GRAIN BOUNDARY RECOMBINATION PROCESSES AND CARRIER TRANSPORT IN POLYCRYSTALLINE SEMICONDUCTORS UNDER OPTICAL ILLUMINATION

Transcription

1 CHAPTER V GRAIN BOUNDARY RECOMBINATION PROCESSES AND CARRIER TRANSPORT IN POLYCRYSTALLINE SEMICONDUCTORS UNDER OPTICAL ILLUMINATION 5.1 INTRODUCTION P olycrystalline semiconductors are potential candidates for many electronic devices. The electrical transport properties of these materials differ dramatically from mono-crystalline materials due to the presence of grain boundaries. These grain-boundaries generally contain a high density of trapping centers and impurities that have been segregated from grains during growth [196]. Inspite of the extensive studies, however, a fundamental understanding of the electrical properties of these materials has not yet been reached [197]. For instance the exact origin, density and nature of distribution of grain-boundary interface states are still not clear. ESR experiments suggest that major defects in the PX materials are the dangling bonds [196,197]. However, it is still not clear whether only these defects are responsible to influence the electrical properties. Contrary to this, the deep level transient spectroscopy (DLTS) measurements suggest that trap density is generally lower than the impurity concentration [198]. Furthermore it is very difficult to understand the effect of grain-boundaries and impurities accumulated near the grain-boundaries as the interaction between them is very complicated. In some cases, it is observed that the impurities can reduce GB interface states density (passivation) [199,200] and hence can improve the performance of polycrystalline devices.

2 GB interface states are responsible for the formation of space-charge potential barrier qv g at the GB. These barriers control the electrical and photovoltaic properties of PX semiconductor materials. In order to calculate the potential barrier height qv g, the exact knowledge of distribution of grainboundary states is required. The distribution of GB states is determined by the nature of the disorder, dangling bonds and by the local electronic potential fluctuations producing stress fields in the region of structural irregularities [201]. Different types of GB trapping states distributions are observed in PX-Si, depending upon the process of fabrication of the material. A uniform distribution of states is observed in Schottky barriers in PX-Si, an exponential distribution is found in photoconductivity measurements, while a single level (δ-distribution) is found in resistivity and mobility measurements in CVD polysilicon films [37]. Yang et al. [32] explained the electrical properties of PX-GaAs films by considering uniform distribution of GB states. They found that the trap density increases as the free carrier density is increased. They also found that the trap distribution at GB s depends on the substrate material and on the extent preferred orientation of the crystallites in the films [32]. The experimental study of Turner et al. [28] predicted that the GB s in GaAs generally exhibit p-type conduction and these GB s can be passivated by incorporation of Sn. The GB states also act as recombination centers for photogenerated carriers. This recombination of carriers reduces the efficiency of solar cells. In order to study the carrier transport in PX materials under optical illumination, it is necessary to develop a comprehensive conduction model of GB recombination under optical illumination. 99

3 5.2 THEORY OF GRAIN BOUNDARY RECOMBINATION In the present thesis author has considered Gaussian energy distribution of GB states near midgap. It is worthwhile to mention that this energy distribution can be reduced to other distributions as special cases [104,117]. The energy distribution of acceptor-like (or donor-like) midgap states in these materials can be expressed by the following expression [117] n gs (E) = N gs exp[- (E E T ) 2 / 2S 2 ] / [( 2π) S] (5.1) where N gs is the total density of localized states per unit area, S is distribution parameter and E T is the energy position of the mean value of interface states from the valance band edge. If S << kt, then the distribution reduces to δ-distribution as discussed in Chapter-V ASSUMPTIONS The energy band diagram of n-type semiconductors under optical illumination is shown in Fig In this work, following assumptions have been made to study the recombination processes at the grain-boundaries; 1. The polycrystalline semiconductor is composed of identical cubic grains with an average grain size d. 2. Polycrystalline material s transport properties are one dimensional. 3. The grain boundary width is much smaller than d. 4. The only potential barrier in disordered GB region is the space-charge potential barrier qv g created within the grains due to carrier trapping at the GB. The barrier qφ is neglected for GB recombination processes. However both qv g and qφ are considered to explain the electrical conduction. 100

