For hopping transport in disordered materials, the mobility of charge carriers is strongly dependent on temperature and the electric field. Our numerical study shows that both the energy distribution and the mobility of charge carriers in systems with a Gaussian density of states, such as organic disordered semiconductors, can be described by a single parameter - effective temperature, dependent on the magnitude of the electric field. Furthermore, this effective temperature does not depend on the concentration of charge carriers, while the mobility does depend on the charge carrier concentration. The concept of the effective temperature is shown to be valid for systems with and without space-energy correlations in the distribution of localized states.
General properties of the hopping transport of charge carriers in amorphous organic and inorganic materials are discussed. We consider the case where the random energy landscape in the materials is strongly spatially correlated. This is a very typical situation in the organic materials having the Gaussian density of states (DOS) and may be realized in some materials with the exponential DOS. We demonstrate that the different type of DOS leads to a very different functional form of the mobility field dependence even in the case of the identical correlation function of random energy. We provide …
We investigate the effect of a metal plate on the variable range hopping (VRH) conductivity of a two dimensional electron-glass (EG) system. The VRH conductivity is known to have a stretched exponential dependence on temperature, with an exponent $p$ that depends on the shape of the EG's single particle density of states (DOS). For constant DOS $p=1/3$ and for linear DOS $p=1/2$, also known as Mott's and Efros-Shklovskii's VRH respectively. The presence of the plate causes two effects on the EG system, static and dynamic. The well known static effect accounts for the additional screening of th…
Hopping charge transport in amorphous semiconductors having spatially correlated exponential density of states has been considered. Average carrier velocity is exactly calculated for the quasi-equilibrium (nondispersive) transport regime. We suggest also a heuristic approach for the consideration of the carrier velocity for the non-equilibrium dispersive regime.
Qualitatively new transport mechanism is suggested for hopping of carriers according to which the variable-range hopping (VRH) arises from the resonant tunneling between transport states brought into resonance by Coulomb potentials produced by surrounding sites with fluctuating occupations. A semiquantitative description of the hopping transport is given based on the assumption that fluctuations of energies of hopping sites have spectral density 1/f.
We discuss memory effects in the conductance of hopping insulators due to slow rearrangements of many-electron clusters leading to formation of polarons close to the electron hopping sites. An abrupt change in the gate voltage and corresponding shift of the chemical potential change populations of the hopping sites, which then slowly relax due to rearrangements of the clusters. As a result, the density of hopping states becomes time dependent on a scale relevant to rearrangement of the structural defects leading to the excess time dependent conductivity.
We use a mean-field (Hartree-like) approach to study the conductance of a strongly localized electron system in two dimensions. We find a crossover between a regime where Coulomb interactions modify the conductance significantly to a regime where they are negligible. We show that under rather general conditions the conduction obeys a scaling relation which we verify using numerical simulations. The use of a Hartree self-consistent approach gives a clear physical picture, and removes the ambiguity of the use of single-particle tunneling density-of-states (DOS) in the calculation of the conducta…
W. Chr. Germs, K. Guo, R. A. J. Janssen, and M. Kemerink [1] recently measured the temperature and concentration dependent seebeck coefficient in organic thin film transistor and found the seebeck coefficient increases with carrier concentration (corresponding to the gate voltage) in the low temperature regime. They further concluded that this unusual behavior is due to a transition from hopping transport in static localized states to bandlike transport, occurring at low temperatures. This is obviously in contrast to the previous theoretical prediction because it is widely accepted that hoppin…
We reconsider the theory of Hall effect in the systems with hopping conduction. The purpose of the present study is to compare the percolation approach based on the optimal triad model with numerical simulations and recent experimental results. We show that, in the nearest neighbor hopping regime, the results of the percolation theory agree to the simulation. However, in the variable range hopping (VRH) regime, the optimal triad model fails to describe the numerical results. It is related to the extremely small probability to find the optimal triad of sites in the percolation cluster in the VR…
