Variable Range Hopping Theory Research Articles

Probing charge carrier transport regimes in BiFeO3 nanoparticles by Raman spectroscopy

We report a finding of two transport regimes in multiferroic BiFeO3 nanoparticles, which correspond to variable range hopping (VRH) mechanisms. Switch between the two regimes coincides with the transition between antiferromagnetic (AF) and paramagnetic (PM) phase. At low temperatures, it was found that temperature dependence of indirectly extracted resistivity is consistent with Efros-Shklovskii mechanism. As with high temperatures, data fit favourably to Mott VRH theory. Estimated resistivity value of ~ 350mΩcm violates the Mott-Ioffe-Regel criterion, corroborating that the transport does not proceed through conduction bands. Conductivity was probed noninvasively via Raman contactless setup and studied as a function of temperature.

Size effect on high temperature variable range hopping in Al+ implanted 4H-SiC

The hole transport properties of heavily doped 4H-SiC (Al) layers with Al implanted concentrations of 3 × 1020 and 5 × 1020 cm−3 and annealed in the temperature range 1950–2100 °C, have been analyzed to determine the main transport mechanisms. This study shows that the temperature dependence of the resistivity (conductivity) may be accounted for by a variable range hopping (VRH) transport into an impurity band. Depending on the concentration of the implanted impurities and the post-implantation annealing treatment, this VRH mechanism persists over different temperature ranges that may extend up to room temperature. In this framework, two different transport regimes are identified, having the characteristic of an isotropic 3D VRH and an anisotropic nearly 2D VRH. The latter conduction mechanism appears to take place in a rather thick layer (about 400 nm) that is too large to induce a confinement effect of the carrier hops. The possibility that an anisotropic transport may be induced by a structural modification of the implanted layer because of a high density of basal plane stacking faults (SF) in the implanted layers is considered. The interpretation of the conduction in the heaviest doped samples in terms of nearly 2D VRH is supported by the results of the transmission electron microscopy (TEM) investigation on one of the 5 × 1020 cm−3 Al implanted samples of this study. In this context, the average separation between basal plane SFs, measured along the c-axis, which is orthogonal to the carrier transport during electrical characterization, appears to be in keeping with the estimated value of the optimal hopping length of the VRH theory. Conversely, no SFs are detected by TEM in a sample with an Al concentration of 1 × 1019 cm−3 where a 3D nearest neighbor hopping (NNH) transport is observed.

Physical origin of nonlinear transport in organic semiconductor at high carrier densities

The charge transport in some organic semiconductors exhibits nonlinear properties and further universal power-law scaling with both bias and temperature. The physical origin of this behavior is investigated here by using variable-range hopping theory. The results indicate that this nonlinear behavior can be well explained by variable-range hopping theory. In the high temperature regime, charge transport is thermally activated. However, for the low temperature case, field-assisted tunneling will be dominant, which is responsible for the nonlinear characteristics observed by experiment. The comparison between the theoretical calculation and recent experimental data is also presented.

Variable-range-hopping exponents 1/2, 2/5 and 1/4 in HCl-doped polyaniline pellets

We study the variable-range-hopping (VRH) charge transport in polyaniline pellets doped with HCl over a wide range. Increase of the doping not only enhances the electrical conductivity σ but also weakens the disorder, which affects the VRH mechanism. This is revealed in the σ vs temperature ( T) characteristics σ ∝ exp ⁡ [ − ( T 0 / T ) α ] , where α and T 0 carry information on the underlying physics. Below ∼ 200 K, α undergoes a significant disorder dependent change at ∼ 60% of the full doping: α = 1 / 2 for lower doping, whereas more-doped samples exhibit α = 2 / 5 . From ∼ 200 K up to room temperature, most of the samples (i.e., those doped below 80%) exhibit α = 1 / 4 regardless of the corresponding α at low T. These results can be to a large extent explained by the VRH theory of Fogler et al. [M.M. Fogler, S. Teber, B.I. Shklovskii, Phys. Rev. B 69 (2004) 035413] (FTS). In particular, α = 1 / 2 , 2 / 5 , 1 / 4 are therein anticipated for three-dimensional transport together with crossovers between these values in response to changes in disorder and/or T. The doping (disorder) dependence of T 0 for a given α = 1 / 2 or 2 / 5 also follows the FTS predictions reasonably well. The appearance of α = 1 / 4 at high T can be understood from an analysis of energy scales that are relevant to VRH as described by the FTS model.

