© Owned by the authors, published by EDP Sciences, 2012

1 Introduction

Solar cells based on Si micro- or nanowire arrays with radial p-n junction are of great interest as potentially low-cost solutions in solar cell production and have been extensively studied in the last years [1, 2, 3, 4, 5, 6]. Unlike planar structures, photon absorption and minority carrier collection are decoupled as shown in Figure 1, and are not anymore in competition when optimizing the cell dimensions. In fact, the wires can be grown long enough to optimize light absorption while being small enough in radius to collect photo-generated carriers. This last property enables the use of low purity Si substrates with a short minority carrier diffusion length.

Fig. 1 Structures of solar cells based on (a) a radial p-n junction, (b) a planar p-n junction. Light penetration into the cell is characterized by the parameter 1/α, α being the wavelength-dependent absorption coefficient. The diffusion length of the generated minority carriers is given by Ldiff. In the case of the radial junction, light absorption and carrier collection are decoupled.

The vapour-liquid-solid (VLS) technique [7] assisted by gold catalyst is one of the most current methods to grow micro- and nano-wires for photovoltaic applications. Despite many advantages in using Au as VLS catalyst (non toxicity, chemical stability, availability), Au is known to diffuse in the nanowires [8] creating deep-level defects in Si and leading to an increase of the carrier recombination rate. Among the large number of catalyst materials that have been tested to replace Au [9], aluminium is a good candidate since it does not create deep-level defects. Nevertheless, Al is a p-type dopant and the wires can be strongly p-doped. It is not obvious to know whether the performances of the solar device are strongly influenced by such a p-type doping level. In that case, numerical modelling can be of great help. In this modelling work, we have studied one single-wire Si solar cell which consists in a p-type crystalline silicon (c-Si) core surrounded by a n-type hydrogenated amorphous silicon (a-Si:H) thin layer. Cell efficiency, open-circuit voltage (V_oc), short-circuit current (I_sc) and fill factor (FF) under AM 1.5 spectrum have been calculated as a function of the p-type doping density (N_a) in the core. A drastic decrease in V_oc has been observed when N_a reaches values higher than 1 × 10¹⁷ cm^-3. An insight into the transport properties of the wire in the dark can help to understand this behavior.

2 Silicon nanowire modelling : geometry and physical parameters

Simulations of the electrical characteristics of a single-wire c-Si/a-Si:H radial heterojunction solar cell were performed using ATLAS Silvaco software [10]. The geometry of the radial p-n junction studied here is given in Figure 2. The design was defined as a simulation plane in 2-dimensional cylindrical coordinates which models the 3D cylindrical structure. The wire dimensions were chosen according to experimental data of fabricated wire array cells, with a wire length of 25 μm and a radius of 810 nm. The p-type c-Si core (radius of 800 nm) is surrounded by a thin layer (10 nm) of n-type doped a-Si:H. Ohmic contacts were used, one at the bottom of the structure on the crystalline base and another one around the wire on the amorphous layer. At the bottom contact, the electron surface recombination velocity has been taken equal to 50 cm s^-1. For the surrounded contact, flat band conditions were imposed.

Fig. 2 Single-wire geometry of the model.

For the a-Si:H layer, the distribution of the density of states (DOS) is made of two exponential band tails (valence and conduction band tails) and two Gaussian distributions of deep defect states, one donor-like and one acceptor-like. The DOS and doping concentration were adjusted such that the Fermi level at room temperature was set at 0.2 eV below the conduction band edge E_c. The electron and hole mobilities were taken equal to μ_n = 20 cm²  V^-1  s^-1 and μ_p = 5 cm² V^-1  s^-1, respectively. In the c-Si core, we fixed a value of 1 ms for the carrier lifetime which corresponds to a diffusion length much bigger than the wire radius. The doping density N_a was varied between 1 × 10¹⁵ cm^-3 and 1 × 10¹⁹ cm^-3 such that the Fermi level at room temperature moves from 0.25 eV above the valence band edge E_v to a position very close to E_v, respectively.

Fig. 3 Variations of the efficiency (a) and of the open-circuit voltage Voc (b) of the modeled single-wire cell as a function of the p-type doping density Nain the c-Si core. These results are compared to those calculated for an equivalent c-Si(p)/a-Si:H(n) planar structure.

