*Article* **Characterization and Design of Photovoltaic Solar Cells That Absorb Ultraviolet, Visible and Infrared Light**

**Sara Bernardes** <sup>1</sup>**, Ricardo A. Marques Lameirinhas** 1,2,**\*, João Paulo N. Torres** 1,2,3 **and Carlos A. F. Fernandes** 1,2


**\*** Correspondence: ricardo.lameirinhas@tecnico.ulisboa.pt

**Abstract:** The world is witnessing a tide of change in the photovoltaic industry like never before; we are far from the solar cells of ten years ago that only had 15–18% efficiency. More and more, multi-junction technologies seem to be the future for photovoltaics, with these technologies already hitting the mark of 30% under 1-sun. This work focuses especially on a state-of-the-art triple-junction solar cell, the GaInP/GaInAs/Ge lattice-matched, that is currently being used in most satellites and concentrator photovoltaic systems. The three subcells are first analyzed individually and then the whole cell is put together and simulated. The typical figures-of-merit are extracted; all the − curves obtained are presented, along with the external quantum efficiencies. A study on how temperature affects the cell was done, given its relevance when talking about space applications. An overall optimization of the cell is also elaborated; the cell's thickness and doping are changed so that maximum efficiency can be reached. For a better understanding of how varying both these properties affect efficiency, graphic 3D plots were computed based on the obtained results. Considering this optimization, an improvement of 0.2343% on the cell's efficiency is obtained.

**Keywords:** concentrator systems; GaInP/GaInAs/Ge; multi-junction; photovoltaics; solar cells; space; triple-junction

#### **1. Introduction**

The constant search for new energetic solutions to face the ever-demanding world's energy consumption has been one of the main focus amongst researchers in the twenty-first century. At the time this article is being written, a good and affordable alternative seems to be found in the use of renewable energies [1–19]. Even though the world is not yet prepared to switch completely to renewable sources, the installed capacity of these sources is increasing day by day, with the global renewable generation power already surpassing 2300 gigawatts. In 2018, 20% of this total generation capacity came from solar power, that continued to dominate in terms of new power installed, representing an increase of 24% [1].

This global solar expansion mainly derives from the capability of the photovoltaic (PV) industry to face the challenges that have been proposed until now.

In the years to come, PV has the capacity of becoming one of the major energy sources in the world—as the price of fossil fuels continuously rises [6–19], the cost of solar PV has been substantially decreasing over the last two decades, with its LCOE (Levelized Cost Of Energy) being estimated to be within the range of 0.03 to 0.10 \$/kWh by 2020–2022 [2,3,20]. This prophetizes a solid future for the PV industry, especially if it is supported by the decrease in battery prices.

All of this motivates the industry to come up with new and improved solutions; one of those improvements in recent decades is the use of III-V multi-junction solar cells. These photovoltaic devices employ III-V semiconductors (made of elements in groups III and V

**Citation:** Bernardes, S.; Lameirinhas, R.A.M.; Torres, J.P.N.; Fernandes, C.A.F. Characterization and Design of Photovoltaic Solar Cells That Absorb Ultraviolet, Visible and Infrared Light. *Nanomaterials* **2021**, *11*, 78. https:// doi.org/10.3390/nano11010078

Received: 15 November 2020 Accepted: 26 December 2020 Published: 1 January 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

of the Periodic Table), typically in a stacked distribution [4,5,21,22]. These cells have been demonstrating solid results in terms of efficiency, since the first III-V GaInP/GaAs tandem cell was demonstrated by Olson and Kurtz at NREL in 1996, with a record efficiency of more than 30% [21]. Today, III-V cells already hit the mark of 45.7% in concentrator photovoltaics (NREL, 4-junction GaInP/GaAs/GaInAs/GaInAs, 234 suns) [22], demonstrating extraordinary advances in choosing optimal bandgap distributions.

This work will focus on a specific III-V cell, the GaInP/GaInAs/Ge lattice-matched cell, the state-of-the-art cell for both concentrator photovoltaics and space applications. The main objective is then to build a simulation model that allows for a characterization of the subcells that form the whole cell, extracting − curves and external quantum efficiencies, along with the most relevant figures-of-merit, such as fill-factors and efficiencies. A general optimization of the cell will also be attempted; this will be done by altering the thickness and doping of some layers.

It is used a Finite Element Tool, for being capable of simulating a 2D and 3D solar devices by providing a large set of physical models (drift-diffusion, general optoelectronic interactions with ray tracing, Fermi-Dirac statistics, etc.) for semiconductor device simulation.

#### **2. Fields of Application of III-V Solar Cells**

The III-V MJ (multi-junction) solar cells are utilized in the most varied fields of application, the most important two being space applications and concentrator photovoltaic (CPV) systems, as illustrated respectively in Figure 1a,b. These two fields represent very different operating conditions for solar cells, and thus different design approaches for each field must be considered. Record efficiencies of 35.8% (AM0 spectrum) [23] and 46% (AM1.5d spectrum, 508 suns) [22] were already demonstrated for space and CPV applications, respectively.

(**a**) (**b**) **Figure 1.** Some examples of solar cells use in space and terrestrial applications: (**a**) NASA's InSight Lander robot, powered by solar energy, and already owns the off-world record of power generation. (**b**) A HCPV parabolic system that uses high-efficiency multi-junction modules by Solartron Energy Systems.

#### *2.1. Space Applications*

Regarding space applications, III-V cells have become the go-to technology, not only because of their high-efficiency results but also because of their high tolerance to radiation exposure. After being irradiated with high radiation doses, these cells showed an EOL (endof-life) efficiency that was higher than a BOL (beginning-of-life) efficiency of a standard Si solar cell. Of course, this represented a major change for the spacecraft industry, since a good EOL efficiency is intrinsically connected to the weight and cost of the overall system, paramount factors when discussing the launch of a spacecraft, in which the cost is determined by €/kg, as opposed to €/Wp in terrestrial applications.

Therefore, these cells, given their high EOL efficiencies, good radiation tolerance, and high power-to-mass ratios (W/kg), meet the requirements of the majority of the NASA OSS (National Aeronautics and Space Administration Operational Support Services) missions, that call for high specific power values, making them the state-of-the-art cells for the majority of satellites and space vehicles.

Another important aspect concerning missions in space is the temperature at which PV modules must operate in certain harsher environments. Space PV arrays must be prepared to endure both high and low temperatures, depending on the mission's orbit. This leads to the necessity of studying the cell's temperature coefficient (dη/dT) to have a measure on how the performance of the cell will vary with temperature. When under the AM0 spectrum, the normalized temperature coefficient of a Si solar cell is in the range of −3 × 10−<sup>3</sup> /°C to −5 × 10−<sup>3</sup> /°C, while for tandem GaAs/Ge cells the temperature coefficient is approximately −2 × 10−<sup>3</sup> /°C [24].

This notorious difference in temperature coefficients is explained by the variance of bandgap in both cells; solar cells that have in their composition materials with higher bandgap values show lower efficiency losses with temperature [25]. This means that there will be an ideal bandgap for each operating temperature.

#### *2.2. Terrestrial Concentrator Systems*

On Earth, the task of implementing III-V plate modules would represent a heavy cost of production, with the cost of a typical III-V high-efficiency cell being around 10 \$/cm<sup>2</sup> [26]. To counter this problem, solar PV companies developed concentrator photovoltaic systems (CPV), in which sunlight is concentrated with the use of mirror lenses. Usual concentration ratios for III-V cells may go from 500× to 2000× , the latter being commonly called high concentration PV (HCPV).

The increase in irradiance will directly affect the short-circuit current of the cell, increasing it. Resorting to Equation (1), it is easy to see that incrementing affects the open-circuit voltage of the cell, which increases logarithmically by several / factors. This boost in the will be more evident for a multi-junction cell, in which every subcell will contribute for the increase of with concentration, and thus rising the fill-factor of the overall cell [6–19].

$$V\_{OC} = nV\_T \ln{\frac{I\_{SC}}{I\_0}} + 1\tag{1}$$

For this reason, it would be fair to think the higher the concentration ratio, the higher the efficiency of the cell. Alas, in reality, no device is ideal, including solar cells; there are always losses that need to be considered, such as series and shunt resistances that must be taken into consideration. The concentration increase will have a dominant impact on the overall efficiency, diminishing the (Fill Factor), and changing the − characteristic. The greater the concentration ratio, the higher the impact will be on the cell; e.g., for the TJ (triple-junction) GaInP/GaInAs/Ge, when incrementing the series resistance from = 0 to = 0.1 Ω, the is reduced from 90% (1 sun) to 87% at 83 suns, and to 71% at 500 suns [27].

Analyzing this data, it was then evident that some changes in series and shunt resistances had to be made in such a way that cells could operate under high concentration levels so that losses could be, to an extent, negligible. Every concentrator cell has a concentration limit for which the efficiency will start to drop, and several studies are being conducted in this matter. In the work of Steiner et al. [28], three tests were made using the single junction GaAs solar cell to prove the reduction in the and efficiency: three optimized grids for concentrations of = 100, = 450, and = 1000 were tested, and the cell showed a maximum efficiency of 29.09% for a concentration of 450 suns.

#### **3. III-V Solar Cell Design**

For a better understanding of the fundamentals behind a III-V solar cell it is necessary to perceive where they differ from the simple junction cell. It has already been stated that III-V multi-junction cells are top performers in their fields of application, when compared with their single-junction counterparts, given that the latter have their efficiency limited a priori.

#### *3.1. Bandgap versus Efficiency*

In order to grasp why single-junction cells are limited efficiency-wise, one has to fathom how the bandgap is of paramount importance when discussing solar cells.

