USING STANDARD SYSTE - Materials Science Institute of Madrid

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.

I. INTRODUCTION

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

manganites.

Our aim in this paper is tw**of**old. 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.

II. EXPERIMENT

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.

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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

grains. 8

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,

1

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

function.

PRB 61 COULOMB BLOCKADE VERSUS INTERGRAIN . . .

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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 ,

2

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:

where

g 1 g 2

GT,E C

g1E C /3TE C

*/T , 2

E C

*E C e g/2

4

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.

B. Discussion

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

3

5

9552 M. GARCÍA-HERNÁNDEZ et al.

PRB 61

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

reported above.

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.

IV. CONCLUSIONS

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.

ACKNOWLEDGMENTS

We acknowledge financial support from the Spanish

CICyT Project No. MAT97/345 and Project No. PB97/

0875.

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