Microscopic model

Since we are interested in the intrinsic charge-transport properties of organic semiconductor crystals, we assume single crystals of high purity, free of structural disorder and chemical defects. In this case, a tight-binding Hamiltonian is often considered, as it provides a simple but illustrative model for the description of the electronic properties of molecular solids:Reference Coropceanu, Cornil, da Silva Filho, Olivier, Silbey and Brédas^1, Reference Pope and Swenberg^27, Reference Silinsh and Cápek²⁸

(1)$$H_{\rm{e}} = \sum\limits_m {{\rm{\varepsilon }}_m a_m^ + a_m } + \sum\limits_{m \ne n} {t_{mn} a_m^ + a_n } .$$

Here, $a_m^ + $ and a_m denote the creation and annihilation operators, respectively, for an electron at site m; ε_m represents the energy of an electron/hole located on molecule m, and t_mn is the electronic coupling (also often referred to as transfer integral) between molecules m and n.

The site energy ε_m corresponds to the ionization potential in the case of holes and the electron affinity in the case of electrons. It can be roughly divided into an intramolecular component and an intermolecular component. The inter-molecular component, referred to as the electronic polarization energy, arises from the interaction of the electron or hole localized on a given molecule with the induced and permanent multipole moments of the surrounding molecules.Reference Martinelli, Idé, Sánchez-Carrera, Coropceanu, Brédas, Ducasse, Castet, Cornil and Beljonne^19, Reference Pope and Swenberg^27, Reference Norton and Brédas²⁹ The transfer integral, t_mn, describes the ability of the charge carrier to move between molecules m and n and is determined by the overlap of the wave functions corresponding to the electrons (holes) localized on sites m and n. In general, t_mn decays exponentially with increasing monomer-monomer distance.Reference Coropceanu, Cornil, da Silva Filho, Olivier, Silbey and Brédas¹

In the framework of the tight-binding description given in Equation 1, two major electron-phonon coupling mechanisms can be distinguished. The first comes from the modulation by vibrations of the site energy (local or Holstein-typeReference Holstein^30, Reference Holstein³¹ electron-phonon coupling). The second mechanism is due to the dependence of the transfer integral on the distances between adjacent molecules and their relative orientations (nonlocal or Peierls-typeReference Peierls³² electron-phonon coupling). These two contributions are also referred to as diagonal and nondiagonal (or off-diagonal) dynamic disorder mechanisms, respectively.Reference Bässler and Köhler² In the electron-transfer literature for both conventional donor-acceptor systemsReference Berlin, Grozema, Siebbeles and Ratner³³ and complex biological systems,Reference Balabin and Onuchic³⁴ the nonlocal electron-phonon interaction is also expressed as the non-Condon effect, since it explicitly considers the variations in electronic coupling as a function of vibrational coordinates (in the Condon approximation, the dependence of a matrix element such as a transfer integral on vibration coordinates is neglectedReference May and Kühn³⁵).

In the harmonic approximation, the Hamiltonian of the phonon subsystem (lattice vibrations) H _ph and the dependence of the site energies and transfer integrals on vibration coordinates when considering only the linear electron-phonon interactions are given by:

(2)$$H_{ph} = {1 \over 2}\sum\limits_j {\hbar {\rm{\omega }}_j \left({P_j^2 + Q_j^2 } \right)} AU1$$

(3)$${\rm{\varepsilon }}_m = {\rm{\varepsilon }}_m^{(0)} + \sum\limits_j {{\rm{\upsilon }}_{jm} Q_j } + \cdots $$

(4)$$t_{mn} = t_{mn}^{(0)} + \sum\limits_j {{\rm{\upsilon }}_{jmn} Q_j } + \cdots $$

Here, ω_j, Q_j, and P_j denote the frequency and the dimensionless coordinate and momentum of vibration mode j, respectively; ${\rm{\varepsilon }}_m^{(0)} $ and $t_{mn}^{(0)} $ stand for the site energies and transfer integrals obtained at the equilibrium geometry; υ_jmn (υ_jm) represents the nonlocal (local) linear electron-phonon coupling constants; and ħ is the reduced Planck constant.

