arX

iv:1

108.

5263

v1 [

cond

-mat

.str

-el]

26

Aug

201

1

Electronic transport within a quasi two-dimensional model for rubrene single-crystal

field effect transistors

F. Gargiulo,1, ∗ C.A. Perroni,1, 2 V. Marigliano Ramaglia,1, 2 and V. Cataudella1, 2

1Dipartimento di Scienze Fisiche, Univ. di Napoli “Federico II”, I-80126 Italy2CNR-SPIN

Spectral and transport properties of the quasi two-dimensional adiabatic Su-Schrieffer-Heegermodel are studied adjusting the parameters in order to model rubrene single-crystal field effecttransistors with small but finite density of injected charge carriers. We show that, with increasingtemperature T , the chemical potential moves into the tail of the density of states corresponding tolocalized states, but this is not enough to drive the system into an insulating state. The mobilityalong different crystallographic directions is calculated including vertex corrections which give rise toa transport lifetime one order of magnitude smaller than spectral lifetime of the states involved in thetransport mechanism. With increasing temperature, the transport properties reach the Ioffe-Regellimit which is ascribed to less and less appreciable contribution of itinerant states to the conductionprocess. The model provides features of the mobility in close agreement with experiments: rightorder of magnitude, scaling as a power law T−γ , with γ close or larger than two, and correctanisotropy ratio between different in-plane directions. Due to a realistic high dimensional model,the results are not biased by uncontrolled approximations.

PACS numbers:

I. INTRODUCTION

In recent years, the interest in plastic electronics hasgrown considerably. In particular, organic field effecttransistors (OFET) are the most common devices em-ployed to characterize the transport properties of the or-ganic semiconductors. Single-crystal FET constructedusing ultrapure small molecule semiconductors have al-lowed to measure mobilities up to one order of magni-tude larger then those typical of thin film transistors.1,2

Among them, rubrene OFETs are much studied.Transport measurements from low to room tempera-

ture in single crystal OFETs show a behaviour of thecharge carrier mobility µ which can be defined “band-like”(µ ∝ T−γ, with the exponent γ close to two) similar tothat observed in crystalline inorganic semiconductors.2

The observation of the classical Hall effect is also ex-plained in terms of the presence of relatively free chargecarriers, so that the charge conduction is incompatiblewith hopping transport between localized states.2 How-ever, the order of magnitude of mobility is much smallerthan that of pure inorganic semiconductors, and themean free path for the carriers has been theoreticallyestimated to be comparable with the molecular separa-tion with increasing temperature reaching the Ioffe-Regellimit.3

Extended vs. localized features of charge carriers ap-pear also in spectroscopic observations. Angle resolvedphotoemission spectroscopy (ARPES) supports the ex-tended character of states.4,5 Indeed, photoemission ex-periments suggest that the quasi-particle energy disper-sion does exhibit a weak mass renormalization even if thewidth of the peaks increases significantly with tempera-ture. On the other hand, some spectroscopic probes, suchas electron paramagnetic resonance (EPR),6,7 THz,8 andmodulated spectroscopy9 are in favor of states localized

within few molecules. The theoretical explanation of thephysical mechanism underlying the conduction proper-ties is still not fully understood.10

One of the main problems is to conciliate band-likewith localized features of charge carriers.11 In particu-lar, the nature of conduction at room temperature withmobilities close to the Ioffe-Regel limit remains contro-versial. Actually, in contrast with other polyacenes, inrubrene, to ascribe the presence of localized features tosmall polarons is not likely since the electron-phonon (el-ph) coupling is not large enough to justify the polaronformation.10

A model that is to some extent close to theSu-Schrieffer-Heeger (SSH)12 one has been recentlyintroduced.11 It is a one-dimensional (1D) system wherethe effect of the electron-phonon coupling is reduced toa modulation of the transfer integral induced by low fre-quency intermolecular modes. This model applies to themost conductive crystal axis of high mobility systems,such as rubrene.13 It has been studied by using a dy-namic approach where vibrational modes are treated asclassical variables. This approach has been recently gen-eralized to two dimensions.14 Within this method, thecomputed mobility is in agreement with experimental re-sults. However, the role of dimensionality of the system isnot clear: in fact, in the 1D case, one has µ ∝ T−2, while,in the 2D case, the decrease of the mobility with temper-ature is intermediate between µ ∝ T−2 and µ ∝ T−1.Moreover, this approach is not satisfying from differentpoints of view. First, the dynamics of only one chargeparticle is studied neglecting completely the role of thechemical potential. Then, the dynamic disorder on thecharge carrier due to the coupling with vibrational modesis included in a peculiar way: in fact, the electron is notcoupled to classical gaussian fields with time dependentcorrelation functions which are determined by the con-

2

tact of the oscillators with an external bath.15 In thiscase, the total system is composed only by one electron(or hole) and vibrational modes, therefore it is not clearif the corresponding coupled dynamics recovers the ther-mal equilibrium on long times and if the Einstein rela-tion can be properly used to determine the conductivityfrom the diffusivity. Finally, if the low frequency modesare not assumed much slower than the electron dynamicsbut occur in the same timescale, the quantum nature ofphonons cannot be neglected.