We report some basic results regarding transport in disordered reaction-diffusion systems with birth (A->2A), death (A->0), and binary competition (2A->A) processes. We consider a model in which the growth process is only allowed to take place in certain areas--"oases"--while the rest of space--the "desert"--is hostile to growth. In the limit of low oasis density, transport is mediated through rare "hopping" events, necessitating the inclusion of discreteness effects in the model. By first considering transport between two oases, we are able to derive an approximate expression for the average …
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 …
In the previous paper [Nenashev et al., arXiv:0912.3161] an analytical theory confirmed by numerical simulations has been developed for the field-dependent hopping diffusion coefficient D(F) in one-dimensional systems with Gaussian disorder. The main result of that paper is the linear, non-analytic field dependence of the diffusion coefficient at low electric fields. In the current paper, an analytical theory is developed for the field-dependent diffusion coefficient in three- and two-dimensional Gaussian disordered systems in the hopping transport regime. The theory predicts a smooth paraboli…
We discuss the spin excitations in systems with hopping electron conduction and strong position disorder. We focus on the problem in a strong magnetic field when the spin Hamiltonian can be reduced to the effective single-particle Hamiltonian and treated with conventional numerical technics. It is shown that in a 3D system with Heisenberg exchange interaction the spin excitations have a delocalized part of the spectrum even in the limit of strong disorder, thus leading to the possibility of the coherent spin transport. The spin transport provided by the delocalized excitations can be described…
We derive the kinetic equations for the hopping transport that take into account electron spin and the possibility of double occupation. In the Ohmic regime the equations are reduced to the generalized Miller-Abrahams resistor network. We apply these equations to the problem of the magnetic moment relaxation due to the interaction with the random hyperfine fields. It is shown that in a wide range of parameters the relaxation rate is governed by the hops with the similar rates as spin precession frequency. It is demonstrated that at the large time scale spin relaxation is non-exponential. We ar…
We derive the kinetic equations for polaron hopping in organics that explicitly take into account the double occupation possibility and pair intersite correlations. The equations include simplified phenomenological spin dynamics and provide a self-consistent framework for the description of the bipolaron mechanism of the organic magnetoresistance. At low applied voltages the equations can be reduced to effective resistor network that generalizes the Miller-Abrahams network and includes the effect of spin relaxation on the system resistivity. Our theory discloses the close relationship between …
We derive the system of equations that allows to include non-equilibrium correlations of filling numbers into the theory of the hopping transport. The system includes the correlations of arbitrary order in a universal way and can be cut at any place relevant to a specific problem to achieve the balance between rigor and computation possibilities. In the linear-response approximation, it can be represented as an equivalent electric circuit that generalizes the Miller-Abrahams resistor network. With our approach, we show that non-equilibrium correlations are essential to calculate conductivity a…
An exact analytical theory is developed for calculating the diffusion coefficient of charge carriers in strongly anisotropic disordered solids with one-dimensional hopping transport mode for any dependence of the hopping rates on space and energy. So far such a theory existed only for calculating the carrier mobility. The dependence of the diffusion coefficient on the electric field evidences a linear, non-analytic behavior at low fields for all considered models of disorder. The mobility, on the contrary, demonstrates a parabolic, analytic field dependence for a random-barrier model, being li…
Revisiting charge transport in degenerate hopping systems we present a modification to the drift diffusion equation where instead of employing the generalized Einstein relation we add an energy flux term thus solving several inconsistencies. This leads also to the conclusion that, contrary to common belief, a constant quasi-Fermi level does not necessarily ensure zero electrical current. While we revisit the drift diffusion process in the context of degenerate hopping systems, a considerable part of the argumentation put forward can be applied generally to degenerate systems.
In a heavily doped semiconductor with weak spin-orbital interaction the Dyakonov-Perel spin relaxation rate is known to be proportional to the Drude conductivity. We argue that in the case of weak spin-orbital interaction this proportionality goes beyond the Drude mechanism: it stays valid through the metal-insulator transition and in the range of the exponentially small hopping conductivity.
