Anomalous Efros-Shklovskii- Variable range
hopping conduction in composites of polymer
and iron carbide nanoparticle embedded in
carbon
S. Shekhar, V. Prasad and S. V. Subramanyam
Department of Physics, Indian Institute of Science, Bangalore, 560 012 India.
Abstract
Efros - Shklovskii variable range hopping (ES-VRH) conduction mechanism [ρ =
ρ0 exp(TES /T)1/2 ] is observed in composites of polymer and iron carbide nanoparticle embedded in carbon relatively at high temperatures. A crossover from one
ES-VRH type to another ES-VRH type is observed with different ES temperature
(TES ) and prefactor (ρ0 ) as temperature decreases.
Key words: Transport properties; Variable range hopping; Magnetoresistance
PACS: 72.80.Tm;71.30.+h;72.20.-i
1
Introduction
Mott to Efros-Shklovskii variable range hopping (ES-VRH) crossover has been
observed in many localized systems as temperature decreases [1–3]. This kind
of crossover was predicted by Efros and Shklovskii prior to the experimental
observations [4,5]. In principle variable range hopping conduction is possible
at sufficiently low temperatures where activated type of conduction is not possible. Coulomb interaction in hopping regime which produces a gap (coulomb)
in density of states is responsible for ES-VRH type of conduction mechanism.
Mott to ES, VRH crossover is observed as temperature recedes from high to
low because at higher temperature hopping becomes much larger than the
coulomb gap and its effect is suppressed [5]. In few of the system it has been
found that the ES to Mott VRH crossover is due to filling of Coulomb gap in
higher temperature range [8]. The cause of this crossover is due to change in
electronic density of states (DOS). At higher temperatures where Mott law is
obeyed DOS, g(E) is a constant whereas at lower temperatures where EfrosShklovskii VRH is obeyed density of states near Fermi energy (EF ) varies as
Preprint submitted to Elsevier Science
18 August 2006
g(E)=g0 |E-EF |D−1 and takes parabolic form in three dimensions. In intermediately and highly compensated semiconductor DOS is not constant and it
decays in entire energy range (to its peak value at E=0). In those cases in entire VRH temperature region conduction mechanism is governed by ES-VRH
and crossover from Mott to Efros VRH is absent.
In this letter we are reporting the crossover from one ES-VRH dependence
to another ES-VRS dependence of resistivity in two different temperature regions. This is in contrast with existing experimental evidences. The resistivity
behaves as ln(ρ/ρ0 )=(TES /T )1/2 from 300K to 55K and there is a transition
below this temperature. Again in the temperature range 30K-6K ES-VRH is
followed by the composites with different ρ0 and TES .
2
Experimental
Iron-carbide nanoparticles embedded in carbon matrix (soot) are prepared by
pyrolysis of maleic anhydride and ferrocene (2:1 molecular weight ratio) at
9800 C for 5 12 hours [18]. The final product is black color soot which is used
as filler. The nanoparticles are less than 100nm in size and there are few
layers of graphene on it. Composites are prepared by the dispersion of filler
in poly(vinyl chloride) (PVC) in different proportions. PVC is first dissolve
in Tetrahydrofuran (THF) and filler is ultrasonically dispersed in solution to
obtain a good dispersion. The details of sample preparation and characterization is discussed elsewhere [17]. The conventional four probe method is used
to measure the low temperature transport and inside a commercial Janis liquid Helium cryostat with suprerconducting magnet attachment. The contacts
were made by silver paint. Below 15K the temperature was controlled within
the limit of 0.005K where as near to room temperature the fluctuation was
0.2K. To eliminate the error due to temperature fluctuation 20 reading was
taken in each positive as well as negative current directions and averaged.
3
Results and Discussions
Polymer composites are macroscopically disordered systems and conduction
mechanism is well described by hopping conduction. In many of the cases in
those materials temperature dependence of resistivity follows
ρ = BT −p exp(T0 /T )n
(1)
2
100 25
6.25
r (ohm cm)
14
ln
2.8
1.6
1.0
T(K)
12
43K
10
36.4%
26.5%
8
21.9%
4.7K
15.4%
6
4
0.0
0.2
0.4
-1/2
T
0.6
0.8
1.0
-1/2
(K
)
Fig. 1. lnρ vs T−1/2 plots from 300K-1.3K for composites having different proportion
of filler. Percentage of filler in samples are marked near the symbols inside the figure.
