Optimization and benchmarking of the Mercury IBC cell

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Optimization and benchmarking of the Mercury IBC cell

Dorus Dekker

25/11/14

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A thesis submitted for the degree of master of science of the University of Amsterdam Assessors from the UvA:

Wim Sinke & Tom Gregorkiewicz Supervisors from ECN

Agnes Mewe, Teun Burgers & Ilkay Cesar Research work was carried out at ECN Solar Energy,

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Foreword

Fundamental physics underlies almost all the technology in the world. However, in the study physics, little is taught about its application. I believe that fundamental physics is crucial for the development of humanity. But, I would like to see more of the impact of the physics on technology and the world. This is the reason why I searched for a institute like ECN to do my master thesis, since this is a place where they use the knowledge from fundamental physics, and apply it to improve technologies to gain improvements in sustainable energy. My master track is Advanced Matter and Energy Physics, and therefore research at solar cells ts neatly in my studies.

The world population is increasing rapidly, and the demand for energy is growing with it. The fossil fuels that we are now using took millions of years of forming. And the huge demand for energy is consuming these resources within decades. There is a lot of fossil fuels stored, and new techniques are found to gain access to more resources. However, this can not go on for ever, eventually the fossil fuels will be depleted.

Also the amount of consumption is so large, that it changes the atmosphere noticeably. Within these millions of years that the fossil fuels were created, specic atoms where stored, and by burning them, a lot of carbon is released in the air. I do not want to interfere in the discussion what the eect of this increase in carbon concentration in the atmosphere is. But a increase from 270 to 395 particles carbon per million particles in the atmosphere in the past hundred years[13] sounds severe, and as long as we do not know the consequence of this increase, we should be careful.

And lastly, the western world is greatly dependent on the supply from the rest of the world. Nowadays trade with Russia is stopped. Which could lead to great energy shortages. It would be better for the western world to be less dependent on the import of energy.

These three reasons give a good plea to invest in sustainable energy, to replace fossil fuels: because of the shortage, environmental pollution and energy independence. Solar energy is one of the many alternatives as sustainable energy. It converts the solar light directly into electric power. The solar cell can produce everywhere electric power, although more energy could be achieved close to the equator and in areas with few clouds.

Unfortunately, together with many other sustainable energy sources, solar energy is still more expensive than energy from fossil fuels. Therefore research is needed to make the solar panel more ecient, cheaper to produce or both.

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Abstract

In this work the the Mercury solar cell was optimized. The Mercury cell is an IBC front oating emitter (FFE) cell and was compared with an IBC front surface eld (FSF) cell to see the benets. And the emitter contact was improved to increase the eciency of the Mercury cell.

The FSF IBC cells are the most ecient single junction solar cells[1], that nowadays are available. An IBC cell has the contacts at the rear, leaving the front totally free to receive sunlight. Although this design yield high eciencies, they are very expensive. The Mercury cell is an IBC cell design to reduce the cost of the IBC cell at the cost of minimal eciency loss and be producible by industry. This still high eciency, easy manufacturing design should lower the production cost of solar energy, and thereby compete with fossil fuels as energy source.

The Mercury cell design was determined via simulation, using the program Quokka. This program was also used to see the dierence between the FFE and the FSF cell. In this work a method was developed to give a picture of the performance of the cell as function of changing parameters. Hereby a function was tted over multiple simulated points to see the behavior of the solar cell. In this work, mostly the parameter change in dimensions were examined.

Two designs for the Mercury cell, producible by industry, were optimized in this work. The high eciency (HE) cell, with small features but high eciency, up to 21.1%. And the easy manufacturing (EM) design, with larger features with an eciency of 20.6%. The optimization of the EM design result in an emitter half width is 415 µm and BSF half width is 175 µm, for the optimization of the EM design result in both emitter and BSF half widths of 500 µm.

To see the value of the Mercury design, the FFE was compared with an FSF, which are both IBC cells, only with a dierent front. The two Mercury designs, the HE and the EM cell, were used in this comparison. For the HE design, with small features, the eciency was the same for the FFE as for the FSF cell. For the EM design the advantage of the Mercury cell is shown, because the FFE cell is 2% absolute more ecient than the FSF cell. An EM FFE cell is less eective than a HE FFE cell, but the losses are relatively low, much less than for the FSF cells.

Furthermore, one of the biggest losses in a cell is the recombination at the emitter contact. Therefore a deep boron emitter was tested to see whether this improves the cell. The deep boron diusion shields the contacts better from electrons, and thereby reduce the recombination.

Experimentally the deep boron emitter was studied. Cells with the deep boron emitter were made and compared with cells with a regular emitter, and subsequently it was compared with simulated work to underpin the results.

With the deep boron emitter contact the open circuit voltage of a cell, was higher than with a regular boron diusion. Where screen printed contacts contacting a regular boron diusion has a saturation current density of 3000 fA/cm2 _{this deep boron diusion gives a saturation}

current density of only 1000 fA/cm2_{, these values are determined by simulations.}

The deep boron emitter eciently shields the emitter from electrons to reduce recombina-tion, and thereby increase the voltage yield from a solar cell. Quokka simulations show that the HE FFE cell with a deep boron emitter will have an increase of eciency of 0.2 % absolute and the EM FFE cell of 0.1 % absolute.

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Chapter 1

Introduction

At the research group Device Architecture at ECN (in which this research work was conducted) improvements and new designs for cells and modules for the design of the solar cell are inves-tigated. The solar cell is the part in which the sunlight is transformed in electric power. It is the aim of these designs to be producible by the industry, to be produced at large scale and to be competitive with other designs.

Various designs of solar cells exist. One of the most ecient solar cells, for turning sunlight into electric power is the Interdigitated Back Contact solar cells (IBC solar cell). The advantage of this cell, compared with other solar cells, is that the contacts to extract the current from the cell are at the rear. Hereby no metalization is at the front to reect the sunlight which would reduce the sun power reaching the cell. These IBC cells have a very high eciency, however they are expensive and hard to make. Nowadays they are produced at smaller scale.

Two dierent types of IBC cells are the front oating emitter (FFE) cell and the front surface eld (FSF) cell. The eciency record IBC cell is an FSF solar cell[1]. In such a cell, the front repels minority carriers from the bulk, hereby reducing the recombination at the front. Whereas the FFE solar cell, repels the majority carriers at the front from the bulk.

For an FSF cell to work, it needs very small features to be ecient. The FFE cell is also most ecient with very small features. However, the Mercury cell also allows larger features, and is therefore easier to manufacture[2], with minimal eciency loss. The idea is that the eciency of this FFE cell is lower than it would be with smaller features, but not as low as it would be for an FSF cell with larger features. However, this drop in eciency of the Mercury cell should be made up by the cost reduction.

In this thesis the dierence in performance between the FFE and the FSF is examined, therefore looking at both cells with small features and larger features. For this novel Mercury design to work, an optimal design is searched for. Besides this, the contact emitter is improved. Although the cost reduction is key in of this new design, no extensive cost analysis are presented in this work.

Chapter 2: Firstly we treat the basics of the solar cell by an example of a basic solar cell, to get an understanding for the mechanisms referred to later. It is a short description of how light is transformed in to positive and negative carriers, by exciting electrons in a silicon crystal, and how these carriers are separated by doping the surface of the cell, to extract the current. Chapter 3: The principle of the IBC cell is explained, and with it the FFE and the FSF cell. These are IBC cells, which means that the contacts are at the rear. The dierence between the FFE and the FSF cell is mainly the path of the minority carriers within the cell. This is used to give an explanation for expected dierence, why the FFE cell is better for larger cell features.

