Dual-junction a-Si/a-Si (same band-gap tandem) solar cells have lower material cost (no GeH4) than tandem cells using a-SiGe, and have slightly higher efficiencies (0.5-1 % absolute) than a-Si single junction cells. Same band-gap a-Si/a-Si tandems have been in production for decades. Multiband-gap dual junction (a-Si/a – SiGe: Doughty and Gallagher 1990, or a-Si/nc-Si Yamamoto et al. 2004; Meier et al. 2007) and triple-junction (a-Si/a-SiGe/a-SiGe or a-Si/a-SiGe/nc-Si) solar cells use spectrum-splitting. They achieve higher conversion efficiencies, usually over 10 % stabilized for small area <1 cm2 cells (Doughty and Gallagher 1990; Kroll et al. 2007; Fujioka et al. 2006; Mahan et al. 2001). While all-amorphous a-Si (1.8 eV)/a-SiGe (1.6 eV)/a-SiGe (1.4 eV) triple-junction solar cells have produced the most efficient a-Si-based cells today (Yang et al. 1997), triple junctions based on a-Si/a-SiGe/nc-Si or a-Si/nc-Si/nc-Si (no Ge) have produced nearly equivalent stabilized efficiencies (Kroll et al. 2007; Mahan et al. 2001; Yang et al. 2005).

[1]A better fit for the Born repulsion is obtained by the sum of a power and an exponential law:

VBorn = rm + Y exp^ – y^j. (1.2)

Г0 is the softness parameter, listed for ions in Table 1.7. For more sophisticated repulsion potentials, see Shanker and Kumar (1987). в is the force constant (see Eq. (1.1)) and m is an empirical exponent. For ionic crystals the exponent m lies between 6 and 10.

[2]The promotion energy is 4.3, 3.5, and 3.3 eV for C, Si, and a-Sn, respectively. However, when forming bonds by establishing electron bridges to neighboring atoms, a substantially larger energy is gained, therefore resulting in net binding forces. Diamond has the highest cohesive energy in this series, despite the fact that its promotion energy is the largest because its sp3-sp3 C-C bonds are the strongest (see Harrison 1980).

[3]Meaning compounds between one element of group III and one element of group V on the periodic system of elements.

[4]This empirical quantity can be defined in several ways (e. g., as Mohs, Vickers, or Brinell hardness) and is a macroscopic mechanical representation of the cohesive strength of the lattice. In Table 1.9 the often used Mohs hardness is listed, which orders the listed minerals according to the ability of the higher-numbered one to scratch the lower-numbered minerals.

[5]A is the face spun between b and c, B between a and c, and C between a and b.

[6]There are other modifications possible. For example, seven for Si, of which four are stable at room temperature and ambient pressure (see Landolt and Bornstein 1982a, 1982b; Landolt and Bornstein 1987). Only Si I and a-Si are included in this book. Si III is face-centered cubic and a semimetal; Si IV is hexagonal diamond and is a medium gap semiconductor (see Besson et al. 1987). Wurtzite lattice (CdS) are constructed from two intertwined hexagonal sublattices of Cd and S.

[7]The surface of glasses, even at very high magnification, does not show any characteristic structure;

after fracture, glasses show no preferred cleavage planes whatsoever.

[9]When grown from an evaporation source, spiral growth is preferred with a substrate perpendicular to the source (e. g. CdS). This requires that the source remains stationary to produce large crystallites, and makes continuous growth on a band disadvantageous as it tends to produce small crystallites in various orientations.

[10]An isocoric P in a Si lattice can be thought of as “created” by adding to a lattice atom a proton, i. e., a point charge, and an extra electron (the donor electron, Sect. 4.2), thereby creating the most ideal hydrogen-like defect. Any other hydrogen-like donor, e. g., As or Sb in Si, is of different size, causing more lattice deformation and a substantially different core potential 1.

[11]That is, a defect that can act as a donor or acceptor depending on the chemical potential of the lattice (influenced, e. g., by optical excitation or other doping, by compensation).

[12]The notation of charges with respect to the neutral lattice was introduced by Kroger, Vink, and Schottky (see Schottky and Stockmann 1954). This notation should not be confused with the charge identification used in an ionic lattice, e. g., Na+ Cl- Inclusion of a Cd++ instead of a Na+ ion makes the cadmium ion singly positively charged with respect to the neutral lattice; hence it is identified here as CdNa when referenced specifically as a lattice defect.

[13]Electron-beam-induced conductivity, used in a scanning electron microscope (Heydenreich et al. 1981).

[14]The electron affinity of a semiconductor is defined as the energy difference from the lower edge of the conduction band to the vacuum level, i. e., the energy gained when an electron is brought from infinity into the bulk of a crystal, resting at Ec. It should be distinguished from the electron affinity of an atom, which is equal to the energy gained when an electron is brought from infinity to attach to an atom and forms an anion.

[15]The de Broglie wavelength is on the same order of magnitude as the uncertainty distance obtained from Heisenberg’s uncertainty principle Ax < hAp, which has the same form as Adb. This yields uncertainty distances of =10 A for thermal (free) electrons at room temperature.

[16]In one dimension, there are other periodic potentials for which the Schrodinger equation can be integrated explicitly. V(x) = —Vo sech2(y’x) is one such potential, which yields solutions in terms of hypergeometric functions (see Mills and Montroll 1970). The results are quite similar to the Kronig-Penney potential discussed later.

[17]For E > V0, the square root in в becomes imaginary. Introducing у ± ^/[2m0(E — УсО/й2], and with sinh(iy) = і sin(y) and cosh(iA) = і cos(y), we obtain for higher electron energies a similar equation:

sin(Ya2) sin(aa1) + cos(Ya2) cos(aaO = cos(ka).