Taking into consideration a single-junction solar cell with bandgap , only photons with their energy higher or equal to are absorbed. Photons for which the energy is higher than the bandgap , there is a certain amount of energy that is in excess and will be lost, an phenomenon also known as thermalization losses. This means that the energy that will be effectively converted into electric current will be just a portion of the photon's total energy. With this, it is evident that the device will only operate at maximum efficiency when the photon's energy, ℎ, is equal to the bandgap . Alas, when considering the wide spectrum of sunlight, absorbing just the photons of a specific wavelength imposes quite a limitation on the overall efficiency of the cell.

In trying to solve this problem, a few solutions were developed. One of them is broadly used today in the PV industry: the concept of multi-junction solar cells. Instead of trying to make the cell operate only at a specific wavelength, one could try to divide the light spectrum into several spectral sections and associate a subcell with an appropriate bandgap to each one of them. This way, every subcell would have the unique function of absorbing photons of a specific wavelength range.

Now, there are different approaches to solve the problem and split the sunlight's spectrum. The first is a quite intuitive one, called the spatial distribution method, illustrated on Figure 2a, and consists in using a prism to separate a beam of white light into several different wavelengths and spatially arranging subcells with different bandgap values accordingly.

**Figure 2.** Spectral splitting approaches: (**a**) Spatial distribution, with the use of a prism, and (**b**) Stacked distribution of a 3-junction cell.

Even though the spatial distribution is employed in some CPV (concentrator photovoltaic) systems, there are some difficulties associated when using this method. The approach that is broadly used nowadays when designing MJ solar cells is the stacked distribution, as presented in Figure 2b. This method consists in stacking the subcells on top of each other by order of bandgap, so the subcell with a lower bandgap is placed on the bottom of the cell and the one with a higher bandgap is placed on the top. This way, the high energy photons can be absorbed right on top of the cell by subcells with high bandgap values, forcing the low energy photons to penetrate further into the lower layers, where the low bandgap subcells are placed. As a result, the photons will be efficiently distributed and absorbed throughout the stack, increasing overall performance.

For this reason, the choice of bandgap combinations is a decisive step in multi-junction design. Given that III-V semiconductors show high versatility in possible bandgap combinations, they are one of the best choices for designing state-of-the-art solar cells. Bearing this in mind, Fraunhofer ISE developed the etaOpt software, capable of predicting cell efficiencies based on how many p–n junctions they are made of and what are their respective bandgaps. According to the results obtained from etaOpt, the efficiency can increase substantially with the number of p–n junctions, however, this gain is dampened for higher counts, i.e., a jump from 2 to 3 junctions provides a much larger increase than one from 5 to 6 junctions [29]. Knowing this data *a priori* is quite important for manufacturers, since the amount of efficiency gained may not justify higher production costs that derive from augmenting the number of p–n junctions.

#### *3.2. Bandgap versus Lattice Constant*

The choice of an appropriate bandgap does not take into account only the spectral regions, but also the choice of the lattice constant, since one depends on the other. This selection determines the structure of a MJ solar cell—if the materials all have, approximately, the same lattice constant, the cell is said to be lattice-matched; on the contrary, when the materials have different lattice constants, one says that the cell is lattice-mismatched or metamorphic (MM).

This distinction is significant when discussing solar cell design, given that stacked materials with different lattice constants may create dislocations, which can ruin the quality of the material and thus its performance. The production of metamorphic cells has to consider appropriate strategies, such as step-graded buffers that make the transition between two materials with different lattice-constants less abrupt.

#### **4. The GaInP/GaInAs/Ge Solar Cell**

Regarding this work, it seems only relevant to discuss approaches in which the GaInP/GaInAs/Ge solar cells are utilized. The two most relevant examples are the latticematched triple-junction and the upright metamorphic structures [4,5].

#### *4.1. III-V Solar Cell Designs*

At the time this article is being written, the lattice-matched triple-junction Ga0.5In0.5P/ Ga0.99In0.01As/Ge, on Figure 3a, is the state-of-the-art cell for both space and terrestrial concentrator applications. The subcells are all lattice-matched to Ge, assuring that no dislocations are created. The cell itself consists of three main p–n junctions composed of GaInP, GaInAs, and Ge, stacked on top of each other, connected in series. The light falls on the GaInP subcell, which has the higher bandgap, as it was already explained previously. Each one of these subcells is connected through tunnel junctions with low resistance and high optical transmissivity coefficients. However, one of the problems of this approach is that the spectrum splitting is not optimal, resulting in an excessive current in the bottom Ge cell.

One possible way to counter this problem is to increase the absorption of photons in the upper cells, resulting in less current discrepancy. This can be achieved by lowering the bandgaps of the top and middle subcells by increasing the In composition in both Ga In1<sup>−</sup>P and Ga In1<sup>−</sup>As materials. By doing this, the lattice constant also alters, and thus the materials no longer have the same lattice constant, making the cell lattice-mismatched or metamorphic (MM). This type of approach in monolithic structures may derive in dislocations that can harm material quality if no special measures are taken. In the case of the upright metamorphic TJ GaInP/GaInAs/Ge cell, presented in Figure 3b, one of those measures is to implement a GaInAs graded buffer between the middle and bottom cells, so that the lattice constant increases gradually and not abruptly.

**Figure 3.** Simplified schematic of the triple-junction Ga0.5In0.5P/Ga0.99In0.01As/Ge cell: (**a**) latticematched, and (**b**) upright metamorphic approaches.

#### *4.2. Simulating the LM State-Of-The-Art Cell*

(**a**)

In order to simulate this cell, one has to take into account that several companies are currently researching various approaches to its development, the two most important being Fraunhofer ISE and Spectrolab, Inc.

In this work, the approach that was utilized is identical to the one used at Spectrolab, where this cell already demonstrated an efficiency of 32% under 1-sun (AM1.5G spectrum) [30]. Moreover, it is assumed a 1 cm<sup>2</sup> active area. While there is published research of this cell concerning some of its specific structural information, there are not many details available about doping and thickness values and the material compositions of each layer, given that all of these specifics are treated as proprietary information of Spectrolab.

Having as basis the detailed Ph.D. dissertation of Sharma [31], it was possible to put together an accurate model to simulate the cell. Some modifications were made to best adapt the cell to the one demonstrated by Spectrolab in the research paper of King et al. [30]. The simulated cell structure with all its layers is illustrated in Figure 4.

Firstly, to comprehend how the stacked cell works, it is necessary to perceive the role that each subcell plays in the monolithic cell by analyzing the materials that constitute each layer.

#### 4.2.1. The GaInP Top Subcell

Beginning from top to bottom, the first step was to simulate the GaInP top cell. This cell, as stated previously, has to absorb high energy photons, since it is on top of this cell that the light beams will fall onto. The Ga In1<sup>−</sup>P material is then chosen for its bandgap, which is = 1.89 eV for a composition of = 0.5. This is a pretty high value that allows for the first high energy photons to be absorbed.

Besides the main p–n junction being composed of GaInP, the top subcell also contains two extra layers: the back-surface (BSF) and the window or front-surface (FSF) layers.

The window layer acts as an absorber layer, and thus it will have to have a high bandgap, small thickness, and a low series resistance. The material chosen can be the AlInP since it has a pretty high bandgap value and it is capable of being lattice-matched to the rest of the cell.

In contrast, the BSF layer exists to boost the short-circuit current of the cell, given that sharing the applied voltage across the n–p–p+ junctions minimizes the reflection of minority carriers and therefore leads to the decrease of the dark current. The material that is chosen for this is the quaternary AlGaInP.

**Figure 4.** Detailed schematic of the simulated LM Ga0.5In0.5P/Ga0.99In0.01As/Ge solar cell.

#### 4.2.2. The GaInAs Middle Subcell

The second subcell to be simulated is the middle GaInAs cell, which is based in the more simple GaAs solar cell. It is lattice-matched to all the components that form the whole monolithic cell, with the main ternary compound, Ga In1<sup>−</sup>As, having the composition = 0.99 since its lattice constant corresponds to an exact-match to Ge's.

The subcell also has window and back-surface layers that are composed of highlydoped GaInP (composition of = 0.5) given the high optical output of this material.

#### 4.2.3. The Ge Bottom Subcell

Finally, the bottom subcell is made of a Ge substrate, instead of the typically used GaAs. This has two major advantages; firstly, Ge is cheaper than GaAs, and secondly, since Ge has a very low bandgap ( = 0.66 eV) the thickness of the subcell can be reduced from around 300 μm for the GaAs substrate to 170 μm for the Ge substrate.

Apart from a GaInP window layer similar to the middle cell one, the subcell also has a buffer layer made of highly-doped n-GaInAs (composition of = 0.99) in order to reduce the ohmic contact between the bottom cell and the tunnel junction.

#### 4.2.4. I–V Characteristic of the Stacked Cell

With the subcells already demonstrated, the next step was to try and assemble all of them in a monolithic cell.

Besides stacking the subcells on top of each other and separating them with appropriate tunnel junctions (AlGaAs–GaAs and AlGaAs–AlGaAs), it was also necessary to emulate the resistivity between subcell–tunnel diode and tunnel diode p–n junctions. This is made by establishing ohmic contacts with extremely high resistances that act as boundary conditions.

Two simulation models were tested: the first one, the cell was simulated in a Finite Element Tool without the metal grid (MG) on top, and the front contact had the same horizontal extension of the rest of the cell layers. This approach is a 1-D model since the structure only varies in one direction (vertical). The results were, then, artificially high since the contact effects were not being considered; the second method was employed so that the model would consider contact effects of the metal grid. Ergo, the cathode (top electrode) became smaller and a cap layer made of n+-GaAs was put below it with good ohmic contact formation in mind. Since there is this variation in the horizontal axis now, the model is a 2-D model.

