1 Introduction

Solar cell modelling is fundamental to a detailed understanding of the device operation, and a comprehensive model requires a detailed knowledge of the material parameters. A brief overview of the semiconductor properties relevant to solar cell operation is given in this chapter, including the semiconductor band structure and carrier statistics, transport and optical properties, recombination processes, material aspects of radiation damage to solar cells in space and effects observed under heavy doping, with special attention given to the properties of hydrogenated amorphous silicon. The principal semiconductor parameters encountered in photovoltaic applications are summarised in Tables 1 and 2. The refractive indices of materials used for antireflection coating can be found in Table 3.

TABLE 1 Properties of the Principal Semiconductors with Photovoltaic Applications (all at 300 K). Bandgap: d  =  Direct; i = Indirect. Crystal Structure: Dia = Diamond, Zb = Zinc Blende, Ch = Chalcopyrite. The Refractive Index is given at the Wavelength 590 nm (2.1 eV) Unless Otherwise Stated. Principal Sources of Data: c-Si [20]; GaAs [3]; InP [4,41]; a-Si [5] and Section 10; CdTe [6]; CIS [7]; Al_xGa_1-xAs [9]. Details of the Absorption Coefficient and Refractive Index, Including the Wavelength Dependence, can be Found in References [27] and [28]. X is the Electron Affinity and TEC Stands for Thermal Expansion Coefficient

^∗At 600 nm.

TABLE 2 The Energy gap E_g, Refractive Index n and the Electron Affinity X of Transparent Conducting Semiconductors Used as Window Layers in Thin-Film Solar Cells

TABLE 3 The Refractive Index at 590 nm (2.1 eV) of the Common Materials used for Antireflection Coating. The full-Wavelength Dependence of Most of these Substances can be Found in References [27] and [28]

Numerous computer programs that use the material parameters to model solar cell operation have been developed over the years, and several are now available commercially:

Some programs are available free of charge. These include, for example, SimWindows which can be downloaded from www-ocs.colorado.edu/SimWindows/simwin.html.

A discussion of the main principles of the numerical techniques can be found in specialised texts that deal with the modelling of solar cells (see, for example, the review [1]) or with the modelling of semiconductor devices in general [2].

2 Semiconductor Band Structure

The energy gap (or band gap) E_g and its structure as a function of the wave vector are key characteristics of the semiconductor material and of fundamental importance to the operation of the solar cell (see Figure 1). The principal features of interest are the temperature variation of the band-gap energy E_g and the magnitude of wave vector associated with low-energy transitions.

The variation of the band gap with temperature can be described by an expression originally suggested by Varshni [10]

(1)

where T is the absolute temperature and the parameters α and β are given in Table 4.

TABLE 4 The Parameters E_g0, α, and β in Equation (1). Sources of Data: (a) Thurmond [11]; (b) Varshni [10]

The current produced by solar cells is generated by optical transitions across the band gap. Two types of such transitions can be distinguished: (1) direct transitions where the momentum of the resulting electron–hole pair is very close to zero, and (2) indirect transitions where the resulting electron–hole pair has a finite momentum. The latter transitions require the assistance of a phonon (quantum of lattice vibration). Thus, there are two types of semiconductors:

The optical absorption in indirect-gap semiconductors is considerably weaker than in direct-gap semiconductors, as shown in Figure 2 on the example of silicon and gallium arsenide.

3 Carrier Statistics in Semiconductors

In thermal equilibrium, the temperature and the electrochemical potential (the Fermi level, denoted by E_F) are constant throughout the device. The product of the electron concentration n and the hole concentration p is then independent of doping and obeys the mass action law

(2)

where k_B is the Boltzmann constant, n_i is the electron (or the hole) concentration in the intrinsic semiconductor, and the effective densities of states N_c and N_v are given by

(3)

Here, h is the Planck constant and m_e and m_h are the electron and hole density-of-states effective masses (see Table 5). In Equation (3), the effective mass m_e includes a factor that allows for several equivalent minima of the conduction band in indirect-gap semiconductors (see Figure 1).

