Improving External Quantum Efficiency by Subwavelength Nano Multi-Layered Structures for Optoelectronic Devices

Based on thin film optics (TFO) and finite element method (FEM), we have theoretically investigated improving external quantum efficiency (EQE) by anti-reflection (AR) films constructed from subwavelength nano multi-layers (NML) of low and high index materials, where the low and high index materials are MgF2 and Ta2O5, respectively. This kind of NML dielectric structures have the advantages of low-cost and flexible. Three kinds of substrates have been studied here, which are glass, polyethylene naphthalate (PEN), and polyethylene terephthalate (PET), respectively. TFO theory has been used to obtain the transmittance and reflectance, which agrees well with FEM results. For NML structure, TFO is more efficient than FEM in terms of calculation time and accuracy. The average AR effects (AAREs) are about 3.69%, 3.46% and 3.07% on glass substrate, about 6.15%, 5.56% and 5.02% on PEN substrate, and about 5.06%, 4.62% and 4.17% on PET substrate, for AR bands (ARBs) 350~800 nm, 350~1100 nm and 350~1500 nm, respectively. The results also reflect that wider AR bandwidth needs more NML layers. In practice, this kind of AR films can be widely applicable to enhance the EQE of the optoelectronic devices (OEDs).


I. INTRODUCTION
Transparent conductive substrates (TCSs) are characterized by their high transmission of visible light and simultaneously very high electrical DC (direct current) sheet conductivity [1]- [6]. TCSs can be applied as the transparent electrodes, which are the key component of optoelectronic devices (OEDs) for information (displays) and energy (photovoltaics, window glass and architectural) technologies [7]- [9]. External quantum efficiency (EQE) is an important physical quantity to evaluate the performance of OEDs. Besides optimizing the electronic characteristics of each structure of OEDs and reducing the recombination of electrons and holes, EQE also takes into account the optical performance of the device, which contains transmissivity, absorptivity and reflectivity. Introducing optical methods and The associate editor coordinating the review of this manuscript and approving it for publication was Kin Kee Chow. applying anti-reflection (AR) layer is an important way to improve optical performance and then enhance EQE of the OEDs. Generally, nanoscale periodic structures are adopted to achieve AR layer in most works [10]- [12]. However, such nanoscale periodic structures are manufactured skillfully by costly techniques such as lithography and nano-imprint lithography. Therefore, high efficiency and low-cost AR layers are very meaningful for OEDs in both academia and industry.
One-dimensional Bragg photonic crystals (BPCs) with the Bragg back reflector structures has been used in OEDs to reduce the reflection of incident light [13]- [15]. BPCs are characterized by periodic medium structures with photonic band gap (PBG), which can control the path of light [16], [17]. PBG can be obtained by electromagnetic simulation on a determined PBC. The electromagnetic simulation methods generally include FDTD (finite difference time domain) and FEM (finite element method) [18], [19]. However, these simulation methods (FDTD and FEM) are time-consuming, especially for the case that the structure size is much smaller than the wave length.
In this article, the theory of thin film optics (TFO) [20] is applied to study one-dimensional BPC, which is characterized as nano multi-layers (NML). It is worth noting that simulation based on TFO takes much less time than FDTD and FEM. Therefore, it is easier to design the corresponding BPC structure from the desired PBG structure by TFO. Finally, FEM has also been used as a contrast to simulate the NML, and the results were consistent with TFO theory. However, for NML structure TFO is more efficient than FEM in terms of calculation time and accuracy. For FEM, the more accurate the result is, the denser the mesh is, resulting in a longer computation time.
This kind of NML AR film can be fabricated by lowcost techniques, for example magnetron sputtering, electron beam evaporation, and thermal deposition process. Commercial software based on TFO is widely used to guide the industrial production of optical film, such as optical film design of lenses. TFO can also be used to guide the research activities of design, analysis, and monitoring of optical film coatings [1], [21]- [24].
We have studied the NML AR films of TiO 2 and SiO 2 in the early work [25], where two grades AR films have been investigated. The 1-st NML AR films involve the interface of substrate and air, and the 2-nd NML AR films involve the interface of transparent conductive film and substrate. In Ref. [25], the AR effect of the 2-nd AR is much smaller than the 1-st AR. In view of the practical fabrication of OEDs only affecting the 2-nd AR, we will focus on the need for the NML AR film of MgF 2 and Ta 2 O 5 at the interface between substrate and air. It is worth noting that the previous work concentrates on the AR ability of a specific band range. However, this article concentrates on the possibility of the AR band extension to different bandwidths. In addition, FEM is used as a contrast. Finally, it reflects that wider AR bandwidth means more NML layers, and TFO is more efficient than FEM in terms of calculation time and accuracy. There are previous published works also about using MgF 2 and Ta 2 O 5 NML as AR coating [26,27]. Their NML AR films are much more than 30 layers and obey some periodic rules. Our NML films break the periodic rules, and are only dozen layers or less with better AR effect.