[18]In an infinite crystal the electron (when not interacting with a localized defect) is not localized and is described by a simple wavefunction (i. e., having one wavelength and the same amplitude throughout the crystal). The probability of finding it is the same throughout the crystal (<хФ2). When localized, the electron is represented by a superposition of several wavefunctions of slightly different wavelengths. The superposition of these wavefunctions is referred to as a wave packet.

[19]For the electron behavior, only expectation values can be given. In order to maintain Newton’s second law, we continue to use hk [Eq. (6.15)], which is no longer an electron momentum. It is well-defined within the crystal and is referred to as crystal momentum. We then separate the electron properties from those of the crystal by using d2E/dk2 to define its effective mass.

[20]In theory, the electron will continue to accelerate in the opposite direction to the field and lose energy, thereby descending in the band, and the above-described process will proceed in the reverse direction until the electron has reached the lower band edge, where the entire process repeats itself. This oscillating behavior is called the Bloch oscillation. Long before the oscillation can be completed, however, scattering interrupts the process. Whether in rare cases (e. g., in narrow minibands of superlattices or ultrapure semiconductors at low temperatures) such Bloch oscillations are observable, and whether they are theoretically justifiable in more advanced models (Krieger and Iafrate 1986) is still controversial. In three-dimensional lattices, other bands overlap and transitions into these bands complicate the picture.

[21]This concept must be used with caution, since k is a good quantum number only when electrons can move without scattering over at least several lattice distances. That is certainly not the case

AB-compounds containing these elements are referred to as pnictides or chalcogenides.

[23]In a quantum-mechanical picture any E(k) state represents a certain mass and velocity. In a filled band, all of them add up to zero.

[24]A positive charge was arbitrarily related to the charge of a glass rod, rubbed with silk (by Benjamin Franklin); this charge was not caused by an added electron as it became known later, but by a missing electron on the glass rod. This electron was removed by the silk.

[25]In this and all chapters dealing with electrical conductivity, the electric field is identified as F and the energy as E.

[26]The difference between metals, where the overlap range is allowed, and semiconductors, where the overlap range is forbidden (band gap), depends on Wigner’s rules (Wigner 1959), which state that eigenstates belonging to different symmetry groups of the Hamiltonian cannot mix (metals). In semiconductors they do mix, yielding sp3-hybridization for Si.

[27]Each band contains a large number of energy levels. Increasing numbers of electrons first fill the levels at the lowest energy (for T = 0) and then successively higher and higher energies. This process is referred to as band filling.

[28]There are also Dresselhaus parameters L, M, and N (Dresselhaus et al. 1955) to describe the valence band. They are related to the Luttinger parameters by Eq. (7.14).

[29]Here cyclotron resonance is discussed within the same band, and quantum effects are neglected. This can be justified when, neglecting scattering, each electron describes full circles which have to be integers of its De Broglie wavelength [Eq. (7.6)]. This integer represents the quantum number nq of the circle; and for the magnetic induction discussed here it is a large number. Resonance means absorption (or emission) of one quantum ha>c, hence changing nq by Anq =±1, which is the selection rule for cyclotron transitions. Since Anq ^ nq, a change in circle diameter is negligible; hence the classical approach is justified. At higher fields the circles become smaller; and when approaching atomic size, the quantum levels (Landau levels) become wider-spaced and a quantum mechanical approach is required. For reviews, see Lax (1963), Mavroides (1972), McCombe and Wagner (1975).

[30]The circle diameter is typically of the order of 10-3 cm for a magnetic induction of 10 kG; here vn is the thermal velocity of an electron. In metals, however, one also has to consider the skin penetration of the probing electromagnetic field. The skin depth of a metal is usually a very small fraction of the circle diameter, so that the probing ac-field can interact only at the very top part of each electron cycle close to the surface. This enhances information about near-surface behavior in metals, while in semiconductors, probing extends throughout the bulk.

[31]The conventional term alloy of metals also encompasses crystallite mixtures of non-intersoluble metals, such as lead and tin (solder). Here, however, only materials within their solubility ranges are discussed. The Hume-Rothery rule identifies these metals as having similar binding character, similar valency, and similar atomic radii (Hume-Rothery 1936). Corresponding guidelines apply to the intersolubility of cations or anions in compounds.

[32]The alternating potential shown in Fig. 7.2a is of type I, i. e., a minimum of Ec(x) coincides with a maximum of Ev(X). Both minima and maxima coincide in a type-I superlattice (Fig. 7.27b). An example for type II is the Ga^ In1_f As and GaAs^ Sb1_f superlattice. For values of Z and n below 0.25, the valence band of the former extends above the conduction band of the latter, resulting in quasimetallic behavior. For a review of type-II superlattices, see Voos and Esaki (1981).

[33]However, with an important condition missing—the periodic barrier of similar thickness and height.

[34]A tight-binding model, however, provides information only about the gross properties of a semiconductor (crystalline or amorphous).

[35] Or cyclic boundary conditions; here energy and particle number are conserved by demanding, that with the passage of a particle out of a surface, an identical one enters from the opposite surface (Born-von Karman boundary condition). The two conditions are mathematically equivalent.

[36]Here and in several of the following figures, the energy axis is plotted vertically in order to facilitate comparison with the band model even though g(E)dE is the dependent and E the independent variable.

[37] This can easily be seen at the Г-point for k() = 0: Here we have Ф(k = 0, r) = u(0, r) exp(i0 ■ r) = u(0, r).

[38]In semiconductors with several equivalent minima (Si, Ge), the wavefunction becomes a sum of contributions from each of the minima

J2ajFjc(r)ujc(kJ0, r).

