The Physics of Industrial Crystalline Silicon Solar Cells

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CHAPTER ONE

Otwin Breitenstein Max Planck Institute of Microstructure Physics, Halle, Germany

Contents 1. Introduction and Chapter Methodology 1 2. Basic Theory of Solar Cells 4 2.1 Solar cell in thermal equilibrium 4 2.2 Biased solar cell 8 2.3 Analysis of the bulk lifetime 13 2.4 Depletion region recombination 18 2.5 Illuminated solar cell 20 2.6 Reverse current 23 3. Theory Versus Experiment 24 4. Origins of Nonideal Characteristics 26 4.1 The depletion region recombination (second diode) current 27 4.2 The diffusion (first diode) current 37 4.3 The ohmic current 44 4.4 The reverse current 48 4.5 Relation between dark and illuminated characteristics 59 5. Summary and Outlook 67 Acknowledgments 70 References 70

1. INTRODUCTION AND CHAPTER METHODOLOGY Solar cells made from silicon wafers are the oldest type of solar cells, which were developed in Bell Laboratories in the 1950s for space applica- tions. While the first silicon solar cell made in 1953 had an energy conver- sion efficiency of 6%, already in 1958 the “Vanguard 1” satellite was powered by 108 silicon solar cells having an efficiency of 10.5% (http:// en.wikipedia.org/wiki/Solar_cell#History_of_solar_cells). Today, the

2 Otwin Breitenstein world efficiency record for crystalline silicon solar cells is at 25% (Green et al., 2012), and typical industrial cells are already approaching 20% (Song et al., 2012). This impressive advancement was only possible based on a deep understanding of the physics underlying these solar cells. Note that semiconductor physics is a relatively young science. The theory of a p–n junction was developed only in 1949 (Shockley, 1949), the papers describing the Shockley–Read–Hall (SRH) recombination statistics appeared in 1952 (Hall, 1952; Shockley and Read, 1952), and in 1957 the diode theory became extended to generation and recombination processes in the depletion region (Sah et al., 1957). Until now, these are the basic papers for understanding the physics of solar cells. Today, this theory is an integral part of textbooks on semiconductor physics and technology (see, e.g., Sze and Ng, 2007). This chapter will not replace such a textbook. For understanding it, basic knowledge in solid state and semiconductor physics is required. This chapter basically consists of two parts. In the first part, the established theory of the operation of solar cells is reviewed. Here the most important relations describing a solar cell are derived and made physically clear. Then the predictions of this theory are compared with typically measured solar cell characteristics, which reveal significant deviations from the theory. The main focus of the second part of this chapter is to point on the reasons for these deviations and explain their physical origins. The widely accepted model electrically describing silicon solar cells is the so-called two-diode model, which will be discussed in the following section. However, as mentioned above, the current–voltage (I–V ) characteristics of industrial silicon solar cells show significant deviations from the classical two-diode model predictions. This holds particularly for cells made from multicrystalline material, which contain high concentrations of crystal defects like grain boundaries, dislocations, and precipitates, fabricated by the so-called vertical gradient freeze (Trempa et al., 2010) or Bridgman method (Mu¨ller et al., 2006). Even the characteristics of industrial monocrystalline cells, which do not contain these crystal defects, deviate from the theoretical predictions. In particular, the so-called depletion region recombination current or second diode current is usually several orders of magnitude larger than expected, and its ideality factor is significantly larger than the expected value of two. This nonideal behavior was observed already very early and tentatively attributed to the existence of metallic precipitates or other defects in the depletion region (Queisser, 1962). In that work (Queisser, 1962) it was already suspected that local leakage currents could be responsible for the nonideal diode behavior, and it was speculated that Author's personal copy

The Physics of Industrial Crystalline Silicon Solar Cells 3 the edge region of a cell could significantly contribute to these nonideal currents. Later on, this nonideal behavior was attempted to be explained also under the assumption of a homogeneous current flow by attributing it to trap-assisted tunneling (Kaminski et al., 1996; Schenk and Krumbein, 1995). However, in crystalline silicon solar cells, the defect levels responsible for this effect could never be identified. There were attempts to explain the large ideality factors solely by the influence of the series resistance (McIntosh, 2001; van der Heide et al., 2005). As will be shown in Section 4.1, this explanation is not sufficient for interpreting large ideality factors in well-processed cells. It has turned out that the key for a detailed understanding of the dark characteristic of solar cells is the spatially resolved mapping of the local current density of solar cells in the dark. Until now, all textbooks dedicated to solar cells still generally assume that a solar cell behaves homogeneously, e.g., (Green, 1998; Wu¨rfel, 2005). Until 1994, there was no experimental technique available that could map the dark forward current of a solar cell with sufficient accuracy. In principle, this current can be mapped by infrared (IR) thermography (Simo and Martinuzzi, 1990). However, since silicon is a good conductor of heat, the thermal signals are generally weak and the images appear blurred. Therefore, conventional IR thermography is only able to image breakdown currents under a reverse bias of several Volts, and the obtained spatial resolution is very poor (several mm, see Simo and Martinuzzi, 1990). The first method enabling a sensitive imaging of the dark forward current with a good spatial resolution was the “Dynamic Precision Contact Thermography” (DPCT) method (Breitenstein et al., 1994, 1997). Here a very sensitive miniature temperature sensor was probing the cell surface point-by-point in contact mode, and in each position the cell bias was square-pulsed and the local surface temperature modulation was measured and evaluated over some periods according to the lock-in principle. This technique already reached a sensitivity in the 100 mK range (standard thermography: 20–100 mK), and, due to its dynamic nature, the spatial resolution was well below one mm. Its only limitation was its low speed; taking a 100 100 pixel image took several hours. Therefore, DPCT was later replacedÂ by IR camera-based lock-in thermography (LIT). This technique was developed already before it was introduced to photovoltaics (Kuo et al., 1988), and since then it was mainly used in nondestructive testing, hence for “looking below the surface of bodies” (Busse et al., 1992). In the following, LIT was also used for inves- tigating local leakage currents in integrated circuits (Breitenstein et al., 2000) Author's personal copy