Variable range hopping conduction and percolation networks in the pellets formed from pristine and boron-doped carbon nanohorn particles

Resistivity of pelletized carbon nanohorn particles (CNHs) decreased immediately as the pressure for making the pellet increased. This behavior can be fitted by a power scaling relation with quasi-2-dimensional percolating conducting channels. Similarly, the resistivity of a pelletized boron-doped CNHs (B-CNHs) also decreased rapidly as a function of boron concentration in the region of low boron contents (<∼0.5%). On the other hand, the characteristic temperature for variable range hopping (VRH) conduction indicated different behavior, showing almost constant values for pristine CNHs and drastic changes for B-CNHs. By analyzing these experimental results using VRH theory it was found that the resistivity drop for B-CNHs can be explained by the increase of the electronic density of states at the Fermi level of nanohorns due to substitutional doping of boron in the sp2 bonded carbon network which is unlike the percolation for pristine CNHs.

DC/AC unified OTFT compact modeling and circuit design for RFID applications

Organic materials and devices are gaining more and more attention in microelectronics. They are dedicated to low cost applications and easy in fabrications. Device structure and organic material characteristics are key points to achieve targeted products. Considering technology and product specifications, compact organic device modeling and circuit design are necessary to evaluate technology performance in RFID tags. This model is based on variable range hopping theory (VRH), i.e., a carrier may either hop over a small distance with a high activation energy or hop over a long distance with a low activation energy. The model has been developed using a physical basis where the model parameters can easily be extracted and it improves convergence in circuit simulations. It is also suitable for computer aided design (CAD) applications.

Understanding Mott’s law from scaling of variable-range-hopping currents and intrinsic current fluctuations

We have used the master equation to simulate variable-range hopping (VRH) of charges in a strongly disordered $d$-dimensional energy landscape $(d=1,2,3)$. The current distribution over hopping distances and hopping energies gives a clear insight into the difference between hops that occur most frequently, dominate quantitatively in the integral over the mobility distribution, or are critical ones that still need to be considered in that integral to recover the full low-temperature mobility. The recently reported scaling with temperature of the VRH-current distribution over hopping distances and hopping energies is quantitatively analyzed in 1D and 2D, and accurately confirmed. Based on this, we present an analytical scaling theory of VRH, which distinguishes between a scaling part of the distribution and an exponential tail, separated by critical currents that set the scale and that follow self-consistently at each temperature. This naturally renders Mott's law for the low-temperature mobility, in a way and with a physical picture different from that of the established critical-percolation-network approach to VRH. We argue that current fluctuations as observed in simulations are intrinsic to VRH and play an essential role in this distinction.

Non-Fermi liquid theory of quantum Hall effects

Within the Grassmannian U(2N)/U(N) × U(N) nonlinear σ-model representation of localization, one can study the low-energy dynamics of both a free and interacting electron gas. We study the crossover between these two fundamentally different physical problems. We show how the topological arguments for the exact quantization of the Hall conductance are extended to include the Coulomb interaction problem. We discuss dynamical scaling and make contact with the theory of variable range hopping.