4 5. There is no segregation of impurity atoms from grains to the GB, keeping in view that certain portion of dopant atoms might be segregated at grainboundaries at high doping levels [21,65]. 6. The parameters N gs, E T and S are assumed to be independent of grain size and doping concentration. 7. All GB states have equal capture cross sections whatever be their origin. For a recombination center there are two capture cross sections, one for the capture by neutral centre (σ N ) and the other for capture by Coulombic attractive centre (σ c ). Thus if σ p and σ n are the hole and electron capture cross sections of the GB interface states, then σ p = σ c and σ n = σ N. 8. The grain-boundary space-charge region is depleted of free carriers and its width W g is much smaller than the grain size [82,97,100, ,112,117,202]. This assumption is valid under both low and high excitation condition. At low excitation levels the GB space-charge region will be depleted of free carriers due to the presence of high electric field. On the other hand W g will be negligible as compared to grain size at high illumination levels. 9. Photogeneration of electrons and holes is uniform throughout the volume of the specimen [105,111,114,117,124] under long wavelength illumination only. This approximation is also not good enough for large thickness films. 10. Under sufficient illumination condition the dark hole concentration is negligible. 11. Under sufficient illumination condition the quasi-fermi level of the majority carriers E Fn is flat everywhere but the quasi-fermi level of minority carriers E Fp is allowed to vary with distance [82]. 101

5 5.2.2 GRAIN BOUNDARY POTENTIAL BARRIER HEIGHT Under optical illumination the photogenerated minority carriers of PXsemiconductors drift towards the grain-boundary surface due to high electric field present in the space-charge region. As a result of this, recombination of the excess minority carriers with the trapped majority carriers at GB is enhanced and a new interface charge is established at the GB through S-R-H capture and emission processes [45]. In this way, the GB space-charge potential barrier height is reduced from its dark value qv go and the Fermi level splits into electron and hole quasi-fermi levels E Fn and E Fp respectively. By equating the total charge accumulated in the GB interface states to the charge in the two neighboring depletion regions one can obtain qv g under optical illumination: Ec(o) (8ε N qv g ) 1/2 = q n gs (E) f(e) de (5.2) Ev(o) where the permittivity of PX material is ε and N is the doping density. The occupation function f(e) is given by f(e) = σ N n(o) + σ c n i β -1 / [σ N n(o) + σ N n i β + σ c p(o) + σ c n i β -1 ]. (5.3) where n(o) and p(o) are electron and hole densities at the GB, respectively, n i is intrinsic carrier concentration, β = exp[(e - E i )/kt], and E i is the intrinsic Fermi level. The electron and hole concentrations at the GB s are given by n(o) = N exp (- qv g / kt) (5.4) p(o) = (n 2 i / N) exp(qv g / kt) exp ( E F (o) / kt) (5.5) where E F (o) = E Fn (o) - E Fp (o) is the separation of quasi-fermi levels at the grain-boundary. 102

6 Fig. 5.1 The Energy Band diagram of n-type PX-Semiconductor under optical illumination depicting Gaussian energy distribution of midgap states. 103

7 5.2.3 RECOMBINATION VELOCITY Neglecting the generation and recombination of minority carriers the current density at any point x in the region can be expressed by; J p (x) = q ( D p / kt) p(x) d/dx [E Fp (x)], (5.6) where D p is the diffusion coefficient for holes and p(x) is the minority carrier concentration in bulk part at distance x from the interface. p(x) = n i exp[{e Fp (x) E i (x) } / kt] (5.7) The derivative of E i w.r.t. x defines the electric field. The total minority carrier current density flowing into the GB from the two adjacent depletion regions can be expressed as; J p (x) = J r (o) / 2 + J r (W g ) / 2 (5.8) where J r (o) is the GB recombination current density and J r (W g ) is recombination current density at the edge of the depletion region. The recombination current density is given by J r (W g ) = (qw g / 2τ b ) [(n 2 i /N){exp(qV g / kt) 1} exp( E F (o) / kt) - (J r (o) / 4qD p ) W g (π kt /qv g ) 1/2 erf(η)] [1 + (qw g / 2τ b ) (W g / 4qD p ) (π kt /qv g ) 1/2 erf(η)] -1 (5.9) where τ b is the minority carrier lifetime in the bulk part of the grain. If we consider the Joshi and Bhatt recombination model [117,118] in GB space-charge region (0 x W g ), the minority carrier concentration at the depletion edge can be expressed as P(W g ) = p(o) exp (-qv g / kt) + [{J r (o) + J r (W g )} [π kt/ qv g ] 1/2 (W g / 4qD p ) erf(η)] (5.10) where η = (qv g / kt) 1/2. 104

