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Lateral current effects on the voltage distribution in the emitter of solar cells under concentrated sunlight
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Solar Energy 83 (2009) 456–461 www.elsevier.com/locate/solener
Lateral current effects on the voltage distribution in the emitter of solar cells under concentrated sunlight Arturo Morales-Acevedo * CINVESTAV-IPN, Electrical Engineering Department, Avenida IPN No. 2508, 07360 Me´xico, DF, Mexico Received 10 November 2006; received in revised form 3 September 2008; accepted 5 September 2008 Available online 9 October 2008 Communicated by: Associate Editor E. Stefanakos
Abstract The design of the grid contact in silicon solar cells is one of the most important steps for the optimization and fabrication of these energy conversion devices. The voltage drop due to the lateral flow of current towards the grid fingers can be a limiting factor causing the reduction of conversion efficiency. For low current levels this voltage drop can be made small, for typical values of sheet resistance in the emitter, but for solar cells made to operate at high sun concentrations this efficiency loss can be important, unless there is a clear vision of the current and voltage distribution so that the emitter and grid design can be improved. Hence, it is important to establish and solve the current and voltage distribution equations for solar cells with a grid contact. In this work, first these equations are established and then they are solved in order to show the effects that the lateral current flow in the emitter cause on the voltage distribution, particularly at high illumination levels. In addition, it will be shown that the open circuit voltage is significantly reduced due to the lateral current flow as compared to the value predicted from a simple equivalent circuit with a lumped resistance model. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Solar cells; Grid contacts; Solar concentration
1. Introduction Silicon solar cells (single and polycrystalline) currently have the biggest portion of the market for photovoltaic systems in the world. The maximum silicon solar cell efficiencies achieved under one sun radiation are close to 25% (Green et al., 2008), and these will be very difficult to improve further. However, solar cells operating under concentrated solar radiation still have some room to be improved, making it possible to use sunlight concentration together with solar cells designed to operate under these conditions. For this purpose, it is fundamental to design the structure and the grid to be used in such solar cells.
*
Tel.: +52 5557473781. E-mail address: [email protected]
0038-092X/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.solener.2008.09.005
The conventional silicon solar cell structure is the one shown in Fig. 1. On top of the emitter, a grid contact is deposited to have simultaneously a low shadowing factor and a low power loss due to the Joule dissipation caused by the lateral current flow from the point of generation to the finger contact. Given the finite resistivity in the emitter, the bigger the finger separation, the bigger will be the Joule power losses, but at the same time the shadowing losses will be smaller. In other words, there is a compromise which implies the optimization of this parameter (finger separation). In diverse articles and textbooks (Handy, 1967; Scharlack, 1979; Morales-Acevedo, 1985; Green, 1982), the design of the grid is made by making the approximation that the current in this region is uniform and constant at any point of the emitter; but the variation of the potential caused by the lateral current flow implies that at each point of the junction this current density will be non-uniform.
A. Morales-Acevedo / Solar Energy 83 (2009) 456–461
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that current density is constant everywhere in the emitter, but obviously this assumption is not correct, particularly for solar cells operating under concentrated sunlight. The current flow from the junction towards the fingers causes a voltage drop at every point so that the current density itself is not uniform in the emitter as shown in Fig. 2. The current density at any point x of the emitter of a solar cell where there is a potential V(x) will be V ðxÞ 1 ð1Þ J ðxÞ ¼ J L J 0 exp VT
Fig. 1. Typical structure for a p–n junction solar cell with a grid contact on top. Two consecutive fingers of the grid are located at x = 0 and x = S.
