J. Phys. Chem. C 2009, 113, 4679–4686
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Interaction of Charge Carriers with Lattice Vibrations in Organic Molecular Semiconductors: Naphthalene as a Case Study Veaceslav Coropceanu,*,† Roel S. Sa´nchez-Carrera,† Pavel Paramonov,#,† Graeme M. Day,‡ and Jean-Luc Bre´das*,† School of Chemistry and Biochemistry and Center for Organic Photonics and Electronics, Georgia Institute of Technology, 901 Atlantic DriVe NW, Atlanta, Georgia 30332-0400, and Department of Chemistry, UniVersity of Cambridge, Lensfield Road, Cambridge, U.K., CB2 1EW ReceiVed: January 7, 2009
Recent theoretical studies suggest that the modulation of the electronic couplings (transfer integrals) between adjacent molecules by lattice vibrations, i.e., the so-called nonlocal electron-phonon coupling, plays a key role in the charge-transport properties of molecular organic semiconductors. However, a detailed understanding of this mechanism is still missing. Here, we combine density functional theory calculations and molecular mechanics simulations and use a chemistry-based insight to derive the nonlocal electron-phonon coupling constants due to the interaction of charge carriers with the optical lattice vibrations in the naphthalene crystal. The results point to a very strong coupling to both translational and librational intermolecular vibrational modes as well as to intramolecular modes. Along some crystal directions, the nonlocal interactions are found to be dominated by nontotally symmetric vibrational modes which lead to an alternation (Peierls-type dimerization) pattern. Importantly, we introduce two parameters that can be used: (i) to quantify the total strength of the nonlocal electron-vibration mechanism in the form of a reorganization energy term; and (ii) to define the extent of the thermal fluctuations of the electronic couplings. Interestingly, zero-point fluctuations are seen to be very significant. 1. Introduction Reaching a complete understanding of the charge-transport mechanism in molecular organic semiconductors still represents a major challenge.1,2 In the absence of chemical and physical defects, the nature of charge transport depends on a subtle interplay between electronic and electron-vibration (phonon) interactions. In organic semiconductors, there are two major electron-phonon mechanisms: The first comes from the modulation of the energies of the relevant molecular levels (roughly speaking, the LUMO energies for electrons and the HOMO energies for holes; these are referred to, in general, as the site energies) by vibrations and is described as local coupling. The second mechanism arises from the modulation of the electronic couplings (transfer integrals) between adjacent molecules by intermolecular vibrations and is referred to as nonlocal coupling.1 The local electron-vibration coupling is the key interaction considered in conventional electron-transfer theory and in Holstein’s molecular crystal model.3,4 Microscopic theories derived from a density matrix approach,5 a generalized master equation approach,6 dynamical mean-field theory,7 or more recently a finite-temperature variational method8 generally provide a good understanding of the role of local electron-phonon interactions on the charge-transport properties. The overall strength of this coupling is expressed by the relaxation or polaron binding energy Epol, or, in the context of electron-transfer theory, by the reorganization energy λ (≈ 2Epol).1,3,4 It consists * Corresponding authors. E-mail:
[email protected]; jean-luc.bredas@ chemistry.gatech.edu. † Georgia Institute of Technology. ‡ University of Cambridge. # Present address: Department of Physics, University of Akron, Akron, OH 44325-4001.
