Exponential Density Of States Research Articles

Carrier mobility in organic semiconductors with an exponential density of states under the framework of admittance spectroscopy

The carrier transport in amorphous organic semiconductors with an exponential density of states is systematically studied under the framework of admittance spectroscopy. Due to the exponential distribution, the slope of the double logarithmic curve of current density versus voltage is expressed as b+2, where b relates to the width of the density of states. And the mobility is proportional to nb, where n is the density of carrier. In this case, the peak frequencies of the negative differential susceptance and imaginary part of impedance turn out to be functions of parameter b. Therefore, the relation between transit time and mobility will be a linear function of parameter b. Applying our model to interpret experimental data of both small molecular and polymeric materials are illustrated. The contributions of electric field and of carrier density to mobility are discussed.

Spectral (in)dependence of nonequilibrium charge carriers lifetime and density of states distribution in the vicinity of the band gap edge in F8BT polymer

Photoelectric properties of the F8BT polymer are investigated by the Constant photocurrent method (CPM) based on the photocurrent spectra Δjph(hν) and absorption spectra αCPM(hν). The negligibly weak dependence of charge carriers lifetime on the photon energy is found for hν = 2.2–2.7 eV. The peculiarities of the absorption spectra and the photoconductivity spectra are revealed to be determined mainly by the Gaussian distribution of the density of states (DOS). The density of trap states in the F8BT polymer thin films in the exponential DOS is estimated to be 103 times higher than in the Gaussian DOS.

A Novel Density of States (DOS) for Disordered Organic Semiconductors.

In this work, we proposed a novel theory of DOS for disordered organic semiconductors based on the frontier orbital theory and probability statistics. The proposed DOS has been verified by comparing with other DOS alternatives and experimental data, and the mobility calculated by the proposed DOS is closer to experimental data than traditional DOS. Moreover, we also provide a detailed method to choose the DOS parameter for better use of the proposed DOS. This paper also contains a prediction for the DOS parameters, and it has been verified by the experimental data. More importantly, the physical meaning of the proposed DOS parameter has been explained by equilibrium energy theory and transport energy theory to make this proposed model more rational. Compared with the improved DOS based on Gaussian and exponential DOS, this work is a new attempt to combine probabilistic theory with physical theory related to DOS in disordered organic semiconductors, showing great significance for the further investigation of the properties of DOS.

Power-Law Density of States in Organic Solar Cells Revealed by the Open-Circuit Voltage Dependence of the Ideality Factor.

The density of states (DOS) is fundamentally important for understanding physical processes in organic disordered semiconductors, yet hard to determine experimentally. We evaluated the DOS by considering recombination via tail states and using the temperature and open-circuit voltage (V_{oc}) dependence of the ideality factor. By performing Suns-V_{oc} measurements, we find that the energetic disorder increases deeper into the band gap, which is not expected for a Gaussian or exponential DOS. The linear dependence of the disorder on energy reveals the power-law DOS in organic solar cells.

A Unified Distribution Form of the Density of States in Disordered Organic Semiconductors

An accurate density of states (DOS) is essential for understanding charge carrier transport properties and then better developing the electronic and optoelectronic devices. Currently, Gaussian and exponential forms are the two most general DOS, which can be unified in the tail state of the state. However, Gaussian DOS at high concentrations is not applicable, as well as exponential DOS at low concentrations. To avoid the disadvantage of Gaussian and exponential form, based on the Poisson flow, a unified distribution form of the DOS have been developed to discuss the charge carrier transport. The proposed DOS is more consistent with the actual situation and exhibits better rationality under the whole range of carrier density. Otherwise, we for the first time introduced the entropy to analyze the disorder parameters. Based on the proposed DOS, the concentration and temperature dependence of mobility has been discussed in detail.

Modeling of Grain Boundary Barrier Height for Undoped Polycrystalline Silicon Channel in Macaroni MOSFETs

In this article, a physically based explicit analytical model for grain boundary (GB) barrier height ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi _{\text {B}}$ </tex-math></inline-formula> ) near the polycrystalline Si (poly-Si) channel/gate oxide interface is proposed for macaroni MOSFETs, the unit cell of 3-D NAND Flash memory. The model is derived by considering a cylindrical coordinate system and is expressed as the Lambert W function and the cosine integral. To verify our model, a computer-aided design (TCAD) simulation tool, including a carrier transport model for poly Si channel, is used and calibrated against experimental data of 3-D NAND string current. The validity of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi _{\text {B}}$ </tex-math></inline-formula> -model at <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${V}_{\text {GS}} &gt; {V}_{\text {FB}}$ </tex-math></inline-formula> is demonstrated by comparing the model with simulation data extracted from calibrated exponential density of states (DOS) distribution for grain boundary traps (GBTs). In the verification stage, simulations are also implemented under various exponential DOS distributions and channel hole radius, and a good agreement between the model and the simulation data is achieved.

