Thin Film CIGS Solar Cells , Photovoltaic Modules , and the Problems of Modeling

Starting from the results regarding a nonvacuum technique to fabricate CIGS thin films for solar cells by means of single-step electrodeposition, we focus on the methodological problems of modeling at cell structure and photovoltaic module levels. As a matter of fact, electrodeposition is known as a practical alternative to costly vacuum-based technologies for semiconductor processing in the photovoltaic device sector, but it can lead to quite different structural and electrical properties. For this reason, a greater effort is required to ensure that the perspectives of the electrical engineer and thematerial scientist are given an opportunity for a closer comparison and a common language. Derived parameters from ongoing experiments have been used for simulation with the different approaches, in order to develop a set of tools which can be used to put together modeling both at single cell structure and complete module levels.


Introduction
Thin film solar cells based on a compound of the elements Copper, Indium, Gallium, and Selenium, that is, CIGS semiconductors, are considered as highly promising light-toelectricity converters thanks to their direct bandgaps which can be efficiently matched to the solar spectrum [1].Among different fabrication methods suitable for the absorber layer, electrodeposition represents an important alternative to the expensive vacuum-based technologies.Using a single-step electrodeposition, several groups have fabricated CIGS films [2], and in some case an efficiency exceeding 15% has been reported [3], not too far from top performances made possible by the most studied vacuum-based three-stage process [1].
One of the most important requirements for successful application of the one-step electrodeposition here considered, for CIGS film production, is to control composition of deposited films in a reliable and reproducible way (it is noteworthy that we found several technological requirements in common with optical applications we had previously developed in different research contexts, to which we address the reader [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]).
Furthermore, film formation must be manufacturing friendly, possibly eliminating the selenization step, which is environmentally undesired due to the toxicity of Se.
In some recent works [20,21], we described our first results on the option to fabricate CIGS-based solar cells directly using electrodeposited films by adjusting the concentration of the solution and eliminating the unwanted selenization step.Within the limits of the present discussion, it will be enough to mention that (1) electrodeposition of CIGS, with a controlled and reproducible Cu/(In +Ga) and Ga/(In +Ga) molar ratios, has been presented and (2) Cu-In-Se-Ga content and formation of secondary phases before and after the annealing process have been investigated [4,6].
Also, in the context of our research project, it is required to derive parameters from ongoing experiments on the layered junction structures and use them for simulation with different approaches, in order to develop a set of tools which can be used to put together modeling both at single cell and module levels, what we will describe here.To complete the International Journal of Photoenergy overall picture of the ongoing activity, we will mention that we are adapting a previous sensor measurement setup to map solar cell performance with a light-beam-induced current (LBIC) configuration [22][23][24].
In particular, from real data, we have derived the fiveparameter model and the complete characteristic of a CIGS module.Afterwards, from a structural model, setting real CIGS cells parameters, we have derived the cell characteristic.A circuital model-able to describe the cells interconnections and packaging-took into account the simulated cells data, has been finally compared with the previously mentioned module simulation.This comparison could be very useful to determine the internal parameters that extend the structural cells model to the module characteristics, so that the different perspectives of the electrical engineer and the material scientist are given an opportunity for a closer collaboration and a common language we are actively looking for [25].
Hence, this paper is organized as follows.In Section 2 we describe a general introduction to model simulation methods and the five-parameter analysis of a real CIGS module.Then, structural cell simulation is described in Section 3 together with a parasitic network resistance modeling and a comparison with the module data.Finally, conclusions are drawn in Section 4.

