1 Introduction

The cumulative capacity of solar photovoltaic power in the global markets continues to increase in popularity. From its optical, electronic, and chemical properties, CdS is considered the best-suited n-type heterojunction partner to CdTe for high efficiency and low cost, with a maximum theoretical efficiency of about 22.1% for such cells ^[1].

Photovoltaic science and technology have been improving simultaneously, and the development of their applications has been outstanding. In China, the total annual solar cell and module production capacity may increase from 361 GW at the end of last year to up to 600 GW at the end of 2022, according to the Asia Europe Clean Energy (solar) advisory. The cumulative worldwide capacity of solar photovoltaic power continues to increase ^{[1, 2]}.

In 2020 there was a world record in efficiency obtained with solar cells manufactured with thin-film technology at 23.4% ^[3], and maximum efficiency of 22.1% in CdTe thin-film solar cells was recently reported ^[4]. CdTe is one of the best options for developing better and more efficient thin-film solar cells due to its excellent band gap (1.5eV) at room temperature, high absorption coefficient, very robust, and highly chemically stable ^[5]. On the scientific and technological side, excellent results have been achieved ^{[6, 7]}. Creating new ones and improving the existing materials for photovoltaic conversion has been one of the most studied areas in the last years ^{[8, 9]}. The plasmonic effect is a crucial ingredient for implementing most photovoltaic materials; this is responsible for the most exciting phenomena that such materials present, for instance, negative refractive index, superlensing, and cloaking ^{[10, 11]}.

The surface plasmons phenomenology has recently been investigated in various systems ^{[12, 13, 14, 15, 6]}, which aim to optimize the trapping of photons and absorption processes. In addition, plasmons increase the capacity to generate electron-hole pairs, allowing higher performance. This phenomenon is present in the interfaces of semiconductor thin films. Therefore, its study is necessary to understand photovoltaic energy’s gain or loss mechanisms to improve the efficiency of solar cells.

In this work, we were able to measure the solar cell capacitance by applying an alternating voltage of frequency ω, considering the system at room temperature. With these assumptions, it was possible to observe an increase of free charge carriers in each material that makes up the heterojunction; thus, we can consider the semiconductor as a cloud of electrons and holes ^{[17, 18]}. When the frequency increases, some properties of these materials change, such as the electric permittivity ^{[19, 20]}. We report a theoretical study based on the presence of plasmons in the interlayer, considering the frequency dependence of the applied voltage on the permittivity.

The Capacitance-Voltage (C-V) graph is compared with the experimental results of the CdS/CdTe solar cells and previous theoretical models. For instance, Castillo et al. ^{[21, 22]} consider in their model a direct voltage and a constant electrical permittivity; however, in the present report, an alternating voltage which is more consistent with the experimental conditions, and a dependence on the electrical permittivity with the frequency of the applied voltage is also assumed. As a result, we obtained a better fitting to the C-V graph experimental data by using these premises as well as consistency for values of the discontinuity band and interface charge density, respectively.

2 Theoretical method

2.1 Dependence of the dielectric permittivity when applying a variable electric field to a semiconductor

The theoretical model C-V for CdS/CdTe solar cells was improved as shown in Fig. 1. To obtain the relation of the dielectric permittivity with the frequency of the applied voltage during the experimental measurement process, it was considered the model of electron gas applying a positive background, where the charge density is given by ρ0=n|e|, where n is the number of electrons per unit of volume, and e is the fundamental electric charge.

Figure 1 CdS / CdTe polycrystalline solar cell (second generation), where the CdS is used as window material and CdTe as the active material. 

If longitudinal waves are applied to the electron gas, dilatations and contractions appear, destroying the charge neutrality, and the presence of Coulomb restitution forces occurs. We obtained the plasmon dispersion relation ^[23]

ωk2=ωp2+αmnk2, (1)

with

ωp2=4πne2mε0, (2)

where ωp is the plasma frequency, m the electron mass, and k the wave vector.

If k = 0 in (1), the displacement x of the electron gas is related to the polarization as

P=nex=-ne2mω2. (3)

The dielectric permittivity can be written in terms of the electric susceptibility as

ετ=1+χ, (4)

where

χ=PE, (5)

with P the polarization and E the electrical field.

Using Eqs. (3), (4), and (5), the frequency dependence is obtained for the electric permittivity as

ετ(ω)=1+4πPE=1-4πne2mω2=1-ωp2ω2, (6)

with this expression, we can consider the following cases

• • If ω>ωp, then ε(ω)>0, which means that the wave propagates.

• • If ω<ωp, then ε(ω)<0, which implies that the wave decays exponentially.

Since the field frequency is smaller than the CdS and the CdTe thin films plasma frequencies, the permittivity is negative ^[24].

3 Results

Using Eqs. (2) and (6), the electrical permittivity and the plasma frequency for each semiconductor were determined (see Table I). The permittivity values obey the dispersion relation in Eq. (6) for a wave vector close to the plasma frequency. The fact that the values of the plasma frequency ωp are less than the frequency ω makes the permittivity negative, causing the waves to be reflected and preventing them from propagating in the medium. Semiconductors, especially CdS and CdTe, have low densities of free carriers, so the plasma frequency is between the mid and near-infrared ^[29].

The Capacitance Voltage technique was used to compare the theoretical results with the experimental ones and determine the heterojunction’s physical properties. The C-V measurements were done at a frequency of 0.1 MHz from 0 to 40 mV. From Eqs. (7) and (14), the capacitance values have been obtained by varying the voltage in 0.05mV intervals. The parameters used in the calculation are based on the experimental data and parameters reported in previous works ^{[21, 22, 26]}. Figure 2 compares the experimental data (black spheres), the theoretical values considering only one ternary layer (Castillo et al. red spheres), and the theoretical results our model predict (blue spheres).

Figure 2 C-V plot. Comparison of experimental data (black spheres), interface ternary layer (red spheres), and the consideration of plasmon effect in permittivity (blue spheres). 

A Schottky barrier in the contacts exists for the applied voltage from 0 to 19 mV, inducing a non-ohmic behavior. So, there is a significant disagreement between theory and experiment in that voltage range. Furthermore, fitting the experimental data allows getting the valence band discontinuity value of ΔEv=0.89 eV and the interface charge density σ=1×1013 m^-2 results consistent with those previously obtained ^[21]

4 Conclusions

Our results offer insight into understanding physical phenomena involved in semiconductor interfaces; such findings could provide tools for improving solar cells. Considering the frequency of the alternating incoming voltage, we found that plasmons in the interface are formed with a plasma frequency greater than the frequency of the applied voltage, which makes a negative permittivity; thus, electromagnetic waves are reflected and therefore decay exponentially in space. Moreover, the plasmons frequency found is in the infrared region, hence the plasmon does not affect the solar cell efficiency. The capacitance versus the applied voltage of CdS/CdTe thin-film solar cells has been calculated. Comparing to previously reported theoretical results where frequency dependency over the electric permittivity is not considered, a better fitting with our model is achieved.