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Calculation of the resistance of the diffused top layer in a photovoltaic cell
Solar Cells, 19 (1986 - 1987) 1 - 7
1
CALCULATION OF THE RESISTANCE OF THE DIFFUSED TOP LAYER IN A PHOTOVOLTAIC CELL A. K. ABOUL SEOUD and H. AMER
Faculty of Engineering, University of Alexandria, Alexandria (Egypt) (Received March 18, 1985; accepted in revised form May 30, 1985)
Summary The resistance of the top layer of a photovoltaic cell is calculated using a numerical solution of Poisson's equation. The boundary conditions of the side of the slab facing the depletion layer are determined using a distributed model for the photovoltaic cell. The series resistances in the model are obtained from the actual resistances of the flow line paths between mesh points and the top collector. The exact junction voltage at that mesh point is then used to obtain the flow line path. Iteration between the results of Poisson's equation and the junction voltage converged after three to four trials.
1. Introduction In this work we are interested in evaluating the series resistance r s in the top thin layer of the photovoltaic cell. Evaluation of this parameter was based on the measurement of the sheet resistance of the top layer using practical means such as the four-point probe. Several papers have reported optimization o f the metallic collector geometry based on fixed sheet resistance values and other effects, such as collector shadowing and bus bar resistances [1 - 6]. Simulations of photovoltaic cells then give the actual representation by use of a distributed model. The series resistance in these models has always been represented as a lumped element based on sheet resistance measurements [7 - 10]. The effects of crowding and increase in the potential gradient along the top of the depletion layer facing illumination become substantial at high photoconcentration. Very little work has been done in this area [11]. In this paper a two-dimensional model is proposed using the distributed transmission line model; firstly the depletion layer potential distribution is determined and secondly the potential distribution in the thin top layer of the junction is determined by numerical means. Knowing the potential distribution one can determine the flow lines and the resistance of each flow line path. These data are then reinserted into the first part to obtain 0379-6787/86/$3.50
© Elsevier Sequoia/Printed in The Netherlands
the potential boundary conditions at the depletion layer. These two problems are iterated successively till the solution is stabilized. The o u t c o m e gives the effective p o w e r loss in terms of the contact geometry, the photocurrent, the dark saturation current and the load voltage.
2. Mathematical solution Calculations are made for several mesh dimensions and several doping profiles in a two
(1)
where a is the conductivity. For a uniformly d o p e d semiconductor V o = 0 and in this case V 2v = 0
(2)
When the semiconductor is d o p e d b y diffusion, the impurity profile takes the form of a complementary error function [12] varying in a direction normal to the surface. Then o(z) = Aerfc(az)
where A and a are scaling factors, and V a = ~o(z)/~z
Then V 2v = ( - ~ v / a z . ~ a ( z ) / a z ) / ~ ( z )
(3)
The boundary conditions for both equations are determined using certain features o f the photocell. (1) The distance between each pair of collectors and the collector width are constant. (2) Due to repetitive patterns there is always a symmetry around lines along the z-axis midway between each t w o successive collectors, as well as along the mid-point of the collector. These lines of symmetry also apply to the mathematical solution." Therefore we can confine our solution to the zone abcd as shown in Fig. l(a), since the drop voltage in the bulk and contact are neglected. The flow lines should be tangential to a - d and b - c , so that the symmetry criterion in the solution is satisfied. The segment (a-e) is an equipotential surface assumed to be a perfect c o n d u c t o r and is kept at the load voltage. The segment ( e - b ) is the interface between the semiconductor and vacuum; therefore the flow lines have to be tangential there. The boundary conditions along the segment ( c - b - e ) can be treated as the potential distribution between the points c and e. The drop voltage is controlled b y the value o f the resistivity along the path. As seen before, the flow lines must be
Contact
pWI2"P-LG12-~ ='-I t
Top layer
1
L
- Depletion layer
!
•..-Lines of symmetry
(a) LGI2
~-W 12 ;;~
L
N
~21 x
1 (b)
Fig. i. Geometrical symmetry in the cell: (a) lines of symmetry around a collector; (b) the basic repetitative unit in the solution due to symmetry.
along the line of s y m m e t r y (c-b), then tangential to the segment (b-e). Similarly the boundary condition along the segment (d-a) is taken as the potential distribution depending on the conductance along that path and hence a tangential flow line. Finally (d-c) is divided into two segments (f-c) and (d-f). In these two segments the potential at the boundary is determined using the transmission line model. The equivalent circuit of the photovoltaic cell is used assuming that the shunt resistance r p = co. The series resistance rs is taken as the resistance between the point at the top o f the depletion layer and the collector, bearing in mind that both the bulk and the metallic resistances are neglected. The equivalent circuit is an element in the distributed model in the segment (f-c). The value of vi at a particular point is determined in terms of rs as
Ip - - I 0 eqJ'l/let
= (Pj - - P L ) / r s
(4)
All parameters are known except r, and ~j. If we assume that r, is known, then we can determine vj in the segment (f-c). The equivalent circuit in the segment ( d - f ) of Fig. 2 is I0 e qvi/k T ffi (VL - - Pj)/rs since Ip = 0 due to shadowing by the collector. r
Fig. 2. E q u i v a l e n t c i r c u i t f o r p - n j u n c t i o n .