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Chapter 4: Bulk and surface properties are discussed, they have much inuence on the performance of the cell. The design of the dimensions is one aspect of the cell, but bulk and surface properties give limits to the performance. The eect of the bulk and surface properties is more elaborated later, but the principles and manner of measurement is discussed.

Chapter 5: A solar cell under illumination gives a current with a certain voltage. The I-V curve is discussed, on this curve there is one point at which the power output is maximum. In this section the eect of the bulk and surface properties on the current, voltage and shape of the I-V curve is shown, and therefore the eect on the eciency is shown.

Chapter 6: In this chapter the simulations are described. First the optimization for the Mercury cell is done, for as well a design with small features for high eciency as well as larger features for easy manufacturing. Furthermore a comparison is done between the FFE and the FSF cell. This is done with dierent cell properties, varying front passivation and cell thickness. Chapter 7: Lastly, a deep boron diusion is tested. The idea of this diusion is that the emitter contacts improve, thereby making the Mercury cell into a better one. This deep boron diusion shields the emitter contact better from the negative charge carriers, and this decreases the recombination current. Mainly the eect on the Voc was studied.

Chapter A: In the end is a appendix for some derivations and extra pictures. These deriva-tions are relevant, but they are long and would complicate the reading of this thesis.

For this work many used sources are the article "Mercury: A Novel Design for a back junction back contact cell with front oating emitter for high eciency and simplied processing" by I Cesar, the theses "Surface, emitter and bulk recombination in silicon and development of silicon nitride passivated solar cells" by M. J. Kerr and "High-eciency back contact back-junction silicon solar cells" by F. Granek and the books "Solid state physics" by J. R Hook and H. E. Hall and "Device electronics for integrated circuits" by R. S. Muller

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Chapter 2

The solar cell principle

The solar cell is a device that transforms sun power into electric power. The cell consists of a semiconductor that absorbs the light, which creates carriers by the photoelectric eect. By making use of doping, an electric eld is created within the cell which separates electrons and holes, the carriers can be extracted to create electric power.

2.1 Solar cell bulk material

A semiconductor material is not really conductive like a metal, but it is more conductive than an isolator like glass. In a semiconductor are barely electrons in the conduction band, and the valence band is almost completely lled. The energy dierence between the valence and conduction band is the band gap, for a semiconductor this band gap is is around 1 eV , much smaller than for an isolator, where the band gap is more then 5 eV .

An electron which receives more energy, for example from heat or a photon, than the band gap goes from the valence band to the conduction band, these are called excited electrons, this is shown in gure 2.1. At room temperature some electrons are excited. These excited electrons can move to neighboring atoms. Hereby, the excited electron is a negative charge that can move freely through the cell. Also the vacant electron spot, called hole, can move through the crystal as a positive charge, as an electron from neighboring atoms now takes the place of the excited electron. The density of carriers at room temperature is called intrinsic excess carrier density (ni).

For most solar cells silicon is used. This material is abundant on the earth, and the processing of silicon is perfected because the chip industry also works with silicon crystals. The valence band of a silicon atom consists of four electrons. The valence electrons bound to neighboring silicon atoms, and form a diamond cubic crystal structure.

The conductivity of a semiconductor is dependent on the excited electrons. When a photon is absorbed by an electron in the valence band, it will use the energy of the photon to bring that electron in an excited state, this will enhance the conductivity of the crystal. This free electron and hole can move freely through the cell. This is shown schematically in a potential plot in gure 2.1 . However, it is only a matter of time before these electrons and holes nd each other and will recombine. To get current from such a cell, electrons and holes have to be separated, and extracted via contacts.

2.2 Doping

To separate the carriers the potential eld of the conduction and valence band is adjusted by adding doping. These are atoms that replace a silicon atom in the crystal. To t neatly in the

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Figure 2.1: The valence band is completely lled with electrons, where the conduction band is lacking of electrons, except for the excited electrons. These exited electrons will make conductive electrons in the conduction band and conducting holes in the valence band. these carriers are free to move through the crystal, but the holes and the electrons can nd each other and recombine

crystal, these atoms have to show resemblance, therefore only atoms which are close to silicon (Si) in the periodic table are used. In gure 2.2 the periodic table is shown, and a zoom in of silicon and its surrounding. The atoms in the columns next to silicon in the periodic table are mainly dierent in that they have one electron more or less in their valence band.

Hereby, the atoms right of silicon have an extra electron. This electron is loosely bound and can move freely to neighboring atoms, hereby leaving a positive nucleus. These atoms are called donors, because they give an electron to the system. On the other hand, atoms at the left side of silicon have one electron less, therefore having a hole that can move through the silicon bulk. These atoms are called acceptors, because they can receive an electron from the system. A region with mainly donors is called a n-type region, because the carriers are mostly negative, whereas a region with mainly acceptors is called a p-type region, because the carriers are mainly positive.

The complete bulk is lowly doped to add a bit conductivity, making it either n-type or p-type. At the surface of the cell extra doping is added. This doping could be of the same polarity or the opposite. At the bulk boron and phosphorus are used to dope it with, for the surface also aluminum could be used. This doping at the surface is used to contact the cell. Positive carriers are extracted at the p-type region, whereas the negative carriers are extracted at the n-type region. In gure 2.3 is a schematic picture of an example of a solar cell. Here the bulk is p-type. The contacts at the front contact an n-type region, this region is the opposite polarity as the bulk, and therefore called emitter. At the rear the contact are alloyed with a region of the same polarity as the bulk, therefore called surface eld (SF).

The carriers within the doped regions can move through the crystal, and when an n-type region is neighboring a p-type region, the carriers can even enter each others region. As Fick's law states, the concentration dierence of the electrons and holes will drive the two to each others region. However, when, for example, an electron is leaving its n-type region, this region will become positively charged, and the region it enters negatively. Therefore, there will be a electric eld driving the carriers back to their region. Finally the concentration gradient and electric power will be in equilibrium. These carriers in the other region can make the electron conguration of the doped atoms similar to the silicon, this is lowering the energy level of the carriers and is therefore favorable for the carriers. In gure 2.4 is a schematic picture of how an electron from a donor (P) moved to a acceptor atom (B) within a silicon crystal, hereby

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Figure 2.2: The periodic table of elements and a zoom in a of silicon and its surrounding. The columns indicate the number of electrons in the valence orbital. Silicon has four valence electrons, atoms in the column left of silicon have only three valence electrons, where atoms right of silicon have ve electrons per atom. those atoms could replace a silicon atom in a crystal structure, thereby changing electric properties.

Figure 2.3: Standard solar cell. The bulk is p-type, as is the rear. At the rear the holes are extracted. The front is n-type, and the electrons are extracted via the ngers. The front is textured to reduce reection.

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Figure 2.4: Phosphorus is a donor, making its region n-type, while Boron is a acceptor, leading to a p-type region. In the depletion region electrons from the donors provide a bond between the acceptor and the silicon. Because of this shift of electrons the depletion region is partly positively and partly negatively charged.

creating a positive and negative region.

