SUPPLEMENTARY INFORMATION - Nature Research · The ensemble of tight-binding models deﬁned in Fig. 1a is studied via the transient localisation theory as described in [1, 2]. For

In the format provided by the authors and unedited.

A map of high mobility molecular semiconductors- Supporting Information -

S. Fratini1, S. Ciuchi2, D. Mayou1, G. Trambly de Laissardiere3 & A. Troisi4

1 Institut Neel CNRS and Universite Grenoble Alpes,

F-38042 Grenoble, France2 Department of Physical and Chemical Sciences University of L’Aquila,

Via Vetoio, I-67100 L’Aquila,

Italy & CNR-ISC Via dei Taurini,

I-00185 Rome, Italy3 Laboratoire de Physique Theorique et Modelisation,

CNRS, Universite de Cergy-Pontoise,

F-95302 Cergy-Pontoise, France

and4 Dept. Chemistry, University of Liverpool,

Liverpool L69 7ZD, United Kingdom

Contents

I. Theoretical models for charge transport 2A. Transient localisation theory 2

Factors controlling the mobility 2Methods 4Relation with IPR 4

B. Non-adiabatic hopping 5C. Semiclassical band theory 6

II. Comparison with experiments 7A. Mobility data from the literature 7B. Experimental validation of transient localisation theory 8

III. ab initio determination of microscopic parameters 8A. Evaluation of J and ∆J/J 8B. Correlations between Js 10C. Fluctuation time τ 11

References 11

1

A map of high-mobility molecular semiconductors

© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NMAT4970

NATURE MATERIALS | www.nature.com/naturematerials 1

http://dx.doi.org/10.1038/nmat4970

I. THEORETICAL MODELS FOR CHARGE TRANSPORT

A. Transient localisation theory

The ensemble of tight-binding models defined in Fig. 1a is studied via the transientlocalisation theory as described in [1, 2]. For each choice of the pristine parameters Ja, Jb, Jc,the first step consists in setting up a disordered lattice by sampling the transfer integrals froma gaussian distribution of width ∆J around these average values, and computing the timedependent quantum spread ∆X2(E, t) = 〈[X(t)−X(0)]2〉E for states of energy E. From thisquantity 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 oscillationtime τ is defined as �2τ (E) = (1/τ)

∫∆X2(E, t)e−t/τdt. This is illustrated in Fig. 1b of

the main manuscript together with the density of states (DOS) of the random lattice for agiven choice of microscopic parameters. The transient localisation length is defined by thestatistical average L2

τ = 〈�2τ (E)〉, assuming Boltzmann statistics for the carriers (since weare considering holes, the dominant contributions to the average come from the top edge ofthe band). The mobility is then expressed as in Eq. (1) of the main text, that we reporthere for convenience

µ =e

kBT

L2τ

2τ. (1)

The dependence of the transient localisation length on the fluctuation time is illustrated inFig. S1(a). In the explored time range, this quantity increases as a power law L2

τ ∝ τα withα <∼ 1, following the well-known subdiffusive behaviour of localised systems at intermediatetimes. Convergence to the actual localisation length of the statically disordered lattice isvery slow for 2D isotropic band structures (estimated to L2 � 103 in the example shown,via a logarithmic extrapolation of the diffusivity at long times [3]) but becomes much fasterfor anisotropic lattices where localisation effects are stronger.

Factors controlling the mobility

We now analyse all possible factors that control the mobility within transient localisationtheory. The different parameters of the theory are: the absolute values of the transfer inte-grals 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 summarisedin Figs. 1c and 1d. Our results show that the mobility is very much affected bythe relative values of the Js, but only weakly by their absolute values. This is insharp contrast with both hopping-like (µ ∝ J2) 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 oneof the most salient features of high-mobility organic semiconductors: the coexistenceof highly conducting extended band states together with poorly conducting tail statesthat are generated by disorder near the band edges [6]. As illustrated in Fig. 1b of themain manuscript, the squared localisation length varies from values around one lattice

2





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L2⌧/⌧

Table I

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a b

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 Ja = Jb = Jc = 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 L2τ/τ that enters in Eq. (1). (b) The quantity L2

τ/τ across the ensemble,

evaluated at three different values of the fluctuation time τ = 10, 20, 40h/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, becausethe population of electronic states is controlled — via the Boltzmann distribution —by the ratio kBT/J , the proportion of extended states that can be thermally populatedand participate to transport is progressively reduced [1]. This causes a reduction ofthe statistically averaged L2

τ , cancelling the benefits of increasing J that would bepredicted in the case of semiclassical transport.

