Issue 
EPJ Photovolt.
Volume 2, 2011



Article Number  20101  
Number of page(s)  12  
Section  Modelling  
DOI  https://doi.org/10.1051/epjpv/2011025  
Published online  17 November 2011 
https://doi.org/10.1051/epjpv/2011025
Factors limiting the opencircuit voltage in microcrystalline silicon solar cells
^{1}
Energy Research Unit, Indian Association for the Cultivation of
Science, 700 032
Kolkata,
India
^{2}
Laboratoire de Physique des Interfaces et des Couches Minces,
École Polytechnique, CNRS, 91128
Palaiseau,
France
^{a}
email: madhumitanath21@yahoo.co.in
Received: 18 March 2011
Accepted: 22 September 2011
Published online:
17
November
2011
In studying photovoltaic devices made with silicon thin films and considering them according to their grain size, it is curious that as the crystalline fraction increases, the opencircuit voltage (V_{oc}) – rather than approaching that of the singlecrystal case – shows a decline. To gain an insight into this behavior, observed in hydrogenated microcrystalline silicon (μcSi:H) solar cells prepared under a variety of deposition conditions, we have used a detailed electricaloptical computer modeling program, ASDMP. Two typical μcSi:H cells with low (~79%) and higher (~93%) crystalline volume fractions (F_{c}), deposited in our laboratory and showing this general trend, were modeled. From the parameters extracted by simulation of their experimental current density – voltage and quantum efficiency characteristics, it was inferred that the higher F_{c} cell has both a higher band gap defect density as well as a lower band gap energy. Our calculations reveal that the proximity of the quasiFermi levels to the energy bands in cells based on highly crystallized μcSi:H (assumed to have a lower band gap), results in both higher free and trapped carrier densities. The trapped hole population, that is particularly high near the P/I interface, results in a strong interface field, a collapse of the field in the volume, and hence a lower opencircuit voltage. Interestingly enough, we were able to fabricate fluorinated μcSi:H:F cells having 100% crystalline fraction as well as very large grains, that violate the general trend and show a higher V_{oc}. Modeling indicates that this is possible for the latter case, as also for a crystalline silicon PN cell, in spite of a sharply reduced band gap, because the lower effective density of states at the band edges and a sharply reduced gap defect density overcome the effect of the lower band gap.
© EDP Sciences 2011
1 Introduction
When considering the full spectrum of thin film Sibased photovoltaic technologies and ranking them according to their crystalline volume fraction (F_{c}), it is remarked that the opencircuit voltage (V_{oc}) decreases with increasing F_{c}. This has been observed by various groups working with cells fabricated from hydrogenated microcrystalline silicon (μcSi:H) films deposited by a variety of deposition techniques [1, 2, 3, 4] (Fig. 1). This fact hinders the further development of thinfilm silicon photovoltaics as the loss in V_{oc} offsets any gain in the shortcircuit current density (J_{sc}), prompting Mai et al. [1, 2] to remark that the optimum μcSi:H solar cells are always obtained with intermediate crystallinity. The physical reason behind this loss in V_{oc} is not immediately clear, as the maximum impact of the most obvious factor (the smaller band gap) is diminished by the example of singlecrystal silicon (cSi), which possesses the smallest band gap but a higher V_{oc} than cells with smaller grains (Fig. 1). Since the observation is a decrease of V_{oc} with increasing F_{c} of the μcSi:H absorber from 60% to over 90%, while a monocrystalline silicon PN cell has an appreciably higher V_{oc}, it is expected that a missing link maybe found – a limiting case of a very wellcrystallized and large grained μcSi:H material – which when used as the absorber layer in a solar cell, should yield an opencircuit voltage intermediate between that of a cell based on a highlycrystallized μcSi:H absorber and the considerably higher V_{oc} of a monocSi PN cell. The parameters characterizing such a cell, as deduced from modeling, should then help us understand how the opencircuit voltage of μcSi:H solar cells can be made to approach the higher V_{oc} that characterizes single crystal silicon cells.
As a matter of fact we have found that solar cells based on fluorinated μcSi:H [5] (μcSi:H:F), with a very high F_{c} as well as a significant fraction of large grains (F_{lg}), appear to violate the general rule (Fig. 1 – in fact this figure shows, besides the typical V_{oc} of classical diffused junction cSi solar cells [6], also the V_{oc}’s of “heterojunction with intrinsic thin layer (HIT)” cells on Ptype [7] and Ntype [8] cSi substrates, for completeness). A material containing large grains is a dense material with low oxygen content, which can be obtained when using SiF_{4} based plasma processes [5]. We then chose two of a series of typical (that follow the general rule of decreasing V_{oc} with increasing F_{c} – Fig. 1) μcSi:H cells, as well as a cell based on μcSi:H:F, for modeling using the detailed electricaloptical computer code ASDMP [9, 10]. Using parameters extracted by simulating the output experimental characteristics of the μcSi:H thin film solar cells with absorber layers of different degrees of crystallinity, we identify in this article, the parameters responsible for the general decline of V_{oc} in such cells with increasing crystalline volume fraction (Fig. 1). Moreover by simulating the output characteristics of the μcSi:H:F solar cells, we could identify the critical parameters which prevent μcSi:H solar cells in achieving the higher V_{oc} attainable in cSi PN cells.
Fig. 1 Results taken from various literature sources [1, 2, 3, 4] indicate the general trend of a decrease in the opencircuit voltage of μcSi:H thin film PIN cells, with increase of the Raman crystallinity of the films. We also show an exception where the V_{oc} actually increases [5] in a highly crystallized μcSi:H:F cell, specially after interface treatment (indicated by the arrow). Also shown in the figure are typical V_{oc}’s of classical diffused junction cSi solar cells (open circles – 5, Green’s ref), of a “heterojunction with intrinsic thin layer (HIT)” cell on a Ptype cSi substrate [7] and of a HIT cell on Ntype cSi [8] (closed circles). 
Comparison between the measured and simulated solar cell output parameters of the PIN devices having low and intermediate crystalline volume fraction (F_{c})μcSi:H as the intrinsic layer. F_{lg} gives the fraction of large grains in the Ilayer. Also compared are the measured and modeled output parameters of the highly crystallized, large grained μcSi:H:F cell.
