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Optimal design of front-contact metallization for photovoltaic solar cells
Solid-Slate Electronics Vol. 24. pp, 79--83 © PergamonPress Ltd., 1981. Printed in Great Britain
0038--110118110101--0079/102.0010
OPTIMAL DESIGN OF FRONT-CONTACT METALLIZATION FOR PHOTOVOLTAIC SOLAR CELLS MARIO CONTI SGS-ATES Componenti Elettronici SpA, Castelletto di Settimo Milanese. Milano, Italy (Received 13 October 1979; in revisedform 4 June 1980)
Abstraet--A new approach to the design of the front metal grid in standard photovoltaic solar cells is presented which allows the synthesis of the optimal grid profile for an arbitrary cell geometry. An expression for the total power loss is derived taking into account ohmic losses in the metal grid, losses in the front diffused layer of the cell and losses due to grid shadowing. The loss is minimized by a variational method solving the differential Euler equation and the optimal grid profile is obtained; then the expression for the minimum total power loss is derived. Applications to rectangular and circular solar cells are presented. The photocurrent generated under a solar light flux d,(x) can be expressed as a product of d~(x) and a proportionality coefficient, C(x), referred to as the current factor.This coefficient accounts for the quantum efficiency of the cell and for the local bias conditions which vary along x as a result of the voltage drop in the finger. Hence L is given by:
INTRODUCTION
To collect the current generated by solar irradiation, conventional photovoltaic cells are provided with a front-contact grid. A high-conductivity material is normally employed, typically silver, gold or aluminum, to reduce the ohmic power losses[l]. Since these materials are opaque, the area covered by the grid is shadowed and the light energy falling on this area is not converted into electric current. Shadowing losses can be avoided by using transparent conductors[2] such as tin oxide, ITO, etc. but the conductivities of these materials are orders of magnitude lower than those of the metals and the deposition of a reliable layer of sufficiently low resistance has yet to be achieved. This is of particular importance in solar cells operating under concentrated sunlight in which the metal sheet resistance must be in the order of a few milliohms. A solar cell provided with an opaque metal grid achieves the maximum efficiency when the total power loss introduced by shadowing and grid resistance is minimized. Many published papers consider the minimum total loss for a particular geometry, typically parallel-stripe rectangular [3] and two-layer rectangular[4] geometries. This paper describes a new approach to the problem, deriving a method to obtain the optimal grid profile for any given geometry. This is illustrated by specific applications.
L = fo L &(x). C(x). y(x) dx
(1)
where h and L are the finger width and length respectively. The current l(x) through the metal finger at the abscissa x is: l(x) -- fo x &(sc)C(~c)Z(~) d~
\.
(2)
/"
\
OPTIMAL DESIGN
Assume that the solar cell consists of a periodic structure in which n identical elements are connected in parallel and send current to a bus bar of negligeable resistance (Fig. 1). The reference co-ordinate system is orthogonal with the x-axis along the finger. The power loss caused by shadowing can then be expressed as the product of the maximum-power output voltage, Vm, and the current lost by shadowing, Is.
I
i
I
o
P~=V,~'L. SSE VoL 24, No. I--F
Fig. I. Rectangular and circular geometries. 79
80
MARIOCONTI
where Z(x) is the collection width. The power loss in a finger element of length dx is given by: (12(x) deR = ~ ~ dx (3) where ,o and T are the resistivity and thickness of the metal grid respectively (both assumed uniform). Extending this to the non-uniform case is straightforward; the power dissipated is then obtained by integrating (3) over the whole finger.
o (L l"(x).
Pu : - ~ J ° y - ~ ox. An additional power loss results from the flow of photogenerated current in the heavily doped front region. To a first approximation this can be expressed as: -f- dx pF = foL ~ p~3[dp(x) . C(x) . _.~ ] " . -Z(x)
= Or:z I-2,foL C2(a2Z 3 dx where t ~ is the sheet resistance of the front region. Hence the total power loss is given by: P, =
f L
12 p I (V,, " da " C" y(X)+y--(~"~+-f~ d~2C~Z3)dx. (4)
To solve the problem a function y,,(x) must be found which minimizes P,: this can be found mathematically by solving the differential Euler equation[5]:
dx Oy
,gy
associated with the function:
v,.
c.
