Simulation of a New CZTS Solar Cell Model with ZnO/CdS Core-Shell Nanowires for High Efficiency

Abstract

The numerical modeling of Cu₂ZnSnS₄ solar cells with ZnO/CdS core-shell nanowires of optimal dimensions with and without graphene is described in detail in this study. The COMSOL Simulation was used to determine the optimal values of core diameter and shell thickness by comparing their optical performance and to evaluate the optical and electrical properties of the different models. The deposition of a nanolayer of graphene on the layer of MoS₂ made it possible to obtain a maximum absorption of 97.8% against 96.5% without the deposition of graphene.The difference between generation rates and between recombination rates of electron–hole pairs of models with and without graphene is explored.The electrical parameters obtained, such as the filling factor (FF), the short-circuit current density (Jsc), the open-circuit voltage (Voc), and the efficiency (EFF) are, respectively, 81.7%, 6.2 mA/cm², 0.63 V, and 16.6% in the presence of graphene against 79.2%, 6.1 mA/cm², 0.6 V, and 15.07% in the absence of graphene. The suggested results will be useful for future research work in the field of CZTS-based solar cells with ZnO/CdS core-shell nanowires with broadband light absorption rates.

1. Introduction

Recent studies on thin-film solar cells have shown that Cu–Zn–Sn–S, i.e., Cu₂ZnSnS₄ (CZTS), is a very good absorber [1,2,3]. In addition to its high absorption coefficient of around 10⁴ cm⁻¹ and its refractive index of 2.72, CZTS has an interesting bandgap of around 1.45 to 1.65 eV [4,5,6,7,8]. On the other hand, the application of nanotechnology to the solar cell improves its performance (high absorption, low reflection, etc.) and reduces the cost of nanowire materials [9,10,11].

In 2013, Research on the fabrication of MEH-PPV [2-methoxy-5-(2′-ethylhexyloxy)-p-phenylenevinylene]/Al:ZnO nanorod arrays based on ordered bulk heterojunction hybrid solar cells) has resulted in high performance (Jsc = 5.32 mA/cm², Voc = 195 mV, FF = 27.71%,and EFF = 0.287%), an average optical transmission of approximately 78%, and an improved absorption coefficient compared to the catalyst layer seeded [12]. On the other hand, in 2014, an Experimental study of aligned ZnO nanorods on ZnO thin films doped with 2% Sn yielded high transmittance in the visible region, a large area with longer aligned ZnO nanorods, and improved performance (short-circuit current density Jsc = 2.840 mA/cm², open-circuit voltage Voc = 0.552, fill factor FF = 0.381, and the power conversion efficiency of 0.599%) compared to the undoped film [13]. Also, in 2015, Jalal Rouhi et al. showed that vertically aligned ZnO–ZnS core-shell nanocone arrays can be useful for optoelectronic devices due to a good surface-to-volume ratio, good electrical performance (Jsc = 12.2 mA/cm², Voc = 0.68 V, and FF = 0.49), better light trapping, and longer life than the other structures [14].

Recent photocatalytic studies indicate that ZnO/CdS core-shell nanowire arrays exhibit high cell separation efficiency compared to ZnO nanorods [15]. Experimental studies have also proven that the photocatalytic and photoelectrochemical activities of the ZnO/CdS core-shell structure are significantly higher than those of the bare ZnO structure [15,16]. This means that the choice of the diameter of the ZnO core and the thickness of the CdS shell plays an important role in improving the performance of the solar cell. Although cadmium (Cd) is a toxic product, CdSremains a very good junction zone reducer [17], which improves its interface with the absorber. Note also that in 2015, by Ansoft HFSS14 simulation on a periodic array of double-shell nanowire structures (ZnO/CdS/CZTS) embedded in a thin multilayer film, Qian Liu et al. obtained an absorption rate of about 90% in a range of wavelengths between 400 nm and 1000 nm, i.e., a frequency range of 300 to 750 THz [18,19].As molybdenum (Mo) is widely used as a back contact in CZTS-based cells considering its good electrical properties [20], due to its chemical reaction with CZTS particles, a MoS₂ layer is formed, and voids are observed. Although the formed layer of MoS₂ has a positive effect, the growth of the MoS₂ layer near the Mo/CZTS interface during the sulfurization process can negatively impact the back contact cell parameters (series resistance and fill factor) depending on thickness or quality of MoS₂ [21,22]; it can also be noted that the voids appearing in the MoS₂/CZTS interlayer generally lead to a degradation of the performance of the solar cell [23].The use of graphene in the Mo/CZTS interface could help us to remedy this problem thanks to these properties [24]. Graphene, although it is not a good absorber, is increasingly used in applications as a two-dimensional material to improve surface contacts due to its unique large area properties, electron mobility high, and its superior catalytic capacity [25,26,27,28,29,30]. The graphene is a zero-bandgap semimetal [31,32] and has a thermal conductivity (3000–5000 Wm⁻¹ K⁻¹) [33]. It has also been shown that under certain conditions, the contact of graphene with the single layer of MoS₂ effectively improves photon absorption and photocatalytic efficiency [34,35,36,37]. Chonge Wang et al. showed by simulation that the introduction of a graphene nanolayer between the ZnO core and the CdS shell could lead to an improvement in the optical absorption of the cell [38]. This shows that it is interesting to study the optical and electrical performance of the model in which graphene is sandwiched between the core and the shell and also deposited on the MoS₂ layer. Transmission Electron Microscope (TEM) investigations of the two structures, Mo/CZTS and Mo/Graphene/CZTS, were carried out, and the presence of a layer of graphene between Mo and CZTS led to a reduction in thickness from 25% to 33% of the layer of MoS₂ formed; the suppression of the growth of secondary phases near the MoS₂/CZTS interface has also been observed [24]. Note that very recent research on the growth of MoS₂-graphene heterostructures at low temperatures has made it possible to know that by depositing a seed layer of a thin film of Mo on graphene followed by an H₂S sulphurization process, it was possible to obtain an atomic ratio of Mo on S, lower than the stoichiometric value 2 of the MoS₂ standard and, therefore, an improved catalytic performance [39]. In addition, Xiaolong Liu et al. were able to observe by chemical vapor deposition that MoS₂ develops preferentially with a network aligned with the epitaxial graphene [40]. It is therefore found that graphene, used as a Mo/CZTS interlayer, can serve as an agent for controlling the growth of MoS₂. Note that the performance of these cells will be improved with optimally sized nanowires. Indeed, previous scientific studies have established guidelines for the choice of the diameter of the nanowire and the pitch of the network of nanowires [41,42,43,44,45,46,47,48,49]. It has been shown that for an optimized array with better optical performance and ultimate efficiency, increasing the length of the nanowires leads to a large increase in the pitch of the nanowires, but that the absorption of solar energy in the array InP Nanowire, for example, exhibits local maxima at nanowire diameters of approximately D1 = 170 and D2 = 410 nm, regardless of the length of the nanowires, knowing that the ultimate efficiency of D1 is greater than that of D2 [50]. Further, Goutam Kumar Dalapati et al. showed that with a thickness of CZTS of 1000 nm, obtained by the co-sputtering or sequential sputtering method during 15 to 30 min of the process of growing the precursor films, the thickness formed of MoS₂ varies on average by 170 nm [51]. Also, Farjana Akter Jhuma et al. have shown that the appropriate thickness of CdS for CZTS is between 50 and 150 nm [52]. However, research has shown that the optical absorption of suspended monolayer graphene at a normal incidence is very low, around 2.3% in the visible and near-infrared regions [53].