**[39]**nq is the principal quantum number, describing the entire energy spectrum for a simple hydrogen atom. All other states are degenerate. Therefore, in a pure Coulomb potential, this quantum number is the only one that determines the energy of a hydrogen level. When deviations from this spherical potential appear in a crystal, the D = l(l + 1) = n2q degeneracy of each of these levels is removed, and the energy of the і-, p-, d-states are shifted according to RH/(nq +1)2. To further lift the remaining degeneracies of the magnetic quantum number, a magnetic field must act.

[40]The anisotropy is observed because of the satellite band ellipsoid resulting in an anisotropy of the effective masses with mt and mt in the main axis.

[41]For instance, when aqH ~ 50 A for the 1s state, it is 200 A for the 2s and 450 A for the 3s states, making the hydrogenic effective mass approximation a much improved approximation. In addition, in semiconductors where e/m is already very large, e. g., in GaAs with є^m.0/mn = 192.5, resulting in aqH = 101.9 A. 18a, this approximation is quite good for the 1s state. In GaAs, it results in EqH = 5.83 meV, while the experimental values vary from 5.81 to 6.1 mV for GaAs:Si and GaAs:Ge. For more comparisons between theory and experiment, see Bassani et al. (1974).

[42] Deep levels also appear in narrow band gap materials (see Lischka 1986).

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[43]The atomic electronegativity is defined as the difference between the s – energy of host and impurity atoms for donors, and the respective p-energy for acceptors.

[44]However, in actuality, deformations of the surrounding lattice result, with consequent lowering of the symmetry.

[45]The use of lower – and upper-case letters to describe the states gives a good example to distinguish between one – and multi-electron states. The latter includes electron-electron interaction.

[46]F-centers (“Farb” centers: German for color centers) were the first lattice defects correctly identified and described by their electronic structure by Pohl and coworkers (see the review by Pohl 1938). Later associates of two, three, or four F-centers were observed and referred to as M-, R – and N-centers (N1 in planar and N2 in tetrahedral arrangement)—see Schulman and Compton (1962).

[47]Quenching from high temperature is not fast enough to freeze-in measurable densities of vacancies (Watkins 1986).

[48]The A1 state of the acceptor is always strongly bound and lies within or below the valence band, while the T2 states may emerge from the conduction band into the gap, to which both approach asymptotically with increasing depth of the binding potential of the electron or hole in the impurity.

[49]Due to the thermal nature of incorporation, i. e, during a heat treatment between 350 and 555 °C in Czochralski-grown Si.

[50]With parallel spins there are less alternatives to populate states of equal energy (less degeneracy). With antiparallel spin orientation, there is more degeneracy, giving the opportunity to the Jahn – Teller splitting to produce an even lower level. For transition metal impurities, Hund’s rule, and for Si-vacancies, the Jahn-Teller effect produces the lower ground state (Zunger 1983).

[51] / 2mn 3/2 fE.

g(E)dE = n E – eV f (V)dV, (12.3)

[52] ( 2mn 3/2

That is, high within a band, there are no changes in the density of states compared with the ideal lattice. For E < Ec, however, the density of states is modified to

[54]For a self-consistent determination of the screening, which depends on the carrier density, which in turn depends on the level density, which again is influenced by the screening length—see Hwang and Brews (1971).

[55]For instance, in InSb with mn = 0.0116, the effective density of states is Nc = 3 • 1016 cm 3;

[56]Bipolar devices made from such materials are capable of switching at high speed from a low to a high conducting state at a critical bias (Ovshinsky 1968).

[57]It should be noted that the definition of a band state is related to the coherence length of an electron

wave, which is essentially the same as the mean free path X (see Sect. 13.3.1 for more details).

[59] Strictly speaking, steady-state carrier transport is due to external forces only. The diffusion current originates from a deformed density profile due to external forces, and is a portion of the conventionally considered diffusion component. The major part of the diffusion is used to compensate the built-in field, and has no part in the actual carrier transport: both drift and diffusion cancel each other and are caused by an artificial model consideration.

K. W. Boer, Handbook of the Physics of Thin-Film Solar Cells,

DOI 10.1007/978-3-642-36748-9_15, © Springer-Verlag Berlin Heidelberg 2013

[60]The rms (root mean square) velocity, which is more commonly used, should be distinguished

from the slightly different average velocity and from the most probable speed. Their ratios are

vrm : (|v|) : vmp = v/3 : П : Д = 1.2247 : 1.1284 : 1, as long as the carriers follow Boltzmann

statistics. For a distinction between these different velocities, see Fig. 15.1.

[64]This motion resembles a random walk (Chandrasekha 1943).

[65]In chapters dealing with carrier transport, F is chosen for the field, since E is used for the energy.

[66]Often a minimum scattering angle of 90° is used to distinguish scattering events with loss of memory from forward scattering events.

[67]Major deviations from linearity of Eg with composition are observed when the conduction band minimum lies at a different point in the Brillouin zone for the two end members. One example is the alloy of Ge and Si. Other deviations such as bowing are observed when the alloying atoms are of substantially and

[68]prc-junctions are the best studied intentional space-charge regions. Inhomogeneous doping distributions-especially near surfaces, contacts, or other crystal inhomogeneities are often unintentional and hard to eliminate.

[69]This argument no longer holds with a bias, which will modify the space – charge; partial heating occurs, proportional to the fraction of external field. This heating can be related to the tilting of the quasi-Fermi levels (Boer 1985a).

