A Typical Solar Cell Engineering Essay

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In this paper, we investigate the losses associated with gridlines and busbars and how to minimize these losses. We use a square 10cm by 10cm cell with the following initial parameters: a short circuit current density of 40 mA/cm2, an open circuit voltage of .65 V, a fill factor of 80%, an efficiency of 20.8%, and an emitter resistance of 100 ohms/square. We were told to design the top grid for this solar cell, such that the losses would not exceed 8%. By using multiple busbars, tapered contacts, and double printed contacts, we were able to design a grid with losses of 7.73%.File:Solar cell.png

Figure 1: A typical solar cell, showing gridlines and busbars (courtesy of the U.S. Department of Energy)


Figure 2: A cross-sectional view of a typical solar cell (image courtesy of PVeducation.org) At its simplest, a solar cell is just a p-n junction exposed to sunlight. Solar cells operate by absorbing photons of light from the sun and using the energy from the photons to generate electron-hole pairs. In order to generate the electron-hole pairs, the incident photons of light must have an energy level higher than the band gap of the semiconductor used in the p-n junction. Therefore, it is important to pick a semiconductor that has an appropriate bandgap and is easy to manufacture. The image at the top of the next page, illustrates the band diagram for a silicon solar cell. The image shows an amorphous to crystalline silicon hetero-junction; whereas, we will be using a crystalline silicon homo-junction. However, the result is very similar. Conventionally, silicon has been chosen as the semiconductor of choice for solar cell design. http://pvlab.epfl.ch/files/content/sites/pvlabnew/files/groups/PV-LAB-unit/public/Pictures%20and%20pdf/commun/HIT%20group/Band%20diagram%20HIT.jpgFigure 1: A cross-sectional view of a typical solar cell (image courtesy of PVeducation.org)

Figure 3: Band diagram of a solar cell (image courtesy of http://pvlab.epfl.ch/heterojunction_solar_cells)Silicon is chosen because its band gap absorbs a wavelength of light with a naturally strong intensity. Another reason to use silicon is the ease of fabrication. The only problem with silicon is, it has an indirect bandgap, illustrated in the image on the bottom right of this page. This makes the process of generating the extra charge carriers more difficult. Either, the photon must have more energy to cause a completely vertical transition, or something must account for the change in momentum associated with horizontal movement. Usually lattice vibrations, which are modeled as a quasi-particle, the phonon, account for this change. Fortunately, phonon assisted transitions are beyond the scope of this paper, and we can simply accept the parameters given to us in the design specifications, without going in-depth into the physics behind them.http://large.stanford.edu/courses/2007/ap273/hellstrom1/images/III_1_small.gif

Figure 4: E-k diagram of silicon (courtesy of Stanford University)Once sunlight strikes the solar cell, the generated charge carriers flow through the circuit and produce a current. In order to collect these carriers and form a completed circuit, there must be some sort of electrical contacts on the semiconductor. We are investigating the best way to implement the front contact. Because any contact placed on the front of the solar cell will block sunlight, there will be a shadowing loss associated with the area of the front contact. In addition, there will be resistive losses between the contact and the semiconductor, as well as resistive losses in the contacts themselves. The losses due to shadowing will be minimized when the area of the front grid is minimized; however, a smaller area leads to a higher resistance, and the higher resistance causes more losses. Therefore, both losses must be considered when designing an optimal grid pattern. One way to reduce the area covered by the contacts, without increasing the resistive losses, is to taper the gridlines and busbars. This is effective because the current, at the extremities of the gridlines and busbars is less than the current at the middle and connecting ends of the gridlines and busbars. Since the current will be lower at the extremities, we can have a slightly higher resistance without increasing the losses. Because it is an easy way to increase efficiency, we decided to taper our busbars and gridlines. In fact, by tapering the gridlines and busbars, we reduced the combined losses due to shadowing and resistance to current flow within the gridlines and busbars by 15% [1].

Conventional wisdom suggests the most efficient way to apply metallic contacts onto solar cells is to screen-print conductive paint onto the solar cells. When the ink dries, it forms the grid that we will be optimizing in this paper. Many methods exist to optimize the design of a grid to capture the generated electrons. We have derived equations to model the losses associated with the various components of our front grid.


In order to reduce the complexity of the necessary calculations, we have divided our solar cell into M unit cells. Each unit cell is identical. It has an equivalent number of fingers and half of a busbar. Therefore each busbar is fed by two unit cells. This will make our calculations much easier because we can now solve for the power loss in each unit cell and multiply the result by M. The following calculations are per unit cell, except for the contact loss. This loss is dependant only on the spacing of the fingers and the resistivity of the emitter layer. The reason for this is we are assuming no current flows directly from the solar cell to the busbar. All current in the busbar is assumed to have come from the gridlines.

