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 solution of the puzzle concerning the disorder effect on the mobility temperature dependence has been suggested.
1. Introduction 2. Molecular dynamics simulation of amorphous carbon structures 3. Atomistic simulation of the bombardment process during the BEN phase of diamond CVD 4. Growth of amorphous silicon 5. One-dimensional hopping in disordered organic solids
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 …
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…
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 this article, we present an alternative method for simulating charge transport in disordered organic materials by using a buffer lattice at the boundary. This method does not require careful tracking of carrier's hopping pattern across boundaries. Suitability of this method is established by reproducing the field dependence of mobility, carrier relaxation and carrier diffusion in disordered organic systems obtained by simulating the charge transport for the full length of the systems along the field direction without and boundary condition. The significance of the buffer lattice is emphasiz…
It is still under debate whether the classical Einstein relation in disordered organic semiconductors is valid. We investigated Einstein relation in disordered organic semiconductors theoretically. The results show that, the classic Einstein relation deviate dramatically with disorder and electric field, even in the case of thermal equilibrium.
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 …
A recently introduced percolative theory of unipolar organic magnetoresistance is generalized by treating the hyperfine interaction semiclassically for an arbitrary hopping rate. Compact analytic results for the magnetoresistance are achievable when carrier hopping occurs much more frequently than the hyperfine field precession period. In other regimes, the magnetoresistance can be straightforwardly evaluated numerically. Slow and fast hopping magnetoresistance are found to be uniquely characterized by their lineshapes. We find that the threshold hopping distance is analogous a phenomenologica…
We provide a phenomenological formula which describes the low-frequency optical absorption of charge carriers in disordered systems with localization. This allows to extract, from experimental data on the optical conductivity, the relevant microscopic parameters determining the transport properties, such as the carrier localization length and the elastic and inelastic scattering times. This general formula is tested and applied here to organic semiconductors, where dynamical molecular disorder is known to play a key role in the transport properties. The present treatment captures the basic ide…
We performed direct calculations of carrier hopping rates in strongly disordered conjugated polymers based on the atomic structure of the system, the corresponding electronic states and their coupling to all phonon modes. We found that the dependence of hopping rates on distance and the dependence of the mobility on temperature are significantly different than the ones stemming from the simple Miller-Abrahams model, regardless of the choice of the parameters in the model. A new model that satisfactorily describes the hopping rates in the system and avoids the explicit calculation of electron-p…
A systematic study of the transport energy in disordered organic semiconductors based on variable range hopping theory has been presented here. The temperature, electric field, material disorder and carrier concentration dependent transport energy is extensively discussed. We demonstrate here, transport energy is not a general concept and invalid even in low electric field and concentration regime.
We suggest universal expressions for the rates of charge transition between molecules in organic disordered semiconductors, which differ between electrons and holes. The donor and acceptor molecules (monomers) are represented in terms of their frontier orbitals, with asymptotic tails reflecting the different asymptotic potential wells which bind different charge carriers. This model predicts an asymmetry of the tunnel factors and the interaction with phonon field with respect to electrons and holes, leading to different formulas for the transition rates and the thermal mobility of the differen…
Charge carrier mobility in disordered organic materials is being actively studied, motivated by several applications such as organic light emitting diodes and organic field-effect transistors. It is known that the mobility in disordered organic materials depends on the chemical potential which in turn depends on the carrier concentration. However, the functional dependence of chemical potential on the carrier concentration is not known. In this study, we focus on the chemical potential in organic materials with Gaussian disorder. We identify three cases of non-degenerate, degenerate and satura…
At low temperature using thermodynamics of irreversible processes the general expressions for the temperature dependence of the thermopower in the case of the hopping conductivity for disordered materials are found. The account of influence of impurity levels degeneration on the thermopower is lead. In a view of received results experimental data of the thermopower in amorphous and impurity semiconductors are discussed.
