Momar Diaw et al. / International Journal of Engineering Science and Technology (IJEST)
3D study to improve the IQE of the bifacial
polycrystalline silicon solar cell from the
grain’s geometries and the applied
magnetic field
MOMAR DIAW
Laboratory of Semiconductors and Solar Energy, Department of Physics, Faculty of Science and Technology,
Cheikh Anta Diop University, Dakar, SENEGAL SECOND AUTHOR
Bernard Zouma
Laboratory of environnemnt materials, Physics departement, Unit Training and Research in Applied Science
(UFR), University of Ouagadougou, BURKINA FASO
AHMETH SERE
Laboratory of environnemnt materials, Physics departement, Unit Training and Research in Applied Science
(UFR), University of Ouagadougou, BURKINA FASO
Senghane Mbodji
Laboratory of Semiconductors and Solar Energy, Department of Physics, Faculty of Science and Technology,
Cheikh Anta Diop University, Dakar, SENEGAL
Department of physics, University Alioune Diop of Bambey, P.O Box 30 Bambey, SENEGAL; 00 221 77 522
0101, msenghane@yahoo.fr - senghanem@gmail.com
Aminata Gueye Camara
Laboratory of Semiconductors and Solar Energy, Department of Physics, Faculty of Science and Technology,
Cheikh Anta Diop University, Dakar, SENEGAL
Grégoire Sissoko
Laboratory of Semiconductors and Solar Energy, Department of Physics, Faculty of Science and Technology,
Cheikh Anta Diop University, Dakar, SENEGAL
00221 77 632 90 41; gsissoko@yahoo.com
Abstract : A study aiming to an improved internal quantum efficiency of silicon solar cell is presented. The
efficiency depends on several internal and external parameters. Especially the polycrystalline silicon solar cell,
which is the most used for the terrestrial applications of photovoltaic solar energy, has the lowest efficiency.
Because there are many grains and grain boundaries with much recombination of the carriers at the grain
boundaries, thus the structure of the grain as its geometry becomes an internal key parameter for this efficiency.
The external key parameter taken into account here is the magnetic field. Although its effect on the carriers is
well known, its impact on the recombination’s phenomena into the two different regions of the cell like the
emitter and the base needs to be known. Before doing it, two classical geometries of the grain are retained to
investigate through simulation an adequate structure providing a high efficiency. Thus an appropriated 3D
model with new assumptions is used to describe the bifacial polycrystalline silicon solar cell and new
expressions of carrier densities, photocurrent and quantum efficiency are established.
Keywords: Quantum efficiency - grain geometry - grain boundary - Magnetic field.
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Momar Diaw et al. / International Journal of Engineering Science and Technology (IJEST)
Nomenclature
B

