We find that the phase coherence factor derived from Hall effect measurements of single-crystal thin-film field-effect transistors of pentacene, which relates the intrinsic charge transport with the phase coherence, has a strong correlation with the thermal fluctuations of transfer energies between neighboring molecules. This observation also holds true for other organic semiconductors such as tetracene, dianthrathiophene (DAT)-V, and dinaphtho[2,3-b:2′,3′-f]thieno[3,2-b]thiophene (DNTT). This gives us clues for constructing flexible molecular systems with high carrier mobility.

The recent progress in the fabrication technology of organic single-crystal semiconductors and thin-film field-effect transistors with very high carrier mobility up to 40 cm^{2}/V $\xb7$ s requires us to elucidate the mechanisms of carrier transport in organic semiconductors, which are assemblies of *π*-conjugate molecules weakly bonded by van der Waals interactions.^{1,2} Observations of the crossover from the hopping transport of localized carriers to bandlike transport with a diffusive nature are expected to provide us with clues allowing us to reveal the carrier transport mechanisms.

In this direction, Uemura *et al.* recently showed from measurements of Hall conductivity that the carrier coherence factor defined by $\alpha \u2261C/[\u2202(1/RH)/\u2202VG]$ reveals the bandlike nature of charge transport.^{3} Here, *C* is the capacitance and *R _{H}* is the Hall coefficient. As the gate voltage

*V*induced carrier number $Q=CVG$ and $1/RH$ provides us with information on the density of mobile carriers for electron and hole, their ratio

_{G}*α*gives the relative number of carriers contributing to Hall transport. Since the Hall current is related to the wave number

*k*through the kinetic term $(\u210fk+qA)2/2m*$,

*α*is considered to be related to the coherent wave nature of carriers. Uemura

*et al.*measured the Hall effect in organic field-effect transistors (OFETs) to obtain $\alpha =0.5$ for pentacene and

*α*= 1 for rubrene, where rubrene has higher mobility. Therefore,

*α*plays an important role in the search for suitable organic semiconductor materials exhibiting high mobility in bandlike transport.

In this paper, we show evidence of a strong correlation between the carrier coherence factor *α* and the thermally induced fluctuations of transfer energies with neighboring molecules. The thermal fluctuation effects of molecular motion have been discussed for transport properties of organic semiconductors.^{4–10} We present numerical data of transfer energy fluctuations of pentacene at various temperatures and pressures, and we compare them with experimental observations of *α*. As a result, we find a strong correlation between these two factors. We apply this method to other organic semiconductors and obtain consistent results with those of transport experiments.

We extract parameters for the transport properties of organic semiconductors from calculations based on density functional theory (DFT), with van der Waals interactions included in the framework of the DFT Dispersion (DFT-D) approach.^{11} The calculation procedure is as follows. First, we determine the equilibrium positions and energies for neighboring molecules and compute the total energy shift due to the translational and librational motion of rigid molecules.^{12} Then, the thermal fluctuations of molecular motion from equilibrium positions are determined for a specific temperature using the equipartition theorem. To verify the accuracy of the present van der Waals energy correction $Edisp$, we compare the calculation results with experimental data obtained by the translation-libration-screw (TLS) analysis.^{13,14} The left panel of Fig. 1(a) shows a schematic diagram of a pentacene molecule coupled with neighboring molecules,^{15} which affect the translational and librational motion. The right panel of Fig. 1(a) shows the translational and librational motion, which interact with neighboring molecular systems. The obtained librational motion of pentacene using the DFT-D approach is shown in Fig. 1(b) as a function of temperature, which is in good agreement with the results of the TLS analysis of crystal structures in the present and the previous^{16} experimental observations.

The effective transfer energies *γ _{ij}* for the highest occupied molecular orbital (HOMO) states are extracted from the DFT-D calculations of the electronic states as follows.

