grain boundary recombination processes and carrier...

CHAPTER – V

GRAIN BOUNDARY RECOMBINATION PROCESSES AND CARRIER TRANSPORT IN POLYCRYSTALLINE

SEMICONDUCTORS UNDER OPTICAL ILLUMINATION

5.1 INTRODUCTION

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.

P

99

GB interface states are responsible for the formation of space-charge

potential barrier qVg at the GB. These barriers control the electrical and

photovoltaic properties of PX semiconductor materials. In order to calculate the

potential barrier height qVg, the exact knowledge of distribution of grain-

boundary 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.

100

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]

ngs(E) = Ngs exp[- (E – ET)2 / 2S2] / [(√2π)⋅S] (5.1)

where Ngs is the total density of localized states per unit area, S is distribution

parameter and ET 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.

5.2.1 ASSUMPTIONS

The energy band diagram of n-type semiconductors under optical

illumination is shown in Fig. 5.1. 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 qVg created within the grains due to carrier trapping at the

GB. The barrier qφ is neglected for GB recombination processes. However

both qVg and qφ are considered to explain the electrical conduction.

101

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 grain-

boundaries at high doping levels [21,65].

6. The parameters Ngs, ET 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 (σN) and the other for capture by Coulombic

attractive centre (σc). Thus if σp and σn are the hole and electron capture

cross sections of the GB interface states, then σp = σc and σn = σN.

8. The grain-boundary space-charge region is depleted of free carriers and its

width Wg is much smaller than the grain size [82,97,100,102-

103,112,117,202]. 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 Wg 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 EFn is flat everywhere but the quasi-Fermi level of minority carriers

EFp is allowed to vary with distance [82].

102

5.2.2 GRAIN BOUNDARY POTENTIAL BARRIER HEIGHT

Under optical illumination the photogenerated minority carriers of PX-

semiconductors 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 qVgo and the Fermi level splits into electron

and hole quasi-Fermi levels EFn and EFp respectively. By equating the total

charge accumulated in the GB interface states to the charge in the two

neighboring depletion regions one can obtain qVg under optical illumination:

Ec(o) (8ε N qVg)1/2 = q ∫ ngs (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) = σNn(o) + σcniβ-1/ [σNn(o) + σNniβ + σc p(o) + σcniβ-1]. (5.3)

where n(o) and p(o) are electron and hole densities at the GB, respectively, ni is

intrinsic carrier concentration, β = exp[(E - Ei)/kT], and Ei is the intrinsic Fermi

level. The electron and hole concentrations at the GB’s are given by

n(o) = N exp (- qVg / kT) (5.4)

p(o) = (ni2 / N) exp(qVg / kT) exp ( ∆EF(o) / kT) (5.5)

where ∆EF(o) = EFn(o) - EFp(o) is the separation of quasi-Fermi levels at the

grain-boundary.

103

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

104

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;

Jp(x) = q ( Dp / kT) p(x) d/dx [EFp(x)], (5.6)

where Dp 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) = ni exp[{EFp(x) – Ei(x) } / kT] (5.7)

The derivative of Ei 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;

Jp(x) = Jr(o) / 2 + Jr(Wg) / 2 (5.8)

where Jr(o) is the GB recombination current density and Jr(Wg) is recombination

current density at the edge of the depletion region. The recombination current

density is given by

Jr(Wg) = (qWg / 2τb) [(ni2 /N){exp(qVg / kT) – 1} exp(∆EF(o) / kT)

- (Jr(o) / 4qDp ) Wg⋅ (π kT /qVg)1/2 erf(η)]

⋅[1 + (qWg / 2τb) (Wg / 4qDp ) ⋅ (π kT /qVg)1/2 erf(η)]-1 (5.9)

where τb 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 ≤ Wg), the minority carrier concentration at the depletion edge can be

expressed as

P(Wg) = p(o) exp (-qVg / kT) + [{Jr(o) + Jr(Wg)}⋅[π kT/ qVg]1/2⋅ (Wg/ 4qDp) erf(η)]

(5.10)

where η = (qVg / kT)1/2.

105

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 quasi-

neutral 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 > Wg

of the grain is equal to the effective recombination velocity S(Wg) 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(Wg) = [Jr(Wg) + Jr(o) ] / 2q{P(Wg) – po}

= [Jr(Wg) + Jr(o) ] / {2qP(Wg)} (5.11)

where po 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) = Jr(o) / q p(o) (5.12)

5.2.4 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(Wg), then the density of these carriers in this part

of the grain can be given by [7,82]

P(x) = p(∞) – [{S(Wg)Lb / Dp} / {1+ S(Wg) Lb/ Dp }]⋅p(∞) exp {(-x + Wg)/ Lb}

(5.13)

where p(∞) is the concentration of holes in the field free region (x >> Wg) of the

grain. p(∞) can be determined from the above Eqn. by assuming that x = Wg.

