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1 In the format provided by the authors and unedited. DOI: /NMAT4970 A map of high-mobility molecular semiconductors S. Fratini 1, S. Ciuchi 2, D. Mayou 1, G. Trambly de Laissardière 3 & A. Troisi 4 1 Institut Néel CNRS and Université Grenoble Alpes, F Grenoble, France 2 Department of Physical and Chemical Sciences University of L Aquila, Via Vetoio, I L Aquila, Italy & CNR-ISC Via dei Taurini, I Rome, Italy 3 Laboratoire de Physique Théorique et Modélisation, CNRS, Université de Cergy-Pontoise, F Cergy-Pontoise, France and 4 Dept. Chemistry, University of Liverpool, Liverpool L69 7ZD, United Kingdom Contents I. Theoretical models for charge transport 2 A. Transient localisation theory 2 Factors controlling the mobility 2 Methods 4 Relation with IPR 4 B. Non-adiabatic hopping 5 C. Semiclassical band theory 6 II. Comparison with experiments 7 A. Mobility data from the literature 7 B. Experimental validation of transient localisation theory 8 III. ab initio determination of microscopic parameters 8 A. Evaluation of J and J/J 8 B. Correlations between Js 10 C. Fluctuation time τ 11 References 11 NATURE MATERIALS 1

2 I. THEORETICAL MODELS FOR CHARGE TRANSPORT A. Transient localisation theory The ensemble of tight-binding models defined in Fig. 1a is studied via the transient localisation theory as described in [1, 2]. For each choice of the pristine parameters J a,j b,j c, the first step consists in setting up a disordered lattice by sampling the transfer integrals from a gaussian distribution of width J around these average values, and computing the time dependent quantum spread X 2 (E,t)= [ ˆX(t) ˆX(0)] 2 E for states of energy E. From this quantity on can then directly calculate the mobility of the actual (dynamically disordered) material via the Kubo formalism, using the relaxation time approximation (RTA) of Ref. [1]. The spread attained by states of energy E after a typical inter-molecular oscillation time τ is defined as l 2 τ(e) = (1/τ) X 2 (E,t)e t/τ dt. This is illustrated in Fig. 1b of the main manuscript together with the density of states (DOS) of the random lattice for a given choice of microscopic parameters. The transient localisation length is defined by the statistical average L 2 τ = l 2 τ(e), assuming Boltzmann statistics for the carriers (since we are considering holes, the dominant contributions to the average come from the top edge of the band). The mobility is then expressed as in Eq. (1) of the main text, that we report here for convenience µ = e L 2 τ k B T 2τ. (1) The dependence of the transient localisation length on the fluctuation time is illustrated in Fig. S1(a). In the explored time range, this quantity increases as a power law L 2 τ τ α with α < 1, following the well-known subdiffusive behaviour of localised systems at intermediate times. Convergence to the actual localisation length of the statically disordered lattice is very slow for 2D isotropic band structures (estimated to L in the example shown, via a logarithmic extrapolation of the diffusivity at long times [3]) but becomes much faster for anisotropic lattices where localisation effects are stronger. Factors controlling the mobility We now analyse all possible factors that control the mobility within transient localisation theory. The different parameters of the theory are: the absolute values of the transfer integrals J, the ratios between the Js along different bond directions, their relative fluctuations J/J, the fluctuation time τ, the temperature T and the lattice parameters. Effect of the Js. This is the main focus of the main manuscript and is summarised in Figs. 1c and 1d. Our results show that the mobility is very much affected by the relative values of the Js, but only weakly by their absolute values. This is in sharp contrast with both hopping-like (µ J 2 ) and semiclassical band-like transport (µ 1/m J). The weak dependence of the mobility on the magnitude of J can be traced back to one of the most salient features of high-mobility organic semiconductors: the coexistence of highly conducting extended band states together with poorly conducting tail states that are generated by disorder near the band edges [6]. As illustrated in Fig. 1b of the main manuscript, the squared localisation length varies from values around one lattice NATURE MATERIALS 2