Hence, it is convenient to establish and solve the equations that describe the voltage and current distributions in the emitter of a solar cell with a grid contact on top. In the following sections, first the equations for current and voltage are established and then they are solved, in such a way that a better visualization of their distribution in the emitter is achieved. It is shown that for solar cells operating under concentrated sunlight and operating at a voltage close to the open circuit voltage there will be regions in the emitter that generate current while other regions act as a sink for such a generated current. This phenomenon is precisely what determines the actual open circuit voltage in a solar cell under concentrated sunlight since, as it will be shown, due to this effect the open circuit voltage will be reduced with respect to the predicted one when the series resistance is concentrated in a single resistance element Rs, as described in many papers and textbooks using a simple equivalent circuit for the solar cell. 2. Equations for the voltage and current distributions in the emitter The calculation of the power losses due to the lateral current flow are often made based on the approximation
where J0 is the saturation current density at the junction, JL is the photo-current density caused by the collection of the carriers generated by the incident radiation and VT is the thermal potential kT/q at the absolute temperature T. The total current at point x that flows towards the finger is Z S=2 J ðyÞdy ð2Þ IðxÞ ¼ b x
where b is the arbitrary length of a finger element on the surface of the emitter. S is the separation between two consecutive fingers (see Fig. 1). Let us consider a differential element dx between x and x + dx where a voltage drop dV(x) will be caused as a consequence of the current flow through the differential resistance element in this region. Hence Z S=2 dx J ðyÞdy dx ð3Þ dV ðxÞ ¼ qs IðxÞ ¼ qs b x where qs is the sheet resistivity in the emitter. Therefore, the potential V(x) satisfies the following differential equation: d2 V ðxÞ ¼ qs J ðxÞ: dx2
This equation can be re-written for the potential normalized with respect to the thermal potential U(z) = V(z)/VT, as a function of the position normalized to half the finger distance z (=2x/S), i.e. d2 U ðzÞ qs S 2 ¼ J ðzÞ: 4V T dz2
Fig. 2. Schematics of the reference unit cell element. It shows the lateral current flux from the junction border towards a finger in the grid. By the symmetry of the problem it is only necessary to study the region between x = 0 and x = S/2, which can be considered a unit element of the emitter to be analyzed. The length of the grid finger is considered to be of arbitrary length (b).
ð4Þ
ð5Þ
The boundary conditions in this case correspond to initial conditions for the potential and the electric field at z = 0 (finger contact). We shall assume that the potential at the finger is the cell operating potential Vc (Uc = Vc/VT), and the electric field at this contact is zero as should be for an ideal ohmic contact. A simple approximate solution of Eq. (5) can be obtained when the current density is assumed to be independent of the position, i.e., J(x) = J, as it is typically assumed in many textbooks. The solution in this case is U ðzÞ ¼
qs S 2 J 2 z þ U c: 8V T
ð6Þ
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For example, for a solar cell short circuited (Vc = 0), the total voltage drop due to the laminar current flow is DV ¼
qs S 2 J L : 8
ð7Þ
In these conditions, the contribution to the specific series resistance would be Rss = qs (S2/8). For example, for a solar cell in these conditions with finger separation of 0.1 cm and sheet resistivity of 40 X per square, operating in short circuit, the associated series resistance would be only 0.05 X cm2, which can be considered small when the cell operate under illumination equivalent to one sun. However, it can be seen that the solution expressed by Eq. (6) is not the correct solution for Eq. (5), in general, since the current density varies according to what is expressed by Eq. (1). Eq. (5) is non-linear, and therefore a numerical solution is required as discussed in the following section. 3. Exact model results and discussion For silicon solar cells, typical values for the cell parameters are: J0 = 1 1012 A/cm2, JL = 0.035 A/cm2, S = 0.1 cm and qs = 40 X per square. Therefore, the ideal open circuit voltage would be Voc = 0.626 V. When the photocell is short circuited (the potential at the contact Vc = 0), the exact solution will be the one expressed by Eq. (6) since the dark current in this case will be zero. In other words, the approximate solution and the exact one are identical in this case. However, when the operating bias of the cell is close to the open circuit voltage, or close to the maximum power point, there will be a difference between these two solutions which is usually small for low illumination levels. Eq. (5) can be solved numerically, for example with a Runge-Kutta method (Burden and Faires, 1985), using computer programs such as MAPLE or MATHEMATI-