2 Experimental details
Microcrystalline PIN solar cells having the structure textured ZnO/PμcSi:H/IμcSi:H/NaSi:H/Aluminum have been deposited in a multiplasma monochamber radiofrequency plasmaenhanced chemical vapor deposition (RFPECVD) reactor [11]. Two sets of μcSi:H solar cells have been deposited with ~60 mW/cm^{2} of RFpower at 175 °C. We employed different ratios of silane to hydrogen flow rates during the intrinsic layer deposition for the two sets – SiH_{4}:H_{2} = 6:200 (cell A) and SiH_{4}:H_{2} = 4:200 (cell B) [4], which results in different total crystalline volume fraction (F_{c}) and large grain fraction (F_{lg}) in the samples (Tab. 1). A third set has a fluorinated μcSi:H intrinsic (I)layer (Tab. 1), where the Ilayer was deposited from a SiF_{4}, H_{2} and Ar gas mixture [5]. The respective flow rates of the above gases were 3, 3 and 70 sccm respectively. The deposition pressure was 2.7 torr, the RF power 440 mW cm^{2} and the substrate temperature 200 °C. The μcSi:H Player and the aSi:H Nlayer were deposited in a second multichamber reactor, using TMB and PH_{3} as dopant gases, and thus necessitating a vacuum break between the doped and intrinsic layers. A hydrogen plasma treatment (after both air breaks) was applied towards passivation of the P/I and I/N defects. The P and Nlayers were deposited using similar conditions for all cells. It maybe pointed out that a series of cells having similar output characteristics were deposited during each run and that the values cited in Table 1 are representative of each series.
The composition of the films was obtained from the Bruggeman effective medium approximation modeling [12] of the pseudodielectric function of the films, deduced from spectroscopic ellipsometry measurements. This approach has been shown to be well adapted to the modeling of μcSi films [13]. In this approach we used as components for the model the dielectric functions of (i) amorphous silicon, (ii) voids to take into account the porosity of the films; (iii) small grain μcSi and (iv) large grain μcSi material produced by Chemical vapor deposition (CVD) at ~650 °C [14]. In Table 1 we report the values of the total crystalline fraction F_{c} (which is the sum of the small grain and large grain fractions) and the large grain fraction F_{lg}, for the two sets of μcSi:H samples and those of μcSi:H:F. Indeed, achieving a high value of F_{lg} in the case of μcSi:H:F, indicates that the films obtained by PECVD at 175 °C have a grain size similar to that of polycrystalline layers produced by CVD at ~650 °C [14]. The greatly improved spectral response at long wavelengths also confirms the very high crystalline volume fraction of μcSi:H:F [5].
3 Simulation model
The onedimensional electricaloptical model ASDMP [9, 10] (amorphous semiconductor device modeling program) used in this study solves the Poisson’s equation and the two carrier continuity equations under steady state conditions for the given device structure, and yields the resulting JV characteristics and the quantum efficiency. The electrical part of the modeling program is described in references [15, 16]. The expressions for the free and trapped charges, the recombination term, the boundary conditions and the solution technique in this program are similar to the AMPS computer program developed by McElheny et al. [17].
The gap state model used in these calculations consists of the tail states, as well as a donorlike and an acceptorlike set of Gaussian distribution functions to simulate the deep dangling bond states. The contact barrier heights for a cell with the Player in contact with the ZnO at x = 0 and the Nlayer in contact with aluminum at x = L, are taken to be 1.11 eV and 0.2 eV, respectively. Since the activation energies of our PμcSi:H and NaSi:H layers are 0.09 eV (PμcSi:H band gap is 1.2 eV) and 0.2 eV respectively, this implies that no effective surface band bending has been assumed at the ZnO/PμcSi:H contact and the NaSi:H/Al contact is ohmic. The assumption of no band bending at the ZnO/PμcSi:H interface originates from the fact that we had to assume an extremely high Player doping density (3 × 10^{19} cm^{3}), therefore also a very high Player defect density, to simulate these cells. This means that when the TCO and the Player are put in contact, the trapped electrons on the PμcSi:H side are confined to a very thin layer on the Pside, resulting in a very high surface band bending that however does not appreciably extend into the bulk of the Player. For the purpose of calculating the builtin potential (V_{bi}) and V_{oc}, the bulk activation energy of the Player is then already achieved almost at the TCO/P interface, and the bandbending does not extend to any appreciable thickness of the Player.
The generation term in the continuity equations has been calculated using a semiempirical model [18], that has been integrated into the modeling program [9, 10]. Both specular interference effects and diffused reflectances and transmittances due to interface roughness are taken into account. It is now wellrecognized that TCO texture is a key issue in increasing cell efficiency, as it reduces optical reflection loss and greatly increases light diffusion. Weakly absorbed radiation, when scattered, can be partly absorbed in a high refractive index layer, such as amorphous or microcrystalline Si (n ≈ 3.7), due to total internal reflection at the interfaces giving rise to optical confinement. However because of the broad distribution of grain size and shape, the interaction between the multilayer device and incident light is very complex and so, rather than a rigorous threedimensional electromagnetic treatment of the diffused radiation, a rather sophisticated semiempirical model [18], was integrated [9, 10] with the electrical model [15, 16]. Here diffused reflectances and transmittances, are derived from angular resolved photometric measurements, and used as input parameters. In the model, the electromagnetic field’s specular reflection and transmission are assumed proportional to the Fresnel coefficients, the proportionality factor depending on the amount of total diffused light. In the specular part light coherence is kept, resulting in interference effects when the TCO is more or less flat. However in the diffused part, light coherence with the incident light is assumed lost, so, the point where light is diffused, is considered as a new source emitting in several directions in the stack. The latter light in each direction is assumed to be a plane wave, and each wave, when it meets the next rough interface, is again divided into specular and diffused components. Instead of calculating and successively adding each of these components, the total electromagnetic field is directly derived by the matrix method of Abeles [19, 20, 21]. In the model it is possible to consider up to two rough interfaces. These are taken to be the TCO/P and N/metal interfaces in the present case. The complex refractive indices for each layer of the structure are also required as input, and have been measured inhouse by spectroscopic ellipsometry. These are presented in Figure 2.
