Competing tunneling and capacitive paths in Co-ZrO 2 granular thin films

The ac electrical response is studied in thin films composed of well-defined nanometric Co particles embedded in an insulating ZrO 2 matrix which tends to coat them, preventing the formation of aggregates. In the dielectric regime, ac transport originates from the competition between interparticle capacitive Cp and tunneling Rt channels, the latter being thermally assisted. This competition yields an absorption phenomenon at a characteristic frequency 1/( RtCp), which is observed in the range 10–10 000 Hz. In this way, the effective ac properties mimic theuniversalresponse of disordered dielectric materials. Temperature and frequency determine the complexity and nature of the ac electrical paths, which have been successfully modeled by an Rt-Cp network.

Granular films composed of magnetic nanoparticles embedded in an insulating matrix are currently the object of very active research because of their magnetic and transport properties, which have promising technological applications. 1In addition, the rapid development of several film preparation techniques provides granular systems with a well controlled nanostructure, in which the interparticle transport processes and magnetic behavior of nanoparticles can be studied.In particular, dc transport properties of granular thin films have been widely investigated in the last decades 2 and explained in the dielectric regime at low electrical fields in terms of thermally assisted tunneling. 3Although studies of the frequency behavior of tunneling junctions have been undertaken, 4,5 the ac response of granular metals in the dielectric regime is not yet properly characterized in the limit where the admittances rising from interparticle tunneling processes and interparticle capacitances become of similar magnitude.][8][9][10] In these cases, a random network of capacitive and resistive bonds, which represent the insulating and metal components, respectively, has successfully modeled the ac response of granular metals. 6,7][9] In this paper, we report ac electrical measurements in granular thin films with competing tunneling and capacitive channels in the dielectric regime.Most of the observed ac properties mimic the universal response of disordered dielectric materials, but at much lower frequencies.In particular, as a direct consequence of the competition between tunneling and capacitive conduction, an absorption phenomenon is observed in the range 10-10 000 Hz, which is very much lower as compared to that corresponding to the classical optical properties in granular media. 10This ac behavior is explained in terms of an R-C network in which each bond contains a parallel combination of randomly distributed values of a resistor ͑R͒ and a capacitor (C), where R is the tunneling resistance and C the interparticle capacitance.
Below the percolation threshold, the ac electrical transport between isolated metallic particles embedded in an insulating matrix is mainly due to two processes: ͑i͒ quantum electron tunneling between grains with separations of the order or less than one nanometer and ͑ii͒ the ac conduction by virtue of the capacitance between charged surfaces of neighbor particles, which can be thought of as electrodes of a complex network of capacitors.At low electrical fields, the first mechanism is temperature dependent, since it requires the thermal generation of charge carriers overcoming the Coulomb charging energy of the particles. 3Assuming that the volume fraction x of the metallic component, is uniform within a region of a few particle diameters d it is reasonable to expect that l/d is a constant, l being the mean interparticle distance.Consequently, at each temperature the percolation path chiefly extends over the smallest particles whose Coulomb energies can be thermally activated.This is due to l/d being constant and the height of the tunnel barrier that increases exponentially with the interparticle separation. 3ontrarily, the capacitance is temperature independent and is always present between neighboring particles with their faced surfaces oppositely charged, no matter what their separation is.In fact, the highest capacitance occurs between the largest particles in spite of their large separation, since the capacitance for particles with similar size increases linearly with d ͑their capacitance is proportional to d 2 /l with d/l being approximately constant throughout the system͒.Furthermore, ac transport between the largest particles is only possible by capacitance, since their corresponding large separation yields very high tunnel barriers.