CA. Using this algorithm with an adaptive step, Eq. (5) was solved, and the results for a contact potential equal to 95% of the ideal open circuit voltage Voc (626 mV) is shown in Fig. 3. The exact solution is compared with the potential distribution obtained by means of the approximate solution when the current is assumed to be constant as described by Eq. (6). It can be observed that the difference is small, as expected. Now, let us see what happens if the same solar cell is under concentrated sunlight equivalent to 100 suns. The illumination current density JL would become JL = 3.5 A/cm2. Again, it will be assumed that the operating voltage is 95% of the ideal open circuit voltage which now becomes 0.745 V. The result is shown in Fig. 4, where it is compared also with the approximate case of a constant current. It can be seen that the behavior in both cases are now totally different, and the approximate solution would be erroneous, predicting a monotonous growth of the voltage, up to the point x = S/2 where the voltage would be 0.883 V. Instead of this behavior, the exact solution shows that the voltage reaches a maximum of 0.77 V, around z = 0.9, and then it decays towards 0.767 V at z = 1 (x = S/2). The total potential drop would be only 0.06 V, which is very different to the total voltage drop predicted by the approximate solution of 0.175 V. Hence, the constant current approximation gives overestimated values with respect to the exact solution of Eq. (5). In order to complement our visualization, the current distribution for these operating conditions is shown in Fig. 5. It should be noticed in this figure that the operation of the cell under 100 suns with a voltage around 95% of the ideal open circuit voltage (i.e., Vc = 0.708 V) causes that a portion of the region under analysis acts as a sink for the current generated at the junction portion close to the contact. This would give a net current density in the opposite sense than the illumination current, and this can only occur if the cell is operating at a potential which is higher than the real open circuit voltage. In other words, in these con-
Fig. 3. Comparison between the exact solution for a typical solar cell under 1 sun radiation and the approximate solution when the potential at the contacts is 0.95 times the ideal open circuit voltage (Vc = 0.95 VT log[JL/J0 + 1]).
A. Morales-Acevedo / Solar Energy 83 (2009) 456–461
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Fig. 4. Same comparison, as for the preceding figure, when the solar cell is operating at 100 suns with a voltage equal to 0.95 times the ideal open circuit voltage at this illumination level.
Fig. 5. The current density distribution at each point in the emitter of the solar cell of the preceding figure, at 100 suns of illumination.
ditions an external applied voltage higher than the open circuit voltage would be required in order to sustain the current going into the region acting as a current sink. Without an external potential at the cell, i.e., under selfpolarization conditions, the maximum voltage observed at the contact will be the real open circuit voltage of the device, and this will occur when the net current is zero. In other words, the observed total current at the contact will be zero, but locally, at each point in the emitter the current will be different from zero. A portion of the emitter will be generating current while another portion will act as a sink for this current, so that the net current is zero. Hence, in real open circuit voltage conditions the net current should be zero R S=2 J ðyÞdy ¼ 0: ð8Þ < J >¼ 0 S=2 For the considered solar cell, the calculated real open circuit voltage would be 0.668 V; a value 77 mV smaller than 0.745 V, which is the ideal open circuit voltage that
could be obtained if there were no effects associated to the distributed resistance in the emitter. This value was determined by having the potential at the contact so that the net current density was zero, as explained above. The potential distribution under these conditions is shown in Fig. 6. The total voltage drop in the emitter is 0.116 V, and such a voltage drop causes the real open circuit voltage to be smaller than the ideal value. A better understanding of the importance of the above result can be achieved if we consider the behavior for a cell as expected from the classical concentrated series resistance model (Rs) V þ JRs < J >¼ J L J 0 exp : ð9Þ VT Typically, the effects due to the lateral current flow are considered through the value of the series resistance Rs where all the series resistance components are concentrated. From Eq. (9) we can see that under open circuit conditions
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Fig. 6. Potential distribution in the emitter of the above solar cell at 100 suns operating at the real open circuit voltage. At the contact, the measured voltage would be 668 mV, which can be compared with the ideal value (745 mV) for this solar cell.
