Dependence of charge transfer reorganization energy on carrier localisation in organic molecular crystals

Abstract

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.

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Introduction

Organic molecular crystals (OMCs) hold great potential for cheaper more-versatile alternatives to silicon-based electronic devices. One key to eventual technological success is the realisation of OMC-based devices with high charge carrier mobilities. Recent reports have proven that OMCs can perform to demanding standards with OMC transistor devices showing field effect mobilities of the order of that of amorphous silicon.^1,2 At ambient operating temperatures, although the charge mobility in OMCs is likely to be dominated by intermolecular hopping rather than band-like transport,^3,4 a full understanding of the charge transport mechanism and the nature of charge carriers in this regime is still lacking. In reality, the nature of the carriers and their transport mechanism probably spans a range of intermediate possibilities, which may be exhibited at different conditions for different materials. Although it is tempting to follow the full localisation of carriers from their formally infinite extent in a band-like conductor to highly localised carriers with spatial extent no larger than one molecule, there is no compelling evidence to suggest that carriers in OMCs and related materials reach such acute limits of localisation at ambient temperature. Recent experiments on the transport in the OMCs tetracene and pentacene, for example, support the existence of partially delocalised carriers with significant spatial extent rather than highly localised carriers for device operating temperatures above 250 K.^5,6 The nature of such partially localised carriers and why they should persist with respect to more localised states at such relatively high temperatures is still unresolved. Using detailed electronic structure calculations we attempt to address this central question with respect to dithiophene–tetrathiafulvalene (DT–TTF), an OMC with one of the highest reported field effect mobilities.^2,7 Studying the interdependent influences of local (de)localisation, polarisation, and geometric relaxation on the carrier hopping efficiency we suggest that the high mobilities observed in DT–TTF and related OMCs may be due to these materials sustaining partially delocalised polarons with charge extended over a number of neighbouring molecules. Our results suggest that limited intermolecular charge delocalisation could assist in maintaining high carrier mobilities at higher temperatures where we predict that highly localised carriers would provide less energetically efficient transport of charge. The concept of partially delocalised polaronic carriers may also be useful in understanding the temperature induced band-to-hopping transition in OMCs.

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 transport^9–11 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 (λ_int) as it excludes the influence and energy changes of the molecular environment during charge transfer. Splitting λ into an internal part (λ_int) and an external contribution (λ_ext) is much used in studies of solvated processes, where λ_ext is often greater than λ_int.¹² Most studies of carrier hopping in OMCs have focussed on λ_int as an approximation to λ_total, implicitly assuming that the magnitude of λ_ext is negligible compared to λ_int, and that the dependence of λ_int 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 λ_int being a consequence of their rigid planar conformations. Studies have indicated that λ_int often plays a more important role than t in determining intrinsic charge transfer rates¹⁵ in OMCs and that calculations of λ_int 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 mobilities^2,7,16,17 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 λ_int from small embedded models incorporating a geometrically fixed yet electronically responsive environment yield significantly smaller values than λ_int from an isolated DT–TTF molecule.¹⁸ Although showing the importance of intermolecular interactions in determining λ_int 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 λ_total as the intermolecular degrees of freedom are frozen and thus no intermolecular relaxation can take place. Herein, using the DT–TTF system, we calculate λ_int 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 λ_total.

