Fig. 2(a)]. All films have thickness of the order of the incident light wavelength and the interfaces are considered perfectly smooth. Consequently, the optical wave propagating through them will follow the reflection and transmission law, forming standing waves throughout them. Using the equivalent transmission line circuit of Fig. 2(b), we may calculate the optical intensity reflection coefficient jρj2 and the transmission coefficient jtj2 inside the substrate for each wavelength in the spectrum of operation. Each layer #ii  1 to N is represented by a transmission line with thickness di , refractive index ni , characteristic impedance Zi , and transmission coefficient γ i , whereas the

(a)

(b) Fig. 2. (a) A stack of planar coherent layers and (b) its equivalent transmission line circuit for normal incident of light.

semi-infinite media are represented by their characteristic impedances Z0 and ZN
1 , respectively. The current I 0 represents the normal incident optical electric field, whereas I i is the input current of layer #i and I N
1 the input current of the substrate. The coefficient jρj2 is estimated through the calculation of the transmission line’s input impedance, whereas the coefficient jtj2 is estimated through the calculation of the optical intensity ratio jI N
1 ∕I 0 j2 . Using elementary transmission line theory, the optical intensity reflection coefficient jρj2 , where ρ is the current reflection coefficient, is given by the following expression: ρ

Z0 − Zin ; Z0
Zin

(4)

where Zin is the input impedance of the equivalent transmission lines connected in tandem and is calculated iteratively as follows: Zin  Zin  Z1 1  Z2 Zin 2

Zin
Z1 tanhγ 1 d1  2 Z1


Zin 2

tanhγ 1 d1 

Zin
Z2 tanhγ 2 d2  3 Z2
Zin tanhγ 2 d2  3

;

;

 Zin N  ZN

Zin N
1


ZN tanhγ N dN 

ZN
Zin N
1 tanhγ N dN 

;

(5)

is the input impedance at the #i interface where Zin i and in particular Zin N
1  ZN
1 due to the semiinfinite dimensions of the substrate.

The optical intensity transmission coefficient from the incidence medium #0 into the substrate medium #N
1 is calculated as follows: jtj2  nN
1 ∕n0 jI N
1 ∕I 0 j2 ;

(6)

where I N
1  I 1

N  Y i1

−1 Zin i
1 coshγ i di 
sinhγ i di  ; Zi

(7)

and I 1  1
ρI 0 . For both jρj2 and jtj2 the transmission line calculations consider an infinitely thick glass layer and thus can be considered as an approximation that omits the reflection from the glass–air interface. This omission could be corrected by treating the glass substrate as an incoherent layer of finite thickness. C. Reflection and Transmission Calculation from and through an Incoherent Multilayer Structure

Herein, we consider the general case of a multilayer structure that consists of successive coherent and incoherent layers. In order to calculate the optical intensity reflection from the multilayer structure we use the TLM as it is depicted in Fig. 3. The bulk layers are considered as incoherent, while the thin films with thicknesses of the order of the incident light wavelength are treated as coherent. The reflection from the coherent multilayers and the transmission through them can be calculated using the methodology discussed in Subsection 2.B. Nevertheless, the light propagation through the thick incoherent layers should be calculated using only the optical intensity, and consequently, the phase rotation should not be taken into account [1]. 20 February 2015 / Vol. 54, No. 6 / APPLIED OPTICS

1495

(a)

(b)

(c) Fig. 3. (a) Stack of incoherent bulk layers separated by coherent ones, (b) the equivalent transmission line circuit for the calculation of the transmission inside an incoherent layer and the first reflection from the coherent multilayer stack, and (c) the equivalent transmission line circuit for the calculation of the reflection back to the incident incoherent medium from the bulk incoherent layer.

The application of the equivalent transmission lines follows an iterative procedure, starting from the last incoherent layer #N. In this context, the intensity reflection coefficient jρN−1 j2 is the first one that should be calculated by taking into account the incoherent layer #N. The calculation of jρN−1 j2 will be deduced as the summation of two terms, denoted as jρj2 and jρ− j2 . The first term corresponds to the reflection from the coherent multilayer stack #N − 1 and does not take into account the secondary reflection from the coherent layers #N, due to the incoherent properties of the succeeding incoherent layer #N. This omission is counterbalanced by the second term. Actually, the second term corresponds to the optical intensity that is transmitted into the incoherent layer #N, through the coherent multilayer stack #N − 1, and next is reflected from the coherent multilayer stack #N back to the incoherent layer #N − 1. In the above-mentioned context, the optical intensity reflection coefficient jρj2 can be calculated coherently using the equivalent transmission line circuit of Fig. 3(b) and following the iterative calculation scheme of Eqs. (4) and (5). Of course, due to the slightly different symbol notation of Fig. 3(b), the characteristic impedance Z0 in Eq. (4) should be replaced by ZN−1. Furthermore, in the last of the 1496

