Abstract
Charge carrier transport in organic semiconductors is at the heart of many revolutionary technologies ranging from organic transistors, lightemitting diodes, flexible displays and photovoltaic cells. Yet, the nature of charge carriers and their transport mechanism in these materials is still unclear. Here we show that by solving the timedependent electronic Schrödinger equation coupled to nuclear motion for eight organic molecular crystals, the excess charge carrier forms a polaron delocalized over up to 10–20 molecules in the most conductive crystals. The polaron propagates through the crystal by diffusive jumps over several lattice spacings at a time during which it expands more than twice its size. Computed values for polaron size and charge mobility are in excellent agreement with experimental estimates and correlate very well with the recently proposed transient localization theory.
Introduction
Organic semiconductors (OSs) differ from inorganic semiconductors in two important aspects: they are made of small or polymeric molecules that are held together by weak vanderWaals interactions rather than covalent bonds. Hence, thermal motions of the molecules around their lattice positions is very pronounced and leads to large fluctuations of electronic coupling, also termed offdiagonal electron–phonon coupling. Secondly, the static dielectric constant of OSs is typically very small and, as a consequence, the reorganization energy λ or local electron–phonon coupling is small, too (0.2 eV or less). These two material properties place charge transport in OSs in a regime that challenges traditional transport descriptions^{1,2,3,4,5,6}. The banddescription asserts the existence of “Bloch” states and typically breaks down at ambient temperatures where the mean free path of the scattered carrier becomes smaller than the intermolecular lattice spacing^{3,7,8}. The polaronic band models of Ortmann and Hannewald successfully reconciles some of the effects of coupled nuclear and electronic motion^{9} but they too become problematic at ambient temperature^{3}. Charge hopping models on the other hand assert the existence of a finite free energy barrier for charge hopping and a separation of time scales between charge transfer and the molecular motions coupled to it^{3,5,6,10}. For common OSs like rubrene, pentacene and C_{60} one or both requirements are not met^{11,12}. In recent years, the community has pursued the development of either more advanced theories, e.g., transient localization theory^{4,13,14,15}, or approximate quantum dynamical direct propagation methods^{16,17,18,19,20}. Yet, the latter were mostly limited to simple displaced harmonic oscillator model Hamiltonians and often the socalled back reaction from the electronic to the nuclear degrees of freedom was not accounted for.
What is needed to advance our understanding of charge transport in OSs are numerically efficient, reliable and practical direct propagation schemes for electronnuclear motion that are free of limiting model assumptions, that seamlessly bridge the gap between different mechanistic regimes and that support assumptions of alternative transport theories^{4,13,14,15}. Here we show that mixed quantumclassical nonadiabatic molecular dynamics in the framework of our recently developed fragmentorbital based surface hopping (FOBSH) method^{21,22,23} is a truly predictive approach in this regard, in particular at ambient and high temperature where nuclear quantum effects are still relatively small^{24}. While previous investigations were limited to charge transport in short chains of small molecules^{23}, latest algorithmic developments now allow us to apply FOBSH for the first time to charge transport in realistic nanoscale systems formed of up to a few hundred mediumsized organic molecules.
Results
Charge carrier transport mechanism in 2D materials
In this work we use the FOBSH methodology to uncover the nature and transport mechanism of charge carriers in eight single crystalline OSs, each exhibiting 2D conductance in their herringbone layers yet a significantly smaller or vanishing conductance in the respective orthogonal direction (see Fig. 1). The systems were chosen to represent low, medium and highly conductive OSs with experimental mobilities spanning three orders of magnitude. The electronic structure calculations of important transport parameters such as electronic couplings and reorganization energies, as well as the subsequent parametrization of the molecular model are described in Methods, with parameters summarized in Table 1. We note in passing that reorganization energy is assumed to be equal to the intramolecular (or “innersphere”) contribution. The intermolecular (or “outersphere”) contribution is typically very small in apolar OSs^{25,26} studied here and is neglected. Details on the FOBSH method and simulation protocols as well as a discussion of important properties including detailed balance, internal consistency (Supplementary Fig. 1) and decoherence (Supplementary Fig. 2) are given in the Methods section.
The initial dynamics of the hole carrier wavefunction Ψ(t) and the polaron size (defined by the inverse participation ratio (IPR) as described in Methods) over the first 100 fs are shown in Fig. 2 for two representative OSs (T = 300 K): panels (a)–(f) for the low mobility OS pMSB and panels (g)–(l) for the high mobility OS pentacene. Starting from an electronic wavefunction that is initially localized on a single pMSB molecule (Fig. 2b), we observe frequent hops of the electron hole, each involving rapid delocalization of the hole carrier wavefunction over a few molecules (Fig. 2c(e)) and relocalization on a single molecule that is one or a few molecular spacings (0.3–0.5 nm) apart (Fig. 2d(f)). The average IPR is equal to 1.7 and the rootmeansquare fluctuation σ equals 0.9 (see Table 2). The observed hole hopping mechanism is not unexpected for this OS because the thermal average of electronic coupling between the molecules, V = 〈H_{kl}^{2}〉^{1/2}, is significantly smaller than reorganization energy, ξ = 2V/λ = 0.1. However, the mechanism differs from the Marcus picture often used to model small polaron hopping in OSs^{6}, in that several molecules bridging initial donor and final acceptor may come simultaneously into energetic resonance resulting in hole transfer to a molecule beyond the nearest neighbor in a single hopping event. This is more reminiscent of the flickering resonance mechanism recently proposed by Skourtis and Beratan for hole transport in DNA^{27}.
