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Journal des Sciences
3D STUDY OF BIFACIAL SILICON SOLAR CELL UNDER INTENSE
LIGHT CONCENTRATION AND UNDER EXTERNAL CONSTANT
MAGNETIC FIELD: EFFECT OF MAGNETIC FIELD AND BASE DEPTH
ON EXCESS MINORITY CARRIER GENERATION.
M. ZOUNGRANA1, I. LY2, B. ZOUMA1, F. I. BARRO3, G. SISSOKO4
1
Laboratoire des Semi-conducteurs et d’Energie Solaire, Département de Physique,
Faculté des Sciences et Techniques, Université Cheikh Anta Diop de Dakar, Sénégal
2
Département Génie Electro-Mécanique, Ecole Polytechnique de Thiès, Université de Thiès, Sénégal
3
Département Energie, Institut de Recherche en Sciences Appliquées et Technologies(IRSAT),
Centre National pour la Recherche Scientifique et Technologique (CNRST), Burkina Faso
4
Laboratoire des Semi-conducteurs et d’Energie Solaire, Département de Physique,
Faculté des Sciences et Techniques, Université Cheikh Anta Diop de Dakar, Sénégal
Abstract:
Many studies on recombination and electrical parameters of
solar cells under constant illumination neglecting the bulk
electric field have been proposed with one [1-3] or three [4]
dimensional approach. This work present a three-dimensional
study of bifacial silicon solar cell under intense light
concentration (more than 50 suns) [5] and under constant
magnetic field. This approach is based on the resolution of the
minority continuity equation, taking into account the
distribution of the electric field in the bulk evaluated as a
function of both majority and minority carrier densities [5]. In
this approach, news analytical expressions of diffusion length,
diffusion coefficient and excess minority carrier density was
established for front illumination. The effects of base depth
and magnetic field [1-3] on excess minority carrier generation
and profile are then analysed.
Résumé:
Plusieurs études portant sur les paramètres de recombinaison
et les paramètres électriques des photopiles ont été faites à
une[1-3] ou 3 [4] dimensions en négligeant cependant le
champ électrique interne (cristallin). Ce travail est une étude
d’une photopile bifaciale sous concentration (au delà de 50
soleils) [5] et en présence d’un champ magnétique externe
constant ; la résolution de l’équation de continuité est faite ici
en exprimant le champ électrique cristallin comme fonction à
la fois des densités de porteurs minoritaires et majoritaires[5].
Avec cette approche, de nouvelles expressions du coefficient
de diffusion et de la densité de porteurs minoritaires en excès
sont proposées pour un éclairement par la face avant de la
photopile. Les effets de l’épaisseur de la base ainsi que du
champ magnétique [1-3] sur la densité des porteurs
minoritaires en excès sont alors analysés.
Keywords : Intense light concentration, carrier generation, Mots clés : concentration, génération de porteurs, champ
electric field
électrique
I . INTRODUCTION
The output of a solar cell is closely bound to its electronics properties mainly governed by electrons
and holes diffusion length, diffusion coefficient and lifetime. However, some externals factors as
magnetic field, external electric field, intense light [6] and internals factors as grain size, grain
boundary recombination velocity [7], electric field in the bulk due to carrier concentration [5] can
influence the solar cell quality. Numerous investigations on the effects of these externals and internals
factors on solar cells have been proposed in one dimensional study.
In this work, we propose a three dimensional study of bifacial solar cell under intense light
concentration (more than 50 suns) and under external variable magnetic field. The calculations were
carried out in the case of a polycrystalline device with columnar grain orientation and front side
illumination. This approach is based on the resolution of the continuity equation, taking into account
the distribution of the electric field in the bulk evaluated as a function of both majority and minority
carrier densities [5]. In this approach news analytical expressions of carrier diffusion length, diffusion
coefficient, excess minority carrier density was established for a front side illumination. The effect of
magnetic field on carrier diffusion length and carrier diffusion coefficient, and the effects of base
depth and magnetic field [1-3] on excess minority carrier generation and profile are then analysed.