APPLIED OPTICS / Vol. 54, No. 6 / 20 February 2015

Eqs. (5), ZN and Zin N
1  ZN
1 should be replaced by Zk and ZN , respectively. In order to estimate the second term of the optical intensity reflection coefficient, i.e., jρ− j2 , we should first calculate the optical intensity transmitted to the incoherent layer #N through the coherent multilayer stack #N − 1. The former transmitted optical intensity can be estimated using again the transmission line of Fig. 3(b) but this time using Eq. (6) to calculate jtN−1 j2 . Due to the different symbol notation of Fig. 3(b) the expression in Eq. (6) should be modified as follows: jtN−1 j2  nN ∕nN−1 jI k ∕I 0 j2 :

(8)

Next, the optical intensity transmitted into the incoherent layer #N is reflected back toward the coherent multilayer stack #N − 1, providing up to this point an optical intensity coefficient jρN j2 jtN−1 j2 . The reflection coefficient ρN of the coherent multilayer stack #N is estimated using Eqs. (4) and (5). Finally, by using the transmission line of Fig. 3(c), the optical intensity transmission coefficient jt0N−1 j2 from the incoherent layer #N, through the coherent multilayer stack #N − 1, and back to the incoherent layer #N − 1 can be estimated as in Eq. (8) using the following expression:

jt0N−1 j2  nN−1 ∕nN jI 0k ∕I 00 j2 ; where similar to Eq. (7) I 0k  I 01

1  Y

I 0k

coshγ i di 


ik

(9)

is given as follows:

Zin i−1 Zi

−1 sinhγ i di 

I 01  1
ρ00 I 00 :

(10)

In Eq. (10) k is the number of coherent layers in  ZN−1 and ρ00 is the input reflecstack #N − 1, Zin 0 tion coefficient of the entire transmission line of Fig. 3(c) given by

where is calculated iteratively as described in Eq. (5) with minor differences due to the different symbol notation of the layers:

Zk
Zin k−1 tanhγ k dk 

:

(12)

According to the above-discussed calculation procedure the reflection coefficient jρ− j2 is finally given by the following expression: jρ− j2  jρN j2 jtN−1 j2 jt0N−1 j2 :

(13)

It should be mentioned that the estimation of the coefficient jρ− j2 , as calculated by Eq. (13), is taking into account only one reflection from the interface between the incoherent layer #N and the coherent layers #N. This is an approximation for the case where the reflection coefficients jρN j2 and jρ00 j2 ≪ 1. A more accurate approximation with all possible reflections from the two interfaces of the #N incoherent layer will modify Eq. (13) as follows: jρ− j2 

jρN j2 jtN−1 j2 jt0N−1 j2 : 1 − jρN j2 jρ00 j2

(14)

Additionally, if incoherent medium #N is lossy with an attenuation coefficient αN , then Eq. (14) should be modified again as follows: jρ− j2 

e−4aN dN jρN j2 jtN−1 j2 jt0N−1 j2 : 1 − jρN j2 jρ00 j2 e−4aN dN

(15)

Nonetheless, keeping the transparency of all layers high, the losses should be very low and Eq. (15) is approximated sufficiently by Eq. (13) or Eq. (14). Concluding, for the calculation of the spectral reflectance from the #N − 1 interface, the initially estimated coefficient jρj2 and the secondary reflection term jρ− j2 should be added resulting to the following expression:

(16)

Equation (16) is identical with the expression (15) of [1], which has been derived by applying the TMM approach, and not only verifies TLM’s accuracy, but also confirms that (15) of [1] correctly embodies the multiple reflections between the interfaces of the incoherent layer #N. Furthermore, the transmission coefficient jtN j2 inside the semi-infinite substrate could be calculated as follows:

(11)

Z0in

Zin k−1
Zk tanhγ k dk 

e−4aN dN jρN j2 jtN−1 j2 jt0N−1 j2 : 1 − jρN j2 jρ00 j2 e−4aN dN

jtN j2  e−2aN dN jtj2 jtN−1 j2

Z − Z0in ρ00  N ; ZN
Z0in

 Zk Z0in  Zin k

jρN−1 j2  jρj2


1 ; 1 − jρN j jρ00 j2 e−4aN dN 2

(17)

whereas the transmission coefficient jtj2 is calculated from the equivalent transmission line circuit of the coherent layers #N using Eq. (6). Equation (17) is also equivalent to the TMM-based expression (16) of [1]. The multilayer structure that consists of the incoherent layer #N, together with its neighbor stacks of coherent layers #N − 1 and #N, now represent the new interface between the incoherent layer #N − 1 and the semi-infinite substrate. Using the previously derived coefficients jρN−1 j2 and jtN j2 , the aforesaid calculation procedure can be followed iteratively up to the leftmost #1 interface. D.