Modified charge carrier density for organic semiconductors modeled by an exponential density of states

Charge-transport models are usually developed by first finding an appropriate density of states (DOS) which is extracted from experimental data. For organic materials, two of the more common ones include the Gaussian density of states and the exponential density of states (EDOS). This article will focus on the latter. Charge-transport models which employ an EDOS have been extensively researched, and many articles are still being published. However, from an analytical point of view, only approximate mathematical expressions for the charge carrier density are ever used. This, in general, forces a charge-transport model to be valid only within a limited temperature range. This article illustrates a more mathematically exact way to handle an organic semiconductor whose DOS can be represented by an exponential function.

Diffusion of a particle in the spatially correlated exponential random energy landscape: Transition from normal to anomalous diffusion.

Diffusive transport of a particle in a spatially correlated random energy landscape having exponential density of states has been considered. We exactly calculate the diffusivity in the nondispersive quasi-equilibrium transport regime for the 1D transport model and found that for slow decaying correlation functions the diffusivity becomes singular at some particular temperature higher than the temperature of the transition to the true non-equilibrium dispersive transport regime. It means that the diffusion becomes anomalous and does not follow the usual ∝ t1/2 law. In such situation, the fully developed non-equilibrium regime emerges in two stages: first, at some temperature there is the transition from the normal to anomalous diffusion, and then at lower temperature the average velocity for the infinite medium goes to zero, thus indicating the development of the true dispersive regime. Validity of the Einstein relation is discussed for the situation where the diffusivity does exist. We provide also some arguments in favor of conservation of the major features of the new transition scenario in higher dimensions.

Dispersive and steady-state recombination in organic disordered semiconductors

Charge carrier recombination in organic disordered semiconductors is strongly influenced by the thermalization of charge carriers in the density of states (DOS). Measurements of recombination dynamics, conducted under transient or steady-state conditions, can easily be misinterpreted when a detailed understanding of the interplay of thermalization and recombination is missing. To enable adequate measurement analysis, we solve the multiple-trapping problem for recombining charge carriers and analyze it in the transient and steady excitation paradigm for different DOS distributions. We show that recombination rates measured after pulsed excitation are inherently time dependent since recombination gradually slows down as carriers relax in the DOS. When measuring the recombination order after pulsed excitation, this leads to an apparent high-order recombination at short times. As times goes on, the recombination order approaches an asymptotic value. For the Gaussian and the exponential DOS distributions, this asymptotic value equals the recombination order of the equilibrated system under steady excitation. For a more general DOS distribution, the recombination order can also depend on the carrier density, under both transient and steady-state conditions. We conclude that transient experiments can provide rich information about recombination in and out of equilibrium and the underlying DOS occupation provided that consistent modeling of the system is performed.

Understanding mobility degeneration mechanism in organic thin-film transistors (OTFT)

Mobility degradation at high gate bias is often observed in organic thin film transistors. We propose a mechanism for this confusing phenomenon, based on the percolation theory with the presence of disordered energy landscape with an exponential density of states. Within a simple model we show how the surface states at insulator/organic interface trap a portion of channel carriers, and result in decrease of mobility as well as source/drain current with gate voltage. Depending on the competition between the carrier accumulation and surface trapping effect, two different carrier density dependences of mobility are obtained, in excellent agreement with experiment data.

Extraction of the sub-band gap density of states of Nb doped ZnO thin film transistors using C-V measurements

The sub-band gap density of states (DOS) of Nb doped ZnO thin film transistors were extracted using a multi-frequency capacitance-voltage (C-V) method. The results can be represented by a two-term exponential DOS, representing the tail and deep states. The parameters for the tail and deep states are Ntail=1.6×1019cm−3, Ttail=540K, Ndeep=6.5×1016cm−3 and Tdeep=4058K respectively. Furthermore, the DOS from C-V provides a good fit with current-voltage characteristics, using the multiple trap and release model.

Hopping charge transport in amorphous organic and inorganic materials with spatially correlated random energy landscape

The general properties of the hopping transport of charge carriers in amorphous organic and inorganic materials are discussed. The case where the random energy landscape in the material is strongly spatially correlated is considered. This situation is typical of organic materials with the Gaussian density of states (DOS) and may also be realized in some materials with the exponential DOS. It is demonstrated that the different DOS types can lead to very different functional forms of the mobility field dependence even for the identical correlation function of random energy. Important arguments are provided in favor of the significant contribution of the local orientational order to the total magnitude of energetic disorder in organic materials. A simple but promising model of charge transport in highly anisotropic composites materials is proposed.