8 It has already been demonstrated by experimental studies that the effect of grain-boundaries exists not only in the depletion region but also in the quasineutral and neutral regions of the adjacent grains [100,107,137,203]. Therefore the recombination velocity of minority carriers at any point in the bulk part x > W g of the grain is equal to the effective recombination velocity S(W g ) for all grain sizes. The recombination velocity is the most important parameter and can be experimentally accessible. It relates the GB recombination to the bulk recombination. This velocity can be expressed by S(W g ) = [J r (W g ) + J r (o) ] / 2q{P(W g ) p o } = [J r (W g ) + J r (o) ] / {2qP(W g )} (5.11) where p o is the minority carrier density at equilibrium. There is one more recombination velocity for PX-semiconductor materials known as surface recombination velocity S(o). This represents the recombination of minority carriers at GB and is defined as S(o) = J r (o) / q p(o) (5.12) MINORITY CARRIER DENSITY IN THE BULK PART If the minority carrier recombination velocity at any point in the bulk part of the grain is represented by S(W g ), then the density of these carriers in this part of the grain can be given by [7,82] P(x) = p( ) [{S(W g )L b / D p } / {1+ S(W g ) L b / D p }] p( ) exp {(-x + W g )/ L b } (5.13) where p( ) is the concentration of holes in the field free region (x >> W g ) of the grain. p( ) can be determined from the above Eqn. by assuming that x = W g. Hence we get p( ) = P(W g ) + J r (W g ) L b / qd p. (5.14) 105

9 From Eqns. (5.13) and (5.14), we find that P(x) = P(W g ) + [J r (W g ) L b /qd p ][1 exp{(-x + W g ) /L b }] (5.15) This equation is valid under the condition that W g << d RECOMBINATION CURRENT DENSITY If it is assumed that the transition rate of a carrier bound to a trapping center to the adjacent center is much smaller than the rate to the conduction or valance band where many quantum states are available for transition [204], and neglect the recombination of carriers in the valance and conduction bands, then the recombination current density at the GB can also be obtained by using S-R- H theory [45]. The total steady state recombination current density at the GB surface can be obtained by integrating the expression for the recombination current density at a single interface energy gap between the limits E v (o) and E c (o), i.e. J r (o) = q σ c σ N V th n 2 i {exp[ E F (o) / kt] -1} Ec(o) n gs (E) de / [σ N n(o) + σ N n i β + σ c p(o) + σ c n i β -1 ] Ev(o) = 0 (5.16) where V th is the thermal velocity of the carriers. Under the optical illumination, the minority carriers are generated inside the grain. These photogenerated minority carriers recombine with the majority carriers in the bulk region, space-charge region and at the surface of the grainboundaries. Considering these recombination processes and the assumptions made earlier, the total photogenerated minority carrier current inside a cubic grain is expressed as, qg d 3 = 3 d 2 [ J r (o) + 2J r (W g ) ] + J b (d - 2W g ) 2 (5.17) 106

10 where G is the uniform photo generation rate of electron-hole pairs and J b is the recombination current density in the bulk part of the grain (x > W g ) and is given by d/2 J b = 2q / τ b p(x) dx (5.18) Wg If the condition of uniform illumination is not satisfied, then G can be considered as G = α I o exp(-αx) (5.19) where α is the absorption coefficient and I o is the illumination at the semiconductor surface. Thus under non-uniform illumination the above equation reduces to d d/2 αqd 2 I o exp(-αx) dx = 3d 2 [J r (o) + 2J r (W g )] + (d-2w g ) 2 2q / τ b p(x)dx 0 Wg (5.20) From, Eqns. (5.10, 5.17 and 5.18), the GB recombination current density can be expressed as J r (o) = qm [Gd {(d 6W g ) p(o) / τ b } exp(-qv g / kt)] (5.21) where M = M 1 [1 + exp(-ν o qv g / kt)] -1 (5.22) and M 1 = [3 +{(d -6W g )(W g / 4L b 2 )} [π kt / qv g ] 1/2 erf(η) + (d -6W g ) / 2L b - (1-4W g / d) {1- exp [-(d -2W g )/ 2L b ] }] -1 (5.23) From Eqn. (5.21), we note that J r (o) is mainly controlled by the grain size, bulk diffusion length, illumination level, absorption coefficient and the separation of quasi-fermi levels at GB. 107