The simulation is based on the resolution of Poisson’s equation and electron and hole continuity equations. The Boltzmann statistics is used with the drift-diffusion model in ATLAS, and Shockley-Read-Hall recombination was considered. The AM1.5 solar spectrum was used for the optical generation, the wire being under vertical illumination (see Fig. 2). Current-voltage characteristics (I(V)) were calculated in the dark and under standard one-sun illumination conditions. The open-circuit voltage, the short-circuit current, and the efficiency of the cell were deduced from the I(V) curves under illumination.

3 Simulation results and discussion

The variations of the efficiency and of the open-circuit voltage of the single-wire device is given as a function of the doping density N_a in Figures 3a and 3b, respectively. These variations are compared to those of an equivalent planar solar cell. The planar structure was defined as a thin a-Si:H(n) layer (10 nm) on a 25 μm-thick c-Si(p) wafer. The material properties and the illuminated surface are strictly the same in both structures. Moreover, whatever the doping density in the studied range, the short-circuit current density was found constant and equal to 32.1 mA/cm² for both cylindrical and planar geometries.

It can be observed on Figures 3a and 3b that the efficiency and V_oc increase with log(N_a) when N_a increases until an optimum value N_aopt which differs in both structures. In the case of the wire cell, N_aopt is found equal to 1 × 10¹⁷ cm^-3. Beyond N_aopt, there is a strong decrease of the efficiency which is mainly due to a heavy drop in the V_oc, and this behavior is more important in the silicon wire than in the planar device.

The V_oc is given by the relationship : V_oc = nkT/q ln(I_sc/I_o), where n is the diode ideality factor, k the Boltzmann’s constant, T the temperature, q the electronic charge and I_o the dark reverse saturation current. Since I_sc remains constant with N_a, the variation of the V_oc is related to the dark current. Thus, for the single-wire structure, the I(V) characteristics in the dark (I_dark) were calculated for different values of N_a between 1 × 10¹⁵ cm^-3 and 1 × 10¹⁹ cm^-3 and the results are presented in Figure 4.

Fig. 4 Forward dark current characteristics Idark(V) for different doping density Na in the c-Si(p) core of the wire. For cm-3, the fits of the curves with the exponential law I0exp(qV/kT) corresponding to an ideal diode behavior is also given (dash lines).

At low doping concentration (cm^-3), the trend of the curves remains the same and I_dark decreases when N_a increases, which is a classical behavior in a p-n junction. A change in the slope of the curves can be observed, which indicates two current regimes : at high forward bias (V), the I_dark(V) curves can be fitted by the expression I_dark = I₀exp(qV/kT) as shown in Figure 4, which corresponds to an ideal diode behavior with a diffusion current. The values of I₀ deduced from these fits are found inversely proportional to N_a, which explains the variations of the V_oc previously described in this range of N_a. At low forward bias, the dark current seems to be dominated by a recombination/generation current in the a-Si:H layer. At high doping density (N_a > 1 × 10¹⁷ cm^-3), no part of the I_dark(V) characteristic at forward bias can be fitted by the previous exponential law and the recombination current in the a-Si:H shell becomes predominant all over the forward bias range. This is illustrated in Figure 5 where I_dark(V) characteristics are calculated for various defect density N_defect in the a-Si:H layer and a given value of N_a equal to 1 × 10¹⁹ cm^-3. It can be observed that the dark current in the wire decreases when N_defect increases.

Fig. 5 Forward dark current characteristics Idark(V) for various values of the defect density Ndefect in the a-Si:H layer and a given doping density Na in the c-Si(p) core equal to 1 × 1019 cm-3.

Fig. 6 Mapping of the dark current density distribution near the c-Si(p)/a-Si:H(n) heterointerface of the wire for Na = 1 × 1017 cm-3 and Na = 1 × 1019 cm-3. Most of the current is concentrated at the bottom interface for the lowest value of Na whereas it is homogeneously spread all over the p-n junction for the highest one.