Having as a reference the structure of the cell used at Spectrolab, the model developed in the Finite Element Tool was identical to the one depicted in Figure 4. In Figure 5, the obtained − characteristics are presented for both simulation models: 1-D model (without the MG) and 2-D model (with the MG). The most important figures of merit obtained from the simulated results are shown in Table 1; for comparison purposes, the experimental results from Spectrolab, Inc. [30] are also presented.

**Figure 5.** Simulated − characteristics of the stacked cell. All curves were obtained using the AM0 spectrum. Figures of merit are presented in Table 1.


**Table 1.** Comparison of experimental and simulation values for the stacked cell.

\* metal grid.

Analyzing the results, one can see that the best model to emulate the original cell's behavior is the 2-D model, in which some of the device's losses are being considered. The cell was emulated successfully to some extent: both the open-circuit voltage and short-circuit current were fairly replicated, which means that the overall structure (region materials, thickness, doping, etc.) was correctly modeled. Alas, both the fill-factor and efficiency were not consistent with the experimental results from Spectrolab. One explanation for this may be that losses were not properly accounted for in the final model, even taking the metal grid under consideration.

#### 4.2.5. External Quantum Efficiencies

The final test was to obtain the External Quantum Efficiency (EQE) from each subcell when stacked. This analysis provides a frequency response of the cell, which can be precious information to understand and further optimize solar cells.

Resorting to the optical bias method, it was possible to extract the individual EQE of each subcell. This method consists of saturating all the subcells simultaneously, except the one under test, so that the saturated junctions will not limit the current, while that the cell that is not saturated (the one under study) will determine the current value, and thus its EQE can be computed.

When computing the EQE, it is necessary to have in mind that each cell will only absorb in a very specific wavelength range, that strongly depends on the bandgap of the other subcells. This dependence is due to the fact that the light spectrum is being split by the stacked distribution. In the lattice-matched approach, the GaInP top cell absorbs photons with energy ℎ > 1.89 eV, the GaInAs middle cell will absorb between the range of 1.89 > ℎ > 1.41 eV and, finally, the Ge bottom cell will absorb photons with energy 1.41 > ℎ > 0.661 eV. All of this is well illustrated in Figure 6, in which the simulated results in the Finite Element Tool are presented.

**Figure 6.** Simulated External Quantum Efficiencies of each subcell.

#### *4.3. Temperature Effects*

Temperature, naturally, is one of the most important factors when studying the behavior of semiconductors. This way, solar cells are usually tested for a nominal operating cell temperature (NOCT) of 25 °C, which is generally approximated to = 300 K in absolute temperature values.

However, the photovoltaic cells under study have to be designed to withstand the extreme temperatures that only space can bestow. These temperatures can go from very high temperatures (HIHT (high intensity high temperature) missions) and deep-space temperatures like −170 °C, which is the cell temperature for Saturn-orbit missions. Therefore, it makes sense to try and emulate the cell under these conditions.

#### High and Low Temperatures

To try and perceive how high temperatures affect solar cell performance, a simulation was run first for = 300 K and then for higher temperatures, in intervals of 50 K, to the final temperature of = 500 K. Besides studying the cell's behavior at high temperatures, it is also important to understand how they perform at temperatures below 0 °C. Even if

some parameters variances can be expected, namely the increase in the open-circuit voltage and overall efficiency, there is some interest in how they vary for low temperatures.

The extracted − curves for both ranges of temperature are illustrated in Figure 7, with the most relevant figures-of-merit from both plots (Figure 7a,b) being registered in Table 2.

(**b**)

**Figure 7.** Tandem cell's − characteristics obtained for two different intervals of temperature: (**a**) high temperature range, from = 300 K to = 500 K, and (**b**) low temperature range, from = 250 K to = 300 K. All curves were obtained using the AM0 spectrum.

**Table 2.** Measured values for both temperature ranges: high and low temperatures, from the respective plots (a) and (b), displayed in Figure 7.


Both the open-circuit voltage and short-circuit current behave as expected: has an insignificant variance whereas the decreases substantially as temperature increases.

As for the efficiency and fill-factor, they both decrease as temperature rises, however, this is only valid to a certain point. As the array temperature gets colder, the variance in certain parameters begins to be non-linear. This is because, as temperature decreases, carriers start to enter the state of "freeze-out", in which there is not enough thermal energy

for the dopants to be fully ionized, and thus there will be a shortage of charge carriers. Another issue is the phenomenon called "broken-knee" or "double-slope", in which the − characteristic becomes degraded, generating a great reduction in the fill-factor and efficiency—this can be seen in the obtained curve for = 230 K.

Notwithstanding, colder environments, to a certain extent, are good for solar cells since there is a boost in the overall performance; the obtained results confirm the need for some PV panels to have cooling systems installed so that the power conversion efficiency is maximized.

#### **5. Cell Optimization**

With the lattice-matched GaInP/GaInAs/Ge solar cell properly reproduced and simulated, an overall optimization of the cell is attempted. In order to do this, two studies on how thickness and doping affect the overall performance of the cell were made. The first study takes into account the top and middle subcells and their respective thicknesses. The second study will take into account the doping of the GaInP top subcell. The properties of the whole cell were maintained constant with the default, previously simulated parameter values.

Considering that the cell that was being simulated up to this point was optimized for CPV (concentrator photovoltaic) applications, this work will attempt to perform an optimization for space applications in LEO (low-Earth orbit, <1000 km) missions. The spectrum utilized was the AM0 and the cell temperature was = 300 K.

#### *5.1. Thickness Variation*

When varying the cell thickness, it is necessary to select which layers are going to be altered. Since the photocurrent of the entire cell is determined by the top cell, the first layers to be chosen were the GaInP- base and emitter layers. The BSF and FSF layers were not altered, since their values were already at the minimum possible. The main goal of this study is to choose thickness values that establish a compromise between efficiency and size of the cell, given that the less cell bulkiness the better.

The first test consisted in varying both base and emitter thicknesses of the top cell and evaluate the efficiency, , improvement. Other parameters like short-circuit current, , open-circuit voltage, , fill-factor, , and the variation in efficiency, ΔEff., were also registered. Both default (gray) and best (green) obtained results for the first test are displayed in Table 3. The best efficiency achieved was 31.80% which in comparison to the initial value of 31.76% corresponds to an improvement of +0.1107%.


**Table 3.** First study, first test: Top GaInP subcell thickness variation of the p-base and n-emitter layers.

Default values ; Best obtained values .

The second test was analogous to the first, except it was made considering only the middle subcell thickness. Once again, the BSF and FSF layers were not altered, varying only the thickness of both GaInAs- base and emitter layers. The top GaInP layers' thicknesses were the initial ones, without employing the optimization of the first test. The default values along with the best-obtained results are presented in Table 4. Alas, in this case, the best-obtained results (in green) correspond to a thickness increase of 0.5 μm in the base thickness. Since a bulkier cell is not the desired outcome, the second-best results (red) that achieved an efficiency of 31.8036% were chosen. This efficiency value corresponds to an improvement of +0.1098%.


**Table 4.** First study, second test: Middle GaInAs subcell thickness variation of the p-base and n-emitter layers.

Default values ; Best obtained values ; Second best values .

All of the obtained results for both tests are illustrated in two 3D plots, in which one can observe how cell the layers' thicknesses affect the overall performance of the cell. The 3D surface plots are presented in Figure 8 and were made resorting to the Curve Fitting Tool of MATLAB©.

With this visual aid, it fairly clear that for the first test (Figure 8a), the efficiency depends on both GaInP- base and emitter thicknesses, being apparent that higher efficiencies concentrate in a range of values that are roughly in the center of the plot.

Similarly, analyzing the 3D plot for the second test (Figure 8b) it is evident that the higher the middle subcell's base thickness, the higher the efficiency. Unlike the first test, the GaInAs-base thickness is predominant in how the efficiency varies.

Finally, the best results from both tests were simulated, so that both subcell optimizations could be taken into account. The obtained parameters were: = 18.5943 mA, = 2.6276 V, = 88.98% and an efficiency of = 31.8431%, which translates in an improvement of 0.2343%, in comparison with the initial value.

(**a**) **Figure 8.** *Cont*.

#### (**b**)

**Figure 8.** Graphic display of the obtained results for the: (**a**) first test: top GaInP subcell, and (**b**) middle GaInAs subcell. Both 3D plots were obtained using the 3D fitted surface with the cubic method of MATLAB Curve Fitting Toolbox.

#### *5.2. Doping Variation*

The second and final study was designed to evaluate how doping alters the performance of the cell. This last simulation is run with the best thickness values obtained in the first study.

Only the top cell's base and emitter layers are going to be contemplated in this study. Once more, Table 5 shows the best-obtained results of doping variation for the GaInP- base and emitter layers. The best obtained efficiency was 33.0194%, which corresponds to a total improvement of +3.9368% of the very first efficiency value that was = 31.7687% (refer to Tables 3 and 4).

**Table 5.** Second study: doping variation of the p-base and n-emitter layers in the top subcell.


Default values ; Best obtained values .

The doping values that were simulated were carefully chosen, given that the higher the doping, the lower the potential barrier to be overcome, making higher efficiencies possible to achieve. However, this efficiency increase can not be indefinite, since the minority carrier lifetime and diffusion length decrease with doping increase [32]. Hence, searching for the optimal doping value that increases efficiency without degrading the electronic properties of the semiconductor is of paramount importance. Values past 2.00 × 1018 cm−<sup>3</sup> for the base and 1 × 1019 cm−<sup>3</sup> were not chosen, given that simulations run with doping values higher than these resulted in deterioration of the − curve.

