PHYSICAL REVIEW B
VOLUME 61, NUMBER 14
1 APRIL 2000-II
Coulomb blockade versus intergrain resistance in colossal magnetoresistive
manganite granular films
M. García-Hernández, F. Guinea, A. de Andrés, J. L. Martínez, C. Prieto, and L. Vázquez
Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, E-28049 Madrid, Spain
Received 14 December 1999
The problem of the low-temperature resistance of granular films of manganites with nanometric grain sizes
is addressed. The observed upturn in the low-temperature resistance is understood in terms of charging effects.
However, the standard Coulomb blockade scenario does not describe the observed dependence of the charging
energies on the applied magnetic field. We propose a theoretical framework in which a distribution of charging
energies exists, due mainly to the randomness in the intergranular conductances and not in the grain diameters.
The increase of the conductances with the magnetic field induces a renormalization of the charging energies
that explain the experimental observations.
The conductance in granular metals has been the focus of
many studies since the pionering work of Helman and
Abeles in the 1970s. 1,2 It is well established that the electrical
conduction in granular metals results from the transport
of electrons or holes from charged to neutral grains. This
requires the generation of a charge carrier by removing an
electron from a neutral grain and placing it in a neighboring
hitherto neutral grain at the expense of the electrostatic
charging energy E c . At low temperature and for small
grains, it is increasingly difficult to activate such a mechanism,
and the situation may evolve to a point in which transport
could be effectively blocked. This constitutes the socalled
Coulomb blockade CB, which results in an upturn in
the low-temperature resistivity. This phenomenon has been
reported in all granular metallic films.
Recently, in order to explain the upturn observed in the
low-temperature resistance of nanometric oxide powders, the
existence of intergranular CB has been proposed: Coey et al.
invoke a spin-dependent Coulomb gap to explain the conductance
of half metalic ferromagnetic CrO 2 diluted into an
insulating antiferromagnetic matrix of Cr 2 O 3 ; 3 Sun et al.
conjecture with its existence in a trilayer junction of
La 2/3 Sr 1/3 MnO 3 -SrTiO 3 -La 2/3 Sr 1/3 MnO 3 , due to tiny metallic
inclusions of La 2/3 Sr 1/3 MnO 3 LSMO at the interfaces; 4 Balcells
et al. report measurements on a series of ball milled
nanometric powders of LSMO whose smallest members lack
the metallic behavior expected for bulky manganite. 5 Finally,
it cannot be overlooked that these effects are not restricted to
nanometric particles since evidence of the above-mentioned
behavior is already present in bulk ceramics of
La 2/3 Ca 1/3 MnO 3 LCMO when doped with Y, 6 where the
upturn in R(T) is found to be larger as Y doping increases.
In this context, it seems in order to address the question as to
whether a pure CB mechanism and/or other effects such as
intergrain connectivity and structural disorder are at the root
of the reported resistivity divergences in magnetoresistive
Our aim in this paper is twofold. First, from the experimental
viewpoint, we study nanometric samples that exhibit
the same transport and magnetic properties of bulk LCMO
manganites. We will also follow the dependence of the electronic
transport properties on the magnetic field so as to assess
the nature of the observed enhacement of the lowtemperature
resistance. Second, we discuss models based on
the current understanding of CB processes 1 and propose an
approach to the analysis of charging effects in granular films
that incorporates our experimental evidence.
We have developed a protocol to grow LCMO thin films
whose grain size is controlled by film thickness. Following
classical ceramic techniques La 2/3 Ca 1/3 MnO 3 was prepared
by mixing a stoichometric amount of La 2 O 3 , MnO 2 , and
CaCO 3 followed by the the annealing of the mixture for 72 h
at 1400 °C and quenching of the sample in air. In order to
conform the target, the powder was subsequently pressed and
sintered at 1200 °C for another 10 h. No steps were taken to
further densify the target. X-ray diffractograms, analyzed in
terms of a pseudocubic structure, render a value for the lattice
constant of 5.466 Å, implying an actual composition of
La 0.73 Ca 0.27 MnO 3 .
The films were deposited by dc-magnetron sputtering on
Si100 substrates. The vacuum chamber allows a vacuum
base pressure of 10 7 mbar. The sputtering proccess takes
place at room temperature in a mixed atmosphere of Ar and
O 2 flowing at a ratio of 4:1, resulting in a total pressure of
5.410 3 mbar. The deposition rate was verified by smallangle
x-ray diffraction to be linear in time and equal to 250
Å/h. The as-grown films were found to be amorphous and, in
order to reproduce the properties found in the bulk material,
they needed to be annealed to allow a polycrystal to be
formed. We have tried several annealing sequences so as to
approach the magnetic and transport properties of the bulk
while minimizing the grain size. The best results were
achieved for samples annealed under continuous O 2 flow at
1220 K for 10 min. X-ray diffraction patterns do not show
any preferential orientation of the manganite on Si100.