Reflection Calculation from a Partial Coherent Interface

The analysis of coherent multilayer thin-film structures in Subsection 2.B considers the interfaces between the layers as ideally smooth. However, the techniques that are used for the films’ deposition may leave the interfaces rough with a randomly deployed roughness. As long as the rms height h of the rough surface is small compared to the wavelength, the reflected light is scattered predominately in the specular direction. In such cases, the rough surface does not scatter considerable light out of the specular beam and in this sense the influence of the rough surface is not evident or significant enough to be taken into account in nonspecular directions. However, the diffused or scattered light in those directions reduces overall the specularly reflected light. Thus, the existence of roughness is more evident in the specular direction acting as an antireflection coating that should be taken into account. The reduction of the specular reflection coefficient mitigates the interference effects by comparison with a perfectly smooth interface and in this sense justifies the characterization of the rough interface as a partial coherent interface. In the above-mentioned context, the calculation of the reduced reflection coefficient of a rough surface with small but not negligible rms roughness h provides more accurate results and can be easily integrated into a coherent calculation scheme. If an estimation of the rms roughness h is available through measurements, then it is possible to use various 20 February 2015 / Vol. 54, No. 6 / APPLIED OPTICS

1497

models to calculate the mitigation of the reflection coefficient of a rough interface inside a multilayer structure and overall the mitigation of the reflection coefficient from the entire multilayer structure. The choice of the appropriate reflection coefficient calculation model depends on the length scale of the surface irregularities. When the irregularities are comparable or greater than the wavelength, then roughness is assumed to be macroscopic and the reflection coefficient calculations are typically performed using the modified Fresnel reflection and transmission coefficients [2]. On the other hand, when the length scale of the irregularities is small compared to the wavelength, then roughness is considered to be microscopic [24]. The effective media approximation (EMA) [25] is a widely deployed method to calculate the rough interface reflection coefficient, when the roughness is considered microscopic. In the multilayer structure of Fig. 4, the interface between layer #i − 1 and layer #i is partially coherent due to a macroscopic roughness of rms depth h. Thus, the smooth interface reflection coefficient ρi−1 will be replaced by the modified one ρ0i−1 [1,26]. In particular, the standing wave reflection coefficient from a rough partial coherent interface could be calculated using the modified Fresnel transmission coefficient by considering the multiple passes in both directions of the optical waves through the rough interface. An approximate form of the reflection coefficient when the rough interface is followed by a multilayer structure is then expressed as follows: ρ0i−1  ρi−1 exp−k20 ni − ni−1 2 h2 ;

(18a)

where the exponential factor in Eq. (18a) is the mitigation factor of the smooth interface transmission

coefficient after a round trip through the rough interface [1,26]. Furthermore, when the rough interface is followed by a semi-infinite substrate, then the modified Fresnel reflection coefficient is adopted: ρ0i−1  ρi−1 exp−2k20 n2i−1 h2 :

(18b)

In the above-mentioned context, the reflection coefficient ρi−1 can be expressed according to the elementary transmission line theory as ρi−1 

Zi−1 − Zin i

Zi−1
Zin i

;

(19)

where Zin is calculated recursively from layer #i up i to the rightmost layer #N similar to Eq. (5): Zin i

 Zi

Zin i
1
Zi tanhγ i di 

Zi
Zin i
1 tanhγ i di 

:

The equivalent transmission lines of layers #1 up to the layer #i − 1 are connected in tandem and now are terminated at an equivalent load Z0in , which i embodies the macroscopic surface roughness by considering the modified reflection coefficient ρ0i−1 :  Zin Z0in i i

1 − ρ0i−1 : 1
ρ0i−1

(b) Fig. 4. (a) Stack of planar coherent layers with one rough interface and (b) its equivalent transmission line circuit. APPLIED OPTICS / Vol. 54, No. 6 / 20 February 2015

(21)

Next, the structure’s input impedance taking into account the roughness at the #i − 1∕#i interface is calculated iteratively as follows:

(a)

1498

(20)

Z0in  Z0in  Z1 1 Z0in  Z2 2

Z0in
Z1 tanhγ 1 d1  2

A. Thin-Film Thickness Measurement

Z1
Z0in tanhγ 1 d1  2

Z0in
Z2 tanhγ 2 d2  3

Z2
Z0in tanhγ 2 d2  3 

Z0in i−1  Zi−1

Z0in i


Zi−1 tanhγ i−1 di−1 

tanhγ i−1 di−1  Zi−1
Z0in i

:

(22)

Consequently, the reflection coefficient at the #0/#1 interface, taking into account the macroscopic roughness, is given by the following expression: ρ

Z0 − Z0in : Z0
Z0in

(23)

In the case of a microscopic roughness, the small length scale could be treated using an effective medium approximation. Essentially, EMA is introducing an intermediate layer of thickness 2h and effective index value that lays between the ones of layers #i − 1 and #i. The insertion of this effective layer attempts to capture the “smoother” transition of light across the rough interface due to the penetration of one layer into the other, which in turn results in a reduced reflection coefficient when compared with the perfectly smooth interface. The EMA method is reasonably accurate, when the difference of the refractive indices between the two layers is quite small. When the EMA is adopted, the equivalent TLM circuit degenerates into the coherent multilayer structure of Subsection 2.B with the addition of one intermediate layer at the position of the rough interface. 3. Numerical Results and Applications

In this section, the proposed TLM is going to be applied on two different types of problems. The first one concerns the application of the method to measure the thicknesses of OLED layers during deposition. This is an experimental method based on the measurement of reflected light from a multilayer of thin films deposited on a thick incoherent glass substrate. Using spectral measurements of the incident and reflected light, the thickness of the films could be derived by fitting measurements to TLM estimations. The second problem under consideration, concerns the application of the TLM for the calculation of the EQE of an OPV with a rough interface. The roughness between two organic layers that are deposited with the spin-coating technique or between an organic layer and a metal oxide is not negligible, and its influence on the EQE is going to be examined. The output of this study will be a contour chart depicting the short circuit photocurrent for different thicknesses of the layers with rough interfaces.