Electronic coupling

The electronic coupling is a key factor that determines the extent of charge delocalization in the crystal. In organic systems, it is also the main source of anisotropy in the charge-carrier mobility. A number of approaches exist to evaluate the electronic couplings (transfer integrals), which have been reviewed extensively in the literature.Reference Newton^36–Reference Kaduk, Kowalczyk and Van Voorhis³⁹ In the mean-field approximation (where electron-electron interactions are treated in an average fashion), a simple approach to evaluate the transfer integrals in a single-component system is to make use of Koopman’s theorem (according to which the ionization energy of a system can be approximated as the energy of the frontier orbital)Reference Koopmans⁴⁰ and to estimate t_mn for holes (electrons) as half the energy splitting of the highest two occupied molecular orbitals ε_H and ε_H−1 (lowest two unoccupied molecular orbitals, ε_L+1 and ε_L) of the molecular dimers based on molecules m and n: $t_{mn} = {{\left[ {{\rm{\varepsilon }}_{{\rm{H}}\left({{\rm{L}} + 1} \right)} - {\rm{\varepsilon }}_{{\rm{H}} - 1\left({\rm{L}} \right)} } \right]} \mathord{\left/$.Reference Coropceanu, Cornil, da Silva Filho, Olivier, Silbey and Brédas^1, Reference Valeev, Coropceanu, da Silva Filho, Salman and Brédas¹⁶ However, this approach works properly only in the case of symmetric dimers.Reference Valeev, Coropceanu, da Silva Filho, Salman and Brédas¹⁶ In a general case, the transfer integral for any given dimer can be evaluated by using a method that defines the dimer orbitals in terms of molecular (fragment) orbitals.Reference Valeev, Coropceanu, da Silva Filho, Salman and Brédas^16, Reference Senthilkumar, Grozema, Bickelhaupt and Siebbeles⁴¹ Alternatively, band-structure calculations, in particular those based on contemporary density functional theory (DFT) methods (see Figures 1 and 2),Reference Coropceanu, Cornil, da Silva Filho, Olivier, Silbey and Brédas^1, Reference Zhu, Yi, Li, Kim, Coropceanu and Brédas^13, Reference Coropceanu, Sánchez-Carrera, Paramonov, Day and Brédas^17, Reference Sánchez-Carrera, Paramonov, Day, Coropceanu and Brédas^20, Reference Li, Coropceanu and Brédas⁴² can be performed on the crystals to evaluate the electronic couplings.

Figure 1.

Calculated band structure, density of states (DOS), effective masses, and electronic couplings in the pentacene crystal.Reference Sánchez-Carrera, Paramonov, Day, Coropceanu and Brédas^20, Reference Li, Coropceanu and Brédas⁴² (a) Chemical structure of pentacene. (b) Band structure (the conduction band and valence band are labeled as CB and VB, respectively) of the pentacene crystal determined at the density functional theory (DFT) level using the B3LYP functional and the 6-31G basis set. The points of high symmetry in the first Brillouin zone used for the calculations of the band dispersion are labeled as follows: Γ = (0, 0, 0), X = (0.5, 0, 0), Y = (0, 0.5, 0), Z = (0, 0, 0.5), U = (0, 0.5, 0.5), V = (0.5, 0, 0.5), T = (0.5, 0.5, 0), and R = (0.5, 0.5, 0.5), all in crystallographic coordinates. The zero of energy corresponds to the top of the VB. C and H atoms are given in gray and white, respectively. (c) Illustration of the most important charge-transport pathways for holes and electrons in pentacene. The band edges for holes and electrons located at V′ = (0.375, 0.5, 0.075) and R = (0.5, 0.5, 0.5), respectively. The approximate position of the V′ point is shown with the arrow. The numbers in blue circles label the molecules used for electronic couplings (t_mn) calculations; the red solid line shows the direction of the smallest effective mass component m ₁. m ₀ is the mass of a free electron in vacuum.