The situation is strikingly different for pentacene. The initially localized electronic wavefunction Ψ(t) (Fig. 2h) rapidly spreads over many molecules (Fig. 2i) to form a polaron that is preferentially delocalized along the T_{1} direction where πorbital overlap and hence electronic coupling between neighboring molecules is the largest (Fig. 2j). On average, the polaron is delocalized over 18 molecules (σ = 10.3) in excellent agreement with estimates based on experimental electron spin resonance data, 17 molecules at 290 K^{28}. Delocalization occurs because electronic coupling is now on the same order of magnitude as reorganization energy, ξ = 2.2, which brings several molecules simultaneously into energetic resonance at any point in time. Yet, disorder in the site energies and electronic couplings prevent the wavefunction from further delocalization. In FOBSH this effect is born out by the wavefunction Ψ(t) projecting on the ground or low energy electron hole eigenstates (i.e., states close to the valence band edge, see Supplementary Fig. 1), which are delocalized over no more than a dozen molecules. The motion of the polaron within the herringbone layer of pentacene is particularly intriguing. Neighboring clusters of molecules frequently come into energetic resonance with the polaron causing Ψ(t) to expand to about twice its size for short durations of time (Fig. 2k). At this point Ψ(t) projects on higherlying electron hole eigenstates (i.e., states closer to the middle of the valence band), which are more extensively delocalized, typically over 20–50 molecules. Some of these sudden bursts of the wavefunction are successful, meaning Ψ(t) returns to a lowlying electron hole eigenstate that is localized on a neighboring cluster of molecules (Fig. 2l).
The dynamics at longer times, up to a few picoseconds, is shown in Fig. 3 for both materials. We find that the average duration of a “resonance”, defined here by the time it takes for the IPR to exceed and subsequently return below 〈IPR〉 + σ is 7 and 12 fs for pMSB and pentacene, respectively, see Fig. 3a and b, which is close to the characteristic oscillation time of intramolecular vibrations and site energy fluctuations. The average time between two resonances is about an order of magnitude larger, 52 fs for pMSB and 114 fs for pentacene. Similar values are obtained for the other compounds, see Table 2. These resonances give rise to spatial displacements as described qualitatively above and shown in Fig. 3 by way of projecting Ψ(t) on the crystallographic directions b and T_{1} of pMSB (Fig. 3c) and pentacene (Fig. 3d), respectively. Yet, significant displacements along these directions occur at somewhat longer times than the time between two resonances, more characteristic of the oscillation time of the electronic coupling fluctuations, τ = 159 and 202 fs rad^{−1} for pMSB and pentacene, respectively, see Fig. 3c and d. Hence, as one would expect, only a fraction of the resonances (estimated to be about 0.2–0.5) leads to a successful displacement. Notably, the wavefunction displacements in pentacene are over several lattice spacings at a time, 3–5 nm, that is about an order of magnitude larger than the (mostly nearestneighbor) displacements in pMSB. As we will see in the following, this difference gives rise to a ≈50fold higher charge mobility in pentacene relative to pMSB.
Charge mobility and wavefunction delocalization
For the calculation of charge mobility we run 1000 FOBSH trajectories for each system to obtain the meansquare displacement (MSD) of the center of Ψ(t) as a function of time (see Methods section). After a short initial relaxation period we observe a linear increase of the MSD with time, implying that the Einstein diffusion approximation is valid (Supplementary Fig. 3). The charge mobilities obtained from the Einstein relation are shown in Fig. 4a (data in blue). They are in excellent agreement with experiments or within the experimental error bars where uncertain, with typical deviations of less than a factor of two for mobilities spanning 3 orders of magnitude. We find that charge mobility correlates very well with both: average polaron size, as defined by the inverse participation ratio (IPR) (Fig. 4b), and the order parameter ξ (Fig. 4c) determining the existence and height of the free energy barrier for charge transfer between nearest neighbors, as illustrated in Fig. 4d. As discussed below, traditional hopping and band models fail to provide a uniformly good description of charge transport in the OSs investigated.
FOBSH mobilities up to ≈1 cm^{2} V^{−1} s^{−1} including the one for pMSB (ξ < 0.2), are well reproduced by a chemical master equation for small polaron hopping between nearest neighbors with hopping rates from electron transfer (ET) theory, as described in Methods, (data in dark green in Fig. 4a), despite our observation above for pMSB that the actual mechanism is more intricate than simple nearest neighbor hopping. For OSs with larger mobilities, ≈1–5 cm^{2} V^{−1} s^{−1} (0.2 < ξ < 1), the free energy barrier is small, causing the polaron to delocalize over 2–5 molecules according to FOBSH simulations. Hence, in this regime, the small polaron hopping model assuming nearest neighbor hops of a fully localized charge carrier is no longer a good physical model of the charge transport process. Nonetheless, if one solves the chemical Master equation with nearest neighbor hopping rates from ET theory, the resultant mobilities are in good agreement with FOBSH and experimental values (data in shaded green). This agreement appears to be coincidental as the small polaron hopping mechanism bears no resemblance with the actual mechanism obtained from FOBSH. Indeed, it is well known that a small polaron hopping model may give the same order of magnitude in mobility or current as a larger polaron model^{29}  agreement with the experimental mobility gives no sufficient information on the mechanism.