The thickness of each film of a solution-processed OLED device is a critical parameter, which depends strongly on the rotation speed during the spin-coating process, its duration and solution concentration. Therefore, each film’s thickness should be measured right after the spin-coating process, starting from the transparent ITO-coated glass substrate. A simple though accurate technique for film thickness measurement can be used based on the reflection spectral measurement at the normal direction. A fiber reflection probe with a broadband light source and a spectrometer could be deployed, where the light from the source illuminates the multilayer structure at the normal direction, while the reflection is analyzed by the spectrometer. The comparison between the spectrum of the source and the reflected one can be used for the determination of the layer’s thickness. In particular, using the spectral distribution of the complex refractive indices of each film, their thickness could be estimated by fitting TLM estimations to the measured reflection spectrum. The method is repeated each time we spin-coat a new layer upon the previous ones and consider its thickness as a new fitting parameter (Fig. 5). The spectrum of the illuminating light source and the reflection spectra of the multilayer ITO/glass/air and PEDOT:PSS/ITO/glass/air, respectively (Fig. 6), have been measured using the spectrometer and the fiber reflection probe described above. The complex refractive indices of the involved materials have been taken from the literature considering an anisotropic PEDOT:PSS’s refractive index [27]. Considering the thickness D of the coherent layer of ITO as a fitting parameter, the spectral reflectance measurements [solid line in Fig. 7(a)] have been fitted numerically using the TLM estimations described in Subsection 2.C [dotted line in Fig. 7(a)]. In particular, according to fitting with respect to the maximum

Fig. 5. Individual films’ thickness measurement setup during the spin-coating process of the successive layers, starting from the transparent ITO-coated glass substrate. 20 February 2015 / Vol. 54, No. 6 / APPLIED OPTICS

1499

Fig. 6. Incidence light source spectrum (dashed line), reflection spectra form ITO/glass/air (red line), and from PEDOT:PSS/ ITO/glass/air (blue line).

interference point, where the measurements are more sensitive with respect to the layer thickness, we conclude that the actual thickness of ITO is 92 nm. Moreover, the dotted line in Fig. 7(a) represents the reflected light from the equivalent smooth ITO

Fig. 7. (a) Fitting with 92 nm ITO thickness at the reflection spectrum of air/ITO/glass/air with partial coherent air/ITO interface (dashed line) and the corresponding reflection with smooth air/ ITO interface (dotted line). (b) Fitting with 52 nm PEDOT:PSS thickness at the reflection spectrum of PEDOT:PSS/ITO/glass/ air interfaces. 1500

APPLIED OPTICS / Vol. 54, No. 6 / 20 February 2015

surface, although the measured one is illustrated by the solid line in the same figure. The difference in the level of optical intensity may potentially be justified by the macroscopic roughness of the air/ ITO interface. By taking into account the macroscopic partial coherence in the air/ITO interface described in Subsection 2.D and considering the rms roughness h as an additional fitting parameter, the dashed line of Fig. 7(a) is the optimum when h  8 nm. This fitting result appears to be overestimated, if we consider the rms roughness values that have been reported using atomic force microscopy [28,29]. This deviation may be accounted for by inevitable experimental setup errors, such as a small deviation from the normal incidence, along with typical calibration errors that may exist. The same procedure has been repeated for the successive PEDOT:PSS layer, as illustrated in Fig. 7(b). In this case, the fitting process estimation is advantageous, due to destructive interference phenomena appearing in 450–500 nm, which in general demonstrate a strong resonant behavior, and thus provide more accurate results. In the particular case of the PEDOT:PSS layer the optimum fitting thickness is estimated to be 52 nm. By repeating the method and using the spectral refractive index and extinction coefficient of each layer, the thickness of each layer for the specific device can be estimated. Concluding, it is worthwhile to mention that the accuracy of the fitting depends strongly on the constructive and destructive interference phenomena that could be revealed in a range of wavelengths, which next could be attributed to the films’ thickness. In general, destructive interference phenomena demonstrate a more resonant behavior and establish a more explicit relation with the layer thickness. For example, the destructive interference effects in the range of 450–500 nm of Fig. 7(b) allow thickness estimation within a few nanometers. On the other hand, constructive interference phenomena do not demonstrate such selectivity with respect to the thickness of the dielectric layers. Thus, the thickness estimation, which is based upon the constructive interference phenomena around 550 nm in Fig. 7(a), provides a more ambiguous result. The thickness measurement accuracy depends strongly on the light source wavelength spectrum in comparison with the order of the film thickness. Particularly, it is possible to increase the accuracy of the measurement by extending the light source’s spectrum toward smaller wavelengths. Thus, more fringes will appear within the spectrum and the fitting will be more accurate. However, for thin films of thickness less than 100 nm this is possible only with UV light sources together with an adequate spectrometer. Although it has not been demonstrated experimentally here, the TLM can also provide estimations of the transmitted spectra through the glass substrate and into the air; thus, it can also facilitate experimental estimations of both refractive index and thickness of

each dielectric layer, as has been reported elsewhere [30]. B. Calculation of the External Quantum Efficiency of an OPV with Rough Interfaces