Five Lumped Parameters Simulation
A photovoltaic (PV) module mathematical model is a very useful tool to evaluate electrical energy production during the operative working of PV plants [26].Moreover, such models are important to perform economic evaluations and compare different PV plants implementations [27][28][29][30].
In the literature, seldom parametric mathematical models have been proposed to simulate the solar cell functioning on the basis of distributed parameters equivalent circuits, while lumped parameters equivalent circuits are largely more common [31][32][33][34][35].Among the lumped parameters models, there are different implementations that consider four, five, or six parameters with various threshold accuracies.Model parameters can be evaluated on the basis of the numerical data from the manufacturers [31-34, 36, 37].Finally, fitting methods are used to match - characteristic curves with different temperature and irradiance values.
Among lumped parameters equivalent circuit models, the five-parameter one has been here taken into account.The lumped parameters single diode model (Figure 1), thanks to the shunt resistor ( sh ), is able to simulate, with good accuracy, the solar cell working condition.It is also able to work correctly even at low irradiance conditions and it shows a lower computational burden in comparison, for example, to the double diode model.
The five lumped cell parameters are described herein: (i) the photoelectric current that depends on temperature, radiation, and structural characteristics (  ); (ii) the diode saturation current that depends on temperature and structural characteristics ( 0 ); (iii) the diode quality factor that depends on the cell junction characteristics (); (iv) the series resistance that depends on the structural characteristics of the cell (  ); (v) the shunt resistance that depends on the structural characteristics of the cell ( sh ).
The photocurrent depends on cell irradiance, temperature, and structural characteristics.Moreover, it is directly proportional to the irradiancy.The diode, in the equivalent circuit, represents the recombination current in the quasi-neutral region of the p-n junction [32,38].The standard Shockley equation for the circuit in Figure 1 is as follows: where (iii)  is the Boltzmann constant [J/K], (iv)  is the cell output voltage [V], (v)  is the cell output current [A].
The diode thermal voltage (  /) is also indicated with   .The series resistance   represent internal losses of different kind; more relevant are [39,40] (i) front and back electrode resistances; (ii) lead resistance; (iii) metal semiconductor ohmic contact.
The shunt resistance takes into account the overall photovoltaic cell dispersion currents.
Eventually, the analytic relation of the cell characteristic is It is fortunate that the same model can be extended to represent even the photovoltaic module when identical cellsalso at the same temperature and irradiation-are connected either in series or in parallel.For example, for the series of two cells, using the fiveparameter model, the equivalent circuit is that depicted in Figure 2 [41,42].
The aforementioned two cells scheme, in the previous hypothesis, is equivalent in the final analysis to the following simpler circuit of Figure 3 [41,43,44].The extension to  CS cells, connected in series, is obvious and the analytic relation for the module is then In order to evaluate the five-parameter values, the method hereafter reported has been adopted [33,45,46].Equation ( 3) is written by the three characteristic points: maximum power, short-circuit, and open-circuit points, obtaining (4), (5), and (6), respectively.Moreover, the / = 0 condition by the maximum power point is derived, giving (7).Finally, the fifth equation is obtained by calculating the / value calculated in the short-circuit point, which results in (8).
Equations ( 4) and ( 6) allow to evaluate  ref 0 (9).Equations ( 9) and ( 5) are developed to build (10), where the three unknowns   ,  sh , and   have to be evaluated.Equation (10) together with ( 7) and ( 8) allow to get (12) and (13).  ,  sh and   values are determined from (10), (12), and (13) through the iterative method hereafter reported: where In the equations detailed previously, standard test conditions, that is, STC, are indicate with "ref " as apex.To summarize the adopted iterative method, a block diagram is traced in Figure 4 [46].At the first stage, the   and  sh starting values are arbitrarily chosen and put into (10), in order to evaluate   .Subsequently, ( 13) is used to evaluate  sh .The new  sh value is put into (12) in order to check the / = 0 condition.If / = 0, the latter  sh value is the right one and (2) can be used to evaluate the  form factor; otherwise, the  sh initial value has to be changed and the iteration goes on.The previously mentioned equations can be adopted in that form only under standard conditions.For this reason, it is necessary to take into account the changes of the  0 ,   ,  SC , and  OC parameters with respect to temperature and irradiation.According to [46,47], (i)  0 does not depend on irradiation but only on temperature; (ii)   depends only on temperature; (iii)  SC depends both on temperature and irradiation; (iv)  OC depends only on irradiation.
The following equations describe what was stated previously: where (i)   SC is the short-circuit current temperature coefficient; (ii)   OC is the open-circuit voltage temperature coefficient.
Therefore, ( 14), ( 15), (19), and ( 20) represent the fiveparameter mathematical model under changing temperature and radiation conditions.Finally, the PV module output voltage can be expressed to good purpose by (21), derived from (1): For example, considering a commercial module with the following nominal data: In the following picture (Figure 5), the behavior of the commercial module characteristic obtained from the fiveparameter model in depicted, which will be used later on for a paired comparison.

