Tunneling magnetoresistance in Co-ZrO 2 granular thin films

Granular films composed of well defined nanometric Co particles embedded in an insulating ZrO2 matrix were prepared by pulsed laser depositon in a wide range of Co volume concentrations 0.15 x 0.43 . High-resolution transmission electron microscopy TEM showed very sharp interfaces between the crystalline particles and the amorphous matrix. Narrow particle size distributions were determined from TEM and by fitting the low-field magnetic susceptibility and isothermal magnetization in the paramagnetic regime to a distribution of Langevin functions. The magnetic particle size varies little for Co volume concentrations x 0.32 and increases as the percolation limit is approached. The tunneling magnetoresistance TMR was successfully reproduced using the Inoue-Maekawa model. The maximum value of TMR was temperatureindependent within 50–300 K, and largely increased at low T, suggesting the occurrence of higher-order tunneling processes. Consequently, the tunneling conductance and TMR in clean granular metals are dominated by the Coulomb gap and the inherent particle size distribution.


I. INTRODUCTION
The enormous amount of research effort during the last 50 years on the magnetic behavior of nanostructured systems consisting of small particles is a sign of its continuing technological and scientific interest. 1In particular, granular films with a very well controlled nanostructure have very promising technological applications as a result of technological advances in several film preparation techniques.From a scientific point of view, these technological advances make possible to prepare model granular solids with well controlled particle size and shape, thereby facilitating interpretation of their properties.][4][5][6] Finite-size effects dominate the magnetic behavior of individual nanoparticles, increasing their relevance as the particle size decreases.Among these, the phenomenon of superparamagnetism has probably provoked the largest amount of research.At low concentrations, the blocking of the magnetization of each particle is expected to occur at low enough T, this process becoming a collective phenomenon at high concentrations for which the freezing of the whole system of magnetic moments due to interparticle interactions is observed.However, there are still some open questions, e.g., the influence of the particle microstructure on the magnetization reversal.The superparamagnetic limit has relevant implications in the thermal and time stability of the bit in magnetic recording media. 7,85][16][17][18] In the dielectric regime, granular FM metals show giant tunneling magnetoresistance, caused by spin-dependent tunneling of the conduction electrons at the magnetic-insulator interfaces.Electrons with spin parallel to the particle magnetization have larger probability for tunneling than those electrons with antiparallel spin, leading to an overall lower resistance when all magnetic particle moments are aligned than in the case of randomly orientated moments.This magnetoresistive effect in granular materials is directly affected by magnetic correlations, since the latter determine the magnetic configuration of the system and, consequently, the magnetotransport properties.0][21][22][23][24] Some results indicate that tunneling magnetoresistance ͑TMR͒ is almost temperature independent in a wide temperature range, in agreement with the Inoue-Maekawa model. 25However, the relation between magnetism and magnetotransport in granular metals is still an open question, since there are systems showing strong temperature dependence of TMR. 19,26,27Some of these examples show the distinctive 1 / T decay of TMR of the Helman-Abeles model. 20Moreover, it is well known that imperfect metal-insulating interfaces and spin-flip scattering, due to, e.g., surface spin disorder and paramagnetic spins in the matrix, lead to an additional temperature decrease in TMR.