Analogously to the first study, a 3D fitted cubic surface of the results was plotted and it is illustrated in Figure 9. It is clear that the base doping is predominant in efficiency variation; as it increases, efficiency values increase, reaching a peak region in which the

efficiency is the highest possible. Beyond those values, there is an abrupt drop in the short-circuit current and open-circuit voltage, resulting in an efficiency reduction.

**Figure 9.** Graphic display of the obtained results of the second study. The 3D fitted surface with cubic method (MATLAB©).

This concludes the optimization of the cell for operation under the AM0 spectrum, at the nominal temperature of = 300 K. As it has already been mentioned, temperatures in space can oscillate from extremely low to very high temperatures (sometimes in the same mission), and so each PV array must be optimized in accordance with the conditions it is planned to operate at.

#### **6. Conclusions**

The main aim of this work was to create a model so that a triple-junction state-ofthe-art solar cell could be emulated and then analyzed with accuracy, without the need to resort to more advanced, and expensive, simulation technologies.

Comparing the simulated results with the actual experimental results by Spectrolab, Inc. one could say that the main goal was achieved, and the cell was emulated successfully. Both the open-circuit voltage and short-circuit current were fairly replicated, which means that the region materials, thickness, and doping were correctly modeled.

However, both the fill-factor and efficiency were not consistent with their experimental counterparts; this may have to do with the fact that losses were not properly accounted in the modeled cell since the only loss mechanisms present were the metal grid and the back/front contacts, and the fact that complex refractive indices were used in simulation (the imaginary part accounts for losses). Experimental values are calculated by appropriate measuring devices, such as multimeters, connecting them in series/parallel to a resistor, which in turn is connected to both terminals of the cell. This results in part of the losses not being accounted for in the simulation.

Other relevant differences are the external quantum efficiencies that were obtained for each subcell, in contrast with the experimental curves from Spectrolab, Inc. [33]. This is due mainly to the use of refractive indexes that do not correspond to the exact composition of a certain material. For instance, the most obvious difference is between the simulated and experimental frequency responses in the middle cell; this discrepancy may reside in the fact that the only refractive index available (from the databases) is not a rigorous match for the composition of = 0.99 in Ga0.99In0.01As. This explanation is valid for other ternaries used as well.

Furthermore, bearing in mind that temperature plays a significant part in semiconductor performance, a test to evaluate how temperature influences the cell was also conducted. Most of the published research on how the GaInP/GaInAs/Ge solar cell behaves under different temperatures only has into consideration the higher range of temperature, given its paramount use in concentrator photovoltaic systems. With spacecraft implementation in sight, it was thought to be relevant to verify how the cell behaviors at low temperatures. Even though some plausible results were obtained, simulations in temperatures below 230 K did not obtain convergence, considering that the cell's design was not prepared for such low-temperature environments.

Finally, an optimization of the GaInP/GaInAs/Ge LM cell was also conducted. In this optimization, certain cell parameters were tweaked so it could reach its maximum potential for a 1-AM0 incidence. This could prove of some value for the photovoltaic industry that is dedicated to the manufacturing of solar cells for space applications, given that the doping can significantly boost the cell's efficiency.

Regarding the simulation times, depending on the mesh fineness and the voltage step that are being employed, the whole model takes roughly six minutes to simulate with Newton's method. This may be an advantage over more complex and detailed ways of simulation that are more time-consuming if the main objective is simply to obtain the major figures-of-merit of the cell.

**Author Contributions:** S.B. was responsible to write the original draft, J.P.N.T. and C.A.F.F. are her supervisors, J.P.N.T. and R.A.M.L. analysed the results and they were responsible to review and editing the final manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** This work was supported in part by FCT/MCTES through national funds and in part by cofounded EU funds under Project UIDB50008/2020.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


### *Article* **Optical Nanoantennas for Photovoltaic Applications**

#### **Francisco Duarte 1, João Paulo N. Torres 1,2,3, António Baptista 1,4 and Ricardo A. Marques Lameirinhas 1,2,\***


**Abstract:** In the last decade, the development and progress of nanotechnology has enabled a better understanding of the light–matter interaction at the nanoscale. Its unique capability to fabricate new structures at atomic scale has already produced novel materials and devices with great potential applications in a wide range of fields. In this context, nanotechnology allows the development of models, such as nanometric optical antennas, with dimensions smaller than the wavelength of the incident electromagnetic wave. In this article, the behavior of optical aperture nanoantennas, a metal sheet with apertures of dimensions smaller than the wavelength, combined with photovoltaic solar panels is studied. This technique emerged as a potential renewable energy solution, by increasing the efficiency of solar cells, while reducing their manufacturing and electricity production costs. The objective of this article is to perform a performance analysis, using COMSOL Multiphysics software, with different materials and designs of nanoantennas and choosing the most suitable one for use on a solar photovoltaic panel.

**Keywords:** nanoantennas; optics; optoelectronic devices; photovoltaic technology; rectennas

#### **1. Introduction**

In the last decade, the advances in the nanoscale dimension enabled the development of new devices, such as nanoantennas or optical antennas, due to the emergence of a new branch of science known as nanooptics, which studies the transmission and reception of optical signals at the nanoscale. These devices have been the object of intense research and development activity, with the goal to reach the captivating possibility of confining the electromagnetic radiation in spatial dimensions smaller than the wavelength of light.

The transmission through a metal plane with subwavelength-sized holes can be drastically increased if a periodic arrangement of holes is used. This phenomenon is widely known as Extraordinary Optical Transmission [1]. The usage of nanoantenas with apertures smaller than the light wavelength can locally enhance light–matter interaction. Thus, nanoantennas are devices that have the ability to manipulate and control optical radiation at subwavelength scales.

Nanoantennas are a nanoscale version of radio-frequency (RF) or microwave antennas. However, throughout this article it will be proven that in the process of sizing the nanoantennas, it will not be enough to reduce the size of the RF antennas to the optical domain, mainly because of the unique material properties of metals that influence the behavior of antennas at the nanoscale: the existence at the interface between metals and dielectrics of surface plasmon-polariton electromagnetic waves, which gives rise to resonant effects not available at RF [1,2].

The use of optical antennas for solar energy harvesting has received significant interest as they represent a viable alternative to the traditional energy harvesting technologies.

**Citation:** Duarte, F.; Torres, J.P.N.; Baptista, A.; Marques Lameirinhas, R.A. Optical Nanoantennas for Photovoltaic Applications. *Nanomaterials* **2021**, *11*, 422. https://doi.org/10.3390/ nano11020422

Academic Editor: Antonio Di Bartolomeo Received: 5 January 2021 Accepted: 3 February 2021 Published: 7 February 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Economical large-scale fabrication of nanoantenna devices would support applications such as building integrated photovoltaics and supplementing the power grid [3].

#### **2. Rectenna System for Solar Energy Harvesting**

The nanoantenna itself does not convert the collected AC current into DC current, and so it needs to be complemented with a rectifying element. The whole structure is commonly referred to as a rectenna [4].

A rectenna is a circuit containing an optical antenna, filter circuits, and a rectifying diode or bridge rectifier for the conversion of electromagnetic energy propagating through space (solar energy) into DC electric power (through the photovoltaic effect).

A schematic representation of a nano-rectenna system is depicted in Figure 1.

**Figure 1.** Representation of the nano-rectenna system (adapted from the work in [5]).

First, electromagnetic radiation is collected by the nanoantenna device. However, the output obtained from a single nanoantenna element is not enough to drive the rectifier and to provide DC power to an external load. The efficiency of a single optical antenna is generally low and its functionality is limited. Therefore, nanoantennas are arranged into arrays to increase their signal. The total field captured by the array is the addition of the fields captured by each nanoantenna [6].

The AC current generated in the nanoantenna arrays is collected and rectified into DC current by the rectifier system. This system has different rectifiers whose outputs can be DC coupled together, allowing arrays of nanoantennas to be networked to further increase output power [3].

As optical radiation requires high-speed rectification, high frequency metal–insulator– metal (MIM) diodes—also known as tunneling diodes—are commonly used forthis purpose.

According to Moddel and Grover, the MIM diodes must have three key characteristics in order to have an efficient rectifier system [6]: high responsivity, that is, a measure of the rectified DC voltage or current as a function of the input radiant power; low resistance, in order to have a good impedance matching between the antenna and the diode; and asymmetry in the I–V curve, so the diode must have asymmetric characteristics for the rectenna be operated without applying an external DC bias.

Examples of material combinations used for diode rectifiers include Ni/NiO/Ni, Nb/Nb2O5/Pt, Nb/TiO2/Pt, Cu/TiO2/Pt, Nb/MgO/Pt, and Nb/Al2O3/Nb [6].

#### **3. Experimentally Studied Nanoantenna Materials and Designs**

A large variety of nanoantenna geometries has been researched for multiple potential applications. Currently, nanoantennas structures are mainly made of plasmonic materials, i.e., specially designed metal (usually gold or silver) nanoparticles with unique optical properties. Plasmonic materials exhibit strong light absorption in the visible region of the spectrum [4].

The advances in the manufacturing techniques allowed the construction of different formats of nanoantennas. The main types of plasmonic nanoantennas that have been proposed and investigated experimentally are represented in Figure 2.

**Figure 2.** Main types of plasmonic nanoantennas (adapted from the work in [7]).

#### *3.1. Plasmonic Monopole Nanoantenna*

The most basic type of plasmonic nanoantenna, a monopole, is a single metallic nanoparticle that can enhance the electromagnetic field strength in its surrounding area upon excitation of plasmon resonances. Monopole nanoantennas have advantages over other geometries, because they are easier to engineer and are well isolated from interference due to the ground plane. Their characteristics are dependent on the shape, size, material, and dielectric environment of the nanoparticle [7].

