Design of thin film solar cells based on a unified simple analytical model | Journal of Applied Research and Technology. JART

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Piñón, José Manuel Juárez García, Omar Serrano Pérez, Fernando Juárez López" "autores" => array:6 [ 0 => array:2 [ "nombre" => "Edgar Serrano" "apellidos" => "Pérez" ] 1 => array:2 [ "nombre" => "Javier Serrano" "apellidos" => "Pérez" ] 2 => array:2 [ "nombre" => "Fernando Martínez" "apellidos" => "Piñón" ] 3 => array:2 [ "nombre" => "José Manuel Juárez" "apellidos" => "García" ] 4 => array:2 [ "nombre" => "Omar Serrano" "apellidos" => "Pérez" ] 5 => array:2 [ "nombre" => "Fernando Juárez" "apellidos" => "López" ] ] ] ] ] "idiomaDefecto" => "en" "EPUB" => "https://multimedia.elsevier.es/PublicationsMultimediaV1/item/epub/S1665642317300986?idApp=UINPBA00004N" "url" => "/16656423/0000001500000006/v1_201801120630/S1665642317300986/v1_201801120630/en/main.assets" ] "en" => array:20 [ "idiomaDefecto" => true "cabecera" => "Original" "titulo" => "Design of thin film solar cells based on a unified simple analytical model" "tieneTextoCompleto" => true "paginas" => array:1 [ 0 => array:2 [ "paginaInicial" => "599" "paginaFinal" => "608" ] ] "autores" => array:1 [ 0 => array:4 [ "autoresLista" => "Armando Acevedo-Luna, Roberto Bernal-Correa, Jorge Montes-Monsalve, Arturo Morales-Acevedo" "autores" => array:4 [ 0 => array:3 [ "nombre" => "Armando" "apellidos" => "Acevedo-Luna" "referencia" => array:1 [ 0 => array:2 [ "etiqueta" => "a" "identificador" => "aff0005" ] ] ] 1 => array:3 [ "nombre" => "Roberto" "apellidos" => "Bernal-Correa" "referencia" => array:1 [ 0 => array:2 [ "etiqueta" => "b" "identificador" => "aff0010" ] ] ] 2 => array:3 [ "nombre" => "Jorge" "apellidos" => "Montes-Monsalve" "referencia" => array:1 [ 0 => array:2 [ "etiqueta" => "c" "identificador" => "aff0015" ] ] ] 3 => array:4 [ "nombre" => "Arturo" "apellidos" => "Morales-Acevedo" "email" => array:1 [ 0 => "amorales@solar.cinvestav.mx" ] "referencia" => array:2 [ 0 => array:2 [ "etiqueta" => "a" "identificador" => "aff0005" ] 1 => array:2 [ "etiqueta" => "*" "identificador" => "cor0005" ] ] ] ] "afiliaciones" => array:3 [ 0 => array:3 [ "entidad" => "Centro de Investigación y de Estudios Avanzados del IPN, Electrical Engineering Department, Avenida IPN # 2508, 07360 Mexico, D.F., Mexico" "etiqueta" => "a" "identificador" => "aff0005" ] 1 => array:3 [ "entidad" => "Grupo DEMA, Facultad de Ciencias e Ingeniería, Universidad del Sinú, Montería, Colombia" "etiqueta" => "b" "identificador" => "aff0010" ] 2 => array:3 [ "entidad" => "Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Colombia, Manizales, Colombia" "etiqueta" => "c" "identificador" => "aff0015" ] ] "correspondencia" => array:1 [ 0 => array:3 [ "identificador" => "cor0005" "etiqueta" => "⁎" "correspondencia" => "Corresponding author." ] ] ] ] "resumenGrafico" => array:2 [ "original" => 0 "multimedia" => array:7 [ "identificador" => "fig0070" "etiqueta" => "Fig. 14" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr14.jpeg" "Alto" => 1125 "Ancho" => 1473 "Tamanyo" => 107662 ] ] "descripcion" => array:1 [ "en" => "External and internal quantum efficiency for the CdS/CdTe solar cell." ] ] ] "textoCompleto" => "1IntroductionThere is the expectation that thin film solar cells will be the alternative to silicon solar cells which is the dominant technology at the photovoltaic market today. Improved efficiencies and lower costs of thin film solar cells are required for this goal to become a reality. Among the most developed thin film solar cells we have CdS/CdTe and CdS/CIGS which recently have attained efficiencies above 20% (<a class="elsevierStyleCrossRef" href="#bib0050">Green, Emery, Hishikawa, Warta, & Dunlop, 2015</a>). Each one of these technologies needs solving some problems related to the quality of the deposited materials (by different techniques), but as it will be shown here, reducing the film thickness and achieving higher open circuit voltages are required for both kind of solar cells. However, in general, CdTe and CIS solar cells are studied independently of each other without having a more integral vision.It must be observed that from the structural point of view, CdTe and CIS solar cells are similar and therefore their physical behavior is also identical. The change is in the absorbing material (with their own properties) and the technological steps followed to make the solar cells. For example, CdTe is typically obtained by closed space vapor transport (CSVT) or Rf-sputtering (<a class="elsevierStyleCrossRef" href="#bib0095">Morales-Acevedo, 2006</a>), while CIS solar cells are obtained by co-evaporation or Rf-sputtering among other different techniques (<a class="elsevierStyleCrossRef" href="#bib0130">Singh & Patra, 2010</a>). CdTe typically cannot be p-type doped above 1015cm−3, and cells with small CdTe thickness cannot be easily done because pinholes cause the device degradation. Then, present CdTe minimum thickness is around 4–6μm. In the case of CIS solar cells, the CIS (or CIGS) thickness is around 3–4μm.It will be shown that good cells can be designed with absorber thickness around 1μm or below. Hence, the technological challenge is achieving CdTe and CIS materials with good properties, but without pinholes using the present or new deposition techniques. In such a case, the amount of material would be reduced by about 60–75% with a corresponding decrease in the total cost of the cells.In this paper, we shall describe a simple analytical model that can be used for the above mentioned solar cells (CdTe and CIS), so that they can be designed easily. This model is complete in the sense that it includes carrier transport limited by diffusion and by generation–recombination at the space charge region. It also takes into account that the space charge region width is dependent upon the operating voltage and therefore the superposition principle is no longer valid. In other words, the current density due to illumination is not a constant with respect to the applied voltage, but it has some dependence upon this variable. Similarly, the total dark saturation current density will not be independent of the operating voltage, as it is usually assumed.Having a simple model that considers the above effects for thin film solar cells is very important because, in general, they are not taken into account because for conventional cells, such as those made with silicon, these effects are negligible. Conventional junction silicon solar cells are typically made with absorber thickness of more than 200μm, so that the space charge region effects on good solar cells are small because the depletion region thickness is of the order of 1μm.For thin film solar cells, the space region effects become important because the total volumes of the space charge region and the quasi-neutral regions are of the same order, particularly for very thin film solar cells. Furthermore, under some situations (which depends upon the equilibrium majority carrier concentration and the thickness of the absorbing material) the depletion region may extend along the whole length of the absorbing material. In this case, the recombination in the depletion region will limit the total dark current, but at the same time the photo-generated carriers will be collected efficiently because of the presence of the high electric field in this region. And this electric field will be larger for thinner solar cells. Hence, there is a complex relation between majority carrier concentration (Na), absorbing thickness, lifetime and mobility of minority carriers for the whole operating voltage range of the solar cell. The main objective for this work will be to have a simple model that takes into consideration all the above effects.It will be shown that the model predicts that if the recombination at the back surface is reduced, for example by having a p+ region before the back contact, so that surface recombination velocities are small (102cm/s or less) compared to the high recombination velocities obtained at ohmic contacts (above 107cm/s), then very thin cells can achieve improved efficiencies than those with thick absorbers because of the reduction of bulk recombination, although there might be some loss of photo-current density.Finally, in order to have a complete model for designing thin film solar cells, the optical design was also considered by using the optical matrix method applied to all the films that are part of the cells. The electrical and optical calculations were applied to CdTe and CIS solar cells, as an example of the application of the models described here.2I–V modelingThe reference structure of a thin film solar cell is shown in <a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a>. It is formed by three main regions: The transparent conducting oxide (TCO) which allows the light passing to the hetero-junction, the window semiconductor layer (typically CdS), and the absorbing semiconductor material. Ohmic contacts are both made at the back of the absorbing layer and at the front TCO layer in order to connect the cell to the external circuit. We shall assume that the back is covered by a metallic contact which reflects totally those photons that pass the absorbing layer without being absorbed causing a second pass of such photons through this layer.<elsevierMultimedia ident="fig0005"></elsevierMultimedia>As explained above, an important parameter that determines the electric and photoelectric properties in a solar cell is the thickness of the space charge region (depletion region) in both the p-type and n-type sides of the heterojunction by means of the following expressions:<elsevierMultimedia ident="eq0005"></elsevierMultimedia><elsevierMultimedia ident="eq0010"></elsevierMultimedia>where ¿p is the relative permittivity of the p-type material and ¿n is that of the n-type one, Na and Nd are the acceptor and donor concentrations at each region, respectively, V is the bias voltage and Vbi is the built-in potential, calculated as:<elsevierMultimedia ident="eq0015"></elsevierMultimedia>where ΔEc and ΔEv are the conduction and valence band discontinuities, respectively. kb is the Boltzmann constant, ni,p and ni,n are the intrinsic carrier concentrations of the p and n-type materials. Nc,p and Nc,n are the effective densities of states in the conduction band of the p and n-type materials and finally Nv,p and Nv,n are the effective densities of states in the valence band of the p and n-type materials.In some cases, xp as calculated by Eq. <a class="elsevierStyleCrossRef" href="#eq0005">(1)</a> might be larger than the thickness of the absorber layer itself (Wp). In this case, the depletion region would extend over the entire material, so that the depletion region would be limited by the thickness of the absorber layer (i.e., xp=Wp and xn=Wp(Na/Nd)). This condition could be changed as the forward operating voltage is increased because the depletion thickness is reduced as the forward voltage is increased.To determine the total photocurrent density, there are two contributions to be considered. The one limited by carrier diffusion and the one due to the electric field drift of the generated carriers at the space charge region. For each wavelength λ, it is necessary to consider the transmittance T(λ) in the TCO layer, and the total reflectance R(λ) due to the multilayer system, as will be determined in the next section.The photocurrent density limited by ambipolar diffusion at the quasi-neutral regions are given by <a class="elsevierStyleCrossRef" href="#bib0070">Kosyachenko (2010)</a> and <a class="elsevierStyleCrossRef" href="#bib0105">Nelson (2003)</a>:<elsevierMultimedia ident="eq0095"></elsevierMultimedia><elsevierMultimedia ident="eq0100"></elsevierMultimedia>In addition, the photocurrent density due to the carriers generated (a 100% collection efficiency is assumed under the influence of the high electric field) at the space charge region will be:<elsevierMultimedia ident="eq0105"></elsevierMultimedia>Ln and Lp are the minority carrier diffusion lengths, Wp and Wn are the p and n layer thickness, respectively, and Sp and Sn are the respective surface recombination velocities. In all cases, the photon density flux N0(λ) corresponds to the AM1.5G standard solar spectrum. The absorption coefficients of the window layer α1(λ) and the absorber layer α2(λ) were considered in the same wavelength range up to the wavelength that can be absorbed by a specific absorber material. Photons with larger wavelengths will not produce any photo-current.The total current density is the sum of each of the above current densities (J′p, J′n and J′scr) and integrated in the range between λmin (300nm) and λmax (850nm for CdTe and 1200nm for CIS, which correspond to the absorption edges of the CdTe and CIS layers, respectively). It is given by the following expression:<elsevierMultimedia ident="eq0035"></elsevierMultimedia>The dark current components will also be limited either by diffusion or by generation–recombination at the space-charge region (<a class="elsevierStyleCrossRef" href="#bib0140">Sze, 2008</a>).