[70]With bias, the Fermi level in a junction is split into two quasi-Fermi levels which are tilted, however, with space-dependent slope. Regions of high slope within the junction region will become preferentially heated. The formation of such regions depends on the change of the carrier distribution with bias and its contribution to the electrochemical potential (the quasi-Fermi level). Integration of transport-, Poisson-, and continuity equations yields a quantitative description of this behavior (Boer 1985a).

[71] In the following sections the magnetic induction B is used, which is connected to the magnetic field H by B = xXqH, with xq the permeability of free space and /xL the relative permeability. Occasionally, the magnetization M is used, which, similar to the polarization, is introduced via B = xqH + M with M = Xm ■ Xqxx H and /x/x = 1 + Xm1 with Xm the magnetic susceptibility.

K. W. Boer, Handbook of the Physics of Thin-Film Solar Cells,

DOI 10.1007/978-3-642-36748-9_16, © Springer-Verlag Berlin Heidelberg 2013

[72]If v x B is on the order of F, its influence becomes negligible since, from Eq. (16.2), a term v • (v x B) would appear in Eq. (31.4) which vanishes, since v is orthogonal to the vector v x B.

[73]Although the electric and magnetic fields act as external forces, and one has — (F + v x B) as total force, the scalar product of (v x B) ■ j is zero since the vectors v x B and j are perpendicular to each other; in first approximation, there is no energy input into the carrier gas from a magnetic field.

[74]In the kz direction there are subbands; in the kx and ky directions, there are discrete levels in

[75]In contrast to the case of vanishing magnetic induction where the motion proceeds in the x direction and, without scattering, is accelerated.

[76]Electron injection relates to electrode properties not discussed in this book. It provides an experimental means of increasing the carrier density by simply increasing the bias, thereby injecting more carriers from an appropriate electrode. For a review, see Rose (1978b).

[77]The velocity of light contained in a is the best known of the three constants.

[78]An error up to 20 % can occur when applying Eq. (30.1) because of nonlinearities, interaction of different scattering events, as shown by Rode and Knight (1971).

[79]The experimentally observed exponent of T is -1.67 for Ge (Conwell 1952) and not -1.5. The

exponent of T for Si is still larger (^2.5). Inserting actual values for SiQ ci = L56 • 1012 dyn/cm2,

[81]The deformation potential is defined as the change in band gap per unit strain, and is typically on the order of 10 eV. For a listing, see Table 17.1.

mn — 0.2 m0 and S — 9.5 eV), one obtains xn — 5,900 cm2/Vs, a value that is larger by a factor

of 4 than the measured [in — 1,500 cm2/V s at 300 K.

4K2 can be expressed as the ratio of the mechanical to the total work in a piezoelectrical mate

rial: K2 — (е^.і/сі)/[єє0 + epz/ci], with epz the piezoelectric constant (which is on the order of

[86]-5 As/cm2), and ci the longitudinal elastic constant (relating the tension T to the stress S and the electric field F as T — ci S — epzF ).

[87]Here, Fop ~ 1 + в ln(e + 1) + 1/(в + 1), with в — (2|к|Ад)2, and Xp the Debye length given in Sect. A.3.

[88]More sophisticated estimates (e. g., Fuchs 1938; Sondheimer 1952) are applied for thin metal layers, where surface-induced effects are less complex. These yield results on the same order of magnitude as given here.

[89]The electron is accelerated in the electric field and occupies a range of k states, thereby losing translational invariance, i. e., resulting in the loss of k as a good quantum number. Different k-state mixing can describe the behavior and causes k-state broadening.

[90]One can understand this by equating damping with transfer of energy into heat, and optical absorption with extraction of this energy from the radiation field. Such absorption occurs even outside a specific electronic or ionic resonance absorption.

[91]A Brewster angle is defined as the angle under which no component Eu is reflected. Here sin Фі = cos Фі hence

[92]Reflectance (etc.) is used rather than reflectivity since it is not normalized to the unit area; this is similar to the use of the word resistance (not normalized) vs. resistivity, distinguishing between the suffixes – ance and – ivity.

[93]It is interesting to see that Eq. (22.23) can be rewritten, using the quasi-hydrogen energy EqH, as

[94]R = Roentgen; 1 R is the amount of X-ray (or у – ray) irradiation that produces 1 esu of charges (2.08 • 109 ions) per cm3 of air. Natural background radiation is 120 mR per year. Diagnostic single X-ray exposures lie between 50 and 500 mR per exposure. Skin reddening occurs near 500 R.

[95]This can be done by a preceding, incomplete glowcurve run, which proceeds from run to run to progressively higher temperatures, followed by cooling to the starting temperature for the following run.

[96]That is, the Hall effect measured for photogenerated majority carriers.

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DOI 10.1007/978-3-642-36748-9_24, © Springer-Verlag Berlin Heidelberg 2013

[97]Except when carrier excitation takes place from filled trap levels.

[98]For an extrinsic photoconductor with an absorption constant of 10 cm 1 this is equivalent to a photon flux of 1016 cm-2 s-1 for photons with an energy of 1 eV, this is also equivalent to an illumination of 1 mW/cm2, i. e., about 1 % of solar radiation of 100 mW/cm2; this example should be interpreted to obtain a rough estimate only, since monochromatic light is inappropriately compared here with polychromatic sunlight.

[99]This recharging of recombination centers is given as a simple example of the sensitizing or de

sensitizing of a photoconductor. In actuality, deep centers with different degrees of lattice coupling

[102] These relate to the basic Maxwell’s equation with its condition for the conservation of electrons

dp

V-j = —£; (24.59)

for equilibrium, with Sp/dt = 0, it follows V-j = 0. In semiconductors, with the introduction of holes, one has two types of currents, jn and jp, and expects with gn = gp and rn = rp that V • (jn + jp) = 0; the sign dilemma in comparing this equation with Eq. (24.58) can be resolved by replacing the conventional e = e with — e for electrons and +e for holes. This is the condition for the conservation of charges. In actuality, however, only a fraction of the electrons and holes are mobile, others are trapped and do not contribute to the currents while participating in the total neutrality account.

