Validity of the Einstein relation in disordered organic semiconductors

Quantum acoustics unravels Planckian resistivity

Strange metals exhibit universal linear-in-temperature resistivity described by a Planckian scattering rate, the origin of which remains elusive. By employing an approach inspired by quantum optics, we arrive at the coherent state representation of lattice vibrations: quantum acoustics. Utilizing this nonperturbative framework, we demonstrate that lattice vibrations could serve as active drivers in the Planckian resistivity phenomenon, challenging prevailing theories. By treating charge carriers as quantum wave packets negotiating the dynamic acoustic field, we find that a competition ensues between localization and delocalization giving rise to the previously conjectured universal quantum bound of diffusion, [Formula: see text], independent of temperature or any other material parameters. This leads to the enigmatic T-linear resistivity over hundreds of degrees, except at very low temperatures. Quantum diffusion also explains why strange metals have much higher electrical resistivity than typical metals. Our work elucidates the critical role of phonons in Planckian resistivity from a unique perspective and reconsiders their significance in the transport properties of strange metals.

Unified Entropy-Ruled Einstein's Relation for Bulk and Low-Dimensional Molecular-Material Systems: A Hopping-to-Band Shift Paradigm

We present a unified paradigm on entropy-ruled Einstein's diffusion-mobility relation (μ/D ratio) for 1D, 2D, and 3D free-electron solid state systems. The localization transport in the extended molecules is well approximated by the continuum time-delayed hopping factor within our unified entropy-ruled transport method of noninteracting quantum systems. Moreover, we generalize an entropy-dependent diffusion relation for 1D, 2D, and 3D systems as defined by D d , h e f f = D d , h e f f = 0 ⁡ exp ( ( d - 1 ) h e f f d + 2 ) , where heff and d are the effective entropy and dimension (d = 1, 2, 3), respectively. This generalized relation is valid for both equilibrium and nonequilibrium transport systems since the parameter heff is closely connected with the nonequilibrium fluctuation theorem-based entropy production rule. Importantly, we herein revisit the Boltzmann approach using an entropy-ruled method for mobility calculation for the universal quantum materials that is expressed as μ d = [ ( d d + 2 ) q d h e f f d η ] v F 2 τ 2 , where v F 2 τ 2 is the diffusion constant for band transport systems and η is the chemical potential. According to our entropy-ruled μ/D relation, the Navamani-Shockley diode equation is transformed.

Enhanced Exciton Drift Transport through Suppressed Diffusion in One-Dimensional Guides

Drift-diffusion dynamics is investigated in a one-dimensional (1D) exciton guide at room temperature. Spatial engineering of the exciton energy in a WSe2 monolayer is achieved using local strain to confine and direct exciton transport. An unexpected and massive deviation from the Einstein relation is observed and correlated to exciton capture by defects. We find that the capture reduces exciton temperature and diffusion so much that drift transport visibility improves to 38% as excitons traverse asymmetrically over regions with occupied defect states. Based on measurements over multiple potential gradients, we estimate the exciton mobility to be 169 ± 39 cm2/(eV s) at room temperature. Experiments at elevated exciton densities reveal that the exciton drift velocity monotonically increases with exciton density, unlike exciton mobility, due to contributions from nonequilibrium many-body effects.

Quantum-Classical Transition Analogy of the Diffusion-Mobility Relation for Hopping and Band Transport Systems: Molecules to Materials

We propose a quantum-classical transition analogy for Einstein's diffusion-mobility (D/μ) relation to reveal electron-hole dynamics in both the degenerate and nondegenerate molecular and material systems. Here, one-to-one variation between differential entropy and chemical potential (Δη/Δh_s) is the proposed analogy, which unifies quantum and classical transport. The degeneracy stabilization energy on D/μ decides whether the transport is quantum or classical; accordingly, the transformation occurs in the Navamani-Shockley diode equation.

Direct Tracking of Charge Carrier Drift and Extraction from Perovskite Solar Cells by Means of Transient Electroabsorption Spectroscopy

The best perovskite solar cells currently demonstrate more than 25% efficiencies, yet many fundamental processes that determine the operation of these devices are still not fully understood. In particular, even though the device performance strongly depends on charge carrier transport across the perovskite layer to selective electrodes, information about this process is still very controversial. Here, we investigate charge carrier motion and extraction from an archetypical CH3NH3PbI3 (MAPI) perovskite solar cell. We use the ultrafast electric-field-modulated transient absorption technique, which allows us to evaluate the electric field dynamics from the time-resolved electroabsorption spectra and to visualize the motion of charge carriers with subpicosecond time resolution. We demonstrate that photogenerated holes drift across the mesoporous TiO2/perovskite layer during hundreds of picoseconds. On the other hand, their extraction into the spiro-OMeTAD hole transporting layer lasts for more than 1 nanosecond, suggesting that the hole extraction is limited by the perovskite/spiro-OMeTAD interface rather than by the hole transport through the perovskite layer. Additionally, we use the ultrafast time-resolved fluorescence technique that reveals fluorescence decay during tens of picoseconds, which we attribute to the spatial separation of electrons and holes.

