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Equivalent Electric Model of the Junction
Recombination Velocity limiting the Open Circuit
of a Vertical Parallel Junction Solar Cell under
Frequency Modulation
1
Fatimata BA, 2Boureima SEIBOU, 3Mamadou WADE, 1Marcel Sitor DIOUF, 3Ibrahima LY and 1Grégoire
SISSOKO
1
Faculty of Science and Technology, University Cheikh Anta Diop, Dakar
2
Ecole des Mines de Niamey-Niger
3
Ecole Polytechnique of Thies, EPT, Thies, Senegal
ABSTRACT
In this paper, a study of the junction recombination velocity limiting the open circuit (SFoc) of a silicon solar cell under
polychromatic illumination, infrequency modulation is presented. From the continuity equation, the density of minority charge
carriers in the base and the photovoltage are determined. The study of this photovoltage, according to the junction
recombination velocity, allows us to determine the junction recombination velocity limiting the open circuit. An equivalent
electric model is proposed by means of representations Bode and Nyquist diagrams of SFoc.
Keywords: Junction recombination velocity, Photovoltage, Bode and Nyquist diagrams, Impedance spectroscopy
1. INTRODUCTION
Among the parameters that affect the performance of a solar cell include the junction recombination velocity. Some
technical characterization developed in static regime [1] or in dynamic regime (frequency or transient) [2],are intended
to identify recombination parameters of minority charge carriers photogenerated solar cells [3]- [5], which are
electronic order (diffusion length, lifetime, recombination velocity at surfaces and interfaces of free charge carriers) and
electrical (shunt resistance, series resistance, ...).And other more advanced have led to the technical determination of
the junction recombination velocity initiating the short-circuit (SFsc) [6]of a bifacial solar cell in static regime under
multispectral illumination, the technical determination of the effective diffusion length and intrinsic recombination
velocity of a multi junction solar cell [7], the influence of the electric field on the junction recombination velocity
limiting the open circuit (SFoc) of a bifacial solar cell [8],.
In our case, we propose to study the effect of the angular frequency on the junction recombination velocity SFoc of a
typical parallel vertical junction solar cell n+-p-n+[9] -[10],by means of Bode and Nyquist diagrams.
2. THEORY
A solar cell considered is n+-p-n+ and its structure is shown in Figure1:
Figure 1: Parallel vertical junction solar cell
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Where SCR is the space charge zone H the total thickness of the base and z the depth of illumination.
Under the effect of an excitation light, absorption phenomena, creating electron-hole pairs, diffusion and recombination
of minority carriers, are noted in the base of the solar cell. These phenomena result in the following continuity
equation:
 2  ( x, t )  ( x, t )
 ( x, t )
D ( )

 g ( z, t ) 
2
x

t
(1)
Where:
-D (ω) is the complex diffusion coefficient of the minority carriers in the base of the solar cell [11] - [12] given by:

1  ( ) 2
 1  ( ) 2
D ( )  D 


2 2 2
2
(1   2 2 ) 2  ( 2 ) 2
 (1    )  ( 2 )

j

(2)
With D the diffusion coefficient constant.
- τ the minority carriers lifetime
- δ(x, t) the global minority carriers density at position x and time t
- g(z, t) the global generation rate of minority carriers at the depth in the base z and time t.
δ(x, t) and g(z, t) can be written in the form as follows [13] -[14]:
 ( x, t )   ( x). exp( jt )
g ( z, t )  g ( z ). exp( jt )
(3)
(4)
Where δ(x) is the minority carriers density according to the thickness x; ω the angular frequency; g(z) the minority
carriers generation is written as follows[15]:
3
g ( z )   ai . exp(bi z )
(5)
i 1
ai and bi are coefficients deduced from modeling of the generation rate considered for the overall solar radiation
spectrum [16].
Solving equation (1) leads to the general solution of the minority carriers density:
 ( x)  A sinh(
With
Ki 
3
x
x
)  B cosh(
)   k i . exp(bi z )
L( )
L( )
1
(6)
ai
L2 ( )
D ( )
L (ω) is the diffusion length according to the frequency:
L ( )   .D .
1  j
1  ( ) 2
(7)
Using the boundary conditions, the coefficients A and B have been determined [17] - [18] given by:
- At the junction(x=0)
D( ).
 ( x )
x
Volume 4, Issue 7, July 2016
x0
 Sf . ( x) x 0
(8)
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The excess minority carriers undergo recombination events to the emitter-base junction(x = 0) of the solar cell. This
recombination activity in the junction is characterized by Sf (junction recombination velocity) [19]-[21].
Sf is the sum of the intrinsic recombination velocity denoted Sfo which is the intrinsic recombination velocity induces
by shunt resistance and depending only on the intrinsic parameters of the solar cell and the recombination velocity Sfm
which is the junction recombination velocity relates to the external load imposing the operating point of the solar cell.
Sf  Sf 0  Sf m
(9)
At the middle of the base(x =H / 2)
 ( x )
x
x
H
2
0
(10)
From the excess minority carriers density, we can deduce the photovoltage across the junction, according to the
Boltzmann’s relation as follow:
 Nb