of both intra- and intermolecular contributions; the former reflect the changes in the geometry of individual molecules and the latter in the polarization of the surrounding molecules, upon going from the neutral to the charged state and vice versa. In the case of nonpolar molecules, the local electron-phonon coupling is dominated by intramolecular contributions. Extensive studies we recently performed on this issue show that quantumchemical methods, in particular those based on density functional theory, provide a reliable description of the local electronphonon coupling constants and the associated reorganization energies.1,9–13 However, transport theories based solely on the original Holstein molecular model cannot fully describe the chargetransport mechanisms in organic materials. There is a growing consensus in the literature that both local and nonlocal electron-vibration couplings have to be taken into account in order to obtain a comprehensive description of charge transport.1,14–22 The nonlocal coupling constitutes the major interaction considered in so-called Peierls-type models,23 such as the Su-Schrieffer-Heeger Hamiltonian that has been largely applied to conducting polymers.24,25 This mechanism, referred to as the non-Condon effect (since it explicitly considers the variations in the electronic coupling as a function of vibrational coordinates), is also an important player in electron-transfer processes in conventional donor-acceptor26 and more complex biological systems.27 While the importance of the nonlocal coupling in the description of charge transport in organic molecular semiconductors has been discussed in the literature since more than four decades ago,28–34 by and large its role is much less well studied and understood than that of local coupling. A major problem in the development of generalized nonlocal (polaronic) models is related to the fact that, in contrast to the local electron-phonon
10.1021/jp900157p CCC: $40.75 2009 American Chemical Society Published on Web 02/20/2009
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interactions, a systematic investigation of the nonlocal coupling constants is still lacking. In this work, we take a critical step in this direction by determining the nonlocal electron-phonon coupling constants that arise from the interaction of charge carriers with all the optical lattice vibrations present in the naphthalene crystal. We combine density functional theory (DFT) calculations and molecular mechanics (MM) simulations to investigate the lattice dynamics and to derive the nonlocal coupling constants. The results also allow us to discuss the effect of nonlocal coupling on the vibration (phonon)-assisted contribution to charge transport. 2. Methodology In order to describe the basic concepts of electron-vibration coupling, it suffices to rely on a simple tight-binding (Hu¨ckellike) approximation. In this context, the electronic properties of the system can be described by the following Hamiltonian:
H)
∑ emam+am + ∑ tmnam+an m
(1)
mn
Here, em is the electron site energy (or Coulomb integral R in a Hu¨ckel context) and tmn is the transfer integral or electronic + (am) is the creation coupling (Hu¨ckel resonance integral β); am (annihilation) operator for an electron on site m. The nonlocal electron-vibration (phonon) interaction is obtained by expanding the transfer integrals, tmn, in a power series of the vibrational (phonon) coordinates. Here, we consider the modulations of the transfer integrals by all optical phonons. As has been the case in previous investigations,17,18 the dispersion of the optical modes with momentum is neglected, and we estimate the nonlocal electron-phonon couplings using the normal-mode coordinates derived at the Γ-point (i.e., at the center of the crystal unit cell in reciprocal space). The Hamiltonian of the lattice vibrations and the dependence of the transfer integrals on normal-mode coordinates are given by:
H)
1 2
∑ pωj(Pj2 + Qj2)
(2)
∑ VjmnQj +
(3)
j
tmn ) t(0) mn +
· · ·
j
Here, ωj, Qj, and Pj denote the frequency and the dimensionless coordinate and momentum of vibrational mode j, respectively; (0) terms represent the transfer integrals obtained at the the tmn equilibrium geometry and Vjmn are the linear nonlocal electronphonon coupling constants (note that the impact of quadratic and higher-order terms is usually considered to be negligible). The determination of the coupling constants Vjmnconstitutes the main focus of the present work; we underscore that they play, in the case of nonlocal electron-vibration interactions, a role similar to the Huang-Rhys factors in the case of local electron-vibration interactions. The geometric structure and electronic properties of the naphthalene crystal were computed at the DFT level, using the Perdew-Burke-Ernzerhof exchange-correlation functional with the plane-wave basis set (300 eV cutoff) and projector augmented wave potentials.35,36 The self-consistent calculations were carried out with an 8 × 8 × 8 k-point mesh. The Γ-point lattice phonons were derived by means of numerical derivatives using a 0.03 Å atomic displacement step. All DFT crystal-structure calculations were carried out using the VASP code.37 While the DFT geometry optimizations were performed by constraining the cell parameters to the experimental values38 (a ) 8.098 Å, b ) 5.953 Å, c ) 8.652 Å, R ) 90°, β ) 124.4°,
Figure 1. The crystal structure of naphthalene and the molecular pairs with the largest transfer integrals considered in the calculations.
