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Volume 2, Issue 9, September 2014
SILICON SOLAR CELL UNDER BACK SIDE
ILLUMINATION: EFFECT OF MAGNETIC
FIELD
1
Khady FAYE, 1Idrissa GAYE, 1Sega GUEYE, 2Mamadou WADE, 1Grégoire SISSOKO
1
Department of Physics, 1Laboratory of Semiconductors and Solar Energy, Physics Department, Faculty of Science and
Technology University Cheikh Anta Diop – Dakar – SENEGAL.
2
Electromechanical Engineering Department, Polytechnic School of Thies – SENEGAL
ABSTRACT
A theoretical study, in static regime, of a bifacial solar cell under back side illumination and under the influence of a constant
magnetic field, is presented in one dimension model. An analysis of magnetic field effects on the minority carriers density, the
photocurrent, the photovoltage, the capacity, the short-circuit photocurrent density, the open circuit photovoltage, is made. The
I-V characteristic, series and shunt resistances are determined and studied according to the magnetic field.
Keywords: silicon solar cell - back illumination - magnetic field.
1. INTRODUCTION
The electric conversion of solar energy is obtained with semiconductor materials. The most used material is the
crystalline silicon (monocrystalline, polycrystalline or amorphous) [1]-[3]. However with the development of
technologies, we attend the phenomenon of electromagnetic pollution (exhibition of electronics devices to
electromagnetic fields). This is how we suggested studying the effect of magnetic field on the solar cell. On this
relatively big theme we chose to study the effect of magnetic field applied in the base of a solar cell, under
polychromatic illumination. We are limited this study to back side illumination. This is where the phenomena of
recombination are very important, we looked at first of the influence of magnetic field on the minority carriers density,
then we are interested at effects on the photocurrent density, the photovoltage, the capacity, the short-circuit
photocurrent and the open circuit photovoltage. About the study of I-V characteristic, we also looked for the effect of
magnetic field on the series and shunt resistances.
2. THEORY
An n+-p-p+ solar cell type [2]-[4] is schematized at figure 1:
Figure 1: schema of a bifacial solar cell under magnetic field
When the solar cell is enlightened, various processes take place in the base: It is about the generation, recombination
and diffusion of excess minority carriers. All these processes can be represented by the following continuity equation.
 2 ( x )  ( x )
G ( x)


2
2
D*
x
L*
(1)
G(x) is the carrier generation rate. When we illuminated the back surface of the solar cell by a polychromatic light in
static regime, the expression generation rate is given by the following relation
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G ( x )  n   ai  exp(bi ( H  x ))
(2)
i 1
ai and bi are coefficients deduced from solar radiation under AM 1,5 spectrum [5]; we set n = 1 (Number of sun).
D*(B) indicates diffusion coefficient of electrons generated in the base. It depends on the magnetic field. It is given by
the relation 3. [6]-[7]:
D * ( B) 
D0
1  (   B) 2
(3)
D0 is diffusion constant without magnetic field and μ is the electron mobility.
L* is the diffusion length of excess minority carriers depending on the magnetic field. It’s given by the following
expression:
L* ( B)  D * ( B)  
(4)
The Profile of diffusion length versus magnetic field is represented on figure 2:
Figure 2: Diffusion length versus magnetic field (logarithmic scale); (Nb = 1017 cm-3 ; µ = 1350 cm2.V.S-1
H = 0,03 cm ; D0 = 35 cm2.S-1)
For low values of the magnetic field, the diffusion length remains constant. But, for high values of magnetic field, the
diffusion length decreases gradually. The magnetic field deflects or slows down the minority carriers.  is the minority
carriers lifetime
The general solution of the continuity equation is given by expression (5):
3
x
x
n.ai .L *2
(5)
 ( x )  A. cosh( * )  B. sinh( * )  
. exp( bi .( H  x ))
L
L
i 1 D * .(bi .L * 1)
Coefficients A and B can be determined through the following boundary conditions.