• Effect of the transfer integral fluctuations ∆J/J . It is well known by now that im-proved materials can be designed if the dynamical off-diagonal disorder ∆J/J is overallreduced. This was proposed theoretically [1, 2] on the basis of Eq. (1), and verifiedexperimentally in Refs. [4, 5]. Fig. 4 of the main manuscript shows that a reductionof dynamical disorder by 20% causes an increase of the mobility by a factor of about1.5 − 2 for all compounds in the studied ensemble. Values of the transfer integralfluctuations calculated from first principles for a number of compounds are reportedin Table I. For all molecular compounds studied, the ∆J/J averaged over all bonddirections lies within the range 0.3 − 0.6 (TIPS-pentacene being an exception, withanomalously large molecular disorder). Because our focus is on how different elec-tronic structures respond to a given level of disorder, in the manuscript we have takena 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 L2

τ increases with fluctuation timein the subdiffusive regime of relevance here, as shown in Fig. S1(a), which partly

3





cancels the explicit factor τ in the denominator. With L2τ ∝ τα and α = 0.7 in the

example shown, the ratio L2τ/τ depends on the fluctuation time as a weak power law

τ−0.3. Analogous exponents arise throughout the ensemble of models except for strictlyone dimensional systems, where α drops to zero. Moreover, τ does not change verymuch 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 mainmanuscript.

• Effects of the lattice parameters. Finally, Eq. (2) shows that the mobility has atrivial explicit dependence on the inter-molecular distances. Because the dependenceis 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 bondlengths are the same. The effects of different lattice structures can be trivially includedin the calculations by restoring the appropriate lengths as provided by Xray diffractionexperiments. Interestingly, all compounds studied here can be separated in two clearlydistinct categories. In systems where the molecules are standing perpendicular tothe high mobility plane, the denser allowed packing leads to generally shorter bonddistances. This is exemplified in Table I where we report the corresponding areaper molecule: pentacene (24.28 A2), C10-DNTT (22.83), C10-DNBDT (23.62). In allother compounds, the molecules are organised with the long axis lying close to the highmobility plane, leading to larger inter-molecular distances. The corresponding areasper molecule in this second class are typically larger by a factor of 2 or more: rubrene(51.84 A2), 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 (albeitquantitatively important) factors, in Fig. 2 we have reported the experimental valuesof the mobility divided by the corresponding area per molecule, which appropriatelyscales as the average bond distance squared.

Methods

The numerical simulations of the dynamical spread ∆X2(E, t) performed in this workuse a real-space method based on orthogonal polynomials of the Hamiltonian. The methodtakes into account exactly all quantum processes, including Anderson localisation effectscaused by lattice randomness, and allows to treat efficiently system sizes of 107 sites andabove, which were not attainable with the method used in Refs. [1, 2]. Historically it hasbeen 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 areperformed on clusters consisting of up to 320000 molecules, which is sufficient for convergenceat the considered levels of disorder and temperature.

