Copper Zinc Tin Sulfide-Based Thin-Film Solar Cells

Part I

Introduction

1

An Overview of CZTS-Based Thin-Film Solar Cells

Kentaro Ito

Department of Electrical and Electronic Engineering, Shinshu University, 1-17-4 Wakasato, Nagano 380-8553, Japan

1.1 Introduction

This book deals with the thin-film solar cell with optical absorber layer composed of the copper-zinc-tin-sulphide-based quaternary semiconductor represented by chemical formula Cu2ZnSnS4 or related compound semiconductors. Throughout this book, we abbreviate the quaternary compound as CZTS. The concept of CZTS thin-film solar cells is based on the following principles. The compound semiconductor meets two necessary conditions for efficient solar cells. One is the direct nature of the band gap and the other is its width within a certain optimal range for photovoltaic cells. Because the pre-factor of absorption coefficient for the CZTS thin film is large enough the layer of just micron thickness is able to absorb sunlight sufficiently, and the use of it as an absorber does not have any damaging effects on photocurrents. The probability of radiative recombination in the film is able to exceed that of non-radiative recombination if both absorption and emission of photons are caused by an allowed direct transition of carriers between valence and conduction bands without any intermediaries such as crystal defects and phonons. It is therefore possible for cell efficiency to approach the theoretical limit if Shockley–Read–Hall-type recombination centers, which play a role in bypassing the direct recombination, are diminished and at the same time a device structure to confine excited electrons in the CZTS base layer is implemented. The CZTS semiconductor is potential candidate material for terawatt (TW) -scale photovoltaic energy conversion: a fractional amount of the elemental constituents produced annually is sufficient to fabricate CZTS thin-film solar cells which can supply renewable energy on a scale comparable to the world’s electricity consumption. The multiplicity of the compound is advantageous in designing the semiconductor material for photovoltaic devices, because we can control its physical properties depending on a substitution of the cation or anion included in the fundamental tetrahedron for another cation or anion and we can also avoid the undesirable use of rare or toxic elements. The incomplete (9%) substitution of sulfur for selenium is a typical example, which has lead to the achievement of alloy thin-film solar cells with over 10% efficiency [1, 2].

The physics of the photovoltaic effect are described in Section 1.2, including: the spectral irradiance of solar radiation and the influence of the Earth’s atmosphere on it; the upper limit of conversion efficiency of a single-junction solar cell which is evaluated on the basis of a detailed balance model; an optimal range of energy band gaps for photovoltaic energy conversion; optical absorption in semiconductor thin films and the estimation of the thickness of the absorber layer required for an efficient thin-film solar cell; and important roles of semiconductor pn- (positive or negative) homo- and hetero-junctions in the photovoltaic effect. In Section 1.3 we describe the pursuit of an optimal semiconductor for photovoltaic applications which have a band gap within the optimal range. The history of the thin-film solar cell is first discussed, including studies on some mono-crystalline semiconductor materials and their photovoltaic applications and the development of a chalcopyrite-type thin-film solar cell for comparison. We then describe how the concept of CZTS technology originated. Finally, we describe our synthesis and characterization of the CZTS absorber and n-type buffer layers to conclude the chapter.

1.2 The Photovoltaic Effect

1.2.1 Solar Radiation

1.2.1.1 Extra-terrestrial Radiation

At the core of the sun, nuclear fusion of hydrogen releases massive heat. The sun is surrounded by a thin atmosphere which consists mostly of hydrogen atoms. This is the so-called photosphere that absorbs the heat and emits electromagnetic radiation into outer space with almost the same spectral radiation as that of a black body in thermal equilibrium at a high temperature T S. According to Planck’s formula, the power emitted per unit projected area of the black body into a unit solid angle per unit frequency interval is given by the spectral irradiance Lν (T S), defined

(1.1)

where ν is the frequency of radiation, c is the light speed, h is the Plank constant, and k B is the Boltzmann constant. The photon energy of electromagnetic oscillation at frequency ν is given by hν. The solid angle ΩS of the sun (in steradians) which is seen from the Earth is calculated as:

(1.2)

where r is the radius of the sun (i.e. 6.96 × 10⁵ km) and R is the mean orbital radius of the Earth rotating around the sun (i.e. 1.496 × 10⁸ km).