-
y
where y' is the partial derivative Oy/Ox. The solution can be found easily:
I(x)
![
P
Y" = v'lC(x) • ,/,(x)) ~/~ v-O-~-7-T, TT'
(5)
Expression (5) allows the design of the optimal finger profile when (2) for I is taken into account. The solar flux profile ,h(x), the current factor C(xl and the collection width Z(x) are assumed known. in practice, however, Z is not known whereas the total width Z + y is, and depends on the chosen cell geometry. Consequently the collection width depends on ym which is not a rigorously known function. This approximation can be justified because y is a small fraction of the total width, Z + y, in efficient solar cells, thus the current l(x) is not greatly influenced by it. It also makes the problem mathematically tractable.
The expression for the minimal total power loss. Pt, can be obtained from t4) and If). P, = 2"
l(x)~ fcb(x) " C(x)) dx
foLd,,xi2. C,x)" Z3(xldx.
(6)
Since Pt is a monotonic function of Z, it reaches a minimum when Z(x)=0. Obviously this solution is meaningless practically so it can be concluded that Z should be as low as possible without affecting the feasibility of the optimal profile, y,,,, given by (5). A similar analytical approach can be followed when other variables are to be minimized, such as the voltage drop between 0 and L. RECTANGULARGEOMETRY For this widely-used geometry, if we assume that the light flux, C(xl, and Zfx) are both uniform, we can see from (2) that I is propbrtional to x. From (6) the optimal profile y,, is also proportional to x. This supports the validity of the observations on linear-taper metal fingers reported by Serreze[4]. A slightly less trivial situation is found in solar cells employed in linear concentrator systems, such as the linear parabolic trough. In this case the concentrated solar flux d,(x) is far from uniform. Again, expression (5} is useful in the determination of the optimal profile, y,., for the individual fingers under a local light flux ~h. CIRCULARGEOMETRY Circular solar cells used in concentrators with Fresnel lenses are provided with a peripheral conducting ring to collect the current contributions of the individual fingers. As a result of the circular geometry the total collection width, Z, is proportional to x. If the light irradiation is uniform and C(x) constant, I(x) is a parabolic function of x. Consequently the optimal finger profile is also parabolic. This profile was employed in a 2" n + p silicon solar cell designed to feature maximum efficiency at a concentration of 50X[~]. The layout, shown in Fig. 2, consists of a set of rectangles delineated by linear-taper metal fingers. Clearly, the total radial with, integrated over a circle of radius x, is proportional to x 2. The alternative layout, shown in Fig. 3, was also evaluated. It features an overall finger width proportional to x" like the previous example and the performance is practically equal. For concentrator solar cells a linear profile is sometimes used[7]. Expression (6) gives us that in this case the total loss, Pt, is 12.5% greater than with a parabolic profile with the same shadowing area. In this comparison the sheet resistance is assumed to be zero. The solar flux concentrated by commercial acrylic Fresnel lenses is far from uniform and consequently d~(x) is not constant. This was studied in detail by James and Williams[8]. In the concentration range 30-100X the light flux is roughly uniform in the central area of the cell
Front-contact metallization for solar cells
Fig. 2. Front metal grid for a 2 in. solar cell designed to operate at 50X (SPV-050),
81
82
MARIO CONTI
Fig. 3. The SPV--050 solar cell with an alternative metal grid.
83
Front-contact metallization for solar cells but noticeably non-uniform at the periphery as a result of chromatic dispersion in the lens. Thus al the periphery the optimal grid profile y,~ increases substantially since the term ¢b in the denominator is a rapidly decreasing function of x while l(x) increases monotonically.
