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.
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 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
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.
100
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 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 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 EFn is flat everywhere but the quasi-Fermi level of minority carriers
EFp is allowed to vary with distance [82].
101
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.
102
Fig. 5.1
The Energy Band diagram of n-type PX-Semiconductor under optical
illumination depicting Gaussian energy distribution of midgap states.
103
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 + J r(Wg) / 2
(5.8)
where Jr(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
Jr(Wg) = (qWg / 2τb) [(ni2 /N){exp(qVg / kT) – 1} exp(∆EF(o) / kT)
- (Jr(o) / 4qDp ) W g⋅ (π kT /qVg)1/2 erf(η)]
⋅[1 + (qW g / 2τb) (W g / 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 ≤ W g), the minority carrier concentration at the depletion edge can be
expressed as
P(W g) = p(o) exp (-qVg / kT) + [{Jr(o) + Jr(Wg)}⋅[π kT/ qVg]1/2⋅ (W g/ 4qDp) erf(η)]
(5.10)
where η = (qVg / kT)1/2.
104
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 > Wg
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) = [Jr(W g) + Jr(o) ] / 2q{P(Wg) – po}
= [Jr(W g) + 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)
5.2.4
(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)Lb / Dp} / {1+ S(W g) Lb/ Dp }]⋅p(∞) exp {(-x + W g)/ Lb}
(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 = Wg.
Hence we get
p(∞) = P(Wg) + Jr(W g) Lb / qDp.
(5.14)
105
From Eqns. (5.13) and (5.14), we find that
P(x) = P(Wg) + [J r(W g) Lb /qDp][1 – exp{(-x + W g) /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-RH 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 grainboundaries. 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 [ J r(o) + 2Jr (W g) ] + Jb (d - 2W g)2
106
(5.17)
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
αqd 2 ∫ Io exp(-αx) dx = 3d2 [Jr(o) + 2Jr(Wg)] + (d-2W g)2 2q / τb
0
∫
p(x)dx
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 – 6W g) p(o) / τb} exp(-qVg / kT)]
where M = M1 [1 + exp(-ν o qVg / kT)]-1
and
(5.21)
(5.22)
M1 = [3 +{(d -6W g)(Wg / 4Lb2)} [π kT / qVg]1/2 erf(η)
+ (d -6W g) / 2Lb - (1- 4Wg / d)⋅{1- exp [-(d -2W g)/ 2Lb] }]-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
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 J r(o) ,
P(W g), S(W g) 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
108
0.45
Ref [82]
0.4
qV g (eV)
Theory
0.35
0.3
0.25
1015
1016
1017
1018
1019
1020
Illumination Level G(cm -3S -1)
(a)
0.3
Ref [98]
0.25
qVg (eV)
Theory
0.2
0.15
0.1
0.05
1015
1016
1017
1018
1019
1020
Illumination Level G(cm-3S -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
0.35
d=10 µm
qV g (eV)
0.3
102
0.25
0.2
103
0.15
0.1
1015
1016
1017
1018
1019
1020
1019
1020
Illumination Level G(cm -3S -1)
(a)
0.6
0.55
0.5
10
qV g (eV)
0.45
0.4
0.35
d=104µm
0.3
0.25
0.2
1015
1016
1017
1018
Illumination Level G(cm -3s-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
0.4
qV g (eV)
0.3
G=1018
(cm3s-1)
0.2
1020
0.1
0
1
10
100
1000
10000
Grain size d(µ m)
(a)
qV g (eV)
0.6
0.5
G=1016 (cm3/s)
0.4
1018
0.3
1020
0.2
1
10
100
1000
10000
Grain size d(µ m)
(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
∆EF(o) on ‘d’ and ‘G’ can be explained by considering the variation of Jr(o) with
these parameters. At low illumination levels J r(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(W g) for PX-GaAs is 2 - 3 times larger than that
for PX-Si.
2.
In the larger grain size range Jr(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
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.
112
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(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)
= 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)
113
0.5
EF (o) (eV)
0.4
0.3
d=103 µm
0.2
102
10
0.1
0
1015
1016
1017
1018
1019
1020
Illumination Level G(cm -3S -1)
(a)
0.8
0.7
d=104µm
EF (o) (eV)
0.6
0.5
10
0.4
0.3
0.2
1016
1017
1018
1019
1020
Illumination Level G(cm -3s-1)
(b)
Fig. 5.5
Variation of ∆EF(o) with illumination level at different grain sizes for
(a) PX-Si and (b) PX-GaAs.
