Research Journal of Applied Sciences, Engineering and Technology 4(17): 2967-2972, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: December 20, 2011
Accepted: January 12, 2012
Published: September 01, 2012
External Electric Field Influence on Charge Carriers and Electrical Parameters
of Polycrystalline Silicon Solar Cell
1
M. Zoungrana, 2B. Dieng, 3O.H. Lemrabott, 1F. Toure, 1M.A. Ould El Moujtaba,
1
M.L. Sow and 1G. Sissoko
1
Faculty of Science and Technology, University Cheikh Anta Diop, Dakar, Senegal
2
University of Bambey, UFR SATIC, UBI, Physics Section, Bambey, Senegal
3
Higher Multinational School of Telecommunications, ESMT, Dakar, Senegal
Abstract: This study deals with external electric field influence on polycrystalline silicon solar cell behavior.
We study an n-p-p+ solar cell under electric field resulting from a polarization and under constant multispectral
illumination. Taking into account this electric field, we establish news expressions of continuity equation,
photocurrent density and back surface recombination velocity. On the basis of these equations, we studied
electric field effect on charge carrier’s distribution in the bulk of the base, on photocurrent behaviour and on
the charge carrier’s recombination at the rear zone of the base.
Keywords: Electric field, electric polarization, multispectral light illumination, photocurrent, polycrystalline
silicon, recombination velocity
equation for the distribution of charge carriers in the bulk
of the base is given by Eq. (1):
INTRODUCTION
Many researches on solar cells are designed to
improve their performance. To do this we try to find
manufacturing technology (Neamen, 2003).
or operating conditions that minimize the effects
limiting the performance of solar cells, that is to say
photogenerated minority carrier’s recombination in the
bulk of the base (Shockley-Read-Hall, Auger and
radiative (Sze and Kwok, 2007; Sinton and Swanson,
1987) and surface, shading effects and resistive losses.
We present in this study, a study of electric
polarization effect on crystalline silicon solar cell under
multispectral illumination and static conditions
(Madougou et al., 2007). This study deals with external
electric field effect on photogenerated carrier’s behaviour
in the base.
PRESENTATION OF THE MODEL AND
CONTINUITY EQUATION DETERMINATION
We present in Fig. 1 a n-p-p+ (Le Quang et al., 1992).
Solar cell under multispectral illumination. In order to
study external electric field influence on charge carrier’s
behaviour in the bulk of solar cell base, we polarize by
applying a potential difference and we study in the theory
of Quasi Neutral Base (QNB) (Furlan and Amon, 1985).
External polarisation creates an internal electric field
that influences charge carriers total movement. This
electric field is the sum of the external electric field
resulting from polarization and of the solar cell internal
r r
r
electric field ( E = E ext + E int ). Under these conditions the
∂ δ ( x) d ⎡ 1
1
⎤
=
( − . J ) − ( − . J x + dx ) ⎥ − R( x ) + G ( x ) (1)
∂t
dx ⎢⎣ e x
e
⎦
3
“e” is the elementary charge, G( x ) = ∑ ai . e −b x is the
i
i =1
carrier generation rate at position x (Mohammad, 1987),
ai and bi are coefficients deduced from modeling of the
generation rate considered for overall the solar radiation
spectrum when AM=1.5 (Sissoko et al., 1996), R(x) = *(t)
/ J represents minority carrier recombination rate at the
position x:
*(x) is the photogenerated minority carriers density
at the depth x in the base and t is the lifetime of these
carriers.
The distance dx is infinitesimal, we can do a
development of J(x + dx) in Taylor series: J(x + dx) = j(x)
+ Mj(x) / Mx dx. Equation (1) becomes:
∂δ ( x ) 1 ∂J ( x )
= .
+ G( x ) − R ( x )
∂t
e ∂x
(2)
J(x) is the current density resulting of both
conduction current Jc = *(x).e. :. E and diffusion current
Jd = e . Dn . M*(x) / Mx along (Ox) axis, m is the electron
mobility and Dn the diffusion coefficient of electrons
photogenerated in the base. Jx therefore expressed as:
Corresponding Author: M. Zoungrana, Faculty of Science and Technology, University Cheikh Anta Diop, Dakar, Senegal
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Res. J. App. Sci. Eng. Technol., 4(17): 2967-2972, 2012
Fig. 1: n-p-p+silicon solar cell under external electric field
J x = J c + J d = δ ( x ) . e. µ . E + e. Dn .