Variable range hopping transport has been investigated in amorphous YSi from 30 mK to 2 K as a function of the temperature T and electric field F. For F=0, where conduction is along sinuous paths (isotropic percolation), the conductance G depends on T according to an Efros-Shklovskii law. The nonlinear I-V's were studied up to very high fields for which G no longer depends on T (directed percolation : current paths along F). We show that heating effects are negligible. Then, we show that for low F values, ln(G(F,T))=ln(G(0,T))-eFL/kT. The order of magnitude (5-10 nm) and the T dependence of L …
A new method for investigating relaxation phenomena for charge carriers hopping between localized tail states has been developed. It allows us to consider both charge and energy {\it dispersive} transport. The method is based on the idea of quasi-elasticity: the typical energy loss during a hop is much less than all other characteristic energies. We have investigated two models with different density of states energy dependencies with our method. In general, we have found that the motion of a packet in energy space is affected by two competing tendencies. First, there is a packet broadening, i…
The Dirac fermion in the random chiral models is studied which includes the random gauge field model and the random hopping model. We focus on a connection between continuum and lattice models to give a clear perspective for the random chiral models. Two distinct structures of density of states (DoS) around zero energy, one is a power-law dependence on energy in the intermediate energy range and the other is a diverging one at zero energy, are revealed by an extensive numerical study for large systems up to $250\times 250$. For the random hopping model, our finding of the diverging DoS within …
We investigate theoretically the effect of a finite electric field on the resistivity of a disordered one-dimensional system in the variable-range hopping regime. We find that at low fields the transport is inhibited by rare fluctuations in the random distribution of localized states that create high-resistance ``breaks'' in the hopping network. As the field increases, the breaks become less resistive. In strong fields the breaks are overrun and the electron distribution function is driven far from equilibrum. The logarithm of the resistance initially shows a simple exponential drop with the f…
We present a study of the temperature (T) dependence of the dc electrical conductivity of polyaniline pellets doped over a wide range. A crossover between low-T variable-range hopping and high-T nearest-neighbor hopping has been found below room temperature for most of the samples, which is atypical for conducting polymers. This allows us to construct an experimental diagram that accounts for different regimes in the hopping electron transport and closely resembles predictions of the Fogler-Teber-Shklovskii model in three dimensions.
In the conventional theory of hopping transport the positions of localized electronic states are assumed to be fixed, and thermal fluctuations of atoms enter the theory only through the notion of phonons. On the other hand, in 1D and 2D lattices, where fluctuations prevent formation of long-range order, the motion of atoms has the character of the large scale diffusion. In this case the picture of static localized sites may be inadequate. We argue that for a certain range of parameters, hopping of charge carriers among localization sites in a network of 1D chains is a much slower process than …
Variable-range hopping transport along short one-dimensional wires and across the shortest dimension of thin three-dimensional films and narrow two-dimensional ribbons is studied theoretically. Geometric and transport characteristics of the hopping resistor network are shown to depend on temperature T and the dimensionality of the system. In two and three dimensions the usual Mott law applies at high T where the correlation length of the network is smaller than the sample thickness. As T decreases, the network breaks into sparse filamentary paths, while the Mott law changes to a different T-de…
Effect of energetic disorder on charge carrier transport in organic materials has been reexamined. A reliable method for mobility calculation and subsequent evaluation of relevant disorder parameters has been discussed. This method is well suited for a direct calculation of the magnitude of dipolar disorder $σ_\textrm{dip}$ in polar organic materials from the current transients. Calculation of $σ_\textrm{dip}$ for several transport materials with varying concentration of polar dopants gives concentration dependences that are in reasonable agreement with theoretical predictions. A possible solu…
We investigate transport in a disordered reaction-diffusion (RD) model consisting of particles which are allowed to diffuse, compete with one another (2A->A), give birth in small areas called "oases" (A->2A), and die in the "desert" outside the oases (A->0). This model has previously been used to study bacterial populations in the lab and is related to a model of plankton populations in the oceans. We first consider the nature of transport between two oases: in the limit of high growth rate, this is effectively a first passage process, and we are able to determine the first passage time probab…
The transport characteristics of a disordered MoS$_2$ nanoflake in the insulator regime are studied by electrical and magnetotransport measurements. The layered MoS$_2$ nanoflake is exfoliated from a bulk MoS$_2$ crystal and the conductance $G$ and magnetoresistance are measured in a four-probe setup over a wide range of temperatures. At high temperatures, we observe that $\log_{10}G$ exhibits a $-T^{-1}$ temperature dependence and the transport in the nanoflake dominantly arises from thermal activation. At low temperatures, where the transport in the nanoflake dominantly takes place via varia…