114
100
10-1
Jr (o) (Acm-2)
10-2
d=103µm
10-3
10-4
102
10-5
10-6
10
10-7
10-8
1015
1016
1017
1018
1019
1020
Illumination Level G(cm -3s-1)
(a)
10-1
10-2
Jr (o) (A/cm2)
10-3
d=103 µm
10-4
10-5
10
10-6
10-7
1015
1016
1017
1018
1019
1020
Illumination Level G(cm-3s-1)
(b)
Fig. 5.6
Variation of Jr(o) with illumination level at different grain sizes for
(a) PX-Si and (b) PX-GaAs.
115
100
10-1
Jr(o) (Acm-2)
10-2
G=1018
(cm3/s)
10-3
10-4
1016
10
-5
10-6
10-7
10-8
1
10
100
1000
10000
grain size d(µ m)
(a)
10
0
G=1020
(cm3/s)
10 -1
Jr (o) (A/cm2)
10 -2
10 -3
1018
10 -4
10 -5
1016
10 -6
10 -7
10 -8
1
10
100
1000
10000
Grain size d(µ m)
(b)
Fig. 5.7
Variation of Jr(o) with grain size at different illumination levels for
(a) PX-Si and (b) PX-GaAs.
116
7x104
d=104µm
6x104
S(o) (cm/s)
5x104
4x104
102
3x104
2x104
1x104
1015
1016
1017
1018
1019
1020
Illumination Level G(cm -3S-1)
(a)
5
10
S(o) (cm/s)
104
d=103 µm
103
10
102
101
1015
1016
1017
1018
1019
1020
Illumination Level G (cm-3s-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
105
G=1018(cm3/s)
S(o) (cm/s)
104
1016
103
102
1
10
100
1000
10000
1000
10000
Grain size d(µ m)
S(o) (cm/s)
(a)
105
1020
104
G=1018
(cm3/s)
103
1016
102
101
1
10
100
Grain size d(µ m)
Fig. 5.9
(b)
Variation of S(o) with grain size at different illumination levels for
(a) PX-Si and (b) PX-GaAs.
118
10-5
10-6
Jr (W g ) (A/cm2)
d=103µm
10-7
10-8
10-9
10-10
1015
1016
1017
1018
1019
1020
Illumination Level G(cm -3S -1)
(a)
Jr (Wg ) (A/cm2)
10-3
d=103 µm
10-4
10-5
10-6
1015
1016
1017
1018
1019
Illumination Level G(cm-3s-1)
(b)
Fig. 5.10
Variation of Jr(Wg) with illumination level for
(a) PX-Si and (b) PX-GaAs.
119
1020
10-7
Jr (W g ) (A / cm 2)
10-8
G=1018(cm3/s)
10-9
1016
10-10
10-11
1
10
100
1000
Grain size d(µ m)
(a)
10
-3
G=1020
(cm3/s)
Jr (W g ) (A/cm2)
1018
10-4
1016
10-5
10-6
1
10
100
1000
10000
Grain size d(µ m)
(b)
Fig. 5.11
Variation of Jr(Wg) with grain size at different illumination levels for
(a) PX-Si and (b) PX-GaAs.
120
6x106
S(W g ) (cm/s)
5x106
d=102µm
4x106
3x106
103
2x106
1x106
1015
1016
1017
1018
1020
1019
Illumination Level G(cm -3s-1)
(a)
2.070x106
d=10 µm
S(W g) (cm/s)
2.068x106
102
2.066x106
103
2.064x106
1015
1016
1017
1018
1019
1020
-3 -1
Illumination Level G(cm s )
(b)
Fig. 5.12
Variation of S(Wg) with illumination level at different grain sizes for
(a) PX-Si and (b) PX-GaAs.
121
8.2x106
S(W g ) (cm/s)
6.2x106
G=1018 (cm 3/s)
4.2x106
1020
6
2.2x10
2x105
1
10
100
1000
10000
Grain size d(µ m)
(a)
2.070x106
G=1016 (cm3/s)
S(W g ) (cm/s)
2.068x106
G=1018
2.066x106
G=1020
2.064x106
2.062x106
1
10
100
1000
10000
Grain size d(µ m)
(b)
Fig. 5.13
Variation of S(Wg) with grain size at different illumination levels for
(a) PX-Si and (b) PX-GaAs.
122
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(2W g + δ)
= [(2π m* kT)1/2] / [q2 (2W g + δ) n*] [1/{ TFE + TFES + TE + TST }]
(5.27)
The average resistivity of the sample is given by
ρ* = Va / Jd
= ρb (2W g + δ) /d + ρc [d - (2W g + δ)] /d
= ρgb (δ / d) + ρo (2W g / 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
123
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 J 4
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 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 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 J 4 components. A good agreement of the present theory with the
124
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
125
10
Theory
Resistivity (Ohm-cm)
Ref. [123 ]
1
0.1
2
3
4
5
6
7
1000/T (K-1)
Fig. 5.14(a)
Variation of resistivity with inverse temperature for PX-Si
under 1 SUN illumination, Ref. [123].
q φ ( eV )
1
0.1
0.01
2
3
4
5
6
7
-1
1000/T (K )
Fig. 5.14(b) Variation of qφ with inverse temperature to match the
experimental data.