At even higher mobilities, ⪆5 cm^{2} V^{−1} s^{−1} (ξ ≥ 1), the free energy barrier disappears completely and polarons are delocalized over several to many molecules, as observed above for pentacene. In this regime band theory does not give an adequate description either: experimental mobilities are overestimated due to strong thermal motions violating basic assumptions of this theory (data in shaded red; only for still higher mobilities this theory becomes valid). By contrast, FOBSH describes all regimes relevant to OSs accurately and seamlessly bridges the gap between small polaron hopping and band transport.
Our results obtained from explicit time propagation of the electronnuclear dynamics can be used to test more recent theoretical models of charge transport in OSs, e.g., the transient localization theory (TLT) proposed by Fratini and Ciuchi^{4,13,14,15}. This theory is based on the observation that electronic coupling fluctuations on the time scale τ = 0.1 − 1 ps rad^{−1}, cause a transient localization of the charge carrier, in agreement with what we observe for pentacene in Fig. 3b. The main result of TLT is that the mobility is related to the squared transient localization length of the carrier wavefunction, \(L_\tau ^2\), \(\mu _{{\mathrm{TLT}}} = eL_\tau ^2/(2k_{\mathrm{B}}T\tau )\), where e is the unit charge, k_{B} the Boltzmann constant and T the temperature. We have calculated μ_{TLT} from \(L_\tau ^2\) using the electronic Hamiltonians sampled along present FOBSH trajectories and setting the site energies to zero (see Table 2 for values of τ). We find that TLT gives indeed a good prediction of experimental values in the high mobility regime (see Fig. 5a, data in green). If site energy fluctuations are retained in the electronic Hamiltonian, TLT also captures the hoppinglike regime, albeit concomitant with a slight increase in deviation for the high mobility regime (data in red). We find also a good correlation between our IPR and the localization length \(L_\tau ^2\) divided by the area per molecule within the herringbone layer, as shown in Fig. 5b.
Charge mobility limiting factors
An important objective in the discovery process of efficient OSs is the understanding of the aspects limiting polaron delocalization and ultimately charge mobility. According to Troisi and coworkers the major limiting factor are the thermal fluctuations of electronic coupling between the molecules leading to localization of the electronic eigenstates and hence to reduction in mobility^{3,13,14,16}. Indeed several attempts have already been made experimentally to reduce offdiagonal disorder, with some successes^{30,31}. To estimate the maximum possible boost in charge mobility that one could achieve via complete removal of offdiagonal electron–phonon coupling, we carried out FOBSH simulations with electronic couplings frozen to their mean values (Fig. 4c, data in olive). While in the small polaron hopping regime (ξ < 0.2) the mobility slightly decreases, as one would expect from nonadiabatic ET theory, in the medium and large polaron regime (ξ > 0.2) the mobility increases significantly, by up to a factor of 7 for rubrene. Yet, the charge carrier is still polaronic due to the thermal fluctuations of the site energies (diagonal electronphonon coupling). If the latter are frozen as well, the polaron fully delocalizes and the band transport regime is reached.
Discussion
In conclusion, we have demonstrated that FOBSH enables unprecedented insight into the elusive, intricate and longdebated nature and dynamics of charge carrier transport in crystalline OSs, based on rigorous physical principles. In contrast to traditional transport theories, it provides a sound description of the notoriously challenging but practically important charge transport regime of room temperature highmobility OS materials. Although FOBSH is a nonadiabatic molecular dynamics (MD) method, it is fairly computationally inexpensive, with a cost per MD step that is typically 2–35 times higher than that for a classical molecular dynamics simulation on systems with a few hundreds to a thousand molecules, respectively. Therefore we expect this methodology to become a practical tool for the computeraided design of nextgeneration highmobility OS materials, and more generally for the realistic prediction of charge transport mechanisms in “soft” condensed matter including wet biological molecules.