The dotted vertical lines show the eye estimated temperatures where crossovers
occur.
in a large temperature range (300K-4.2K) [9,10] and the pre exponential factor
(BT−p =ρ0 ) has weak temperature dependence. The value of n varies widely
between 1/4 and 1 depending on the different physical situations. In fact the
exponent can take any value (<1) depending on the form of DOS. If DOS varies
as g(E)=g0 |E-EF |m then n is given by (m+1)/(m+4) [11]. Superlocalized
model of variable range hopping also predicts different values of exponent
[12,13]. Lien et. al [14] have reported the 2D Mott VRH dependence (lnR
α T −1/3 ) from 300K to 60K in amorphous indium oxide and nickel-silicon
films and a crossover to harder soft gap (lnR α T −n ) with exponent (n) 0.77.
A transition in resistivity from soft gap (ES-VRH) to hard gap (activated
type) is also reported as temperature goes down [15,16]. In fact many type
of crossovers have been observed namely, Mott-VRH to ES-VRH, ES-VRH
to activated type, Mott-VRH to some arbitrary VRH and a comprehensive
theoretical model is missing. The mathematical model suggested by Aharony
et. al.[3] and theoretical model by Lien et. al.[14] is quite a useful in describing
the different type of crossovers.
Fig.1 shows the plot of lnρ vs T−1/2 in temperature range 300K-1.3K and
change in slope is found nearly at 45K. For clarity lnρ vs. T−1/2 is plotted
in temperature range 300K-10K (fig. 2) and a crossover is observed (hump in
between 55K-35K). ES-VRH is followed in between 300K-55K and 30K-6K
on either side of hump. Mott and other VRH expressions have been tried but
ES-VRH was found to be best least square fit. The magnetotransport properties have been studied from 300K to 8K and the response of the samples
] is negative at
too, confirms crossover. The magnetoresistance [MR= ρ(H)−ρ(0)
ρ(0)
300K in entire range of magnetic field (0-11T). Its value gradually decreases
(magnetoconductance MC increases) up to 55K and below this temperature
3
177.8
10
19.8
11.1
0.225
0.300
44.4
T (K)
36.4%
26.5%
21.9%
15.4%
8
7
ln
r (ohm cm)
9
6
54K
34K
5
0.075
0.150
-1/2
T
-1/2
(K
)
Fig. 2. lnρ vs T−1/2 plots from 300K-10K. The temperature range between the
vertical dotted line is crossover range. The arrowed line is freely hand drawn straight
line between the extreme points. There is a noticeable change in the slope after the
crossover. The crossover region is independent of the samples.
44
r (ohm cm)
12.0
11
20
7
5
T (K)
36.4%
26.5%
ln
10.5
9.0
21.9%
7.5
0.150
(a)
0.225
0.300
-1/2
T
625
ln
r (ohm cm)
8.25
7.50
278
15.4%
156
100
34.4%
0.375
0.450
-1/2
(K
)
69
51
39
T (K)
26.5%
6.75
6.00
21.9%
5.25
(b)
15.4%
0.04 0.06 0.08 0.10 0.12 0.14 0.16
-1/2
T
-1/2
(K
)
Fig. 3. (a) lnρ vs T−1/2 plots from 30K-6K.(b) lnρ vs T−1/2 plots from 300K-55K The
symbols are experimental data and solid lines are best linear fit to the corresponding
data. The value of TES and ρ0 depicted in table 1 are calculated from the slopes
and intercepts of the linear fits.
it starts increasing (MC decreasing) and finally it becomes positive at somewhere near to 15K. Below this temperature MR monotonically increases. At
low temperatures and low fields ln[ρ(B)/ρ(0)] is proportional to B2 . In simplest picture the positive MR at low temperatures is observed in strongly
4
MR (%)
0.0
-0.3
300K
125K
-0.6
70K
55K
-0.9
MR (%)
0.0
-0.3
40K
25K
-0.6
MR (%)
3
15
2
10
08
1
0
0
2
4
6
8
10
12
Magnetic Field (T)
Fig. 4. Magnetoresistance plot of composite (filler:21.9%). The MR is negative at
higher temperatures. The MR at 25K retraces 40K. The MR starts increasing above
40K and below 15K it is positive.
Table 1
Table shows the value of TES (ES temperature), ξ (localization length), ρ0 (pre
factor) in different ES-VRH (temperature) regions, ρresidual (resistivity at T=0)
and dρ/dT in 2.4K-1.3K region.