<elsevierMultimedia ident="eq0110"></elsevierMultimedia>where T is the temperature, k is the Boltzmann constant, and J0 and J00 are the dark saturation currents due to diffusion and generation–recombination, respectively. They are given by<elsevierMultimedia ident="eq0115"></elsevierMultimedia>where<elsevierMultimedia ident="eq0120"></elsevierMultimedia><elsevierMultimedia ident="eq0125"></elsevierMultimedia>and<elsevierMultimedia ident="eq0130"></elsevierMultimedia>ni,n and ni,p are the intrinsic carrier densities, p0 and n0 are the minority carrier concentrations, and τp and τn are the minority carrier lifetimes, in the n and p semiconductors, respectively.Therefore, all the dark saturation current density components depend upon the operating voltage because xp and xn are functions of this variable, as given by Eqs. <a class="elsevierStyleCrossRefs" href="#eq0005">(1) and (2)</a>. The total current density of the cell is then given by:<elsevierMultimedia ident="eq0065"></elsevierMultimedia>From this J–V dependence the maximum generated power can be determined by assuming the series and shunt resistance losses to be zero. The efficiency will be given by<elsevierMultimedia ident="eq0070"></elsevierMultimedia>where Jm and Vm are the current density and voltage at the maximum power point, respectively, and Pinc is the incident radiation power density.Notice that the above model uses well known expressions, but they unified into a simple full analytical model for including the effects mentioned in the preceding section. Including these effects is important for quantifying in a more complete manner the behavior of thin film solar cells.3Optical modelA simple way to evaluate the optical losses for a thin film solar cell is to use the optical matrix method. The optical matrix is expressed in terms of Fresnel coefficients for a system of m layers, where the mth and (m−1)th are layers with complex refractive index nm=nm−ikm and nm−1=nm−1−ikm−1.The optical matrix for the mth layer is defined by the following expression (<a class="elsevierStyleCrossRef" href="#bib0055">Heavens, 1954</a>):<elsevierMultimedia ident="eq0075"></elsevierMultimedia>where each of the elements are calculated in accordance to the following set of equations:<elsevierMultimedia ident="eq0080"></elsevierMultimedia>gm and hm are functions that depend on both the refractive index and the extinction coefficient. The values of α and γ can be calculated accordingly to<elsevierMultimedia ident="eq0085"></elsevierMultimedia>where λ is the wavelength and d is the respective layer thickness. The reflectance R(λ) for a system of four layers on a substrate is calculated as<elsevierMultimedia ident="eq0090"></elsevierMultimedia>where t1,4, u1,4, p1,4 and q1,4 are the equivalent elements of the product of the optical matrices for each layer.Based on this model, as well as on the values of refractive index n(λ) and extinction coefficient k(λ) reported in the literature (see <a class="elsevierStyleCrossRef" href="#fig0010">Fig. 2</a>) (<a class="elsevierStyleCrossRefs" href="#bib0045">Filmetrics, 2016; Mclntosh, 2016; Pan, Zhang, Teng, Li, & Li, 2008; Polyanskiy, 2016; Treharne et al., 2011</a>), values of R(λ) were determined for the solar cells studied in this paper, as will be discussed in the next section.<elsevierMultimedia ident="fig0010"></elsevierMultimedia>4Application of the model to CIS solar cellsWith the purpose of determining the efficiency of a CIS solar cell, schematically shown in <a class="elsevierStyleCrossRef" href="#fig0015">Figure 3</a>, the relevant physical parameters were obtained from previous reports (<a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>). The value of the CIS absorber layer thickness was varied between 500nm and 3000nm in order to obtain the solar cell efficiency. The band diagram of the structure under equilibrium was built by applying Anderson's rule (<a class="elsevierStyleCrossRef" href="#fig0020">Fig. 4</a>) (<a class="elsevierStyleCrossRefs" href="#bib0015">Anderson, 1960; Sharma & Purohit, 1974</a>), in which the bandgap values and characteristic electron affinities are observed.<elsevierMultimedia ident="fig0015"></elsevierMultimedia><elsevierMultimedia ident="tbl0005"></elsevierMultimedia><elsevierMultimedia ident="fig0020"></elsevierMultimedia>In <a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a>, the result of R(λ) for the ZnO:Al/CdS/CIS/Mo/glass structure in the 300–1200nm region (CIS absorption edge) is observed. In the graph, a characteristic behavior with oscillations due to interference effects at different wavelengths is evident. This behavior is expected because of the thickness and refractive index difference of the AZO material compared to that of CdS and the other layers below (<a class="elsevierStyleCrossRef" href="#bib0080">Mahdjoub & Hadjeris, 2013</a>).<elsevierMultimedia ident="fig0025"></elsevierMultimedia>A complete calculation of the solar cell efficiency is made taking into account the transmittance as a function of wavelength T(λ) for the above cell structure. In this case, a transmittance average around 80% (above the ZnO:Al absorption edge) is obtained, which is in good agreement with other reported theoretical and experimental results (<a class="elsevierStyleCrossRefs" href="#bib0025">Bernal-Correa, Morales-Acevedo, Montes-Monsalve, & Pulzara-Mora, 2016; Kuwahata & Minemoto, 2014</a>).Based on the values shown in <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>, as well as the results for R(λ), T(λ), and taking into account the spectra for α1(λ) and α2(λ), Jph was calculated using the model described in Section <a class="elsevierStyleCrossRef" href="#sec0010">2</a>. The photons not absorbed in the first pass of light through the cell structure are reflected at the back contact, so that these photons must be included for the calculation of the light current density because they have a second opportunity to be absorbed by the absorber layer. This additional photo-current has also been calculated, although it is not given explicitly in the expressions for Jph, above.The values of Jsc as a function of the CIS thickness are shown in <a class="elsevierStyleCrossRef" href="#fig0030">Figure 6</a>. For a high recombination velocity at the back (107cm/s), a rapid decrease of the Jsc (mA/cm2) is observed for absorber layer thicknesses lower than 3μm, in accordance with the behavior reported in the published literature for CIS solar cells (<a class="elsevierStyleCrossRefs" href="#bib0120">Ibdah et al., 2014; Matin-Bhuiyan, Shafkat-Islam, & Jyoti-Datta, 2012; Tivanov, Astashenok, Fedotov, & Węgierek, 2012</a>). However, for a low back recombination velocity (102cm/s), there is an optimum thickness below 1000nm.<elsevierMultimedia ident="fig0030"></elsevierMultimedia><a class="elsevierStyleCrossRef" href="#fig0035">Figure 7</a> shows the results of internal quantum efficiency IQE(λ) and external quantum efficiency EQE(λ) as functions of the photon wavelength for a CIS solar cell with absorber layer thickness of 3000nm and high recombination velocity at the back (107cm/s). In <a class="elsevierStyleCrossRef" href="#fig0035">Figure 7</a>, we can see clearly that the difference between IQE and EQE is due to the optical effects associated to the optical transmittance and reflectance for the given solar cell structure (shown in <a class="elsevierStyleCrossRef" href="#fig0025">Fig. 5</a>). Particularly for wavelengths below 450nm, the EQE is greatly reduced as a consequence of the reduced transmittance at the front layer. The J–V and P–V characteristic curves were also calculated, from which Jm, Vm and Voc, FF and efficiency (η) were determined, as shown in <a class="elsevierStyleCrossRef" href="#fig0040">Figure 8</a>.<elsevierMultimedia ident="fig0035"></elsevierMultimedia><elsevierMultimedia ident="fig0040"></elsevierMultimedia>Results for solar cell efficiency when varying thickness of the absorber layer are shown in <a class="elsevierStyleCrossRef" href="#fig0045">Figure 9</a>. Notice that when the surface recombination velocity is low at the back, the optimum CIS thickness is small (around 750nm), which is in agreement to what was explained in Section <a class="elsevierStyleCrossRef" href="#sec0005">1</a>. Then, the material required can be reduced with the respective reduction of cost, and at the same time a high cell efficiency can be expected. Hence, it is convenient to have the reduction of the surface recombination velocity at the back, while achieving high quality absorber thin films without pinholes. These are objectives to be pursued in future CIS or CIGS and CdTe solar cells, as well.<elsevierMultimedia ident="fig0045"></elsevierMultimedia>5Efficiency of a CdS/CdTe solar cellThe developed model was also applied to determine the expected efficiency for CdS/CdTe solar cells, like the one illustrated in <a class="elsevierStyleCrossRef" href="#fig0050">Figure 10</a> with the band diagram shown in <a class="elsevierStyleCrossRef" href="#fig0055">Figure 11</a>. The physical parameters used in the simulations are displayed in <a class="elsevierStyleCrossRef" href="#tbl0010">Table 2</a>. In this case, the CdTe absorber layer thickness was varied between 500nm and 4000nm. Thickness for the CdS layer was the same as for the CIS cell (100nm). In addition, the thickness of the SnO2 layer was assumed to be 500nm (similar to the ZnO:Al layer thickness in the CIS cell).<elsevierMultimedia ident="fig0050"></elsevierMultimedia><elsevierMultimedia ident="fig0055"></elsevierMultimedia><elsevierMultimedia ident="tbl0010"></elsevierMultimedia>The calculated reflectance R(λ) and transmittance T(λ) of the structure depicted in <a class="elsevierStyleCrossRef" href="#fig0050">Figure 10</a> is shown in <a class="elsevierStyleCrossRef" href="#fig0060">Figure 12</a> in the range of 300–850nm. The range of wavelength is shorter than for the CIS cell since the absorption edge of CdTe is smaller than for CIS films. It is worth mentioning that the average reflectance in this structure is smaller than for the above CIS structure (less than 6%).<elsevierMultimedia ident="fig0060"></elsevierMultimedia><a class="elsevierStyleCrossRef" href="#tbl0010">Table 2</a> contains the physical parameters for CdS and CdTe used in the calculation of the solar cell mentioned above. The calculation was also made for two values of back recombination velocity Sn. The low recombination velocity value can be achieved by having a specially treated CdTe layer. For example, an etching at the back may cause a tellurium rich surface, achieving a p+ region. In addition, <a class="elsevierStyleCrossRef" href="#bib0100">Morales-Acevedo (2014)</a> has suggested the inclusion of an additional back p+ layer of another material, such as ZnTe, forming a back heterojunction that blocks electrons flowing to the back contact. This minority carrier blocking should reduce strongly the recombination velocity at the back.The short-circuit current density results are shown in <a class="elsevierStyleCrossRef" href="#fig0065">Figure 13</a>. It can be observed that its highest value is obtained at low recombination velocities; nevertheless, it does not correspond to the thinnest CdTe layer, but to an intermediate value. This can be explained by acknowledging that in a very thin layer the depletion region (and the associated electrostatic field) can extend over its entirety but it absorbs less light than a thick one; on the other hand, in a very thick layer the incident light is practically absorbed in its totality, but the depletion region extends only over a narrow part of the material and hence the electron-hole pairs photo-generated outside this depletion region have a lower probability to be collected, which in turn is reflected in a decrease of the short-circuit current.<elsevierMultimedia ident="fig0065"></elsevierMultimedia><a class="elsevierStyleCrossRef" href="#fig0070">Figure 14</a> shows the external and internal quantum efficiency for the CdS/CdTe solar cell with a CdTe thickness of 1μm and back recombination velocity of 102cm/s, which corresponds to the cell with the best performance as will be seen in <a class="elsevierStyleCrossRef" href="#fig0080">Figure 16</a>.<elsevierMultimedia ident="fig0070"></elsevierMultimedia>In <a class="elsevierStyleCrossRef" href="#fig0075">Figure 15</a>, the expected J–V and P–V curves (current density and power as functions of voltage) of the solar cell with a CdTe thickness of 1μm and back recombination velocity of 102cm/s are presented. J–V and P–V curves of solar cells with different thicknesses were also calculated, nevertheless this was the solar cell with the highest efficiency.<elsevierMultimedia ident="fig0075"></elsevierMultimedia><a class="elsevierStyleCrossRef" href="#fig0080">Figure 16</a>(a)–(c) shows the output parameters of the solar cell as functions of the CdTe thickness. From a quick glance at these graphs, the Voc and FF have their highest values at 0.5μm and they decrease as the CdTe layer becomes thicker for Sn=102cm/s; this behavior has been explained in Section <a class="elsevierStyleCrossRef" href="#sec0005">1</a> and seems to be in good agreement with other works (<a class="elsevierStyleCrossRef" href="#bib0100">Morales-Acevedo, 2014</a>) and it is due to the reduction of the surface recombination velocity at the back contact. The highest value obtained for Voc is 947mV and for the fill factor, 0.847 when the CdTe thickness is around 500nm. Regarding the efficiency, the highest value obtained is 18.66% and it is once again achieved with a low back recombination velocity (Sn=102cm/s) and with a CdTe thickness of 1μm.