Diffusion Coefficient of a Brownian Particle in Equilibrium and Nonequilibrium: Einstein Model and Beyond

The diffusion of small particles is omnipresent in many processes occurring in nature. As such, it is widely studied and exerted in almost all branches of sciences. It constitutes such a broad and often rather complex subject of exploration that we opt here to narrow our survey to the case of the diffusion coefficient for a Brownian particle that can be modeled in the framework of Langevin dynamics. Our main focus centers on the temperature dependence of the diffusion coefficient for several fundamental models of diverse physical systems. Starting out with diffusion in equilibrium for which the Einstein theory holds, we consider a number of physical situations outside of free Brownian motion and end by surveying nonequilibrium diffusion for a time-periodically driven Brownian particle dwelling randomly in a periodic potential. For this latter situation the diffusion coefficient exhibits an intriguingly non-monotonic dependence on temperature.

Injection-limited and space charge-limited currents in organic semiconductor devices with nanopatterned metal electrodes

Charge injection at metal-organic interfaces often limits the electric current in organic light-emitting diodes without additional injection layers. Integrated nanopatterned electrodes may provide a way to overcome this current injection limit by local field enhancements leading to locally space charge-limited currents. We compare electrical characteristics of planar and nanopatterned hole-only devices based on the charge transport material NPB with different thicknesses in order to investigate the nanopattern's effect on the current limitation mechanism. Integration of a periodic nanograting into the metal electrode yields a current increase of about 1.5-4 times, depending on thickness and operating voltage. To verify the experimental results, we implement a finite element simulation model that solves the coupled Poisson and drift-diffusion equations in a weak form. It includes space charges, drift and diffusion currents, nonlinear mobility, and charge injection at the boundaries. We find in experiment and simulation that the planar devices exhibit injection-limited currents, whereas the currents in the nanopatterned devices are dominated by space charge effects, overcoming the planar injection limit. The simulations show space charge accumulations at the corners of the nanopattern, confirming the idea of locally space charge-limited currents.

Generalization on Entropy-Ruled Charge and Energy Transport for Organic Solids and Biomolecular Aggregates

Herein, a generalized version of the entropy-ruled charge and energy transport mechanism for organic solids and biomolecular aggregates is presented. The effects of thermal disorder and electric field on electronic transport in molecular solids have been quantified by entropy, which eventually varies with respect to the typical disorder (static or dynamic). Based on our previous differential entropy (h _s )-driven charge transport method, we explore the nonsteady carrier energy flux principle for soft matter systems from small organic solids to macrobiomolecular aggregates. Through this principle, the synergic nature of charge and energy transport in different organic systems is addressed. In this work, entropy is the key parameter to classify whether the carrier dynamics is in a nonsteady or steady state. Besides that, we also propose the formulation for unifying the hopping and band transport, which provides the relaxation time-hopping rate relation and the relaxation time-effective mass ratio. The calculated disorder drift time (or entropy-weighted carrier drift time) for hole transport in an alkyl-substituted triphenylamine (TPA) molecular device is 9.3 × 10^-7 s, which illustrates nuclear dynamics-coupled charge transfer kinetics. The existence of nonequilibrium transport is anticipated while the carrier dynamics is in the nonsteady state, which is further examined from the rate of traversing potential in octupolar molecules. Our entropy-ruled Einstein model connects the adiabatic band and nonadiabatic hopping transport mechanisms. The logarithmic current density at different electric field-assisted site energy differences provides information about the typical transport (whether trap-free diffusion or trap-assisted recombination) in molecular devices, which reflects in the Navamani-Shockley diode equation.

Enhanced Spin Transport of Conjugated Polymer in the Semiconductor/Insulating Polymer Blend

Conjugated polymers are of high potential in the development of spintronic devices. In this paper, we report systematic studies on spin transport properties of a semiconducting polymer PBDTTT-C-T in the permalloy/polymer/Pt trilayer using the spin pumping method. Pure spin current with long spin relaxation time is observed via the inverse spin Hall effect (ISHE) measurements. Furthermore, spin current is also found to propagate through the blend film consisting of a small amount of PBDTTT-C-T in an insulating matrix. The polymer blend exhibits a remarkably enhanced spin relaxation length (56 nm) and carrier mobility compared to pristine PBDTTT-C-T. From film microstructural characterizations, we propose that the enhanced spin/carrier transport properties are attributed to the formation of interlinked nanonetwork comprising of the PBDTTT-C-T chain bundles in the inert matrix to afford efficient intrachain charge conduction pathway. Temperature- dependent ISHE measurements support the spin-orbit coupling dominated spin relaxation mechanism.