3 a L 2 L 2 / b L 2 / Table I FIG. S1: (a) Dependence of the statistically averaged transient localisation length on the fluctuation time τ, calculated at T =0.25J for an isotropic electronic structure with J a = J b = J c = J/ 3, J =0.1eV and disorder amplitude J/J =0.5 (length in units of the lattice parameter a, time in units of h/j). The gray arrow indicates the localisation length attained in the static disorder limit and the red bar is the interval of τj/ h ratios derived from the data in Table I. The dashed line is the ratio L 2 τ /τ that enters in Eq. (1). (b) The quantity L 2 τ /τ across the ensemble, evaluated at three different values of the fluctuation time τ = 10, 20, 40 h/j, corresponding to h/τ =2.5, 5, 10 mev. unit in the tails, to values 100 times larger within the band. As J increases, because the population of electronic states is controlled via the Boltzmann distribution by the ratio k B T/J, the proportion of extended states that can be thermally populated and participate to transport is progressively reduced [1]. This causes a reduction of the statistically averaged L 2 τ, cancelling the benefits of increasing J that would be predicted in the case of semiclassical transport. Effect of the transfer integral fluctuations J/J. It is well known by now that improved materials can be designed if the dynamical off-diagonal disorder J/J is overall reduced. This was proposed theoretically [1, 2] on the basis of Eq. (1), and verified experimentally in Refs. [4, 5]. Fig. 4 of the main manuscript shows that a reduction of dynamical disorder by 20% causes an increase of the mobility by a factor of about for all compounds in the studied ensemble. Values of the transfer integral fluctuations calculated from first principles for a number of compounds are reported in Table I. For all molecular compounds studied, the J/J averaged over all bond directions lies within the range (TIPS-pentacene being an exception, with anomalously large molecular disorder). Because our focus is on how different electronic structures respond to a given level of disorder, in the manuscript we have taken a constant value J/J =0.5 throughout the ensemble. Effect of τ. The dependence of the mobility on the molecular fluctuation time is weak, as illustrated in Fig. S1(b). This happens because L 2 τ increases with fluctuation time in the subdiffusive regime of relevance here, as shown in Fig. S1(a), which partly NATURE MATERIALS 3

4 cancels the explicit factor τ in the denominator. With L 2 τ τ α and α =0.7 in the example shown, the ratio L 2 τ/τ depends on the fluctuation time as a weak power law τ 0.3. Analogous exponents arise throughout the ensemble of models except for strictly one dimensional systems, where α drops to zero. Moreover, τ does not change very much across compounds as reported in section III.B and Table I of this SI. Effect of T. The temperature dependence is illustrated in Fig. 3a of the main manuscript. Effects of the lattice parameters. Finally, Eq. (2) shows that the mobility has a trivial explicit dependence on the inter-molecular distances. Because the dependence is quadratic, doubling all bond lengths will result in a fourfold increase in mobility. In the main manuscript we have considered for simplicity a structure where all bond lengths are the same. The effects of different lattice structures can be trivially included in the calculations by restoring the appropriate lengths as provided by Xray diffraction experiments. Interestingly, all compounds studied here can be separated in two clearly distinct categories. In systems where the molecules are standing perpendicular to the high mobility plane, the denser allowed packing leads to generally shorter bond distances. This is exemplified in Table I where we report the corresponding area per molecule: pentacene (24.28 Å 2 ), C10-DNTT (22.83), C10-DNBDT (23.62). In all other compounds, the molecules are organised with the long axis lying close to the high mobility plane, leading to larger inter-molecular distances. The corresponding areas per molecule in this second class are typically larger by a factor of 2 or more: rubrene (51.84 Å 2 ), TESADT (47.37), dif-tesadt (46.34), TMTES-Pn (54.80), TIPS-Pn (58.24). To compare all systems in a single universal plot and include these trivial (albeit quantitatively important) factors, in Fig. 2 we have reported the experimental values of the mobility divided by the corresponding area per molecule, which appropriately scales as the average bond distance squared. Methods The numerical simulations of the dynamical spread X 2 (E,t) performed in this work use a real-space method based on orthogonal polynomials of the Hamiltonian. The method takes into account exactly all quantum processes, including Anderson localisation effects caused by lattice randomness, and allows