<elsevierMultimedia ident="fig0080"></elsevierMultimedia>As a general observation, the best results are achieved with a low back surface recombination velocity and small thickness. It is also important to notice that for thick CdTe layers, the recombination velocity does not have a significant effect on the outcome of the cell parameters.Notice that reported record efficiencies for CdTe solar cells are above those calculated here, but the results shown here correspond to the expected efficiencies when the transport parameters are those given in <a class="elsevierStyleCrossRef" href="#tbl0010">Table 2</a>. These parameters are typical for most of the reported cells in different laboratories around the world. Record efficiencies above 20% might correspond to an absorber layer which is not pure CdTe, as can be observed from quantum efficiency measurements made on record cells (<a class="elsevierStyleCrossRef" href="#bib0050">Green et al., 2015</a>), and possibly the material has been prepared under conditions that have allowed the parameters such as electron diffusion length or majority carrier concentration (Na) above those assumed here. Unfortunately, this fact cannot be confirmed because these measurements have not been reported for the record cells and nor the exact composition of the absorber layer (some alloy), so that the calculations should not be compared directly with experimental results unless all the required parameters are already known.In summary, the model here pretends to be of help when designing thin film cells when the parameters given in <a class="elsevierStyleCrossRefs" href="#tbl0005">Table 1 or 2</a> are well known, for the respective absorber material. In addition, in accordance to the discussion given in Section <a class="elsevierStyleCrossRef" href="#sec0005">1</a>, this model shows that thinner cells are better than thicker ones, when the surface recombination at the back is low (102cm/s) compared to the high values (107cm/s) obtained at ohmic contacts. The model can be applied for the design of other cells such as those based on CIGS, CZTS and even Perovskite materials when the respective parameter values are used.6ConclusionWe have presented a unified simple model for determining the expected J–V curves for poly-crystalline thin film solar cells. The model takes into consideration the voltage dependence of the illumination current density which can be limited either by diffusion in the neutral region or by electric field-drift at the space charge region. Similarly, it also takes into consideration that recombination under dark conditions depends on the applied voltage. For very thin cells, these current components might be determined by a space charge region extending for the whole absorber length. In addition, the optical properties of the layers were considered for calculating the reflectance and transmittance for a given cell structure. The model was applied to CIS and CdTe solar cells. For both cases, it was shown that very thin cells can be highly efficient when the recombination at the back is low (less than 102cm/s). This also explains the high efficiencies achieved by very thin perovskite solar cells, and it poses a challenge to develop deposition methods for very thin CdTe and CIGS absorber materials without pinholes, so that improved efficiencies are obtained when the surface recombination velocity is made small at the back, by having a p+ or an electron blocking region in contact with the absorber material before the ohmic contact.Conflict of interestThe authors have no conflicts of interest to declare." "textoCompletoSecciones" => array:1 [ "secciones" => array:11 [ 0 => array:3 [ "identificador" => "xres970554" "titulo" => "Abstract" "secciones" => array:1 [ 0 => array:1 [ "identificador" => "abst0005" ] ] ] 1 => array:2 [ "identificador" => "xpalclavsec941220" "titulo" => "Keywords" ] 2 => array:2 [ "identificador" => "sec0005" "titulo" => "Introduction" ] 3 => array:2 [ "identificador" => "sec0010" "titulo" => "I–V modeling" ] 4 => array:2 [ "identificador" => "sec0015" "titulo" => "Optical model" ] 5 => array:2 [ "identificador" => "sec0020" "titulo" => "Application of the model to CIS solar cells" ] 6 => array:2 [ "identificador" => "sec0025" "titulo" => "Efficiency of a CdS/CdTe solar cell" ] 7 => array:2 [ "identificador" => "sec0030" "titulo" => "Conclusion" ] 8 => array:2 [ "identificador" => "sec0120" "titulo" => "Conflict of interest" ] 9 => array:2 [ "identificador" => "xack328689" "titulo" => "Acknowledgements" ] 10 => array:1 [ "titulo" => "References" ] ] ] "pdfFichero" => "main.pdf" "tienePdf" => true "fechaRecibido" => "2017-01-28" "fechaAceptado" => "2017-08-03" "PalabrasClave" => array:1 [ "en" => array:1 [ 0 => array:4 [ "clase" => "keyword" "titulo" => "Keywords" "identificador" => "xpalclavsec941220" "palabras" => array:5 [ 0 => "Solar cells" 1 => "Thin film" 2 => "CdTe" 3 => "CIS" 4 => "Analytical model" ] ] ] ] "tieneResumen" => true "resumen" => array:1 [ "en" => array:2 [ "titulo" => "Abstract" "resumen" => "Polycrystalline thin film solar cells made with absorber materials such as CdTe, CIGS, CZTS and metalorganic halides (perovskites) are currently important alternatives for the silicon solar cell technology, which still dominates the photovoltaic market. Then, it is important to have tools which can be used to design this kind of solar cells. For this purpose, we have developed a unified simple analytical model that can be applied to thin film solar cells. The model is based on the basic physics of hetero-junction devices, but it takes into consideration that the space charge region can extend along the major part of the cell length, particularly for very thin cells, causing important effects that typically are not observed in conventional junction devices. Photo-generated carriers are collected by electric field-drift instead of diffusion, and simultaneously strong recombination at this region may dominate the electrical I–V characteristic of the cell. Since the space-charge region width varies with the applied voltage, the illumination current density and the saturation dark current density are no longer independent of the voltage as is assumed for conventional solar cells. When the model is applied to CIS and CdTe solar cells as examples, it is found that it is possible to design very thin film solar cells (absorber less than 1μm thick) with high efficiencies, whenever the recombination velocity at the back surface becomes small (102cm/s), instead of the high recombination velocities present at ohmic contacts (107cm/s). This fact implies the cost reduction of thin film solar cells by reducing absorber material thickness, and therefore it poses a challenge to develop deposition methods for very thin CdTe and CIGS absorber materials without pinholes, so that improved efficiencies are obtained when the surface recombination velocity is made small at the back by having a p+ or an electron blocking region before the ohmic contact. This result also explains the high efficiencies achieved by very thin perovskite solar cells." ] ] "NotaPie" => array:1 [ 0 => array:1 [ "nota" => "Peer Review under the responsibility of Universidad Nacional Autónoma de México." ] ] "multimedia" => array:36 [ 0 => array:7 [ "identificador" => "fig0005" "etiqueta" => "Fig. 1" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr1.jpeg" "Alto" => 1805 "Ancho" => 1250 "Tamanyo" => 80266 ] ] "descripcion" => array:1 [ "en" => "Typical structure of a thin film solar cell." ] ] 1 => array:7 [ "identificador" => "fig0010" "etiqueta" => "Fig. 2" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr2.jpeg" "Alto" => 1024 "Ancho" => 2479 "Tamanyo" => 161047 ] ] "descripcion" => array:1 [ "en" => "(a) The refractive index and (b) the extinction coefficient for ZnO:Al, Mo, CIS, CdS, SnO2 and CdTe films." ] ] 2 => array:7 [ "identificador" => "fig0015" "etiqueta" => "Fig. 3" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr3.jpeg" "Alto" => 931 "Ancho" => 1250 "Tamanyo" => 80290 ] ] "descripcion" => array:1 [ "en" => "Scheme of a Mo/CIS/CdS/TCO/solar cell structure." ] ] 3 => array:7 [ "identificador" => "fig0020" "etiqueta" => "Fig. 4" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr4.jpeg" "Alto" => 1183 "Ancho" => 1469 "Tamanyo" => 85334 ] ] "descripcion" => array:1 [ "en" => "Energy band diagram of a CdS/CIS solar cell under equilibrium condition." ] ] 4 => array:7 [ "identificador" => "fig0025" "etiqueta" => "Fig. 5" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr5.jpeg" "Alto" => 1012 "Ancho" => 1480 "Tamanyo" => 107838 ] ] "descripcion" => array:1 [ "en" => "Reflectance of a CIS solar cell with a TCO layer and transmittance of a TCO layer of 500nm." ] ] 5 => array:7 [ "identificador" => "fig0030" "etiqueta" => "Fig. 6" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr6.jpeg" "Alto" => 1186 "Ancho" => 1442 "Tamanyo" => 77638 ] ] "descripcion" => array:1 [ "en" => "Current density of the CIS solar cell as a function of the absorber layer thickness." ] ] 6 => array:7 [ "identificador" => "fig0035" "etiqueta" => "Fig. 7" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr7.jpeg" "Alto" => 1075 "Ancho" => 1498 "Tamanyo" => 98342 ] ] "descripcion" => array:1 [ "en" => "Comparison of the external and internal quantum efficiency of CIS solar cell." ] ] 7 => array:7 [ "identificador" => "fig0040" "etiqueta" => "Fig. 8" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr8.jpeg" "Alto" => 1110 "Ancho" => 1499 "Tamanyo" => 99720 ] ] "descripcion" => array:1 [ "en" => "Power and current density versus voltage curves for CIS solar cell." ] ] 8 => array:7 [ "identificador" => "fig0045" "etiqueta" => "Fig. 9" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr9.jpeg" "Alto" => 2112 "Ancho" => 2506 "Tamanyo" => 233388 ] ] "descripcion" => array:1 [ "en" => "Output parameters of the CIS solar cell as a function of the absorber layer thickness for different back surface recombination velocities." ] ] 9 => array:7 [ "identificador" => "fig0050" "etiqueta" => "Fig. 10" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr10.jpeg" "Alto" => 1226 "Ancho" => 1268 "Tamanyo" => 61703 ] ] "descripcion" => array:1 [ "en" => "CdS/CdTe solar cell structure." ] ] 10 => array:7 [ "identificador" => "fig0055" "etiqueta" => "Fig. 11" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr11.jpeg" "Alto" => 1210 "Ancho" => 1497 "Tamanyo" => 87363 ] ] "descripcion" => array:1 [ "en" => "Energy band diagram of a CdS/CdTe solar cell under equilibrium condition." ] ] 11 => array:7 [ "identificador" => "fig0060" "etiqueta" => "Fig. 12" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr12.jpeg" "Alto" => 1080 "Ancho" => 1488 "Tamanyo" => 107178 ] ] "descripcion" => array:1 [ "en" => "Reflectance of a CdTe solar cell with a TCO layer and transmittance of a TCO layer of 500nm." ] ] 12 => array:7 [ "identificador" => "fig0065" "etiqueta" => "Fig. 13" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr13.jpeg" "Alto" => 1176 "Ancho" => 1479 "Tamanyo" => 82921 ] ] "descripcion" => array:1 [ "en" => "Short-circuit current density vs CdTe thickness." ] ] 13 => array:7 [ "identificador" => "fig0070" "etiqueta" => "Fig. 14" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr14.jpeg" "Alto" => 1125 "Ancho" => 1473 "Tamanyo" => 107662 ] ] "descripcion" => array:1 [ "en" => "External and internal quantum efficiency for the CdS/CdTe solar cell." ] ] 14 => array:7 [ "identificador" => "fig0075" "etiqueta" => "Fig. 15" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr15.jpeg" "Alto" => 1076 "Ancho" => 1495 "Tamanyo" => 107280 ] ] "descripcion" => array:1 [ "en" => "Current density and power as functions of voltage for the 1μm thick CdTe solar cell with Sn=102cm/s." ] ] 15 => array:7 [ "identificador" => "fig0080" "etiqueta" => "Fig. 16" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr16.jpeg" "Alto" => 2124 "Ancho" => 2500 "Tamanyo" => 238624 ] ] "descripcion" => array:1 [ "en" => "Output parameters of the CdTe solar cell as a function of the absorber layer thickness for different back surface recombination velocities." ] ] 16 => array:8 [ "identificador" => "tbl0005" "etiqueta" => "Table 1" "tipo" => "MULTIMEDIATABLA" "mostrarFloat" => true "mostrarDisplay" => false "detalles" => array:1 [ 0 => array:3 [ "identificador" => "at1" "detalle" => "Table " "rol" => "short" ] ] "tabla" => array:1 [ "tablatextoimagen" => array:1 [ 0 => array:2 [ "tabla" => array:1 [ 0 => """ <table border="0" frame="\n \t\t\t\t\tvoid\n \t\t\t\t" class=""><thead title="thead"><tr title="table-row"><th class="td" title="table-head " align="" valign="top" scope="col" style="border-bottom: 2px solid black"> \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">CuInSe2-p \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">CdS-n \t\t\t\t\t\t\n \t\t\t\t</th></tr></thead><tbody title="tbody"><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Eg (eV) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1.02 