Hence we get

p(∞) = P(Wg) + Jr(Wg) Lb / qDp. (5.14)

106

From Eqns. (5.13) and (5.14), we find that

P(x) = P(Wg) + [Jr(Wg) Lb /qDp][1 – exp{(-x + Wg) /Lb}] (5.15)

This equation is valid under the condition that Wg << d.

5.2.5 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 Ev(o) and

Ec(o), i.e.

Jr(o) = q σcσN Vth ni2 {exp[∆EF(o) / kT] -1}

Ec(o)

× ∫ ngs(E) dE / [σN n(o) + σN ni β + σc p(o) + σc ni β-1] Ev(o) = 0

(5.16)

where Vth 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 grain-

boundaries. Considering these recombination processes and the assumptions

made earlier, the total photogenerated minority carrier current inside a cubic

grain is expressed as,

qG d3 = 3 d2 [ Jr(o) + 2Jr (Wg) ] + Jb (d - 2Wg)2 (5.17)

107

where G is the uniform photo generation rate of electron-hole pairs and Jb is the

recombination current density in the bulk part of the grain (x > Wg) and is given

by

d/2 Jb = 2q / τb ∫ p(x) dx (5.18) Wg

If the condition of uniform illumination is not satisfied, then G can be considered

as

G = α Io exp(-αx) (5.19)

where α is the absorption coefficient and Io is the illumination at the

semiconductor surface. Thus under non-uniform illumination the above equation

reduces to

d d/2

αqd2 ∫ Io exp(-αx) dx = 3d2 [Jr(o) + 2Jr(Wg)] + (d-2Wg)2 2q / τb ∫ 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

Jr(o) = qM [Gd – {(d – 6Wg) p(o) / τb} exp(-qVg / kT)] (5.21)

where M = M1 [1 + exp(-νo qVg / kT)]-1 (5.22)

and M1 = [3 +{(d -6Wg)(Wg / 4Lb2)} [π kT / qVg]1/2 erf(η)

+ (d -6Wg) / 2Lb - (1- 4Wg / d)⋅{1- exp [-(d -2Wg)/ 2Lb] }]-1 (5.23)

From Eqn. (5.21), we note that Jr(o) is mainly controlled by the grain size, bulk

diffusion length, illumination level, absorption coefficient and the separation of

quasi-Fermi levels at GB.

108

By equating the Eqns. (5.16) and (5.21) and using Eqn. (5.1) and (5.2),

the dependence of ∆EF(o), and hence qVg 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 Jr(o) ,

P(Wg), S(Wg) and S(o) can be computed.

5.2.6 DISCUSSION

The computed variation of qVg as a function of optical illumination level is

shown in Fig. 5.2. 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 qVg on illumination level for different values of grain

size for PX-Si and PX-GaAs materials is shown in Fig. 5.3. These plots show

that as illumination level increases qVg decreases. It is also observed that as

grain size increases the dependence of qVg on illumination level also increases.

It is further noted that in the low illumination range the dependence of qVg 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 qVg on grain size for PX-Si is more

as compared to that of PX-GaAs (Fig. 5.4). Present computations show that for

d >> Lb, qVg is independent of grain size whatever the illumination level is. The

dependence of separation between quasi-Fermi levels at GB; ∆EF(o), on grain

size and illumination level is shown in Fig. 5.5. We again note that for PX-GaAs

∆EF(o) is independent of grain size at large grain sizes. The dependence of

109

Theory

Ref [82]

1019 102010181017101610150.25

0.3

0.35

0.4

0.45

Illumination Level G(cm -3S-1)

qVg

(eV)

(a)

Theory

Ref [98]

1015 1016 1017 1018 10201019

0.05

0.1

0.15

0.2

0.25

0.3

Illumination Level G(cm-3S-1)

qVg

(eV)

(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].

110

103

1019 10201018101710161015

0.1

0.15

0.2

0.25

0.3

0.35


qVg

(eV)

102

d=10 µm

(a)

1015 1016 1017 1018 10201019

10

d=104µm

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Illumination Level G(cm -3s-1)

qVg

(eV)

(b)

Fig. 5.3 Variation of GB potential barrier with illumination level at different

grain sizes, for (a) PX-Si and (b) PX-GaAs.

