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CHAPTER ONE
The Physics of Industrial
Crystalline Silicon Solar Cells
Otwin Breitenstein
Max Planck Institute of Microstructure Physics, Halle, Germany
Contents
1. Introduction and Chapter Methodology
2. Basic Theory of Solar Cells
2.1 Solar cell in thermal equilibrium
2.2 Biased solar cell
2.3 Analysis of the bulk lifetime
2.4 Depletion region recombination
2.5 Illuminated solar cell
2.6 Reverse current
3. Theory Versus Experiment
4. Origins of Nonideal Characteristics
4.1 The depletion region recombination (second diode) current
4.2 The diffusion (first diode) current
4.3 The ohmic current
4.4 The reverse current
4.5 Relation between dark and illuminated characteristics
5. Summary and Outlook
Acknowledgments
References
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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 applications. While the first silicon solar cell made in 1953 had an energy conversion 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
Semiconductors and Semimetals, Volume 89
ISSN 0080-8784
http://dx.doi.org/10.1016/B978-0-12-381343-5.00001-X
#
2013 Elsevier Inc.
All rights reserved.
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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 (Müller 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
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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; Würfel, 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 replaced 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 investigating local leakage currents in integrated circuits (Breitenstein et al., 2000)
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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 a solar cell is closely related to its dark characteristic, 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.
“nþ–p” means that the n-side is much more highly doped (up to 1020 cm#3)
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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 ¼ metal),
not to scale.
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Otwin Breitenstein
than the p-side of the junction (typically 1016 cm#3), 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 (dashdotted 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
!
"
!
"
EF # E c
Ev # EF
n ¼ Nc exp
, p ¼ Nv exp
kT
kT
ð1:1Þ
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 ¼ ND and in the neutral p-region p ¼ NA holds, with ND being the donor
doping concentration at the n-side and NA the acceptor doping
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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
ð1:2Þ
EF # Ev ¼ xp ¼ kT ln
NA
For a typical base doping concentration of 1016 cm#3, 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
for ND ¼ 2 ! 1020 cm#3 to xn ' #50 meV, hence the Fermi level is actually
lying in the conduction band 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 derivative of the potential to x, hence to the bending of the potential. 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:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ee0 ðVr þ Vd Þ
W¼
ð1:3Þ
eNA
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:
Vd ¼
Eg # x p # x n
e
ð1:4Þ
Also at the ohmic contacts depletion regions exist. In some older textbooks, ohmic contacts are still described as carrier accumulation regions,
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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. Second, 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 schematically 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
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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 highenergy 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
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
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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 ¼ n2i /NA (ni ¼ intrinsic carrier concentration,
NA ¼ acceptor concentration), leading to an expression for the thermal generation rate G:
np ¼ Gt ¼
n2i
n2
, G¼ i
NA
t NA
ð1:5Þ
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
pffiffiffiffiffiffiffi
generated within one diffusion length Ld ¼ De t (De ¼ electron diffusion
constant in the p-region). This horizontal current density can be expressed
regarding (1.5) as
pffiffiffiffiffi
en2i Ld n2i e De
pffiffi
J01 ¼ GeLd ¼
ð1:6Þ
¼
tNA
NA t
Since this current density is independent of the 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 nto 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 exponentially with increasing forward bias V, since correspondingly more electrons
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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
Jdiff ¼ J01 exp
ð1:7Þ
kT
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 ¼ J01 exp
#1
ð1:8Þ
kT
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 Würfel (2005). Then it can easily be understood that each single
recombination channel (e.g., bulk and surface recombination) leads to a separate and independent contribution to the corresponding total J01. The same
physics works for the hole exchange between the emitter and the base,
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Otwin Breitenstein
which is not shown in Fig. 1.2, leading to Je01 (“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:
bulk
back
þ J01
¼
J01 ¼ J01
en2i d
en2 sback
en2 d
þ i
¼ i
NA
NA tbulk
NA teff
ð1:9Þ
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
values of the base contribution of J01 of about 500 and 1000 fA/cm2, 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
pffiffi
length Ld ) t. 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 volume is constant, the recombination rate and thus also the
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pffiffiffiffiffiffi
diffusion current is proportional to 1/teff and not to 1= teff , as Eq. (1.9)
shows. A saturation density of 1000 fA/cm2 corresponds according to
Eq. (1.9) for d ¼ 200 mm and NA ¼ 1016 cm#3 to an effective lifetime of
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-steadystate 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 infore
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 parameters, 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
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Otwin Breitenstein
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 parameters, 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
¼ cn Nt ,
¼ cp Nt
tn
tp
ð1:10Þ
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 þ (for being occupied by an electron or not), it is a donor, and if the charge state changes
between 0 and # (for being occupied by a hole or not), it is an acceptor.