In the Helman-Abeles model, it is assumed that magnetic exchange energy has to be added to the Coulomb charging energy in order to account for TMR, so that these two energies are comparable.In contrast, Inoue and Maekawa 25 assume that the tunneling conductance in the dielectric regime can be derived from the classical expression for the spindependent tunneling in FM/I/FM junctions given by Landauer 28 and Maekawa and Gäfvert. 29Inoue and Maekawa neglected the exchange energy as being small when compared to the Coulomb energy.TMR is then temperature independent.We note that the experimental Coulomb energies are very high, for example in Co-Al-O samples with a high degree of intermixing ͑from 1. 6  Besides, there is the open question of the strong increase of TMR at low temperature 24 which was not predicted by any of the two models, and needed the occurrence of higherorder tunneling processes.Granular solids display a broad particle size distribution such that the gross simplification of the Inoue-Maekawa model 25 is not to take into account tunneling between grains of different sizes.Mitani et al. 24 extended the model by assuming that the natural consequence of the particle size distribution is the successive onset, as the temperature is decreased, of tunneling processes between large grains through small ones.Mitani et al. 24 referred to these processes as "higher order" since they involve more than two particles.These processes are negligible at high temperature, where tunneling takes place through particles of similar sizes: electrons cannot tunnel either to a smaller particle, since the Coulomb charging energy is larger, or to a larger particle, which is further away, since the tunnel resistance is larger.This was already noted by Abeles et al. 4 In this paper, we present a study of the magnetic and magnetotransport properties of granular Co-ZrO 2 thin films prepared by pulsed laser deposition ͑PLD͒ within a wide volume concentration range of Co ͑0.15Ͻ x Ͻ 0.43͒.The samples display well defined Co nanoparticles and sharp particle-matrix interfaces.The size distribution of magnetic nanoparticles deduced from magnetic measurements is in a good agreement with the observation from high-resolution transmission electron microscopy ͑TEM͒ and allows us to fit the T and H dependences of the tunneling magnetoresistance ͑TMR͒.The particles are uniform in size almost for all Co concentrations and they are very small until percolation where big Co aggregates begin to appear.This is probably a consequence of the PLD used in the preparation of the samples.Finally, the maximum value of TMR is mostly T independent, except for very low T, at which it increases, suggesting the onset of higher-order tunneling process.
Our work suggests that tunneling conductance in clean granular metals is dominated by the Coulomb gap and that the higher-order tunneling processes between large particles mediated by small particles are intrinsic of granular metals, due to the inherent particle size distribution.

II. EXPERIMENTAL DETAILS
Co-ZrO 2 granular films were grown by KrF pulsed laser deposition ͑wavelength of 248 nm, pulse duration of =34 ns͒.The samples were deposited at room temperature in a vacuum chamber with rotating composite targets made of sectors of ZrO 2 and pure cobalt.Different surface ratio of target components led to different volume fraction x of Co, ranging from metallic to dielectric regime ͑0.15Ͻ x Ͻ 0.43͒.The distance between target and substrate was fixed at 30 mm.The laser fluency was typically 2 J / cm 2 .
Average compositions were determined by microprobe analyses.The films were structurally characterized with x-ray diffraction ͑XRD͒ and high-resolution transmission electron microscopy ͑TEM͒ ͑Fig.1͒.The particle size distributions were also obtained by fitting the low-field magnetic susceptibility ͑zero-field cooling͒ ͑Fig.2͒ and hightemperature isothermal magnetization curves M͑H͒ ͑Fig.3͒ to a distribution of Langevin functions which model the superparamagnetic behavior of the particle size distribution.
The substrates for TEM experiments were silicon nitride membrane windows allowing observation of as-deposited samples.As an example, Fig. 1  In a zero-field-cooled ͑ZFC͒ experiment, the sample is initially at a high temperature.Then the sample is cooled down to the lowest measuring temperature, a small magnetic field ͑typically 100 Oe͒ is applied, and the magnetization is measured as temperature is increased.The field-cooled ͑FC͒ experiment is similar, but a magnetic field is applied at a high temperature and the sample is cooled under the applied magnetic field.The magnetization curves M͑H͒ were recorded up to 50 kOe.
Magnetoresistance ͑MR͒ measurements were carried out by a four-probe technique using a lock-in amplifier and an ac constant current source at very low frequencies ͑57 or 111 Hz͒.The magnetoresistance curves MR͑T , H͒ = R͑H , T͒ − R͑H =0,T͒, being R͑H , T͒ the H-and T-dependent resistance, were measured within 30-300 K and up to either 10 or 50 kOe.The films prepared for transport measurements were deposited onto insulating glass substrates.After deposition, four parallel copper strips were evaporated on the sample surface.The strips were approximately 1 mm wide, and were separated by a distance of 0.7 mm in order to minimize the direct ͑geometric͒ electrode contribution to the capacitance.