and γ ) 90°), the unit cell was allowed to vary during the MM geometry optimizations, resulting in the following parameters: a ) 8.146 Å, b ) 6.033 Å, c ) 8.716 Å, R ) 90°, β ) 122.47°, and γ ) 90°. The lattice dynamics at the Γ-point was also investigated by means of MM simulations using the DMAREL program.39 An empirical Buckingham (exp-6) model40 was used for the repulsion-dispersion interactions, and the atomic point charges were determined from DFT B3LYP/6-31G** calculations, using the CHelpG algorithm as implemented in the Gaussian03 package.41 The MM calculations were carried out in the framework of the rigid-body approximation.42,43 The transfer integrals for selected nearest-neighbor pairs of molecules were evaluated at both DFT-optimized and MMoptimized crystal geometries by using a fragment orbital approach in combination with a basis set orthogonalization procedure.44 These calculations were performed with the PW91 functional and Slater-type triple-ζ plus polarization basis sets for all atoms using the ADF package.45 3. Results and Discussion Γ-Point Lattice Phonons. The crystal structure of naphthalene (see Figure 1) belongs to the monoclinic P21/a space group with two molecules in the unit cell. The MM and DFT estimates of the first nine low-energy (intermolecular) modes derived at the Γ-point are given in Table 1 along with the experimental values46 (a complete listing of the DFT frequencies is given in the Supporting Information, SI). The DFT and MM estimates are in very good mutual agreement and also compare very well with the experimental data. There is a total of 105 optical modes; in the rigid-body approximation, these can be classified as 96 intramolecular modes and 9 intermolecular modes comprising three translational and six librational modes. The translational modes are of odd-symmetry type, either Au or Bu; in each mode, the translationally nonequivalent molecules move out-of-phase along one of the molecular symmetry axes. All librational modes are of even symmetry, either Ag or Bg; they represent either a
Interaction of Charge Carriers with Lattice Vibrations
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TABLE 1: Frequencies of the Intermolecular Optical Modes in the Naphthalene Crystal from MM and DFT Normal-Mode Calculationsa type
MM: ω
DFT: ω
expt: ω
translation (Au) libration (Bg) libration (Ag) translation (Bu) libration (Bg) libration (Ag) translation (Au) libration (Ag) libration (Bg)
50 54 59 63 82 99 117 152 153
51 56 63 76 97 102 121 126 150
44 56 67 75 83 88 106 121 141
a The experimental frequencies from ref 46 are also given. All values are in cm-1.
(0) TABLE 2: DFT Estimates of the Transfer Integrals tmn (in meV) in the Naphthalene Crystal Using DFT- and MM-Optimized Crystal Geometries
DFT t(0) b t(0) ab t(0) abc
MM
hole
electron
hole
electron
-36 -12 18
14 -35 -6
-31 -20 13
17 -27 -6
symmetric or antisymmetric combination of molecular rotations of translationally nonequivalent molecules around a given molecular axis.43 In the monoclinic P21/a space group, the molecules lie on the crystallographic inversion centers; therefore, the translational modes and librational modes do not mix.43 The DFT calculations, in agreement with previous MM calculations,47 indicate some modest mixing between the intermolecular and intramolecular modes. For instance, while at the DFT level, the first nine low-energy modes are very similar to those derived in the framework of the rigid-body approximation and are thus intermolecular in nature; some deviations from the fully planar geometry of the free molecule are also observed. Transfer Integrals. In the framework of the tight-binding Hamiltonian, eq 1, and assuming interactions only between nearest-neighbor molecules located along crystal axes [Rm Rn ) (a, (b, (c; here, vector Rm(n) indicates the position of site m(n)] and on unit-cell face centers [Rm - Rn ) ((a ( b)/2, ((a ( b + 2c)/2], the dependence of the conduction (valence) band energy on the electron wave vector k is given by:48–50
E(k) ) ε0 + 2ta cos ka + 2tb cos kb + 2tc cos kc (
{ [ k(a 2+ b) + cos k(a 2- b) ] + k(a + b + 2c) k(a - b + 2c) t [cos + cos ]} (4) 2 2
2 tab cos abc
The DFT calculations show that the largest transfer integrals for both electrons and holes are found along the short crystal b-axis (tb, pair 1 in Figure 1) and the diagonal directions in the ab plane (tab, pair 2). The other significant terms are the transfer integrals (tabc, pair 3) along the Rm - Rn ) ((a ( b + 2c)/2 directions (see Figure 1). The related values of the transfer integrals derived at both DFT and MM equilibrium geometries are collected in Table 2. The slight discrepancies between the values obtained using the DFT- and MM-optimized geometries are due to the well-known sensitivity of the transfer integrals to the exact packing configurations.1,51 Electron-Phonon (vibration) Couplings. The nonlocal electron-phonon coupling constants (Vjmn ) ∂tmn/∂Qj) were