Models and computational methodology

Molecules in DT–TTF lie face-to-face with two adjacent neighbours in one-dimensional stacks, where the interfacial interaction is by far the main intermolecular interaction.¹⁶ We use four increasingly realistic representations of the molecular environment (Fig. 1) for estimating λ_total: one fully relaxed molecule (S1), one molecule restricted to relax in the x–y plane (S2), a stack of three molecules with only the central molecule (CM) allowed to freely relax (S3), a stack of five molecules with the central CM and its two local neighbouring molecules allowed to freely relax (S4). For S3 and S4 the two terminal molecules were fixed at their experimentally determined crystal geometries in their crystallographic positions¹⁹ ensuring the structural integrity of the stack. In S3 and S4, the CMs, when relaxed using each of the methods described below, all tilt slightly away from the angle adopted in the real crystal by ∼5° due to absence of weak inter-stack interactions in our models. This small difference is very unlikely to strongly influence t,^4,11 and since the same relaxed inclination difference (within 1°) is found in both charged and neutral relaxed stacks, there will be negligible contribution to λ. From our previous preliminary study,¹⁸ we noted that the fixed embedding relaxed molecules in a small stack (S3) promotes two effects reducing λ_int: (i) relative reduction in the geometric freedom of neutral or charged molecules induced by the steric hindrance of the embedding, (ii) reduction in the geometric change of a charged molecule via charge delocalisation. In general, λ may also be affected by intermolecular polarisation (both of a molecular and electronic nature), dependent on difference between the polarisation energy of the positively charged molecule in its relaxed cationic state and the same positively charged molecule being in a relaxed neutral configuration. The polarisation in each case is dependent on the geometry, position and orientation of the positive molecule with respect to its environment. The degree of polarisation is also dependent on the degree of localisation of the charge such that it is only significant for relatively localised carriers and for fully-delocalised carriers the contribution to λ will tend to zero. In the case of strongly localised carriers, the single charged molecule distorts towards its ionic conformation influencing its interactions and thus geometries of its molecular neighbours through short-range intermolecular interactions. This molecular polarisation gives rise to a corresponding (nearly) small molecular polaron.⁴ Localised charge carriers also electronically polarise neighbouring molecules, which can, in principle, be over much longer length scales. The energy estimation of a so polarised molecular crystal is important with respect to the electronic and transport properties of many OMCs.⁴ Due to difficulties in allowing for molecular polarisation and delocalisation of charge over many molecules, calculations of long range polarisation are currently limited to OMCs with vanishing intermolecular overlap.²⁰ These approaches are probably unsuitable for TTF-derivatives, such as the DT–TTF system under study, which have significant S⋯S overlap. In this study we concentrate on effects due to local molecular and electronic polarisation. We note that, although we know of no other attempt to incorporate the effects of local molecular and electronic polarisation in the evaluation of λ for OMCs, such considerations are relatively common in studies of electron transfer in solvated/biological systems.²¹ It is also noted that in totally neutral systems, electronic polarisation from the local molecular environment can be important for evaluating on-site energies in OMCs.²² Although such effects are likely to be small in our system where all molecules in the perfect crystal stack are symmetrically equivalent, our use of extended models should give a good account of any such polarisation effects that may possibly arise upon energy-minimising the stack geometries.

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.

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.

Results and discussion

Fig. 3 and Table 1 show how the reorganisation energy varies in carrier models S1–S4[L, SL, D]. From S1 to S2 the decrease in λ_int is due to the geometric restriction of the neutral molecule to be closer to its planar relaxed charged state. This structural restriction is similarly effective at reducing λ_int with respect to S1 in all cases. From S2 to S3 the geometric restriction is not explicitly enforced but is more realistically induced by fixed nearest neighbour molecules allowing for energy changes due to charge delocalisation and polarisation. The S3[L] model localises the hole almost solely on the CM, S3[D] displays an almost totally delocalised solution, and S3[SL] a weakly-localised polaronic state formed over all three molecules. For each different carrier type (L, SL, D) a similar order of magnitude drop in λ_int is observed (lower branches in Fig. 3) from S2 to S3 (L: −0.288 eV, SL: −0.233 eV, D: −0.196 eV). From S2 to S3, in all cases, the geometry of the relaxed neutral state molecule (CM in S3) remains relatively unchanged with respect to bond distances (<0.004 Å) and planarity (<1.8°) showing that the change in λ_int is mainly due to differences in the relaxed charged state molecular conformation and/or due to non-geometry-perturbing electronic effects. In the S3[D] relaxed charged system, the CM relaxed geometry is close to that of the corresponding neutral CM due to charge delocalisation explaining the reduction in λ_int.¹⁸ For S3[L], where the positive charge is localised on the CM and its geometry is hardly changed from the isolated charged molecule, the drop in λ_int is due to electronic polarisation of the geometrically fixed but electronically responsive local environment. To quantitatively evaluate the contributions of charge delocalisation and polarisation in each S2 to S3 λ_int decrease, we may partition λ_int of S3 in two complementary ways: (i) considering energy differences between the charged CM in the neutral and relaxed geometries from the S3 calculations without the embedding molecules, and (ii) considering the energy of an isolated charged molecule and two isolated embedding molecules from S3 (from the relaxed and neutral calculations) and comparing with the corresponding energies of the combined S3 system (bars in Fig. 3 summarise the partitioning). The mechanism of λ_intreductionvia delocalisation is only dominant for S3[D]. As the localisation of the carrier is increased in S3[SL], polarisation becomes the main mechanism of λ_totalreduction; responsible for 81% of the decrease with respect to S2. In the highly localised S3[L] model, delocalisation accounts for only 4% of the λ_int drop from S2 to S3[L]. Estimating the electronic polarisation energy of a charge localised on the CM in S3 via the energy drop in placing a point positive charge between two neutral DT–TTF molecules gives a very similar value (1.125 ± 0.03 eV) for all models indicating that polarisation effects are invariant to the Hamiltonian employed. For all S3 calculations (L, SL, D), the CM geometry in the relaxed neutral stack is similarly close to experiment (0.062 ± 0.008 Å RMS difference) showing a consistent geometric description. Due to the fixed embedding molecules, the spacing between molecules in S3 also matches that from experiment and is the same in each S3 calculation (3.99 Å). We take the intermolecular distance as the average between the four atoms laying on the C=C axis of each neighbouring molecule, so as to avoid the influence of the small tilt described above.