A. dc susceptibility
The ZFC and FC curves of samples with different Co volume concentration well within the dielectric regime ͑0.15Ͻ x Ͻ 0.37͒ are shown in Fig. 2 ͑circles͒.All the curves exhibit the characteristic features of a narrow distribution of small particles.The ZFC peak at the temperature T p is observed at low temperatures for all concentrations, indicating a small mean particle size.The ZFC peak position moves from T p = 5 K for x = 0.15 to 18 K for x = 0.32, which is a sign of a slight increase of the mean particle volume, as the Co volume concentration rises.From the values of T p , the average anisotropy energy barrier ͗KV͘ can be estimated as ͗KV͘Ϸln͑t/ 0 ͒k B ͗T B ͘, where k B is the Boltzmann constant, ͗T B ͘ is the mean blocking temperature, which is directly proportional to T p ͑T p = ␣T B , ␣ = 2 for a broad distribution 30 ͒, t is the time window ͑typically t Ϸ 6 s for the superconducting quantum interference device measurements͒ and 0 is the mean time between consecutive attempts to overcome the energy barrier which is about 0 Ϸ 10 −11 s.The values for T p and ͗KV͘ are given in Table I.
Above T p , the ZFC and FC curves join at T = T rev , i.e., T rev stands for the T above which the M͑T͒ dependence becomes reversible ͑Table I͒.In this regime, the temperature dependence of the low-field susceptibility = M / H is given by the Curie-Weiss law ͑T͒ = C / ͑T + ͒.The value of C increases when the ferromagnetic concentration is increased, as expected for particles that become slightly larger as x increases, approaching the percolation threshold in the range x p ϳ 0.45 ͑Table I͒.The Curie-Weiss temperature is almost zero for x ഛ 0.27, indicating negligible interactions between particles.The values for both x = 0.32, 0.37 have a positive sign, which suggests dominant antiferromagnetic character of the interparticle interactions.Table I also summarizes the values of C and .
For temperatures below T p , the FC curve keeps rising steeply except for the highest concentration x = 0.37.Even for this concentration, the FC curve does not become completely flat, as has been observed for similar concentrations in Co-SiO 2 granular films 15 and in metallic CoFe-Ag alloys. 31,32This feature seems to indicate the absence of strong magnetic interactions between particles for Co concentrations up to about x = 0.30, as may also be deduced from the relatively small values of .Sankar et al. 33 reported = −29 K and = −34 K for Co-SiO 2 granular films with x = 0.32 and x = 0.41, respectively.These values are opposite in sign and much larger than the ones we found and may be related to stronger interactions in granular Co-SiO 2 than in Co-ZrO 2 , which support the suggestion of our samples being clean samples with a very reduced degree of particle-matrix intermixing.
Experimental ZFC magnetization curves have been fitted ͑solid line in Fig. 2͒ to a distribution of Langevin function ͑L͑x͒ = tanh −1 ͑x͒ −1/x͒ with a temperature-dependent cut-off which models the blocking of the superparamagnetic behavior of the Co particles as the temperature is reduced: where V is the particle volume and H is the applied magnetic field.It was assumed that all particles have the bulk value for M S ͑M S = 0.15 B /Å 3 , where B is the Bohr magneton͒.The particles are considered to have a linear-logarithmic size distribution f͑D͒, where D is the particle diameter.The thresh-old diameter D B and the blocking temperature T B are directly related according to D B 3 = ␣ B T B , where ␣ B is a fitting parameter ͓2.5Ͻ ␣ B Ͻ 3.5 nm 3 / K for Co ͑Ref.34͔͒.M res accounts for any temperature-independent magnetization that may arise from ferromagnetic impurities ͑droplets͒ and/or diamagnetic contribution from the substrate or sample holder.The contribution of the blocked particles ͑particles with diameters D Ͼ D B ͒ has been neglected, since the ZFC curves were measured under a low magnetic field.