J L ¼ J 0 exp
V oc : VT
ð10Þ
Then, according to this model there should be no effect on the open circuit voltage Voc due to the series resistance. However, the results described above show that this model is not correct since there is a reduction of the open circuit voltage with respect to the ideal one. This voltage reduction is associated to the lateral current flow going from one part of the cell to another region, where this current is consumed, in such a way that the net current out of the grid contact is zero under the open circuit condition. This effect is small for cells operating under low illumination levels with a well designed grid, but it may become an important effect at high illumination levels. In order to reduce this potential drop at the emitter, it would be necessary to design the emitter and the grid contact so that the resistive losses due to the lateral current flow are diminished. For example, if the sheet resistivity could be reduced to 20 X per square, and ideally the parameters such as JL and J0 did not change (it actually would not be so), the real open circuit voltage at 100 suns would increase close to the ideal value of 0.745 V. Alternatively, if the finger separation were reduced to half the assumed value, i.e., S = 0.05 cm, the open circuit voltage should also reach up to 0.745 V. However, the latter change would also affect the short circuit current density since the covered area by the grid on the surface would increase, and therefore a re-design process would be needed in order to achieve the maximum conversion efficiency. Any of these two forms for reducing the voltage drop in the emitter could help improve the behavior of the solar cell at high illumination levels. All of the above discussion implies that establishing Eq. (5) for the distributed potential in the emitter, and having a solution method for it, allows a calculation for the open circuit voltage of a given solar cell, particularly when operating at high illumination levels (more than 100 suns), and it also helps in visualizing what mod-
ifications will be required to have a more efficient cell in such conditions. 4. Conclusions The non-linear differential equation for the voltage in the emitter of a solar cell with a grid contact was established. The solution of this equation for the distributed voltage in the emitter of a solar cell is useful in order to visualize in a clear way what happens when the cell is operating both at low and high illumination levels. It has been shown that the exact solution coincides with the approximate solution, which assumes a constant current at each point of the emitter, under short circuit conditions, but there may be appreciable differences when the cell operates close to the open circuit voltage, under high illumination levels. The non-linear effects associated to the lateral voltage drop causes not only a non-uniform current distribution, but it also causes that some portion of the emitter under illumination acts as a source while another portion acts as a sink for the generated current. Therefore, a new (dynamic) definition of open circuit voltage is needed so that at this voltage the net current at the contact is zero, although the current at each point of the emitter is not zero. This non-uniform current and voltage distributions cause a smaller open circuit voltage than the one predicted by simple textbook models, for which the lumped series resistance has no effect on the open circuit voltage. In other words, the non-linear behavior caused by the lateral current flow in the emitter of solar cells should be taken into account in order to design appropriately the grid of solar cells, particularly when operating at high illumination levels. References Burden, R., Faires, D., 1985. Numerical Analysis, third ed. Wadsworth Inc., Belmont.
A. Morales-Acevedo / Solar Energy 83 (2009) 456–461 Green, M.A., 1982. Solar Cells, Operating Principles, Technology and System Applications. Prentice-Hall, Englewood Cliffs, NJ. Green, M.A., Emery, K., Hishikawa, Y., Warta, W., 2008. Solar cell efficiency tables (version 32). Progress in Photovoltaics: Research and Applications 16, 435.
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Handy, R.J., 1967. Theoretical analysis of the series resistance of a solar cell. Solid-State Electronics 10, 765. Morales-Acevedo, A., 1985. Optimum concentration factor for silicon solar cells. Solar Cells 14, 43. Scharlack, R.S., 1979. The optimal design of solar cell grid lines. Solar Energy 23, 199.