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">2.42 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Dn (cm2s−1) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1.05 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1.25 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Dp (cm2s−1) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">0.21 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">0.084 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">W (cm) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">0.6–3×10−4 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">0.1×10−4 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Ln (cm) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">2.3×10−4 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">2.5×10−4 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Lp (cm) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">4.6×10−5 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">2.9×10−6 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Nd (cm−3) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">– \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1×1017 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Na (cm−3) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">2×1016 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">– \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">¿ \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">13.6 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">10 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Xe (eV) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">4.3 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">4.5 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Nc (cm−3) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">2.2×1018 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">2.2×1018 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Nv (cm−3) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1.8×1019 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1.8×1019 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Sn (cm/s) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">102, 107 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">– \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Sp (cm/s) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">– \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">107 \t\t\t\t\t\t\n \t\t\t\t</td></tr></tbody></table> """ ] "imagenFichero" => array:1 [ 0 => "xTab1641942.png" ] ] ] ] "descripcion" => array:1 [ "en" => "Values for the parameters used in the calculations of CIS solar cell (<a class="elsevierStyleCrossRefs" href="#bib0010">Amin, Tang, & Sopian, 2007; Benmira & Aida, 2013; Busacca et al., 2014; Daza et al., 2017; Kasap & Capper, 2006; Slonopas et al., 2016; Touafek, Aida, & Mahamdi, 2012</a>)." ] ] 17 => array:8 [ "identificador" => "tbl0010" "etiqueta" => "Table 2" "tipo" => "MULTIMEDIATABLA" "mostrarFloat" => true "mostrarDisplay" => false "detalles" => array:1 [ 0 => array:3 [ "identificador" => "at2" "detalle" => "Table " "rol" => "short" ] ] "tabla" => array:1 [ "tablatextoimagen" => array:1 [ 0 => array:2 [ "tabla" => array:1 [ 0 => """ <table border="0" frame="\n \t\t\t\t\tvoid\n \t\t\t\t" class=""><thead title="thead"><tr title="table-row"><th class="td" title="table-head " align="" valign="top" scope="col" style="border-bottom: 2px solid black"> \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">CdTe-p \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">CdS-n \t\t\t\t\t\t\n \t\t\t\t</th></tr></thead><tbody title="tbody"><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Eg (eV) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1.5 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">2.42 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Dn (cm2s−1) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">2.585 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1.25 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Dp (cm2s−1) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1.55 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">0.084 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">W (cm) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">0.5–4×10−4 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">0.1×10−4 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Ln (cm) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1.017×10−4 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">2.5×10−4 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Lp (cm) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1.24×10−3 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">2.9×10−6 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Nd (cm−3) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">– \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1×1017 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Na (cm−3) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1×1015 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">– \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">¿ \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">10.3 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">10 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Xe (eV) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">4.28 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">4.5 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Nc (cm−3) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">8×1017 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">2.2×1018 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Nv (cm−3) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1.8×1019 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1.8×1019 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Sn (cm/s) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">102, 107 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">– \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top">Sp (cm/s) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">– \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">107 \t\t\t\t\t\t\n \t\t\t\t</td></tr></tbody></table> """ ] "imagenFichero" => array:1 [ 0 => "xTab1641943.png" ] ] ] ] "descripcion" => array:1 [ "en" => "Parameters used in the calculations of the CdS/CdTe solar cell (<a class="elsevierStyleCrossRefs" href="#bib0005">Amin, Sopian, & Konagai, 2007; Amin, Tang, et al., 2007; Benmira & Aida, 2013; Busacca et al., 2014; Daza et al., 2017; Fardi & Buny, 2013; Kasap & Capper, 2006; Khosroabadi, Keshmiri, & Marjani, 2014; Slonopas et al., 2016; Touafek et al., 2012</a>)." ] ] 18 => array:6 [ "identificador" => "eq0005" "etiqueta" => "(1)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "xp(V)=2εpεnNd(Vbi−V)qNa(εnNd+εpNa)1/2" "Fichero" => "STRIPIN_si1.jpeg" "Tamanyo" => 3788 "Alto" => 44 "Ancho" => 236 ] ] 19 => array:6 [ "identificador" => "eq0010" "etiqueta" => "(2)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "xn(V)=2εpεnNa(Vbi−V)qNd(εnNd+εpNa)1/2" "Fichero" => "STRIPIN_si2.jpeg" "Tamanyo" => 3747 "Alto" => 44 "Ancho" => 236 ] ] 20 => array:6 [ "identificador" => "eq0015" "etiqueta" => "(3)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Vbi=ΔEc−ΔEv2+kbT lnNaNdni,pni,n+kbT2lnNc,pNv,nNc,nNv,p" "Fichero" => "STRIPIN_si3.jpeg" "Tamanyo" => 5518 "Alto" => 92 "Ancho" => 278 ] ] 21 => array:6 [ "identificador" => "eq0095" "etiqueta" => "(4)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "J′p(λ)=dJpdλ =qN0(λ)(1−R(λ))T(λ)α1(λ)Lp(α1(λ)2Lp2−1)SpLpDp+α1(λ)Lp−e−α1(λ)(Wn−xn)SpLpDpcoshWn−XnLp+senhWn−XnLpSpLpDpsenhWn−XnLp+coshWn−XnLp−α1(λ)Lpe−α1(λ)(Wn−xn)" "Fichero" => "STRIPIN_si7.jpeg" "Tamanyo" => 17730 "Alto" => 153 "Ancho" => 855 ] ] 22 => array:6 [ "identificador" => "eq0100" "etiqueta" => "(5)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "J′n(λ)=dJndλ=qN0(λ)(1−R(λ))T(λ)α2(λ)Ln(α2(λ)2Ln2−1)e−(α1(λ)(Wn)+α2(λ)(xp))α2(λ)Ln−SnLnDncosh(Wp−xp)Ln−e−α2(λ)(Wp−xp)+senh(Wp−xp)Ln+α2Lne−α2(λ)(Wp−xp)SnLnDnsenh(Wp−xp)Ln+cosh(Wp−xp)Ln" "Fichero" => "STRIPIN_si8.jpeg" "Tamanyo" => 14797 "Alto" => 112 "Ancho" => 706 ] ] 23 => array:6 [ "identificador" => "eq0105" "etiqueta" => "(6)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "J′scr(λ)=dJgendλ=qN0(λ)(1−R(λ))T(λ)e−α1(λ)(Wn−xn)[(1−e−α1(λ)xn)+e−α1(λ)xn(1−e−α2(λ)xp)]" "Fichero" => "STRIPIN_si9.jpeg" "Tamanyo" => 6376 "Alto" => 90 "Ancho" => 370 ] ] 24 => array:6 [ "identificador" => "eq0035" "etiqueta" => "(7)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Jph(V)=∫λminλmaxJ′n(λ)+J′p(λ)+J′scr(λ) dλ" "Fichero" => "STRIPIN_si13.jpeg" "Tamanyo" => 3401 "Alto" => 42 "Ancho" => 305 ] ] 25 => array:6 [ "identificador" => "eq0110" "etiqueta" => "(8)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Jdark≈J0eqVkT−1+J00eqV2kT−1" "Fichero" => "STRIPIN_si14.jpeg" "Tamanyo" => 2689 "Alto" => 30 "Ancho" => 260 ] ] 26 => array:6 [ "identificador" => "eq0115" "etiqueta" => "(9)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "J0(V)=J0p(V)+J0n(V)" "Fichero" => "STRIPIN_si15.jpeg" "Tamanyo" => 1623 "Alto" => 17 "Ancho" => 170 ] ] 27 => array:6 [ "identificador" => "eq0120" "etiqueta" => "(10)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "J0p=qDpp0LpSpLpDpcosh(Wn−xn)Lp+senh(Wn−xn)LpSpLpDpsenh(Wn−xn)Lp+cosh(Wn−xn)Lp" "Fichero" => "STRIPIN_si16.jpeg" "Tamanyo" => 6758 "Alto" => 59 "Ancho" => 339 ] ] 28 => array:6 [ "identificador" => "eq0125" "etiqueta" => "(11)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "J0n=qDnn0LnSnLnDncosh(Wp−xp)Ln+sinh(Wp−xp)LnSnLnDnsenh(Wp−xp)Ln+cosh(Wp−xp)Ln" "Fichero" => "STRIPIN_si17.jpeg" "Tamanyo" => 6652 "Alto" => 59 "Ancho" => 337 ] ] 29 => array:6 [ "identificador" => "eq0130" "etiqueta" => "(12)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "J00(V)=qxnni,nτp+xpni,pτn" "Fichero" => "STRIPIN_si18.jpeg" "Tamanyo" => 2523 "Alto" => 40 "Ancho" => 206 ] ] 30 => array:6 [ "identificador" => "eq0065" "etiqueta" => "(13)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Jcell(v)=Jph(v)−Jdark(v)" "Fichero" => "STRIPIN_si19.jpeg" "Tamanyo" => 1673 "Alto" => 17 "Ancho" => 180 ] ] 31 => array:6 [ "identificador" => "eq0070" "etiqueta" => "(14)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "η=JmVmPinc" "Fichero" => "STRIPIN_si20.jpeg" "Tamanyo" => 1044 "Alto" => 36 "Ancho" => 73 ] ] 32 => array:6 [ "identificador" => "eq0075" "etiqueta" => "(15)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Cm=pm+iqmrm+ismtm+iumvm+iwm" "Fichero" => "STRIPIN_si21.jpeg" "Tamanyo" => 3018 "Alto" => 49 "Ancho" => 216 ] ] 33 => array:6 [ "identificador" => "eq0080" "etiqueta" => "(16)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "pm=eαm−1 cos γm−1qm=eαm−1 sin γm−1rm=eαm−1(gm cos γm−1−hm sin γm−1)sm=eαm−1(hm cos γm−1+gm sin γm−1)tm=e−αm−1(gm cos γm−1+hm sin γm−1)um=e−αm−1(hm cos γm−1−gm sin γm−1)vm=e−αm−1 cos γm−1wm=−e−αm−1 sin γm−1" "Fichero" => "STRIPIN_si22.jpeg" "Tamanyo" => 13458 "Alto" => 223 "Ancho" => 291 ] ] 34 => array:6 [ "identificador" => "eq0085" "etiqueta" => "(17)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "αm−1=2πkm−1dm−1λ;  γm−1=2πnm−1dm−1λ" "Fichero" => "STRIPIN_si23.jpeg" "Tamanyo" => 3129 "Alto" => 34 "Ancho" => 333 ] ] 35 => array:6 [ "identificador" => "eq0090" "etiqueta" => "(18)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "R(λ)=t1,42+u1,42p1,42+q1,42" "Fichero" => "STRIPIN_si24.jpeg" "Tamanyo" => 2052 "Alto" => 46 "Ancho" => 128 ] ] ] "bibliografia" => array:2 [ "titulo" => "References" "seccion" => array:1 [ 0 => array:2 [ "identificador" => "bibs0015" "bibliografiaReferencia" => array:31 [ 0 => array:3 [ "identificador" => "bib0005" "etiqueta" => "Amin et al., 2007a" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:2 [ "titulo" => "Numerical modeling of CdS/CdTe and CdS/CdTe/ZnTe solar cells as a function of CdTe thickness" "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:3 [ 0 => "N. Amin" 1 => "K. Sopian" 2 => "M. Konagai" ] ] ] ] ] "host" => array:1 [ 0 => array:2 [ "doi" => "10.1002/smll.201703142" "Revista" => array:8 [ "tituloSerie" => "Solar Energy Materials & Solar Cells" "fecha" => "2007" "volumen" => "91" "numero" => "13" "paginaInicial" => "1202" "paginaFinal" => "1208" "link" => array:1 [ 0 => array:2 [ "url" => "https://www.ncbi.nlm.nih.gov/pubmed/29319230" "web" => "Medline" ] ] "itemHostRev" => array:3 [ "pii" => "S0022534712047155" "estado" => "S300" "issn" => "00225347" ] ] ] ] ] ] ] 1 => array:3 [ "identificador" => "bib0010" "etiqueta" => "Amin et al., 2007b" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:2 [ "titulo" => "Numerical modeling of the copper–indium–selenium (CIS) based solar cell performance by AMPS-1D" "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:3 [ 0 => "N. Amin" 1 => "M. Tang" 2 => "K. Sopian" ] ] ] ] ] "host" => array:1 [ 0 => array:1 [ "LibroEditado" => array:4 [ "titulo" => "The 5th student conference on research and development" "paginaInicial" => "1" "paginaFinal" => "6" "serieFecha" => "2007" ] ] ] ] ] ] 2 => array:3 [ "identificador" => "bib0015" "etiqueta" => "Anderson, 1960" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:2 [ "titulo" => "Germanium–gallium arsenide heterojunctions" "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:1 [ 0 => "R.L. 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"vista" => "all" ] ] ] "idiomaDefecto" => "en" "url" => "/16656423/0000001500000006/v1_201801120630/S1665642317300950/v1_201801120630/en/main.assets" "Apartado" => array:4 [ "identificador" => "41800" "tipo" => "SECCION" "en" => array:2 [ "titulo" => "Articles" "idiomaDefecto" => true ] "idiomaDefecto" => "en" ] "PDF" => "https://static.elsevier.es/multimedia/16656423/0000001500000006/v1_201801120630/S1665642317300950/v1_201801120630/en/main.pdf?idApp=UINPBA00004N&text.app=https://www.elsevier.es/" "EPUB" => "https://multimedia.elsevier.es/PublicationsMultimediaV1/item/epub/S1665642317300950?idApp=UINPBA00004N"]