D
e, q
0
G
gx
gy
H
IQE
J
Jcc
L
n, p
SB
SE
SF
Sg


W
*
Intensity of the magnetic field (T)
Density of carriers (cm-3)
Diffusion Coefficient (cm.s-2)
Elementary charge (C)
Incidental beam of photons
(W.m-2)
Generation rate
Length of the grain following the x-axis (cm)
Length of the grain following the y-axis (cm)
Thickness of the base region (µm)
Internal quantum efficieny (%)
Photocurrent density (A. m-2)
Short circuit photocurrent density (A. m-2)
Diffusion length of the carriers (µm)
Indices relating to the electron and the hole
Electron rear surface recombination velocity in the base (cm.s-1)
Hole front surface recombination velocity in the emitter (cm.s-1)
Junction recombination velocity (cm.s-1)
Grain boundaries recombination velocity (cm.s-1)
Lifetime of the carriers (s)
Mobility (cm2.V-1.s-2)
Thickness of the emitter region (µm)
Indicate the dependence to the magnetic field
1. Introduction
The internal quantum efficiency (IQE) is a fundamental parameter of the silicon solar cells. Its knowledge
allows determining the recombination’s parameters of the solar cells as the effective diffusion length, the
effective lifetime and the recombination velocities, because it depends on the generation and the recombination
phenomena in the solar cell. Several studies [Basu, et al. (1994)] [Benmohamed and Remram (2007)] [Betser, et
al. (1995)] [Cuevas, et al. (2002)] [Diallo, et al. (2008)] [Dieng, et al. (2007)] have led to determine these above
parameters in order to characterize the solar cells.
The polycrystalline silicon solar cells still mainly used to convert sunlight to electricity for the terrestrial
applications of the photovoltaic solar energy. These cells are low-cost and have a low efficiency, which is
generally under to 18%. One of the raisons of this low efficiency is the existing of many grains [Dugas (1994)]
[Fedrorov, et al. (2002)] in the solar cell so many grain boundaries recombination [Gall, et al. (2006)]
[Macdonald, et al. (2001)]. The grains are welded and the number of established junctions by one grain depends
to its geometry, then the number of sides equal the number of grain boundaries for one grain. Increasing the
efficiency of polycrystalline solar cell needs to reduce the grain boundaries the most possible. The four-side
geometry appears as a judicious choice for the grains to reduce the recombinations in the cell.
About the recombination’s phenomena in the solar cells, the external influences are not taken into account in the
most time. Nevertheless, the Hall Effect with the magnetic field on the carriers is well known and the
geomagnetic field exists continuously. This reality led certain researchers to integrate the magnetic field in their
works [Madougou, et al. (2007)] [Madougou, et al. (2005)] [Madougou, et al. (2004)] [Mbodji, et al. (2009)]
[Pisarkiewicz, et al. (2004)] [Saritas, et al. (1988)] [Spiegel, et al. (2000)] .
In this study, we are going to investigate from an improved expression of IQE, an adequate geometry of the
grains that increase the efficiency in the first hand and the impacts of an external magnetic field on the
recombinations into the emitter and the base regions in the second hand. In this fact, a three dimensional (3D)
model of the bifacial polycrystalline silicon solar cell is described under a constant field magnetic field. The
benefit on this model is the introduction of the grain size and the grain boundaries recombination velocity like
the recombination parameters.
2. Theoretical model
The polycrystalline silicon solar is manufactured from many grains. The size of the grain in this kind of solar
cell is set 1 µm to 1 mm. In this part, we are going to describe one grain in the 3D model, which is placed in a
constant magnetic field. The direction of the magnetic field is parallel to the junction as we can see on the figure
1 below.
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Momar Diaw et al. / International Journal of Engineering Science and Technology (IJEST)
Fig 1.a: 3D model of a grain of the bifacial polycrystalline silicon solar cell under a constant magnetic field.
Fig 1.b: 3D model of a grain of the bifacial polycrystalline silicon solar cell under a constant magnetic field.
Fig 1.c: 3D model of a grain of the bifacial polycrystalline silicon solar cell under a constant magnetic field.
The grain can be illuminated on front side (emitter) and on rear side (base). The illumination direction is
perpendicular to the junction. This direction is also the carrier’s generation. The sample of the study (c) has four
sides so the surface is given by the product gx.gy.
The grain boundaries recombination velocity (Sg) is the same for the base and the emitter regions.
The ratio of the minority carrier generation is definite for front side illumination by the equation (1) and for rear
side illumination by the equation (2).
(1)
G z,   1 R eWz
1
0
G 2 z,     1  R   0  e  H z 
(2)
In these equations, the penetration depth z varies from –W to zero in the emitter region and from zero to H in the
base region. The thickness of the heavy-doped region is neglected.
 and R are respectively the absorption and the reflection coefficients, they are function of the wavelength of
the incident light . The values of these coefficients are taking at solar spectrum AM1.5.
The considered magnetic field is one-dimensional, constant and directed along y-axis and then we can write:
(3)
B  Bj
Considering the electric field E  0 , the magneto transport theory [Van Sark, et al. (2005)] gives the
following relations:
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Momar Diaw et al. / International Journal of Engineering Science and Technology (IJEST)




(4)
J n  eD n   n   n J n  B




(5)
J p   eD p   p   p J p  B
The development of the equations above with some mathematical theorems, leads to the continuity equations in
the base and the emitter regions. The new continuity equations with the magnetic field are given respectively for
the electrons in the base region and for the holes in the emitter region.
For the electrons in the base region:
 2n
x
2
 qn 
 2n
y

2
 2n
z 2

n
D *n  t n

Gn
(6.a)
D *n
where
2n  1   n B2
D*n 
(6.b)
(6.c)
Dn
1   n B2
For the holes in the emitter region:
 2p
x
2
 qp 
 2p
y