Computer simulation of the hopping charge transport in disordered organic materials has been carried out explicitly taking into account charge-charge interactions. This approach provides a possibility to take into account dynamic correlations that are neglected by more traditional approaches like mean field theory. It was found that the effect of interaction is no less significant than the usually considered effect of filling of deep states by non-interacting carriers. It was found too that carrier mobility generally increases with the increase of carrier density, but the effect of interaction…
In hopping magnetoresistance of doped insulators, an applied magnetic field shrinks the electron (hole) s-wave function of a donor or an acceptor and this reduces the overlap between hopping sites resulting in the positive magnetoresistance quadratic in a weak magnetic field, B. We extend the theory of hopping magnetoresistance to states with nonzero orbital momenta. Different from s-states, a weak magnetic field expands the electron (hole) wave functions with positive magnetic quantum numbers, m > 0, and shrinks the states with negative m in a wide region outside the point defect. This togeth…
Charge carriers that execute multi-phonon hopping generally interact strongly enough with phonons to form polarons. A polarons sluggish motion is linked to slowly shifting atomic displacements that severely reduce the intrinsic width of its transport band. Here a means to estimate hopping polarons bandwidths from Seebeck-coefficient measurements is described. The magnitudes of semiconductors Seebeck coefficients are usually quite large (greater than 86 microvolts/K) near room temperature. However, in accord with the third law of thermodynamics, Seebeck coefficients must vanish at absolute zero…
In granular superconductors, individual grains can contain bound Cooper pairs while the system as a whole is strongly insulating. In such cases the conductivity is determined by electron hopping between localized states in individual grains. Here we examine a model of hopping conductivity in such an insulating granular superconductor, where disorder is assumed to be provided by random charges embedded in the insulating gaps between grains. We use computer simulations to calculate the single-electron and electron pair density of states at different values of the superconducting gap $Δ$, and we …
A Flickering Resonance Degeneracy on Entropy-Ruled Charge Transport in the Extended Hopping Sites for Biological Systems: Role of Static-to-Dynamic Disorder Transition
KPR Institute of Engineering and Technology（KPR工程学院与技术学院）
Using analytical arguments and computer simulations we show that the dependence of the hopping carrier mobility on the electric field $μ(F)/μ(0)$ in a system of random sites is determined by the localization length $a$, and not by the concentration of sites $N$. This result is in drastic contrast to what is usually assumed in the literature for a theoretical description of experimental data and for device modeling, where $N^{-1/3}$ is considered as the decisive length scale for $μ(F)$. We show that although the limiting value $μ(F \rightarrow 0)$ is determined by the ratio $N^{-1/3}/a$, the de…
We present a theory of the Anomalous Hall Effect (AHE) in ferromagnetic (Ga,Mn)As in the regime when conduction is due to phonon-assisted hopping of holes between localized states in the impurity band. We show that the microscopic origin of the anomalous Hall conductivity in this system can be attributed to a phase that a hole gains when hopping around closed-loop paths in the presence of spin-orbit interactions and background magnetization of the localized Mn moments. Mapping the problem to a random resistor network, we derive an analytic expression for the macroscopic anomalous Hall conducti…
Charge transport in amorphous semiconductors having spatially correlated exponential density of states (DOS) has been considered for the arbitrary behavior of the correlation function of random energies. Average carrier velocity is exactly calculated for the quasi-equilibrium (nondispersive) transport regime. For the symmetric exponential DOS with exponential tails for low and high energies and non-positive correlation function the temperature for the transition to the dispersive transport regime depends on correlation properties and becomes greater than the traditional estimation based on the…
In many organic molecules the strong coupling of excess charges to vibrational modes leads to the formation of polarons, i.e., a localized state of a charge carrier and a molecular deformation. Incoherent hopping of polarons along the molecule is the dominant mechanism of transport at room temperature. We study the far-from-equilibrium situation where, due to the applied bias, the induced number of charge carriers on the molecule is high enough such that charge correlations become relevant. We develop a diagrammatic theory that exactly accounts for all many-particle correlations functions for …
Far tails of the density of state (DOS) are calculated for the simple models of organic amorphous material, the model of dipolar glass and model of quadrupolar glass. It was found that in both models far tails are non-Gaussian. In the dipolar glass model the DOS is symmetric around zero energy, while for the model of quadrupolar glass the DOS is generally asymmetric and its asymmetry is directly related to the particular geometry of quadrupoles. Far tails of the DOS are relevant for the quasi-equilibrium transport of the charge carriers at low temperature. Asymmetry of DOS in quadrupolar glass…
The charge transport in some organic semiconductors demonstrates nonlinear properties and further universal power-law scaling with both bias and temperature. The physical origin of this behavior is investigated here using variable range hopping theory. The results shows, this universal power-law scaling can be well explained by variable range hopping theory . Relation to the recent experimental data is also discussed.