Fig. 2 Values of the complex refractive indices (a) real part, n and (b) imaginary part, κ for low (79%), intermediate (93%) crystalline volume fraction (F_{c})μcSi:H, and for μcSi:H:F, F_{c} ~ 100%, compared to the respective values of aSi:H (amorph) and cSi (crys). 
4 Experimental results and analysis
Since the aim of this article is to understand the general trend in μcSi:H solar cells, which is that the opencircuit voltage (and the fill factor (FF) to a certain extent) decreases with increasing crystalline volume fraction, as also to understand why the most ordered cSi PN cell has a higher V_{oc} (Fig. 1), we need to model a variety of output characteristics of μcSi:H cells in order to extract parameters that characterize a representative cell of each series. The first 2 cells of Table 1 follow the general trend, as is obvious from the appreciably lower V_{oc} of the cell with the intermediate crystalline volume fraction. However the fluorinated μcSi:H cell with a very high value of the crystalline volume fraction (F_{c}) violates this general trend (Tab. 1) and exhibits both higher V_{oc} and J_{sc}. It may be noted that this cell has a particularly high large grain fraction, which may have a bearing to its exceptional behavior. The interest in modeling the latter cell is to gain an insight into the possible reasons why the most ordered monocSi PN cell has a higher V_{oc} than μcSi:H cells.
Fig. 3 Calculated dark JV characteristics of the intermediate F_{c} and low F_{c}μcSi:H cells at 30 °C, compared to experimental results. The lines are our modeling results, while symbols represent experimental measurements. 
The experimental solar cell output parameters for the “low F_{c}” cell (device A, F_{c} ~ 79%, no large grain fraction detected), the “intermediate F_{c}” cell (device B, F_{c} ~ 93%, with large grain fraction ~27%), and the fluorinated μcSi:H cell (F_{c} ~ 100%, F_{lg} ~ 50%) are compared to the modeling results in Table 1. The dark current density vs. voltage (JV) characteristics of the first two types of cells at 30 °C have been both measured and simulated, and the good agreement between these results can be noted from Figure 3. Brammer and Stiebig [22, 23] have also observed that the dark forward current at low forward voltages is a strong function of the silane concentration (SC) and is lower for higher values of SC, in other words, lower for the lower F_{c} cell, A. This comparison has been done for temperatures from 10 to 50 °C, and the modeled and experimental dark saturation current density (J_{0}) and diode ideality factor (n) of these cells are presented in Figures 4a and 4b. The external quantum efficiency (EQE) curves of all three types of solar cells are compared to their experimental counterparts in Figures 5a–5c. The match with experiments appears to be satisfactory.
Fig. 4 Calculated values (open symbols) of (a) the reverse saturation current density (J_{0}) and (b) the diode ideality factor (n) for low and intermediate F_{c}μcSi:H solar cells, compared to experiments (closed symbols) at temperatures from 10 °C to 50 °C. The lines are guides to the eye. 
Fig. 5 Calculated external quantum efficiency (EQE) curves under AM1.5 bias light and shortcircuit conditions for the (a) low F_{c}μcSi:H cell A, (b) intermediate F_{c}μcSi:H cell B and (c) high F_{c}, large grained μcSi:H:F cell, compared to experimental results. 
All experimental results of a particular type of cell – μcSi:H cell A, μcSi:H cell B and of the fluorinated μcSi:H cell – have been simulated with the same set of input parameters, which are given in Table 2. It may be mentioned that the electron and hole mobilities shown in Table 2 and used as input to the modeling program, are the band microscopic mobilities. The drift mobilities measured in actual experiments are the band microscopic mobilities reduced by trapping and detrapping of carriers, and are therefore one to two orders of magnitude lower. Moreover, we have assumed in general that the carrier mobilities are higher as the material becomes more crystallized. To accurately model all aspects of the experimentally measured solar cell performance, we had to assume that the more crystallized μcSi:H cell B (with however a large grain fraction that is considerably lower than in the highly crystallized fluorinated μcSi:H cell) has a lower band gap, higher carrier mobilities, a higher midgap defect density, and broader band tails (Tab. 2) relative to the μcSi:H cell A. We further infer from modeling that the μcSi:H:F cell (F_{c} ~ 100%) has an even lower band gap and higher carrier mobilities. This dense material with low oxygen content and very high large grain fraction [5], should have fewer grain boundaries and modeling indicates that it has a sharply reduced dangling bond defect density and effective density of states at the band edges, the latter similar to cSi. (Tab. 2). Only its valence and conduction band tails appear to be fairly broad (the characteristic energies are 40 m eV and 20 m eV respectively for the valence and conduction band tails, as in the case of the intermediate F_{c}μcSi:H cell B). This implies that this highly crystallized material nevertheless has a strained lattice. Justification of the parameters inferred from modeling that characterize the different types of cells (Tab. 2) will be presented in Section 6.
Parameters that characterize intrinsic μcSi:H of different degree of crystallinity (as extracted by modeling). The quantities in brackets marked with asterisks in the column of parameters of μcSi:H:F correspond to the values extracted by modeling similar μcSi:H samples (refs TSF, JAP of Rubinelli).
However, it maybe relevant at this point to compare the parameters extracted by the present modeling (Tab. 2) to some other modeling results in the literature. Strengers et al. [24] and Sturiale et al. [25] have modeled the dark and illuminated JV characteristics of μcSisolar cells, where the intrinsic layer is deposited by the hot wire CVD (HWCVD) technique. They have observed [24] that in these samples the optical absorption in the red region is much higher than in amorphous silicon. Since the optical absorption is related to the imaginary part of the complex refractive index (κ) and our Figure 2b indicates that this is the case for our μcSi:H:F samples, it is reasonable to compare the parameters used in their modeling of HWCVD deposited μcSi cells to those of our highest F_{c}μcSi:H:F cell. The comparison is shown in Table 2 in brackets with asterisks. We note that the values of references [24, 25] are quite close to what we have assumed for our μcSi:H:F Ilayer. In particular the effective DOS in the valence and conduction bands; as well as the tail prefactors G_{D0}, G_{A0} are similar to ours and an order of magnitude lower than those normally assumed for hydrogenated amorphous silicon and also assumed for the low (A) and intermediate (B) F_{c}μcSi:H Ilayers here, for reasons to be justified in Section 6. The mid gap defect density is ~10^{15} cm^{3}, also like our value for μcSi:H:F and the band gap of 1.25 eV is close to ours, and higher than that of cSi (1.12 eV). Only the capture crosssections of the dangling bond states are more than two orders of magnitude higher than our case. Probably this had to be assumed for the HWCVD μcSi Ilayers [24, 25] since the current density from these devices (13−16 mA cm^{2} − 22) is much lower than ours (23.20 mA cm^{2}) for comparable values of the absorption coefficients. On the other hand, our average mid gap defect density is the same and their capture crosssections in the highest F_{c}μcSi:H Ilayer are much closer to the values extracted from modeling the dark JV characteristics of PECVD μcSi I layers in reference [22].