Recently, the transport properties of the 1D SSHmodel have been analyzed within a different adiabaticapproach.16 Due to the geometry of OFET, in which theorganic semiconductor is on the gate, one can assumethat the oscillators are at the thermodynamic equilib-rium, therefore the problem is mapped onto that of a sin-gle quantum particle in a random potential (generalizedAnderson problem).17 However, in order to have finiteresults of conductivity, vertex corrections have been com-pletely neglected. Very recently, some of us have made asystematic study of this 1D model including the vertexcorrections.18 While finite frequency quantities are prop-erly calculated in this 1D model, the inclusion of vertexcorrections leads to a vanishing mobility unless an "ad-hoc" broadening of the energy eigenvalues is assumed.

In this paper, we generalize a previous work19 in whichsome of usf have analyzed the properties of electrons cou-pled to local modes within the Holstein model by meansof the adiabatic approximation.20,21. We formulate anextension of the 1D SSH model to the quasi 2D casesince this is the relevant geometry for OFETs geometry.Moreover, the use of a high dimensional realistic modelallows to include the physical anisotropy as an essentialfeature and to overcome the difficulties due to the lo-calization of all the states in reduced dimensions.17 Asdiscussed in Appendix, the proper scaling towards thethermodynamic limit is performed for the quasi 2D mod-els investigated in this paper.

An important characteristic of our study is the smallbut finite carrier density with the introduction of thechemical potential. With increasing temperature, thechemical potential goes towards the tail of localizedstates. Therefore, all the experimentally quantitiesstrongly dependent on the position of the chemical posi-tion will probe the features of localized states. The studyof spectral properties clarify that the states that mainlycontribute to the conduction process have low momen-tum and are not at the chemical potential. The mobilityµ is studied as a function of the el-ph coupling and parti-cle density. The behavior of mobility is in agreement withexperiments. Not only the order of magnitude and theanisotropy ratio between different directions are right,but also the temperature dependence of µ is correctlyreproduced since it scales as a power law T−γ with γclose or larger than two. The inclusion of vertex correc-tions is relevant in the calculation, in particular to geta transport lifetime one order smaller than the spectrallifetime of the states involved in the transport mecha-

nism. Therefore, high order corrections in two-particlecorrelation functions enhance the interaction effects onthe properties of charge carriers. Actually, with increas-ing temperature, the Ioffe-Regel limit is reached sincethe contribution of itinerant states to the conduction be-comes less and less relevant.

The paper is organized as follows. In section II, themodel and approach are discussed in detail. The spectraland transport properties are discussed in section III andIV, respectively. Finally, section V contains conclusionsand further discussions.

II. MODEL AND CALCULATION METHOD

The model studied in this paper include the anisotropyof small molecule organic semiconductors due to its highdimensionality. The following Hamiltonian defines themodel:

H =∑

Ri

1

2ku2

Ri+∑

Ri

1

2Mu2

Ri+Hel, (1)

where uRiis the classical oscillator displacement at the

site in the position Ri, uRithe classical oscillator velocity

at the site in the position Ri, M the oscillator mass, kthe elastic constant. The whole lattice dynamic is due toa single effective phononic mode whose frequency ω0 =√

K/M is assumed of the order of 5− 6meV .13

Since we are mostly interested in dc conductivity andlow frequency spectral properties, we need to determinean effective Hamiltonian for the electron degrees of free-dom Hel in Eq. (1 valid for low energy and particle den-sity. We start from the orthorhombic lattice of rubrenewith two molecules per unit cell.10 The dispersion lawof the lowest among the two highest occupied molecularorbital (HOMO) derived bands is

ε (k) = −2ha cos (kaa)− 2hb cos (kbb)− 2hc cos (kcc)

−2h a+b2

cos

(

kaa+ kbb

2

)

− 2h a−b2

cos

(

kaa− kbb

2

)

(2)

dropping the site energy diagonal terms. The quanti-ties a, b, c represent the lattice parameter lengths alongthe three crystallographic vectors of the conventional cell.We assume a = 7.184, b = 14.443, c = 26.897.10 We fol-low the experimental paper by Ding et al. in order toextract the transfer integrals.4 The lowest HOMO de-rived band is measured through ARPES yielding thehopping parameters ha = 62.5meV , hb = 12.5meV ,h a+b