Thus, to improve the light absorption of graphene in the visible and near-infrared regions, there have been several approaches. The one that caught our attention is that of the bound state in the continuum (BIC) [54]. This mechanism appears to be efficient and easier to implement. The method is based on confinement and therefore trapping of light. It was also shown that when the graphene monolayer is sandwiched between two dielectric layers, just like absorption, an increase in the electric field intensity on the upper and lower faces of the graphene layer is observed. Add to this that with the tuning of the geometric parameters, this device can serve as a modulator and chemical/biochemical sensor [55].

The particularity of this model is that a graphene layer is not only sandwiched between the CZTS and MoS₂ layers but also placed between the ZnO core and its CdS shell. Additionally, these ZnO/CdScore-shell nanowire arrays are embedded in the CZTS absorber and MoS₂ layer; the assembly is located between a Mo layer below and the CdS/ZnO/ITO structure above.

This shows the importance of using aZnO/CdS core-shell nanowire in CZTS-based cells with the incorporation, or not, of graphene. Thus, several studies have been carried out over the last ten years on these cells and have obtained encouraging results. Indeed, the experiments performed by R. Suet al. on ZnO/CdS core-shell nanowires gave Jsc = 3.35 mA/cm² and FF = 32%, and on ZnO/CdS/CuSbS₂, core-shell nanowires gave results such as Jsc = 6.48 mA and EFF = 52% [56]. Additionally, YoungjoTak et al., in another experiment on ZnO-CdSZnO/CdS core/shell nanowire heterostructure arrays, obtained Jsc = 7.23 mA/cm² and Voc = 1.55 V and an efficiency of conversion of 3.53% [57]. Other studies also performed on the solar cell model based on ZnO/CdS core-shell nanowire arrays resulted in a maximum absorption of 92% at a wavelength of 500 nm [58]. Chonge Wang et al., by simulation on ZnO/CdS core-shell nanowire arrays, by including graphene in the CZTS-based solar cell structure, obtained Jsc = 6.39 mA/cm², Voc = 0.63 V, EFF = 16.8%, and a maximum absorption and reflection of 97% and 40%, respectively [38].

Although such a model is difficult to achieve due to the growth deposition of nanowires through the graphene layer, models based on the same principle have been studied and implemented, and have shown the improved optical and electrical performance of the solar cell [55,59]. Experimentally, ZnO nanowires are grown by electrochemical process and a CdS shell is synthesized on ZnO nanowires using the hydrothermal method [15,52,55,59]. This same hydrothermal growth approach can be used to push the nanowires through the graphene layer [15].

In order to study optical and electrical performance, the ZnO core and CdS shell nanowires, after having obtained by simulation the optimum core radius and shell thickness, are embedded in a CZTS absorber and inserted into the model.

These solar cell designs appear to have a promising future, especially if used with optimally sized core-shell nanowires and an improved light trapping system. These new models of the solar cell, little known to researchers, are the subject of our study.

2. Materials and Methods

To obtain the optical and electrical performance of the different models, we used the COMSOL Multiphysics simulation software for the different models of nanowire solar cells. COMSOL uses the finite element method to solve for the electromagnetic fields within the modeling domains. It is also able to calculate a couple of several physical domains. This study focuses on two main components (semiconductor and electromagnetic wave and frequency domain) coupled with studies in the wavelength domain, semiconductor equilibrium, and stationary [60,61,62].