The results obtained for pentaceneReference Sánchez-Carrera, Paramonov, Day, Coropceanu and Brédas^20, Reference Li, Coropceanu and Brédas⁴² and DMQtT–F₄TCNQReference Zhu, Yi, Li, Kim, Coropceanu and Brédas¹³ using both approaches (i.e., the evaluations of the transfer integrals between dimers and the calculations of the crystal electronic bands) are shown in Figures 1 and 2. In the case of pentacene (Figure 1), the largest electronic couplings for both electrons and holes (about 90 meV) are obtained along the diagonal directions (along which the molecules pack in a herringbone fashion) within the ab-plane. In addition, significant couplings are found as well along the a-direction. The band-structure calculations indicate that the smallest effective masses are about 1.6 m ₀ and 1.5 m ₀ for holes and electrons, respectively. The effective mass for holes estimated by means of ARUPS measurements is about 1.3m _0.Reference Kakuta, Hirahara, Matsuda, Nagao, Hasegawa, Ueno and Sakamoto⁴³

Figure 2.

Calculated band structure, density of states (DOS), effective masses, and electronic couplings in the dimethylquaterthiophene-7,7,8,8-tetracyano-2,3,5,6-tetrafluoroquinodimethane (DMQtT-F₄TCNQ) crystal.Reference Zhu, Yi, Li, Kim, Coropceanu and Brédas¹³ (a) Chemical structures of DMQtT and F₄TCNQ; C, H, S, N, and F atoms are given in gray, white, yellow, dark blue, and light blue colors, respectively. (b) Band structure of the DMQtT-F₄TCNQ crystal determined at the density functional theory level using the B3LYP functional and the 6-31G basis set (DFT-B3LYP/6–31G). The points of high symmetry in the first Brillouin zone used for the calculations of the band dispersion are labeled as follows: Γ = (0, 0, 0), X = (0.5, 0, 0), Y = (0, 0.5, 0), Z = (0, 0, 0.5), U = (0, 0.5, 0.5), V = (0.5, 0, 0.5), T = (0.5, 0.5, 0), and R = (0.5, 0.5, 0.5), all in crystallographic coordinates. The zero of energy corresponds to the top of the valence band (VB). Note: CB, conduction band. (c) Illustration of the most important charge-transport pathways for holes and electrons in DMQtT-F₄TCNQ. The effective transfer integrals (t ^eff) along the stacking direction have a super-exchange nature. The numbers in blue circles label the molecules used for electronic couplings (t_mn) calculations; the red solid lines show the direction of two smallest effective mass components m ₁ and m ₂. This compound is characterized by very small effective masses and large electronic couplings for both holes and electrons.

Large electronic couplings of about 100 meV are found for both holes and electrons along the stacking direction in DMQtT–F₄TCNQ (Figure 2).Reference Zhu, Yi, Li, Kim, Coropceanu and Brédas¹³ This result might be surprising at first sight. In fact, in contrast to pentacene, the electronic couplings between D molecules or between A molecules have a super-exchange nature (these transfer integrals are denoted as t ^eff in Figure 2) (i.e., the electronic coupling for holes [electrons] results from the mixing of the frontier orbitals of two closest donor [acceptor] molecules with the orbitals of the “bridging” acceptor [donor] molecule). The effective mass along the stacking direction of DMQtT–F₄TCNQ (this component is labeled m ₁ in Figure 2) is calculated to be remarkably small for both holes (0.2 m ₀) and electrons (0.26 m ₀). This result can be explained by the fact that the effective mass also depends on the effective hopping distance ($m = \hbar ^2 /2ta^2 $, a— distance); this distance along the stacking direction in the DMQtT–F₄TCNQ is much larger than (essentially twice as large as) the characteristic distances between adjacent, strongly interacting molecules in pentacene.