Methods
Fragment orbitalbased surface hopping (FOBSH)
FOBSH is a mixed quantumclassical fewest switches surface hopping technique that permits efficient simulation of charge and exciton transport in condensed phase materials^{21,22,23,32}. In FOBSH, it is assumed that the complicated manybody dynamics of an excess electron or electron hole can be effectively described by a timedependent oneparticle wavefunction, Ψ(t). The latter is expanded in a basis of fragment or site orbitals, here the frontier orbitals of the charge mediating molecules, that is, the highest occupied molecular orbitals (HOMO) for hole transport or the lowest unoccupied molecular orbitals (LUMO) for electron transport,
where R(t) denotes the timedependent nuclear positions. The wavefunction is propagated according to the electronic Schrödinger equation in the timedependent potential due to nuclear motion,
where H_{kl} = 〈ϕ_{k}Hϕ_{l}〉 and \(d_{kl} = \langle \phi _k\dot \phi _l\rangle\) are the electronic Hamiltonian matrix elements and nonadiabatic coupling elements (NACEs) in the (quasidiabatic) site orbital basis {ϕ_{l}}. The diagonal matrix element of the electronic Hamiltonian or site energy, H_{kk}, is the total electronic energy of the system when the charge carrier is localized on molecule k while all other molecules k ≠ l are charge neutral. The offdiagonal matrix elements, H_{kl}, are often referred to as electronic couplings or transfer integrals. Both, site energies and electronic couplings fluctuate due to nuclear motion (note dependence on R(t) on the right hand side of Eq. (2)), which is referred to as diagonal and offdiagonal electronphonon coupling. The nuclei propagate on a single (“active”) adiabatic electronic potential energy surface at any time (as obtained by diagonalization of H_{kl}) and hop stochastically between different surfaces according to Tully’s hopping probability^{33} (in the current context, not to be confused with the charge carrier hopping mechanism). Trivial or nonavoided crossings between the dense adiabatic PESs in our systems are dealt with using a recently implemented state tracking algorithm^{23}. The electronic decoherence is corrected by exponential damping of all except the active adiabatic electronic states using the Heisenberg principlebased decoherence time^{22,23}. The decoherence correction occasionally leads to artificial longrange charge transfer that is removed with a projection algorithm as detailed in ref. ^{23}.
A key feature of FOBSH is that explicit electronic structure calculations of the elements H_{kl} and their nuclear derivatives are avoided during time propagation, which allows us to investigate large systems and long time scales. The site energies H_{kk} and gradients ∇_{R}H_{kk} are approximated with a classical force field, while electronic couplings H_{kl}, coupling derivatives ∇_{R}H_{kl} and NACEs d_{kl} between the site orbitals are calculated using the analytic overlap method (AOM)^{34}. The site energies and AOM couplings are parametrized from DFT calculations, see section Molecular model below. The electronic Hamiltonian and the nuclear derivatives are calculated every MD time step in the site basis and transformed to the adiabatic basis for propagation of the nuclei and calculation of the hopping probabilities. For further details of the method we refer to our recent publications^{21,22,23}.
Molecular model
Each molecule of the simulated systems can exist in two charge states: neutral and charged. The intra and intermolecular interaction terms for the neutral state are taken from the Generalized Amber Force Field (GAFF)^{35}. For the charged state, the equilibrium bond lengths of the molecule are displaced with respect to the neutral state so that the reorganization energy λ obtained from the force field is equal to the value obtained from DFT calculations,
where E_{C(N)}(R_{N(C)}) is the energy of the charged (neutral) molecule in the optimized neutral (charged) state and E_{C(N)}(R_{C(N)}) is the energy of charged (neutral) molecule in the optimized charged (neutral) minimum. The geometry of charged and neutral molecules were optimized with the B3LYP functional and the 6–311g(d) basis set using the Gaussian program^{36}. The reorganization energies obtained are summarized in Table 1 and the displacement of equilibrium bond lengths are shown in Supplementary Fig. 4. Hybrid functionals are known to give good equilibrium structures and better energies for bond stretching than GGA functionals, which is important for the calculation of reorganization energies^{11,37}. Taking anthracene as example, we obtain similar values for two of the most popular hybrid functionals, λ = 142.1 meV for B3LYP and 149.9 meV for PBE0, but a smaller value for the GGA functional BLYP, 102.4 meV, due to the well known deficiency of the latter functional to underestimate the energy for bond stretching. The results are well converged with respect to the basis set used. Only very small changes in λ are obtained as the basis set is increased: 138.1, 142.1, and 138.5 meV for the 6–31G(d), 6–311G(d) and 6–311G+(d, p) basis sets, respectively, using the B3LYP functional.
The force field equilibrium bond lengths of the charged state was adjusted by scaling the DFT displacements until force field and DFT reorganization energy matched. The scaling constant β, summarized in Supplementary Fig. 4, is close to unity for all systems, which means that the displacements in the force field and in DFT are almost identical. All other intra and intermolecular parameters were chosen to be the same as for the neutral state. The site energies H_{kk} and nuclear gradients ∇_{R}H_{kk} are obtained by assigning molecule k the force field parameters for the charged state and all other molecules l ≠ k the parameters for the neutral state. For the systems investigated, electrostatic interactions in the form of fixed point charges do not significantly alter the energetics of the charge localized states because only the charged molecule carries a net charge while the other molecules are charge neutral and apolar. Hence, for the purpose of computational efficiency, electrostatic interactions were switched off. Therefore, the very small contribution in reorganization energy due to intermolecular modes^{6} (also termed outersphere reorganization in the chemistry literature) is neglected. We expect that this is no longer a good approximation for crystals formed of polar or hydrogen bonded molecules. In this case the full electrostatics including electronic polarization of the molecules should be included, as it is well known that site energy fluctuations and hence reorganization free energies are overestimated for fixed pointcharge models^{38,39}.