The Structural Model from the Cell to the Module
In parallel to the previously described approach, usually the first choice for the electrical engineer, it is worthwhile to follow as well the microscopic point of view, more familiar to the material scientist.In order to generate a current, energy derived from the sun must impact on a photovoltaic device having a preexisting electrical field such that free electrons can succeed in separating from holes.Nowadays, numerical simulation offers advantages to the design, performance prediction, and comprehension of the fundamental phenomena ruling the operation of complex devices, such as solar cells, also allowing to investigate the physics of their inner processes.Our tool of choice was wxAMPS, a software capable of representing the electrical transport phenomena and the optical response of a wide variety of layered structures like those typical of solar cells.For this reason, it was also applied to simulate the behavior of the solar cell discussed in this work.This software is an updated version, rewritten in C++, of the one-dimensional simulation program Analysis of Microelectronic and Photonic Structures (AMPS-1D) that was initially developed by Fonash et al. at Pennsylvania State University [48,49].Various layers of stacked materials, producing homojunctions, heterojunctions, and multijunctions, can be studied by appropriately selecting characteristic parameters.Thus, wxAMPS is a modern, often updated, solar cell simulator for modeling one-dimensional devices composed of several materials.It accepts the same input parameters as AMPS and is based on similar physical principles and numerical descriptions of defects and recombinations [49], while adding tunnelling effects based on trap-assisted and intraband tunnelling models [50,51].This program incorporates a new algorithm combining the Gummel and Newton methods.In fact, Gummel's method alone is of limited usefulness when simulating devices with very high defect densities, while Newton's method alone works poorly when determining intraband tunnelling current.The advantages of the combination are in terms of better stability and more consistent convergence in problems in which intraband International Journal of Photoenergy tunnelling is critical in the determination of precise solutions [52].
With wxAMPS, a hypothetically unlimited number of layers can be modeled; this added flexibility is suited to tailoring designs of devices with parameters, at any given depth profile, so as to establish optimal degrees of efficiency conversions for solar cells.The program allows data input through three main windows that represent the ambient conditions, the properties of different materials for the individual layers, and the resolving model (trap-assisted tunnelling or intraband).You can use the simulation parameters provided by the University of Illinois, Engineering Wiki [53], or you can set custom data as needed.Operating temperature, solar spectrum, quantum efficiency, front and back contact data, surface recombination velocity, and bias voltages are the environmental parameters which set device working conditions.Once stored, ambient conditions are defined for environment simulation, but new settings can be edited by users loading a properly tailored file.In detail, it is possible to take into account the values of Φ  (the barrier height defined as   −   ) for the front and back contacts and reflection coefficients [54].
The material properties of each layer are divided into four groups accessed by separate tabs: electrical, defect, optical, and advanced.A grid allows users to control and edit the overall structure of the device.Tables organize material properties and these are provided by a Wiki [53] website or are editable using common worksheets.Likewise, absorption coefficients can be edited directly by users or be loaded from external data files.
Once all required input is entered, it is possible to start data analysis.The results are displayed by an efficient graphical layout that presents them.Output data can be provided in two main forms: directly through the graphical user interface or indirectly through files readable by common spreadsheet programs.
A CIGS thin film solar cell structure, with parameters derived from our experiments, was simulated after specifying, as the input data, the material parameters for each individual layer of the stacked device structures.Specifically, several layers including the top contact, bottom contact, intrinsic ZnO layer, CdS buffer layer, high-recombination interface, surface defect layer on top of the CIGS film, and CIGS absorber defined the CIGS solar-cell structure.The employed thicknesses of structural layers and the material parameters, which by the way fall into the acceptable ranges reported in the literature [55], are shown in Table 1.
As usual, room temperature and standard sunlight (AM 1.5 G) were the assumed working conditions.Furthermore, the material parameters used in the simulations were kept unchanged.Tables 2 and 3 summarize material Gaussian defect states and contact parameters, respectively; Table 4 provides an explanation of the symbols used in the previous tables.The front and back contacts are solely defined by their work functions Φ  .
Considering that an accurate optical assessment of a photovoltaic cell is hardly trivial, in order to facilitate the discussion and describe the considered simplifications, we will try to follow and illustrate reflections and absorptions to  which photons from incident sunlight impacting on cells are exposed in their path.As illustrated in Figure 6, incident rays reflect at the air-glass (1), glass-encapsulant (3), and encapsulant-cell (5) interfaces.In the latter case, reflection is often diffuse, leading International Journal of Photoenergy to some of the reflected light being totally internally reflected at the glass-air interface, remaining within the cell.Furthermore, incident rays are absorbed by the glass (2), the encapsulant (4), and the cell's antireflection coating or metal fingers (6).In addition, further losses arise from incident rays within interspace between adjacent cells (7).
These seven interactions depend on the light's incident wavelength and angle [56].