2

= h a−b2

= 57.5meV . The dispersion law at kc = 0

is drawn in figure (1). For small values of ka and kb, thisgraph strictly resembles a paraboloid. It is clear that aneffective dispersion law

ε (k) = −2ta cos (kaa)− 2tb cos (kbb)

is capable to fit well the dispersion law (2) for smallvalues of ka and kb and kc = 0. Fit estimates give

3

−0.4

−0.2

0

0.2

0.4−0.4

−0.2

0

0.2

0.4

−380

−375

−370

−365

−360

−355

−350E

nerg

y (m

eV)

ka*akb*b

Ene

rgy

(meV

)

−380

−375

−370

−365

−360

−355

Figure 1: Dispersion law ε (k) for rubrene at kc = 0.

ta = 118.6meV and tb = 68.6meV . For tc there is noexperimental measure but theoretical estimates seem toagree that it should be small compared with other direc-tions and it is likely owing to the large interplanar sep-aration of rubrene. In the following we assume tc muchsmaller than ta and tb. As we will discuss later, vary-ing its value seems not too affect much the calculatedmobilities.

The electronic part Hel is then

Hel = −∑

Ri,~δ

t~δ (Ri)[

a+Ri+

~δaRi

]

. (3)

t~δ (Ri) = t~δ − α~δ

(

uRi− u

Ri+~δ

)

~δ = ~a,~b,~c (4)

In the last formula a+Ri

(aRi) represents the creation (de-

struction) of a charge carrier in the site Ri, t~δ is the bare

effective transfer integral in the direction ~δ, α~δthe el-ph

parameter that controls the effect on the transfer inte-

gral of the ion displacements in the direction ~δ. Oncefixed αa, we impose αb/αa = tb/ta and, in the same way,αc/αa = tc/ta.

The dynamic contained in the term Hel is fully quan-tum, while we remind that the lattice obeys a classicaldynamic. This assumption is justified by the adiabaticratio ~ω/ta ≃ 0.05. Therefore, the results discussed inthis paper are valid starting from temperatures such thatT > ~ω/kB, with kB Boltzmann constant. In the adia-batic limit the dimensionless quantity λ

λ =α2a

4kta(5)

is the only relevant parameter to quantify the el-ph cou-pling strength. An ab-initio estimate13 of λ0 = 0.087has been given for rubrene and it has been substan-tially confirmed later.22,23 This represents our startingpoint, but a study of the mobility behaviour when λraises has been performed and is reported in the fol-lowing sections. In fact, if λ0 is claimed to be a goodvalue for the coupling in one dimension, keeping in

mind the increase of the kinetic energy with dimension-ality, the equivalent λ should be higher in three di-mensions. Simulations will focus on temperature sig-nificantly lower than ta/kB because this is the typicalrange investigated experimentally in OFETs. Moreover,in most OFETs the induced charge carrier concentrationc = number of carriers/number of sites is rarely higherthan 0.01 and this is the upper limit for the concentra-tions in our simulations. As shown in a previous pa-per dealing with a very similar but 1D model,18 for thisrange of concentrations, the probability distribution ofdisplacements is only slightly renormalized. Therefore,it is acceptable to refer to a free-oscillator (Gaussian)distribution for the displacements:

P ({uRi}) =

(

2π

βK

)L2

exp

(

−βK

2

∑

i

u2Ri

)

(6)

where L = La ∗ Lb ∗ Lc is the finite size of the latticeand β = (kBT )

−1. We fix Lc = 2 (i.e. we consider two

crystalline layers) because in OFETs the effective channelof conductions cover very few planes.24 Moreover, it isworth reminding that tc is very small.

We use exact diagonalization to investigate both staticand dynamical properties for the hamiltonian (1). Atfixed configuration ({uRi

}), one has to diagonalize theHamiltonian yielding L eigenvalues Er, with r = 1, .., L.The eigenvector components Ui,r, with i = 1, .., L, aregiven through the unitary matrix U which diagonalizesthe electronic problem. Each observable results froman average on lattice configurations. For example, theelectron part of the partition function can be calcu-lated as 〈Zel ({uRi

})〉, where Zel ({uRi}) is the quan-

tum partition function of the electron subsystem at agiven displacement configuration {uRi

}.25 Similarly, thiswould provide approximation-free evaluation of observ-ables such as mobility and spectral function in the semi-classical limit.26 The real limitation is the accessible finitesize of the lattice. The computing time for each diagonal-ization is of order L3 and this constraints our analyses upto La = Lb = 24. In order to reduce the finite size effect,we use periodic boundary conditions. We have checkedthat, for all the static and dynamic quantities studiedin this paper, the thermodynamic limit is reached. Theaverages are performed by means of a Monte-Carlo pro-cedure. Actually, we generate a sequence of random num-bers distributed according to P ({uRi

}). For the systemsinvestigated in this paper, to get a good accuracy even fordynamic quantities we perform the averages on a numberof iterations up to ten thousands.

