Diffusion Coefficient in Silicon Solar Cell with Applied Magnetic Field and under Frequency: Electric Equivalent Circuits

In this paper, a theory on the determination of the diffusion coefficient of excess minority carriers in the base of a silicon solar cell is presented. The diffusion coefficient expression has been established and is related to both frequency modulation and applied magnetic field; the study is then carried out using the impedance spectroscopy method and Bode diagrams. From the diffusion coefficient, we deduced the diffusion length and the minority carriers’ mobility. Electric parameters were derived from the diffusion coefficient equivalent circuits.


Introduction
Photovoltaic conversion is ensured by a solar cell whose conversion efficiency depends on the nature of the semiconductor structure, its manufacturing technique and processes.This study deals with the minority carriers' diffusion coefficient in the base.Several studies have been done on the minority carriers' diffusion coefficient [1] [2] and diffusion length [3]- [6] in the goal of improving solar cells quality for a better conversion efficiency.Authors determine the diffusion coefficient that depends on both frequency modulation and magnetic field in one hand or combine local diffusion length calculated from the laser beam induced current and bulk lifetime ob-tained from microwave-detected photo conductance decay measurements in another hand.They also determine the diffusion length using electroluminescence emission measurements of multicrystalline silicon solar cells at near-bandgap wavelengths which coincides with the one calculated from spectrally quantum efficiency methods.Our study consists on the determination of the diffusion coefficient by using the DRÜDE model, when the solar cell is under a polychromatic illumination in frequency modulation and under an applied magnetic field.Expression of the diffusion coefficient is established as a complex number according to the frequency and the magnetic field.This implies that capacitive and/or inductive effects in the diffusion phenomena directly are related to the solar cell behavior.Considering the minority carrier diffusion in the base as an electric phenomenon; an electric-diffusion analogy, by means of the Bode and Nyquist diagrams [7] of the diffusion coefficient, is pointed out.Equivalent circuits of the diffusion coefficient are then deduced and the related parameters are calculated.

Theory
On Figure 1, a simplified n + -p-p + silicon solar cell type [8] [9] is presented: Where d is the thickness of the emitter and H 0 the thickness of the solar cell.
The solar cell illumination, in frequency modulation, generates electron-hole pairs in the base where a constant magnetic field = B Cte is applied perpendicularly [10] [11].The excess minority carriers in a p-base (electrons) are subjected to the electrical force e F , magnetic force m F and frictional force f F during their diffusion in the crystal lattice.By use of the basic dynamics principle applied on electrons, one obtains Equation (1): where m n , q and τ n are respectively the effective mass of the electron, the elementary charge and the average lifetime of electrons in the base; V is the electron velocity; E the electric field resulting from the base pola- rization.
Equation (1) can be rewritten as follows: The electron velocity and the electric field can be expressed as: where o V and o E are respectively the velocity and the electric field amplitudes; t is the time, 2πf ω = is the angular frequency and f the frequency; i is the imaginary number (i 2 = -1).
Substituting Equation (3) in Equation ( 2), one obtains: where Equation ( 4) becomes: In one hand, using the vectorial product for each members, and in the other hand the scalar product of o V and 2 K , one gets: and ( ) where we substitute Equations ( 6)- (9) in Equation ( 5), and we get: Equation ( 10) can be rewritten in the form: (11) with c n qB m ω = the cyclotron frequency [12] [13] of the electron.
The term is the parallel component of the electron velocity and is neglected because magnetic energy according to this direction is neglected.So Equation (11) becomes: We rewrite Equation ( 12) as: where is the complex mobility of electrons according to the frequency modulation and the magnetic field.
From the Einstein relation , where K B is the Boltzmann constant, T the absolute temperature and * n D the complex diffusion coefficient of the electrons according to the magnetic field and the frequency modulation [14] [15], we obtain: where D n is the intrinsic diffusion constant of the electrons without applied magnetic field.

Diffusion Coefficient Profile
Figure 2 presents the diffusion coefficient of the minority carriers in base of the solar cell in 3D view according to the frequency modulation and the applied magnetic field: One can observe that diffusion coefficient decreases with the frequency modulation; effectively, for higher frequency, carriers cannot relaxate so that they cannot move properly in the material.Given that diffusion coefficient expresses the ability of carriers to diffuse in the material, frequency modulation increase leads to a direct decrease of the diffusion coefficient.Diffusion coefficient increases with magnetic field until a certain magnetic field value B o from which the diffusion coefficient decreases.This maximum value of the diffusion coefficient corresponds to a resonance phenomenon.Above B o , minority carriers trajectories are more and more incurvated so that they cannot move in the base: this corresponds to a decrease of the diffusion coefficient with magnetic field.The decrease of the diffusion coefficient while increasing magnetic field and frequency modulation degradates the intrinsic properties of the solar cell by affecting basically the energy bands.