to treat efficiently system sizes of 10 7 sites and above, which were not attainable with the method used in Refs. [1, 2]. Historically it has been developed in several stages and applied to various models such as amorphous systems [7, 8], quasicrystals [9, 10] and graphene-based systems (see Ref. [11] and references therein). It is applied here for the first time to organic semiconductors. The present simulations are performed on clusters consisting of up to molecules, which is sufficient for convergence at the considered levels of disorder and temperature. Relation with IPR Because the calculation of L 2 τ in large samples requires non-trivial numerical methods as described above, we also propose here a simpler approach for a faster qualitative screening of NATURE MATERIALS 4

5 FIG. S2: Comparison of the squared transient localisation length L 2 τ at τ =0.05 h/j and the static IPR of disorder-induced tail states, calculated for the ensemble of models considered in the main manuscript, indexed by the angle θ. devices, which consists in calculating the inverse participation ratio (IPR) of the parent statically disordered lattice. This is defined as IPR= ( i i ψ i 4 ) 1, with ψ the wavefunction and i the state localized on each lattice point, and can be accessed straightforwardly by performing exact diagonalisations of the ensemble of tight-binding models defined in Fig. 1a assuming a static gaussian distribution of transfer integrals of width J. The diagonalisations shown in Fig. S2 are performed on clusters of 3200 molecules, sufficiently large not to affect the IPR at the edge region of the DOS (note instead that full convergence of the IPR within the band requires much larger sample sizes). The reported IPR is the value calculated at the Fermi energy for a fixed density of 1 carrier per 125 molecules, i.e. well within the disorder-induced tails. While this quantity does not allow to make quantitative predictions for the mobility (because, at variance with the L 2 τ introduced above, it does not contain information on the dynamics of molecules), the IPR of the tail states near the band edge is conceptually related to the (dynamical) transient localisation length introduced above (gray arrow in Fig. S1 (a)) and the main trends are correctly reproduced across the ensemble, as shown in Fig. S2. B. Non-adiabatic hopping We provide here further details on the calculations reported in Fig. 2a of the main manuscript, where we computed the relative variation of the charge mobility in the case of non-adiabatic hopping between nearest neighbours, which include for example Marcus or Jortner formulations of the charge hopping rate [12]. The hopping time should be much slower than the vibrational relaxation time for any hopping theory to be consistent. As the vibrational relaxation time is typically 1psin liquids and 10 ps in organic solids [13, 14] one concludes that the fluctuations of the transfer integrals with characteristic timescale of 1 ps are faster than the hopping time. In NATURE MATERIALS 5

6 this limit the hopping rate at a constant temperature is proportional to the squared average of the hopping integrals (in all non-adiabatic theories). In systems like liquid crystals or amorphous phases where the fluctuation time can become slower than the hopping rate one needs to proceed differently [15]. For the model considered in this work with a constant standard deviation σ = J/J =0.5 we have J 2 a = J a 2 + σ 2 =1.25J 2 a (and similarly for the couplings J b and J c ). Taking the hopping rates to be proportional to the squared coupling in all directions, we now evaluate the diffusivity tensor by performing a convolution of the diffusion in the three directions, as described in the SI of Ref. [16]. In brief, for a 2D hexagonal lattice, where the three hopping vectors form angles θ 1 = 0, θ 2 = π/3,θ 3 =2π/3 with respect to the x axis, the diffusivity tensor D can be expressed in terms of the diffusion coefficient along the three hopping directions D 1, D 2, D 3 (each proportional to the squared average transfer integral) as: where D 1 = V 1 (V 1 + V 2 ) 1 V 2 (V 1 (V 1 + V 2 ) 1 V 2 + V 3 ) 1 V 3 [ ] 1/D1 0 V 1 = 0 1/ε [ ][ ][ cos θ2 sin θ V 2 = 2 1/D2 0 cos θ2 sin θ 2 cos θ 2 0 1/ε sin θ 2 ] sin θ 2 cos θ 2 [ ][ ][ cos θ3 sin θ V 3 = 3 1/D3 0 cos θ3 sin θ 3 cos θ 3 0 1/ε sin θ 3 ] sin θ 3 cos θ 3 and ɛ is an arbitrary small number. The main manuscript reports the trace of the diffusivity tensor (proportional to the mobility tensor) multiplied by an arbitrary constant. C. Semiclassical band theory For the band theory calculations