We had also attempted to model the experimental characteristics of the “intermediate F_{c}” cell B, without decreasing the mobility gap, and thus by increasing the gap state defect density alone. This is because, Yan et al. [26] have mentioned without employing detailed modeling to support their statement, that this decrease of V_{oc} is due to distorted bonds in the grain boundary regions that lead to increased band gap defects. Modeling however reveals that a Gaussian defect density of some ~6 × 10^{17} cm^{3} must be assumed to match the low V_{oc} in this case, which also causes a sharp fall in the shortcircuit current density (J_{sc}) and fill factor (FF), resulting in our determination that all aspects of the solar cell output characteristics of cell B cannot be matched by increasing the band gap defect density alone.
Modeling of the EQE curves in Figure 5a and 5b suggests that the Player in the intermediate F_{c} cell B is thinner than that in the low F_{c} cell, although the Players in all types of cell were deposited under the same experimental conditions. This means that some etching of this layer occurred during the subsequent Ilayer deposition for the intermediate F_{c} case. This is not surprising, since to obtain the more crystallized μcSi:H layer a higher hydrogen dilution was employed. Also, the P/I interface layer is both thinner and demonstrates lower capture crosssections than in the case of the low F_{c}μcSi:H cell. These factors combine to yield the very high blue response in the case of the intermediate F_{c}μcSi:H solar cell (Fig. 5b). An interface layer is expected when the Ilayer is deposited on top of the Player, as occurs in a PIN device deposition process. Hence, physically one might expect a thinner interfacial layer in the intermediate F_{c} Ilayer case, since the band gap mismatch between its Ilayer (E_{g} = 1.33 eV) and the highly crystallized Player (E_{g} = 1.2 eV, deposition parameters same for both cells) is smaller than for the low F_{c}μcSi:H solar cell (Ilayer band gap 1.4 eV).
5 Discussion
As already stated in the previous section the intermediate F_{c}μcSi:H cell B has higher carrier mobilities, a higher dangling bond defect density, broader band tails and lower band gap than μcSi:H cell type A, having the lowest crystalline volume fraction (Tabs. 1, 2). It is a combination of these factors that results in cell B having a higher current density but lower V_{oc}, FF and conversion efficiency relative to cell A. In the following we will study how the higher gap defect density and lower band gap affect cell performance. In studying the sensitivity of μcSi:H cell performance to each of the abovementioned parameters, all other parameters are held constant at the values of the μcSi:H cell, type A having the lowest F_{c}, which we may call our reference case.
5.1 Effect of changes in the Ilayer band gap defects on the photovoltaic response
5.1.1 Sensitivity to the characteristic energy of the band tails
The μcSi:H cell A (Tab. 2) presents values for the characteristic energy of the valence and conduction band tails of E_{D} = 20 m eV and E_{A} = 10 m eV, respectively. To model the intermediate F_{c} cell B, one of the changes that we had to assume was broader band tails, as characterized by E_{D} = 40 m eV and E_{A} = 20 m eV. Figure 6a depicts the illuminated JV characteristics for two cases differing only in the characteristic energy of the band tails and indicating a fall of V_{oc} by ~0.03 V when the band tails broaden. Figure 6b plots the electric field in the two devices, with the high field at the P/I interface shown in a different scale in the inset. We find that higher photogenerated hole trapping in the broader valence band tail near the P/I interface, where the quasiFermi level lies close to the valence band, results in a stronger electric field near this interface, and a consequent fall of the field in the volume of the absorber layer. This fact is known [15], to bring down the opencircuit voltage.
Fig. 6 (a) Sensitivity of the illuminated JV characteristic to the valence and conduction band tail characteristic energies and (b) the electric field in the device when the band tails are sharp (characteristics energies of band tails E_{D},E_{A} = 20 me V, 10 m eV respectively, characteristic of μcSi:H cell type (A) and when they are broader (E_{D},E_{A} = 40 me V, 20 m eV respectively, characteristic of cell type (B). The electric field over the P/I interface region is plotted on a different scale in the inset. 
5.1.2 Sensitivity to the Gaussian defect density
Again, using as reference the case of the lowest F_{c}μcSi:H cell type A with parameters as given in Table 2, we have studied the impact of an increase in only the Gaussian defect density starting from a value of 4 × 10^{16} cm^{3}, characteristic of cell A, to 8 × 10^{16} cm^{3}, inferred from modeling of the μcSi:H cell B. This leads to a decrease in V_{oc} from 0.54 V down to 0.52 V and an increase in the dark saturation current J_{0} from 1.15 × 10^{5} mA/cm^{2} up to 8.35 × 10^{5} mA/cm^{2}. We again find that a higher field near the P/I interface due to higher photogenerated hole trapping for the case having higher mid gap DOS is responsible for the collapse in the bulk electric field and the fall in V_{oc}.
We may confirm our inferences above (obtained by detailed modeling using ASDMP) regarding a lower V_{oc} in the cell having a higher defect density by considering the approximate analytical formula which ignores the shunt and series resistances: (1)Here n is the diode ideality factor, J_{0} the dark saturation current density, q the electronic charge and T the absolute temperature. It indicates that as the dark reverse saturation current density J_{0} increases, V_{oc} decreases. We indeed find that J_{0} is increased by a higher mid gap defect density, and that this corresponds to a lowerV_{oc}, in agreement with equation (1).