We report exact multielectronic ground-states dependent on electron concentration for quantum mechanical two-dimensional disordered two-band type many body models in the presence of disordered hoppings and disordered repulsive finite Hubbard interactions, in fixed lattice topology considered provided by Bravais lattices. The obtained ground-states loose their eigenfunction character for independent electron approximation, perturbatively are not connected to the non-interacting but disordered case, and describe a localization-delocalization transition driven by the electron concentration, being…
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 …
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…
A model for magnetoresistance in positionally disordered organic materials is presented and solved using percolation theory. The model describes the effects of spin flips on hopping transport by considering the effect of spin dynamics on an effective density of hopping sites. Faster spin-flip transitions open up `spin-blocked' pathways to become viable conduction channels and hence produces magnetoresistance. The magnetoresistance can be found analytically in several regimes, including when the spin-flip time is slower than the hopping time. The ratio of hopping time to the hyperfine precessio…
Two models involving particles moving by ``hopping'' in disordered media are investigated: I) A model glass-forming liquid is investigated by molecular dynamics under (pseudo-) equilibrium conditions. ``Standard'' results such as mean square displacements, intermediate scattering functions, etc. are reported. At low temperatures hopping is present in the system as indicated by a secondary peak in the distribution of particle displacements during a time interval 't'. The dynamics of the model is analyzed in terms of its potential energy landscape (potential energy as function of the 3N particle…
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 ordered regions (micro crystallites and aggregates) in the other wise disordered polymer host matrix on field and temperature dependence of mobility (μ) has been simulated. Increase in concentration of ordered regions leads to increase in magnitude of mobility and in high field regime the saturation of the mobility occurs at lower electric field strength. The influence of different mean and standard deviation of Gaussian density of states (DOS) of ordered regions on the field dependence of mobility was studied and found to be significant only at higher concentrations. Weak inf…
A recent theory of magnetoresistance in positionally disordered organic semiconductors is extended to include exchange and dipolar couplings between polarons. Analytic results are discovered when the hyperfine, exchange, and dipolar interactions have little time to operate between hopping events. We find an angle-of-field dependence of the magnetoresistance that agrees with previous experiments and numerical simulations. In addition we report new magnetoresistive behavior that critically depends upon the amount of anisotropy in the dipolar interaction.
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.
Wetzelaer, Koster, and Blom [PRL 107, 066605] recently observed that the classic Einstein relation $\frac{D}μ=\frac{kT}{q}$ is still valid in disordered semiconductors in thermal(quasi)equilibrium by studying the diffusion-driven currents of single-carrier diodes. This is in contrast to the previous theoretical prediction [2] and experiment observation from Leo et.al. [3] where the Einstein relation increases with 1/T. We will discuss the reason for this discrepancy here and suggest that one-carrier diodes experiment might nor be suitable for verifying Einstein relation in organic semiconducto…
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…
The single-parameter scaling hypothesis predicts the absence of delocalized states for noninteracting quasiparticles in low-dimensional disordered systems. We show analytically and numerically that extended states may occur in the one- and two-dimensional Anderson model with a nonrandom hopping falling off as some power of the distance between sites. The different size scaling of the bare level spacing and the renormalized magnitude of the disorder seen by the quasiparticles finally results in the delocalization of states at one of the band edges of the quasiparticle energy spectrum. The deloc…
We study localization in two- and three channel quasi-1D systems using multichain tight-binding Anderson models with nearest-neighbour interchain hopping. In the three chain case we discuss both the case of free- and that of periodic boundary conditions between the chains. The finite disordered wires are connected to ideal leads and the localization length is defined from the Landauer conductance in terms of the transmission coefficients matrix. The transmission- and reflection amplitudes in properly defined quantum channels are obtained from S-matrices constructed from transfer matrices in Bl…
We develop a theory of magnetoresistance based on variable-range hopping. An exponentially large, low-field and necessarily positive magnetoresistance effect is predicted in the presence of Hubbard interaction and spin-dynamics under certain conditions. The theory was developed with the recently discovered organic magnetoresistance in mind. To account for the experimental observation that the organic magnetoresistance effect can also be negative, we tentatively amend the theory with a mechanism of bipolaron formation.