11 By equating the Eqns. (5.16) and (5.21) and using Eqn. (5.1) and (5.2), the dependence of E F (o), and hence qv g on the various parameters such as grain size, illumination level, doping density, bulk diffusion length and temperature can be studied. From these calculations the corresponding J r (o), P(W g ), S(W g ) and S(o) can be computed DISCUSSION The computed variation of qv g as a function of optical illumination level is shown in Fig The values of the parameters used to compare theory with the available experimental data are listed in Table 5.1. Values of different selected parameters are reasonable and agree with several studies [97-117]. A good agreement is observed between theory and available experimental results [82,98]. The dependence of qv g on illumination level for different values of grain size for PX-Si and PX-GaAs materials is shown in Fig These plots show that as illumination level increases qv g decreases. It is also observed that as grain size increases the dependence of qv g on illumination level also increases. It is further noted that in the low illumination range the dependence of qv g on illumination level for PX-GaAs is more as compared to that of PX-Si. In contrast, in large grain size range the dependence of qv g on grain size for PX-Si is more as compared to that of PX-GaAs (Fig. 5.4). Present computations show that for d >> L b, qv g is independent of grain size whatever the illumination level is. The dependence of separation between quasi-fermi levels at GB; E F (o), on grain size and illumination level is shown in Fig We again note that for PX-GaAs E F (o) is independent of grain size at large grain sizes. The dependence of 108

12 Ref [82] Theory qvg (ev) Illumination Level G(cm -3 S -1 ) (a) Ref [98] Theory qvg (ev) Illumination Level G(cm -3 S -1 ) (b) Fig. 5.2 Variation of GB potential barrier with illumination level for PX-Si Experimental points are taken from (a) [82] and (b) [98]. 109

13 d=10 µm qvg (ev) Illumination Level G(cm -3 S -1 ) (a) qvg (ev) d=10 4 µm Illumination Level G(cm -3 s -1 ) (b) Fig. 5.3 Variation of GB potential barrier with illumination level at different grain sizes, for (a) PX-Si and (b) PX-GaAs. 110

14 G=10 18 (cm 3 s -1 ) qvg (ev) Grain size d(mm) (a) 0.5 G=10 16 (cm 3 /s) qvg (ev) Grain size d(mm) (b) Fig. 5.4 Variation of GB potential barrier with grain size at different illumination levels for (a) PX-Si and (b) PX-GaAs. 111

15 E F (o) on d and G can be explained by considering the variation of J r (o) with these parameters. At low illumination levels J r (o) is very small Fig. 5.6, and consequently E F (o) is approximately zero for Si, but it is appreciable for GaAs. Recombination current J r (o) increases with increasing illumination level, as a result of this more GB states act as recombination centers and hence E F (o) increases. It is also observed that in the large grain size range J r (o ) becomes independent of grain size Fig Note that in this grain size range qv g is independent of grain size. The dependence of recombination velocity S(o) on illumination level and grain size is shown in Figs. 5.8 and 5.9. From these plots we note that the dependence of S(o) on illumination level is different for PX-Si than for PX-GaAs. In the case of Si, S(o) increases with increasing illumination level, attains a maximum and then starts decreasing in the high illumination range. On the other hand, S(o) for PX-GaAs goes on increasing with increasing illumination level. The computed variation of space-charge recombination current density J r (W g ) with illumination level and grain size is shown in Figs and From these figures we note that: 1. The order of magnitude of J r (W g ) for PX-GaAs is 2-3 times larger than that for PX-Si. 2. In the larger grain size range J r (W g ) for PX-GaAs becomes independent of grain size while that for PX-Si goes on increasing slowly with increasing d. The reason for this different behavior of two materials is the small value of L b for PX-GaAs. 3. In the low grain size range J r (W g ) for PX-Si is less sensitive towards grain size as compared to PX-GaAs. 112