Fig. 3 Change in λ for S1–S4[D, SL, L]). Open symbols show λ_int of pentacene. Bars show the percentage contribution of intermolecular polarisation (light)versus delocalisation (dark) in the S2–S3 decrease in λ (lower branches).

Table 1 Reorganisation energies (eV) calculated using various charge carrier models for DT-TTF (S1-S4[D,SL,L]) and for pentacene using a single molecule. The values follow the lower branches of each series of points (S1–S4) in Fig. 3. (Note: the (S1-S3)[D] values differ by a very small amount (≤0.006 eV) to those in ref. 18 due to a slightly improved integration grid in the present study)

	[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

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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 λ_int, 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 λ_int 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 λ_int 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 λ_intreduction by 0.063 eV. For S3[L] the effect is severe and the predicted values of λ_int range up to slightly above that of the respective S2 value showing the high sensitivity of λ_int 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, λ_int 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, λ_int, 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 λ_total. 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]λ_total 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 λ_total(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, λ_total 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]λ_total values are also very sensitive to local environmental fluctuations (e.g. via thermal molecular vibrations) thus making it unlikely that the lowest λ_int 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 λ_total 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 λ_int 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 λ_total/low temperature) band-like picture and (high λ_total/high temperature) small polarons, where short-range intermolecular delocalisation provides sufficiently thermally resistant low λ_total 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.

Conclusion

In summary, our calculations show: (i)λ in OMCs may be strongly reduced by the local environment through both molecular and electronic polarisation, (ii) the extent of the reduction depends on the degree of localisation of the charge carriers and how they interact with their local environment. Although our analysis has concentrated on DT–TTF, giving indirect evidence to support the fact that it possesses partially delocalised carriers, we believe that our results are likely to be relevant to the general understanding of transport in high mobility OMCs. We note, for example, that rubrene, possessing one of the highest measured mobilities of any OMC thus far discovered, has a rather high value of λ with respect to that of pentacene based upon calculations considering only the isolated molecule.³⁴ We suggest that, following a similar analysis as outlined in the present study, the effects of the local molecular environment will prove to be very important in reducing this bare molecule λ estimate, thus contributing to explaining the observed high carrier mobilities in rubrene. In highlighting the importance of local molecular environment we hope that our results may help suggest new OMCs for organic field effect devices. In future extensions of our method (e.g. including more extended and thermally activated molecular environments) we aim to improve upon the present models and to provide more accurate insights into the charge carriers in OMCs and their dependence on the molecular environment.

Acknowledgements

Financial support from the Spanish Ministerio de Ciencia y Tecnologia (projects NBA05-33-001 (F.I.), CTQ2005-08459-CO2-01 (F.I. and S.T.B.), CTQ2006-06333/BQU (M.M), project-EMOCIONa (M.M)) and, in part, from the Generalitat de Catalunya (Project 2005SGR-00697 and Distinció per a la Promoció de la Recerca Universitària de la Generalitat de Catalunya granted to F.I.) is fully acknowledged.

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