4 Otwin Breitenstein and in solar cells (Breitenstein et al., 2001). Meanwhile, LIT is a widely used standard imaging method for characterizing solar cells, which is commercially available. Details to its basics, realization, and application are given in (Breitenstein et al., 2010a). Since the illuminated I–V characteristic of asolarcelliscloselyrelatedtoitsdarkcharacteristic,LIT can even be used for performing a detailed local analysis of the efficiency of inhomogeneous solar cells (Breitenstein, 2011, 2012). In the last years, in addition to LIT, also camera-based electroluminescence (EL) and photoluminescence (PL) imaging methods have been developed for the local characterization of inhomogeneous solar cells. An overview over these methods and their comparison to LIT-based methods can be found in Breitenstein et al. (2011a). The topics covered in this chapter are as follows: • Section 2: Basic Theory of Solar Cells – Section 2.1: Solar cell in thermal equilibrium – Section 2.2: Biased solar cell – Section 2.3: Analysis of the bulk lifetime – Section 2.4: Illuminated solar cell – Section 2.5: Reverse current • Section 3: Theory Versus Experiment • Section 4: Origins of Nonideal Characteristics – Section 4.1: The depletion region recombination (second diode) current – Section 4.2: The diffusion (first diode) current – Section 4.3: The ohmic current – Section 4.4: The reverse current – Section 4.5: Relation between dark and illuminated characteristics • Section 5: Summary and Outlook

2. BASIC THEORY OF SOLAR CELLS 2.1. Solar cell in thermal equilibrium

Figure 1.1A shows qualitatively the band scheme of an nþ–p junction, including its ohmic contacts, as it is present in usual industrial solar cells, in thermal equilibrium. Particularly, the x-axis is not to scale, in reality the emitter thickness is a factor of 500 smaller than the base thickness. 20 3 “nþ–p” means that the n-side is much more highly doped (up to 10 cmÀ ) Author's personal copy

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Figure 1.1 Schematic band diagram (A), profile of the space charge density (B), and profile of the electric fields (C) in an nþ–p junction, assuming homogeneous nondegenerate emitter doping, including its ohmic contacts (ME metal), ¼ not to scale. Author's personal copy

6 Otwin Breitenstein

16 3 than the p-side of the junction (typically 10 cmÀ ), which holds for typical P-diffused p-base solar cells. Moreover, for keeping the explanations simple, the nþ-type emitter in Fig. 1.1 is assumed to be homogeneously and nondegenerately doped, in contrast to real diffused emitters. Figure 1.1B shows the local space charge densities and (C) the electric fields in this device, both also not to scale. For homogeneous doping, the volume of the n- and of the p-material does not contain any space charge or any fields. Therefore, these regions are called the neutral material. In reality, the inhomogeneously doped emitter contains certain fields. At the metallurgical p–n junction, holes have been diffused into the n-material and electrons into the p-material, which has led to an electrostatic potential difference between both regions. The electrostatic potential in n-material is more positive compared to that in the p-material. It is sometimes hard to understand why this potential difference cannot be equilibrated by closing an electric contact between both sides. Here it must be known that the electrostatic potential in a semiconductor is not a measurable voltage, as it is in a metal. In a semiconductor, the measurable voltage is the position of the chemical potential of the electrons, which is the Fermi level. By definition, a slope of the Fermi level is equivalent to a current flow in the device (Sze and Ng, 2007). Therefore, in thermal equilibrium, the Fermi level (dash- dotted line in Fig. 1.1A) crosses the device horizontally, hence no current flows. In the two metal contacts, the Fermi level is at the same energy as in the semiconductor, therefore closing these contacts does not lead to any current flow and particularly not to an equilibration of the different electrostatic potentials in the n- and in the p-region. As long as the Fermi level is lying sufficiently deep (several kT) in the band gap (nondegenerate doping condition), the electron resp. hole concentrations n and p can be described by

E E E E n N exp F À c , p N exp v À F 1:1 ¼ c kT ¼ v kT ð Þ

Here Nc and Nv are the effective densities of states in the conduction resp. valence band, EF is the Fermi energy position, Ec and Ev are the energy positions of the conduction resp. valence band, and kT is the thermal energy. At room temperature, the shallow donors and acceptors are completely ionized, hence, in the absence of any compensation, in the neutral n-region n N and in the neutral p-region p N holds, with N being the donor ¼ D ¼ A D doping concentration at the n-side and NA the acceptor doping Author's personal copy

The Physics of Industrial Crystalline Silicon Solar Cells 7 concentration at the p-side. Thus, via Eq. (1.1) the doping concentrations govern the position of the Fermi level in the neutral p-material:

Nv EF Ev x kT ln 1:2 À ¼ p ¼ N ð Þ A 16 3 For a typical base doping concentration of 10 cmÀ , Eq. (1.2) leads to xp 190 meV at RT. Note that Eqs. (1.1) and (1.2) only hold for the p-material, since the n-doped emitter is usually degenerately doped (metallic-like behavior). Here Fermi statistics has to be applied, leading 20 3 for ND 2 10 cmÀ to xn 50 meV, hence the Fermi level is actually lying in¼ theÂ conduction bandÀ there. In the depletion region of the p–n junction, n and p are negligibly small; therefore, the two doping concentrations ND and NA govern the positive resp. negative ionic space charge densities there. According to the Poisson equation (Sze and Ng, 2007), the space charge density is proportional to the second deriv- ative of the potential to x,hencetothebendingofthepotential.Therefore,in the depletion region, for homogeneous doping concentration, the bands have a parabolic shape. If ND NA holds, as shown here, most of the potential drop in equilibrium and under reverse bias occurs within the more lowly doped p-region, leading to the relation for the depletion region width:

2ee0 Vr Vd W ðÞþ 1:3 ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieNA ð Þ

Here Vr is the applied reverse voltage (zero in Fig. 1.1), Vd is the equilibrium barrier height, also called the diffusion voltage, ee0 is the permittivity of the material, and e is the electron charge. Note that (1.3) only holds in total depletion approximation, hence by neglecting the so-called edge regions of the junction, where the free carrier concentration gradually decreases toward zero. According to the discussion above concerning the Fermi energy position, the diffusion voltage Vd equals the gap energy Eg minus the sum of the energy distances between the Fermi energy and the neighbored band edges in the p- and the n-material xp and xn, divided by e for converting the energy into a voltage:

Eg x x V À p À n 1:4 d ¼ e ð Þ Also at the ohmic contacts depletion regions exist. In some older textbooks, ohmic contacts are still described as carrier accumulation regions, Author's personal copy

8 Otwin Breitenstein based on the idea that the barrier height of a metal contact is just the difference between the electron affinities of the semiconductor and the metal. It is claimed that, for certain metal–semiconductor combinations, this could lead to a negative barrier height and thus to an accumulation contact. This theory is wrong, see e.g., Sze and Ng (2007). In reality, the barrier height is always dominated by interface states, and the electron affinity plays only a minor role. These interface states capture free carriers and are therefore in n-material negatively and in p-material positively charged (see Fig. 1.1B). Therefore in both cases the metal–semiconductor contact implies a depletion region, the metal type and the doping concentration only govern the barrier height. According to Eq. (1.3), which also holds for metal– semiconductor contacts, the doping concentration influences the depletion region width W. For highly doped material, W comes into the nm region. Then the carriers may tunnel through the barrier, which is the mechanism how ohmic contacts work. Note that the p-contact region is also highly doped, which has two reasons: first, only the high doping concentration enables ohmic contact formation, as for the nþ contact described above. Sec- ond, the higher electrostatic potential in the p-contact region reduces under illumination the electron (minority carrier) concentration in this region, thereby reducing the recombination rate at the contact. At the junction between the p- and the pþ-region, holes have diffused from the pþ- into the p-region. This leads to a positive space charge at the p- and a negative at the pþ-edge. This is the origin of the so-called back surface field (BSF) in this region, which repels minority carriers (see Fig. 1.1C). For a diffused BSF region, due to the concentration depth profile, this field extends up to the p-contact. The same holds for a highly doped diffused emitter region.

