This article will be included, in edited form, as a chapter of the forthcoming book “Physics of Organic Semiconductors”, edited by W. Brütting and C. Adachi (Wiley-VCH, Weinheim, 2012), ISBN 978-3-527-41053-8.
This review is focused on understanding of the charge-transport physics of high-mobility organic semiconductors at a molecular level. We review recent high-mobility small-molecule and conjugated polymer materials with a focus on crystalline materials that have been able to exceed mobilities of 0.5–1 cm2/V s. We discuss some of the main, competing factors that govern charge transport in these materials and present theoretical approaches that have been developed to describe systems in which moderately strong intermolecular electronic interactions and strong electron–phonon interactions are present. Finally, we review recent experimental results that have aimed to address the important question of whether at room-temperature charge carriers in these high-mobility organic semiconductors are in fact simply extended Bloch electrons that undergo occasional scattering processes or are localized on individual molecules and move by hopping.
There has been tremendous progress in discovering new classes of organic semiconductors, that provide field-effect mobilities, µ, above 1 cm2/V s and allow addressing increasingly demanding thin-film electronic applications 1. For many years, the performance of organic field-effect transistors (OFETs) seemed to be inherently lower than that of their inorganic counterparts, in particular, compared to amorphous silicon (a-Si) and polycrystalline FETs which have characteristic mobilities of the order of 0.5–1 and 100 cm2/V s, respectively. Due to extensive materials development and evaluation of different classes of organic semiconductors there is now a broad range of organic semiconductors, both vacuum and solution processible, as well as small-molecule and conjugated polymer based, which are able to reach mobility values exceeding that of a-Si. The processing characteristics of these materials make them suitable for applications that cannot easily be addressed by many inorganic materials. Organic semiconductors are inherently low-temperature materials. Because of the absence of covalent bonding between molecules, they can be processed at temperatures below typically 100–150 °C from either vapor phase or solution into high-quality thin films that exhibit a low density of electronic defects. This makes them suitable for applications with a limited temperature window during manufacturing, such as those that require a flexible plastic substrate, or applications that require integration with other low-temperature materials such as biological molecules. It also makes these materials compatible with direct-write printing-based manufacturing techniques.
As a result of these improved performance and unusual processing characteristics organic FETs are now in advanced stages of development for commercial applications in flexible active matrix displays, such as active matrix, electrophoretic displays on plastic substrates. OFET-driven, active matrix OLED displays on plastic substrates have been demonstrated 2. Circuit applications in RFID tags and ambient intelligent devices are also being pursued. Very recently, a first 8-bit microprocessor was realized by an OFET integrated circuit 3. For an excellent, recent review on applications of organic FETs please refer to Ref. 4.
In spite of extensive research efforts there is still considerable controversy about the correct physical description of the charge carriers and the mechanism of charge transport in organic semiconductors, particularly in high-mobility systems. In low-mobility, disordered organic semiconductors, it is well established that charge transport is limited by static, energetic, and positional disorder and proceeds by hopping of localized charge carriers in an energetically disordered landscape of sites. This transport regime is very well investigated from studies on amorphous organic semiconductors as used in xerographic applications 5. In contrast, charge transport in high-mobility molecular semiconductors with mobilities of 1 cm2/V s or higher is intermediate between the classical band-like, Bloch electron conduction of crystalline inorganic semiconductors and the molecular hopping conduction in disordered, low-mobility organic semiconductors. At least at room temperature (RT) a description in terms of band transport remains controversial because the mean free path of charges is estimated to be of the order of the molecular distance. However, a clear understanding of the nature of the charge carriers, their charge-transport mechanism and the factors that limit charge-carrier mobilities in this high-mobility regime is essential for providing clear guidelines to materials chemists for design and synthesis of materials that allow even higher levels of performance to be achieved.
This review is focused on understanding the charge-transport physics of high-mobility organic semiconductors at a molecular level. There is an extensive literature already on charge transport in molecular crystals following the pioneering work of Karl and others in the 1970s, which has clarified the influence of key factors, such as purity and structural anisotropy, on the temperature-dependent charge-transport properties of molecular crystals, such as naphthalene. For an excellent review of this work see, for example, Ref. 6. Here, we focus on more recent, molecular-level insight into the charge-transport physics of molecular semiconductors that has been gained from investigating a wider range of heteroatom, molecular structures and from applying in-depth experimental characterization and state-of-the-art theoretical modeling techniques. The review is less concerned with other important aspects of the device physics of organic FETs, such as the role of the gate dielectric 7, the importance of efficient charge injection from the source–drain contacts as well as the nature of defect states and their role in device stability that have been reviewed elsewhere 8, 9. In Section 2 we review recent high-mobility small-molecule and conjugated polymer materials with a focus on crystalline materials that have been able to exceed mobilities of 0.5–1 cm2/V s. In Section 3 we discuss some of the main, competing factors that govern charge transport in these materials and present theoretical approaches that have been developed to describe systems in which moderately strong intermolecular electronic interactions and strong electron–phonon interactions are present. In Section 4 we review recent experimental results that have aimed to address the important question of whether at RT charge carriers in these high-mobility organic semiconductors are in fact simply extended Bloch electrons that undergo occasional scattering processes or are localized on individual molecules and move by hopping.
2 Review of recent high-mobility small-molecule and polymer organic semiconductors
The field-effect mobility of an organic FET is critically dependent on the local intramolecular structure and the intermolecular packing between the molecules. These determine the reorganization energy and the transfer integral, which are the main parameters that govern charge transport 10. For the design of a high-mobility material as a rule of thumb the reorganization energy should be small and the transfer integral large. However, equally important is how uniform the molecular environment is along the interface. Static structural and energetic disorder encountered by the charges moving along the interface results in percolation motion with certain sites becoming rate limiting for the transport along the interface. The effect that such sites have on mobility can easily mask the more intrinsic dependence of mobility on molecular structure and packing 8. The disorder might also be dynamic due to fluctuations in the intermolecular packing that occur on the same timescale as carrier motion and require charge carriers to remain localized on particular lattice sites until a favorable molecular configuration for charge transfer occurs dynamically 11–13. The transport physics of organic semiconductors will be discussed in more detail below. In this section we first discuss recently discovered small-molecule semiconductors that provide field-effect mobilities >1 cm2/V s (Fig. 1) and then review recent high-mobility conjugated polymers (Fig. 2). The discussion will be focused on materials that exhibit high-hole mobilities in FET configurations. Due to space constraints we are unable to discuss the wide range of materials for n-type FETs. We only state here that similarly high-carrier mobilities exceeding 1 cm2/V s are also achievable for electron transport and we refer to the recent excellent review by Anthony et al. 14 on this subject.
A focus area for new materials development continues to be the class of acene and fused heteroacene materials. The extensive work on systems such as pentacene and rubrene has established such fused, extended π-systems to be good candidates for achieving high mobility. Recent synthetic efforts have been devoted to developing acene-based materials with better chemical stability and solution processibility. An important class of solution-processible acenes were developed by Anthony et al. 15 based on the excellent solubility characteristics of the trialkylsilylethynyl group, such as triisopropyl-silylethynyl (TIPS), substituted in the central 6- and 13-position of a pentacene core. TIPS pentacene (TIPS-P) shows a high degree of crystallinity, even in spin-coated films, and exhibits a two-dimensional (2D), brickwall, and a cofacial π–π-stacking motif in the plane of the films with the side chains oriented near normal to the substrate surface. Mobilities exceeding 1 cm2/V s can be obtained in films of neat TIPS-P provided that care is taken to carefully adjust the growth and substrate conditions to control the nucleation of the crystalline grains 16. A variety of heteroacene derivatives, such as triethylsilylethynyl anthradithiophene (TESADT) 17 or difluorinated (di-F) TESADT 18 exhibit similarly high levels of performance. Some derivatives substituted with electron-withdrawing groups, such as 7,8,9,10-tetrafluoro-5,12-bis(TIPSethynyl)tetraceno[2,3-b]thiophene 19 have been shown to exhibit high-mobility ambipolar transport with electron and hole mobilities exceeding 0.1 cm2/V s. Anthony's group has investigated the structure–property relationships of a large number of solution-processible, trialkylsilylethynyl substituted acenes, and has studied the correlation between the crystal-packing motif and carrier mobility. It was concluded that generally one-dimensional (1D), π–π stacking tends to result in lower mobility than 2D, brickwall packing 20. An interesting derivative, which possibly does not follow this simple rule as discussed below, is 1,4,8,11-tetramethyl-6,13-triethylsilylethynyl pentacene (TMTES-P), which exhibits 1D π–π stacking, but allows reaching field-effect mobilities up to 3 cm2/V s 21. Several groups have reported more straightforward control of film growth and more uniform electrical characteristics when blending the TIPS-P molecules with a polymer binder 22. In blend films of TIPS-P and diF-TESADT, respectively, Hamilton et al. 23 reported mobilities of 1 and 2.4 cm2/V s, respectively. The high performance is associated with a segregation of the small molecule to the surface and interface of the films.