111

G=1018

(cm3s-1)

1020

0

0.1

0.2

0.3

0.4

1 10 100 1000 10000

Grain size d(µ m)

qVg

(eV)

(a)

G=1016 (cm3/s)

1020

1018

0.2

0.3

0.4

0.5

0.6

1 10 100 1000 10000

Grain size d(µ m)

qVg

(eV)

(b)

Fig. 5.4 Variation of GB potential barrier with grain size at different illumination levels for (a) PX-Si and (b) PX-GaAs.

112

∆EF(o) on ‘d’ and ‘G’ can be explained by considering the variation of Jr(o) with

these parameters. At low illumination levels Jr(o) is very small Fig. 5.6, and

consequently ∆EF(o) is approximately zero for Si, but it is appreciable for GaAs.

Recombination current Jr(o) increases with increasing illumination level, as a

result of this more GB states act as recombination centers and hence ∆EF(o)

increases. It is also observed that in the large grain size range Jr(o) becomes

independent of grain size Fig. 5.7. Note that in this grain size range qVg 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

Jr(Wg) with illumination level and grain size is shown in Figs. 5.10 and 5.11.

From these figures we note that:

1. The order of magnitude of Jr(Wg) for PX-GaAs is 2 - 3 times larger than that

for PX-Si.

2. In the larger grain size range Jr(Wg) 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

Lb for PX-GaAs.

3. In the low grain size range Jr(Wg) for PX-Si is less sensitive towards grain

size as compared to PX-GaAs.

113

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. 5.12 and 5.13. It is also observed that for PX-GaAs, S(Wg) 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) = qVgo + 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 qVgo under optical illumination. Thus under illumination Eqn. (5.24) can be

expressed as

qV(x) = qVg + qφ = qHL for 0 ≤ x ≤ δ/2. (5.25)

If illumination level is very high then qVg ≤ qφ. Thus under such conditions at

room temperature the maximum barrier height in the GB region is given by

qHL = qφ. (5.26)

114

10

102

d=103 µm

1019 10201018101710161015

0

0.1

0.2

0.3

0.4

0.5


E F(o

) (e

V)

(a)

1019 1020101810171016

10

d=104µm

0.2

0.3

0.4

0.5

0.6

0.7

0.8


E F(o

) (eV

)

(b)

Fig. 5.5 Variation of ∆EF(o) with illumination level at different grain sizes for

(a) PX-Si and (b) PX-GaAs.

115

d=103µm

102

10

1019 10201018101710161015

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100


J r(o

) (A

cm-2

)

(a)

1019 10201018101710161015

d=103 µm

10

10-7

10-6

10-5

10-4

10-3

10-2

10-1

Illumination Level G(cm-3s-1)

J r(o

) (A

/cm

2 )

(b)

Fig. 5.6 Variation of Jr(o) with illumination level at different grain sizes for


116

1016

G=1018

(cm3/s)

100

10-1

10-2

10-3

10-8

10-7

10-6

10-5

10-4

1 10 100 1000 10000

grain size d(µm)

J r(o

) (A

cm-2

)

(a)

G=1020

(cm3/s)

1018

1016

100

10-1

10-2

10-3

10-4

10-5

10-6

10-7

10-8

1 10 100 1000 10000

Grain size d(µm)

J r(o

) (A

/cm

2 )

(b)

Fig. 5.7 Variation of Jr(o) with grain size at different illumination levels for


117

d=104µm

102

1x104

2x104

3x104

7x104

6x104

5x104

4x104

1019 10201018101710161015


S(o)

(cm

/s)

(a)

10

d=103 µm

105

104

103

102

101

102010191018101710161015

Illumination Level G (cm-3s-1)

S(o)

(cm

/s)

(b)

Fig. 5.8 Variation of S(o) with illumination level at different grain sizes for


118

1016

G=1018(cm3/s)

105

104

103

102

1 10 100 1000 10000

Grain size d(µ m)

S(o)

(cm

/s)

(a)

1016

G=1018

(cm3/s)

1020105

104

103

102

101

1 10 100 1000 10000

Grain size d(µm)

S(o)

(cm

/s)

(b)

Fig. 5.9 Variation of S(o) with grain size at different illumination levels for (a) PX-Si and (b) PX-GaAs.