General properties of charge carrier transport in disordered organic materials are discussed. Spatial correlation between energies of transport sites determines the form of the drift mobility field dependence. Particular kind of spatial correlation in a disordered material depends on its nature. Mobility field dependences have to be different in polar and nonpolar materials. Different methods of mobility calculation from the shape of photocurrent transient are analyzed. A widely used method is very sensitive to the variation of the shape of the transient and sometimes produces results that eff…
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…
In crystalline semiconductors, absorption onset sharpness is characterized by temperature dependent Urbach energies. These energies quantify the static, structural disorder causing localized exponential-tail states, and dynamic disorder from electron-phonon scattering. Applicability of this exponential-tail model to disordered solids has been long debated. Nonetheless, exponential fittings are routinely applied to sub-gap absorption analysis of organic semiconductors. Herein, we elucidate the sub-gap spectral line-shapes of organic semiconductors and their blends by temperature-dependent quant…
We observe a complex change in the hopping exponent value from 1/2 to 1/3 as a function of disorder strength and electron density in a sodium-doped silicon MOSFET. The disorder was varied by applying a gate voltage and thermally drifting the ions to different positions in the oxide. The same gate was then used at low temperature to modify the carrier concentration. Magnetoconductivity measurements are compatible with a change in transport mechanisms when either the disorder or the electron density is modified suggesting a possible transition from a Mott insulator to an Anderson insulator in th…
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…
A theory is introduced for spin relaxation and spin diffusion of hopping carriers in a disordered system. For disorder described by a distribution of waiting times between hops (e.g. from multiple traps, site-energy disorder and/or positional disorder) the dominant spin relaxation mechanisms in organic semiconductors (hyperfine, hopping-induced spin-orbit, and intra-site spin relaxation) each produce different characteristic spin relaxation and spin diffusion dependences on temperature. The resulting unique experimental signatures predicted by the theory for each mechanism in organic semicondu…
Transport properties of a series of poly(3-hexylthiophene) organic field effect transistors with Cr, Cu and Au source/drain electrodes were examined over a broad temperature range. The current-voltage characteristics of the injecting contacts are extracted from the dependence of conductance on channel length. With reasonable parameters, a model of hopping injection into a disordered density of localized states, with emphasis on the primary injection event, agrees well with the field and the temperature dependence of the data over a broad range of temperatures and gate voltages.
We study the thermopower of a disordered nanowire in the field effect transistor configuration. After a first paper devoted to the elastic coherent regime (Bosisio R., Fleury G. and Pichard J.-L. 2014 \textit{New J. Phys.} \textbf{16} 035004), we consider here the inelastic activated regime taking place at higher temperatures. In the case where charge transport is thermally assisted by phonons (Mott Variable Range Hopping regime), we use the Miller-Abrahams random resistor network model as recently adapted by Jiang et al. for thermoelectric transport. This approach previously used to study the…
In disordered systems, the hopping conductivity regime is usually realized at low temperatures where spin-related phenomena differ strongly from the case of delocalized carriers. We develop the unified microscopic theory of current induced spin orientation, spin-galvanic and spin-Hall effects for the two-dimensional hopping regime. We show that the corresponding susceptibilities are proportional to each other and determined by the interplay between the drift and the diffusion spin currents. Estimations are made for realistic semiconductor heterostructures using the percolation theory. We show …
Nonlinear conductivity effects are studied experimentally and theoretically for thin samples of disordered ionic conductors. Following previous work in this field the {\it experimental nonlinear conductivity} of sodium ion conducting glasses is analyzed in terms of apparent hopping distances. Values up to 43 Åare obtained. Due to higher-order harmonic current density detection, any undesired effects arising from Joule heating can be excluded. Additionally, the influence of temperature and sample thickness on the nonlinearity is explored. From the {\it theoretical side} the nonlinear conductivi…
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…
Nodal semimetals (e.g. Dirac, Weyl and nodal-line semimetals, graphene, etc.) and systems of pinned particles with power-law interactions (trapped ultracold ions, nitrogen defects in diamonds, spins in solids, etc.) are presently at the centre of attention of large communities of researchers working in condensed-matter and atomic, molecular and optical physics. Although seemingly unrelated, both classes of systems are abundant with novel fundamental thermodynamic and transport phenomena. In this paper, we demonstrate that low-energy field theories of quasiparticles in semimetals may be mapped …
We numerically investigate the link between the delocalization-localization transition and entanglement in a disordered long-range hopping model of spinless fermions by studying various static and dynamical quantities. This includes the inverse participation ratio, level-statistics, entanglement entropy and number fluctuations in the subsystem along with quench and wave-packet dynamics. Finite systems show delocalized, quasi-localized and localized phases. The delocalized phase shows strong area-law violation whereas the (quasi)localized phase adheres to (for large subsystems) the strict area …