Another class of molecules that has recently attracted significant attention are end-substituted phenylene-thiophene, selenophene, or thiazine fused ring systems. In vacuum-sublimed films of unsubstituted diphenyl-benzothieno-benzothiophene and dinaphto-chalcogenopheno-chalcogenophenes based on thiophene (DNTT) or selenophene (DNSS) mobilities of 1–3 cm2/V s were reported 24, 25. Single crystals of DNTT laminated onto a Cytop gate dielectric showed mobilities up to 8.3 cm2/V s 26. In vapor-grown single crystals of pyrrolobisbenzothiazine with cofacial, 2D π–π stacking mobilities of up to 3.6 cm2/V s were reported on octadecyltrichlorosilane (OTS) modified SiO2. Derivatives substituted with C8H17 on the central nitrogen position exhibited more 1D π–π stacking and reduced mobilities up to 0.4 cm2/V s 27. Interestingly, in 3,10-dimethyl substituted DNTT, an unusual 3D-herringbone stacking was observed due to slippage of one of the molecules in the unit cell along the c-direction. Surprisingly, this 3D-herringbone stacking still supports only slightly degraded mobility of 0.8 cm2/V s compared to that of DNTT.
Spin-coated films of di-alkyl end-substituted derivatives of benzothienobenzothiophene (BTBT) exhibit a highly ordered, crystalline microstructure with alternating layers of aliphatic side chains and conjugated layers parallel to the substrate and herringbone stacking within the conjugated layers. High mobilities of 1–3 cm2/V s were initially reported for Cn-BTBT (n = 11–13). Cn-BTBT with longer sides chains tended to exhibit slightly higher mobility than Cn-BTBT with shorter side chains, because the stronger hydrophobic interaction induced by the longer side chains enhances the conjugated molecular overlap. By drop casting onto an inclined substrate the film morphology for C8-BTBT could be optimized and bottom-gate FETs with mobilities of 5 cm2/V s were reported 28. Attempts to replace sulfur by selenium surprisingly resulted in poorer device performance as thin films of Cn-BSBS adopt a microstructure with larger inclination of the molecular axis from the substrate normal and poorer electronic overlap than in the corresponding Cn-BTBT films. Attempts to incorporate the BTBT conjugated core into a conjugated polymer were initially unsuccessful, but a closely related copolymer based on naphtodithiophene-bithiophene (PNDBT) with semicrystalline, lamellar microstructure exhibits high-hole mobilities up to 0.54 cm2/V s 29. In contrast to Cn-BTBT di-alkylated Cn-DNTT exhibits only moderate solubility that is too low for forming films by spin coating, but shows excellent device performance with mobilities up to 7.6 cm2/V s in vacuum-sublimed thin films of C10-DNTT 30. By using a drop-casting method from hot solution with growth directed by an inclined surface solution-deposited films of C10-DNTT were incorporated into both bottom as well as top-gate FETs with mobilities up to 11 cm2/V s 31.
Hu and coworkers 32 compared linear-shaped dibenzothienodithiophene (DBTDT) and sickle-shaped bisbenzothienothiophene (BBTT) 33 and found that while DBTDT adopts a herringbone π–π stacking with mobility of 0.5 cm2/V s in vacuum-deposited thin films, BBTT has significantly higher solubility, adopts a cofacial π–π stacking, and solution-grown crystals exhibit field-effect mobilities of 0.2–0.6 cm2/V s in bottom-gate FETs on OTS-modified SiO2. Gao et al. 34 investigated end-substituted di-alkyl-dithienobenzodithiophene with cofacial, 2D π–π stacking and demonstrated mobilities of 1–1.7 cm2/V s in drop-cast films on untreated SiO2.
Recent discoveries of molecules with high-carrier mobilities >1 cm2/V s have by no means been confined to acenes or fused thiophene–phenylene systems. Due to space constraints we cannot provide a complete account, but as examples we quote titanylphtaolocyanine (TiOPc), which has a nonplanar, square pyramid shape, and exhibits mobilities of up to 3.6 cm2/V s in vacuum-grown films on OTS-modified SiO235. Derivatives of tetrathiafulvalene (TTF) have also attracted significant attention. In solution-grown crystals of dithiophene-TTF field-effect mobilities of 3.6 cm2/V s were reported for bottom-gate FETs on hexamethyldisilazane (HMDS) or OTS-modified SiO236. In solution-grown crystals of hexamethylene-TTF (HMTTF) contacted with TTF-TCNQ source-drain electrodes and a parylene gate dielectric mobilities of 10 cm2/V s were reported. The molecules adopt a 2D, cofacial, brick-wall π–π stacking motif 37.
We now shift our discussion to high-mobility conjugated polymers. Although it is possible to achieve reasonably high mobilities of 10−3–10−2 cm2/V s using amorphous materials, as long as the polymer is designed to minimize energetic disorder 8, 38, 39 it is easier to ensure this in semicrystalline materials. It appears that many of the recently discovered high-mobility polymers adopt a lamellar microstructure similar to that found in P3HT 40 or poly(3,3′-dialkyl-quaterthiophene) (PQT) 41. Through the formation of alternating layers of conjugated backbones separated by layers of flexible side chains with the layers oriented in the plane of the film efficient charge transport in the plane of the film along the active interface of the FET is facilitated, because the motion of charges is not impeded by the presence of the inevitable flexible, insulating side chains. One focus of recent attention has been on thienothiophene-based copolymers after the discovery of high mobilities of up to 1.1 cm2/V s in poly(2,5-bis(3-alkylthiophen-2-yl)thieno(3,2-b)thiophene) (PBTTT) 42, 43. The lower density of side chains in thiophene–thienothiophene copolymers such as PBTTT not only slightly increases the ionization potential and improves the stability, but allows for better side-chain interdigitation and more highly crystalline structures. This work has recently been reviewed by McCulloch et al. 44.
A number of groups have successfully investigated different classes of donor–acceptor copolymers 45. A copolymer of benzothiadiazole and cyclopentadithiophene (CDT-BTZ) was recently shown to exhibit mobilities up to 1.4 cm2/V s. It was originally claimed that thin films of this polymer had an amorphous microstructure 46, but it was later discovered in dip-coated films of higher molecular weight than the polymer is in fact semicrystalline 47. It adopts an anisotropic lamellar structure, again resembling that of P3HT, with the polymer backbone and the direction of π–π stacking in the plane of the film. The exceptionally high-mobility values achieved in this polymer have been attributed to its donor–acceptor character, which increases the interchain attraction and result in a small π–π stacking distance of 3.7–3.9 Å. The Bao group reported several low bandgap, donor–acceptor polymers 48, 49, including poly(didodecylquaterthiophene-alt-didodecylbithiazole) (PQTBTz-C12) 50, which adopts a highly crystalline lamellar microstructure and allows mobilities of 0.3 cm2/Vs and excellent bias stress stability comparable to that of a-Si thin film transistors to be achieved. Guo et al. 51 copolymerized phtalimide acceptor units with thiophene (Ph-BT12) and achieved a small π–π stacking distance of 3.6 Å and field-effect mobilities of 0.2 cm2/V s. Liu et al. 52 showed that a high degree of crystallinity is not always a prerequisite for high mobility. They reported copolymers based on N-alkyl-dithienopyrrole and thiophene (DTP-BT) and showed that mobilities of up to 0.2 cm2/V s could be achieved in as-spun films of the polymer with a seemingly amorphous microstructure. Rieger et al. 53 investigated copolymers of thiophene and benzodithiophene (BDT-BT) with a curved backbone that affords a good compromise between solubility and aggregation and exhibits mobilities up to 0.5 cm2/V s in top-gate devices with polystyrene gate dielectric.
