Modelling Study of Magnetic Field ’ s Effects on Solar Cell ’ s Transient Decay

Experimental setup of transient decay which occurs between two steady state operating points is recalled. The continuity equation is resolved using both the junction dynamic velocity (Sf) and back side recombination velocity (Sb). The transient excess minority carriers density appears as the sum of infinite terms. Influence of magnetic field on the transient excess minority carriers density and transient photo voltage is studied and it is demonstrated that the use of this technique is valid only when the magnetic field is lower than 0.001 T.

method to measure electron diffusion coefficient (D) and electron lifetimes of dye-sensitized solar cells (DCS) [14].
Respecting to the transient state operating between two steady state real operating points, it has been carried out on a solar cell placed in a fast-switch-interrupted circuit and submitted to a constant multispectral illumination.The transient decay occurs between two steady states through operating points depending on two variables resistors; this technique allows obtaining a transient decay at any operating point of the solar cell I-V curve (from the short circuit to the open one).By combining experimental and calculation results of the photo voltage transient decay response, the eigen value ω 0 of the fundamental decay mode and the decay time constant, the minority carrier lifetime (τ) is determined.The effective minority carrier diffusion length (L eff ), the junction dynamic velocity (Sf) and the backside recombination velocity (Sb) are then deduced [10]- [17].
One of the external factors affecting efficiency of the solar cell, is the magnetic field which has various origins [19].Depending on the position of the solar cell and the emitting source of the magnetic field, one can note some effects on the solar cells devices.That is why some works concerning influence of magnetic field on solar cells are done [12] [19] [20].It is shown that, the peak power (P m ), the short-circuit photocurrent, the photo voltage and the solar cell diffusion capacitance which is reduced to the Shockley's depletion model decrease with the magnetic field when the solar cells is operating in steady state [21].It has been also proved that under frequency modulation, the electron intrinsic mobility (µ), the diffusion coefficient (D), and the diffusion length (L) decrease when the silicon solar cell is both under applied magnetic field and frequency illumination mode [12].
In this present work, we are interesting of solar cell under magnetic field and in transient decay which occurs between two steady state operating points.We examine magnetic field effects on the eigenvalue ω 0 of the fundamental decay mode, the excess minority carriers and the photo voltage decay.

Experimental Setup and Working Principle of the Transient Decay Occurring between Two Steady State Operating Points
The experimental setup (Figure 1) includes a square signal generator (BRI8500) which pilots a Mosfet transistor type RFP50N06, two variables resistors R 1 and R 2 , a silicon solar cell submitted to a constant multispectral illumination, a digital oscilloscope, and a microcomputer.The transient decay occurs according to the following procedure, described below.
At time t < 0 the solar cell is in parallel with the resistor R 2 giving the potential V 2 corresponding to the steady state operating point F 2 .At time t = 0, the fast switch begins turning on and at time t > 0 is completely turned on.It then connects the resistor R 2 in parallel with the solar cell and the resistor R 1 .The voltage V 2 drops from V 2 to V 1 corresponding to the new steady state operating point F 1 as it can be seen on Figure 2.
This experience give an experimental photo voltage decay V(t) which occurs between F 2 and F 1 .This experience is innovative because not involving light pulse and also does not consider only the open circuit photo voltage or the short-circuit photo current conditions as usually done in the OCVD and both the OCDV and PCD measurement techniques [13].
Figure 2 shows also how the I-V curves and then, the operating points are affected by the magnetic field as shown in [21].

Theory
A schematic diagram of the considered silicon solar cell under a magnetic field with its different coordinates is given in Figure 3.
The illumination was assumed to be uniform, such that the carrier generation rate depended only on depth (z) in the base and was expressed as: Coefficients a i and b i are deduced from modelling of the generation rate considered for over all the solar radiation spectrum [17].Also, the contributions of emitter and space charge region were neglected, the injection level is constant and the analysis was limited to the base region of thickness, H = 200 μm.
Hence, the transient excess minority carrier distribution ( ) , z t δ in the cell base was derived by solving the 1D continuity equation [10] [16]: where, the diffusion constant under magnetic field was expressed as [22]: µ is the mobility of excess minority carriers in the base [22].
D and τ are respectively, solar cell minority carrier diffusion constant coefficient and carrier lifetime [10].
Equation ( 2) is resolved using below boundary conditions -And at the back side of the silicon solar cell, Sf is defined as the Junction dynamic velocity [23].It is the sum of two terms which traduce losses at junction and the collected excess minority carriers.Sf was called junction recombination velocity in our previous papers [ [22].Sb is the back surface recombination velocity which traduces excess minority carriers lost at rear side of the solar cell [1]- [3] [8]- [10] [16] [17].
Equations ( 4) and ( 5) form a STURM Liouville's system of equations [9] whose solutions are separable variables of space and time.Thus, we can set: Z(z) is the space function and T(t), the time function are written, respectively: and with: The boundary conditions give: and ( ) with: n is a natural number and: Equation ( 10) is transcendental equation (of infinite number of positive roots), and was solved by Newton-Raphson numerical method [24] to obtain the allowed solutions ω n .In other works which are based to OCDV method, solution from the work of Rose and Weaver [6] are adopted.
ω 0 is the eigenvalue of the fundamental decay mode and ω n are the eigenvalues of the harmonic of order n decay mode, when n > 0.
Hence, we can see that A and B have discrete values and were finally calculated by normalization and Fourier transform [24].Thus, the transient excess minority carriers density appears as the sum of infinite terms δ n (z;t).Each term, δ n (z;t), is the contribution of order n to the transient excess minority carriers density.
When n is equal to zero, we have the first term, δ 0 (z;t), corresponding to fundamental mode which is characterized by ω 0 and when n > 0, δ n (z;t) corresponds to harmonic of order n characterized by ω n .
δ n (z;t) is expressed as: The transient excess minority carriers density is: The magnetic field effects on the eigenvalue of the fundamental decay mode and on the eigenvalues of the harmonic of order n decay mode are given on Tables 1-3.
Tables 1-3: range values of eigenvalue ω 0 of the fundamental decay mode, eigenvalues ω n (n > 0) of the harmonic of order n decay modes and corresponding decay time constant modes: We can note that, the eigenvalue ω 0 of the fundamental decay mode, the eigenvalues ω n (n > 0) of the harmonic of order n decay mode decrease with the magnetic field, conversely, the different corresponding decay time constants modes increase with the magnetic field.