Hence, donor-like levels are, in ionized state (if occupied by a hole),
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electrostatically attractive for electrons, hence for them cn ( cp holds, and
acceptor-like levels are, if occupied by an electron, attractive for holes, leading to cp ( cn. Therefore, the often-made assumption cn ¼ cp (or tn ¼ tp) for a
mid-gap level (e.g., made in McIntosh, 2001) is actually unrealistic. 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 steadystate 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:
!n ¼
cp p
cn n
, !p ¼
, ! þ !p ¼ 1
cn n þ cp p n
cn n þ cp p
ð1:11Þ
For optical excitation always Dn ¼ Dp holds, since electron–hole-pairs
are generated. Taking p0 ¼ NA, Eqs. (1.10) and (1.11) lead to the carrier
dependence of the excess carrier lifetime in p-material if governed by a single
deep SRH level (Sze and Ng, 2007):
1
NA þ Dn
¼ cn Nt !p ¼
tSRH
tp Dn þ tn ðNA þ DnÞ
ð1:12Þ
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
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Otwin Breitenstein
concentrations Dn, tSRH ¼ tn always holds. This is the low excitation limit of
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 ¼ cn/cp. For tn ( tp (acceptor-like center) in
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 ¼ tp þ tn holds, then the lifetime is governed by the larger of the
two lifetimes. 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 surrounded by a p–n junction. Alternatively, a complete solar cell prior to metallization 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
¼
þ
þ
þ
þ
teff tSRH trad tAuger tsurf tp#n
ð1:13Þ
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 ¼ bulk thickness):
1
tsurf
¼
s
d
ð1:14Þ
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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 Vocconditions 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 passivating 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
e
e
ð1:15Þ
Jdiff ¼ J01 exp # 1
VT
Only the bulk diffusion current injects electrons back into the base, but the
emitter diffusion current injects holes into the base, where they recombine.
Therefore, for a floating p–n junction, this current contribution J ediff 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
np ¼ DnðNA þ DnÞ ¼ n2i exp
V
e
e DnðNA þ DnÞ
, Jdiff
¼ J01
VT
n2i
ð1:16Þ
Now the excess electron loss at the floating p–n junction may be
expressed in terms of a recombination velocity sp–n:
e
e
¼ eDn sp#n , sp#n ¼ J01
Jloss ¼ Jdiff
NA þ Dn
en2i
ð1:17Þ
This leads together with Eq. (1.14) to the final result:
sp#n
1
e NA þ Dn
¼ J01
¼
d
tp#n
edn2i
ð1:18Þ
This means that, for low excess carrier concentrations, the influence
of the emitter on the lifetime is independent of Dn and is described by
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Otwin Breitenstein
1/tp–n ¼ J e01NA/(edn2i ). If Dn comes into the order of NA (toward high injection), 1/tp–n increases proportional to Dn; hence tp–n decreases proportional
to 1/Dn with the proportionality factor being proportional to J e01 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,
which is the most common method for measuring J e01 independent of
J b01 (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:
tðt Þ ¼
@Dnðt Þ=@t
Dnðt Þ
ð1:19Þ
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
one single Dn(t) transient allows the measurement of J e01 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. As Fig. 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
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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:
Gdr ¼
ni
tbulk
, J02 ¼ Gdr ew ¼
eni w
tbulk
ð1:20Þ
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
horizontally; therefore, in any position of the junction np ¼ n2i exp(eV/kT)
holds. This recombination occurs in the middle of the depletion region,
pffiffiffiffi
therefore in this region n ¼ p ¼ np ¼ ni exp(V/2VT) holds. Since the depletion region recombination current is proportional to n resp. p in this region,
this leads together with Eq. (1.20) to the expression for the depletion region
current:
!
!
"
"
eV
Jrec ¼ J02 exp
#1
ð1:21Þ
2kT
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 recombinationgeneration 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 ¼ 1.5 O cm, which is
typical for solar cells, and for a lifetime of about 40 ms, the expected value
of J02 should be about 5 ! 10#11 A/cm2. Note that this is significantly larger
than J01, which is expected to be 10#12 A/cm2 here (1000 fA/cm2). Therefore, at low forward bias, the recombination current always dominates over
the diffusion current, but at higher forward bias the diffusion current
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Otwin Breitenstein
10-7
r bulk = 5 W. cm
10-8
r bulk = 1 W. cm
J0DR (A/cm2)
10-9
r bulk = 0.2 W. cm
10-10
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
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can be described for a thick cell as Jph ¼ GeLd. Like the thermal generation
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:
J ¼ J01
!