In order to take into account overall losses due to reflection and absorption, we have set to 0.2 the reflection coefficient for light impinging on the uppermost surface.Instead, for the back surface, the reflection coefficient was set to 0.9.
As well known, a CIGS solar cell presents a composite structure and the study is difficult due to this stratified window, consisting of a thin Al doped ZnO layer, an un-doped ZnO, a CdS buffer layer (or using Cd-free buffer layers).In order to predict the behavior of solar cells with a complex structure, as in the case in analysis, it is important to use specific and detailed physical models which are implemented to the computer, so that the effect of considered input material parameters can be defined and assessed quantitatively.
Figure 7 shows the - characteristic (per cm 2 ), calculated using wxAMPS again, of the CIGS solar cell structure having material parameters listed in Tables 1-4, at room temperature and standard sunlight (AM 1.5 G).Now, we will recall a brief qualitative description of the - characteristic curve of a common solar cell.In fact, a normal - curve presents a smooth shape in which it is possible to distinguish three distinct voltage regions: first, above 0 V, a slightly sloped region; second, below  OC , a steeply sloped and, in the region of the maximum power point, a "knee." As shown in Figure 7, normally the three regions are smooth and continuous, but the position of the knee depends on cell technology and manufacturer.In detail, crystalline silicon cells show sharper knees whereas thin film solar cells have gradual knees.The slopes of the curve in the first two regions are caused by parasitic effects.
Truth be told, several causes lead to the power dissipation in solar cells, but among these the most important is power dissipation due to parasitic resistances.Figure 8 represents the one diode equivalent circuit for a p-n-junction solar cell and, in this model, solar cells are described as a current generator in parallel with a diode and a shunt resistance, internal  sh of cell which all are connected in series with another resistance, internal   of cell.  is the photogenerated current and  diode is the current within a p-njunction solar cell diode whereas  sh and  cell are the currents into parasitic resistances (i.e., internal  sh and   ).In order to design a performing solar cell, internal   should be as low as possible and  sh as high as possible.
For a typical CIGS cell, formation of internal series resistance (  ) of the cell, is mainly due to a ZnO:Al layer film resistance (typically 15 Ω/◻), resistances of various cell layers, and resistance of top and back contacts.In particular, the contact resistance between CIGS absorber layer and Molybdenum (back contact) can be ohmic or junction-like depending on type and CIGS doping level.
It is assumable that in a common solar cell,  sh arises from current leaking within the cell in high-conducting paths cross the p-n-junction or around the edges.Shunt paths, that is, high-conducting paths, within the material can be a result of presence of the impurities or crystal damage during the production phases [57].
The simulation program wxAmps takes into account overall volume resistances and the two resistances of back and top contact that arise in different device layers.In particular, Figure 7 shows - curve obtained from simulations with wxAmps.This curve shows that the simulated cell has an internal series resistance equal to 1.8 [Ω × cm 2 ] and an internal shunt resistance equal to 890 [ Ω × cm 2 ], respectively.
In order to determine the electrical behaviour of the module, starting from individual cells, other resistive contributions to those simply derivable from - characteristic of elementary cell must be added: sheet resistance of ZnO:Al, grid contributions, and interconnection resistance between module cells (i.e., metallic interconnections and contacts) [58].Such contributions shall be considered separately because wxAmps is a one-dimensional simulator and device's crosssectional dimensions and interconnections between cells are neglected.Similarly, we must also consider additional shunt resistance of overall module mainly due to the fact that shunt paths exist in photovoltaic solar cell interconnects and module  sc mismatch.
In detail, this additional shunt resistance is due to defects in materials or degradation of interconnections between cells monolithically integrated in a module.Dielectric material inhomogeneities and defects in thin-film solar cells or crystal damage that can produce a variation of the shunt resistance, extending to cells (shunt currents through the p-n junction), have been observed experimentally [59].Unintentional partial shorting of cell near edges can cause a highly ohmic shunt path, for example, connection between front-front and the back-back contacts of all cells.Shunt currents between the front and back contacts of cells are other possible causes.
The set of all these additional resistances are simplified considering electrical model of Figure 9 obtained adding the resistors  sh external and  s external .The electric model was properly evaluated to switch from single cell to complete module.The module considered in our study consists of 72 cells in series and each having an area of around 230 cm 2 .
This step complicates the discussion by introducing (as we have seen) a network of parasitic resistances.
If  sh external and  s external are taken into account, according to scheme in Figure 9, -, equation of module can be written as a function of single cell: sh external *  cell  sh external + ( sh external /2) −  mod /  sh external + ( sh external /2) . ( s external and  sh external are detrimental to solar cell and module performance, in fact, both reduce fill factor as is illustrated in Figures 10(a) and 10(b).The current  sh leaking through the  sh of module decreases current output,  module , to intended load.The smaller  sh , the greater  sh , and lower  module for a given voltage, V.A very low  sh reduces  OC but does not affect  sh .  gives rise to a potential drop, which reduces the voltage output, V, but leaves  OC unaffected, whereas a very high   reduces  SC .
International Journal of Photoenergy 9 I module (A) V module (V)