To choose the optimal diameter of the ZnO core and the optimal thickness of the CdS shell of the nanowire, we relied on the model of the CZTS solar cell containing ZnO/CdS core-shell nanowires immersed in a CZTS absorber. Figure 1a shows CZTS solar cell structures with core model ZnO/CdS core-shell nanowire arrays. The model is based on a 3 µm × 3 µm square base. The principle is to compare the optical performance of models with different values of ZnO core diameter (D) and CdS shell thickness (Sh).

It is known that for minimum reflection, antireflection coatings with refractive indices n1 and n2, of thicknesses d1, d2, of wavelength $\lambda$, satisfy the following condition: n1d1 = n2d2 = $\lambda$/4 [63,64]. The dimensions of the models were obtained by taking these criteria into account and choosing $\lambda$ = 888 nm [21].

Each nanowire has a length of 1000 nm and a ZnO core covered with a CdS shell. The space between the nanowires is filled with MoS₂/CZTS films of respective thicknesses of 170 nm and 830 nm. The top, along the z-axis, is formed of layers of CdS/ZnO/In₂O₃-SnO₂ (Indium tin oxide or ITO) structures with respective thicknesses of 100 nm, 140 nm, and 170 nm. Below the nanowires, along the z-axis, is the layer of Mo with a thickness of 200 nm. An air thickness of 500 nm has been added above the model only to take into account the air parameters as shown in Figure 1b. The device as presented is composed from bottom to top of a rear contact layer in Mo, an absorbent layer in CZTS, a buffer layer in CdS, a window layer in ZnO, and a layer of transparent oxide in ITO. The substrate and the grid contact are not represented in the model. This structure was chosen to take into account the properties of the materials listed above.

By respecting the conditions and geometric constraints cited below in the model, we chose nanowires 160 nm in diameter, including the diameter (D) of the core and the thickness (Sh) of the shell, such that D + 2 × Sh = 160 nm. The core diameter and the shell thickness of the three models to be compared are, therefore: 70 nm and 45 nm, 100 nm and 30 nm, and 120 nm and 20 nm. For a more efficient nanowire array, the fill factor (nanowire diameter/nanowire pitch) should be between 0.4 and 0.5 [65]. This factor, like many other studies, allowed us to choose a nanowire pitch of 360 nm [15,43,66].

The study of this simulation proceeds in two phases. The first is a comparative study based on the calculation of the electric field, absorption, reflection, and transmission in a single nanowire for each of the three models. The results of this study will make it possible to choose an efficient model with optimal dimensions. Figure 1b shows the CZTS solar cell structures with ZnO/CdS core-shell nanowire arrays of a single nanowire model. This model is based on a 360 nm × 360 nm square base.

We considered the Solar Irradiance Spectra Air Mass (AM1.5) [67] with a power density of 100 mW/cm² for solar lighting, whose solar zenith angle is 48.2°.

The absorption, $A(\lambda )$, as a function of the wavelength can be calculated by Equation (1) [68,69], but in our simulation, the absorption, $A(\lambda )$, was calculated using Equation (2):

$$A\left(\lambda \right)=\frac{1}{2}\int \frac{2\pi {\epsilon }_{0}c}{\lambda }{\left|E\left(\lambda \right)\right|}^{2}Im(n^2\left(\lambda \right))dV$$

(1)

where E($\lambda$) is a vector of the electric field as a function of the wavelength, ε₀ = 8.85 × 10⁻¹² Fm⁻¹ is the vacuum permittivity, c = 299,792,458 m/s is the speed of light in vacuum, $\lambda$ is the photon wavelength, and n is the complex refractive index. It is recalled that the absorption, A($\lambda$), the transmitted incident light, T($\lambda$), and the reflection, R($\lambda$), in the cell must satisfy the relation below [65]:

$$R\left(\lambda \right)+A\left(\lambda \right)+T\left(\lambda \right)=1$$

(2)

In the magnetic propagation wave, the diffusion parameters are determined by the parameter S in terms of the electric field. The electromagnetic and electric fields acting on the ports are, therefore, calculated after the excitation of the port1. The electric fields E₁ and E₂, respectively at ports 1 and 2, are transformed into a complex scalar number corresponding to the voltage. We assumed that the fields are normalized with respect to the integral of the energy flow through each cross-section of the ports. The electric field E_C calculated at port1 also contains the reflected field. The coefficients S₁₁ and S₂₁ are calculated automatically according to Equations (3) and (4) [62]. These coefficients respectively determine the reflection, R($\lambda$), and the transmission, T($\lambda$). The absorption is calculated by the Equation (2). The results obtained by Equation (1) can be used for verification.

$$S_{11}=\frac{{\displaystyle \underset{Port1}{\int }\left(\left(E_c-E_1\right)*E_1^{*}\right)d{A}_1}}{{\displaystyle \underset{Port1}{\int }(E_1*E_1^{*})d{A}_1}}$$

(3)

$$S_{21}=\frac{{\displaystyle \underset{Port2}{\int }\left(E_c*E_2^{*}\right)d{A}_2}}{{\displaystyle \underset{Port2}{\int }(E_2*E_2^{*})d{A}_2}}$$

(4)

where A₁ and A₂ are the areas of port 1 and port 2, respectively.

The refractive indices of materials are available from Ref [70]. The refractive index of CZTS was taken from the experience of Nabeel A. Bakr et al. thanks to the WebPlotDigitizer simulator [6,71].