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Design of thin film solar cells based on a unified simple analytical model

Armando Acevedo-Lunaa, Roberto Bernal-Correab, Jorge Montes-Monsalvec, Arturo Morales-Acevedoa,

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amorales@solar.cinvestav.mx

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a Centro de Investigación y de Estudios Avanzados del IPN, Electrical Engineering Department, Avenida IPN # 2508, 07360 Mexico, D.F., Mexico

b Grupo DEMA, Facultad de Ciencias e Ingeniería, Universidad del Sinú, Montería, Colombia

c Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Colombia, Manizales, Colombia

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    "textoCompleto" => "<span class="elsevierStyleSections"><span id="sec0005" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">1</span><span class="elsevierStyleSectionTitle" id="sect0015">Introduction</span><p id="par0005" class="elsevierStylePara elsevierViewall">There is the expectation that thin film solar cells will be the alternative to silicon solar cells which is the dominant technology at the photovoltaic market today&#46; Improved efficiencies and lower costs of thin film solar cells are required for this goal to become a reality&#46; Among the most developed thin film solar cells we have CdS&#47;CdTe and CdS&#47;CIGS which recently have attained efficiencies above 20&#37; &#40;<a class="elsevierStyleCrossRef" href="#bib0050">Green&#44; Emery&#44; Hishikawa&#44; Warta&#44; &#38; Dunlop&#44; 2015</a>&#41;&#46; Each one of these technologies needs solving some problems related to the quality of the deposited materials &#40;by different techniques&#41;&#44; but as it will be shown here&#44; reducing the film thickness and achieving higher open circuit voltages are required for both kind of solar cells&#46; However&#44; in general&#44; CdTe and CIS solar cells are studied independently of each other without having a more integral vision&#46;</p><p id="par0010" class="elsevierStylePara elsevierViewall">It must be observed that from the structural point of view&#44; CdTe and CIS solar cells are similar and therefore their physical behavior is also identical&#46; The change is in the absorbing material &#40;with their own properties&#41; and the technological steps followed to make the solar cells&#46; For example&#44; CdTe is typically obtained by closed space vapor transport &#40;CSVT&#41; or Rf-sputtering &#40;<a class="elsevierStyleCrossRef" href="#bib0095">Morales-Acevedo&#44; 2006</a>&#41;&#44; while CIS solar cells are obtained by co-evaporation or Rf-sputtering among other different techniques &#40;<a class="elsevierStyleCrossRef" href="#bib0130">Singh &#38; Patra&#44; 2010</a>&#41;&#46; CdTe typically cannot be p-type doped above 10<span class="elsevierStyleSup">15</span><span class="elsevierStyleHsp" style=""></span>cm<span class="elsevierStyleSup">&#8722;3</span>&#44; and cells with small CdTe thickness cannot be easily done because pinholes cause the device degradation&#46; Then&#44; present CdTe minimum thickness is around 4&#8211;6<span class="elsevierStyleHsp" style=""></span>&#956;m&#46; In the case of CIS solar cells&#44; the CIS &#40;or CIGS&#41; thickness is around 3&#8211;4<span class="elsevierStyleHsp" style=""></span>&#956;m&#46;</p><p id="par0015" class="elsevierStylePara elsevierViewall">It will be shown that good cells can be designed with absorber thickness around 1<span class="elsevierStyleHsp" style=""></span>&#956;m or below&#46; Hence&#44; the technological challenge is achieving CdTe and CIS materials with good properties&#44; but without pinholes using the present or new deposition techniques&#46; In such a case&#44; the amount of material would be reduced by about 60&#8211;75&#37; with a corresponding decrease in the total cost of the cells&#46;</p><p id="par0020" class="elsevierStylePara elsevierViewall">In this paper&#44; we shall describe a simple analytical model that can be used for the above mentioned solar cells &#40;CdTe and CIS&#41;&#44; so that they can be designed easily&#46; This model is complete in the sense that it includes carrier transport limited by diffusion and by generation&#8211;recombination at the space charge region&#46; It also takes into account that the space charge region width is dependent upon the operating voltage and therefore the superposition principle is no longer valid&#46; In other words&#44; the current density due to illumination is not a constant with respect to the applied voltage&#44; but it has some dependence upon this variable&#46; Similarly&#44; the total dark saturation current density will not be independent of the operating voltage&#44; as it is usually assumed&#46;</p><p id="par0025" class="elsevierStylePara elsevierViewall">Having a simple model that considers the above effects for thin film solar cells is very important because&#44; in general&#44; they are not taken into account because for conventional cells&#44; such as those made with silicon&#44; these effects are negligible&#46; Conventional junction silicon solar cells are typically made with absorber thickness of more than 200<span class="elsevierStyleHsp" style=""></span>&#956;m&#44; so that the space charge region effects on good solar cells are small because the depletion region thickness is of the order of 1<span class="elsevierStyleHsp" style=""></span>&#956;m&#46;</p><p id="par0030" class="elsevierStylePara elsevierViewall">For thin film solar cells&#44; the space region effects become important because the total volumes of the space charge region and the quasi-neutral regions are of the same order&#44; particularly for very thin film solar cells&#46; Furthermore&#44; under some situations &#40;which depends upon the equilibrium majority carrier concentration and the thickness of the absorbing material&#41; the depletion region may extend along the whole length of the absorbing material&#46; In this case&#44; the recombination in the depletion region will limit the total dark current&#44; but at the same time the photo-generated carriers will be collected efficiently because of the presence of the high electric field in this region&#46; And this electric field will be larger for thinner solar cells&#46; Hence&#44; there is a complex relation between majority carrier concentration &#40;Na&#41;&#44; absorbing thickness&#44; lifetime and mobility of minority carriers for the whole operating voltage range of the solar cell&#46; The main objective for this work will be to have a simple model that takes into consideration all the above effects&#46;</p><p id="par0035" class="elsevierStylePara elsevierViewall">It will be shown that the model predicts that if the recombination at the back surface is reduced&#44; for example by having a p&#43; region before the back contact&#44; so that surface recombination velocities are small &#40;10<span class="elsevierStyleSup">2</span><span class="elsevierStyleHsp" style=""></span>cm&#47;s or less&#41; compared to the high recombination velocities obtained at ohmic contacts &#40;above 10<span class="elsevierStyleSup">7</span><span class="elsevierStyleHsp" style=""></span>cm&#47;s&#41;&#44; then very thin cells can achieve improved efficiencies than those with thick absorbers because of the reduction of bulk recombination&#44; although there might be some loss of photo-current density&#46;</p><p id="par0040" class="elsevierStylePara elsevierViewall">Finally&#44; in order to have a complete model for designing thin film solar cells&#44; the optical design was also considered by using the optical matrix method applied to all the films that are part of the cells&#46; The electrical and optical calculations were applied to CdTe and CIS solar cells&#44; as an example of the application of the models described here&#46;</p></span><span id="sec0010" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">2</span><span class="elsevierStyleSectionTitle" id="sect0020"><span class="elsevierStyleItalic">I</span>&#8211;<span class="elsevierStyleItalic">V</span> modeling</span><p id="par0045" class="elsevierStylePara elsevierViewall">The reference structure of a thin film solar cell is shown in <a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a>&#46; It is formed by three main regions&#58; The transparent conducting oxide &#40;TCO&#41; which allows the light passing to the hetero-junction&#44; the window semiconductor layer &#40;typically CdS&#41;&#44; and the absorbing semiconductor material&#46; Ohmic contacts are both made at the back of the absorbing layer and at the front TCO layer in order to connect the cell to the external circuit&#46; We shall assume that the back is covered by a metallic contact which reflects totally those photons that pass the absorbing layer without being absorbed causing a second pass of such photons through this layer&#46;</p><elsevierMultimedia ident="fig0005"></elsevierMultimedia><p id="par0050" class="elsevierStylePara elsevierViewall">As explained above&#44; an important parameter that determines the electric and photoelectric properties in a solar cell is the thickness of the space charge region &#40;depletion region&#41; in both the p-type and n-type sides of the heterojunction by means of the following expressions&#58;<elsevierMultimedia ident="eq0005"></elsevierMultimedia><elsevierMultimedia ident="eq0010"></elsevierMultimedia>where <span class="elsevierStyleItalic">¿</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span> is the relative permittivity of the p-type material and <span class="elsevierStyleItalic">¿</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span> is that of the n-type one&#44; <span class="elsevierStyleItalic">N</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">a</span></span> and <span class="elsevierStyleItalic">N</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">d</span></span> are the acceptor and donor concentrations at each region&#44; respectively&#44; <span class="elsevierStyleItalic">V</span> is the bias voltage and <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">bi</span></span> is the built-in potential&#44; calculated as&#58;<elsevierMultimedia ident="eq0015"></elsevierMultimedia>where &#916;<span class="elsevierStyleItalic">E</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">c</span></span> and &#916;Ev are the conduction and valence band discontinuities&#44; respectively&#46; <span class="elsevierStyleItalic">k</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">b</span></span> is the Boltzmann constant&#44; <span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span>&#44;<span class="elsevierStyleItalic">p</span></span> and <span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span>&#44;<span class="elsevierStyleItalic">n</span></span> are the intrinsic carrier concentrations of the p and n-type materials&#46; <span class="elsevierStyleItalic">N</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">c</span>&#44;<span class="elsevierStyleItalic">p</span></span> and <span class="elsevierStyleItalic">N</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">c</span>&#44;<span class="elsevierStyleItalic">n</span></span> are the effective densities of states in the conduction band of the p and n-type materials and finally Nv&#44;p and Nv&#44;n are the effective densities of states in the valence band of the p and n-type materials&#46;</p><p id="par0065" class="elsevierStylePara elsevierViewall">In some cases&#44; <span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span> as calculated by Eq&#46; <a class="elsevierStyleCrossRef" href="#eq0005">&#40;1&#41;</a> might be larger than the thickness of the absorber layer itself &#40;<span class="elsevierStyleItalic">W</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span>&#41;&#46; In this case&#44; the depletion region would extend over the entire material&#44; so that the depletion region would be limited by the thickness of the absorber layer &#40;i&#46;e&#46;&#44; <span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">W</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span> and <span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">W</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span>&#40;<span class="elsevierStyleItalic">N</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">a</span></span>&#47;<span class="elsevierStyleItalic">N</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">d</span></span>&#41;&#41;&#46; This condition could be changed as the forward operating voltage is increased