TABLE 5 The Densities of States in the Conduction and Valence Band (N_c and N_v, Respectively), the Intrinsic Carrier Concentration n₁ and the Density-of-States Effective Masses m_e and m_h (all at 300 K) in Si, GaAs and InP. m₀ Denotes the Free-Electron Mass.

Data from References [20,35]; Intrinsic Carrier Concentration in Si from Reference [12]

Equation (2) does not hold for a solar cell in operation. Current flow as well as electron transitions between different bands or other quantum states are induced by differences and gradients of the electrochemical potentials, and temperature gradients may also exist. It is then usual to assign a quasi-Fermi level to each band that describes the appropriate type of carriers. Thus, electrons in the conduction band will be described by the quasi-Fermi level E_Fn, and holes by E_Fp. It is convenient to define also the appropriate potentials Φ_n and Φ_p by

(4)

The use of quasi-Fermi levels to describe solar cell operation was already mentioned in Section 4.1 of Chapter Ia-1. The formalism leads to expressions for electron and hole concentrations, which are summarised in Table 6.

TABLE 6 Carrier Concentration in Degenerate and Nondegenerate Semiconductors. F_1/2 Denotes the Integral

4 The Transport Equations

The electron and the hole current densities J_n and J_p are governed transport by the transport equations

(5)

where n and p are the electron and the hole; the concentrations μ_n and μ_p are the electron and the hole mobilities; D_n and D_p are the electron and the hole diffusion constants; and ε is the electric field. The first term in each equation is due to drift in the electric field ε, and the second term corresponds to carrier diffusion. With the help of the quasi-Fermi levels, the equations in (5) can be written as

(6)

The equations in (6) are valid for a semiconductor with a homogeneous composition and when position-dependent dopant concentration is included. A generalisation to semiconductors with position-dependent band gaps can be found, for example, in [13].

In a region where space charge exists (for example, in the junction), the Poisson equation is needed to link the electrostatic potential ψ with the charge density ρ:

(7)

with

(8)

(9)

where ε is the static dielectric constant of the semiconductor (see Table 1).

In nondegenerate semiconductors, the diffusion constants are related to mobilities by the Einstein relations

(10)

The generalisation of the Einstein relations to degenerate semiconductors is discussed, for example, in [14].

The conservation of electrons and holes is expressed by the continuity equations

(11)

where G and U are the generation and recombination rates, which may be different for electrons and holes if there are transitions into or from localised states.

5 Carrier Mobility

In weak fields, the drift mobilities in Equations (5) represent the ratio between the mean carrier velocity and the electric field. The mobilities—which are generally different for majority and minority carriers—depend on the concentration of charged impurities and on the temperature. For silicon, these empirical dependencies are generally expressed in the Caughey–Thomas form [15]:

(12)

The values of the various constants for majority and minority carriers are given in Tables 7 and 8. A full model that includes the effects of lattice scattering, impurity scattering, carrier–carrier scattering, and impurity clustering effects at high concentration is described in [16], where the reader can find further details.

TABLE 7 The Values of Parameters in Equation (12) for Majority-Carrier Mobility in Silicon

(from Reference [17]; T_n = T/300)

TABLE 8 The Values of Parameters in Equation (12) for Minority-Carrier Mobility in Silicon

(from references [18,19])

TABLE 9 Parameters in Equation (12) that were used to Fit the Room-Temperature Electron and Hole Mobilities in InP

In strong fields, the mobility of carriers accelerated in an electric field parallel to the current flow is reduced since the carrier velocity saturates. The field dependence of the mean velocity v (only quoted reliably for majority carriers) is given by

(13)

where μlf is the appropriate low field value of the mobility (Equation (12)), the parameter β = 2 for electrons and β = 1 for holes, and vsat is the saturation velocity, identical for electrons and holes [20]:

(14)

In gallium arsenide, the empirical fitting of the mobility data is of a more complex nature, and the reader is referred to the reference [21] for more information regarding the majority-carrier mobility. The temperature and dopant concentration dependences of the minority-carrier mobility are shown in Figures 3 and 4.

