Electron-electron interaction in percolative
network of polymer - amorphous carbon
composites
S. Shekhar, V. Prasad and S. V. Subramanyam
Department of Physics, Indian Institute of Science, Bangalore, 560 012 India.
Abstract
The polymer-amorphous carbon composites show a negative magnetoconductance
which varies as B2 at low fields which changes to B1/2 at sufficiently high fields. The
magnetoconductance gives the evidence of electron-electron interaction in composites whose conductivity follows thermal fluctuation induced tunneling and falls in
the critical regime.
Key words: Magnetoresistance; Electron-electron interaction; Transport properties
PACS: 72.80.Tm; 72.20.Mf; 72.15.Gd
1
Introduction
The effect of weak localization (WL) and electron-electron interaction in two
dimensional as well as in three dimensional systems are well studied and documented [1,2]. These two effects individually substantially alter the transport
and magnetotransport behavior of the disordered systems. The weak localization effect has been observed in wide range of disordered materials including
metallic glasses, granular metals, semiconductors and amorphous alloys [4–6].
In many of the systems such as copper films, ion implanted polymers, conducting polymers and disordered carbon electron-electron interaction effect
have been realized [11–13,21]. It is very natural to expect the WL and e-e
effects simultaneously affect the transport properties. Their combined effects
have been reflected in many disordered systems [7,15,16]. The effect of WL
and e-e interaction are observed in percolating network, too [7,8]. In this letter
we are reporting the result of magnetoconductance extensively carried out on
a percolative network of polymer-carbon composites in the temperature range
1.3K to 50K.
Preprint submitted to Elsevier Science
11 September 2006
2
Theoretical Background
The weak localization effect is often associated with spin-orbit scattering and
in case it is significant it changes MC drastically. In three dimensional system
the MC due to combined effect of weak localization along with spin-orbit
contribution is given [3,4] as:
∆σ = [σ(B) − σ(0)] = α
1
e2 eB 1/2 3
B
B
) − ( + β)f3 ( )]
( ) [ f3 (
2
2π ~ ~
2 Bso
2
Bi
(1)
−1
where Bi =~/4eDτi and Bso =~/4eD(τi−1 + 2τso
); D, τi and τso are electron
diffusion coefficient, inelastic collision induced relaxation rate and spin-orbit
scattering relaxation rate respectively. The function f3 (x) is given by
x3/2
,
48
f3 (x) = {0.605,
x¿1
xÀ1
(2)
The term β (known as Maki-Thompson contribution) in eq.1 is the contribution from the quenching of superconducting fluctuation due to electron pairing
in Cooper channel and effective at temperatures far away from Tc . In non superconducting materials this effect can be safely neglected. The MC depends
on the relative strength of inelastic relaxation rate τi−1 and spin orbit scatter−1
ing relaxation rate τso
. When there is very weak or no spin orbit scattering
−1
−1
(i.e. τso ¿ τi ) the eq.1 reduces to the case of pure weak localization effect and one gets positive magnetoconductance which is due to dephasing of
back scattered charge carriers. In opposite limit when spin-orbit scattering is
−1
stronger (τso
À τi−1 ⇒ Bso À Bi ) eqn.1 gives
∆σ ' −α
e2
e 1/2 B 2
(
)
3/2
192π 2 ~ ~
Bi
∆σ ' −α
∆σ ' +α
(B ¿ Bi )
(3)
0.605e2 eB 1/2
( )
4π 2 ~ ~
(Bi ¿ B ¿ Bso )
(4)
0.605e2 eB 1/2
( )
2π 2 ~ ~
(B À Bso )
(5)
The electron-electron interaction effects too, produce positive MR and the
type of dependence mentioned above. The exact dependence of MC on field
in electron electron interaction is written as
∆σ = −
e2 F kT 1/2 gµB B
(
) g3 (
)
4π 2 ~ 2~D
kT
(6)
2
where F is the interaction constant and the function g3 in low field and high
field limit is given by
g3 (h) ' 0.084h2
√
g3 (h) ' h − 1.3
(h ¿ 1)
(h À 1)
(7)
(8)
Eq. 7 and 8 define the limit on the MC dependence as B2 or B1/2 .