[103]The left-hand region is identified with a parameter index 1, the right-hand region with an index 2.

[104]When an inhomogeneous semiconductor is held between two electrodes and, after a current has passed, it is dropped into the beacher of an electrometer without the electrodes, these net charges can be measured and give an indication of the inhomogeneities (Boer and Kummel 1957).

[105]Three boundary conditions for Eqs. (25.6)—(25.9) in region 1, and three conditions for the same equations in region 2.

within this range, the electron density remains larger than the free hole density.

[107]Its use is permissible also in an nn+-junction and to some extent throughout some Schottky

barriers, provided that the carrier depletion in the Schottky barrier remains “moderate”, i. e., even

[109]See Eqs. (25.7) and (25.8).

[110]This holds strictly only for electron-fields, but, except for graded band gap semiconductors or highly doped regions with inhomogeneous dopant distribution, this distinction need not be made.

[111]The Fermi-level is often described as the electrochemical potential.

[112]Its distribution is slightly deformed, according to the changes in n(x) with applied bias, while its amplitude remains nearly unchanged. This reflects the fact that at a first approximation the electron distribution is pushed “sideways” (in the x-direction) with changing bias, with little deformation at the point of its steepest slope, as shown Fig. 25.2a.

[113]This argument no longer holds with a bias, which will modify the space-charge; partial heating occurs, proportional to the fraction of external field. This heating can be related to the tilting of the quasi-Fermi levels (Boer 1985a).

[114]We are using the denotation as an nn+-junction somewhat loosely in this chapter, merely to indicate that the electron density in the n+-region is orders of magnitude larger than in the n- region.

[115]We will see later that the distance to which such carriers can be swept into the lowly doped region is given by the diffusion length or at higher fields by the drift length (Sect. 5.3.2) which may be smaller than the device thickness.

[116]Though modified by the scattering of electrons in the semiconductor.

[117]This is distinctly different from the influence of a Schottky barrier or a ри-junction on the current voltage characteristic, which introduce new, more highly resistive regions that expand or contract with bias.

[118]Neglecting the voltage drop in the highly conducting region 2. A somewhat better approximation yields F(-d1)(3/2)V/d1, yielding 9/8 as numerating factor in Eq. (25.41).

[119]It is an expression coined to indicate similarity to the current in a vacuum diode.

[120]Even though the electron density inside a metal is much higher than in the semiconductor, at its boundary to the semiconductor this density is substantially reduced according to its effective work function. It is this electron density which causes a reduction of n in the semiconductor at the interface.

[121]A similar Schottky barrier appears in p-type semiconductors near a metal electrode with low work function, again when the hole density near the electrode is much smaller than in the bulk. Here the space-charge region is negatively charged and the resulting field is positive.

[122]This is slightly different from Nc within the semiconductor bulk (see Eq. (25.26)) because of a

different effective mass at the interface.

[124]The error encountered at the boundary of this range (8 ■ 10-6 cm) seem to be rather large (factor 2) when judging from the plot in linear scale of Fig. 26.1. The accumulative error, when integrating from the metal/semiconductor interface, however, is tolerable, as shown in Fig. 26.2. The substantial simplification in the mathematical analysis justifies this seemingly crude approach.

[125]A comparison with the previously discussed example of majority carrier injection, in which n ^ Nd, presents the other alternative for the two cases for which the discussion of this one-carrier space-charge distribution can be drastically simplified.

[126]We have rewritten the first, the transport equation as a function of dn/dx to identify this set as a set of three differential equations that need to be solved.

[127]That is, the maximum field which lies in this approximation at the metal/semiconductor boundary (neglecting image forces).

[128]However, at higher doping densities, especially close to the metal interface, tunneling fields may

be reached when Nd > 1018 cm—3. This often is desired to make a contact “ohmic” and such

increased defect density can be reached, e. g. by gas discharge treatments (Butler 1980).

[131]We have introduced here a shifted coordinate system (x1 ,n). The amount of the shift in x is determined by the boundary condition, as will be discussed later in this section.

[132]The first term of Eq. (26.29) is identical with Eq. (25.12) when replacing fn, D using Eq. (26.15).

[133]Since its pre-exponential factor is the drift current, which for a large reverse bias (i. e. for a vanishing exponential) is the limiting current.

[134]The formalism used here is similar to the one used to develop the expression for the diffusion currents inside a semiconductor with gradually varying carrier density. However, the rather abrupt (in less than a mean free path) change in carrier density at both sides of the surface interlayer justifies the use of the Richardson-Dushman electron emission relation here.

[135]We assume that nc (at the metal side of the junction) remains constant and is given by Eq. (26.1).

[136]This approach is mathematically correct; however, one should recognize that, even though the drift velocity is limited to approximately the rms velocity in bulk semiconductors (Boer 2002, Chap. 26) resulting in a factor 1/2 in Eq. (26.50), conditions at the thin boundary layer are more complex, and need detailed studies to also become physically appropriate.

[137] Here we have used a general velocity v and a general field to indicate the type of relationship rather than the specific one explained in this section.

[138]Therefore this range is also referred to as the square root range.

[139]Since F(x) increases linearly with decreasing x, the product n(x)F(x) must remain constant in the DRO-range; namely nF = jn/efin and jn = j = const.