5.2 Effect of the band gap on the opencircuit voltage
As has already been stated, in order to model all aspects of the dark and illuminated characteristics of the intermediate F_{c}μcSi:H cell B (Tab. 1), one of our assumptions was a lower mobility band gap in this material (Tab. 2) relative to case A having a lower crystalline volume fraction. In Table 4 we compare the JV parameters of two μcSi:H solar cells having exactly the same parameters as the lowest F_{c}μcSi:H cell A (Tab. 2), except that in one case (named “high E_{g}”) the Ilayer band gap is 1.4 eV (the value used to model the lowest F_{c} cell A, Tab. 2), while in the other case (named “low E_{g}”) the I μcSi:H band gap is 1.33 eV, the value required to simulate the intermediate F_{c} cell B (Tab. 2). As a result we observe a drop in V_{oc} from 0.54 V down to 0.50 V, accompanied by a factor of 10 increase in J_{o}, while the J_{sc} and FF are practically not affected.
The solar cell output of the fluorinated μcSi:H solar cell compared to the output of a hypothetical cell having higher effective DOS at the band edges and higher exponential tail prefactors.
The solar cell output parameters of two μcSi:H cells having exactly the same parameters as the lowest F_{c}μcSi:H cell A , except the band gap of the intrinsic material.
As mentioned already, all parameters for the low and high “E_{g}” cells, as also the light absorbed in every segment of the cells, have been assumed the same for the two cases that differ only in the band gap. Nevertheless, modeling reveals that to accommodate the difference in the band gap between the two cells, both the quasiFermi level separation and the distance of the Fermi levels from the band edges are less for the case of the cell with a lower intrinsic layer band gap. This latter fact results in a higher free carrier density in the bands as quantified by the following expressions of the free carrier densities: The trapped carrier density at any point in the device depends on the corresponding free carrier density, the defect density at that location as well as the relative values of the charged and neutral capture crosssections of these defect states. Hence all else remaining the same, an increase of the free carrier density due to a lowering of the band gap, results in increased carrier trapping in the defect states. As the quasiFermi level for holes is closer to the valence band of the intrinsic layer at the P/I interface, the higher free hole density (from Eq. (3)) in the “low E_{g}, intermediate F_{c}” cell, results in particularly high hole trapping and hence electric field over the P/I interface region, that in a manner similar to Figure 6b for the band tail case, leads to a fall of the electric field in the volume of the absorber layer. The latter fact is known [15] to bring down the opencircuit voltage, and explains the lower V_{oc} of the cell having μcSi:H of lower band gap, higher F_{c}.
In Sections 5.1 and 5.2 we have analyzed the reasons why a solar cell having a higher crystalline volume fraction μcSi:H intrinsic layer may have a lower V_{oc} by examining specific sets of models. However, as seen in the experimental results (Tab. 1), a μcSi:H cell with a higher F_{c} will show a higher J_{sc}, primarily because of higher free carrier mobilities (Tab. 2). Nevertheless, as Table 1 indicates, the energy conversion efficiency for the higher F_{c}μcSi:H cell B is ultimately lower than that of cell A, due to the accompanying reduction in V_{oc} and FF. In other words, due to three interacting effects – higher mid gap and tail defect density and lower band gap – the intermediate F_{c}μcSi:H cell B (Tab. 1) will show a fall in V_{oc} significant enough to cancel the advantage of a higher J_{sc} due to higher carrier mobilities.
We now examine how it is possible for the fluorinated μcSi:H cell having an even higher F_{c}, as well as a very high largegrain fraction, to have a higher V_{oc} than the intermediate F_{c} cell B, in spite of a further reduction of its band gap, as predicted by modeling.
5.3 V_{oc} in the highly crystallized large grained μcSi:H:F cell
So far we have compared the parameters as deduced from modeling (Tab. 2) that characterize the μcSi:H solar cells A (F_{c} = 79%, no large grains detected) and the more crystallized cell B (F_{c} = 93%, F_{lg} = 27%), and explained in Sections 5.1 and 5.2, why the latter shows a lower opencircuit voltage and fill factor than the former. This is normal behavior for μcSi:H cells, as evidenced from numerous experimental observations (Fig. 1) and is the reason why the best μcSi:H solar cells have so far been produced close to the aSi:H/μcSi:H transition. However experiments indicate an allround improvement in the output characteristics of the highly crystallized large grained fluorinated μcSi:H (F_{c} ~ 100%, F_{lg} ~ 50%) cell relative to type B μcSi:H cells, indicating that the output properties of this cell violates the general trend of decreasing V_{oc} with increasing crystalline volume fraction (Fig. 1). By modeling its output characteristics (Tab. 1 and Fig. 5c), we have extracted the parameters that characterize this material (Tab. 2). The salient features of the μcSi:H:F cell parameters are: (a) a reduced band gap compared to μcSi:H cells A and B, (b) higher carrier mobilities, (c) sharply reduced dangling bond defect density, (d) reduced effective DOS at the band edges, that match those of monocrystalline silicon and (e) higher absorption over a large portion of the longer visible wavelengths compared to low and intermediate F_{c}μcSi:H (Fig. 2b, the absorption coefficient is proportional to the imaginary part of the complex refractive index κ). The characteristic energies of its band tails however are similar to those of cell B, indicating that this highly crystallized material nevertheless has a strained lattice. The higher absorption and, to a smaller extent the larger carrier mobilities, are responsible for the high current density in this cell (Tab. 1). Assumption of the lowest band gap for this material, having the highest crystalline volume fraction, follows the general trend of the parameters inferred by modeling the output characteristics of the three types of μcSi:H cells (Tab. 2). What is surprising however is that this fact does not lead to a further fall in V_{oc} following the general rule in μcSi:H solar cells (Fig. 1). One reason for this is the sharp fall in the dangling bond defect density in this case that, as described in Section 5.1.2, leads to an improvement of the electric field over the intrinsic layer and increased V_{oc}. This fact partially cancels the negative effect of a lower band gap on V_{oc}. In the following subsection we examine the effect of reduced effective DOS at the band edges for the case of the fluorinated μcSi:H Ilayer (Tab. 2) and address how this can also partly explain the observed improvement in V_{oc} for this type of cells.