3
Experimental
The composites have been prepared by mixing amorphous carbon (a-C) flake
in different proportion to Poly(vinyl chloride) with the help of magnetic stirrer
[9]. The composites are disordered in nature. The composites filled with higher
concentration of a-C are more conducting. The details have been discussed in
reference [9]. The magnetoconductance has been measured by conventional
(dc) four probe technique inside a commercial bath type cryostat (Janis) with
supreconducting magnet with maximum attainable field of 12T. The magnetoconductance of the composites having a-C content higher than percolation
threshold (conducting samples) have been measured at different temperatures
varying from 1.3K-50K. The fluctuation in temperature was less than 0.002K
below 10K where as above 10K the fluctuation was 0.01K. The value of MC
does not vary much from sample to sample but the higher resistive samples
show slightly lower magnetoresistance (Higher MC). The change in the direction of field has no effect on the magnetotransport properties. Above 25K a
negligibly small magnetoresistance (MR) is observed.
4
Results and Discussion
The temperature dependence of resistivity for disordered metals at low temperatures due to electron-electron interaction-weak localization (ee-WL) model
follows [10,12]
σ(T ) = σ0 + αT 1/2 + βT p/2
(9)
The first term is a constant related to residual resistivity at T=0, the second term arises due to electron-electron interaction and third term takes care
of weak localization effect on transport properties. The interaction effect is
dominant at low temperatures and WL is effective relatively at higher temperatures. But electron-electron interaction and WL model are applicable to
3
0
-3
1.3K
-6
2.5K
4.2K
-9
6.0K
-12
8.0K
a-C 33.3%
MC (%)
0
-3
1.3K
-6
2.5K
4.2K
-9
6.0K
-12
8.0K
a-C 25%
0
-3
1.3K
-6
2.5K
4.2K
-9
6.0K
8.0K
-12
0
2
a-C 6.4%
4
6
8
10
12
Field (Tesla)
Fig. 1. Low temperature normalized magnetoconductance ( σ(B)−σ(0)
)% of polymer
σ(0)
composites
disordered metals only. In those materials ρ1.3 /ρ300 is slightly greater than 1.
In polymer composites the given ratio is nearly 2 for 33% of loading and increases gradually to 6 for 6.4% which is more towards the critical regime (near
to metal insulator boundary) of insulating side. Moreover fits of eq.9 were very
unphysical. The ratio ρ1.3 /ρ300 suggests that the samples are in critical regime.
In the critical regime the conductivity follows [14]
σ(T ) ∼
e2 pF T β
(
)
~2 E F
(10)
where PF is the Fermi momentum and e is the electronic charge. The β lies
in the range 1/3< β <1. Again the fit was found to be extremely poor for all
the samples for the specified value of β.
The composites follow thermal fluctuation induced tunneling (Sheng’s model
[17])
ρ = ρ0 exp[T1 /(T0 + T )]
(11)
at temperatures below 50K-60K [9]. This theory takes care of large size of the
particles which forms a junction with insulating material. In fact in the cases
4
0.0
-0.5
-1.0
-1.5
10K
-2.0
15K
-2.5
20K
-3.0
25K
a-C 33.3 %
MC (%)
0.0
-0.5
-1.0
10K
-1.5
15K
20K
-2.0
a-C 25 %
25K
0.0
-0.5
-1.0
10K
15K
-1.5
20K
-2.0
a-C 6.4 %
25K
0
2
4
6
8
10
12
Field (Tesla)
Fig. 2. Normalized Magnetoconductance ( σ(B)−σ(0)
)% of polymer composites in
σ(0)
temperature range 10K-25K.
Table 1
Magnetic field dependence range of composites. The variation in the range is ±0.1
from sample to sample.
1.3K
2.5K
4.2K
6K
8K
>10K
B2
< 1.4T
< 3.1T
< 4.8T
< 7.4T
< 10.9T
12T and high
B 1/2
> 2.35T
> 3.3T
> 5.8T
> 8.1T
À 11T
À 12T
where filler particle size is larger experimental results follow this conduction
mechanism [18,19]. Here T1 and T0 are energy required for charge carriers
to tunnel and low temperature, temperature independent resistivity respectively[17]. Near to 250K the resistivity of composites exhibit a minimum (as
clearly seen in fig.3). The difference in thermal expansion coefficient of PVC
and a-C are responsible for this behavior. As temperature increases the matrix
expands and the separation between a-C particles increase resulting in increase
of resistance at higher temperature. Not much have been studied about the effect of magnetic field on the Sheng’s model. In this letter we have investigated
the effect of magnetic field on Sheng’s conduction mechanism.