Another great utility of the monopole optical antenna is when it is integrated in a Near-field Scanning Optical Microscopy (NSOM), to be used as a near-field probe for measurements [8]. An effective nanoantenna can be used in spectroscopy: it needs to interact strongly with incident electromagnetic radiation in order to measure its intensity.

#### *3.2. Plasmonic Dipole Nanoantenna*

Dipole configurations are widely used in radio frequency and microwave ranges. Therefore, it is not a surprise that analogs of such antennas also appeared in the optical range. This type of optical antenna is widely used in near-field optical probes, just like the monopole. The dipole optical antenna is constituted either by dimers or two monopoles separated by a small space (gap). Usually, there is a high field confinement in the gap between the two metallic nanoparticles [7].

The design of plasmonic nanoantennas may rely on the same principles used in RF antennas. For example, the length of the dipole RF antenna is approximately half the wavelength of the incident radio waves, whereas the length of the dipole plasmonic nanoantenna is smaller than the wavelength, *λ*, of incident light in free space [9].

#### *3.3. Plasmonic Bowtie Nanoantenna*

Another typical structure is the bowtie nanoantenna, consisting of two triangular shape nanoparticles aligned along their axes and forming the feed gap with their tips. These optical antennas are a variant of the dipole nanoantennas. Such geometry ensures a wider bandwidth together with large field localizations in the feed gap compared to the straight dipole.

The bowtie topology is considered to be one of the most efficient nanoantenna geometries for solar energy harvesting. According to Sen Yan [5], in their study it is shown that the bowtie topology can increase the total radiation efficiency and rectenna efficiency compared to the straight dipole by a considerable 10%.

#### *3.4. Plasmonic Yagi-Uda Nanoantenna*

RF Yagi–Uda type antennas are usually used to receive TV signals from remote stations, due to their high directivity. Their plasmonic counterparts consist of a reflector and one or several directors.

Yagi–Uda optical antennas can be useful in many applications: in wireless communications, in the fields of biology and medicine, in nanophotonic circuits, in quantum information technology, in data storage (as an optical chip), in photodetectors, and in photovoltaic (PV) systems.

#### *3.5. Plasmonic Spiral-Square Nanoantenna*

This design of nanoantenna allows the electromagnetic radiation to be harvested in one specific point in its structure—the gap (feed point) between two metallic arms, as presented on Figure 3. Thus, this topology has a wider angle of incidence exposure in comparison to other formats, which makes it an ideal geometry for solar energy harvesting.

**Figure 3.** Geometry of a square-spiral nanoantenna (sourced from the work in [10]).

They also demonstrate a high directivity that can be further improved by increasing the number of arms.

#### *3.6. Dielectric Nanoantennas*

A new research direction of optical antennas has recently been suggested with the introduction of dielectric nanoantennas. Optical antennas constructed with dielectric materials have several advantages over their metallic counterparts due to unique features not found in plasmonic nanoantennas [4].

Dielectric nanoantennas are fabricated from optically transparent materials that have low dissipative losses at optical frequencies. Unlike gold or silver, dielectric nanoantennas are usually made from silicon nanoparticles which are widely used in nanoelectronics to fabricate transistors and diodes. Furthermore, silicon has a high permittivity and exhibits very strong electric and magnetic resonances at the nanoscale, and thus improves radiation efficiency and antenna directivity, expanding the range of applications for nanoantenna structures [4,11].

The authors of [11] used silicon nanoparticles to demonstrate the performance of all-dielectric nanoantennas. They have analyzed an all-dielectric analog of the plasmonic Yagi–Uda nanoantenna consisting of an array of nanoelements: four directors and one reflector particle made of silicon. In this type of structure, the optimal performance is obtained when the director nanoparticles sustain a magnetic resonance and the reflector nanoparticle sustains an electric resonance [4]. schematic representation of this antenna is shown in Figure 4.

The dipole source is placed equally from the reflector and the first director surfaces at the distance D. The separation between surfaces of the neighboring directors is also equal to D.

The operational regime of a dielectric Yagi–Uda nanoantenna strongly depends on the distance between its elements. According to Krasnok [11], in their study it was verified that the radiation efficiency of the dielectric Yagi–Uda nanoantenna slowly decreased with decreasing distance between its elements, while the radiation efficiency of a plasmonic antenna of similar design and dimensions was greatly affected by the decrease in distance between particles. This is due to increased metal losses caused by proximity of adjacent metallic nanoparticles.

However, for larger separation distances, D, the radiation efficiencies of both types of nanoantennas were very identical. Although dissipation losses of silicon are much smaller than those of silver, the dielectric particle absorbs the EM energy by the whole spherical volume, while absorption only occurs at the surface of metallic particles. As a result, there is no substantial difference in the performance of these two types of nanoantennas for relatively large distances between its elements.

To sum up, based on the results of this study, a conclusion could be made that alldielectric nanoantennas demonstrate major advantages over their metallic counterparts: much lower Joule losses and strong optically induced magnetization [11].

#### *3.7. Aperture Nanoantennas*

There is another type of optical antenna that is interesting for the topic of this article: aperture optical antennas. Light passing in a small aperture is the subject of intense scientific interest since the very first introduction of the concept of diffraction by Grimaldi in 1665 [12].

The first theory of diffraction due to a slit, that is much less than the light wavelength, in a thin metal layer was developed by Bethe. This theory predicted that the power transmitted by the slit would decrease as the slit diameter decreased relative to the wavelength of the EM radiation. This theory proved to be incorrect when Ebbesen, in 1998, observed the extraordinary optical transmission phenomenon (EOT) [1]. The EOT is an optical phenomenon, in which a structure containing subwavelength apertures transmits more light than might naively be expected. Ebbesen et al. observed that when focusing a light beam in a thick metallic film where there was a subwavelength hole array, a large increase of incident electromagnetic wave transmission occurs, i.e., a periodic array of subwavelength holes, as presented in Figure 5, transmits more light than a large macroscopic hole with the same area as the sum of all the small holes [1,2,13–19].

**Figure 5.** Schematic view of 200 nm diameter aperture arrays with 1 um period (adapted from the work in [12]).

This discovery would be fundamental, as it not only allowed great technological developments during the last decade, but also allowed a better understanding of the diffraction by small slits in relation to the light wavelength [20,21].

According to Wenger, there are three main types of aperture antennas [12]: single subwavelength aperture, single aperture surrounded by shallow surface corrugations, and subwavelength aperture arrays.

#### **4. Surface Plasmon Resonance**

As referred in the introduction, incident light on the optical antenna causes the excitation of free electrons in metallic particles. More precisely, EM waves induce timevarying electric fields in the nanoantenna that apply a force on the gas of electrons inside the device, causing them to move back and forth at the same frequency range as the incoming light. This phenomenon is known as surface plasmon. At specific optical frequencies the nanoantenna resonates at the same frequency as the incoming light which enables the absorption of the incoming radiation [4,15–19].

It should be taken into account that, at optical frequencies, metals do not act as perfect conductors: their conductivity changes dramatically, and so they are unable to respond to the time-varying electric field immediately. The wave propagation within the material is affected, which means that the penetration of EM radiation into metals can no longer be neglected.

Thus, at optical frequencies an antenna no longer responds to external wavelength but to a shorter effective wavelength that depends on the material properties [20].

EM radiation penetrates the metal of the nanoantenna and gives rise to oscillations of the free-electron gas. These electron oscillations can give rise to plasmon resonances, depending on the size, shape, and index of refraction of the particle as well as the optical constants of its surrounding [21].

When these oscillations are optimized, i.e., when the metal structure is sized to achieve the resonance condition, it is called Surface Plasmon Resonance (SPR). It is also important to mention that there are two types of surface plasmons [15–19,22]: Surface Plasmon Polariton (SPP), when the EM waves strike a metallic film and are confined to the surface of this film, and Localized Surface Plasmon (LSP), when the coupling is made with a metal nanoparticle with a diameter much smaller than the incident wavelength.

Surface plasmons are highly confined energy fields made by the oscillation of electrons on the surface of nanoantennas. When a metallic nanoparticle is illuminated by light, surface plasmons will be coupled with the photons of incident light in the form of a propagating surface wave [23].

SPPs are infrared or visible frequency EM waves trapped at or guided along metal– dielectric interfaces [24]. This coupling of plasmons—either SPPs or LSPs—and photons results in charge oscillation in the visible and infrared regimes depending on the metal used. SPPs are shorter in wavelength than the incident light (photons). Therefore, SPPs provide a significant reduction in effective wavelength and have tighter spatial confinement and higher local field intensity [24].

Recent development of nanofabrication techniques enabled construction of a variety of metal structures at the subwavelength scale and opened the research area called plasmonics, a subfield of nanophotonics studying the manipulation of light coupled to electrons at the nanoscale.

The properties of optical antennas are still under the intensive study and so research efforts to relate plasmonics with subwavelength optical antenna are in a developing stage [23].

#### **5. Efficiency**

The radiation efficiency of nanoantennas is a key parameter for solar energy harvesting. It is the first factor in the total efficiency product by which nanoantennas can convert incident light to useful energy. This efficiency depends directly on the type of metal used as conductor and the dimensions of the nanoantenna [6].

The main advantage of this type of technology in comparison to the conventional solar photovoltaic cells is its far greater efficiency by which the transformation of electromagnetic energy into DC electric power is performed. Typical efficiencies for traditional silicon cells are in the order of 20%, whereas nanoantennas go from a stunning 70% for silver nanodipoles [25] to a more realistic 50% for aluminum dipoles [26]. Most solar radiation is in the visible and infrared (IR) wavelength region, and so nanoantennas need to be designed for this part of the spectrum, with the aim of being an alternative to conventional solar photovoltaic cells.

