Back Surface Recombination Velocity Modeling in White Biased Silicon Solar Cell under Steady State

In this paper, we extend the concept of back surface recombination through a study of a silicon mono facial solar cell in static regime and under polychromatic illumination. Back surface recombination velocities noted Sbe , Sbj and Sbr are determined for which respectively we derived, the power, the fill factor and the conversion efficiency, that become constant whatever the thickness of the solar cell. We have then obtained the expression of the minority carrier’s density in the base from the continuity equation. We then have determined the photocurrent density, the photo voltage, the power, the fill factor and finally the conversion efficiency.


Introduction
Today one of the main objectives in the domain of the photovoltaic energy is to improve the solar cell efficiency and to reduce the manufacturing cost.Until now, the conversion efficiency in photovoltaic solar is in the order of 20% in the laboratories but only in the order of 17% to the commercial level.This objective has to optimize the solar cell thickness (H) and the recombination parameters in the bulk and surfaces.
n L and n D are linked by the well-known Einstein expression: n τ is the excess minority carrier lifetime n G is the excess minority carriers generation rate.
The n G is in the form of a whole series [11] as it is expressed in following where 0 λ (=0.3 μm) and g λ (=1.2 μm) are respectively the minimum and the maximum wavelength in the polychromatic source.0 F λ is the incident photon flux; 0λ α is the monochromatic absorption coef- ficient of the material; 0 r λ is the monochromatic reflection coefficient at the surface of the material; The solution of the continuity equation with the conventional method can be written as: ( ) A and B are determined by the boundary conditions.The junction at x = 0 ( ) ( ) where Sf is the minority carrier recombination velocity at the junction [12] [13] [14].
Sf is expressed as the sum of two terms: At the rear face: H is the solar cell base thickness and Sb is the excess minority carrier recombination velocity at the back surface [15] [16] and yields to low values for sample with back surface field (BSF).H is the solar cell base thickness.

Photocurrent Density
The solar cell photocurrent density is determined from the minority carrier density by the following relationship:

Photovoltage
The Boltzmann relation gives us the following photovoltage expression as: , ln 1 , , With: n i is the intrinsic carrier density, N B is the base doping density, V T is the thermal voltage at a given temperature T.

Power
The output electric power is an essential parameter for a solar cell.It indicates the capacity of the solar cell to provide an electricity to the external load.The electrical power produced by the solar cell under polychromatic illumination constant and for a given operating point is determined by: With: J is the photocurrent density in the external load, Jd represents the dark current density [14] [17], and , 0 n : carrier density at equilibrium.

Photocurrent Density
We plot on the curves 4 and 5, the profile of the photocurrent density versus junction recombination velocity ( Sf ) respectively for a given thickness H and for different Sb values On Figure 2, for a given curve, we observed three zones [17]:      photocurrent density is higher for the low values of the back surface recombination velocity whereas when the recombination velocity is high, it is more important for thick solar cells.

Photo Voltage
We have plotted on Figure 4 and Figure 5 respectively the effect of Sb and thickness on the photo voltage versus junction recombination velocity ( Sf ).
On Figure 4, for each curve we observe two zones [17]: i) in the interval

Power
Recognizing the expression of the power, we represent respectively on Figure 6       This maximum power therefore corresponds to a junction recombination velocity optimum that can be also obtained by the following equation Then for 4 4 10 cm s Sf ≥ × , it tends towards the short circuit condition, then the photovoltage tends to nil value [17].This then causes a decrease of the power which cancels itself for large Sf values.
Figure 8 always shows that the power decreases with the increasing back surface recombination velocity.This is from to the recombination lost factor is negligible for small Sb values.But when the back surface recombination velocity gradually increases, the power decreases because for Sb values all the carriers generated are lost via recombination.
On the contrary, Figure 7 shows that the power increases with the thickness of the solar cell.Because the photocurrent density increases with the thickness causing then an increase of the power.
In Figure 8, we notice that for a given curve, the power has a horizontal stage .As Sb reflects the lost factor of recombination, when it is weak, the power reaches its most important value, then the power decreases when Sb increases to finally take its smallest value for the large Sb value.The curves show that for Sb Sbe ≤ the power decreases with the increase of the thickness.Whereas for Sb Sbe ≥ , it increases with the thickness Figure 9 shows that for Sb Sbe = , all the curves are confused.This shows that Sbe is a back surface recombination velocity which allows obtaining the same power whatever the solar cell thickness.
We can conclude that, for solar cells of different thicknesses, the most important power is obtained with the weakest solar cell in the case of lower back surface recombination velocity.But for high back surface recombination velocity, the most important power is obtained with the thinnest solar cell.
It is recognized that there is a back surface recombination velocity ( Sbe ) which allows to obtain the same power for all thicknesses.