2.2. Biased solar cell In the following section, the equations describing the current–voltage (I–V) characteristic of a p–n junction will be derived. Figure 1.2 shows schemat- ically the band structure of a p–n junction (a) in thermal equilibrium, (b) under reverse, and (c) under forward bias. The dashed line in the middle of the gap symbolizes a mid-gap SRH recombination center, which governs the excess carrier lifetime t in the neutral material. The physics of a p–n junction can only be understood by considering horizontal and vertical thermally induced processes, which are symbolized in Fig. 1.2 by arrows. For clarity only electron processes are indicated, the same processes also hold for holes, where the energy scaling is inverted. Even in thermal equilibrium Author's personal copy

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Figure 1.2 Schematic band diagram of a p–n junction (A) in thermal equilibrium, (B) under reverse bias, and (C) under forward bias, only electron currents are shown The dashed line represents a deep SRH recombination center governing the excess carrier lifetime in the p-region. After Breitenstein (2013), by courtesy of Springer. there is spontaneous thermal carrier generation (upward arrows) and recombination (downward arrows), and there is horizontal carrier movement. Note that the free carriers not only exist close to the band edges, as it is often displayed in such schemes, but also deep in the bands. They follow the Fermi statistics, which, if the Fermi level is lying within the band gap, corresponds to Maxwell–Boltzmann statistics. These electrons deep in the band are characterized by a large kinetic energy. Therefore, they may be called high- energy or “hot” electrons, though they are in thermal equilibrium with all other electrons and with the lattice. With increasing energy distance DE to the band edges, the free carrier concentration decreases essentially proportional to exp( DE/kT) (in reality also the density of states plays a À role). Since the Fermi energy is going horizontally through Fig. 1.2A (not shown there, but in Fig. 1.1), the concentration of electrons in the p-side (in Fig. 1.2 left) essentially equals that in the n-side (right) having an energy above the position of the conduction band edge in the p-side. Only these “hot” electrons have sufficient kinetic energy to overcome the decelerating electric field in the depletion region and to enter the p-side. The two driving forces for horizontal carrier movement are the concentration gradient, leading to the so-called diffusion current, and the electric field, leading to the field current. The consideration of these two current contributions independently, with only the sum of both being a measurable net current, is called the detailed balance principle. In thermal equilibrium, across the whole depletion region, these two horizontal currents balance each other (Sze and Ng, 2007). Then also the net horizontal current across the p–n junction is zero. A similar detailed balance principle holds for recombination and thermal generation. In any position, under thermal equilibrium, thermal carrier generation is balanced by carrier recombination. For Author's personal copy

10 Otwin Breitenstein any kind of homogeneous carrier generation, the equilibrium electron (minority carrier) concentration at the p-side np can be expressed by the generation rate G (given in units of generated carriers per cm3 and second) multiplied by the excess carrier lifetime t. This relation is the base of all quasi-static lifetime measurement techniques. It also holds for the equilibrium thermal carrier generation in the neutral volume sketched in Fig. 1.2. On the other hand, the electron concentration in the p-material can be expressed by np n2/N (n intrinsic carrier concentration, ¼ i A i ¼ NA acceptor concentration), leading to an expression for the thermal generation¼ rate G:

n2 n2 np Gt i , G i 1:5 ¼ ¼ NA ¼ t NA ð Þ An interesting point here is to understand why the equilibrium minority carrier concentration is independent of the lifetime. It might be expected that, in low lifetime regions, the excess carrier concentration is lower, as it holds, for example, under light excitation condition. However, in these regions the rate of spontaneous thermal carrier generation is also correspondingly higher, leading to the same excess carrier concentration as in high lifetime regions, if the net doping concentration is the same. Under reverse bias (Fig. 1.2B), the concentration of “hot” electrons at the n-side, which have sufficient kinetic energy to overcome the barrier, is reduced. Therefore, the diffusion current of electrons from the n- to the p-side becomes negligibly small. Now the current across the p–n junction is dominated by the flow of thermally generated electrons from the p-side to the n-side. The electric field of the junction drains all electrons generated within one diffusion length Ld pDet (De electron diffusion constant in the p-region). This horizontal¼ current density¼ can be expressed regarding (1.5) as ffiffiffiffiffiffiffiffi

2 2 eni Ld ni epDe J01 GeLd 1:6 ¼ ¼ tNA ¼ NApt ð Þ ffiffiffiffiffiffi Since this current density is independent of theffiffiffi reverse bias, it is called a saturation current density. Under zero bias, this thermally generated current is exactly balanced by the diffusion current of electrons running from the n- to the p-side, see the horizontal arrows in Fig. 1.2A. When a forward bias is applied (Fig. 1.2C), the magnitude of this diffusion current rises exponen- tially with increasing forward bias V, since correspondingly more electrons Author's personal copy

The Physics of Industrial Crystalline Silicon Solar Cells 11 have enough kinetic energy to overcome the energy barrier. Since the diffusion current at 0 V equals (1.6), its bias dependence can be described as

eV J J exp 1:7 diff ¼ 01 kT ð Þ The net dark current is Eq. (1.7) minus the thermal generation current (1.6), leading to Shockley’s diode equation (Shockley, 1949):