2
 2p
z 2

p
D *p  t p

Gp
D *p
(7.a)
where
D*n 
D*p 
(7.b)
Dn
1   n B
2
(7.c)
Dp
 
1  pB
2
In these equations above, the density of the carriers is a function of the special coordinates x, y, z and the
wavelength .
The solutions of these continuity equations are the densities of minority carriers, they have given by the
following equation where the parameter  =1 for front side illumination and  =2 for the rear side illumination
according to the works of Dugas [Veschetti, et al. (2005)]
-
For the electrons in the base region
 
 C nj 
 n  x , y , z ,      Z n  i , j z ,   cos C ni x  cos 
y 
i  0 j 0
 n 
 z
Z n 1i , j z ,    A n 1i , j cosh  
 L ni , j


  B  sinh
n 1i , j


 z
Z n 2 i , j z ,    A n 2 i , j cosh  
 L ni , j

and
Kni, j
 1  R0 
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Dni, j   2





 z

 Lni , j


  K   e    W  z 
ni , j


 z


  B
n 2 i , j sinh
 Lni , j




(8.a)
(8.b)

  K   e    H  z 
ni , j


(8.c)
(8.d)
2
1/ Lni, j 

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The constants A ni , j and Bni, j are determined with the following boundaries conditions:
 x, y,0,  
 n x, y, z,  
 S Fn n 
z
Dn
z 0
(9.a)
 x, y, H,  
 n x, y, z,  
 S B n
z
D n
z H
(9.b)
The coefficients C ni and C nj are determined with the following grain boundaries conditions:
n (x, y, z, )
 Sg
g
x
x  x
n (
2
 n  ( x , y , z ,  )
y
-
  Sg
gy
y
gx
, y, z, )
2
2Dn
 n ( x , 
gy
2
2Dn
(10.a)
(10.b)
, z,  )
2
For the hole in the emitter region
 C pj
  Z p  i , j z ,   cos C pi x  cos 

 p  x , y , z ,   

 p
i  0 j 0

y


(11.a)
 z 
 z 
Zp1i, j z,   Ap1i, j cosh    Bp1i, j sinh    Kpi, j eWz
L 
L 
 pi, j 
 pi, j 
Zp 2i, j
z,   

z
Ap 2i, j cosh 
L
 pi, j
(11.b)



  B sinh z   K  e    H  z 
p 2i , j
pi, j

 L 

 pi, j 
(11.c)
and


2
Kpi, j  1  R0  Dpi, j 2  1/ Lpi, j 

 (11.d)
The constants A pi , j and Bpi, j are determined with the following boundaries conditions:
At the junction :
 p x, y, z,  
 S Fp
z
p x, y, z,  
z
 SE
 Sg
x 
(12.b)
Dp
C pi and C pj are determined with the following grain boundaries conditions:
The coefficients
p (
gx
gx
, y, z, )
2
2Dp
(13.a)
gy
(13.b)
2
 p ( x , y, z,  )
y
p x, y,W,  
z W
p ( x, y, z, )
(12.a)
D p
z 0
At the front side:
x
 p x, y,0,  
y
gy
 Sg
 p ( x , 
2
2 Dp
, z,  )
2
The expressions of these densities above were calculated in our previous work [Madougou, et al. (2004)].
From the minority carrier densities, the photocurrent densities are calculated.
g
(14)
qDn  g    n x, y, z,   
J n   
 g 2  g2 
 dydx
gxg y
J p   
 qDp