We have used modern supercomputer facilities to carry out extensive Monte Carlo simulations of 2D hopping (at negligible Coulomb interaction) in conductors with the completely random distribution of localized sites in both space and energy, within a broad range of the applied electric field $E$ and temperature $T$, both within and beyond the variable-range hopping region. The calculated properties include not only dc current and statistics of localized site occupation and hop lengths, but also the current fluctuation spectrum. Within the calculation accuracy, the model does not exhibit $1/f$ n…
The electronic transport in polypyrrole thin films synthesized chemically from the vapor phase is studied as a function of temperature as well as of electric and magnetic fields. We find distinct differences in comparison to the behavior of both polypyrrole films prepared by electrochemical growth as well as of the bulk films obtained from conventional chemical synthesis. For small electric fields F, a transition from Efros-Shklovskii variable range hopping to Arrhenius activated transport is observed at 30 K. High electric fields induce short range hopping. The characteristic hopping distance…
We study electronic transport in Anderson insulators with strong Coulomb interactions in dimensions d>=2. Close to the metal insulator transition where the single particle localization length is much larger than interparticle-distance, the interactions lead to a strongly correlated quantum glass phase. Even though single particle excitations are localized and the system is insulating, there are collective electronic modes which remain delocalized down to parametrically small energies. These collective excitations serve as a continuous bath which can provide the activation energy for variable r…
At low energy scales charge transport in the insulating Si:P is dominated by activated hopping between the localized donor electron states. Thus, theoretical models for a disordered system with electron-electron interaction are appropriate to interpret the electric conductivity spectra. With a newly developed technique we have measured the complex broadband microwave conductivity of Si:P from 100 MHz to 5 GHz in a broad range of phosphorus concentration $n/n_c$ from 0.56 to 0.95 relative to the critical value $n_c=3.5\times 10^{18}$ cm$^{-3}$ corresponding to the metal-insulator transition dri…
General properties of the transport of charge carriers (electrons and holes) in disordered organic materials are discussed. It was demonstrated that the dominant part of the total energetic disorder in organic material is usually provided by the electrostatic disorder, generated by randomly located and oriented dipoles and quadrupoles. For this reason this disorder is strongly spatially correlated. Spatial correlation directly governs the field dependence of the carrier drift mobility. Shape of the current transients, which is of primary importance for a correct determination of the carrier mo…
The low-temperature Hall resistivity ρ_{xy} of La_{2/3}A_{1/3}MnO_3 single crystals (where A stands for Ca, Pb and Ca, or Sr) can be separated into Ordinary and Anomalous contributions, giving rise to Ordinary and Anomalous Hall effects, respectively. However, no such decomposition is possible near the Curie temperature which, in these systems, is close to metal-to-insulator transition. Rather, for all of these compounds and to a good approximation, the ρ_{xy} data at various temperatures and magnetic fields collapse (up to an overall scale), on to a single function of the reduced magnetizatio…
We have extended our supercomputer-enabled Monte Carlo simulations of hopping transport in completely disordered 2D conductors to the case of substantial electron-electron Coulomb interaction. Such interaction may not only suppress the average value of hopping current, but also affect its fluctuations rather substantially. In particular, the spectral density $S_I (f)$ of current fluctuations exhibits, at sufficiently low frequencies, a $1/f$-like increase which approximately follows the Hooge scaling, even at vanishing temperature. At higher $f$, there is a crossover to a broad range of freque…
Despite a large amount of data and numerous theoretical proposals, the microscopic mechanism of transport in thick film resistors remains unclear. However, recent low temperature measurements point toward a possible variable range hopping mechanism of transport. Here we examine how such a mechanism affects the gauge factor of thick film resistors. We find that at sufficiently low temperatures $T$, for which the resistivity follows the Mott's law $R(T)\sim \exp(T_0/T)^{1/4}$, the gauge factor GF is proportional to $(T_0/T)^{1/4}$. Moreover, the inclusion of Coulomb gap effects leads to ${\rm GF…
Randomness and disorder have strong impact on transport processes in quantum systems and give rise to phenomena such as Anderson localization [1-3], many-body localization [4] or glassy dynamics [5]. Their characteristics thereby depend on the strength and type of disorder. An important class are hopping models, where particles or excitations move through a system which has randomized couplings. This includes, e.g., spin glasses [5], coupled optical waveguides [6], or NV center arrays [7]. They are also key to understand excitation transport in molecular and biological systems, such as light h…