The spectral photon irradiance which is defined by the number Nν (T S) of incident photons per square meter per second per Hertz arriving at the top of the Earth’s atmosphere is therefore expressed:

(1.3)

The smooth curve shown in Figure 1.1 is the theoretical plot of Nν (T S) versus photon energy hν, evaluated by substituting the effective temperature T S of the photosphere (which is assumed to be 5772 K) into Equation (1.3). The accuracy of this assumption is confirmed in Figure 1.1 as the theoretical curve for the electromagnetic wave emitted from a black body at T S agrees quite well with the observed spectra of extra-terrestrial radiation in photon energy of 1.15–1.72 eV. The two curves also cross at 3.11 and 0.44 eV. The latter spectra are derived from those of AM0 radiation measured as a function of wavelength [3]. However, there is a significant difference in peak irradiance between the two curves.

Figure 1.1 Spectral photon irradiance of AM0, AM1.5D and black body radiation.

The total power of electromagnetic radiation which is perpendicularly incident on a unit area is called the solar constant C S and is thus defined by:

(1.4)

where σ is the Stefan-Boltzmann constant equal to 5.67 × 10–8 W m–2 K–4. The effective surface temperature T S (= 5772 K) of the Sun is determined such that the theoretical value of C S given by Equation (1.4) agrees with the solar constant 1.3608 kW m–2, which was recently measured by NASA’s Solar Radiation and Climate Experiment Satellite [4].

1.2.1.2 Terrestrial Radiation

Figure 1.2 depicts how solar radiation reaches the Earth’s surface after passing through the atmosphere of effective thickness d (=8.4 km). The radiation is assumed to be perpendicularly incident on a planar solar cell inclined by the solar zenith angle θ (the angle between the local zenith and the line of the sight from that place to the Sun). The light pathlength s given by d/cosθ depends both on time and place on the Earth. The index air mass (AM), which is proportional to the pathlength s, is defined:

(1.5)

Figure 1.2 Schematic path length of sunlight through the atmosphere, given by d/cosθ

Since the AM is equal to the light pathlength s normalized to the thickness d, the extra-terrestrial radiation described above is often referred to as AM0 radiation. Figure 1.1 shows the standard reference spectra of AM1.5D and AM0 rays together with black body radiation as a function of photon energy [3]. The AM1.5D rays are directly incident with a zenith angle θ of 48.2°. The total incident power P i which is evaluated by integrating hνNν (T S) for the AM1.5D spectra over the whole photon energy range amounts to 0.90 kW m–2, that is, 34% of AM0 radiation is absorbed and scattered by the atmosphere.

The total power directly incident on the Earth’s surface I D is estimated as:

(1.6)

where R E is the radius of the Earth (6.4 × 10³ km), θ is the zenith angle between the vertical at a place on the Earth and the direct light beam from the sun, and τ a is the optical transmittance of the atmosphere which is experimentally determined by [5]:

(1.7)

According to Equations (1.5–1.7), I D is estimated as 0.30πR E ² C S = 1.04 × 10¹⁷ W. If this amount is accumulated during one year, its total is equivalent to 50,000 times the recent annual world electricity consumption of 20 PWh. This wide margin could be reduced by 3–4 orders of magnitude when we take into account the scarcity of the land available for module installation, the limitation imposed on solar cell efficiency, climate conditions and the peak demand of electricity. However, the margin thus modified still indicates that photovoltaic technology is an effective means of coping with the limiting supplies of fossil fuel in the foreseeable future.

The validity of Equation (1.7) can be approximately confirmed in the following. When the air mass is equal to 1.5, the transparency of the atmosphere given by Equation (1.7) takes a value of 0.63. The absorbance of the atmosphere is therefore estimated as 37%, which is comparable to the above-mentioned absorption loss of extra-terrestrial radiation in the atmosphere.

Despite the disturbance due to the surrounding atmosphere, the photon irradiance of AM1.5D radiation exhibits a peak at the same photon energy of about 0.77 eV as that of AM0 sunlight. The horizontal axis for the curve shown in Figure 1.1 represents the photon energy in electron-volts, which is converted to Joules if multiplied by electronic charge q. Since the wavelength of the photon is given by c/ν (see the upper horizontal axis of the figure), the peak photon energy corresponds to 1.6 μm which falls within a near-infrared range. The spectral sensitivity of the retina does not match this peak, however; instead it has evolved to cover the shorter-wavelength range in which solar irradiance is hardly weakened by the Earth’s atmosphere.