∆σ varies as B2 at lower fields. As the temperature increases the B2 dependence extends to a larger field. Above 10K in entire 12T range B2 dependence
is observed. At sufficiently low temperature and high field ∆σ makes a tran5
0
50
100
150
200
250
300
250
300
5.4
a-c 25 %
4.8
4.2
3.6
3.0
2.4
160
r (ohm-cm)
a-C 7.4 %
120
80
40
4.2
a-C 33.3 %
3.6
3.0
2.4
0
50
100
150
200
T (K)
Fig. 3. Plot shows temperature dependence of a-C composites in temperature range
1.3K-300K. In the temperature range 1.3K-50K the composites follow thermal fluctuation induced tunneling conduction mechanism [9].
0
Temp=4.2K
MC (%)
-2
-4
a-C
33.3%
25.0%
-6
16.7%
06.4%
-8
-12
-9
-6
-3
0
3
6
9
12
Field (Tesla)
Fig. 4. Magnetoconductance [MC= σ(B)−σ(0)
] of composites in both positive and
σ(0)
negative field. No appreciable change in MC is observed.
sition from B2 to B1/2 dependence as shown in the fig.5. The range of B2 and
B1/2 dependence are depicted in Table 1. This kind of behavior is observed in
weak localization with spin-orbit interaction or electron-electron (e-e) interaction model. Neither PVC nor a-C contain heavy elements, so the effect of
spin-orbit scattering is negligible. Moreover in weak localization model ∆ρ/ρ α
ρ so the higher resistive samples show higher MR (lower magnetoconductance)
[20]. The behavior of magnetoconductance observed here is in well agreement
with the electron-electron interaction model. Here the MR decreases as one
goes to higher temperatures suggested by electron-electron interaction model.
In e-e model the B2 dependence should be observed in the limit given by
eq.7 (gµB B/κT ¿ 1). So at very low temperatures only in very low fields B2
6
6.3
4.5
0
0.0
8.9
7.7
0
Ds (S/m)
Ds (S/m)
-0.3
-0.6
Filler: 33.3%
-0.9
5.5
9.5
7.7
0.0
B (T)
11
B (T)
-0.3
-0.6
-0.9
Filler: 33.33%
Temp. : 4.2K
Temp. : 6.0K
-1.2
0
20
40
2
60
0
80
30
60
9.9
11.9
5.8
7.2
-0.3
-0.9
Filler: 33.3%
2.4
10.9
13
-0.6
-0.9
Temp. : 4.2K
-1.5
2.1
150
B (T)
Ds (S/m)
Ds (S/m)
9.0
B (T)
-0.6
-1.2
120
2
B (T )
8.1
6.5
5
-0.3
90
2
2
B (T )
Filler: 33.33%
Temp. : 6.0K
2.7
3.0
1/2
B
(T
3.3
3.6
1/2
)
-1.2
2.4
2.7
1/2
B
3.0
(T
1/2
3.3
3.6
)
Fig. 5. ∆σ=[σ(B) − σ(0)]100. ∆σ shows B2 dependence at low fields and B1/2
dependence at higher fields.
dependence should be observed where as at high temperatures at very high
field B2 dependence should be observed. Similar kind of behavior is observed
in composites. Above 10K in entire 12T range B2 dependence is observed. In
high field limit (gµB B/κT À 1, i.e. at low temperature and high field) at low
temperatures B1/2 dependence has been noticed. So the e-e theory is precisely
explaining the magnetoconductance behavior in the composites except for the
fact that the low temperature transport does not exactly follow the T1/2 dependence (e-e model). In critical regime the different mechanism for transport
and magnetotransport has been observed earlier also [21]. It is worth to mention here that the a-C too (the filler), shows the e-e interaction effects both in
terms of transport and magnetotransport properties [11]. But the MR values
is much small compared to composites and moreover the MR of the system
not necessarily depends on its constituents.
5
Conclusions
The negative magnetoconductance has been observed in polymer- amorphous
carbon composites. At low fields the MC dependence on field is B2 which
changes to B1/2 at higher fields. The range of low field and high field limits
strongly depend on temperature. The MC behavior indicates the presence of
electron-electron interaction effects. The transport properties of the composites at low temperatures are governed by thermal fluctuation induced tunneling (Sheng’s model). It is very interesting to observe the different mechanism
7
for transport properties and magnetotransport properties in critical regime.
Besides that the effect of magnetic field in Sheng’s model has been studied
extensively for which a theoretical model is not well established.
Acknowledgments:We would like to thank the DST, New Delhi for providing
the National High Magnetic Field Facility where this work has been carried
out. One of us (S.S.) expresses his gratitude to Madan Mithra for some useful
discussions and providing several related references.
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