The heavy drop in V_oc at high doping density of the c-Si(p) core is directly related to a strong increase of the dark current which, at such N_a values, mainly becomes a recombination current in the defect-states rich a-Si:H layer. However, according to Figure 3b, this behavior is much more pronounced for the radial junction than for the planar one (it has to be reminded that this comparison is presented for the same illuminated area in both structures). A mapping of the dark current components near the c-Si(p)/a-Si:H(n) heterointerface of the single wire is presented in Figure 6 for N_a = 1 × 10¹⁷ cm^-3 and N_a = 1 × 10¹⁹ cm^-3. It can be observed that for the lowest N_a value, most of the current is concentrated at the bottom interface of the structure whereas it is homogenously spread all over the junction for N_a = 1 × 10¹⁹ cm^-3. Simulations have shown that this difference in the current distribution appears around N_a = 1 × 10¹⁷ cm^-3.

Fig. 7 Forward dark current characteristics Idark(V) for two wirelength values (L = 25 μm and L = 250 μm) at Na = 1 × 1017 cm-3 and Na = 1 × 1019 cm-3.

Consequently, the active junction surface that really contributes to the dark current becomes wider when N_a increases and it is expected that the ratio between the radius R of the wire (related to the illuminated area) and the wirelength L (related to the p-n junction) will affect the photovoltaic performance of the wire solar cell as soon as high doping density in the core is reached. This is also illustrated in Figure 7 where the dark current characteristics are plotted for two wirelength values L = 25 μm and L = 250 μm. For N_a = 1 × 10¹⁷ cm^-3, the high forward bias dark current does not depend on the wirelength whereas it increases with L for N_a = 1 × 10¹⁹ cm^-3 due to an homogeneous current distribution all over the wirelength. Thus, at high values of N_a, we should obtain a comparable decrease in V_oc between a planar and a radial junction for an aspect ratio (ratio between the illuminated area and the junction area) of the wire equal to 1. In the wire structures we have studied, the junction area is bigger than the illuminated area (ratio R/L much smaller than 1) such that the drop in V_oc with an increasing doping density is much more pronounced for the single-wire cell than for the planar one.

Fig. 8 Mapping of the dark current density distribution near the c-Si(n)/a-Si:H(p) heterointerface of the wire with ΔEv = 0.42 eV and ΔEv = 0.15 eV. For the highest value of ΔEv, most of the current is concentrated at the bottom interface in the crystalline core. For ΔEv = 0.15 eV, an homogeneous current distribution is obtained all along the heterointerface.

Fig. 9 Mapping of the dark current density distribution near the interface of a silicon crystalline homojunction wire with a doping density of the core Na = 1 × 1017 cm-3 and Na = 1 × 1019 cm-3.

At low doping density, the current is dominated by the minority carriers diffusion and it mainly runs through a small active surface of the junction at the bottom of the structure as shown in Figure 6. This specific behavior seems to be related to the presence of a highly conductive electron interface layer at the c-Si(p) surface. Indeed, the band offsets between a-Si:H and c-Si can provide a strong band bending which results in a strong inversion layer at the c-Si surface [11, 12, 13]. For c-Si(p)/a-Si:H(n) structures, it has been shown from numerical calculations that the inversion layer occurs even at low values of the conduction band offset ΔE_c and that the electron concentration in the interface region of c-Si strongly increases with ΔE_c as soon as eV [12]. In the case of c-Si(n)/a-Si:H(p) structures, a lower limit of the valence band offset ΔE_v has been found equal to 0.25 eV [13] such that ΔE_v can be varied in a wider range than ΔE_c to observe or not an interface inversion layer. Thus, in order to establish a link between the current distribution at the wire heterointerface and the presence-or not of an inversion surface layer, we have made some calculations on a c-Si(n)/ a-Si:H(p) radial cell for different values of the valence band offset. A mapping of the dark current components near the c-Si(n)/a-Si:H(p) interface is given in Figure 8 for ΔEv = 0.42eV. and ΔEv = 0.15eV. It can be observed that the current distribution is homogeneous along the wire for ΔEv = 0.15eV whereas it is concentrated at the bottom of the cell for ΔEv = 0.42eV which can be explained by a charge accumulation in the inversion layer at the interface. Furthermore, calculations performed on a single wire crystalline homojunction also show an homogeneous current distribution all along the wire interface whatever the doping density in the p-type core as shown in Figure 9. The I_dark(V) characteristics follow the exponential law given above with a dependence on the wirelength (not shown here). To come back to our c-Si(p)/a-Si:H(n) wire structures, high values of the doping density in the crystalline core lead to a modification of the band bending such that no strong inversion layer occurs. An homogeneous current distribution is thus observed along the wire for cm^-3. Further studies have to be done to explain more precisely the location of the “hot spot” at the bottom of the interface in the case of a non homogeneous current distribution. In particular, the geometry and position of the contacts can play a important role in that feature.

