Site Energy Disorder Research Articles

Entropy-ruled nonequilibrium charge transport in thiazolothiazole-based molecular crystals: a quantum chemical study.

The charge and energy fluctuations in molecular solids are crucial factors for a better understanding of charge transport (CT) in organic semiconductors. The energetic disorder-coupled molecular charge transport is still not well-established. Moreover, the conventional Einstein's diffusion (D)-mobility (μ) relation fails to explain the quantum features of organic semiconductors, including nonequilibrium and degenerate transport systems, where kB is the Boltzmann constant, T is the temperature and q is the electric charge. To overcome this issue, a unified version of the entropy-ruled D/μ relation was proposed by Navamani (J. Phys. Chem. Lett., 2024, 15, 2519-2528) for hopping and band transport systems as where d, η and heff are the dimension (d = 1, 2, 3), chemical potential and effective entropy, respectively. Within this context, we investigate the CT properties of 2,5-bis(4-methoxyphenyl)thiazolo[5,4-d]thiazole (MOP-TZTZ) and 2,5-bis(2,4,5 trifluorophenyl)-thiazolo[5,4-d]thiazole (TFP-TZTZ) molecular solids using electronic structure calculations and the entropy-ruled method. The CT key parameters such as charge transfer integral and site energy are computed by matrix elements of the Kohn-Sham Hamiltonian. Using Marcus theory, the charge transfer rate is numerically calculated for MOP-TZTZ and TFP-TZTZ molecular crystals under different site energy disorder (ΔEij(E⃑)) situations. Using our entropy-ruled method, the exact diffusion-mobility (D/μ) and other transport quantities such as thermodynamic density of states, conductivity, and current density are calculated for these derivatives at different applied electric field values via the site energy disorder. The theoretical results show that the molecule TFP-TZTZ has good hole mobility (∼0.012 cm2 V-1 s-1) at a site energy disorder value of 90 meV. The obtained ideality factor from the Navamani-Shockley diode current density equation categorizes the typical transport as either the Langevin-type or Shockley-Read-Hall mechanism in the studied molecular solids. Our analysis clearly shows that both the electron and hole transport in these MOP-TZTZ and TFP-TZTZ molecules follow the trap-free Langevin mechanism, which is indeed ideal for designing charge-transporting molecular devices.

Extracting the excitonic Hamiltonian of a chlorophyll dimer from broadband two-dimensional electronic spectroscopy.

We apply Frenkel exciton theory to model the entire Q-band of a tightly bound chlorophyll dimer inspired by the photosynthetic reaction center of photosystem II. The potential of broadband two-dimensional electronic spectroscopy experiment spanning the Qx and Qy regions to extract the parameters of the model dimer Hamiltonian is examined through theoretical simulations of the experiment. We find that the local nature of Qx excitation enables identification of molecular properties of the delocalized Qy excitons. Specifically, we demonstrate that the cross-peak region, where excitation energy is resonant with Qy while detection is at Qx, contains specific spectral signatures that can reveal the full real-space molecular Hamiltonian, a task that is impossible by considering the Qy transitions alone. System-bath coupling and site energy disorder in realistic systems may limit the resolution of these spectral signatures due to spectral congestion.

Correlated spectral fluctuations quantified by line shape analysis of fifth-order two-dimensional electronic spectra.

Correlated spectral fluctuations were suggested to coordinate excitation transport inside natural light harvesting complexes. We demonstrate the capacities of 2D line shapes from fifth-order coherent electronic signals (R5-2D) to report on such fluctuations in molecular aggregates and present a stochastic approach to fluctuations in correlated site and bi-exciton binding energies in the optical dynamics of Frenkel excitons. The model is applied to R5-2D line shapes of a homodimer, and we show that the peak tilt dynamics are a measure for site energy disorder, inter-site correlation, and the strength of bi-exciton binding energy fluctuations.

Static Disorder in Excitation Energies of the Fenna-Matthews-Olson Protein: Structure-Based Theory Meets Experiment.

Inhomogeneous broadening of optical lines of the Fenna–Matthews–Olson (FMO) light-harvesting protein is investigated by combining a Monte Carlo sampling of low-energy conformational substates of the protein with a quantum chemical/electrostatic calculation of local transition energies (site energies) of the pigments. The good agreement between the optical spectra calculated for the inhomogeneous ensemble and the experimental data demonstrates that electrostatics is the dominant contributor to static disorder in site energies. Rotamers of polar amino acid side chains are found to cause bimodal distribution functions of site energy shifts, which can be probed by hole burning and single-molecule spectroscopy. When summing over the large number of contributions, the resulting distribution functions of the site energies become Gaussians, and the correlations in site energy fluctuations at different sites practically average to zero. These results demonstrate that static disorder in the FMO protein is in the realm of the central limit theorem of statistics.

