Study of the Intrinsic Recombination Velocity at the Junction of Silicon Solar under Frequency Modulation and Irradiation

In this study, a method for determining the intrinsic recombination velocity at the junction of a silicon solar cell is presented. The expression of intrinsic recombination velocity at the junction was established under irradiation in frequency modulation. Based on this expression, an electrical model of the intrinsic recombination velocity at the junction is presented.


Introduction
Photovoltaic conversion is provided by a solar photovoltaic cell whose conversion efficiency depends on the nature and the semiconductor structure, its manufacturing technique and operation.Given the low efficiency of these solar cells, researchers have been involved in various research works by offering several characterization techniques of the semiconductor material and in particular on the design of solar cells.Among the most important parameters in the different techniques characterization, one can cite: the diffusion coefficient we propose in this paper, a method for determining the intrinsic recombination velocity at the junction of a silicon solar cell monofaciale under monochromatic illumination in frequency dynamic regime under irradiation and finally an electric circuit model equivalent was presented [6] [7].

Theoretical Study
The photovoltaic cell considered is of type n + pp + [8] [9] and its structure is presented at the Figure 1.
Under the effect of an excitation (optical or electric) of the charge carriers are generated in the base the photovoltaic cell.They can either pass through the space charge region where they participate in the external current, or they undergo recombination.The latter are due to defects (grain boundaries, impurity atoms...) related to the manufacture of the light silicon solar.Taking into account the generation phenomena, diffusion and recombination in the solar cell, the equation of continuity the minority carrier charge in the base at x frequency in dynamicregime is of the form: where D * is the diffusion coefficient G and the global rate of generation of carriers.The determination of the diffusion coefficient and the global rate of generation proves fundamental for the study of a solar cell in static regime or frequency dynamic regime.Indeed, several studies have been done on the global rate of generation and the diffusion coefficient [10] in order to improve the performance of conversion.Indeed, this last describes the diffusive nature of minority carriers in material, in other words it expresses the capacity of the carriers to be spread within material.Different expressions of the global rate of generation and diffusion coefficient are proposed: a) Global rate of generation under monochromatic illumination [11].
The expression of the global rate of generation under monochromatic illumination in frequency dynamic regime and static mode is given by the relation: with g n (x) the space component and e i t ω time component.According to the illumination mode the generation rate g n (x) as follows: For illumination of the front face: (n = 1) ( ) ( ) For illumination the back face: (n = 2) For simultaneous illumination of two faces: α(λ) is the absorption coefficient at the wavelength λ; I 0 is the flux incident monochromatic light; R(λ) is the reflection coefficient at the wavelength λ; H is the thickness of the solar cell.
b) Global rate of generation under polychromatic illumination [12].
The expression of the global rate of generation under polychromatic illumination is given by: According to the illumination mode, the generation rate g n (x) is as follows: For illumination of the front face: ( ) For illumination the backface: For a simultaneous illumination of two face: a i and b i are coefficients deduced from solar radiation under AM 1,5 spectrum; we set n = 1 (Number of sun).c) Diffusion coefficient according to the rate of doping [13].
The doping of semiconductors is an important element for the manufacture of electronic components.Indeed, a pure semiconductor (intrinsic) is almost insulating at the ambient temperature, with a valence band almost entirely full, and a tape of conduction almost entirely empty.Therefore, the interesting electronic properties are related to the possibility "of doping" material by the introduction of adequate impurities making it possible to introduce free carriers.The diffusion coefficient depends on the doping level is given by the equation: where T V k T q = ⋅ is the thermal tension, K the Boltzmann constant, q elementary charge and Nb is the rate of doping in the base.d) Diffusion coefficient according to the magnetic field [14].
The expression of the diffusion coefficient D * as a function of magnetic field in the static mode is given by the relationship: D 0 is diffusion constant without magnetic field, μ is the electron mobility and B is the intensity of the magnetic field.
e) Diffusion coefficient according to the rate of doping and the magnetic field [15].The expressions of the coefficient of diffusion and diffusion length according to the magnetic field and rate of doping on a photovoltaic cell with static rate were proposed.It is given by the relation:  [16].The expression of the diffusion coefficient as a function of frequency is given by the following relationship: With D is the coefficient of minority carrier diffusion in the base without of pulsation, radiation and magnetic field.g) Diffusion coefficient according to the frequency and the rate of doping [17].
Under the effect of doping rate, the expression of the diffusion coefficient in frequency dynamic regime is of the following form: h) Diffusion coefficient according to the frequency and the magnetic field [18].Frequency in the dynamic regime and under the effect of a magnetic field, the expression of the diffusion coefficient is given by the following relationship: is the cyclotron frequency and m n the mass of the electron.
i) Diffusion coefficient according to the frequency, the damage coefficient and radiation energy [10].
The terms of the diffusion coefficient and diffusion length depending on the radiation energy and damage coefficient by frequency dynamic regime are given respectively by the following equations: ( ) , p D kl ϕ is the diffusion coefficient according to the damage coefficient and radiation energy.