Relation with IPR

Because the calculation of L2τ in large samples requires non-trivial numerical methods as

described above, we also propose here a simpler approach for a faster qualitative screening of

4





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FIG. S2: Comparison of the squared transient localisation length L2τ at τ = 0.05h/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 stat-ically disordered lattice. This is defined as IPR= (

∑i |〈i|ψi〉|4)−1, with |ψ〉 the wavefunction

and |i〉 the state localized on each lattice point, and can be accessed straightforwardly byperforming exact diagonalisations of the ensemble of tight-binding models defined in Fig. 1aassuming a static gaussian distribution of transfer integrals of width ∆J . The diagonalisa-tions shown in Fig. S2 are performed on clusters of 3200 molecules, sufficiently large not toaffect the IPR at the edge region of the DOS (note instead that full convergence of the IPRwithin the band requires much larger sample sizes). The reported IPR is the value calculatedat the Fermi energy for a fixed density of 1 carrier per 125 molecules, i.e. well within thedisorder-induced tails. While this quantity does not allow to make quantitative predictionsfor the mobility (because, at variance with the L2

τ introduced above, it does not containinformation on the dynamics of molecules), the IPR of the tail states near the band edge isconceptually related to the (dynamical) transient localisation length introduced above (grayarrow in Fig. S1 (a)) and the main trends are correctly reproduced across the ensemble, asshown in Fig. S2.

B. Non-adiabatic hopping

We provide here further details on the calculations reported in Fig. 2a of the mainmanuscript, where we computed the relative variation of the charge mobility in the case ofnon-adiabatic hopping between nearest neighbours, which include for example Marcus orJortner formulations of the charge hopping rate [12].

The hopping time should be much slower than the vibrational relaxation time for anyhopping theory to be consistent. As the vibrational relaxation time is typically ∼ 1 ps inliquids and ∼ 10 ps in organic solids [13, 14] one concludes that the fluctuations of thetransfer integrals with characteristic timescale of 1 ps are faster than the hopping time. In

5





this limit the hopping rate at a constant temperature is proportional to the squared averageof the hopping integrals (in all non-adiabatic theories). In systems like liquid crystals oramorphous phases where the fluctuation time can become slower than the hopping rate oneneeds to proceed differently [15]. For the model considered in this work with a constantstandard deviation σ = ∆J/J = 0.5 we have 〈J2

a〉 = 〈Ja〉2 + σ2 = 1.25J2a (and similarly for

the couplings Jb and Jc).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 threedirections, as described in the SI of Ref. [16]. In brief, for a 2D hexagonal lattice, where thethree hopping vectors form angles θ1 = 0, θ2 = π/3, θ3 = 2π/3 with respect to the x axis,the diffusivity tensor D can be expressed in terms of the diffusion coefficient along the threehopping directions D1, D2, D3 (each proportional to the squared average transfer integral)as:

D−1 = V1(V1 +V2)−1V2(V1(V1 +V2)

−1V2 +V3)−1V3

where

V1 =

[1/D1 00 1/ε

]

V2 =

[cos θ2 sin θ2− sin θ2 cos θ2

] [1/D2 00 1/ε

] [cos θ2 − sin θ2sin θ2 cos θ2

]

V3 =

[cos θ3 sin θ3− sin θ3 cos θ3

] [1/D3 00 1/ε

] [cos θ3 − sin θ3sin θ3 cos θ3

]

and ε is an arbitrary small number. The main manuscript reports the trace of the diffusivitytensor (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 ensembleof tight-binding models and evaluate the mobility to lowest-order in the fluctuations ofthe transfer integral as described in the SI of Ref. [6]. We first express the carrier-lattice

interaction Hamiltonian in second quantisation as, HI = (1/N)∑

k,q

∑i=a,b,c g

(i)k,qc

+k+qckx

(i)q ,

with N the number of molecules, c+k , ck 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) · δi], with δi the vectors connecting nearest-neighbours as shown inFig. 1a and dJ/dx the sensitivity of the transfer integrals to inter-molecular deformations.

The semiclassical mobility is expressed as

µ(T ) = (e/kBT )〈v2kτk〉 =e

nkBT

∑k

v2kτke+(εk−µ)/kBT (2)

where τk and vk are respectively the scattering time and the band velocity for states ofmomentum k, µ is the chemical potential and n =

∑k e

+(εk−µ)/T the thermally populatedcarrier density (the + sign in the exponent is for holes). In the quasi-elastic limit where the

6





intermolecular vibration frequency sets the smallest energy scale in the problem, hω0 � T, Jthe scattering time is obtained to second order in HI as

1/τk =2kBT

hω0

∫dq g2k,k+q δ(εk − εk+q). (3)

Note that in the above expression we have omitted for simplicity the correction factor fromthe scattering angle between incoming and outgoing states that defines the transport scat-tering time, (1 − cos θk,k+q), which however only amounts to a numerical correction factorof order 1 [6].