Multiscale simulations of singlet and triplet exciton dynamics in energetically disordered molecular systems based on many-body Green's functions theory

We present a multiscale model based on many-body Green’s functions theory in the GW approximation and the Bethe–Salpeter equation (GW-BSE) for the simulation of singlet and triplet exciton transport in molecular materials. Dynamics of coupled electron–hole pairs are modeled as a sequence of incoherent tunneling and decay events in a disordered morphology obtained at room temperature from molecular dynamics. The ingredients of the rates associated to the events, i.e. reorganization energies, site energies, lifetimes, and coupling elements, are determined from a combination of GW-BSE and classical polarizable force field techniques. Kinetic Monte Carlo simulations are then employed to evaluate dynamical properties such as the excitonic diffusion tensor and diffusion lengths. Using DCV5T-Me(3,3), a crystalline organic semiconductor, we demonstrate how this multiscale approach provides insight into the fundamental factors driving the transport processes. Comparing the results obtained via different calculation models, we investigate in particular the effects of charge-transfer mediated high exciton coupling and the influence of internal site energy disorder due to conformational variations. We show that a small number of high coupling elements indicative of delocalized exciton states does not impact the overall dynamics perceptively. Molecules with energies in the tail of the excitonic density of states dominate singlet decay, independent of the level of disorder taken into account in the simulation. Overall, our approach yields singlet diffusion lengths on the order of 10 nm as expected for energetically disordered molecular materials.

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.

Relating free energy and open-circuit voltage to disorder in organic photovoltaic systems.

Efficient exciton dissociation into mobile charge carries a crucial factor underscoring the performance of organic polymer-based bulk-heterojunction photovoltaic devices. In this paper, we compute the energies of charge-transfer (CT) states of the model donor-acceptor lattice system with varying degrees of structural disorder to investigate how fluctuations in the material properties affect electron-hole separation. We also demonstrate how proper statistical treatment of the CT energies recovers the experimentally observed "hot" and "cold" exciton dissociation pathways. Using a quantum mechanical model for a model heterojunction interface, we recover experimental values for the open-circuit voltage at 50 and 100 meV of site-energy disorder. We find that energetic and conformational disorder generally facilitates charge transfer; however, due to excess energy supplied by photoexcitation, highly energetic electron-hole pairs can dissociate in unfavorable directions, potentially never contributing to the photocurrent while "cold" excitons follow the free energy curve defined at the operating temperature of the device.

Diffusive and arrested transport of atoms under tailored disorder

Ultracold atoms in optical lattices offer a unique platform for investigating disorder-driven phenomena. While static disordered site potentials have been explored in a number of experiments, a more general, dynamical control over site-energy and off-diagonal tunnelling disorder has been lacking. The use of atomic quantum states as synthetic dimensions has introduced the spectroscopic, site-resolved control necessary to engineer more tailored realisations of disorder. Here, we present explorations of dynamical and tunneling disorder in an atomic system by controlling laser-driven dynamics of atomic population in a momentum-space lattice. By applying static tunnelling phase disorder to a one-dimensional lattice, we observe ballistic quantum spreading. When the applied disorder fluctuates on time scales comparable to intersite tunnelling, we instead observe diffusive atomic transport, signalling a crossover from quantum to classical expansion dynamics. We compare these observations to the case of static site-energy disorder, where we directly observe quantum localisation.

Origin of non-Gaussian site energy disorder in molecular aggregates

Gaussian site energy disorder is an ad hoc concept usually implemented in simulations of excitation dynamics in molecular systems. In this letter we suggest a mechanism which may cause correlated static energy disorder in a broad range of molecular systems. Our approach leads to non-Gaussian site energy distribution, which strongly affects statistical properties of exciton wavefunctions and consequently changes material functional characteristics.

A n-vector model for charge transport in molecular semiconductors.

We develop a lattice model utilizing coarse-grained molecular sites to study charge transport in molecular semiconducting materials. The model bridges atomistic descriptions and structureless lattice models by mapping molecular structure onto sets of spatial vectors isomorphic with spin vectors in a classical n-vector Heisenberg model. Specifically, this model incorporates molecular topology-dependent orientational and intermolecular coupling preferences, including the direct inclusion of spatially correlated transfer integrals and site energy disorder. This model contains the essential physics required to explicitly simulate the interplay of molecular topology and correlated structural disorder, and their effect on charge transport. As a demonstration of its utility, we apply this model to analyze the effects of long-range orientational correlations, molecular topology, and intermolecular interaction strength on charge motion in bulk molecular semiconductors.