Local electron-phonon coupling

The local electron-phonon coupling represents a key interaction in Holstein-type polaron models.Reference Holstein³¹ The overall strength is given by the polaron binding energy E _pol (or relaxation energy) at a given site m ($E_{m,{\rm{pol}}} = \sum\nolimits_j {{{{\rm{\upsilon }}_{jm}^2 } \mathord{\left/$). It consists of both intramolecular (internal) and intermolecular (external) contributions. The internal component reflects the changes, upon going from the neutral state to the charged state or vice versa, in the geometry of an individual molecule (see Figure 3a); the external component reflects the variations in the polarization and positions of the surrounding molecules. We note that the reorganization energy (that enters the electron-transfer rate equation in Marcus’ electron transfer theoryReference Marcus^44, Reference Barbara, Meyer and Ratner⁴⁵) roughly corresponds to twice the polaron binding energy as it considers the relaxations of both the molecule to which the electron is transferred and the molecule from which the electron originates.

Figure 3.

Intramolecular contribution to the polaron binding energy.Reference Coropceanu, Cornil, da Silva Filho, Olivier, Silbey and Brédas^1, Reference Zhu, Yi, Li, Kim, Coropceanu and Brédas^13, Reference Coropceanu, Malagoli, da Silva Filho, Gruhn, Bill and Brédas⁴⁶ (a) Illustration of the site energy (ε⁽⁰⁾) and polaron binding energy (E _pol). Q is the coordinate of a molecular vibrational mode. (b) Intramolecular component of the polaron binding energy for holes and electrons in the pentacene, DMQtT (dimethylquaterthiophene), and F₄TCNQ (7,7,8,8-tetracyano-2,3,5,6-tetrafluoroquinodimethane) molecules, calculated at the density functional theory level using the B3LYP functional and the 6-31G(d,p) basis set.

The extensive studies we have performed on this issue indicate that quantum-chemical methods, in particular those based on DFT, provide a reliable description of the intramolecular contributions to the local electron-phonon coupling constants and associated reorganization energies.Reference Coropceanu, Cornil, da Silva Filho, Olivier, Silbey and Brédas¹ The calculated E _pol values for holes and electrons in pentacene are about 49 meV and 66 meV, respectively.Reference Coropceanu, Malagoli, da Silva Filho, Gruhn, Bill and Brédas⁴⁶E _pol for electrons in F₄TCNQ and for holes in DMQtT is estimated to be 128 meV and 146 meV, respectively. An interesting aspect is that the contributions to the polaron binding energies coming from each vibrational mode can be deduced, as shown in Figure 3b. We see from the figure that both holes and electrons mainly interact with high-frequency intramolecular vibrational modes.

The external contribution to the local electron-phonon coupling has in general been much less studied than the intramolecular contribution. Several approaches, based on molecular mechanicsReference Brovchenko^47, Reference McMahon and Troisi⁴⁸ or hybrid quantum-mechanics/molecular mechanicsReference Norton and Brédas²⁹ methodologies and, more recently, a valence-bond/Hartree–Fock technique,Reference Martinelli, Idé, Sánchez-Carrera, Coropceanu, Brédas, Ducasse, Castet, Cornil and Beljonne¹⁹ have been used to evaluate the external reorganization energies in oligoacenes. These calculations indicate that the external contribution to the polaron binding energy is much smaller (a few percent) than the internal contribution.

Interestingly, the polaron binding energies calculated for both pentacene and DMQtT-F₄TCNQ are smaller than the half-widths of the conduction and valence bands (given by 2t along the direction corresponding to the herringbone packing in the case of pentacene and to the D-A stacks in DMQtT-F₄TCNQ). These results imply that the formation of small molecular polarons in these systems is not favorable.

Nonlocal electron-phonon coupling

The nonlocal electron-phonon coupling constitutes the major interaction considered in so-called Peierls-type models, such as the Su-Schrieffer-Heeger (SSH) HamiltonianReference Peierls^32, Reference Su, Schrieffer and Heeger⁴⁹ that has been largely applied to conjugated polymers. The overall strength of the nonlocal electron-phonon coupling can be quantified by means of the relaxation energies $L_{mn} $ ($L_{mn} = \sum\nolimits_j {{{{\rm{\upsilon }}_{jmn}^2 } \mathord{\left/$) that we introduced in Reference Reference Coropceanu, Sánchez-Carrera, Paramonov, Day and Brédas17. These relaxation energies can be considered to play a role in the nonlocal electron-phonon mechanism similar to E _pol in the case of local coupling.Reference Coropceanu, Sánchez-Carrera, Paramonov, Day and Brédas¹⁷