Soot
TES
ξ
ρ0
TES
ξ (Å)
ρ0 (ohm)
ρresidual (ohm)
%
K
Å
ohm-cm
K
Å
ohm-cm
ohm-cm
300K-55K
30K-6K
dρ/dT
2.4K-1.3K
36.4
196.8
395.1
73.0
115.0
676.3
128.4
173920
-54170
26.5
217.8
357.0
119.4
130.7
595.4
202.5
265519
-81831
21.9
248.9
312.3
200.3
145.9
533.1
381.4
651669
-166394
15.4
238.7
325.8
343.0
186.9
416.0
430.6
1830364
-533155
11.7
319.5
243.4
1009.6
203.0
383.0
1283.2
5637530
-1450080
localized system is due to shrinkage of impurity orbitals wave function which
lead to sharp decrease in overlap of wave-functions resulting in positive MR
[5,19]. The quadratic dependence is consistent with the theory developed for
field dependence on transport properties of localized system [5]. This kind of
dependence is observed when hopping length (Rhop ) is quite large compare to
localization length ξ. This relation (Rhop ¿ ξ) holds good at low temperature
as Rhop ≈ T −1/2 [6]. At higher temperatures where Rhop ∼ ξ the backscat5
6
1.2x10
36.4%
26.5%
21.9%
15.4%
r (ohm cm)
5
9.0x10
5
6.0x10
5
3.0x10
0.0
0.9
1.2
1.5
1.8
2.1
2.4
2.7
Temperature (K)
Fig. 5. ρ vs T plots from 2.4K-1.3K. The experimental data and their least square
linear fits are shown. The slope (dρ/dT ) and the intercept (ρresidual ) are mentioned
in table 1.
tering becomes important and weak localization effect dominates which does
not come in picture at low temperatures due to larger hopping length [7]. It is
weak localization which is responsible for positive MC at higher temperatures.
To calculate the different parameters of the ES-VRH, straight lines have been
fitted in lnρ vs. T−1/2 plots of two different ES-VRH regions (300K-55K and
30K-6K) as shown in the Fig. 3. The value of TES and ρ0 are calculated from
the slope and intercept respectively of the straight line fit. The localization
length ξ is related to TES as follows.
TES =
2.8e2
4π²0 κξkB
(2)
The soot inside the PVC matrix makes a percolative path and after the threshold concentration (∼2.2% by weight) matrix becomes conducting. Composites
having higher concentrations of soot are more conducting due to larger number of conducting paths. Also the value of prefactor ρ0 decreases consistently
for conducting samples. The localization length ξ value is large (of the order
of 100 atomic distance), ξ (T0 ) is larger (smaller) for conducting samples and
localization length increases in lower temperature region ES-VRH for a particular sample. The conducting sample shows larger localization length which
is expected. The localization length (ξ) increases in low temperature ES-VRH
region implies that the samples are more delocalized on lower temperature ESVRH region. This fact is reflected in fig. 1 where the slopes of the resistivity
curve becomes less steeper as one goes to lower temperature region. The MC
data too confirms the change in localization length. The MC starts decreasing
below 55K due to large Rhop but in the region 40K-25K there is no change in
MC due to the fact that ξ also suddenly takes larger value and that cancels
the effect of increase in Rhop . So this behavior of MC indicates the change in
localization length (i.e. TES ).
6
The granular metal shows ES-VRH type dependence (ρ = ρ0 exp[2(C0 /κB T )1/2 ])
in wide temperature range [5,20] with same exponent. In granular metal the
conduction takes place due to tunneling of charge carriers between conducting
grains. This conduction mechanism is valid below the percolation threshold
where the conducting particles do not touch each other and are separated by
dielectric (insulating) material. In the present case the samples are far above
the threshold value. The possibility of granular conduction has been discarded.
In VRH conduction as we discussed earlier that the exponent value changes
with nature of density of states [11]. As such there is no change in nature
of DOS in crossover. So we believe that this crossover is due to the different
coulomb gaps ∆C on either VRH and the magnetotransport data gives that
indication (TES is related with Coulomb gap as ∆C ≈ KB TES /10 ). There
is a possibility that thermal contraction of polymer at low temperature is
responsible for change in coulomb gap.