At the junction (x = 0):
D *
 ( x )
x
x 0
 Sf   ( x )
x 0
(6)
At the back surface (x = H):
D *
 ( x )
x
x H
  Sb   ( x ) x H
(7)
Where H is the total thickness of the solar cell, Sf and Sb are respectively junction recombination velocity and back side
recombination velocity; Back side recombination velocity depends on magnetic field. To determine Sb we use the
following relation:
Jph
0
Sf
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Where Jph is the photocurrent density. After calculation we find the following expression:
H
H
bi.L*(B).ebi .H sinh(* ) bi.L*(B).cosh(* )
D (B)
L (B)
L (B)
Sb(B)  * .(
)
H
H
L (B) i1
bi .H
*
e  cosh(* ) bi.L (B).sinh(* )
L (B)
L (B)
*
3
(9)
We present on figure 3 the module of back side recombination velocity versus magnetic field:
Figure 3: Module of back side recombination velocity versus magnetic field
Figure 3 shows that the module of back side recombination velocity decreases for increasing values of magnetic field.
The determination of the coefficients A, B allows determining completely the minority carriers density in the base [8][10].
3. RESULTS AND DISCUSSIONS
3-1 Excess minority carriers density
We present on figure 4 the minority carriers density versus depth x in the base for various applied magnetic field
values:
1) B=0T
2) B=410-4 T
3) B=910-4 T
4) B= 10-3 T
Figure 4: Carriers density versus base depth for various magnetic field (Sf = 4.104 cm.s-1)
The magnetic field entails an increase of the minority carriers density in the base: It deflects or blocks these minority
carriers.
3.2 Photocurrent density
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The photocurrent is obtained by the gradient of minority carriers at junction and is given by the following expression:
J ph  q  D * 
 ( x)
x
x 0
(10)
Where q is the elementary charge. On figure 5, we represented the profile of photocurrent density versus junction
recombination velocity for various magnetic field values:
1) B=0 T
2) B=410-4 T
3) B=910-4 T
4) B= 10-3 T
Figure 5: Photocurrent density versus junction recombination velocity for various magnetic field values
The photocurrent increases with junction recombination velocity Sf and presents two situations: One at low values of Sf
and one other at high values of Sf. The first state translates a situation of open circuit whereas the second translates a
short-circuit. We observe that photocurrent density decreases with magnetic field. The magnetic field strength deflects
carriers or slows down them. So there are few photogenerated carriers who cross the junction.
3.2.1 Short-circuit photocurrent
The profile of short-circuit photocurrent density is given to the following figure:
4.1
Figure 6: Short-circuit photocurrent density versus magnetic field
The figure 6 shows that short-circuit photocurrent Jphcc decreases for high values of magnetic field [11].
3.3 The photovoltage: Open circuit photovoltage
The photovoltage of the solar cell is given by the Boltzmann relation:
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V ph  VT  ln(1 
VT 
Nb
  ( x) x  0 )
2
ni
Kb  T
q
(11)
(12)
Where
VT is the thermal voltage;
ni is the intrinsic concentration of minority carriers
Kb is the Boltzmann constant
T is the absolute temperature.
We represent on figure 7 the profile of photovoltage versus junction recombination velocity for various magnetic field
values:
1) B=0T
2) B=410-4 T
3) B=910-4 T
4) B= 10-3 T
Figure 7: Photovoltage versus junction recombination velocity for various magnetic field values
Figure 7 shows that for low values of junction recombination velocity (Sf), the photovoltage is maximal, and it
decreases for high values of Sf. Under back side illumination, for low values, the photovoltage increases with magnetic
field. When the magnetic field becomes important (10-3), the photovoltage decreases. This, where the recombination
velocity at back surface Sb become important.
Figure 8 shows the profile of open circuit photovoltage versus magnetic field:
Figure 8: Open circuit photovoltage versus magnetic field
We observe that open circuit photovoltage increases with low magnetic field values and it decreases however with high
magnetic field values.
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3.4 I-V Characteristic
Figure 9 shows The I-V characteristic for various magnetic field values:
1) B=0T
2) B=410-4 T
3) B=910-4 T
4) B= 10-3 T
Figure 9: I-V Characteristic for various magnetic field values
We note in this figure 9 that, Near short-circuit, the current is there almost constant and corresponds at short-circuit
current. When the photovoltage aims towards open circuit photovoltage, photocurrent decreases to nullify.