because the depletion thickness is reduced as the forward voltage is increased&#46;</p><p id="par0070" class="elsevierStylePara elsevierViewall">To determine the total photocurrent density&#44; there are two contributions to be considered&#46; The one limited by carrier diffusion and the one due to the electric field drift of the generated carriers at the space charge region&#46; For each wavelength <span class="elsevierStyleItalic">&#955;</span>&#44; it is necessary to consider the transmittance <span class="elsevierStyleItalic">T</span>&#40;<span class="elsevierStyleItalic">&#955;</span>&#41; in the TCO layer&#44; and the total reflectance <span class="elsevierStyleItalic">R</span>&#40;<span class="elsevierStyleItalic">&#955;</span>&#41; due to the multilayer system&#44; as will be determined in the next section&#46;</p><p id="par0075" class="elsevierStylePara elsevierViewall">The photocurrent density limited by ambipolar diffusion at the quasi-neutral regions are given by <a class="elsevierStyleCrossRef" href="#bib0070">Kosyachenko &#40;2010&#41;</a> and <a class="elsevierStyleCrossRef" href="#bib0105">Nelson &#40;2003&#41;</a>&#58;<elsevierMultimedia ident="eq0095"></elsevierMultimedia><elsevierMultimedia ident="eq0100"></elsevierMultimedia>In addition&#44; 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In all cases&#44; the photon density flux <span class="elsevierStyleItalic">N</span><span class="elsevierStyleInf">0</span>&#40;<span class="elsevierStyleItalic">&#955;</span>&#41; corresponds to the AM1&#46;5G standard solar spectrum&#46; The absorption coefficients of the window layer <span class="elsevierStyleItalic">&#945;</span><span class="elsevierStyleInf">1</span>&#40;<span class="elsevierStyleItalic">&#955;</span>&#41; and the absorber layer <span class="elsevierStyleItalic">&#945;</span><span class="elsevierStyleInf">2</span>&#40;<span class="elsevierStyleItalic">&#955;</span>&#41; were considered in the same wavelength range up to the wavelength that can be absorbed by a specific absorber material&#46; Photons with larger wavelengths will not produce any photo-current&#46;</p><p id="par0090" class="elsevierStylePara elsevierViewall">The total current density is the sum of each of the above current densities &#40;J&#8242;p&#44; J&#8242;n and J&#8242;scr&#41; and integrated in the range between <span class="elsevierStyleItalic">&#955;</span><span class="elsevierStyleInf">min</span> &#40;300<span class="elsevierStyleHsp" style=""></span>nm&#41; and <span class="elsevierStyleItalic">&#955;</span><span class="elsevierStyleInf">max</span> &#40;850<span class="elsevierStyleHsp" style=""></span>nm for CdTe and 1200<span class="elsevierStyleHsp" style=""></span>nm for CIS&#44; which correspond to the absorption edges of the CdTe and CIS layers&#44; respectively&#41;&#46; It is given by the following expression&#58;<elsevierMultimedia ident="eq0035"></elsevierMultimedia>The dark current components will also be limited either by diffusion or by generation&#8211;recombination at the space-charge region &#40;<a class="elsevierStyleCrossRef" href="#bib0140">Sze&#44; 2008</a>&#41;&#46;<elsevierMultimedia ident="eq0110"></elsevierMultimedia>where <span class="elsevierStyleItalic">T</span> is the temperature&#44; <span class="elsevierStyleItalic">k</span> is the Boltzmann constant&#44; and <span class="elsevierStyleItalic">J</span><span class="elsevierStyleInf">0</span> and <span class="elsevierStyleItalic">J</span><span class="elsevierStyleInf">00</span> are the dark saturation currents due to diffusion and generation&#8211;recombination&#44; respectively&#46; They are given by<elsevierMultimedia ident="eq0115"></elsevierMultimedia>where<elsevierMultimedia ident="eq0120"></elsevierMultimedia><elsevierMultimedia ident="eq0125"></elsevierMultimedia>and<elsevierMultimedia ident="eq0130"></elsevierMultimedia><span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span>&#44;<span class="elsevierStyleItalic">n</span></span> and <span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span>&#44;<span class="elsevierStyleItalic">p</span></span> are the intrinsic carrier densities&#44; <span class="elsevierStyleItalic">p</span><span class="elsevierStyleInf">0</span> and <span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">0</span> are the minority carrier concentrations&#44; and <span class="elsevierStyleItalic">&#964;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span> and <span class="elsevierStyleItalic">&#964;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span> are the minority carrier lifetimes&#44; in the <span class="elsevierStyleItalic">n</span> and <span class="elsevierStyleItalic">p</span> semiconductors&#44; respectively&#46;</p><p id="par0120" class="elsevierStylePara elsevierViewall">Therefore&#44; all the dark saturation current density components depend upon the operating voltage because <span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span> and <span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span> are functions of this variable&#44; as given by Eqs&#46; <a class="elsevierStyleCrossRefs" href="#eq0005">&#40;1&#41; and &#40;2&#41;</a>&#46; The total current density of the cell is then given by&#58;<elsevierMultimedia ident="eq0065"></elsevierMultimedia>From this <span class="elsevierStyleItalic">J</span>&#8211;<span class="elsevierStyleItalic">V</span> dependence the maximum generated power can be determined by assuming the series and shunt resistance losses to be zero&#46; The efficiency will be given by<elsevierMultimedia ident="eq0070"></elsevierMultimedia>where <span class="elsevierStyleItalic">J</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">m</span></span> and <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">m</span></span> are the current density and voltage at the maximum power point&#44; respectively&#44; and <span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">inc</span></span> is the incident radiation power density&#46;</p><p id="par0135" class="elsevierStylePara elsevierViewall">Notice that the above model uses well known expressions&#44; but they unified into a simple full analytical model for including the effects mentioned in the preceding section&#46; Including these effects is important for quantifying in a more complete manner the behavior of thin film solar cells&#46;</p></span><span id="sec0015" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">3</span><span class="elsevierStyleSectionTitle" id="sect0025">Optical model</span><p id="par0140" class="elsevierStylePara elsevierViewall">A simple way to evaluate the optical losses for a thin film solar cell is to use the optical matrix method&#46; The optical matrix is expressed in terms of Fresnel coefficients for a system of <span class="elsevierStyleItalic">m</span> layers&#44; where the <span class="elsevierStyleItalic">m</span>th and &#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span>1&#41;th are layers with complex refractive index <span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">m</span></span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">m</span></span><span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">ik</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">m</span></span> and <span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">m</span>&#8722;1</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">m</span>&#8722;1</span><span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">ik</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">m</span>&#8722;1</span>&#46;</p><p id="par0145" class="elsevierStylePara elsevierViewall">The optical matrix for the <span class="elsevierStyleItalic">m</span>th layer is defined by the following expression &#40;<a class="elsevierStyleCrossRef" href="#bib0055">Heavens&#44; 1954</a>&#41;&#58;<elsevierMultimedia ident="eq0075"></elsevierMultimedia>where each of the elements are calculated in accordance to the following set of equations&#58;<elsevierMultimedia ident="eq0080"></elsevierMultimedia><span class="elsevierStyleItalic">g</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">m</span></span> and <span class="elsevierStyleItalic">h</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">m</span></span> are functions that depend on both the refractive index and the extinction coefficient&#46; The values of <span class="elsevierStyleItalic">&#945;</span> and <span class="elsevierStyleItalic">&#947;</span> can be calculated accordingly to<elsevierMultimedia ident="eq0085"></elsevierMultimedia>where <span class="elsevierStyleItalic">&#955;</span> is the wavelength and <span class="elsevierStyleItalic">d</span> is the respective layer thickness&#46; The reflectance <span class="elsevierStyleItalic">R</span>&#40;<span class="elsevierStyleItalic">&#955;</span>&#41; for a system of four layers on a substrate is calculated as<elsevierMultimedia ident="eq0090"></elsevierMultimedia>where <span class="elsevierStyleItalic">t</span><span class="elsevierStyleInf">1&#44;4</span>&#44; <span class="elsevierStyleItalic">u</span><span class="elsevierStyleInf">1&#44;4</span>&#44; <span class="elsevierStyleItalic">p</span><span class="elsevierStyleInf">1&#44;4</span> and <span class="elsevierStyleItalic">q</span><span class="elsevierStyleInf">1&#44;4</span> are the equivalent elements of the product of the optical matrices for each layer&#46;</p><p id="par0170" class="elsevierStylePara elsevierViewall">Based on this model&#44; as well as on the values of refractive index <span class="elsevierStyleItalic">n</span>&#40;<span class="elsevierStyleItalic">&#955;</span>&#41; and extinction coefficient <span class="elsevierStyleItalic">k</span>&#40;<span class="elsevierStyleItalic">&#955;</span>&#41; reported in the literature &#40;see <a class="elsevierStyleCrossRef" href="#fig0010">Fig&#46; 2</a>&#41; &#40;<a class="elsevierStyleCrossRefs" href="#bib0045">Filmetrics&#44; 2016&#59; Mclntosh&#44; 2016&#59; Pan&#44; Zhang&#44; Teng&#44; Li&#44; &#38; Li&#44; 2008&#59; Polyanskiy&#44; 2016&#59; Treharne et al&#46;&#44; 2011</a>&#41;&#44; values of <span class="elsevierStyleItalic">R</span>&#40;<span class="elsevierStyleItalic">&#955;</span>&#41; were determined for the solar cells studied in this paper&#44; as will be discussed in the next section&#46;</p><elsevierMultimedia ident="fig0010"></elsevierMultimedia></span><span id="sec0020" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">4</span><span class="elsevierStyleSectionTitle" id="sect0030">Application of the model to CIS solar cells</span><p id="par0175" class="elsevierStylePara elsevierViewall">With the purpose of determining the efficiency of a CIS solar cell&#44; schematically shown in <a class="elsevierStyleCrossRef" href="#fig0015">Figure 3</a>&#44; the relevant physical parameters were obtained from previous reports &#40;<a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>&#41;&#46; The value of the CIS absorber layer thickness was varied between 500<span class="elsevierStyleHsp" style=""></span>nm and 3000<span class="elsevierStyleHsp" style=""></span>nm in order to obtain the solar cell efficiency&#46; The band diagram of the structure under equilibrium was built by applying Anderson&#39;s rule &#40;<a class="elsevierStyleCrossRef" href="#fig0020">Fig&#46; 4</a>&#41; &#40;<a class="elsevierStyleCrossRefs" href="#bib0015">Anderson&#44; 1960&#59; Sharma &#38; Purohit&#44; 1974</a>&#41;&#44; in which the bandgap values and characteristic electron affinities are observed&#46;</p><elsevierMultimedia ident="fig0015"></elsevierMultimedia><elsevierMultimedia ident="tbl0005"></elsevierMultimedia><elsevierMultimedia ident="fig0020"></elsevierMultimedia><p id="par0180" class="elsevierStylePara elsevierViewall">In <a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a>&#44; the result of <span class="elsevierStyleItalic">R</span>&#40;<span class="elsevierStyleItalic">&#955;</span>&#41; for the ZnO&#58;Al&#47;CdS&#47;CIS&#47;Mo&#47;glass structure in the 300&#8211;1200<span class="elsevierStyleHsp" style=""></span>nm region &#40;CIS absorption edge&#41; is observed&#46; In the graph&#44; a characteristic behavior with oscillations due to interference effects at different wavelengths is evident&#46; This behavior is expected because of the thickness and refractive index difference of the AZO material compared to that of CdS and the other layers below &#40;<a class="elsevierStyleCrossRef" href="#bib0080">Mahdjoub &#38; Hadjeris&#44; 2013</a>&#41;&#46;</p><elsevierMultimedia ident="fig0025"></elsevierMultimedia><p id="par0185" class="elsevierStylePara elsevierViewall">A complete calculation of the solar cell efficiency is made taking into account the transmittance as a function of wavelength <span class="elsevierStyleItalic">T</span>&#40;<span class="elsevierStyleItalic">&#955;</span>&#41; for the above cell structure&#46; In this case&#44; a transmittance average around 80&#37; &#40;above the ZnO&#58;Al absorption edge&#41; is obtained&#44; which is in good agreement with other reported theoretical and experimental results &#40;<a class="elsevierStyleCrossRefs" href="#bib0025">Bernal-Correa&#44; Morales-Acevedo&#44; Montes-Monsalve&#44; &#38; Pulzara-Mora&#44; 2016&#59; Kuwahata &#38; Minemoto&#44; 