II. PRINCIPLE
Generally, OEDs structure is based on TCSs, such as FTO (fluorine-doped tin oxide) glass, ITO (indium tin oxide) glass, FTO PEN (polyethylene naphthalate) plastic, ITO PEN, FTO PET (polyethylene terephthalate), ITO PET, and so on [1]- [9], [25]. TCS is constructed of substrate and transparent conductive film. As shown in Fig. 1, our NML AR film is on the outside of substrate, and OEDs are manufactured on transparent conductive film. Light is coming in from NML AR film, goes through the TCS, and then finally enters into the OEDs. In this article, our NML AR film is at the interface between substrate and air. This kind of NML AR films are easy to fabricate by physical vapor deposition on the existing TCSs, which can realize mass production in business without knowing the craftsmanship of transparent conductive film.
In the following sections, we will investigate the NML AR films on three kinds of substrates, which are glass, PEN, and PET. It is worth noting that the PEN and PET plastic is flexible, which can be used for flexible OEDs [28]- [31].
In TFO design, low and high index materials are necessary to achieve the desired optical target. The materials should have low chromatic dispersity and no absorption. In this study, the low index material is MgF 2 , and the high index material is Ta 2 O 5 . Once the material is selected, the thickness of every layer of NML will determine the optimized result. Encouragingly, these thicknesses can be obtained by optical target optimization simulation of Essential Macleod or other commercial software.
Optical target optimization simulation requires the knowledge of optical parameters of the materials. The refractive index n and extinction coefficient k dependence on the wavelength λ (nm) for all the materials in this study can be obtained from the pioneer published works [32]- [34], as shown in Fig. 2. However, every different fabrication method with different condition will influence the optical parameters of material, and then influence the final designs. If researchers want to compare the experimental data with the theoretical results, they must obtain the optical property of every layer material in their experiment condition by ellipsometer.
Here we have ignored the magnetic effect, and the relative permeability of all above materials is assumed to be 1. However, there are two kinds of glass, which are white glass and electron-grade glass. In usual, the white glass is used for ITO glass, and the electron-grade glass is used for FTO glass. Optical properties of these two glasses are different. To simplify the problem, the substrate glasses in our following study are all white glass. Generally, the relative permittivity of air is 1.     Fig. 3, the improvement of optical performance is obvious in corresponding ARB (∼800nm, ∼1100nm and ∼1500nm). For transmittance, as shown in sub- Figs. 3(a) and (b), the maximum transmittance of ARBs ∼800nm, ∼1100nm and ∼1500nm are about 95.5%, which are about 3.5% bigger than the no NML AR situation. For reflectance, as shown in sub-Figs. 3(c) and (d), the minimum reflectance of ARBs ∼800nm, ∼1100nm and ∼1500nm are about 4.5%, which are about 3.5% smaller than the no NML AR situation.

