2

Photovoltaic Fundamentals

2.1 Introduction 2-1

2.2 Market Drivers 2-2

2.3 Optical Absorption 2-3

Introduction • Semiconductor Materials • Generation of EHP by Photon Absorption

2.4 Extrinsic Semiconductors and the pn Junction 2-6

Extrinsic Semiconductors • pn Junction

2.5 Maximizing Cell Performance 2-10

Externally Biased pn Junction • Parameter Optimization • Minimizing Cell Resistance Losses

2.6 Traditional PV Cells 2-20

Introduction • Crystalline Silicon Cells • Amorphous Silicon Cells • Copper Indium Gallium Diselenide Cells • Cadmium Telluride Cells • Gallium Arsenide Cells

2.7 Emerging Technologies 2-24

New Developments in Silicon Technology • CIS-Family-Based Absorbers • Other III–V and II–VI Emerging Technologies • Other Technologies

2.8 PV Electronics and Systems 2-26

Introduction • PV System Electronic Components

2.9 Conclusions 2-28

References2-28

Roger A. Messenger

Florida Atlantic University

2.1 Introduction

Becquerel [1] first discovered that sunlight can be converted directly into electricity in 1839, when he observed the photogalvanic effect. After more than a century of theoretical work, the first solar cell was not developed until 1954, by Chapin, Fuller, and Pearson, when sufficiently pure semiconductor material had become available. It had an efficiency of 6%. Only 4 years later, the first solar cells were used on the Vanguard I orbiting satellite.

As is often the case, extraterrestrial use of a technology has led to the terrestrial use of the technology. For extraterrestrial cells, the power:weight ratio was the determining factor for their selection, since weight was the expensive part of the liftoff equation. Reliability, of course, was also of paramount importance. Hence, for extraterrestrial power generation, the cost of the photovoltaic (PV) cells was secondary, provided that they met the primary objectives.

A new objective for PV cells emerged following the organization of the petroleum exporting countries (OPEC) petroleum embargo of the early 1970s, which highlighted the need to seek out alternative energy sources. In addition, the observation that fossil fuel sources pollute the atmosphere, water and soil with a number of pollutants such as SOx, NOx, particulates, and CO2 suggested that alternative sources should involve minimal air or water pollution. And, of course, perhaps the most desirable aspect of any new source would be an unlimited supply of fuel, as opposed to the observed dwindling supplies of domestic petroleum.

Solar energy became the obvious choice for alternative energy, either in the form of electricity via the PV effect or via solar thermal heating of water or other substances. But it would be necessary to significantly reduce the cost of capturing the solar resource before it could be used as a replacement for the relatively inexpensive fossil sources.

Since the events of the mid-1970s, the cost per watt of PV cells has declined steadily. Figure 2.1 shows an average decline of approximately 7%/year between 1980 and 2010, which represents a 50% drop every 10 years. In fact, between 2005 and 2010, the price dropped by 50% in a 5 year period. Installations have grown from tens of kilowatts to thousands of megawatts per year, and continue to grow at a rapid rate. Recent rapid growth and accompanying declines in prices can be accounted for by a number of aggressive incentive programs around the world, such as in Germany, Spain, China, and California. For example, Germany now produces more than 20% of its electrical energy by renewable sources. Costa Rica will soon produce nearly 100% of its electrical energy needs from renewable sources, but in this case, hydroelectric and wind will be the major contributors to the renewable portfolio.

2.2 Market Drivers

Price, efficiency, performance, warranties, incentives, interest rates, money availability, space availability, and sometimes aesthetics have been the primary market drivers for current PV technologies. In fact, the decision to install generally involves consideration of all these factors.

Certainly price is important, but obviously if an inexpensive module will cost half as much as another module, but will last only 25% as long, the more expensive module will have a more attractive lifecycle cost. In many instances, where space is limited, the system owner will want to install as much power as possible in the available space in order to generate as much energy as possible. Where space is a consideration, then the metric for PV system selection is based upon kWh/m2/year/$ for the energy produced by the system. And, in fact, generally the monetary consideration is only secondary to the energy production consideration.

Probably the most important driver among the remaining factors is incentive programs. It has been clearly demonstrated in Germany, Spain, California, New Jersey, Ontario, and a number of other countries, states, and cities that an attractive incentive program will encourage people to install PV systems on their homes and businesses. Incentive programs may take the form of onetime rebates or tax credits or they may take the form of a guaranteed premium price paid over a guaranteed time period for the energy produced by the renewable system. In the final analysis, the owner of a system has a greater incentive as the payback time of the system decreases. Generally, if a system will pay for itself with the value of the energy it produces within a 6 year period, it is a very attractive investment for a homeowner and an acceptable investment for many businesses. The trade-off must be the impact on utility ratepayers, since, generally the funding for the incentive comes either directly from the utility or indirectly via government loans or grants, which, ultimately, are paid for by the taxpayer, who is most likely a ratepayer. Incentive programs must be carefully crafted to ensure acceptance by those who wish to install systems, but not to impose an excessive cost burden on the general population. Just as the energy produced must be sustainable, the incentive program must also be sustainable.

2.3 Optical Absorption

2.3.1 Introduction

When light shines on a material, it is reflected, transmitted, or absorbed. Absorption of light is simply the conversion of the energy contained in the incident photon to some other form of energy, typically heat. Some materials, however, happen to have just the right properties needed to convert the energy in the incident photons to electrical energy.

2.3.2 Semiconductor Materials

Semiconductor materials are characterized as being perfect insulators at absolute zero temperature, with charge carriers being made available for conduction as the temperature of the material is increased. This phenomenon can be explained on the basis of quantum theory, by noting that semiconductor materials have an energy band gap between the valence band and the conduction band. The valence band represents the allowable energies of valence electrons that are bound to host atoms. The conduction band represents the allowable energies of electrons that have received energy from some mechanism and are now no longer bound to specific host atoms.

As temperature of a semiconductor sample is increased, sufficient energy is imparted to a small fraction of the electrons in the valence band for them to move to the conduction band. In effect, these electrons are leaving covalent bonds in the semiconductor host material. When an electron leaves the valence band, an opening is left which may now be occupied by another electron, provided that the other electron moves to the opening. If this happens, of course, the electron that moves in the valence band to the opening, leaves behind an opening in the location from which it moved. If one engages in an elegant quantum mechanical explanation of this phenomenon, it must be concluded that the electron moving in the valence band must have either a negative effective mass and a negative charge or, alternatively, a positive effective mass and a positive charge. The latter has been the popular description, and, as a result, the electron motion in the valence band is called hole motion, where “holes” is the name chosen for the positive charges, since they relate to the moving holes that the electrons have left in the valence band.

What is important to note about these conduction electrons and valence holes is that they have occurred in pairs. Hence, when an electron is moved from the valence band to the conduction band in a semiconductor by whatever means, it constitutes the creation of an electron-hole pair (EHP). Both charge carriers are then free to become a part of the conduction process in the material.

2.3.3 Generation of EHP by Photon Absorption

The energy in a photon is given by the familiar equation,

$\text{E=h}\nu \text{=}\frac{\text{hc}}{\lambda }(\text{J}),$ (2.1a)

where

h is Planck’s constant (h = 6.63 × 10−34 J s)

c is the speed of light (c = 2.998 × 108 m/s)

ν is the frequency of the photon in Hertz

λ is the wavelength of the photon in meters

Since energies at the atomic level are typically expressed in electron volts (1 eV = 1.6 × 10−19 J) and wavelengths are typically expressed in micrometers, it is possible to express hc in appropriate units, so that if λ is expressed in μm, then E will be expressed in eV. The conversion yields

$\text{E=}\frac{\text{1}\text{.24}}{\lambda }(\text{eV}).$ (2.1b)

The energy in a photon must exceed the semiconductor bandgap energy, Eg, to be absorbed. Photons with energies at and just above Eg are most readily absorbed because they most closely match bandgap energy and momentum considerations. If a photon has energy greater than the bandgap, it still can produce only a single EHP. The remainder of the photon energy is lost to the cell as heat. It is thus desirable that the semiconductor used for photoabsorption have a bandgap energy such that a maximum percentage of the solar spectrum will be efficiently absorbed.

Now, note that the solar spectrum peaks at λ ≈ 0.5 μm. Equation 2.1b shows that a bandgap energy of approximately 2.5 eV corresponds to the peak in the solar spectrum. In fact, since the peak of the solar spectrum is relatively broad, bandgap energies down to 1.0 eV can still be relatively efficient absorbers, and in certain special cell configurations, even smaller bandgap materials are appropriate.

The nature of the bandgap also affects the efficiency of absorption in a material. A more complete representation of semiconductor bandgaps must show the relationship between bandgap energy as well as bandgap momentum. As electrons make transitions between conduction band and valence band, both energy and momentum transfer normally take place, and both must be properly balanced in accordance with conservation of energy and conservation of momentum laws.

Some semiconducting materials are classified as direct bandgap materials, while others are classified as indirect bandgap materials. Figure 2.2 [6] shows the bandgap diagrams for two materials considering momentum as well as energy. Note that for silicon, the bottom of the conduction band is displaced in the momentum direction from the peak of the valence band. This is an indirect bandgap, while the GaAs diagram shows a direct bandgap, where the bottom of the conduction band is aligned with the top of the valence band.