Within the same framework, it is also possible to cal-culate the spectral functions, the density of states, andthe conductivity. At fixed configuration, the density ofstates is D(ω) ({uRi

}):

D(ω) ({uRi}) =

1

L

∑

i

ARi,Ri(ω) ({uRi

}) , (7)

where ARi,Ri(ω) ({uRi

}) represents the diagonal term of

4

the spectral function

ARi,Rj(ω) ({uRi

}) =∑

r

b∗i,rbj,rδ (Er − ω) . (8)

The spectral function in momentum representationAk(ω) can be determined after performing the averagesover the lattice configurations. In the case of conductiv-ity, for each configuration, we calculate the exact Kuboformula27

Re [σρ,ρ (ω) ({uRi})] =

π(

1− e−βω)

Lω

∑

r 6=s

pr (1− ps)

|〈r |Jρ| s〉|2δ (Es − Er + ω) ,(9)

where ρ = a, b, c, pr is the Fermi distribution

pr =1

1 + exp (β (Er − µp))(10)

corresponding to the exact eigenvalue Er at any chemicalpotential µp and 〈r |Jρ| s〉 is the matrix element of thecurrent operator Jρ along the direction eρ, defined as

Jρ = ie∑

~Ri,~δ

t~δ (Ri)(

~δ · eρ

)

c†~Ri

c~Ri+~δ, (11)

with e electron charge. We notice that, in contrast withspectral properties, the temperature enters the calcula-tion not only through the displacement distribution, butalso directly for each configuration through the Fermi dis-tributions pr. The numerical calculation of the conduc-tivity is able to include the vertex corrections discardedby previous approaches.

Finally, the mobility is defined as

µρ = limω→0+

Re [σρ,ρ (ω)]

ec. (12)

For finite lattice sizes, the delta function appearing inspectral and transport properties has to be replaced witha Lorentzian, thus introducing a finite broadening η

δ (E + ω) = limη→0+

1

π

η

(E + ω)2+ η2

. (13)

Clearly, the limit procedure means that the value of ηshould be kept as small as possible. This does not giveany problem for quantities at finite frequency, as far asω > η. In the case of mobility, this procedure has tobe correctly implemented since there is the limit ω → 0.The correct expression is obtained performing the limitsin the following order: ω 7→ 0, η 7→ 0 and L 7→ ∞. Theselimits are not trivial, so that in the Appendix we havereported in detail the “calibration procedure” followed tofix the right η for the calculation of the mobility. Incontrast with the 1D case, in our system the mobilitydependence on the broadening η exhibits a quasi-plateauwith a maximum whose location gets closer to the zero

as the size of the lattice increases. This makes possi-ble to set η as the lowest energy scale of the system.In addition, it emerges quite clearly that, for the sizesconsidered in this work, the system is very close to thethermodynamic limit. This shows that the broadeningη is a parameter of the system which does not influencethe mobility estimates. A detailed analysis has allowedus to obtain values for ωmin and ηmin that assure the de-termination of the correct value of the mobility for anytemperature. For the mobilities presented in this paper,we used ωmin = 10−3ta, ηmin of the order of 2 ∗ 10−2ta.

In the following, we will measure energies in units of ta.We will use units such that Boltzmann constant kB = 1and Planck constant ~ = 1. We will consider the hoppingparameters for rubrene derived in this section, thereforetb = 0.58. Along the c axis, we assume tc = 0.18. Thesmall effects due to a change in the value of tc will be dis-cussed in the end of section about transport properties.Next spectral properties are analyzed.

III. SPECTRAL PROPERTIES

The study of spectral properties is important to indi-viduate the states that mainly contribute to the conduc-tion process. In Fig. 2, the Density Of States (DOS)is shown for different temperatures at λ = 0.12 andc = 0.002. As shown in logarithmic scale, the DOS hasa tail with a low energy exponential behavior. This re-gion corresponds to localized states.28 We have checkedanalyzing the wave functions extracted from exact diago-nalizations that, actually, states with energies deep in thetail are strongly localized (one or two lattice parametersalong the different directions as localization length). Onthe other hand, beyond the shoulder (see Fig. 2 for T1,T2 and T3), the itinerant nature of states is clearly ob-tained. This analysis gives rough indications for the mo-bility edge energy which can be located very close to theband edge for free electron Ec (in our case, Ec = −3.52).

