New Approach to Determine Vertical Parallel Junction Silicon Solar Cell Parameters

The aim of this work is to study the electrical behavior of a vertical parallel junction silicon solar cell by mean of impedance spectroscopy. By solving diffusion-recombination equation, the excess minority carrier density expression was established and the photocurrent and the photo voltage were deduced. We plotted Nyquist diagram of the solar cell impedance and gave corresponding electrical circuits. We then introduced impedance spectroscopy method to access for the first time to some electrical parameters of a vertical junction solar cell (series resistance, parallel resistance, capacitance and inductance).


INTRODUCTION
The increasing demand for clean energy and the largely untapped potential of the sun as an energy source is making solar energy conversion technology increasingly important.As a result, the demand for solar cells, which convert sunlight directly into electricity, is growing.Solar or Photovoltaic (PV) cells are made up of semiconductor materials that absorb photons from sunlight and then release electrons, causing an electric current to flow when the cell is connected to a load (charge).A variety of measurements are used to characterize a solar cell's performance (Green, 2008;Dieng et al., 2011;Kumar et al., 2001).In this study, we will lay stress on the impedance spectroscopy (Scofield, 1995;Chenvidhya et al., 2005) as a tool for vertical junction electrical characterization; to the best of our knowledge, this is the first time impedance spectroscopy is applied to a vertical junction solar cell.Based on the continuity equation, the excess minority carrier density is derived; from that excess minority carrier density, both photocurrent density and photovoltage are calculated.The vertical junction solar cell impedance is then deduced.By help of Nyquist diagram, electrical circuits of the vertical junction are proposed and from impedance spectroscopy theory, series resistance, parallel resistance, capacitance and inductance are extracted.• The space charge region and the emitter thicknesses are very small compared to that of the base; their contributions are then neglected so that only the contribution of the base is taken into account (Sissoko et al., 1998).• The emitter and the base regions are quasi-neutrals (Mathieu and Fanet, 2009); there is no internal electric field without the space charge region.• The electrical and optical properties are identical at any point in the base; this enables us to use the Cartesian coordinates.We also took into account the reflection on the silicon material.
• The temperature effect on the performance of the photovoltaic cell is not taken into account.• The excitation is done with a monochromatic frequency modulated light.The incident photons flux ɸ (λ) is taken for AM 1.5 solar spectrum (Green, 2008).This case is characteristic of the whole terrestrial applications independently of the processing site.Excess minority carrier density: Taking into account generation, recombination and diffusion phenomena in the base, the equation governing the variation of the minority carriers density δ (x, y, z, t) under frequency modulation (Dieng et al., 2011) is: D (ω) and τ are, respectively the excess minority carrier diffusion coefficient and lifetime.The excess minority carrier density can be written as: The excess minority carrier generation rate is given by Dieng et al. (2011): ) x = The base depth along x axis ω = The angular frequency θ = The incidence angle z = The base depth according to the vertical axis λ = The illumination wavelength If we replace Eq. ( 2) into (1), the temporal part is eliminated and we obtain: The solution of this equation is in the form: ( ) Coefficients A and B is determined through the following boundary conditions (Dieng et al., 2011): Sf is related to the carrier flow through the junction (Diallo et al., 2008).• At the middle of the base (x = H/2): Photocurrent density: The main contribution to the photocurrent Jph comes from diffusion; Jph can be written in the form: q is the elementary charge.
The factor 2 results from the two junctions around the base and the fact that they are connected in a parallel manner.

Photo voltage:
According to the Boltzmann's relation, the photo voltage is given by Sze and Ng. Kwok (2007) and Hu (2010): With V T the thermal voltage, Nb the base doping density and n 0 the intrinsic carrier density.