[140]We neglect here pre-breakdown effects which cause a steep increase of the current at still higher reverse bias.

[141]This tilting is too small to be visible in Fig. 26.6.

[142]The following analogy may help to remember the formula: the change in population is given by the birth rate (g) minus death rate (= population over life expectancy) plus the drop-off from travelers through the region (change in current multiplied by — 1/e).

[143]We are using here the term recombination somewhat loosely be fore defining the distinction between trapping and recombination in Sect. 27.1.3.

[144]Capital letters are consistently used to identify the density of states, lower case letters to identify the density of electrons or holes in these states.

[145]In corresponding photon energy ranges of indirect band gap semiconductors, the absorption coefficient is roughly three orders of magnitude smaller.

[146] After reflection is subtracted.

[147]AM 1 stand for air mass 1 and indicates the optical absorption by an air column when the sun stands at the zenith. In total power, this absorption amounts to 26.6 %, namely from 140 mW/cm2 above the earth’s atmosphere to 100 mW/cm2 at AM 1. With decreased elevation ф the light path through the atmosphere becomes longer as 1/ cos(90° – ф) which is used as the corresponding air mass value. E. g., for ф = 42° one has sunlight of AM 1.5, a value often use as more realistic for solar cell calibration in solar simulators.

[148]Except for high mobility semiconductors at low temperatures where impact ionization competes favorably.

[149]With increased path length in an electric field more energy is accumulated. The ionization rate per unit path length is measured in cm-1.

[150]The exact relation contains the Fermi integrals F1/2 (see Boer 2000). The approximation only holds for the non-degenerate case, i. e., for Ec — EFn > kT.

[151]The inequality of Eq. (27.23) holds for optical excitation but not for shifted distribution in pn – junctions in reverse bias (see Sect. 31.2).

[152]In good photoconductors, however, the majority quasi-Fermi level is also substantially changed.

[153]For estimating Si and Sj, one needs to know the center’s cross section, which may be estimated from the center’s charge and bonding character. For example, a center that is neutral without an electron in it has a cross section for an electron on the order of 10-16 cm2. After it has captured the electron it is negatively charged; thus its capture cross section for a hole has increased to ^10-14 cm2. For this example, sn/sp = 10-2 and Sj = -0.12 eV will be used. For hole traps the charge character may turn from neutral to positive after hole capture, making sn/sp = 100 and Sj = +0.12 eV.

[154]A diffusion current of each carrier is exactly compensated by an opposing drift current.

[155]In order to emphasize the equilibrium values of n and p, we have attached a subscript zero.

[156]Since these centers are more important when they become recombination centers, they are identified here with the subscript r.

[157]Here both n(x) and p(x) have decreased below the equilibrium distribution, while the space charge region has widened.

[158]Here both n(x) and p(x) have increased above the equilibrium values.

[159]This assumption is not a very realistic one since the charge character of the center changes when capturing a carrier (see Sect. 27.2.2). However, the qualitative behavior deduced from Eq. (27.33) will remain valid.

[160]We are using here the notation of an incremental current since in some of the devices only a fraction of the total electron or hole current is influenced, as will be described below.

[161]In an inhomogeneous semiconductor, the determination of the divergence-free electron or hole current is a bit more involved.

[162]Equation (27.47) can be obtained from Eqs. (27.30), (27.45), and (27.46) with n = n0 + 8n and p = p0 + 8n and using napa = n2, 8n = 8p, when traps can be neglected, since electrons and holes are mutually created, and for n as minority carrier, assuming 8n ^ nQ.

[163]These conditions can be realized by optical excitation with intrinsic light that is absorbed close to the surface (carrier injection), or for the opposite case by excessive carrier recombination at the surface and with carrier generation or recombination, must also follow the current continuity equation (see Sect. 27.3.2).

[164]This approximation results from the fact that in the bulk the majority carrier density n ^ {p, 2ni cosh[ (Ei – Et)/(kT)]}.

[165]We have used here pjD to indicate that the hole density at the boundary may depend on the current

[167]In order to separate the effects of a bias controlled pjD and a surface – recombination-controlled ps, we have chosen consistently the left surface as being bias-controlled and the right surface as being recombination-controlled. In actuality, the conditions are interwoven, as shown in Sect. 29.2 and the relevant subsections.

[168] Recombination always tends to restore thermal equilibrium. Therefore p(th) is contained in Eq. (28.32) and not p(o).

[169]Equation (28.35) can be verified by differentiating p(x) with B given in Eq. (28.9) and setting dp/dx = 0.

[170]Design parameters for good solar cells are s <d_p0 and Lp > 3d1.

[171]Since such fields can extend by many Debye lengths beyond a Schottky barrier or junction, one must consider such field-influence on the diffusion in much thicker device slabs.

[172] p(d1) is kept constant in forward and in reverse bias in order to simplify the following discussion.

[173]The selection of d1 here is due to the specific example in which we assumed a neutral electrode at d1 with a flat-band (no space charge) connection to the semiconductor/metal interface. When a space-charge layer is also present at d1, the identification of jpi is more involved (see Sect. 29.2.2).

[174]For justification of this unconventional approach see Sect. 29.3.

[175]Even though Eq. (29.6) should contain the space-dependent minority carrier densities, we have replaced these by the constant pjD and later in Eq. (29.7) by nj; this is justified to an improved approximation with reverse bias as can be seen from the computed result of a step like U(x).

[176]For simplicity of the mathematical description we have chosen modified carrier densities p* and p* rather than the modified minority carrier lifetime Tp [given in Eq. (28.49)], which would result in a somewhat longer expression.

One of the continuity equations can be replaced by the total current equation j = jn + jp.