The case of an organic photovoltaic will be examined as an application of the TLM for multilayer structures with partial coherent interfaces. In particular, a hybrid BHJ OPV device with P3HT:PCBM acting as the photoactive material, will be modeled through the equivalent transmission lines of Fig. 8. Zglass , ZITO , ZMoO , Zactive , and ZAl are the characteristic impedances of glass, ITO, MoO3 , photoactive layer, and aluminum, whereas dITO  100 nm, dMoO  30 nm, and D are their thicknesses, respectively. We assume that the deposition techniques of all layers, namely spin-coating and vacuum thermal deposition, may leave all interfaces rough with maximum equivalent roughness h  25 nm, which also embodies the potential nonparallelism of the plane interfaces. From all the interfaces, the one between MoO3 and the photoactive layer is of specific interest, as the metal oxides may have the deepest roughness after their deposition [31], and also due to its mixing with the photoactive layer. Considering the roughness of this interface microscopic as well as being dominant compared to the roughness of other interfaces, we may deploy the EMA and TLM to simulate the multilayer structure. According to the EMA, at least one effective layer with length 2h should be inserted between MoO3 and the photoactive layer. Due to the insertion of this layer, we assume that the thickness of both MoO3 and photoactive layer is reduced by h; however, if there is evidence of another size proportion another assumption can be made as appropriate. As shown in Fig. 8, the intermediate layer is represented by a transmission line with characteristic impedance and transmission coefficient ZEMA and γ EMA , respectively. Its complex refractive index is considered to be the mean value between the complex refractive indices of the successive media; however, any other assumption deemed necessary can be made. Due to the former assumption the extinction coefficient of the intermediate layer will be half of that of the photoactive layer’s. Consequently,

it is expected that it will contribute to the production of the device’s photocurrent but with reduced efficiency. For the calculation of the short circuit photocurrent of the aforementioned device, the integration of the optical electric field distribution along the photoactive and the effective layer should be performed in order to take into account the light absorption from the dispersive p-n junctions throughout the photoactive material. The optical electric field distribution along the photoactive and effective layer can be obtained through the calculation of the current distribution along the corresponding equivalent transmission lines of Fig. 8. Using the input impedin in in ances Zin 1 , Z2 , Z3 , Z4 , and ZAl that are provided by following the recursive calculation scheme of Eq. (5), the input current at the photoactive layer and effective layer interfaces are given by Eqs. (24a) and (24b), respectively: −1  Zin I 3  1
ρI 0 coshγ ITO dITO 
2 sinhγ ITO dITO  ZITO  −1 in Z × coshγ MoO d0MoO 
3 sinhγ MoO d0MoO  ; ZMoO (24a)  I4  I3

−1 Zin 4 coshγ EMA dEMA 
sinhγ EMA dEMA  ; ZEMA (24b)

where the layer thicknesses d0MoO  dMoO − h and dEMA  2h correspond to the thickness of the MoO3 just before the roughness and the equivalent thickness of the effective layer due to roughness, respectively. Next, the current distribution along the effective and photoactive layer (z0 is the distance from the EMA/active layer interface and z is the distance from aluminum) is given by Eqs. (25a) and (25b), respectively: I EMA z0 

 I3 

  Zin 4 0 0 sinhγ EMA z  coshγ EMA z 
ZEMA

Zin coshγ EMA dEMA 
4 sinhγ EMA dEMA  ZEMA

;

(25a)  ZAl sinhγ active z coshγ active z
Zactive ; I active z  I 4  Z coshγ active D0 
Al sinhγ active D0  Zactive (25b) 

Fig. 8. Equivalent transmission line for an OPV structure (ITOMoO3 -P3HT:PCBM-Al) where the interface between the MoO3 and P3HT:PCBM layers demonstrates a rough profile.