The origin of the unusual Qy red shift in LH1–RC complexes from purple bacteria Thermochromatium tepidum as revealed by Stark absorption spectroscopy

Native LH1–RC of photosynthetic purple bacteria Thermochromatium (Tch.) tepidum, B915, has an ultra-red BChl a Qy absorption. Two blue-shifted complexes obtained by chemical modification, B893 and B882, have increasing full widths at half maximum (FWHM) and decreasing transition dipole oscillator strength. 77K Stark absorption spectroscopy studies were employed for the three complexes, trying to understand the origin of the 915nm absorption. We found that Tr(∆α) and |∆μ| of both Qy and carotenoid (Car) bands are larger than for other purple bacterial LH complexes reported previously. Moreover, the red shifts of the Qy bands are associated with (1) increasing Tr(∆α) and |∆μ| of the Qy band, (2) the red shift of the Car Stark signal and (3) the increasing |∆μ| of the Car band. Based on the results and the crystal structure, a combined effect of exciton-charge transfer (CT) states mixing, and inhomogeneous narrowing of the BChl a site energy is proposed to be the origin of the 915nm absorption. CT-exciton state mixing has long been found to be the origin of strong Stark signal in LH1 and special pair, and the more extent of the mixing in Tch. tepidum LH1 is mainly the consequence of the shorter BChl–BChl distances. The less flexible protein structure results in a smaller site energy disorder (inhomogeneous narrowing), which was demonstrated to be able to influence |∆μ| and absorption.

Influence of intra-pigment vibrations on dynamics of photosynthetic exciton.

We have numerically investigated the effect of an underdamped intra-pigment vibrational mode on an exciton's quantum coherence and energy transfer efficiency. Our model describes a bacteriochlorophyll a pigment-protein dimer under the conditions at which photosynthetic energy transfer occurs. The dimer is modeled using a theoretical treatment of a vibronic exciton, and its dynamics are numerically analyzed using a non-Markovian and non-perturbative method. We examined the system's response to various values of the Huang-Rhys factor, site energy difference, reorganization energy, and reorganization energy difference. We found that the inclusion of the intra-pigment vibronic mode allows for long-lived oscillatory quantum coherences to occur. This excitonic coherence is robust against static site-energy disorder. The vibrational mode also promotes exciton transfer along the site-energy landscape thus improving the overall energy transfer efficiency.

Transition from Thermodynamic to Kinetic-Limited Excitonic Energy Migration in Colloidal Quantum Dot Solids

We investigate the temperature dependence of excitonic energy migration in CdSe/ZnCdS core/shell colloidal quantum dot (QD) solids using spectrally resolved transient photoluminescence spectroscopy. In line with previous studies, we observe that as excitons hop among different energy sites within the inhomogeneously broadened QD ensemble, the photoluminescence spectrum transiently shifts toward the red. Using shape-engineered nanocrystals having a temperature-independent radiative lifetime, the magnitude of this transient red-shift is found to vary nonmonotonically with temperature. Near room temperature, the magnitude of this transient red-shift is determined by thermal equilibrium within the site energy distribution. As the sample temperature is reduced, the site-to-site hopping rate slows down and excitons become kinetically trapped at local minima in the global energy landscape. For a more homogeneous QD ensemble, reduction in site energy disorder causes the transition from thermodynamic to kinetic-limited behavior to shift to lower temperatures. These results have implications for the design of colloidal QD optoelectronic devices and advance our understanding of exciton dynamics and energy transport in disordered systems.

Monte Carlo modeling of transport in PbSe nanocrystal films

A Monte Carlo hopping model was developed to simulate electron and hole transport in nanocrystalline PbSe films. Transport is carried out as a series of thermally activated hopping events between neighboring sites on a cubic lattice. Each site, representing an individual nanocrystal, is assigned a size-dependent electronic structure, and the effects of particle size, charging, interparticle coupling, and energetic disorder on electron and hole mobilities were investigated. Results of simulated field-effect measurements confirm that electron mobilities and conductivities at constant carrier densities increase with particle diameter by an order of magnitude up to 5 nm and begin to decrease above 6 nm. We find that as particle size increases, fewer hops are required to traverse the same distance and that site energy disorder significantly inhibits transport in films composed of smaller nanoparticles. The dip in mobilities and conductivities at larger particle sizes can be explained by a decrease in tunneling amplitudes and by charging penalties that are incurred more frequently when carriers are confined to fewer, larger nanoparticles. Using a nearly identical set of parameter values as the electron simulations, hole mobility simulations confirm measurements that increase monotonically with particle size over two orders of magnitude.

Distinguishing Spin Relaxation Mechanisms in Organic Semiconductors

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 intrasite 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 semiconductors provide a prescription for determining the dominant spin relaxation mechanism.

