Stefan T.
Bromley
*ab,
Francesc
Illas
a and
Marta
Mas-Torrent
c
aDept. de Química Física i Institut de Recerca de Química Teòrica i Computacional, Universitat de Barcelona, E-08028, Barcelona, Spain
bInstitutió Catalana de Recerca i Estudis Advançats (ICREA), 08100 Barcelona, Spain
cInstitut de Ciència de Materials de Barcelona (CSIC), Campus UAB, 08193, Bellaterra, Spain
First published on 13th November 2007
Taking the organic molecular material dithiophene–tetrathiafulvalene (DT–TTF) as an example of a high mobility organic molecular material, we use density functional calculations to calculate the dependency of the reorganization energy associated with charge carrier transport on: (i) the geometric and electronic responsiveness of the local molecular crystal environment, and, (ii) the local spatial extent of the charge carrier. We find that in our most realistic extended models the charge transfer reorganization energy is strongly dependent on carrier localization. In particular, whereas highly localized carriers are found to be highly susceptible to their charge transfer efficiency being affected by changes in the local crystal environment, more delocalized carriers are better able to maintain their low reorganization energies. Considering that maintaining a relatively small charge transfer reorganization energy magnitude is an important factor in achieving high carrier mobilities, we suggest that those materials better able to sustain carriers with short-range thermally resistant intermolecular delocalisation should be sought for device applications.
In semiclassical models, two parameters describe the temperature dependent hopping rate of localised carriers in OMCs:8,9 the intermolecular electronic coupling constant (t), and the reorganisation energy (λ). t measures the orbital overlap between neighboring neutral molecules in an OMC crystal and it is known to be sensitive to the relative orientations and positions of adjacent molecules, affecting carrier transport9–11 In addition to maximising the intermolecular coupling, in order to ensure high carrier mobilities in OMCs the energy changes occurring during transport, measured by λ, should also be minimised. From calculations employing only one isolated molecule, λ is estimated from twice the energy difference between the molecular cation at its relaxed neutral geometry and the molecular cation in its fully relaxed state. This measure of λ is better termed the internal reorganisation energy (λint) as it excludes the influence and energy changes of the molecular environment during charge transfer. Splitting λ into an internal part (λint) and an external contribution (λext) is much used in studies of solvated processes, where λext is often greater than λint.12 Most studies of carrier hopping in OMCs have focussed on λint as an approximation to λtotal, implicitly assuming that the magnitude of λext is negligible compared to λint, and that the dependence of λint on the external environment is very weak. Such assumptions are likely to be quite accurate for molecules such as pentacene;13,14 the relatively low calculated values of λint being a consequence of their rigid planar conformations. Studies have indicated that λint often plays a more important role than t in determining intrinsic charge transfer rates15 in OMCs and that calculations of λint and how it varies with chemical substitutions may provide a guide to finding high mobility candidate molecules.14 Although useful, we stress that such a theoretical filter inevitably omits many important molecules, some of which have already been crystallised into high mobility materials. Recently, a number of TTF-derivative OMCs have been shown to have high mobilities2,7,16,17 but which, unlike pentacene, are made of relatively flexible molecules with significantly different conformations in the charged and neutral state. For DT–TTF, displaying a very high field effect mobility of 1.4–3.6 cm2V−1 s−1,2,7 we have previously shown that estimates of λint from small embedded models incorporating a geometrically fixed yet electronically responsive environment yield significantly smaller values than λint from an isolated DT–TTF molecule.18 Although showing the importance of intermolecular interactions in determining λint and thus the mobility of the material, the model assumed highly delocalised carriers, which is probably less appropriate for comparison with ambient temperature data. Moreover the use of a rigid environment precludes any estimates of λtotal as the intermolecular degrees of freedom are frozen and thus no intermolecular relaxation can take place. Herein, using the DT–TTF system, we calculate λint using a range of hole charge carrier models allowing for varying degrees of: (i) the influence of the local polarisability, (ii) the charge carrier localisation. For our largest models we also allow for local molecular relaxation to take place thus providing a first order estimate of λtotal.
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Fig. 1 Upper: DT–TTF molecule, lower: crystal stack models: (S1) relaxed single molecule, (S2) planar constrained single, (S3) central relaxed molecule, outer two molecules fixed, (S4) central three relaxed molecules, outer two molecules fixed. |
Further to varying the molecular environment (S1–S4) we also use three Hamiltonians, each with a different proportion of Hartree–Fock exchange (HFE), in order to influence the degree of charge localisation in the resulting charge carrier model.23 Specifically, we use: HF (100 percent HFE), density functional theory (DFT) using a hybrid meta-generalised gradient approximation functional BB1K24(42 percent HFE), and DFT using the hybrid exchange–correlation functional B3LYP25(20 percent HFE). We examine the hole carriers using a double-zeta 6-31G(d,p) basis set with polarisation on all atoms, which is known to provide converged calculations of molecular overlap,26 and the GAMESS-UK code.27 The resulting charge carrier types may be classified as local (L), semi-local (SL), and delocalised (D) for Hamiltonians incorporating 100, 42, and 20 percent HFE, respectively. Fig. 2 shows the molecular polarisation (a) and charge localisation (b) in S4[L], S4[SL] and S4[D] models showing the small polaronic, semi-localised polaronic and delocalised nature of each, respectively.