Atomic force microscopy AFM images of the films reveal
quite homogeneous samples consisting of spheres
whose average diameters d range from 12 to 80 nm depend-
0163-1829/2000/6114/95494/$15.00 PRB 61 9549 ©2000 The American Physical Society
9550 M. GARCÍA-HERNÁNDEZ et al.
FIG. 1. AFM images of a representative film at two different
scales corresponding to the sample with average grain diameter d
40 nm as seen in 1 m1 m and 250 nm250 nm scales.
ing on the film thickness see Fig. 1. Note that, due to the
granular morphology of our samples after annealing, any dependence
of the film strain at the interface on the thickness
of the initial deposit should be discarded.
A Quantum Design superconducting quantuminterference
device SQUID magnetometer was used for
magnetization measurements under a static field of 0.1 T.
The measured hysteresis cycles are compatible with samples
built of single-domain spheres with the magnetic field at a
random distribution of angles to the easy axis. Further, the
Curie temperatures around 245 K and coercitive field
around 600 Oe are very similar for all the films, which
leads to the conclusion that from the magnetic point of view
all of our films are very similar to each other.
The electrical resistance and its field dependence were
measured under a magnetic field up to 9 T using a four-probe
method with a Quantum Design physical properties measurement
system PPMS. In a previous work the transport data
of polycrystalline manganites have been fitted to a model
where the important parameter is the effective section for
metallic conduction, which, in turn, depends on the connectivity
between grains and not on their size. 7 The steady increase
of the high-field magnetoresistance MR was explained
considering that this effective section and therefore
the conductivity increase linearly with the applied field
which aligns the blocked Mn spins at the surface of the
III. RESULTS AND DISCUSSION
Figure 2 shows the resistance for different magnetic fields
corresponding to our smallest grain size film (d12 nm.
The inset shows also the resistance at zero field normalized
to the room-temperature value for representative films in the
set. All of them share the same basic features: a metalinsulator
MI transition at a temperature slightly below the
ferromagnetic transition and an upturn of the resistance that
starts to develop around 40 K, which could be assigned to
the existence of a Coulomb gap. The latter is more prominent
as d decreases. As expected, a decrease of the resistance is
observed upon application of a magnetic field. 9
In order to get some insight into the physics of the lowtemperature
resistance we have analyzed the portion of the
curves, up to T150 K, as the sum of two contributions
RTAR 0 B exp T,
FIG. 2. Resistance vs temperature for the smallest particle size
explored (d12 nm at several magnetic fields. Inset: resistance vs
temperature for other films in the series at 0 T. Continuous lines are
best fit to the data.
where R 0 stands for the resistence of the bulky ceramic,
which does not present the upturn below 40 K, A is an amplitude
factor, and the second term describes the Coulomb
blockade effect. Regarding the latter contribution, we mention
that fits improve assuming a T 1/2 dependence in the
argument of the exponential instead of the T 1 dependence
postulated for a pure CB effect. B is an amplitude factor and
is a fitting parameter that is related to E C . The model
reproduces the data below 150 K, as can be seen in the inset
of Fig. 2 a closeup is found in Fig. 5. Figure 3a shows the
results for the parameter corresponding to the best fit of
our R(T) curves measured under magnetic fields of 0Tand
9 T versus the average grain diameter. For all our samples an
approximate 80% decrease of the magnitude (T,H
0 T)(T,H9 T)/(T,H0 T is observed when a
magnetic field of 9Tisapplied.
So far, our model basically agrees with that used to analyze
ceramic samples. 5 At this point, it shoud be recalled that
the T 1/2 model accounts for the low-temperature resistance
of granular metals embedded within an insulating matrix,
and it is only valid in the tunneling regime, where typical
intergrain resistances are much larger than h/e 2 12
10 3 . 1 It also assumes that the energy barriers are inversely
proportional to the grain radius. Here, we deal with
an altogether different kind of system: it is a half metallic
oxide, grains are in close contact and the grain size distribution
does not relate to the energy barriers. Therefore, it is not
FIG. 3. a Best fit results for the parameter vs the average
grain diameter of the films. b Magnetic field dependence of for
the film with d12 nm. The full line shows the fit to an exponential
PRB 61 COULOMB BLOCKADE VERSUS INTERGRAIN . . .
FIG. 4. Sketch of the connections of the small grains that block
the current path. The grains with good contacts do not show CB
effects, which are restricted to the unit within the square.
a surprise that effects such as the observed variation of the
charging energies with the applied field cannot be explained
by a standard CB process.