In all cases, a good agreement is achieved between the experimental data and the theoretical curves.The average particle diameter, D m = D 0 exp͑ 2 /2͒, is for small values of such as those listed in Table I, almost equal to D 0 .The fitting parameters, summarized in Table I, demonstrate that the average particle size is almost constant up to x ϳ 0.32.The distribution width seems to become slightly larger for higher x with the exception of the x = 0.37 sample, which appears to have a quite narrow size distribution.This is the reason why x = 0.32 exhibits a higher T p as compared to x = 0.37 ͑Fig.2͒.We note that the estimation of the average particle size from magnetic data requires the aprioristic assumption of the Co magnetization ͑per unit volume͒, which is usually taken as the bulk value ͑as it is done in our case͒.However, it is well known that the bulk value is an overestimation of the particle magnetization in fine particle systems, due to strong surface and finite-size effects. 34,35In our case, around 50-60 % of Co atoms are affected by the lowcoordination number at the surface, giving rise to smaller M S as compared to bulk Co.Consequently, this method leads to an underestimation of the average particle size 34,35 as compared to that obtained from TEM.
The observed mean particle diameters are smaller and their growth with increasing concentration is less pronounced as compared to the data reported by Abeles and coworkers in co-sputtered Ni-Al 2 O 3 ͑D varies from 1.5 nm for x = 0.08 to 4 nm for x = 0.44͒ and Ni-SiO 2 ͑D = 3.5 nm for x = 0.40͒ granular systems, 30,36 15,33 This may be related to the choice of the matrix oxide: the ZrO 2 seems to coat the Co nanoparticles well confining them in small globules, thereby avoiding the growth of large clusters.The fact that our films have been prepared by means of laser ablation and not with the usual cathode sputtering may also explain the observed different microstructure, although similar sizes and concentration dependences have also been observed in co-sputtered Co-AlO x films by Ohnuma and co-workers, who reported an average particle diameter of about 2 nm almost independent of the Co concentration in a range from 20 up to 50 at.%. 38 We note that our results are also in accordance with other works on systems containing ͑various͒ discontinuous ͑less than one monolayer͒ Co layers embedded in Al 2 O 3 . 39,40

B. Magnetization
The magnetization as a function of the applied field for several Co concentrations and various temperatures in the superparamagnetic regime is given in Fig. 3͑a͒.As in the case of ZFC curves, M͑H͒ can be fitted ͓solid line in Fig. 3͑a͔͒ to a distribution of Langevin functions as The last term in Eq. ͑2͒ corresponds to a residual susceptibility, which is necessary to account for paramagnetic and/or diamagnetic residual contributions, coming essentially from the substrate and the matrix.Equation ͑2͒ is valid in the superparamagnetic regime, i.e., for temperatures well above the blocking temperature.Equation ͑2͒ points out that the magnetization only depends on the reduced parameter H / T, as long as the last term has been subtracted from experimental data.This implies that in the superparamagnetic regime the magnetization curves should scale when M is plotted as a function of H / T. We observe that the magnetization curves scale for x ഛ 0.37 ͓Fig.3͑b͔͒; a fact which confirms the absence of any significant interparticle interactions for these concentrations.
It is worth noting that the magnetic behavior of the samples becomes increasingly softer as the Co content increases, which should be a direct consequence of the growth in size of the Co particles observed as x increases.However, the particle diameters are quite similar for all concentrations and, only close to percolation, large aggregates start to form leading to an overall increase of the mean particle size.
The width of the particle-size distribution obtained from the fit of M͑H͒ to Eq. ͑2͒ becomes gradually broader as Co fraction increases ͑see Table II͒.It should be noted though, that the sample with x = 0.37 displays a somewhat different behavior: it seems to have a narrower particle size distribution in spite of its relatively higher Co content, which manifests itself in a more Langevin-like M͑H͒ dependence.On the other hand, those samples with a large exhibit a steeper rise in M͑H͒ for low fields as well as a slower but steadier increase for high magnetic fields.As we have already noted for the ZFC curves, the actual preparation conditions ͑energy and fluency of the beam, substrate position, etc.͒ produce some variation in the mean particle size and width of the distribution, even for samples with close nominal Co contents.The microstructure of the samples thus depends to some extent on the synthesis conditions.