3.5 Study of series and shunt resistances
3.5.1 Series resistance
The series resistance Rs is a fundamental parameter which depends on nature of substratum, on temperature, on used
technology and is very important about the quality of a solar cell. It characterizes the contacts resistance On I-V
characteristic we have two operating points, It is about open circuit situation where the photovoltage is maximal and
short-circuit situation where the photocurrent is maximal. In this study, for determining the series resistance Rs, we
propose the equivalent electrical model of the solar cell in open circuit [12]-[13].
By applying the law of meshs to the circuit of figure 10, we find the expression of
series resistance:
circuit of the solar
RS 
V phco  V ph ( Sf )
(13)
J ph ( Sf )
RS and Rch are respectively series and charge resistance. Sf aims towards Sfco which is the junction recombination
velocity limiting the open circuit. The profile of series resistance versus junction recombination velocity for various
magnetic field values is given on figure11:
1) B=0 T
2) B=410-4 T
3) B=910-4 T
4) B= 10-3 T
Figure 11: Series resistance versus junction recombination velocity for various magnetic field values
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We note that at open circuit situation, series resistance increases gradually with junction recombination velocity. We
also notice an increase of the series resistance with the magnetic field: This effect is called magneto-resistance and is
due to an increase of the resistivity of the material then the degradation of diffusion coefficient.
3.5.2 Shunt resistance
The shunt resistance characterizes current losses at the junction. In this study, for determining the shunt resistance Rsh,
we propose the equivalent electrical model of the solar cell in short circuit [14]. The illustrative model of this device is
given to the figure 12:
: Equivalent electrical model of the solar cell in short circuit
By applying the nodes lawVto (the
circuit 12, we find the expression of the shunt resistance datum in the relation 14:
ph Sf )
(14)
R Sh 
J phcc  J ph ( Sf )
Where Jphcc is the short-circuit photocurrent; it’s determined from the following relation:
J phcc  lim J ph
(15)
x 
RSh and Rch are respectively the shunt and charge resistance. Sf aims towards Sfcc which is the junction recombination
velocity initiating the short circuit. We represent at figure 13 the shunt resistance versus junction recombination
velocity for various magnetic field B values:
1) B=0 T
2) B=410-4 T
3) B=910-4 T
4) B= 10-3 T
Figure 13: Shunt resistance versus junction recombination velocity for various magnetic field values
We note an increase of the shunt resistance with the recombination velocity at junction. We can also notice that the
shunt resistance increase with magnetic field. It means a drop of current which crosses this one, thus an improvement
of the current delivered by the solar cell.
3.6 The capacity
The capacity is obtained from the following expression:
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C
Q
V ph
(16)
After calculation we find the following expression:
C  C0 
q   (0)
VT
(17)
With
ni 2
Nb
VT
q
C0 
(18)
C0 is the intrinsic capacity without illumination. Figure 14 presents the profile of the capacity versus junction
recombination velocity for various magnetic field values:
1) B=0 T
2) B=410-4 T
3) B=910-4 T
4) B= 10-3 T
Figure 14: Capacity versus junction recombination velocity for various magnetic field values
We observe on this figure a decrease of the capacity with junction recombination velocity Sf. When Sf increases, the
minority carriers cross the junction to participate at the photocurrent. So, the storage of these carriers near the junction
decreases. We also note that the capacity increases with magnetic field B because it entails storage of minority carriers
in the base. On figure 15, we present the profile of the capacity versus photovoltage for various magnetic field values
1) B=0 T
2) B=410-4 T
3) B=910-4 T
4) B= 10-3 T
Figure 15: Logarithm of the capacity versus photovoltage for various magnetic field values
The logarithm of the capacity versus photovoltage is a straight of slope 1/VT whose originally corresponds at value
ln(C0). We have a single value of C0 because the intrinsic capacity without illumination does not depend on the
magnetic field B. We find C0 = 1,24.10-7 F/cm2.