In Figure 1.1 there are several distinct bands in which the photons are absorbed by the atmosphere. A narrow band which appears at 1.63 eV is due to absorption by oxygen molecules. Water vapor is mainly responsible for optical absorption in the atmosphere at the low photon energy range of 0.46–1.7 eV. In this molecule there are two chemical bonds as expressed by H-O-H. Their vibration has numerous quantized states, which can be excited by photons. Their optical absorption bands are always broad and often strong compared with that of oxygen molecules. For example, the absorption band which is centered around 0.90 eV, denoted Ω, is assigned to a combined vibration mode associated with symmetric stretching and bending of the bonds [6]. We later refer to the case where optical absorption bands of water vapor have a significant influence on the efficiency of a solar cell.

1.2.2 Upper Limit of Conversion Efficiency in a Single-Junction Solar Cell

One of the most important criteria for solar cell materials is the band gap of semiconductors. We explain analytically why it may be of a direct nature and within a certain range of energy. It should be stressed that the compound of CZTS with which this book deals has an optimal direct energy gap. We define the absorber material as ideal if it possesses a direct band gap within the optimal range and, at the same time, is completely free from any sort of non-radiative recombination of carriers. In the strict sense of the word, mono-crystalline Si as an absorber could not be ideal because its band gap is of an indirect nature despite the fact that it has an optimal band gap of 1.12 eV. We discuss the silicon solar cell for comparison in Section 1.2.5.

Based on the detailed balance model which holds thermodynamically between the Sun’s radiation and light emission from a solar cell, Shockley and Queisser first predicted the efficiency limit of a single junction solar cell, which was later called the SQ limit [7]. An extended version of their theory was reported by Yablonovitch et al. [8]. It was assumed that incident photons whose energy is higher and slightly lower than the band gap E g are capable of exciting an electron and a hole which eventually recombine to emit a photon spontaneously. Both groups of authors incorporated the idealistic assumption that the mobility of carriers is infinitely large.

In a p-type semiconductor at thermal equilibrium there are a lot of free holes with concentration p 0 which is approximately equal to the density of doped acceptor impurities, giving rise to the p-type conductivity. The hole concentration p 0 is given by

(1.8)

where N v is the density of states of a valence band, E v is the energy level at the top of the band, E F is the Fermi level and T c is the ambient temperature [9]. As the acceptor density increases, E F tends towards E v. In addition, there is a very small amount of free electrons with concentration n 0 given by

(1.9)

where N c is the density of states of a conduction band and E c is the energy level at the bottom of the band. The product p 0 n 0 is independent of both the type and the density of impurities, since E c–E v is the forbidden band gap E g of the semiconductor, that is, an intrinsic property. The square root of the product is therefore referred to as intrinsic concentration, denoted n i and defined:

(1.10)

When the cell consisting of a planar pn-junction is excited by solar radiation, an open-circuit voltage V oc appears between the p-type and the n-type semiconductors. The open-circuit condition corresponds to the steady state of a thermodynamic system comprising the Sun, Earth and a solar cell. In this state the production rate of entropy reaches a minimum and a chemical potential difference qV oc arises at the semiconductor pn-junction. We make several assumptions, including: the planar junction with a unit area emits light within a hemisphere as a Lambertian source and hence reflects light completely at a rear electrode; and the cell is in thermal equilibrium at 298 K. Here we will simply assume that solar radiation with photon energy hν ≥ E g excites the cell to emit black body radiation, but that of hν < E g does not. The quasi-Fermi level E Fn for electrons in the p-type semiconductor moves upwards by qV oc in reference to the Fermi level E F (see Fig. 1.3). As long as the cell is in thermal equilibrium, and hence not illuminated by solar radiation, the Fermi level E F is constant everywhere. However, the minority carrier concentration n increases due to solar radiation as follows:

(1.11)

Figure 1.3 Band diagram of the pn-junction that is of open-circuit and illuminated by sunlight

The concentration of photo-excited excess electrons (n–n 0) is equal to that of photo-excited excess holes (p–p 0) because they are produced in pairs. This amount is much smaller than the majority carrier (hole) concentration p 0 in thermal equilibrium when the p-type semiconductor of a solar cell is taken into account. In contrast, this amount (n–n 0) exceeds the minority carrier (electron) concentration n 0 at thermal equilibrium, as expressed by the exponential factor of exp(qV oc/k B T c) in the right-hand side of Equation (1.11). The excited electrons diffuse from the vicinity of the n-type emitter towards the right edge of the semiconductor. In the idealistic solar cell, the diffusion length of minority carriers is infinitely large. If the right surface of the p-type absorber has an extremely low recombination velocity, which a back surface field might bring about, the quasi-Fermi level E Fn would be almost constant throughout the region.