It is worth noting that the fitted particle size distributions ͑Table II͒ are in perfect agreement with those values found by fitting ZFC curves.Using these diameters and the ͗KV͘ values from Table I, the effective anisotropy constant K can be estimated ͑see Table II͒.The anisotropy constant is fairly constant ͑K ϳ 16-20 erg/ cm 3 ͒ for all concentrations except for x = 0.37 for which is significantly smaller ͑K ϳ 8 erg/ cm 3 ͒.
For bulk fcc Co, the effective magnetocrystalline anisotropy constant is given by K eff =K 1 /4, 30,41 where K 1 is the first-order anisotropy constant whose value is approximately K 1 Ϸ 2.6ϫ 10 6 erg/ cm 3 for Co. 42,43 This leads to K eff Ϸ 6.5 ϫ 10 5 erg/ cm 3 .In comparison to this value, our values of K are larger, strongly suggesting that surface anisotropy gives an additional contribution for such small particles and raises the total effective anisotropy.Luis et al. 40 recently reported a size-dependent effective anisotropy constant of sputtered Co cluster in Al 2 O 3 that went down from 2.4ϫ 10 7 erg/ cm 3 for ͗D͘ = 0.8 nm to 3 ϫ 10 5 erg/ cm 3 for ͗D͘ = 5.2 nm and showed a linear dependence on the logarithm of the mean diameter.Jamet et al. 34 report a slightly smaller effective anisotropy constant of 2 ϫ 10 6 erg/ cm 3 for Co particles with a 3-nm diameter in a niobium matrix.Our results are in good agreement with these experimental findings.
From the volume anisotropy K of the nanoparticles and that of their bulk counterpart K V , the value of surface anisotropy K S can be estimated for spherical particles 1 with diameter D, as K = K V + ͑6/D͒K S .Taking K ϳ 18ϫ 10 6 erg/ cm 3 , D ϳ 1.5 nm and K V ϳ 6.5ϫ 10 5 erg/ cm 3 , we obtain K S ϳ 0.43 erg/ cm 2 , which is in agreement with previous results for Co ͑K S ϳ 0.2-0.4erg/ cm 2 in Ref. 44͒ and other nanoparticle systems. 35

C. Magnetoresistance
In the dielectric regime, the temperature dependence of the dc resistivity follows an exponential decay with T −1/2 for all samples below the percolation threshold, 45,46 which is characteristic of insulating granular systems with a distribution of nanometric particle sizes in which conduction takes place by thermally assisted tunneling. 4he field dependence of the tunneling magnetoresistance ͑TMR͒ of a Co-ZrO 2 sample with x = 0.34 is plotted in Fig. 4͑a͒.Figure 4͑b͒ displays a zoom for low fields, which shows a finite coercive field H C of about 300 Oe at 10 K.These features correlate with the M͑H͒ behavior, as shown in Figs.4͑c͒ and 4͑d͒.The maximum of the resistance-field R͑H͒ curve takes place at the coercive field, which corresponds to the state of maximum disorder in the orientation of the neighboring magnetic particle moments.Therefore the field dependence of the resistance, which is related to the alignment between particle moments, reaches a maximum.We also observe that for high fields ͑H Ͼ 10 kOe͒ the resis- tance keeps decreasing with an almost linear field dependence.This high-field magnetoresistance is probably caused by some spin disorder at the particle-matrix interfaces, which are magnetically very hard, and/or any residual magnetic impurities inside the oxide matrix.