Transient Excess Minority Carriers Density
For the simulation process, all the key equations were input into MathCAD software.
With base thickness and doping density fixed, the eigenvalues ω n were determined by numerical solution of Equations ( 9) and ( 10) using Newton-Raphson method [23].Compared to both OCDV and PCD techniques [13], where eigenvalues are calculated considering two operating points (open circuit and short-circuit), our method does not related to any operating point [10]- [16].
For the simulated modeling analyses, the transient excess minority carriers density was expressed as a function of magnetic field B, the junction dynamic velocity (Sf), the back side recombination velocity (Sb), the harmonic of order n, and the time (t).
Figures 4-6 show plots of the excess minority carriers density versus both the time and the magnetic field B which takes 0, 0.001 T and 0.01 T, respectively.Without magnetic field, it has been remarked that the excess minority carriers density is obtained by the fundamental mode.The contributions of the harmonic of order n is neglected as found by [9] [10]- [16].
Figure 5 is the plot of the transient excess minority carriers density when B = 0.001 T.
When the applied magnetic field is B = 0.001 T, we noted two situations.The first is when t < 2 µs, the harmonics of order 1 and 2 aren't null but would be neglected while the fundamental mode reaches to the transient excess minority carriers density.When t > 2 µs, the transient excess minority carriers density is equal to the contribution of fundamental's mode.
In Figure 6, we show the variation of the transient excess minority carriers as the function of time when B = 0.01 T.
For B = 0.01 T, the excess minority carriers density cannot be taken to be equal to the contribution of the fundamental mode.The contribution of all terms are important.
Hence, for low magnetic field (B < 10 −3 T) we concluded that the transient excess minority carriers density is given by the contribution of the fundamental mode.Thus, the transient excess minority carriers density, at the junction, is expressed as: For high values of magnetic field (B > 10 −3 T), the transient excess minority carriers density is calculated taking into account for all terms.So, both the OCDV and PCD method [13] which is used to determine the lifetime and the surface recombination velocity and the transient decay technique which occurs between two steady state operating conditions [9] [10]- [15] developed for investigation of recombination parameters wouldn't be valid in this range values of magnetic field.

Transient Photo Voltage Decay
Since the excess minority carrier density and the charge carriers gap ( ) δφ at the junction are known, from the Boltzmann relation we can derive the photo voltage transient decay across the junction as: δφ is charge carriers gap at the junction [10] and V T is the thermal voltage.
The transient photo voltage decay was also simulated by analysing dependence on magnetic field.Figure 7 shows plot of the transient photo voltage decay, at H = 200 µm, τ = 4.6 µs and ∆V = 0.01 V, with the magnetic field taking 0 T, 0.001 T, 0.01 T and 0.1 T, respectively.In Figure 7, it is shown that, the transient photo voltage decay increases with magnetic field.The increase of magnetic field corresponds to low diffusion coefficient and diffusion length as demonstrated by [12].Hence, excess minority carriers don't cross the junction and some of them are deflected leading to low transient photocurrent delivered by the solar cell as shown by [9].

Conclusions
The transient photo voltage decay occurring between two steady state operating points of a solar cell under magnetic field is studied.The corresponding experimental setup is recalled and the theoretical calculations of both the transient excess minority carriers density which is the sum of an infinite terms and photo voltage decay are made.Studies of effect of the magnetic field showed that eigenvalues decreased with magnetic field, conversely, to decay time constants which increased with magnetic field.
When the magnetic field B is lower than 0.001 T, the transient excess minority carriers density is equal to the contribution of the fundamental mode.The transient photo voltage decay can be used for the determination of recombination parameters.But for magnetic field greater than 0.001 T, the excess minority carriers density is the sum of all terms and the transient decay method is not recommended for the characterization of the solar cell.

Figure 3 .
Figure 3. Illuminated silicon solar cell under magnetic field.
decay time constant and where ω > 0.