Since the considered band structures are anisotropic we consider v2k and consequently µas tensors. To appropriately compare the semiclassical results with the RTA calculations wewrite the classical (thermal) fluctuation of the transfer integrals as (∆J)2 = (dJ/dx)2kBT/Kusing 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 themain manuscript for each material. Given the large spread of values that can be foundin the literature, we have considered only materials for which a mobility decreasing withtemperature has been demonstrated and showing good reproducibility between differentgroups, as these are as close to the ideal intrinsic behaviour as can be achieved today. Asummary is given in Table I.

Rubrene. Charge mobility data from different groups are similar despite different fabricationtechniques. Considering few recent measurements, the Frisbie group [17] reported mobilityin the 10−15 cm2/Vs range (for a number of samples) along the high mobility direction withan air gap connection with the gate. The Batlogg group [18] reported an identical rangewith cytop as dielectric material. To compare across materials we have to consider theaverage mobility in the high mobility plane. The anisotropy ratio (µxx/µyy) for rubrene wasconsistently estimated around 2.7 (see Ref. [19] and references therein). Thus a reasonableexperimental mobility averaged over the plane is (12.5 ± 2.5) × (1 + 1/2.7)/2 = 8.6 ± 1.7cm2/Vs. This value is very similar to the first high mobility of rubrene reported by Podzorovet al.[20].Pentacene. Although this material is less broadly used now there are also fairly consistentreports of its FET mobility. On SiO2 gate the values reported are ∼ 1 cm2/Vs (Ref. [21]),in the range of 0.6− 2.3 cm2/Vs depending on the orientation [22] and 2.2 cm2/Vs in Ref.[23]. Solution processed pentacene devices are reported to have mobilities in the 0.6 − 2.3cm2/Vs range [24]. Similar values coupled to Hall effect measurements (proving the close-to-intrinsic nature of charge transport in pentacene) are reported in [25]. A fairly representativeexperimental range of mobility used to compare with the theory is 1.45± 0.85 cm2/Vs.TIPS-Pentacene (6,13-bis(triisopropylsilylethynyl)pentacene). The reported values are con-sistent in range. Ref. [4] reports µ < 1 cm2/Vs; Ref. [26] reports ∼ 0.7 cm2/Vs; Ref. [27]reports 0.202±0.012 cm2/Vs. A range that captures these observations is therefore 0.6±0.4cm2/Vs.

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TMTES-Pentacene (1,4,8,11-Tetramethyl-6,13-triethylsilylethynylpentacene) . We use 2.5±0.5 cm2/Vs, considering the range 2−3 cm2/Vs given in Ref. [4] and the value of 2.5 cm2/Vsfrom a separate measurement of the same authors [28].C10-DNTT (2,9-di-decyl-dinaphtho-[2,3-b:20,30-f]-thieno-[3,2-b]thiophene). We have usedthe value (reported with no error) of 8.5 cm2/Vs from Ref. [29].C10-DNBDT (3,11-didecyldinaphto[2,3-d:2’,3’-d’]benzo[1,2-b:4,5-b’]dithiophene). TheTakeya group reports 12.1 cm2/Vs with standard deviation 1.4 cm2/Vs in Ref. [30] and9.7 cm2/Vs in Ref. [5]. We use the range with two standard deviations 12.1± 2.8 cm2/Vs.TESADT (5,11-bis(triethylsilylethynyl) anthradithiophene). We take the value µ = 1.5±0.5cm2/Vs from Ref. [4].diF-TESADT (2,8-Difluoro-5,11-bis(triethylsilylethynyl) anthradithiophene), which is thefluorinated analogue of TESADT. We take the value µ = 3.5 ± 0.5 cm2/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 transferintegrals are beneficial to the mobility of organic semiconductors. In doing so, and to addressthe whole class of materials on the same footing, we have kept a number of microscopicparameters fixed across the ensemble: the absolute scale J of the transfer integrals, theirfluctuations ∆J/J with respect to the mean value, the molecular fluctuation time τ and thelattice geometry. The effects of varying the disorder strength ∆J/J and the lattice geometryhave been investigated in the main text. Concerning the remaining parameters, their limitedeffect on the mobility was rationalised by the arguments presented in Sec. I A.