What these diagrams show is that the allowed energies of a particle in the valence band or the conduction band depend on the particle momentum in these bands. An electron transition from a point in the valence band to a point in the conduction band must involve conservation of momentum as well as energy. For example, in Si, even though the separation of the bottom of the conduction band and the top of the valence band is 1.1 eV, it is difficult for a 1.1 eV photon to excite a valence electron to the conduction band because the transition needs to be accompanied with sufficient momentum to cause displacement along the momentum axis, and photons carry little momentum. The valence electron must thus simultaneously gain momentum from another source as it absorbs energy from the incident photon. Since such simultaneous events are unlikely, absorption of photons at the Si bandgap energy is several orders of magnitude less likely than absorption of higher energy photons.

Since photons have so little momentum, it turns out that the direct bandgap materials, such as gallium arsenide (GaAs), cadmium telluride (CdTe), copper indium diselenide (CIS), and amorphous silicon (a-Si:H) absorb photons with energy near the material bandgap energy much more readily than do the indirect materials, such as crystalline silicon. As a result, the direct bandgap absorbing material can be several orders of magnitude thinner than indirect bandgap materials and still absorb a significant part of the incident radiation.

The absorption process is similar to many other physical processes, in that the change in intensity with position is proportional to the initial intensity. As an equation, this becomes

$\frac{\text{dI}}{\text{dx}}\text{=}-\alpha \text{I,}$ (2.2)

with the solution

$\text{I=}{\text{I}}_{\text{o}}{\text{e}}^{-\alpha \text{x}}\text{,}$ (2.3)

where

I is the intensity of the light at a depth x in the material

Io is the intensity at the surface

α is the absorption constant

The absorption constant depends on the material and on the wavelength. At energies above the bandgap, the absorption constant increases relatively slowly for indirect bandgap semiconductors and increases relatively quickly for direct bandgap materials. Equation 2.3 shows that the thickness of material needed for significant absorption needs to be several times the reciprocal of the absorption constant. This is important information for the designer of a PV cell, since the cell must be sufficiently thick to absorb the incident light.

In any case, when the photon is absorbed, it generates an EHP. The question, then, is what happens to the EHP?

2.4 Extrinsic Semiconductors and the pn Junction

2.4.1 Extrinsic Semiconductors

At T = 0 K, intrinsic semiconductors, i.e., semiconductor materials with no impurities, have all covalent bonds completed with no leftover electrons or holes. If certain impurities are introduced into intrinsic semiconductors, there can be leftover electrons or holes at T = 0 K. For example, consider silicon, which is a group IV element, which covalently bonds with four nearest neighbor atoms to complete the outer electron shells of all the atoms. At T = 0 K, all the covalently bonded electrons are in place, whereas at room temperature, about one in 1012 of these covalent bonds will break, forming an EHP, resulting in minimal charge carriers for current flow.

If, however, phosphorous, a group V element, is introduced into the silicon in small quantities, such as one part in 106, four of the valence electrons of the phosphorous atoms will covalently bond to the neighboring silicon atoms, while the fifth valence electron will have no electrons with which to covalently bond. This fifth electron remains weakly coupled to the phosphorous atom, readily dislodged by temperature, since it requires only 0.04 eV to excite the electron from the atom to the conduction band [6]. At room temperature, sufficient thermal energy is available to dislodge essentially all of these extra electrons from the phosphorous impurities. These electrons thus enter the conduction band under thermal equilibrium conditions, and the concentration of electrons in the conduction band becomes nearly equal to the concentration of phosphorous atoms, since the impurity concentration is normally on the order of 108 times larger than the intrinsic carrier concentration.

Since the phosphorous atoms donate electrons to the material, they are called donor atoms and are represented by the concentration, ND. Note that the phosphorous, or other group V impurities, do not add holes to the material. They only add electrons. They are thus designated as n-type impurities.

On the other hand, if group III atoms such as boron are added to the intrinsic silicon, they have only three valence electrons to covalently bond with nearest silicon neighbors. The missing covalent bond appears the same as a hole, which can be released to the material by applying a small amount of thermal energy. Again, at room temperature, nearly all of the available holes from the group III impurity are donated to the conduction process in the host material. Since the concentration of impurities will normally be much larger than the intrinsic carrier concentration, the concentration of holes in the material will be approximately equal to the concentration of p-type impurities.

Historically, group III impurities in silicon have been viewed as electron acceptors, which, in effect, donate holes to the material. Rather than being termed hole donors, however, they have been called acceptors. Thus, acceptor impurities donate holes, but no electrons, to the material and the resulting hole density is approximately equal to the density of acceptors, which is represented as NA.

An interesting property of free electrons and holes is that they like each other. In fact, when they are close to each other, they have a strong tendency to recombine. This observation can be expressed as [6]

${\text{n}}_{\text{o}}{\text{p}}_{\text{o}}\text{=}{\text{n}}_{\text{i}}^{\text{2}}\text{(T)=K}{\text{T}}^{\text{3}}{\text{e}}^{-{\text{E}}_{\text{g}}\text{/}\text{kT}}\text{,}$ (2.4)

where

no represents the thermal equilibrium concentration of free electrons at a point in the semiconductor crystal

po represents the thermal equilibrium concentration of holes

ni represents the concentration of intrinsic charge carriers

T is the temperature in K

K is a constant that depends upon the material

Note that in intrinsic material, no ≅ po = ni since, by definition, the intrinsic material has no impurities that affect the electrical properties of the material.

In Si, for example, which has approximately 1022 covalent bonds/cm3, approximately 1 in 1012 bonds creates an EHP at room temperature. This means that at room temperature for Si, ni ≅ 1010/cm3.

Equation 2.4 shows that adding a mere one part per million of a donor or acceptor impurity can increase the free charge carrier concentration of the material by a factor of 106 for silicon. Equation 2.4 also shows that in thermal equilibrium, if either an n-type or a p-type impurity is added to the host material, that the concentration of the other charge carrier will decrease dramatically, since it is still necessary to satisfy (2.4) under thermal equilibrium conditions.

In extrinsic semiconductors, the charge carrier with the highest concentration is called the majority carrier and the charge carrier with the lowest concentration is called the minority carrier. Hence, electrons are majority carriers in n-type material, and holes are minority carriers in n-type material. The opposite is true for p-type material.

If both n-type and p-type impurities are added to a material, then whichever impurity has the higher concentration, will become the dominant impurity. However, it is then necessary to acknowledge a net impurity concentration that is given by the difference between the donor and acceptor concentrations. If, for example, ND > NA, then the net impurity concentration is defined as Nd = ND − NA. Similarly, if NA > ND, then Na = NA − ND.

2.4.2 pn Junction

2.4.2.1 Junction Formation and Built-In Potential

Although n-type and p-type materials are interesting and useful, the real fun starts when a junction is formed between n-type and p-type materials. The pn junction is treated in gory detail in most semiconductor device theory textbooks. Here, the need is to establish the foundation for the establishment of an electric field across a pn junction and to note the effect of this electric field on photo-generated EHPs.

Figure 2.3 shows a pn junction formed by placing p-type impurities on one side and n-type impurities on the other side. There are many ways to accomplish this structure. The most common is the diffused junction.

When a junction is formed, the first thing to happen is that the conduction electrons on the n-side of the junction notice the scarcity of conduction electrons on the p-side, and the valence holes on the p-side notice the scarcity of valence holes on the n-side. Since both types of charge carrier are undergoing random thermal motion, they begin to diffuse to the opposite side of the junction in search of the wide open spaces. The result is diffusion of electrons and holes across the junction, as indicated in Figure 2.3.

When an electron leaves the n-side for the p-side, however, it leaves behind a positive donor ion on the n-side, right at the junction. Similarly, when a hole leaves the p-side for the n-side, it leaves a negative acceptor ion on the p-side. If large numbers of holes and electrons travel across the junction, large numbers of fixed positive and negative ions are left at the junction boundaries. These fixed ions, as a result of Gauss’ law, create an electric field that originates on the positive ions and terminates on the negative ions. Hence, the number of positive ions on the n-side of the junction must be equal to the number of negative ions on the p-side of the junction.

The electric field across the junction, of course, gives rise to a drift current in the direction of the electric field. This means that holes will travel in the direction of the electric field and electrons will travel opposite the direction of the field, as shown in Figure 2.3. Notice that for both the electrons and for the holes, the drift current component is opposite the diffusion current component. At this point, one can invoke Kirchhoff’s current law to establish that the drift and diffusion components for each charge carrier must be equal and opposite, since there is no net current flow through the junction region. This phenomenon is known as the law of detailed balance.

