Universality of the electrical transport in granular metals

Abstract

The universality of the ac electrical transport in granular metals has been scarcely studied and the actual mechanisms involved in the scaling laws are not well understood. Previous works have reported on the scaling of capacitance and dielectric loss at different temperatures in Co-ZrO2 granular metals. However, the characteristic frequency used to scale the conductivity spectra has not been discussed, yet. This report provides unambiguous evidence of the universal relaxation behavior of Pd-ZrO2 granular thin films over wide frequency (11 Hz-2 MHz) and temperature ranges (40-180 K) by means of Impedance Spectroscopy. The frequency dependence of the imaginary parts of both the impedance Z″ and electrical modulus M″ exhibit respective peaks at frequencies ωmax that follow a thermal activation law, ωmax ∝ exp(T(1/2)). Moreover, the real part of electrical conductivity σ' follows the Jonscher's universal power law, while the onset of the conductivity dispersion also corresponds to ωmax. Interestingly enough, ωmax can be used as the scaling parameter for Z″, M″ and σ', such that the corresponding spectra collapse onto single master curves. All in all, these facts show that the Time-Temperature Superposition Principle holds for the ac conductance of granular metals, in which both electron tunneling and capacitive paths among particles compete, exhibiting a well-characterized universal behavior.

Figure 1. Plot of real part of the conductivity σ′ as a function of frequency in a log-log scale in the temperature range of 40–280 K.

The inset shows the dc resistivity ρ_dc = 1/σ_dc (obtained from extrapolation of σ′ to zero frequency) versus T^−1/2 in a semi-log scale. The solid lines are linear fits.

Figure 2

(a) Imaginary part of impedance Z″ as a function of frequency in a semi-log scale, (b) Imaginary part of the electrical modulus M′ as a function of frequency in a semi-log scale. Two peaks are observed in the spectra, at low and high frequency, respectively.

Figure 3. Simplistic sketch of the ac electrical conductance model in Pd-ZrO₂ granular thin film in the dielectric regime.

At low frequency, most of the smallest Pd particles are electrically connected by the dc tunneling backbone , whereas, at ~1 kHz, an additional contribution of assisted tunneling resistive paths among smaller particles, initially isolated at low frequencies, improves the electrical conductance. The polarized bigger particles only contribute to the capacitive conductance C_i due to the large separation from each other.

Figure 4

(a) Scaling plot versus ω/ω_max in a log-log scale. The inset shows the variation of the frequency of the peak in Z″ as a function of T^−1/2 in a semi-log scale and (b) Scaling plot versus ω/ω_max in a log-log scale. The inset shows τ_1M and τ_2M as a function of T^1/2 in a semi-log scale. τ_1M is the low frequency and τ_2M is the high frequency relaxation times, respectively, obtained from the peaks in the main figure. The solid lines are linear fits.

Figure 5. Scaling plot of σ′/σ_dc versus ω/ω_max in a log-log scale in the temperature range 40–180 K.

The inset shows a log-log scale of σ′ versus frequency in the frequency range of 11 Hz–10 kHz. Solid lines indicate the fit of the data to a power law with a fractional exponent n.