SUMMARY AND CONCLUSIONS
The approach presented here facilitates the design of the optimal front metal grid; minimizing the total power lost through ohmic losses and shadowing when an opaque conducting material is used. The non-uniformity of the solar flux was taken into consideration because it influences the optimal profile. A locally dependent current factor C(x) can also be considered although its variation is normally unknown: it depends on the actual voltage distribution along the metal finger which in turn depends on C(x) through y,,, (see[6]). To solve this problem a complete simulation of the cell should be carried out, but a useful indication of
C(x) can be inferred from the voltage distribution on practical cells, which can be measured easily[9]. REFERENCES
I. H.J. Hovel, Solar Cells, Semiconductors and Semimetals, Vol. I I. Academic Press, New York (1975). 2. J. J: Bachmann, W. R. Sinclair, F. A. Thiel, H. Schreiber Jr., P. H. Schmidt, E. G. Spencer, W. L. Fcldmann and E. Buehler, i3th Phot. Spec. Conf. Washington, p. 524 (1978). 3. D. Redfield, RCA Rev. 38,475 0977). 4. H. B. Serreze, 13th IEEE Phot. Spec. Conf. Washington, p. 609 (1978): A. Flat and A. G. Milnes, Solar Energy 23, 289 (1979); A. R. Moore. RCA Rev. 40, 140 (1979). 5. P. M. Morse and H. Feshbach, Methods of Theoretical Physics. McGraw-HillNew York (1953). 6. M Conti, A. Modelli and G. Vento, 2nd E.C. Phot. Sol. En. Conf. Berlin. p. 800 11979). 7. E. L. Burgess and J. G. Fossum, IEEE Trans Electron DeL,. ED-24,433 (1977L 8. C. W. James and J. K. Williams, 13th Phot. Spec. Conf., Washington, p. 673 (1978). 9. A. M. Mazzone, Alia Frequenza, 47, 285E (1978); P. U. Calzolarari and A. M. Mazzone. 13th Phot. Spec. Conf., Washington, p. 320 (1978).
0038--110118110101--0079/102.0010
OPTIMAL DESIGN OF FRONT-CONTACT METALLIZATION FOR PHOTOVOLTAIC SOLAR CELLS MARIO CONTI SGS-ATES Componenti Elettronici SpA, Castelletto di Settimo Milanese. Milano, Italy (Received 13 October 1979; in revisedform 4 June 1980)
Abstraet--A new approach to the design of the front metal grid in standard photovoltaic solar cells is presented which allows the synthesis of the optimal grid profile for an arbitrary cell geometry. An expression for the total power loss is derived taking into account ohmic losses in the metal grid, losses in the front diffused layer of the cell and losses due to grid shadowing. The loss is minimized by a variational method solving the differential Euler equation and the optimal grid profile is obtained; then the expression for the minimum total power loss is derived. Applications to rectangular and circular solar cells are presented. The photocurrent generated under a solar light flux d,(x) can be expressed as a product of d~(x) and a proportionality coefficient, C(x), referred to as the current factor.This coefficient accounts for the quantum efficiency of the cell and for the local bias conditions which vary along x as a result of the voltage drop in the finger. Hence L is given by:
INTRODUCTION
To collect the current generated by solar irradiation, conventional photovoltaic cells are provided with a front-contact grid. A high-conductivity material is normally employed, typically silver, gold or aluminum, to reduce the ohmic power losses[l]. Since these materials are opaque, the area covered by the grid is shadowed and the light energy falling on this area is not converted into electric current. Shadowing losses can be avoided by using transparent conductors[2] such as tin oxide, ITO, etc. but the conductivities of these materials are orders of magnitude lower than those of the metals and the deposition of a reliable layer of sufficiently low resistance has yet to be achieved. This is of particular importance in solar cells operating under concentrated sunlight in which the metal sheet resistance must be in the order of a few milliohms. A solar cell provided with an opaque metal grid achieves the maximum efficiency when the total power loss introduced by shadowing and grid resistance is minimized. Many published papers consider the minimum total loss for a particular geometry, typically parallel-stripe rectangular [3] and two-layer rectangular[4] geometries. This paper describes a new approach to the problem, deriving a method to obtain the optimal grid profile for any given geometry. This is illustrated by specific applications.