Charge transport in polyaniline doped with 3‐nitro‐1,2,4‐triazol‐5(4H)‐one

AbstractPolyaniline (PANI) base was protonated in aqueous solutions of an organic acid, 3‐nitro‐1,2,4‐triazol‐5(4H)‐one (NTO). The temperature dependence of DC conductivity of PANI‐NTO seems to correspond to the theory of variable range hopping (VRH) in three dimensions. The frequency dependence of AC conductivity also reflects the hopping nature of mobile charges. The activation energy for the polymers with protonation degree above 0.12 remains constant with increasing dopant concentration and DC conductivity. The value of this constant may correspond to the energy needed for the ionization of dopant counterion. The fit of the electric relaxation function to the stretched exponential function ϕ(t) = exp[−(t/τ)β] gives the stretch parameter β about 0.35, which shows that the distribution of relaxation times is broad and indicates a high inhomogeneity in the distribution of a dopant.

Hopping conductivity in the quantum Hall effect: revival of universal scaling.

We have measured the temperature dependence of the conductivity sigma(xx) of a two-dimensional electron system deep into the localized regime of the quantum Hall plateau transition. Using variable-range hopping theory we extract directly the localization length xi from this experiment. We use our results to study the scaling behavior of xi as a function of the filling factor distance /deltanu/ to the critical point of the transition. We find for all samples a power-law behavior xi equivalent to /deltanu/(-gamma) in agreement with the theoretically proposed universal exponent gamma = 2.35.

Variable-range hopping in the critical regime

Mott's theory of variable range hopping (VRH) is extended to the critical regime approaching the metal-insulator transition where the hopping energy ${\ensuremath{\Delta}}_{h}(T)l\mathrm{kT}.$ The Miller-Abrahams impedance between a pair of sites is minimized for an arbitrary value of ${\ensuremath{\Delta}}_{h}/kT.$ The theory features spatial dispersion of the dielectric response function $\ensuremath{\epsilon}(R,n),$ which introduces a new length scale ${r}_{s}$ into the Miller-Abrahams resonance energy. In the regime ${\ensuremath{\Delta}}_{h}\ensuremath{\ll}\mathrm{kT}$ the Mott exponent 1/4 changes to 2/7 and the numerical factor [18.1] in the Mott ${T}_{0}$ is reduced to 1.51. Efros-Shklovskii VRH is also modified by the inclusion of $\ensuremath{\epsilon}(R,n)$ and the ${T}_{0}^{\ensuremath{'}}$ depends critically on the spatial dispersion of $\ensuremath{\epsilon}(R,n)$ and this feature allows an explanation of the Ge:Ga data of Watanabe et al., which exhibits ${T}_{0}^{\ensuremath{'}}\ensuremath{\propto}(1\ensuremath{-}{n/n}_{c}{)}^{1.0}.$ The new result for the Mott case allows the Si:As and Si:P results to be explained over a range of ${10}^{6}$ in Mott ${T}_{0}$ values encompassing the crossover from high-temperature Mott VRH to conventional low-temperature Mott VRH. The reduced density dependence of the Mott VRH prefactor ${\ensuremath{\sigma}}_{0}(n)$ is also satisfactorily explained. In the critical regime for $1\ensuremath{-}{n/n}_{c}l0.05,$ Si:As and Si:P yield the localization length exponent $\ensuremath{\nu}\ensuremath{\sim}1.$

Transport of heavily boron-doped synthetic semiconductor diamond in the hopping regime

We report electrical transport measurements of synthetic diamonds doped with boron about 100 ppm. The resistivity has been measured in the temperature range 20--300 K by the van der Pauw method. We have also investigated infrared absorption coefficient at room temperature. Furthermore, we have studied the effect of ${\mathrm{P}}^{+9}$ ion irradiations with 150 MeV which introduce donor defects and also the influence of the annealing after the irradiation. The key features of the present measurement are as follows: (i) The observed temperature dependence of resistivity shows characteristic features of the variable-range-hopping (VRH). A crossover from ${T}^{\ensuremath{-}1/4}$ behavior of Mott VRH to the ${T}^{\ensuremath{-}1/2}$ form of Efros VRH is found to occur at 100 K. (ii) Below 50 K we have observed the hard gap ${T}^{\ensuremath{-}1}$ form. The width of the hard gap is about 10 meV. (iii) The experimental results are quantitatively explained by the theory of VRH. (iv) The observed width of the optical hard gap is equal to that of the hard gap obtained from resistivity. (v) The annealing effects appear only in the hard gap region and do not appear in the variable-range-hopping region. This result leads to the speculation that the hard gap is weakened by the lattice defects.

