Disordered Hopping System Research Articles

Carrier mobility in two-dimensional disordered hopping systems

A model of the equilibrium 2D hopping mobility in a disordered organic semiconductor is formulated for arbitrary charge carrier densities and arbitrary temperatures. The calculated dependence of the 2D mobility upon inverse temperature is compared with experimental data obtained on 2D carrier transport in poly(3-hexylthiophene) thin film field-effect transistors.

Injection-limited contact in bottom-contact pentacene organic thin-film transistors

The contact effect in bottom-contact pentacene-based thin-film transistor, with Au source/drain electrode, is investigated. By separating the voltage drop at the source contact from the source-drain voltage, the current–voltage characteristic of the injection contact is obtained. The contact characteristics are in good agreement with a model of carrier injection from a metal electrode into a disordered hopping system.

Charge Injection into Bottom-Contact Pentacene Thin-Film Transistors

ABSTRACTWe investigate the contact of bottom-contact pentacene-based thin-film transistors, with Au source/drain electrode. By separating the voltage drop at the contact from the source-drain voltage, the current-voltage characteristic of the injection contact is obtained. The contact characteristics are in a good agreement with a model of carrier injection from a metal electrode into a disordered hopping system.

Spin transport in disordered two-dimensional hopping systems with Rashba spin-orbit interaction

The influence of Rashba spin-orbit interaction on the spin dynamics of a topologically disordered hopping system is studied in this paper. This is a significant generalization of a previous investigation, where an ordered (polaronic) hopping system has been considered instead. It is found that in the limit where the Rashba length is large compared to the typical hopping length, the spin dynamics of a disordered system can still be described by the expressions derived for an ordered system, under the provision that one takes into account the frequency dependence of the diffusion constant and the mobility (which are determined by charge transport and are independent of spin). With these results we are able to make explicit the influence of disorder on spin related quantities as, e.g., the spin lifetime in hopping systems.

Effect of doping on the density-of-states distribution and carrier hopping in disordered organic semiconductors

The effect of doping on the density-of-states (DOS) distribution and charge-carrier transport in a disordered hopping system is considered analytically. It is shown that doping such a system produces a random distribution of dopant ions, which Coulombically interact with carriers localized in intrinsic hopping sites. This interaction further increases the energy disorder and broadens the deep tail of the DOS distribution. Therefore, doping of a disordered organic semiconductor, on the one hand, increases the concentration of charge carriers and lifts up the Fermi level but, on the other hand, creates additional deep Coulombic traps of the opposite polarity. While the former effect facilitates conductivity, the latter strongly suppresses the carrier hopping rate. A model of hopping in a doped disordered organic semiconductor is suggested. It is shown that the doping efficiency strongly depends upon the energy disorder and external electric field.

Two-dimensional electron-hole capture in a disordered hopping system

We model the two-dimensional recombination of electrons and holes in a system where the mean free path is short compared with the thermal capture radius. This recombination mechanism is relevant to the operation of bilayer organic light-emitting diodes (LED's), where electrons and holes accumulate on either side of the internal heterojunction. The electron-hole recombination rate can be limited by the time taken for these charge carriers to drift and diffuse to positions where electrons and holes are directly opposite to each other on either side of the interface, at which point rapid formation of an emissive neutral state can occur. In this paper, we use analytical and numerical techniques to find the rate of this two-dimensional electron-hole capture process. Where one species of carrier is significantly less mobile than the other, we find that the recombination rate depends superlinearly on the density of the less mobile carrier. Numerical simulations allow the effects of disorder to be taken into account in a microscopic hopping model. Direct solution of the master equation for hopping provides more efficient solutions than Monte Carlo simulations. The rate constants extracted from our model are consistent with efficient emission from bilayer LED's without requiring independent hopping of electrons and holes over the internal barrier at the heterojunction.