L = fo L &(x). C(x). y(x) dx
(1)
where h and L are the finger width and length respectively. The current l(x) through the metal finger at the abscissa x is: l(x) -- fo x &(sc)C(~c)Z(~) d~
\.
(2)
/"
\
OPTIMAL DESIGN
Assume that the solar cell consists of a periodic structure in which n identical elements are connected in parallel and send current to a bus bar of negligeable resistance (Fig. 1). The reference co-ordinate system is orthogonal with the x-axis along the finger. The power loss caused by shadowing can then be expressed as the product of the maximum-power output voltage, Vm, and the current lost by shadowing, Is.
I
i
I
o
P~=V,~'L. SSE VoL 24, No. I--F
Fig. I. Rectangular and circular geometries. 79
80
MARIOCONTI
where Z(x) is the collection width. The power loss in a finger element of length dx is given by: (12(x) deR = ~ ~ dx (3) where ,o and T are the resistivity and thickness of the metal grid respectively (both assumed uniform). Extending this to the non-uniform case is straightforward; the power dissipated is then obtained by integrating (3) over the whole finger.
o (L l"(x).
Pu : - ~ J ° y - ~ ox. An additional power loss results from the flow of photogenerated current in the heavily doped front region. To a first approximation this can be expressed as: -f- dx pF = foL ~ p~3[dp(x) . C(x) . _.~ ] " . -Z(x)
= Or:z I-2,foL C2(a2Z 3 dx where t ~ is the sheet resistance of the front region. Hence the total power loss is given by: P, =
f L
12 p I (V,, " da " C" y(X)+y--(~"~+-f~ d~2C~Z3)dx. (4)
To solve the problem a function y,,(x) must be found which minimizes P,: this can be found mathematically by solving the differential Euler equation[5]:
dx Oy
,gy
associated with the function:
v,.
c.
-
y
where y' is the partial derivative Oy/Ox. The solution can be found easily:
I(x)
![
P
Y" = v'lC(x) • ,/,(x)) ~/~ v-O-~-7-T, TT'
(5)
Expression (5) allows the design of the optimal finger profile when (2) for I is taken into account. The solar flux profile ,h(x), the current factor C(xl and the collection width Z(x) are assumed known. in practice, however, Z is not known whereas the total width Z + y is, and depends on the chosen cell geometry. Consequently the collection width depends on ym which is not a rigorously known function. This approximation can be justified because y is a small fraction of the total width, Z + y, in efficient solar cells, thus the current l(x) is not greatly influenced by it. It also makes the problem mathematically tractable.
The expression for the minimal total power loss. Pt, can be obtained from t4) and If). P, = 2"
l(x)~ fcb(x) " C(x)) dx
foLd,,xi2. C,x)" Z3(xldx.