The electronic coupling matrix elements H_{kl}, k ≠ l, are calculated using AOM^{34}. The first step involves the calculations of reference electronic couplings. They are obtained from FODFT^{25,39,40} calculations on a set of molecular dimer geometries that is comprised of all nearest neighbor dimers in the crystal structure and in selected structures obtained from molecular dynamics simulation of the crystal at room temperature. The FODFT calculations are carried out with the CPMD program package^{41} using the PBE exchange correlation functional. Core electrons are described by Troullier–Martins pseudo potentials, and the valence electron states are expanded in plane waves with a reciprocal space plane wave cutoff of 90 Ry. The dimers are centered in the simulation box, and a vacuum of 4 Å was applied in each dimension. Using the same functional and basis set, the accuracy of FODFT couplings was benchmarked before on the HAB11^{42} and HAB7 databases^{43} for electronic coupling for hole and electron transfer in πconjugated organic dimers. While the mean relative unsigned error with respect to highlevel abinitio reference values was found to be reasonably small (27.9%), the values were slightly but uniformly underestimated. Hence, as recommended in the previous studies, we scaled the FODFT couplings for hole and electron transfer systems by a factor of 1.348 and 1.325 to obtain best estimates.
In the second step for calculation of H_{kl}, the DFT molecular frontier orbital (HOMO for hole transfer, LUMO for electron transfer) is projected on a minimum Slater basis of p orbitals with optimized Slater decay coefficients taken from ref. ^{34} (completeness of projection ≥ 0.98). In this minimum Slater basis the overlap between the HOMO (LUMO) orbitals of two monomers forming a dimer, \(\bar S_{kl}\), can be calculated analytically and is extremely fast due to the small number of basis functions involved. For π–conjugated systems, it is usually sufficient to include only one optimized Slater p–orbital per atom contributing to π–conjugation, in this case:
where i and j run over all π–conjugated atoms in molecules k and l, respectively, and p_{π,i} is the Slater type orbital p on atom i, c_{pπ,i} is the corresponding expansion coefficient obtained by projection of the DFT molecular frontier orbital. Importantly, we find good linear correlation between \(\bar S_{kl}\) from Eq. (4) and H_{kl} from FODFT, see Supplementary Fig. 5, which allows us to estimate H_{kl} from \(\bar S_{kl}\) very rapidly for any geometry sampled along the trajectories. We fit a simple linear function for each OS, \(H_{kl} = C\bar S_{kl}\), which we refer to as AOM couplings, and C is a constant of proportion. The fits of the scaling factor C in Supplementary Fig. 5 are done either by minimization of residuals of log(H_{kl}) to weight the error of couplings over all orders of magnitude uniformly (giving C = C_{log}), or by minimization of residuals of H_{kl} to weight more strongly the error of the largest couplings, which determine mobility (giving C = C_{lin}). In most systems both methods give very similar results with mean relative unsigned errors of 36% (C_{log}) and 44% (C_{lin}) for AOM couplings with respect to FODFT couplings (average error for all systems in Supplementary Fig. 5). The largest difference between C_{log} and C_{lin}, obtained for pMSBh^{+}, still results in a rather small uncertainty in the nonadiabatic ET rate (\(k_{{\mathrm{ET}}} \propto H_{kl}^2\)), of a factor of \(C_{{\mathrm{lin}}}^2/C_{{\mathrm{log}}}^2 = 2.5\). Each MD time step the HOMO (LUMO) on each molecule is updated, as described in detail in ref. ^{21} and H_{kl} between molecular pairs is estimated from \(\bar S_{kl}\) via the above linear relationship. The nuclear derivatives ∇_{R}H_{kl} and the NACE d_{kl} are obtained from finite differences of the overlap Eq. (4) with respect to nuclear displacements and time increments, respectively^{21}.
Possible shortcomings of AOM arise from the fact that, although atomic orbitals comprising the HOMO (LUMO) follow the motion of the atoms during the dynamics, the expansion coefficients in Eq. (4) are frozen otherwise. However, our checks indicated that this is a very good approximation, especially for rigid molecules, where orbitals are stable against intermolecular vibrations. More sophisticated interpolation schemes, or machine learning techniques could be used in future to improve reconstruction of the orbitals along the dynamics. Another source of inaccuracy could be the minimal basis set employed in Eq. (4), where only a single orbital per atom is considered. Although the validity of this approximation has been successfully tested before^{34}, one could use a larger basis set to improve the orbitals representation.