Effect of site energy fluctuation on charge transport in disordered organic molecules

Effect of dynamics of site energy disorder on charge transport in organic molecular semiconductors is not yet well-established. In order to study the relationship between the dynamics of site energy disorder and charge transport, we have performed a multiscale study on dialkyl substituted thienothiophene capped benzobisthiazole (BDHTT-BBT) and methyl-substituted dicyanovinyl-capped quinquethiophene (DCV5T-Me) molecular solids. In this study, we explore the structural dynamics and correlated charge transport by electronic structure calculations, molecular dynamics, and kinetic Monte-Carlo simulations. We have also proposed the differential entropy dependent diffusion and charge density equations to study the electric field drifted diffusion property and carrier density. In this investigation, we have addressed the transformation mechanism from dynamic to static disorder in the extended stacked molecular units. Here, the decrease in the charge transfer rate due to site energy fluctuations reveals the dispersion transport along the extended π-stacked molecules. Furthermore, the calculated current density for a different set of site energy difference values shows the validity and the limitations of the Einstein relation. Based on the calculated ideality factor, we have classified the charge transport in these molecules as either the Langevin or the Shockley-Read-Hall type mechanism. Through the calculated mobility, current density, and ideality factor analysis, we categorize the applicability of molecules of interest for photovoltaic or light emitting diode applications.

Unexpected Outstanding Room Temperature Spin Transport Verified in Organic-Inorganic Hybrid Perovskite Film

Fundamental understanding on the spin transport properties of semiconducting organic-inorganic hybrid perovskites (OIHP) is of great importance for advancing their applications for spin-optoelectronic devices. Herein, the study of spin-pumping induced inverse spin Hall effect in Ni₈₀Fe₂₀(Py)/CH₃NH₃PbCl_3-xI_x/Pt trilayers with different OIHP spacer thicknesses concludes the spin diffusion length in CH₃NH₃PbCl_3-xI_x as large as 61 ± 7 nm at room temperature. In addition, spin-valves with a structure of Ni₈₀Fe₂₀(Py)/CH₃NH₃PbCl_3-xI_x/AlO_x/Co was fabricated as well. Using a ∼160 nm-thick CH₃NH₃PbCl_3-xI_x spacer, the present spin valve exhibits a positive magnetoresistance (MR) of 0.57% at 10 K. Thus, the present spin related results demonstrate that electrical spin-polarized carrier injection, transport, and detection, which are essential in spintronic devices, can be successfully established in OIHP films. The outstanding spin transport in the present CH₃NH₃PbCl_3-xI_x should be owing to its highly out-of-plane oriented crystalline texture and Rashba spin splitting at domain boundaries between crystallographic orientations. These results demonstrate OIHP as very attractive materials for spintronics.

Higher order effects in organic LEDs with sub-bandgap turn-on

Spin-dependent nonlinear processes in organic materials such as singlet-fission and triplet-triplet annihilation could increase the performance for photovoltaics, detectors, and light emitting diodes. Rubrene/C60 light emitting diodes exhibit a distinct low voltage (half-bandgap) threshold for emission. Two origins for the low voltage turn-on have been proposed: (i) Auger assisted energy up-conversion, and (ii) triplet-triplet annihilation. We test these proposals by systematically altering the rubrene/C60 interface kinetics by introducing thin interlayers. Quantitative analysis of the unmodified rubrene/C60 device suggests that higher order processes can be ruled out as the origin of the sub-bandgap turn-on. Rather, band-to-band recombination is the most likely radiative recombination process. However, insertion of a bathocuproine layer yields a 3-fold increase in luminance compared to the unmodified device. This indicates that suppression of parasitic interface processes by judicious modification of the interface allows a triplet-triplet annihilation channel to be observed.

Diffusion of a particle in the Gaussian random-energy landscape: Einstein relation and analytical properties of average velocity and diffusivity as functions of driving force

We demonstrate that the Einstein relation for the diffusion of a particle in the random-energy landscape with the Gaussian density of states is an exclusive one-dimensional property and does not hold in higher dimensions. We also consider the analytical properties of the particle velocity and diffusivity for the limit of weak driving force and establish a connection between these properties and dimensionality and spatial correlation of the random-energy landscape.