( )
, p L kl ϕ is the diffusion length according to the damage coefficient and radiation energy.
L 0 is the diffusion length without of pulsation, radiation and magnetic field.
( ) is the diffusion length according on the damage coefficient of radiation flux and magnetic field.j) Diffusion coefficient according to the frequency, the damage coefficient, the radiation energy and the magnetic field [20].
The expression of the diffusion coefficient as a function of the modulation frequency and the intensity of the magnetic field is given by equation k) Diffusion coefficient according to the recombination velocity at the grain boundaries and grain size [21].
The diffusion coefficient in function of the recombination velocity at the grain boundaries and the grain size is obtained from the solution of the equation of continuity of the three-dimensional base.By using the boundary conditions, we obtain two transcendental equations.The expressions transcendental equations and the diffusion coefficient are given respectively by the following Equations ( 21), ( 22) and (23).
g x is the width of the grain, g y the length of the grain S g b effective recombination velocity at grain boundaries, C k and C J are the eigen values of transcendental equations.From these equations, we will draw the expressions of eigen value C K and C J based on the recombination rate at the joints and of the grain size.Thus the expression of the diffusion coefficient as a function of the grain size g and the recombination velocity at grain boundaries is given by equation ( 24): l) Diffusion coefficient according to the temperature [22].
The diffusion coefficient based on the static state temperature is given by the relationship Albert Einstein which is written in the form: ( ) [23] is the mobility of minority carrier's according to the temperature.
m) Diffusion coefficient according to the electric field [24].Under the effect of an electric field and static regime the diffusion coefficient of minority carriers in the base is given by the following expression: where L E is a factor that accounts for the phenomena of migration in the base of the solar cell.
After having pointed out some expressions of global generation and diffusion coefficient, we can solve a priori the continuity equation.For the resolution of the continuity equation, the overall rate of generation and minority carrier's density can be expressed in the following form, respectively.
( ) ( ) ( ) Thus Equation ( 1) can be put in the form: The Equation ( 30) being a differential of the second degree with second member thus the general solution is: For the determination of the coefficients A and B we are used the boundary conditions [25] following.At the junction (x = 0) ( ) ( ) At the back surface (x = H) ( ) ( ) where Sf and Sb are respectively the recombination velocity of minority charge carriers at the junction and at the back face; Sf is the sum of two contributions: [26] 0 Sf 0 is the intrinsic recombination velocity at the junction induced by the shunt resistor and the Sf j is the recombination velocity related to the load imposing the point of operation on the photovoltaic cell.

Photocurrent Density
The photocurrent is obtained by the gradient of minority carriers at junction and is given by the following expression: , , , , , ,  , , , , , , where q is the elementary charge.On Figure 2, we represented the profile of photocurrent density versus the  The shape of this curve shows that for the photovoltaic cells of the type BSF (low Sb), the density of photocurrent is maximum for finally decrease until reaching a constant value for the great Sb values modelling solar cells of an ohmic type.

Expressions of the Intrinsic Recombination Velocities at the Junction SF 0 .
The intrinsic recombination velocity at the junction recombinatoires characterizing certain phenomena of excess minority carriers in solar cells surfaces, is investigated to check their qualities.Thus, studies have been conducted on the determination of intrinsic expressions of recombination velocities at the junction into static regimes, transient and frequency dynamic in monochromatic or polychromatic illumination.Indeed, the shape of the curve of variation of the photocurrent density of the modulus versus the recombination rate at the back side Sb shows that at high values of the recombination rate at the back side Sb, these curves show each horizontal level.The gradient of the photocurrent compared to Sb tends then towards zero.Thus, the intrinsic recombination velocity at the junction is obtained by derivation of the density of the photocurrent (Jph) compared to the recombination velocity at the back face Sb.
Depending on the system studied and the manner of illumination of the solar cell, a number of expressions of the intrinsic recombination velocity at the junction Sf 0 have been proposed.a) Static regime [27] [28] When the bifacial solar cell is illuminated by its front or its rear face with a monochromatic light, the intrinsic recombination velocity at the junction is given by the following expressions.
For illumination of the front face: ( ) For illumination the back face: exp exp exp exp 1 For simultaneous illumination of two face: ( ) When the solar cell is under polychromatic illumination via its front or its rear face, expressions intrinsic recombination velocity at the junction are given by: For illumination of the front face: ( ) ( ) For illumination of the back face: ( ) For simultaneous illumination of two face: Parameter b i is coefficient deduced from solar radiation under AM 1,5 [29].
In the three-dimensional model, the intrinsic recombination velocity at the junction which takes account of recombination velocity at the grain boundaries is given by: Under polychromatic illumination: , 1 sinh cosh exp cosh cosh sinh cosh Under monochromatic illumination: ( ) k, j: directions related indexes x and y respectively.L kj : Username scattering length associated with modes j and k in the database.b) Transient dynamic regime [30].
When the transient dynamic regime is obtained by varying the operating point of a solar cell under constant multispectral illumination, the intrinsic recombination velocity at the junction depends on the eigenvalues of the transcendental equation is given by: ( ) The expression of the intrinsic recombination velocity at the junction is: In the presence of magnetic field, it has the same expression but with D which depends on the magnetic field.We have: c) Frequency Dynamic regime [31].
When illuminates the bifacial solar cell with a multispectral light frequency modulation, new expressions of the intrinsic recombination speeds are obtained: • For illumination from the front side, the intrinsic recombination velocity at the junction is given by: ( ) • For illumination the back face, the intrinsic recombination velocity at the junction is given by: ( ) • If a simultaneous illumination of both sides, the intrinsic recombination velocity at the junction is given by: In these expressions ( ) represents the diffusion length according of frequency.D ω is the complex scattering coefficient according of frequency.The profile of each intrinsic recombination velocity considered, shows an increase in speed as a function of the modulation frequency of the illumination; with the result that it y' has a very significant recombination of excess minority carriers in the junction of the photovoltaic cell.When one varies the incident wavelength, in addition to the modulation frequency, the same expressions of intrinsic recombination velocities above are obtained.These recombination rates decrease as a function of wavelength.