x
y
x
y
2

gx


2

gy
z

z 0
  p x, y, z,   
 dydx
z
 z 0
 2 2
gxg y  gx  g y 
2
2
(15)
The short-circuit photocurrent densities are obtained for the high values of the junction recombination velocities
SF from the photocurrent densities above, then we can write:
(16)
Jcc n    J n   S
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Fn  4.10
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Jcc p    J p  
(17)
SFp  4.10 4
The internal quantum efficiency is the ratio of the short circuit photocurrent density by the product of the
fraction of the photons beam really having have penetrated in the solar cell and generated the carriers which take
part this short circuit photocurrent density and the elementary charge [Warta, et al. (2002)].
Now, by considering the contribution of the emitter region to the IQE, we can write:
(18)
Jcc p    Jcc n  
IQE    
q1  R   0  
In spite of its weak width, the emitter region plays an important role in the recombination’s process. When the
emitter is illuminated, it becomes a recombination’s zone of the electrons from to the base region [Madougou, et
al. (2004)].
According to equation (18), IQE is a function of the wavelength. The curve of IQE versus the wavelength
commonly called spectral response [Zouma, et al. (2009)] [Zoungrana, et al. (2007)] .
3. Simulations and discussions
3.1 Investigation on the grains geometry
Three different grain sizes like gx.gy with three different grain boundaries recombination velocities (Sg), have
been accepted to investigate an adequate geometry of the grain in the polycrystalline silicon solar cell from its
internal quantum efficiency.
Plunged into 5.10-5 T of the magnetic field intensity, the main regions of the grain like the base region and the
emitter region are respectively 299,9 µm and 0,1 µm thick. The diffusion lengths are respectively 100 µm and
0,04 µm for the electrons and the holes. The grain is illuminated firstly on the front side and secondly on the
rear side.
The simulated curves of the internal quantum efficiency versus the wavelength pointing out the influence of
the grain geometry for the front side and for the rear side illuminations are respectively presented on the figure 2
and on the figure 3 below.
Fig 2: curve of internal quantum efficiency versus the wavelength for different geometries of the grain at three different values of the grain
boundaries recombination’s velocity for the front side illumination. SE=103 cm.s-1, SB=103 cm.s-1, Dn=26 cm2.s-1, Dp=4 cm2.s-1, Ln=100 µm
and Lp=0,04 µm.
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Fig 3: curve of internal quantum efficiency versus the wavelength for different geometries of the grain at three different values of the grain
boundaries recombination’s velocity for the rear side illumination. SE=103 cm.s-1, SB=103 cm.s-1, Dn=26 cm2.s-1, Dp=4 cm2.s-1, Ln=100
µm and Lp=0,04 µm.
The effect of the grain size is strongly depending to the boundaries recombination’s velocity. For small values
of Sg like Sg ≤ 102 cm.s-1, the three curves are practically identical for the front side illumination (fig. 2) then the
effect of the grain’s geometry cannot establish. However, from 103 cm.s-1 of Sg the effect of the geometry is
clear for the two sides’ illumination.
Generally, the internal quantum efficiency increases with the grain size [Madougou, et al. (2004)]. More there
are many carriers’ recombinations at the grain boundaries more the IQE is sensitive to the geometry of the
illuminated grain. The grain with the geometry gx.gy = (2×6).10-6cm2 provides the lowest IQE while those
provided by the grains of the geometries gx.gy = (6×6).10-6cm2 and gx.gy = (10×6).10-6cm2 are practically the
same.
When the grain of the polycrystalline silicon solar cell, which is described on the fig. 1c above, is illuminated,
the maximum of generated electrons in the base region is localized at the centre of the grain. Then more the
product gx.gy is bigger, more the grain boundaries are far from the centre, so there are many electrons that can
participate to the short circuit current.
For the big grain sizes, the geometry has not a big impact on IQE in spite of a strong grain boundaries
recombination. The rectangular surface (gx≠gy) is the geometry that provides the high efficiency compared to the
squared surface (gx=gy), but both of them can be an adequate geometry in the polycrystalline silicon solar cells.
From these curves, the influences of the grain’s geometry and the grain boundaries recombination on the
efficiency are clearly established. The known lowest efficiency of the polycrystalline silicon solar cells can be
justified. In these solar cells, the grains are small and do not have necessarily a classical geometry like the
squared surface or the rectangular surface. There are several grains having a none-classical geometry that is due
to the manufacture’s defaults and then there are more grain boundaries. The grain boundaries are the regions of
the carrier’s recombination and a high recombination velocity at grain boundaries is very baneful to the
efficiency of solar cell that seen on the two figures above; the IQE is strongly reduced for the two sides’
illuminations.