The number of photons emitted from the cell increases by the same factor as above and should balance the incident photons which are able to contribute to the re-emission of radiation. We therefore obtain:

(1.12)

where θ is the zenith angle between an emitted light flux and the direction perpendicular to the cell in a spherical coordinate system, φ is the azimuth angle of the flux, Lν (T c) is the spectral irradiance that is re-emitted from the cell and is considered equal to blackbody radiation at T c, and Nν (T S) is either the measured or the theoretical spectrum of solar photon irradiance. The radiative recombination efficiency η r for re-emission is defined as follows. The radiative (non-radiative) recombination probability per unit time is equal to the inverse of radiative (non-radiative) recombination lifetime τ r (τ nr). Since total probability of recombination of minority carriers is equal to the sum of the radiative recombination probability and the non-radiative recombination probability, η r is defined:

(1.13)

where τ is the total lifetime. The lifetime τ r is given by ϕ/Bp 0 where B is the bimolecular recombination coefficient due to band-to-band recombination and ϕ is the photon recycling factor, that is, the inverse probability of the photon escape through the boundaries of the optical absorber without re-absorption [10, 11]. The factor ϕ is larger than 1 as long as there is enough space for re-emitted photons to excite electron-hole pairs in the absorber. Lifetime τ nr is usually governed by crystal defects such as impurities. According to the simplified Shockley–Read–Hall (SRH) model, τ nr is given by the reciprocal of the product of the density N t of recombination centers, their capture cross-section σ n and the thermal velocity v th of electrons [12]. When p 0 is very high, Auger recombination prevails so that the non-radiative lifetime is inversely proportional to the square of p 0 [13].

Equation (1.12) can be rearranged to yield

(1.14)

where J sc is the short-circuit current density given by Equation (1.15) and J 0 is defined by Equation (1.16):

(1.15)

(1.16)

Energy band gap E g would be much larger than thermal energy k B T c (=0.026 eV) if we take the energy range of E g (c. 1 eV) suitable for a solar cell into consideration. Each term in Equation (1.16) is a rapidly decreasing function of integer m. To evaluate the theoretical performance of solar cells, we later use an equation which is approximated only by the first term (i.e. m = 1).

If the cell is kept at thermal equilibrium in the dark, J sc and V oc should decrease to zero. This inevitable relation could not be seen if we use Equation 1.14 which is derived by neglecting the strict conservation law that the total number of photons incident both from the Sun and the Earth’s atmosphere balances that of re-emitted photons. By taking account of this, a strict expression for V oc is

(1.17)

The open-circuit voltage increases with the radiative recombination efficiency η r. In a real solar cell there might be a photonic loss due to the non-radiative recombination of carriers via crystal defects which leads to a decrease in V oc. There might also be an optical loss at the rear surface of the cell. These unfavorable circumstances lead to a degradation of efficiency.

The fill factor (FF) of a solar cell is empirically determined by [14]:

(1.18)

If the solar cell is short-circuited, the current density J sc can be obtained from the cell as given by Equation (1.15). The value of J sc in a non-idealistic solar cell is less than that given by Equation (1.15) above because the minority carriers generated in the semiconductor are not collected completely; they are partly lost due to their short diffusion length before arriving at the interface of the pn-junction (see Section 1.2.5). An optical loss due to reflection of solar radiation at the incident surface also causes a decrease in J sc.

The efficiency η of the solar cell is defined as the product of the open-circuit voltage V oc, short-circuit current density J sc and fill factor FF, divided by the incident power P i of solar radiation, that is:

(1.19)

The cell efficiency is calculated under various illumination conditions according to the above equations in the following section.