^{17}When we have the orbitals $\Psi i$ for the HOMO and HOMO-1 states from the interaction of two molecules at a distance

*d*and an angle

*θ*, the effective orbital energies are described by the 2 × 2 secular equation $det|H\xaf\u2212\epsilon S\xaf|=0$. Here, $H\xaf$ and $S\xaf$ are expressed by $ei=\u27e8\Psi i|H\xaf|\Psi i\u27e9$ and unity for the diagonal elements and by $Jij=\u27e8\Psi i|H\xaf|\Psi j\u27e9$ and $Sij=\u27e8\Psi i|\Psi j\u27e9$ for the off-diagonal elements, respectively. Note that the matrix elements of

*e*and

_{i}*J*have the same physical meaning as on-site and transfer integrals, respectively, although these two sets are not identical. Using Lödin's symmetric transformation to obtain an orthogonal basis sets that maintain the initial character of the orbitals as much as possible, the effective on-site and transfer energies are evaluated using

_{ij}In this way, the secular equation becomes the standard form and the resulting energy splitting between the HOMO and HOMO-1 levels becomes $\Delta \epsilon 12=(e1eff\u2212e2eff)2+(2\gamma 12)2$. In Fig. 1(c), we show an example of the transfer energy *γ*_{12} as a function of librational angle *θ*. The data for the evolution of *γ _{ij}* are useful for evaluating the charge transport energies of organic semiconductors, which have flexible structures.

Owing to the weak bonding between molecules, organic semiconductor materials are flexible with large molecular displacements, which affect the intermolecular charge transfer. Indeed, we see in Fig. 1(c) that the transfer energy *γ _{ij}* between molecules is comparable to the room temperature of 25 meV. In this situation, charge transport behaviors are strongly affected by the thermal fluctuations of molecular displacements and thus the charge transfer energy. It is well known that random fluctuations induce the localized behavior of wavefunctions, with carrier transport changing from a diffusive (bandlike) to hopping nature.

^{18}Thus, it is important to find the relationship between the phase coherence factor observed experimentally by performing Hall effect measurements and the effect of thermal fluctuations on charge transfer energies obtained from calculations. Here, $\Delta \gamma ij$ is obtained from the thermal fluctuation of molecular motion, such as in Fig. 1(c), and we consider the transfer energy fluctuation using by the dimensionless quantity $\Delta \gamma /\gamma $. We take the root mean square values for all the translational and librational motion as $\Delta \gamma /\gamma =1/N\u2211ij,k(\Delta \gamma ij/\gamma ij)2$, where

*ij*are the indexes of six combinations of neighboring molecules, and

*k*denotes the index of a translational

*T*or librational $Lk\u2032k\u2032$ mode.

_{kk}Figure 2 (top) presents the DFT-D calculation results for the thermally induced fluctuation of the transfer energy $\Delta \gamma /\gamma $ of pentacene as a function of temperature for various reduction rates of the lattice constant under pressures. The corresponding experimental data of the pressure-induced rate of reduction are shown in the inset of Fig. 2 (bottom). We see that the fluctuation $\Delta \gamma /\gamma $ increases monotonically with temperature,^{19} while it decreases upon applying pressures owing to the confinement of the molecular motion. Note that the rate of decrease in the fluctuation under pressure becomes small with increasing volume reduction.

In Hall effect measurements, the four-terminal conductivity $\sigma =I/(V2\u2212V1)\xb7l/W$ and the Hall voltage *V _{H}* are measured simultaneously as a function of gate voltage

*V*and applied magnetic field

_{G}*B*, where

*l*and

*W*represent the channel distance and the width of the sample, respectively. Then the Hall coefficient is obtained as $RH=\Delta VH/(I\Delta B)$ and the field-effect mobility is evaluated using $\mu FET=\sigma /Q=\sigma /C(V2\u2212V1)$ with charge density

*Q*and unit-area capacitance

*C*. $1/RH$ increases with decreasing

*V*negatively owing to charge accumulation at the interface of the thin-film device, which shows that $1/RH$ is proportional to the carrier density. Interestingly, the ratio of

_{G}*Q*to $1/RH$ does not depend on extrinsic effects such as interface traps and grain boundaries and is thus considered as an intrinsic property of single-crystal organic semiconductors. Since the magnetic field couples with the wave vector

*k*of the coherent carriers, this enables us to define the dimensionless parameter

*α*as the ratio $Q/(1/RH)=C/[\u2202(1/RH)/\u2202VG]$, which is considered to represent the degree of the intrinsic carrier coherent nature over the organic molecules and is thus called the phase coherence factor.

