Simple Hopping Model Research Articles

Hopping dynamics of ions and polarons in disordered materials: On the potential of nonlinear conductivity spectroscopy

The potential of nonlinear conductivity spectroscopy for obtaining new information on the hopping dynamics of mobile charge carriers in disordered materials is analyzed from a theoretical as well as from an experimental point of view. The nonlinear conductivity spectra of simple hopping models are calculated by means of analytical methods and Monte Carlo simulations. It is shown that the nonlinearity of the high-frequency conductivity is strongly influenced by the local asymmetry of the potential landscape, while the nonlinearity of the low-frequency conductivity is sensitive to the structure of the long-range diffusion pathways. Furthermore, experimental results for the nonlinear conductivity of ion conducting glasses and polymers are reviewed.

Thermally Activated Low Frequency Dielectric Dispersion of Antiferroelectric Pb 1.075 La 0.025 Zr 0.95 Ti 0.05 O 3 Thin Film

Antiferroelectric Pb 1.075 La 0.025 Zr 0.95 Ti 0.05 O 3 (PLZT) thin films on Pt/Ti/SiO 2 /Si substrate have been fabricated by pulsed laser deposition technique. The temperature dependent dielectric constant and polarization-electric field (P-E) hysteresis loops of PLZT film demonstrate the phase transition from antiferroelectric to ferroelectric. The Curie temperature (T c ) and dielectric constant at T c are 196C and 2065, respectively. The maximum saturated polarization reaches nearly 50 w C/cm 2 , corresponding to a maximum induced charge of 12.5 nC. The complex dielectric constants and ac conductivities of PLZT thin film are measured as a function of frequency (0.01Hz∼1 MHz) and temperature (25∼200C) in order to investigate dielectric dispersions and conduction mechanisms. We also observe the strong low-frequency dielectric dispersion at high temperature above 125C due to the migration of thermally activated oxygen vacancies. The estimated activation energy for conduction is 0.76 eV. The conduction mechanism seems more complicated than simple hopping model. Consequently, understanding of oxygen vacancy transfer in the antiferroelectric PLZT thin film is important for practical applications in high charge storage and actuator devices.

Thermally Activated Low Frequency Dielectric Dispersion of Antiferroelectric Pb 1.075 La 0.025 Zr 0.95 Ti 0.05 O 3 Thin Film

Antiferroelectric Pb 1.075 La 0.025 Zr 0.95 Ti 0.05 O 3 (PLZT) thin films on Pt/Ti/SiO 2 /Si substrate have been fabricated by pulsed laser deposition technique. The temperature dependent dielectric constant and polarization-electric field (P-E) hysteresis loops of PLZT film demonstrate the phase transition from antiferroelectric to ferroelectric. The Curie temperature (T c ) and dielectric constant at T c are 196C and 2065, respectively. The maximum saturated polarization reaches nearly 50 w C/cm 2 , corresponding to a maximum induced charge of 12.5 nC. The complex dielectric constants and ac conductivities of PLZT thin film are measured as a function of frequency (0.01Hz∼1 MHz) and temperature (25∼200C) in order to investigate dielectric dispersions and conduction mechanisms. We also observe the strong low-frequency dielectric dispersion at high temperature above 125C due to the migration of thermally activated oxygen vacancies. The estimated activation energy for conduction is 0.76 eV. The conduction mechanism seems more complicated than simple hopping model. Consequently, understanding of oxygen vacancy transfer in the antiferroelectric PLZT thin film is important for practical applications in high charge storage and actuator devices.

On the diffusion of Li + defects in LiCoO 2 and LiNiO 2

NMR spectra of 7Li nucleus have been measured in LiCoO 2 and LiNiO 2. Line widths of the spectra decreased with increasing temperature above 400 K. The decrease obviously indicates the motional narrowing originating from diffusing Li + defects. A simple ion hopping model revealed that the diffusion constant of LiCoO 2 with an activation energy of 0.3 eV is larger than that of LiNiO 2 with the activation energy of 0.6 eV. This suggests that lithium ions are easier to diffuse in LiCoO 2 than in LiNiO 2 at high temperatures.

Diffusion and rheology in a model of glassy materials

We study self-diffusion within a simple hopping model for glassy materials. (The model is Bouchaud's model of glasses (J.-P. Bouchaud, J. Phys. I France 2, 1705 (1992)), as extended to describe rheological properties (P. Sollich, F. Lequeux, P. Hébraud, M.E. Cates, Phys. Rev. Lett. 78, 2020 (1997)).) We investigate the breakdown, near the glass transition, of the (generalized) Stokes-Einstein relation between self-diffusion of a tracer particle and the (frequency-dependent) viscosity of the system as a whole. This stems from the presence of a broad distribution of relaxation times of which different moments control diffusion and rheology. We also investigate the effect of flow (oscillatory shear) on self-diffusion and show that this causes a finite diffusivity in the temperature regime below the glass transition (where this was previously zero). At higher temperatures the diffusivity is enhanced by a power law frequency dependence that also characterises the rheological response. The relevance of these findings to soft glassy materials (foams, emulsions etc.) as well as to conventional glass-forming liquids is discussed.