16 The effective recombination velocity at the depletion edge of GB for both the materials is found to be decreasing rapidly with increasing illumination level Figs and It is also observed that for PX-GaAs, S(W g ) is more sensitive to grain size and illumination level as compared to PX-Si, especially in the small grain size range. 5.3 CARRIER TRANSPORT IN POLYCRYSTALLINE MATERIALS UNDER OPTICAL ILLUMINATION In this chapter author has studied the electrical properties of PX-Si and PX-GaAs materials under optical illumination. As mentioned in Chapter-III of this thesis, in order to study the carrier transport across the GB s in these materials, one must consider the disordered nature of the GB material. The effect of GB material is represented by a rectangular potential barrier of height qφ and width δ. The potential barrier for the charge carriers in the GB region for n-type polycrystalline semiconductor in the dark is represented by Eqn. (3.11). qv(x) = qv go + qφ = qh(o) for 0 x δ/2. (5.24) As mentioned earlier in this chapter, the potential barrier is reduced from its dark value qv go under optical illumination. Thus under illumination Eqn. (5.24) can be expressed as qv(x) = qv g + qφ = qh L for 0 x δ/2. (5.25) If illumination level is very high then qv g qφ. Thus under such conditions at room temperature the maximum barrier height in the GB region is given by qh L = qφ. (5.26) 113

17 EF(o) (ev) d=10 3 µm Illumination Level G(cm -3 S -1 ) (a) d=10 4 µm EF(o) (ev) Illumination Level G(cm -3 s -1 ) (b) Fig. 5.5 Variation of DE F (o) with illumination level at different grain sizes for (a) PX-Si and (b) PX-GaAs. 114

18 Jr(o) (Acm -2 ) d=10 3 µm Illumination Level G(cm -3 s -1 ) (a) Jr(o) (A/cm 2 ) d=10 3 µm Illumination Level G(cm -3 s -1 ) (b) Fig. 5.6 Variation of J r (o) with illumination level at different grain sizes for (a) PX-Si and (b) PX-GaAs. 115

19 Jr(o) (Acm -2 ) G=10 18 (cm 3 /s) grain size d(mm) (a) G=10 20 (cm 3 /s) Jr(o) (A/cm 2 ) Grain size d(mm) (b) Fig. 5.7 Variation of J r (o) with grain size at different illumination levels for (a) PX-Si and (b) PX-GaAs. 116

20 7x10 4 d=10 4 µm 6x10 4 S(o) (cm/s) 5x10 4 4x10 4 3x x10 4 1x Illumination Level G(cm -3 S -1 ) (a) 10 4 d=10 3 µm S(o) (cm/s) Illumination Level G (cm -3 s -1 ) (b) Fig. 5.8 Variation of S(o) with illumination level at different grain sizes for (a) PX-Si and (b) PX-GaAs. 117

21 10 5 G=10 18 (cm 3 /s) 10 4 S(o) (cm/s) Grain size d(mm) (a) G=10 18 (cm 3 /s) S(o) (cm/s) Grain size d(mm) Fig. 5.9 (b) Variation of S(o) with grain size at different illumination levels for (a) PX-Si and (b) PX-GaAs. 118

22 d=10 3 µm Jr(W g) (A/cm 2 ) Illumination Level G(cm -3 S -1 ) (a) Jr(Wg) (A/cm 2 ) d=10 3 µm Illumination Level G(cm -3 s -1 ) (b) Fig Variation of J r (W g ) with illumination level for (a) PX-Si and (b) PX-GaAs. 119

23 Jr(Wg) (A / cm 2 ) G=10 18 (cm 3 /s) Grain size d(mm) (a) 10-3 G=10 20 (cm 3 /s) Jr(Wg) (A/cm 2 ) Grain size d(mm) (b) Fig Variation of J r (W g ) with grain size at different illumination levels for (a) PX-Si and (b) PX-GaAs. 120

24 6x10 6 5x10 6 d=10 2 µm S(Wg) (cm/s) 4x10 6 3x x10 6 1x x10 6 Illumination Level G(cm -3 s -1 ) (a) d=10 µm 2.068x10 6 S(Wg) (cm/s) 2.066x x Illumination Level G(cm -3 s -1 ) (b) Fig Variation of S(W g ) with illumination level at different grain sizes for (a) PX-Si and (b) PX-GaAs. 121

25 8.2x x10 6 S(Wg) (cm/s) 4.2x10 6 G=10 18 (cm 3 /s) 2.2x x Grain size d(mm) (a) 2.070x10 6 G=10 16 (cm 3 /s) 2.068x10 6 G=10 18 S(Wg) (cm/s) 2.066x10 6 G= x x Grain size d(mm) (b) Fig Variation of S(W g ) with grain size at different illumination levels for (a) PX-Si and (b) PX-GaAs. 122