A. Reformulation of the model
In the following, we extend the model to situations where
good contacts between grains can be established, and conductances
are of the order of Ne 2 /h, where N is the number
of channels at the contact. In this regime, the conductance is
not simply thermally activated, and other effects, like
cotunneling, 10 play a role. 11–13 Furthermore, the capacitances
of individual grains are renormalized by coupling to the
other grains and, therefore, are no longer determined solely
by the grain radius. The following hypothesis marks the central
differences with respect to the standard CB model: the
conductance is limited by the grains with the poorest contacts,
which we shall denote as weak links in our model, and
grains with good contacts show no CB effects, even if their
radii are small. Grain capacitances mainly depend on the
quality of the contacts established with the neighboring
grains and therefore on the connectivity of the system.
In this context, connectivity should also be understood in
a broad sense. It does not only refer to the quality and section
of the physical contacts between grain boundaries 7,14 but also
to any alteration of the effective conductance. Realizations of
such microscopic weak links are misaligned Mn spins at the
surface, 8,13,15,16 distortions of the Mn-O-Mn angles due to
structurally unbalanced enviroments at the grain surface, and
impurities or defects. 17 The latter are good candidates to explain
the observations in the Y-doped manganites 6 as well as
recent results on LCMO single crystals. 18
With this rationale behind us, we now proceed to reformulate
the standard CB model. Given the quite homogeneous
grain size distributions we can neglect the variation in
radii within a sample. There is, however, a distribution of
effective charging energies, as in Ref. 1, but in this case is
due to different intergrain conductances. We assume that the
main source of randomness in our model stems from the
contacts. The distribution of intergrain conductances, g i in
units of e 2 /h), is modeled by the function
Pge g ,
FIG. 5. Upper curves: Fitting to the experimental results (H
0 T for the film with d12 nm. Open circles: experimental data.
Broken line: RAe /T) BR 0 . Full line: our model in the text.
Lower curves: Same symbols as above and same film but with
applied magnetic field (H9 T.
where is an adjustable parameter. The basic unit of our
model is shown in Fig. 4, which is equivalent to a single
electron transistor. 19 The conductance as function of temperature,
intergrain conductances (g 1 and g 2 ), and charging
energies are not completely defined, although the main features
are known: there is a strong renormalization of the
charging energy if g1, cotunneling dominates at low temperature,
and there is a well-defined high-temperature expansion
for the conductance. We ignore the effects due to spin
accummulation in the central grain that may modify the
field-dependent properties. 20,21 We now interpolate between
these known temperature limits, and the conductance of the
basic unit reads:
g 1 g 2
g1E C /3TE C
*/T , 2
*E C e g/2
is the renormalized effective charging energy at low temperature,
and gg 1 g 2 .
From comparison to other theoretical analyses, 22 we estimate
that the error in this interpolation is no larger than
20–30 % over the relevant range of parameters. The total
contribution of the intergrain contacts to the conductance is
given by a combination, in parallel, of the conductances
given in Eq. 3:
G inter Tdg 1 dg 2 GT,E C Pg 1 Pg 2 .
As when fitting to a exp(/T) dependence, the total resistance
is defined as
RTAR 0 TBG 1 inter T, 6
where R 0 is the resistance of the ceramic sample.
The model has, therefore, four adjustable parameters, E C ,
which defines the distribution of conductances, A, and B.
The fitting to the experimental curve for the sample of average
particle size 12 nm is shown in Fig. 5. Besides the
constant A, which just fixes the overall scale of intergrain
effects, the two other physical parameters used are 0.5
and E C 13 K. Thus, the average intergrain conductance, in
9552 M. GARCÍA-HERNÁNDEZ et al.
the absence of charging effects, is g 1 2e 2 /h, which
is the typical conductance due to two quantum channels, or,
alternatively, to a contact area k 2 F , where k F is the Fermi
wave vector. The deduced value of g, of order 2e 2 /h, implies
that the model is consistent with the assumed modification
of charging effects due to the intergrain conductances.
It is interesting to compare the two fitting formulas describing
reasonably well the observations. In the model proposed
in Ref. 1, the energy scale depends both on the
charging energies of the grains and the intergrain barrier. If
the transmission between grains is given by T(R)exp
(2m*V/ 2 R), where V is the barrier height, R is the
grain radius, and the charging energy is e 2 /R, then
e 2 2m*V/ 2 . This energy is, roughly, the charging energy
of grains of radius R such that T(R)1. Thus, even if a
broad distribution of capacitances is assumed, the lowtemperature
behavior (k B T) is determined by the contacts
with conductances of order e 2 /h. For these contacts, the
expression in Eq. 3 should be more accurate, as it incorporates
cotunneling and the renormalization of the charging
energies due to the coupling to other grains. Moreover, the
radius of the grains that determine the conductance at temperature
T scale as R(T)e 4 /(k B T). Thus, in order to
explain the observations in the range 4–80 K, the radii of the
grains within a sample should vary by, at least, a factor of 4.