(5)
Equations (4) and (5) give the potential distribution along the depletion layer in terms of the variable rs. The voltage at point f in Fig. l(a) is determined by extrapolating the potentials vn _ 2 and vn _ 1 toward f. The loss per unit area (W m -2) is calculated using the expression n
1/(n -- 1) ~ ( v j
-
-
pL)2/rj + ((L - - x ) / L ) ( p n - - p L ) 2 / r n
2
and the total input p o w e r per unit area (W m -2) is 1/(n -- 1) ~
yj(vj --VL)/r~ + ((L - - x ) / L ) V n ( V n -- VL)/r n
2
where N is the total number of mesh points along d - f - c , L = ( W + L G ) / (N -- 1), X is the distance from the mesh point N to the point f and n is the number of mesh points just before the point f as in Fig. l ( b ) .
3. Results and discussion Different runs using the program were carried o u t assuming the foUowing conditions: (i) load voltage = 0.48 V; (ii) conductivity at the surface o0 = 1000 mho cm-1; (iii) a diffusion pattern taking the form a ( z ) = Oo(Z) erfc (az)
where a = 1 / 2 ( D t ) 1/2, taking 2 ( D t ) 1/2= 0.2, 0.4, 0.6, 0.8 and 1; (iv) an effective layer where the collector current flows with an assumed thickness t = 0.3/~m. Different values of the electrode width W varying from 3/zm to 97 Izm were used in the calculation. The interelectrode distance L G was taken to vary from 30/~m to 970 #m. The layer efficiency is plotted as a function of W / ( W + L G ) for various values of W + L G ; Figs. 3 - 7 show various patterns of diffusion. The significance of shadowing decreases with the decrease of the coefficient of the erfc for the doping profile. The rate of change of efficiency with shadowing W / ( W + L G ) increases with the increase in the sharpness of the doping profile, as is clear when Figs. 3 - 7 are compared. However, the efficiency is highly sensitive to the value of ( W / L G ) , since the increase in the path length will increase the loss and consequently decrease the efficiency.
4. Conclusions
There should be a potential gradient along the boundaries of the depletion layer. The field there stays almost constant because the density of the current is moderate.
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Fig. 7. Effect of shadowing percentage on collection efficiency for 2(Dr) 1/2 = 1.0. I t is e x p e c t e d t h a t t h e n o n l i n e a r i t y in t h e p o t e n t i a l a l o n g t h e b o u n d a r y l a y e r will b e severe a t high light c o n c e n t r a t i o n s . I t is also e x p e c t e d t h a t t h e e f f i c i e n c y will c h a n g e m o r e drastically w i t h t h e c h a n g e in ( W + L G ) as t h e i n t e n s i t y o f t h e p h o t o c u r r e n t is i n c r e a s e d . A p a p e r o n t h e use o f t h e s a m e t e c h n i q u e t o c a l c u l a t e series losses f o r v a r i o u s i n c i d e n t light intensities is in p r e p a r a t i o n .
References 1 A. R. Moore, R C A Rev., 38 (1977) 486. 2 D. Redfield, RCA Rev., 38 (1977) 463. 3 N . C. Wyeth, Solid-State Electron., 20 (1977) 629. 4 D. Redield, RCA Rev., 38 (1977) 475. 5 A . R. Moore, RCA Rev., 40 (1979) 140. 6 M . Conti, Solid-State Electron., 24 (1981) 79. 7 H. Murmann and D. Widmann, IEEE Trans. Electron. Devices, 16 (1969) 1022. 8 H. H. Berger, Solid-State Electron., 15 (1972) 145. 9 I. F. Chang, J. Electrochem. Soc., 117 (1970) 368. 10 Y. K. Fang, C. Y. Chang and Y. K. Su, Solid-State Electron., 22 (1979) 933. 11 S. B. Schuldt, Solid-State Electron., 21 (1978) 715. 12 R. M. Burger and R. P. Donavan, Fundamentals o f Silicon Integrated Device Technology, Vol. I, Prentice-Hall, Englewood Cliffs, NJ, 1967, p. 190.