By setting the sum of the electron diffusion current and the electron drift current equal to zero, it is possible to solve for the potential difference across the junction in terms of the impurity concentrations on either side of the junction. Proceeding with this operation yields

${\text{V}}_{\text{j}}\text{=}\frac{\text{kT}}{\text{q}}\text{ln}\frac{{\text{n}}_{\text{no}}}{{\text{n}}_{\text{po}}}\text{.}$ (2.5a)

It is now possible to express the built-in potential in terms of the impurity concentrations on either side of the junction by recognizing that nno ≅ ND and npo ≅ (ni)2/NA. Substituting these values into (2.5a) yields finally

${\text{V}}_{\text{j}}\text{=}\frac{\text{kT}}{\text{q}}\text{ln}\frac{{\text{N}}_{\text{A}}{\text{N}}_{\text{D}}}{{\text{n}}_{\text{i}}^{\text{2}}}\text{.}$ (2.5b)

At this point, a word about the region containing the donor ions and acceptor ions is in order. Note first that outside this region, electron and hole concentrations remain at their thermal equilibrium values. Within the region, however, the concentration of electrons must change from the high value on the n-side to the low value on the p-side. Similarly, the hole concentration must change from the high value on the p-side to the low value on the n-side. Considering that the high values are really high, i.e., on the order of 1018/cm3, while the low values are really low, i.e., on the order of 102/cm3, this means that within a short distance of the beginning of the ionized region, the concentration must drop significantly below the equilibrium value. Because the concentrations of charge carriers in the ionized region are so low, this region is often termed the depletion region, in recognition of the depletion of mobile charge carriers in the region. Furthermore, because of the charge due to the ions in this region, the depletion region is also often referred to as the space charge layer. For the balance of this chapter, this region will simply be referred to as the junction.

The next step in the development of the behavior of the pn junction in the presence of sunlight is to let the sun shine in and see what happens.

2.4.2.2 Illuminated pn Junction

Equation 2.3 governs the absorption of photons at or near a pn junction. Noting that an absorbed photon releases an EHP, it is now possible to explore what happens after the generation of the EHP. Those EHPs generated within the pn junction will be considered first, followed by the EHPs generated outside, but near the junction, and then by EHPs generated further from the junction boundary.

If an EHP is generated within the junction, as shown in Figure 2.4 (points B and C), both charge carriers will be acted upon by the built-in electric field. Since the field is directed from the n-side of the junction to the p-side of the junction, the field will cause the electrons to be swept quickly toward the n-side and the holes to be swept quickly toward the p-side. Once out of the junction region, the optically generated carriers become a part of the majority carriers of the respective regions, with the result that excess concentrations of majority carriers appear at the edges of the junction. These excess majority carriers then diffuse away from the junction toward the external contacts, since the concentration of majority carriers has been enhanced only near the junction.

The addition of excess majority charge carriers to each side of the junction results in either a voltage between the external terminals of the material or a flow of current in the external circuit or both. If an external wire is connected between the n-side of the material and the p-side of the material, a current, I1, will flow in the wire from the p-side to the n-side. This current will be proportional to the number of EHPs generated in the junction region, which, in turn, will be proportional to the intensity of the incident light (irradiation).

If an EHP is generated outside the junction region, but close to the junction (with “close” yet to be defined, but shown as point A in Figure 2.4), it is possible that, due to random thermal motion, either the electron, the hole, or both will end up moving into the junction region. Suppose that an EHP is generated in the n-region close to the junction. Then suppose that the hole, which is the minority carrier in the n-region, manages to reach the junction before it recombines. If it can do this, it will be swept across the junction to the p-side and the net effect will be the same as if the EHP had been generated within the junction, since the electron is already on the n-side as a majority carrier. Similarly, if an EHP is generated within the p-region, but close to the junction, and if the minority carrier electron reaches the junction before recombining, it will be swept across to the n-side where it is a majority carrier. So what is meant by close?

Clearly, the minority carriers of the optically generated EHPs outside the junction region must not recombine before they reach the junction. If they do, then, effectively, both carriers are lost from the conduction process, as in point D in Figure 2.4. Since the majority carrier is already on the correct side of the junction, the minority carrier must therefore reach the junction in less than a minority carrier lifetime, τn or τp.

To convert these times into distances, it is necessary to note that the carriers travel by diffusion once they are created. Since only the thermal velocity has been associated with diffusion, but since the thermal velocity is random in direction, it is necessary to introduce the concept of minority carrier diffusion length, which represents the distance, on the average, which a minority carrier will travel before it recombines. The diffusion length can be shown to be related to the minority carrier lifetime and diffusion constant by the formula [7]

${\text{L}}_{\text{m}}\text{=}\sqrt{{\text{D}}_{\text{m}}{\tau }_{\text{m}}}\text{,}$ (2.6)

where m has been introduced to represent n for electrons or p for holes. It can also be shown that on the average, if an EHP is generated within a minority carrier diffusion length of the junction, the associated minority carrier will reach the junction. In reality, some minority carriers generated closer than a diffusion length will recombine before reaching the junction, while some minority carriers generated farther than a diffusion length from the junction will reach the junction before recombining.

Hence, to maximize photocurrent, it is desirable to maximize the number of photons that will be absorbed either within the junction or within a minority carrier diffusion length of the junction. The minority carriers of the EHPs generated outside this region have a higher probability of recombining before they have a chance to diffuse to the junction. If a minority carrier from an optically generated EHP recombines before it crosses the junction and becomes a majority carrier, it, along with the opposite carrier with which it recombines, is no longer available for conduction. The engineering design challenge then lies in maximizing α as well as maximizing the junction width and minority carrier diffusion lengths. Additional information about the optimization of cell performance emerges when the performance of the cell under external bias is explored.

2.5 Maximizing Cell Performance

2.5.1 Externally Biased pn Junction

Figure 2.5 shows a pn junction connected to an external battery with the internally generated electric field direction included. If (2.5a) is recalled, taking into account that the externally applied voltage, V, with the exception of any voltage drop in the neutral regions of the material, will appear as opposing the junction voltage, Equation 2.5a becomes

${\text{V}}_{\text{j}}-\text{V=}\frac{\text{kT}}{\text{q}}\text{ln}\frac{{\text{n}}_{\text{n}}}{{\text{n}}_{\text{p}}}\text{,}$ (2.7)

where now nn and np are the total concentrations of electrons on the n-side of the junction and on the p-side of the junction, no longer in thermal equilibrium. Thermal equilibrium will exist only when the externally applied voltage is zero, meaning that $np=n_i^2$ , is only true when V = 0. However, under conditions known as low injection levels, it will still be the case that the concentration of electrons on the n-side will remain close to the thermal equilibrium concentration. For this condition, (2.7) becomes

${\text{V}}_{\text{j}}-\text{V=}\frac{\text{kT}}{\text{q}}\text{ln}\frac{{\text{N}}_{\text{d}}}{{\text{n}}_{\text{p}}}\text{.}$ (2.8)

The next step is solving (2.8) for np and then going through a detailed analysis of the spatial dependence of np between the junction edge and the contact on the p-side of the junction. Then, a similar analysis is done for the spatial dependence of pn on the p-side of the junction. Finally, it is noted that current flow in the neutral regions of the device is primarily due to diffusion of the higher concentrations of excess charges near the junction toward the contact areas where the minority carrier concentrations are essentially zero. Solving the diffusion equations for np and pn in the neutral regions leads to [8]

$\text{I=}{\text{I}}_{\text{n}}\text{+}{\text{I}}_{\text{p}}\text{=qA}\left(\frac{{\text{D}}_{\text{n}}{\text{n}}_{\text{po}}}{{\text{L}}_{\text{n}}}\text{ctnh}\frac{{\text{w}}_{\text{p}}}{{\text{L}}_{\text{n}}}\text{+}\frac{{\text{D}}_{\text{p}}{\text{p}}_{\text{no}}}{{\text{L}}_{\text{p}}}\text{ctnh}\frac{{\text{w}}_{\text{n}}}{{\text{L}}_{\text{p}}}\right)\left({\text{e}}^{\text{qV}/\text{kT}}-\text{1}\right)\text{,}$ (2.9)

where

A is the cross-sectional area of the junction

Dn and Dp are the diffusion coefficients of the minority carriers in the neutral regions

Ln and Lp are the diffusion lengths of the minority carriers in the neutral regions

wn and wp are the physical widths of the neutral regions

npo and pno are the equilibrium concentrations of minority carriers in the neutral regions

Note that the current indicated in (2.9) flows in the direction opposite to the optically generated current described earlier. Letting qA (nasty expression) = Io and incorporating the photocurrent component into (2.9) finally yields the complete equation for the current in the PV cell to be

$\text{I=}{\text{I}}_{\text{l}}-{\text{I}}_{\text{o}}\left({\text{e}}^{\text{qV}/\text{kT}}-\text{1}\right)\text{,}$ (2.10)

which is the familiar diode equation with a photocurrent component. Note that the current of (2.10) is directed out of the positive terminal of the device, so that when the current and voltage are both positive, the device is delivering power to the external circuit. The detailed derivation of this expression can be found in Chapter 10 of [8].

The open circuit voltage and the short circuit current of the PV cell can be determined from (2.10). Setting I = 0, solving for V and noting that I1 >> Io under normal cell illumination conditions gives the cell open circuit voltage

${\text{V}}_{\text{OC}}\text{=}\frac{\text{kT}}{\text{q}}\text{ln}\frac{{\text{I}}_{\text{l}}\text{+}{\text{I}}_{\text{o}}}{{\text{I}}_{\text{o}}}\underset{¯}{\sim }\frac{\text{kT}}{\text{q}}\text{ln}\frac{{\text{I}}_{\text{l}}}{{\text{I}}_{\text{o}}}\text{.}$ (2.11)

Setting V = 0 and solving for I yields the simple result that ISC = Il. In other words, the cell short circuit current is the photocurrent of the cell.