1.2.3 Optimal Band Gap for Solar Cells

The spectral absorbance Aν of the Earth’s atmosphere at hν is defined as the natural logarithm of AM1.5D irradiance divided by AM0 irradiance. Based on the Beer–Lambert law, we first scrutinize the absorbance which is caused by the two processes predominant in a high-energy range of photons:

(1.20)

where ρ is the density of gas molecules responsible for preventing rays from reaching a solar cell both fully and directly, σ is the cross-section of molecules, the subscript R represents air particles contributing to Rayleigh scattering [15], and the subscript o indicates the ozone molecules that absorb photons. The effective path length ξ in the right-hand side of the equation is equivalent to ρ d / n Lcosθ in which n L corresponds to the Loschmidt constant 2.69 × 10²⁵ m–3, that is, the density of gas molecules at 273 K and 1013 hPa. The value of ξ is estimated as 8.4 km and 3.2 mm for air and ozone molecules, respectively.

The absorbance is plotted in Figure 1.4 as a function of the fourth power of hν. There is a photon energy region extending from 2.5 to 3.8 eV where the spectral absorbance increases linearly as the fourth power of hν increases, indicating that the extra-terrestrial radiation is scattered by particles in atmospheric air, the diameter of which is much smaller than the related light wavelength [15]. Since the absorbance at hν = 2.6 eV is 0.41, the cross-section σ R of air is estimated as1.2 × 10–30 m². This value is 30% higher than that for nitrogen which was observed at the same photon energy in a laboratory [16].

Figure 1.4 Relationship between absorbance of the Earth’s atmosphere and the fourth power of photon energy

Despite the fact that the effective path length ξ o of ozone is extremely small, its absorbance at photon energy higher than 4.1 eV is predominant. A very slightly elevated broad peak of the absorbance curve centered at around 2.06 eV might be superimposed on the absorbance curve due to Rayleigh scattering and coincide with a broad peak of the cross-section σ o which was observed in laboratory absorption spectra of ozone [17].

We now discuss the effect of the Earth’s atmosphere on the efficiency limit. Figure 1.5 shows the theoretical efficiency of a solar cell based on a detailed balance as a function of semiconductor band gap under various radiation conditions at T c = 298 K. The radiative recombination efficiency η r is assumed to be equal to 1. In other words, electrons and holes do not recombine non-radiatively, which means τ r/τ nr = 0. The cell under AM1.5G, AM1.5D and AM0 radiation exhibits the maximum efficiency η max of 33.8, 33.4 and 30.4% at band gap E g,max of 1.34, 1.14 and 1.24 eV, respectively. AM1.5G radiation includes both direct and diffuse radiation. The appearance of double peaks in the former two curves is caused by the third optical absorption band ρστ due to water vapor.

Figure 1.5 Theoretical efficiency versus band gap curves of an ideal solar cell under various radiation conditions: (a) AM1.5G and black body radiation; and (b) AM1.5D and AM0 radiation

The curve for the cell illuminated by blackbody radiation at 5772 K is also included in the figure for reference. As far as solar cells are concerned, the black-body radiation is almost equivalent to AM0. Abrams et al. showed that η max = 29.83% and E g,max = 1.34 eV using a different formulation for black body radiation and the two characteristic values of T S = 6000 K and T c = 300 K [18]. Shockley and Queisser showed much earlier that η max = 30% and E g,max = 1.1 eV by adopting these characteristic temperatures [7]. The latter value (1.1 eV) is significantly lower than that calculated by the former authors and implied that the silicon crystal, an indirect semiconductor, could be the most suitable material for solar cells. If we adopted these two characteristic temperatures instead of T S = 5772 K and T c = 298 K (see Table 1.1), this would yield η max = 31.03% and E g,max = 1.304 eV. The small discrepancies between the estimations made by Abrams et al. and the current authors are due to a difference in approximations used.

Table 1.1 Effects of the Earth’s atmosphere on the performance of an ideal solar cell at 298 K. Theoretical calculations for black body radiation at 5772 K are also shown for reference

The characteristic parameters under these four types of radiation are summarized in Table 1.1. The band gap which makes it possible for cell efficiency to exceed 95% of the maximum efficiency η max ranges from the narrow end E g,n of 1.06 eV to the wide end E g,w of 1.50 eV under AM1.5G. It is quite interesting that CZTS and an alloy, Cu2ZnSn(S,Se)4 (abbreviated as CZTSSe), possess the band gaps within this range. The values of E g,max, E g,n and E g,w shown above agree very well with those reported by Yablonovitch et al. [8]. Under AM1.5D, E g,max and E g,w are significantly smaller than under AM0, indicating that the photon number in high-energy spectra is decreased by Rayleigh scattering. The significant blue shift of E g,max, which is theoretically predicted above when solar radiation is switched from AM1.5D to AM1.5G, is solely due to this scattering. The band gap of 1.56 eV is still optimal for extra-terrestrial applications of a solar cell, but not for terrestrial applications. Since high-energy photons scattered by air can be partly incident on a solar cell under AM1.5G, cell efficiency with a band gap wider than 1.1 eV is always higher than that under AM1.5D. Consistent with Rayleigh scattering, the relative difference between the two becomes large as the band gap is increased to 3.5 eV.