Polaron effects on thermally stimulated photoluminescence in disordered organic systems

Thermally stimulated photoluminescence (TSPL) in disordered organic materials subjected to the polaron formation was studied both theoretically and experimentally. Since TSPL measurements were performed after some dwell time on samples, which have been photoexcited at helium temperature, the initial energy distribution of polarons was formed in the course of low-temperature energy relaxation. In order to calculate this distribution, an analytic model of low-temperature energy relaxation of polarons in a disordered hopping system was formulated. This model proves that energy relaxation of polarons is much slower than that of charge carriers in a similarly disordered but rigid hopping system. At moderate values of the polaron binding energy the suppressed rate of low-temperature relaxation leads to a shift of TSPL curves to lower temperatures as compared to what could be expected in the absence of the polaronic effects. This shift is both obtained from analytic calculations and observed experimentally in some polysilylene derivatives.

Effect of backward carrier jumps on variable-range hopping in disordered materials

A variable-range hopping model is described that accounts for the effect of back-and-forth carrier jumps between occasionally close hopping sites isolated from the rest of the hopping network. It is shown that accounting for this effect for constant density of states (DOS) does not change the Mott ${T}^{\ensuremath{-}1/4}$ law but yields a modified coefficient that can be considered as a percolation threshold for dc conductivity in a positionally and energetically disordered hopping system. In a system with an exponential DOS distribution, the backward jumps significantly reduce dc conductivity but only slightly affect its functional dependence upon temperature.

Effective transport energy versus the energy of most probable jumps in disordered hopping systems

An analytic expression for the effective transport energy in a positionally random and energetically disordered hopping system is obtained. It is shown that multiple carrier jumps within pairs of occasionally close localized states strongly affect the position of the effective transport level on the energy scale and lead to a noticeable difference between the effective transport energy and the energy of most probable jumps. In a hopping system with a Gaussian density-of-states energy distribution, the equilibrium carrier mobility is found to be an almost factorized function of temperature and concentration of localized states.

Equilibrium carrier mobility in disordered hopping systems

Abstract It is shown that the charge carrier mobility in a positionally and energetically disordered hopping system can be evaluated by averaging either the hopping rates or the hopping times over the thermal equilibrium energy distribution of localized carriers. However, at variance with averaging the hopping rates, averaging the hopping times can be correct only if the energy dependence of the carrier energy relaxation time is also taken into consideration. The temperature and concentration dependences of the equilibrium carrier mobility were calculated by averaging the hopping rates. The consideration was based on the variable-range hopping approach with full account for the interplay between jump distance and energy difference. However, the obtained results prove that, in good quantitative agreement with both Monte Carlo simulations and experimental data, the mobility can be approximated as a product of two functions. The first function depends almost solely upon the temperature and reveals only a wea...

Equilibrium carrier mobility in disordered hopping systems

Equilibrium carrier mobility in disordered hopping systems

Onsager approach to the universal cross-over in variable-range hopping with Coulomb interactions

A model describing low-temperature dc conductivity in a disordered hopping system is formulated. The model considers one-particle excitations as a variable range hopping with a constant density of states near the Fermi level and accounts for Coulomb interaction of charge carriers on the basis of Onsager dissociation probability. The resulting temperature dependence of the conductivity reveals a universal crossover from Mott's T −1/4 to Efros–Shklovskii's T −1/2 laws with decreasing temperature and increasing density of localized states.

Current injection from a metal to a disordered hopping system. III. Comparison between experiment and Monte Carlo simulation

We have performed electric-field and temperature-dependent electron injection studies in an aluminum/tris(8-hydroxy-quinolinolato)aluminum/magnesium:silver single-layer organic light-emitting diode. Analysis of the observed injection currents in terms of the classic Fowler-Nordheim (FN) tunneling or Richardson-Schottky (RS) thermionic emission proved to be inadequate. Whereas, the FN-type behavior at high-electric fields must be considered accidental, the injection currents qualitatively resemble those of the RS concept. However, quantitative differences are observed concerning the RS coefficient, the prefactor current, and the temperature dependence. On the other hand, the experimental data are in excellent agreement with a recently presented Monte Carlo simulation [U. Wolf et al., Phys. Rev. B 59, 7507 (1999)] of carrier injection from a metal to an organic dielectric with random hopping sites.