∂δ ( x )
∂x
Let us LE = : .EJ and introduce electrons diffusion
length in this expression, we obtain the following
expression: LE = : .E . L2n / Dn . By replacing LE = : .E.
L2n / Dn in Eq. (6), we obtain the Eq. (7) follow:
(3)
By injecting (3) in (2), we have:
∂ 2δ ( x ) LE ∂δ ( x ) δ ( x ) G( x )
+ 2 .
− 2 +
=0
Dn
∂x 2
L n ∂x
Ln
∂ δ ( x) 1 ∂ ⎛
∂δ ( x ) ⎞
= .
⎜ δ ( x ) . e . µ . E + e . Dn .
⎟ + G( x ) = R ( x )
∂t
e ∂x⎝
∂x ⎠
finally:
∂δ ( x )
∂δ ( x )
∂E
∂ 2δ ( x )
= µ.E .
+ µ .δ ( x )
+ Dn .
+ G ( x ) − R( x )
∂t
∂x
∂x
∂x 2
(7)
The solution of this differential equation without
second member is given by:
(4)
*(x) e $x . [A.ch(" .x) + B .sh(" .x)] with
(L
1
2
Equation (4) is the general continuity equation of
minority charge carriers in the base of a semiconductor
polarized. In the case of our study, the polarization is
constant, we have ME / Mx = 0. Over the solar cell is also
under constant multispectral illumination in static
conditions, we have M*(x) / Mt = 0
The continuity equation for minority charge carriers
in the base of the solar cell under such conditions can be
summarized thus:
α=
E + 4. L2 n) 2
2. L2 n
and $ = !LE / 2. L2n
The general solution of the differential equation with
second member is given by the following expression:
3
δ ( x ) = e βx .[ A . ch(α . x ) + B . sh(α . x )] + ∑ ci . e − b x
i
i =1
(8)
with
∂δ ( x )
∂ δ ( x)
+ µ.E .
+ G ( x ) − R( x ) = 0
∂x
∂ x2
2
Dn .
(5)
ci = −
Solving the continuity equation: Equation (5) is a
differential equation of second order with constant
coefficients and second member which may be released as
follows, after replacing R(x) by its expression:
Dn .[ L n .b 2 i − LE .bi − 1]
Coefficients A and B can be determined through the
boundary conditions given by:
C
∂ δ ( x ) µ . E ∂δ ( x ) δ ( x ) G( x )
.
+
−
+
=0
Dn
∂x
τ . Dn
Dn
∂ x2
α i . L2 n
2
at the junction (x = 0):
2
(6)
SF =
2968
Dn ∂δ ( x )
.
x =0
∂x
δ ( 0)
(9)
Res. J. App. Sci. Eng. Technol., 4(17): 2967-2972, 2012
13
14
1
2
3
15.10
1:E = 0V/cm
2:E = 2V/cm
3:E = 5V/cm
4:E = 8V/cm
5:E = 10V/cm
Change carriets density (cm-3)
Change carriets density (cm- 3)
1.10
12
1.10
4
5
0
0
0.02
0.01
Base depth x (cm)
Change carriets density (cm-3)
8
8
8
8
25.10
B=
8
26.10
2
4
6
8
Electric field E (V/cm)
at the back surface (x = H):
∂δ ( x )
∂ x x=H
(10)
Parameters SF and SB represent respectively the junction
recombination velocity and the back surface
recombination velocity (Diallo et al., 2008; Mbodji et al.,
2010).
Expressions of coefficients A and B are given by Eq.
(11) and (12):
⎡
− H ( β + b) ⎤
−⎥
⎢ Dn .α .( Dn . bi − S B ). e
⎥
⎢
⎥
⎢ ( Dn . bi S F ).( Dn . β S B ).
⎥
⎢ sh α . H − D b + S .
(
)
(
)
n i
F
⎥
⎢
3 ⎢ D .α . ch α . H
⎥
(
)
n
⎥
A= ⎢
⎢ Dn .α .( S B + S F )ch(α . H ) + ⎥
i =1 ⎢
⎥
⎥
⎢ ⎡ D 2 .α 2 − ( D . β + S ) ⎤
n
B ⎥
⎥
⎢ ⎢ n
.
