International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 12, December 2013)
Effect of the Boundary Recombination Velocity and the Grain
Size at the Phenomenological Parameters of the Monofacial
Solar Cells under Multispectral Illumination in Steady State
S. Gueye1, H. Ly. Diallo2, M. Ndiaye3, M. M. Dione4, G. Sissoko5
1,3,4,5
Faculte des Sciences et Technologies, Universite Cheikh Anta Diop de Dakar
BP 5005, Dakar-Fann, Senegal
2
University of Thies, UFR SET, Thies, Senegal
Abstract— This paper deals with a 3 dimensional modeling
of a polycrystalline silicon solar cell in steady state under
constant multispectral illumination. After the resolution of the
continuity equation the expression of photocurrent density
and photovoltage are presented and the impact of the
recombination velocity at grain boundaries and grain size on
these parameters will also be considered.
Keywords— recombination velocity at grain boundariesgrain sizes.
I. INTRODUCTION
The recombination velocity at grain boundary Sgb and
grain size are two parameters that strongly influence the
efficiency of the solar cell polycrystalline with many
impurities that it contains. Their control contributes to the
improvement of photovoltaic conversion efficiency. To
better understand their effects, many studies have also been
carried out in three dimensions and have established their
influence on some parameters are: diffusion capacity
[S.Mbodj et al 2010, M. M Deme et al 2010], junction
surface and back recombination [M, M Dione and al 2010],
the parameters of recombination [HL Diallo et al 2006], the
electrical parameters [A Dieng et al 2009] and the space
charge region [M. Deme et M al 2009]. In our work, we
study their influence on the minority carrier’s charge
density, the photocurrent and photovoltage. The study will
be three-dimensional.
Figure 1 Model of the solar cell solar columnar
In this paper, we use a fibrously oriented columnar
model (Dugas, 1994; Deme et al., 2010; Diallo and al.,
2008) presented below (figure.1 (1 to 3)) with the
following assumptions:
• The grains have square cross section (gx= gy= g;
0.002cm  g  0.2cm ) and their electrical
properties are homogeneous. We can then use the
Cartesian coordinates;
• The illumination is uniform. We then have a
generation rate depending only on the depth in the
base z and wavelength .
• The grain boundaries are perpendicular to the junction
and their recombination velocities independent from
generation rate under an illumination AM1.5. So the
boundary conditions of continuity equation are linear;
• The contribution of the emitter and space charge
region is neglected (Dugas, 1994), so this analysis is
only developed in the base region;
II. THEORY
The polycrystalline silicon is composed of several grains
of various shapes and sizes (between 1 micron and 1 mm)
Fig 1-1. So use a columnar model (Figure 2) where the
grain will be represented by a parallelepiped fig.1-2. [J.
Thongpron et al., 2006] facilitate our study. In this model it
is possible to analyze the distribution of minority carrier
charge in three dimensions on the grain.
1
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 12, December 2013)
• The thickness H and the base doping level Nb are
130m
and 1017 cm-3 (Linda et al., 1998)
respectively.
This type of solar cell solar cell is called back surface
field (BSF) [Linda M. and Koschier et al., 1998]. We
proceed in our study as follows:
- We neglect the contribution of the emitter [Barro F.I et
al., 2008] whose thickness is very small compared to the
base;
- We assume that the grain is square face which implies
gx = gy;
- We assume the recombination velocity at grain
boundaries is the same for all plans.
The phenomena of charge carrier generation, diffusion
and conduction in the grain are related by a mathematical
equation called the continuity equation:
At the grain boundaries
c k  tan(c k 
gx
Sgb
)
2
D
(6)
 2 ( x, y, z )  2 ( x, y, z )  2 ( x, y, z )  ( x, y, z )
1



   G( z )
x 2
y 2
z 2
L2
D
c j  tan(c j 
gy
Sgb
)
2
D
(7)
  ( x, y, z ) 
Sgb
gy

 gy   D   ( x, 2 , z )

y

 y  2
(5)
Sgb: is the grain boundaries recombination velocity at
the while gx and gy represent the size of the grain. From
equations (4) and (5) we can deduce the equations (6) and
(7)
The first three terms on the left reflect the distribution
while the fourth conduction and the right term the
generation of charge carriers. δ (x, y, z) is the density of
minority carriers charge, L is the diffusion length of charge
carriers into the base, the diffusion coefficient D of the
carriers in the base and G (z) the rate of generation of
carriers under illumination multispectral constant at depth z
in the base region. Its expression [J Dugas et al., 1996 F.
Ahmed and S. Garg, 1986]
2
(4)
2
(1)
G( z )    ai exp(bi  z )
Sgb
gx
  ( x, y, z ) 