The nonlocal electron-phonon coupling constants have been evaluated for several organic crystalline semiconductors but, in nearly all instances, on the basis of the normal vibrational modes estimated at the Γ-point (center of the Brillouin zone).Reference Coropceanu, Sánchez-Carrera, Paramonov, Day and Brédas^17, Reference Girlando, Grisanti, Masino, Bilotti, Brillante, Della Valle and Venuti^18, Reference Sánchez-Carrera, Paramonov, Day, Coropceanu and Brédas^20, Reference Hannewald, Stojanović, Schellekens, Bobbert, Kresse and Hafner^50–Reference Girlando, Masino, Bilotti, Brillante, Della Valle and Venuti⁵² While these investigations have provided good insight, more recent studies based on a supercell approachReference Yi, Coropceanu and Brédas⁵³ have allowed us to take into account the dispersion of the phonon modes.Reference Li, Yi, Coropceanu and Brédas^21, Reference Yi, Coropceanu and Brédas⁵³ Our results indicate that the nonlocal electron-phonon couplings are significantly underestimated when they are evaluated just from Γ-point normal-mode calculations. For instance, in comparison with Γ-point estimates, when considering phonon dispersion, the converged values (obtained by systematically increasing the supercell size) of L in pentacene are larger by about 30–100% and 220–280% for the herringbone dimers and parallel dimers, respectively.Reference Li, Yi, Coropceanu and Brédas^21, Reference Yi, Coropceanu and Brédas⁵³

We note that in pentacene, the values of L_mn remain smaller than ${{L_{mn} } \mathord{\left/$, with the ratios $t_{mn}^{(0)} $ about 0.1–0.2.Reference Yi, Coropceanu and Brédas⁵³ However, it is useful to recall that L_mn is directly related to the variance of the thermal fluctuations of the corresponding transfer integral.Reference Coropceanu, Sánchez-Carrera, Paramonov, Day and Brédas¹⁷ The value of L = 20 meV, estimated for the herringbone dimer in pentacene, results at room temperature in a standard deviation of the transfer integral of about 30 meV (i.e., about 30% of the value of the transfer integral). This result underlines that at elevated temperatures, the nonlocal coupling leads to a substantial dynamical disorder. A small value for L along the stacking direction has been estimated for DMQtT-F₄TCNQ.Reference Zhu, Yi, Li, Kim, Coropceanu and Brédas¹³ However, the description of the nonlocal interactions in DA bimolecular crystals, due to the super-exchange nature of the electronic coupling, represents a very complex problem, and a more comprehensive study of this mechanism is still needed.

The nonlocal electron-phonon couplings can also be obtained through molecular dynamics simulations.Reference Yi, Coropceanu and Brédas^53, Reference Troisi and Orlandi⁵⁴Figure 4 shows the evolution of the transfer integrals with sampling time and the derived probability distribution of the transfer integrals at 300 K for the parallel and herringbone dimers (t ₃₄ and t ₂₃ in Figure 1) in the pentacene crystal. L can be computed from the width of the probability distribution, while the Fourier transform of the autocorrelation function $F_{mn} (t) = \left\langle {{\rm{\delta }}t_{mn} \left(0 \right){\rm{\delta }}t_{mn} \left(t \right)} \right\rangle $, where ${\rm{\delta }}t_{mn} \left(t \right) = t_{mn} \left(t \right) - \left\langle {t_{mn} \left(t \right)} \right\rangle $, yields the frequency dependence of L. As seen from Figure 4, only low-frequency modes contribute to L, with significant couplings for phonons with energies smaller than 20 cm⁻¹.Reference Yi, Coropceanu and Brédas^53, Reference Troisi and Orlandi⁵⁴ Interestingly, the energy of the lowest optical phonon in pentacene is estimated to be about 20 cm⁻¹;Reference Sánchez-Carrera, Paramonov, Day, Coropceanu and Brédas²⁰ thus, these results underline the importance of the role that acoustic phonons play in the charge-carrier mobilities in the crystal.