Losstotal = (Lossshadowing + Lossgridlines + Lossbusbars )*M + Losscontact

The loss due to shadowing is the easiest to calculate. It is simply the shadowed area divided by the total area.

Lossshadowing =

Total Area = 100 cm2

Shadowed Area = (Nb2WbLb + Ng2WgLg)/2

The ½ term is introduced because of the triangular shape of the busbars and gridlines.

Shadowed Area = NbWbLb + NgWgLg

Nb = number of busbars 2Wb = Width of busbars at originating point

Wb = Width of busbar half-way along its length Lb = Length of busbars

Ng = number of gridlines 2Wg = Width of gridlines at originating point

Wg = Width of gridline half-way along its length Lg = Length of gridlines

For the loss in the gridlines, we assume all current flows to the nearest gridline, and the current flows in a direction perpendicular to the gridline.

S = distance from center to center of gridlines A = dimension of unit cell parallel to busbar

B = dimension of unit cell perpendicular to busbar J0 = current density with zero shadowing

V0 = open circuit voltage with zero shadowing ρsm = sheet resisitivity of the metal

Figure 6: Details around a single busbar in a unit cell (Fig 5 & 6 courtesy of [1])

Figure 5: Details around a single gridline in a unit cell (Wf corresponds to Wg because we are using the term gridline, which is interchangeable with the term finger)

Resistive losses in gridlines:

Lossgridlines = dPg = I2dR

dPg = (J0Sx)2 ( * ρsmdx)

Pg = B3S2ρsmJ02/(4Wg)

Since there are A/S gridlines per unit cell,

Pg = AB3SρsmJ02/(4Wg) Watts/unit cell

Resistive losses in busbars:

Lossbusbar = Pb = I2dR

Pb = (J0By)2 ( * ρsmdy)

Pb = A3B2ρsmJ02/(4Wb)

From [5], we know that the optimum busbar width occurs when the resisitive losses in the busbar equals the optical losses from busbar shadowing. Also, smaller S and Wg will lead to smaller losses. This is because as Wg decreases, the distance between gridlines, can be decreased (thereby increasing the number of gridlines), without increasing shadowed area. This decreases the contact resistance between the semiconductor and the metal contacts, while holding other losses constant [1]. Therefore, Wg should be chosen by fabrication limits, and S can be found from a simplification of the above equations.

Wg = (SB/2)

S = (2Wg)/ (

Alternatively, in order to get a whole number of gridlines per unit cell, a S can be chosen such that Wg is feasible and the number of gridlines will be a whole number.

Losses due to contact resistance:

Losscontact =

Losscontact =

Equations derived with assistance from [1]-[4].

With these equations, we will be able to calculate the losses associated with our screen-printed contacts.

Overview of Screen Printing:

Figure 7: Rotary style screen-printing. Used in many large scale industrial applications (image courtesy of [6])Solar cell contacts are usually applied via screen-printing because of low cost, highly flexible grid patterns, and high speed. In screen-printing, the customer determines what pattern of gridlines and busbars they want for their front contact. This pattern is then used to create a mask of wire mesh. This mask is placed over the cell, preventing ink from contacting the cell, except where the gridlines and busbars will be placed. Then, a squeegee forces the ink through the open portions of the mask and onto the solar cell. The final thickness of the line is dependent on how far the mask is kept away from the solar cell, the force of the squeegee, and the speed of the process. After the ink application, the solar cell passes through an oven, designed to dry and solidify the ink on the solar cell [6]. Below are several figures depicting the process.

Figure 8: straight-line screen-printing process and two small-scale screen-printers (image courtesy of [6])

Figure 9: Illustration of squeegee operation (image courtesy of [7])

Figure 10: Large-scale screen-printing process with drying ovens (image courtesy of [6])

One of the key factors involved in screen-printing is aspect ratio. Aspect ratio is the height of a line divided by its width. A higher aspect ratio will maintain nearly constant shadowing, but significantly decrease the series resistance. The shadowing is assumed constant because the gridlines are so thin that even if their thickness were increased by an order of magnitude, the increase in shadowing is negligible. In other words, the gridlines are so thin, they can be approximated as two-dimensional. The only area being shadowed is the area physically under the contacts [7]. One way to increase aspect ratio is to double print lines. Because there is a maximum feasible thickness per print cycle, the only way to increase the height is to print multiple times. The feasible thickness per print cycle is given by:

t = Vscreenkp

Where Vscreen is volume of the screen, kp is the pick out ratio, c is the concentration of solid in the ink, and ρ is the density of the material in the final film [6]. By double printing the lines, as long as there is no mismatch between the first line and the second line, the thickness can be increased [7].

