D
Journal of Energy and Power Engineering 7 (2013) 903-906
DAVID
PUBLISHING
Three Dimensional Study of Spectral Response of
Polycrystalline Silicon Solar Cells: Vertical Junction
Frequency Modulation Scheme
Nouhou Bako Zeïnabou1, 2, Hawa Ly Diallo2, Aminata Gueye Camara2, Moustapha Thiam2, Dan Maza
Abouzeidi1, Madougou Saïdou3 and Grégoire Sissoko2
1. Département de Physique, FST, Université de Maradi, Maradi 65110, Niger
2. Laboratoire des Semiconducteurs et d’Energie Solaire, Département de Physique, FST, Université Cheikh Anta Diop, Dakar,
Sénégal
3. Département de Physique, ENS, Université Abdou Moumouni de Niamey, Niamey10 963, Niger
Received: August 08, 2012 / Accepted: November 19, 2012 / Published: May 31, 2013.
Abstract: In this paper, the modeling of a bifacial polycrystalline silicon solar cells vertical junction is presented. The study in dynamic
frequency is limited to wavelengths from 400 nm to 1100 nm. The dependence of solar cell spectral response on wavelengths for
several modulation frequencies was evaluated by using solar cell internal quantum efficiency.The objective is to characterize the
polycrystalline silicon in 3D. The effect of frequency modulation pulsation on the phase of internal quantum efficiency was presented
as well as values of shunt and series resistance for various grains size values. The results show that the value of maximum internal
quantum efficiency is about 50% with a wavelength of 0.82 nm and a frequency of 103 rad/s under monochromatic illumination.
Key words: Solar cell vertical junction, polycrystalline silicon, frequency modulation, internal quantum efficiency, wavelength.
1. Introduction
The polycrystalline silicon solar cells are of great
importance in research.
This is mainly due to two factors: the cost and the
control of material properties. Among these
characteristics we have size and orientation of grain
responsible of recombination center [1, 2]. These
contribute to reducing the rate of energy induction of
cells [3, 4].
A technique for determining the electrical
parameters of the polycrystalline silicon solar cell is
developed in this paper, to characterize the solar cell,
based on the Nyquist diagram. Thus, the frequency
band of the cell is scanned and spectral analysis
Corresponding author: Nouhou Bako Zeïnabou, Dr.,
research field: photovoltaic solar energy. E-mail:
zeinabou.bako@univ-maradi.ne.
performed. The approach consists of a three
dimensional representation of the grain of the
polycrystalline silicon solar cells vertical junction in
frequency modulation. For the study, only the
contribution of the base in the photogenerated excess
minority carriers’ is taken into account [5].
2. Theoretical Study of the Cell
The three dimensional representation of a grain of
the polycrystalline silicon solar cell is shown in Fig. 1
[6].
The continuity equation related to photogenerated
excess minority carriers’ in the base region of the cell
under monochromatic light modulation frequency is
given by:
  x, y, z, t
  x, y, z, t 
Dn  2n x, y, z, t   n
 Gz, t   n
(1)
n
t
904
Three Dimensional Study of Spectral Response of Polycrystalline Silicon Solar Cells:
Vertical Junction Frequency Modulation Scheme
At the back side relative to the incident surface of
illumination, it is given by:
 z  gz
   x , y , z 
D
n
 - Sar   n  x, y , z  z  gz (9)
 n

z
z  gz

Fig. 1 Model of a grain of solar cell.
The photogenerated excess minority carriers’
density  ( x , y , z , t ) illuminated by a monochromatic
light incident wavelength (λ) and in modulation
frequency ω is expressed by:
 n  x , y , z , t    n  x , y , z   e i   t
(2)
The optical generation rate G(z, t) of minority
carriers in the base region is a time and position
dependent and it is given by the expression [7]:
G z , t   g z   e i    t
(3)
with the spacial part and temporal part.
For the front side illumination of the cell, the optical
generation rate g(z) function of depth is given by the Eq.
(4):
g ( z )   ( ) I 0 (1  R( )) exp( ( ) z )
(4)
The diffusion length of minority carriers is defined by:
L 2n   n D n
(5)
Boundary conditions are set [8, 9] to solve Eq. (1).
Thus, at the junction of the grain of solar cell, they
have:
 gy

 y 2


 n ( x, y , z )
Dn
y

y
gy
2
 S f   n ( x, y, z )
y
 gy
2
(6)
At the back side of the grain of solar cell the equation
is given by:
gy

 y 2


 n ( x , y , z )
 Dn
y

y
gy
2
 - Sb   n ( x , y , z )
y
gy
2
(7)
At the front side of the grain of solar cell, the
equation is given by:
z  0