26 Considering the GB recombination processes and scattering effects of GB s the net current density through the GB for small applied voltage can be expressed by the help of Eqn. (3.24) RESISTIVITY AND MOBILITY The grain-boundary resistivity of the bulk part of the semiconductor under optical illumination at V o >> 2kT/q can be expressed as ρ b = V o / J(2W g + δ) = [(2π m * kt) 1/2 ] / [q 2 (2W g + δ) n * ] [1/{ TFE + TFES + TE + TST }] (5.27) The average resistivity of the sample is given by ρ * = V a / J d = ρ b (2W g + δ) /d + ρ c [d - (2W g + δ)] /d = ρ gb (δ / d) + ρ o (2W g / δ) + ρ c [ d - (2W g + δ)] /d (5.28) where ρ gb, ρ o and ρ c are GB barrier resistivity, space-charge barrier resistivity and single crystal resistivity respectively. The effective carrier mobility of the carriers in the PX-semiconductor materials is given by µ * = 1 / (q ρ * n * ) (5.29) where n * is the effective density of mobile majority carriers. 5.4 DISCUSSION Fig. 5.14(a) represents the variation of experimental and computed resistivity with temperature for PX-Si under strong illumination (1 SUN). An excellent agreement is observed between computed and experimental data [123]. This fact demonstrates the validity of author s conduction model 123

27 developed in Chapter-III and GB recombination model developed in this chapter. Values of GB potential height qφ at different temperatures used to match present theory with experimental data are plotted in Fig. 5.14(b). From this plot we note that qφ decreases with decreasing temperature. A similar type of dependence of qφ on temperature is also observed in dark condition (Chapter-III). The decrease in qφ with decreasing temperature is due to the decreasing disorderliness at the GB. The present work once again demonstrates that the GB material is of disordered nature. The values of parameters used to explain the above mentioned variations are given in Table 5.2. The decrease in resistivity with decreasing temperature under illumination is due to the decrease in qv g. The dependence of resistivity of PX-Si on temperature for different values of grain size and illumination level is shown in Fig As grain size or illumination level decreases the resistivity tends to increase and approaches its corresponding dark value. The variation of the ratio of components of current density J 1, J 2, J 3 and J 4 to the total current density with temperature is shown in Fig It can be noted that the conduction in PX-Si under illumination is controlled by components J 2, J 3 and J 4. However, as temperature decreases the contribution of J 4 component increases and at very low temperature its contribution may be greater than that of J 3. This fact predicts that to explain the electrical properties of PX-semiconductors under solar illumination the component J 4 plays an important role. The same importance of J 4 is found in Chapter-III (under dark condition). It should be noted that in low temperature range qv g is much smaller than qφ at high illumination levels. Consequently the conduction is dominated by J 2, J 3 and J 4 components. A good agreement of the present theory with the 124

28 experimental results suggests that the possibility of hopping of charge carriers [67,117] in GB states near the Fermi level at low temperature is nil. The computed and available experimental data [123] for the variation of effective Hall mobility as a function of temperature for PX-Si under 1 SUN illumination is shown in Fig. 5.17(a). From this plot we note that the computed results are in agreement analytically with the experimental results. It should be noted that author has calculated drift mobility whereas the experimental data is for Hall mobility. From this plot, it can be noted that µ * is approximately proportional to T in low temperature range. The variation of effective mobility µ * with grain size at different temperature is shown in Fig. 5.17(b). The dependence of effective resistivity of PX-GaAs material on temperature under solar illumination (1 SUN) are shown in Fig. 5.18(a), by considering the parameter values listed in Table 5.3. When we compare Figs and Fig. 5.18, we note that the resistivity of PX-GaAs decreases with decreasing temperature in a wide temperature range as compared to that for PX-Si. This is because qv g for GaAs is greater than that for Si. The values of qφ for different temperatures are also shown in this Fig. 5.18(b). The variations of effective resistivity with grain size and illumination level are shown in Fig From this figure we note that the resistivity of this material decreases with increasing illumination level and grain size. The variations of effective mobility with temperature, and with grain size at different temperatures under 1 SUN illumination are shown in Fig In Fig author has also shown the variation of different components of current density with temperature for 125

29 10 Theory Ref. [123 ] Resistivity (Ohm-cm) /T (K -1 ) Fig. 5.14(a) Variation of resistivity with inverse temperature for PX-Si under 1 SUN illumination, Ref. [123]. 1 q f ( ev) /T (K -1 ) Fig. 5.14(b) Variation of qf with inverse temperature to match the experimental data. 126