For more precision in the performance of the models, we calculated the ultimate absorption efficiency, η, of each model using the following relation [72,73]:

$$\mathsf{\eta }=\frac{{\displaystyle {\int }_{{\lambda }_l}^{{\lambda }_g}I(\lambda )A\left(\lambda \right)\frac{\lambda }{{\lambda }_g}d\lambda }}{{\displaystyle {\int }_{{\lambda }_l}^{{\lambda }_u}I\left(\lambda \right)d\lambda }}$$

(5)

where I$\left(\lambda \right)$ is the solar spectral irradiance, and ${\lambda }_l$ = 300 nm and ${\lambda }_u$ = 4000 nm are, respectively, the negligible solar irradiance and the upper limit of the available data for the solar spectrum, and ${\lambda }_g$ = 900 nm is the photon wavelength corresponding to the average bandgap of the materials of the model. The short circuit current and open-circuit voltage give a large place to evaluate the electrical performance of the solar cell.

By using Shockley’s diode equation as a function to voltage, the current density, J, is defined by [62]

$$J(V)=J_0[exp(\frac{e}{KT}V)-1]-J_{sc}$$

(6)

where $J_0$ is saturation current density, V is the applied bias voltage, KT/e is the thermal voltage equal to 25.69 mV at 25 °C, and the short current density condition under illumination is $J_{sc}$.

Table 1. The values of the electrical and thermal parameters used in COMSOL as partially reported in Refs [1,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92].

Parameters	Mo	Graphene	MoS2	CZTS	CdS	ZnO	ITO
W (nm)	200	1	170	1000	100	140	150
εr	-	3.3	10	13.6	9	9	9
Nc (cm−3)	-	1 × 1018	0.7 × 1018	2.2 × 1018	2.2 × 1018	2.2 × 1018	2.2 × 1018
Nv (cm−3)	-	1 × 1019	0.3 × 1019	1.8 × 1019	1.8 × 1019	1.8 × 1019	1.8 × 1019
μn (cm2/V·s)	-	1 × 109	100	100	30	100	100
χ (eV)	-	3.5	4.14	4.2	4.2	4.4	3.6
Eg (eV)	1.96	0–0.25	1.1	1.45	2.4	3.3	3.6
CB (cm−3)	2.2 × 1018	3 × 109–3 × 1022	2.2 × 1018	2.2 × 1018	2.2 × 1019	2.2 × 1019	4 × 1019
VB (cm−3)	1.8 × 1019	3 × 109–3 × 1022	1.8 × 1019	1.8 × 1019	1.8 × 1018	1.8 × 1019	1 × 1019
NA (cm−3)	-	-	1 × 1015	1 × 1016	1 × 1013	1 × 1013	1 × 1013
ND (cm−3)	-	1 × 1019	5 × 1013	5 × 1013	5 × 1017	5 × 1018	1 × 1019
CB (cm−3)	2.2 × 1018	1 × 1022	2.2 × 1018	2.2 × 1018	2.2 × 1019	2.2 × 1019	2.2 × 1019
VB (cm−3)	1.8 × 1019	1 × 1022	1.8 × 1019	1.8 × 1019	1.8 × 1018	1.8 × 1019	1.8 × 1019
Tvn (cms−1)	1 × 107	1 × 107	1 × 107	1 × 107	1 × 107	1 × 107	1 × 107
Tvp (cms−1)	1 × 107	1 × 107	1 × 107	1 × 107	1 × 107	1 × 107	1 × 107
D (g·cm−1)	4.69	2.328	6.9	4.56	4.82	5.61	7.12
K (W/m·K)	138	140	103	2.95	6.2	23.4	10
Cp (Jg–1 K–1)	0.277	0.7	0.12	0.41	0.21	0.404	0.36
τn/τp (ns)	-	0.03/0.03	2.4/2.4	5.4/5.4	0.005/0.005	0.003/0.003	0.003/0.003
h (W/m2K)	363 × 106	30 × 102	605 × 106	14 × 104	124 × 104	468 × 106	33 × 106
h (W/m2K)	363 × 106	30 × 102	605 × 106	14 × 104	124 × 104	468 × 106	33 × 106
Cn (cm6 s−1)	0	0	0	1 × 10−29	1 × 10−29	1 × 10−29	1 × 10−29
Cp (cm6 s−1)	0	0	0	1 × 10−29	1 × 10−29	1 × 10−29	1 × 10−29
C (cm3 s−1)	0	0	0	5 × 10−9	5 × 10−9	5 × 10−9	5 × 10−9

W: Thickness. εr: Relative permittivity. Nc: Effective conduction band density. Nv: Effective valence band density. μn: Electron mobility. χ: Electron Affinity. Eg: Bandgap. CB: effective density of states. VB: effective density of states. N_A: Acceptor concentration. N_D: Donor concentration. CB: Effective conduction band density. VB: Effective valence band density. Tvn: Thermal velocity of electrons. Tvp: Thermal velocity of holes. D: Density. K: Thermal Conductivity. Cp: Specific heat. τn/τp: Lifetime electron/hole. h: Heat transfer coefficient. C_n: Auger recombination coefficient for electron. C_p: Auger recombination coefficient for hole. C: Direct band-to-band recombination coefficient.

contains the values of parameters used in our simulation COMSOL. The initial temperature of the cell is assumed to be very close to the ambient temperature (T = 300 K).

To emphasize the credibility of our choice, in the study of CZTS solar cells with a ZnO core diameter of 100 nm, the CdS shell thickness of 30 nm received special attention [15,21].