III. DESIGNS OF THE NML AR FILMS
For PEN with the NML AR films in Fig. 4, the improvement of optical performance is obvious in corresponding ARB (∼800nm, ∼1100nm and ∼1500nm). For transmittance, as shown in sub-Figs. 4(a) and (b), the maximum transmittance of ARBs ∼800nm, ∼1100nm and ∼1500nm  are about 94%, which are about 5.5% bigger than the no NML AR situation. For reflectance, as shown in sub-Figs. 4(c) and (d), the minimum reflectance of ARBs ∼800nm, ∼1100nm and ∼1500nm are about 5.5%, which are about 5% smaller than the no NML AR situation.
For PET with the NML AR films in Fig. 5, the improvement of optical performance is obvious in corresponding ARB (∼800nm, ∼1100nm and ∼1500nm). For transmittance, as shown in sub- Figs. 5(a) and (b), the maximum transmittance of ARBs ∼800nm, ∼1100nm and ∼1500nm are about 94.5%, which are about 4.5% bigger than the no NML AR situation. For reflectance, as shown in sub-Figs. 5(c) and (d), the minimum reflectance of ARBs ∼800nm, ∼1100nm and ∼1500nm are about 5%, which are about 4.5% smaller than the no NML AR situation.
Then we use the FEM (COMSOL Multiphysics) as a contrast of TFO to simulate the NML. In order to simulate the physical model, 3D geometry is selected, Radio Frequency (RF) module is chosen, and two Electromagnetic Waves (emw1 and emw2) Frequency Domains are used. The first physical domain emw1 is used to simulate the incident electromagnetic wave propagation in air, and the second physical domain emw2 is used to simulate AR NML situation with the electric field distribution of the first physical domain as the initial conditions. The incident electromagnetic wave follows the negative z direction, and the polarization of the electric field follow the x direction. Therefore, the electric field distribution of the first physical domain emw1.Ex represents the incident electromagnetic wave, and the electric field distribution of the second physical domain emw2.Ex represent the superposition of incident and reflected electromagnetic waves.
In COMSOL, the incidence electromagnetic field is TM (Transverse Magnetic) mode wave, as shown in Fig. 6(a).  Ei and Hi are the electric field and magnetic field of the incident wave, while Er and Hr are the electric field and magnetic field of the reflected wave. Red arrows and yellow cross, respectively describe the directions of the electric field and magnetic field. Yellow cross means that the direction is perpendicular to the plane of the paper and inside it.  Consequently, emw1.Ex = Ei, and emw2.Ex = Ei -Er. Integration can reduce the numerical noise error. Then the reflectance of the incident electric field ρ and energy flux R can be expressed as where S is the incident plane as shown in Fig. 6(a).
Because the thickness aberration of the hundreds nm films and 1mm substrate is too large, the mesh division in COMSOL is too close to affect the simulation speed. Therefore, the thickness of the substrate is ignored in the model simulation by COMSOL. So we are only thinking about the reflection on the front face of the substrate. In practice, the thickness of the substrate and end face both have effects on the reflectivity of the system. As shown in sub-Figs. 6(b) (c) and (d), the solid and dotted lines are the results from TFO and FEM, respectively. Sub-Figs. 6(b) (c) and (d) are reflectance dependence on the wavelength of glass (b) PEN (c) and PET (d) substrates with 6 (red lines) 8 (yellow lines) and 12 (green lines) layers NML AR films by TFO (solid lines) and FEM (dotted lines) methods. These results reflect that the two methods TFO and FEM agree well with each other. However, the obtained data for each of these curves by FEM takes about four hours, while the data by TFO only takes about a few seconds. It is worth noting that the difference between FTO and FEM results increases with the number of layers. That is because as the number of layers increases, the larger the mesh becomes, the greater the numerical error becomes.
Finally, we shall define a physical quantity AARE (shorted for average AR effect) to describe the average AR of different ARBs, where T 1 is the transmittance of NML AR substrate, and T 2 is the transmittance of none AR substrate. Combining Eq. (3) with the data in Figs. (3)-(5), we can get the AARE of different substrates with different ARBs, as shown in Table 4. Finally, AAREs are about 3.69%, 3.46% and 3.07% on glass substrate, about 6.15%, 5.56% and 5.02% on PEN substrate, and about 5.06%, 4.62% and 4.17% on PET substrate, for AR bands (ARBs) 350∼800 nm, 350∼1100 nm and 350∼1500 nm, respectively.

IV. CONCLUSION
We have theoretically investigated the AR films based on NML of MgF 2 and Ta 2 O 5 on three kinds of substrates (glass, PEN and PET) for three kinds of ARBs (350∼800 nm, 350∼1100 nm, and 350∼1500 nm). This kind of low and high index dielectric NML structures have the advantages of low-cost and flexible. Such NML AR films can be used to improve the reflectance of air and substrate interface for TCS. The present method, i.e. using TFO to design NML AR films, will help to guide development of many kinds of OEDs.