The total efficiency of a rectenna consists of two parts: (1) the efficiency by which the light is captured by the nanoantenna and brought to its terminals, also known as radiation efficiency, *ηrad total*, and (2) the efficiency by which the captured light is transformed into low frequency electrical power by the rectifier, *ηmat total*.

According to Kotter, the total radiation efficiency could be given by expression 1 [25], where *λ* is the wavelength of the incident light and the upper and lower integration limits *λstart* and *λstop* should cover the optical bandwidth for the solar energy harvesting.

$$\eta\_{total}^{rad} = \frac{\int\_{\lambda\_{start}}^{\lambda\_{stop}} P\_{inc}(\lambda) \eta^{rad}(\lambda) d\lambda}{\int\_{\lambda\_{start}}^{\lambda\_{step}} P\_{inc}(\lambda) d\lambda} \tag{1}$$

Furthermore, *Pinc*(*λ*) is a function of the wavelength that follows Planck's law for black body radiation according to expression 2, with T being the absolute temperature of the black body that in this case is the temperature of the surface of the sun, h the Planck's constant, c the speed of light in vacuum, and k the Boltzmann constant.

$$P\_{\rm inc}(\lambda) = \frac{2\pi hc^2}{\lambda^5} \frac{1}{e^{\frac{hc}{\lambda kT}} - 1} \tag{2}$$

*ηrad* is the radiation efficiency of the antenna as a function of the wavelength that is given by expression 3, where *Prad*, *Pinj*, and *Ploss* are the radiated power, the power injected at the terminals, and the power dissipated in the metal of the nanoantenna, respectively.

$$\eta^{rad} = \frac{P^{rad}}{P^{inj}} = \frac{P^{rad}}{P^{rad} + P^{loss}} \tag{3}$$

In order to generate DC power in the load, a rectifier is connected to the input port to rectify the current flowing in the antenna's structure that oscillates around hundreds of THz. Like the total radiation efficiency, it is also possible to define the total matching efficiency as described on expression 4, where *ηmat*is the matching efficiency of the nanoantenna rectifier system given by expression 5, with *Zrec* being the impedance of the rectifier and *Zant* the

input impedance of the nanoantenna. Moreover, *Rrec* is the real part of the impedance of the rectifier and *Rant* the real part of the nanoantenna input impedance.

$$\eta\_{total}^{\text{mat}} = \frac{\int\_{\lambda\_{\text{start}}}^{\lambda\_{\text{stpr}}} P\_{\text{inc}}(\lambda) \eta^{\text{rad}}(\lambda) \eta^{\text{mat}}(\lambda) d\lambda}{\int\_{\lambda\_{\text{stpr}}}^{\lambda\_{\text{stpr}}} P\_{\text{inc}}(\lambda) \eta^{\text{rad}}(\lambda) d\lambda} \tag{4}$$

$$\eta^{rad} = \frac{4R\_{rcc}R\_{ant}}{|Z\_{rcc} + Z\_{ant}|^2} \tag{5}$$

All these quantities are marked in Figure 6, an equivalent circuit of the total rectenna system, where both the transmitting and receiving processes can be easily described.

**Figure 6.** Equivalent circuit for the rectenna system (adapted from the work in [5]).

*Vopen* is the voltage generated by the receiving antenna at its open terminals, while *Vrec* is the voltage seen at the terminals when a current is flowing to the rectifier. The useful power is the power going to the impedance of the rectifier *Zrec* and it is given by expression 6.

$$P\_{\rm rec} = \frac{R\_{\rm rec}}{2} \frac{V\_{\rm open}^2}{|Z\_{\rm rec} + Z\_{\rm ant}|^2} \tag{6}$$

This power is maximal under optical matching conditions, i.e., *Zrec* = *Zant*∗, leading to expression 7.

$$P\_{\rm rec} = \frac{V\_{\rm open}^2}{8R\_{\rm ant}}\tag{7}$$

Finally, to define the total rectenna efficiency, *ηrec total*, presented on expression 8, is just needed to sum expressions 1 and 4.

$$
\eta\_{total}^{rcc} = \eta\_{total}^{rad} \eta\_{total}^{mat} \tag{8}
$$

#### **6. Model: Solar Cell**

A solar cell, shown in Figure 7, is a PIN structure device with no voltage directly applied across the junction. The solar cell converts light into electrical power and delivers this power to a load. This process requires a material that can absorb the light photons. The interaction of an electron with a photon leads to the promotion of an electron from the valence band into the conduction band leaving behind a hole, i.e., the absorption of a photon by a semiconductor material results in the generation of an electron–hole pair. After an electron–hole pair is created, the electron and the hole move from the solar cell into an external circuit, producing a photocurrent I. The electron then dissipates its energy in the external circuit and returns to the solar cell [26–42].

Some processes illustrated in Figure 7 are (1) absorption of a photon leads to the generation of an electron–hole pair; (2) recombination of electrons and holes; (3) electrons and holes can be separated with semipermeable membranes; (4) the separated electrons can be used to drive an electric circuit; and (5) after all electrons passed through the circuit, they will recombine with holes.

**Figure 7.** Simple model a solar cell connected to a load (sourced from the work in [43]).

That solar cell is a PIN junction, also illustrated in Figure 8. The PIN structure consists of a p region andanregion separated by an intrinsic layer. The p region and n region have different electrons concentration: the n-type has an excess of electrons while the p-type has an excess of holes, i.e., positive charges. The intrinsic layer width W is much larger than the space charge width of a normal PN junction [28–42].

**Figure 8.** Circuit of a PIN junction, where W is the intrinsic layer width.

Absorption of light occurs in the intrinsic zone. A voltage *VR* is applied so that there is an electric field in the intrinsic zone large enough so when the photons are absorbed, an electron–hole pair is created, i.e., a negative charge, electron, goes to the conduction band of the semiconductor and in the valence band a positive charge is going to move on the action of the electric field [28–42]. Therefore, there is an electric field that immediately separates the positive from the negative charge (the negative goes to one of the terminals and the positive one goes to the other).

The output of the PV cell is often represented with the relation between the current and voltage. This is known as the current–voltage curve (I–V curve). The I–V curve, represented in Figure 9, is a snapshot of all the potential combinations of current and voltage possible from a cell under standard test conditions (STC) [28–44]: (i) cell temperature: 25 ºC (298.16 K); (ii) incident irradiance on the cell: *G* = 1000 W m<sup>−</sup>2; and (iii) spectral distribution of solar radiation: AM 1.5 spectrum.

**Figure 9.** I–V curve (red) and power curve (blue) of a solar cell (sourced from the work in [45]).

The point in the I–V curve at which the maximum power is attainable is called Maximum Power Point (MPP), being that power calculated by expression 9 [28–42].

$$P\_{MP} = V\_{MP} \times I\_{MP} \tag{9}$$

The representation of equipment through equivalent electrical circuits is a technique used in the field of electrical engineering. In order to study the PV equipment, a simplified electrical model is presented in Figure 10 [28–42].

**Figure 10.** Equivalent circuit of a PV cell (1 diode and 3 parameters model-1M3P).

This model has three parameters: *Is*, *I*0, and *n*.

*Is*, also known as *Ipv*, represents the electric current generated by the beam of light radiation, consisting of photons, upon reaching the active surface of the cell. The level of this current depends on the irradiance [28–42].

The PIN junction functions as a diode that is traversed by an internal unidirectional current *Id* which depends on the voltage *V* at the terminals of the cell and on the parameters *I*<sup>0</sup> and *n*, as it is possible to verify from expression 10 [28–42].

$$I\_d = I\_0 \left( e^{\frac{V}{n^{v\_T}}} - 1 \right) \tag{10}$$

Then, *I*<sup>0</sup> is the he reverse saturation current of the diode, n is the diode ideality factor and *vT* is the thermal voltage for a given temperature, determined using expression 11, from the Boltzmann's constant, *k*, and electron charge value, *q* [28–42].

$$
v\_T = \frac{kT}{q} \tag{11}$$

Using the Kirchhoff's Current Law (KCL) on that internal node, expression 12 is revealed [28–42].

$$I = I\_s - I\_d = I\_s - I\_0 \left( e^{\frac{V}{n \cdot v}} - 1 \right) \tag{12}$$

However, the simplified model of 1 diode and 3 parameters is not a strict representation of the PV cell. It is necessary to take into account the voltage drop in the circuit up to the external contacts, which can be represented by a series resistance *Rs* and also the leakage currents, which can be represented by a parallel resistance, *Rp*. The influence of these parameters on the I–V characteristic of the solar cell can be studied using the equivalent circuit presented on Figure 11 [28–42].

**Figure 11.** Equivalent circuit of a PV cell (1 diode and 5 parameters model-1M5P).

The model parameters are *Is*, *I*0, *n*, *Rs*, and *Rp*, and thus the output current can be related to the output voltage based on expression 13 [28–42].

$$I = I\_s - I\_d - I\_{R\_p} = I\_s - I\_d = I\_s - I\_0 \left(e^{\frac{V}{\pi \varepsilon\_T}} - 1\right) - \frac{V + R\_s I}{R\_p} \tag{13}$$

#### **7. Simulation Results**

In this section, a set of simulations are going to be presented. The main software used for this study was COMSOL Multiphysics®. It is generally used for modeling and simulation of real-world multiphysics systems.

First, we begin to module a PIN junction (solar cell). The purpose of the simulation is to study the propagation of light inside the semiconductor device. The incident light, an EM wave with a wavelength of 530 nm in the visible band, hits a silicon PIN junction with dimensions 150 nm, length of the p-junction; 2 um, length of the intrinsic layer; and 80 nm, length of the n-junction. The width is 0.5 um, while the PIN junction depth is 640 nm. These values are representative for a 0.35 um CMOS process.