Figure 4.

Investigation of non-local electron-phonon coupling by means of molecular dynamics simulations:Reference Yi, Coropceanu and Brédas^53, Reference Troisi and Orlandi⁵⁴ (upper panels) the time modulation of the transfer integrals (t) due to thermal motion, (middle panels) the distribution of the transfer integrals (P), and (lower panels) frequency dependence of the relaxation energy (L(ω)) obtained by a Fourier transform of the autocorrelation function (F) of the transfer integrals for holes (dark red) and electrons (dark green) at 300 K for the parallel (pair 34, left panels) and herringbone (pair 23, right panels) dimers (see Figure 1) in the pentacene crystal.

Charge-transport models

Until recently, interpretation of charge transport in organic semiconductors was made most often in the framework of the Holstein small-polaron theory.Reference Holstein^30, Reference Holstein³¹ According to this model, (1) at low temperatures, charge transport can exhibit a conventional band-like behavior with the mobility decreasing in a power-law fashion (${\rm{\mu }} \propto T^{ - n} $, $n > 0$) with increasing temperature (T); and (2) at high temperatures, as a result of strong polaron localization effects (band narrowing), the motion of the carriers could be described as a sequence of uncorrelated hops, with the mobility showing an Arrhenius-like (activated) temperature dependence.Reference Coropceanu, Cornil, da Silva Filho, Olivier, Silbey and Brédas^1, Reference Bässler and Köhler² However, experimental data obtained on high-quality single crystals indicate that the appearance of activated transport is in many instances more likely due to the presence of structural disorder and chemical defects rather than the intrinsic signature of polaron formation.Reference Anthony^6–Reference Jurchescu, Baas and Palstra¹⁰

As pointed out previously and in more detail elsewhere,Reference Coropceanu, Cornil, da Silva Filho, Olivier, Silbey and Brédas¹ in most organic systems of current interest such as pentacene and rubrene, the polaron binding energy due to local electron-phonon coupling is of the same magnitude as, or even (much) smaller than, the width of the conduction or valence band. In addition, the largest part of the polaron binding energy arises from high-frequency intramolecular vibrations whose excited levels are thermally inaccessible even at room temperature. Therefore, a local-type Holstein mechanism taken alone cannot provide a comprehensive description of the charge-transport properties of organic semiconductors.

An attempt to extend the microscopic transport theory for the case where both local and nonlocal electron-phonon couplings are operative was made initially by Munn and SilbeyReference Munn and Silbey^55, Reference Munn and Silbey⁵⁶ and later by Hannewald and co-workers.Reference Hannewald, Stojanović, Schellekens, Bobbert, Kresse and Hafner^50, Reference Ortmann, Bechstedt and Hannewald⁵⁷ However, these contributions, as well as more recent work by Zhao and co-workers,Reference Chen, Ye, Zhang and Zhao⁵⁸ make use of a canonical (unitary) transformationReference Mahan⁵⁹ in the spirit of small polaron theory and omit specific terms. In the case of the Holstein model, such a methodology works only in the case of very weak electronic couplings.Reference Coropceanu, Cornil, da Silva Filho, Olivier, Silbey and Brédas^1, Reference Holstein^30, Reference Holstein³¹ The range of application of this approach in the case of combined Holstein-Peierls models remains an open question.

In order to study the impact of large nonlocal electron-phonon couplings on charge transport, Troisi and OrlandiReference Troisi and Orlandi²⁶ used an approach in which the vibrations were treated classically. This work as well as other studies based on similar approachesReference Wang, Chen, Zheng, Wang and Shi^60–Reference Zhong and Zhao⁶² underscored the importance of the dynamical disorder arising from these effects. Recently, Fratini and CiuchiReference Ciuchi and Fratini^22, Reference Fratini and Ciuchi²⁵ investigated the charge transport in high-mobility organic crystals by treating the nonlocal coupling in the framework of a quasi-static approximation (justified when the electronic processes are much faster than the lattice vibrations). The authors concluded that coherent band-like states and incoherent localized states induced by the thermal disorder coexist in the electronic spectrum, and both contribute to the charge-transport processes.