The states available in the tail increase with temper-ature. It is important to analyze the role played by thechemical potential µp with varying the temperature. Ac-tually, µp enters the energy tail and will penetrate intoit with increasing temperature. At fixed particle densityc = 0.002, for T = 0.12, one has µp = −3.49, while,for T = 0.24, µp = −3.88 (see the inset of Fig. 2).One important point is that the quantity Ec and theclose mobility edge are always over µp. Therefore, inthe regime of low density relevant for OFET, the itiner-ant states are not at µ but at higher energies. We willshow that these states are relevant for the conductionprocess. Therefore, the analysis of the properties of ahigh dimensional model points out that both localizedand itinerant states are present in the system. This is aclear advantage of our work over previous studies in lowdimensionality11,14,16 in which there are states that aremore localized at very low energy and just less localized

5

-0.4 -0.2 0.0 0.2 0.4

0.01

0.1

Ec-

p(T

3)E

c-

p(T

2)

D

OS

p

T1=0.12 T2=0.18 T3=0.24

Ec-

p(T

1)

0.10 0.15 0.20 0.25

-3.8

-3.6

-3.4

T

Figure 2: The DOS as a function of the frequency (measuredfrom the chemical potential µp) at λ = 0.12 and c = 0.002 fordifferent temperatures. Ec is the band edge for free electrons.

-3.40 -3.35 -3.30 -3.25 -3.200

5

10

15

20

0.12 0.16 0.20 0.24

0.04

0.06

0.08

Ak=

0

T=0.10 T=0.16 T=0.20 T=0.24

T

Figure 3: The spectral function at k = 0 as a function of thefrequency at λ = 0.12 and c = 0.002 for different tempera-tures. In the inset, the widths at half height of the spectralfunction is plotted as function of the temperature.

close to the free electron edge. Apparently, in our sys-tem, with increasing temperature, more localized statesbecome available close to µp, and so the itinerant statesbecome statistically less effective due to the behavior ofthe chemical potential. Eventually, the effect of pene-tration of µp in the tail will overcome the enlargementof available itinerant states due to the Fermi statistics.We have checked that the trend of the chemical potentialtowards the energy region of the tail is enhanced withincreasing the el-ph coupling λ.

The density of states can be calculated as the sumof the spectral functions Ak over all the momenta k.

We have checked that the spectral functions with lowmomentum are more peaked, while, with increasing k,they tend to broaden.19 The tail in the DOS is due to amarked width of the high momentum spectral functions.In Fig. 3 we report the spectral function at k = 0 forthe same model parameters as the previous figure. Thelow damped states close to k = 0 will keep their itiner-ant character, and they will be involved into the diffusiveconduction process. Actually, we notice that the spectralfunction gives a negligible contribution to the weight ofDOS at the chemical potential for all the temperatures.For example, at T = 0.20, the spectral weight is concen-trated in an energy region higher than that in which thechemical potential is located (µp = −3.74 for c = 0.002).Actually, the spectral weight is in the region betweenµp + 2T and µp + 3T . Finally, we point out that, withincreasing temperature, the peak position of the spec-tral function is only poorly renormalized in comparisonwith the bare one, in agreement with results of 1D SSHmodel.18

It is interesting to estimate the lifetime of the statesat low momentum. The spectral functions at k = 0 shiftand broaden with increasing temperature due to the en-hanced role of the el-ph coupling. We have estimated thewidths at half height in the inset of Fig. 3. At intermedi-ate temperatures, the width is of the order of 0.05t, thatmeans of the order of 6 meV . Therefore, the lifetime ofthese states is of the order of 22 fs. In the next section,this lifetime will be compared with the transport lifetimederived from the mobility.

The analysis of the spectral properties has clarifiedimportant features of the chemical potential and of thestates participating at the transport process. Next, weinvestigate the "transition rules" between states involvedinto the conduction and analyze the mobility as a func-tion of the particle density and the el-ph coupling.