Bode Diagram of the Intrinsic Recombination Velocity
The expression of the intrinsic recombination velocity at the junction of a photovoltaic cell in monofociale frequency dynamic mode under monochromatic illumination in the presence of magnetic field and under irradia-tion energy is given by the following expression: ( ) The phase of the intrinsic recombination velocity at the junction Sf 0 versus the logarithm of frequency for different values of the flow is shown in Figure 3.
It is noted that the quasi-static regime Sf phase is negative, so the capacitive effects predominate, by dynamic system comprises a frequency, the phase Sf is both negative and positive.In this case, simultaneously inductive effects and capacitive effects.

Nyquist Diagram of Intrinsic Recombination Velocity at the Junction and Model Electrical Equivalent
In Figure 4 we represent the Nyquist diagram of the recombination rate at the junction for different values of the irradiation energy.
To better interpret Figure 4, there is shown in Figure 5 the imaginary part as a function of the real part of the intrinsic recombination velocity at the junction to a value of the irradiation energy.Then the imaginary part of the intrinsic recombination velocity at the junction Sf 01 is plotted against the logarithm of the modulation frequency in Figure 6.
Graphically, we find that    (related to Sf 0,p ) where short circuit of the solar cell by using calibration curves ( ) [19].For that, it is necessary to project the value of Sf 0 from the calibration curve for the corresponding resistor in each case for various irradiation energy.
The Figure 7 shows the equivalent electrical circuit of the intrinsic recombination velocity at the junction when using the Nyquist diagram (Figure 4), characterized the capacitive and inductive phenomena observed in the representation of the frequency depending on the phase (Figure 3) where C is the capacitance; L is the inductance and the parallel resistance R p characterized carriers flight resistance at the base-rear contact interface Knowing the values of the parallel resistor R p and cutoff frequency value [34] we can deduce the capacity from the relation ship: From this Table 1, we find that when the value of the irradiation energy increases, the resistance decreases R P which is due to the slowdown in minority carrier diffusion caused by irradiation.Consequently, the intrinsic properties of the solar cell are damaged; this implies a lower quality of the cells [32] [33].

Conclusion
A theoretical study has been performed on the intrinsic recombination velocity at the junction of a Sf 0 monofaciale silicon solar cell and under monochromatic illumination modulation frequency and irradiation.This study allowed us to remember the expressions of the global generation rate depending on the type of illumination and expressions of the diffusion coefficient according to some electrical parameters.In addition, the study of the intrinsic recombination velocity at the junction has allowed us to highlight the different relationships of the intrinsic recombination velocity at the junction following the plans and the type of illumination.Indeed, this study Imaginary of SF ( cm/s ) log ( ω ) also allowed us to show that frequency dynamic regime, the model of the recombination rate at the junction Sf 0 is ideal.Moreover with an applied irradiation energy, recombination of the minority carriers is slowed.Taking into account these results, an equivalent electrical model of the intrinsic recombination velocity at the junction has been proposed.

Figure 1 .
Figure 1.An n+-p-p+ type of a silicon solar cell under applied magnetic field.
, B) is the diffusion length of the minority carrier in the base under the influence of the magnetic field and the doping level.It is given by the following expression: coefficient according to the frequency

Figure 5 .
Figure 5. Imaginary part of Sf versus its real part.

Table 1 .
Electric and intrinsic parameters of the intrinsic recombination velocity at the junction Sf 0 .