Abstract
Considering recent modifications on ntype highly doped silicon parameters, an emitter optimization was made based on onedimensional models with analytical solutions. In order to get good accuracy, a fifth order approximation has been considered. Two kinds of emitters, homogeneous and nonhomogeneous, with phosphorus Gaussian profile emitter solar cells were optimized. According to our results: homogeneous emitter solar cells show their maximum efficiencies (<FONT FACE="Symbol">h @ 21.6021.74%)</FONT>with doping levelsnus = 1x10(19)  5x10(18) (cm3) and (1.22.0) mum emitter thickness range. Nonhomogeneous emitter solar cells provide a slightly higher efficiency (eta = 21.8221.92%), with Ns = 1x10(20) (cm3) with 2.0 mum thickness under metalcontacted surface and Ns = 1x10(19)  5x10(18) (cm3) with (1.22.0) mum thickness range, (sheet resistance range 90100 <FONT FACE="Symbol">W/ <img SRC="http:/img/fbpe/mr/v4n2/n2a17fou.gif" BORDER="0"></FONT>) under passivated surface. Although nonhomogeneous emitter solar cells have a higher efficiency than homogeneous emitter ones, the required technology is more complex and their overall interest for practical applications is questionable.
solar cell efficiencies1; homogeneous emitter2; nonhomogeneous emitter3; Gaussian profile4
Studies of Phosphorus Gaussian Profile Emitter Silicon Solar Cells
N. Stem^{*}, M. Cid^{**}
Laboratório de Microeletrônica, Depto. de Engenharia de Sistemas Eletrônicos
Escola Politécnica da Universidade de São Paulo, C.P. 61548
05424970 São Paulo  SP, Brazil
email: *nstem@lme.usp.br, **mcid@lme.usp.br
Received: January 30, 2001; Revised: April 4, 2001
Considering recent modifications on ntype highly doped silicon parameters, an emitter optimization was made based on onedimensional models with analytical solutions. In order to get good accuracy, a fifth order approximation has been considered. Two kinds of emitters, homogeneous and nonhomogeneous, with phosphorus Gaussian profile emitter solar cells were optimized. According to our results: homogeneous emitter solar cells show their maximum efficiencies (h @ 21.6021.74%) with doping levels N_{s }= 1x10^{19}  5x10^{18} (cm^{3}) and (1.22.0) mm emitter thickness range. Nonhomogeneous emitter solar cells provide a slightly higher efficiency (h = 21.8221.92%), with N_{s} = 1x10^{20} (cm^{3}) with 2.0 mm thickness under metalcontacted surface and N_{s} = 1x10^{19}  5x10^{18} (cm^{3}) with (1.22.0) mm thickness range, (sheet resistance range 90100 W/ ) under passivated surface. Although nonhomogeneous emitter solar cells have a higher efficiency than homogeneous emitter ones, the required technology is more complex and their overall interest for practical applications is questionable.