Low-field hopping among randomly-distributed sites with uncorrelated energetic disorder

The low-field mobility μ of a small concentration of charge carriers hopping among a random distribution of transport sites is studied, as a function of the mean interparticle spacing ρ and the temperature T, for model systems having different site-energy distribution functions. For a uniform density of states our calculations show that the mobility obeys empirical scaling laws similar to those found in the theory of variable-range hopping. For a binary distribution of site energies we observe a crossover as a function of site density between trap-limited conduction and trap-mediated conduction. For a Gaussian density of states our results confirm the quadratic inverse temperature dependence of lnμ found in Monte Carlo studies, although quantitative characterization of this dependence is found to depend sensitively on the degree of spatial disorder in ways that could impact the extraction of microscopic parameters from experimental data.

Quantum-conductance-fluctuation correlation range and amplitude as a parameter-independent probe of the localized-to-extended-state transition in narrow Si MOSFET's.

As a function of gate voltage and device conductance, we observe a dramatic one order of magnitude jump in the correlation range for quantum conductance fluctuations in narrow Si metal-oxide-semiconductor field-effect transistors at low temperatures. This increase in the gate-voltage range over which the fluctuations are correlated is observed to occur near the conductance value where the inelastic diffusion length equals the localization length---the condition at which Thouless predicts a change in the character from localized to extended states. We analyze both the correlation length and the amplitude of the fluctuations to provide a parameter-independent investigation of this transition. We find behavior in reasonable agreement with the theory of variable-range hopping on the localized side and diffusive-regime quantum conductance fluctuations on the extended side.

Transport properties of FeCl3-doped polypyrroles at different dopant concentrations

The electrical conductivities of FeCl3-doped polypyrrole samples with different dopant concentrations have been measured in the temperature range 25-300 K. Thermoelectric power measurements have been performed on two polypyrrole samples at low doping levels between 50 and 300 K. The variation in DC conductivity can be explained by Mott's variable-range hopping (VRH) model. Fitting with Mott's VRH theory, the conductivity data provide the values of the density N( epsilon F) of states at the Fermi level and the hopping distance R. The low value and the sublinear character of the thermoelectric power is consistent with the VRH theory.

Electrical properties and stoichiometry in La 2NiO 4

The electrical properties of La 2NiO 4 have been studied with respect to the stoichiometry of the material. The conductivity of reduced compositions has been measured between 20 and 1000 K and compared to that of the air-prepared ones. Nonstoichiometry is present in both cases and produces disorder leading to Anderson localization and to the definition of a mobility edge in the σ x 2− y 2 itinerant band. Air-prepared compounds contain in addition a large number of Ni 3+ states which may overlap the itinerant σ x 2− y 2 band. For reduced materials containing small amounts of Ni 3+, the electrical properties can be described below 200 K by a hopping conductivity at the Fermi level within a sharply peaked density of states. The results are well described within the frame of the Mott theory of variable range hopping. Above 200 K highly reduced materials exhibit direct excitation of holes from Ni 3+ states to the mobility edge in the itinerant band. Under conditions appropriate to air-prepared materials, the Fermi level is shifted toward the itinerant band and a major contribution to the conductivity arises from hopping at the Fermi level. At high temperature a progressive excitation of carriers from the localized states is anticipated with an eventual exhaustion region. This last assumption is corroborated with a shift of the conductivity maximum to higher temperature for increasingly reduced materials.