(6)
Since Pt is a monotonic function of Z, it reaches a minimum when Z(x)=0. Obviously this solution is meaningless practically so it can be concluded that Z should be as low as possible without affecting the feasibility of the optimal profile, y,,,, given by (5). A similar analytical approach can be followed when other variables are to be minimized, such as the voltage drop between 0 and L. RECTANGULARGEOMETRY For this widely-used geometry, if we assume that the light flux, C(xl, and Zfx) are both uniform, we can see from (2) that I is propbrtional to x. From (6) the optimal profile y,, is also proportional to x. This supports the validity of the observations on linear-taper metal fingers reported by Serreze[4]. A slightly less trivial situation is found in solar cells employed in linear concentrator systems, such as the linear parabolic trough. In this case the concentrated solar flux d,(x) is far from uniform. Again, expression (5} is useful in the determination of the optimal profile, y,., for the individual fingers under a local light flux ~h. CIRCULARGEOMETRY Circular solar cells used in concentrators with Fresnel lenses are provided with a peripheral conducting ring to collect the current contributions of the individual fingers. As a result of the circular geometry the total collection width, Z, is proportional to x. If the light irradiation is uniform and C(x) constant, I(x) is a parabolic function of x. Consequently the optimal finger profile is also parabolic. This profile was employed in a 2" n + p silicon solar cell designed to feature maximum efficiency at a concentration of 50X[~]. The layout, shown in Fig. 2, consists of a set of rectangles delineated by linear-taper metal fingers. Clearly, the total radial with, integrated over a circle of radius x, is proportional to x 2. The alternative layout, shown in Fig. 3, was also evaluated. It features an overall finger width proportional to x" like the previous example and the performance is practically equal. For concentrator solar cells a linear profile is sometimes used[7]. Expression (6) gives us that in this case the total loss, Pt, is 12.5% greater than with a parabolic profile with the same shadowing area. In this comparison the sheet resistance is assumed to be zero. The solar flux concentrated by commercial acrylic Fresnel lenses is far from uniform and consequently d~(x) is not constant. This was studied in detail by James and Williams[8]. In the concentration range 30-100X the light flux is roughly uniform in the central area of the cell
Front-contact metallization for solar cells
Fig. 2. Front metal grid for a 2 in. solar cell designed to operate at 50X (SPV-050),
81
82
MARIO CONTI
Fig. 3. The SPV--050 solar cell with an alternative metal grid.
83
Front-contact metallization for solar cells but noticeably non-uniform at the periphery as a result of chromatic dispersion in the lens. Thus al the periphery the optimal grid profile y,~ increases substantially since the term ¢b in the denominator is a rapidly decreasing function of x while l(x) increases monotonically.
SUMMARY AND CONCLUSIONS
The approach presented here facilitates the design of the optimal front metal grid; minimizing the total power lost through ohmic losses and shadowing when an opaque conducting material is used. The non-uniformity of the solar flux was taken into consideration because it influences the optimal profile. A locally dependent current factor C(x) can also be considered although its variation is normally unknown: it depends on the actual voltage distribution along the metal finger which in turn depends on C(x) through y,,, (see[6]). To solve this problem a complete simulation of the cell should be carried out, but a useful indication of
C(x) can be inferred from the voltage distribution on practical cells, which can be measured easily[9]. REFERENCES
I. H.J. Hovel, Solar Cells, Semiconductors and Semimetals, Vol. I I. Academic Press, New York (1975). 2. J. J: Bachmann, W. R. Sinclair, F. A. Thiel, H. Schreiber Jr., P. H. Schmidt, E. G. Spencer, W. L. Fcldmann and E. Buehler, i3th Phot. Spec. Conf. Washington, p. 524 (1978). 3. D. Redfield, RCA Rev. 38,475 0977). 4. H. B. Serreze, 13th IEEE Phot. Spec. Conf. Washington, p. 609 (1978): A. Flat and A. G. Milnes, Solar Energy 23, 289 (1979); A. R. Moore. RCA Rev. 40, 140 (1979). 5. P. M. Morse and H. Feshbach, Methods of Theoretical Physics. McGraw-HillNew York (1953). 6. M Conti, A. Modelli and G. Vento, 2nd E.C. Phot. Sol. En. Conf. Berlin. p. 800 11979). 7. E. L. Burgess and J. G. Fossum, IEEE Trans Electron DeL,. ED-24,433 (1977L 8. C. W. James and J. K. Williams, 13th Phot. Spec. Conf., Washington, p. 673 (1978). 9. A. M. Mazzone, Alia Frequenza, 47, 285E (1978); P. U. Calzolarari and A. M. Mazzone. 13th Phot. Spec. Conf., Washington, p. 320 (1978).