computed numerically. This was achieved by distorting first the crystal along all normal-mode coordinates and computing the related transfer integrals for each distorted geometry; then, the derivatives of the transfer integrals with respect to normal coordinates were evaluated. The results obtained using the MM normal modes are given in Table 3; the coupling constants based on the DFT vibrational coordinates are illustrated in Figure 2 (while the numerical values are given in the Supporting Information). The data reveal that there are significant couplings to both translational and librational modes. In an optical translational mode, all translationally equiValent molecules such as in pair 1 (see Figure 1) move in phase; therefore, in this case, the translational modes do not contribute to the nonlocal coupling. As seen from Table 3, in the rigidbody approximation, the related couplings for pair 1 are identically zero; the minor couplings obtained at the DFT level (see Supporting Information) are due to the hybridization between intra- and intermolecular vibrations. In contrast, in the case of translationally nonequiValent molecules (pairs 2 and 3 in Figure 1), both librational and translational modes contribute to the nonlocal interactions. The largest vibrational couplings in the case of holes are obtained along the diagonal directions in the ab plane (pair 2), while in the case of electrons the couplings along the b-axis and ab-plane, diagonals are very similar. The DFT calculations also underscore a significant coupling to intramolecular vibrations.22 As seen from Figure 2, a vibration around 1600 cm-1 yields, in fact, the largest coupling constant for both electrons and holes. A pictorial representation of this vibrational mode is given in Figure 3. The DFT calculations allow us to understand this intriguing result; they show that a distortion along this mode significantly affects the electronic density pattern of the frontier molecular orbitals and, as a consequence, has a major effect on the intermolecular overlap integrals and ultimately on the electronic and vibrational couplings (see Supporting Information for additional results). The nonlocal couplings for optical phonons in naphthalene were also estimated by Hannewald et al.17 and Wang et al.18 The transfer integrals and electron-phonon couplings were derived by projecting the band-structure results into the tightbinding model given by eq 4. In these studies, only the coupling with three intermolecular (totally symmetric librational) modes was considered (we note that eq 4 assumes P21/a symmetry and is not applicable when the crystal is distorted along a nontotally symmetric vibrational coordinate). Nonlocal coupling to only totally symmetric modes was also suggested by Della Valle et al.15 It is, therefore, important to emphasize that, from a group-theory standpoint, the nonlocal coupling is actually not restricted to totally symmetric lattice vibrations. In fact, as seen from Table 3 (see also Supporting Information), all nine optical intermolecular vibrations are coupled. Moreover, both the MM and DFT results highlight that along some crystal directions the nonlocal interaction is actually dominated by nontotally symmetric phonons. The main effect of the symmetry can be brought to light when considering the interaction among translationally nonequivalent molecules. The totally symmetric vibrations keep the same pattern for the electronic interactions; for instance, the electronic couplings between a molecule in the ab plane and its four diagonal neighbors remain equal to each other when the crystal is distorted along any totally symmetric coordinate (as illustrated in Figure 3d). This picture is completely different for antisymmetric phonons. The example of the translational mode at 51 cm-1, shown in Figure 3e, reveals that a distortion along such a coordinate leads to an alternation, or Peierls-type dimeriza-
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Figure 2. DFT estimates of the nonlocal hole-phonon and electron-phonon coupling constants as a function of vibration (phonon) energy. The insets highlight more clearly the contributions due to intermolecular vibrations.