The geometry consists of two parts: the first part is air (in gray), whose edge on top is used as the source for the EM wave that arrives to the solar cell, and the second part, in blue, is the PIN junction (from top to bottom, n-junction, intrinsic zone, and the p-junction).

The results are obtained through the simulations performed on COMSOL Multiphysics®, which uses the finite element method (FEM). This is a numerical method for solving problems of engineering and mathematical physics. To solve a problem, it subdivides a large system into smaller, simpler parts called finite elements.

In this case, FEM is used to calculate the electric field, so that the program needs to define a mesh to solve the system of equations.

A customized mesh with triangular elements and a maximum element size of 10 nm was defined, as presented on Figure 12. The basic condition is that the mesh size should be lower than wavelength, in order not to have numerical errors in the calculation of the solution.

The parameters used for the mesh on the different simulations are represented on Table 1.

**Figure 12.** Schematic representation of a PIN junction on COMSOL Multiphysics®, as well as its mesh.


**Table 1.** Mesh parameters COMSOL Multiphysics®.

The mesh settings determine the resolution of the finite element mesh used to discretize the model. A higher value results in a finer mesh in narrow regions. In this example, because the geometry contains small edges and faces, an extremely fine mesh was designed. This will better resolve the variations of the stress field and give a more accurate result. Refining the mesh size to improve computational accuracy always involves some sacrifice in speed and typically requires increased memory usage [46].

This study is focus on the Transverse Electric (TE) polarization. TE polarized light is characterized by its electric field being perpendicular to the plane of incidence. For TE light, the magnetic field lies in the plane of incidence, thus its always perpendicular to the electric field in isotropic materials. On the other hand, Transverse Magnetic (TM) polarized light is characterized by its magnetic field being perpendicular to the plane of incidence [47].

In this case, the electric field has only one component along the z-direction (horizontal axis).

The PIN junction was tested for different values of *λ* (light wavelength): 400 nm (blue), 530 nm (green), and 800 nm (IR), as observed on Figure 13.

For a light wavelength of 400 nm, in the blue region, the photons are absorbed mainly in the top of the intrinsic region. The electric field is zero in the bottom part of the intrinsic region.

For a light wavelength of 530 nm, the electric field is stronger in the n-junction and decreases along the intrinsic zone, due to the fact that the photons are absorbed mainly in this area.

**Figure 13.** Normalized Electric field, z-component of the cross section of a PIN junction.

For a light wavelength of 800 nm, it is observed that the electric field practically does not decrease along the intrinsic zone. Thus, it is concluded that there is almost no absorption of photons for this wavelength.

When a nanoantenna with an array of air slits or apertures is introduced on top of the silicon PIN junction, the behavior of the electric field changes.

The main purpose of the simulations with a nanoantenna is to observe the difference between a PIN junction without nanoantenna and with a nanoantenna. Furthermore, it is our interest to analyze the evolution of the diffraction pattern as the number of air slits increases, namely, a three-slit, a seven-slit, and a fifteen-slit array, and to compare the simulation results with the results expected by the classical theory [48].

The simulation environment used is similar to that of Figure 12, where an incident light wave hits the PIN junction by propagating through the air slit arrays and absorbed along the intrinsic region.

Various experiments were performed, where the electric field was normalized to E(0), that is, the incident electric field. The incident light wave has an electric field whose amplitude is registered. This amplitude is constant for all the simulated cases and thus it will serve for normalization. It is necessary to have a normalization constant in order to better compare the electric field values for the cases when there is a nanoantenna on top of the PIN junction and when there is no nanoantenna (the structure will be different).

When light hits the surface, the electric field is no longer the incident field. It is the incident field plus the reflected field, and the reflected field varies whether or not there is a nanoantenna.

In these experiments, it was considered that the dimensions of the air slits and their spacing had subwavelength dimensions as well as the metal thickness. Furthermore, for four different values of the light wavelength, four particular cases were considered: (i) nanoantenna metal thickness, *λ*/10 and air slit width, *λ*/10; (ii) nanoantenna metal thickness, *λ*/100 and air slit width, *λ*/2; (iii) nanoantenna metal thickness, *λ*/100 and air slit width, *λ*/5; and (iv) nanoantenna metal thickness, *λ*/100 and air slit width, *λ*/10.

For each case, on top of the PIN junction a three-slit, a seven-slit, and a fifteen-slit array were tested. The procedures required to study and simulate a fifteen-slit array are identical to those used to simulate a three-slit or a seven-slit array. The parameters are the same, differing only in the number of slits. The maximum absolute values of the normalized electric field along the intrinsic region for an aluminum nanoantenna were registered on Table 2.


**Table 2.** Maximum absolute values of the normalized electric field along the intrinsic region.

When the total electric field is normalized by the incident field, it is possible to immediately check whether the radiation through the intrinsic region is higher or lower than the incident radiation. In other words, if any numerical value obtained by the different simulations is greater than 1, it means that the structure itself has the capacity to transmit more light than its incidence, which indicates the occurrence of the Extraordinary Optical Transmission phenomenon. The results highlighted in green indicate the occurrence of the EOT phenomenon.

The metal thickness *λ*/100 proved to be more efficient and thus more simulations were performed with this size. This metal thickness was the most efficient as it can be in part attributed to the fact that aluminum, for very small film thicknesses, has a very large transmission coefficient and a low reflection coefficient. Meanwhile, for a metal thickness of *λ*/10 there was no occurrence of the EOT phenomenon. Contrary to what happens in the previous case, in this case practically everything is reflected and little transmitted.

The results obtained from the simulations indicate that (i) if the nanoantenna metal thickness is much smaller in relation to the wavelength, the stronger will be the electric field intensity in the intrinsic region, and (ii) the smaller the air slit width in relation to the wavelength, the smaller the intensity of the electric field in the intrinsic region, as expected given the classical theories of diffraction.

These results are confirmed by the classical theory as EOT is observed mainly due to the constructive interference of SPPs propagating between the slits of the nanoantenna, where they can be coupled from/into radiation.

The shape, dimensions, and the spacing between apertures are fundamental parameters that must be carefully sized to allow the propagation of SPPs and the occurrence of the EOT phenomenon. With the aid of MATLAB software, a 1D plot was made to compare the

values of the normalized electric field along the intrinsic zone for the light wavelength of 400 nm with an aluminum nanoantenna and without nanoantennas.

It is observed in all cases that the electric field is stronger in the n-junction and then rapidly reaches the zero value in the middle of the intrinsic zone.

By analyzing Figure 14, it is observable that the normalized electric field is stronger without nanoantennas. For this light wavelength, the results for other parameters of metal thickness and air slit width in Table 2 are quite identical, and thus for a light wavelength of 400 nm, the introduction of nanoantennas for solar harvesting does not contribute for a bigger efficiency of the solar cell.

**Figure 14.** 1D Plot of the normalized electric field, z-component, along the intrinsic zone for a light wavelength of 400 nm: Si PIN junction (blue); 3-slit array Al nanoantenna (red); 7-slit array Al nanoantenna (black); and 15-slit array Al nanoantenna (magenta).

Given a light wavelength of 530 nm, according to Table 2 for a metal thickness of *λ*/100 and an air slit width of *λ*/5 the EOT phenomenon barely occurs. Like in the previous case, a 1D plot was made on MATLAB and it is presented on Figure 15.

**Figure 15.** 1D Plot of the normalized electric field, z-component, along the intrinsic zone for a light wavelength of 530 nm: Si PIN junction (blue); 3-slit array Al nanoantenna (red); 7-slit array Al nanoantenna (black); and 15-slit array Al nanoantenna (magenta).

It is observable on Figure 15 that the results obtained for the normalized electric field with and without an aluminum nanoantenna are very similar. Therefore, one can conclude that for 530 nm of light wavelength the introduction of nanoantennas for solar harvesting barely contributes for a bigger efficiency of the solar cell.

For a light wavelength of 800 nm, the EOT phenomenon does not occur if the metal thickness is *λ*/10. For a metal thickness of *λ*/100, the EOT phenomenon occurs for every case and thus it is concluded that the nanoantennas are indeed efficient for this wavelength where *λ*/100 is the optimum thickness (see Table 2).

Below in Figures 16–18, the cases where the EOT phenomenon occurs are represented. By analyzing Figure 16, although the 15-slit array nanoantenna has recorded the maximum absolute value of the normalized electric field, the seven-slit array is the most efficient nanoantenna type, as the normalized electric field is higher along the entire intrinsic zone.

**Figure 16.** 1D Plot of the normalized electric field, z-component, along the intrinsic zone for a light wavelength of 800 nm (metal thickness: *λ*/100; air slit width: *λ*/2).

By analyzing Figure 17, one can conclude that a seven-slit array is the most efficient along the intrinsic zone, and from Figure 18, it is concluded that it is the three-slit array.

**Figure 17.** 1D Plot of the normalized electric field, z-component, along the intrinsic zone for a light wavelength of 800 nm (metal thickness: *λ*/100; air slit width: *λ*/5).

**Figure 18.** 1D Plot of the normalized electric field, z-component, along the intrinsic zone for a light wavelength of 800 nm (metal thickness: *λ*/100; air slit width: *λ*/10).

Even though there is the occurrence of EOT, the nanoantennas are less efficient for this air slit width and thus there is no visible advantage on their implementation.

The procedures that are necessary to carry out the study and simulation of an array of slits with different material types are identical to those used previously. The metals that will be considered in the following simulations using the COMSOL Multiphysics® software are Gold (Au) and Platinum (Pt).