In order to check further the validity of our assumptions,
we have used the same model without changing the average
grain capacitance to fit the high-field resistance, as shown in
Fig. 5. The only parameter modified is , which determines
the average intergrain conductance. The value used is
0.2 instead of 0.5 for H0). The average conductance
changes from g2e 2 /h at H0 tog4.5e 2 /h at
H9 T. If one assumes that the conductance between grains
i and j goes as g ij g 0 ij cos 2 ( ij /2), where ij is the angle
between the magnetizations of grains i and j, then g(H
→)2g(H0), which is consistent with the variation
in the parameter needed to fit the experimental data, as
Thus, our model accounts for the observed magnetic field
dependence of the resistance without invoking a variation of
the charging energies E C as a function of the applied field.
As the field increases, the coupling between grains is enhanced,
leading to delocalization of the charges to neighboring
grains. The outcome of this process can be understood as
a reduction in the effective capacitance of the grains, induced
by the field. Our model allows us to quantify this effect,
which is shown to be independent of a particular distribution
of grain sizes, or the existence of correlations between the
grain radii and the energy barriers. Finally, due to the linear
increase of the conductance with the applied field, 7,8 the exponential
dependence of parameter on the field Fig. 3b
can be inferred by the renormalization of the charging energy
as defined in Eq. 2.
In conclusion, we have investigated the low-temperature
resistance and its magnetic field dependence in manganite
granular films. Charging effects are observed below 50 K
and found to be sensitive to the application of a magnetic
field. A simple model of CB, in which the grain capacitances
are not changed by the applied field, but where variations in
the intergrain conductances lead to a reduction of the CB
effects, describes well the experimental findings.
We acknowledge financial support from the Spanish
CICyT Project No. MAT97/345 and Project No. PB97/
1 J. S. Helman and B. Abeles, Phys. Rev. Lett. 37, 1429 1976.
2 P. Sheng, Philos. Mag. B 65, 357 1992.
3 J. M. D. Coey, A. E. Berkowitz, Ll. Balcells, F. F. Putris, and A.
Barry, Phys. Rev. Lett. 80, 3815 1998.
4 J. Z. Sun, D. W. Abraham, K. Roche, and S. S. P. Parkin, Appl.
Phys. Lett. 73, 1008 1998.
5 Ll. Balcells, J. Fontcuberta, B. Martinez, and X. Obradors, Phys.
Rev. B 58, R14 697 1998.
6 A. Maignan, C. Simon, V. Caignaert, and B. Raveau, J. Appl.
Phys. 79, 7891 1996.
7 A. de Andrés, M. García-Hernández, and J. L. Martinez, Phys.
Rev. B 60, 7328 1999.
8 A. de Andrés, M. García-Hernández, J. L. Martinez, and C.
Prieto, Appl. Phys. Lett. 74, 3884 1999.
9 J. M. D. Coey, M. Viret, and S. von Molnar, Adv. Phys. 48, 167
10 D. V. Averin and Y. V. Nazarov, Phys. Rev. Lett. 65, 2446
11 S. Takahashi and S. Maekawa, Phys. Rev. Lett. 80, 1758 1998.
12 S. Mitani, S. Takanashi, K. Takanashi, K. Yakushiji, S.
Maekawa, and H. Fujimori, Phys. Rev. Lett. 81, 2799 1998.
13 F. Guinea, Phys. Rev. B 58, 9212 1998.
14 H. Y. Hwang, S. W. Cheong, N. P. Ong, and B. Batlogg, Phys.
Rev. Lett. 77, 2041 1996.
15 M. J. Calderón, L. Brey, and F. Guinea, Phys. Rev. B 60, 6698
16 F. Ott, S. Barberan, J. G. Lunney, J. M. D. Coey, P. Berthet, A.
M. de Leon-Guevara, and A. Revcolevschi, Phys. Rev. B 58,
17 N. D. Mathur, G. Burnell, S. P. Isaac, T. J. Jackson, B. S. Teo, L.
L. MacManus-Driscoll, L. F. Cohen, J. E. Evetts, and M. G.
Blamire, Nature London 387, 266 1997.
18 M. Fath, S. Freisem, A. A. Menovsky, Y. Tomioka, J. Aarts, and
J. A. Mydosh, Science 285, 1540 1999.
19 Single Electron Tunneling, edited by M. H. Devoret and H. Grabert
Plenum Press, New York, 1992.
20 J. Barnaś and A. Fert, Europhys. Lett. 44, 851998.
21 A. Bratass, Y. V. Nazarov, J. Inoue, and G. E. W. Bauer, Phys.
Rev. B 59, 931999.
22 J. König, H. Schoeller, and G. Schon, Phys. Rev. Lett. 78, 4482