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Fig. 2 S4[L, SL, D]: Upper graph: central C–C bond length with respect to the neutral stack where the molecules are numbered from one end of the S4 stack. Lower graph: Mulliken atomic sulfur charges where the numbering relates to six sulfur atoms of each molecule going through the S4 stack from one end, (1–6), (7–12), (13–18), (19–24), (25–30), where the outer two numbers in each sub-range relate to the two sulfur atoms furthest from the central C–C bond. |
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Fig. 3 Change in λ for S1–S4[D, SL, L]). Open symbols show λint of pentacene. Bars show the percentage contribution of intermolecular polarisation (light)versus delocalisation (dark) in the S2–S3 decrease in λ (lower branches). |
[L] / HF | [SL] / BB1K | [D] / B3LYP | |
---|---|---|---|
DT-TTF (S1) | 0.925 | 0.628 | 0.568 |
DT-TTF (S2) | 0.544 | 0.287 | 0.237 |
DT-TTF (S3) | 0.256 | 0.054 | 0.041 |
DT-TTF (S4) | 0.359 | 0.057 | 0.023 |
Pentacene (S1) | 0.307 | 0.139 | 0.096 |
When charged, the degree of geometric relaxation differs in each S3 model leading to a variable geometric difference between the embedding and relaxed CM. To see how such differences affect λint, we repeated the S3 calculations with a range of slightly distorted embedding molecules placed in the crystal positions and orientations. For alternative molecular embedding geometries we employed both (i) neutral molecules relaxed using each of the three calculation methods (i.e.HF, BB1K, B3LYP) rather than the experimental crystal geometries, and (ii) molecules with atom positions randomly distorted from their experimentally-determined neutral crystal positions by up to 0.05 Å. The maximum change in λint compared to the experimental crystal molecular embedding is shown in the upper branches of the lines (S2–S3[L,SL,D]) in Fig. 3. For S3[D] the effect is small and λint does not vary more than 0.025 eV with respect to the crystal embedding. For S3[SL] the effect is slightly more pronounced, but only up to the extent of diminishing the S2–S3 λintreduction by 0.063 eV. For S3[L] the effect is severe and the predicted values of λint range up to slightly above that of the respective S2 value showing the high sensitivity of λint to the local environment for highly localised carriers. In the real DT–TTF material at ambient conditions the geometries and positions of the constituent molecules will be susceptible to small thermal motions. Careful examination of the thermal dependence of the intermolecular coupling (t) in pentacene and related OMCs using a tight-binding method, has shown that the tendency of such fluctuations is to induce localisation of the charge carriers,10 leading to a transport mechanism based on the thermal modulation of t.28 As noted above, λint values for pentacene and similar molecules are intrinsically very small due to their internal high structural rigidity and are largely independent of the local environment in the respective OMCs.18 In such cases, models of OMC charge transport can safely omit explicit consideration of λ and how it is affected by carrier delocalisation as assumed in the model of ref. 28. In cases where the value of λ is strongly dependent on the molecular environment and on the (de)localisation of the charge carrier, as in the case of DT–TTF, a more general model of the charge transport in OMCs will probably require incorporation of the reorganisation energy and a treatment of more delocalised charge carriers. Herein we show explicitly that small geometric fluctuations in the local molecular environment of DT–TTF molecules is more disruptive to the charge transfer reorganisation energy, λint, for localised charge carriers, but, conversely, hardly affects the carrier transfer efficiency for more delocalised states. We consider that the intrinsic localisation length scale of the carriers is the key factor. With respect to the transfer integral modulation mechanism28 we may consider that an OMC with an intrinsic tendency to have more delocalised carriers is likely able to maintain reasonable magnitudes of molecular overlap, t, with increasing temperatures (and thus higher potential mobilities) with respect to an OMC that favours intrinsically more localised carriers under the same conditions. Our results further suggest that, at least for some OMCs, more delocalised carriers will better maintain charge transfer efficiency (i.e. relatively low λ values) with respect to environmental disturbances (e.g., thermal vibrations) over intrinsically localised carriers. Although it is difficult to predict the intrinsic tendency of any OMC to (de)localise polaronic-like carriers with current theoretical methods, using our approach of employing a range of Hamilitonians within a consistent methodology, each providing ground state solutions with a different degree of charge localisation, we are able to probe the effects of varying the nature of the charge carrier.