The samples used in this study were 400 nm thick films composed of nanometric Co particles embedded in an insulating matrix of ZrO 2 stabilized with 7 mol.% Y 2 O 3 .The films were deposited at room temperature onto glass substrates by laser ablation of rotating targets comprised of sec-tors of zirconia and pure cobalt.Samples were reproducible as long as the preparation conditions were the same and the experimental parameters ͑e.g.energy fluence, size of the spot, and roughness of the target surface͒ were carefully controlled.Parts of the films were annealed at 575 °C for several hours.The average composition was determined by x-ray photoelectron spectroscopy and microprobe analysis.High resolution transmission electron microscopy ͑TEM͒ indicated the existence of crystalline Co particles ͑see the detail in Fig. 1͒ of round shape embedded in amorphous zirconia.The particle size distribution was determined by TEM as well as by fitting the low-field magnetic susceptibility ͑zero field cooling͒ and the high-temperature isothermal magnetization curves, to a distribution of Langevin functions.TEM showed sharp interfaces between the Co particles and insulating matrix with no detectable interdiffusion boundaries.Moreover, the amorphous matrix tends to coat the metallic particles, preventing the formation of aggregates even at relatively high Co concentrations.Thus, the medium is partially correlated, a fact which raises the effective percolation threshold.This allows us to obtain samples with a high metallic concentration consisting of nanometric isolated particles with small interparticle separations, avoiding the formation of filamentary clusters as observed in samples prepared by other methods near the percolation threshold. 8he existence of metal-matrix interfaces of high quality is also confirmed by the observation of a quasitemperature independent magnetoresistance of about 7% in samples with Co-volume fractions close to xϭ0.40. 11This value is in agreement with those found elsewhere for different Co-based granular systems 12 and corresponds to a spin polarization of 0.27, close to the expected value of 0.33Ϫ0.35. 13The ac electrical measurements were carried out by a four probe technique using a lock-in amplifier and an ac constant current source in the frequency range of 10 to 6500 Hz.Four parallel strips of copper separated by 0.7 mm were deposited on the sample's top surface as electrodes.
A deep insight in the ac electrical properties is gained by analyzing the phase of the voltage drop across the sample as a function of the frequency .The dependence of the () curves on the Co-volume fraction and annealing temperature will be published elsewhere. 11In this paper, we focus on a sample with a Co-volume fraction xϭ0.37.Linearlogarithmic size distributions were obtained from low-field susceptibility, magnetization curves, and TEM images.For the as-deposited samples the mean diameter is dϭ3.0Ϯ0.5 nm with a width ϭ0.17, whereas for the annealed samples dϭ17Ϯ2 nm and ϭ0.37. Figure 2 shows (), measured at different temperatures for the as-deposited sample.At high temperatures ͑around 290 K͒ and low frequencies, is roughly zero, corresponding to the current circulating essentially by interparticle tunneling through a resistive path.As is increased the admittance associated with the interparticle capacitance increases and the current progressively flows through capacitive paths which are in series and/or parallel configurations with respect to the tunneling channels, diminishing the phase of the voltage drop.At intermediate temperatures ͑see curves for 134.7, 90.3, 59.7, and 40.5 K in Fig. 2͒, a constant phase regime ͑CPR͒ is reached after a first phase drop similar to that observed at higher temperatures.The onset of this plateau occurs at frequencies at which the admittance associated with the capacitance between large particles becomes comparable to the tunneling conductance.In this way, capacitive short cuts are randomly inserted in the tunneling path, improving the electrical connectivity and thus favoring the current circulation through the resistive backbone.This regime is lost at higher frequencies, when the admittance due to the capacitance between small particles becomes comparable to the tunneling conductance and a significant part of the total current is flowing through this capacitive channel.The CPR shifts towards lower frequencies as the temperature is reduced because of the effective decrease of the tunneling conductance, as the amount of charge carriers is reduced.In particular, at 29 K, CPR has moved to frequencies below 10 Hz ͑see Fig. 2͒.
All the above features are also found in the ac response of a cubic random R-C network, 14 in which a fraction of the electrical bonds x r are occupied by a parallel combination of a resistor R t and a capacitor C p ͑interparticle tunneling resistance and capacitance, respectively͒, and the rest by capacitors only, of higher capacitance C p Ј ͑capacitance between the largest particles not electrically connected by tunneling͒.For 10 nm particles with a separation of 1 nm, R t is estimated from experimental data on tunneling junctions to be in the range 10 14 Ϫ10 12 ⍀ as the temperature varies from 30 to 290 K, 15 C p is about 10 Ϫ17 F ͑Ref. 16͒ and C p Ј is assumed to be 50 C p .The inset of the Fig. 2 shows () for a cubic lattice with 2900 electrical bonds, which was calculated with the WINSPICE program. 17Three values of x r and R t have been used to simulate the effect of temperature, leading to a very good qualitative agreement with the experimental results.The thermal variation of the mean value of corresponding to the CPR can be understood from the logarithmic mixing rule 18 ln *ϭ ͚ n ln n * ,