This image of charged regions is schematically shown in gure 2.5. By using the Maxwell equation

E = ˆ

ρ

dx (2.1)

the electric eld (E) can be obtained. And with this electric eld the potential dierence in voltage (V ) can be calculated using:

V = e ˆ Edx = e ¨ ρ dx (2.2)

Here is ρ the charge density and the permittivity. The electric eld and the potential dierence are also shown in gure 2.5. This gure shows that the potential over the junction makes a jump. By applying this energy jump at the band structure of gure 2.1 the potential band structure at the pn-junction is obtained, shown in schematically in gure 2.6.

Carriers at this junction will be forced to go in one direction. Electrons to the n-type region, and holes to the p-type region. The doping is diused at the surface, this has three functions. Firstly, either electrons or holes will be scarce at the surface, depending on the junction, and therefore reduce the recombination since both electrons and holes are needed to recombine. Secondly, the other carrier type will be present in abundance. A contact at such a region will extract these carriers, and thereby determine the polarity of the contact. Furthermore, these higher doped surfaces are much more conductive than the bulk. Hereby carriers can conduct laterally though the surface diusion to the contacts more easily.

2.3 Cell design

In gure 2.3 is a picture of a solar cell. In the bulk electron hole pairs are created. In this example the bulk is p-type. Therefore the number of electrons are determined by the number of absorbed photons, minus the number of recombined electron hole pairs. Whereas, the number of holes is the same as the number of electrons, because they are created together, plus the number of acceptor doping in the bulk. The carriers are extracted via metal contacts. The

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Figure 2.5: By the use of integrals, the charge graph leads to the electric eld and the a potential graph. This potential jump will separate the electrons and holes.

Figure 2.6: Schematically view of the band structure at a pn-junction. Because of the charged regions in the depletion zone, a potential jump is created. This will separate the charge carriers.

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cell in gure 2.3 has contact ngers at the front, in this way a part of the front of the cell is blocked from sunlight. Underneath these ngers the silicon is doped with donors, thereby the carriers at this surface will mainly consist of electrons. This doped region is called emitter, for the polarity is opposite to the bulk. Because of this doping, the contacts at the front will extract the electrons from the cell.

In this cell, the surface at the rear is doped with acceptors, a bit heavier than the bulk, this is called back surface eld (BSF). This extra doping will repel the electrons from the surface. The complete rear is covered with metal to extract the holes from the cell.

In this example the bulk is p-type, this means that initially there are a lot of free holes present, and few electrons. Therefore they are called majority and minority carriers respectively. For a minority carrier the chance is very high to encounter a majority carrier and recombine. For the majority carrier the chance to nd a minority carrier is much smaller, therefore, the lifetime of a minority carrier will be much less than the lifetime of a majority carrier.

Many more solar cell designs exist, varying for example in n or p-type bulk, dierent posi-tioning of doping contacts, varying thickness and quantity of doping. One of these designs is the IBC cell and in particular the Mercury cell. This cell is discussed more extensively in next chapter.

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Chapter 3

IBC cell principle

An IBC cell (interdigitated back contact cell), is a solar cell with all the contacts at the back. A picture of the rear of such a cell as it is made at ECN is shown in gure 3.1A . At the back the ngers that extract the positively and negatively charged carriers from the wafer are alternated. This is shown in the zoom in, in gure 3.1B.

The bulk of the IBC cell is n-type, this is because n-type silicon is less sensitive for impurities for Shockly, Read, Hall recombination and therefore the lifetime of the carriers in a n-type wafer is higher than for a p-type region[8]. The holes are the minority carriers, whereas the electrons are the majority carriers. The biggest dierence between these two carriers in the bulk is that the chance of a hole to encounter an electron with which it can recombine is bigger than the change of an electron to meet a hole where it can recombine with. Therefore, the path length of the minority carriers is of great importance for eciency of the cell. A long path length means more recombination and less current. The exact background of the carrier lifetime is discussed extensively in chapter 4.1.

This minority carrier mean path length from its creation to the contact through the bulk is dependent on the design of the cell. A schematic cross section of an IBC cell is shown in gure 3.1C. The width of the emitter and BSF at the rear are very important. Also whether the front is a front oating emitter (FFE) or front surface eld (FSF) is very important.

A small BSF width results in a short path length of the hole minority carriers to the emitter. Within the emitter, the holes are the majority carriers and can be transported to the contact. This reduction of time the holes spend in the bulk will result in less recombination, and more current. However, by making the BSF as small as possible, the alignment of the metal is critical and would require expensive patterning methods. To increase the BSF width, without losing too much current, the Mercury cell is designed.

The Mercury cell is an FFE IBC cell, where the minority carriers in the bulk get collected by this FFE. Since this emitter is not contacted, the holes are not drained. The number of minority carriers entering the front region, because of pn-junction attraction, will be as many as the number of minority carriers leaving the FFE due to the concentration gradient over the junction. However, the minority carriers in the bulk above the emitter will also be collected at the rear emitter and extracted from the cell. Therefore, the concentration of minority carriers above the emitter in the bulk is less than the concentration above the BSF, and more minority carriers from the FFE will be injected into the bulk above the emitter than above the BSF. With reasonable conductivity of the FFE, the concentration dierence will lead to a current through the FFE from above the BSF to above the rear emitter. This is called the pumping eect, the ow of the holes is schematically shown in gure 3.1C. This eect reduces the path length of the minority carriers through the bulk, and therefore reduces the recombination and increases the current.

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A B C

Figure 3.1: A: Picture of an IBC cell from the rear, at which the contacts are. B: Zoom in on the metalization of the IBC cell. Busbars are vertical, ngers are perpendicular, circles are for the contacting with the module. C: cross section of a unit cell of the wafer. Left is a cell with an FFE and right is a cell with an FSF. The top indicate the front at which the light is shone in the cell, while at the rear the contacts extract the carriers. The black arrows indicate the minority carriers current.

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Chapter 4

Wafer properties

We can subdivide the wafer properties in bulk properties and surface properties. The bulk can roughly be described by the bulk lifetime, bulk resistance and polarity. More complicated are the surface properties. They depend greatly on the doping prole and passivation.

By doping the surface of an n-type wafer a pn or n-type high-low junction is created at the surface, this is the interface of the surface diusion and the bulk. At such junctions one carrier type will be repelled from the surface. In this manner recombination at the surface will be reduced because both an electron and a hole are needed to recombine. However doping is pollution of the crystal which will become spots for the carriers to get stuck and recombine. Hence heavy doping will decrease the lifetime of the carriers. Furthermore, the surface of the silicon crystal has a lot of defects. This is because the diamond structure of the silicon, like in the bulk, can not be maintained, since at the surface the silicon atoms are not completely surrounded with other silicon atoms. To remove defective dangling bonds of the silicon at the surface, a passivation layer is used. As an extra feature, the refractive index of this passivation layer can be tuned. The index can be tuned reduce reection on the sunny side of the cell, or increase reection at the rear.

Since in an IBC cell both emitter and BSF are at the rear, they also have a junction at the very narrow interface of the two diusions, named p+_n+ _{junction. In the simulations}

recombination at this junction is not taken in to account. However, this region is only a small fraction of the surface, and it is expected that it will not have to much impact.

In this work, the saturation current density (J0) and the sheet resistance (Rsheet) are

considered to be the crucial parameters that determine the surface properties. Other (optical) parameters will not be studied in detail. To determine the J0 the wafer lifetime and excess

carrier density have to be measured.