A new class of low-bandgap conjugated polymers that has recently attracted significant attention are low-bandgap copolymers of diketopyrrolopyrrole (DPP). Bürgi et al. 54 were first to report a low-bandgap copolymer based on DPP and thiophene, which not only exhibited high-hole mobilities of 0.1 cm2/V s, but also electron mobilities of the same magnitude and could be operated as an ambipolar, infrared emitting light-emitting transistor. Nelson et al. 55 demonstrated that in semicrystalline, spin-coated films of a copolymer of electron-donating dithienopyrrole (DTP) and electron-accepting DPP mobilities of 0.3–0.4 cm2/V s on OTS-modified SiO2 can be achieved. Li et al. 56 investigated copolymers of DPP and thienothiophene and demonstrated high-hole mobilities of up to 0.94 cm2/V s in spin-coated films annealed at 200 °C with a semicrystalline, lamella microstructure on OTS-modified SiO2. Electron mobilities were much lower. In bottom-gate OFETs with SiO2-modified SiO2 copolymers of DPP with electron-accepting thiophene-benzothiadiazole-thiophene (TBT) exhibit ambipolar transport with high-electron mobilities of 0.4 cm2/V s and hole mobilities of 0.35 cm2/V s were observed 57. In top-gate FETs with PMMA gate dielectric and copolymers of DPP and silaindacenodithiophene electron mobilities of 0.1 cm2/V s and hole mobilities of 0.4 cm2/V s were found 58. Even higher mobilities have recently been reported for top-gate FETs with PMMA gate dielectric and copolymers of DPP and thienothiophene. Record hole mobilities for a conjugated polymer FET with electron and hole mobilities both exceeding 1 cm2/V s were achieved 59, 60, which demonstrates the potential for reaching with conjugated polymers similarly high-charge-carrier mobilities as in the small-molecule systems described above.
3 General discussion transport physics/transport models of organic semiconductors
The charge-transport physics of a specific organic semiconductor system can be considered to be governed by several key physical parameters: the intrinsic bandwidth J due to the intermolecular interactions in a perfectly periodic molecular solid, the static diagonal and nondiagonal disorder parameters, σ and Σ, respectively, that characterize the distribution in site energies and transfer integrals due to static disorder in the real molecular solid, the reorganization energy λreorg, which is a measure of the strength of the local electron–phonon/electron–ion interaction and the strength of the nonlocal electron–phonon coupling which reflects phonon-induced, dynamic fluctuations in the transfer integrals. The origin of these parameters can be seen from the Hamiltonian of the electron–phonon system in a molecular solid:
Here, are creation (annihilation) operators for an electron on site j. are creation (annihilation) operators for a phonon with wavevector and polarization index s and frequency . εj is the energy of a charge sitting on a molecule on site j. The transfer integral characterizes the strength of the intermolecular, electronic interaction between neighboring molecules on sites i and j of the crystal causing charges to move between sites. and are the respective local and nonlocal electron–phonon coupling constants that measure the strength of the interaction between electrons and intramolecular and intermolecular vibrations. As can be seen from Eq. (1) the local electron–phonon coupling gives rise to a modification of the site energies, while the nonlocal electron–phonon coupling modulates the transfer integrals due to fluctuations in molecular distances and/or orientation.
3.1 Transfer integral Jij
If there was no static disorder in the values of εi and Jij and the electron–phonon coupling could be ignored, one would expect the electron states of a molecular crystal comprising a perfectly periodic array of molecules to be simply Bloch waves with a defined wavevector , just like in crystalline inorganic semiconductors. The only difference would be that in a tight-binding description these Bloch states would be constructed not from the atomic orbitals of isolated atoms, but from the molecular orbitals of isolated molecules. Jij determines the bandwidth B of the energy-momentum dispersion of these Bloch states. A simple tight-binding model of a 1D molecular stack would predict, for example, a bandwidth B = 4J. The transfer integral can be calculated from
where and are the relevant HOMO and LUMO orbitals in the case of hole and electron transport, respectively, of the isolated molecules and Hel is the effective one-electron Hamiltonian of the crystal. It can often simply be estimated as half the energy splitting between the HOMO and HOMO−1 (LUMO + 1 and LUMO) of a dimer of the molecule computed in the intermolecular packing configuration encountered in the full crystal 61.
Jij has been calculated for a broad range of organic molecules using quantum chemical or density functional theory (DFT) techniques (for a detailed discussion see Ref. 61). From these calculations several important insights have been gained, in particular regarding the expected magnitude of Jij and its dependence on intermolecular packing. Since the intermolecular distances in van-der-Waals-bonded molecular solids are significantly larger than in covalently bonded inorganic semiconductors, such as silicon, the typical bandwidth in molecular crystals is comparatively small, but not negligible. Typical values predicted by band-structure calculations are of the order of several 100 meV (Fig. 3). Experimental investigations of the band structure of thin, single crystals of pentacene by angle-resolved ultraviolet photoemission spectroscopy (AR-UPS) have shown that the bandwidth at 140 K in the direction of the surface Brillouin zone is indeed of the order of 330 meV 62 (Fig. 4).
The bandwidth is, however, very sensitive to intermolecular separation and decreases approximately exponentially with intermolecular distance. Within a range of π–π stacking distances of 3.4–4 Å typically encountered in organic semiconductors the transfer integral can vary by a factor 3–4. For the different molecular pairs in the crystal structure of pentacene values for the hole transfer integrals in the range of 35–85 meV have been estimated from DFT calculations 63. These values are comparable in magnitude to typical intra-/intermolecular vibrational modes. Values for electrons and holes can exhibit significant differences reflecting the different wavefunctions of the HOMO and LUMO states. These calculations have also demonstrated that the transfer integral is very sensitive to the exact intermolecular stacking arrangement. Small relative shifts of intermolecular positions can cause large variations in the magnitude and sign of the transfer integral reflecting changes in the overlap between the peaks and nodes of the molecular wavefunctions (Fig. 7a) 61. This is why transport properties in molecular crystals can depend sensitively on polymorph structure, but it also suggests that nondiagonal electron–phonon coupling might be particularly relevant in the transport physics of these materials, as will be discussed below.
If static disorder and electron–phonon coupling were sufficiently weak that the standard Bloch electron description of covalently bonded semiconductors could indeed be applied to molecular crystals one would expect the charge-carrier mobility to exhibit a characteristic band-like temperature dependence µ = eτ(T)/m, with effective mass m*. The temperature-dependent scattering time τ(T) should exhibit a decrease with increasing temperature, reflecting thermally induced excitation of the various scattering mechanisms, such as phonons. However, even in the purest and highest mobility molecular crystals the electron–phonon coupling might be too strong in order to be taken into account merely as an occasional scattering mechanism. Much of the theoretical effort in the field has been devoted to methods of treating and understanding the potential effects of the electron–phonon interaction terms in the above Hamiltonian.