2014</a>&#41;&#46;</p><p id="par0190" class="elsevierStylePara elsevierViewall">Based on the values shown in <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>&#44; as well as the results for <span class="elsevierStyleItalic">R</span>&#40;<span class="elsevierStyleItalic">&#955;</span>&#41;&#44; <span class="elsevierStyleItalic">T</span>&#40;<span class="elsevierStyleItalic">&#955;</span>&#41;&#44; and taking into account the spectra for <span class="elsevierStyleItalic">&#945;</span><span class="elsevierStyleInf">1</span>&#40;<span class="elsevierStyleItalic">&#955;</span>&#41; and <span class="elsevierStyleItalic">&#945;</span><span class="elsevierStyleInf">2</span>&#40;<span class="elsevierStyleItalic">&#955;</span>&#41;&#44; <span class="elsevierStyleItalic">J</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ph</span></span> was calculated using the model described in Section <a class="elsevierStyleCrossRef" href="#sec0010">2</a>&#46; The photons not absorbed in the first pass of light through the cell structure are reflected at the back contact&#44; so that these photons must be included for the calculation of the light current density because they have a second opportunity to be absorbed by the absorber layer&#46; This additional photo-current has also been calculated&#44; although it is not given explicitly in the expressions for <span class="elsevierStyleItalic">J</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ph</span></span>&#44; above&#46;</p><p id="par0195" class="elsevierStylePara elsevierViewall">The values of <span class="elsevierStyleItalic">J</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">sc</span></span> as a function of the CIS thickness are shown in <a class="elsevierStyleCrossRef" href="#fig0030">Figure 6</a>&#46; For a high recombination velocity at the back &#40;10<span class="elsevierStyleSup">7</span><span class="elsevierStyleHsp" style=""></span>cm&#47;s&#41;&#44; a rapid decrease of the <span class="elsevierStyleItalic">J</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">sc</span></span> &#40;mA&#47;cm<span class="elsevierStyleSup">2</span>&#41; is observed for absorber layer thicknesses lower than 3<span class="elsevierStyleHsp" style=""></span>&#956;m&#44; in accordance with the behavior reported in the published literature for CIS solar cells &#40;<a class="elsevierStyleCrossRefs" href="#bib0120">Ibdah et al&#46;&#44; 2014&#59; Matin-Bhuiyan&#44; Shafkat-Islam&#44; &#38; Jyoti-Datta&#44; 2012&#59; Tivanov&#44; Astashenok&#44; Fedotov&#44; &#38; W&#281;gierek&#44; 2012</a>&#41;&#46; However&#44; for a low back recombination velocity &#40;10<span class="elsevierStyleSup">2</span><span class="elsevierStyleHsp" style=""></span>cm&#47;s&#41;&#44; there is an optimum thickness below 1000<span class="elsevierStyleHsp" style=""></span>nm&#46;</p><elsevierMultimedia ident="fig0030"></elsevierMultimedia><p id="par0200" class="elsevierStylePara elsevierViewall"><a class="elsevierStyleCrossRef" href="#fig0035">Figure 7</a> shows the results of internal quantum efficiency IQE&#40;<span class="elsevierStyleItalic">&#955;</span>&#41; and external quantum efficiency EQE&#40;<span class="elsevierStyleItalic">&#955;</span>&#41; as functions of the photon wavelength for a CIS solar cell with absorber layer thickness of 3000<span class="elsevierStyleHsp" style=""></span>nm and high recombination velocity at the back &#40;10<span class="elsevierStyleSup">7</span><span class="elsevierStyleHsp" style=""></span>cm&#47;s&#41;&#46; In <a class="elsevierStyleCrossRef" href="#fig0035">Figure 7</a>&#44; we can see clearly that the difference between IQE and EQE is due to the optical effects associated to the optical transmittance and reflectance for the given solar cell structure &#40;shown in <a class="elsevierStyleCrossRef" href="#fig0025">Fig&#46; 5</a>&#41;&#46; Particularly for wavelengths below 450<span class="elsevierStyleHsp" style=""></span>nm&#44; the EQE is greatly reduced as a consequence of the reduced transmittance at the front layer&#46; The <span class="elsevierStyleItalic">J</span>&#8211;<span class="elsevierStyleItalic">V</span> and <span class="elsevierStyleItalic">P</span>&#8211;<span class="elsevierStyleItalic">V</span> characteristic curves were also calculated&#44; from which <span class="elsevierStyleItalic">J</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">m</span></span>&#44; <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">m</span></span> and <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">oc</span></span>&#44; FF and efficiency &#40;<span class="elsevierStyleItalic">&#951;</span>&#41; were determined&#44; as shown in <a class="elsevierStyleCrossRef" href="#fig0040">Figure 8</a>&#46;</p><elsevierMultimedia ident="fig0035"></elsevierMultimedia><elsevierMultimedia ident="fig0040"></elsevierMultimedia><p id="par0205" class="elsevierStylePara elsevierViewall">Results for solar cell efficiency when varying thickness of the absorber layer are shown in <a class="elsevierStyleCrossRef" href="#fig0045">Figure 9</a>&#46; Notice that when the surface recombination velocity is low at the back&#44; the optimum CIS thickness is small &#40;around 750<span class="elsevierStyleHsp" style=""></span>nm&#41;&#44; which is in agreement to what was explained in Section <a class="elsevierStyleCrossRef" href="#sec0005">1</a>&#46; Then&#44; the material required can be reduced with the respective reduction of cost&#44; and at the same time a high cell efficiency can be expected&#46; Hence&#44; it is convenient to have the reduction of the surface recombination velocity at the back&#44; while achieving high quality absorber thin films without pinholes&#46; These are objectives to be pursued in future CIS or CIGS and CdTe solar cells&#44; as well&#46;</p><elsevierMultimedia ident="fig0045"></elsevierMultimedia></span><span id="sec0025" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">5</span><span class="elsevierStyleSectionTitle" id="sect0035">Efficiency of a CdS&#47;CdTe solar cell</span><p id="par0210" class="elsevierStylePara elsevierViewall">The developed model was also applied to determine the expected efficiency for CdS&#47;CdTe solar cells&#44; like the one illustrated in <a class="elsevierStyleCrossRef" href="#fig0050">Figure 10</a> with the band diagram shown in <a class="elsevierStyleCrossRef" href="#fig0055">Figure 11</a>&#46; The physical parameters used in the simulations are displayed in <a class="elsevierStyleCrossRef" href="#tbl0010">Table 2</a>&#46; In this case&#44; the CdTe absorber layer thickness was varied between 500<span class="elsevierStyleHsp" style=""></span>nm and 4000<span class="elsevierStyleHsp" style=""></span>nm&#46; Thickness for the CdS layer was the same as for the CIS cell &#40;100<span class="elsevierStyleHsp" style=""></span>nm&#41;&#46; In addition&#44; the thickness of the SnO<span class="elsevierStyleInf">2</span> layer was assumed to be 500<span class="elsevierStyleHsp" style=""></span>nm &#40;similar to the ZnO&#58;Al layer thickness in the CIS cell&#41;&#46;</p><elsevierMultimedia ident="fig0050"></elsevierMultimedia><elsevierMultimedia ident="fig0055"></elsevierMultimedia><elsevierMultimedia ident="tbl0010"></elsevierMultimedia><p id="par0215" class="elsevierStylePara elsevierViewall">The calculated reflectance <span class="elsevierStyleItalic">R</span>&#40;<span class="elsevierStyleItalic">&#955;</span>&#41; and transmittance <span class="elsevierStyleItalic">T</span>&#40;<span class="elsevierStyleItalic">&#955;</span>&#41; of the structure depicted in <a class="elsevierStyleCrossRef" href="#fig0050">Figure 10</a> is shown in <a class="elsevierStyleCrossRef" href="#fig0060">Figure 12</a> in the range of 300&#8211;850<span class="elsevierStyleHsp" style=""></span>nm&#46; The range of wavelength is shorter than for the CIS cell since the absorption edge of CdTe is smaller than for CIS films&#46; It is worth mentioning that the average reflectance in this structure is smaller than for the above CIS structure &#40;less than 6&#37;&#41;&#46;</p><elsevierMultimedia ident="fig0060"></elsevierMultimedia><p id="par0220" class="elsevierStylePara elsevierViewall"><a class="elsevierStyleCrossRef" href="#tbl0010">Table 2</a> contains the physical parameters for CdS and CdTe used in the calculation of the solar cell mentioned above&#46; The calculation was also made for two values of back recombination velocity <span class="elsevierStyleItalic">S</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span>&#46; The low recombination velocity value can be achieved by having a specially treated CdTe layer&#46; For example&#44; an etching at the back may cause a tellurium rich surface&#44; achieving a p&#43; region&#46; In addition&#44; <a class="elsevierStyleCrossRef" href="#bib0100">Morales-Acevedo &#40;2014&#41;</a> has suggested the inclusion of an additional back p&#43; layer of another material&#44; such as ZnTe&#44; forming a back heterojunction that blocks electrons flowing to the back contact&#46; This minority carrier blocking should reduce strongly the recombination velocity at the back&#46;</p><p id="par0225" class="elsevierStylePara elsevierViewall">The short-circuit current density results are shown in <a class="elsevierStyleCrossRef" href="#fig0065">Figure 13</a>&#46; It can be observed that its highest value is obtained at low recombination velocities&#59; nevertheless&#44; it does not correspond to the thinnest CdTe layer&#44; but to an intermediate value&#46; This can be explained by acknowledging that in a very thin layer the depletion region &#40;and the associated electrostatic field&#41; can extend over its entirety but it absorbs less light than a thick one&#59; on the other hand&#44; in a very thick layer the incident light is practically absorbed in its totality&#44; but the depletion region extends only over a narrow part of the material and hence the electron-hole pairs photo-generated outside this depletion region have a lower probability to be collected&#44; which in turn is reflected in a decrease of the short-circuit current&#46;</p><elsevierMultimedia ident="fig0065"></elsevierMultimedia><p id="par0230" class="elsevierStylePara elsevierViewall"><a class="elsevierStyleCrossRef" href="#fig0070">Figure 14</a> shows the external and internal quantum efficiency for the CdS&#47;CdTe solar cell with a CdTe thickness of 1<span class="elsevierStyleHsp" style=""></span>&#956;m and back recombination velocity of 10<span class="elsevierStyleSup">2</span><span class="elsevierStyleHsp" style=""></span>cm&#47;s&#44; which corresponds to the cell with the best performance as will be seen in <a class="elsevierStyleCrossRef" href="#fig0080">Figure 16</a>&#46;</p><elsevierMultimedia ident="fig0070"></elsevierMultimedia><p id="par0235" class="elsevierStylePara elsevierViewall">In <a class="elsevierStyleCrossRef" href="#fig0075">Figure 15</a>&#44; the expected <span class="elsevierStyleItalic">J</span>&#8211;<span class="elsevierStyleItalic">V</span> and <span class="elsevierStyleItalic">P</span>&#8211;<span class="elsevierStyleItalic">V</span> curves &#40;current density and power as functions of voltage&#41; of the solar cell with a CdTe thickness of 1<span class="elsevierStyleHsp" style=""></span>&#956;m and back recombination velocity of 10<span class="elsevierStyleSup">2</span><span class="elsevierStyleHsp" style=""></span>cm&#47;s are presented&#46; <span class="elsevierStyleItalic">J</span>&#8211;<span class="elsevierStyleItalic">V</span> and <span class="elsevierStyleItalic">P</span>&#8211;<span class="elsevierStyleItalic">V</span> curves of solar cells with different thicknesses were also calculated&#44; nevertheless this was the solar cell with the highest efficiency&#46;</p><elsevierMultimedia ident="fig0075"></elsevierMultimedia><p id="par0240" class="elsevierStylePara elsevierViewall"><a class="elsevierStyleCrossRef" href="#fig0080">Figure 16</a>&#40;a&#41;&#8211;&#40;c&#41; shows the output parameters of the solar cell as functions of the CdTe thickness&#46; From a quick glance at these graphs&#44; the <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">oc</span></span> and FF have their highest values at 0&#46;5<span class="elsevierStyleHsp" style=""></span>&#956;m and they decrease as the CdTe layer becomes thicker for <span class="elsevierStyleItalic">S</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>10<span class="elsevierStyleSup">2</span><span class="elsevierStyleHsp" style=""></span>cm&#47;s&#59; this behavior has been explained in Section <a class="elsevierStyleCrossRef" href="#sec0005">1</a> and seems to be in good agreement with other works &#40;<a class="elsevierStyleCrossRef" href="#bib0100">Morales-Acevedo&#44; 2014</a>&#41; and it is due to the reduction of the surface recombination velocity at the back contact&#46; The highest value obtained for <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">oc</span></span> is 947<span class="elsevierStyleHsp" style=""></span>mV and for the fill factor&#44; 0&#46;847 when the CdTe thickness is around 500<span class="elsevierStyleHsp" style=""></span>nm&#46; Regarding the efficiency&#44; the highest value obtained is 18&#46;66&#37; and it is once again achieved with a low back recombination