eV J J exp 1 1:8 ¼ 01 kT À ð Þ This net current is traditionally also called “diffusion current” (Sah et al., 1957), since for V>kT/e (the thermal voltage VT, about 26 mV at room temperature) it is dominated by Eq. (1.7). The electrons, which are injected under forward bias into the p-region, recombine there basically within one diffusion length Ld, indicated by the thick downward arrows in Fig. 1.2C. This is the same region, which is responsible for the generation current (1.6). This means that J01 is a measure of the bulk recombination rate within one diffusion length; the stronger the bulk recombination (low t), the larger is J01. The name “diffusion current” suggests that this current contribution would be governed by transport properties, which is actually misleading. It is wrong to imagine the p–n junction as a kind of valve, where the magnitude of the current flow is only governed, for example, by the barrier height. Instead, it must be considered that, both under zero and under forward bias, the lateral carrier exchange between the n- and the p-side due to the thermal carrier movement is so strong that the magnitude of the net current is determined by the speed at which the carriers are recombining on the other side. Therefore, in some research groups, Eq. (1.8) is called “recombination current,” which is traditionally used for the depletion region recombination current, which will be described below. This still increases the linguistic confusion. Therefore, throughout this chapter, Eq. (1.8) will be called “diffusion current.” The correct way to imagine this current is to consider the quasi Fermi levels of electrons and holes as chemical potentials, which essentially horizontally cross the p–n junction, as it is described, for example, by Wu¨rfel (2005). Then it can easily be understood that each single recombination channel (e.g., bulk and surface recombination) leads to a sep- arate and independent contribution to the corresponding total J01. The same physics works for the hole exchange between the emitter and the base, Author's personal copy

12 Otwin Breitenstein

e which is not shown in Fig. 1.2, leading to J01 (“e” for emitter). Since the emitter (donor) doping concentration ND in the denominator of Eq. (1.5) is very high, the emitter contribution to J01 is often neglected compared to the base contribution. It will be shown in the next section and in Section 4.2 that this is not generally justified. Equation (1.6) actually only holds for a cell having a thickness much larger than the minority carrier diffusion length in the material. This is the case, for example, in the positions of recombination-active grain boundaries in multicrystalline solar cells. In good regions of these cells and also in monocrystalline cells, the base thickness is not large but rather small compared to the diffusion length, which is the presupposition for collecting most of the generated excess carriers. Then also the recombination at the back surface and/or the back contact contributes significantly to the bulk recombination and thus influences J01. In the limit of infinite diffusion length, the quasi Fermi level crosses the bulk horizontally, leading for the bulk contribution of J01 to the simple relation:

2 2 2 bulk back eni d eni sback eni d J01 J01 J01 1:9 ¼ þ ¼ NAtbulk þ NA ¼ NAteff ð Þ

Here sback is the recombination velocity of the backside and d is the bulk thickness. For the general case of arbitrary diffusion length, other expressions for J01 can be derived, which contain the influence of a finite bulk thickness and the front and back surface recombination velocities, see appendix of the PVCDROM (http://pveducation.org/pvcdrom). The use of these expressions is often avoided by replacing t in Eq. (1.6) or (1.9) by an effective bulk lifetime teff, which also includes the back surface recombination. A typical average value for teff of a monocrystalline silicon solar cell in today’s standard technology implying a full-area Al back contact is about 160 ms, and for a multicrystalline cell it is about 40 ms, leading after Eq. (1.6) to expected 2 values of the base contribution of J01 of about 500 and 1000 fA/cm , respectively. It must be reminded that the use of teff in combination with Eq. (1.6) only holds for low lifetime regions like grain boundaries. It suggests that the diffusion current is proportional to the inverse square root of the lifetime. For an infinitely thick cell, this square root dependence only stems from the fact that the recombination volume is proportional to the diffusion length Ld pt. Within this volume, the recombination rate is proportional to 1/t. In the same way, if the cell is thinner than Ld and the recombination volumeffiffiffi is constant, the recombination rate and thus also the Author's personal copy

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diffusion current is proportional to 1/teff and not to 1=pteff , as Eq. (1.9) shows. A saturation density of 1000 fA/cm2 corresponds according to 16 3 ffiffiffiffiffiffiffi Eq. (1.9) for d 200 mm and NA 10 cmÀ to an effective lifetime of ¼ ¼ 32 ms, compared to 40 ms according to Eq. (1.6). As Fig. 1.2C shows, the diffusion current under forward bias involves only the high-energy fraction of the electrons on the n-side. When these carriers arrive at the p-side, they have lost their kinetic energy due to the decelerating field, without having dissipated any heat. Due to this current flow, the mean temperature of the electron gas decreases, which instantly leads to a decrease of the crystal temperature. This is the physical reason for the Peltier cooling effect, which occurs at the p–n junction under forward bias (Breitenstein and Rakotoniaina, 2005).

2.3. Analysis of the bulk lifetime Excess carrier lifetime measurements are the base of semiconductor material characterization. The most popular method for doing this is quasi-steady- state photoconductance (QSSPC) (Sinton et al., 1996). Here a wafer or a device is exposed to a light pulse with slowly varying intensity, and the change of the conductance is measured inductively as a function of the light intensity. Alternatively, for high lifetimes, also the transient of the conductance after short pulse excitation may be measured and converted into a lifetime. If the electron and hole mobilities are known, the conductivity change may be converted into an excess carrier density, which is, for a given generation rate, proportional to the excess carrier lifetime. The dependence of this lifetime on the excess carrier density delivers valuable information on the recombination mechanisms. As a rule, this technique is used to investigate the bulk lifetime in surface-passivated wafers. However, it can also be used to investigate devices containing a p–n junction, if these devices are not metallized. If applied to wafers, the result of a QSSPC analysis describes the recombination properties of the bulk material and the surfaces. If QSSPC is applied to devices containing a p–n junction, the results also deliver infor- e mation on the emitter-part of the diffusion current J01, which describes the recombination in the emitter. Since this type of measurement is a standard tool for characterizing bulk and emitter recombination, its physical base will be reviewed here. A SRH recombination center is basically characterized by three param- eters, which are its energy position in the gap, usually expressed as the energy distance to the nearest band edge, and the capture coefficients for electrons Author's personal copy

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and for holes cn and cp, respectively. These capture coefficients, having the unit of cm3/s, are often, in analogy to particle physics, expressed as products of a capture cross section and the thermal velocity. Knowing these param- eters, the thermal emission rates (probabilities) for electrons and holes may be derived (Sze and Ng, 2007). For a center concentration of Nt, the capture coefficients cn and cp govern the lifetime of excess electrons or holes, under the condition that the center is occupied by a hole or an electron, respectively:

1 1 cnNt, cpNt 1:10 tn ¼ tp ¼ ð Þ

For many device analysis methods [e.g., in the popular solar cell simulation software PC1D (http://www.pv.unsw.edu.au/info-about/ our-school/products-services/pc1d)], instead of Nt, cn, and cp, only the two lifetimes tn and tp defined by Eq. (1.10) are given. Note, however, that these are not always real excess carrier lifetimes. The real lifetime still depends on the occupancy state of the center, as it will be described below. Since a center may only be occupied either by a hole or by an electron, the lifetimes (1.10) never hold both at the same time. For example, in p-type material all deep levels are occupied by holes. Hence, an incoming hole finds the center already occupied by a hole and this center does not reduce the hole lifetime. Therefore, in p-material, the hole lifetime is close to infinite, even if the material contains a high concentration of recombination centers and is described by a low tp according to Eq. (1.10). It is usually assumed that a center has two charge states, one if occupied by an electron and one if not, or one if not occupied by a hole and one if occupied, which is the same. This duality of electron and hole occupancy is only a definition, which also holds for energy bands: A band totally occupied by electrons (e.g., a valence band in an n-doped semiconductor) does not contain a significant amount of holes, and a band totally occupied by holes (e.g., a conduction band in a p-doped semiconductor) does not contain a significant amount of electrons. Independent of the energy position of an SRH center (in the upper or in the lower half of the gap) the charge state of the two possible occupation states governs whether a center is donor- or acceptor-like. If the charge state changes between 0 and (for being occupied by an electron or not), it is a donor, and if the chargeþ state changes between 0 and (for being occupied by a hole or not), it is an acceptor. À Hence, donor-like levels are, in ionized state (if occupied by a hole), Author's personal copy

The Physics of Industrial Crystalline Silicon Solar Cells 15 electrostatically attractive for electrons, hence for them c c holds, and n p acceptor-like levels are, if occupied by an electron, attractive for holes, leading to cp cn. Therefore, the often-made assumption cn cp (or tn tp) for a mid-gap level (e.g., made in McIntosh, 2001) is actually¼ unrealistic.¼ Deep centers may also have more than one occupancy state and correspondingly several energy levels, which will not be considered here. Since for shallow levels, like B and P in Si, at room temperature the thermal emission rates are higher than the capture rates, they are ionized at room temperature. For so-called deep levels (lying more than about 200 meV distant to the band edge), at room temperature and in the neutral material, the thermal emission may be neglected compared to thermal carrier capture. In the following section, we will concentrate on deep levels, hence we will neglect all terms related to thermal emission. In thermal equilibrium, the occupancy state of a center is given by the energy position relative to the Fermi level: All levels lying below the Fermi level are occupied by an electron (at 0 K), and all lying above the Fermi level are occupied by a hole. Under steady- state excitation condition, only the centers lying above the electron quasi-Fermi level are generally occupied by a hole and that lying below the hole quasi-Fermi level are generally occupied by an electron. For the deep centers lying between the quasi-Fermi levels, the electron and hole occupancy factors n and p depend on the ratio of the capture rates for electrons and holes and on the electron and hole concentrations n and p: c n c p n , p , 1 1:11 n ¼ c n c p p ¼ c n c p n þ p ¼ ð Þ n þ p n þ p For optical excitation always Dn Dp holds, since electron–hole-pairs ¼ are generated. Taking p0 NA, Eqs. (1.10) and (1.11) lead to the carrier dependence of the excess carrier¼ lifetime in p-material if governed by a single deep SRH level (Sze and Ng, 2007):

1 NA Dn cnNtp þ 1:12 tSRH ¼ ¼ tpDn tn NA Dn ð Þ þ ðÞþ This formula only holds for deep levels, where the thermal emission probability can be neglected. If also more shallow levels are considered, an analog formula also contains the concentrations n1 and p1, which are the electron and hole concentrations if the Fermi level coincides with the energy position of the levels (Sze and Ng, 2007). For deep levels, these concentrations are negligibly small. In the limit of small excess carrier Author's personal copy

16 Otwin Breitenstein

concentrations Dn, tSRH tn always holds. This is the low excitation limit of ¼ tSRH. It coincides with Eq. (1.10) because for low excitation intensity any deep level in p-material is occupied by a hole. If Dn increases, the behavior depends on the ratio of tp/tn cn/cp. For tn tp (acceptor-like center) in ¼ p-material also for high excitation levels tSRH tn holds, since the center remains mainly occupied by a hole. For tp tn (donor-like center), however, with increasing Dn the hole occupancy factor according to Eq. (1.11) decreases. Then tSRH increases with increasing Dn. If the lifetimes strongly deviate from each other, in a certain excess carrier concentration regime tSRH increases proportional to Dn. In the limit of high excitation, if tSRH tp tn holds, then the lifetime is governed by the larger of the two lifetimes.¼ þ This makes the center less recombination-active to incoming electrons. This effect is called “saturation of a SRH center.” Another interesting point is the influence of a p–n junction on the effective bulk lifetime. For example, the lifetime may be investigated in a wafer directly coming out of the POCl3-diffusion, which is completely sur- rounded by a p–n junction. Alternatively, a complete solar cell prior to met- allization may be investigated. In both cases, the electrically floating p–n junction influences the measured lifetimes. Since the emitter is very thin, such a QSSPC lifetime measurement only reflects the bulk properties. As a rule such lifetime investigations are performed as a function of the excess carrier concentration Dn (in p-type material). The effective bulk lifetime may be influenced by SRH recombination, by radiative and Auger recombination, by a recombination-active surface, and by the presence of a p–n junction: 1 1 1 1 1 1 1:13 teff ¼ tSRH þ trad þ tAuger þ tsurf þ tp n ð Þ À

The injection-level dependence of tSRH has been discussed above and trad plays only a minor role in silicon. However, since lifetime investigations are usually performed up to high-injection levels, tAuger has to be considered. For this mechanism, several parameterizations are available, which have been compared for example in Reichel et al. (2012a). The surface recombination is described by the surface recombination velocity s, which is often assumed to be independent of Dn and is related to the lifetime via (d bulk thickness): ¼ 1 s 1:14 tsurf ¼ d ð Þ Author's personal copy

The Physics of Industrial Crystalline Silicon Solar Cells 17

We will now discuss how an electrically floating p–n junction influences the bulk lifetime for various excess carrier concentrations. Under illumination, excess electrons are attracted by the emitter and flow into it, leading to a bias-independent photocurrent Jph; see Section 2.5. If the p–n junction is not shunted, this biases the emitter into forward direction until Voc- conditions are established. Then the emitter injects most of the electrons back to the base, which reduces the net electron current into the base. This is the reason why floating p–n junctions have already been used for passiv- ating surfaces. However, not all electrons are injected back. The photocurrent is balanced by the complete dark current, which is the sum of the bulk diffusion current described by Eq. (1.8) and the emitter diffusion current e characterized by J01: V Je Je exp 1 1:15 diff ¼ 01 V À ð Þ T Only the bulk diffusion current injects electrons back into the base, but the emitter diffusion current injects holes into the base, where they recombine. e Therefore, for a floating p–n junction, this current contribution Jdiff represents the loss of excess carriers due to recombination in the emitter. For the double emitter (sandwich) geometry or the single emitter plus passivated backside geometry mentioned above, it is usually assumed that the excess carrier concentration Dn is essentially homogeneous across the bulk thickness. Then, for V>VT, the exponential term containing V in Eq. (1.15) may be expressed by the excess carrier concentration, leading to