M. Zoungrana et al / J. Sci.Vol. 7, N° 4 (2007) 73 – 81
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II . MODEL AND ASSUMPTION
II . 1 Analytical formulation
We consider a bifacial polycrystalline Si-solar cell operating under a concentrated polychromatic
light. This kind of solar cell, because of its manufacture technique is constituted of several grains of
different sizes and forms separated by grain boundaries that are very great recombination centers.
Macroscopics and microscopics parameters of the solar cell are deeply dependent on grain boundaries.
These grains boundaries are either grains junctions or either local distortions caused by solar cell
manufacture foulness. With no loss in generality, we consider a columnar model in where the solar
cell is modelled as a regular array of parallelepipedic grains connected in parallel [7].
Each grain has a square section gx = gy and a thickness H as shown in Fig 1. The recombination
planes are assumed to be thin surfaces between two adjacent grains and are located at x = ±
y=±
gy
2
gx
and
2
; respectively, perpendicular to the x- and y-axis of a cartesian coordinates (O; x; y; z)
[7]. The grains boundaries are perpendicular to the junction and their effective recombination velocity
Sgb is constant.
The cell is excited uniformly on the emitter surface by an intense light concentration (more than 50
suns). The light penetrates the junction at the plane z = 0 and the back surface is situated at z=H .
Because of the light intensity, carrier concentration in the base is not uniform. So, we take into
account the electric field E(z) due to the difference of carrier concentration on Z axis [5].
To investigate the magnetic field influence on carrier behaviour and photocurrent, we applied a
variable external magnetic field with the induction B parallel to the surface of the n-p junction [8].
Figure 1: Theoretical model of square grain with an external
magnetic field and the internal electric field
To put in evidence the electric field analytical expression on every axis, we considered a one
dimensional study of a solar cell under intense light concentration [5]. If we take into account the
electric field E(z) in the base, the expression of the excess minority carrier continuity equation is [5]:
Dn ⋅
∂ 2δ ( z )
∂ ( E ( z ) ⋅ δ ( z )) δ ( z )
+ µn ⋅
−
= −C ⋅ G ( z )
2
∂z
τn
∂ ( z)
Dn and µn represent respectively diffusion coefficient and electrons mobility, δ(z) is the excess
minority carrier density in the solar cell base and G(z) the generation rate at the distance z of
junction, C is the light concentration (in suns). Dn and µn are not position dependent.
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The expressions of electrons and holes current densities respectively Jn (z) and Jp (z) are:
J n ( z ) = q ⋅ µ n ⋅ E ( z ) ⋅ δ n ( z ) + q ⋅ Dn ⋅
and
J p ( z) = q ⋅ µ p ⋅ E( z) ⋅ δ p ( z) − q ⋅ D p ⋅
∂δ n ( z )
∂z
∂δ p ( z )
∂z
Under open circuit condition, the total current density J (z)= Jn (z)+ Jp (z) is equal to zero. Using the
charge neutrality condition, in the p-region δn(z)+ n0 = δp(z) ( n0 is the doping level) [5], and the very
high injection condition δ n ( z ) n0 we have: δ n ( z ) δ p ( z ).
So, the electric field E(z) on Z axis reads:
E( z) D p − Dn
µ p + µn
⋅
∂δ n ( x, y, z )
1
δ n ( x, y , z )
∂z
Introducing this expression of the electric field in the model of figure 1 and taking into account our
assumptions, the expression of excess minority carriers distribution in the solar cell is:
∂δ ( x, y , z ) 1
= ⋅ ∇.J n + G ( z ) − R( z )
e
∂t
G(z) is the excess minority carrier generation at position z , and R(z) the recombination rate at this
position: for front side illumination [1,2]
3
G ( z ) = C ⋅ ∑ ai ⋅ e −bi z
i =1
a i and b i are deduced from modelling of the generation rate considered for all the solar radiation
spectrum [1,2].
Recombination rate expression is R ( z ) =
δ n ( x, y , z )
where τ n is electron lifetime in the base, and
τn
the excess minority carrier density in the base of our illuminated solar cell.
The expression of J n is given by the equation of transportation phenomenon [9]:
J n = J d + J In + J c
In this expression:
- J d represent carriers diffusion current on X,Y,Z axis:
J d = e.Dn .∇δ n ( x, y, z )
- J In represent the Y axis magnetic field induced current:
J In = − µ n .J n Λ B with B = B j , B ( x) = B ( z ) = 0
- J c represent conduction current on Z axis.