Solar Energy 83 (2009) 456–461 www.elsevier.com/locate/solener
Lateral current effects on the voltage distribution in the emitter of solar cells under concentrated sunlight Arturo Morales-Acevedo * CINVESTAV-IPN, Electrical Engineering Department, Avenida IPN No. 2508, 07360 Me´xico, DF, Mexico Received 10 November 2006; received in revised form 3 September 2008; accepted 5 September 2008 Available online 9 October 2008 Communicated by: Associate Editor E. Stefanakos
Abstract The design of the grid contact in silicon solar cells is one of the most important steps for the optimization and fabrication of these energy conversion devices. The voltage drop due to the lateral flow of current towards the grid fingers can be a limiting factor causing the reduction of conversion efficiency. For low current levels this voltage drop can be made small, for typical values of sheet resistance in the emitter, but for solar cells made to operate at high sun concentrations this efficiency loss can be important, unless there is a clear vision of the current and voltage distribution so that the emitter and grid design can be improved. Hence, it is important to establish and solve the current and voltage distribution equations for solar cells with a grid contact. In this work, first these equations are established and then they are solved in order to show the effects that the lateral current flow in the emitter cause on the voltage distribution, particularly at high illumination levels. In addition, it will be shown that the open circuit voltage is significantly reduced due to the lateral current flow as compared to the value predicted from a simple equivalent circuit with a lumped resistance model. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Solar cells; Grid contacts; Solar concentration
1. Introduction Silicon solar cells (single and polycrystalline) currently have the biggest portion of the market for photovoltaic systems in the world. The maximum silicon solar cell efficiencies achieved under one sun radiation are close to 25% (Green et al., 2008), and these will be very difficult to improve further. However, solar cells operating under concentrated solar radiation still have some room to be improved, making it possible to use sunlight concentration together with solar cells designed to operate under these conditions. For this purpose, it is fundamental to design the structure and the grid to be used in such solar cells.
*
Tel.: +52 5557473781. E-mail address: [email protected]
0038-092X/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.solener.2008.09.005
The conventional silicon solar cell structure is the one shown in Fig. 1. On top of the emitter, a grid contact is deposited to have simultaneously a low shadowing factor and a low power loss due to the Joule dissipation caused by the lateral current flow from the point of generation to the finger contact. Given the finite resistivity in the emitter, the bigger the finger separation, the bigger will be the Joule power losses, but at the same time the shadowing losses will be smaller. In other words, there is a compromise which implies the optimization of this parameter (finger separation). In diverse articles and textbooks (Handy, 1967; Scharlack, 1979; Morales-Acevedo, 1985; Green, 1982), the design of the grid is made by making the approximation that the current in this region is uniform and constant at any point of the emitter; but the variation of the potential caused by the lateral current flow implies that at each point of the junction this current density will be non-uniform.
A. Morales-Acevedo / Solar Energy 83 (2009) 456–461
457
that current density is constant everywhere in the emitter, but obviously this assumption is not correct, particularly for solar cells operating under concentrated sunlight. The current flow from the junction towards the fingers causes a voltage drop at every point so that the current density itself is not uniform in the emitter as shown in Fig. 2. The current density at any point x of the emitter of a solar cell where there is a potential V(x) will be V ðxÞ 1 ð1Þ J ðxÞ ¼ J L J 0 exp VT
Fig. 1. Typical structure for a p–n junction solar cell with a grid contact on top. Two consecutive fingers of the grid are located at x = 0 and x = S.