Taking into account the properties cited above of graphene, we propose in the second phase of the simulation to study a second model, this time containing graphene to arrive at an even more efficient model. We will call it, “the model with graphene interlayer”, designating the first model as “the model without graphene interlayer”. The simulation results of these two models will be compared. We chose the thickness of 1 nm of graphene in the simulation by referring to the experiment in which Maria Jabeen et al. showed that by choosing 1 nm thick graphene deposited on the SiO2 layer [90], the optical properties of the solar cell improved [93,94].

Figure 2 shows the basic 3D model of CZTS solar cell structures with ZnO/CdS core-shell nanowire arrays and a graphene interlayer.

Almost perfect absorption of single-layer graphene can be achieved due to the high light confinement and near-field enhancement of CZTS slabs. This absorption is calculated using Equation (1) and is very dependent on the electromagnetic frequency of light and medium. In practice, the CZTS thin film is produced by the co-sputtering technique of Cu, SnS, and ZnS for the deposition of precursors, and followed by sulfurization in an H₂S atmosphere since sputter deposition is used to deposit thin-films, elemental material films, complex alloy films, etc. [95]. However, the deposited film may have a different composition from the source. Thus, after having covered the soda-lime glass with a layer of molybdenum, CZTS is deposited there. On the other hand, the chemical bath deposition being at a low temperature is simpler and more reliable for the deposition of thin layers; its use is suitable for nanomaterials, but not for semiconductors, due to the small size of the deposited crystals [96]. Thus, the p–n junction was formed by the chemical bath deposition of CdS. ZnO nanowires-based CZTS devices are fabricated by a reverse process that is used for standard thin-film solar cells, where ZnO nanowire windows are first prepared on ITO glass by the electrochemical method followed by CdS deposition, then CZTS [21].It is known that chemical vapor deposition (CVD) is useful in the process of depositing extremely thin layers of material, with high performance and high purity; therefore, the graphene monolayer is deposited directly on the Mo layer using the CVD [97,98,99].This deposition technique is chosen in this context, taking into account the fact that despite evaporation at a very high temperature of the precursors, the duration of the deposition remains very short, due to a very low thickness of the graphene.

In the visible and near-infrared regions, graphene is modeled as an ultra-thin dielectric film of a thickness of $t_g= 1\mathrm{nm}$, and its new permittivity is written as ${\epsilon }_g=1+\frac{j{\sigma }_g}{\omega {\epsilon }_0{t}_g}$, and according to Kubo’s relation, the graphene surface conductivity, ${\sigma }_g$, is calculated as the sum of intra band ${\sigma }_{\mathrm{int}\mathrm{ra}}$ and inter band ${\sigma }_{\mathrm{int}\mathrm{er}}$ conductivity of graphene, and given by [55]:

$${\sigma }_{\mathrm{int}\mathrm{ra}}(\omega )=-j\frac{e^2{k}_{B}T}{\pi {\hslash }^2(\omega -2j\Gamma )}[\frac{{\mu }_c}{k_{B}T}+2\ln (e^{-\frac{{\mu }_{{}_c}}{K_{B}T}}+1)]$$

(7)

$${\sigma }_{\mathrm{int}\mathrm{er}}(\omega )=-j\frac{e^2}{4\pi {\hslash }^2}\ln [\frac{2\left|{\mu }_c\right|-(\omega -j2\Gamma )\hslash }{2\left|{\mu }_c\right|+(\omega -j2\Gamma )\hslash }]$$

(8)

where $\omega$ is the angular frequency of the photon, ${\epsilon }_0$ is the vacuum permittivity, $\hslash$ = 6.582 × 10⁻¹⁶ eVs is the reduced Plank’s constant, e = 1.6 × 10⁻¹⁹C is the electron charge, $k_B$ is the Boltzmann constant, µ_c = 0.15 eV is the chemical potential, $\tau$ = 0.5 ps is the momentum relaxation time, T = 300 K is the operating temperature, and $\Gamma =1/2\tau$ is the phenomenological scattering rate. K_BT = 0.0256 eV, $\Gamma$ = 0.11 eV, and ω = 2πf, where f is frequency. To achieve a very good approximation, the frequency must be in the range of 1 to 300 THz [99].

Thus, in the model with the graphene interlayer, the core-shell nanowires of ZnO/CdS are above the Mo/MoS₂/Graphene structure. To minimize the complexity of the work, the continuation of the simulation was carried out in 2D since the difference between the results is not very large compared to 3D. In addition to the electric field and optical properties, the study was interested in determining the total rates of generation and recombination, the short-circuit current density (Jsc), the open-circuit voltage (Voc), and efficiencies. Figure 3 shows in 2D the two structural models with the graphene interlayer (Figure 3a), without graphene interlayer (Figure 3b) of the CZTS solar cell with arrays of ZnO/CdS core-shell nanowires. Both models are based on a 3µm × 3µm square base.

Also, in this model, the top, along the Z-axis, is formed by the layers of CdS/ZnO/In₂O₃-SnO₂ (Indium tin oxide, ITO) structures with respective thicknesses of 100 nm, 140 nm, and 170 nm. Under the nanowires, along the Z-axis, is a layer of Mo with a thickness of 200 nm. Between the nanowires is the MoS₂/Graphene/CZTS structure with respective thicknesses of 170 nm, 1 nm, and 829 nm. An air thickness of 500 nm has been added above each model only to take into account the air parameters.