IV. TRANSPORT PROPERTIES

In this section, the focus will be on the Kubo formulafor the conductivity given in Eq. (9). We will start fromthe analysis of the square modulus matrix elements ofthe current operator |〈r |Ja| s〉|

2 along the direction a be-tween exact eigenstates r and s. In Fig. 4, we reportthe contour plot of the current matrix elements in whichthe two axes are linked to the energies of the eigenstates.We consider two different temperatures T = 0.12 andT = 0.24. The dark color point towards high values of thematrix elements, the bright color to low ones. The dif-ference in the intensity between bright and dark regionscorrespond to square modulus matrix elements differingfor about two orders of magnitude. The most importantfeatures emerging from the upper plot is that, in the en-ergy range important for low particle densities, there isa narrow region from about −3.4 to −3.3 in which thematrix elements give an appreciable contribution. More-over, states close in energy do not repel, but have sizable

6

−3.5 −3 −2.5 −2 −1.5

Er

−3.5

−3

−2.5

−2

−1.5

Es

0

50

100

150

200

250

−3.5 −3 −2.5 −2 −1.5

Er

−3.5

−3

−2.5

−2

−1.5

Es

0

20

40

60

80

100

120

Figure 4: Contour plot of square modulus matrix elements (inunits of e

2a2/~2) of the current operator along the direction a

between exact eigenstates at λ = 0.12 and T1 = 0.12 (upperpanel), T2 = 0.24 (lower panel) .

overlapping. These features are clearly ascribed to therole of itinerant states.28 We notice that there are sev-eral gaps in the intensities between −3.3 and −2.6, then,very high matrix elements occur going toward the cen-ter of the electronic band. At higher temperature (Fig.4, lower panel), the contour map is broader, the gapsare almost completely removed and the absolute magni-tude of the matrix elements is lower. We point out thethese "transition rules" are not strongly dependent onthe particle density. We stress that, only for density clarger than 0.3 − 0.4, the matrix elements are sizeable,and the conduction states are far from the tail of theDOS, therefore one would get a "metallic state" typicalof an inorganic compound. Indeed, both low density andel-ph interaction contribute to reduce the conducibilitiesof OFET in comparison with inorganic systems.

The set of parameters used for the calculation of themobility are discussed in Appendix A. Before calculatingthe mobility as a function of the temperature, we ana-lyze the mobility along the a direction as a function ofthe energy of the states which contribute to it. More pre-

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

−3.6 −3.4 −3.2 −3 −2.8 −2.6 −2.4

∆σ/∆

ω

ω

T=0.12

T=0.18

T=0.24

Figure 5: The conducibility contribution along the a directionas function of the energies of the "ingoing states" at λ = 0.12and c = 0.002 for different temperatures. The unit of theconducibility is µ0 where µ0 = ea2/~ = 5.16cm2/(V s).

cisely, in Eq. (9), we sum over the "outgoing states" sand analyze the results in terms of the "ingoing states"r. We again investigate the regime of low density rele-vant for OFET. In Fig. 5, we plot ∆σ/∆ω as a functionof ω. At low temperature T = 0.12, the "initial" statesrelevant for the conduction are in the energy range wherethe current matrix elements are larger, i.e. from aboutω = −3.4. Furthermore, they are in the range betweenµp+T and µp+4T , so that they are itinerant. We, finally,note that the curve reproduces the same gaps as showedin Fig. 4. With increasing temperature, the itinerantstates become more diffusive and distant from the chem-ical potential, and, within the energy range close to µp,more and more localized states are present. This is thereason why the mobility gets lower at room temperaturewhere it is nearly flat as function of the energy.

In Fig. (6), the mobility along the a direction as afunction of the temperature is reported at fixed couplingλ = 0.1 and for different concentrations c. The upperpanel shows that the absolute magnitude of the mobil-ity substantially agrees with the experimental estimatesbeing µ ≃ 10cm2/(V s) at room temperature, while, inthe inset, where the scale is logarithmic, the mobilityexhibits a quite linear dependence on the temperaturethat means a “band-like” behaviour with inverse power-law µ ∝ T−γ . The exponent γ is evaluated as the slopeof the straight lines drawn in the lower panel and the re-sults that we obtain from the fit are in the range 2− 2.4,where the highest value is related to the lower concen-tration. This trend is in agreement with experimentalmeasures that for rubrene establish γ ≃ 2 for tempera-tures T > 170−180K.2 A feature of our model is that themobility decreases for higher concentrations of carriers.This trend has been already found in the one-dimensionalmodel.18 Actually, due to the "selection rules" discussedpreviously, increasing density can bring the system intoan energy range where the current matrix elements get

7

150 200 250 300 3500

5

10

15

20

25

30

35

40

45

50

55

60

160 200 24010

20

30

405060

(c

m2 /V

s)

T (K)

c=0.001 c=0.002 c=0.005 c=0.008

T (K) (cm

2 /Vs)

Figure 6: Mobility along the a direction as a function of thetemperature for different values of the concentration c. Thecoupling strength is fixed at λ = 0.1. The inset of the lowerpanel reports the parameters of the fits via the function µ ∝

T−γ .

lower.In Fig. (7) the mobility as a function of the temper-

ature is reported at fixed concentration c and for differ-ent couplings starting from the value λ = 0.087. Wejust notice two essential features emerging from Fig. (7).Quite obviously, the mobility decreases when the cou-pling strength is higher but, differently from the one-dimensional value which predicts the formation of bondpolaron at λ = 0.12,18 in this case this does not occureven at higher values of λ for which the “band-like” be-haviour still holds. To conclude this point, we notice thatthe effect of the coupling on γ is less relevant then thatof the particle density c.