Keywords: solar cell efficiencies1, homogeneous emitter2, nonhomogeneous emitter3, Gaussian profile4
1. Introduction
In order to improve solar cell efficiencies, several theoretical optimizations of emitter region have been proposed throughout the years. In the eighties a breakthrough occurred in the emitter design philosophy aimed to the development of efficient silicon solar cells^{1}. It was shown that the best emitters should be thick and moderately doped. According to A. Cuevas and M. Balbuena^{2}, solar cells with homogeneous emitters with N_{s} = 1x10^{19}  5x10^{18} (cm^{3}) and thickness of 1.02.0 (mm) are the best ones. In 1990, based on these works, A. Aberle et al. optimized LDD (locally deep diffused) and homogeneous emitters, getting the conclusion that LDD emitters are better than homogeneous ones^{3}. Despite the required technology to process these emitters are complex, nonhomogeneous emitters has still been adopted in research laboratories, and they are found in the solar cells with record efficiencies. However a question must be asked: are nonhomogeneous emitters significantly better? Moreover, recent modifications of highly doped ntype parameters made new theoretical optimizations of emitters imperative. Therefore, this work has two important motivations: comparison between the two kinds of emitters and the necessity of optimizing emitters considering updated parameters^{4}. Theoretical models with analytical solutions have been used to study n^{+} emitter region^{5,6}. Admitting a Gaussian profile, two kinds of emitters, homogeneous and nonhomogeneous, were optimized and compared considering different kinds of surfaces, emitter doping levels and thicknesses. In order to calculate theoretical solar cell efficiency, a complete n^{+}pp^{+} structure was considered. To point out the effect of emitter optimization, the base and p^{+} region parameters have been considered constant. The base region has been admitted to have a resistivity of r = 1 W.cm, a diffusion length 1350 mm and a thickness 290 mm. A structure with back surface field (BSF) has been also considered, adding a uniform profile and a rear surface recombination velocity S_{n} = 200 (cm/s)^{7}. The photon absorption within the BSF region was neglected. Neither light trapping effects nor surface reflection have been taken into account. The metal grid shadowing factor was assumed as F_{m} = 3 (%). The output parameters (shortcircuit density (J_{sc}), opencircuit voltage (V_{oc}) and efficiency (h)) were calculated using wellknown relationships^{8}. The solar cells efficiencies were obtained using the fill factor given by Greens expression^{9}.
2. Theoretical Emitter Model
The transport equations for minority carriers in an ntype emitter under steady condition are:
where, J_{p} is the minority carriers current density; m_{p}, minority carrier mobility; p, minority carrier concentration; p_{o}, minority carrier equilibrium concentration; E_{p}, electric field; q, Coulomb charge; D_{p}, diffusion coefficient; G, carrier generation and t_{p}, minority carrier lifetime.
As it is well known, some effects such as band gap narrowing, Fermi level degeneracy and changes of behavior of minority carrier lifetime and mobility occur when a region is highly doped, as for solar cell emitters. As the parameters, band gap narrowing, minority carrier mobility and lifetime, are position dependent, an analytical solution to Eqs. (1) and (2) is required. Table 1 shows the internal parameters and expressions used for the emitter model.
Considering the apparent band gap notation , the minority carrier equilibrium concentration (p_{o}) can be written according to Eq. (3).
where, n_{io} is the intrinsic carrier concentration; k, Boltzmann constant; T, temperature in Kelvin and N(x) is the dopant profile. As it was said previously, in this work only Gaussian profiles were considered. The apparent band gap narrowing takes into account the Fermi degeneracy when the surface doping level is higher than 1.4x10^{17} (cm^{3}). In order to simplify the calculations, the normalized hole concentration (p(x)), normalized current density (J(x)) and normalized carrier generation (G(x)) were defined (see Eqs. (4), (5) and (6)).
Looking for the solution for Eqs. (1) and (2), two boundary conditions were considered: one is at the depletionregion boundary x = 0 and the other is at the emitter surface x = W_{E}, as it can be seen in Eqs. (7) and (8).
where S_{p }is surface recombination velocity.
Therefore the transport equations could be rewritten as:
and
After some substitutions using Eqs. (9) and (10), the normalized current density and the minority carrier equilibrium concentration are obtained, though Eqs. (11) and (12) respectively. These equations are written as functions of infinite series of three coefficients A(x), B(x) and C(x)  see Eqs. (13), (14) and (15) respectively. An excellent accuracy was assured considering a fifth order approximation, as it is shown by A. Cuevas et al^{10}. The emitter current density is formed by two components: saturation current density (first member) and the collected current density (second member). When the generation (G(x)) comes to zero, the emitter is in dark and the coefficient C(x) is zero too, so the emitter current density (J(0)) is equal to the saturation current density (J_{oE}). The base region could be described by these expressions too, if minority carrier lifetime and mobility were considered constant and adequate expressions were used for parameters.
and
where
The emitter collection efficiency, expressed in (16), is the ratio between the photogenerated current density and the total emitter generated current density. The total emitter generated current density was calculated using the standard spectrum AM1.5G from ASTM 89287.
In order to calculate the theoretical efficiency of a complete structure and looking for only the effect of emitter region, the base and the p^{+} emitter region parameters have been considered constant as previously referred.