In Tables 3 and 4 are registered the maximum absolute values of the normalized electric field along the intrinsic region for a nanoantenna of gold and for another of platinum, respectively, on top of a silicon PIN junction.


**Table 3.** Maximum absolute values of the normalized electric field along the intrinsic region for a gold nanoantenna.


**Table 4.** Maximum absolute values of the normalized electric field along the intrinsic region for a platinum nanoantenna.

From the observation of both tables above, and comparing the results with an aluminum nanoantenna in Table 2, one can verify that the EOT phenomenon is present in all material types. In addition, it is possible to observe that the EOT phenomenon is stronger with an aluminum nanoantenna, as maximum absolute values of the normalized electric field along the intrinsic region of 10 × the incident field were registered for a three-slit array.

For a gold or a platinum nanoantenna, the results obtained show that the EOT phenomenon is mostly present for the light wavelengths of 800 nm and 1550 nm. These results show clear evidence of the EOT phenomenon and constitute an interesting result for the implementation of an aperture nanoantenna, as the electric field in the near-field region is strongly enhanced.

In Figure 19, the case where the maximum absolute value of the normalized electric field along the intrinsic region for an aluminum nanoantenna on top of a Si PIN junction had the highest value, as compared with the other nanoantenna material types. From the observation of Figure 19, it is clearly visible the difference of the normalized electric field along the intrinsic zone for the aluminum nanoantenna and the other material types.

**Figure 19.** 1D Plot of the normalized electric field, z-component, along the intrinsic zone for different nanoantenna material types: Si PIN junction (blue), Al nanoantenna (red), Au nanoantenna (black), and Pt nanoantenna (magenta) (3-slit array; light wavelength: 1550 nm; metal thickness: *λ*/100; and air slit width: *λ*/5).

For the same parameters, in Figures 20 and 21 are represented the simulation results for a seven-slit array and a fifteen-slit array nanoantenna, respectively, for the three materials types and the Si PIN junction.

**Figure 20.** 1D Plot of the normalized electric field, z-component, along the intrinsic zone for different nanoantenna material types: Si PIN junction (blue), Al nanoantenna (red), Au nanoantenna (black), and Pt nanoantenna (magenta) (7-slit array; light wavelength: 1550 nm; metal thickness: *λ*/100; air slit width: *λ*/5).

**Figure 21.** 1D Plot of the normalized electric field, z-component, along the intrinsic zone for different nanoantenna material types: Si PIN junction (blue), Al nanoantenna (red), Au nanoantenna (black), and Pt nanoantenna (magenta) (15-slit array; light wavelength: 1550 nm; metal thickness: *λ*/100; air slit width: *λ*/5).

By analyzing these figures, it is verified that the aluminum and the gold nanoantenna have by far a stronger normalized electric field along the entire intrinsic region compared to the platinum nanoantenna and the case without any nanoantennas. Comparing the three cases above, one can conclude that aluminum is the most appropriate material for the application of an optical antenna.

#### **8. Study of the Short-Circuit Current and the Open-Circuit Voltage on the Solar Cell**

As previously referred, a solar cell can be modeled using the single diode and 3 parameters model, that includes the I–V and the P–V characteristics of a typical module.

The problem of modeling a PV system is further compounded by the fact that the I–V curve of a PV module is dependent on the irradiance and temperature, which are continuously changing. Consequently, the parameters required to model a PV module must be adjusted according to the ambient temperature and irradiance [43]. Two main parameters that are used to characterize the performance of a solar cell are the short-circuit current, *Isc*, and the open-circuit voltage, *Voc*.

In order to prove that the model used during the simulations on COMSOL Multiphysics® is indeed a solar cell, the short-circuit current and the open-circuit voltage were measured upon variation of the irradiance and temperature. As the software does not simulate directly the short-circuit current in the cell, a simulation of the current density norm, *Jsc*, was made. In this simulation, the solar cell was short-circuited as depicted in Figure 22.

**Figure 22.** Short-circuited solar cell model (gray: Si PIN junction; dark blue: aluminum; light blue: Air).

According to Ibrahim, the complete equation for the short-circuit current, taking into account that it varies with the irradiance and the temperature on the solar cell, is described by expression 14, where *αSTC* is the thermal coefficient of the short-circuit current, measuring the variation of *Isc* with an increase of 1 ºC of temperature T [49].

$$I\_{\rm sc}(G, T) = \frac{G}{G\_{STC}} [I\_{\rm sc\_{STC}} + \mathfrak{a}\_{\rm sc}(T - T\_{STC})] \tag{14}$$

Although COMSOL can simulate the variation in the temperature, during the simulations the temperature T on the PV cell is considered to be constant and equal to STC. Considering that the software does not simulate directly the short-circuit current in the cell, but the current density norm, given by expression 15.

$$J\_{\rm sc} = \frac{G}{G\_{\rm STC}} J\_{\rm sc,STC} \tag{15}$$

Table 5 contains the average values of the current density norm with the input irradiance.


**Table 5.** Values of the current density norm with the input irradiance.

From Figure 23, the current density norm varies almost linearly with the input irradiance for this range of values on the PV cell. The slight nonlinearity can be attributed to the resistance of the material (in this case, aluminum). The resistance of any material is a function of the material's resistivity, *ρ*, and the material's dimensions, and it is given by expression 16, where L, t, and W are the length, the thickness, and the width of the material, respectively [43].

$$R = \frac{\rho L}{t \, W} \tag{16}$$

As presented on Figure 22, there are 7 blocks or sections of aluminum surrounding the solar cell (in order to perform a short-circuit of the PV cell). Based on expression 16, it is possible to determine the value of that blocks resistance, which is presented on Table 6, for a *<sup>ρ</sup>*(*<sup>T</sup>* = 25 °C) = 2.70 × <sup>10</sup>−8<sup>Ω</sup> m and *<sup>t</sup>* = 640 nm.

**Figure 23.** 1D Plot of the variation of *Jsc* with the irradiance (*x*-axis: Irradiance [W/m2]; *y*-axis: Average value of the current density norm, *Jscavg* [×10<sup>6</sup> A m<sup>−</sup>1].


**Table 6.** Aluminum sections dimensions and resistance value.

By analyzing the values of the resistance for each block of aluminum, one can conclude that the resistance is an important factor to consider. All the values obtained for the resistance of each block are in agreement with the 0.35 um CMOS process. Therefore, the nonlinearity of the current density norm with the input irradiance is explained.

Similar to the procedure to the *Isc*, it is possible to verify how *Voc* varies with the irradiance, as verified on Table 7.


**Table 7.** Values of the open-circuit voltage with the input irradiance.

The open-circuit voltage has a steady value equal to 0.8121 V, for different values of the irradiance, G, leading to the conclusion that it is not dependent on the irradiance. This value is the maximum voltage the solar cell on this model can deliver.

1000 −4.5189 −5.3311 0.8121 1200 −4.5189 −5.3311 0.8121

The open-circuit voltage varies with the irradiance by expression 17, leading to the conclusion that this variations is not very significant, because it follows a logarithmic function, where *Ns* is the number of series-connected cells in a PV module (if it is a single PV cell, this value is equal to 1) and *αoc* is the thermal coefficient of the open-circuit voltage.

$$V\_{\rm oc}(G, T) = V\_{\rm 0c\_{STC}} + \frac{N\_s k \, T}{q} \ln(G) + a\_{\rm ac} (T - T\_{STC}) \tag{17}$$

To conclude, the short-circuit current, *Isc*, varies nonlinearly with irradiance and its variation with temperature is fairly small depending on its temperature coefficient. When determining the dependence of the open-circuit voltage, *Voc*, on temperature and irradiance, it is found that it is strongly dependent only on the temperature. It has been observed that the results obtained in this study are is accordance with what is expected by the classical theory of a photovoltaic cell and so the model that was tested on COMSOL software is valid.

#### **9. Conclusions**

The main objective of this article is the study and simulation of the behavior of an optical antenna with subwavelength dimensions for solar harvesting on PV panels. To perform such study, the COMSOL Multiphysics® software was used, to obtain the simulation numerical results of the studied structures.

It has been demonstrated with several simulations in different conditions that the EOT phenomenon was always confirmed on nanoantennas with three materials: aluminum (Al), gold (Au), and platinum (Pt). Thus, it means that these structures have the capacity to transmit more light than its incidence, in orders of magnitude greater than predicted by standard

aperture theory. These experiments provide evidence that these unusual optical properties are due to the coupling of light with SPPs on the surface of the metallic nanoantennas.

Additionally, it has been verified with the simulation results that optimum results were obtained for light wavelengths of 800 nm and 1550 nm. These results constitute an interesting result for the implementation of an aperture nanoantenna, as they cover a wide range of the spectrum: the EOT phenomenon was verified on almost the entire visible region as well as the IR region. Typical silicon solar cells have proven to be inefficient at these wavelengths.

Although most of the researchers use gold or silver to fabricate the optical antennas, the results obtained in this article show that aluminum can have even better results than the other material types, mainly due to its transmission and reflection coefficients. Furthermore, among all metals analyzed, aluminum has the smallest skin depth in the visible spectrum, as well as being cheaper than gold or platinum. However, aluminum is unstable. It oxidizes quickly, and the optical properties are lost. Therefore, aluminum has to be coated with an antioxidant compound.

**Author Contributions:** F.D. was responsible to write the original draft, J.P.N.T. and A.B. are his supervisors, J.P.N.T. and R.A.M.L. analyzed the results and they were responsible to review and editing the final manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** This work was supported in part by FCT/MCTES through national funds and in part by cofounded EU funds under Project UIDB50008/2020.

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **References**