126
100
Resistivity (Ohm-cm)
d=102µm
10
103
1
104
0.1
2
3
4
5
6
7
-1
1000/T (K )
(a)
1000
(Ohm-cm)
100
G=1018 cm3/s
10
1019
1020
1
0.1
2
3
4
5
6
7
-1
1000/T (K )
(b)
Fig. 5.15
Computed variation of resistivity with inverse temperature for
(a) different grain sizes and (b) different illumination levels.
127
102
J1/J, J 2/J, J 3/J, J 4/J
100
10-2
10-4
10-6
J1/J
J2/J
J3/J
10-8
J4/J
10-10
2
3
4
5
6
7
1000/T (K-1).
Fig. 5.16 Variation of the ratio of components of current to total current density
with inverse temperature.
128
10000
Effective mobility (cm 2/ V-s)
Theory
Single
crystal
1000
100
100
1000
T (K)
(a)
10000
Effective mobility (cm 2/ V-s)
1000
100
10
167 K
300 K
1
500 K
400 K
0.1
0.01
1
10
100
1000
10000
Grain size d (µ m)
(b)
Fig. 5.17
Computed variation of effective mobility with (a) temperature and
(b) grain size at different temperatures.
129
Resistivity (ohm-cm)
1.5
1
0.5
0
2
3
4
5
6
7
6
7
1000/T (K-1)
(a)
0.2
0.1
q
(eV)
0.15
0.05
0
2
3
4
5
1000/T (K-1)
(b)
Fig. 5.18
Variation of (a) resistivity and (b) qφ with inverse temperature
for PX-GaAs under 1 SUN illumination.
130
Resistivity (ohm-cm)
1.5
1
d=103µm
0.5
104
0
2
3
4
5
6
7
5
6
7
1000/T (K-1)
(a)
Resistivity (ohm-cm)
1.5
1
G=1018
(cm3/s)
0.5
1020
0
2
3
4
1000/T (K-1)
(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.
131
Effective mobility (cm 2/V-s)
10000
1000
100
100
1000
T (K)
(a)
5
10
Effective mobility (cm 2/V-s)
103
101
10-1
150 K
10-3
200 K
300 K
-5
10
410 K
500 K
-7
10
10-9
10-11
1
10
100
1000
10000
Grain size d(µ m)
(b)
Fig. 5.20
(a) Variation of effective mobility with temperature
(b) Effective mobility with grain size at different temperatures.
132
10-2
J1/J, J 2/J, J 3/J, J 4/J
10-3
10-4
10-5
10-6
J1/J
-7
J2/J
10
J3/J
J4/J
10-8
10-9
2
3
4
5
6
7
1000/T (K-1)
Fig. 5.21 Variations of ratio of components of current to total current density
with inverse temperature for PX-GaAs under 1 SUN illumination.
133
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 J2 and J 3 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 J r(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
J r(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.
134
(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
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)
1.5×1012
9×1011
2.5×1012
1.2×1012
N (cm-3)
1.3×1016
3×1016
3×1016
3×1016
ET (eV)
0.56
0.64
0.63
0.36
3
d (µm)
10
10
(1 - 10 )
(1 - 104)
Lb(µm)
2.44
100
100
6
τb (s)
4.76×10-6
7.69×10-6
7.69×10-6
3.47×10-9
5.5kT
1kT
1kT
5kT
S (eV)
3
4
σC (cm )
10
10
10
10-13
σN (cm2)
8×10-15
1.3×10-16
1.3×10-16
8×10-15
Dn(cm2s -1)
13.5
13.5
13.5
6.475
2
-13
-14
-14
136
Table 5.2
Parameters used for theoretical computations for Figs. 5.14 – 5.17.
Parameter
PX-Si
N (cm-3)
5×1015
Ngs(cm-2)
5.45×1011
ET (eV)
0.63
d(µm)
103
S (eV)
5kT
σc (cm )
2×10-14
σN (cm2)
10-15
Lb(µm)
100
τb (s)
5.23×10-6
2
Table 5.3
Parameters used for theoretical computations for Figs. 5.18 – 5.21.
Parameter
PX-GaAs
-3
N (cm )
3×1016
Ngs (cm-2)
9×1011
ET (eV)
0.06
d (µm)
104
S (eV)
5kT
σC (cm )
10-13
σN (cm2)
8×10-15
Lb (µm)
6
τb (s)
3.47×10-9
2
137