119

d=103µm

1015 1016 1017 1018 1020101910-10

10-9

10-8

10-7

10-6

10-5


J r(W

g)

(A/c

m2 )

(a)

1019 10201018101710161015

d=103 µm

10-3

10-4

10-5

10-6


J r(W

g) (A

/cm

2 )

(b)

Fig. 5.10 Variation of Jr(Wg) with illumination level for


120

10-11

10-10

10-9

10-8

10-7

1 10 100 1000

Grain size d(µ m)

J r(W

g) (

A /

cm2 ) G=1018(cm3/s)

1016

(a)

G=1020

(cm3/s)

1018

1016

10-3

10-4

10-5

10-6

1 10 100 1000 10000

Grain size d(µm)

J r(W

g) (A

/cm

2 )

(b)

Fig. 5.11 Variation of Jr(Wg) with grain size at different illumination levels for


121

103

d=102µm

1015 1016 1017 1018 10201019

1x106

2x106

3x106

4x106

5x106

6x106


S(W

g) (

cm/s

)

(a)

103

102

d=10 µm

102010191018101710161015

2.070x106

2.068x106

2.066x106

2.064x106


S(W

g) (c

m/s

)

(b)

Fig. 5.12 Variation of S(Wg) with illumination level at different grain sizes for


122

2x105

2.2x106

4.2x106

6.2x106

8.2x106

1 10 100 1000 10000Grain size d(µm)

S(W

g) (

cm/s

) G=1018 (cm3/s)

1020

(a)

G=1020

G=1018

G=1016 (cm3/s)2.070x106

2.068x106

2.066x106

2.064x106

2.062x106

1 10 100 1000 10000

Grain size d(µm)

S(W

g) (c

m/s

)

(b)

Fig. 5.13 Variation of S(Wg) with grain size at different illumination levels for


123

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).

5.3.1 RESISTIVITY AND MOBILITY

The grain-boundary resistivity of the bulk part of the semiconductor under

optical illumination at Vo >> 2kT/q can be expressed as

ρb = Vo / J(2Wg + δ)

= [(2π m* kT)1/2] / [q2 (2Wg + δ) n*] [1/{ TFE + TFES + TE + TST }]

(5.27)

The average resistivity of the sample is given by

ρ* = Va / Jd

= ρb (2Wg + δ) /d + ρc [d - (2Wg + δ)] /d

= ρgb (δ / d) + ρo (2Wg / δ) + ρc [ d - (2Wg + δ)] /d (5.28)

where ρgb, ρo and ρc 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

124

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 qVg. The

dependence of resistivity of PX-Si on temperature for different values of grain

size and illumination level is shown in Fig. 5.15. 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 J1, J2, J3 and J4

to the total current density with temperature is shown in Fig. 5.16. It can be

noted that the conduction in PX-Si under illumination is controlled by

components J2, J3 and J4. However, as temperature decreases the contribution

of J4 component increases and at very low temperature its contribution may be

greater than that of J3. This fact predicts that to explain the electrical properties

of PX-semiconductors under solar illumination the component J4 plays an

important role. The same importance of J4 is found in Chapter-III (under dark

condition). It should be noted that in low temperature range qVg is much smaller

than qφ at high illumination levels. Consequently the conduction is dominated by

J2, J3 and J4 components. A good agreement of the present theory with the

125

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.

5.14 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 qVg 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. 5.19.

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. 5.20. In Fig. 5.21 author has also shown the

variation of different components of current density with temperature for

126

0.1

1

10

2 3 4 5 6 7

1000/T (K-1)

Res

istiv

ity (O

hm-c

m)

TheoryRef. [123 ]

Fig. 5.14(a) Variation of resistivity with inverse temperature for PX-Si

under 1 SUN illumination, Ref. [123].

0.01

0.1

1

2 3 4 5 6 7

1000/T (K-1)

q φ

(eV

)

Fig. 5.14(b) Variation of qφ with inverse temperature to match the experimental data.

127

104

103

d=102µm

0.1

1

10

100

2 3 4 5 6 7

1000/T (K-1)

Res

istiv

ity (O

hm-c

m)

(a)

1020

1019

G=1018 cm3/s

0.1

1

10

100

1000

2 3 4 5 6 7

1000/T (K-1)

(Ohm

-cm

)

(b)

Fig. 5.15 Computed variation of resistivity with inverse temperature for (a) different grain sizes and (b) different illumination levels.

128

102

100

10-2

10-4

10-6

10-8

10-10

2 3 4 5 6 71000/T (K-1).

J 1/J

, J 2

/J,

J 3/J

, J 4

/J

J1/JJ2/JJ3/JJ4/J

Fig. 5.16 Variation of the ratio of components of current to total current density

with inverse temperature.

129

100

1000

10000

100 1000T (K)

Effe

ctiv

e m

obili

ty (c

m2 / V

-s)

Theory

Singlecrystal

(a)

0.01

0.1

1

10

100

1000

10000

1 10 100 1000 10000

Grain size d (µ m)

Effe

ctiv

e m

obili

ty (c

m2 / V

-s)

167 K300 K500 K400 K

(b)

Fig. 5.17 Computed variation of effective mobility with (a) temperature and

(b) grain size at different temperatures.