A one-dimensional driven lattice gas with disorder in the particle hopping probabilities is considered. It has previously been shown that in the version of the model with random sequential updating, a phase transition occurs from a low density inhomogeneous phase to a high density congested phase. Here the steady states for both parallel (fully synchronous) updating and ordered sequential updating are solved exactly and the phase transition shown to persist in both cases. For parallel dynamics and forward ordered sequential dynamics the phase transition occurs at the same density but for backw…
Motivated by neutral excitations in disordered electronic materials and systems of trapped ultracold particles with long-range interactions, we study energy-level statistics of quasiparticles with the power-law hopping Hamiltonian $\propto 1/r^α$ in a strong random potential. In solid-state systems such quasiparticles, which are exemplified by neutral dipolar excitations, lead to long-range correlations of local observables and may dominate energy transport. Focussing on the excitations in disordered electronic systems, we compute the energy-level correlation function $R_2(ω)$ in a finite syst…
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 concentrati…
In one-dimensional electronic systems with strong repulsive interactions, charge excitations propagate much faster than spin excitations. Such systems therefore have an intermediate temperature range [termed the "spin-incoherent Luttinger liquid'" (SILL) regime] where charge excitations are "cold" (i.e., have low entropy) whereas spin excitations are "hot." We explore the effects of charge-sector disorder in the SILL regime in the absence of external sources of equilibration. We argue that the disorder localizes all charge-sector excitations; however, spin excitations are protected against ful…
We report measurements of disordered graphene probed by both a high electric field and a high magnetic field. By apply a high source-drain voltage Vsd, we are able to study the current-voltage relation I-Vsd of our device. With increasing Vsd, a crossover from the linear I-Vsd regime to the non-linear one, and eventually to activationless-hopping transport occurs. In the activationless-hopping regime, the importance of Coulomb interactions between charged carriers is demonstrated. Moreover, we show that delocalization of carriers which are strongly localized at low T and at small Vsd occurs wi…
Monte Carlo simulation of the charge carrier transport in disordered nonpolar organic materials has been carried out. As a suitable model we considered the model of quadrupolar glass. A general formula for the temperature and field dependence of the mobility was suggested. A comparison with experimental data has been carried out.
Using Gaussian integral transform techniques borrowed from functional-integral field theory and the replica trick we derive a version of the coherent-potential approximation (CPA) suited for describing ($i$) the diffusive (hopping) motion of classical particles in a random environment and ($ii$) the vibrational properties of materials with spatially fluctuating elastic coefficients in topologically disordered materials. The effective medium in the present version of the CPA is not a lattice but a homogeneous and isotropic medium, representing an amorphous material on a mesoscopic scale. The tr…
We investigate the effect of coupling Anderson localized particles in one dimension to a system of marginally localized phonons having a symmetry protected delocalized mode at zero frequency. This situation is naturally realized for electrons coupled to phonons in a disordered nano-wire as well as for ultra-cold fermions coupled to phonons of a superfluid in a one dimensional disordered trap. To determine if the coupled system can be many-body localized we analyze the phonon-mediated hopping transport for both the weak and strong coupling regimes. We show that the usual variable-range hopping …
We introduce a new real space super cell approximation method for treating the electronic states of disordered systems. This method is general and allows both randomness in the on-site energies and in the hopping integrals. In the special case of randomness in the on-site energies only, this method is equivalent to the Non Local Coherent Potential Approximation (NLCPA) derived previously.
We analyse the phase diagram of ultra-cold bosons in a one-dimensional superlattice potential with disorder using the time evolving block decimation algorithm for infinite sized systems (iTEBD). For degenerate potential energies within the unit cell of the superlattice loophole-shaped insulating phases with non-integer filling emerge with a particle-hole gap proportional to the boson hopping. Adding a small amount of disorder destroys this gap. For not too large disorder the loophole Mott regions detach from the axis of vanishing hopping giving rise to insulating islands. Thus the system shows…
We investigate the effect of diagonal disorder on bosons in an optical lattice described by an Anderson-Hubbard model at zero temperature. It is known that within Gutzwiller mean-field theory spatially resolved calculations suffer particularly from finite system sizes in the disordered case, while arithmetic averaging of the order parameter cannot describe the Bose glass phase for finite hopping $J>0$. Here we present and apply a new \emph{stochastic} mean-field theory which captures localization due to disorder, includes non-trivial dimensional effects beyond the mean-field scaling level and …
We study the one-dimensional tight-binding models which include a slowly varying, incommensurate off-diagonal modulation on the hopping amplitude. Interestingly, we find that the mobility edges can appear only when this off-diagonal (hopping) disorder is included in the model, which is different from the known results induced by the diagonal disorder. We further study the situation where the off-diagonal and diagonal disorder terms (the incommensurate potential) are both included and find that the locations of mobility edges change significantly and the varying trend of the mobility edge becom…