5.3.1 Sensitivity of V_{oc} to the effective DOS at the band edges
Table 3 compares the solar cell output parameters of the fluorinated μcSi:H cell with that of a hypothetical cell D, having identical parameters as the former, except that the effective DOS at the band edges in cell D are like those in hydrogenated amorphous silicon (aSi:H) or the other μcSi:H cells A and B. In other words N_{c},N_{v} for cell D are 2 × 10^{20} cm^{3}, while they are 2.8 × 10^{19} cm^{3} and 1.04 × 10^{19} cm^{3} for N_{c},N_{v} respectively for fluorinated μcSi:H (Tab. 2). Note that the exponential band tail prefactors are related to the N_{c}, N_{v} via the relations: (4)so that a fall in N_{c}, N_{v} automatically reduces G_{A0} and G_{D0}. This impacts on the tail defect density according to the relations: where g_{A},g_{D} are the tail defect densities (cm^{3}eV^{1}) at energy locations E and E′ respectively from the conduction and valence band edges; and E_{A} and E_{D} are the characteristic energies of the respective band tails. Thus reduced N_{c(v)} lead to reduced band tail defect density, even for the same values of the characteristic energies of the band tails. We thus note that although the fluorinated μcSi:H cell has the same values of E_{D} and E_{A} as the μcSi:H cell B (Tab. 2), the band tail defect density is smaller for the former. This then is one reason for the higher opencircuit voltage for this case relative to cell B (Tab. 1), according to the arguments presented in Section 5.1.1.
The position of the Fermilevel is determined by the relaxation, trapping, and recombination dynamics of the photogenerated carriers, and thus for a given density of freecarriers, a greater quasiFermi level separation can be achieved with a lower N_{c(v)} (from Eqs. (2), (3)) and a higher V_{oc} will result. Also lower N_{c(v)} means lower free – and therefore trapped carrier densities – for a given quasiFermi level separation (Eqs. (2), (3)). Lower values of trapped carrier densities lead to lower P/I interface field and hence more field penetration into the bulk of the device (as explained in Sect. 5.2); therefore [15] an improved V_{oc}. Table 3 indicates large improvements in V_{oc} and FF possible as a consequence of the fall in N_{c}, N_{v}. We thus conclude that the fluorinated μcSi:H cell has a higher V_{oc} and FF relative to the μcSi:H cell B, because the combined effect of a lower dangling bond DOS, a lower effective DOS at the band edges and lower exponential tail prefactors, overcome the negative influence of its reduced band gap.
5.4 V_{oc} in crystalline silicon PN solar cells
The original aim of this article was to investigate the general trend in μcSi:H solar cells, which is that their opencircuit voltage decreases with increasing crystalline volume fraction (Fig. 1) and our modeling has indicated that one of the principal reasons for this is the lower energy band gap (Sect. 5.2) in more crystallized material. Crystalline silicon (cSi) has a band gap of only 1.12 eV, so all else being equal, it should produce cells with an even lower V_{oc}. However, the typical V_{oc} of cSi solar cells is higher than in those of good quality μcSi:H and typically lies between 0.55 V and 0.6 V [6], while world record cSi cells possess a V_{oc} above 0.7 V [27]. Fortunately we have been able to produce in our laboratory a series of μcSi:H solar cells – the fluorinated μcSi:H series of cells – that violate the observed general trend in μcSi:H cells (Fig. 1) and exhibit a higher V_{oc}, in spite of having a lower energy band gap. This series has therefore provided the necessary insight to explain why the limiting case of cSi solar cells can possess higher V_{oc} in spite of a strongly reduced band gap. In the previous subsection we have shown for the case of μcSi:H:F, that the higher V_{oc} in spite of a reduced band gap was made possible by sharply reduced dangling bond (DB) and tail defects, as well as reduced effective DOS at the band edges. We have assumed the effective DOS at the band edges in μcSi:H:F to be similar to the low values of cSi. Moreover, cSi has a DB DOS that is three orders of magnitude lower than even the relatively low midgap DOS of μcSi:H:F (Tab. 2), and the tail states are absent. Therefore it is now only to be expected that cSi solar cells will have higher V_{oc} and FF than those of the μcSi:H:F cells, which are indeed far superior to those observed in highly crystalline μcSi:H cells (example cell B in Tabs. 1 and 2), obeying the general trend of V_{oc} as a function of the crystalline volume fraction (Fig. 1).
6 Justification of the parameters deduced by modeling in the three types of μcSi:H solar cells studied
Table 2 shows the parameters that characterize the intrinsic layers of the low F_{c}μcSi:H series of cells A, the intermediate F_{c} cell series B and the large grained high F_{c} fluorinated μcSi:H solar cells as inferred by modeling their output characteristics. These indicate that the intermediate F_{c}μcSi:H cell B, has a lower band gap, higher carrier mobilities, and both higher midgap defect density and broader band tails as compared to the cell having low F_{c}μcSi:H A. In order to simulate the improved cell performance (Tab. 1) of the μcSi:H:F cell, we had to assume even higher carrier mobilities, lower midgap defects and lower effective DOS at the band edges (similar to crystalline silicon), while its band gap was assumed to be smaller than that of cells of type B. We have throughout assumed higher carrier mobilities for more crystallized materials. The presence of a significant fraction of large grains in a material produced by PECVD at 175 °C has been shown to correlate with improved transport properties of the films [28], and in particular we have shown elsewhere that the electron mobility as measured by time resolved microwave conductivity , increases with the fraction of large grains [28].
We have assumed a decreasing band gap with increasing crystalline volume fraction. This assumption is wellsupported by reports from the literature. For example, Delley and Steigmeier [29] – who computed the band gap of μcSi:H as a function of cluster diameter using the density functional approach for finite structures – have shown that the band gap of μcSi:H increases as the cluster size decreases. A higher band gap for lesscrystallized μcSi:H was also previously measured by Hamma and Roca i Cabarrocas [30] using in situ Kelvin probe analysis and the “Flat Band Heterojunction” technique. Merdzhanova et al. [31] have studied the photoluminescence (PL) in thin film μcSi:H PIN solar cells deposited by the Hot wire chemical vapor deposition technique [32] and have observed that the PL band shifts to higher energy with decreasing crystalline volume fraction.