3.2 Reorganization energy λ
To get an understanding of the effects of diagonal electron–phonon coupling it is helpful to discuss the limit in which disorder, intermolecular interactions and nonlocal electron–phonon coupling can be considered weak. Holstein showed in a series of papers in 1959 64, 65 that if the transfer integral was strictly zero, the local electron–phonon coupling would result in full localization of the charge onto a single site. The energy of the localized charge is stabilized by the so-called polaron binding energy
associated with a local molecular distortion induced by the charge. Eb and the so-called reorganization energy λ, which is twice the polaron binding energy, play an important role in the theories of electron transfer in a transport regime in which the reorganization energy can be considered to be large compared to the transfer integral. The reorganization energy can also be expressed in terms of the energies of the neutral (E) and charged (E+) molecule computed for both the ground-state nuclear configuration of the neutral (M) and charged molecule (M+) or in terms of the difference in the configuration coordinates between the neutral and charged ground-state configuration, (neglecting the dispersion of the phonon modes):
Reliable values of reorganization energy have been computed for a range of molecules by taking into account the contributions from all individual vibrational and rotational modes (Fig. 5; for a review see Ref. 61). The intramolecular modes correspond to local molecular distortions associated with charge injection. However, intermolecular lattice modes involving the relative displacement of molecules also make contributions to the charge-carrier polarization energy, which should not be ignored for a reliable estimate of λ. The reorganization energy depends sensitively on molecular structure and substitution and also on the polarity of the charge carrier. λ tends to decrease with increasing conjugation length and increase in the presence of polar substitution groups. Hole charge carriers in pentacene have been found to exhibit a particularly low value of λ = 90 meV, while sexithiophene has a significantly higher value of 320 meV. It should be noted that the value of λ for pentacene is in excellent agreement with gas-phase ultraviolet photoelectron spectroscopy (UPS) measurements 66.
Holstein 65 considered a model for charge transport in which the nondiagonal electron–phonon coupling is neglected and the diagonal electron–phonon coupling constant g to optical vibrational modes is large. Although Holstein's perturbation theory approach for taking into account the effect of finite J is limited to systems with small electronic bandwidth, and is probably not applicable to systems like pentacene, where λ is of comparable magnitude to the estimated transfer integral, we discuss it here because its identification of two different transport regimes, an extended polaron band-like transport regime at low temperature and a localized, small polaron hopping regime at high temperatures is widely believed to be more generally valid. As stated above in the limit J = 0 the charge is localized on an individual lattice site p and is accompanied with a vibrational state characterized by occupation numbers, Nk, for each vibrational mode. For finite J Holstein applied perturbation theory to calculate the probability for transitions to a neighboring site (p → p ±1). At low temperatures this transition probability is dominated by diagonal transitions in which none of the Nk are altered and Bloch-type extended polaron states are formed. The polaron bandwidth is renormalized and is expected to be significantly smaller than the electron bandwidth B due to a dependence on a vibrational overlap integral. The polaron bandwidth is also predicted to drop rapidly with increasing temperature. Such narrowing of the bandwidth has recently been observed experimentally in ARUPS measurements on thin films of pentacene on graphite 67. In this regime the effect of nondiagonal transitions can be taken into account as a scattering mechanism and the mobility drops rapidly with increasing temperature. For a 1D system with lattice constant a and single optical vibration mode with frequency ω0 the mobility in this polaron band transport regime can be expressed as:
Holstein estimated that at a transition temperature Tt of around the Debye temperature, i.e., typically , the lifetime of these states becomes so short that the associated energy uncertainty becomes comparable to the bandwidth. At higher temperature nondiagonal transitions, in which some of the Nk are altered as a result of the electron transfer, dominate and transport can be described as a hopping process of small polarons that are localized on individual lattice sites and hop between pairs of sites with a mobility:
At high temperatures this reduces to an Arrhenius temperature behavior with an activation energy Ea = λ/4, which is identical to the one predicted by Marcus electron transfer theory. At very high temperatures the mobility is expected to drop again with a temperature dependence. Holstein also established that for transfer integrals that exceed a critical value perturbation theory can no longer be applied and an adiabatic treatment of the electron-transfer process is required. The DFT and QC calculations discussed above suggest that in systems such as pentacene λ is of comparable magnitude to J, which suggests that simple, nonadiabatic hopping models for electron transfer are probably not applicable.
Several authors have developed approaches to solving the Holstein Hamiltonian for a broader range of parameters, while still ignoring nondiagonal couplings. Silbey and Munn 68 showed using a density matrix approach that the existence of two transport regimes, a polaron band transport regime at low temperatures with mobility decreasing rapidly with increasing temperature and a high temperature, small polaron hopping regime is still expected over a wider range of parameter values for J and g than those accessible by Holstein's approach. As g increases the overall mobility in the band regime is reduced, but the hopping contribution is enhanced (Fig. 6). In the high-temperature hopping regime a thermally activated regime as predicted by Holstein is only observed for very large values of g > 3, that are probably not realized in many molecular systems, as they would imply values for λ that are too large. For intermediate and small values of g, the effect of the contribution from hopping transport is to slow down the decrease of mobility with increasing temperature undergone by the band transport contribution and lead to a relatively weak temperature dependence of the mobility near room-temperature mobility. Kenkre et al. 69 used a generalized master equation approach to extend the Holstein model to higher dimensions and to fit the theoretical mobility expression to the experimental data of naphthalene molecular crystals. They estimated a ratio of for diagonal coupling to a vibrational mode with . More recently, Cheng and Silbey 70 used a generally applicable variational method to study the crossover from band-like to hopping transport. A different approach to explain the band-like temperature dependence of the mobility near RT in molecular crystals was recently proposed by Nan et al. 71 who considered a quantum-mechanical expression of the charge-transfer rate in the localized hopping regime conventionally described by Marcus theory and showed that a mobility decreasing with increasing temperature can also be obtained in a hopping transport regime.
3.3 Nonlocal electron–phonon coupling constant
The nonlocal electron–phonon coupling modulates the transfer integrals. It arises mainly from lattice modes that modulate the molecular distances, relative positions, and/or orientation. A number of authors have developed approaches to include these terms in the treatment of the Hamiltonian. Although an exact diagonalization of the Hamiltonian as in the limit considered by Holstein is not possible, Munn and Silbey 72, 73 showed using a unitary transformation method that in the presence of nonlocal coupling it is still possible to distinguish clearly between a band-like polaron low-temperature regime and a small-polaron hopping regime at higher temperatures. In the band-like regime the inclusion of nonlocal coupling increases the scattering rate and decreases the mobility. However, in the high-temperature hopping regime nonlocal couplings give rise to phonon-assisted currents and increase the mobility. Overall, inclusion of nonlocal coupling was predicted to further weaken the temperature dependence of the mobility in the hopping regime and lead to a near temperature-independent diffusion coefficient and a mobility temperature dependence . More recently, Hannewald and Bobbert 74, 75 used a similar unitary transformation method with ab inito estimates of the various coupling constants to calculate the mobility in the Kubo formalism of linear response theory. They showed in particular, that nondiagonal coupling give rise to contributions to the mobility that exhibit a pronounced anisotropy in their temperature dependence. For an excellent review covering nonlocal electron–phonon coupling see Ref. 76.
A different approach to investigating the effects on nondiagonal electron–phonon coupling was recently proposed by Troisi and Orlandi 77, 78. The approach is motivated by the realization from quantum-chemical computations that the transfer integrals in molecular crystals are extremely sensitive to the relative molecular positions and orientations. Small translations of two acene molecules along their long axis by only 0.5–1 Å can result in large changes of the transfer integrals by more than an order of magnitude 61 (Fig. 7a). The origin of this high sensitivity of J to intermolecular packing is the complex shape of the HOMO and LUMO wavefunctions. Furthermore, it appears that in many common molecular crystals the equilibrium molecular stacking configuration does not correspond to a maximum or minimum of the transfer integral, such that small displacements of any molecule from the equilibrium structure are associated with significant changes in the relevant transfer integrals. This is illustrated in Fig. 7b for the case of different solution-processible pentacene derivatives derived from TIPS pentacene, in which π–π stacking does not occur in the conventional herringbone motif, but in a cofocial manner 79. The 2D contour map shows the expected variation of the transfer integral as a function of relative translations of two pentacene molecules in the x- and y-directions. The contour line labeled with an asterisk corresponds to a zero value of J. The plot also shows the actual relative positions that are found in the equilibrium structure of five different pentacene derivatives (Fig. 7a–e). It is clear from this plot that (Fig. 7a) the transfer integral varies strongly as a function of relative position and (Fig. 7b) that the equilibrium structures are located in regions in the vicinity of which significant variations of transfer integral as a function of position are expected.