3. Results
3.1. Optimization
In order to optimize the emitter region, saturation and collected current densities were studied. The saturation current density as a function of emitter thickness with different doping levels is shown in Fig. 1, considering two kinds of surfaces, low and high surface recombination velocities. Passivated surfaces were considered having their recombination velocities dependent on N_{s} (S_{p} = 10^{16} N_{s }cm/s) while for metalcontacted surfaces and nonpassivated surfaces constant values were assumed S_{p} = 3x10^{6} (cm/s) and S_{p }= 2x10^{5} (cm/s)^{4}. A metalcontacted surface requires high doping levels (N_{s} @ 1x10^{20} cm^{3}) and the thickness value (@ 2.0 mm) is selected by technological constrictions. For passivated surfaces "low" doping levels (such as N_{s} = 1x10^{19} cm^{3} and N_{s} = 5x10^{18} cm^{3}) must be chosen.
According to this figure the thick and moderately doped passivated emitters have a low contribution for the total recombination density. For instance a homogeneous emitter with surface doping level N_{s} = 1x10^{19} (cm^{3}) and thickness 1.2 (mm) presents 6.2x10^{14} (A/cm^{2}) as the total emitter recombination density (passivated and nonpassivated regions).
The emitter collection efficiency, obtained from expression (16), with different doping levels as a function of emitter thickness and kinds of surfaces (passivated and nonpassivated) can be observed in Fig. 2. The surface recombination velocity of nonpassivated surface is S_{p} = 2x10^{5} cm/s^{4}.
Analyzing Fig. 2, one can observe that the emitter collection efficiencies (h_{E}) of a passivated surface are higher than nonpassivated ones in most cases. According to this figure, the highly doped and shallow emitters can provide emitter collection efficiencies as high as moderately doped and thick emitters, since they have Gaussian profiles.
3.2. Homogeneous emitters
In order to compare the emitter effects over the complete structure, the base and p^{+} regions have been considered constant. Thus, the output parameters of complete solar cells with homogeneous emitters have been calculated.
The shortcircuit current density (J_{sc}) as function of emitter thickness and surface doping level is shown in Fig. 3. In this figure, it can be seen that the maximums of shortcircuit current density curves are approximately 38.4 (mA/cm^{2}) to all doping levels. However, the J_{sc} of higher doping level emitter solar cells are more sensitive than lower doping level emitter ones when the emitter thickness increases. This occurs because the emitter collection efficiency (h_{E}) decreases significantly due to bandgap narrowing and Auger recombination effects, as it is shown in Fig. 2. As it can be seen, low doping emitters are the best choice to form this region since they provide both high current density and a wide range of optimum thickness.
The internal quantum efficiency of a complete solar cell was calculated as the ratio between the total collected current density and the total photogenerated current density for each wavelength. A complete analysis of the internal quantum efficiency for each surface doping level and the correspondent optimized thickness has been made for wavelengths (l = 3501100 nm) and it has been verified that there is no significant difference among them since the emitters have Gaussian profile.
In Fig. 4 the internal quantum efficiency as function of the wavelength (l) is shown only for an optimized emitter with surface doping level N_{s} = 1x10^{19} (cm^{3}) and emitter thickness W_{E }= 1.2 (mm). The base and BSF regions were previously considered.
According to Fig. 4 the homogeneous emitter presents a high internal quantum efficiency for the emitter wavelength range of interest l = (350600) nm; for instance, for l = 400 nm the internal quantum efficiency is Q_{i}(l) @ 0.98.
Figures 5 and 6 show open circuit voltage (V_{oc}) and efficiency (h) as functions of emitter thickness respectively, considering different emitter doping levels. It can be seen that the limiting factor to achieve high efficiencies is open circuit voltage, which is determined by the saturation current density.