J c = e.µ n .δ n .( x, y, z ).E
where, according to our model, E is expressed by:
E = − E ( z )k
When the solar cell is front illuminated, we suppose an uniform carriers distribution along X and Y
axis at any base depth. Therefore, conduction current doesn't exist according these axis, and so that:
∂J c ∂J c
=
=0
∂x
∂y
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On the basis of these assumptions, we deduce the expressions of electrons density current J n
and
then, the components of ∇.J n are given by the following expressions:
∂J dx
∂J
∂J
+ µ n .B. dz + µ n .B. cz
∂J nx
∂x
∂x
= ∂x
2 2
1 + µn B
∂x
∂J ny ∂J dy
=
∂y
∂y
∂J dz
∂J
∂J
− µ n .B. dx + µ n .B. cz
∂J nz
∂z
∂z
= ∂z
2 2
1 + µn B
∂z
Taking into account the coordinates of J n , equation 5 can be written in permanent regime
(
∂δ ( x, y, z )
= 0) as:
∂t
Cx ⋅
∂ 2 δ ( x, y , z )
∂ 2 δ ( x, y , z ) ∂ 2 δ ( x, y , z ) G ( z ) δ ( x, y , z )
+
⋅
+
+ ∗ −
= 0 with
C
y
D
L∗2
∂x 2
∂y 2
∂z 2
Cx =
D∗ =
Dn ( µ p + µ n )
2.Dn .µ n + Dn .µ p − µ n .D p
(Dn − µ n⋅ A) , L∗2 = τ
1+ µ ⋅ B
2
n
2
n
and C y =
⋅ D ∗ with A =
Dn ( µ p + µ n ).(1 + µ n2 B 2 )
2.Dn .µ n + Dn .µ p − µ n .D p
D p − Dn
µ p + µn
and R( z ) =
and
δ ( x, y , z )
τn
Equation 13 is the continuity equation of a solar cell under constant multispectral illumination and
without magnetic field. Our model lead us to the news expressions of carrier diffusion length (L*) and
carrier diffusion coefficient (D*), which depend on magnetic field, electrons and holes mobility, and
D p − Dn
characterize the
diffusion coefficients (Dn, Dp). In these expressions, the coefficient A =
µ p + µn
approximation made on the electric field and Cm =1+µn2 ⋅ B2 the one made on the magnetic field.
II . 2 Solution of the continuity equation
Equations 13 is a partial derivative differential equation for which the general shape of the solution
can be written [3] as:
δ ( x, y, z ) = ∑ ∑ Z j , k ( z ) ⋅ cos(C xj x) ⋅ cos(C yk y ), with C xj =
j
k
Cj
Cx
and C yk =
Ck
Cy
The coefficients Cj and Ck are determinate by the boundary conditions on the grain boundaries for
x=±
gx
and
2
y=±
gy
2
respectively:
g
Sgb
⎡ ∂δ ( x, y, z ) ⎤
= ± ∗ ⋅ δ ( ± x , y, z )
⎢
⎥
∂x
2
D
⎣
⎦ x =± g x
2
gy
⎡ ∂δ ( x, y, z ) ⎤
Sgb
and ⎢
= ± ∗ ⋅ δ ( x, ± , z )
⎥
2
∂y
D
⎣
⎦ y =± g y
2
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The coefficients Cxj and Cyk are given by the transcendental equations:
gy
gx
Sgb
Sgb
) = ∗ and C yk tan(C yk ⋅ ) = ∗
2
2
D
D
While taking into account that cos(C xj x) (and cos(C yk y ) ) are orthogonal functions, the z
C xj tan(C xj ⋅
dependence of δ(x,y,z) can be expressed for a front side illumination as [3]:
⎛ z
Z jk ( z ) = A jk cosh⎜
⎜ L j, k
⎝
⎞
⎛
⎟ + B jk sinh⎜ z
⎟
⎜ L j, k
⎠
⎝
⎞ 3
⎟ − ∑ K i e −bi⋅z with :
⎟ i =1
⎠
1
⎡ 2
1
1 ⎤2
2
= ⎢C j + C k + 2 ⎥ and
L j, k ⎣
Ln ⎦
gy
g