Figure 11: Illustration of the increase in aspect ratio by highly accurate double printed lines (courtesy of [7])

Of course, this means the solar cell must go through the printing process twice. However, considering the highly automated nature of the process, this does not seem like an issue. A more pressing concern is, the second line must match up exactly with first line. By using new technologies, the mismatch is kept below 10 µm. Furthermore, this mismatch will not negatively affect cell performance [7]. We believe this is a good method to increase our aspect ratio [7].

Another technique to increase the aspect ratio of the gridlines and busbars is the hotmelt process. This process uses an ink that is solid at room temperatures and must be heated to 50-90 degrees Centigrade to be printed. This means that all parts of the process must be hot, including the screen, the squeegee, and the print nest. However, because the ink is a solid at room temperature, it does not need to be fired after it is printed. This alternative technique, when paired with a more highly concentrated silver ink, can result in gridline that is twice as tall as a conventionally printed gridline [8]. This effectively eliminates the need for double printed lines, and it eliminates the problem of mismatch. Unfortunately, the increase in silver content in the paste will increase the cost. In addition, the requirement to heat anything that comes in contact with the ink makes the fabrication process more complicated. It is unknown whether this technology will prove efficient enough to justify the increased costs involved in the fabrication process. An investigation into the potential merits of this technology is outside the scope of this paper, but will undoubtedly be conducted by other groups investigating ways to increase solar cell efficiency.

Another important factor to consider, in the screen-printing process, is ink selection. Because there are a wide variety of inks on the market, we needed to investigate the different properties associated with each ink. We decided to focus on the varieties of ink produced by the Dupont Corporation. We assembled the properties of a wide range of Dupont products in the table below. We have also included an ink from a paper presented at a Photovoltaic Specialists Conference. We wanted to find a commercially available ink that would match the ink from that paper (reference [5]). By choosing an ink with specifications that match the ink from the paper, we can use the actual values for resistivity presented in the paper, and we will still be following the requirement that we only use commercially available materials. This will allow us to be much more accurate in our final calculations.




Drying Specs.



Height (μm)


Width (μm)




Solamet PV410



for 5-60 min.






Solamet PV412



for 5-60 min.






Solamet PV414



for 5-60 min.






Solamet PV416



for 5-60 min.






Solamet PV430


170⁰C for 30 min.




Straight line


Solamet PV17A



for 10 min.




Double Printing


Solamet PV17D



for 10 min.




Double Printing


Solamet PV17F



for 10 min.




Double Printing


Silver Paste


715⁰C for 45 sec.




Straight line


Table 1: list of specifications for several different conductive inks.

Figure 12: Firing profile for Solamet PV17 series (courtesy of [14]-[16])

By comparing the silver paste, from reference [5], to the PV17 series, it appears the authors of [5] used a similar product to one of the PV17 series inks. Therefore, we will also be using a PV17 series ink in our final design.

Design Selections:

Figure 13: Image of our final front grid design. (Courtesy of [1], alterations made to accommodate our dimensions)

For our design, we elected to use a straight-line screen printer. This makes more sense for us because we will not be manufacturing a large amount of solar cells; instead, we are doing a proof of concept design to see if we can effectively match customer specifications. To achieve positive results, we have also chosen to use the PV17A Dupont ink. This ink is very similar to the ink described in reference [5], and its properties seem ideally suited to achieving a low loss profile. Because of the similarities between the inks, we believe the authors of [5] most likely used a PV17 or comparable ink. This will allow us to use the resistivity data from [5] when we do our final calculations. In addition, the PV17A ink is designed for double printing. We should be able to get a height near 20 μm with a width around 70 μm. This would give us an aspect ratio of 28%.

Originally, we were going to use a single busbar with the interconnect point in one corner of the cell. Unfortunately, this resulted in very large losses. We tried to mitigate this by finding a very conductive ink. However, we were unable to find a commercially available ink that would reduce the losses below 8%. Because we were unable to make this design work, we had to scrap our one busbar plan. We found we could increase efficiency by reducing the distance the current had to travel in the thin, and relatively highly resistive, gridlines. To reduce gridline length, we decided to use multiple busbars.