To evaluate the behavior of charge carriers and, therefore, the density of the short-circuit current (Jsc) in the model as a function of the height, we determined through simulation the generation g(x) and recombination r(x) rates, as well as the total generation rates, G, and the total recombination rates, R, of the hole–electron pairs according to the relationships below [100,101,102]:

$$g(x)={\displaystyle {\int }_{{\lambda }_i}^{{\lambda }_f}[1-R^2(\lambda )]I(\lambda )\alpha (\lambda )e^{-\alpha x}d\lambda }$$

(9)

$$r(x)=\frac{n(x)p(x)-n_i^2}{{\tau }_p[n(x)+n_i]+{\tau }_n[p(x)+n_i]}$$

(10)

$$G={\displaystyle {\int }_0^{d}g(x)dx}$$

(11)

$$R={\displaystyle {\int }_0^{d}r(x)dx}$$

(12)

where 𝑛 = 𝑛₀ + Δ𝑛, 𝑝 = 𝑝₀ + Δ𝑝, 𝑛₀, and 𝑝₀ are the densities of carriers at equilibrium, Δ𝑛 and Δ𝑝 are the densities of excess carriers, 𝜏_n and 𝜏_p are the lifetimes of the carriers at pendant bond state, and 𝑛_i is the intrinsic density of carriers; d is the thickness of the model, and ${\lambda }_i$ = 200 nm and ${\lambda }_f$ = 2000 nm (wavelength study interval).

The current density (J) as a function of voltage (V) is obtained from Equation (6). The ultimate absorption efficiency is calculated using Equation (5).

Since the model is composed of several materials, we were interested in the effective permittivity, but as the system is a bit complex, in this simulation, we adopted a simplified method by calculating the average permittivity. This average permittivity can be calculated according to the following relationship [103]:

$${\epsilon }_{moy}={\displaystyle \sum _{i=1}^7\epsilon i\times fi}$$

(13)

$$fi=\frac{Vi}{{\displaystyle \sum _1^{7}Vj}}$$

(14)

with

$${\displaystyle \sum _1^{7}fi}=1$$

(15)

where ${\epsilon }_1$, ${\epsilon }_2$,${\epsilon }_3$, ${\epsilon }_4$, ${\epsilon }_5$, ${\epsilon }_6$, and ${\epsilon }_7$ are relative permittivity of air, ITO, ZnO, CdS, CZTS, MoS₂, and, Mo, respectively; $f_1,f_2,f_3,f_4,f_5,f_6,$ and $f_7$ are volume fractions of air, ITO, ZnO, CdS, CZTS, MoS₂, and Mo, respectively. According to Table 1, the permittivity of graphene is much lower than that of MoS₂ and CZTS.

The values of the average permittivities with and without the graphene interlayer found, according to Equation (13), are ${\epsilon }_{moy}(with\_gra\_\mathrm{int})=9.08$ and ${\epsilon }_{moy}(without\_gra\_\mathrm{int})=9.1$, respectively.

3. Meshing Geometric Models

For simulation calculations, it is mandatory to define the mesh of each of the geometric models. Figure 4 shows the schematic mesh structures of the single nanowire of the CZTS solar cell with ZnO/CdS core-shell nanowire arrays.

The transverse magnetic field (TM) source is determined by setting the port limit as a “periodic port”. Incident light (port wave excitation) enters through the top port while the bottom port is located under the models. Two periodic conditions are necessary for the simulation. The periodic type is that of the “Floquet period”, and its vector, k, comes from the periodic port. The simulation region is defined as follows: “swept mesh” for the air region and “freely divided tetrahedral mesh” for the other regions. For this physical interface, the maximum size of the cells of the grid are preferably less than a certain fraction of the wavelength. The incident light wavelength range of this simulation model is 200 nm to 2000 nm, so the maximum cell size is 60 nm and the minimum cell size is 4 nm. All simulation models use the same grid cell size to maintain computational rigor. In “Parameters1” of the global definitions, we have chosen to add a layer of 500 nm above the model to take into account the air parameters in the simulation.

Figure 5 shows the schematic mesh structures of the models with the graphene interlayer (Figure 5a) and without the graphene interlayer (Figure 5b) of the CZTS solar cell with ZnO/CdS core-shell nanowire arrays.

The bibliographical references of the equations used in the simulation are given in the Supplementary Materials (Table S1).

4. Results and Discussion

The simulation study on the model with a single nanowire gave us the following results:

Figure 6, Figure 7 and Figure 8 show the evolution of the electric field, the absorption, and the reflection as a function of the wavelength in the different models with a single nanowire.

The synthesis of these three results is given in

Table 2. Comparative results of electric fields, absorptions, and reflections of different models with a single nanowire.

Single Nanowire Model-(Core Diameter/Shell Thickness)	Maximum ElectricField in the Nanowire (V/m)	Maximum Absorption in the Nanowire	Maximum Reflection in the Nanowire
70 nm/45 nm	1000	89% at 390 nm	64% at 1820 nm
100 nm/30 nm	1300	96% at 500 nm	43% at 1800 nm
120 nm/20 nm	1200	94% at 420 nm	62% at 1810 nm

.

Through this summary, Table 2, we see that the optimal values of the electric field, absorption, and reflection, respectively, of 1000 V/m, 96%, and 43% are obtained by the model of the CZTS solar cell with ZnO/CdS core-shell nanowire whose the diameter of the ZnO core of is 100 nm and the thickness of the CdS shell is 30 nm.

Figure 9 shows the electric field along with the height of the model with an optimal core diameter and shell thickness values in CZTS solar cells with ZnO/CdS core-shell nanowire arrays; Figure 9a is shown with the graphene interlayer, and Figure 9b without the graphene interlayer.