Even with all the details incorporated in the development of (2.9), what has been left out is the fact that when high current flows in the cell, the ohmic resistance of the neutral regions creates an additional voltage drop beyond the drop across the junction. This series voltage drop in the neutral regions causes the I–V relationship of (2.9) to be modified as shown in Figure 2.6. In this figure, one can observe that for each cell illumination level, only one point on the curve represents maximum power production, Pm, as is shown in the figure. The fill factor of a cell is defined as FF = Pm/VOCISC.

2.5.2 Parameter Optimization

2.5.2.1 Introduction

Equation 2.10 indicates, albeit in a somewhat subtle manner, that to maximize the power output of a PV cell, it is desirable to maximize the open-circuit voltage, short-circuit current, and fill factor of a cell. Noting Figure 2.6, it should be evident that maximizing the open-circuit voltage and the short-circuit current will maximize the power output for an ideal cell characterized by (2.10). Real cells, of course, have some series resistance, so there will be power dissipated by this resistance, similar to the power loss in a conventional battery due to its internal resistance. In any case, recalling that the open-circuit voltage increases as the ratio of photo current to reverse saturation current increases, a desirable design criteria is to maximize this ratio, provided that it does not proportionally reduce the short-circuit current of the device.

Fortunately, this is not the case, since maximizing the short-circuit current requires maximizing the photocurrent. It is thus instructive to look closely at the parameters that determine both the reverse saturation current and the photocurrent. Techniques for lowering series resistance will then be discussed.

2.5.2.2 Minimizing the Reverse Saturation Current

Beginning with the reverse saturation current as expressed in (2.9), the first observation is that the equilibrium minority carrier concentrations at the edges of the pn junction are related to the intrinsic carrier concentration through (2.4). Hence,

${\text{p}}_{\text{no}}\text{=}\frac{{\text{n}}_{\text{i}}^{\text{2}}}{{\text{N}}_{\text{D}}}\text{and}{\text{n}}_{\text{po}}\text{=}\frac{{\text{n}}_{\text{i}}^{\text{2}}}{{\text{N}}_{\text{A}}}\text{.}$ (2.12)

Thus far, no analytic expression for the constant, K, in (2.4) has been developed. Such an expression can be obtained by considering Fermi levels, densities of states, and other quantities that are discussed in solid-state devices textbooks. Since the goal here is to determine how to minimize the reverse saturation current, and not to go into detail of quantum mechanical proofs, the result is noted here with the recommendation that the interested reader consult a good solid-state devices text for the development of the result. The result is

${\text{n}}_{\text{i}}^{\text{2}}\text{=4}{\left(\frac{\text{2}\pi \text{kT}}{{\text{h}}^{\text{2}}}\right)}^{\text{3}}{\left({\text{m}}_{\text{n}}^{\text{*}}{\text{m}}_{\text{p}}^{\text{*}}\right)}^{\text{3}/2}{\text{e}}^{-{\text{E}}_{\text{g}}/\text{kT}}\text{,}$ (2.13)

where

$m_n^{\text{*}}$ and $m_p^{\text{*}}$ are the electron effective mass and hole effective mass in the host material

Eg is the bandgap energy of the host material

These effective masses can be greater than or less than the rest mass of the electron, depending on the degree of curvature of the valence and conduction bands when plotted as energy versus momentum as in Figure 2.2. In fact, the effective mass can also depend on the band in which the carrier resides in a material. For more information on effective mass, the reader is encouraged to consult the references listed at the end of the chapter [6,7].

Now, using (2.6) with (2.12) and (2.13) in (2.9) the following final result for the reverse saturation current is obtained:

${\text{I}}_{\text{o}}\text{=}\left(\text{4qA}\left(\frac{\text{2}\pi \text{kT}}{{\text{h}}^{\text{2}}}\right){\left({\text{m}}_{\text{n}}^{\text{*}}{\text{m}}_{\text{p}}^{\text{*}}\right)}^{3/2}{\text{e}}^{-{\text{E}}_{\text{g}}\text{}\text{/}\text{kT}}\right)\times \left(\frac{\text{1}}{{\text{N}}_{\text{A}}}\sqrt{\frac{{\text{D}}_{\text{n}}}{{\tau }_{\text{n}}}}\text{ctnh}\frac{{\text{l}}_{\text{p}}}{\sqrt{{\text{D}}_{\text{n}}{\tau }_{\text{n}}}}\text{+}\frac{\text{1}}{{\text{N}}_{\text{D}}}\sqrt{\frac{{\text{D}}_{\text{p}}}{{\tau }_{\text{p}}}}\text{ctnh}\frac{{\text{l}}_{\text{n}}}{\sqrt{{\text{D}}_{\text{p}}{\tau }_{\text{p}}}}\right)\text{.}$ (2.14)

Since the design goal is to minimize Io while still maximizing the ratio Iℓ:Io, the next step is to express the photocurrent in some detail so the values of appropriate parameters can be considered in the design choices.

2.5.2.3 Optimizing Photocurrent

In Section 2.4.2.2, the photocurrent optimization process was discussed qualitatively. In this section the specific parameters that govern the absorption of light and the lifetime of the absorbed charge carriers will be discussed, and a formula for the photocurrent will be presented for comparison with the formula for reverse saturation current. In particular, minimizing reflection of the incident photons, maximizing the minority carrier diffusion lengths, maximizing the junction width, and minimizing surface recombination velocity will be discussed. The PV cell designer will then know exactly what to do to make the perfect cell.

2.5.2.3.1 Minimizing Reflection of Incident Photons

The interface between air and the semiconductor surface constitutes an impedance mismatch, since the electrical conductivities and the dielectric constants of air and a PV cell are different. As a result, part of the incident wave must be reflected in order to meet the boundary conditions imposed by the solution of the wave equation on the electric field, E, and the electric displacement, D.

Those readers who are experts at electromagnetic field theory will recognize that this problem is readily solved by the use of a quarter-wave matching coating on the PV cell. If the coating on the cell has a dielectric constant equal to the geometric mean of the dielectric constants of the cell and of air and if the coating is one-quarter wavelength thick, it will act as an impedance-matching transformer and minimize reflections. Of course, the coating must be transparent to the incident light. This means that it needs to be an insulator with a bandgap that exceeds the energy of the shortest wavelength light to be absorbed by the PV cell. Alternatively, it needs some other property that minimizes the value of the absorption coefficient for the material, such as an indirect bandgap.

It is also important to realize that a quarter wavelength is on the order of 0.1 μm. This is extremely thin, and may pose a problem for spreading a uniform coating of this thickness. And, of course, since it is desirable to absorb a range of wavelengths, the antireflective coating (ARC) will be optimized at only a single wavelength. Despite these problems, coatings have been developed that meet the requirements quite well.

An alternative to ARCs now commonly in use with Si PV cells is to manufacture the cells with textured front and back surfaces, as shown in Figure 2.7. Textured surfaces promote reflection and refraction of the incident photons to extend the photon path lengths within the material, thus enhancing the probability of generating EHPs within a minority carrier diffusion length of the junction. The bottom line is that a textured surface acts to enhance the capture of photons and also acts to prevent the escape of captured photons before they can produce EHPs. Furthermore, the textured surface is not wavelength dependent as is the ARC.

2.5.2.3.2 Maximizing Minority Carrier Diffusion Lengths

Since the diffusion lengths are given by (2.6), it is necessary to explore the factors that determine the diffusion constants and minority carrier lifetimes in different materials. It needs to be recognized that changing a diffusion constant may affect the minority carrier lifetime, so the product needs to be maximized.

Diffusion constants depend on scattering of carriers by host atoms as well as by impurity atoms. The scattering process is both material dependent and temperature dependent. In a material at a low temperature with a well-defined crystal structure, scattering of charge carriers is relatively minimal, so they tend to have high mobilities. Figure 2.8 illustrates the experimentally determined dependence of the electron mobility on temperature and on impurity concentration in Si. The Einstein relationship shows that the diffusion constant is proportional to the product of mobility and temperature. This relationship is shown in Figure 2.8. So, once again, there is a trade-off. While increasing impurity concentrations in the host material increases the built-in junction potential, increasing impurity concentrations decreases the carrier diffusion constants.

The material of Figure 2.8 is single crystal material. In polycrystalline or amorphous material, the lack of crystal lattice symmetry has a significant effect on the mobility and diffusion constant, causing significant reduction in these quantities. However, if the absorption constant can be made large enough for these materials, the corresponding decrease in diffusion length may be compensated for by the increased absorption rate.

When an electron and a hole recombine, certain energy and momentum balances must be achieved. Locations in the host material that provide for optimal recombination conditions are known as recombination centers. Hence, the minority carrier lifetime is determined by the density of recombination centers in the host material.

One type of recombination center is a crystal defect, so that as the number of crystal defects increases, the number of recombination centers increases. This means that crystal defects reduce the diffusion constant as well as the minority carrier lifetime in a material.