4.1 Carrier lifetime

The lifetime of a carrier is limited by the recombination. Recombination is the process at which the electron-hole pairs, created by a photon, encounter each other and the electron relaxes to the valence band. The excess energy of the electron is released in the form of either photons, phonons or a combination of the two.

The lifetime (τ) is given by the concentration of electron hole pairs (∆n) divided by the recombination per second per volume (U):

τ ≡ ∆n

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4.1.1 Dierent recombination mechanisms

Within a semiconductor, there are three methods of recombination: Radiative-, Auger- and Shockley - Read - Hall (SRH) recombination[11]. The lifetime can be calculated from the recombination rate using equation 4.1 The total recombination rate is given by the sum of all individual recombination rates. Therefore the eective lifetime is given by:

1 τef f = 1 τRad + 1 τAuger + 1 τSRH (4.2) Hereby, the shortest lifetime is dominating the eective lifetime. Therefore the behavior of dierent lifetimes are studied under high and low injection level. The denition whether the cell has a high or low injection level, is stated by whether there are the electron-hole pair concentration is higher than the doping concentration (high injection) or the other way around (low injection).

First the individual recombination mechanisms are discussed. To simplify the functions, they are approached at high injection (hi) and low injection (li) level, and in conclusion they are compared with each other.

4.1.1.1 Radiative recombination

This recombination occurs when an electron and hole encounter each other, and the electron relaxes to the valence band. However, silicon has an indirect band gap. This means that the electron has an other momentum than the hole and therefore the energy of the excited state can not simply be given to a single photon, because a single photon can not contain impulse. Therefore at the annihilation of the electron-hole pair at least one photon and one phonon is created. This four particle interaction makes it much less likely to occur[9].

The radiative recombination is given by

U = npσvtnP = npB (4.3)

B ≡ σvtnP (4.4)

With n and p the electron and hole concentration, σ the interaction cross section, vtn the

velocity of the carriers, P the chance the interacted electron and hole annihilate and B the recombination coecient which is a constant and only material dependent. The calculated value for B in silicon is 2 · 10−15_cm3_/s_{[10]. However the measured value is a bit higher, stated}

at 9.5 · 10−15_cm3_/s_[12].

At low injection level, the minority carrier concentration equals the excess carrier density, and the majority carrier concentration is approximately the doping concentration. There fol-lows, as shown in equation 4.5, that the radiative lifetime at low injection level is constant.

τrad,li = ∆n U = ρmin BρminNdop = 1 BNdop (4.5) At high injection level, the excess carrier concentration is far more than the doping. There-fore the doping can be neglected, and the electron concentration is as high as the hole concen-tration. ∆n = n = p (4.6) τrad,hi = ∆n U = ∆n Bnp = 1 B∆n (4.7)

At high injection level, the lifetime caused by radiative recombination decreases linearly for higher injection level.

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4.1.1.2 Auger recombination

Auger recombination is the annihilation of an electron-hole pair, at which the excess energy is given to an excited electron or hole. In this manner, only three particles are needed. The recombination rate can be calculated as in equation 4.3. However, with Auger recombination two electrons or two holes are needed to interact, leading to:

U = n2pσvtnPn+ np2σvtnPp= Cnn2p + Cpnp2 (4.8)

In which the used values for Cnand Cpare mostly those determined by Dziewior and Schmid

as 2.8 · 10−31_cm6_/s_{and 0.99 · 10}−31_cm6_/s _{respectively.}

At low injection level, the concentration of minority carriers is very low. Therefore the chance two minority carriers interact with a majority carrier is negligible. Together with the assumption that the majority carrier concentration is given by the doping, leads to a Auger lifetime of: τAuger,li= ∆n U = ∆n Cnn2p + Cpnp2 = 1 CmajNdop2 (4.9) This lifetime is only depending on the doping type and concentration. Since the Auger coecient for electrons is almost three times as high as the Auger coecient for holes, the lifetime of an n-type material is almost three times as high as for p-type material, under low injection.

At high injection, the dierence between electron and hole concentration is negligible and therefore the dierence between n-type and p-type material disappears.

τAuger,hi= ∆n U = ∆n Cnn2p + Cpnp2 = 1 (Cn+ Cp) ∆n2 (4.10)

However, these derivations are little simplistic, in reality Auger recombination is more com-plicated. But it gives a good understanding how the Auger recombination works. Experimental work[8] shows the following relations for the Auger lifetime for low injection.

τAuger,li=

1 Cmaj· Ndop1.65

(4.11) With Cn = 1.8 · 10−24cm6/s and Cp= 0.6 · 10−24cm6/s. And for high injection

τAuger,hi=

1

3 · 10−27_{· ∆n}1.8 (4.12)

4.1.1.3 Shockley - Read - Hall recombination

This recombination occurs through a defect in the semiconductor crystal. Owing to this defect, a discrete energy level within the band gap can exist. This defect level can function as a trap for the conducting carriers via which recombination can occur in a two step process.

In thermal equilibrium, these defect states are occupied by electrons or holes according to the Fermi Dirac distribution. In gure 4.1 are the four possible carrier exchanges with the defects. r1 an electron relaxes from the conduction band to the defect, r2 is the rate at which

an electron is excited from the defect state to the conduction band. r3 is the relaxation rate

of a hole, where r4 is the excitation rate of a hole to the defect. To get a good understanding

of this process, the four exchange mechanisms of carriers with the defect state are discussed individually, and later combined to get a picture of the total SRH recombination.

For an electron to relax to the defect state an excited electron has to encounter an empty defect state and interact. The fraction of empty defects state is given by the function F (E),

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Figure 4.1: Trapping and detrapping of carriers in defect stats within the band gab. which is the Fermi Dirac distribution at equilibrium. The fraction of occupied states for µ−Et

kBT can be written as,

fD(Et) =

1

1 + e(µ−Et)/kBT ≈ e

(Et−µ)/kBT (4.13)

Here Etis the level of the defect and µ the Fermi level. The rate r1 is then given as:

r1= nNt[1 − f (Et)] vthσn= τno−1n [1 − f (Et)] (4.14)

With Ntthe defect density,vththe thermal velocity of charge carriers , σn the cross section of

an electron to interact with an empty defect and τno ≡ 1/Ntvnoσn the fundamental electron

lifetime.

In equilibrium the excitation and relaxation of the electrons in the conduction band are the same. The rate at which electrons from defect states excite to the conduction band is only proportional to the number of electrons in the defect state, being,

r2= Ntf (Et)en= r1,i (4.15)

With this equation the spontaneous excitation factor en can be calculated and with it redene

r2. en= r1,i Ntf (Et) = vthσnnie(Ei−Et)/kBT (4.16) r2= Ntf (Et) · vtnσnnie(Et−Ei)β = τno−1f (Et)nie(Et−Ei)β (4.17)

For recombination rate r3and r4 the same process can be used.

r3= τpo−1pf (Et) (4.18)

r4= τpo−1[1 − f (Et)] · nie(Ei−Et)/kBT (4.19)

When the cell is illuminated, it is not at equilibrium. The recombination rate via the defects is the dierence in rate between the relaxing electrons from the conduction band to the defect and the spontaneous excitation of the electrons in the defects to the conduction band. This also applies for the holes, which gives:

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By using r1− r2 = r3− r4 the Fermi factor f(Et) can be calculated and remove from the

denition of the SRH recombination. τno−1n−τno−1 h n + nie(Et−Ei)/kBT i f (Et) = −τpo−1nie(Ei−Et)/kBT+τpo−1 h p + nie(Ei−Et)/kBT i f (Et) (4.21) f (Et) = τ_no−1n + τ_po−1nie(Ei−Et)/kBT τno−1n + nie(Et−Ei)/kBT + τpo−1p + nie(Ei−Et)/kBT = τ_no−1n + τ_po−1nie(Ei−Et)/kBT χ (4.22) χ ≡ τno−1 h n + nie(Et−Ei)/kBT i + τpo−1 h p + nie(Ei−Et)/kBT i (4.23) USRH = r1− r2= τ_no−1n · χ χ − τ −1 no h n + nie(Et−Ei)β iτ_no−1n + τ_po−1n_ie(Ei−Et)/kBT χ (4.24) USRH= np − n2 i τpon + nie(Et−Ei)/kBT + τnop + nie(Ei−Et)/kBT (4.25) τSRH = ∆n USRH =τpon + nie (Et−Ei)/kBT ∆n + τ nop + nie(Ei−Et)/kBT ∆n (np − n2 i) (4.26) The electron concentration is the intrinsic concentration plus the excitation n ≡ n0+ ∆n, as is

for the hole p ≡ p0+ ∆n. The product of the intrinsic electron concentration and the intrinsic

hole concentration is the square of the intrinsic carrier concentration

n2_i ≡ n0p0 (4.27)

By applying this to the lifetime in equation 4.26, the lifetime will result in:

τSRH=

τpon0+ ∆n + nie(Et−Ei)/kBT + τnop0+ ∆n + nie(Ei−Et)/kBT

n0+ p0+ ∆n (4.28)

It is hard to see what the behavior of this function is, therefore some specic cases are studied. First we assume the energy level of the defect is in the middle of the band gap, whereby e(Et−Ei)/kBT _{= 1}. Second the material is doped, making the intrinsic minority carries

concentration and the intrinsic carrier concentration negligible. For low injection level the excess carrier concentration is negligible as well, the lifetime of equation 4.28will become:

τSRH,li= τmin,o (4.29)

For high injection, the excess carrier concentration participate in equation 4.28 and the lifetime is given by:

τSRH,hi= τpo+ τno (4.30)

Where τno= 20msand τpo= 1ms.

4.1.1.4 Dominating lifetime

The radiative, Auger and SRH recombination occur all within a silicon wafer. The shortest lifetime mostly determines the eective lifetime. Since the three recombination mechanisms are dierent depending on the injection level, the dominant lifetime is varying. In gure 4.2 the curves for the dierent lifetimes are shown, based on the calculation. Although this gure is only an example, and per cell the values are dierent, the trend is for the general case.

The graph of gure 4.2 is based on the derived equations and input parameters stated in table 4.1. At low injection level the lifetime is independent of the injection concentration. In

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Figure 4.2: Lifetime curves for three recombination mechanisms as function of the injection level

Recombination _{Low injection level}Lifetime_{High injection level} Parameters[8] Radiative τrad,li= _BN1 dop τrad,hi= 1 B∆n B = 9.5 · 10 −15_cm3_/s Auger τAuger,li=_C 1

majNdop2 τAuger,hi=

1 (Cn+Cp)∆n2

Cn= 1.8 · 1024cm6/s

Cn= 0.6 · 1024cm6/s

SRH τSRH,li= τmin,o τSRH,hi= τpo+ τno τno = 20ms& τpo= 1ms.

Table 4.1: Injection dependence of the lifetime for various recombination mechanisms this example graph, the SRH recombination is dominant at low injection level. This is of a p-type material. The lifetime of electrons is much higher than those of holes. Therefore equation 4.29 shows that for an n-type material the τSRH,hi is than for a p-type material and the SRH

lifetime is not dominant for an n-type material.

At high injection level the radiative lifetime is inversely linearly proportional to the injection level, the Auger lifetime in inversely quadratically proportional to the injection level and the SRH lifetime increases and becomes independent again. Therefore the Auger recombination will be dominant at high injection level.

4.1.2 Lifetime measurements

To determine the passivation level of the cell surface, the conductivity of the wafer is measured with QSSPC, with Sinton WCT-120 QSSPC lifetime tool. In gure 4.3 is a schematic view of the setup. The coil underneath the sample induces a magnetic eld which changes by the conductivity of the sample, therewith the conductivity of the sample is monitored during a light

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Figure 4.3: The schematic QSSPC lifetime measurement setup. A lamp ashes light to excite electrons to make conducting electrons and holes. By means of a coil the conductivity is measured. The relation between the excited carriers and the conductivity is used to calculate the carrier concentration, lifetime, saturation current density and implied open circuit voltage. ash. The increase in conductivity is linearly proportional to the excess carrier density:

∆σ = q∆n (µe+ µp) W (4.31)

With ∆σ the increase in conductivity, ∆n the excess carrier density, µe, µpthe electron and

hole mobility respectively, and W the wafer thickness.

The electron hole pair generation during the ash is deduced through the current output of a reference solar cell. An example of such a measurement of conductivity and light intensity is shown in gure 4.4.

To deduce the lifetime from a measurement of the conductivity and the illumination as in gure 4.4, three approaches are possible, depending on the rate of the carrier lifetime of the sample and the ash time. When the lifetime of the sample is much longer than that of the ash time the generation and recombination can be approached as if all the generation takes place at one moment and the excess carrier density decays exponentially. This is called the transient mode, the following function can be tted over the excess carrier density curve to deduce the lifetime:

∆n(t) = ∆n(0)e−t/τef f (4.32)

This leads to a equation for the lifetime: τef f =

∆n

∂∆n ∂t

(4.33)

When the carrier lifetime of the sample is much shorter than the ash time, the generation and recombination can be approached as a quasi steady state. The generation, recombination

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Figure 4.4: Plot of measured illumination and conductance of the cell with the QSSPC. The illumination is determined with a reference cell and the conductance is measured with a coil-induced magnetic eld. The light induces carriers which will decay exponentially. With this measurement the lifetime, saturation current density and the implied open circuit voltage can be determined.

and excess carrier density can be treated as constantly in balance, which will lead to:

G = R (4.34) R = ∆n τef f (4.35) τef f = ∆n G (4.36)

With G the generation and R the recombination.

In the case that the lifetime and the ash time are of the same order, a combination of both methods is used, called generalized mode:

τef f =

∆n G −∂∆n

∂t

(4.37) If the lifetime of the excess carriers would be long, it could be assumed that the excess carrier density is rather stable. Therefore, ∂∆n

∂t would be relative small, and can be neglected

with respect to G. It can be seen that in this limit equation 4.37 will congure to equation 4.36. In the limit of a relatively short lifetime ∂∆n

∂t becomes rather large and G can be neglected.

Hereby equation 4.37 congures to equation 4.33.

4.2 Saturation current density

The saturation current density (J0) is a measure for the recombination. It is the current entering

a region to recombine there. In most cells the recombination is dominated at the surface. In this work there is a focus on the J0 of the junctions at the surface of the solar cell. This J0

at the junction, is the drift of the minority carriers from the neutral region over the depletion region, not necessary with illumination. This number of carriers that cross the junction, is the

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number of carriers created in the volume within the range of the travel length of a carrier[5]. For the electrons this will be as

J0,e = generation−rate · carriers−within−range = e

np

τe,p

(Le) (4.38)

By using Lp=pDpτp and np= n2i/NA equation 4.38 will result in:

Jo,e= eDen2i LeNA =r Dn τn n2 i NA (4.39) With e the electric charge, Lp, Lnthe diusion length of the holes and electrons respectively,

Dp, Dn the diusion coecient of the holes and electrons respectively, τp, τnthe holes' and

elec-trons' lifetime, ND and NA the doping concentration of the donors and acceptors respectively

and ni the intrinsic carrier concentration.