130

0

0.5

1

1.5

2 3 4 5 6 7

1000/T (K-1)

Res

istiv

ity (o

hm-c

m)

(a)

0

0.05

0.1

0.15

0.2

2 3 4 5 6 7

1000/T (K-1)

q (e

V)

(b)

Fig. 5.18 Variation of (a) resistivity and (b) qφ with inverse temperature

for PX-GaAs under 1 SUN illumination.

131

104

d=103µm

0

0.5

1

1.5

2 3 4 5 6 7

1000/T (K-1)

Res

istiv

ity (o

hm-c

m)

(a)

1020

G=1018

(cm3/s)

0

0.5

1

1.5

2 3 4 5 6 7

1000/T (K-1)

Res

istiv

ity (o

hm-c

m)

(b)

Fig. 5.19 Computed variation of resistivity of PX-GaAs with inverse

temperature for different values of (a) grain size and (b) illumination level.

132

100

1000

10000

100 1000T (K)

Effe

ctiv

e m

obili

ty (c

m2 /V

-s)

(a)

10-11

10-9

10-7

10-5

10-3

10-1

101

103

105

1 10 100 1000 10000

Grain size d(µm)

Effe

ctiv

e m

obili

ty (c

m2 /V

-s)

150 K200 K300 K410 K500 K

(b)

Fig. 5.20 (a) Variation of effective mobility with temperature

(b) Effective mobility with grain size at different temperatures.

133

10-2

10-3

10-4

10-5

10-6

10-7

10-8

10-9

2 3 4 5 6 7

1000/T (K-1)

J 1/J

, J 2

/J,

J 3/J

, J 4

/J

J1/JJ2/JJ3/JJ4/J

Fig. 5.21 Variations of ratio of components of current to total current density with inverse temperature for PX-GaAs under 1 SUN illumination.

134

PX-GaAs films. It can be seen that in high temperature range, J3 has higher

contribution as compared to the other components. As temperature decreases,

the contribution of J2 and J3 increases and becomes comparable to that of J3. In

the very low temperature range, current is determined by J4 component. Thus

like PX-Si films, the electrical properties of PX-GaAs films under solar

illumination are controlled by J2, J3 and J4 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) The distribution of GB states is of Gaussian type in both the materials.

(ii) The behaviour of Jr(Wg) 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

Jr(o) on grain size decreases.

(v) The electrical properties of polycrystalline semiconductors under solar

illumination are not only controlled by J2 and J3 components of current but

also by an additional component J4. The contribution of J4 component

increases rapidly as temperature decreases.

135

(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.

136

Table 5.1

Parameters used in theoretical computations to match the experimental

data for different materials.

δ (Å) =20

Vth(cm2/s) =107

T (K) =300

G (1Sun=cm3/s) = 1020

Parameters PX-Si PX-GaAs

Ref. [82] Ref. [98] Figs.5.3-5.13 Figs.5.3-5.13

Ngs (cm-2)

N (cm-3)

ET (eV)

d (µm)

Lb(µm)

τb (s)

S (eV)

σC (cm2)

σN (cm2)

Dn(cm2s-1)

1.5×1012

1.3×1016

0.56

103

2.44

4.76×10-6

5.5kT

10-13

8×10-15

13.5

9×1011

3×1016

0.64

103

100

7.69×10-6

1kT

10-14

1.3×10-16

13.5

2.5×1012

3×1016

0.63

(1 - 104)

100

7.69×10-6

1kT

10-14

1.3×10-16

13.5

1.2×1012

3×1016

0.36

(1 - 104)

6

3.47×10-9

5kT

10-13

8×10-15

6.475

137

Table 5.2

Parameters used for theoretical computations for Figs. 5.14 – 5.17.

Parameter PX-Si

N (cm-3)

Ngs(cm-2)

ET (eV)

d(µm)

S (eV)

σc (cm2)

σN (cm2)

Lb(µm)

τb (s)

5×1015

5.45×1011

0.63

103

5kT

2×10-14

10-15

100

5.23×10-6

Table 5.3

Parameters used for theoretical computations for Figs. 5.18 – 5.21.

Parameter PX-GaAs

N (cm-3)

Ngs (cm-2)

ET (eV)

d (µm)

S (eV)

σC (cm2)

σN (cm2)

Lb (µm)

τb (s)

3×1016

9×1011

0.06

104

5kT

10-13

8×10-15

6

3.47×10-9

grain boundary recombination processes and carrier...

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