Fill Factor (FF)
It represents the fraction of the lost power in the semiconductor material either by resistance effect or by recombination phenomena of the carrier's photogenerated.It is given by the relationship: max P is the maximum power value which can be extracted, Voc represents the open circuit photovoltage and Jsc represents the short circuit photocur- rent density.In Figure 10, we represent the curve of the fill factor (FF) versus back surface recombination velocity for different thicknesses H. the fill factor reaches its maximum value.Because for small Sb values, the loss factor of recombination is low therefore the lost power is low (Figure 8).When Sb rises (   ⋅ , the fill factor (FF) reaches its minimum value.In this interval, the lost factor reaches its maximum value leading thus to a minimum fill factor because corresponding to a minimum power (Figure 8).The curves show that the fill factor (FF) decreases with the increase of the thickness for Sb Sbj ≤ .Whereas for Sb Sbj ≥ the fill factor increases with the thickness.Sbj corresponds to a back surface recombination velocity which gives the same

Conversion Efficiency
The conversion efficiency is the fraction between electrical power generated by the solar cell and the power of the incident flux ( Pinc ) received by this solar cell.
The maximum efficiency is, the fraction between the maximum power charged in and the forward power ( 21 sun 100 mW cm Pinc = = ).The maximum conversion efficiency is given by the following relationship: We present in Figure 11, the profile of the conversion efficiency versus back surface recombination velocity for different thicknesses H.
The conversion efficiency curves versus back surface recombination velocity have the same profile.
For a given thickness H, the conversion efficiency is constant for

Conclusions
In this work, we have studied first the profile of photocurrent density, the profile of the photo voltage and of the power.The knowledge of these parameters has allowed then to study the fill factor and the conversion efficiency.
For the photocurrent density, solar cells generate a larger photocurrent when the junction recombination velocity is high (

Figure 1 .
Figure 1.Solar cell illuminated by its front side.

0
Sf is related to the shunt resistance.It represent the intrinsic junction recombination velocity.m Sf characterizes the photocurrent collected through the external load m R connected to the solar cell.

i) for the interval 10
cm s Sf ≤ , the photocurrent is zero.The minority carriers are blocked and stored at the junction.Any carrier crosses the junction: this is the open circuit condition.constant and retains its maximum value corresponding to the short-circuit photocurrent.All the photogenerated carrier crosses the junction: this is the short circuit.The effect of the back surface recombination velocity on the short-circuit photocurrent density isreflected by an increase of the photocurrent density when Sb decreases (BSF effect).On Figure3(a) and Figure3(b), it is observed that for slight Sb values, the effect of the thickness on the photocurrent is very low.But the photocurrent decreases slightly with the increase of the solar cell thickness (Figure3(b)).On the contrary for large Sb values, the effect of the thickness on the photocurrent density is only visible for large Sf values ( , the short-circuit photocurrent density increases with the thickness (Figure 3(b)).For 0.02 cm H ≥ , the photocurrent density becomes nearly constant.Therefore for thin photovoltaic solar cell (low thickness), the short-circuit

Figure 2 .Figure 3 .
Figure 2. Photocurrent density versus junction recombination velocity for different Sb values: Photocurrent density versus junction recombination velocity for different H values and for slight Sb value.
photovoltage is constant and corresponds to the open circuit photovoltage Voc .Here the carriers photo gener- ated are blocked and cannot cross the junction, their accumulations explain the maximum photovoltage value.ii) in the interval photovoltage decreases rapidly up to cancel itself.This is because all excess minority carriers photo generated gradually cross the junction until there are charges no more: this is the short circuit.The curves show that for low Sb values (Figure 5(a)), Voc decreases with the increase of the solar cell thickness whereas high Sb , Voc increases with the thickness (Figure 5(b)).The open-circuit photo voltage is all the more slight that the solar cell is thin for the low back surface recombination velocity.Whereas for the thick solar cells the open-circuit photo voltage is low for high back surface recombination velocity values.

Figure 5 .
Figure 5. (a) Photo voltage versus junction recombination velocity for different H values and for low Sb values: 1 200 cm s; Sb = values

Figure 6 .
Figure 6.Power versus junction recombination velocity for different Sb values:

Figure 7 .
Figure 7. Power versus junction recombination velocity for different H values:

Figure 8 .
Figure 8. Power versus back surface recombination velocity for different H values:

Figure 9 .
Figure 9. Power versus junction recombination velocity for different H values, with Sb Sbe = :

Figure 10
Figure10shows that for a given curve, the fill factor (FF) is constant in the interval loss factor of recombination increases.The power decreases then (Figure8) thus causing the decrease of the fill factor (FF).

Figure 10 .
Figure 10.Fill factor versus junction recombination velocity for different H values:

⋅
. The curves show that for Sb Sbr ≤ , the conversion efficiency decreases with the increase of the thickness, while for Sb Sbr ≥ , it increases with the thickness.This allows to conclude that the thin solar cells are DOI: 10.4236/jmp.2018.92012200 Journal of Modern Physics

Figure 11 .
Figure 11.Conversion efficiency versus junction recombination velocity for different H values: the thick one with low Sb values.On the other hand, in the range of high Sb values, thick solar cell are more advantageous.Sbr is a back surface recombination velocity which allows to get the same conversion efficiency with the different thicknesses of the solar cell.
The thick solar cells (H = 0.025 cm) are more beneficial because they generate a higher photocurrent density.For the photo voltage, the open circuit photo voltage decreases with Sb .It also decreases with the thickness for the small Sb values but increases with the thickness for the high back surface recombination velocity.For the power, fill factor and conversion efficiency, we have found Sbe , Sbj and Sbr respectively, the back surface recombination velocities for which they are invariable whatever the solar cell thickness is.