3.3. Nonhomogeneous emitters
Like it was made in the former case, the output parameters for complete solar cell were analysed. Nonhomogeneous emitters are made up of two regions: a passivated one (moderately doped) and a metalcontacted one (highly doped). By examining Fig. 1, it can be seen that to reach low saturation current density (J_{oE}) under metalcontacted surface emitters, this region must be highly doped. In this case, the thicker the emitter is, the lower J_{oE} is. So in order to optimize nonhomogeneous emitters, the surface doping level of N_{s }= 1x10^{20} cm^{3} and 2 mm thickness (due to technological constrictions) were chosen. Shortcircuit current densities (J_{sc}) are the same as homogeneous emitter solar cell ones, since it was considered the same metal grid shadowing factor F_{m} (3%) for both kinds of emitters. Opencircuit voltage (V_{oc}) behavior of nonhomogeneous emitter solar cells is quite different of homogeneous emitter solar cells  compare Figs. 5 and ^{7} . Nonhomogeneous emitters have their maximum V_{oc} higher than the homogeneous ones.
Their maximum efficiencies are (h @ 21.8221.92%) with N_{s} = 1x10^{19}  5x10^{18} (cm^{3}) and (1.22.0) (mm) thickness for passivated surface, as it can be seen in Fig. 8. Since the approximation of a constant (r_{s}) has been made, efficiencies related to emitter sheet resistances higher than 100 W/ are overestimated.
Table 2 shows a comparison between the optimum electrical output parameters for homogeneous and nonhomogeneous emitter solar cells, and the total emitter recombination as well.
It can be seen that the efficiencies in general are better for nonhomogeneous emitter solar cells, the maximum efficiencies related to each different doping level are slightly higher, about 0.20 (%), than homogeneous ones.
As it has been commented previously, the opencircuit voltage is the limiting factor to reach higher efficiencies. When the kind of emitter is changed from homogeneous into nonhomogeneous the emitter recombination current density decreases from 6.2x10^{14} (A/cm^{2}) to 3.9x10^{14} (A/cm^{2}), resulting in increase of approximately 1% in the opencircuit voltage for a passivated region with N_{s} = 1x10^{19 }(cm^{3}) and W_{E }= 1.2 (mm). Therefore, the nonhomogeneous emitter solar cells are still responsible for the record efficiency laboratory solar cells, having more flexibility to obtain good ohmic contacts. However, the complexity of the involved technology points the attention to the studies of Gaussian profile homogeneous emitter ones and their applications.
4. Conclusions
In this work a theoretical optimization of highly doped ntype region with Gaussian profile and complete solar cells have been made as functions of surface recombination velocity, emitter thickness and surface doping level. Furthermore, updated internal parameters have been used to take into account recent published changes. Considering these new conditions, homogeneous and nonhomogeneous were studied. Homogeneous emitter solar cells show the maximum range of solar cell efficiency, h = 21.6021.74% for N_{s} = 1x10^{19}  5x10^{18} cm^{3} with 1.22.0 mm emitter thickness range, corresponding to emitter sheet resistances 90100 W/ respectively. Nonhomogeneous emitter solar cells provide their best efficiencies h = (21.8221.92%) with surface doping levels N_{s }= 1x10^{20} cm^{3} with 2.0 mm thickness under metalcontacted surface and N_{s} = 1x10^{19}  5x10^{18} cm^{3} with (1.22.0) mm thickness respectively under passivated surface. The nonhomogeneous emitter efficiencies are slightly better (0.20%) than homogeneous one, this relatively small difference is due to the fact that recombination in the base region also contributes to limit the cell voltage. Despite nonhomogeneous emitters are more complex, they are still relevant for ultra high efficiency solar cells. However, in practical silicon solar cells, the difference between the two kinds of emitters vanishes almost completely; since the contribution from the base region is still larger than that assumed in this paper, nonhomogeneous provide more flexibility to obtain good ohmic contacts. The use of optimized homogeneous emitters with screenprinting metallization still awaits innovative ideas and experimental demonstration.
Acknowledgments
This work was supported by FAPESP under contract n° 95/094350 and by Rhae/CNPq under contract n° 610157/94.
FAPESP helped in meeting the publication costs of this article
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Publication Dates

Publication in this collection
11 July 2001 
Date of issue
2001
History

Received
30 Jan 2001 
Reviewed
04 Apr 2001