16 ⋅ sin(C xj ⋅ x ) ⋅ sin(C yk ⋅ )
1
2
2
= ∗
D j , k D ⋅ sin(C xj ⋅ g x ) + C xj ⋅ g x ⋅ sin(C yk ⋅ g y ) + C yk ⋅ g y
[
][
]
The expressions of the coefficients Ajk and Bjk are obtained by solving equation 14, with the following
boundary conditions at the two edges of the base region [10,11]
- the junction interface at x = 0 :
Sf =
D∗
⎡ ∂δ ( x, y, z ) ⎤
⋅⎢
⎥⎦
∂z
δ ( x, y, 0) ⎣
z =0
- the rear side at x=H
Sb = −
D∗
⎡ ∂δ ( x, y, z ) ⎤
⋅⎢
⎥⎦
∂z
δ ( x, y , H ) ⎣
z=H
This lead to:
3
A jk = ∑ K i ⋅
i =1
1
L j, k
⎛ Sf
⎛ Sf
⎞
⎞
⋅ ⎜⎜ ∗ − bi ⎟⎟ ⋅ exp(− bi ⋅ H ) + β j , k ⋅ ⎜⎜ ∗ + bi ⎟⎟
⎝D
⎠
⎝D
⎠
Sf
1
⋅ β j,k +
⋅α j, k
∗
L j, k
D
and
S f ⎛ Sb
⎛ Sf
⎞
⎞
⎜⎜ ∗ + bi ⎟⎟
(
)
exp
⋅
−
⋅
−
⋅
−
α
⋅
bi
bi
H
⎜
⎟
j
,
k
∗
∗
3
D ⎝D
⎠
⎝D
⎠
B jk = ∑ K i ⋅
Sf
1
i =1
⋅ β j, k +
⋅α j, k
∗
D
L j,k
with:
α j,k =
⎛ H
1
⋅ sinh ⎜
⎜ L j, k
L j, k
⎝
⎞ Sb
⎛ H
⎟+
⋅ cosh⎜
∗
⎟ D
⎜ L j, k
⎠
⎝
⎞
⎟ and
⎟
⎠
β j,k =
⎛ H
1
⋅ cosh⎜
⎜ L j, k
L j, k
⎝
⎞ Sb
⎛ H
⎟+
⎜
⋅
sinh
⎟ D∗
⎜ L j, k
⎠
⎝
⎞
⎟
⎟
⎠
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III . SIMULATION RESULTS AND DISCUSSIONS
We present in this part a 3-D study of magnetic and electric field effects, on the excess minority
carriers generation in the bulk of the base of a bifacial solar cell front illuminated.
This study in modelling is made on the basis of the following parameters: base depth: H = 300 µm;
doping level:
coefficient:
N A = 1016 cm −3 ;
electron diffusion length:
Ln = 200 µm ; electrons diffusion
Dn = 26 cm 2 .s −1 ; holes diffusion coefficient: D p = 4 cm 2 .s −1 ; electron mobility:
µ n = 1350 cm.s −1 ; hole mobility: µ p = 150 cm.s −1 .
III .1 Carriers generation in the base
We study in this part, base depth and magnetic field influences on minority carriers generation in the
bulk of base. We stand in a context where the photogenerated carriers in base bulk are blocked (open
circuit condition: S f = S fo ).
a. Base depth effect
The following curves put in evidence the excess minority carrier photogeneration in the solar cell at
different base depth.
Fig. 2 (a, b and c) give respectively the excess minority carriers photogenerated at Z = 0.0001
(region close to the junction) ; Z = 0.015 (in the middle of the base) and Z = 0.020cm (rear side of
the base) of a bifacial silicon solar cell front illuminated.
Figures 2: Excess minority carriers profile in the base of a front photoexcited
cell in versus grain width ( X×Y) and base depth ( Z ) for: C = 200 suns;
B = 5. 10-⁵ T; Sf = 1.5·101.5 cm.s-¹; Sb = 104 cm.s-¹; Sgb = 100 cm.s-¹
M. Zoungrana et al / J. Sci.Vol. 7, N° 4 (2007) 73 - 81
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The observed carriers densities exhibit clearly that excess minority carriers photogenerated in the base
is extensively tributary not only on the width of the grain base surface ( X × Y ) but also on the
position of this surface.