Starting from the mobility, we can determine the scat-tering time from the relation µ = eτtr/m. Since themass is weakly renormalized from the el-ph interaction,one can assume m as the bare mass at k = 0. In theinset of Fig. 8, the scattering time as a function of thetemperature is shown. We point out that is on the scaleof the fs, so that it is one order of magnitude lower thanthe damping time of the states important for the spec-tral properties (on the scale of ten fs). Therefore, thetransport processes amplify the effects of the el-ph inter-action and the vertex corrections introduced within ourapproach are fundamental to take into account this effect.

From the scattering time, one can deduce the meanfree path as ltr = vavτtr, where vav is an average velocityof the charge carriers. We estimate vav from the averagekinetic energy as follows:

vav =

√

2

m〈t~a〉. (14)

The operator 〈. . .〉 has to be intended both as a thermicaverage and as an average over the configurations {uRi

}.

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=0.087 =0.100 =0.120 =0.150

=2.275 =2.298 =2.327 =2.322

Figure 7: Mobility along the a direction as a function of thetemperature for different values of the coupling strength. Theconcentration c is fixed at c = 0.002. The inset reports theparameters of the fits via the function µ ∝ T−γ .

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Figure 8: The mean free path (in units of the lattice param-eter a) as a function of the temperature at λ = 0.12 andc = 0.002. The inset reports the scattering time as functionof the temperature for the same model parameters of the mainplot

The quantity ltr is reported in Fig. 8. It is always on thescale of a few lattice parameters. The most importantfeature is its temperature behavior. As a consequenceof the el-ph interaction effects, close to room tempera-ture, it becomes of the order of half lattice parameter a.This means that the Ioffe-Regel limit is reached.29 Thedecrease of the mobility in the Ioffe-Regel limit is not dueto a mass renormalization (dynamic and/or static) butis due to a reduction of the available itinerant states (theonly ones able to transport current) with the tempera-ture. We remark that this result is due to the fundamen-tal role played by vertex corrections in the calculation ofthe mobility.

8

Another important quantity is the anisotropy of thetransport properties along different in-plane directions.Up to now, we have discussed the mobility only along thea direction, since the corresponding quantity along the baxis can be roughly reproduced taking into account theanisotropy factors tb/ta = 0.58. Actually, the mobilityalong b direction roughly scales as the square of the ratiotb/ta, therefore it is about 0.33 times the mobility alongthe a direction. This value is in good agreement withexperimental results.2

Finally, we briefly discuss the role of hopping parame-ter tc on the mobility features. Switching from tc = 0.18to tc = 0.10 produces a relative reduction of the mobilitythat lies between 10% and 15% depending on the con-centration of carriers c. Although the sensitivity of themobility to tc is less relevant than other parameters, tccan’t properly be treated as a “dead parameter”. Theremay be two reasons for this effect: first reducing the in-terplanar coupling yields to a system more bidimensionaland many theoretical predictions state that a purely bidi-mensional system should be an insulator,28then, when tcbecomes comparable with the broadening η (we keep ηin the range 0.1 − 0.4tc, see Appendix) the finiteness ofthe lattice appears partially invalidating the procedure.Probably, both effects give their contribution to the re-duction of the mobility.

V. SUMMARY AND CONCLUSIONS

Spectral and transport properties of the quasi two-dimensional adiabatic Su-Schrieffer-Heeger model havebeen discussed with reference to rubrene single-crystalfield effect transistors. An important ingredient of ourmodel is the small but finite carrier density, therefore theinteresting behavior of the chemical potential as functionof temperature has been investigated. Actually, with in-creasing temperature, the chemical potential always en-ters the tail of the density of states corresponding to lo-calized states. Therefore, all the experimentally quanti-ties strongly dependent on the position of the chemicalposition will probe the features of localized states. Thecombined study of spectral and transport properties isfundamental to shed light on the intricate conductionprocess of these materials characterized by small particledensity and intermediate el-ph coupling. From this anal-ysis it emerges that the states that mainly contribute tothe conduction process are the itinerant ones which arein the energy range between µp + T and µp + 4T . Withincreasing temperature, these states are affected by en-hanced interaction effects and are more distant from µp,and, at the same time, more and more localized statesbecome available in the energy range close to the chemi-cal potential. Therefore, close to room temperature, thetransport properties reach the Ioffe-Regel limit. More-over, the transport lifetime is almost one order of mag-nitude smaller than the spectral lifetime of the statesinvolved in the transport mechanism. In order to get

this result, it is important to include vertex correctionsinto the calculation of the mobility. The mobility as afunction of temperature is in good agreement with ex-periments: indeed, the mobility has the right order ofmagnitude, it scales as a power law T−γ , with the γ closeor larger than two, and has the correct anisotropy ratiobetween different in-plane directions. The use of a real-istic quasi 2D model is fundamental for the analysis ofthis paper providing reliable results.