^{3}

Figure 2 (bottom) shows experimental observations of pentacene as a function of temperature at two pressures. The data are taken from the Hall effect measurements of single-crystal pentacene thin-film OFETs. We see that $\alpha =0.5$ at room temperature under atmospheric pressure, which increases with decreasing temperature to $\alpha =0.65$ at 160 K. *α* grows significantly with pressure from 0.7 at room temperature and approaches 1 at 200 K and 1.1 GPa.

Here, we consider the relevance of the thermal fluctuation of the transfer energy $\Delta \gamma /\gamma $ obtained from the calculations, which determines the extent of the localized nature of wavefunctions, to the phase coherence factor *α* obtained from the experiments, which describes the extent of coupling between partially extended electronic states and the external electromagnetic field. Using the experimental data for the lattice constant under pressures in the inset of Fig. 2 (bottom), we obtain the transfer energy fluctuation $\Delta \gamma /\gamma $ at specific temperatures and pressures. Figure 3 shows $\Delta \gamma /\gamma $ obtained by the calculation as a function of the carrier coherence factor *α* obtained experimentally. A correlation can be seen between the two quantities. This is evidence that the coherent bandlike carrier transport is related to the thermal fluctuations of transfer energies between organic molecules.

We apply this analysis to organic semiconductors other than pentacene and compare the results with those of experiments on carrier transport. Here, we choose tetracene, dinaphtho[2,3-b:2′,3′-f]thieno[3,2-b]thiophene (DNTT), and dianthrathiophene (DAT)-V.^{20} The structures of these molecules are shown at the bottom of Fig. 4. Note that the crystal structures of these molecular systems are the same as that of pentacene, as shown in Fig. 1(a), although the distances between two neighboring molecules at equilibrium positions are different for each molecular system. Therefore, we perform the same calculations based on the DFT-D approach for the four organic semiconductor materials, considering various directions shown in Fig. 1(b) while varying the changing distances and angles from the equilibrium configurations, and we analyze the thermal transfer energy fluctuations.

Figure 4 (top) gives the thermal transfer energy fluctuations. The magnitudes of $\Delta \gamma /\gamma $ increase in the order

which shows that the effects of thermal fluctuations are smallest in DNTT molecules among these molecular systems. Since the phase coherence factor *α* is strongly correlated with thermal fluctuations and *α* is related to the bandlike transport with high carrier mobility, we expect that the DNTT molecular system has the highest mobility for carrier transport. Actually, in experiments, it was found that $\alpha \u22481$ for DNTT^{21} compared with $\alpha \u22480.5$ for pentacene,^{3} and correspondingly DNTT has very high mobility for charge transport, much higher than that of pentacene.

Since the carrier coherence factor is related to the fluctuations of transfer energies between neighboring molecules and thus the main source obstructing carrier transport, the search for single-crystal structures of organic molecules with reduced thermal fluctuations is the key essence for the realization of high-mobility organic systems. In this respect, comparing the present four molecules, we find that it is preferable to have zigzag structures such as DNTT to straight structures to reduce the thermal fluctuation. These observations provide important information and clues for the possible material design and construction of flexible molecular systems exhibiting high carrier mobility.^{22}

In summary, we find that the phase coherence factor *α* derived from Hall conductivity measurements has a significant correlation with the thermal fluctuations of transfer energies between neighboring molecules in pentacene molecular systems. This shows evidence that for pentacene single-crystal thin-film field-effect transistors, the bandlike carrier transport is strongly affected by thermal fluctuations. We find that this observation holds true for other organic semiconductors such as tetracene, DAT-V, and DNTT. This gives us clues for constructing flexible molecular systems with high carrier mobility.

We would like to acknowledge M. Tsukuda and H. Tamura for valuable comments and suggestions. This work was supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan. H. I. acknowledges financial support from JST-PRESTO.

## References

The data for the TLS analysis of crystal structures of organic semiconductors are obtained from the experiments using the SPring-8 third-generation synchrotron radiation facility in Kobe, Japan.

The distances between two pentacene molecules at equilibrium positions are $d=6.06\u2009\xc5$ between P1–P4 and P1–P7, $d=5.15\u2009\xc5$ between P1–P2 and P1–P5, and $d=4.80\u2009\xc5$ between P1–P3 and P1–P6, respectively.

_{6}H

_{5}-functional groups to the main tetracene structures, exhibit extreme high mobility.