  ( , y, z )


gx
x
D
2
 x 
Relations (6) and (7) are called transcendental equations,
resolution graph shows the values.ck and cj.
Putting equation (3) into equation (1), replacing the
expressions generated by appropriate values and terms and
taking into account the orthogonallity of the cos (c kx) and
cos (cjx) family, we obtain [Dugas et al., 1994]:
(8)
(2)
1
1
 c k2  c 2j  2
2
L
Lk , j
i 0
III. STUDY OF THE DENSITY OF CHARGE CARRIERS
The solution of the equation of continuity is the density
of charge carriers which can be expressed as
 ( x, y, z )    Z kj ( z )  cos(ck  x)cos(c j  y )
Dk , j 
and
(3)
k j
D  sin(ck  gx)  ck  gx   sin(c j  gy )  c j  gy 
gx
gy
16  sin(ck  )  sin(ck  )
2
2
(9)
(10)
Lk ;j and Dkj are respectively the effective diffusion
length and the effective diffusion coefficient of the solar
cell.
Figures 2a and 2b below show respectively the effects of
recombination velocity at grain boundaries and grain size
on the effective diffusion coefficient.
Zkj (z) is a function of the carrier density; it depends on
the depth z. cj and ck are eigen values are obtained from
the N, following boundary condition
2
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 12, December 2013)
Where Sf represent the junction recombination velocity
At the Back Surface (z = Wb)
Sb
  ( x, y, z ) 
    ( x, y,Wb)



z
D

 z Wb
(14)
Where Sb is the surface recombination velocity of
charge carriers in the rear area in excess of the base. By
replacing its value in equations (13) and (14) we are left
with a system of equations that allows us to draw the
constants Ak,j and Bk,j their values are given by (15) and
(16)
Figure 2 a: Effective Diffusion coefficient cm2/s versus
recombination velocity Sgb (cm/s), g = 0,01cm
1
Sb
Sf
(
 bi )  exp(bi  gz )  Yk , j  (
 bi )
Lk , j D
D
Ak , jav   Ki 
Sf  Yk , j X k , j

D
Lk , j
Sf Sb
Sf
 (  bi )  exp(bi  gz )  X k , j  (  bi )
D
Bk , jav   Ki  D D
Sf  Yk , j X k , j

D
Lk , j
(15)
(16)
And whis:
(17)
Figure 2 b : Effective Diffusion coefficient cm2/s versus grain size
Sgb = 103cm/s
The
effective
diffusion
coefficient
decreases
exponentially when the boundary recombination velocity
and the grain size increases.
By solving the equation (10) we find
Z k , j ( z )  Ak , j .ch(
2
z
z
)  Bk , j .sh(
)   K i . exp( bi .z )
Lk , j
Lk , j
i 0
A.
Influence of Recombination Velocity at Grain
Boundaries on the Profile of the Carrier’s Density
Figure 3 below illustrates the influence of recombination
at grain boundaries on the distribution of charge carriers in
the volume of the base region.
(11)
Lk , j  ai
2
Ki 
Dk , j (bi  Lk , j  1)
2
2
(12)
Where
Ak,j et Bk,j are constants which are determined from the
following boundary conditions:[A.Dieng et al 2007,Sissoko
G et al 1998]
At the Junction (z = 0)
Sf
  ( x, y, z ) 

  ( x, y,0)


z
z 0 D
(13)
Figure 3 Minority carries density versus base depth for different
boundary recombination velocity, g = 0,01cm D = 26cm2/s, Sf
=103cm/s, x = y = 0,01cm, AM 1, 5
3
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 12, December 2013)
It is observed in this curve that the carrier density
increases slightly in the region near the junction and a peak
corresponding to a region where the carriers are stored. It
also decreases towards the inside of the base for a fixed
grain size. Similarly, it appears that increasing the
recombination velocity joints leads to a decrease in carrier
density this is explained by the fact that increasing the
speed at grain boundaries is synonymous with electrical
activity stronger seals, thus trapping most of the minority
carriers supported at these centers traps. This reflects the
effect of the activity of recombinant grain boundaries.
We also note that with the increase of the grain boundaries
recombination velocity, there is a displacement of the point
of maximum density to the junction, which reflects a
reduction in the number of carriers involved in the
production of photocurrent. [H. L. Diallo et al., 2006, A
Dieng et al., 2009]
This phenomenon is explained by the fact that a decrease
in grain size reflects an increase the recombination centers
in the grain boundaries and thus the potential
recombination.
B. Influence of Grain Size on the Minority Carrier Density
Figure 4 shows the effect of grain size on the distribution
of minority charge carriers in the volume of the base
Figure 5: 3D curve of minority carries density versus grain size and
boundary recombination velocity, x = y = 0cm; Sf = 104cm/s; D =
26cm2/s, AM = 1, 5
IV. PHOTOCURRENT DENSITY
The expression of the photocurrent density is given by
the following equation [J. Ducas et al., 1998]
J ph 
q  D gx2 gy2   ( x, y, z ) 
 gx  gy
  dx  dy
gx  gy  2  2 
z
 z 0
(18)
q is the electron charge and  ( x, y, z ) represents the
carrier density substituting the expression  ( x, y, z) in (18)
and integrating, we obtain
 Bk , jav 2