Simple Solutions of the Torrey–Bloch Equations in the NMR Study of Molecular Diffusion

A simple technique for solving the Torrey–Bloch equations appearing in the calculation of the NMR signal under gradient fields is presented. It is applicable to arbitrary time dependence of the gradient field to arbitrary initial distribution of spins, and to spin motion on discrete lattices as well as in the continuum under conditions of unrestricted diffusion. Known results are recovered as particular cases and new results are presented. The discrete lattice results are shown to be similar to known results for restricted diffusion in the continuum. Also presented is a surprising equivalence between results for a simple two-site hopping model and earlier expressions for the NMR signal for spins undergoing restricted diffusion in a continuum.

Quantum interference effects of electrons scattered by antidots in magnetic fields

We investigate the quantum transport phenomena for a quasi-one-dimensional quantum wire with antidots in magnetic fields. The calculated magnetoconductance shows a quasi-periodic structure of the group of dips. The characteristic properties of the magnetoconductance can be understood through a simple hopping model, where the electron transfers between the neighboring pinned orbits. We also discuss the interference effects on the conductance in connection with local electronic states.

Model and theory for the determination of diffusion coefficients by Auger electron spectroscopy measurements and an application to oxygen diffusion along the [0001] and [101¯0] axes in single crystal zirconium

A simple hopping model of the diffusion of adsorbed species from a surface into the bulk of a material has been formulated and solved mathematically. The difference in the energy barriers for an atom moving between the atomic layers at the surface and in the bulk are explicitly considered. This model is also capable of describing the initial stages of diffusion, something that conventional solutions of the continuum diffusion equation cannot handle. Auger electron spectroscopy has been used to measure the dissolution rate of oxygen from Zr(0001) and Zr(101¯0) surface into the bulk. Satisfactory results were obtained by applying our model to the diffusion data for these two zirconium surfaces for two different heating schedules: (i) rapid temperature ramp-and-hold and (ii) continuous linear heating with respect to time. The resulting Arrhenius expressions for diffusion are: D = (0.115 ± 0.031)exp[(−44.45 ± 4.82)kcal/ RT]cm 2/s along Zr[0001] and D = (1.07 ± 0.26)exp[(−46.18 ± 4.22)kcal/RT]cm 2/s along Zr[101¯0].

AC conductivity of amorphous indium-selenium films

The electrical conductivity of flash-evaporated InSex films was measured in the frequency range 103 to 107 Hz and in the temperature range 200 to 300 K. Studies of electrical properties of films grown from different stoichiometries of flash-source materials are reported in this work. The observed behaviour is described in terms of a simple hopping model. The temperature dependence of sigma AC can be well explained by a correlated barrier hopping model. The density of localized states N(EF) is also estimated for these materials.

A Dynamic Monte Carlo Simulation of Sorbate Mobility in Zeolites: The Effects of Molecular Crowding on Sorbate Mobility

AbstractA finite lattice of adsorption sites, as shown by Monte Carlo simulation, is used to develop a simple hopping model of small molecules within the alpha cage of zeolite NaA. A two body attractive energetic interaction is employed for occupied pairs of nearest neighbor sites. A many body repulsive interaction term accounts for the crowding associated with site saturation. This term becomes important when the site-site spacing is less than the van der Waals diameter of the adsorbate. The dynamic Monte Carlo method is used to evaluate site to site hopping frequencies as a function of loading based on this potential energy function. As the sorbate-sorbate attractive interaction is increased (or, equivalently, as the temperature is reduced), mobility minima occur at certain lattice occupancies which may be explained by the formation of energetically favorable clusters on the cubocathedral lattice. In other words, molecular crowding can cause sorbate mobility to suffer minima as loading is increased. This prediction is in agreement with recent Xe NMR measurements.

High-field and low-field transport in amorphous semiconductors

Field and temperature dependence of transits in amorphous semiconductors are tackled using two different methods. We use analytical and Monte-Carlo simulation techniques. We compare with existing data on hopping systems. The model does seem to account reasonably well for the observed temperature and exp (−η Fα) field dependence of transit times in amorphous semiconductor multilayers. The anomaly in the low field temperature dependence observed by the Dundee group however cannot be produced by the simple hopping model. The reasons are discussed.