2 V e e DnNA Dn np DnNA Dn ni exp , Jdiff J01 ðÞ2þ 1:16 ¼ ðÞ¼þ VT ¼ ni ð Þ Now the excess electron loss at the floating p–n junction may be expressed in terms of a recombination velocity sp–n:

e e NA Dn Jloss Jdiff eDnsp n, sp n J01 þ2 1:17 ¼ ¼ À À ¼ eni ð Þ This leads together with Eq. (1.14) to the final result:

1 sp n e NA Dn À J01 þ2 1:18 tp n ¼ d ¼ edni ð Þ À This means that, for low excess carrier concentrations, the influence of the emitter on the lifetime is independent of Dn and is described by Author's personal copy

18 Otwin Breitenstein

e 2 1/tp–n J01NA/(edni ). If Dn comes into the order of NA (toward high injec- ¼ tion), 1/tp–n increases proportional to Dn; hence tp–n decreases proportional e to 1/Dn with the proportionality factor being proportional to J01 according to Eq. (1.18). In Section 4.2, an example of a lifetime analysis will be introduced. Equation (1.18) is the base of the so-called Kane–Swanson method, e which is the most common method for measuring J01 independent of b J01 (Kane and Swanson, 1985). Note that, if the lifetime depends on Dn, the measured excess carrier transient is not exponential anymore. Then the definition of the lifetime is the slope of the Dn-transient related to Dn at this time: @Dnt=@t t t ðÞ 1:19 ðÞ¼ Dnt ð Þ ðÞ This definition is used in most time-dependent (non-steady-state) methods for measuring lifetimes, and it is also the base for quasi-steady-state lifetime measuring methods like QSSPC and PL imaging. In the limit of high injection (Dn NA), a 1/t transient forms instead of an exponential one (Kane and Swanson, 1985). In any case, already the evaluation of e one single Dn(t) transient allows the measurement of J01 according to Eqs. (1.18) and (1.19), but the quasi-steady-state methods are equivalent.

2.4. Depletion region recombination In Fig. 1.2, generation and recombination are considered not only in the bulk, but also in the depletion region. This generation and recombination is most effective for mid-gap levels and is then locally confined to a narrow region in the middle of the depletion region, where in thermal equilibrium the Fermi level crosses the defect level (Sze and Ng, 2007). The effective width of this region should be w.AsFig. 1.2A shows, at zero bias in this region recombination and thermal generation occur at the same time in the same place, as in the neutral material. However, they occur with a significantly higher rate per volume, since the electron occupancy state of the mid-gap level is 1/2 here, whereas in the neutral p-material it is very small. Therefore, this current is often called “recombination-generation current” (Sah et al., 1957). As explained above, in this sense also J01 is a “recombination-generation current” of the neutral material and the surfaces. Again, under reverse bias (Fig. 1.2B), the depletion region generation current dominates over recombination, and under forward bias (Fig. 1.2C) recombination dominates over generation. The thermal generation rate Author's personal copy

The Physics of Industrial Crystalline Silicon Solar Cells 19 in the middle of the depletion region Gdr is calculated in analogy to Eq. (1.5) by replacing np by ni. The saturation current density J02 for the depletion region current is then calculated in analogy to Eq. (1.6), leading to:

dr ni dr eniw G , J02 G ew 1:20 ¼ tbulk ¼ ¼ tbulk ð Þ

Here tbulk is the bulk excess carrier lifetime, which should be larger than teff since it does not contain any surface contribution. Under forward bias, the quasi Fermi energies in a silicon cell are usually crossing the p–n junction 2 horizontally; therefore, in any position of the junction np ni exp(eV/kT) holds. This recombination occurs in the middle of the depletion¼ region, therefore in this region n p pnp ni exp(V/2VT) holds. Since the deple- ¼ ¼ ¼ tion region recombination current is proportional to n resp. p in this region, this leads together with Eq. (1.20)ffiffiffiffiffito the expression for the depletion region current: eV J J exp 1 1:21 rec ¼ 02 2kT À ð Þ The number “2” in the denominator of Eq. (1.21) is called the ideality factor of the depletion region current. Following the originally given name (Sah et al., 1957) and the convention in most textbooks (e.g., Sze and Ng, 2007), throughout this chapter we will call Eq. (1.21) the “recombination current” and Eq. (1.8) the “diffusion current” contribution of the dark current. Unfortunately, the effective recombination layer width w in Eq. (1.20) is not exactly known. In fact, the occupancy state of the mid-gap level is strongly position-dependent, and the extension of the recombination- generation region also depends on the bias V. This is the reason why, even for a mid-gap level, the ideality factor of the recombination current is expected to be slightly smaller than 2 (McIntosh, 2001; McIntosh et al., 2000; Sah et al., 1957). In McIntosh (2001), the graph shown in Fig. 1.3 was published ( J02 called J0DR here), which is based on realistic numerical device simulations using the same assumption of a mid-gap level as done here. It shows that, for a bulk conductivity of r bulk 1.5 O cm, which is ¼ typical for solar cells, and for a lifetime of about 40 ms, the expected value 11 2 of J02 should be about 5 10À A/cm . Note that this is significantly larger Â 12 2 2 than J01, which is expected to be 10À A/cm here (1000 fA/cm ). There- fore, at low forward bias, the recombination current always dominates over the diffusion current, but at higher forward bias the diffusion current Author's personal copy

20 Otwin Breitenstein

10-7

r bulk = 5 W. cm 10-8

r bulk = 1 W. cm 10-9

) r = 0.2 W. cm

2 bulk

10-10 (A/cm 0DR J 10-11

10-12

10-13 10-7 10-6 10-5 10-4 10-3 Carrier lifetime (s)

Figure 1.3 Numerical simulation of a diffused silicon junction. J0DR was only slightly affected by variation in the emitter profile. From McIntosh (2001). By courtesy of K.R. McIntosh. dominates. In the absence of ohmic currents, the expected effective ideality factor at low voltages should be about two, and at higher voltages it should be unity, as long as the base stays in low-injection condition, hence as long as n p holds there, and if the series resistance does not play a role yet. The bias, at which this transition occurs, strongly depends on the magnitudes of J01 and J02. In our example, it is expected to be about 0.2 V. Hence, at the maximum power point (mpp) of a solar cell, which typically is close to 0.5 V, the theoretically expected characteristic should not be influenced by the recombination current anymore.