Hence, it is convenient to establish and solve the equations that describe the voltage and current distributions in the emitter of a solar cell with a grid contact on top. In the following sections, first the equations for current and voltage are established and then they are solved, in such a way that a better visualization of their distribution in the emitter is achieved. It is shown that for solar cells operating under concentrated sunlight and operating at a voltage close to the open circuit voltage there will be regions in the emitter that generate current while other regions act as a sink for such a generated current. This phenomenon is precisely what determines the actual open circuit voltage in a solar cell under concentrated sunlight since, as it will be shown, due to this effect the open circuit voltage will be reduced with respect to the predicted one when the series resistance is concentrated in a single resistance element Rs, as described in many papers and textbooks using a simple equivalent circuit for the solar cell. 2. Equations for the voltage and current distributions in the emitter The calculation of the power losses due to the lateral current flow are often made based on the approximation
where J0 is the saturation current density at the junction, JL is the photo-current density caused by the collection of the carriers generated by the incident radiation and VT is the thermal potential kT/q at the absolute temperature T. The total current at point x that flows towards the finger is Z S=2 J ðyÞdy ð2Þ IðxÞ ¼ b x
where b is the arbitrary length of a finger element on the surface of the emitter. S is the separation between two consecutive fingers (see Fig. 1). Let us consider a differential element dx between x and x + dx where a voltage drop dV(x) will be caused as a consequence of the current flow through the differential resistance element in this region. Hence Z S=2 dx J ðyÞdy dx ð3Þ dV ðxÞ ¼ qs IðxÞ ¼ qs b x where qs is the sheet resistivity in the emitter. Therefore, the potential V(x) satisfies the following differential equation: d2 V ðxÞ ¼ qs J ðxÞ: dx2
This equation can be re-written for the potential normalized with respect to the thermal potential U(z) = V(z)/VT, as a function of the position normalized to half the finger distance z (=2x/S), i.e. d2 U ðzÞ qs S 2 ¼ J ðzÞ: 4V T dz2
Fig. 2. Schematics of the reference unit cell element. It shows the lateral current flux from the junction border towards a finger in the grid. By the symmetry of the problem it is only necessary to study the region between x = 0 and x = S/2, which can be considered a unit element of the emitter to be analyzed. The length of the grid finger is considered to be of arbitrary length (b).
ð4Þ
ð5Þ
The boundary conditions in this case correspond to initial conditions for the potential and the electric field at z = 0 (finger contact). We shall assume that the potential at the finger is the cell operating potential Vc (Uc = Vc/VT), and the electric field at this contact is zero as should be for an ideal ohmic contact. A simple approximate solution of Eq. (5) can be obtained when the current density is assumed to be independent of the position, i.e., J(x) = J, as it is typically assumed in many textbooks. The solution in this case is U ðzÞ ¼
qs S 2 J 2 z þ U c: 8V T
ð6Þ
458
A. Morales-Acevedo / Solar Energy 83 (2009) 456–461
For example, for a solar cell short circuited (Vc = 0), the total voltage drop due to the laminar current flow is DV ¼
qs S 2 J L : 8
ð7Þ
In these conditions, the contribution to the specific series resistance would be Rss = qs (S2/8). For example, for a solar cell in these conditions with finger separation of 0.1 cm and sheet resistivity of 40 X per square, operating in short circuit, the associated series resistance would be only 0.05 X cm2, which can be considered small when the cell operate under illumination equivalent to one sun. However, it can be seen that the solution expressed by Eq. (6) is not the correct solution for Eq. (5), in general, since the current density varies according to what is expressed by Eq. (1). Eq. (5) is non-linear, and therefore a numerical solution is required as discussed in the following section. 3. Exact model results and discussion For silicon solar cells, typical values for the cell parameters are: J0 = 1 1012 A/cm2, JL = 0.035 A/cm2, S = 0.1 cm and qs = 40 X per square. Therefore, the ideal open circuit voltage would be Voc = 0.626 V. When the photocell is short circuited (the potential at the contact Vc = 0), the exact solution will be the one expressed by Eq. (6) since the dark current in this case will be zero. In other words, the approximate solution and the exact one are identical in this case. However, when the operating bias of the cell is close to the open circuit voltage, or close to the maximum power point, there will be a difference between these two solutions which is usually small for low illumination levels. Eq. (5) can be solved numerically, for example with a Runge-Kutta method (Burden and Faires, 1985), using computer programs such as MAPLE or MATHEMATI-