The mean-size dependence of the exchange narrowing in molecular J-aggregates

The effect of segment-size fluctuations on exchange narrowing in a molecular J-aggregate of site-energy disordered distributions is studied using a one-dimensional Frenkel-exciton model. It is found that the segment-size disorder leads to the width of the absorption spectra deviating from the scaling law, σ 4/3 of the site-energy disordered standard deviation σ, being suitable for the system only with the site-energy disorder. In larger σ, the segment-size disorder has little influence on the linear absorption spectra. With increasing segment mean-length, the absorption line width monotonically increases, and then approaches a saturated value. By comparing a system of larger mean-length segment with a smaller one, both with the same segment-size disorder, it is found that the absorption line width of the former is broadened, and the exchange narrowing effect is reduced. The present result shows that the correlation effect can be partially maintained for the system with larger mean-length segment.

Polarizable Atomistic Calculation of Site Energy Disorder in Amorphous Alq3

A polarizable molecular dynamics simulation and calculation scheme for site energy disorder is presented in amorphous tris(8-hydroxyquinolinato)aluminum (Alq(3)) by means of the charge response kernel (CRK) method. The CRK fit to the electrostatic potential and the tight-binding approximation are introduced, which enables modeling of the polarizable electrostatic interaction for a large molecule systematically from an ab initio calculation. The site energy disorder for electron and hole transfers is calculated in amorphous Alq(3) and the effect of the polarization on the site energy disorder is discussed.

Nonlinear hopping transport in ring systems and open channels

We study the nonlinear hopping transport in one-dimensional rings and open channels. Analytical results are derived for the stationary current response to a constant bias without assuming any specific coupling to the external fields. It is shown that anomalous large effective jump lengths, as observed in recent experiments by taking the ratio of the third order nonlinear and the linear conductivity, can occur already in ordered systems. Rectification effects due to site energy disorder in ring systems are expected to become irrelevant for large system sizes. In open channels in contrast, rectification effects occur already for disorder in the jump barriers and do not vanish in the thermodynamic limit. Numerical solutions for a sinusoidal bias show that the ring system provides a good description for the transport behavior in the open channel for intermediate and high frequencies. For low frequencies temporal variations in the mean particle number have to be taken into account in the open channel, which cannot be captured in the more simple ring model.

Influence of disorder on the magnetism of graphene bilayers

We numerically calculate the impact of site-energy disorder on the magnetism of biased bilayer graphene formed with the Bernal stacking. By using the mean-field method, we approximately solve the Hubbard Hamiltonian and calculate the average magnetization of the two layers, which indicates that the disorder does not change the nature of the first-order phase transition between paramagnetism and ferromagnetism, but may increase the critical interaction strength Uc of the Hubbard Hamiltonian for the system to become the ferromagnetic phase. We also calculate the dependencies of relevant physical quantities on temperature. The implications of the results are discussed.

Influence of alkyl chain length on charge transport in symmetrically substituted poly(2,5-dialkoxy-p-phenylenevinylene) polymers

We report on the hole transport characteristics, as measured by time of flight, of a family of symmetrically substituted dialkoxy poly($p$-phenylenevinylene) polymers with different side-chain length. As side-chain length is decreased, the magnitude of the hole mobility ${\ensuremath{\mu}}_{h}$ increases while the field dependence of ${\ensuremath{\mu}}_{h}$ becomes more positive and the temperature dependence of ${\ensuremath{\mu}}_{h}$ becomes stronger. For the shortest side-chain derivative studied, ${\ensuremath{\mu}}_{h}$ exceeds ${10}^{\ensuremath{-}4}\text{ }{\text{cm}}^{2}\text{ }{\text{V}}^{\ensuremath{-}1}\text{ }{\text{s}}^{\ensuremath{-}1}$ at electric fields greater than ${10}^{5}\text{ }\text{V}\text{ }{\text{cm}}^{\ensuremath{-}1}$. The trend in magnitude of ${\ensuremath{\mu}}_{h}$ with side-chain length is consistent with the expected increase in electronic wave-function overlap as interchain separation decreases, while the trends in electric-field and temperature dependences of ${\ensuremath{\mu}}_{h}$ are consistent with increasing site energy disorder. We show that the electrostatic contribution to the site energy difference for pairs of oligomers follows the observed trend as a function of interchain separation, although the pairwise contribution is too small to explain the data quantitatively. Nonresonant Raman spectroscopy is used to characterize the microstructure of our films. We construct spatial maps of the Raman ratio ${I}_{1280}/{I}_{1581}$ and confirm an expected decrease in average film density with side-chain extension. The structural heterogeneity in the maps is analyzed but no clear correlation is observed with transport properties, suggesting that the structural variations relevant for charge transport occur on a length scale finer than the resolution of $\ensuremath{\sim}1\text{ }\ensuremath{\mu}\text{m}$.