4 Conclusion

We have studied the potential performance of a single c-Si(p)/a-Si:H(n) wire based solar cell through 2D numerical modelling. We have noticed in particular a drastic drop of the open-circuit voltage when the p-type doping density N_a of the wire core increases beyond an optimum value equal to 1 × 10¹⁷ cm^-3. This loss in V_oc is linked to an increase of the dark current with N_a, the transport in the forward bias region being dominated by recombination current in the a-Si:H layer at high values of N_a. Moreover, the decrease in V_oc is less pronounced for an equivalent planar c-Si(p)/a-Si:H(n) heterojunction, the illuminated area of the planar and the radial modelled structures being the same. This difference can be explained by a dependence of the dark current in the wire with the wirelength. Thus, for cm^-3 the wire based solar cell performance strongly depend on the ratio between the radius, which is linked to the illuminated area, and the wirelength, which is related to the p-n junction area. In conclusion, this study shows that the doping density N_a of the c-Si core should be kept below an optimum value of 1 × 10¹⁷ cm^-3 and that any VLS catalyst which introduces a higher p-type doping density in the core should be avoided.

Acknowledgments

This work has been supported by French Research National Agency (ANR) through Habitat intelligent et solaire photovoltaïque program (Projet Siflex No. ANR-08-HABISOL-010).

References

All Figures

	Fig. 1 Structures of solar cells based on (a) a radial p-n junction, (b) a planar p-n junction. Light penetration into the cell is characterized by the parameter 1/α, α being the wavelength-dependent absorption coefficient. The diffusion length of the generated minority carriers is given by Ldiff. In the case of the radial junction, light absorption and carrier collection are decoupled.
In the text

	Fig. 2 Single-wire geometry of the model.
In the text

	Fig. 3 Variations of the efficiency (a) and of the open-circuit voltage Voc (b) of the modeled single-wire cell as a function of the p-type doping density Nain the c-Si core. These results are compared to those calculated for an equivalent c-Si(p)/a-Si:H(n) planar structure.
In the text

	Fig. 4 Forward dark current characteristics Idark(V) for different doping density Na in the c-Si(p) core of the wire. For cm-3, the fits of the curves with the exponential law I0exp(qV/kT) corresponding to an ideal diode behavior is also given (dash lines).
In the text

	Fig. 5 Forward dark current characteristics Idark(V) for various values of the defect density Ndefect in the a-Si:H layer and a given doping density Na in the c-Si(p) core equal to 1 × 1019 cm-3.
In the text

	Fig. 6 Mapping of the dark current density distribution near the c-Si(p)/a-Si:H(n) heterointerface of the wire for Na = 1 × 1017 cm-3 and Na = 1 × 1019 cm-3. Most of the current is concentrated at the bottom interface for the lowest value of Na whereas it is homogeneously spread all over the p-n junction for the highest one.
In the text

	Fig. 7 Forward dark current characteristics Idark(V) for two wirelength values (L = 25 μm and L = 250 μm) at Na = 1 × 1017 cm-3 and Na = 1 × 1019 cm-3.
In the text

	Fig. 8 Mapping of the dark current density distribution near the c-Si(n)/a-Si:H(p) heterointerface of the wire with ΔEv = 0.42 eV and ΔEv = 0.15 eV. For the highest value of ΔEv, most of the current is concentrated at the bottom interface in the crystalline core. For ΔEv = 0.15 eV, an homogeneous current distribution is obtained all along the heterointerface.
In the text

	Fig. 9 Mapping of the dark current density distribution near the interface of a silicon crystalline homojunction wire with a doping density of the core Na = 1 × 1017 cm-3 and Na = 1 × 1019 cm-3.
In the text