At or near RT significant thermal fluctuations in relative molecular positions and orientations are expected due to the soft, van-der-Waals bonding in these materials. This is illustrated in Fig. 7c where the individual dots show fluctuations in molecular configuration that were detected in a molecular dynamics simulation at RT. There is a significant thermal broadening of the values of the transfer integral that are encountered in the array of molecules during the MD simulation and the standard deviation of these fluctuations is comparable to the magnitude of the average transfer integral. By combining DFT calculations with molecular dynamics simulations Coropceanu et al. 80 came to similar conclusions about the strength of thermal fluctuations of the transfer integral of acene crystals and estimated quantitative values for the nondiagonal electron–phonon coupling constants for different inter- and intramolecular modes. The fluctuations of the transfer integral occur on a ps timescale. From the Fourier transform of the corresponding autocorrelation function it can be concluded that there is a distribution of vibrational modes that contribute to the modulation of transfer integrals centered around 40 cm−1, while modes above 160 cm−1 make only negligible contributions 81.
It is clear from these simulations that at and near RT electrons moving through such a solid are expected to experience significant dynamic disorder in the transfer integrals with strong fluctuations occurring on a ps timescale. If one considers a frozen lattice configuration at a specific point in time an electron will see a disordered lattice of molecules. This disorder in the transfer integrals fundamentally affects the nature of the electron states and is expected to result in localization of at least some of the electron states if the disorder is strong enough 82. Troisi and Orlandi 77 found that at RT the disorder is in fact strong enough to localize all the states in the density of states (DOS) within a computed localization length of the order of several tens of Å. States near the edge of the DOS are more strongly localized. The localization length increases as a function of decreasing temperature because of decreasing thermal fluctuations. Troisi's approach allows a microscopic model of the DOS and localization length for such dynamically disordered systems to be built as a function of temperature. The specific molecular and crystal structure can be taken into account and the results can be compared to experimental results, as discussed in more detail below. Fratini and Ciuchi 83 used a Green's function method to consider localization induced by dynamic disorder in a 1D model system and predicted a DOS with a tail of states that are strongly localized by dynamic disorder, while states in the center of the DOS were found to be significantly more delocalized.
Troisi and Orlandi 77 also developed a model for the mobility in such a dynamically disordered system. The model treats the motion of the nuclei classically and assumes one effective vibrational mode of a specific frequency to represent the more complex distribution of modes of the real crystal. It is based on integrating the time-dependent Schrödinger equation for electrons to predict the time evolution of the electron wavefunction starting with an initial electron wavefunction that is localized over a characteristic localization length by the dynamic disorder as discussed above. By calculating the charge-carrier diffusion length as a function of time, Boltzmann averaging over different initial wavefunctions and fitting the averaged diffusion length to obtain a diffusion coefficient a mobility value is predicted from the Einstein relation. Troisi 78 demonstrated that the simple transport model can reproduce the order of magnitude of the mobility observed at RT in rubrene and pentacene. It predicts a band-like temperature dependence of the mobility, , because as the temperature is decreased the dynamic disorder is reduced and the localization length of the electronic states is increased.
Troisi's original model has been criticized for treating the phonons classically, for approximating the complex dynamics with a single effective mode to include in the Hamiltonian and for being a 1D model. Dynamic disorder might possibly be a strong effect in 1D only, but in 2D one could argue it should affect the electronic states significantly less. However, recent work on an extension of the model to 2D has shown that the basic prediction of the model, that the electronic states become localized by dynamic disorder in the transfer integrals, remains valid also in 2D 84. The model was also recently extended to systems that exhibit dynamical fluctuations over a wide frequency range from typically 0 to 200 cm−185, 86. The classical treatment of the phonons is clearly a limitation of the model and warrants further study. However, at RT the condition , which renders such classical treatment valid, should be valid for the low-energy lattice modes that appear to be responsible for the dynamic disorder.
To better understand charge transport in this regime it is helpful to consider the different timescales that are relevant. The intermolecular interactions with transfer integrals J ≈ 100 meV should in principle give rise to formation of mesoscopically extended Bloch electrons on a timescale . However, as stated above we need to consider the coupling of the electrons to intramolecular vibrational modes () as well as intermolecular librations and other acoustic and optical phonon modes () 87. Intramolecular reorganization occurs on a typical timescale comparable to the timescale of charge delocalization and might therefore be considered to lead primarily to renormalization of the transfer integral 88. Intermolecular reorganization, on the other hand, occurs on a slower timescale of and it is therefore expected that on a sub-100 fs timescale charge carriers see a dynamically disordered potential landscape into which they can extend over only a small length scale. A short time later the lattice configuration has changed and the charges then extend into the new configuration. In this process the charges can be considered to “surf” on the potential created by the lattice phonon modes. A similar argument for dynamic localization was put forward by Zuppiroli and coworkers 88, who argued that fluctuations in the polarization energy due to intermolecular phonons can also result in similar dynamic disorder driven carrier localization.
3.4 Static disorder parameters σ and Σ
In a real molecular solid structural or chemical defects, such as branching defects, regioregularity defects, conjugation defects, chemical impurities, point or line defects, dipolar disorder due to random orientation of polar groups of the organic semiconductor, or the gate dielectric lead to time-independent, spatial variations of the energy and hopping distances from site to site. Again, according to Anderson 82, in the presence of such static disorder some of the electronic states, particularly those close to the band edge, are no longer delocalized, but become localized. If the disorder is sufficiently strong all states become localized. This regime is encountered in many disordered, amorphous low-mobility organic solids and conjugated polymers. Charge transport can be described by hopping between pairs of localized molecular states. The distribution of site energies can often be approximated by a Gaussian distribution with standard deviation σ. Similarly, the off-diagonal disorder can be described by a Gaussian distribution with standard deviation Σ, reflecting the distribution in hopping distances.
The transport of charges injected into a molecular solid dominated by the effects of disorder, is well understood from the work on molecularly doped polymers and other organic photoconductors used in xerography 5, 89, 90. Assuming a disorder-broadened Gaussian density of transport states with a characteristic width σ Bässler 91 has shown on the basis of Monte–Carlo simulations that an injected carrier hopping through such an otherwise empty DOS relaxes to a dynamic equilibrium energy below the center of the DOS leading to a characteristic temperature dependence of the mobility. The model has been improved by Novikov et al. 92, who showed that the dominant source of diagonal disorder is due to charge–dipole interactions, and that spatial correlations of such interactions need to be taken into account in order to explain the commonly observed Poole–Frenkel dependence of the mobility on the electrical field, and derived an expression for the electric field and temperature dependence of the mobility in a correlated DOS with both diagonal as well as nondiagonal, positional disorder:
The model describes the transport of individual injected carriers at zero/small carrier concentrations, i.e., should in principle not be directly applicable to the relatively high-carrier concentrations p = 1018–1019 cm−3 present in the accumulation layer of FETs. Vissenberg and Matters 93 has developed a percolation model for variable range hopping transport in the accumulation layer of an FET assuming an exponential DOS with width T0. An expression for the field-effect mobility as a function of carrier concentration p was derived:
where σ0 is the prefactor for the conductivity, α the effective overlap parameter between localized states, and is the critical number for onset of percolation. Transport in this model can be effectively described as activation from a gate-voltage-dependent Fermi energy to a specific transport energy in the DOS.
