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 nano-dipoles [
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, , and (2) the efficiency by which the captured light is transformed into low frequency electrical power by the rectifier, .
According to Kotter, the total radiation efficiency could be given by expression 1 [
25], where
is the wavelength of the incident light and the upper and lower integration limits
and
should cover the optical bandwidth for the solar energy harvesting.
Furthermore,
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.
is the radiation efficiency of the antenna as a function of the wavelength that is given by expression 3, where
,
, and
are the radiated power, the power injected at the terminals, and the power dissipated in the metal of the nanoantenna, respectively.
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
is the matching efficiency of the nanoantenna rectifier system given by expression 5, with
being the impedance of the rectifier and
the input impedance of the nanoantenna. Moreover,
is the real part of the impedance of the rectifier and
the real part of the nanoantenna input impedance.
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.
is the voltage generated by the receiving antenna at its open terminals, while
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
and it is given by expression 6.
This power is maximal under optical matching conditions, i.e.,
, leading to expression 7.
Finally, to define the total rectenna efficiency,
, presented on expression 8, is just needed to sum expressions 1 and 4.
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,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
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.
That solar cell is a PIN junction, also illustrated in
Figure 8. The PIN structure consists of a p region and a n region 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,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42].
Absorption of light occurs in the intrinsic zone. A voltage
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,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
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,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42,
43,
44]: (i) cell temperature: 25 °C (298.16 K); (ii) incident irradiance on the cell:
; and (iii) spectral distribution of solar radiation: AM 1.5 spectrum.
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,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42].
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,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42].
This model has three parameters: , , and n.
, also known as
, 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,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42].
The PIN junction functions as a diode that is traversed by an internal unidirectional current
which depends on the voltage
V at the terminals of the cell and on the parameters
and
n, as it is possible to verify from expression 10 [
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42].
Then,
is the he reverse saturation current of the diode, n is the diode ideality factor and
is the thermal voltage for a given temperature, determined using expression 11, from the Boltzmann’s constant,
k, and electron charge value,
q [
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42].
Using the Kirchhoff’s Current Law (KCL) on that internal node, expression 12 is revealed [
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42].
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
and also the leakage currents, which can be represented by a parallel resistance,
. 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,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42].
The model parameters are
,
,
n,
, and
, and thus the output current can be related to the output voltage based on expression 13 [
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42].
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.
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
(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.
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, and air slit width, ; (ii) nanoantenna metal thickness, and air slit width, ; (iii) nanoantenna metal thickness, and air slit width, ; and (iv) nanoantenna metal thickness, and air slit width, .
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.
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 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 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.
Given a light wavelength of 530 nm, according to
Table 2 for a metal thickness of
and an air slit width of
the EOT phenomenon barely occurs. Like in the previous case, a 1D plot was made on MATLAB and it is presented on
Figure 15.
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
. For a metal thickness of
, the EOT phenomenon occurs for every case and thus it is concluded that the nanoantennas are indeed efficient for this wavelength where
is the optimum thickness (see
Table 2).
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.
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.
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
Table 3 and
Table 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.
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.
For the same parameters, in
Figure 20 and
Figure 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.
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,
, and the open-circuit voltage,
.
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,
, was made. In this simulation, the solar cell was short-circuited as depicted in
Figure 22.
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
is the thermal coefficient of the short-circuit current, measuring the variation of
with an increase of 1 °C of temperature T [
49].
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.
Table 5 contains the average 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,
, 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].
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
C) = 2.70 ×
and
nm.
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
, it is possible to verify how
varies with the irradiance, as verified on
Table 7.
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.
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
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
is the thermal coefficient of the open-circuit voltage.
To conclude, the short-circuit current, , 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, , 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.