Theoretical modeling of charge transport in triphenylamine–benzimidazole based organic solids for their application as host-materials in phosphorescent OLEDs

The dynamic disorder and electric field effects on charge transport in triphenylamine–benzimidazole based molecular solids have been investigated using electronic structure calculations, molecular dynamics and Monte-Carlo simulations. During the charge propagation, the energy loss of the carrier in each hopping step is monitored by Monte-Carlo simulation. We derive a survival probability correlated momentum–energy distribution for drift-diffusion analysis and we demonstrate the dispersion initiated charge trapping mechanism which is indeed ideal for light emission efficiency via recombination. In the present model, the proposed carrier drift energy–current density expression and Shockley diode current density equation are used to study the current density–voltage characteristics; accordingly the ideality factor (∼1.8–2.0) dictates the deviation of Einstein's classical diffusion–mobility relation (where the ideality factor is unity). The dual mechanism of electric field assisted site energy gap on coherent-like transport and the electric field stretched dispersion on recombination are observed in tris(3′-(1-phenyl-1H-benzimidazole-2-yl)biphenyl-4-yl)amine (TBBI) and tris(4′-(1-phenyl-1H-benzimidazole-2-yl)biphenyl-4-yl)amine (TIBN) molecular systems, which can be used as host materials in organic light emitting diodes (OLEDs). We find the transport going from coherent to incoherent, due to the conversion mechanism of dynamic to static disorder. This can also be a controlled by applied electric field. By adjusting the applied electric field, film thickness and changing the π-stacked molecular aggregation via substitutions, one can fix the dispersive parameter and accordingly calculate the charge transport properties to design efficient host-materials for photovoltaic and light emitting diode devices.

Dynamic disorder and electric field affect the charge (hole and electron) transport in host-materials for OLEDs.

Mobility dependent recombination models for organic solar cells

Modern solar cell technologies are driven by the effort to enhance power conversion efficiencies. A main mechanism limiting power conversion efficiencies is charge carrier recombination which is a direct function of the encounter probability of both recombination partners. In inorganic solar cells with rather high charge carrier mobilities, charge carrier recombination is often dominated by energetic states which subsequently trap both recombination partners for recombination. Free charge carriers move fast enough for Coulomb attraction to be irrelevant for the encounter probability. Thus, charge carrier recombination is independent of charge carrier mobilities. In organic semiconductors charge carrier mobilities are much lower. Therefore, electrons and holes have more time react to mutual Coulomb-forces. This results in the strong charge carrier mobility dependencies of the observed charge carrier recombination rates. In 1903 Paul Langevin published a fundamental model to describe the recombination of ions in gas-phase or aqueous solutions, known today as Langevin recombination. During the last decades this model was used to interpret and model recombination in organic semiconductors. However, certain experiments especially with bulk-heterojunction solar cells reveal much lower recombination rates than predicted by Langevin. In search of an explanation, many material and device properties such as morphology and energetic properties have been examined in order to extend the validity of the Langevin model. A key argument for most of these extended models is, that electron and hole must find each other at a mutual spatial location. This encounter may be limited for instance by trapping of charges in trap states, by selective electrodes separating electrons and holes, or simply by the morphology of the involved semiconductors, making it impossible for electrons and holes to recombine at high rates. In this review, we discuss the development of mobility limited recombination models from the early Langevin theory to state-of-the art models for charge carrier recombination in organic solar cells.

Simulating charge transport in organic semiconductors and devices: a review

Charge transport simulation can be a valuable tool to better understand, optimise and design organic transistors (OTFTs), photovoltaics (OPVs), and light-emitting diodes (OLEDs). This review presents an overview of common charge transport and device models; namely drift-diffusion, master equation, mesoscale kinetic Monte Carlo and quantum chemical Monte Carlo, and a discussion of the relative merits of each. This is followed by a review of the application of these models as applied to charge transport in organic semiconductors and devices, highlighting in particular the insights made possible by modelling. The review concludes with an outlook for charge transport modelling in organic electronics.

Charge carrier thermalization in organic diodes

Charge carrier mobilities of organic semiconductors are often characterized using steady-state measurements of space charge limited diodes. These measurements assume that charge carriers are in a steady-state equilibrium. In reality, however, energetically hot carriers are introduces by photo-excitation and injection into highly energetic sites from the electrodes. These carriers perturb the equilibrium density of occupied states, and therefore change the overall charge transport properties. In this paper, we look into the effect of energetically hot carriers on the charge transport in organic semiconductors using steady state kinetic Monte Carlo simulations. For injected hot carriers in a typical organic semiconductor, rapid energetic relaxation occurs in the order of tens of nanoseconds, which is much faster than the typical transit time of a charge carrier throught the device. Furthermore, we investigate the impact of photo-generated carriers on the steady-state mobility. For a typical organic voltaic material, an increase in mobility of a factor of 1.1 is found. Therefore, we conclude that the impact of energetically hot carriers on normal device operation is limited.