An alternative theoretical framework for understanding the effects of disorder is the multiple trapping model, which is well established for describing transport in amorphous silicon, and has been claimed to be more appropriate for describing the charge transport in microcrystalline polymers such as P3HT 94 and poly(bis(alkyl-thienyl-bithiophene) 95. This model assumes that the static disorder is sufficiently weak that in a certain energy range the DOS becomes high enough that electronic states above the so-called mobility edge are extended, while electronic states below the mobility edge remain localized. The current is assumed to be transported by carriers that are thermally activated into the delocalized states above the mobility edge, while carriers in localized states are effectively trapped. Assuming a specific DOS, and a mobility for carriers above the mobility edge, the FET current can be obtained by first determining the position of the quasi-Fermi level at the interface for a particular gate voltage and then calculating the number of free carriers which are thermally excited above the mobility edge using Fermi–Dirac statistics. A common feature of all transport models that assume that transport is determined by energetic disorder is that they predict a temperature-dependent mobility that increases with increasing temperature and it is generally challenging to distinguish between the different models on the basis of an analysis of this temperature dependence alone.
4 Transport physics of high-mobility molecular semiconductors
In this section, we review recent experimental characterization of the charge-transport physics of high-mobility molecular semiconductor FETs. Our discussion is limited to transport in single crystals or polycrystalline thin films with large grain size that allow the more intrinsic charge-transport properties and electronic structure within single-crystalline grains in the absence of grain boundaries to be studied. The general transport physics of molecular single crystal FETs has been reviewed in an excellent review article by Gershenson et al. 96, which includes discussions on the experimentally determined mobility anisotropy, the role of the gate dielectric, effects observed at high-carrier concentration and understanding of the nature of defect and trap states. When studying charge transport in organic semiconductors using FET configurations one should always keep in mind that the electronic properties of the surface might differ significantly from those in the bulk, not only because the presence of the dielectric can induce significant electronic disorder and polarization effects 8, but also because the molecular structure in the first layer on the surface/interface might differ significantly from that in the bulk, as recently demonstrated for rubrene single crystals by X-ray diffraction 97. In this section, we focus on recent attempts to answer one of the fundamental, remaining open questions, which is whether the transport in highly crystalline molecular semiconductors at or near RT is governed by band transport or hopping transport, i.e., whether the wavefunction of the charge carriers at or near RT is extended over many molecules or confined to a single molecule. As discussed in Section 3 this remains the subject of considerable debate.
Important experimental insight into this question comes from the temperature dependence of the FET mobility in high-mobility rubrene single crystals with room-temperature mobility values of 10–40 cm2/V s. These exhibit a band-like temperature dependence between RT and 200 K, i.e., the mobility decreases with increasing temperature (Fig. 8a) 98. The temperature dependence can be fitted to a power-law dependence. At lower temperature the mobility enters a trap dominated thermally activated regime. A similar temperature dependence has also been observed for transport in the bulk, for example, by transient photoconductivity experiments 99, 100. It is in reasonably good qualitative agreement with the theoretical predictions by Hannewald discussed above. However, as our discussion of theoretical approaches to describe systems with strong electron–phonon coupling has shown, considering the temperature dependence of the mobility alone is not necessarily a very sound basis for distinguishing extended state, band-like transport from localized hopping transport.
Complementary evidence that supports a band-like conduction mechanism includes the observation of a Drude-like optical conductivity induced by the formation of an accumulation layer in rubrene single crystals at RT. The effective mass of the carriers was estimated to be close to that of the free electron mass 101, 102, respectively, suggesting that polaronic effects are absent or too weak to lead to significant mass enhancement. It was also claimed that in the optical spectra range close to the bandgap there are no characteristic charge-induced absorption that are characteristic for isolated radical cations, also suggesting the absence of pronounced polaronic relaxation effects 101. We note, however, that in the raw experimental charge-modulation spectrum of these devices there were strong interference artefacts, which make it difficult to investigate any charge-induced absorptions in the optical spectral range. We argue below that in devices optimized to eliminate such interference artefacts clear evidence for polaron formation is observed, even in high-mobility rubrene single crystals.
Measurements of the Seebeck coefficient S also provide a means of assessing the relevance of electron–phonon coupling in molecular semiconductors. S is a measure of the entropy transported by a thermally excited charge carrier and can be determined from the thermoelectric voltage generated between two electric contacts between which a temperature gradient ΔT is established, S = Vthermoel/ΔT. In inorganic semiconductors with negligible electron–phonon coupling S is determined mainly by a configurational, electronic entropy contribution that depends on the electronic DOS, and the energy-dependent conductivity σ(E) 103:
where Etr (EF) is the transport (Fermi) energy level, i.e., the edge of the valence band, and A is a quantity that depends on the detailed shape of σ(E). For many inorganic semiconductors A is of the order of one. In contrast, in a system with strong polaronic effects, in which the charges lead to significant softening of the vibrational modes, there should be additional vibrational contributions to the Seebeck coefficients that would make S significantly larger than what is expected from the pure electronic configurational entropy term in Eq. (6) 104.
The main challenge of interpreting Seebeck measurements on molecular semiconductors is an accurate knowledge of the energy difference EF–Etr. von Muhlenen et al. 104 reported thermopower measurements on relatively low mobility, polycrystalline pentacene FETs on SiO2 and found relatively high values of the Seebeck coefficient of the order of 240–500 µV/K. No significant dependence on the gate voltage was detected. EF–Etr was estimated from a comparison of the activation energy of the thin-film conductivity and the activation energy of the field-effect mobility. From this analysis the contribution to the Seebeck coefficient from the configurational entropy term was found to be small and it was claimed that the main reason for the large Seebeck coefficient stems from the vibrational entropy contribution due to phonon-mode softening associated with charge formation. Pernstich et al. 105 reported contradictory results. In their thermopower measurements on rubrene single crystals and pentacene thin films DOS modeling of the temperature dependence of FET characteristics was used to estimate and it was concluded that the magnitude of S is well within what could be expected for nonpolaronic transport (A = 2–4) and suggestions that there is no significant charge–vibration coupling. In this study a significant decrease of with increasing carrier concentration/gate voltage was also detected.
Electron spin resonance (ESR) provides a microscopic probe for distinguishing between transport regimes that involve hopping of localized charges and those that involve delocalized electrons 106. In a standard ESR measurement a DC magnetic field applied to the sample causes a Zeeman splitting of the energy levels of spin-1/2 charge carriers, the magnitude of which is measured by the absorption of a microwave field. The position of the ESR resonance allows extraction of the g-factor of the electron and its anisotropy as a function of direction of the applied field. The ESR linewidth provides information about the hyperfine coupling between the electron spin density distribution and the protons of the molecule. For localized electrons hyperfine interactions should in principle give rise to a broad, Gaussian lineshape, while for delocalized electrons the motional narrowing effect tends to result in narrow, Lorentzian lineshape. Kuroda and coworkers 107 used field-induced ESR measurements on pentacene FETs to determine the anisotropy of the g-factor. For thin-film pentacene FETs with a relatively low mobility of 0.01 cm2/V s they claimed a Gaussian, temperature-independent lineshape, indicating that the charge carriers were static and displayed no motional narrowing. From an analysis of the linewidth broadening due to hyperfine coupling () the number of molecules N over which the charge-carrier wavefunction is extended was estimated to be of the order of 10. Matsui et al. 108 used similar pentacene FETs, but with higher mobility up to 0.6 cm2/V s. They claimed that the ESR spectra exhibit a Lorentzian lineshape, which was found to narrow considerably at high temperatures and high-gate voltages, providing experimental evidence for motional narrowing due to polaron diffusion. According to the theory of motional narrowing in a system in which transport is determined by a multiple trapping and release the full width at half-maximum of the ESR spectrum, , is related to the typical residence time of charges in a trap state, , through where γ is the gyromagnetic ratio and is the inhomogeneity of the local magnetic field due to the hyperfine interaction. Values for pentacene films of tc = 0.7–2 ns were extracted. A similar analysis was also applied to ESR measurement on single-crystal rubrene FETs 109 for which shorter values of tc = 60–700 ps were extracted reflecting the higher mobility of 1–8 cm2/Vs compared to thin-film pentacene. A correlation between ESR linewidth and mobility was reported for a rubrene device with different chemical modification of the gate dielectric/rubrene interface. Recently, Matsui et al. 110 presented a careful analysis on pentacene FETs to explain why the ESR spectrum retains a near-Lorentzian lineshape even at low temperatures. From saturation experiments and an absence of linewidth broadening with increasing microwave power at low temperatures it was concluded that below a temperature of 50 K the charges become strongly trapped and no motional narrowing effect is present, i.e., that the trap residence time becomes larger than . They were able to explain the near-Lorentzian lineshape at low temperature by assuming the existence of three different trap states in which the trapped charge is distributed over N = 1.5, 5, and 6–20 molecules, each trap state contributing a Gaussian-shaped peak with a linewidth (Fig. 9). A theoretical Holstein polaron analysis of the expected binding energy for these trap states predicted binding energies of 140 meV (N = 1–5), 22 meV (N = 5), 5–15 meV (N = 6–20). At RT this estimate of N does not apply because of the linewidth being determined by motional narrowing.