Another property that had to be assumed in order to model the experimental characteristics are lower band tail characteristic energies in the μcSi:H film of the lowest F_{c}, the absorber layer in cell A (Tab. 2). This assumption is also supported by experimental evidence, such as the previously cited work of Merdzhanova et al. [31]. Their assumption that PL originates from transitions between localized band tail states indicates that lesscrystallized μcSi:H has sharper band tails. Additionally, μcSi:H films with lower F_{c} are expected to have a larger fraction of hydrogenated amorphous silicon (aSi:H), which encourages structural relaxation of the μcSi:H network [31], thus giving rise to less strained films with sharper band tails. Since aSi:H is also expected to passivate grain boundary defects, our additional assumption of a lower Gaussian defect density for the least crystallized μcSi:H in cell A (Tab. 2), compared to the more crystallized Ilayer in cell B, also appears to be justified.
However it may be noted that while the intermediate F_{c}μcSi:H Ilayer B is predicted to have a higher DB defect density, compared to the less crystallized Ilayer A, modeling the highly crystallized fluorinated solar cell requires a sharp decrease of this defect density. To understand this, we first note that although the intermediate F_{c} (F_{c} = 93%) Ilayer B (Tab. 2) has a high crystalline volume fraction, its large grain fraction F_{lg} is considerably lower than in the case of μcSi:H:F. High F_{c} with low F_{lg} implies the presence of a large number of tiny crystallites and hence many grain boundaries, with a higher probability of defects. Additionally this material B, with a high F_{c}, possesses a low amorphous fraction (F_{a}). The latter is known to passivate grain boundary defects. Thus a large number of grain boundaries, together with a low F_{a} would necessarily lead to a high number of DB defects. Also more crystallized μcSi:H (case B, Tab. 2) is known to have a fairly large oxygen content [33]. One possibility is that this oxygen occupies substitutional sites in the μcSi:H lattice and produces Ntype doping. However any appreciable doping of an intrinsic layer of a solar cell containing band gap defects where carriers can be trapped, would lead to sharp interface fields, weak penetration of field and flat bands in the volume of the intrinsic layer resulting in high recombination that should strongly reduce the current. This is not observed in the μcSi:H cells studied (Tab. 1), thus excluding the possibility of oxygen producing any appreciable doping in the μcSi:H cell B. These oxygen atoms, which are probably located at grain boundaries, therefore produce defects in this material leading to a higher DB DOS in the more crystallized Ilayer B (Tab. 2). On the other hand fluorinated μcSi:H has not only a high crystalline volume fraction but also a high fraction of large grains resulting in fewer grain boundaries and therefore less grain boundary defects. Also, the use of SiF_{4} as gas precursors allows to reduce the concentration of oxygen leading to a dense large grained material with low defects [28] and justifies the sharply reduced defect density in this material as inferred from modeling (Tab. 2), that is consistent with the enhanced electronic properties of this material [28].
6.1 Bandedge effective DOS (Nc, Nv) in amorphous and microcrystalline solids
We have underlined in Sections 5.3.1 and 5.4 that a key factor for improving the opencircuit voltage in the large grained fluorinated μcSi:H and cSi PN cells is the reduced band edge effective DOS (N_{c(v)}), lower by nearly an order of magnitude compared to amorphous and disordered μcSi:H (Ilayers A and B, Tab. 2). However, the reason behind the higher N_{c(v)} in disordered silicon has not yet been addressed. To do so, it must first be noted that the effective DOS at the valence band edge (N_{v}) is calculated from the relationship satisfying: (7)and similarly for N_{c}, assuming that the Fermi level E_{F} is sufficiently far away from the valence band to approximate the Fermi distribution with the exponential relationship shown. Because N_{v} is calculated using the product of the true DOS with the Fermi distribution of holes, the states closer to the bandedge influence the value of N_{v} more strongly than those far from the edge. For this reason, the shifting of states towards the bandedge will increase N_{v}, though the absolute number of states may not change.
Theoretical evidence strongly suggests that such a shifting of states closer to the band edges occurs when a tetrahedrally bonded crystalline solid is amorphized. Solving the appropriate Bethe lattice for amorphous silicon, Joannopoulos [34, 35] showed a significant shifting of the DOS towards the bandedge. The tightbinding Hamiltonian used showed that the shift was particularly great for the Plike states at the valence band edge [34, 35] and also accounted for the steepening of the valence band edge densityofstates with disorder as observed in Xray photoemission experiments [36]. Using a Continuous Random Network (CRN) model, Singh [37] has also shown that dihedralangle and topological disorders lead to an increase of the DOS at the valence and conduction band edge respectively. The effect is equally seen when the effect of hydrogen is included. The strain in stretched SiSi bonds may partially be released by hydrogen incorporation (resulting in hydrogenated amorphous or micro crystalline silicon, the materials of interest in PIN solar cells), and hence removed from the tails [38]. The addition of hydrogen thus widens the band gap, moving the peak in the valence band DOS closer to the band edge, and increasing N_{v} at the band edge. Monte Carlo calculations examining mixed phase nanocrystallineamorphous silicon showed that the location of the peak in the valence band DOS is little affected by the addition of a considerable volume fraction of crystalline material [39], although the midgap and lower energy DOS were significantly modified. This result indicates that the amorphized fraction has a dominant effect on the location of this peak, and that the assumption of higher N_{c} and N_{v} at the band edges, is justified even for microcrystalline silicon with a considerable crystalline volume fraction (in other words for the μcSi:H Ilayers A and B, Tab. 2). It should be noted that in going from cSi to aSi:H, the peak in the valence band DOS (for example) is not shifted up in absolute terms, but only relative to the relevant mobility/energy gap. It is this relative shift that is important in determining N_{c(v)}.