Impurities also generally make good recombination centers, especially those impurities with energies near the center of the bandgap. These impurities are thus different from the donor and acceptor impurities that are purposefully used in the host material, since donor impurities have energies relatively close to the conduction band and acceptor impurities have energies relatively close to the valence band.

Minority carrier lifetimes also depend on the concentration of charge carriers in the material. An approximation of the dependence of electron minority carrier lifetime on carrier concentration and location of the trapping energy within the energy gap is given by [8]

${\tau }_{\text{n}}\text{=}\frac{\text{n′}\left[\text{n+p+2}{\text{n}}_{\text{i}}\text{cosh((}{\text{E}}_{\text{t}}-{\text{E}}_{\text{i}})\text{/}\text{kT)}\right]}{\text{C}{\text{N}}_{\text{t}}\text{(np}-{\text{n}}_{\text{i}}^{\text{2}}\text{)}}\text{,}$ (2.15a)

where

C is the capture cross section of the impurity in cm3/s

Nt is the density of trapping centers

Et and Ei are the energies of the trapping center and the intrinsic Fermi level

In most materials, the intrinsic Fermi level is very close to the center of the bandgap. Under most illumination conditions, the hyperbolic term will be negligible compared to the majority carrier concentration and the excess electron concentration, as minority carriers in p-type material will be much larger than the electron thermal equilibrium concentration. Under these conditions, for minority electrons in p-type material, (2.15a) reduces to

${\tau }_{\text{n}}\simeq \frac{\text{1}}{\text{C}{\text{N}}_{\text{t}}}\text{.}$ (2.15b)

Hence, to maximize the minority carrier lifetime, it is necessary to minimize the concentration of trapping centers and to be sure that any existing trapping centers have minimal capture cross sections.

2.5.2.3.3 Maximizing Junction Width

Since it has been determined that it is desirable to absorb photons within the confines of the pn junction, it is desirable to maximize the width of the junction. It is therefore necessary to explore the parameters that govern the junction width.

An expression for the width of a pn junction can be obtained by solving Gauss’ law at the junction, since the junction is a region that contains electric charge. Solution of Gauss’ law, of course, is dependent upon the ability to express the spatial distribution of the space charge in mathematical, or, at least, in graphical form. Depending on the process used to form the junction, the impurity profile across the junction can be approximated by different expressions. Junctions formed by epitaxial growth or by ion implantation can be controlled to have impurity profiles to meet the discretion of the operator. Junctions grown by diffusion can be reasonably approximated by a linearly graded model. The interested reader is encouraged to consult a reference on semiconductor devices for detailed information on the production of various junction impurity profiles.

The junction with uniform concentrations of impurities is convenient to use to obtain a feeling for how to maximize the width of a junction. Solution of Gauss’ law for a junction with uniform concentration of donors on one side and a uniform concentration of acceptors on the other side yields solutions for the width of the space charge layer in the n-type and in the p-type material. The total junction width is then simply the sum of the widths of the two sides of the space charge layer. The results for each side are

${\text{W}}_{\text{n}}\text{=}{\left[\frac{\text{2}\epsilon {\text{N}}_{\text{A}}}{\text{q}{\text{N}}_{\text{D}}\text{(}{\text{N}}_{\text{A}}+{\text{N}}_{\text{D}}\text{)}}\right]}^{1/2}{({\text{V}}_{\text{j}}-\text{V})}^{1/2}$ (2.16a)

and

${\text{W}}_{\text{p}}\text{=}{\left[\frac{\text{2}\epsilon {\text{N}}_{\text{D}}}{\text{q}{\text{N}}_{\text{A}}\text{(}{\text{N}}_{\text{D}}\text{+}{\text{N}}_{\text{A}}\text{)}}\right]}^{1/2}{({\text{V}}_{\text{j}}-\text{V})}^{1/2}\text{.}$ (2.16b)

The overall width of the junction can now be determined by summing Equations 2.16a and 2.16b to get

$\text{W=}{\left[\frac{\text{2}\epsilon \text{(}{\text{N}}_{\text{D}}\text{+}{\text{N}}_{\text{A}}\text{)}}{\text{q}{\text{N}}_{\text{A}}{\text{N}}_{\text{D}}}\right]}^{\text{1/2}}{({\text{V}}_{\text{j}}-\text{V})}^{\text{1/2}}\text{.}$ (2.17)

At this point, it should be recognized that the voltage across the junction due to the external voltage across the cell, V, will never exceed the built-in voltage, Vj. The reason is that, as the externally applied voltage becomes more positive, the cell current increases exponentially and causes voltage drops in the neutral regions of the cell, so only a fraction of the externally applied voltage actually appears across the junction. Hence, there is no need to worry about the junction width becoming zero or imaginary. In the case of PV operation, the external cell voltage will hopefully be at the maximum power point, which is generally between 0.5 and 0.6 V for silicon.

Next, observe that, as the external cell voltage increases, the width of the junction decreases. As a result, the absorption of photons decreases. This suggests that it would be desirable to design the cell to have the largest possible built-in potential to minimize the effect of increasing the externally applied voltage. This involves an interesting trade-off, since the built-in junction voltage is logarithmically dependent on the product of the donor and acceptor concentrations (see 2.5b), and the junction width is inversely proportional to the square root of the product of the two quantities. Combining (2.5b) and (2.17) results in

$\text{W=}{\left[\frac{\text{2}\epsilon \text{(}{\text{N}}_{\text{D}}\text{+}{\text{N}}_{\text{A}})}{\text{q}{\text{N}}_{\text{D}}{\text{N}}_{\text{A}}}\right]}^{\text{1/2}}{\left(\frac{\text{kT}}{\text{q}}\text{ln}\frac{{\text{N}}_{\text{A}}{\text{N}}_{\text{D}}}{{\text{n}}_{\text{i}}^{\text{2}}}-\text{V}\right)}^{\text{1/2}}\text{.}$ (2.18)

Note now that maximizing W is achieved by making either NA >> ND or by making ND >> NA.

Another way to increase the width of the junction is to include a layer of intrinsic material between the p-side and the n-side as shown in Figure 2.9. In this pin junction, there are no impurities to ionize in the intrinsic material, but the ionization still takes place at the edges of the n-type and the p-type material. As a result, there is still a strong electric field across the junction and there is still a built-in potential across the junction. Since the intrinsic region could conceivably be of any width, it is necessary to determine the limits on the width of the intrinsic region.

The only feature of the intrinsic region that degrades performance is the fact that it has a width. If it has a width, then it takes time for a charge carrier to traverse this width. If it takes time, then there is a chance that the carrier will recombine. Thus, the width of the intrinsic layer simply needs to be kept short enough to minimize recombination. The particles travel through the intrinsic region with a relatively high drift velocity due to the built-in electric field at the junction. Since the thermal velocities of the carriers still exceed the drift velocities by several orders of magnitude, the width of the intrinsic layer needs to be kept on the order of about one diffusion length.

2.5.2.3.4 Minimizing Surface Recombination Velocity

If an EHP is generated near a surface, it becomes more probable that the minority carrier will diffuse to the surface. Since photocurrent depends on minority carriers’ diffusing to the junction and ultimately across the junction, surface recombination of minority carriers before they can travel to the junction reduces the available photocurrent. When the surface is within a minority carrier diffusion length of the junction, which is often desirable to ensure that generation of EHPs is maximized near the junction, minority carrier surface recombination can significantly reduce the efficiency of the cell.

Surface recombination depends on the density of excess minority carriers; in this case, as generated by photon absorption, and on the average recombination center density per unit area, Nsr, on the surface. The density of recombination centers is very high at contacts and is also high at surfaces in general, since the crystal structure is interrupted at the surface. Imperfections at the surface, whether due to impurities or to crystal defects, all act as recombination centers.

The recombination rate, U, is expressed as number/cm2/s and is given by [8]

${\text{U = cN}}_{\text{sr}}\text{m′,}$ (2.19)

where

m′ is used to represent the excess minority carrier concentration, whether electrons or holes

c is a constant that incorporates the lifetime of a minority carrier at a recombination center

Analysis of the dimensions of the parameters in (2.19) shows that the units of cNsr are cm/s. This product is called the surface recombination velocity, S. The total number of excess minority carriers recombining per unit time and subsequent loss of potential photocurrent is thus dependent on the density of recombination centers at the surface and on the area of the surface. Minimizing surface recombination thus may involve reducing the density of recombination centers or reducing the density of minority carriers at the surface.

If the surface is completely covered by a contact, then little can be done to reduce surface recombination if minority carriers reach the surface, since recombination rates at contacts are very high. However, if the surface is not completely covered by a contact, such as at the front surface, then a number of techniques have been discovered that will result in passivation of the surface. Silicon oxide and silicon nitrogen passivation are two methods that are used to passivate silicon surfaces.

Another method of reducing surface recombination is to passivate the surface and then only allow the back contact to contact the cell over a fraction of the total cell area. While this tends to increase series resistance to the contact, if the cell material near the contact is doped more heavily, the ohmic resistance of the material is decreased and the benefit of reduced surface recombination offsets the cost of somewhat higher series resistance. Furthermore, an E-field is created that attracts majority carriers to the contact and repels minority carriers.