Hall effect measurements have also been used because they might provide a more direct way of probing the intrinsic charge-transport properties of organic semiconductors than measurements of the longitudinal conductivity. In a Hall experiment on a FET a magnetic field is applied perpendicular to the plane of the accumulation layer and a transverse voltage on opposite sides of the channel is measured as a longitudinal current is flowing along the channel. The main quantities that are measured are the Hall resistance RH and the longitudinal conductivity σ of the accumulation layer. These define a Hall carrier concentration nH as well as a Hall mobility µH according to:
UH is the transverse Hall voltage measured by two voltage probes on opposites sides of the channel separated by the channel width W, V the longitudinal voltage drop associated with the current I measured between two voltage probes along the channel separated by a distance L*, and the ± sign applies to electrons and hole charge carriers, respectively.
For an ideal, wide-bandgap semiconductor or metal with an isotropic band structure and no trap states the Hall carrier concentration should be equal to the total carrier concentration n and the Hall mobility should be equal to the longitudinal drift mobility µ. This situation we will refer to as the ideal Hall effect. In a FET configuration it is possible to check whether a Hall signature is ideal since the total carrier concentration is known accurately from . In fact, one of the main applications of Hall measurements for systems where the carrier n is not known, is the independent determination of n. If the system has some trap states, such that at any point in time a fraction of charges reside in localized trap states, these charges do not have a velocity and, consequently, do not experience a Lorentz force. In this situation nH can be interpreted as the fraction of the total carrier concentration, which is in mobile band states and are able to contribute to the Hall voltage. µH continues to provide a measure of the band mobility, while the drift mobility extracted by dividing the longitudinal conductivity by the total carrier concentration would be reduced, reflecting the influence of the traps.
In a hopping system the situation is more complicated. A simple interpretation of nH is not possible and the key quantity that needs to be computed in order to interpret the Hall effect in such systems is the transverse conductivity in the presence of the magnetic field. A number of authors have developed theoretical approaches to calculate the Hall mobility for different hopping transport models 111. The magnetic field manifests itself through the interference of the transition amplitudes associated with the carrier moving between two sites via different paths. Emin 112 considered the Holstein polaron model in the high-temperature hopping limit and showed that the key quantity that determines the transverse conductivity and the Hall mobility are three-site hopping probabilities. Generally, in hopping systems µH and RH differ significantly from µ and (en)−1, respectively. In this sense, Hall effect measurements should constitute a powerful, qualitative test for whether a system exhibits hopping transport or whether the charge carriers are associated with well-defined velocities.
Hall measurements in organic semiconductors are challenging due to the high-longitudinal sheet resistance associated with the relatively low-field-effect mobilities, high-contact resistance and threshold voltage and current instabilities on the timescale of the Hall measurements. They were first reported for high-mobility FETs based on rubrene single crystals 113, 114. Podzorov et al. 114 showed that at RT rubrene FETs with an air-gap dielectric exhibit an ideal Hall effect with and µH = µFET, suggesting that all carriers induced by the gate voltage have well-defined velocity and contribute to the Hall voltage (Fig. 8b). At temperatures <250 K nH and µFET were found to exhibit thermally activated behavior as a result of charge-carrier trapping, while µH exhibited a band-like temperature dependence from RT to 170 K below which the Hall signature could not be measured due to increased noise. Takeya et al. 115 showed for rubrene FETs in contact with a relatively high-k gate dielectric of polyvinylphenol (PVP) that the Hall mobility drops sensitively with increasing gate voltage, suggesting that high-mobility charge transport is possible, while at the interface the mobility is reduced by surface scattering. The first Hall measurements on thin-film OFETs were reported for pentacene bottom-gate OFETs with a mobility of 0.4 cm2/V s 116. The inverse Hall resistance was a factor 2–3 larger than the gate-induced charge-carrier concentration in the temperature range from 250 to 380 K 117. Below 250 K this difference increased further. This cannot be interpreted in terms of a trapping effect as discussed above, which would lead to but rather indicates that charge transport in such relatively low-mobility polycrystalline OFETs exhibits characteristics of hopping transport. Different behavior was recently observed for vacuum-evaporated polycrystalline FETs based on DNTT with slightly higher field-effect mobility of 0.4–1.2 cm2/V s. Here, the Hall effect was observed to be ideal at RT as well as low temperatures down to 160 K suggesting that the presence of grain boundaries alone does not prevent the observation of an ideal Hall effect in organic semiconductors 118. The extracted Hall mobility in the thin films was found to be thermally activated while reference measurements on single crystals of DNTT revealed a near temperature-independent Hall mobility. This was interpreted in terms of a contribution from grain-boundary resistance to the Hall mobility of the thin films.
One of the important questions that was left unanswered by these Hall effect studies is an understanding of to what extent observation of an ideal or nonideal Hall effect system reflects any microscopic charge-transporting states in a particular materials system. It remained unclear, for example, why a molecular semiconductor such as unsubstituted pentacene shows a nonideal Hall effect, while DNTT is able to exhibit an ideal Hall effect, while the field-effect mobilities of the two systems are quite comparable. In a recent series of papers 119, 120 we have proposed an experimental methodology to address this question by combining temperature-dependent FET measurements with Hall effect measurements, optical spectroscopy of charges that provides a direct spectroscopic assessment of the presence of polaronic relaxation effects and theoretical simulations of dynamic disorder for the specific molecular structures investigated experimentally. In the following, we illustrate this methodology by comparing two solution-processible pentacene derivatives, 6,13-bistriisopropyl-silylethynyl pentacene (TIPS-P) 15 and 1,4,8,11-tetramethyl-6,13-triethylsilylethynyl pentacene (TMTES-P) 21. Both systems exhibit high-charge-carrier mobilities in spin-coated thin films in contact with a low-k, Cytop gate dielectric with room-temperature values of 1.5–2 cm2/V s for TIPS-P and even higher values of 2.6–3.5 cm2/V s for TMTES-P. The measurements are performed on thin, polycrystalline films, but the grain size is large, >10 µm, typically, such that it should be possible to study more intrinsic charge-transport properties in these devices. The main advantage of using thin films compared to single crystals is that it is easier to extend the temperature range of electrical and optical measurements to temperatures close to 4 K. With single-crystal FETs this has proven challenging due to problems with mismatch of thermal expansion coefficients between the crystals and the substrate/other layers.
The temperature dependence of TIPS-P top-gate FETs with mobilities of 1.5–2 cm2/V s was found to similar to that observed for rubrene single crystal FETs 119. Between RT and 200 K the mobility increases with decreasing temperature, providing clear evidence that even in solution-processed molecular thin films a band-like temperature dependence of the mobility can be observed. At lower temperature, again similarly to single crystals, the mobility becomes thermally activated with a relatively small activation energy of 5–15 meV. The nature of the static, shallow traps states that are most likely responsible for the temperature-activated behavior is unknown, but they could be associated with dislocations, point defects, or other structural defects incorporated into the film during solution growth.