presented in the text we consider the same ensemble of tight-binding models and evaluate the mobility to lowest-order in the fluctuations of the transfer integral as described in the SI of Ref. [6]. We first express the carrier-lattice interaction Hamiltonian in second quantisation as, H I = (1/N ) k,q i=a,b,c g(i) k,q c+ k+q c kx (i) q, with N the number of molecules, c + k,c k the creation and annihilation operators for carriers, x (i) q the deformation mode corresponding to a given bond direction. Straightforward algebra allows to write the interaction matrix elements for uncorrelated bond disorder as [g (i) k,q ]2 = 4(dJ/dx) 2 cos[(k q/2) δ i ], with δ i the vectors connecting nearest-neighbours as shown in Fig. 1a and dj/dx the sensitivity of the transfer integrals to inter-molecular deformations. The semiclassical mobility is expressed as µ(t )=(e/k B T ) v 2 kτ k = e nk B T vkτ 2 k e +(ɛ k µ)/k B T where τ k and v k are respectively the scattering time and the band velocity for states of momentum k, µ is the chemical potential and n = k e+(ɛk µ)/t the thermally populated carrier density (the + sign in the exponent is for holes). In the quasi-elastic limit where the k (2) NATURE MATERIALS 6

7 intermolecular vibration frequency sets the smallest energy scale in the problem, hω 0 T,J the scattering time is obtained to second order in H I as 1/τ k = 2k BT dq gk,k+q 2 δ(ɛ k ɛ k+q ). (3) hω 0 Note that in the above expression we have omitted for simplicity the correction factor from the scattering angle between incoming and outgoing states that defines the transport scattering time, (1 cos θ k,k+q ), which however only amounts to a numerical correction factor of order 1 [6]. Since the considered band structures are anisotropic we consider vk 2 and consequently µ as tensors. To appropriately compare the semiclassical results with the RTA calculations we write the classical (thermal) fluctuation of the transfer integrals as ( J) 2 =(dj/dx) 2 k B T/K using the equipartition principle, with K the spring constant, which univocally fixes dj/dx and g (i) k,q for a given value of J/J. II. COMPARISON WITH EXPERIMENTS A. Mobility data from the literature In this section we justify the selected range of experimental mobility reported in the main manuscript for each material. Given the large spread of values that can be found in the literature, we have considered only materials for which a mobility decreasing with temperature has been demonstrated and showing good reproducibility between different groups, as these are as close to the ideal intrinsic behaviour as can be achieved today. A summary is given in Table I. Rubrene. Charge mobility data from different groups are similar despite different fabrication techniques. Considering few recent measurements, the Frisbie group [17] reported mobility in the cm 2 /Vs range (for a number of samples) along the high mobility direction with an air gap connection with the gate. The Batlogg group [18] reported an identical range with cytop as dielectric material. To compare across materials we have to consider the average mobility in the high mobility plane. The anisotropy ratio (µ xx /µ yy ) for rubrene was consistently estimated around 2.7 (see Ref. [19] and references therein). Thus a reasonable experimental mobility averaged over the plane is (12.5 ± 2.5) (1+1/2.7)/2 =8.6 ± 1.7 cm 2 /Vs. This value is very similar to the first high mobility of rubrene reported by Podzorov et al.[20]. Pentacene. Although this material is less broadly used now there are also fairly consistent reports of its FET mobility. On SiO 2 gate the values reported are 1 cm 2 /Vs (Ref. [21]), in the range of cm 2 /Vs depending on the orientation [22] and 2.2 cm 2 /Vs in Ref. [23]. Solution processed pentacene devices are reported to have mobilities in the cm 2 /Vs range [24]. Similar values coupled to Hall effect measurements (proving the close-tointrinsic nature of charge transport in pentacene) are reported in [25]. A fairly representative experimental range of mobility used to compare with the theory is 1.45 ± 0.85 cm 2 /Vs. TIPS-Pentacene (6,13-bis(triisopropylsilylethynyl)pentacene). The reported values are consistent in range. Ref. [4] reports µ<1 cm 2 /Vs; Ref. [26] reports 0.7 cm 2 /Vs; Ref. [27] reports 0.202±0.012 cm 2 /Vs. A range that captures these observations is therefore 0.6±0.4 cm 2 /Vs. NATURE MATERIALS 7