The predictive power of transient localisation theory can however be demonstrated toa 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 allthe microscopic parameters of the theory for a number of compounds (see Table I and SectionIII) and correspondingly determined the quantity L2

τ/τ that governs the mobility. On theexperimental side, in order to minimize the well known variations in mobility observedin devices realised with different methods, on different substrates [32] and measured bydifferent groups [33], we have selected a homogeneous set of materials of the same chemicalclass (second block in Table I), that have been purified and measured in virtually the sameway by Illig and collaborators [4]. The mobilities are thought by the authors to be close tointrinsic 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, bycombining ab initio methods and transient localisation theory one can reach an excellentquantitative agreement with the transport experiments in field effect transistors when theuncertainties 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

8





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TMTES-Pn

diF-TESADT

TESADT

TIPS-Pn

FIG. S3: Test of the proportionality between measured mobility and computed L2τ/τ using a

homogeneous set of measurements on a group of molecules of the same chemical class appeared in

Ref. [4]. L2τ 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] andthe 6-31G* basis set.

The comparison with experiments presented in Fig. S3 for a homogeneous set of ex-perimental data with additional material-specific parameters required the evaluation of thetransfer integral fluctuations ∆J in all bond directions. This was performed as describedseveral times in the past [36, 37] by carrying out a classical MD simulation using the MM3force field, at 300 K, and at constant volume with lattice parameters taken from the crys-tallographic structure. For the molecules considered in Figure S3 the following supercellswere considered 4 × 3 × 3 for TMTES-Pn and diF-TESADT, and 4 × 4 × 3 for TESADTor TIPS-Pn. To evaluate the characteristic fluctuations time τ , the transfer integrals havebeen computed for MD snapshots separated by 0.2 ps (0.12 for TIPS-P) along a dynamics of12 ps (these are reported in Figure S5). For a more accurate evaluation of ∆J , the transferintegrals have been computed for a longer dynamics of 40 ps or more with snapshots takenevery 1 ps (0.6 ps for TIPS-Pn). The transfer integral for pentacene (at the same levelB3LYP/6-31G*) derives from ref. [38], while ∆J/J was evaluated with the semiempiricalmethod 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 3× 3× 3and 4× 3× 3 supercells respectively and sampling the MD trajectory every 1 ps.

To verify that the transfer integrals (specifically their relative magnitude and ∆J/J) arenot too dependent on the computational details we have performed test calculations on asubset of structures for all molecules computed in this work with a much larger basis set (6-

9





TABLE I: Summary of the materials considered, the experimental range of mobility (in cm2/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 A2).