The total J0 is that of the electrons and the hole. This current of the holes works exactly

the same as the electrons, which will result in J0= e s Dp τp n2_i ND +r Dn τn n2_i NA ! (4.40) It seems that a heavier doped surface would lead to a lower J0. However, with extra

dop-ing, the Auger recombination will increase, and thereby decrease the lifetime of the carriers. In practice the lowest J0 at a passivated surface is obtained for the lightest doping.

More-over, the surface passivation is also very important for this J0 value, because it decreases the

recombination through defects in the band gap.

This J0 is an important property, because it is a limit for the voltage that can be obtained.

Under illumination, carriers are created, and because of the potential step at the junctions, electric current will ow. When these carriers can not leave the cell without resistance, a voltage is built up. Because of this potential dierence over the junction, a reverse current will ow. The ease at which this reverse current ows, is proportional with the J0.

You could compare a cell with a beach ball with a constant ow of air through the valve. This airow is an analogue to the the carrier generation by the illumination. The tension (analogue of the voltage) on the surface of the ball increases with the airow. However, when the ball is porous air will escape. This escaping airow, which can be compared with the reverse current in a solar cell, increases with increasing tension. The maximum tension feasible with a certain airow input is restricted by the escaped air ow. When the input ow is equal to the escape airow, the maximum tension is obtained. The case where this reverse current is as big as the forward current coming from the illumination is called open circuit.

The voltage in this open circuit case, called open circuit voltage (Voc), xed by the J0 is

deduced extensively in section 5.3. To determine the J0, the lifetime of the wafer is measured,

how this lifetime measurement works is explained in the previous section. With this measured lifetime and excess carrier density, the J0 is calculated as noted in equation 4.41 which is

derived in the appendix, section A.1. Instead of the denition in equation 4.40, the denition of equation 4.41 can be measured with the QSSPC measurement, with a measurement as in gure 4.4. J0s= W qn2_i 2 ∂_τ1 ef f − 1.66 · 10 −30_{· ∆n}2 ∂∆n = W qn2_i 2 ∂ 1 τef f ∂∆n − 1.66 · 10 −30∂∆n2 ∂∆n ! (4.41) When the lifetime and the excess carrier density is measured, the complete wafer is measured from rear to front. Therefore the the J0 is for both surfaces. By measuring a symmetric wafer,

the J0 of one side is half of the measured value. At an asymmetric wafer, the measured J0 is a

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4.3 Diusion prole and sheet resistance

In a semiconductor in the dark practically all the conductivity is due to the doping. Therefore, the surface resistance is a measurement for the doping quantity. Or to be more specic, the active doping quantity. The active doping is the doping that is integrated in the crystal lattice, whereas the inactive doping is situated between the lattice. The rst is responsible for the conductivity, the second not, this only reduces the lifetime. Note that a passivated surface with a high conductivity has a lot of doping and therefore also has a lot of Auger recombination. This will reduce the lifetime at the surface, and thereby increase the J0, as stated in equation

4.40. Therefore, it is a trade-o between good passivation and good conductivity[8]. The sheet resistance is an indication for the J0, however inactive doping is not measured, thus the sheet

resistance will not guarantee anything.

At a contacted surface, there are already a lot of defects, therefore the reduction of minority carriers is more important.

A shere scan is a measurement of the sheet resistance of a wafer. In this work a shere scan of the surface of the wafer consist of 45 measurements to get a spatial picture of the sheet resistance of the diused layer throughout the wafer surface, within reasonable time. Since the pn-junction does not conduct in dark, a four point probe resistance measurement, with the contacts on a surface with a pn-junction, will only measure the conductance through the surface. Therefore, it is not possible to measure the sheet resistance of a diusion of the same type as the bulk independently from the bulk, because there is no pn-junction to isolate the surface from the bulk. To measure the sheet resistance of such a diusion, wafers of the opposite type get the same treatment, on which this sheet resistance can be measured.

This sheet resistance only gives insight in the quantity of doping. To get a picture of the shape and depth of the doping prole an Electrochemical Capacitance Voltage (ECV) measurement is done. This is a measurement that determines the concentration of active doping at the surface by measuring the capacitance and voltage. By alternating between etching and measuring of the active doping concentration as function of depth, a prole can be made.

To measure the prole of all doping, including the inactive doping, SIMS can be uses. This method measures the mass of the atoms that are released by bombarding the surface.

With the measured active doping concentration prole the sheet resistance can be calculated by integration. The technique of ECV is calibrated for polished wafers. Since the wafers are textured, an area factor is used to compensate for it. This area factor is dened by the ratio of the total surface areas, and by checked if the shere scan Rsheet matches with the calculated ECV Rsheet, this area factor can be tuned. The surface is etched during the measurement. This will change the area factor during the measurement, because etching does not exactly follow the pyramids structure, it will atten the surface. However, a constant area factor during the measurement is calibrated for the complete measurement.

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Chapter 5

Cell performance and

characterization methods

A solar cell is a device that, when light is shone on it, will supply electric power. The quality of a cell is given by its eciency, this is the fraction of solar power that is converted from light into electric power. Even though the cell would be perfectly optimized, a single junction 1 sun silicon solar cell will not achieve a higher eciency than 29% in sunlight, due to the Shockley-Queisser limit. This is because the band gap of silicon is 1.1eV , whereby photons in the solar spectrum with lower energy can not be converted to electron-hole pairs by the silicon. At the other hand, photons with more energy can be absorbed by the silicon, and excite an electron, but all the energy of the photon exceeding the band gap will be dissipated in heat.

The biggest other losses are due to recombination, Ohmic losses due to resistance, shunt current, passivation, optical losses (reection, absorption, transmission). In a cell design, these losses can be minimized. The current and voltage give the performance of the cell. Therefore I-V curves are measured for the cell, ranging from open circuit to short circuit.

5.1 Eciency

The eciency of a solar cell is also called η (eta). This is the percentage of solar power that is converted into electric power, stated in the rst part of equation 5.1. This is obtained under standard test conditions (STC), which is at 25°C, irradiation of 1kW/m2_{, an air mass of 1.5}

(AM1.5) and according to a certain wavelength spectrum, comparable with the sun spectrum. These conditions correspond to a clear day with sunlight incident upon a sun-facing 37°tilted surface with the sun at an angle of 41.81° above the horizon.

In a power measurement the voltage is sweep from zero to maximum voltage, and the current is measured. The point that obtains the highest power output is called maximum power point (MPP), with corresponding the current Impp and the voltage Vmpp. By taking the size of the

surface of the cell into account, the current density Jmpp can be calculated. The eciency

is given by the current and the voltage at maximum power point, divided by the irradiation power. This is shown in equation 5.1.

ef f icency = Pelectric Psun = Jmpp(A/m 2_)V mpp(V ) 1000(W/m2₎ (5.1)

In the PV word the unit in which the current and voltage are expressed is mA/cm2 _{and V}

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5.2 Short circuit current and internal quantum eciency

The power extracted from the cell is a current times the voltage which it supplies. The maximum current extracted from a cell is when the contacts are shorted. In this case, the voltage is zero and the current is as large as the number of photons that are absorbed minus the number of electron hole pairs that are recombined, per second. Therefore the current will go up when more light is absorbed in the cell, and the current will decrease with more recombination. To compare solar cells, the light spectrum and intensity of the light source is standardized, and referred to as "1 sun". The short circuit current (Isc) is an indication of the recombination and

absorbed fraction.