͑1͒
where * is the complex conductance of the system and n * is the conductance of the nth electric bond of the network.At intermediate frequencies at which the contribution of C p can be neglected, a CPR of the conductance is obtained

͑2͒
Therefore, in the CPR is simply Ϫ(1Ϫx r ) /2.From the experimental results ͑see Fig. 2͒ (1Ϫx r ) can be estimated to vary from about 0.05 at 29 K to about 0.13 at 290 K, indicating that as the temperature increases the fraction of effective resistive bonds is reduced.This is because higher thermal activation allows tunneling through smaller particles with higher charging energy, which reduces the total length of the resistive path.
The inset of Fig. 3 shows the thermal dependence of ϭ 0 Ϫ1 , obtained from the experimental data by extrapolating the real part of the conductance () to zero frequency.It is evident that except for a slight deviation at temperatures above 220 K, (T) varies as exp(2 ͱB/k B T) with B ϭ25.3 meV.Such a dependence is predicted 3 for thermally assisted tunneling and corroborates the existence of a tunneling path through the sample.In Fig. 3, the frequency dependence of ()/ 0 is depicted at different temperatures, which according to Eq. ͑2͒ follows a potential law with an exponent roughly within 0.05Ϫ0.13 in the CPR frequency range.
It is worth noting that these electrical properties share with dielectric materials most of the features known as universal response. 19In Fig. 4, it is shown that the total capaci- tance of the R-C network falls at intermediate frequencies following a power law and approaches the total interparticle capacitance of the system.At the same time, the dielectric loss defined as ͓()Ϫ 0 ͔/ develops a peak at the onset of the capacitance fall.For the experimental data, the peak position of the dielectric loss and the power law regime of the capacitance shift towards lower frequencies as the temperature is reduced.This is caused by the large thermal variation of the interparticle tunneling resistance originated from the change in the charge carrier amount.This thermal effect also changes the length of the resistive path, since at high temperatures the charging energy of the small particles may be overcome, creating new electrical paths with low tunneling barriers, and in this way, improving the system connectivity.In fact, the thermal dependence of the peak position of the dielectric loss is found to follow an Arrhenius law ͓see the inset of Fig. 5͑a͔͒, as expected for a thermal activation process, and takes place at a characteristic frequency of the order of 1/(R t C p Ј).The activation energy deduced from this dependence is Wϭ18.6 meV, a value which is comparable to that of the constant B in (T).By using W/k B T as a scaling parameter, all the curves corresponding to the capacitance and the dielectric loss at different temperatures collapse onto two master curves ͑see Fig. 5͒.These curves remarkably resemble those corresponding to the universal dielectric response, 19 but taking place in a lower frequency range than what is usually observed in dielectric materials.Moreover, both sides of the dielectric loss peak follow a power law, as expected for a universal response, with exponents close to 0.5.
No evidence of charge carrier trapping in the interfaces between metallic particles and the amorphous matrix has been observed.However, this effect only gives an additional contribution to the real part of the admittance, as previously observed in tunnel junctions, 5,20 but does not produce a loss peak.We cannot discard the occurrence of this phenomenon in our samples, but if present it would be hidden by the dominant contribution of the R-C competition.
In conclusion, we have showed that the ac absorption phenomenon in Co-ZrO 2 granular films originates from the competition between interparticle tunneling resistance and capacitance which takes place throughout a complex threedimensional random R-C network.The inherent disorder of this network produces an effective universal response which is very similar to that observed in many disordered dielectric materials, but at much lower frequencies.
Financial support of the Spanish CICYT through the MAT2000-0858 and MAT99-0984 projects and the Generalitat of Catalonia through the 2000SGR00025 project are gratefully recognized.

FIG. 1 .FIG. 2 .
FIG. 1. Bright field TEM image for an annealed sample, displaying a distribution of Co particles in an Y 2 O 3 stabilized ZrO 2 matrix.The inset shows parallel interference patterns inside the Co particles.

FIG. 3 .
FIG. 3. Normalized conductance as a function of frequency.Temperatures are 59.7, 90.3, 134.7, 214.7, and 290.1 K from bottom to top.Straight lines indicate the CPR.The inset shows the zero frequency resistivity versus T Ϫ1/2 .