Bode Diagram of the Diffusion Coefficient
The logarithm of the diffusion coefficient module and its phase are represented according to the logarithm of the frequency for various magnetic fields on Figure 3.

Figure 2. Diffusion coefficient module versus frequency modulation f and magnetic field intensity B.
On can note on Figure 3(a)) that without magnetic field, the diffusion coefficient doesn't depend significantly on frequency in quasi steady state (ωτ n  1) and the associated phase (Figure 3(b)) is practically zero: carriers diffusion follows their generation in the base of the solar cell.For higher frequencies, the diffusion coefficient decreases very markedly: the diffusion process is slowed down compare to the generation process; this fact is confirmed by the phase plot with a negative value of the phase.The solar cell behaves as a capacitance from the point of view of the diffusion process.The application of a magnetic field induces resonance peaks of the diffusion coefficient for higher frequencies.This resonance phenomenon arises when the frequency modulation is equal to the cyclotron frequency (the electron frequency on its orbit in the presence of an applied magnetic field).For these higher frequencies, the phase increases to a maximum positive value until a certain frequency corresponding to the resonance phenomenon.Above that frequency, the phase decreases rapidly to a negative value for the remaining frequencies (Figure 3(b)).We then have three different behaviors of the solar cell from the point of view of the diffusion phenomenon:  a resistive effect with phase practically zero. an inductive effect from the cut-off frequency to the resonance frequency. a capacitive effect above the resonance frequency.

Nyquist Diagram and Equivalent Circuit Model of the Diffusion Coefficient
We, now, present the Nyquist diagram of the diffusion coefficient with the associated electrical model for three magnetic field values: 1) B = 0 T The obtained Nyquist diagram is presented on Figure 4(a); this diagram shows three particular points that correspond to: This plot is semi-circular with only negative values of Im(D * ).Given also the Bode diagram, the associated electric equivalent circuit can be presented as on Figure 4(b) [16]- [21].We have a resistance R P in parallel with a capacitance C; R P characterizes resistive effect in the diffusion process (low frequency) and C characterizes capacitive effect at high modulation frequencies; f is the beginning of the resonance and 4 f the end of the resonance.From 0 f = to   we have a resistive and an inductive behavior where both resistance R P2 and inductance L are in while from We can see the Nyquist plot is a circle shape.We have both inductive and resistive behavior from 0 a resistive and capacitive behavior.We have here, resistance R P2 , in- ductance L and capacitance C in parallel.The corresponding electric equivalent circuit is given on Figure 6(b).
We note that resistance R P1 is now neglected because the photogenerated minority carriers do not diffuse any more under the effect of the intense applied magnetic field.

Results and Discussion
Without any magnetic field, the diameter of the semi-circle obtained is equal to R P [17]

Conclusion
A theoretical study has been made on the diffusion coefficient of a crystalline silicon solar cell under polychromatic illumination, in frequency modulation and under magnetic field.With an applied magnetic field, the diffusion of the minority carriers is slowed down because carriers are deviated from their normal trajectories on side surfaces of the base; what involves a poor quality solar cell.Bode and Nyquist diagrams show that the resonance effect appears with the frequency and the magnetic field which change the cell behavior: capacitive, resistive and inductive.For the frequencies and magnetic fields, we electrical equivalent circuits of the diffusion process.

Figure 1 .
Figure 1.An n + -p-p + type of a silicon solar cell scheme under applied magnetic field.
to the cut-off frequency.2) B = 10 -6 T Figure 5(a) and Figure 5(b) show respectively the Nyquist diagram and the equivalent circuit of the diffusion coefficient:The Nyquist diagram has a spiral shape with positive and negative imaginary part.There are now, five particu- Figure 3.

Figure 4 .Figure 5 .
Figure 4. Nyquist diagram and electric equivalent circuit of * n D without magnetic field.(a) Nyquist diagram; (b) Electric equivalent circuit.
to f → ∞ we have now resistive and capacitive behavior.The corresponding equivalent circuit is presented on Figure5(b).3)B = 10 -4 T In Figure6(a) Nyquist plot (Figure 6(a)) and an equivalent circuit model (Figure 6(b)) are presented: [18] which is a resistance associated to the steady state diffusion process characterized by D n .The corresponding diffusion length is deduced from the relation * the Einstein relation, the minority carriers' mobility is also

Table 1 .
Electric and intrinsic parameters for *