2.5.2.3.5 Final Expression for the Photocurrent

An interesting exercise is to calculate the maximum obtainable efficiency of a given PV cell. Equation 2.3 indicates the general expression for photon absorption. Since the absorption coefficient is wavelength dependent, the general formula for overall absorption must take this dependence into account. The challenge in design of the PV cell and selection of appropriate host material is to avoid absorption before the photon is close enough to the junction, but to ensure absorption when the photon is within a minority carrier diffusion length of either side of the junction.

The foregoing discussion can be quantified in terms of cell parameters in the development of an expression for the photocurrent. Considering a monochromatic photon flux incident on the p-side of a p+n junction (the + indicates strongly doped), the following expression for the hole component of the photocurrent can be obtained. The expression is obtained from the solution of the diffusion equation in the neutral region on the n-side of the junction for the diffusion of the photon-created minority holes to the back contact of the cell [7].

$\Delta {\text{I}}_{\text{lp}}\text{=}\frac{\text{qA}{\text{F}}_{\text{ph}}\alpha {\text{L}}_{\text{p}}}{{\alpha }^{\text{2}}{\text{L}}_{\text{p}}^{\text{2}}-\text{1}}\left[\frac{\text{S}\text{cosh(}{\text{w}}_{\text{n}}\text{/}{\text{L}}_{\text{p}}\text{)+(}{\text{D}}_{\text{p}}\text{/}{\text{L}}_{\text{p}}\text{)sinh(}{\text{w}}_{\text{n}}\text{/}{\text{L}}_{\text{p}}\text{)+}(\alpha {\text{D}}_{\text{p}}-\text{S}){\text{e}}^{-\alpha {\text{w}}_{\text{n}}}}{\text{S}\text{sinh(}{\text{w}}_{\text{n}}\text{/}{\text{L}}_{\text{p}}\text{)+(}{\text{D}}_{\text{p}}\text{/}{\text{L}}_{\text{p}}\text{)cosh(}{\text{w}}_{\text{n}}\text{/}{\text{L}}_{\text{p}}\text{)}}-\alpha {\text{L}}_{\text{p}}\right]\text{.}$ (2.20)

It is assumed that the cell has a relatively thin p-side and that the n-side has a width, wn. In (2.20), Fph represents the number of photons per cm2 per second per unit wavelength incident on the cell. The effect of the surface recombination velocity on the reduction of photocurrent is more or less clearly demonstrated by (2.20). The mathematical whiz will immediately be able to determine that small values of S maximize the photocurrent, and large values reduce the photocurrent, while one with average math skills may need to plug in some numbers.

Equation 2.20 is thus maximized when α and Lp are maximized and S is minimized. The upper limit of the expression then becomes

$\Delta {\text{I}}_{\text{lp}}\text{=}-\text{qA}{\text{F}}_{\text{ph}}\text{,}$ (2.21)

indicating that all photons have been absorbed and all have contributed to the photocurrent of the cell.

Since sunlight is not monochromatic, (2.21) must be integrated over the incident photon spectrum, noting all wavelength-dependent quantities, to obtain the total hole current. An expression must then be developed for the electron component of the current and integrated over the spectrum to yield the total photocurrent as the sum of the hole and electron currents. This mathematical challenge is clearly a member of the nontrivial set of math exercises. Yet, some have persisted at a solution to the problem and have determined the maximum efficiencies that can be expected for cells of various materials. Table 2.1 shows the theoretical optimum efficiencies for several different PV materials.

2.5.3 Minimizing Cell Resistance Losses

Any voltage drop in the regions between the junction and the contacts of a PV cell will result in ohmic power losses. In addition, surface effects at the cell edges may result in shunt resistance between the contacts. It is thus desirable to keep any such losses to a minimum by keeping the series resistance of the cell at a minimum and the shunt resistance at a maximum. With the exception of the cell front contacts, the procedure is relatively straightforward.

Most cells are designed with the front layer relatively thin and highly doped, so the conductivity of the layer is relatively high. The back layer, however, is generally more lightly doped in order to increase the junction width and to allow for longer minority carrier diffusion length to increase photon absorption. There must therefore be careful consideration of the thickness of this region in order to maximize the performance of these competing processes.

If the back contact material is allowed to diffuse into the cell, the impurity concentration can be increased at the back side of the cell. This is important for relatively thick cells, commonly fabricated by slicing single crystals into wafers. The contact material must produce either n-type or p-type material if it diffuses into the material, depending on whether the back of the cell is n-type or p-type.

In addition to reducing the ohmic resistance by increasing the impurity concentration, the region near the contact with increased impurity concentration produces an additional electric field that increases the carrier velocity, thus producing a further equivalent reduction in resistance. The electric field is produced in a manner similar to the electric field that is produced at the junction.

For example, if the back material is p-type, holes from the more heavily doped region near the contact diffuse toward the junction, leaving behind negative acceptor ions. Although there is no source of positive ions in the p-region, the holes that diffuse away from the contact create an accumulated positive charge that is distributed through the more weakly doped region. The electric field, of course, causes a hole drift current, which, in thermal equilibrium, balances the hole diffusion current. When the excess holes generated by the photoabsorption process reach the region of the electric field near the contact, however, they are swept more quickly toward the contact. This effect can be viewed as the equivalent of moving the contact closer to the junction, which, in turn, has the ultimate effect of increasing the gradient of excess carriers at the edge of the junction. This increase in gradient increases the diffusion current of holes away from the junction. Since this diffusion current strongly dominates the total current, the total current across the junction is thus increased by the heavily doped layer near the back contact.

At the front contact, another balancing act is needed. Ideally, the front contact should cover the entire front surface. The problem with this, however, is that if the front contact is not transparent to the incident photons, it will reflect them away. In most cases, the front contact is reflecting. Since the front/top layer of the cell is generally very thin, even though it may be heavily doped, the resistance in the transverse direction will be relatively high because of the thin layer. This means that if the contact is placed at the edge of the cell to enable maximum photon absorption, the resistance along the surface to the contact will be relatively large.

The compromise, then, is to create a contact that covers the front surface with many tiny fingers. This network of tiny fingers, which, in turn, are connected to larger and larger fingers, is similar to the configuration of the capillaries that feed veins in a circulatory system. The idea is to maintain more or less constant current density in the contact fingers, so that as more current is collected, the cross-sectional area of the contact must be increased.

Finally, shunt resistance is maximized by ensuring that no leakage occurs at the perimeter of the cell. This can be done by nitrogen passivation or simply by coating the edge of the cell with insulating material to prevent contaminants from providing a current path across the junction at the edges.

2.6 Traditional PV Cells

2.6.1 Introduction

Traditional PV cells are based on the theoretical considerations of Sections 2.3 through 2.5. Cells currently commercially available are based on crystalline, multicrystalline, and amorphous (thin film) silicon (Si-C, Si, a-Si:H); copper indium gallium diselenide (CIGS) thin films; CdTe thin films; and III-V compounds such as gallium arsenide (GaAs). They all have pn junctions and all are subject to the optimization considerations previously discussed. This section will present a brief summary of the structures of each cell along with current (2011) performance information. For the interested reader, reference [8] considers all of the cells in this section in greater detail.

2.6.2 Crystalline Silicon Cells

Crystalline silicon cells can be either monocrystalline or polycrystalline. The monocrystalline cells are generally somewhat more efficient, but are also somewhat more energy intensive to produce. The cells are composed of approximately 200 μm thick slices of p-type single crystal ingots grown from a melt, with their circumferences squared up by slicing the round cross section into an approximately square cross section, similar to the way that lumber is processed from logs. The junction is formed by diffusing n-type impurities to a depth of approximately 1/α, where α is the absorption constant. The surfaces are then polished, textured and contacts are attached on front and back of the cell. The resulting cell cross section is essentially that of Figure 2.7. An adaptation of the structure of Figure 2.7 involves connecting the top layer of the cell through to the back of the cell so all contacts of the cell will be on the back. This increases cell efficiency by eliminating reflection of incident photons from front contacts. Typical cell conversion efficiencies for cells with front contacts can approach 17% and conversion efficiencies of back contact cells can exceed 20%.

By pouring molten silicon into a quartz crucible with a rectangular or square cross section under carefully controlled temperature conditions, it is possible to form a bar of silicon that consists of crystalline domains, but is not monocrystalline. The advantage of this polycrystalline silicon ingot is that it does not have to be “squared up,” thus saving a processing step and some energy as well. The disadvantage is that the crystal boundaries reduce mobilities and provide trapping centers, so the efficiency of the cell is somewhat reduced to the 14%–15% range.

The interesting point, however, is that since monocrystalline silicon cells have rounded corners, when they are assembled into a module to produce more power and more voltage, the module has voids at the corners of the cells, such that no electricity is produced at these locations. The polycrystalline cells, on the other hand, being rectangles, have less wasted module space and the overall module efficiency approximates the efficiency of a monocrystalline module, except for the back contact versions. Figure 2.10 shows photos of front contact monocrystalline cells, front contact polycrystalline cells, and back contact monocrystalline cells.