We then observe that the electric field is not uniform with the height of the model. It increases on average towards the top of the model by considering Equation (16) [104].

$$\left|E\right|=\frac{q}{4\pi \epsilon d^2}$$

(16)

where q is the charge of a carrier whose absolute value is 1.602 × 10⁻¹⁹ C, therefore, of an electron, ε is the permittivity of the medium, and d is the distance from the top of the model.

Note that the average permittivity for the graphene interlayer model is lower than that for the model without a graphene interlayer. According to Equation (16), a decrease in the permittivity (ε) leads to an increase in the electric field. Therefore, the electric field in the model with the graphene interlayer increases on average compared to that of the model without the graphene interlayer. Note that the maximum electric field is 2093.352 V/m at a distance of 332.352 nm from the top of the cell for the model with graphene interlayer, and 2020 V/m at a distance of 259.128 nm from the top of the cell for the model without an interlayer of graphene. The average electric field is 687 V/m for the model with a graphene interlayer and 658 V/m for the model without a graphene interlayer. The addition of a 1 nm thick layer of graphene results in a slight increase in the depth, d, from the MoS₂/CZTS interlayer; therefore, there is a slight decrease in the electric field from the MoS₂/CZTS interlayer to the bottom of the model. Taking into account the low thickness and the low permittivity of graphene, this reduction is not too visible.

In the model with the graphene interlayer, according to Equation (1), absorption increases on average with increasing electric field. Figure 10 shows absorption as a function of wavelength (from 200 nm to 2000 nm) for models with a graphene interlayer (green curve) and without graphene interlayer (blue curve). The angle of incidence is equal to 0°. We note that in half of the ultraviolet range (between 250 and 300 nm), the two models have almost the same absorption. Note that, thanks to the unique large area properties of graphene, a slight improvement in absorption is observed in the graphene interlayer model. The maximum absorption is 97.8% for the model with the graphene interlayer against 96.5% for the model without graphene interlayer. These values were reached at a wavelength of 500 nm. By comparison with a single nanowire, an improvement in maximum absorption of 0.005% = (96.5 − 96)/96 is observed for the model without a graphene interlayer with a nanowire pitch of 360 nm. This explains the fact that with an increasing number of nanowires, the light ray trapping factor increases, thus leading to a decrease in reflection and consequently an improvement in absorption.

The results found in the literature are close to those of our model with ZnO/CdS core-shell nanowires [18,19], but the deposition of the graphene on the MoS₂ layer enabled us to obtain an absorption rate of 97.8%. This core-shell nanowire model proposed in this work can therefore play a promising role in the fabrication of broadband light absorption rate solar cells [19]. This model is very cost-effective in the visible spectrum regions [105].

Figure 11 shows the reflection as a function of the wavelength (from 200 nm to 2000 nm) for the models with a graphene interlayer (green curve) and without graphene interlayer (blue curve). The angle of incidence is equal to 0°. Note that the model without a graphene interlayer has, on average, the greatest reflection, largely due to the presence of air in the MoS₂/CZTS. According to Figure 11, the maximum reflection is obtained in the model without a graphene interlayer, which is 42% against 40% for the model with a graphene interlayer. By comparing the models with all the nanowires and with a single nanowire of the solar cell without a graphene interlayer, a decrease of 0.047% = (43 − 42)/43 is observed, due to an increasing number of trapped light rays due to the neighboring nanowires, and thus causing a decrease in the reflection rate.

Figure 12 and Figure 13, respectively, show the evolution of the total generation rates and the total recombination rates of the models with and without a graphene interlayer as a function of the height of the model. Taking into account the acquired values of the generation and recombination rates, the scale of the axis of these rates (𝑦 axis) is configured as a decimal logarithmic axis. It should be noted that the total generation rate of the model with a graphene interlayer is, on average, higher than that of the model without a graphene interlayer. The reverse is observed for the recombination curves. Like all the generations of charge carriers, the maximums of the generation rate totals are at the surface of the incident light side; this process is explained by Equations (9) and (11). The difference between the curves of the totals of the generation rate and that between the curves of the totals of the recombination rate varies on average by 2% and becomes almost zero at the rear contact (Mo). This difference is mainly due to the extinction coefficient, the low bandgap of the graphene, the lifetime of the carriers, and the densities of carriers in the materials. We also notice that there is a very low rate of generation and recombination of electron–hole pairs near the Mo layer.

Figure 14 shows the current density as a function of the voltage under light illumination in the two models (with and without graphene interlayer). The model without a graphene interlayer has a short-circuit current density (Jsc) of 6.1 mA/cm², while that with a graphene interlayer Jsc is equal to 6.2 mA/cm². This slight advantage of the model without a graphene interlayer can be explained through Equation (17) [106].

$$J_{sc}=qG(Ln+Lp)$$

(17)

where q is the amount of charge of the carriers, G total of the generations of the charge carriers, Ln and Lp are the electron and hole scattering lengths respectively. It is known that the generation and recombination rates are respectively high and low in the model with graphene interlayer thanks to the presence of graphene in the MoS₂/CZTS interface. On the other hand, a decrease in the rate of recombination leads to an increase in Ln and Lp [107].