20 February 2015 / Vol. 54, No. 6 / APPLIED OPTICS

1501

where D0  D − h corresponds to the thickness of the active layer just after the roughness. The EQE is represented by the photocurrent equivalent parameter Q: 2πcε0 n0EMA n00EMA jI EMA z0 ; λj2 ; λ

(26a)

2πcε0 n0active n00active jI active z; λj2 ; λ

(26b)

Qz0 ; λ 

Qz; λ 

where the range of spatial positions z and z0 is given by 0 ≤ z0 ≤ dEMA and 0 ≤ z ≤ D0 , respectively, whereas nEMA  n0EMA − jn00EMA  0.5nactive
0.5nMoO and nactive  n0active − jn00active are the wavelengthdependent complex refractive indices of the effective layer and the active layer, respectively. The normalized Q factor is determined for a wavelength and for a specific active layer thickness with roughness normalized to the corresponding one with infinite thickness: Z 0  Zd D EMA 0 0 ¯ Qλ  Qz; λdz
Qz ; λdz 0 0 Z  ∞ Qz; λdz : 0

(27)

Using Eq. (27) and the solar spectral power density, the short circuit photocurrent can be estimated by considering internal quantum efficiency IQE  1 [5,6]. For the calculation of the EQE the wavelength has been selected at 530 nm, since the photoactive material demonstrates the maximum absorption n00active . The complex refractive indices of the involved materials have been taken from the literature [32,33]. Figure 9 depicts the normalized Q versus active layer thickness for ideally smooth and parallel interfaces (black line) as well as for rough interface between MoO3 and active layer with 15 and 25 nm

Fig. 9. Q-normalized versus photoactive layer thickness for smooth interfaces (black line), for rough MoO3 -photoactive layer interface with 15 nm roughness (blue line), and 25 nm roughness (red line). 1502

APPLIED OPTICS / Vol. 54, No. 6 / 20 February 2015

roughness (blue and red line). The illustrated results show that the interface roughness weakens the interference phenomena inside the photoactive layer by reducing the intensity of both constructive and destructive interference effects. As a result, it is anticipated that the photocurrent should be reduced for a range of active layer thicknesses (see curves’ maxima around 90 nm in Fig. 9), while increased for another (see curves minima around 140 nm in Fig. 9). It is also evident that there is a small range of thicknesses that appear to be insensitive to roughness, e.g., thicknesses about 50 nm. Figure 10 shows the normalized field distribution along MoO3 and photoactive layer with thicknesses dMoO  30 nm and D  120 nm, respectively, and considering both smooth and rough interfaces (dEMA  0, 15, and 25 nm). According to the illustrated results, the field distribution appears to be concentrated in the active layer, as the interface roughness increases. This result agrees with the illustrated destructive interference phenomena that are evident at 120 nm in Fig. 9. Furthermore, in the same figure, the accuracy of the one-layer EMA for roughness simulation is examined. In particular, the full-wave commercial software CST Studio Suite has been applied in order to simulate the optical field distribution along the OPV’s layers. The position of square symbols in Fig. 10 shows a perfect match with the TLM simulation, confirming the validity of the one-layer EMA as well as of its integration in the TLM. Using also the aforementioned TLM calculation procedure, the short circuit photocurrent can be obtained by considering an AM 1.5 G 100 mW∕cm2 illumination source spectrum. In Fig. 11(a), the short circuit photocurrent is depicted in a contour line form, whereas the thickness of the photoactive layer varies between 30 and 180 nm, while the MoO3 varies from 30 nm up to 100 nm. In this case, all interfaces between the layers

Fig. 10. Optical field intensity distribution along the MoO3 and the photoactive layer for smooth interfaces (black line) and for rough MoO3 -photoactive layer interface with 15 nm roughness (blue line), and for 25 nm roughness (red line). The MoO3 layer thickness and the active layer thickness have been selected equal to 30 and 120 nm, respectively.

between the two layers that modifies the photocurrent contour diagram according to Figs. 11(b) and 11(c). It is evident that for a photoactive layer thickness around 100 nm and MoO3 thickness less than 50 nm, the level of the photocurrent is drastically reduced with the increase of roughness. However, for the photoactive layer thickness in the range of 140 nm the roughness may contribute to the photocurrent, while also keeping the thickness of the MoO3 less than 40 nm. Concluding, in the range of 100 nm photoactive layer thickness the smooth interface is being more beneficial from an EQE point of view; however, should the IQE require higher photoactive layer thicknesses, e.g., around 140 nm, then the roughness may compensate a part of the EQE reduction. The proposed TLM model can be even adopted for the evaluation of inclined optical wave incidence, but in a rather more complicated calculation scheme [6]. Partial incoherence that originates from the potential impurities inside the material of each layer [26] and/or from the consideration that the incident light is incoherent [34], could also be treated by the proposed method and will be presented in future work. 4. Conclusions

Fig. 11. (a) Short circuit photocurrent (mA∕cm2 ) for perfectly smooth interfaces, (b) for 15 nm rough interface between MoO3 and the photoactive layer, (c) for 25 nm rough interface between MoO3 and the photoactive layer.

are considered perfectly smooth. As a result, high photocurrent values are achieved only for active layer thickness greater than 200 nm. This is a nonfeasible result because the thick photoactive layers present higher resistance and the IQE decreases drastically. However, there is a small range between 85 and 115 nm, where for thin MoO3 films (less than 45 nm) the photocurrent is up to 12 mA∕cm2 . Next, we take into account the roughness of the interface