velocity &#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>10<span class="elsevierStyleSup">2</span><span class="elsevierStyleHsp" style=""></span>cm&#47;s&#41; and with a CdTe thickness of 1<span class="elsevierStyleHsp" style=""></span>&#956;m&#46;</p><elsevierMultimedia ident="fig0080"></elsevierMultimedia><p id="par0245" class="elsevierStylePara elsevierViewall">As a general observation&#44; the best results are achieved with a low back surface recombination velocity and small thickness&#46; It is also important to notice that for thick CdTe layers&#44; the recombination velocity does not have a significant effect on the outcome of the cell parameters&#46;</p><p id="par0250" class="elsevierStylePara elsevierViewall">Notice that reported record efficiencies for CdTe solar cells are above those calculated here&#44; but the results shown here correspond to the expected efficiencies when the transport parameters are those given in <a class="elsevierStyleCrossRef" href="#tbl0010">Table 2</a>&#46; These parameters are typical for most of the reported cells in different laboratories around the world&#46; Record efficiencies above 20&#37; might correspond to an absorber layer which is not pure CdTe&#44; as can be observed from quantum efficiency measurements made on record cells &#40;<a class="elsevierStyleCrossRef" href="#bib0050">Green et al&#46;&#44; 2015</a>&#41;&#44; and possibly the material has been prepared under conditions that have allowed the parameters such as electron diffusion length or majority carrier concentration &#40;Na&#41; above those assumed here&#46; Unfortunately&#44; this fact cannot be confirmed because these measurements have not been reported for the record cells and nor the exact composition of the absorber layer &#40;some alloy&#41;&#44; so that the calculations should not be compared directly with experimental results unless all the required parameters are already known&#46;</p><p id="par0255" class="elsevierStylePara elsevierViewall">In summary&#44; the model here pretends to be of help when designing thin film cells when the parameters given in <a class="elsevierStyleCrossRefs" href="#tbl0005">Table 1 or 2</a> are well known&#44; for the respective absorber material&#46; In addition&#44; in accordance to the discussion given in Section <a class="elsevierStyleCrossRef" href="#sec0005">1</a>&#44; this model shows that thinner cells are better than thicker ones&#44; when the surface recombination at the back is low &#40;10<span class="elsevierStyleSup">2</span><span class="elsevierStyleHsp" style=""></span>cm&#47;s&#41; compared to the high values &#40;10<span class="elsevierStyleSup">7</span><span class="elsevierStyleHsp" style=""></span>cm&#47;s&#41; obtained at ohmic contacts&#46; The model can be applied for the design of other cells such as those based on CIGS&#44; CZTS and even Perovskite materials when the respective parameter values are used&#46;</p></span><span id="sec0030" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">6</span><span class="elsevierStyleSectionTitle" id="sect0040">Conclusion</span><p id="par0260" class="elsevierStylePara elsevierViewall">We have presented a unified simple model for determining the expected <span class="elsevierStyleItalic">J</span>&#8211;<span class="elsevierStyleItalic">V</span> curves for poly-crystalline thin film solar cells&#46; The model takes into consideration the voltage dependence of the illumination current density which can be limited either by diffusion in the neutral region or by electric field-drift at the space charge region&#46; Similarly&#44; it also takes into consideration that recombination under dark conditions depends on the applied voltage&#46; For very thin cells&#44; these current components might be determined by a space charge region extending for the whole absorber length&#46; In addition&#44; the optical properties of the layers were considered for calculating the reflectance and transmittance for a given cell structure&#46; The model was applied to CIS and CdTe solar cells&#46; For both cases&#44; it was shown that very thin cells can be highly efficient when the recombination at the back is low &#40;less than 10<span class="elsevierStyleSup">2</span><span class="elsevierStyleHsp" style=""></span>cm&#47;s&#41;&#46; This also explains the high efficiencies achieved by very thin perovskite solar cells&#44; and it poses a challenge to develop deposition methods for very thin CdTe and CIGS absorber materials without pinholes&#44; so that improved efficiencies are obtained when the surface recombination velocity is made small at the back&#44; by having a p&#43; or an electron blocking region in contact with the absorber material before the ohmic contact&#46;</p></span><span id="sec0120" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0045">Conflict of interest</span><p id="par0510" class="elsevierStylePara elsevierViewall">The authors have no conflicts of interest to declare&#46;</p></span></span>"
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          "titulo" => "Introduction"
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          "identificador" => "sec0010"
          "titulo" => "I&#8211;V modeling"
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          "titulo" => "Application of the model to CIS solar cells"
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          "titulo" => "Efficiency of a CdS&#47;CdTe solar cell"
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          "identificador" => "sec0030"
          "titulo" => "Conclusion"
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        8 => array:2 [
          "identificador" => "sec0120"
          "titulo" => "Conflict of interest"
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        9 => array:2 [
          "identificador" => "xack328689"
          "titulo" => "Acknowledgements"
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          "titulo" => "References"
        ]
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    "fechaRecibido" => "2017-01-28"
    "fechaAceptado" => "2017-08-03"
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        "resumen" => "<span id="abst0005" class="elsevierStyleSection elsevierViewall"><p id="spar0005" class="elsevierStyleSimplePara elsevierViewall">Polycrystalline thin film solar cells made with absorber materials such as CdTe&#44; CIGS&#44; CZTS and metalorganic halides &#40;perovskites&#41; are currently important alternatives for the silicon solar cell technology&#44; which still dominates the photovoltaic market&#46; Then&#44; it is important to have tools which can be used to design this kind of solar cells&#46; For this purpose&#44; we have developed a unified simple analytical model that can be applied to thin film solar cells&#46; The model is based on the basic physics of hetero-junction devices&#44; but it takes into consideration that the space charge region can extend along the major part of the cell length&#44; particularly for very thin cells&#44; causing important effects that typically are not observed in conventional junction devices&#46; Photo-generated carriers are collected by electric field-drift instead of diffusion&#44; and simultaneously strong recombination at this region may dominate the electrical <span class="elsevierStyleItalic">I</span>&#8211;<span class="elsevierStyleItalic">V</span> characteristic of the cell&#46; Since the space-charge region width varies with the applied voltage&#44; the illumination current density and the saturation dark current density are no longer independent of the voltage as is assumed for conventional solar cells&#46; When the model is applied to CIS and CdTe solar cells as examples&#44; it is found that it is possible to design very thin film solar cells &#40;absorber less than 1<span class="elsevierStyleHsp" style=""></span>&#956;m thick&#41; with high efficiencies&#44; whenever the recombination velocity at the back surface becomes small &#40;10<span class="elsevierStyleSup">2</span><span class="elsevierStyleHsp" style=""></span>cm&#47;s&#41;&#44; instead of the high recombination velocities present at ohmic contacts &#40;10<span class="elsevierStyleSup">7</span><span class="elsevierStyleHsp" style=""></span>cm&#47;s&#41;&#46; This fact implies the cost reduction of thin film solar cells by reducing absorber material thickness&#44; and therefore it poses a challenge to develop deposition methods for very thin CdTe and CIGS absorber materials without pinholes&#44; so that improved efficiencies are obtained when the surface recombination velocity is made small at the back by having a p&#43; or an electron blocking region before the ohmic contact&#46; This result also explains the high efficiencies achieved by very thin perovskite solar cells&#46;</p></span>"
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                  \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top"><span class="elsevierStyleItalic">L</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span> &#40;cm&#41;&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">4&#46;6<span class="elsevierStyleHsp" style=""></span>&#215;<span class="elsevierStyleHsp" style=""></span>10<span class="elsevierStyleSup">&#8722;5</span>&nbsp;\t\t\t\t\t\t\n
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                  \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top"><span class="elsevierStyleItalic">N</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">a</span></span> &#40;cm<span class="elsevierStyleSup">&#8722;3</span>&#41;&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">2<span class="elsevierStyleHsp" style=""></span>&#215;<span class="elsevierStyleHsp" style=""></span>10<span class="elsevierStyleSup">16</span>&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">&#8211;&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top"><span class="elsevierStyleItalic">¿</span>&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">13&#46;6&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">10&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top"><span class="elsevierStyleItalic">X</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">e</span></span> &#40;eV&#41;&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">4&#46;3&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">4&#46;5&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top"><span class="elsevierStyleItalic">N</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">c</span></span> &#40;cm<span class="elsevierStyleSup">&#8722;3</span>&#41;&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">2&#46;2<span class="elsevierStyleHsp" style=""></span>&#215;<span class="elsevierStyleHsp" style=""></span>10<span class="elsevierStyleSup">18</span>&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">2&#46;2<span class="elsevierStyleHsp" style=""></span>&#215;<span class="elsevierStyleHsp" style=""></span>10<span class="elsevierStyleSup">18</span>&nbsp;\t\t\t\t\t\t\n
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                  \t\t\t\t</td></tr><tr title="table-row"><td class="td-with-role" title="table-entry ; entry_with_role_rowhead " align="left" valign="top"><span class="elsevierStyleItalic">W</span> &#40;cm&#41;&nbsp;\t\t\t\t\t\t\n
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Article information

ISSN: 16656423
Original language: English

DOI:10.1016/j.jart.2017.08.002

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2024 April	117	16	133
2024 March	141	12	153
2024 February	160	27	187
2024 January	181	19	200
2023 December	140	13	153
2023 November	198	16	214
2023 October	180	32	212
2023 September	151	20	171
2023 August	130	13	143
2023 July	131	22	153
2023 June	119	24	143
2023 May	185	18	203
2023 April	108	24	132
2023 March	171	35	206
2023 February	110	24	134
2023 January	135	15	150
2022 December	189	31	220
2022 November	134	27	161
2022 October	124	44	168
2022 September	138	33	171
2022 August	136	26	162
2022 July	108	35	143
2022 June	111	24	135
2022 May	120	28	148
2022 April	94	31	125
2022 March	132	27	159
2022 February	121	38	159
2022 January	246	39	285
2021 December	118	21	139
2021 November	127	41	168
2021 October	114	44	158
2021 September	120	30	150
2021 August	165	18	183
2021 July	109	20	129
2021 June	106	25	131
2021 May	101	18	119
2021 April	253	74	327
2021 March	153	27	180
2021 February	99	33	132
2021 January	137	28	165
2020 December	114	21	135
2020 November	109	24	133
2020 October	82	19	101
2020 September	92	25	117
2020 August	91	17	108
2020 July	67	22	89
2020 June	74	16	90
2020 May	70	17	87
2020 April	91	8	99
2020 March	73	31	104
2020 February	88	18	106
2020 January	79	28	107
2019 December	94	15	109
2019 November	64	22	86
2019 October	83	15	98
2019 September	68	10	78
2019 August	61	10	71
2019 July	72	19	91
2019 June	77	34	111
2019 May	143	61	204
2019 April	110	35	145
2019 March	48	25	73
2019 February	73	11	84
2019 January	45	15	60
2018 December	55	9	64
2018 November	62	16	78
2018 October	97	7	104
2018 September	80	14	94
2018 August	35	12	47
2018 July	29	8	37
2018 June	30	6	36
2018 May	31	15	46
2018 April	38	9	47
2018 March	38	7	45
2018 February	35	4	39
2018 January	23	11	34
2017 December	0	1	1

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