In Figs.5͑a͒-5͑c͒, we show magnetoresistance curves at different temperatures and for several Co concentrations ͑circles͒.All samples exhibit a negative tunneling magnetoresistance, typical of a system consisting of small magnetic particles.As in the case of the magnetization, the magnetoresistance drop becomes much more abrupt at low temperatures, in accordance with the temperature dependence observed in the magnetization curves ͑Fig.3͒.A maximum magnetoresistance of about 7% at 50 kOe, which is almost temperature independent, is reached for a concentration around x = 0.37, in very close agreement with TMRϳ 6.8% at 77 K for Co 38 ͑SiO 2 ͒ 62 . 15Sankar et al. 15 argue that a T-independent value of the magnetoresistance is due to reduced intermixing and negligible spin flipping.The magnetoresistance diminishes for very low and high Co concentrations and vanishes completely when the percolation threshold is reached, as expected. 1he MR curves, in the superparamagnetic regime, have been fitted to the expression 25  where f͑D͒ is the linear-logarithmic distribution function.
The values of the fitting parameters D 0 and are listed in Table III.If the proposed model were to be valid, the magnetoresistance curves corresponding to different temperatures should scale in a unique master curve when plotted as a function of H / T, since the field and temperature dependence enter in Eq. ͑3͒ through the Langevin function L͑H , T͒.At low Co concentrations, the MR curves ͓Figs.5͑d͒-5͑f͔͒ indeed scale fairly well, while for x = 0.43 the scaling becomes worse, indicating that the field dependence of the magnetoresistance starts to deviate from Eq. ͑3͒ due to magnetic interactions between particles, as we approach the percolation threshold.However, even for these relatively high concentrations, the fitting is still acceptable giving rise to reasonable values of the parameters.The maximum of the magnetoresistance is almost temperature independent above about 100 K, as best shown in Fig. 5͑b͒, in the panel corresponding to x = 0.37.However, an enhanced MR is observed at low temperature ͑see Fig. 6͒.These observations are in accordance with Mitani et al. 21,24 and Zhu et al. 47 in Co-AlO x and Fe-AlO x , respectively.Mitani and co-workers 24 explained their results in terms of the occurrence of higher-order tunneling processes between large particles mediated by small particles as T decreases, and fitted their experimental data to the equation where n * = ͑B /2 * sT͒ 1/2 , * = k, k being a constant defined in Ref. 24 and = ͱ 2m * ⌽ / ប 2 , ⌽ being the effective barrier height with respect to the Fermi level E F , s being the separation between particles, and B being the prefactor in the well-known expression R͑T͒ = R 0 exp͓−͑B / T͒ 1/2 ͔. 4 The increase in the maximum MR at very low temperature has also been tentatively discussed in terms of the effect of the cluster-surface and matrix interactions leading to a dominant d character of the tunneling electrons. 23ur experimental data have also been fitted to Eq. ͑4͒ ͑solid line in Fig. 6͒ using a spin polarization of P = 0.25± 0.02, * s = 97± 10, and a barrier B = 25.3 meV.The factor B was obtained from a linear regression of ln͓R͑T −1/2 ͔͒ data, 45,46 while the parameters P and * s were chosen to fit the experimental data.We note that the obtained spin polarization P = 0.25 coincides rather well with the value P = 0.27 obtained by fitting the field dependence of the MR, and it is reasonable agreement with the experimental 48 ͑P = 0.35͒ and theoretical 49 ͑P = 0.33͒ values for Co.From the previous analysis, as the effect of intermixing and spin flipping are expected to be limited, the reduced values of P suggest the role of the insulating matrix in determining the actual value of P, as suggested by Slonczewski. 50It is worth noting that the experimental results in Fig. 6 cannot be fitted to the 1 / T behavior of the Helman-Abeles model ͑Ref.20͒ in any temperature range.
At high temperature the experimental MR ratio diminishes a bit more rapidly as compared to the theoretical prediction.This behavior has also been observed in Co-AlO x ͑Ref.25͒ and Fe-Al 2 O 3 ͑Ref.47͒ granular thin films.Zhu and Wang 47 attributed this decrease of the magnetoresistance to spin-flip scattering processes due to magnetic impurities contained in the tunneling barriers.