CA. Using this algorithm with an adaptive step, Eq. (5) was solved, and the results for a contact potential equal to 95% of the ideal open circuit voltage Voc (626 mV) is shown in Fig. 3. The exact solution is compared with the potential distribution obtained by means of the approximate solution when the current is assumed to be constant as described by Eq. (6). It can be observed that the difference is small, as expected. Now, let us see what happens if the same solar cell is under concentrated sunlight equivalent to 100 suns. The illumination current density JL would become JL = 3.5 A/cm2. Again, it will be assumed that the operating voltage is 95% of the ideal open circuit voltage which now becomes 0.745 V. The result is shown in Fig. 4, where it is compared also with the approximate case of a constant current. It can be seen that the behavior in both cases are now totally different, and the approximate solution would be erroneous, predicting a monotonous growth of the voltage, up to the point x = S/2 where the voltage would be 0.883 V. Instead of this behavior, the exact solution shows that the voltage reaches a maximum of 0.77 V, around z = 0.9, and then it decays towards 0.767 V at z = 1 (x = S/2). The total potential drop would be only 0.06 V, which is very different to the total voltage drop predicted by the approximate solution of 0.175 V. Hence, the constant current approximation gives overestimated values with respect to the exact solution of Eq. (5). In order to complement our visualization, the current distribution for these operating conditions is shown in Fig. 5. It should be noticed in this figure that the operation of the cell under 100 suns with a voltage around 95% of the ideal open circuit voltage (i.e., Vc = 0.708 V) causes that a portion of the region under analysis acts as a sink for the current generated at the junction portion close to the contact. This would give a net current density in the opposite sense than the illumination current, and this can only occur if the cell is operating at a potential which is higher than the real open circuit voltage. In other words, in these con-
Fig. 3. Comparison between the exact solution for a typical solar cell under 1 sun radiation and the approximate solution when the potential at the contacts is 0.95 times the ideal open circuit voltage (Vc = 0.95 VT log[JL/J0 + 1]).
A. Morales-Acevedo / Solar Energy 83 (2009) 456–461
459
Fig. 4. Same comparison, as for the preceding figure, when the solar cell is operating at 100 suns with a voltage equal to 0.95 times the ideal open circuit voltage at this illumination level.
Fig. 5. The current density distribution at each point in the emitter of the solar cell of the preceding figure, at 100 suns of illumination.
ditions an external applied voltage higher than the open circuit voltage would be required in order to sustain the current going into the region acting as a current sink. Without an external potential at the cell, i.e., under selfpolarization conditions, the maximum voltage observed at the contact will be the real open circuit voltage of the device, and this will occur when the net current is zero. In other words, the observed total current at the contact will be zero, but locally, at each point in the emitter the current will be different from zero. A portion of the emitter will be generating current while another portion will act as a sink for this current, so that the net current is zero. Hence, in real open circuit voltage conditions the net current should be zero R S=2 J ðyÞdy ¼ 0: ð8Þ < J >¼ 0 S=2 For the considered solar cell, the calculated real open circuit voltage would be 0.668 V; a value 77 mV smaller than 0.745 V, which is the ideal open circuit voltage that
could be obtained if there were no effects associated to the distributed resistance in the emitter. This value was determined by having the potential at the contact so that the net current density was zero, as explained above. The potential distribution under these conditions is shown in Fig. 6. The total voltage drop in the emitter is 0.116 V, and such a voltage drop causes the real open circuit voltage to be smaller than the ideal value. A better understanding of the importance of the above result can be achieved if we consider the behavior for a cell as expected from the classical concentrated series resistance model (Rs) V þ JRs < J >¼ J L J 0 exp : ð9Þ VT Typically, the effects due to the lateral current flow are considered through the value of the series resistance Rs where all the series resistance components are concentrated. From Eq. (9) we can see that under open circuit conditions
460
A. Morales-Acevedo / Solar Energy 83 (2009) 456–461
Fig. 6. Potential distribution in the emitter of the above solar cell at 100 suns operating at the real open circuit voltage. At the contact, the measured voltage would be 668 mV, which can be compared with the ideal value (745 mV) for this solar cell.