Material µexp Ja (∆Ja/Ja) Jb (∆Jb/Jb) Jc (∆Jc/Jc) θ 2πτ area

Rubrene 8.6± 1.7 140.7 (0.282) 21.2 (0.497) 21.2 (0.497) 0.21 0.7 51.84

Pentacene 1.45± 0.85 51.0 (0.327) -74.3 (0.330) 130.6 (0.234) 2.54 - 24.28

C10−DNTT 8.5 109.8 (0.210) -62.5 (0.647) -62.5 (0.647) 0.678 - 22.83

C10−DNBDT 12.1± 2.8 76.8 (0.380) 59.1 (0.592) 59.1 (0.592) 0.83 - 23.62

TIPS-pentacene 0.6± 0.4 -0.8 (0.569) 8.6 (3.25) 10.3 (5.07) 1.63 0.9 58.24

TMTES-pentacene 2.5± 0.5 258.0 (0.264) 26.9 (0.458) 27.3 (0.355) 0.15 1.4 54.80

TESADT 1.5± 0.5 3.84 (0.516) -165 (0.351) -48.6 (0.862) - 1.1 47.37

diF-TESADT 3.5± 0.5 1.86 (0.60) -185 (0.261) -55.9(0.650) - 0.9 46.34

311+G*) obtaining extremely consistent results (correlation r2 between the transfer integralswas in the 0.989-0.999 range for all materials). A similar comparison with calculationsperformed with the 6-31G* basis set and the PBE [39] functional also yields an excellentcorrelation (r2=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 theinaccuracies 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 haveconsidered a model where the transfer integrals in the different bond directions are essentiallyuncorrelated, which we explain in the following. Different modes may have a positive ornegative correlation, as illustrated with visually simple examples in Ref. [41]. If the numberof modes is very large it seems reasonable that, due to the presence of both positive andnegative correlation, the overall correlation is small. To carefully check this assumptionwe computed the r2 correlation coefficient for transfer integrals between pairs of moleculessharing a common molecule for the four molecules considered in Figure S3. The largest valueswe 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 correlatedtransfer 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 inFig. S4.

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-300

-250

-200

-150

-100

-50

-120 -80 -40 0 40J 1

3 / m

eV

J12 / meV

FIG. S4: Instantaneous correlations between the transfer integrals in different bond directions and

involving a common molecule, calculated for TESADT from the quantity 〈J12(t)J13(t)〉 along a

given MD trajectory.

C. Fluctuation time τ

On the basis of previous studies showing a distribution of low frequency modes typicallypeaked at hω0 ≈ 5 meV in several materials, the characteristic time of the transfer integralfluctuation was set to a constant τ = 1/ω0 = 0.13 ps in the main manuscript (this valuecorresponds to a period of molecular oscillation 2πτ = 0.82 ps). To validate this assump-tion we have computed a time series of transfer integrals along a short portion of an MDsimulation for 4 different compounds (transfer integrals computed every 0.2 ps, sufficient toaddress the fluctuation spectrum in the relevant frequency range). The time evolution ofthe 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 thefrequency of the highest peak in the power spectrum for each molecule (for TESADT theaverage frequency between the two maxima of similar height was considered).

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crystalline organic materials. Adv. Funct. Mater. 26, 2292–2315, (2016).

[3] Ciuchi, S., Fratini, S. & Mayou, D. Transient localization in crystalline organic semiconductors.

Phys. Rev. B 83, 081202(R) (2011).

[4] Illig, S. et al. Reducing dynamic disorder in small-molecule organic semiconductors by sup-

pressing large-amplitude thermal motions. Nature Comms 7, 10736 (2016).

[5] Kubo, T. et al. Suppressing molecular vibrations in organic semiconductors by inducing strain

Nature Comms 7, 11156 (2016).

[6] Fratini, S. & Ciuchi, S. Bandlike motion and mobility saturation in organic molecular semi-

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-300

-250

-200

-150

-100

-50

0

0 2 4 6 8 10 12

J / m

eV

t / ps

TESADT

-60

-40

-20

0

20

40

60

80

0 2 4 6 8 10 12

J / m

eV

t / ps

TIPS-Pn

-280

-260

-240

-220

-200

-180

-160

-140

-120

-100

-80

0 2 4 6 8 10 12J

/ meV

t / ps

diF-TESADT

100

150

200

250

300

350

400

450

0 2 4 6 8 10 12

J / m

eV

t / ps

TMTES-Pn

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Inte

nsity

/ a.

u.

(2πτ)-1 / ps-1

diF-TESADTTESADTTIPS-Pn

TMTES-Pn

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.

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[42] In the main manuscript the experimental mobilities are compared with the prediction of our

theory for a generic material with hopping integrals [Ja, Ja, Jb] 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.

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SUPPLEMENTARY INFORMATION - Nature Research · The ensemble of tight-binding models deﬁned in Fig. 1a is studied via the transient localisation theory as described in [1, 2]. For