Forth-back oscillated charge carrier motion in dynamically disordered hexathienocoronene molecules: a theoretical study

Electronic structure calculations were performed to investigate the charge transport properties of hexathienocoronene (HTC) based molecules. The effective displacement of the charge carrier along the π-orbital of nearby molecules is calculated by monitoring the forth and back oscillations of the charge carrier through kinetic Monte Carlo simulation. The charge transport parameters such as charge transfer rate, mobility, hopping conductivity, localized charge density, time average effective mass and degeneracy pressure are calculated and used to study the charge transport mechanism in the studied molecules. The existence of degeneracy levels facilitates the charge transfer and is analyzed through degeneracy pressure. Theoretical results show that the site energy difference in the dynamically disordered system controls the forth-back oscillation of charge carrier and facilitates the unidirectional charge transport mechanism along the sequential localized sites. The ethyl substituted HTC has good hole and electron hopping conductivity of 415 and 894 S cm(-1), respectively, whereas unsubstituted HTC has the small hole mobility of 0.06 cm(2) V(-1) s(-1) which is due to large average effective mass of 1.42 × 10(-28) kg.

Consistent double Gaussian model with non-symmetric potential barriers at contacts for organic diodes

A transport model with double Gaussian density of state (DOS) for organic semiconductors is proposed, with one Gaussian DOS for free carriers and one for trapped carriers.

Visualizing charge separation in bulk heterojunction organic solar cells

Solar cells based on conjugated polymer and fullerene blends have been developed as a low-cost alternative to silicon. For efficient solar cells, electron-hole pairs must separate into free mobile charges that can be extracted in high yield. We still lack good understanding of how, why and when carriers separate against the Coulomb attraction. Here we visualize the charge separation process in bulk heterojunction solar cells by directly measuring charge carrier drift in a polymer:fullerene blend with ultrafast time resolution. We show that initially only closely separated (<1 nm) charge pairs are created and they separate by several nanometres during the first several picoseconds. Charge pairs overcome Coulomb attraction and form free carriers on a subnanosecond time scale. Numerical simulations complementing the experimental data show that fast three-dimensional charge diffusion within an energetically disordered medium, increasing the entropy of the system, is sufficient to drive the charge separation process.

Unraveling the Electronic Heterogeneity of Charge Traps in Conjugated Polymers by Single-Molecule Spectroscopy

Charge trapping is taken for granted in modeling the characteristics of organic semiconductor devices, but very few techniques actually exist to spectroscopically pinpoint trap states. For example, trap levels are often assumed to be discrete in energy. Using the well-known keto defect in polyfluorene as a model, we demonstrate how single-molecule spectroscopy can directly track the formation of charge and exciton traps in conjugated polymers in real time, providing crucial information on the energetic distribution of trap sites relative to the polymer optical gap. Charge traps with universal spectral fingerprints scatter by almost 1 eV in depth, implying that substantial heterogeneity must be taken into account when modeling devices.

Diffusion-limited current in organic metal-insulator-metal diodes

An analytical expression for the diffusion current in organic metal-insulator-metal diodes is derived. The derivation is based on the classical diffusion theory of Schottky, with adaptations to account for the absence of doping, a built-in voltage due to asymmetric contacts, and band bending at the Ohmic contact. The commonly observed deviation of the ideality factor from unity (~1.2) is characteristic of diffusion-limited currents in undoped organic semiconductors. Summing with the classical space-charge limited current provides a full analytic description of the current as a function of voltage, temperature and layer thickness.

Solution-processed organic photovoltaic cells based on a squaraine dye

In this work, 2,4-bis[4-(N,N-diisobutylamino)-2,6-dihydroxyphenyl] squaraine (SQ) was systematically studied as an electron donor in solution processed photovoltaic cells, showing power conversion efficiency of >4.0% under AM1.5G 1 sun illumination at room temperature. Low mobilities were found to limit charge transport in the bulk heterojunctions. Efficiency was thus improved to 5.1% at 80 °C mainly due to improvement of photocurrent extraction. We also demonstrated that the SQ compound synthesized via a simple method has high purity, and thus can be used in photovoltaic cells without further purification. Our results suggest the huge potential of SQ and its analogs in organic photovoltaic applications.