1
CALCULATION OF THE RESISTANCE OF THE DIFFUSED TOP LAYER IN A PHOTOVOLTAIC CELL A. K. ABOUL SEOUD and H. AMER
Faculty of Engineering, University of Alexandria, Alexandria (Egypt) (Received March 18, 1985; accepted in revised form May 30, 1985)
Summary The resistance of the top layer of a photovoltaic cell is calculated using a numerical solution of Poisson's equation. The boundary conditions of the side of the slab facing the depletion layer are determined using a distributed model for the photovoltaic cell. The series resistances in the model are obtained from the actual resistances of the flow line paths between mesh points and the top collector. The exact junction voltage at that mesh point is then used to obtain the flow line path. Iteration between the results of Poisson's equation and the junction voltage converged after three to four trials.
1. Introduction In this work we are interested in evaluating the series resistance r s in the top thin layer of the photovoltaic cell. Evaluation of this parameter was based on the measurement of the sheet resistance of the top layer using practical means such as the four-point probe. Several papers have reported optimization o f the metallic collector geometry based on fixed sheet resistance values and other effects, such as collector shadowing and bus bar resistances [1 - 6]. Simulations of photovoltaic cells then give the actual representation by use of a distributed model. The series resistance in these models has always been represented as a lumped element based on sheet resistance measurements [7 - 10]. The effects of crowding and increase in the potential gradient along the top of the depletion layer facing illumination become substantial at high photoconcentration. Very little work has been done in this area [11]. In this paper a two-dimensional model is proposed using the distributed transmission line model; firstly the depletion layer potential distribution is determined and secondly the potential distribution in the thin top layer of the junction is determined by numerical means. Knowing the potential distribution one can determine the flow lines and the resistance of each flow line path. These data are then reinserted into the first part to obtain 0379-6787/86/$3.50
© Elsevier Sequoia/Printed in The Netherlands
the potential boundary conditions at the depletion layer. These two problems are iterated successively till the solution is stabilized. The o u t c o m e gives the effective p o w e r loss in terms of the contact geometry, the photocurrent, the dark saturation current and the load voltage.
2. Mathematical solution Calculations are made for several mesh dimensions and several doping profiles in a two
(1)
where a is the conductivity. For a uniformly d o p e d semiconductor V o = 0 and in this case V 2v = 0
(2)
When the semiconductor is d o p e d b y diffusion, the impurity profile takes the form of a complementary error function [12] varying in a direction normal to the surface. Then o(z) = Aerfc(az)
where A and a are scaling factors, and V a = ~o(z)/~z
Then V 2v = ( - ~ v / a z . ~ a ( z ) / a z ) / ~ ( z )
(3)
The boundary conditions for both equations are determined using certain features o f the photocell. (1) The distance between each pair of collectors and the collector width are constant. (2) Due to repetitive patterns there is always a symmetry around lines along the z-axis midway between each t w o successive collectors, as well as along the mid-point of the collector. These lines of symmetry also apply to the mathematical solution." Therefore we can confine our solution to the zone abcd as shown in Fig. l(a), since the drop voltage in the bulk and contact are neglected. The flow lines should be tangential to a - d and b - c , so that the symmetry criterion in the solution is satisfied. The segment (a-e) is an equipotential surface assumed to be a perfect c o n d u c t o r and is kept at the load voltage. The segment ( e - b ) is the interface between the semiconductor and vacuum; therefore the flow lines have to be tangential there. The boundary conditions along the segment ( c - b - e ) can be treated as the potential distribution between the points c and e. The drop voltage is controlled b y the value o f the resistivity along the path. As seen before, the flow lines must be
Contact
pWI2"P-LG12-~ ='-I t
Top layer
1
L
- Depletion layer
!
•..-Lines of symmetry
(a) LGI2
~-W 12 ;;~
L
N
~21 x
1 (b)
Fig. i. Geometrical symmetry in the cell: (a) lines of symmetry around a collector; (b) the basic repetitative unit in the solution due to symmetry.
along the line of s y m m e t r y (c-b), then tangential to the segment (b-e). Similarly the boundary condition along the segment (d-a) is taken as the potential distribution depending on the conductance along that path and hence a tangential flow line. Finally (d-c) is divided into two segments (f-c) and (d-f). In these two segments the potential at the boundary is determined using the transmission line model. The equivalent circuit of the photovoltaic cell is used assuming that the shunt resistance r p = co. The series resistance rs is taken as the resistance between the point at the top o f the depletion layer and the collector, bearing in mind that both the bulk and the metallic resistances are neglected. The equivalent circuit is an element in the distributed model in the segment (f-c). The value of vi at a particular point is determined in terms of rs as
Ip - - I 0 eqJ'l/let
= (Pj - - P L ) / r s
(4)
All parameters are known except r, and ~j. If we assume that r, is known, then we can determine vj in the segment (f-c). The equivalent circuit in the segment ( d - f ) of Fig. 2 is I0 e qvi/k T ffi (VL - - Pj)/rs since Ip = 0 due to shadowing by the collector. r
Fig. 2. E q u i v a l e n t c i r c u i t f o r p - n j u n c t i o n .