TABLE 3: Estimates of the Nonlocal Electron (Hole)-Phonon Coupling Constants (in meV) Based on MM Normal-Mode Calculations holes
electrons
ω (cm-1)
type
Vjb
Vjab
Vjabc
Vjb
Vjab
Vjabc
50 54 59 63 82 99 117 152 153
translation (Au) libration (Bg) libration (Ag) translation (Bu) libration (Bg) libration (Ag) translation (Au) libration (Ag) libration (Bg)
0.0 6.8 8.9 0.0 -4.8 -0.8 0.0 4.9 -5.2
10.0 -6.2 -6.1 0.1 2.9 -4.7 -2.3 3.4 4.2
3.3 -0.4 1.0 -1.0 -0.1 0.2 0.8 0.4 -0.3
0.0 -14.0 -17.0 0.0 8.9 3.0 0.0 -1.8 1.8
11.8 -9.9 4.7 -5.7 4.4 -12.9 4.3 0.0 2.0
-1.2 3.2 0.3 -2.2 -2.4 2.5 1.1 -1.1 -0.5
tion,23 in the intermolecular distances among translationally nonequivalent molecules. As a result, an increase in the transfer integral between a molecule and its left neighbor is accompanied by a decrease in the transfer integral with its right neighbor; this mechanism is referred to in the literature as antisymmetric nonlocal electron-phonon coupling.33,34 The consequences of such an antisymmetric nonlocal coupling mechanism both on the system energetics31,33,34 and on charge transport32 have been previously studied to some extent only in the case of onedimensional systems. We also note that antisymmetric nonlocal
couplings can be present among translationally equivalent molecules because of both acoustical and optical modes; however, the investigation of such couplings would require the extension of the present approach beyond the Γ-point approximation. In order to better quantify the nonlocal electronphonon coupling and demonstrate the global impact of all modes, we propose here to introduce the following two parameters:
Interaction of Charge Carriers with Lattice Vibrations
G2mn )
∑ j
Lmn )
∑ j
2 Vjmn 2
2 Vjmn 2pωj
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(5)
(6)
As we show below, G and L determine the degree of thermal fluctuations of the transfer integrals. L can be considered to play in the nonlocal electron-phonon mechanism a role similar to the reorganization energy (polaron binding energy)1,3,4 in the case of local coupling. To illustrate this point, we take an isolated dimer formed by two equivalent molecules in (for the sake of simplicity) the rigid-body approximation. When an excess charge is added to the dimer, according to the tightbinding model (eq 1), two delocalized states Ψ+ and Ψ- appear with energies E+ and E-. In the Condon approximation, i.e., when the dependence of the transfer integral on the vibration coordinates is neglected, the energies E+ and E- are given by:
E+ ) e + E- ) e +
∑
(7)
j
pωjQj2 (0) +t 2
∑
pωjQj2 (0) -t 2
(8)
j
When the dependence of the transfer integrals on vibrational coordinates is taken into account (non-Condon effect), we obtain:
E+ ) e + E- ) e +
∑
∑ VjQj
(9)
j
pωjQj2 (0) +t + 2
∑
pωjQj2 (0) -t 2
∑ VjQj
(10)
j
From eqs 7-10, it follows that, after adding a charge to the dimer, its intermolecular coordinates relax to new equilibrium positions and, as a consequence, the energies of both Ψ+ and
Figure 3. Pictorial representation of (a) the translational normal mode at 51 cm-1, (b) the librational mode at 56 cm-1, and (c) the intramolecular mode at 1634 cm-1, as derived from DFT calculations. (d and e) The change in the pattern of electronic couplings as a result of crystal deformations along the translational mode at 51 cm-1 with d representing the equilibrium situation.
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Figure 4. Sketch of the adiabatic potential energies (and corresponding geometry relaxations) related to the ionization of a molecular dimer: (a) in the Condon approximation; (b) in the non-Condon case.