⎥
⎢ ⎢.( Dn . β − S F )
⎥
⎣
⎦
⎥
⎢
⎥
⎢ sh(α . H )
⎦
⎣
∑
3
∑ Ci
i =1
10
Fig. 3: Junction carriers density profile versus electric field in
short circuit situation: (L: 0.02 cm; SB: 104 cm/s; SF:
8.108 cm/s; H: 0.03 cm; D: 26 cm2 /s; :: 103 cm2/V.s)
.
5.10
3
4
5
0.02
0.01
Base depth x (cm)
D n . α ( D n . bi + S F ) . sh(α . H ) +
26.10
Dn
2
13
Fig. 4: Carrier density profile versus solar cell base depth x in
open circuit situation for different values of the electric
field: (L: 0.02cm; SB: 104cm/s; SF: 0 cm/s; H: 0.03 cm;
D: 26 cm2/s; :: 103 cm2/V.s)
27.10
δ(H)
1
0
28.10
0
SB = −
14
1.10
0
0.03
Fig. 2: Carriers density versus base depth for different values of
the electric field; (L: 0.02 cm; SB: 104 cm/s; SF: 108 cm/s;
H: 0.03 cm; D: 26 cm2 /s; :: 103 cm2/V.s)
C
1:E = 0V/cm
2:E = 2V/cm
3:E = 5V/cm
4:E = 8V/cm
5:E = 10V/cm
(11)
( Dn . bi + S F ) . ( Dn . β + S B )
. ch(α . H ) − ( D n . bi − S B ) .
( Dn . β − S F ) . e − H ( β + b )
D n .α . ( S B + S F ) . ch(α . H ) +
⎡ Dn 2 .α 2 − ( Dn . β + S B ) ⎤
⎥ . sh(α . H )
⎢
⎥
⎢. ( D n . β − S F )
⎦
⎣
i
(12)
Carrier density profile in the base: Figure 2 shows
carrier density variation versus depth x for five values of
external electric field. On Fig. 3 we observe the junction
carriers dependence of the electric field in short circuit
situation.
We observe on the curves in Fig. 2, a first zone where
carrier’s density gradient is positive. All carriers in that
part of the curve can be returned to the junction to
participate in photocurrent. We also note that the different
curves peaks move toward the junction when electric field
increases. This phenomenon is interpreted as a base depth
reduction (Sissoko et al., 1998). Indeed for a solar cell,
it’s the carriers located in the first region (region with
positive gradient) that can cross the junction and
contribute to the photocurrent. The base depth appears to
be limited to the region between the junction and the peak
maxima for each value of the electric field. A shift of
peaks towards the junction characterizes the reducing of
the usable base depth. This phenomenon is accentuated
with increasing electric field.
We also note that with the increase of electric field,
carrier’s density maxima decrease but their positive slopes
increase. The decrease of the carrier’s density maxima
reflects a reduction of photogenerated carriers in the bulk
of the base. The depletion enforcement (Fig. 3) traduces
an increase of the carriers returned to the junction under
the influence of the electric field; there is an increasing
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Res. J. App. Sci. Eng. Technol., 4(17): 2967-2972, 2012
ELECTRIC FIELD EFFECT ON
PHOTOCURRENT DENSITY AND
PHOTOVOLTAGE
14
Photo current dens i ty (A/cm 2 )
15.10
5
4
3
2
14
1.10
Electric field effect on photocurrent density: The
general expression of photocurrent density is given by
(Sinton and Swanson, 1987):
1:E = 0V/cm
2:E = 2V/cm
3:E = 5V/cm
4:E = 8V/cm
5:E = 10V/cm
13
5.10
1
0
2
0
10
4
6
8
10
10
10
Recombination velocity Sf (cm/s)
J
10
10
Photo current density (A/cm 2 )
Fig. 5: Photocurrent density versus junction recombination
velocity for different values of electric field: (SB: 4.104
cm/s; D: 26 cm2/s; L: 0.02 cm; H: 0.03 cm; ::
103cm2/V.s)
ph
⎡
∂δ ( x )
= q ⎢ Dn .
∂x
⎣
x=0
+ µ . E .δ ( 0)
(13)
By injecting Eq. (9) in (13), we have Eq. (14):
J ph = q. *(0) . (SF + : . E)
(14)
By replacing *(0) by its value in Eq. (14), we obtain the
expression of Jph:
0.036
0.034
J
0.030
2
4
6
8
Electric field E (V/cm)
10
Fig. 6: Photocurrent density versus electric field in short circuit
situation: (L: 0.02 cm; SB: 104 cm/s; SF: 108 cm/s; H:
0.03 cm; D: 26 cm2/s; :: 103 cm2/V.s)
carrier concentration at the junction with electric field. It
is as if the electric field resulting from the external bias
accelerates charge carriers so that they can reach the
junction.