(5)
Equations (4) and (5) give the potential distribution along the depletion layer in terms of the variable rs. The voltage at point f in Fig. l(a) is determined by extrapolating the potentials vn _ 2 and vn _ 1 toward f. The loss per unit area (W m -2) is calculated using the expression n
1/(n -- 1) ~ ( v j
-
-
pL)2/rj + ((L - - x ) / L ) ( p n - - p L ) 2 / r n
2
and the total input p o w e r per unit area (W m -2) is 1/(n -- 1) ~
yj(vj --VL)/r~ + ((L - - x ) / L ) V n ( V n -- VL)/r n
2
where N is the total number of mesh points along d - f - c , L = ( W + L G ) / (N -- 1), X is the distance from the mesh point N to the point f and n is the number of mesh points just before the point f as in Fig. l ( b ) .
3. Results and discussion Different runs using the program were carried o u t assuming the foUowing conditions: (i) load voltage = 0.48 V; (ii) conductivity at the surface o0 = 1000 mho cm-1; (iii) a diffusion pattern taking the form a ( z ) = Oo(Z) erfc (az)
where a = 1 / 2 ( D t ) 1/2, taking 2 ( D t ) 1/2= 0.2, 0.4, 0.6, 0.8 and 1; (iv) an effective layer where the collector current flows with an assumed thickness t = 0.3/~m. Different values of the electrode width W varying from 3/zm to 97 Izm were used in the calculation. The interelectrode distance L G was taken to vary from 30/~m to 970 #m. The layer efficiency is plotted as a function of W / ( W + L G ) for various values of W + L G ; Figs. 3 - 7 show various patterns of diffusion. The significance of shadowing decreases with the decrease of the coefficient of the erfc for the doping profile. The rate of change of efficiency with shadowing W / ( W + L G ) increases with the increase in the sharpness of the doping profile, as is clear when Figs. 3 - 7 are compared. However, the efficiency is highly sensitive to the value of ( W / L G ) , since the increase in the path length will increase the loss and consequently decrease the efficiency.
4. Conclusions
There should be a potential gradient along the boundaries of the depletion layer. The field there stays almost constant because the density of the current is moderate.
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. o
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, ,
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~u
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;
o
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~.
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6
W / w + LG
~J
p
0.9']
0,96 4
5
Fig. 7. Effect of shadowing percentage on collection efficiency for 2(Dr) 1/2 = 1.0. I t is e x p e c t e d t h a t t h e n o n l i n e a r i t y in t h e p o t e n t i a l a l o n g t h e b o u n d a r y l a y e r will b e severe a t high light c o n c e n t r a t i o n s . I t is also e x p e c t e d t h a t t h e e f f i c i e n c y will c h a n g e m o r e drastically w i t h t h e c h a n g e in ( W + L G ) as t h e i n t e n s i t y o f t h e p h o t o c u r r e n t is i n c r e a s e d . A p a p e r o n t h e use o f t h e s a m e t e c h n i q u e t o c a l c u l a t e series losses f o r v a r i o u s i n c i d e n t light intensities is in p r e p a r a t i o n .
References 1 A. R. Moore, R C A Rev., 38 (1977) 486. 2 D. Redfield, RCA Rev., 38 (1977) 463. 3 N . C. Wyeth, Solid-State Electron., 20 (1977) 629. 4 D. Redield, RCA Rev., 38 (1977) 475. 5 A . R. Moore, RCA Rev., 40 (1979) 140. 6 M . Conti, Solid-State Electron., 24 (1981) 79. 7 H. Murmann and D. Widmann, IEEE Trans. Electron. Devices, 16 (1969) 1022. 8 H. H. Berger, Solid-State Electron., 15 (1972) 145. 9 I. F. Chang, J. Electrochem. Soc., 117 (1970) 368. 10 Y. K. Fang, C. Y. Chang and Y. K. Su, Solid-State Electron., 22 (1979) 933. 11 S. B. Schuldt, Solid-State Electron., 21 (1978) 715. 12 R. M. Burger and R. P. Donavan, Fundamentals o f Silicon Integrated Device Technology, Vol. I, Prentice-Hall, Englewood Cliffs, NJ, 1967, p. 190.