30 100 d=10 2 µm Resistivity (Ohm-cm) /T (K -1 ) (a) G=10 18 cm 3 /s (Ohm-cm) /T (K -1 ) (b) Fig Computed variation of resistivity with inverse temperature for (a) different grain sizes and (b) different illumination levels. 127

31 J1/J, J2/J, J3/J, J4/J J1/J J2/J J3/J J4/J 1000/T (K -1 ). Fig Variation of the ratio of components of current to total current density with inverse temperature. 128

32 10000 Effective mobility (cm 2 / V-s) 1000 Theory Single crystal T (K) (a) Effective mobility (cm 2 / V-s) K 300 K 500 K 400 K Grain size d (mm) (b) Fig Computed variation of effective mobility with (a) temperature and (b) grain size at different temperatures. 129

33 1.5 Resistivity (ohm-cm) /T (K -1 ) (a) q (ev) /T (K -1 ) (b) Fig Variation of (a) resistivity and (b) qf with inverse temperature for PX-GaAs under 1 SUN illumination. 130

34 1.5 Resistivity (ohm-cm) d=10 3 µm /T (K -1 ) (a) Resistivity (ohm-cm) G=10 18 (cm 3 /s) /T (K -1 ) (b) Fig Computed variation of resistivity of PX-GaAs with inverse temperature for different values of (a) grain size and (b) illumination level. 131

35 10000 Effective mobility (cm 2 /V-s) T (K) (a) Effective mobility (cm 2 /V-s) K 200 K 300 K 410 K 500 K Grain size d(mm) (b) Fig (a) Variation of effective mobility with temperature (b) Effective mobility with grain size at different temperatures. 132

36 J1/J, J2/J, J3/J, J4/J J1/J J2/J J3/J J4/J /T (K -1 ) Fig Variations of ratio of components of current to total current density with inverse temperature for PX-GaAs under 1 SUN illumination. 133

37 PX-GaAs films. It can be seen that in high temperature range, J 3 has higher contribution as compared to the other components. As temperature decreases, the contribution of J 2 and J 3 increases and becomes comparable to that of J 3. In the very low temperature range, current is determined by J 4 component. Thus like PX-Si films, the electrical properties of PX-GaAs films under solar illumination are controlled by J 2, J 3 and J 4 components. 5.5 CONCLUSIONS In this chapter the recombination processes and carrier transport across the GB in PX-Si and PX-GaAs materials under optical illumination are studied. Present study is valid at sufficient illumination level, for partially depleted grains and over a wide temperature range. All four of mechanisms of carrier transport across the grain-boundaries are considered. The important conclusions of present study are mentioned below: (i) (ii) The distribution of GB states is of Gaussian type in both the materials. The behaviour of J r (W g ) in the two materials is different from each other especially in large and small grain size range. (iii) The dependence of recombination velocity S(o) on grain size and illumination level for the two materials is different from each other. (iv) As grain size increases, the dependence of recombination current density J r (o) on grain size decreases. (v) The electrical properties of polycrystalline semiconductors under solar illumination are not only controlled by J 2 and J 3 components of current but also by an additional component J 4. The contribution of J 4 component increases rapidly as temperature decreases. 134

38 (vi) The resistivity of PX-GaAs under optical illumination decreases with decreasing temperature in a wide temperature range as compared to that for PX-Si. (vii) The dependence of qφ on the temperature for PX-GaAs is found to be greater than that for PX-Si. 135

39 Table 5.1 Parameters used in theoretical computations to match the experimental data for different materials. δ (Å) =20 V th (cm 2 /s) =10 7 T (K) =300 G (1Sun=cm 3 /s) = Parameters PX-Si PX-GaAs Ref. [82] Ref. [98] Figs Figs N gs (cm -2 ) N (cm -3 ) E T (ev) d (µm) L b (µm) τ b (s) S (ev) σ C (cm 2 ) σ N (cm 2 ) D n (cm 2 s -1 ) kT kT ( ) kT ( ) kT

40 Table 5.2 Parameters used for theoretical computations for Figs Parameter N (cm -3 ) N gs (cm -2 ) E T (ev) d(µm) S (ev) σ c (cm 2 ) σ N (cm 2 ) L b (µm) τ b (s) PX-Si kT Table 5.3 Parameters used for theoretical computations for Figs Parameter N (cm -3 ) N gs (cm -2 ) E T (ev) d (µm) S (ev) σ C (cm 2 ) σ N (cm 2 ) L b (µm) τ b (s) PX-GaAs kT