We thus justify the assumption of a higher (by ~1 order of magnitude) N_{c(v)} for hydrogenated amorphous and μcSi:H Ilayers A and B, relative to N_{c(v)} of cSi. For the case of the highly crystalline, largegrained μcSi:H:F Ilayer, showing both high J_{sc} and V_{oc}, transport measurements [28] revealed an improved electron mobility that we correlate to a reduced defect density, justifying our assumption of a reduced dangling DOS of 10^{15} cm^{3} for this material (Tab. 2). However our detailed modeling revealed that with a band gap of only ~1.2 eV (Tab. 2), the high values of V_{oc} measured in this case could not be achieved by a DOS of 10^{15} cm^{3} alone, and since this dense largegrained μcSi:H:F, in its properties appears to be very close to monocSi, we assumed N_{c(v)} for this case to be like the latter and were thus able to reproduce all the measured solar cell output (Tab. 2, Fig. 5c). As already discussed similar values of N_{c(v)} have been assumed by other workers in this field [24, 25].
7 Conclusions
In this article we have simulated the dark and illuminated JV and quantum efficiency characteristics of typical solar cells having low (case A, Tabs. 1 and 2) and intermediate crystalline volume fraction μcSi:H (case B, Tabs. 1 and 2). The lower F_{c} material A has been assumed to have a higher energy band gap than the intermediate F_{c} material B. Both experimental and modeling results indicate a higher J_{sc} but lower V_{oc}, FF and conversion efficiency for the solar cell based on intermediate F_{c}μcSi:H B (Tab. 1). In fact the general trend in μcSi:H solar cells is that V_{oc} decreases with increasing F_{c} (Fig. 1). We have analyzed the reasons for this and have found that this can be explained by broader band tails and higher Gaussian defect density in the intermediate F_{c} material B, since structural relaxation of the μcSi:H network and passivation of grain boundary defects cannot properly take place due to the low amorphous silicon content in this wellcrystallized material, as well as due to high oxygen content. Another very important factor is the lower band gap of the intermediate F_{c}μcSi:H B, resulting in higher free carrier density in the bands, due to the proximity of the band edges to the quasiFermi levels. This leads to higher photogenerated hole trapping, especially near the P/I interface, which in turn leads to a collapse of the electric field over the volume and a lower V_{oc}.
We have also shown that high F_{c}, large grained μcSi:H:F, having low oxygen content, is an exception to this general rule (Fig. 1), since its sharply reduced band gap defect density and lower effective DOS at the band edges overcome the negative influence of the lower band gap to produce a higher V_{oc}. In fact fluorinated μcSi:H solar cells serve as a link to explain why cSi PN solar cells, in spite of having a sharply reduced band gap, can have opencircuit voltages considerably higher than μcSi:H solar cells.
Acknowledgments
The work at CNRSLPICM has been partly supported by the European Project “SE Powerfoil” (Project number 038885 SES6). The computer modeling program was developed by P. Chatterjee during the course of a project funded by MNRE and DST, Government of India, and partly during her tenure as Marie Curie fellow at the Laboratoire de Physique des Interfaces et des Couches Minces, Ecole Polytechniuque, Palaiseau, France. E.V. Johnson acknowledges the support of NSERC.
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All Tables
Comparison between the measured and simulated solar cell output parameters of the PIN devices having low and intermediate crystalline volume fraction (F_{c})μcSi:H as the intrinsic layer. F_{lg} gives the fraction of large grains in the Ilayer. Also compared are the measured and modeled output parameters of the highly crystallized, large grained μcSi:H:F cell.
Parameters that characterize intrinsic μcSi:H of different degree of crystallinity (as extracted by modeling). The quantities in brackets marked with asterisks in the column of parameters of μcSi:H:F correspond to the values extracted by modeling similar μcSi:H samples (refs TSF, JAP of Rubinelli).
The solar cell output of the fluorinated μcSi:H solar cell compared to the output of a hypothetical cell having higher effective DOS at the band edges and higher exponential tail prefactors.
The solar cell output parameters of two μcSi:H cells having exactly the same parameters as the lowest F_{c}μcSi:H cell A , except the band gap of the intrinsic material.
All Figures
Fig. 1 Results taken from various literature sources [1, 2, 3, 4] indicate the general trend of a decrease in the opencircuit voltage of μcSi:H thin film PIN cells, with increase of the Raman crystallinity of the films. We also show an exception where the V_{oc} actually increases [5] in a highly crystallized μcSi:H:F cell, specially after interface treatment (indicated by the arrow). Also shown in the figure are typical V_{oc}’s of classical diffused junction cSi solar cells (open circles – 5, Green’s ref), of a “heterojunction with intrinsic thin layer (HIT)” cell on a Ptype cSi substrate [7] and of a HIT cell on Ntype cSi [8] (closed circles). 

In the text 
Fig. 2 Values of the complex refractive indices (a) real part, n and (b) imaginary part, κ for low (79%), intermediate (93%) crystalline volume fraction (F_{c})μcSi:H, and for μcSi:H:F, F_{c} ~ 100%, compared to the respective values of aSi:H (amorph) and cSi (crys). 

In the text 
Fig. 3 Calculated dark JV characteristics of the intermediate F_{c} and low F_{c}μcSi:H cells at 30 °C, compared to experimental results. The lines are our modeling results, while symbols represent experimental measurements. 

In the text 
Fig. 4 Calculated values (open symbols) of (a) the reverse saturation current density (J_{0}) and (b) the diode ideality factor (n) for low and intermediate F_{c}μcSi:H solar cells, compared to experiments (closed symbols) at temperatures from 10 °C to 50 °C. The lines are guides to the eye. 

In the text 
Fig. 5 Calculated external quantum efficiency (EQE) curves under AM1.5 bias light and shortcircuit conditions for the (a) low F_{c}μcSi:H cell A, (b) intermediate F_{c}μcSi:H cell B and (c) high F_{c}, large grained μcSi:H:F cell, compared to experimental results. 

In the text 
Fig. 6 (a) Sensitivity of the illuminated JV characteristic to the valence and conduction band tail characteristic energies and (b) the electric field in the device when the band tails are sharp (characteristics energies of band tails E_{D},E_{A} = 20 me V, 10 m eV respectively, characteristic of μcSi:H cell type (A) and when they are broader (E_{D},E_{A} = 40 me V, 20 m eV respectively, characteristic of cell type (B). The electric field over the P/I interface region is plotted on a different scale in the inset. 

In the text 
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