We also note that the open-circuit voltage (Voc) of the model with graphene interlayer, which is 0.63 V, is higher than that of the model without graphene interlayer, whose value is 0.6V. The explanation for this difference can be deduced from the relation (18) [107].

$$V_{oc}=\frac{KT}{e}\ln (\frac{J_{sc}}{J_0}+1),$$

(18)

where kT/e is the thermal voltage equal to 25.69 mV at 25 °C, Jsc is the short-circuit current density under lighting conditions, and J₀ is the saturation current density, depending on the recombination in the solar cell [106]. In addition, the saturation current, J₀, in the model with the graphene interlayer is the smallest, due to the difference in recombination rates.

Table 3. The numerical value of Jsc, Voc, FF, and EFF in the models with and without a graphene interlayer.

Model of Solar Cell	Voc (V)	Jsc (mA/cm2)	FF (%)	EFF (%)
With graphene interlayer	0.63	6.2	81.7	16.6
Without graphene interlayer	0.6	6.1	79.2	15.07

shows the numerical values of open-circuit voltage (Voc), short current density (Jsc), fill factor (FF), and efficiency (EFF) for models with and without a graphene interlayer. The fill factor (FF) was determined from the data of the relationship between current density (J) and voltage (V) using the relationship below [108]:

$$\mathrm{FF}=\frac{J_{mp}\times V_{mp}}{J_{sc}\times V_{oc}},$$

(19)

where $V_{mp}$ and $J_{mp}$ are, respectively, the voltage and the current density corresponding to the point of the curve where their product is maximum.

Note that the maximum values of Voc, Jsc, EFF, and FF are obtained by the model with a graphene interlayer.

It can be seen that the results of our study are, for the most part, similar to those given in the literature. Thanks to the introduction of graphene and an improved cell design, our model shows the best results on average, as evidenced by the information in

Table 4. Comparative results with previous works.

Physical Model of	Voc (V)	Jsc (mA/cm2)	FF (%)	EFF (%)	Max Absoption (nm)	Wavelength (nm)
W. Sun et al. [21]	0.17	0.7	29.41	0.035	-	500
R. Su et al. [56]	-	6.48	-	52	-	480
Y. Tak et al. [57]	1.55	7.23	64.51	3.53	-	500
C. Wang et al. [38]	0.63	6.39	41.2	16.8	97%	500
M. Malek et al. [12]	0.195	5.32	27.71	0.287
I. Saurdi et al. [13]	0.552	2.840	0.381	0,599
J. Rouhi et al. [14]	0.68	12.2	0.49	4.07
Our study	0.63	6.2	81.7	16.6	97.8	500

. This table gives a summary of the results including those of previous research. Indeed, taking into account the improvement of the contact surface by graphene on the surface of the CZTS and between the core and the shell, and thanks to the significant trapping of light in the nanowires, our model and that of C Wang et al. turned out to be slightly advantageous. We see, on the other hand, that the generation rate is maximum and almost identical to the surface for the models with and without a graphene interlayer, and almost zero at a 1510 nm depth for the model with a graphene interlayer and at 1560 nm depth for the model without a graphene interlayer.

5. Conclusions

The formation of the MoS₂ layer in the Mo/CZTS interface during the sulfurization process, at a certain thickness, can have negative effects on the performance parameters of the cell. This study reports the interest in using the graphene layer to control and improve the contact surface and the growth of the thickness of the MoS₂ interfacial layer formed in the interface between the rear molybdenum (Mo) contact and the CZTS layer. This work was carried out using the COMSOL Multiphysics simulation software based mainly on the study and analysis of the optical and electrical characteristics of CZTS solar cells, based on ZnO/CdS core-shell nanowires with back contact of Mo. After having obtained by simulation the optimal values of the diameter of the ZnO core and of the thickness of the CdS shell, which are respectively 100 nm and 30 nm, two structural models with core-shell nanowires in ZnO/CdS immersed in MoS₂/CZTS thin-film were studied, namely the structure with an interlayer of graphene (Mo/MoS₂/Graphene/CZTS/CdS/ZnO/ITO) and the structure without an interlayer of graphene (Mo/MoS₂/CZTS/CdS/ZnO/ITO). For the model with a graphene interlayer, the graphene light absorption without the prior device is very low; its improvement in the visible and near-infrared regions was obtained thanks to the principle of confinement of light of the bound state in the continuum (BIC). Thus, the best performances were obtained by the model with a graphene interlayer. For this model, the fill factor (FF), the short-circuit current density (Jsc), and the open-circuit voltage (Voc) are, respectively, 81.7%, 630 mV, and 6.2 mA/cm², with a maximum absorption rate of 97.8%; while the model without a graphene interlayer recorded values of FF, Jsc, Voc, and maximum absorption rates of 79.2%, 600 mV, 6.1 mA/cm², and 96.5%, respectively. These performances are due to certain properties of graphene such as the high mobility of the carriers, the higher catalytic capacity, the good optical conductivity, and the capacity to improve the surface contacts. In addition, the rates of recombination and generation of electron–hole pairs are respectively low and high in the model with a graphene interlayer, compared to the model without a graphene interlayer. The lowest maximum reflection, 40%, was obtained in the model with a graphene interlayer. Thanks to the proposed graphene interlayer model, an efficiency of 16.6% was obtained with the model with a graphene interlayer against 15.07% for the model without a graphene interlayer. These simulation results must first be experimentally verified since, in practice, the graphene layer is deposited directly on the Mo layer. These results, once experimentally verified, will serve as an aid for the manufacturers in improving the performance of CZTS solar cells based on ZnO/CdS core-shell nanowire arrays. This core-shell nanowire model presented in this study can be very useful in the fabrication of broadband light absorption rate solar cells. It is also very cost-effective in visible spectrum regions.