RESULTS AND DISCUSSION
We plotted in Fig. 2a and 3a the Nyquist diagram of the vertical junction impedance; the obtained curves are semi-circles of two types: positive one and negative one depending on incidence angle value.For the negative one (Fig. 2a) the parallel vertical junction behaves inductively contrary to the curve in Fig. 3a where capacitance phenomenon predominates.From these two behavior the vertical junction solar cell can be represented by one of the electrical circuit presented on Fig. 2b and 3b Impedance spectroscopy technique: If we consider for example the capacitive behavior the cell with the associated electrical circuit of Fig. 2b, the impedance Z is given by Rs in series with Zc parallel to Rp; Rp = Rsh//R D , Zc = 1/ (jcω).For the circuit of Fig. 3b, the following calculations remain valid if Zc is replaced by Z L = jLω.We then have: If we replace Rp and Zc by their expressions and after some calculations, we obtain: Z is a complex number which real part Re (Z) and imaginary part Im (Z) are given by:  ( ) From the equations above, several particular cases can be considered according to the modulation frequency on one hand and to the ratio of R D and Rsh: The impedance can then is measured in the form: Z = Rs independently of the ratio of Rsh and R D .
Taking into consideration these calculations, if we return to the Nyquist diagram itself, we can associate some particular points depending on the modulation frequency (Dieng et al., 2011) as presented in Fig. 4.
The Nyquist diagram corresponds to a semi-circle of radius Rp/2 and center (Rp/2+Rs, 0).A correspond to low modulation frequency (ω→0) and C to higher modulation frequencies (ω→∞).That is, some electrical parameters can be determined the mean of the technique described above.
To determine the associated capacitance, let us consider the following equation obtained thanks to the cut-off frequency: That leads to C = 5,011.10 - µF.
In the case of an inductive behavior, we have (Table 2).The associated inductance is deduced from Eq. ( 17): This leads to L = 2,505/10 8 µH.

CONCLUSION
Based on an ac-equivalent circuit of a parallel vertical junction solar cell, an expression of the impedance has been established.From Nyquist diagram, we presented two electrical circuits of the solar cell.The impedance spectroscopy technique is then applied to extract for the first time the vertical junction electrical parameters.

Fig. 1 :
Fig. 1: Vertical junction unit cell METHODOLOGY Theoretical development: This study is based on the vertical junction silicon solar cell presented in Fig. 1 with the following assumptions:• The space charge region and the emitter thicknesses are very small compared to that of the base; their contributions are then neglected so that only the contribution of the base is taken into account(Sissoko et al., 1998).• The emitter and the base regions are quasi-neutrals(Mathieu and Fanet, 2009); there is no internal electric field without the space charge region.• The electrical and optical properties are identical at any point in the base; this enables us to use the Cartesian coordinates.We also took into account the reflection on the silicon material.
Fig. 2: (a) Nyquist diagram, (b) electrical circuit of solar cell for capacitive effects Sf: 3.10 3 cm/sec; H: 0.03 cm; Lo: 0.02 cm; z: 0.0001 cm; Do: 26 cm²/sec; λ: 0.5 µm : • C is the total capacitance; C = C T + C D where C D is the diffusion capacitance due to the diffusion of the excess minority carrier density in the base and C T is the transition capacitance due primarily to the fixed atoms ionized in the space charge region.• Rsh models leakage currents existing at the edge of the structure, the whole of the defects in the vicinity of the space charge region (dislocation, recombination centers) and in the base.• R D is the solar cell dynamic resistance, it indicate the local behavior of the solar cell for a given operating conditions.• Rs is the series resistance related to material resistivity and metallization grid.• L is related to the inductive behavior of the cell; it comes from the frequency effects through low resistances particularly the quasi-neutrals regions and metallization.Both capacitive and inductive effects play very important role in determining solar cell parameters (like under flashed irradiance and transient experiments) and in switching circuits with power inverters.

Fig. 4 :
Fig. 4: Nyquist diagram with the particular points Sf: 3.10 3 cm/sec; H: 0.03 cm; Lo: 0.02 cm; z: 0.0001 cm; Do: 26 cm²/sec; λ: 0.5 µm; θ: 48.2° the imaginary part Im (Z) of Z is null and only the real part Re (Z) remains: Re (Z) = Rs + R D •Rsh/ (R D + Rsh) The impedance is rewritten in the form: Z = Rs + R D •Rsh/ (R D + Rsh) o If Rsh>>R D we have Re (Z) = Rs + R D and the impedance Z is reduced to: Z = Rs + R D o If Rsh<<R D then one has Re (Z) = Rs + Rsh and thus Z = Rs + Rsh • If ω→∞ the imaginary part Im (Z) of Z is also null and we have: Re (Z) = Rs

Table 2 :
Series and parallel resistances inductive behavior