8 TMTES-Pentacene (1,4,8,11-Tetramethyl-6,13-triethylsilylethynylpentacene). We use 2.5± 0.5 cm 2 /Vs, considering the range 2 3 cm 2 /Vs given in Ref. [4] and the value of 2.5 cm 2 /Vs from a separate measurement of the same authors [28]. C 10 -DNTT (2,9-di-decyl-dinaphtho-[2,3-b:20,30-f]-thieno-[3,2-b]thiophene). We have used the value (reported with no error) of 8.5 cm 2 /Vs from Ref. [29]. C 10 -DNBDT (3,11-didecyldinaphto[2,3-d:2,3 -d ]benzo[1,2-b:4,5-b ]dithiophene). The Takeya group reports 12.1 cm 2 /Vs with standard deviation 1.4 cm 2 /Vs in Ref. [30] and 9.7 cm 2 /Vs in Ref. [5]. We use the range with two standard deviations 12.1 ± 2.8 cm 2 /Vs. TESADT (5,11-bis(triethylsilylethynyl) anthradithiophene). We take the value µ =1.5±0.5 cm 2 /Vs from Ref. [4]. dif-tesadt (2,8-Difluoro-5,11-bis(triethylsilylethynyl) anthradithiophene), which is the fluorinated analogue of TESADT. We take the value µ =3.5 ± 0.5 cm 2 /Vs from Ref. [4], consistent with the one reported in Ref. [31] B. Experimental validation of transient localisation theory In Figs. 1c and 1d of the main manuscript we have illustrated which patterns of transfer integrals are beneficial to the mobility of organic semiconductors. In doing so, and to address the whole class of materials on the same footing, we have kept a number of microscopic parameters fixed across the ensemble: the absolute scale J of the transfer integrals, their fluctuations J/J with respect to the mean value, the molecular fluctuation time τ and the lattice geometry. The effects of varying the disorder strength J/J and the lattice geometry have been investigated in the main text. Concerning the remaining parameters, their limited effect on the mobility was rationalised by the arguments presented in Sec. I A. The predictive power of transient localisation theory can however be demonstrated to a higher level of accuracy by supplementing it by material-specific ab initio calculations, releasing the assumptions of a constant parameter set. To this aim we have calculated all the microscopic parameters of the theory for a number of compounds (see Table I and Section III) and correspondingly determined the quantity L 2 τ/τ that governs the mobility. On the experimental side, in order to minimize the well known variations in mobility observed in devices realised with different methods, on different substrates [32] and measured by different groups [33], we have selected a homogeneous set of materials of the same chemical class (second block in Table I), that have been purified and measured in virtually the same way by Illig and collaborators [4]. The mobilities are thought by the authors to be close to intrinsic also on the basis of Hall mobility measurements performed on two of the materials (TIPS-Pn and TMTES-Pn) in a similar device configuration [28]. As shown in Fig. S3, by combining ab initio methods and transient localisation theory one can reach an excellent quantitative agreement with the transport experiments in field effect transistors when the uncertainties inherent to the experimental methods are appropriately controlled. III. AB INITIO DETERMINATION OF MICROSCOPIC PARAMETERS A. Evaluation of J and J/J Data on the average transfer integrals used to validate the theory in the main manuscript (Figure 1d) are summarized in Table I. All transfer integrals reported in Table I have been NATURE MATERIALS 8

9 dif-tesadt TMTES-Pn TESADT TIPS-Pn FIG. S3: Test of the proportionality between measured mobility and computed L 2 τ /τ using a homogeneous set of measurements on a group of molecules of the same chemical class appeared in Ref. [4]. L 2 τ was computed using the 2D lattice geometry of the actual crystal. τ was evaluated for each given material as 0.15, 0.22, 0.18 and 0.15 ps for TIPS-Pn, TMTES-Pn, TESADT and dif-tesadt respectively. The values of J and J were computed at the same level of theory (see Table I). This calculation, unlike the spherical map in the main manuscript, takes into account the different parameters J and J/J in different bond directions. computed with the method described in [34] using the B3LYP density functional [35] and the 6-31G* basis set. The comparison with experiments presented in Fig. S3 for a homogeneous set of experimental data with additional material-specific parameters required the evaluation of the transfer integral fluctuations J in all bond directions. This was performed as described several times in the past [36, 37] by carrying out a classical MD simulation using the MM3 force field, at 300 K, and at constant volume with lattice parameters taken from the crystallographic structure. For the molecules considered in Figure S3 the following supercells were considered for TMTES-Pn and dif-tesadt, and for TESADT or TIPS-Pn. To evaluate the characteristic fluctuations time τ, the transfer integrals have been computed for MD snapshots separated by 0.2 ps (0.12 for TIPS-P) along a dynamics of 12 ps (these are reported in Figure S5). For a more accurate evaluation of J, the transfer integrals have been computed for a longer dynamics of 40 ps or more with snapshots taken every 1 ps (0.6 ps for TIPS-Pn). The transfer integral for pentacene (at the same level B3LYP/6-31G*) derives from ref. [38], while J/J was evaluated with the semiempirical method ZINDO in [36] [42]. The data of rubrene have been computed as described in ref.