[178]For educational purposes it is advantageous to use a forward numerical integration of the governing set of differential equation rather than a more conventional finite element method (Snowden 1985). Straightforward integration permits one to study the interaction of different variables and the cause-and-effect relation of changing boundary conditions.

[179]The gr-current is a very small fraction of the currents.

[180]The crossing of p(x) for different bias within the barrier region is a direct result of the control of the boundary concentration pj by the electron density nj via nj pj = n2 required by perfect recombination at the interface. nj, however, is controlled by the dominant majority carrier current which forces a decrease of nj with increased reverse bias, and, in turn, causes an increased pj. This, together with an increased slope of p(x) due to the increased barrier field, results in the crossing of the different p(x) curves in the family of curves of Fig. 29.8. In contrast, n(x) is the dominant variable with an increasing fraction of drift current as the bias is reduced, causing the n(x) profile to widen, the n(x) slope thereby to reduce hence avoiding an n(x)-crossover (the solution curves of the dominant variables must be unique—Fig. 29.8A).

[181]The quasi-Fermi level remains flat wherever drift and diffusion currents are large compared to the net current, i. e., for holes almost in the entire barrier region until x = xd is approached.

[182]The chosen example of a Ge-diode with a relatively high barrier density results in an unfavorable

diode characteristic with high reverse saturation current. A much improved Schottky barrier can be

obtained with a substantially lower nc.

[185]We have introduced the DO-range, which is similar to the DRO-range: i. e., the total carrier current is given by one of the contributing currents only.

[186]The total voltage drop across the entire device is equal to the drop of the majority quasi-Fermi potential: V = [EFn(x = 0) — EFn(d1 )]/e.

[187]In contrast to the entry of the majority carrier Fermi-level into its band, that signifies degeneracy.

[188]In this example only two kinds of levels were assumed: very shallow electron donors and deep recombination centers. In actual practice a larger variety of levels exist, making such an analysis more important.

[189]The only expected change would be in U from being n-controlled to becoming p-controlled; however, this changeover is hidden near nc by n* in the denominator of U(x).

However, the depletion layer approximation becomes inadequate for larger forward bias.

[191]Equation (30.14) is identical to the result of the more general expression.

[192]This can indeed occur for long devices with narrow band gap.

[193]The device extends only slightly beyond the junction region.

Ec — Ed for the n-type side and ^ms = E0 — Ev + Ea for the p-type side (E0 is the vacuum level).

[195] Such a flat band electrode connection requires a metal/semiconductor work function ^ms = E0 —

[196]There are two majority quasi-Fermi levels in a ри-junction, Efp in the p-type region and Бри in the и-type region.

[197]For vanishing current both Boltzmann regions fill the entire device width with s = 5 • 106 cm/s, for a total reverse current contribution of 9.8 mA/cm2.

[198] A fourth step in the highly doped region is not fully developed because of the triangular steep decline of pix).

[199]The difference between the minority carrier density at the outer surface and the equilibrium density [Eqs. (30.56) and (30.57)] that controls the surface recombination current is proportional to the difference at the bulk/junction interface (at ln or lp) that controls the diode current [Eq. (30.18)].

[200]The divergence-free current is now reduced to below 10 8 A/cm2, i. e., to completely negligible values in reverse bias.

[201] Such a scale break results in a break of slopes at the break point. The actual curves, however, have continuous slopes.

[202]Consider the scale break of the figure in your comparison between the DRO and DO ranges.

[203]Such profile can (rarely) be achieved by cross diffusion of dopands for compensation; even though this is usually not symmetric, we will assume such symmetric compensation here.

[204] Since we disregard in this first section the inhomogeneity of the optical absorption constant, this approach is justified only for thin platelets of a thickness of less than the optical absorption constant at the excitation energy. However, for indirect bandgap materials, such as Si this absorption constant is on the order of a typical thickness of such devices (a few hundred |am) this approach is justified.

K. W. Boer, Handbook of the Physics of Thin-Film Solar Cells,

DOI 10.1007/978-3-642-36748-9_31, © Springer-Verlag Berlin Heidelberg 2013

[205]This state is often referred to as low injection state. With sufficient optical generation, e. g., for

solar cells with concentrator, high injection can be reached, in which both carrier densities increase

significantly above the equilibrium concentration.

[208]This effect becomes active only at high enough light intensities, when the majority carrier density increases markedly.

[209]The capture and emission coefficients are identified by c and e respectively with subscripts identifying origin and target level (see Sect. 27.2).

[210]However, when optical generation at longer wavelength is applied, a superlinear branch can be observed, when optical excitation from filled traps increases.

[211]Assuming jp ^ (eipp¥, ipkT dn/dx).

[212]The connection of p(xb) as p (x = 0) with Eq. (31.38) to determine the second boundary condition B, is not clear cut for a simple barrier. A more transparent connection of bulk and barrier is discussed in Sect. 33.1.3 for the d-type high-low heterojunction.

[213] In actuality such simple shift is rarely observed, as often the dark – and photo-diode characteristics cross-over in forward bias.

[214]Such resistances may actually exist in polycrystalline devices as pathes between the electrodes through grain boundaries or as interlayers between grains and the electrodes. But their assumed existence is often overstated and can result in expensive searches for the wrong causes of such deviations from diode curve idealities in solar cell development.

[215]Often the equivalent circuit is extended by other elements, such as another diode and more resistances, providing even more adjustible parameters to produce “better” (and more misleading) agreements with the experiment.

[216]This comparison is easiest seen for curve pair 7.