The TLM has been modified in order to include not only coherent but also incoherent and partial coherent planar layers for the calculation of reflection and transmission coefficients. The method is applied in the experimental determination of individual layer thickness in a solution-processed OLED, where the multilayered structure under consideration includes thick incoherent layers together with thin coherent ones. The method has been used with a typical spectral reflectance experimental technique in order to determine the thickness of coated films by processing the experimental data with a TLM-based numerical fitting technique. The proposed measurement scheme is suitable for the OLEDs’ films thickness determination after their spin-coating process. The accuracy of the method depends strongly on the interference phenomena throughout the multilayer structure. The method has also been applied for the simulation of the EQE of hybrid OPV devices considering IQE  1. Specifically, the photocurrent has been calculated considering the interface between the photoactive layer and the metal oxide layer as rough. The simulated results show that in terms of EQE, the rough interface smoothes out the destructive and constructive interference phenomena, reducing the photocurrent at the constructive regions and vice versa. In future work, more complicated devices, such as an organic opto-coupler, could be modeled taking into account the potential incoherency of the thick coupling medium between the OLED and OPV coherent stacks. 20 February 2015 / Vol. 54, No. 6 / APPLIED OPTICS

1503

This research has been cofinanced by the European Union (European Social Fund–ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)—Research Funding Program: ARCHIMEDES III. Investing in knowledge society through the European Social Fund. References 1. C. C. Katsidis and D. I. Siapkas, “General transfer-matrix method for optical multilayer systems with coherent, partial coherent and incoherent interference,” Appl. Opt. 41, 3978–3987 (2002). 2. C. L. Mitsas and D. I. Siapkas, “Generalized matrix method for analysis of coherent and incoherent reflectance and transmittance of multilayer structures with rough surfaces, interfaces, and finite substrates,” Appl. Opt. 34, 1678–1683 (1995). 3. E. Centurioni, “Generalized matrix method for calculation of internal light energy flux in mixed coherent and incoherent multilayers,” Appl. Opt. 44, 7532–7539 (2005). 4. S. P. Savaidis and N. A. Stathopoulos, “Simulation of light emission from planar multilayered OLEDs, using a transmission-line model,” IEEE J. Quantum Electron. 45, 1089–1099 (2009). 5. N. A. Stathopoulos, L. C. Palilis, S. P. Savaidis, S. R. Yesayan, M. Vasilopoulou, G. Papadimitropoulos, D. Davazoglou, and P. Argitis, “Optical modeling of hybrid polymer solar cells using a transmission line model and comparison with experimental results,” IEEE J. Sel. Top. Quantum Electron. 16, 1784– 1791 (2010). 6. N. A. Stathopoulos, L. C. Palilis, S. R. Yesayan, S. P. Savaidis, M. Vasilopoulou, and P. Argitis, “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,” J. Appl. Phys. 110, 114506 (2011). 7. M. T. Greiner, M. G. Helander, W.-M. Tang, Z.-B. Wang, J. Qiu, and Z.-H. Lu, “Universal energy-level alignment of molecules on metal oxides,” Nat. Mater. 11, 76–81 (2012). 8. M. Vasilopoulou, A. M. Douvas, D. G. Georgiadou, L. C. Palilis, S. Kennou, L. Sygellou, A. Soultati, I. Kostis, G. Papadimitropoulos, D. Davazoglou, and P. Argitis, “The influence of hydrogenation and oxygen vacancies on molybdenum oxide work function and gap states for applications in organic optoelectronics,” J. Am. Chem. Soc. 134, 16178–16187 (2012). 9. L. C. Palilis, M. Vasilopoulou, D. G. Georgiadou, and P. Argitis, “A water soluble inorganic molecular oxide as a novel efficient electron injection layer for hybrid light-emitting diodes (HyLEDs),” Org. Electron. 11, 887–894 (2010). 10. S. H. Park, A. Roy, S. Beaupre, S. Cho, N. Coates, J. S. Moon, D. Moses, M. Leclerc, K. Lee, and A. J. Heeger, “Bulk heterojunction solar cells with internal quantum efficiency approaching 100%,” Nat. Photonics 3, 297–302 (2009). 11. L. M. Chen, Z. Hong, G. Li, and Y. Yang, “Recent progress in polymer solar cells: manipulation of polymer:fullerene morphology and the formation of efficient inverted polymer solar cells,” Adv. Mater. 21, 1434–1449 (2009). 12. F. C. Kebs, “Roll-to-roll fabrication of monolithic large-area polymer solar cells free from indium-tin-oxide,” Sol. Energy Mater. Sol. Cells 93, 1636–1641 (2009). 13. G. Yu, J. Gao, J. C. Hummelen, F. Wudl, and A. J. Heeger, “Polymer photovoltaic cells: enhanced efficiencies via a network of internal donor-acceptor heterojunctions,” Science 270, 1789–1791 (1995). 14. G. Dennler, M. C. Scharber, and C. J. Brabec, “Polymerfullerene bulk-heterojunction solar cells,” Adv. Mater. 21, 1323–1338 (2009). 15. S. Gunes, H. Neurebauer, and N. S. Sariciftci, “Conjugated polymer-based organic solar cells,” Chem. Rev. 107, 1324– 1338 (2007).