IV. CONCLUSION
Magnetic properties and tunneling magnetoresistance of Co-ZrO 2 granular insulating thin films were measured in a series of samples with Co concentrations up to the percolation threshold.The particle size distribution obtained by fitting the low-field magnetic susceptibility, high-temperature isothermal magnetization, and TMR are in close agreement, and match direct TEM observations.The magnetic particle size varied little for Co concentrations x Ͻ 0.32, and the absence of strong magnetic interactions between particles was evidenced.As the percolation limit was approached, the particle diameter increased as a consequence of the formation of larger aggregates, favoring interparticle interactions.The magnetic anisotropy of the Co particles was approximately one order of magnitude higher than that of bulk fcc Co, which is related to the surface contribution to the total anisotropy.The maximum value of TMR was T independent above 50 K and followed the Inoue-Maekawa model, showing that Co-ZrO 2 granular thin films prepared by PLD lead to well-defined Co particles with sharp particle-matrix interfaces.The very high values of the exponent B in the TABLE III.Parameters D 0 and used to fit the magnetoresistance curves in Fig. 5.
temperature dependence of the resistance further supports the high quality of the particle-matrix interfaces in a very wide compositional range.This enables the occurrence of high Coulomb charging energies, so that the Coulomb gap dominates R͑T͒ up to almost room temperature and the magnetic exchange energy ͑in the Helman-Abeles model͒ is negligible.Consequently, TMR is almost temperature independent, except for the strong increase at low temperature, which suggests that higher-order tunneling processes are intrinsic of granular metals, due to inherent particle size distribution.
shows a bright field TEM image of Co-ZrO 2 thin film with x = 0.20.The dark regions indicate the Co particles and the light regions, the amorphous ZrO 2 matrix.The image clearly shows the presence of a regular distribution of Co grains.The particles have very well defined interfaces with the matrix.The left-hand-side inset to Fig. 1 shows a magnification of one particle.The lattice fringes observed in the grains correspond to Co atomic planes, which indicate that the grains are crystalline.The lattice fringes are not present in the ZrO 2 matrix, which suggests that the matrix is amorphous.The size distribution of the particles from TEM ͑right-hand-side inset to Fig. 1͒ has been fitted to a linear-logarithmic function with a mean particle size of D = 2.5 nm and a width of = 0.25 ͑solid line͒.

FIG. 1 .
FIG. 1. TEM images of Co-ZrO 2 granular films with x = 0.20.Left-hand-side inset shows the magnification of one particle.Righthand-side inset shows the histogram of the size distribution obtained from the TEM micrograph.The solid line is the fit to a logarithmic-linear distribution.

FIG. 2 .
FIG. 2. Temperature dependence of the ZFC and FC magnetization of films with various Co concentrations ͑circles͒.Solid lines are the fits of ZFC curves to a distribution of Langevin functions, given by Eq. ͑1͒.
FIG. 4. ͑a͒ Magnetoresistance and ͑c͒ magnetization of a Co-ZrO 2 sample with x = 0.34 at 10 K. ͑b͒ and ͑d͒ shows the low-field behavior.
to 19 meV in Ref. 24͒, such that the Coulomb gap contributes to the overall resistance up to room temperature.In contrast, samples showing 1 / T behavior of TMR display much lower values of the Coulomb energy ͑about 0.29 meV in Ref. 19 and 0.1 meV in Ref. 27; the exchange energy was about 0.31 meV in Ref. 19͒.
the findings of Kodama et al. in sequentially sputtered Co-Al 2 O 3 , 37 and those of Sankar et al. in co-sputtered Co-SiO 2 .

TABLE I .
Experimental values of T p , T rev , the average energy barrier ͗KV͘, Curie-Weiss constant C, and and parameters D 0 , , ␣ B obtained by fitting the ZFC curves in Fig.2for various Co concentrations.

TABLE II .
Parameters D 0 , obtained by fitting the magnetization curves in Fig.3, also as the effective anisotropy constants K and mean volume ͗V͘ for various Co concentrations.