Defect Equilibria and Transport in YBa2Cu3O7‐x at Elevated Temperatures: III, Defect Model and Conductivity Mechanism

Based on the measurements of electrical conductivity and thermopower (Seebeck coefficient) as a function of oxygen partial pressure, the defect structure and corresponding conduction mechanisms at elevated temperatures are considered for the Y‐Ba‐Cu‐O (1:2:3) system. It has been postulated that the simple hopping model is not applicable to YBa2Cu3O7‐x. A modified conduction hopping mechanism has been proposed and equations describing the mobility of charge carriers in the studied system are derived. The most important advantage of the present model, in comparison to previous models, involves considering interactions between both electrons and electron holes and the resulting effect on their mobility terms. The observed departure of experimental data from the Heikes formula is explained by a very high concentration of defects and resulting substantial interactions between the defects which must be taken into account. The proposed transport model exhibits good agreement with experimental data of nonstoichiometry and the presently determined electrical conductivity and thermopower. It has been argued that the charge transfer mechanism depends substantially on the oxygen partial pressure and resulting oxygen content in the oxide lattice.

Nuclear spin relaxation due to the translational diffusion of hydrogen in BCC metals

The authors calculate the spectral density functions required for the theoretical determination of nuclear spin relaxation rates due to magnetic dipolar coupling between diffusing hydrogen spins in BCC metals. Both the like-spin (hydrogen-hydrogen) and unlike-spin (metal-hydrogen) contributions are determined using the Monte Carlo method of Faux, Ross and Sholl (1986). Results are obtained for hydrogen-to-metal ratios of 0.12, 0.3 and 0.6 for the simple hopping model (no multiple occupancy of sites) and for a multiple site-blocking model in which the hydrogen spins block all sites as far as the second or third neighbour. For the simple hopping model, the results are in very good agreement with the multiple-scattering theory of Sankey and Fedders (1980), in the high- and low-temperature limits. For the multiple site-blocking models, it is found that the BPP model (due to Bloembergen, Purcell and Pound, 1948) and Torrey model (1953), can differ significantly from the Monte Carlo result, particularly at higher concentrations. The results obtained for D/T1, where D is the tracer diffusion coefficient and T1 is the spin-lattice relaxation time, are compared to the experimental result on NbH0.6. The Monte Carlo method gives a value which is 2/3 of the experimental value if blocking to either the second or the third neighbour is assumed. Agreement with experiment may be obtained if it is assumed that the hydrogen diffuses by a combination of jumps to nearest-neighbour and second-nearest-neighbour sites.

Diffusion in a 2D bond percolation model. A Monte Carlo simulation

A Monte Carlo computer simulation of a simple stochastic hopping model is presented. The velocity autocorrelation function and the diffusion coefficient are analyzed and compared with the results obtained by using kinetic theory. The theory and the simulation agree within the statistical error.

Hopping conduction in Na-β-alumina: Temperature dependence

The temperature dependence of a.c. conductivity σ(ω), in Na-β-alumina is calculated on the basis of a simple hopping model. Comparison of the calculated results with the experimental data shows reasonably good agreement.

Hopping conduction in Na-β-alumina

The frequency dependence of ionic conductivity, σ(ω), in Na-β-alumina is calculated on the basis of a simple hopping model of conduction. By comparing the calculated results with the experimental data on σ(ω), a possible potential energy-distance diagram is suggested.

Localization of electronic wave functions due to local topology

We present a simple two-dimensional hopping model for independent electrons which has strictly localized states in addition to the extended states. These localized states can exist either in a band gap or within the continuum. The localized states persist if the lattice periodicity is destroyed. Finally, the effect is shown to hold for a much more general class of systems.

Nuclear spin relaxation by translational diffusion in solids: X. Monte Carlo calculation for the simple hopping model

The spectral density functions J(p)( omega ) relevant to nuclear spin relaxation rates due to magnetic dipole coupling between diffusing spins in cubic crystals are calculated, for the simple hopping model, by a Monte Carlo method. The analysis makes use of known exact results in various limits. The statistical fluctuations in the Monte Carlo calculations become more significant at high frequencies but accurate analytic results are known in this regime. Numerical results are therefore obtained over the entire frequency range for all spin concentrations except the monovacancy limit.

Nuclear spin relaxation by translational diffusion in solids: high-frequency limit for the simple hopping model

The high-frequency limit of the correlation functions which determine the nuclear spin relaxation rates due to magnetic dipolar coupling between diffusing spins in a crystal has recently been calculated by MacGillivray and Sholl (1985) for the simple hopping model. It is shown that a modification of the analysis produces more accurate results for the high-frequency limit.

Occupancy correlation functions for diffusion in the high-frequency limit of the simple hopping model

The double Fourier transform S (k, omega ) of the occupancy correlation function, for a particle diffusing on a cubic lattice, has been calculated for all omega and specific k in the limits of low and high particle concentrations by Tahir-Kheli and Elliott (1982). It is shown that their equations may be used to calculate the high-frequency dependence of S(k, omega ) for all concentrations and exact to the power omega -4 in the high-frequency expansion. Results are presented which enable algebraic expressions for S(k, omega ) to be determined, for all k in this limit for the SC, BCC and FCC lattices.