When a band gap is as wide as 4 eV, incident photons terrestrially available for the excitation of electron-hole pairs are reduced both due to Rayleigh scattering by air and optical absorption by ozone, virtually leading to the loss of the entire photovoltaic effect. If a cell is used under an extra-terrestrial condition, the short-circuit current density would approach zero quite rapidly as E g increases further because, under black body radiation, it is defined:

(1.21)

As the band gap is decreased from 4 eV however, the efficiency under AM1.5D and AM1.5G rises rather steeply and exceeds that under AM0 radiation. This is explained by the fact that in the former two cases energy loss caused by excess photon energy (hv–E g) is smaller than in the latter case, and the contribution of the high-energy photons to total incident power is reduced by the presence of air and ozone. The optical absorption by oxygen molecules and the first two absorption bands due to water vapor do not have any significant effect on efficiency.

As the band gap decreases further, cell efficiency falls because the open-circuit voltage decreases. This tendency is amplified when the band gap falls within the energy range of a strong absorption band due to water vapor: the open-circuit voltage drops rapidly as the dark current increases while the short-circuit current density remains almost constant, resulting in a few dips or cliffs in the η versus E g curve. At AM1.5D and AM1.5G the efficiency drops by 5% because of the Ω absorption band. Direct sunlight, which is most likely represented by AM1.5D radiation, can be concentrated by lenses or mirrors. Using a 100× concentrator, for example, we are able to collect solar radiation as if the distance between the Earth and the Sun were made 10 times shorter. As described in Equations (1.2), (1.3) and (1.15), the solid angle ΩS and J sc become 100 times larger, and consequently V oc and FF increase, contributing to the enhancement of cell efficiency. It is desirable to avoid the use of the semiconductor band gap coinciding with an absorption band such as the Ω band, particularly when the multi-junction solar cell is designed for a concentrated solar power system. Sağol et al. showed that the four-junction solar cell with theoretical efficiency 62% could be limited to 59% at best if the band gap of the bottom cell is between 0.8 and 0.9 eV [19].

If the band gap of a semiconductor approaches zero, and hence the semiconductor becomes a nearly perfect black body kept at T c, the open-circuit voltage of the cell based on a detailed balance would exhibit the ultimate open-circuit voltage given by:

(1.22)

Using the five parameters mentioned before, V oc,u is evaluated as 3.7 mV. The cell might therefore be useful for measuring the effective temperature of the sun but is a poor quantum energy converter, with an ultimate efficiency of 0.07%.

1.2.4 Optical Absorption in Semiconductor Thin Films

A semiconductor such as CZTS has the direct forbidden band gap E g shown in Figure 1.6. Electrons at the top of the valence band have the maximum energy when their crystal momentum, that is, the wave number k multiplied by the Planck constant h divided by 2π, is equal to zero. By shedding light on the p-type semiconductor at photon energy of E g, the valence electrons are excited from the top of the valence band to the bottom of the conduction band, generating holes in the valence band and an equal amount of minority carriers (electrons) in the conduction band. Since the conservation law of momentum is satisfied in this direct transition, optical absorption proceeds without difficulty. After electrons are injected to the conduction band, they recombine with the majority carriers (holes) to emit light within their short lifetime τ r. The situation is quite different in an indirect gap semiconductor such as Si. Figure 1.7 demonstrates how an electron excited to the conduction band minimum has a crystal momentum which is nearly equal to π divided by the lattice constant a of silicon, while a hole in the valence band has momentum equal to zero. In order to fulfill the momentum conservation before and after the indirect transition, the third particle must participate such that a phonon with a wavelength of 2a is emitted: the light wavelength pertaining to solar cells is much longer than 2a. The radiative recombination lifetime τ r is therefore very long in silicon. The high absorption coefficient and the large bimolecular recombination coefficient of a direct gap semiconductor make it easier to enhance the energy conversion efficiency of a thin-film solar cell.