We eventually decided to use a five-busbar system. An image of our final design is shown on the previous page. There will be two and a half busbars along the bottom, tapering linearly as they approach the middle of the solar cell. In other words, at half-length the busbars width will be half of its maximum width. There will also be two and a half busbars along the top, tapering linearly as they approach the busbars from the bottom. We will then have five interconnect points on the cell. Each busbar will have its own interconnect point where it meets the edge of the solar cell. This technique provides us with ten 5cm by 2cm unit cells for our final design. Each unit cell will deliver its current to half of a busbar. The current from two individual unit cells will pass through the busbar to the interconnect point. This method minimizes the distance current must travel, while maintaining easy fabrication. In the image above, we highlighted one unit cell. Its vertical dimension is 5cm and its horizontal dimension is 2 cm. Each unit cell is adjacent to half of a busbar, and each interconnect point (marked by a black circle) is fed by two unit cells. The busbar on the right is only half of a busbar to make the internal geometry of the unit cells come out as whole numbers. This allows us to simplify the math, but it does not affect the fabrication process. It is just as easy to screen-print this pattern onto the solar cell as any other pattern. It should be noted that the width of busbars and gridlines are taken at the middle of their respective structure, and the busbar width from our calculations is assumed to be the width of the entire busbar, not just the half seen by one unit cell.

We believe that this grid design will give us a marked increase in efficiency. The tapered gridlines and busbars should reduce shadowing, and the multiple busbar and interconnect points should allow current to get out of the circuit more efficiently than it would in a single busbar system. In fact, this grid design should be approximately three times more efficient than our original single busbar system with the interconnect in one corner of the solar cell [1]. In our final design, the sheet resistivity of the gridlines is ρsmg = 3.57 mΩ, and the sheet resistivity of the busbars is ρsmb = 1.88 mΩ. The resistivity of the busbars is much lower because of the increased area, corresponding to a large width, compared to the gridlines, and an additional 80 µm of silver paint, for interconnection purposes [5]. After we developed our overall grid design and determined our resistivity values, we needed to plug the values into our equations, so we could determine if our design would match the customer specifications.


In order for our results to pass customer specifications, we need our overall losses to be less than or equal to 8%. Using the short circuit current density, the open circuit voltage, the fill factor, and the overall efficiency, we determined the solar cell has an output power of 2.08 Watts. Therefore, an 8% power loss is equivalent to 166.4 mW. If our final design has an output power of greater than 1.91 Watts, we will have succeeded in our goal.

The final fabrication process was a little tricky. Our solar cell needs to be double printed, so it will have to go through the printing process twice. It will also have to be spike fired and dried between each step. Therefore, the ink is applied to the solar cell according to our grid design. It is then fired and dried according to the specifications for PV17A ink in table 1 and fig. 12 on page 10. After it goes through the drying process, the cell goes back to the beginning to get its second layer of paint. This layer is used to increase the thickness to the desired value. The solar cell then goes through the final firing and drying stage. Once it has finished drying, it is ready for the customer. All the customer will have to do is solder the interconnect points to the external load. This is convenient because we chose a solderable paint. Our final design parameters and results are summarized in tables on the next page.



number of busbars


width of busbars

1.076 mm

length of busbars

5 cm

number of gridlines


gridline spacing

5 mm

width of gridlines

74 μm

length of gridlines

see note

height of gridlines

20 μm

aspect ratio


number of unit cells


Note: The parameters busbar and gridline width are measured at the middle of their respective structures. Also, the width of the two half busbars on the right side will be 0.538 mm. This will not introduce more resistive losses because they are only fed by one unit cell each; whereas, the other busbars each carry current from two unit cells. In addition, when determining the length of the gridlines, the position along the busbar must be considered because the tapering will affect the length of the gridline. Please see the formula below.

Lg = 2 - (0.538 - 0.1076*Y) [cm]

Table 2: Final design parametersWhere Y is the position along the busbar and can vary from 0-5cm.


Loss [mW]










% Loss


Note: Loss per element includes shadowing and resistive losses for the gridlines and busbars.

Table 3: Summary of Losses

As shown in the table, we were able to meet the customer specifications by 5.57 mW. We were able to do this because we used highly conductive ink, and because we used a highly optimized front grid design.


After considering many different conductive inks and possible grid designs, we were able to develop a design that achieved the loss profile given to us by the customer. The key technologies we used to achieve these results were: multiple busbars, double printed lines, and tapered gridlines and busbars. We also considered using the hotmelt technology to allow us to skip the double printing step. Unfortunately, we were unable to find enough information about the types of inks used, or the cost efficiency associated with the hotmelt process. As a result, we decided to use a more conventional approach. The largest obstacles, to attaining these low losses, were the shadowing and resistive losses in the busbars. We were able to keep the resistive losses low by applying the solder layer to the top, and by using five busbars. Unfortunately, the extra busbars resulted in more shadowing losses. We had to balance the resistive loss optimization with the shadowing loss optimization. Due to the sheer amount of research centered on reducing shadowing losses, we concluded this is a common limiting condition on efficiency. We believe that the next major advancement in solar cell efficiency will come from this area of research.