The methodologies used by Troisi and OrlandiReference Troisi and Orlandi²⁶ and Fratini and CiuchiReference Fratini and Ciuchi²⁵ are based on the conventional one-dimensional SSH model. In this approximation, the dependence of the transfer integrals on vibration coordinates (Equation 4) is given by $t_{m,m + 1} = t_{m,m + 1}^{(0)} + {\rm{\upsilon }}\left({Q_m - Q_{m + 1} } \right)$. However, Lindenberg and co-workersReference Zhao, Brown and Lindenberg⁶³ have shown that in the framework of the one-dimensional SSH model, a second mechanism exists given by $t_{m,m + 1} = t_{m,m + 1}^{(0)} + {\rm{\upsilon }}_{\rm{s}} \left({Q_m + Q_{m + 1} } \right)$. These two mechanisms are referred to as antisymmetric and symmetric nonlocal electron-phonon couplings, respectively. Our calculations of the nonlocal coupling constants indicate that the strengths of these mechanisms are comparable in oligoacene crystals.Reference Coropceanu, Sánchez-Carrera, Paramonov, Day and Brédas^17, Reference Sánchez-Carrera, Paramonov, Day, Coropceanu and Brédas^20, Reference Li, Yi, Coropceanu and Brédas²¹ In fact, the two coupling mechanisms lead to rather different interaction patterns. For instance, for an electronic band $E_k = - 2t^{(0)} \cos \left({ka} \right)$, where k is the wave vector of the electron, in the case of antisymmetric coupling, the strongest electron-phonon coupling in reciprocal space occurs between electrons at the band center ($k = \pm {\textstyle{{\rm{\pi }} \over {2a}}}$) and short-wavelength phonons; in the case of symmetric coupling, it takes place between electrons at the band edges ($k = 0$ and $ \pm {\textstyle{{\rm{\pi }} \over a}}$) and long-wavelength phonons, as shown in Figure 5. As a result, the corresponding electronic spectra are impacted in very different ways by the two coupling mechanisms. This effect can also be explicitly reflected in the charge-transport properties (see Figure 5e), since the two coupling mechanisms manifest themselves most strongly in different temperature ranges.Reference Li, Yi, Coropceanu and Brédas²¹ When both mechanisms contribute to the scattering processes, a power-law-like behavior is generally observed; however, the exponent strongly depends on the relative contributions of each mechanism and the range of temperatures under consideration.Reference Li, Yi, Coropceanu and Brédas²¹

Figure 5.

Illustration of the impact of symmetric and antisymmetric nonlocal electron-phonon couplings, using parameters relevant for the pentacene crystal: ħω = 50 cm^–1, t ⁽⁰⁾/ħω = 10, L/ħω = 1, $T = 300{\rm{ K}}$. Considering the equation $c = {{L_{\rm{s}} } \mathord{\left/$, the case $c = 1$ corresponds to exclusively symmetric coupling (see (a) and (b)); the case $c = 0$, to exclusively antisymmetric coupling (see (c) and (d)); and the case $c = 0.5$, to equivalent contributions from the two coupling mechanisms. (a) and (c) Strength of the electron-phonon coupling constants in reciprocal space as a function of electron and phonon momentum. (b) and (d) Corresponding electronic spectra and densities of electronic states (DOS). (e) Charge mobility calculated by the Kubo formula.Reference Mahan⁵⁹ (f) Band dispersion extracted from the maximum of the spectral function; the inset gives an enlarged view with the black dashed line providing the band dispersion in an ideal lattice without electron-phonon coupling. Here, ω represents the lattice vibration energy, $t^{(0)} $ the average transfer integral, L = L _s + L _a the total nonlocal relaxation energy with L _s(a) being the contribution from the symmetric (antisymmetric) coupling, $k_{\rm{B}} $ the Boltzmann’s constant, μ⁰ = ea ²/ħ a mobility unit, k the electron wave vector, q the phonon wave vector, and $e$the elementary charge.Reference Li, Yi, Coropceanu and Brédas²¹