J L ¼ J 0 exp
V oc : VT
ð10Þ
Then, according to this model there should be no effect on the open circuit voltage Voc due to the series resistance. However, the results described above show that this model is not correct since there is a reduction of the open circuit voltage with respect to the ideal one. This voltage reduction is associated to the lateral current flow going from one part of the cell to another region, where this current is consumed, in such a way that the net current out of the grid contact is zero under the open circuit condition. This effect is small for cells operating under low illumination levels with a well designed grid, but it may become an important effect at high illumination levels. In order to reduce this potential drop at the emitter, it would be necessary to design the emitter and the grid contact so that the resistive losses due to the lateral current flow are diminished. For example, if the sheet resistivity could be reduced to 20 X per square, and ideally the parameters such as JL and J0 did not change (it actually would not be so), the real open circuit voltage at 100 suns would increase close to the ideal value of 0.745 V. Alternatively, if the finger separation were reduced to half the assumed value, i.e., S = 0.05 cm, the open circuit voltage should also reach up to 0.745 V. However, the latter change would also affect the short circuit current density since the covered area by the grid on the surface would increase, and therefore a re-design process would be needed in order to achieve the maximum conversion efficiency. Any of these two forms for reducing the voltage drop in the emitter could help improve the behavior of the solar cell at high illumination levels. All of the above discussion implies that establishing Eq. (5) for the distributed potential in the emitter, and having a solution method for it, allows a calculation for the open circuit voltage of a given solar cell, particularly when operating at high illumination levels (more than 100 suns), and it also helps in visualizing what mod-
ifications will be required to have a more efficient cell in such conditions. 4. Conclusions The non-linear differential equation for the voltage in the emitter of a solar cell with a grid contact was established. The solution of this equation for the distributed voltage in the emitter of a solar cell is useful in order to visualize in a clear way what happens when the cell is operating both at low and high illumination levels. It has been shown that the exact solution coincides with the approximate solution, which assumes a constant current at each point of the emitter, under short circuit conditions, but there may be appreciable differences when the cell operates close to the open circuit voltage, under high illumination levels. The non-linear effects associated to the lateral voltage drop causes not only a non-uniform current distribution, but it also causes that some portion of the emitter under illumination acts as a source while another portion acts as a sink for the generated current. Therefore, a new (dynamic) definition of open circuit voltage is needed so that at this voltage the net current at the contact is zero, although the current at each point of the emitter is not zero. This non-uniform current and voltage distributions cause a smaller open circuit voltage than the one predicted by simple textbook models, for which the lumped series resistance has no effect on the open circuit voltage. In other words, the non-linear behavior caused by the lateral current flow in the emitter of solar cells should be taken into account in order to design appropriately the grid of solar cells, particularly when operating at high illumination levels. References Burden, R., Faires, D., 1985. Numerical Analysis, third ed. Wadsworth Inc., Belmont.
A. Morales-Acevedo / Solar Energy 83 (2009) 456–461 Green, M.A., 1982. Solar Cells, Operating Principles, Technology and System Applications. Prentice-Hall, Englewood Cliffs, NJ. Green, M.A., Emery, K., Hishikawa, Y., Warta, W., 2008. Solar cell efficiency tables (version 32). Progress in Photovoltaics: Research and Applications 16, 435.
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Handy, R.J., 1967. Theoretical analysis of the series resistance of a solar cell. Solid-State Electronics 10, 765. Morales-Acevedo, A., 1985. Optimum concentration factor for silicon solar cells. Solar Cells 14, 43. Scharlack, R.S., 1979. The optimal design of solar cell grid lines. Solar Energy 23, 199.