V ph  VT ln 1  2  ( x ) x 0 
 n

(11)
With n the intrinsic carriers density, Nb the base doping density and VT the thermal voltage equal to 25.9mV for T
=300K whose expression is:
VT 
kT
q
(12)
Where k is the Boltzmann constant and q the elementary charge electrons (1.610-19C).
By substituting in (11), the expression of minority carriers density, the photovoltage as function of the angular
frequency ω, the junction recombination velocity Sf and the depth in the base z. Therefore, were present the profile of
the photovoltage versus the junction recombination velocity for different angular frequency in Figure2:
Figure 2: Photovoltage versus junction recombination velocity for different angular frequency.
1) ω = 103rad.s-1 ; 2)ω = 105 rad.s-1 ; 3)ω = 107 rad.s-1 ; z = 0.003cm, τ = 10-5s ; H = 0.03µm ; D = 26cm2.s-1.
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For lower junction recombination velocity (Sf 2.102cm.s-1), the photovoltage reaches a maximum level. The value of
this level corresponds to the open circuit photovoltage where there is an accumulation of minority carriers stored across
the junction. This situation is mathematically translated by the following relationship:
V ph
Sf
0
(13)
Several important parameters are used to characterize solar cells: the short-circuit (Jcc), the fill factor (FF), the
efficiency (η) and the open circuit voltage (Voc) which is our study area.
Expression of Voc is then obtained when the junction recombination velocity of photovoltage tends towards values of
SFoc [8].
Voc  lim V ph
(14)
Sf SFoc
Where:
 ( , z )  D( ).sinh(
SFoc ( , z ) 
H
)
2 L( )
(15)
H
L( ). cosh(
)
2 L( )
Nb.D( ).sinh(
With:
 ( , z ) 
2
n (e
VCO
VT
H
)
2 L ( )
 1)
2.1. Bode and Nyquist diagrams
Bode diagram is a method developed to simplify obtaining the frequency response of a complex quantity, while the
Nyquist diagram is the representation of the imaginary part as a function of the real part of a complex quantity[22].
2.1.1. Bode diagram of SFoc
In figure 3(a) and 3(b), are shown the profiles of the module and the phase of SFoc versus the logarithm of angular
frequency:
Figure 3: Module and Phase of SFoc versus logarithm of angular frequency. (z = 0.003cm)
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In Figure 3-a, the module of SFoc is maximum and constant for lower frequencies (ω
ωc).
For ω>ωc, the module of SFoc decreases: it’s the frequency regime. Indeed, for higher frequencies, few minority
carriers are photogenerated, few minority carriers are stored in the emitter-base junction, their diffusion is low and
there is a high recombination in volume of these carriers resulting in low recombination rate SFoc; this fact is
confirmed by the phase plot with a negative value of the phase(figure3-b).
In Figure3-b, the phase is almost zero for frequencies below the cut off frequency (ω<ωc). There is no phase shift
between optical excitation and the recombination of minority carriers when the solar cell is presented as a voltage
generator in static regime. For ω>ωc, the phase decreases. There is a phase shift between optical excitation and there
combination of minority carries in frequency regime: the capacitive effects predominate.
To determine the cut off frequency (ωc) graphically, we project the intercepted point of the two tangents lines to the
curve on the log (ω) axis. This projection allows us to obtain Log (ωc) (see figure3-a) (ωc = 105 rad.s-1).
2.1.1. Nyquist Diagram of SFoc
The imaginary part versus real part of SFoc is shown in figure4:
Figure 4: Im (SFoc) versus Re (SFoc).
In figure 4, the curve representing the imaginary part as a function of the real part of SFoc is a semi-circle where Im
(SFoc) is negative.
This diagram shows three particular points that correspond to: ω=0, ω=ωc and ω→∞.
For zero frequency, the imaginary part of SFoc is zero while its real part corresponds to a constant negative value
(SFoc,s).
For a frequency tending to the cutoff frequency, both real and imaginary parts of SFoc increase .Knowing the value of
the cut off frequency, we can deduce the life time τ of minority carriers photogenerated from the following relationship:

1
c
(16)
Fromτ, we can establish and deduce the capacitance C through the relation:
  ( SFoc, p ).C 
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1
c
(17)
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For a frequency that tends to infiny, the imaginary part of SFoc decreases while its real part tends to zero, allowing us
to determine the SFoc, p diameter of the half-circle. From Nyquist diagram of SFoc, we give the following table:
Table1: Components of SFoc following the Nyquist diagram
ω (rad.s-1)
Nyquist diagram of SFoc
0
SFoc,s
-
-
ωc
-
-
→∞
-
SFoc,s + C //
SFoc,p
-
SFoc,s + SFoc,p
2. 3. Technical determination of the equivalent electric model of SFoc
For this, we consider the center of a semicircle Δ ((SFoc,s+ SFoc,p /2);0) and radius R(SFoc,p /2).
SFoc can be rewritten as the two components (real and imaginary part)
SFoc  Re( SFoc )  j Im( SFoc )
(18) 
(19)
SFoc  X  jY
With X= Re (SFoc); Y=Im (SFoc) and j the complex variable (j=√-1)
To simplify notations, we define the variables s and p; with s = SFoc,s and p=SFoc,p
The terms of SFoc are connected by the equation:
2
p 
p 2

2
 X  ( s  2 )   Y  ( 2 )
(20)
The resolution of the equation leads to the determination of X and Y:
 2 .C 2 .(SFoc, p) 2
X  Re( SFoc)  SFoc, s 
1  (.C.SFoc, p ) 2
.C.(SFoc, p ) 2
Y   Im( SFoc) 
1  (.C.SFoc, p ) 2
(21)
(22)
The characteristics of particular points are summarized by the following equations:
 ω→0
X  Re( SFoc )  SFoc, s
Y   Im( SFoc )  0
(23)
(24)
 ω= ωc
X  Re( SFoc )  SFoc, s 
Y   Im(SFoc ) 
SFoc, p
2
SFoc, p
2
(25)
(26)
 ω→∞
X  Re( SFoc )  SFoc, s  SFoc, p
Y   Im( SFoc )  0
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(27)
(28)
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From these expressions and the figure4, we can establish the following table:
Table2: Module of SFoc for particular points of angular frequency
Drawing on some studies which aim to full process of identifying the electrical parameters of the impedance, we
proceed to the determination of electrical parameters of SFoc [23]:
From the macroscopic point of view, an illuminated photovoltaic cell includes a voltage V across the external load,
crossed by a current I. when the solar cell is in open circuit, we have:
a) V=Voc
b) I=0
c) Sf → SFoc
d) Rch (external load) increases (the impedance Z is higher)
So, there is a relationship between the junction recombination velocity Sf and the external load.
In this following figure, we represent the equivalent circuit which summarizes the functioning solar cell in
photovoltage generator [24]:
Figure 5: Equivalent circuit of the solar cell operating in open circuit
The high impedance (Z), placed in series with the solar cell means that no charge carriers current flows through the
junction. Thus, one can compare the value of SFoc,s in this area to determine the corresponding series resistance(Rs).
From figure 5, we can establish the following relation:
Rs ( , f , z) 
Vco( , z )  Vph( , f , z )
Jph( , f , z )
(29)
With Jph, the photocurrent density given by:
Jph  2.q.D( )
 ( x )
x
x0
(30)
The factor 2 results from the two junctions around the base and the fact that they are connected in a parallel manner.
In figure6,is shown the module of series resistance versus the junction recombination velocity (Rs = f (Sf)). The values
of SFoc,s obtained with the Nyquist representation of SFoc at ω→0 is plotted on the curve. The intersection with the
vertical-axis corresponds to Rs at ω→0.The results obtained for various angular frequencies are consigned in table3.
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Figure 6: Technical determination of Rs. (ω = 0; z = 0.003cm)
Table3: Values of series resistance
ω (rad.s-1)
Rs(Ω.cm2)
0
0.98
103
0.99
5
1.01
10
But however as great as the impedance is, a small current carrying pass into this resistance. Moreover the open circuit
is nothing other than a short- circuit of the solar cell on its own impedance, so there will be always carriers of current
through the junction.
In order to determine the electrical parameters, many researches use the method of impedance spectroscopy [25] - [28],
and al [29] calculated the equivalent complexe impedance to be:
Z  Rs 
 .Rp 2 .Cp 
Rp