Figure 4 below shows the minority carrier density
profile with depth in the base for different values of the
electric field in a situation of open circuit. It is noted here
that the maxima of the carrier density curves in the region
near the junction decrease when the electric field
increases.
We observe also on this figure that all curves have
negative slopes. This traduces the fact that at the open
circuit any carrier can cross the junction to participate to
the photocurrent. The decrease of the maximum carrier
densities with the electric field in both situations (short
and open circuit) characterizes a decrease of
photogeneration and increase of bulk recombination with
the electric field. It means also, a more diffusion of carrier
through the junction with electric field application. This
also reflects an open circuit voltage decrease with the
increase of electric field.
)
⎡ D .α . ( S − D . b ) . ch(α . H ) − e − H ( β + bi ) +
3
n
B
n
i
= ∑ q . Ci . ⎢
⎢
Dn .α . ( S B + S F ) . ch(α . H ) +
i =1
⎣
⎛ D 2 n .α 2 − ( Dn . β + S B ) ⎞ ⎤
⎜
⎟⎥
⎜ . ( D .b + D β ) . sh(α . H )⎟ ⎥
⎝
⎠
n
i
n
⎥ .(S + µ . E )
⎡ Dn 2 .α 2 − ( Dn . β + S B ) ⎤ ⎥ F
⎥ ⎥
⎢
⎢⎣ . ( Dn . β − S F ). sh(α . H )⎥⎦ ⎥
⎦
0.032
0
(
ph
(15)
We note here that when we take into account the
conduction term, we obtain a new expression photocurrent
density.
Figure 5 and 6 below show respectively, photocurrent
density profiles versus junction recombination velocity
and versus external electric field.
In Fig. 5, we notice that each curve presents three
parts: the first part where gradient is almost zero, a second
part where it is positive and the third part where he is also
zero. Without electric polarization (E = 0) we see that the
photocurrent density is practically zero at low SF values
(the carriers are blocked at the junction), the solar cell
operates in this case in open circuit condition. The
photocurrent density increases quickly with SF to finally
stabilize at large SF values. The photocurrent is maximum
and the solar cell therefore operates in short circuit
condition.
When we apply a polarization, we find that the
current at open circuit (SF ÷0) is not zero, but rather is
proportional to the bias induced field, as is the short
circuit. It therefore appears that the electric field has a
great influence on the photocurrent and hence the carrier
diffusion across the junction, as confirmed also in Fig. 6.
2970
Photo cvoltage (V )
0.8
-3
Recombination velocity Sb (cm/s)
Res. J. App. Sci. Eng. Technol., 4(17): 2967-2972, 2012
1:E = 0V/cm
2:E = 2V/cm
3:E = 5V/cm
4:E = 8V/cm
5:E = 10V/cm
0.6
0.4
0.2
0
0
5
10
10
10
Recombination velocity Sf (cm/s)
15
10
Fig. 7: Photovoltage versus junction recombination velocity for
different values of electric field: (SF: 4.104 cm/s; D: 26
cm2/s; L: 0.02 cm; H: 0.03 cm; :: 103 cm2/V.s)
Photo voltage (V)
0.64
5000
4000
3000
2000
1000
0
0
2
4
6
8
Electric field E (V/cm)
10
Fig. 9: Back surface recombination velocity SB versus electric
field: (L: 0.02 cm; H: 0.03 cm; D: 26 cm2/s; :: 103
cm2/Vs)
In addition, some electrons in the p region and a few holes
in the n region, driven by the induced electric field are
able to cross the space charge region even near the open
circuit (SF ÷0). It is this movement which explains
photocurrent presence for low values of junction
recombination velocity SF.
0.62
0.60
0.58
Electric field effect on the photovoltage: The
photovoltage expression is given in the case of the
approximation by Boltzmann:
0.56
0
10
5
Electric field E (V/cm)
15
Fig. 8: Open circuit voltage versus electric field: (SB: 4.104
cm/s; SF: 102 cm/s; D: 26 cm2/s; L: 0.02 cm; H: 0.03 cm;
:: 103 cm2/V.s)
The increasing of the photocurrent obtained near the
open circuit with electrical bias could also be one reason
for the decrease in the carrier density observed on Fig. 4.