2.6.3 Amorphous Silicon Cells

Since a-Si:H is not in a crystalline form, it loses the advantages of high mobility and high diffusion constant. The noncrystalline lattice includes a large number of silicon atoms with outer shell electrons that are not covalently bonded to nearest neighbors. These “dangling” electrons create impurity levels in the bandgap and thus affect the lifetimes of the excited charge carriers. Fortunately, it is possible to apply hydrogen to the material such that the hydrogen fills the dangling bonds, thus passivating these sites in the material and improving the electrical properties. The material is thus described as a-Si:H.

Despite the amorphous nature of this material, it has a very favorable direct bandgap, which enables the material to efficiently absorb photons over a short distance. A 2 μm thickness of the material will absorb most of the incident photons, thus the reason why a-Si:H is considered to be a thin-film PV material.

Figure 2.11 shows several different cell structures. Structures b and c are multijunction structures and thus present important challenges to the cell designer. First of all, each layer tends to act as a current source, such that if current sources are connected in series, each layer must generate the same current as every other layer. Secondly, the layers appear as series diodes. This means that although the generating diode is supplying current, this current must flow across a reverse-biased junction between adjacent layers. This implies that significant current will not flow until the voltage across the reverse-biased junction reaches the reverse breakdown potential. Fortunately, with heavy doping on each side of the junction, the reverse breakdown voltage can be reduced to zero and a tunnel junction is created that allows current to pass unimpeded.

Commercial a-Si:H PV modules are presently available with flexible structures and conversion efficiencies of 8%–10%. An advantage of the technology and module structure is that the module can be applied directly to an approved surface without the need for any additional mounting components.

2.6.4 Copper Indium Gallium Diselenide Cells

Another popular material for thin-film cells is copper indium gallium diselenide (CIGS). This material is also a direct bandgap that absorbs most photons with energies above the bandgap energy within a thickness of about 2 μm. The material is in commercial production at conversion efficiencies in the neighborhood of 12%.

While it is possible to produce both n-type and p-type CIS, homojunctions in the material are neither stable nor efficient. A good junction can be made, however, by creating a heterojunction with n-type CdS and p-type CIS.

The ideal structure uses near-intrinsic material near the junction to create the widest possible depletion region for collection of generated EHPs. The carrier diffusion length can be as much as 2 μm, which is comparable with the overall film thickness. Figure 2.12 shows a basic ZnO/CdS/CIGS/Mo cell structure, which was in popular use in 2004. Again, CIGS technology is advancing rapidly as a result of the thin-film PV partnership program, so by the time this paragraph is read, the structure of Figure 2.12 may be only suitable for history books and general discussion of the challenges encountered in thin-film cell development.

Nearly a dozen processes have been used to achieve the basic cell structure of Figure 2.12. The processes include radio frequency (rf) sputtering, reactive sputtering, chemical vapor deposition, vacuum evaporation, spray deposition, and electrodeposition. Sometimes these processes are implemented sequentially and sometimes they are implemented concurrently. Recently a novel method of manufacturing CIGS modules using cylindrical tubes with spaces in between to allow photons to pass through the module and be reflected back to the cylindrical tubes from the surface upon which the module is attached has been developed. This structure presents minimal wind loading for the module and consequently for most wind zones it can be simply laid on a flat roof and attached to adjacent modules to form the PV array. This module is becoming popular for use with white membrane flat roofing material.

2.6.5 Cadmium Telluride Cells

In theory, CdTe cells have a maximum efficiency limit close to 25%. The material has a favorable direct bandgap and a large absorption constant, allowing for cells of a few μm thickness. By 2001, efficiencies approaching 17% were being achieved for laboratory cells, and module efficiencies had reached 11% for the best large area (8390 cm2) module [11]. Efforts were then focused on scaling up the fabrication process to mass produce the modules, with the result of achieving a production cost of less than $1.00/W in 2008. This cost included an escrow account to be used for recycling the materials at the end of module life.

No fewer than nine companies have shown an interest in commercial applications of CdTe. As of 2001, depending on the fabrication methodology, efficiencies of close to 17% had been achieved for small area cells (≈1 cm2), and 11% on a module with an area of 8390 cm2 [12].

After it was shown that no degradation was observable after 2 years, production-scale manufacturing began. CdTe modules are now being manufactured and marketed at the rate of more than 1 GW annually for utility scale projects and the magic $1.00/W production cost barrier has now been broken for these modules, with an announced 4th quarter 2008 production cost of $0.93/W [8]. Figure 2.13 shows the cross section of a typical CdTe cell. In this figure, antireflective coating (ARC), transparent conducting oxide (TCO) and ethylene vinyl acetate (EVA), which are used to bond the back contact to the glass.

2.6.6 Gallium Arsenide Cells

The 1.43 eV direct bandgap, along with a relatively high absorption constant, makes GaAs an attractive PV material. Historically high production costs, however, have limited the use of GaAs PV cells to extraterrestrial and other special purpose uses, such as in concentrating collectors. Recent advances in concentrating technology, however, enable the use of significantly less active material in a module, such that cost-effective, terrestrial devices may soon be commercially available.

Most modern GaAs cells, however, are prepared by the growth of a GaAs film on a suitable substrate. Figure 2.14 shows one basic GaAs cell structure. The cell begins with the growth of an n-type GaAs layer on a substrate, typically Ge. Then a p-GaAs layer is grown to form the junction and collection region. The top layer of p-type GaAlAs has a bandgap of approximately 1.8 eV. This structure reduces minority carrier surface recombination and transmits photons below the 1.8 eV level to the junction for more efficient absorption.

Cells fabricated with III-V elements are generally extraterrestrial quality. In other words, they are expensive, but they are high-performance units. Efficiencies in excess of 20% are common and efficiencies of cells fabricated on more expensive GaAs substrates have exceeded 34% [11].

An important feature of extraterrestrial quality cells is the need for them to be radiation resistant. Cells are generally tested for their degradation resulting from exposure to healthy doses of 1 MeV or higher energy protons and electrons. Degradation is generally less than 20% for high exposure rates [13].

Extraterrestrial cells are sometimes exposed to temperature extremes, so the cells are also cycled between ~−170°C and +96°C for as many as 1600 cycles. The cells also need to pass a bending test, a contact integrity test, a humidity test, and a high temperature vacuum test, in which the cells are tested at a temperature above 140°C in vacuum for 168 h [13].

Fill factors in excess of 80% have been achieved for GaAs cells. Single cell open-circuit voltages are generally between 0.8 and 0.9 V.

2.7 Emerging Technologies

2.7.1 New Developments in Silicon Technology

While progress continues on conventional Si technology, new ideas are also being pursued for crystalline and amorphous Si cells. The goal of Si technology has been to maintain good transport properties, while improving photon absorption and reducing the material processing cost of the cells. It is likely that several versions of thin Si cells will continue to attract the attention of the PV community, including recent research on thin Si on glass.

Another interesting opportunity for cost reduction in Si cell production is to double up on processing steps. For example, a technique has been developed for simultaneously diffusing boron and phosphorous in a single step, along with growing a passivating oxide layer [14].

As an alternative to the pn junction approach to Si cells, MIS-IL (metal insulator semiconductor inversion layer) cells have been fabricated with 18.5% efficiency [15]. The cell incorporates a point-contacted back electrode to minimize the rear surface recombination, along with Cs beneath the MIS front grid and oxide window passivation of the front surface to define the cell boundaries. Further improvement in cell performance can be obtained by texturing the cell surfaces.

New developments in surface texturing may also simplify the process and result in additional improvement in Si device performance. Discovery of new substrates and methods of growing good quality Si on them is also an interesting possibility for performance improvement and cost reduction for Si cells.

Since new ideas will continue to emerge as interest in Si PV technology continues to grow, the interested reader is encouraged to attend PV conferences and to read the conference publications to stay up-to-date in the field.

2.7.2 CIS-Family-Based Absorbers

Much is yet to be learned about inhomogeneous absorbers and composite absorbers composed of combinations of these various materials. The possibility of multijunction devices is also being explored.

Meanwhile, work is underway to reduce the material usage in the production of CIS modules in order to further reduce production costs. Examples of reduction of material use include halving the width of the Mo contact layer, reduction in the use of H2S and H2Se, and a reduction in ZnO, provided that a minimum thickness can be maintained [16].

2.7.3 Other III–V and II–VI Emerging Technologies

It appears that compound tandem cells will receive appreciable emphasis in the III-V family of cells over the next few years. For example, Ga0.84In0.16As0.68P0.32, lattice matched to GaAs, has a bandgap of 1.55 eV and may prove to be an ideal material for extraterrestrial use, since it also has good radiation resistance [17]. Cells have been fabricated with Al0.51In0.49P and Ga0.51In0.49P window layers, with the best 1 cm2 cell having an efficiency of just over 16%, but having a fill factor of 85.4%.

Cell efficiencies can be increased by concentrating sunlight on the cells. Although the homojunction cell efficiency limit under concentration is just under 40%, quantum well (QW) cells have been proposed to increase the concentrated efficiency beyond the 40% level [18]. In QW cells, intermediate energy levels are introduced between the host semiconductor’s valence and conduction bands to permit absorption of lower energy photons. These levels must be chosen carefully so that they will not act as recombination centers, however, or the gains of EHPs from lower energy incident photons will be lost to the recombination processes. Laboratory cells have shown higher VOC resulting from a decrease in dark current for these cells.