In S3 we could vary the fixed geometry of the electronically polarisable embedding, but in reality the environment will naturally respond to the charge carrier and will be molecularly polarised. To see the effects of a more realistic structurally responsive local molecular environment, together with an increased degree of electronic polarisability, we extended S3 to a stack of five molecules in which both the CM and their immediate neighbours were allowed to relax (S4). In each case the three Hamiltonians employed for the smaller models were used giving rise to carrier models S4[L, SL, D]. We note that these calculations, unlike in the carrier models S1–S3[L, SL, D], also allow for a first order estimate as to the influence of molecular relaxation on the reorganisation energy, i.e. provides an approximate measure of λtotal. For the neutral S4 stack the use of only two fixed embedding molecules still provides a good match with respect to the intermolecular distances down the stack (3.97–4.02 Å S4[L], 3.96–4.04 Å S4[SL], 3.96–4.03 Å S4[D]) as compared with experiment (3.99 Å). Comparing the charged stacks with the neutral stacks, the molecular distortion induced by the charge carrier is mild in the S4[D] and S4[SL] models, gradually diminishing from the CM to zero for the outer molecules. For S4[L] the CM distorts to almost its free ionic geometry forming a small polaronic carrier with neighboring molecules almost unaffected geometrically. The charge-induced molecular distortion in each case, with respect to the lengths of the central C–C bond in each of the molecules, is shown in the upper graph in Fig. 2. The S4[L, SL, D]λtotal values lie between the extremes predicted by the respective S3[L, SL, D]] models using variable fixed embedding environments indicating that carriers in a locally structurally and electronically polarisable molecular environment have neither optimal nor the worst conditions for efficient transport.
To indicate which carrier model may provide the best physical description of the carriers in DT–TTF we may compare both the available experimental data and calculated values of t and λ for DT–TTF, with respect to the same data for the benchmark standard OMC of pentacene. For DT–TTF the only two reports of field effect mobility of single crystals in transistor devices using a silica gate dielectric give 1.42 and 3.67 cm2V−1 s−1. For pentacence single crystals, in a similar experimental set-up, the highest values of field effect mobility range between 1.529 and 2.330 cm2V−1 s−1. As noted above, charge mobility is dependent on both t and λ. Quantitative estimates of the largest t values in the pentacene crystal based on the HOMO (highest occupied molecular orbital) splitting in molecular dimers calculated using the intermediate neglect of differential overlap for spectroscopy (INDO/S) semi-empirical scheme give values of 0.072–0.080 eV.31 Using the same methodology we have previously obtained a intrastack t value of 0.034 eV using two DT–TTF molecules optimised at an AM1 semi-empirical level of theory placed in the experimental crystal positions, following the method in ref. 13. Using the two molecules with geometries directly from the experimental crystal data we obtain very similar t values of 0.038 eV using the INDO/S method, and 0.034 eV using DFT employing the PW91 gradient corrected exchange–correlation functional;32 the latter also recommended for reliable near quantitative results.33 We note further that the three methodologies used in this work (HF, BB1K, B3LYP) all give slightly higher values of the transfer integral with respect to these estimates (based on a dimer taken from the crystal structure) but all give very similar values (0.052 ± 0.002 eV) importantly showing that the reported effects of (de)localisation of the charge carrier on λ is not related to a variable representation of the molecular overlap in the neutral system for each respective model. The fact that the maximum transfer integrals in both pentacene and DT–TTF are of a similar order (although, in fact, lower for DT–TTF) ties in with the current limited experimental data showing that the field effect mobilities are also comparable. Considering that both t and λ are important factors contributing to the mobility in an OMC, we should similarly expect that the reoganisation energy associated with charge be transfer also similar in both materials, if not smaller for DT–TTF. Comparing our most realistic estimates of λtotal(i.e. S4[L, SL, D) for DT–TTF with values for the pentacene molecule at the corresponding level of theory, the differences are: −0.073 eV for S4[D], −0.082 eV for S4[SL] and +0.052 for S4[L](see Fig. 3). For the localised S4[L] carriers, λtotal is significantly above that estimated for pentacene, which, considering the overlap in field effect mobility data for both OMCs, suggests that it may represent a poor carrier model for DT–TTF. The S3[L]λtotal values are also very sensitive to local environmental fluctuations (e.g. via thermal molecular vibrations) thus making it unlikely that the lowest λint values predicted for a near-optimal fixed embedding (lower point of S3[L], Fig. 3) could be sustained under experimental conditions and the mobility would likely be lower. The less localised carrier models, S4[SL] and S4[D] both yield λtotal values lower than that of pentacene, which together with the lower calculated values for t in DT–TTF, tend to support the observed comparable mobilities with respect to pentacene. The reported field effect mobilities are measured at room temperature where molecular vibrations can disrupt long-range delocalisation10 and hopping is assumed to be dominant. The lower sensitivity of λint to environmental disturbances in S3[SL] and S3[D], as noted above, is likely due to the delocalisation in both models. The almost fully-delocalised S4[D] results are effectively band-like in their predictions, which, although suggestive of the importance of delocalisation, are thus probably less appropriate for comparison with room temperature data. The semi-localised polaronic carriers in the S4[SL] model, however, give a reasonable compromise between a (low λtotal/low temperature) band-like picture and (high λtotal/high temperature) small polarons, where short-range intermolecular delocalisation provides sufficiently thermally resistant low λtotal values consistent with the high room temperature field effect mobilities of DT–TTF. Although this theoretical attempt at an approximate characterisation of the carriers in DT–TTF is based on limited experimental data, it should not detract from the strength of our methodology to describe a range of carrier types over a large range of conditions and its ability to give insight into how the nature of the carriers and the environment within which they reside influence their transport properties.
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