 D   n x , y , z 

 n
z
z0
Sav   n  x , y , z  z  0
(8)
At grain boundaries:


gx
x  
2

 n  x, y, z 

 Sg   n  x, y, z  x   gx
 Dn
gx
x
2
x

2

  x, y, z 
 Dn n
 - Sg   n  x, y, z  x   gx
gx
x

2
x 
2

(10)
3. Internal Quantum Efficiency
This section deals specifically with internal quantum
efficiency and its modulation frequency regime phase.
The analysis is spread in the visible range (0.4 µm) to
the infrared (1.1 µm). The modulation frequency varies
from 103 rad.s-1 to 106 rad.s-1 and the recombination
rate at the junction Sf goes to +∞. The expression of
quantum efficiency is given by Eq. (11) [9].
Its profile as a function of wavelength is shown in
Fig. 2 for different values of the pulsation.
Dn

 K n  1  R (  )   I  g  g
o
x
z

g

 g z  x  x , y , z 
n
2
Q    K n *
dx dz
g
0

y
 x
k
j
2

 
(11)
It appears that the short wavelength and small values
of the modulation frequency corresponding to large
values of internal quantum efficiency.
When the modulation frequency increases for a
recombination rate at the junction Sf,, which tends to
infinity and an excess minorities carriers’ velocity at
the grain boundary Sg to 103 cm.s-1, the amplitude of
quantum efficiency decreases. This decrease is not
remarkable from the modulation frequencies of
incident light above 104 Hz [5, 10].
The phase of the internal quantum efficiency is
expressed by Eq. (12) depending on wavelength.
Its evolution is shown in Fig. 3.
 (  )  a tan
 Q 2 ( ) 


 Q 1( ) 
(12)
Three Dimensional Study of Spectral Response of Polycrystalline Silicon Solar Cells:
Vertical Junction Frequency Modulation Scheme
Fig. 2 Spectral response as a function of wavelength for ω
= 103 and 105 rad/s (red curve), ω = 106 rad/s (black curve)
and ω = 107 rad/s (blue curve). gy = 0.03 cm; gz = gx = 0.02
cm ; D = 26 cm2/s ; Sg = 1000 cm/s.
Fig. 4 Nyquist diagram efficiency.
For small pulsation values (   0 ), the real
component of internal quantum efficiency tends to Rs +
Rp. The imaginary component of quantum efficiency
tends to 0.
For the large values of pulsation, (ω tends to infinity),
the reel component of internal quantum efficiency
tends to RS and the imaginary component of internal
quantum efficiency tends to 0.
  

 Re  R s
(13)
 Im  0

The characteristics of maximum points are
summarized by Eq. (14) [11].
Fig. 3 Phase of internal quantum efficiency as a function of
wavelength for different values of modulation frequency. gx
= 90 µm; gz = 200 µm; gy = 120 µm; Sg = 103 cm/s.
Large values of phase correspond to a phase shift
(advance or delay) between the excitation of the solar
cell and the response thereof, while small values of
phases correspond to a simultaneous response.
4. Nyquist Diagram
The complex impedance of the grain of solar cell is
defined by its real and imaginary components which
are a function of frequency ω. These are shown in the
Nquist diagram, in the visible range in order to
characterize the internal quantum efficiency of the
solar cell as shown in Fig. 4.
905

   c

Rp
 Rs
 Re 
2

 Im  R p
2

(14)
5. Results
The particularity of this work is the fact that the
electrical parameters are obtained from the Nyquist
diagram of internal quantum efficiency. The values of
series resistance and shunt resistance according to the
size gy of grain are shown in Table 1.
The decrease of the grain size leads to the
multiplication of recombination centers. The value of
the series resistance increases when gy decreases while
906
Three Dimensional Study of Spectral Response of Polycrystalline Silicon Solar Cells:
Vertical Junction Frequency Modulation Scheme
Table 1 Values of series resistance and shunt resistance of
a solar cell for different grain size.
gy (µm)
70
80
90
100
110
120
Rs (Ω.cm-2)
0.066
0.062
0.056
0.050
0.045
0.04
Rsh (Ω.cm-2)
0.081
0.123
0.161
0.18
0.486
0.602
Sg = 103 cm/s, L = 0.02 cm.
shunt resistances decreases, namely when solar cell
structure tends to polycrystalline.
The theoretical series of results obtained using the
Nyquist diagram help to determine the resistive and
inductive behavior of material.
6. Conclusions
In this paper we have presented a 3D modeling of a
vertical junction solar cell under monochromatic light.
The theoretical study of the solar vertical junction in
frequency modulation has lead to the determination of
the evolution of the internal quantum efficiency under
the influence of the pulsation of the incident light and
modulation frequency. An expression of the internal
has been established which is composed of two parts:
imaginary and real parts that vary with the modulation
frequency. Value of maximum internal quantum
efficiency is about 50% with a wavelength of 0.82 nm
and a frequency of 103 rad/s under monochromatic
illumination.
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