USING STANDARD SYSTE - Materials Science Institute of Madrid

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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 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.

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.

PRB 61

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 . . .

9551

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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