! "
"
!
!
"
"
eV
eV
exp
# 1 þ J02 exp
# 1 # Jph
kT
2kT
ð1:22Þ
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 ¼ 0), these two dark current contributions are zero and J ¼ Jph holds.
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
first diode term, the condition J ¼ 0 leads to the relation for the open-circuit
voltage:
Voc ¼
! "
kT
Jsc
ln
J01
e
ð1:23Þ
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
resistance Rp. Typical values of Rp are between some or some 10 O cm2
(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
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Otwin Breitenstein
a variable named n2. Thus, the final two-diode equation, which is widely
accepted for describing real solar cells, reads:
!
!
"
"
!
!
"
"
V # Rs J
V # Rs J
eV # Rs J
J ¼ J01 exp
# 1 þ J02 exp
#1 þ
# Jsc
VT
n2 VT
Rp
ð1:24Þ
Again VT ¼ kT/e is the thermal voltage being 25.69 mV at T ¼ 25 + C. As
mentioned above, in many cases n2 ¼ 2 is assumed in Eq. (1.24). Note also
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. Therefore, 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
eff
J ðV Þ ¼ J0 exp eff # 1 # Jsc
ð1:25Þ
n VT
In this equation, the influence of ohmic and recombination (second diode)
current contributions is contained in J0eff and neff. 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, with the same
value in the dark and under illumination, is actually wrong and represents only
a coarse approximation. Nevertheless, this approximation is often made. Typical
values for Rs are between 0.5 and 1 O cm2 (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
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lead to a linear characteristic near V ¼ 0. For Rp ¼ 1 the apparent (effective)
parallel resistance is:
Rpeff ¼
1
n2 VT
'
J01
J02
J02
þ
VT n2 VT
ð1:26Þ
The latter relation holds due to the fact that always J02 ( J01 holds. It is
also not correct to measure Rp from the slope of the illuminated characteristic close to V ¼ 0, as this is also often being done. In this case, certain departures from the 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).
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
eni W ni 2eee0 ðVr þ Vd Þ
dr
pffiffiffiffiffiffi
Jr ðVr Þ ¼ eG W ¼
¼
ð1:27Þ
tbulk
tbulk NA
This means that, under reverse bias, the reverse current 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:
MCðV Þ ¼
1
1 # ðVr =Vb Þm
ð1:28Þ
Here Vb is the breakdown voltage, and m is the Miller exponent, often
assumed to be m ¼ 3. For Vr ¼ Vb, MC ¼ 1 holds, which is the basic definition of Vb. For a typical base doping concentration of 1016 cm#3 and a
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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
multiplication only for a base doping concentration above 5 ! 1017 cm#3
(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,
corresponding to J01 ¼ 1000 fA/cm2, will be calculated and compared with
experimentally measured characteristics of a typical industrial cell. Since the
classic diode theory does not explain any parallel resistance, Rp ¼ 1 will be
assumed here. The results are presented in Fig. 1.4. This cell is a typical
156 ! 156 mm2 sized cell made in an industrial production line by the presently (2012) dominating cell technology (50 O/sq emitter, acidic
texturization, full-area Al back contact, 200 mm thickness) from Bridgmantype 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
Jsc ¼ 33.1 mA/cm2 from the experimentally measured illuminated characteristic of this cell was used. The series resistance of Rs ¼ 0.81 O cm2 was calculated from the voltage difference between the measured open-circuit voltage
(0.611 V) and the dark voltage necessary for a dark current equal to the shortcircuit current (0.638 V), which is an often used procedure for measuring Rs:
Vdark ð Jsc Þ # Rs Jsc ¼ Voc
ð1:29Þ
In this case, Rs ¼ 0.81 O cm2 fulfilled condition (1.29). For the bulk lifetime 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
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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
by values of Rp ¼ 44.4 kO cm2, J02 ¼ 5.17 ! 10#8 A/cm2 and n2 ¼ 2.76.
Hence, there is some nonnegligible ohmic conductivity in this cell, J02 is
several orders of magnitude larger than the predicted value of
5 ! 10#11 A/cm2, and its ideality factor n2 is larger than the expected maximum 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
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Otwin Breitenstein
in Section 4.2. Since the dark current was underestimated by theory, also the
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
in the displayed current range for Vr < 50 V (in the nA/cm2 range, sublinearly increasing with Vr), and the breakdown occurs sharply at
Vr ¼ 60 V. In the experimentally measured curve, on the other hand, the
reverse current increases linearly up to Vr ¼ 5 V and then increases superlinearly, showing a typical “soft breakdown” 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