Figure 1.6 E v. k curve of a direct band gap semiconductor

Figure 1.7 E v. k curve of an indirect band gap semiconductor

Table 1.2 shows the photovoltaic characteristics of two types of highly efficient solar cell measured at AM1.5G and 298 K. One consists of a thin film of epitaxial lift-off (ELO) GaAs [20, 21] while the other comprises a bulk silicon crystal with a hetero-junction with intrinsic thin layer (HIT) structure [22–25]. The number in parentheses is the ratio of the measured value to the theoretical value calculated under the above conditions and the assumption that τ r/τ nr = 0 or τ nr = ∞. It should be noted that the former cell achieved 96% of the theoretical open-circuit voltage whereas the latter achieved 85%. The higher achievement ratio in the former could be attributed to the much shorter τ r of GaAs than that of Si. The absorber region of the two solar cells is sandwiched by wide-gap semiconductors, where the hetero-structure is considered effective in reducing the recombination velocity at the interface. Both efficient solar cells share the tendency according to which the open-circuit voltage is a decreasing function of absorber thickness; however, the physics governing each cell is quite different as the discussion in the next section reveals.

Table 1.2 Device characteristics of GaAs and Si solar cells: a comparison between theoretical and experimental

The absorption coefficient of a direct-gap semiconductor is defined [9, 10]:

(1.23)

Only photons with energy higher than or equal to E g can excite the valence electrons and the absorption coefficient increases with photon energy. The absorption coefficient α at hν = 2E g is equal to α 0. According to the van Roosbroeck–Shockley relation for carrier recombination [26], the factor α 0 is proportional to the bimolecular recombination coefficient B. As discussed in the next section, the higher the value of α 0 the shorter the radiative lifetime, the smaller the dark saturation current density, and hence the higher the open-circuit voltage. If the semiconductor is a thin film of thickness t whose rear side has a perfect reflector and there is no reflection at the front surface, the intensity I of light escaping from the front is expressed:

(1.24)

where I 0 is the intensity of incident light. For example, for a value of α 0 of 5 × 10⁴ cm–1 and hν of 1.16E g, with a thickness of 1.7 μm the light intensity absorbed in the film could reach 99.9% of I 0. This is a rough estimate of the thickness theoretically required for the optical absorber layer of a CZTS thin-film solar cell. In the indirect-gap semiconductor of Si however, the value of α 0 is as small as 1 × 10³ cm–1 and the absorption coefficient is proportional to the square of (hν–E g). According to the same estimate as above, a 130 μm thick silicon crystal is required for the absorber.

1.2.5 Semiconductor pn-Junctions

As far as the ideal solar cell which works under the principle of detailed balance is concerned, the electric current J which is supplied from the unit area cell to a load can be expressed:

(1.25)

where V is the voltage drop of the load, R s is the series resistance, R sh is the shunt resistance of the diode, and J d is the dark current density of the cell, defined

(1.26)

Equation (1.25) agrees with Equation (1.17) when R s is zero and R sh is infinite. The equivalent circuit of the cell could therefore be drawn as in Figure 1.8. The circuit model defines J d as the saturation current density in the dark which would be obtained if the pn-junction diode is biased backward (V < 0) and R sh is infinitely large. Since solar cells of course operate under forward bias, J d actually represents the pre-exponential factor of the forward current at a large bias. The value of J d is the sum of two components. One is the current density J 0 which is determined by detailed balance and the other is the component which is proportional to the ratio τ r/τ nr. This ratio is one of the key parameters in obtaining an ideal solar cell because the smaller the value of the ratio, the higher the open-circuit voltage. In an ideal solar cell, τ r/τ nr is equal to zero because there is no indirect recombination.

Figure 1.8 Equivalent circuit model for an idealized solar cell

We next discuss whether the detailed balance model may be applied to the ELO thin-film GaAs or the HIT Si solar cell. Consider a device structure whose base is an n-type semiconductor with the electron concentration n 0. The inherent radiative lifetime τ r/ϕ of GaAs (Si) is estimated to be 50 ns (4.2 ms) by assuming that n 0 is 1 × 10²³ (5 × 10²²) m–3, and the bimolecular recombination coefficient B = 2 × 10–16 (4.73 × 10–21) m³ s–1. Using the theoretical dark current density, J 0 = 5.6 × 10–18 (6.1 × 10–13) A m–2 and experimentally observed V oc and J sc of GaAs (Si) solar cells shown in Table 1.2, the ratio of τ r/τ nr is estimated to be 4.6