TABLE 4: DFT Estimates of the Transfer Integrals t(0), Parameters L and G, and Standard Deviations σQM of the Transfer Integrals at 300 K, Based on the MM- and DFT-Optimized Geometries and Frequency Analyses (all parameters are given in meV) holes pair 1
pair 2
electrons pair 3
t(0) L G σQM (300 K)
-31.0 11.3 10.0 24.3
-19.6 15.8 11.0 28.7
MM 13.0 1.1 2.7 7.4
t(0) L G σQM (300 K)
-35.6 3.6 9.3 14.8
-12.0 7.3 16.0 22.5
DFT 17.9 0.8 5.8 7.8
pair 1
pair 2
pair 3
16.8 38.8 17.0 45.0
-26.8 30.6 15.8 40.0
-6.3 1.8 4.0 9.8
14.1 12.9 15.2 27.6
-35.3 10.0 15.6 25.0
-5.7 1.1 6.0 8.5
Ψ- states are stabilized by L ) ∑jV2j /2pωj; this effect is illustrated in Figure 4. The estimates for G and L based on both DFT and MM normal-mode results are reported in Table 4. While the DFT and MM estimates for G are very similar, the MM calculations yield larger values for L than the DFT calculations. The main reason for this difference is that in DFT the nonlocal interaction is shifted toward high-energy phonons; from the definitions of L and G, higher-energy phonons contribute to G in the same way as low-energy phonons but are scaled down in the case of L. The calculated values of G and L also suggest that, overall, the nonlocal vibrational coupling in the naphthalene crystal is larger for electrons than holes. Phonon-Assisted Charge Transport. As we already underscored, the role of nonlocal coupling on the charge-transport properties is not well understood yet. Microscopic charge-
transport models based on a Holstein-Peierls-type Hamiltonian have been worked out initially by Munn and Silbey31,32 and later by Bobbert and co-workers.17,52 Only the antisymmetric contribution to the nonlocal coupling was considered by the former authors and only the symmetric contribution was considered by the latter. Moreover, both approaches were based on a nonlocal-type canonical transformation (an extension of the small polaron approach) while omitting specific terms. The consequences of these approximations and, therefore, the range of validity of both models are still an open question, although both give results in qualitative agreement with experiment. More recently, Troisi and Orlandi20 have discussed the problem of charge transport in organic semiconductors by using numerical solutions of the time-dependent Schro¨dinger equation. However, in this approach the vibrations were treated classically, which limits the range of validity of the derived results to the hightemperature limit. The complexity of the charge-transport mechanism related to nonlocal coupling arises from its duality. On the one hand, as a scattering mechanism, it works toward decreasing the carrier mobility. On the other hand, as a part of the electronic coupling, it results in an additional, so-called phonon-assisted, contribution to the electrical current.29 In the absence of a comprehensive charge-transport theory, we concentrate in the following discussion on the phonon-assisted contribution. It was shown that this contribution is controlled by the thermal fluctuations of the transfer integrals.29,30 Therefore, we turn to a more detailed discussion of these fluctuations. The variance of the transfer integrals that is taken as a measure of its thermal fluctuations is defined as:
Interaction of Charge Carriers with Lattice Vibrations 2 2 σQM ) 〈t - 〈t 〉QM 〉QM
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(11)
Here, the index QM indicates that the vibrations (phonons) are treated at the quantum-mechanical level; in general, a term such as 〈A〉QM represents the thermal average of an operator A over the lattice phonons:53
〈A〉QM ) Tr[exp(-HL/kBT)A]/Tr[exp(-HL/kBT)] (12) By making use of eqs 2, 3, 11, and 12, we obtain the following results for the average and the variance of the transfer integrals:
〈t〉QM ) t(0) 2 σQM )
Vi2
(13)
∑ 2 coth 2kBTi pω
(14)
i
In the limiting cases of low (pω . kBT) and high (pω , kBT) temperatures, eq 14 reduces to: 2 σQM )
{
pω . kBT G2; 2LkBT; pω , kBT
(15)