J phav  q  D   Rk , j  
  K k , j  bi 
k
j
 Lk , j i 0

Figure 4: Minority carries density versus base depth for different
grains size, Sgb =103 cm/s , Sf = 104 cm/s D = 26cm2/s, AM = 1,5
A.
Effect of the Grain Size
The curves in Figure 6 below show the influence of
grain size on the profile of the photocurrent density.
It is apparent from the curves in Figure 4 that the density
of excess electrons decreases when the grain size decreases.
4
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 12, December 2013)
This shows the negative side of grain boundaries in the
solar cell.
In Figure 8 we represent the following photocurrent
density depending on the grain boundaries recombination
velocity
Figure 6 Photocurrent density versus junction recombination velocity
for different values of g: Sgb = 103cm/s, g = 0.01cm, D = 26cm2, AM =
1, 5
Observed across these curves that the photocurrent
increases with grain size. This is explained by the fact that
the increase in the grain size decreases significantly the
grain boundaries and beyond the possibilities of
recombination recombinant interfaces.
Figure 8 Photocurrent density versus boundary recombination
velocity Sgb pour differences values of Sf : g = 0.01cm, D = 26cm2,
AM = 1, 5
Through Figures 8, we note that the current density is
highest at low speeds recombination at grain boundaries.
But we note that the current density decreases until it
vanishes when the recombination velocity Sgb increases.
Influenced by the recombination velocity at the junction
Sf Indeed, if the recombination velocity at the junction
increases the current density increases.
Figure 9 shows the three-dimensional profile of the
photocurrent density based on the recombination velocity at
the junction Sf and the recombination velocity Sgb.
B. Effect of the Grain Boundary Recombination Velocity
We investigate the effect of velocity of the grain
boundary recombination Sgb on the photocurrent density in
the representative function of recombination velocity at the
junction Sf for different values of the grain boundary
recombination velocity on figure 7
Figure 7 Photocurrent density versus junction recombination velocity
for different values of Sgb : g = 0.01cm,D = 26cm2 et AM = 1,5
As the carrier density the photocurrent is strongly
influenced by the grain boundaries recombination velocity,
but this influence is increasingly felt to Sf is high i.e. in the
vicinity of short circuit. But for different Sgb, the curves
have the same shape. However, we note that if Sgb
increases the amplitude of the current density decreases.
Figure 9: 3D curve of the photocurrent density versus recombination
velocity Sf and boundary recombination velocity Sgb g = 0,01cm D =
26cm2/s AM = 1.5
5
International Journal of Emerging Technology and Advanced Engineering
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Figure 9 illustrates the decrease in the photocurrent
when the recombination velocity at the grain boundaries
increases. The current density increases and reaches a
plateau when the recombination velocity at the junction
increases as the two bearings are visible the first bearing
corresponding to low recombination velocity at the junction
where the current is substantially zero regardless of the
recombination velocity for grain boundaries: the operation
of the solar cell open circuit, a second level where the
current density is maximum and you can clearly see the
effect of Sgb on the photocurrent.
Same considerations in Figure 10 where we have plotted
the carrier density as a function of the recombination
velocity at the junction and the grain size
This figure shows that when the grain size increases the
recombination velocity at the grain boundaries decreases
and tends towards a solar cell monocrystalline structure
where the traps are smaller carriers which increase the
quality of the solar cell.
V. PHOTOVOLTAGE
Photovoltage when the solar cell is illuminated is
determined from the following relationship of Boltzmann:
(20)
V 
k T
q
(21)
Where V is the thermal voltage, k is the Boltzmann
constant, N is the doping level of the base and ni the
intrinsic carrier concentration.
Replacing the density of minority carriers by its
expression we obtain:
Vav  VT  ln(1 
2
N