The focus of this work has been on the effects of lowfrequency intermolecular modes on the transport proper-ties. While the mobility from 100K to 300K is believedto depend almost entirely by this el-ph coupling, closeto room temperature, the effects of other interactions,for example that due to local high frequency modes, cangive a non negligible contribution.30 Actually, when themean free path becomes of the order of the lattice param-eter as a result of the only intermolecular coupling, thelocal coupling can easily give rise to an hopping mecha-nism with an activated mobility. Rubrene is on the vergeof this behavior, that can be accepted as a mechanismfor other poliacenes. As a future work, it could be inter-esting to use the quasi 2D model discussed in this paperin combination with local el-ph coupling in order to bet-ter understand the conduction mechanism close to roomtemperature. Work in this direction is in progress.

Appendix A: “Calibration procedure”

In this Appendix, we report in detail the “calibrationprocedure” followed to fix the right value of the broad-ening η for the calculation of the mobility. According toEqs. (12) and (13), the mobility is obtained performingthe two limits ω → 0+ and η → 0+ together with ther-modynamic limit L → ∞. It is clear that the limit on thesize and on η has to be done at the same time because,actually, they are similar. Moreover, it turns out that, forany fixed choice of the couple (η, L) and for any temper-ature T , the mobility depends slightly on the frequencyω. There is a threshold under which the mobility is notsensitive to further decreasing of ω that means that wehave reached the minimum frequency that our finite sys-tem can resolve. Practically, under the value ω0 = 10−3

the mobility shows a plateau. We have fixed this valuefor all the calculations of mobility.

In Fig. (9), for three different values of temperature,we have plotted the mobility calculated via the Kuboformula (9) to analyze how it depends on η and the sizeL. The strength of coupling is fixed at λ = 0.1 and thecarrier concentration is c = 0.002. At any temperature,the series of data gets closer with increasing the size ofthe lattice. Eventually, for the two biggest sizes, theyare not distinguishable for most of the range in which ηvaries. This allows us to state that, for the maximumsize that we can treat LMAX = La ∗Lb ∗Lc = 24 ∗ 24 ∗ 2,the system actually reaches the thermodynamic limit orat least gets very close to it. For example, in the worst

9

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0.003

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L=24*24*2

0.002

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L=12*12*2

L=16*16*2

L=20*20*2

L=22*22*2

L=24*24*2

Figure 9: Dependence of the conducibility on η for differentsizes L of the lattice. Upper panel T = 0.12. Middle panelT = 0.18. Lower panel T = 0.24. The concentration ofcarriers is c = 0.002 for all the temperatures. The unit of theconducibility is µ0 where µ0 = ea2/~ = 5.16cm2/(V s).

case of low temperature (T = 0.12 equivalent to 165K)the two upper curves split only for η < 0.02tc and it isclear that the value ηMAX for which the highest mobil-ity is reached gets closer to zero as the size rises. Forall the regimes of temperature, the mobility exhibits aquasi-plateau (less definite at low temperatures) and thevalue of ηthreshold under which the mobility collapses isreduced by increasing the size. This trend can be easilyexplained if one notices that, when the size increases, theseparation between two next eigenvalues reduces and thesame happens to the width of the delta function requiredto couple a proper number of close eigenvalues. Anotherfeature that is worth to be noticed is that the curves ofdifferent sizes merge together in better way at high tem-peratures then at low ones but, at the same time, for hightemperatures ηMAX is higher. These arguments lead usto state that, for our model, a correct thermodynamiclimit can be recovered in spite of the analogue 1D modelin which the mobility falls down under a value of η thatin the limit of infinite size converges on a finite η.18. Itseems that in 1D the broadening η is not just a compu-tational need that virtually disappears in the infinite sizelimit but it’s a real missing energy scale without which afinite mobility can’t be obtained. Operatively, for each ofthe three temperature that appear in Fig. (9), we havedetermined the value ηcal (Ti) for which the mobility hasa maximum. Then, at any intermediate temperature T ,the ηcal (T ) is obtained from a quadratic interpolationbased on ηcal (Ti). This procedure allows to calculate amobility very close to that one reached in the thermody-namic limit.

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