Thus, the quantities L and G are directly related to the variance of the transfer integrals in the high and low-temperature limits, respectively. The high-temperature limit can also be evaluated directly by treating the vibrations classically. In this case, the thermal average is given by:53
〈A〉Cl ) f(Q) )
∏i
∫-∞∞ A(Q)f(Q)dQ
(
pωiQi2 pωi exp 2πkBT 2kBT
)
(16) (17)
The classical distribution yields as expected:
〈t〉Cl ) t(0)
(18)
2 σCl ) 2LkBT
(19)
The standard deviations σQM and σCl derived on the basis of the DFT normal-modes are illustrated in Figure 5. Interestingly, at room temperature, despite significant coupling to high-energy phonons, σQM is in general very close to its classical limit (although the two quantities can differ by up to 15%). Importantly, the comparison of the standard deviations with the transfer integral values (see Table 4) underscores that, in some instances, for example for holes along the diagonal directions in the ab plane or electrons along the b-axis, the fluctuation of a transfer integral can be as large as twice its mean Value. The comparison of the values of σQM obtained at room temperature and at 0 K (σQM at 0 K is given by G; see Table 4) indicates that at 0 K the standard deviation is at least 60% of that found at room temperature. Thus, this result suggests that the phonon-assisted contributions to the charge-carrier mobility could be significant across this whole temperature range. However, as was mentioned above, the nonlocal coupling also contributes to carrier scattering; further investigations of this interaction based on more comprehensive models are thus required. It is also of great interest to obtain the distribution probability for the transfer integrals. This is achieved by combining eq 3 and eq 17 (or its quantum-mechanical analogue53):
f(t) )
1
√2πσ2
[
exp -
(t - t(0))2 2σ2
]
Figure 5. Standard deviations σQM and σCl for the transfer integrals tb, tab, and tabc derived on the basis of DFT normal-mode calculations.
(20)
In agreement with the prediction of the central limit theorem for systems with many random variables,54 the f(t) function has
a Gaussian form. The standard deviation σ is given by eq 14 or eq 19 depending on whether the vibrations are treated quantummechanically or classically. This result is important since f(t) can also be derived as a time average by means of molecular (or quantum) dynamics simulations,21 therefore providing an alternative way to estimate the strength of the nonlocal coupling. Conclusions In summary, we have studied the nonlocal electron-vibration (phonon) interactions in the naphthalene crystal. Our results reveal that all nine intermolecular vibrations contribute to the nonlocal electron-phonon and hole-phonon interactions. This finding is in marked contrast to previous results suggesting that only totally symmetric modes are operative. Along some crystal directions, the nonlocal interactions are found to be dominated by nontotally symmetric vibrational modes which lead to a Peierls-type dimerization motif (antisymmetric mechanism). Thus, we believe that further investigations of the symmetriclike and antisymmetric-like nonlocal coupling mechanisms and their impact on charge transport will be of major interest. In order to better understand the role of the nonlocal coupling mechanism on charge transport, we estimated the variance of the transfer integrals due to thermal fluctuations. We have shown that the relaxation energy L, upon adding an excess charge to a dimer, defines the variance of the related transfer integral at high temperature. We have also introduced a parameter G that plays a similar role in the low-temperature limit. While highfrequency vibrations contribute to G in the same way as lowfrequency vibrations, their contribution is scaled down in the case of L. These findings suggest that high-frequency vibrations can have a large impact on the phonon-assisted part of the charge-transport mechanism even at low temperature. Interestingly, the parameter L can be used to quantify the reorganization energy due to the nonlocal electron-vibration coupling.