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4. CONCLUSION
In this theoretical study where the solar cell is enlightened at back surface, we were able to highlight the influence of
the magnetic field applied to the electrical parameters of the solar cell. From this study, we note that the back surface is
very sensitive at magnetic field. The minority carriers density increases with applied magnetic field. The determination
of the minority carriers density in the base of the solar cell, lead at expressions of the photocurrent and the photovoltage
whose the study was made according to junction recombination velocity for various applied magnetic field values. From
the photocurrent and photovoltage, the I-V characteristic was represented. This allowed us to establish the expressions
of series and shunt resistances following two operating points (open circuit situation and short-circuit situation) whose
profiles were studied versus junction recombination velocity for various magnetic field values. The capacity study was
made from profiles of the logarithm of the capacity according to the photovoltage, we determined the intrinsic capacity
without illumination. We showed that the magnetic field entails a charges accumulation near the junction, and it
increase consequently the capacity.
REFERENCES
[1] A .Mouhoub, Alternative Technological processes of Realization of the Crystalline Silicon Solar cells, Rev. Energ.
Ren. Vol. 4 (2001) 11-22
[2] Daniel. L. Meier, Jeong-Mo Hwang, Robert B. Campbell. IEEE Transactions on Electron Devices, vol. ED-35,
No.1, 1988, pp.70 – 78.
[3] Harold J. Hovel semiconductors and semimetals” volume 11 Solar Cells ACADEMIC PRESS New York San
Francisco London 1975
[4] A. Hübner, A.G.Aberle, and R. Hezel 20% Efficient Bifacial Silicon Solar Cells, 14th European PVSEC, (Munich,
2001) pp 1796 – 1798.
[5] S. N. Mohammad An alternative method for the performance analysis of silicon solar cells, J. Appl. Phys. Vol
61.N0 2 767-772 1987
[6] M.I. Ngom, B. Zouma, M. Zoungrana, M. Thiame, Z. N. Bako, A. G. Camara, G. Sissoko Theoretical study of a
parallel vertical multi-junction silicon cell under multispectral illumi-nation: influence of external magnetic field
on the electrical parameters; International Journal of Advanced Technology & Engineering Research (IJATER);
ISSN No 2250-3536 Volume 2, Issue 6, Nov. 2012 pp101-109.
[7] Amadou Diao, Ndeye Thiam, Martial Zoungrana, Gokhan Sahin, Mor Ndiaye, Grégoire Sissoko Diffusion
coefficient in Silicon Solar Cell with applied magnetic field and under frequency: electric equivalent circuits ».
World Journal of Condensed Matter Physics, 2014, 4, 84-92.
[8] G. Sissoko, C. Museruka, A. Corréa, I. Gaye and A. L. Ndiaye, Light spectral effect on recombination parameters
of silicon solar cell. Renewable Energy. 3: 1487-1490. 1996.
[9] G. Sissoko, E. Nanema, Y. L. B. Bocande, A. L. Ndiaye, M. Adj World Renewable Energy Congress (WREC)
1996, part III, pp.1594-1597
[10] H. L. Diallo, A. Wereme, A. S. Maïga and G. Sissoko, 2008. New approach of both junction and back surface
recombination velocities in a 3D modelling study of a polycrystalline silicon solar cell. Eur. Phys. J. Appl. Phys.,
42: 203–11.
[11] S. Madougou, F. Made, M. S. Boukary, G. Sissoko Characteristics for Bifacial Silicon Solar Cell studied under a
Magnetic field, Advanced Materials Research Vols. 18-19 (2007) pp. 303-312
[12] K. Kotsovos, K. Misiakos, 16th European Photovoltaic Solar Energy Conference, 1-5 May 2000, Glasgow, UK
[13] Nd. Thiam, A. Diao, M. Ndiaye, A. Dieng, A. Thiam, M. Sarr, A.S. Maiga and G. Sissoko. Electric Equivalent
Models of Intrinsic Recombination Velocities of a Bifacial Silicon Solar Cell under Frequency Modulation and
Magnetic Field Effect Research Journal of Applied Sciences Engineering and Technology (Published: November
15, 2012). Vol. 4, (22): 4646-4655.
[14] G.sissoko, C. Museruka, A. Corréa, I. Gaye, A. L. Ndiaye. Light spectral effect on recombination parameters of
silicon solar cell. World Renewable Energy Congress (1996), part III, pp.1487-1490
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