The external quantum eciency (EQE) is the ratio of, the current if all the photons would be absorbed to create electron-hole pairs and be collected, and the current of the actual cell that is extracted at short circuit. The ratio of the current if all absorbed photons would create an electron-hole pair which is collected and the current of the actual cell that is extracted at short circuit is called the internal quantum eciency (IQE). To calculate the number of absorbed photons, the number of not reected photons is used, which in practice, for crystalline silicon solar cells, is the same. In an IQE measurement, the EQE and the reection is measured for dierent light wavelengths.

As the Lambert Beer law states, the intensity of light decreases exponentially with increasing depth in a medium, where the derivative of this exponential curve is the absorption. This absorption coecient is dierent for dierent light wavelengths. For instance, blue light is strongly absorbed in silicon and therefore is mostly absorbed at the front of the wafer. However, red light is less strongly absorbed, and therefore is absorbed throughout the wafer, up to the rear. The dierence in IQE in combination with the absolute values of IQE for the dierent wavelengths gives an indication whether the recombination takes place at the front, the bulk or the rear.

5.3 Open circuit voltage and I-V curve

When a cell is illuminated, a current is created, when this current is not directly extracted via a shorted contact, a voltage within the cell is built up. This voltage dierence in the cell will lead to a reversed current. To understand the behavior of the relation between the current and voltage, an ideal case for a solar cell in the dark is considered.

When a p-n junction is in the dark, there is no net current. The band structure at the pn-junction is shown in gure 5.1A. In the n-type region there are a lot of electrons in the conduction band, some of them have enough energy to overcome the energy barrier and drift to the p-type region. This number of electrons is given by the temperature and the statistical Fermi Dirac distribution, equation 5.2. The number of electrons with enough energy to overcome the barrier is proportional to the integration of the Fermi function from the barrier energy (∆µ) to innite energy, which will lead to equation 5.3.

f () = 1

e(−EF)/kBT _{+ 1} ≈ lim:(−EF)/kB T 1

e(EF−)/kBT (5.2)

With EF the Fermi energy level.

Ie0n→p= C

ˆ ∞ Ep

e(EF,−)/kBT_d (5.3)

On the p-type region electrons in the conduction band can drift to the n-type region and cross the junction, but these are the minority in the p-type region. This value is calculated in section 4.2, given by equation 4.39. Since there is no net current, both currents are the same, and called Ie0.

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Figure 5.1: Schematic view of the energy levels in a p-n junction. The lower level is the end of the valence band, dened as 0. EF is the Fermi level. and the upper level is the beginning of

the conductance band divined as EG, the band gap energy. The potential barrier is ∆µ. Left

is in the dark, right under illumination at open circuit.

When a bias voltage is applied the energy barrier is changed with that voltage (eV ). A decrease of the potential barrier will increase the number of electrons in the conduction band in the n-type region with enough energy to drift to the p-type region. In the p-type region the number of electrons in the conduction band is not changed, and therefore the current from p-type region to the n-type region is not changed. The change in current from the n-type region to the p-type region given in equation 5.3 will become:

Ie,n→p(eV ) = C ˆ ∞ Ep−eV e(µ−)/kBT_{d = C} ˆ ∞ Ep e(µ+eV −)/kBT_d = eeV /kBT C ˆ ∞ Ep e(µ−)/kBT_{d = I} e0eeV /kBT (5.4)

Ie(eV ) = Ie,n→p− Ie,p→n= Ie0eeV /kBT − Ie0= Ie0

eeV /kBT _{− 1} (5.5)

This principle holds also true for the holes, but in opposite direction.

The resulting current out of a cell, is the short circuit current, minus the reverse current induced by the voltage dierence within the cell. The current is thus a super position of the dark current with the short circuit current:

I (eV ) = Isc− I0

eeV /kBT _{− 1} (5.6)

This curve is displayed in gure 5.2. As the name open circuit voltage already implies, at this voltage the contact are not connected, and no current will ow. By using equation 5.6 and setting the current to zero, the voltage with open circuit can be calculated.

Voc= kBT q ln Isc I0 + 1 (5.7) We can see in this equation that the voltage is decreasing for higher I0.This I0 or per square

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Figure 5.2: Ideal I-V curve of a solar cell at which Isc is the current at short circuit and Voc is

the voltage at open circuit. The blue line indicates the power output as function of the voltage. [6]

5.4 Fill factor and Ohmic losses

The power harvested from the cell is the product of the current and voltage. This is plotted as a blue line in gure 5.2. For a certain voltage the power output is maximum, this point is called maximum power point (mpp). The power at this point, divided by the open circuit voltage times the short circuit current is called ll factor, short FF.

This value is gives insight on the shape of the I-V curve. This shape is aected by the series and shunt resistance. The electric schedule is shown schematically in gure 5.3 A. The eect of these resistances can be calculated using Ohms law. The voltage loss over a resistance in series is the product of the current and resistance. The current loss over a shunt resistance is the voltage divided by the resistance. This changes equation 5.6 in to:

I (eV ) = Isc− I0

e(eV −I(eV )Rseries)/kBT _{− 1}₋ eV

Rshunt (5.8)

Ideally there is no series resistance and no current runs through the shunt resistance, this is plotted in blue curve in gure 5.3 B, C and D. Increasing series resistance will lead to a drop in voltage, indicated by the red curve in gure 5.3 B, where a decrease in shunt resistance decreases the current, indicated by the red curve in gure 5.3 C. The eect of both resistances is shown in gure 5.3 D, which will reduce the power output at mpp, and thus reduce the ll factor.

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A B

C D

Figure 5.3: Resistance in solar cell. Figure A is schematic view of the electric circuit. In the other graphs the blue line is for a ideal I-V curve, the red line is adjusted by resistances. In graph B is the I-V curve of a circuit with a series resistance of 2 Ωcm2_{. Where graph C is the}

I-V curve with a shunt resistance of 145 Ωcm2_{. Together these resistances will result in a I-V}

curve shown in graph D. The ration of the light and dark gray surfaces is the ll factor. Values in this example are worse than in a standard solar cell to show the eect more clearly. In a standard solar cell the series resistance is lower and shunt resistance is higher.

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Chapter 6

Simulation

To design a solar cell with optimum performance, computer simulation is used. The dimensions and some other parameters of the cell are varied and simulated by Quokka. This program sim-ulates the performance of the cell for selected input parameters. The obtained cell parameters include the short circuit current, open circuit voltage, ll factor and the eciency.

Multiple Quokka simulations with a wide spread quasi random input parameters within certain boundaries were done. By tting a function of the performance as function of these input parameters the trend of these performances can be shown. Hereby making it possible to get the performance for a certain case, nd optima, help making design choices and see clear dierence between an FSF and FFE solar cell.

The input parameters used in the simulations are based on values obtained by measured cells, made at ECN. Hereby input parameters for the simulations could also be used for cells experimentally made. Thereby a comparison between simulations and experimental work could be done.