j

2 
1  (.Rp.Cp ) 2
1  (.Rp.Cp ) 
(31)
With: Rs series resistance; Rp parallel resistance (sometimes referred to as the shunt resistance Rsh) and Cp
capacitance where Rp is parallel to Cp.
In this paper, and from the following relationship:
 .C .( SFoc , p ) 2 
 2 .C 2 .( SFoc , p ) 3
SFoc  SFoc , s 
 j 
2 
1  (.C.SFoc , p ) 2
 1  ( .C.SFoc , p ) 
(32)
Assimilating SFoc,s to Rs and SFoc,p to Rp(parallel resistance),we can rewrite SFoc following the method of
impedance spectroscopy to determine the corresponding electrical parameters. Thus, one obtains the following
relationship:

 2 .C 2 .Rp 3 
Z   Rs 

1  (.C .Rp ) 2 


 .C.Rp 2 
j 
2 
 1  (.C.Rp ) 
(33)
With:
Re( Z )  Rs 
Volume 4, Issue 7, July 2016
 2  C 2  Rp 3 ;
   C  Rp 2
Im(
Z
)

1  (   C  Rp ) 2
1  (   C  Rp ) 2
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The imaginary part versus real part of Z and equivalent electric circuit are shown in figure7:
Figure 8: (a) Nyquist diagram of impedance and (b) equivalent electric circuit of SFoc.(z = 0.003cm)
From Nyquist diagram of complex impedance (see figure 8-a), we have:
The intercepted along the real impedance axis which correspond to Rs when the frequency tends to zero(ω→0) and
Rs+Rp when the fréquence approaches infinity(ω→∞).
The maximum value of │-Im(Z)│occurs at the cut off frequency and corresponds to Rp/2.
The characteristics of these particular points can be summarized by the following equations:
ω→0
Re( Z )  Rs
Im( Z )  0
(34)
(35)
ω→ ωc
Re( Z )  Rs 
Im(Z ) 
Rp
2
(36)
Rp
2
(37)
ω→∞
Re( Z )  Rs  Rp
Im( Z )  0
(38)
(39)
From equivalent electric circuit (figure8-b),C is a capacitance of a capacitor describing the capacitive phenomenon
observed, Rp a parallel resistance which represents the leakage current in the cell and Rs a series resistance which
characterizes the resistive effect of minority carriers to the emitter– base junction.
Based on the preceding relations, we give some results on the table below corresponding to the variations of electrical
parameters for different angular frequencies.
Table4: Electrical parameters
ω (rad.s-1)
Rp (Ω.cm2)
Rs (Ω.cm2)
τ (s)
C (F.cm2)
0
0
0.98
10-5
0
103
4.04
0.99
10-5
2.47 10-6
105
23.48
1.01
10-5
4.25 10-7
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This table shows that with the increase of the angular frequency, the series and shunt resistances increase but
capacitance decrease with the increasing of frequency.
3. CONCLUSION
Atheoretical study of the junction recombination velocity limiting the open circuit of a parallel vertical junction solar
cell has been presented. We note, in static regime the module of SFoc is maximum and constant when the phase which
corresponding to it is almost zero. In frequency regime, as well as the module and the phase of SFoc decreased with the
angular frequency.
From these analyzes and using the method of impedance spectroscopy, an equivalent electric model of the junction
recombination velocity limiting the open circuit (SFoc) has been proposed.
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AUTHOR
Miss FATIMATA BA was born in Dakar-Senegal in 1987. She is working on her Doctorat Thesis
in the Laboraty of Semiconductors and Renewable Energy of the Faculty of Science and Technology
of the University Cheikh Anta Diop in Dakar-Senegal (UCAD-LASES). His research interest is in
the field renewable energy, semiconductor devices characterization and electronics.
Volume 4, Issue 7, July 2016
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