The electric field dependence of carrier mobility in GaAs is different from silicon as the velocity has an ‘overshoot’ that is normally modelled by the following equation [24]:

(15)

where μlf is the appropriate low-field mobility, ε₀ = 4×10³ V/cm, β equals 4 for electrons and 1 for holes, and v_sat is given by [25]

(16)

where T is the temperature in K.

The majority-carrier mobility in indium phosphide reported in reference [26] is shown in Figure 5.

6 Carrier Generation by Optical Absorption

7 Recombination

Recombination processes can be classified in a number of ways. Most texts distinguish between bulk and surface recombination, and between band-to-band recombination as opposed to transitions with the participation of defect levels within the band gap. Recombination processes can also be classified according to the medium that absorbs the energy of the recombining electron–hole pair: radiative recombination (associated with photon emission) or the two principal nonradiative mechanisms by Auger and multiphonon transitions, where the recombination energy is absorbed by a free charge carrier or by lattice vibrations, respectively. An opposite process to Auger recombination (in which an electron–hole pair is generated rather than consumed) is called impact ionisation.

The following sections give a brief summary of the main recombination mechanisms with relevance to solar cell operation, which are summarised schematically in Figure 9.

8 Radiation Damage

Solar cells that operate on board a satellite in an orbit that passes through the Van Allen belts are subjected to fluxes of energetic electrons and protons trapped in the magnetic field of Earth and by fluxes of particles associated with high solar activity [43,44] (see Chapter IId-2). When slowed down in matter, most of the energy of the incident proton or electron is dissipated by interaction with the electron cloud. A relatively small fraction of this energy is dissipated in collisions with the nuclei, resulting in the formation of a lattice defect when energy in excess of a minimum threshold value is transferred to the nucleus [45,46]. A proton with energy in excess of this threshold causes the most damage (typically, of the order of 10–100 atomic displacements) near the end of its range in the crystal. At high energy, on the other hand, a proton creates simple point defects, and the displacement rate decreases with increasing proton energy. In contrast, the displacement rate by electrons increases rapidly with energy and approaches a constant value at higher energies. Electron damage can usually be assumed to be uniform throughout the crystal.

Some defects that are thus created act as recombination centres and reduce the minority-carrier lifetime τ. Thus, it is the lifetime τ (or, equivalently, the diffusion length L) that is the principal quantity of concern when the cell is subjected to the particle radiation in space. Under low injection, the reduction of the diffusion length L is described by the Messenger–Spratt equation (see, for example, [47], p. 151):

(32)

where L₀ is the diffusion length in the unirradiated cell, ϕ is the particle fluence (integrated flux), and KL is a (dimensionless) diffusion-length damage constant characteristic for the material and the type of irradiation.

The damage constant KL generally depends on the type of dopant, and even in single-junction cells, a different damage constant should therefore be introduced for each region of the cell. Moreover, care should be exercised when dealing with low-energy protons (with energy of the order of 0.1–1 MeV), which are stopped near the surface and may create nonuniform damage near the junction. Figures 10 and 11 show the doping concentration and energy dependence of the damage constant KL for electrons and protons in p-type silicon. Details of the damage constants for other materials can be found in references [43], [44], [48], and [49].

Damage constants are usually quoted for 1 MeV electrons and converted to other energies and particles (such as protons) by using the concept of damage equivalent. Conversion tables are available in the Solar Radiation Handbooks that will suit most circumstances. Radiation damage equivalence works well when applied to uniform damage throughout the cell, but care should be exercised in the case of nonuniform damage—for example, for low-frequency protons.