The small value of gx.gy comminuted to the high value of Sg provides the low efficiency of silicon solar cell. The
efficiency of polycrystalline silicon solar cell depends to the grain’s geometry and the grain boundaries
recombination.
From the geometry gx.gy = (6×6).10-6cm2, the efficiencies produced by the grain with a square surface and the
grain with a rectangular surface are high and similar for any value of the grain boundaries recombination
velocity.
This result of simulation is one way among others to improve the efficiency of the polycrystalline silicon solar
cells. Thus, the grain geometry with a squared surface can be a useful description for a sample to investigate
other properties of the silicon solar cell.
3.2. Investigation on the impacts of an external magnetic field
Using the result in the previous paragraph, we are going to point out the effect of the external magnetic field on
the IQE from a grain with the geometry gx.gy = (10×10).10-6cm2. The figure 1c shows the orientation of the
magnetic field vector.
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The grain is respectively front side and rear side illuminated; the results of simulations are presented
respectively on the figures 4 and 5.
Fig 4: Internal quantum efficiency versus wavelength for different magnetic fields in the front side illumination. Sg=102cm.s-1,SE=104 cm.s-1,
SB=103 cm.s-1, Dn=26 cm2.s-1,Dp=4 cm2.s-1, Ln=100 µm and Lp=0,04 µm.
Fig 5: The internal quantum efficiency versus wavelength for different magnetic fields in the rear side illumination. Sg=102cm.s-1,SE=104
cm.s-1, SB=103 cm.s-1, Dn=26 cm2.s-1, Dp=4 cm2.s-1, Ln=100 µm and Lp=0,04 µm.
The terrestrial magnetic field (around to 5.10-5 T) cannot affect seriously the efficiency of the polycrystalline
silicon solar cell in spite of IQE decreases with the magnetic field intensity. However from B = 10-3 T, IQE is
much reduced. When B = 10-2 T, the curve of the internal quantum efficiency versus the wavelength is deformed
for the front and the rear side illuminations. The external magnetic field favors the recombination of the carriers
into the solar cell by reducing their mobility. In the presence of a magnetic field, the diffusion coefficients of the
electrons and the holes are affected according to the equations (6.c) and (7.c); these coefficients decrease with
the magnetic field [Mbodji, et al. (2009)].
The analysis of the IQE curves obtained for the different values of the magnetic field B, shows that the effect of
the magnetic field is emphasized into the base region. The energies transported by the photons whose
wavelengths are bigger than 500 nm, are absorbed in the base region of the solar cell for the front side
illumination. Our study’s sample contains two main regions: the emitter region and the base region where the
minority carriers generated are the electrons, the short wavelengths (< 500 nm) generate the more carriers
(holes) in the emitter region, which becomes the recombination zone [Madougou, et al. (2004)] of the electrons
from the base region. While for the rear side illumination, there are the long wavelengths (> 1000 nm) that
generate the more holes in the emitter region thus the emphasized magnetic field effect starts from the short
wavelengths (figure 5).
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For B ≤ 10-3 T, the impact of the magnetic field on the recombinations into the emitter region is negligible; the
three curves are overcome (fig. 4) and the heavy-impact is localized into the base region only that is
materialized by the shift of the curve.
For the strong magnetic fields (B > 10-3 T), a big shift is observed on all the range of wavelengths. Then this
shift is observed into the emitter region and the base region but it is always emphasized into the second. The
main physical differences between the emitter region and the base region are the thickness and the doping. The
behaviors of IQE into the two regions for different intensities of the magnetic field open the prospects of study
on the optimization of the efficiency of solar cells.
4. Conclusion
From the internal quantum efficiency of a three-dimensional grain of the bifacial polycrystalline silicon solar
cell, we have established the importance of the grain’s geometry as an internal parameter so the grain must have
less grain boundaries. For this, the squared surface and the rectangular surface have been retained and for the
big grain sizes, the geometry with rectangular surface (gx≠gy) provides a high efficiency. As to the external
parameter: the magnetic field, It favors the recombination phenomena of the carriers in the base and the emitter
regions. Its impact is emphasized in the base region nevertheless only the strong magnetic fields (B > 10-3 T)
have a considerable impact in the emitter region. Generally, the magnetic field reduces the efficiency.
Fortunately, the terrestrial magnetic field has a lesser effect on the efficiency of the solar cell.
Apart from the obtained results that improve the efficiency of the solar cells, this study offers some prospects of
study and manufacture of the solar cells.
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