[37] but at the B3LYP/6-31G* level for consistency with the other data presented here. J and J were computed with the same method for C10-DNBDT and C10-DNTT using and supercells respectively and sampling the MD trajectory every 1 ps. To verify that the transfer integrals (specifically their relative magnitude and J/J) are not too dependent on the computational details we have performed test calculations on a subset of structures for all molecules computed in this work with a much larger basis set (6- NATURE MATERIALS 9

10 TABLE I: Summary of the materials considered, the experimental range of mobility (in cm 2 /Vs), the hopping integrals (in mev), the relative disorder, the value of θ used for Fig. 1d of the main text, the fluctuation period 2πτ (in ps, when available) obtained from molecular dynamics simulations and the area per molecule in the plane of high mobility (in Å 2 ). Material µ exp J a ( J a /J a ) J b ( J b /J b ) J c ( J c /J c ) θ 2πτ area Rubrene 8.6 ± (0.282) 21.2 (0.497) 21.2 (0.497) Pentacene 1.45 ± (0.327) (0.330) (0.234) C 10 DNTT (0.210) (0.647) (0.647) C 10 DNBDT 12.1 ± (0.380) 59.1 (0.592) 59.1 (0.592) TIPS-pentacene 0.6 ± (0.569) 8.6 (3.25) 10.3 (5.07) TMTES-pentacene 2.5 ± (0.264) 26.9 (0.458) 27.3 (0.355) TESADT 1.5 ± (0.516) -165 (0.351) (0.862) dif-tesadt 3.5 ± (0.60) -185 (0.261) -55.9(0.650) G*) obtaining extremely consistent results (correlation r 2 between the transfer integrals was in the range for all materials). A similar comparison with calculations performed with the 6-31G* basis set and the PBE [39] functional also yields an excellent correlation (r 2 =0.9964) with the PBE transfer integrals typically smaller by 12% for TIPS- Pn. This level of agreement between electronic structure methods is common (see e.g. ref. [40]) and guarantees that the application of this methodology is not much influenced by the inaccuracies of electronic structure calculations. B. Correlations between Js The variations of the transfer integrals involving a common molecule and due to a localized (e.g. optical) mode are in principle correlated. However, in the main manuscript we have considered a model where the transfer integrals in the different bond directions are essentially uncorrelated, which we explain in the following. Different modes may have a positive or negative correlation, as illustrated with visually simple examples in Ref. [41]. If the number of modes is very large it seems reasonable that, due to the presence of both positive and negative correlation, the overall correlation is small. To carefully check this assumption we computed the r 2 correlation coefficient for transfer integrals between pairs of molecules sharing a common molecule for the four molecules considered in Figure S3. The largest values we found were 0.092, 0.091, 0.055, 0.022, for TESADT, TIPS-Pn, dif-tesadt, TMTES- Pn, suggesting that correlations can be neglected (Figure S4 shows the most correlated transfer integral pair). A similar lack of correlation was found in the past for pentacene [36]. A sample of the instantaneous correlations for TESADT along a MD trajectory is shown in Fig. S4. NATURE MATERIALS 10

11 J 13 / mev J 12 / mev FIG. S4: Instantaneous correlations between the transfer integrals in different bond directions and involving a common molecule, calculated for TESADT from the quantity J 12 (t)j 13 (t) along a given MD trajectory. C. Fluctuation time τ On the basis of previous studies showing a distribution of low frequency modes typically peaked at hω 0 5 mev in several materials, the characteristic time of the transfer integral fluctuation was set to a constant τ =1/ω 0 =0.13 ps in the main manuscript (this value corresponds to a period of molecular oscillation 2πτ =0.82 ps). To validate this assumption we have computed a time series of transfer integrals along a short portion of an MD simulation for 4 different compounds (transfer integrals computed every 0.2 ps, sufficient to address the fluctuation spectrum in the relevant frequency range). The time evolution of the transfer integrals as well as the fluctuation power spectrum are illustrated in Fig. S5. The period 2πτ for the different materials is reported in Table I and was derived from the frequency of the highest peak in the power spectrum for each molecule (for TESADT the average frequency between the two maxima of similar height was considered). [1] Ciuchi, S. & Fratini, S., Electronic transport and quantum localization effects in organic semiconductors. Phys. Rev. B 86, (2012). [2] Fratini, S., Mayou, D. & Ciuchi, S. The transient localization scenario for charge transport in crystalline organic materials. Adv. Funct. Mater. 