Simulation details
Starting from the experimental crystal structures, we built the following supercells for each OS, with number of molecules per supercell included in parenthesis: 14 × 14 × 1 (784) for pMSB; 2 × 2 × 28 (448) for PYR; 10 × 10 × 2 (800) for PER; 12 × 16 × 4 (1536) for NAP; 18 × 28 × 2 (2016) for ANT; 30 × 15 × 1 (1800) for RUB; 25 × 20 × 2 (2000) for DATT; 20 × 30 × 2 (2400) for PEN. Each supercell was equilibrated to a target temperature of 300 K running NVT molecular dynamics simulation for 0.5 ns in a configuration where a single molecule i is in the charged state and all other molecules neutral. The last configuration is used to run an additional 0.5 ns NVE trajectory on the same state to sample initial positions and velocities for the following FOBSH simulations. A subset of the molecules within the herringbone layer (ab plane) or orthogonal to it (c* direction) containing molecule i (as specified in more detail further below) was treated as electronically active, i.e., as “sites” for construction of the electronic Hamiltonian, with their frontier orbital (HOMO or LUMO) contributing to the expansion of the carrier wavefunction Eq. (1). All other molecules of the respective supercell were treated electronically inactive and interacted with the active region only via nonbonded interactions. The charge carrier wavefunction was initialized as the frontier orbital localized on molecule i, Ψ(0) = ϕ_{i}(0). In general, ϕ_{i}(0) is a linear combination of the adiabatic electronic states (i.e., electronic eigenstates) ψ_{j}(0), unless λ is sufficiently large so that ϕ_{i}(0) = ψ_{0}(0); hence the active adiabatic potential energy surface on which the nuclei initially propagate was chosen randomly with a probability proportional to 〈ψ_{j}(0)Ψ(0)〉^{2}. The electronic Schrödinger equation (Eq. (2)) was integrated using the RungeKutta 4th order algorithm and the nuclei were propagated using the velocity Verlet algorithm. The electronic time step was set to be one fifth of the MD time step. The latter is equal to 0.1 fs, as optimized before for similar πconjugated systems^{22}, except for pMSBh^{+}, PERe^{−}c* and PYRe^{−}c*, where a MD time step of 0.05 fs was used. Every MD time step the surface hopping probability and the nonadiabatic coupling vectors (NACVs) are calculated and after a successful hop the velocity component parallel to the NACV is rescaled to conserve total energy^{22}. After an unsuccessful (“frustrated”) hop the sign of the velocity component parallel to the NACV was inverted following Tully’s prescription^{44} which was found to slightly improve internal consistency^{22}. Statetracking for detection of trivial crossings, decoherence correction and a projection algorithm for removal of decoherence correctioninduced artificial longrange charge transfer were applied as described in ref. ^{23}. All surface hopping simulations were carried out in the NVE ensemble using our inhouse implementation of FOBSH in the CP2K simulation package^{45}.
Calculation of IPR
The carrier wavefunction Ψ(t) was used to calculate the two main observables in this work, polaron size defined by the inverse participation ratio (IPR), and charge mobility, μ_{SH}. The IPR is a common measure for the number of molecules over which the carrier wavefunction is delocalized^{12,14,17,20},
where u_{v,n}(t) is the expansion coefficient for the site orbital on molecule v in trajectory n, N_{mol} is the total number of electronically active molecules and N_{traj} the number of FOBSH trajectories. At first we investigated the convergence of the timeaveraged IPR with respect to the number of electronically active molecules within the herringbone layer. While for low mobility OSs a few dozens of electronically active molecules are sufficient to converge the IPR, for medium and high mobility OSs a few hundred molecules within a herringbone layer are required. The convergence for NAPH, PER, ANT, RUB, DATT, and PEN are shown in Supplementary Fig. 6, where each data point is an average over at least 200 FOBSH trajectories of length 1 ps. On this basis, we chose for calculation of the IPR in Figs. 2 and 4 a squareshaped region of the herringbone layer containing the following number of electronically active molecules: 112 for pMSB; 315 for PER; 238 for NAP; 323 for ANT; 783 for RUB, 888 for DATT, 900 for PEN. For PYR and PER in the orthogonal c* directions a 1D chain was selected as detailed below. For each system 600 FOBSH trajectories of length 1 ps were run. After an initial relaxation time of about 200 fs, during which the initially localized polaron expands to its average size, the IPR, Eq. (5), was blockaveraged over the remainder of the trajectories (Fig. 4b).
Calculation of charge mobility
While for the above samples of pMSB, PER, NAP, and ANT the 2D charge mobility tensor within the herringbone layer is converged, for the highmobility OSs: RUB, DATT, and PEN, even larger system sizes would be required, which is currently still unpractical. Importantly, we found that for the former set of systems the charge mobility along the a (b) directions as obtained from the 2D mobility tensor are very well approximated by the charge mobility of a 1D chain of electronically active molecules along the a (b) direction. We expect this correspondence to be even better for RUB, DATT, and PEN because the electronic coupling anisotropy and hence the preference for conduction in a single direction within the herringbone layer is more pronounced than e.g., for NAP and PER. To ensure consistent comparison between the simulated OSs we show in Fig. 4 the mobilities obtained for 1D chains of electronically active molecules in the indicated direction. The following number of molecules were used: 13 for pMSB; 25 for PYR; 20 for PER; 20 for NAP; 36 for ANT; 30 for RUB; 88 for DATT; 76 for PEN. We find that the mobilities are well converged for these system sizes, see Supplementary Fig. 7. For each system 1000 FOBSH trajectories of length 1 ps were run. The meansquare displacement (MSD) of the charge carrier wavefunction was calculated according to Eq. (6),
where x is the position coordinate and x_{v,n}(t) the timedependent position of the center of mass of molecule v in trajectory n along the chain (a, b or c*direction), and x_{0} = 〈Ψ_{n}(0)xΨ_{n}(0)〉 ≈ x_{v=i,n}(0) = 0. The MSDs averaged over FOBSH trajectories are shown in Supplementary Fig. 3. After an initial relaxation time of a few 200 fs (as observed for the IPR), the MSD increases linearly indicative of Einstein diffusion. The Einstein diffusion coefficient is obtained from a linear fit of the data from about 0.5 to 1 ps,
and inserted in the Einstein relation for charge mobility,
where e is the elementary charge, k_{B} the Boltzmann constant and T the temperature (300 K). The mobilities for these systems are well converged with respect to the chain length, as shown in Supplementary Fig. 7. We note that different definitions of MSD have been used in the literature for the calculation of charge mobility from explicit wavefunction propagation. In our previous work the displacement of the center of charge (coc) of the wavefunction was used,
(simply denoted “MSD” in ref. ^{23}), whereas in refs. ^{16,17} the spread of the wavefunction (σ) was used,
with 〈x〉(t) = 〈Ψ_{n}(t)xΨ_{n}(t)〉. We prefer the definition in Eq. (6) because it accounts for both types of diffusion, center of charge motion and spreading of the charge distribution:
Hence, the mobilities can be interpreted in terms of these two contributions as well,
For all systems investigated we find that μ_{SH} ≈ μ_{SH,coc} to a very good approximation since the average size of the polaron remains virtually unchanged after initial relaxation (MSD_{σ} ≈ const). A detailed comparison between the different definitions will be reported in a forthcoming publication. The FOBSH charge mobilities with all electronic couplings frozen to their mean coupling in a given direction, H_{kl}(t) = V = const, were obtained similarly (see Table 1 for values of V) and the results are shown in Fig. 4c.