R
A av   K k , j 
2  k , j  k , j
ni k j
i 0


(22)
A. Effect of the Recombination Velocity at the Grain
Boundaries on the Photovoltage
We show in fig. 12, the impact of the grain boundary
recombination velocity on the photovoltage.
Figure 10 :3D curve of the photocurrent density versus junction
recombination velocity Sf and grain size g : Sgb =103cm /s, D =
26cm2/s AM = 1,5
Also shown the current density as a function of the
recombination velocity at grain boundaries and the grain
size, the profiles are given in Figure 11
Figure 12 Photovoltage versus junction recombination velocity for
different values of boundary recombination velocity aux joints de
grain: D = 26cm2,g = 0.01cm, AM =1,5
Figure 12 shows that the photovoltage depends also on
junction recombination velocity (Sf), and grain boundary
recombination velocity (Sgb).
Figure 11: 3D curve of the photocurrent density versus boundary
recombination velocity Sgb and grain size g: Sf = 10 3cm, D = 26cm2/s
AM = 1,5
6
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 12, December 2013)
The curves of voltage as a function of Sf, for different
Sgb have bearings for low values of recombination velocity
at the junction; in this area the photovoltage is maximum. It
corresponds to the open circuit However, when
recombination velocity exceeds a certain value, the
photovoltage decreases very quickly to cancel the large
recombination velocities Sf: the operation of the solar cell
short-circuit. But the value of Sf on which the short-circuit
is observed is in the order of 1010cm / s, which is a problem
because the speed of the electron in the crystal is of the
order of 106cm/s. Also, when the recombination velocity
Sgb varies the photovoltage varies for low recombination
velocities at the junction corresponding to short circuit Sgb
has no influence on the photovltage that `a from 100000
cm / s, or we get a drop of `amplitude. Thus large values
Sgb correspond to low values of the photovoltage, thus
comprehensively photovoltage decreases the value of open
circuit when the recombination velocity at grain boundaries
increases.
We presents on fig 14 a 3D curve of the photovoltage
versus the recombination velocity and the recombination
velocity at the grain boundary
B.
Figure 14: 3D curve of photovoltage versus Sf and Sgb: g = 0.01cm, D
= 26cm2/s AM = 1, 5
Effect of Grain Size
Figure 13 shows the evolution of the photovoltage as a
function of the recombination velocity Sf different value of
grain sizes
Similarly, we represent the profile of the 3D
photovoltage depending on the recombination velocity Sf
and the grain size at a speed of recombination velocity Sgb
set.
On figure 15 the profile of the photovoltage is presented
versus the grain size and the recombination velocity at the
junction.
Figure 13 Photovoltage versus junction recombination velocity for
different values for different grain size: Sgb = 10 3cm/s, D = 26cm2,
AM1,5
The effect of the size grains is felt that the open circuit
photovoltage namely Sf = 103 cm / s. but beyond this value
the photovoltage is more sensitive to the recombination
velocity at the junction. This allows us to assert that there is
a predominance of recombination at the interfaces when the
solar cell works in open circuit, but when it reaches its
operating short-circuits at the junction recombination
predominates.
Figure 15: 3D curve of photovoltage versus Sf and g: Sgb = 10 3 cm/s,
D = 26cm2/s, Sgb = 103cm/s, AM = 1,5
7
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 12, December 2013)
[4]
H. L. Diallo, I. Ly, M. Zoungrana, Nzonzolo, F. I. Barro, G. Sissoko
3D, (2006) , modeling of a bifacial polycristaline silicon solar cell in
order to exhibite the effect of grain size and grain boundary on the
recombination parameters under a constant white illumination.
Proceedings of the 21st European Photovoltaic Solar Energy
Conference and Exhibition -Dresden, Germany pp. 451-454
[5] A. Dieng, A. Thiam, M. Zoungrana, S. Diallo, F. I. Barro, G.
Sissoko, (2009), Etude à 3D d’une photopile polycrystalline au
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[6] J. Thongpron, K. Kirtikara, C. Jivacate, (2006), Solar Energy
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[10] F. Ahmed and S. Garg, Août 1986, International Centre for
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[11] Dieng. A, Lemrabott. O. H, Maiga. A. S, Diao. A, Sissoko.G, 2007.
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Figures 14 and 15 illustrate more visibility with the
effects of recombination velocity and grain size on the
open-circuit voltage, but the effect of recombination
velocity Sgb is more important than the grain size the open
circuit voltage. They have no impact beyond a certain value
of recombination velocity at the junction.
VI. CONCLUSION
This study high light the effect of the recombination
velocity at grain boundaries and grain size on the
phenomenological parameters polycrystalline solar cell
from a three dimensional modeling. It is clear from this
work that when the grain boundaries recombination
velocity is important photocurrent and photovoltage
decreases and the solar cell loses its quality. This
recombination at the interfaces is more important when the
photocell has small grain sizes. But when the grain size
increases it there's less recombination at the interfaces and
the solar cell has a better quality.
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