Indeed, this phenomenon reflects the fact that a part
of photogenerated carriers in the bulk of the base flows
through the junction. This physical phenomenon could be
interpreted as follows:
The solar cell was reverse biased, the resulting
electric field, oriented from n to p as shown in Fig. 1, will
provide additional energy to the electron-hole pairs of the
base and emitter so that minority carriers can move
toward the junction more easily (Sissoko et al., 1998).
These electrons and holes reinforce some of the
ionized layers of the space charge region, which is the
source of the potential barrier. So there is an increase of
the space charge region’s width and subsequently an
increase of the barrier potential and the electric field at the
junction.
Given that the field at the junction becomes more
intense, the charge carriers sent to this zone flow much
faster, reducing the carrier’s concentration in the bulk of
the base. This could explain the curves in Fig. 3 (decrease
of the maxima and shift to the junction).
⎡ δ m ( 0) ⎤
V phm = VT .ln ⎢
+ 1⎥
⎣ n0
⎦
(16)
VT Thermal voltage VT = k .T /q( VT = 26 mV at T = 300
K)
n0 Electrons density at thermodynamic equilibrium n0 =
ni2/NB
ni Electrons intrinsic concentration for the silicon
NB The doping density at the base (NB = 1016 cm !3) and
k is the Boltzmann constant
Figure 7 and 8 show photovoltages variation versus
junction recombination velocity SF for various electric
field.
We observe in Fig. 7 that large photovoltage values
correspond to small values of SF with a zero gradient
(open circuit voltage) and this gradient becomes negative
and constant whatever electric field value before
vanishing for large SF values.
We also note thattheopen circuit voltage (low values
of SF) decreaseswith the electric fieldaccording to
ourprojections in paragraph(3), as confirmedalsoin Fig. 8.
Electric field effect on back surface recombination
velocity: We study in this section the induced electric
field effect on electron-hole pair generation or
recombination into the solar cell under optical excitation.
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Res. J. App. Sci. Eng. Technol., 4(17): 2967-2972, 2012
By observing photocurrent density profiles (Fig. 5),
we see that the gradient of photocurrent is zero for large
values of SF. In this region of SF, we can write:
MJ ph / MSF = 0
(17)
The solution of Eq. (17) gives two values of back
surface recombination velocity SB:
⎡ Dn 2 .α 2
⎤
⎥ . sh(α . H )
⎢⎣ − ( µ . E + Dn . β ) . Dn β ⎥⎦
Dn .α . ch (α . H )
µ . E . Dn .α . ch(α . H ) − ⎢
S Bo =
− ( µ . E + Dn . β ) . sh (α . H )
REFERENCES
(18)
We note here that SB0 is a function of electric field
and the diffusion coefficient Dn, and independent of the
carrier’s generation terms ai, bi, so it is a diffusion rate.
The second value of SB is:
α .bi . (ch (α . H ) − e − H .( β + b )
i
3
S B = Dn . ∑
i =1
− [α 2 − β . (bi + β )]. sh (α . H )
α .(ch(α . H ) − e − H ( β + b )
i
− (bi + β ). sh(α . H )
minority carriers density, photocurrent and back surface
recombination velocity, all dependent on the electric field.
We studied electric field influence on these parameters. It
appears from this study that the solar cell polarization
decreases back surface recombination velocity and bulk
recombination, increase carrier’s mobility to the junction
and facilitate their crossings of the junction. This
phenomenon comes with the widening of junction space
charge zone, which drive to short circuit current increase
with polarization electric field increase.
(19)
Back surface recombination velocity SB is a function
of generation terms ai, bi, electric field and diffusion
coefficient Dn.
Figure 9 follow illustrates back surface
recombination velocity behavior versus polarization
electric field.
We observe in this figure back surface recombination
decrease with electric field increase. We can therefore say
that solar cell electric polarization reduces carrier’s
recombination at the rear zone of the base and enhances
the BSF (Back Surface Field) effect (Umesh and Jasprit,
2008). These results are in perfect agreement with those
of the preceding paragraphs: increases of photogenerated
carrier’s density at the junction, short-circuit and open
circuit currents with electric field increase. We thus arrive
atthe conclusionthat solar cell electric polarization
reducesthe phenomenon of bulk recombinationin the base
and enhancescarrier’s migrationtothe junctionfor a
possibleparticipation in thephotocurrent.
CONCLUSION
In this study, we established a continuity equation
that depends on the electric field. The resolution of the
continuity equation drove us to new expressions of excess
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