Close to 5K there is another crossover. The VRH and activated type of expressions have been tried below 4.7K but the fit was extremely poor. lnρ vs
T−1/2 shows almost saturation in lnρ below 4K. One can see (Fig.1) that after each crossover the slope becomes less steeper (towards lower temperature)
and it seems the resistivity tends towards a constant value as temperature approaches zero. In between 4.2K and 1.3K the change in the resistance is nearly
a decade and we found metallic behavior below 2.4K to 1.3K (see Fig 5). Resistivity shows linear dependence (Fig 5), in temperature range 2.4K-1.3K with
negative temperature coefficient. Presence of magnetic scatterer (iron carbide
nanoparticle) may cause negative temperature coefficient. Sharp drop in DOS
near Fermi level is too, responsible for negative temperature coefficient. The
1
temperature coefficient of resistivity near to T=0 ( ρresidual
dρ/dT ) is constant
−1
(nearly 0.3K ) for all the samples.Since the data is available only in a limited temperature range (a factor of 2 only) which prevents to make anything
conclusive.
4
Conclusion
In conclusion, this paper has described about multiple crossover in conduction
mechanism as temperature varies from 300K to 1.3K. In first crossover ESVRH is followed on both sides of crossover with different coulomb gaps and
the magnetotransport data imply that. An insulator to metal transition is
observed as a second crossover near to 4.7K. In temperature range 2.4K-1.3K
the behavior is metallic with negative temperature coefficient of resistivity.
Negative temperature coefficient is due to presence of magnetic scatterer and
sharp drop in DOS near Fermi level.
7
Acknowledgments: Our sincere thanks to Dept. of Science Technology, New
Delhi for providing National Low Temperature and High Magnetic Field Facility where this work has been carried out. We would like to thank E. P.
Sajitha for help in preparing the samples and some useful discussions. CSIR
New Delhi is also acknowledged by S. S. for granting scholarship.
References
[1] N. V. Agrinskaya, A. N. Aleshin, Sov. Phys. Solid State 31(1989) 1996.
[2] Y. Zhang, P. Dai, M. Levy, M. P. Sarachik, Phys. Rev.Lett. 64(1990) 2687.
[3] A. Aharony, Y. Zhang, M. P. Sarachik, Phys. Rev. Lett. 68(1992) 3900.
[4] A. L. Efros, B. I. Shklovskii, J. Phys. C: Solid State Phys. 8(1975) L49.
[5] B. I. Shklovskii, A. L. Efros, Electronic properties of doped semiconductors
Springer-Verlag, Berlin 1984.
[6] Y. Zhang and M. P. Sarachik, Phys. Rev. B 43(1991) 7212.
[7] E. Medina and M. Kardar, Phys. Rev. B 46(1992) 9984.
[8] B. Sandow, K. Gloos, R. Rentzsch, A. N. Ionov, W. Schirmacher, Phys. Rev.
Lett. 86(2001) 1845.
[9] D. vander Putten, J. T. Moonen, B. H. Brom, J. C. M. Brokken-Zijp, A. J. M.
Michels, Phys. Rev. Lett. 69(1992) 494.
[10] M. Mehbod, P. Wyder, R. Deltour, C. Pierre, Geuskens, Phys. Rev. B (36)1987
7627.
[11] E. M. Hamilton, Philos. Mag. 26(1972) 1043.
[12] Y. -E. Levy, B. Souillard, Europhys. Lett. 4(1987) 233.
[13] G. Deutscher, Y Levy, B. Souillard, Europhys. Lett. 4(1987) 577.
[14] N. V. Lien, R. Rosenbaum, Phys. Rev. B 56(1997) 14 960.
[15] A. N. Aleshin, A. N. Ionov, R. V. Parfen’ev, I. S. Shlimak, A. Heinrich, J.
Schumann, D. Elefant, Sov. Phys. Solid State 30(1988) 398.
[16] I. Terry, T. Penney, S. vonMolnar, P. Becla, Phys. Rev. Lett. 69(1992) 1800.
[17] S. Shekhar, V. Prasad, S. V. Subramanyam, Mat. Sc. and Engg. B (in press).
[18] E. P. Sajitha, V. Prasad, S. V. Subramanyam, S. Eto, K. Takai, T. Enoki,
Carbon 42(2004) 2815.
[19] J. C. Dawson, C. J. Adkins, J. Phys.: Condens. Matter 8(1996) 8321.
[20] P. Sheng P, B. Abeles, Y. Arie Y, Phys. Rev. Lett. 31(1973) 44.
8