1504

APPLIED OPTICS / Vol. 54, No. 6 / 20 February 2015

16. M. C. Gather, A. Köhnen, and K. Meerholz, “White organic light-emitting diodes,” Adv. Mater. 23, 233–248 (2011). 17. J. Huang, G. Li, E. Wu, Q. Xu, and Y. Yang, “Achieving highefficiency polymer white-light-emitting devices,” Adv. Mater. 18, 114–117 (2006). 18. L. T. Dou, J. B. You, J. Yang, C. C. Chen, Y. J. He, S. Murase, T. Moriarty, K. Emery, G. Li, and Y. Yang, “Tandem polymer solar cells featuring a spectrally matched low-bandgap polymer,” Nat. Photonics 6, 180–185 (2012). 19. Y. M. Sun, G. C. Welch, W. L. Leong, C. J. Takacs, G. C. Bazan, and A. J. Heeger, “Solution-processed small-molecule solar cells with 6.7% efficiency,” Nat. Mater. 11, 44–48 (2012). 20. M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 41),” Prog. Photovoltaics 21, 1–11 (2013). 21. L. A. A. Pettersson, L. S. Roman, and O. Inganas, “Modeling photocurrent action spectra of photovoltaic devices based on organic thin films,” J. Appl. Phys. 86, 487–496 (1999). 22. C. C. Katsidis, A. O. Ajagunna, and A. Georgakilas, “Optical characterization of free electron concentration in heteroepitaxial InN layers using Fourier transform infrared spectroscopy and a 2 × 2 transfer-matrix algebra,” J. Appl. Phys. 113, 073502 (2013). 23. M. Schubert, “Generalized transmission ellipsometry for twisted biaxial dielectric media: application to chiral liquid crystals,” Phys. Rev. B 53, 4265–4274 (1996). 24. C. Chen, C. Ross, N. J. Podraza, C. R. Wronski, and R. W. Collins, “Multichannel Mueller matrix analysis of the evolution of the microscopic roughness and texture during ZnO:Al chemical etching,” in Conference Record of the 31st IEEE Photovoltaic Specialists Conference, (IEEE, 2005), pp. 1524– 1527. 25. J. Springer, A. Poruba, and M. Vanecek, “Improved threedimensional optic model for thin-film silicon solar cells,” J. Appl. Phys. 96, 5329–5337 (2004). 26. M. C. Troparevsky, A. S. Sabau, A. R. Lupini, and Z. Zhang, “Transfer-matrix formalism for the calculation of optical response in multilayer systems: from coherent to incoherent interference,” Opt. Express 18, 24715–24721 (2010). 27. L. A. A. Pettersson, S. Ghosh, and O. Inganas, “Optical anisotropy in thin films of poly(3,4-ethylenedioxythiophene)– poly(4-styrenesulfonate), Org. Electron. 3, 143–148 (2002). 28. Y.-H. Tak, K.-B. Kima, H.-G. Park, K.-H. Lee, and J.-R. Lee, “Criteria for ITO (indium-tin-oxide) thin film as the bottom electrode of an organic light emitting diode,” Thin Solid Films 411, 12–16 (2002). 29. Y. Fukushi, H. Kominami, Y. Nakanishia, and Y. Hatanaka, “Effect of ITO surface state to the aging characteristics of thin film OLED,” Appl. Surf. Sci. 244, 537–540 (2005). 30. M. Flämmich, N. Danz, D. Michaelis, A. Brauer, M. C. Gather, J. H.-W. M. Kremer, and K. Meerholz, “Dispersion model free determination of optical constants: application to materials for organic thin film devices,” Appl. Opt. 48, 1507–1513 (2009). 31. M. Vasilopoulou, L. C. Palilis, D. G. Georgiadou, P. Argitis, S. Kennou, I. Kostis, G. Papadimitropoulos, N. A. Stathopoulos, A. A. Iliadis, N. Konofaos, D. Davazoglou, and L. Sygellou, ‘Tungsten oxides as interfacial layers for improved performance in hybrid optoelectronic devices,” Thin Solid Films 519, 5748–5753 (2011). 32. G. Dennler, K. Forberich, M. C. Scharber, C. J. Brabec, I. Tomis, K. Hingerland, and T. Fromherz, “Angle dependence of external and internal quantum efficiencies in bulkheterojunction organic solar cells,” J. Appl. Phys. 102, 054516 (2007). 33. M. Vasilopoulou, I. Kostis, A. M. Douvas, D. G. Georgiadou, A. Soultati, G. Papadimitropoulos, N. A. Stathopoulos, S. P. Savaidis, P. Argitis, and D. Davazoglou, “Vapor-deposited hydrogenated and oxygen-deficient molybdenumoxide thin films for application in organic optoelectronics,” Surf. Coat. Technol. 230, 202–207 (2013). 34. W. Lee, S.-Y. Lee, J. Kim, S. C. Kim, and B. Lee, “A numerical analysis of the effect of partially coherent light in photovoltaic devices considering coherence length,” Opt. Express 20, A941–A953 (2012).