Exponential-type density of states with clearly cutting tail for organic semiconductors

It was recently demonstrated that the density of states (DOS) is the key factor to determine charge transport, photoelectric and contact properties in disordered organic semiconductors. However, the density of states in organic semiconductors is unclear at present. Although the Gaussian DOS is the most popular, some works support the exponential DOS or combination of both forms. In this paper, we propose three exponential-type DOS, one has complete exponential tail, and other two cuts tails at some places. The variations of mobility with carrier density are obtained through numerically solving variable range hopping (VRH) equations. It is shown that the relationships of mobility with density and Fermi level are very different among results obtained from Gaussian, un-cutting and cutting exponential-type DOS. The results show that the experimental mobility-density data can be well fitted by using single cutting exponential-type DOS in the wide ranges of density, but cannot be fitted by using single Gaussian and un-cutting exponential-type DOS. Instead of the Gaussian and pure exponential DOS, the DOS with exponential core and clearly cutting tail is recommended.

Understanding hopping transport and thermoelectric properties of conducting polymers

We calculate the conductivity $\ensuremath{\sigma}$ and the Seebeck coefficient $S$ for the phonon-assisted hopping transport in conducting polymers poly(3,4-ethylenedioxythiophene) or PEDOT, experimentally studied by Bubnova et al. [J. Am. Chem. Soc. 134, 16456 (2012)]. We use the Monte Carlo technique as well as the semianalytical approach based on the transport energy concept. We demonstrate that both approaches show a good qualitative agreement for the concentration dependence of $\ensuremath{\sigma}$ and $S$. At the same time, we find that the semianalytical approach is not in a position to describe the temperature dependence of the conductivity. We find that both Gaussian and exponential density of states (DOS) reproduce rather well the experimental data for the concentration dependence of $\ensuremath{\sigma}$ and $S$ giving similar fitting parameters of the theory. The obtained parameters correspond to a hopping model of localized quasiparticles extending over 2--3 monomer units with typical jumps over a distance of 3--4 units. The energetic disorder (broadening of the DOS) is estimated to be 0.1 eV. Using the Monte Carlo calculation we reproduce the activation behavior of the conductivity with the calculated activation energy close to the experimentally observed one. We find that for a low carrier concentration a number of free carriers contributing to the transport deviates strongly from the measured oxidation level. Possible reasons for this behavior are discussed. We also study the effect of the dimensionality on the charge transport by calculating the Seebeck coefficient and the conductivity for the cases of three-, two-, and one-dimensional motion.

New insights on traps states in organic semiconductor applying illumination-free transient current method

A transient current measurement technique which does not need light stimulus is developed. This illumination-free transient current (IFTC) technique offers insights on trapping states characteristics such as the capture time of the trapping states, their density of states (DOS) and the attempt-to-escape frequency of trapped carriers. For the analysis of IFTC measurements carried out on metal–insulator–semiconductor (MIS) capacitors, numerical simulation of the transient current were performed as a way around usual assumptions of short transit times which can considerably distort the values from transient current measurements. A combination of experimental data and simulations revealed the need to take into account retrapping events in order to extract the correct band tail density of states and corresponding characteristic parameters. An exponential density of states with a total density of 4·1023m−3 and a width of 50meV was found to be representative of the band tail of spin-coated poly(3-hexlythiophene) (P3HT). A broader exponential distribution with a width of 83meV is extracted for the interface states towards SiO2. Using a multiple trap and release model, the capturing time and the attempt-to-escape frequency of the band tail states were found to be 10-10s and 108s−1, respectively. With this technique, we are able to also pinpoint the energy position of the Fermi level relative to the transport energy level.

Interfaces analysis by impedance spectroscopy and transient current spectroscopy on semiconducting polymers based metal–insulator–semiconductor capacitors

Impedance and transient current measurements on metal–insulator–semiconductor (MIS) capacitors are used as tools to thoroughly investigate the bulk and interface electronic transport properties of semiconducting polymers, i.e. poly(3-hexylthiophene) (P3HT). Distinct features were observed at both interfaces, i.e. metal–semiconductor and semiconductor–insulator. The results revealed a dispersive transport in the bulk due to the band tail of the localized states, presence of interface states at the interface between the insulator and the semiconductor and formation of a less conductive small layer at the interface semiconductor–metal contact due to intrusions of sputtered Au particles. Effects of self-assembled monolayers (SAMs) treatments of the gate insulating dielectric were investigated showing that treating the gate dielectric with either ozone or hexamethyldisilazane (HMDS) or octyltrichlorosilane (OTS) alter not only the interface semiconductor–insulator but the bulk properties as well. An exponential density of states with a width parameter of 38–58meV depending on the surface treatment was found to be representative of the band tail of P3HT. Though both OTS and HMDS treatments slightly increase the density of interface states, only OTS treated samples showed a decrease in disorder parameter of the bulk. The latter fact can be attributed to an increase of the grain size due to a favored π-π stacking film growth. An outcome explaining the already reported increase of the lateral mobility and decrease of the vertical mobility observed upon OTS treatment of the gate insulating dielectric in poly(3-hexylthiophene) based devices.