Current injection from a metal to a disordered hopping system. I. Monte Carlo simulation

Emission-limited charge-carrier injection in the dark from a metal into a random organic dielectric has been studied via Monte Carlo simulations. The dielectric has been modelled in terms of a regular lattice of point sites featuring a Gaussian distribution of energies to represent disorder. The essential input parameters are the zero-field energy barrier for injection $(\ensuremath{\Delta}),$ the variance $(\ensuremath{\sigma})$ of the distribution of the hopping states, electric field, and temperature. By varying the jump distance the unimportance of long-range tunneling transitions has been established. Therefore, Fowler-Nordheim type $j(F)$ charcteristics at high fields have to be considered accidental. The dependence of the injection yield resembles that of Richardson-Schottky (RS) thermionic emission. Quantitative differences are noted, however, concerning the RS coefficient and the temperature dependence. The latter tends to saturate at low temperatures, which is a signature of hopping among sites distributed in energy.

Current injection from a metal to a disordered hopping system. II. Comparison between analytic theory and simulation

Monte Carlo simulations of hopping injection from a metal into a random-organic dielectric, described in the previous paper (I), have been compared to results of analytic theory. Good agreement has been found for the field dependence of the yield of charge carriers as a function of the energy barrier at the interface. This is a crucial test for the validity of the assumptions required in order to render an analytical approach of hopping in a disordered system in the presence of a long-ranged Coulombic potential tractable. The most serious of these is the Onsager-like homogenous medium approach to treat the escape of a charge carrier from the Coulomb well once injected. The primary injection event into the dielectric has been treated in terms of hopping theory implying the concept of transport energy.

The effect of jumps into distant sites on carrier energy relaxation in a disordered hopping system

An analytic model of charge-carrier kinetics in a disordered hopping system accounting for the possibility of jumps to non-nearest hopping neighbours is formulated. The model is applied to the analysis of charge-carrier energy relaxation at both low and finite temperatures. Carrier jumps to distant hopping neighbours are shown to increase the depth of the carrier energy distribution at both low and finite temperatures, and to cause a shift of the effective transport level to deeper states at finite temperatures.

Kinetics of low-temperature charge-carrier recombination in disordered hopping systems

Energetic relaxation and recombination of charge carriers are considered at low temperatures in disordered hopping systems, taking the increasing occupation of deep localized states into account. It is shown that, in the course of carrier relaxation controlled by monomolecular recombination, the occupational density of localized states does not change with time. If bimolecular recombination represents the dominant mode of the carrier-density relaxation, all deep localized states will sooner or later be occupied and the filling effect plays an important role in the process of relaxation.

Low-temperature relaxations of charge carriers in disordered hopping systems

Energetic relaxation, drift, diffusion and recombination of charge carriers are considered at low temperatures in disordered hopping systems. It is shown that, due to its scaling form, an exponential energetic distribution of the density of localized states (DOS) generates very specific transport characteristics which cannot be obtained for other types of DOS functions. In contrast with other distributions, only an exponential DOS function allows (i) the introduction of a low-temperature analogue of the Einstein relation between carrier mobility and diffusivity for some initial time domain of relaxation and (ii) the description of carrier drift in strong electric fields in terms of a field-dependent effective temperature.

Scaling properties of transport parameters in molecularly doped polymers

Charge-carrier transport in an energetically and spatially disordered hopping system is analysed in terms of dimensionless distance and energy variables. The transport characteristics are shown to be described by three dimensionless parameters which in essence represent the concentration, temperature and electric field respectively. Consequently, a relationship is predicted between these parameters and certain scaling properties of the curves of mobility against field and mobility against temperature. This is supported by mobility measurements carried out on a hole transporting compound doped into polycarbonate within the widest concentration range reported so far in the literature.

Langevin‐Type Charge Carrier Recombination in a Disordered Hopping System

AbstractA Monte Carlo simulation study is performed to determine the ratio γ/μ of the rate coefficient for bimolecular charge recombination and the charge carrier mobility in a hopping system with built‐in energetic disorder. The result indicates that γ/μ is independent of disorder but increases at large electric fields at which the charge carrier velocity saturates.