[217]It is important for the understanding of this critical relation of a net junction recombination and the depletion of minority carriers from the adjacent bulk regions, to focus on current continuity that forces the transport of minority carriers to the recombination sink near the center of the junction, and results in a lemniscate shape of jn(x) and jp(x), as shown in Fig. 32.1.

[218]The figure shows the tendency to completely eliminate the junction barrier for a flat band connection at sufficiently high optical generation rates. Such flat band connection can be achieved at even lower optical generation rates in devices with lower doping densities and higher minority carrier life times.

[219]In contrast to the jumps of the quasi-Fermi levels at the metal interface of a Schottky barrier, the jumps for the majority quasi-Fermi levels are negligible in the pn-junction device when contact is made at each side with an appropriate, neutral (or injecting) contact metal.

This no longer holds when d1 or d2 become comparable to Lp and Ln.

[221]The selection of Epp here as the shifted level is due to the chosen boundary condition of keeping Ec(x = d2) = 0.

[222]The reduction of the recombination of a center lying at a greater distance from the center of the gap is due to the more trap-like behavior by partial carrier emission into the nearest band rather than recombination.

[223]Only in a very general approximation one observes the tendency of А ^ 2 with excessive recombination in the space charge region, and of А ^ 1 with dominant recombination in the space charge-free bulk.

[224]Even though the average increase of the recombination center density is only by a factor of 5.5.

[225]This information is complementary to the one given in Sect. 32.2.2; indicating that the solution curves for n and p and the potentials are independent of the distribution of go(x) in thin devices, provided that the total number of absorbed photons remains the same.

[226]The front layer is often referred to as the emitter. We will refrain from doing so, since in solar cells the emission of minority carriers into the junction originates mostly from the much thicker base layer surface. For more see e. g. Lammert and Schwartz (1977), Gray and Schwartz (1984).

[227] This is an artificial condition that is caused by the assumed constant optical generation rate. In actuality go = go(x) and rapidly decreases from left to right. With d2 ^ Ln, averaging of g(x) can no longer be applied. Therefore most of the gr-current flows toward the junction and much less is collected at the right electrode (the light enters from the left).

[228]From Ef — Efp = kT ln[(p20 + Ap)/p 20], one obtains for this adjustment of the majority quasi – Fermi level approximately 10—8 eV.

[229]This split is estimated in the lower doped region (see Sect. 32.2.3).

[230]In actuality, the front is covered by a thin grid electrode, rendering this a three-dimensional problem in which most of the minority carriers a generated more than a diffusion length removed from the actual metal.

[231]The sloping of the density distribution toward the overshoot region is not visible since (d1 ,d2) ^

(Lp, Ln).

1We have consistently referred to this cell as a Cu2S solar cell even though this would indicate that the copper sulfide is a chalcocite, while in actuality it is Djurleite with a stoichiometry closer to 1.98 rather than 2. This also is done in order to indicate that we do not want to use the examples discussed here for more than as possible phenomena rather than staying too close to an actual cell in all the detail discussed here.

K. W. Boer, Handbook of the Physics of Thin-Film Solar Cells,

DOI 10.1007/978-3-642-36748-9_33, © Springer-Verlag Berlin Heidelberg 2013

[233]Even though as a heterojunction cell this type seems to miss the obvious advantage of optical absorption close to the heterojunction, it may still be of technical interest because of the ease of fabrication resulting in relatively inexpensive devices that may still show acceptable conversion efficiencies.

[234]This is a typical characteristic of copper doped CdS by creating a high-field domain.

[235]In actuality, there may be some discontinuities, as discussed in several previous sections, which can be easily introduced but are omitted here to avoid confusion with other effects that are emphasized in this chapter.

[236]Here we assume that the conductivity in the Cu2S is high enough that any voltage drop here can be neglected.

[237]The Schottky barrier approximation is well suited for the CdS part of the heterojunction since in the entire CdS one has p ^ n, and the Cu2S is nearly degenerate, thereby acting in some respects as pseudo-electrode.

[238] When using the relation

[239]Even considering frozen-in steady state for the minority carriers in the dark and reasonable generation rates and lifetimes under sunlight, the minority carrier density within the CdS will remain well below the electron density within the entire barrier region.

[240]For a more precise evaluation of the sequential trap depletion see the corresponding Sect. 27.2.2 that deals with the dark-diode.

[241] Such field quenching is of interest for technical applications, since it permits working with semiconductors of lower purity, allowing less expensive fabrication methods. Without field quenching, such semiconductors would easily be driven into a range of excessive barrier fields, with detrimental influence on performance due to tunneling through the barrier, thereby creating leakage currents.

[242]Only in the bias range between the Boltzmann and the saturation branch can such kinetics be observed. Otherwise the structure of interest becomes hidden in the horizontal current saturation branch.

Observe that we split off an exponential with the diffusion voltage that does not contain A.

[244] For thinner cover layers or topographic layer inhomogeneities the reverse bias saturation current starts to increase again.

[245] Frenkel Poole excitation is caused by the lowering of the Coulomb funnel in field direction by an electric field due to the tilting of the bands in this field, thereby reducing the energy necessary to thermally excite the holes into the valence band.

[246]Saturation can be shifted by adding co-activators, e. g., Al during doping.

[247]This number was calculated for cells of 10.7 % efficiency, yielding 107 W/m2 multiplied with a $1.04 per Watt, yielding $111 perm2.

[248]Recycle First Solar Modules, First Solar.

[249]With CdTe at 5.85 g/cm3 of which Te is 3.1 g/cm3. In 1 m2 of CdS/CdTe cells et 3 |am CdTe thickness this yields a use of 3.1 g/m2 of Te. Therefore the production of 1 GW of such cells with 100 km2 area requires approximately 95 metric tons of Te.