2.7.4 Other Technologies

2.7.4.1 Thermophotovoltaic Cells

To this point, discussion has been limited to the conversion of visible and near infrared spectrum to EHPs. The reason is simply that the solar spectrum peaks out in the visible range. However, heat sources and incandescent light sources produce radiation in the longer infrared regions, and in some instances, it is convenient to harness radiated heat from these processes by converting it to electricity. This means using semiconductors with smaller bandgaps, such as Ge. More exotic structures, such as InAsSbP, with a bandgap of 0.45–0.48 eV have also been fabricated [19].

2.7.4.2 Intermediate Band Solar Cells

In all cells described to this point, absorption of a photon has resulted in the generation of a single EHP. If an intermediate band material is sandwiched between two ordinary semiconductors, it appears that it may be possible for the material to absorb two photons of relatively low energy to produce a single EHP at the combined energies of the two lower energy photons. The first photon raises an electron from the valence band to the intermediate level, creating a hole in the valence band, and the second photon raises the electron from the intermediate level to the conduction band. The trick is to find such an intermediate band material that will “hold” the electron until another photon of the appropriate energy impinges upon the material. Such a material should have half its states filled with electrons and half empty in order to optimally accommodate this electron transfer process. It appears that III–V compounds may be the best candidates for implementation of this technology. Theoretical maximum efficiency of such a cell is 63.2% [20].

2.7.4.3 Super Tandem Cells

If a large number of cells are stacked with the largest bandgap on top and the bandgap of subsequent cells decreasing, the theoretical maximum efficiency is 86.8% [21]. A 1 cm2 four-junction cell has been fabricated with an efficiency of 35.4%. The maximum theoretical efficiency of this cell is 41.6% [22]. Perhaps 1 day one of the readers of this paragraph (or one of their great-great grandchildren) will fabricate a cell with the maximum theoretical efficiency.

2.7.4.4 Hot Carrier Cells

The primary loss mechanism in PV cells is the energy lost in the form of heat when an electron is excited to a state above the bottom of the conduction band of a PV cell by a photon with energy greater than the bandgap. The electron will normally drop to the lowest energy available state in the conduction band, with the energy lost in the process being converted to heat. Hence, if this loss mechanism can be overcome, the efficiency of a cell with a single junction should be capable of approaching that of a super tandem cell. One method of preventing the release of this heat energy by the electron is to heat the cell, so the electron will remain at the higher energy state. The process is called thermoelectronics and is currently being investigated [21].

2.7.4.5 Optical Up- and Down-Conversion

An alternative to varying the electrical bandgap of a material is to reshape the energies of the incident photon flux. Certain materials have been shown to be capable of absorbing two photons of two different energies and subsequently emitting a photon of the combined energy. Other materials have been shown to be capable of absorbing a single high-energy photon and emitting two lower-energy photons. These phenomena are similar to up-conversion and down-conversion in communications circuits at radio frequencies.

By the use of both types of materials, the spectrum incident on a PV cell can be effectively narrowed to a range that will result in more efficient absorption in the PV cell. An advantage of this process is that the optical up- and down-converters need not be a part of the PV cell. They simply need to be placed between the photon source and the PV cell. In tandem cells, the down-converter would be placed ahead of the top cell and the up-converter would be integrated into the cell structure just ahead of the bottom cell [8].

2.7.4.6 Organic PV Cells

Even more exotic than any of the previously mentioned cells is the organic cell. In the organic cell, electrons and holes are not immediately formed as the photon is absorbed. Instead, the incident photon creates an exciton, which is a bound EHP. In order to free the charges, the exciton binding energy must be overcome. This dissociation occurs at the interface between materials of high electron affinity and low ionization potential [22]. Photoluminescence is related to this process. Just to end this section with a little chemistry, the reader will certainly want to know that one material that is a candidate for organic PV happens to be poly{2,5-dimethoxy-1,4-phenylene-1,2-ethenylene-2-methoxy-5-(2-ethyl-hexyloxy)-1,4-phenylene-1,2-ethenylene}, which goes by the nickname M3EH-PPV. Whether M3EH-PPV will dominate the PV market 1 day remains to be seen. So far efficiencies of this very challenging technology have been in the 1% range.

2.8 PV Electronics and Systems

2.8.1 Introduction

Obviously the extent of research and development that has been covered in this chapter may not have been undertaken if a possible market for each technology had not been identified. Prior to 2000, most PV applications were stand-alone applications, such as off-grid cabins. Since that time, however, grid-connected applications have mushroomed and far surpassed stand-alone applications. Grid-connected applications can be either noninteractive or interactive and can be direct grid connect or battery backup. Noninteractive grid-connected systems simply use the grid as a backup source of power when the sun is not shining. Interactive systems are capable of selling energy back to the grid if the host demand is met and excess system output remains available.

Interactive systems require inverters to convert the dc power produced by the PV array into compatible ac power for use at the source as well as for return to the utility. In a utility interactive system, it is necessary for the inverter to shut down if the grid shuts down. However, if a set of batteries are used, then it is possible to energize selected loads when the grid is down as long as the selected loads can be isolated from the grid via switching within the inverter. IEEE Standard 1547 [23] specifies performance parameters for waveform purity and disconnecting from the grid and UL 1741 [24] specifies a testing protocol to ensure compliance with IEEE 1547.

If a system has batteries for backup power, then a battery backup inverter as well as a charge controller will normally be needed.

2.8.2 PV System Electronic Components

2.8.2.1 Inverters

Inverters can have output waveforms ranging from square, for the simplest units, to sine, for the best units. If a unit is to be connected to the grid in a sell mode, it must have no more than 5% total harmonic distortion, with individual harmonic maxima specified by IEEE 1547.

Straight grid-connected, utility interactive inverters are used in the simplest of utility interactive systems. So-called “string inverters” use series/parallel combination of modules that may produce up to 1000 V when open circuited, although installations in the United States must have maximum voltages less than 600 V unless they are on utility-owned property. String inverter power output ranges from the low kW range up to 1 MW, with larger units in the pipeline. String inverters generally have maximum power point tracking (MPPT) circuitry at their inputs, so they can operate the PV array at its maximum power point and thus deliver maximum power to the load. They also incorporate ground fault detection and interruption at their inputs to shut down the PV array if a current-carrying conductor should come in contact with a grounded object. Since they comply with UL1741, they shut down whenever any grid disturbance, such as undervoltage, overvoltage, or frequency error is present on the utility system, even if other inverters are connected to the same system.

Recently, a version of the straight grid-connected inverter, the microinverter, has become very popular. The microinverter is used with either a single module or with a pair of modules. Generally the rated dc module power is between 175 and 240 W. The microinverter is mounted next to the PV module so that ac rather than dc is fed from a rooftop. Furthermore, the ac is connected directly to the utility connection so that if the utility loses power, the microinverter is disabled and no current or voltage is present at its output.

When battery backup is used to create an uninterruptible source, the battery backup inverter must have two separate ac ports. One port is connected to the utility and one is connected to the standby loads. If the grid is operational and the sun is down, the inverter will pass utility power through to the standby loads. If the grid is down, then the inverter provides power to the standby loads via the standby port either directly from the PV array during daytime hours or from the batteries at night, or, for that matter, from a combination of batteries and PV array, depending upon the demand of the standby load. If the grid is down, the grid-connected port is automatically shut down, generally in less than 2 s. Present commercial battery backup inverters are rated at 8 kW or less, with the possibility of connecting series-parallel combinations to deliver up to 80 kW. A 100 kW unit has been developed but is still in the beta testing phase.

2.8.2.2 Charge Controllers

When a battery backup inverter is used in a so-called dc-coupled battery backup configuration, the input of the inverter is connected directly to the system batteries. The PV array is also connected to the batteries, but it is connected through a charge controller. The charge controller is needed for two purposes. The primary purpose is to prevent the batteries from becoming overcharged. The system inverter is set to shut off if the batteries approach the maximum discharge limit. The secondary purpose of the charge controller is to maximize energy transfer from PV array to batteries, assuming that a MPPT charge controller is selected. There is no need for MPPT at the inverter input, since it is always at the battery voltage.

2.9 Conclusions

Regardless of the technology or technologies that may result in low-cost, high-performance PV cells, it must be recognized that the lifecycle cost of a cell depends on the cell’s having the longest possible, maintenance-free lifetime. Thus, along with the developments of new technologies for absorbers, development of reliable encapsulants and packaging for the modules will also merit continued research and development activity.

Every year engineers make improvements on products that have been in existence for many years. Automobiles, airplanes, electronic equipment, building materials, and many more common items see improvement every year. Even the yo-yo, a popular children’s toy during the 1940s and 1950s, came back with better-performing models. Hence, it should come as no surprise to the engineer to see significant improvements and scientific breakthroughs in the PV industry well into the next millennium. The years ahead promise exciting times for the engineers and scientists working on the development of new photovoltaic cell and system technologies, provided that the massive planning and execution phases can be successfully undertaken. This will certainly be the case as the world of nano-devices is explored.

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