6.1 Simulation program Quokka

The program used for the cell simulations in this thesis is Quokka. This program is developed by Andreas Fell and is available at www.pvlighthouse.com.au. Quokka numerically solves the charge carrier transport in one, two or three dimensions in quasi-neutral state. The boundary conditions contain all the properties near the surface of the cell, and those properties are attributed to a layer with thickness zero[14, 3]. The rest of the cell is considered as bulk.

6.1.1 Dierential equations

In the following functions the subscript m can be replaced with n for electrons and p for holes. Because electrical forces are opposite for the electrons and holes we dene sp and snas 1 and -1

respectively. Quokka uses the Poisson equations of the quasi-Fermi potentials for the electrons and the holes separately. By using the unit volts for the energy, the Poisson equation will result in:

~

∇σm∇ϕ~ F m

= smq (g − r) (6.1)

Where σ is the conductivity, ϕF the quasi-Fermi potentials, g the generation of electron

hole pairs and r the recombination of those electron hole pairs. Since using a quasi-neutrality condition the quasi-Fermi potentials can be neglected, and only two dierential equations are needed. The only thing left to do is state the boundary conditions and dene a mesh to solve the dierential equations.

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Figure 6.1: Example of an unit cell with a front oating emitter (left) and front surface eld (right), with one nger contact on the emitter and one nger contacts on the BSF. A symme-try axis; B passivated rear; C passivated emitter; D passivated BSF; E contacted emitter; F contacted BSF

By solving this equation with all the boundary condition, discussed in the next subsection, all the parameters can be obtained with for example the continuity relation:

~

∇~jQm= smq (g − r) (6.2)

6.1.2 Boundary conditions

The boundary conditions are schematically shown in gure 6.1. For the symmetry planes (A), there is no current in the direction normal to the surface. At the not conducting not contacted surface, all the current that goes in, will recombine. This is the surface just between the emitter and the BSF at the rear (B). This boundary should not have a width, since the emitter and the BSF merge in to each other. However, because the simulations sometimes crash when this width is set to zero, the width of this not conducting not contacted surface is set to 1 µm in the simulation. The dierence in outcome of the simulations is minimal.

The diused surface (C and D) have lateral conductance for their majority carriers and surface recombination. The p-type surface (C and E) repels electrons, and collects holes. The n-type surface (D and F) repels the holes and collects the electrons. The contact areas (E and F) have a contact resistance and a corresponding current and voltage.

6.2 Sampling input parameters and its result t

A 2D simulation with Quokka of a unit cell is done within a minute with an Intel i5 processor. Therefore, multiple simulation can be done with varying parameters in short time. Hereby, the behavior of the properties of the cell performance with respect to these parameters can be viewed. Because the cell performance is assumed to change gradually with respect to a small change in input parameters, a function will be tted over the data points. For this t, no physical formula will be used. For most parameters a physical formula would not exist, or it would be too complex. Instead the data is tted with a polynomial function of the fourth power, and it will be made sure this t is valid within the boundaries of the simulations. This will be discussed in section 6.2.2.

Furthermore, it is possible to integrate and normalize one variable. For instance, the bulk resistivity of a silicon ingot is varying throughout the ingot. By integrating the properties of the cells to the bulk resistance within the boundaries of the ingot, the average properties of the wafers from the ingot can be calculated. Hereby, a design can be calculated with its optimum for the complete ingot.

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Figure 6.2: Examples of samplings where two independent parameters are plotted. The bound-aries of the plot give the boundbound-aries of the sampling. Within this boundbound-aries the sampling is random, or quasi random. Figure A is of a pure random sampling, gure B is a LHS design whereas gure C is Orthogonal sampling.

6.2.1 Quasi random sampling

The minimal amount of data points to t a polynomial function up to a certain power p and with d variables, is given by equation 6.3.

n = (d + p)!

d! · p! (6.3)

However, whenever you t a function over a regular grid, the minimal amount of data points needed in one dimension is the polynomial degree plus one. For more dimensions this will lead to a minimal required amount of data point given by function 6.4, which is always more than for given by function 6.3. Therefor a random sampling will require less data points.

n = (p + 1)d (6.4)

However, in a purely random input parameter set, there is no guarantee whether within the boundaries there are big regions without samples and whether a parameter is present throughout its own range. An example of a random sampling is shown in gure 6.2 A.

To make sure these two drawbacks of pure random sampling do not occur, a quasi random sampling method, called Latin hypercube sampling (LHS), could be used. In this method the complete sampling space is divided in a square grid, in which every row and column only have one sample point. Hence, in this method every input parameter is ensured to be present over the whole range. To make sure the input parameters are evenly distributed over the complete space, multiple sampling sets are created and the sampling set at which the distance between the points is largest is used[7]. An example of this LHS design is shown in gure 6.2 B. However, with this method, the properties of the cell, at the boundary of the parameter range, is sometimes established via extrapolation of the t (as would be for the right top corner of gure 6.2 B). Therefore, extra sampling points are added just at the boundary of the input parameter range. A third random sampling method is Orthogonal sampling. With this method, just like the LHS, the space is divided in a square grid, in which every row and column only have one sample point. However, the space is also divided in equal squares, with the requirement that only one sample is in it, as shown in gure 6.2 C. In this way it is ensured that there is no large area without a sample. Instead of taking extra sample point at the boundary of the input parameters range, as with the LHS design, to ensure the reliability, the sampling range is 10% outside the range that is to be examined.

An example of Orthogonal sampling is shown in gure 6.3 A, the color gives the eciency that is simulated with Quokka. The corresponding fourth power polynomial tted function is plotted in 6.3B as a contour plot.

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A B

Figure 6.3: Example of a simulation run, in gure A the eciency is plotted as a result of 100 simulations. In gure B a to the fourth power polynomial function is tted over these point. The range of the simulation is 10% outside that of the plot, the sampling is done with Orthogonal sampling.

6.2.2 Fit reliability

To make sure the t represent the data, the mean square error is calculated to indicate how good the t is. This value is the average of the square of the dierence between the simulated points with respect to the values from t, this is shown in equation 6.5.

sd = s Pn i=1(¯yi− yi) 2 n − 1 (6.5)

With n the number of data points, ¯yi the value of a point calculated with the t and yi the

corresponding simulated value. When the minimal number of data points are used for the t, the t will, per denition, have no error. Therefore, for every run, more than the minimal required samples are taken, at least two times as much.

In gure 6.4 is a plot of the inconsistency of the t with simulated data. The t is made with 100 data points. The plotted points are the dierence between 25 simulations with the corresponding calculated value by the t. These 25 points are not included in the data used for the t. In gure 6.5 is the relative error plotted in a histogram. Here you can see clearly that the error decreases with a t of increasing order. This is explicitly shown in gure 6.6 for polynomial ts of dierent order. The t is more consistent with the point for higher order, but there is a plateau at fourth and fth order. In this thesis a fourth polynomial t is used, having a mean error of only 0.01%.

6.3 Optimizing Mercury cell

The Mercury cell is a novel IBC design with a front oating emitter. Therefore, the optimal dimensions of the cell are yet to be determined. By means of simulations the optimal dimensions are obtained. However, not only the cell eciency is important, the design should also be easy to manufacture. Hence, the size of the emitter and BSF can not be too small. In this stage an extensive costbenet analysis is not made, but two cases are considered. First, highest eciency with smallest design feature that can be made industrially. Second, easy to manufacture with equal widths of rear emitter and BSF.

Optimization and benchmarking of the Mercury IBC cell