4686 J. Phys. Chem. C, Vol. 113, No. 11, 2009 Finally, our results underscore that the strength of the nonlocal electron-phonon coupling in naphthalene is comparable to the strength of the electronic coupling. Therefore, naphthalene and, we expect, oligoacenes in general do not fall in the limiting cases of weak interactions or strong interactions (either electronic or vibronic). Rather, an intermediate coupling regime seems to be more appropriate. The development of new (polaron) models for this complex regime would thus be of great importance. Acknowledgment. This work was primarily supported by the National Science Foundation under the STC (Award DMR0120967) and MRSEC (Award DMR-0212302) Programs, as well as by the Office of Naval Research. Supporting Information Available: A complete list of the DFT frequencies and vibrational couplings, an illustration of the frontier molecular orbitals (pair 1), and the dependence of the corresponding transfer integrals on the distortion along the 1600 cm-1 normal mode. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Coropceanu, V.; Cornil, J.; da Silva Fihlo, D. A.; Olivier, Y.; Silbey, R.; Bredas, J. L. Chem. ReV. 2007, 107, 2165–2165. (2) Bredas, J. L.; Beljonne, D.; Coropceanu, V.; Cornil, J. Chem. ReV. 2004, 104, 4971–5003. (3) Holstein, T. Ann. Phys. (N.Y.) 1959, 8, 325–342. (4) Holstein, T. Ann. Phys. (N.Y.) 1959, 8, 343–389. (5) Silbey, R.; Munn, R. W. J. Chem. Phys. 1980, 72, 2763–2773. (6) Kenkre, V. M.; Andersen, J. D.; Dunlap, D. H.; Duke, C. B. Phys. ReV. Lett. 1989, 62, 1165–1168. (7) Fratini, S.; Ciuchi, S. Phys. ReV. Lett. 2003, 91, 256403. (8) Cheng, Y. C.; Silbey, R. J. J. Chem. Phys. 2008, 128, 114713. (9) Coropceanu, V.; Malagoli, M.; da Silva Fihlo, D. A.; Gruhn, N. E.; Bill, T. G.; Bredas, J. L. Phys. ReV. Lett. 2002, 89, 275503. (10) Malagoli, M.; Coropceanu, V.; da Silva Fihlo, D. A.; Bredas, J. L. J. Chem. Phys. 2004, 120, 7490–7496. (11) Sanchez-Carrera, R. S.; Coropceanu, V.; da Silva Fihlo, D. A.; Friedlein, R.; Osikowicz, W.; Murdey, R.; Suess, C.; Salaneck, W. R.; Bredas, J. L. J. Phys. Chem. B 2006, 110, 18904–18911. (12) Coropceanu, V.; Kwon, O.; Wex, B.; Kaafarani, B. R.; Gruhn, N. E.; Durivage, J. C.; Neckers, D. C.; Bredas, J. L. Chem.sEur. J. 2006, 12, 2073–2080. (13) Kim, E. G.; Coropceanu, V.; Gruhn, N. E.; Sanchez-Carrera, R. S.; Snoeberger, R.; Matzger, A. J.; Bredas, J. L. J. Am. Chem. Soc. 2007, 129, 13072–13081. (14) Masino, M.; Girlando, A.; Brillante, A.; Farina, L.; Della Valle, R. G.; Venuti, E. Macromol. Symp. 2004, 212, 375–380. (15) Della Valle, R. G.; Brillante, A.; Farina, L.; Venuti, E.; Masino, M.; Girlando, A. Mol. Cryst. Liq. Cryst. 2004, 416, 145–154. (16) Stojanovic, V. M.; Bobbert, P. A.; Michels, M. A. J. Phys. ReV. B 2004, 69, 144302. (17) Hannewald, K.; Stojanovic, V. M.; Schellekens, J. M. T.; Bobbert, P. A.; Kresse, G.; Hafner, J. Phys. ReV. B 2004, 69, 075211. (18) Wang, L. J.; Peng, Q.; Li, Q. K.; Shuai, Z. J. Chem. Phys. 2007, 127, 044506. (19) Troisi, A.; Orlandi, G. J. Phys. Chem. A 2006, 110, 4065–4070. (20) Troisi, A.; Orlandi, G. Phys. ReV. Lett. 2006, 96, 086601. (21) Troisi, A. AdV. Mater. 2007, 19, 2000–2004. (22) Kwiatkowski, J. J.; Frost, J. M.; Kirkpatrick, J.; Nelson, J. J. Phys. Chem. A 2008, 112, 9113-9117. (23) Peierls, R. E. Quantum theory of solids; Clarendon Press: Oxford, 1955.
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