The reduction of the diffusion length describes usually the most significant part of the damage, but changes in other parameters on irradiation also sometimes need to be considered. The dark diode current I₀ may increase as a result of compensation by the radiation-induced defects. This, however, occurs usually at a fluence orders of magnitude higher than that which degrades τ or L. Other source of degradation may be defects created in the depletion region, at the interfaces between different regions of the cell, or at the external surfaces.

9 Heavy Doping Effects

At high doping densities, the picture of independent electrons interacting with isolated impurities becomes insufficient to describe the electron properties of semiconductors. As doping concentration increases, an impurity band is formed from the separate Coulomb wave functions located at individual doping impurities. This band gradually merges with the parent band and eventually gives rise to a tail of localised states. Electron–electron interaction becomes important, and the exchange and correlation energy terms have to be taken into account for a satisfactory description of the optical parameters and semiconductor device operation. An early overview of this multifaceted physical problem can be found in [57].

In practical situations, this complex phenomenon is usually described in terms of band-gap narrowing, with a possible correction to effective masses and band anisotropy. Different manifestations of this effect are normally found by optical measurements (including absorption and low-temperature luminescence) and from data that pertain to device operation. Parameters that have been inferred from the latter are usually referred to as apparent band-gap narrowing, denoted by ΔE_g. For n-type material, for example,

(33)

where p₀ is the minority-carrier (hole) concentration and ND is the dopant (donor) concentration.

The experimental data obtained by various methods have been reviewed in [58]. where the reader can find references to much of the earlier work. Jain et al. [59,60] obtain a fit for band-gap narrowing in various materials using a relatively simple and physically transparent expression that, however, is less easy to apply to device modelling. Values for silicon that are now frequently used in semiconductor devices modelling are based on the work of Klaassen et al. [61]. After reviewing the existing experimental data and correcting for the new mobility models and a new value for the intrinsic concentration, Klaassen et al. show that the apparent band-gap narrowing of n- and p-type silicon can be accurately modelled by a single expression:

(34)

where N_dop is the dopant concentration.

For gallium arsenide, Lundstrom et al. [62] recommend the following formula based on fitting empirical data:

where E_F is the Fermi energy, the function F_1/2 is defined in the caption to Table 5, and

10 Properties of Hydrogenated Amorphous Silicon¹

In hydrogenated amorphous silicon (a-Si:H), the effective band gap between the conduction and valence band edges is around 1.8 eV, but a thermal shrinking of the band gap with temperature has been reported [63]:

(35)

For statistical calculations, the corresponding effective densities of states can be approximated by N_c ≈ N_v ≈ 4×10¹⁹ cm⁻³. In contrast to crystalline silicon, the conduction and valence bands show evidence of exponential tails of localised states within the band gap [64]:

(36)

In addition, there is a Gaussian distribution of dangling bond states (states corresponding to nonsaturated silicon bonds) around midgap (Figure 12):

(37)

In device-quality intrinsic a-Si:H films, the density of dangling bonds N_DB ranges from 10¹⁵ to 10¹⁶ cm⁻³.

The carrier mobility in the localised states (band tails and dangling bonds) is negligible. The accepted values for extended states in the valence and conduction band are μ_p = 0.5 cm²/V-s and μ_n = 10 cm²/V-s, respectively. As in other semiconductors, the electron transport is given by the drift-diffusion equations (5) where the Einstein relations (10) apply.

The optical absorption coefficient shows three different zones (Figure 13):

Once the optical absorption coefficient is known, the carrier generation can be easily calculated according to Equation (17), but it is important to note that only photons with energies in excess of the band gap yield useful electron–hole pairs for photovoltaic conversion.

Finally, the dominant recombination mechanism in intrinsic a-Si:H is given by the modified Shockley–Read–Hall equation, as applied to the amphoteric distribution of dangling bonds:

(38)

where the parameters are given in Table 12 [66].

TABLE 12 Recombination Parameters in Amorphous Silicon

Acknowledgements

We are grateful to J. Puigdollers and C. Voz for providing graphs of the absorption coefficient and the density of states in amorphous silicon.

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