Comparison of Models and Conclusions
From the comparison of the five lumped parameters and the structural model of a PV commercial module, we have extracted the values of the local parasitic network resistances to be associated with the structural simulation.Figure 11 shows the good agreement which can be obtained between the two models.
Further refinements are required, but we think that the approach and methods here described can lead to the development of an important tool to evaluate intermediate results during CIGS cell technology development and assessment of its impact on global electrical module performance.This is particularly useful in the course of a research activity which encompasses a variety of contributions from technical subjects such as electrochemistry, semiconductor science, electrical and electronic engineering, and optical measurements.We are trying to keep as linked as possible, during the task of modeling, both the level of single cell structure and that of the complete photovoltaic module.In other words, it is our way to look actively for a common language, so that the different perspectives of the electrical engineer and the material scientist are given an opportunity for a closer collaboration.
We are currently applying this approach to the fabrication of thin film CIGS solar cells by means of single-step electrodeposition, a technique which appears fairly easy and low cost but, at the same time, can lead to quite different structural and electrical properties, so that a reasonable estimation of its impact on module characteristics is required in all the intermediate steps of our project.

Figure 2 :
Figure 2: Equivalent circuit of the two cell series.

Figure 3 :
Figure 3: Equivalent circuit of a two identical cell series.

Figure 6 : 2 )Figure 7 :
Figure 6: Cross-sectional diagram of a conventional photovoltaic module (not in scale), and the optical loss as described in the text.

Figure 8 :
Figure 8: The one diode model for a p-n-junction solar cell.

2 AM = 1 . 5 T
c = 298.15K I-V five lumped parameters module I-V structural simulated module

Figure 11 :
Figure 11: - curve from the five parameter model and structural model with parasitic resistances.

Table 1 :
Simulation material parameters of the CIGS thin-film solar cell.

Table 2 :
Simulation material Gaussian defect for the CIGS solar layers.

Table 3 :
Contact parameters applied to the simulations.

Table 4 :
Explanation of the symbols used to describe the simulation parameters.