Dependence of mobility and IPR on the initial state
As mentioned before, the system was equilibrated with molecule i in the charged state and all other molecules in the neutral state and the charge carrier wavefunction was initialized accordingly as the frontier orbital localized on molecule i, Ψ(0) = ϕ_{i}(0). We also investigated other initializations of the system, e.g., starting from configurations where all molecules were initially equilibrated in the charge neutral state, and the charge carrier wavefunction is initially localized on a randomly chosen single molecule m, Ψ(0) = ϕ_{m}(0). While, obviously, the shorttime dynamics differs in each case, after about 200 fs all of the differently initialized systems relax to the same polaronic state with the same IPR and mobility. In addition, we notice that detailed balance in FOBSH algorithm (see Supplementary Fig. 1) ensures that even when the electronic carrier wavefunction is initialized as a superposition of localized states (for example an eigenstate of the Hamiltonian, namely Ψ(0) = ψ_{n}(0)), after the aforementioned relaxation period the charge carrier forms the same polaronic state and exhibits the same dynamics as in the case of the initially localized charge. This is an important observation because it shows that for calculation of charge mobility it is not necessary to reproduce the (usually unknown) initial state in experiment.
Dependence of mobility on electric field
In the present work the mobilities are calculated for the limit of zero external electric field. We have previously investigated the effect of electric field on carrier mobility in a simple 1D chain of chemically identical molecules^{6}. We found that for a typical set of parameters for OSs, 50 meV coupling and 150 meV reorganization energy, the mobility remains independent on the applied external field up to field strengths of about 10^{6} V cm^{−1}. This is at least an order of magnitude higher than typical field strengths in timeofflight (TOF) measurements (10^{3}–10^{5} V cm^{−1})^{46}. Nonlinear transport behavior may occur at higher field strengths, in which case the (fielddependent) diffusion coefficient in Eq. (8) could be obtained from the drift velocitydrift velocity autocorrelation function or from the derivative of the drift velocity with respect to the electric field strength^{6}.
Detailed balance and internal consistency
Detailed balance and internal consistency are two highly desirable attributes of any surface hopping (SH) simulations^{22}. Detailed balance is achieved when the population of a given adiabatic electronic state i, \(P_i^{{\mathrm{surf}}}\), i.e., the fraction of time the nuclear dynamics runs on adiabatic potential energy surface E_{i},
is equal to the Boltzmann population of this state,
In Eq. (13), δ_{ia,n}(t) = 1 if state i is the active surface a on which the nuclear dynamics is running at time t and equal to zero otherwise, and T_{n} is the length of a trajectory. In Eq. (14), ΔA_{i} is the free energy difference between electronic state i and the electronic ground state i = 0, which can be written as
where ΔE_{i} is the vertical energy gap, ΔE_{i}(R) = E_{i}(R) − E_{0}(R). For each of the 1D chains described above we run 1000 independent equilibrium MD trajectories of length 1 ps in the electronic ground state E_{0} to sample the exponential average, Eq. (15), and compared the resultant Boltzmann population, Eq. (14), to the surface populations obtained from FOBSH, Eq. (13). The results for the first 9 adiabatic electronic states are shown in Supplementary Fig. 1. We find that the surface populations are in excellent agreement with the Boltzmann populations obtained from equilibrium MD simulation. As we have shown for smaller model systems before, the rescaling of the velocity component parallel to the NACV after a successful surface hop is essential to obtain good detailed balance^{22}. The population of excited states within the band that is formed by the frontier orbitals of our molecules is significant for all systems suggesting that thermal excitations of the charge carrier cannot be neglected.
Internal consistency is achieved when the average quantum amplitudes of the electronic wavefunction Ψ(t),
are equal to the surface population \(P_i^{{\mathrm{surf}}}\), Eq. (13). In Eq. (16), c_{i} are the expansion coefficients of Ψ(t) in the adiabatic (i.e., electronic eigenstate) basis, \(\Psi (t) = \mathop {\sum}\nolimits_{i = 1}^M {c_i} (t)\psi _i({\mathbf{R}}(t))\), where c_{i}(t) are related to u_{l}(t) of Eq. (1) by the unitary transformation that diagonalizes the electronic Hamiltonian. We find excellent agreement between the two populations for the lowest five electronic states (down to populations of 10^{−2}), with some minor deviations for higher lying states, see Supplementary Fig. 1. The latter probably arises from the projection algorithm that we use to remove decoherence correctioninduced artificial longrange charge transfer events. Yet, since the small deviations only occur for states with low population this small discrepancy should have no significant effect on our results.