Room-temperature bandlike transport and Hall effect in a high-mobility ambipolar polymer

The advent of a new class of high-mobility semiconducting polymers opens up a window to address fundamental issues in electrical transport mechanism such as transport between localized states versus extended state conduction. Here, we investigate the origin of the ultralow degree of disorder $({E}_{a}\ensuremath{\sim}16\phantom{\rule{0.16em}{0ex}}\mathrm{meV})$ and the ``bandlike'' negative temperature ($T$) coefficient of the field effect electron mobility: ${\ensuremath{\mu}}_{\mathrm{FET}}^{e}(T)$ in a high performance $({\ensuremath{\mu}}_{\mathrm{FET}}^{e}&gt;2.5\phantom{\rule{0.16em}{0ex}}\mathrm{c}{\mathrm{m}}^{2}\phantom{\rule{0.28em}{0ex}}{\mathrm{V}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{s}}^{\ensuremath{-}1})$ diketopyrrolopyrrole based semiconducting polymer. Models based on the framework of mobility edge with exponential density of states are invoked to explain the trends in transport. The temperature window over which the system demonstrates delocalized transport was tuned by a systematic introduction of disorder at the transport interface. Additionally, the Hall mobility $({\ensuremath{\mu}}_{\mathrm{Hall}}^{e})$ extracted from Hall voltage measurements in these devices was found to be comparable to field effect mobility $({\ensuremath{\mu}}_{\mathrm{FET}}^{e})$ in the high $T$ bandlike regime. Comprehensive studies with different combinations of dielectrics and semiconductors demonstrate the effectiveness of rationale molecular design, which emphasizes uniform-energetic landscape and low reorganization energy.

Nonlinear response functions in an exponential trap model

The nonlinear response to an oscillating field is calculated for a kinetic trap model with an exponential density of states above the glass transition temperature T0. For temperatures not too close to T0, the results are similar to those obtained for the model with a Gaussian density of states. The choice of the dynamical variable that couples to the field and in particular its dependence on the trap energies generally has a strong impact on the shape of the dynamic response. The modulus of the frequency dependent third-order response either shows a peak or exhibits a monotonous decay from a finite low-frequency limit to a vanishing response at high frequencies depending on the dynamical variable. If a peak is observed, its height can show different temperature dependencies with the common feature of a scaling behavior near T0. Additionally, in some but not all cases the static nonlinear susceptibility diverges at T0. A recently proposed approximation that relates the cubic response to a four-time correlation function does not give reliable results due to a wrong estimate of the low-frequency limit of the response.

Nongeminate recombination in neat P3HT and P3HT:PCBM blend films

The slow decay of charge carriers in polymer–fullerene blends measured in transient studies has raised a number of questions about the mechanisms of nongeminate recombination in these systems. In an attempt to understand this behavior, we have applied a combination of steady-state and transient photoinduced absorption measurements to compare nongeminate recombination behavior in films of neat poly(3-hexyl thiophene) (P3HT) and P3HT blended with [6,6]-phenyl-C61 butyric acid methyl ester (PCBM). Transient measurements show that carrier recombination in the neat P3HT film exhibits second-order decay with a recombination rate coefficient that is similar to that predicted by Langevin theory. In addition, temperature dependent measurements indicate that neat films exhibit recombination behavior consistent with the Gaussian disorder model. In contrast, the P3HT:PCBM blend films are characterized by a strongly reduced recombination rate and an apparent recombination order greater than two. We then assess a number of previously proposed explanations for this behavior including phase separation, carrier concentration dependent mobility, non-encounter limited recombination, and interfacial states. In the end, we propose a model in which pure domains with a Gaussian density of states are separated by a mixed phase with an exponential density of states. We find that such a model can explain both the reduced magnitude of the recombination rate and the high order recombination kinetics and, based on the current state of knowledge, is the most consistent with experimental observations.

PAUSING-TIME DISTRIBUTION OF ANOMALOUS DIFFUSION OF HYDROGEN WITH RANDOM ENERGY

It has been empirically known that the disordered system exhibits a power-law behavior. We show from calculation the power law t-1-α for the pausing-time distribution of thermal diffusion of hydrogen in the exponential density of states. This power law of the pausing-time distribution is examined by the Monte Carlo simulation. The results agree with those obtained by analytical calculation. We discuss the power law in terms of random walk in fractal structure (Cantor ensemble).