2.5. Illuminated solar cell Until now only the current in the dark was considered. If a solar cell is illuminated and light is absorbed, electron–hole pairs are generated, which are excess carriers. This optical carrier generation acts exactly like the spontaneous thermal carrier generation considered for Fig. 1.2, except that it is many orders of magnitude more intense and is independent of the excess carrier lifetime. Just as for the thermal generation current (1.6), the photocurrent is independent of the bias V. In the case of an infinitely thick solar cell and a homogeneous optical carrier generation rate G, the photocurrent also Author's personal copy

The Physics of Industrial Crystalline Silicon Solar Cells 21 can be described for a thick cell as J GeL . Like the thermal generation ph ¼ d current (1.6), the photocurrent Jph is a reverse current. It superimposes on the bias-dependent dark current described by Eqs. (1.8) and (1.21), which is called the superposition principle, leading to:

eV eV J J exp 1 J exp 1 J 1:22 ¼ 01 kT À þ 02 2kT À À ph ð Þ Here the first diode term with the ideality factor of 1 describes the diffusion current, which is finally due to the recombination in the bulk and emitter material and at the surfaces, and the second diode term with the ideality factor of 2 describes the recombination current, which is due to recombination in the depletion region. If the cell is under short circuit (V 0), these two dark current contributions are zero and J J holds. ¼ ¼ ph Therefore, Jph is called the short-circuit current density Jsc. If the cell is at open circuit, as a rule the first diode term in Eq. (1.22) dominates the dark current. Neglecting the second diode term in Eq. (1.22) and the “ 1” in the À first diode term, the condition J 0 leads to the relation for the open-circuit voltage: ¼

kT J V ln sc 1:23 oc ¼ e J ð Þ 01

This equation shows that, for obtaining a high open-circuit voltage, J01 must be as small as possible. This underlines the importance of the dark current for maximizing the efficiency of a solar cell: By minimizing recombination in the cell, the dark current in a solar cell has to be as small as possible. In fact, the dark current is one of the major enemies of the solar cell maker. The I–V characteristic of real solar cells is also influenced by an inevitable series resistance Rs of the device (being the second major enemy of the solar cell maker), which leads to the fact that the so-called “local voltage” directly at the p–n junction deviates from the voltage V applied to the device. Though the diode theory outlined above does not explain any ohmic conductivity, experience has shown that all solar cells show a noninfinite parallel 2 resistance Rp. Typical values of Rp are between some or some 10 O cm (heavily shunted cells) and some 104 O cm2 (faultless cells, see, e.g., Kaminski et al., 1996). The reasons for this ohmic conductivity will be discussed in Section 4.3. It will be shown in Sections 3 and 4.1 that the ideality factor of the second diode is often larger than two and therefore expressed as Author's personal copy

22 Otwin Breitenstein

a variable named n2. Thus, the final two-diode equation, which is widely accepted for describing real solar cells, reads:

V R J V R J eV R J J J exp À s 1 J exp À s 1 À s J ¼ 01 V À þ 02 n V À þ R À sc T 2 T p 1:24 ð Þ Again VT kT/e is the thermal voltage being 25.69 mV at T 25 C. As mentioned above,¼ in many cases n 2 is assumed in Eq. (1.24)¼. Note also 2 ¼ that, since Eq. (1.24) is a current density, the resistances Rs and Rp are expressed here area-related in units of O cm2. The implications of this approach will be discussed in Section 4.5. Note also that Eq. (1.24) is an implicit equation for J(V), which complicates practical calculations. There- fore, in a limited bias range (usually between the mpp and Voc), it is often simplified to the empirical “one-diode” solar cell equation, again neglecting Rs and containing effective values for J0 and the ideality factor n: V JV Jeff exp 1 J 1:25 ðÞ¼ 0 neff V À À sc ð Þ T In this equation, the influence of ohmic and recombination (second diode) eff eff current contributions is contained in J0 and n . This effective ideality factor neff is that of the whole current and not only of the recombination current. If a real cell characteristic is fitted to Eq. (1.25) for each bias V separately, this leads to the bias-dependent ideality factor n(V), which is very useful for analyzing the conduction mechanism of solar cells (see Section 4.1). The series resistance Rs in Eq. (1.24) contains contributions from the grid lines, from the contact resistances, from the horizontal current flow in the emitter layer, and from the current flow in the base. In a module, contributions from the busbar connection and the interconnection wiring must also be added. It will be shown in Section 4.5 that the use of a constant value for Rs,withthesame value in the dark and under illumination, is actually wrong and represents only acoarseapproximation.Nevertheless,thisapproximationisoftenmade.Typical 2 values for Rs are between 0.5 and 1 O cm (see, e.g., Kaminski et al., 1996). Note that for correctly measuring Rp of a solar cell, the two-diode equation (1.24) has to be fitted to a measured dark or illuminated I–V characteristic. It is not sufficient to evaluate only the linear part of the dark characteristic for low voltages and to interpret the slope as the inverse of Rp, as this is often being done. For low voltages, the two exponential terms in Eq. (1.24) may be developed in a power series. For small values of V, both Author's personal copy

The Physics of Industrial Crystalline Silicon Solar Cells 23 lead to a linear characteristic near V 0. For R the apparent (effective) ¼ p ¼1 parallel resistance is:

eff 1 n2VT Rp 1:26 ¼ J01 J02 J02 ð Þ VT þ n2VT The latter relation holds due to the fact that always J J holds. It is 02 01 also not correct to measure Rp from the slope of the illuminated characteristic close to V 0, as this is also often being done. In this case, certain depar- tures from the¼ superposition principle may lead to erroneous results, which will be discussed in Section 4.5.

2.6. Reverse current Under large reverse bias, Eq. (1.24) is not valid anymore, since any p–n junction breaks down at a certain reverse bias. Moreover, since J02 J01 holds, the thermal carrier generation in the depletion region governs the reverse current. For this case, the second exponential diode term in Eq. (1.24) is only an approximation. Under reverse bias, the generation region widens and becomes nearly homogeneous within the whole depletion width W, which increases with increasing reverse bias according to Eq. (1.3). Therefore, over a rather large bias range, as long as there is no avalanche multiplication yet, considering Eqs. (1.3) and (1.20), the reverse current Jr should increase according to Sze and Ng (2007).