Importance of decoherence correction
In Supplementary Fig. 2 we show results for hole transport in anthracene when the decoherence correction is switched off. First we note that there is no longer internal consistency, quite the opposite: the quantum population is almost the same for all electronic states, i.e., the temperature of the electronic subsystem becomes infinite, the infamous problem of the original Ehrenfest and SH methods^{47}. A consequence of this is that the polaron size (IPR) and the mobility are strongly overestimated because most of the higher lying electronic states that are now occupied are more delocalized than the lower lying states. Even more seriously, the IPR and charge mobility do not converge with system size. For all these reasons it is of utmost importance to apply decoherence correction, otherwise the energy level population and the charge transport dynamics becomes unphysical.
Charge mobility from small polaron hopping
Charge mobilities were calculated for a small polaron model for hopping between nearest neighbors (M) within the 1D chain (green data points in Fig. 4a), e.g., for hole transfer,
where k is the rate constants obtained from electron transfer theory assuming equivalent sites. We adopted the following semiclassical transition state theory formula valid in the nonadiabatic and adiabatic ET regime^{6}
where β = 1/k_{B}T, κ_{el} is the electronic transmission coefficient,
ν_{eff} is the effective nuclear frequency (in our system, taken to be the stretching frequency of an aromatic carbon double bond: 1600 cm^{−1}) and \(\Delta A^{\mathrm{\ddagger }}\) is the activation barrier. For vanishing driving force, ΔA = 0, as is the case here, \(\Delta A^{\mathrm{\ddagger }}\) is given by^{32},
\(\Delta A^{\mathrm{\ddagger }}\) and κ_{el} were evaluated for the same reorganization energy λ used and mean couplings V obtained from FOBSH simulations (Table 1). The time evolution of the population for each site can be found solving the first order differential equation:
where P(t) is a vector containing site populations and \({\Bbb K}\) is the matrix of rate constants. The latter takes the general form:
The solution to Eq. (22) is
where P(0) is the vector of initial populations, in our case the first component P_{1}(0) = 1 and all other components are zero. The MSD is then obtained through
where L is the distance between the center of mass of two neighboring molecules and ν the index of the molecule.
Charge mobility from transient localization theory
We have calculated μ_{TLT} and \(\mu _{{\mathrm{TLT}}}^ \ast\) along the directions specified in Table 3 using the electronic Hamiltonian sampled in FOBSH trajectories. For the calculation of μ_{TLT} onsite energies are set to zero as done in refs. ^{14,15}, whereas for the calculation of \(\mu _{{\mathrm{TLT}}}^ \ast\) the complete Hamiltonian including onsite thermal fluctuations is used. We employed the exact diagonalization method proposed in ref. ^{15} to calculate the squared transient localization length along x and y direction in the 2D herringbone layer of the investigated systems, \(L_{x(y)}^2\), Eqs. (8) and (9) in ref. ^{15}. The intermolecular oscillation time τ = 1/ω_{0} is taken as the inverse of the angular frequency ω_{0} of the highest peak in the power spectrum of the electronic coupling time series evaluated along 5 ps long FOBSH trajectories (summarized in Table 2). Mobilities from TLT are shown in Fig. 5a. The average squared localization length in the 2D plane was divided by the area A per molecule within the herringbone layer to enable comparison with the IPR obtained from FOBSH, \(L_\tau ^2/A = (L_x^2 + L_y^2)/2A\) (see Fig. 5b).
Data availability
All data supporting the findings of this study are available from the corresponding author upon request. All the custom codes used in this study are available from the corresponding author under reasonable request.
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Acknowledgements
We thank Dr. Guido Falk von Rudorff for helpful discussions. S.G., A.C., O.G.Z., and S.Ghosh were supported by the European Research Council (ERC) under the European Union, Horizon 2020 research and innovation programme (grant agreement no. 682539/SOFTCHARGE). Via our membership of the UK’s HEC Materials Chemistry Consortium, which is funded by EPSRC (EP/L000202, EP/R029431), this work used the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk), as well as the UK Materials and Molecular Modeling (MMM) Hub, which is partially funded by EPSRC (EP/P020194), for computational resources. We also acknowledge the use of the UCL Grace High Performance Computing Facility.
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S.G., A.C., M.E., O.G.Z., S. Ghosh enhanced FOBSH code capabilities and accuracy. M.E. developed wavefunction visualization code. S.G., H.Y., and O.G.Z. prepared the systems and parametrized forcefields. S.G. and O.G.Z. performed DFT calculations and setup FOBSH parameters. S.G. and A.C. performed actual simulations and wrote postprocessing tools. J.B. designed the research and helped with data interpretation. S.G. and J.B. wrote the manuscript. All authors reviewed and discussed the manuscript.
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Giannini, S., Carof, A., Ellis, M. et al. Quantum localization and delocalization of charge carriers in organic semiconducting crystals. Nat Commun 10, 3843 (2019). https://doi.org/10.1038/s41467019117759
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