Indeed, one notice that for every level of the base depth, the quantity of the excess minority carriers
increase when we move the grain surrounding outside base surfaces toward the middle of this one,
were, the generation is maximal. This phenomenon is the consequence of grains boundaries
recombination velocities to which grains boundaries are strongly exposed.
We notice also that for the same position of the grain base surface, the excess minority carriers
decrease from the region close to the junction ( Z = 0.0001cm) toward the rear zone of the base
( Z = 0.020cm) . This phenomenon is the consequence of our illumination mode. Indeed, for a front
illumination, absorption intensity decrease from junction toward the rear zone of the solar cell base,
photogenerated carrier in the bulk of the base will vary in the same manner.
b. Magnetic field effect
We present now, magnetic field influence on the excess minority carriers generated in the bulk of the
base for front illumination.
Figure 3: Excess minority carriers profile in the base of a front photoexcited
cell in versus grain width ( X×Y) and magnetic field for: C = 200 suns;
Z=0.001 cm; Sf =1.5·101.5 cm.s⁻ ¹; Sb = 104 cm.s⁻ ¹; Sgb = 100 cm.s⁻ ¹
M. Zoungrana et al / J. Sci.Vol. 7, N° 4 (2007) 73 - 81
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Figure
3
(a,
b
and
c)
show
the
influence
of
magnetic
field
−5
−5
−5
( B = 7.10 T , B = 35.10 T , B = 63.10 T ) on the excess minority carriers near the junction
( Z = 0.0001cm ) for a front illumination. We notice globally that an increase of the applied magnetic
field lead to a very sensitive decrease of the excess minority carriers photogenerated in the base. This
very logical report is the consequence of Hall effect. This result perfectly agrees with diffusion length
and carrier diffusion coefficient decrease with magnetic field increase because of Hall effect on carrier
mobility [12].
III .2 Effect of field on excess minority carrier density profiles
In this paragraph, we study the minority carrier density profiles for a front illumination
mode and with respect to magnetic field. We get in the situation where a part of
photogenerated carrier in the base of the solar cell crosses the junction to participate to the
photocurrent. For that, we work with a junction recombination velocity S f = 10 4 cm.s −1 and
back surface recombination velocity S b = 10 3 cm.s −1 .
We show on Figure 4 (a and b), two observation plans of the base excess minority carrier
density profiles for front illumination and with respect to base depth and magnetic field.
Figure 4: Excess minority carriers profile in the base of a photoexcited cell
versus base depth and magnetic field for front illumination: C = 200 suns;
g=0.003 cm-³; Sf =104 cm.s-¹; Sb = 103 cm.s-¹; Sgb = 100 cm.s-¹
Fig. 4-a and 4-b shows the effect of base depth on the excess minority carriers density profile in the
bulk of the base region for a bifacial solar cell front illuminated.
We observe on figures 4 that for every value of magnetic field, the curves exhibit two characteristic
regions.
- The first region closed to the junction where excess minority carriers photogeneration with respects
to base depth and magnetic field increase. All the carriers photogenerated in this region have more
excess energy to cross junction and participate to current production. Carriers density maxima
displacement toward the junction with respect to magnetic field is discerned like a reduction of the
space-charge region comparatively to the initial diffusion depletion layer [13]
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- The second region where excess minority carriers photogeneration decrease with respect to base
depth and increase with respect to magnetic field. The carriers created in this region of the base don't
have enough energy to surmount the traps and to go back up the slope to participate to current
production. They disappear by recombination in the base.
IV . CONCLUSION
This study introduced a new expression of continuity equation for electrons, which takes into
account magnetic field and electric field due to carriers concentration in the bulk of a solar cell under
intense light concentration. The investigation of carriers generation with respect of base depth and
magnetic field show a strong carriers generation in the region close the junction for front illumination
of solar cell. Under magnetic field effect, excess minority carriers photogeneration increase, but the
quantity of carriers that crosses the junction increases with carriers density curves maxima
displacement toward the junction. The limiting effects of magnetic field on excess minority carriers
density (δ ( x, y, z )) are pronounced for large values of magnetic field.
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