dr eniW ni 2eee0 Vr Vd Jr Vr eG W ðÞþ 1:27 ðÞ¼ ¼ tbulk ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitbulkpNA ð Þ This means that, under reverse bias, the reverse currentffiffiffiffiffiffiffi should be in the order of J02 and should increase sub-proportionally to Vr. If the electric field in the depletion region exceeds a certain limit, the carriers are multiplied by the avalanche effect, leading to a steep increase of the reverse current (breakdown). According to Miller (1957) the avalanche multiplication factor can be described by: 1 MC V 1:28 ðÞ¼1 V =V m ð Þ À ðÞr b Here Vb is the breakdown voltage, and m is the Miller exponent, often assumed to be m 3. For Vr Vb,MC holds, which is the basic def- ¼ ¼ ¼1 16 3 inition of Vb. For a typical base doping concentration of 10 cmÀ and a Author's personal copy

24 Otwin Breitenstein

plane silicon junction, Vb is expected to be about 60 V (Sze and Gibbons, 1966; Sze and Ng, 2007). Thus, the theoretically expected reverse current of a solar cell should be the product of Eqs. (1.27) and (1.28). Band-to-band tunneling under reverse bias (internal field emission, Zener effect) should not play any role for silicon solar cells, since it dominates over avalanche 17 3 multiplication only for a base doping concentration above 5 10 cmÀ Â (Sze and Ng, 2007), which is significantly higher than that used for typical solar cells. However, trap-assisted tunneling may be considered to be responsible for certain pre-breakdown phenomena (see Section 4.4).

3. THEORY VERSUS EXPERIMENT Based on the theoretical predictions summarized in Section 2, now the theoretically expected dark and illuminated I–V characteristics of a typical multicrystalline silicon solar cell with an effective bulk lifetime of 40 ms, 2 corresponding to J01 1000 fA/cm , will be calculated and compared with experimentally measured¼ characteristics of a typical industrial cell. Since the classic diode theory does not explain any parallel resistance, Rp will be assumed here. The results are presented in Fig. 1.4. This cell¼ is1 a typical 156 156mm2 sized cell made in an industrial production line by the pres- Â ently (2012) dominating cell technology (50 O/sq emitter, acidic texturization, full-area Al back contact, 200 mm thickness) from Bridgman- type multicrystalline solar-grade silicon material. The same cell is used for the comparison between dark and illuminated characteristics in Section 4.5. For calculating the theoretical illuminated characteristic, the value of 2 Jsc 33.1 mA/cm from the experimentally measured illuminated character- ¼ 2 istic of this cell was used. The series resistance of Rs 0.81 O cm was calculated from the voltage difference between the measured¼ open-circuit voltage (0.611 V) and the dark voltage necessary for a dark current equal to the short- circuit current (0.638 V), which is an often used procedure for measuring Rs: V J R J V 1:29 darkðÞsc À s sc ¼ oc ð Þ In this case, R 0.81 O cm2 fulfilled condition (1.29). For the bulk life- s ¼ time in Eq. (1.27), as a lower limit the assumed effective bulk lifetime of 40 ms was used. It is visible in Fig. 1.4 that, in the dark forward characteristic (A), the low voltage range (V<0.5 V) shows the strongest deviation between theory and experiment. The measured current in this bias range is governed by the second diode and by ohmic shunting. In the theoretical curve, there was no Author's personal copy

The Physics of Industrial Crystalline Silicon Solar Cells 25

Figure 1.4 Comparison of experimentally measured and theoretically predicted (A) dark forward, (B) illuminated, and (C) dark reverse I–V characteristics. After Breitenstein (2013), by courtesy of Springer. ohmic shunting assumed, and the second diode contribution is so small that it is not visible in the displayed data range. Also in the cell used for these characteristics the ohmic shunting is very low. It will be demonstrated in Section 4.5 that the shown experimental dark characteristic can be described 2 8 2 by values of R 44.4 kO cm , J 5.17 10À A/cm and n 2.76. p ¼ 02 ¼ Â 2 ¼ Hence, there is some nonnegligible ohmic conductivity in this cell, J02 is several orders of magnitude larger than the predicted value of 11 2 5 10À A/cm , and its ideality factor n is larger than the expected max- Â 2 imum value of two. The transition between the J02- and the J01-dominated part of the dark characteristic is close to the mpp near 0.5 V. This proves that in this cell the recombination current already influences the fill factor of this cell, even at full illumination intensity. This result is typical for industrial solar cells and has often been published (see, e.g., Kaminski et al., 1996). The reasons for these discrepancies to the theoretically expected behavior will be discussed in Sections 4.1 and 4.3. Also the experimental value of J01, which governs the dark characteristic for V>0.5 V, is somewhat larger than theoretically expected. The reason for this discrepancy will be discussed Author's personal copy

26 Otwin Breitenstein in Section 4.2.Sincethedarkcurrentwasunderestimatedbytheory,alsothe illuminated characteristics in Fig. 1.4B significantly deviate between theory and experiment. Both the open-circuit voltage Voc and the fill factor are in reality smaller than theoretically estimated. This graph also contains an illuminated characteristic, which was simulated based on the experimental dark characteristic by applying the superposition principle and regarding a constant series resistance of 0.81 O cm2, both in the dark and under illumination. In the dotted curve in Fig. 1.4B Voc is correctly described (it was used to calculate Rs), but the simulated fill factor appears too large. This discrepancy will be resolved in Section 4.5. Finally, Fig. 1.4C shows the theoretical and experimental dark reverse characteristics. Also these curves deviate drastically. The theoretical dark current density is negligibly small 2 in the displayed current range for Vr <50 V (in the nA/cm range, subli- nearly increasing with Vr), and the breakdown occurs sharply at V 60 V. In the experimentally measured curve, on the other hand, the r ¼ reverse current increases linearly up to Vr 5 V and then increases super- linearly, showing a typical “soft breakdown”¼ behavior. A sublinear increase, as predicted by theory, is not visible at all. It will be described in Section 4.4 how this reverse characteristic can be understood. In the following sections, the present state of understanding the different aspects of the nonideal behavior of industrial crystalline silicon solar cells, in particular of cells made from multicrystalline material, will be reviewed, and a selection of experimental results leading to this understanding will be presented. All results regarding the edge region or technological problems (e.g., Al particles at the surface, scratches) hold both for mono- and multicrystalline cells. On the other hand, all results dealing with crystal lattice defects or precipitates only hold for multicrystalline cells. In the last years, “quasi-mono” or “quasi-single crystalline” solar silicon material has also appeared (Gu et al., 2012). This material does not contain large angle grain boundaries, but it may contain a high concentration of dislocations and also low angle grain boundaries, which are basically rows of dislocations. This material is obviously lying anywhere between multi and mono material; therefore, the conclusions from this chapter should also be valid for this kind of material.

4. ORIGINS OF NONIDEAL CHARACTERISTICS In the following subsections, the different aspects of the nonideal behavior of industrial solar cells are separately discussed and the physical origins for this behavior are revealed. It will be shown that all these deviations