26, , (2016). [3] Ciuchi, S., Fratini, S. & Mayou, D. Transient localization in crystalline organic semiconductors. Phys. Rev. B 83, (R) (2011). [4] Illig, S. et al. Reducing dynamic disorder in small-molecule organic semiconductors by suppressing large-amplitude thermal motions. Nature Comms 7, (2016). [5] Kubo, T. et al. Suppressing molecular vibrations in organic semiconductors by inducing strain Nature Comms 7, (2016). [6] Fratini, S. & Ciuchi, S. Bandlike motion and mobility saturation in organic molecular semi- NATURE MATERIALS 11

12 80 60 TIPS-Pn TMTES-Pn J / mev 20 0 J / mev t / ps t / ps 0-50 TESADT dif-tesadt J / mev J / mev t / ps t / ps dif-tesadt TESADT TIPS-Pn TMTES-Pn Intensity / a.u (2πτ) -1 / ps -1 FIG. S5: (top) Time evolution of the transfer integrals along MD trajectories and (bottom) the corresponding power spectrum calculated for TIPS-Pn, TMTES-Pn, TESADT and dif-tesadt. conductors. Phys. Rev. Lett. 103, (2009). [7] Mayou, D. Calculation of the conductivity in the short-mean-free-path regime. Europhys. Lett. 6, (1988). [8] Mayou, D. & Khanna, S.N. A real-space approach to electronic transport. Journal de Physique I 5, (1995). [9] Roche, S. & Mayou, D. Formalism for the computation of the RKKY interaction in aperiodic systems. Phys. Rev. B 60, (1999). [10] Triozon, F., Vidal, J., Mosseri, R. & Mayou, D. Quantum dynamics in two- and threedimensional quasiperiodic tilings. Phys. Rev. B 65, (R) (2002). [11] Trambly de Laissardière, G. & Mayou, D. Conductivity of graphene with resonant and non- NATURE MATERIALS 12

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14 materials with stabilized crystal phase. Adv. Mater. 26, 4546?4551 (2014). [31] Niazi, N. R., et al. Solution-printed organic semiconductor blends exhibiting transport properties on par with single crystals. Nature Communications 6, 8598 (2015). [32] Hulea, I. N. et al. Tunable Fröhlich Polarons in Organic Single-Crystal Transistors. Nature Mater. 5, 982 (2006). [33] McCulloch, I., Salleo, A., & Chabinyc, M. Avoid the kinks when measuring mobility. Science 352, (2106). [34] Troisi, A. & Orlandi, G. The hole transfer in DNA: calculation of electron coupling between close bases. Chem. Phys. Lett. 344, (2001). [35] Becke, A. D. Density-functional thermochemistry. III. The role of exact exchange. J. Chem. Phys (1993). [36] Troisi A. & Orlandi G. Dynamics of the intermolecular transfer integral in crystalline organic semiconductors. J. Phys. Chem. A 110, (2006). [37] Troisi, A. Prediction of the absolute charge mobility of molecular semiconductors: the case of rubrene. Adv. Mater. 19, 2000 (2007). [38] Troisi A. & Orlandi G. Band Structure of the four pentacene polymorphs and effect on the hole mobility at low temperature. J. Phys. Chem. B 109, (2005). [39] Perdew, J. P., Burke, K. and Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett (1996). [40] Huang, J. S., Kertesz, M. Validation of intermolecular transfer integral and bandwidth calculations for organic molecular materials. J. Chem. Phys (2005). [41] Li, Y., Yi, Y., Coropceanu, V. & Brédas, J.-L. Symmetry effects on nonlocal electron-phonon coupling in organic semiconductors. Phys. Rev. B 85, (2012). [42] In the main manuscript the experimental mobilities are compared with the prediction of our theory for a generic material with hopping integrals [J a, J a, J b ] whose coordinate on the semiconductor map is expressed by the coordinate of the angle θ defined in Fig. 1. Any two hopping integrals can be interchanged and/or multiplied (both) by 1 and an angle θ can be defined for all materials for which two hopping integrals are identical or close. For pentacene we have reported the value of θ =2.538 corresponding to the point on the cut that is closest to its actual position on the spherical map. Alternatively we could have used the value of θ =2.494 corresponding to a point on the map with the corresponding transient localisation length. NATURE MATERIALS 14