This observation of band-like transport characteristics in thin films similar to that observed in rubrene single-crystals warranted a more in-depth investigation using Hall effect measurements 119. The sign of the Hall coefficient RH was found to be positive in both materials, consistent with the FETs operating in p-type accumulation mode. 1/RH increases monotonically with |Vg − Vth| (Fig. 10b). In TIPS-P, 1/RH is about two times larger than the gate-induced charge density QCi(Vg – Vth), which is similar to the behavior reported for vacuum-sublimed pentacene 116. The Hall mobility, µH, is about half the field-effect mobility, µFET, at all temperatures from RT to 150 K. In contrast, TMTES-P exhibits a remarkably ideal Hall effect with 1/RH coinciding with Q for |Vg| > 20 V, not only at RT, but for all temperatures from RT to 150 K.
The observation of an ideal Hall effect in TMTES-P suggests unambiguously that, at least in TMTES-P, the charge carriers are not localized on individual molecules and move by hopping, but that their wavefunction must be sufficiently delocalized, such that it is possible to define a wavevector and group velocity that can couple to the magnetic field through a semiclassical Lorentz force expression. The observation of an ideal Hall signature raises the question whether charge carriers in these molecular crystals are in fact best described as conventional Bloch electrons that are only weakly coupled to phonons and delocalized on mesoscopic length scales or in which way the transport properties reflect any molecular reorganization and localization processes.
The degree to which molecular reorganization associated with charge formation is present can directly be investigated by optical charge modulation spectroscopy (CMS) of the charges in the FET accumulation layer as a function of temperature. If molecular reorganization is present, charges in organic semiconductors are expected to have characteristic charge-induced optical absorptions, which differ from those of the neutral molecule and reflect the degree of intramolecular reorganization and polarization associated with charge formation 8. For both TIPS-P and TMTES-P FETs we clearly observed such charge-induced absorption bands (ΔT/T < 0) around 1.2 and 2.6 eV, together with an associated bleaching signal (ΔT/T > 0) of the neutral molecule absorption around 1.8–2.2 eV (Fig. 11) 119. To identify the nature of these transitions we performed quantum-chemical calculations at the semiempirical AM1 level with full configuration interaction (FCI). Calculations were performed on an isolated TIPS-P monomer molecule as well as a dimer. If one takes into account the typical errors in estimating transition energies by quantum-chemical computations in the gas phase, the theoretical spectrum is in good agreement with the experimental CMS spectrum of TIPS-P 119. The calculations on the dimer predict the charge to be fully delocalized over the two molecules. The calculations predict subtle differences between the monomer and the dimer spectrum, which might suggest that the temperature dependence of the CMS spectra can be interpreted as temperature-dependent changes in the degree of charge delocalization. We interpret the broad spectral features in the room-temperature CMS spectrum of TIPS-P as a manifestation of an ensemble average over charges that are delocalized over different numbers N = 2, 3,… of molecules. The sharper CMS spectrum at low temperature is believed to reflect the capturing of the charges in shallow static trap states, which is consistent with the observed temperature dependence of the mobility at low-lateral electric field. In general, the surprisingly good agreement of the experimental charge-induced absorption spectrum with the calculations on small molecular clusters implies that charges in TIPS-P and TMTES-P cannot be in extended Bloch electron states delocalized over 100s or 1000s of molecules, but can only be delocalized over a small number of molecules. For mesoscopically extended states the reorganization would be significantly reduced 121 and we would expect any residual charge-induced absorption features to appear at energies close to those of the neutral system and not result in such strongly spectrally shifted induced absorptions.
To investigate the generality of these results we performed CMS measurements also on rubrene single crystals FETs with mobilities >5 cm2/V s (Fig. 12). As discussed above, it had been concluded from previous CMS measurements 101 that in rubrene no such induced absorptions were present in the optical spectral range. We optimized the device configuration for CMS measurements on rubrene single crystals to remove any interference artefacts in the spectrum. To our surprise, we found that the rubrene CMS spectra are qualitatively very similar to TIPS-P and TMTES-P, with a low-energy charge-induced absorption at 1.6 eV and a higher-lying transition at 2.8 eV. There is no spectroscopic evidence that charge carriers in TMTES-P, TIPS-P, and rubrene single crystals are fundamentally different. Since TMTES-P and rubrene 114 exhibit an ideal Hall effect to which all charge carriers must contribute, we concluded that charges that are only delocalized over a small molecular cluster and result in radical cation-like, charge-induced absorptions, are indeed capable of producing an ideal Hall signature and a band-like temperature dependence of the field-effect mobility.
These observations of limited charge delocalization over only a few molecules emerging from CMS and the ideal Hall effect and band-like temperature dependence of the mobility normally associated with extended-state transport can be rationalized consistently in the dynamic disorder transport model proposed by Troisi and Orlandi 77, 84 discussed above. Fluctuations in the hopping integrals for TIPS-P and TMTES-P were computed using a combination of molecular dynamics and quantum-chemical calculations 78 and the dynamically disordered DOS and localization length at RT were calculated for the crystal structure of the two molecules that had been determined by X-ray diffraction. For both molecules, there is substantial DOS broadening due to thermal motions, accompanied by considerable localization of the wavefunction in the tail of the DOS. Within a few kBT from the DOS edge, the states are localized within a few molecules. The DOS shape for the two molecules is different. In both molecules the π–π stacking is cofacial and the π–π stacking distance are similar, but TMTES-P has a quasi-1D dispersion, reflecting its 1D crystal packing and smaller broadening due to a smaller fluctuation of the coupling, whereas TIPS-P has a 2D brick-wall packing motif (Fig. 13a). When integrating the DOS to a charge concentration of 1019 cm−3 typical of the FET accumulation layer the Fermi level comes to lie in a region with larger delocalization length for TMTES-P than for TIPS-P. This provides a possible explanation for the more ideal Hall signature observed in TMTES-P. The computed isotropic mobilities of 5.1 and 12.4 cm2/V s for TIPS-P and TMTES-P, respectively, are also in a similar ratio as found experimentally for the Hall mobilities and also close to the experimental values in absolute terms. From this analysis it appears that the wavefunction localization length predicted by the dynamic disorder model for systems such as TMTES-P is small enough to provide an explanation to observe the signature of localized states in CMS, but large enough to allow observation of an ideal Hall effect. In fact, Hall transport in organic semiconductors might be qualitatively similar to that in poor metals with weak electron correlation effects, where RH remains ideal when the scattering length approaches the interatomic distance 122.
In this article, we have discussed the materials and charge-transport physics of high-mobility organic semiconductors. In spite of extensive research efforts it has been difficult to answer the simple, but fundamental question whether at or near RT the wavefunctions of the charge carriers in these materials are extended over many molecules and transport is best described in terms of Bloch electrons with occasional scattering, or whether the charge carriers are localized on individual molecules and move by hopping. Over recent years there has been increasing experimental evidence that at least in high-mobility molecular crystals with mobility values exceeding 1–10 cm2/V s a description in terms of Bloch electron transport might be more appropriate to explain the band-like temperature dependence of the mobility or the observation of an ideal Hall effect. However, as we have argued in this review there are also experimental observations that clearly indicate the importance of molecular-relaxation effects, such as the presence of a pronounced optical charge-induced absorption that resembles that of radical cations localized on small molecular clusters. To explain these observations consistently we have proposed that at RT the intermolecular interaction is sufficiently strong and the diagonal electron–phonon coupling sufficiently weak that the charge carriers would in principle be able to form extended Bloch electron states, however, they are hindered in their ability to delocalize fully by nondiagonal electron–phonon coupling due to thermal lattice fluctuations. At RT charges remain localized on a picosecond timescale over small clusters of molecules and they undergo a diffusive motion with the fluctuations in the intermolecular lattice. In such a model it is possible to explain both the band-like temperature dependence of the mobility and the occurrence of an ideal Hall effect, but also the observation of pronounced charge-induced absorption. This description of charge transport points to dynamic disorder as one of the main factors limiting charge-carrier mobilities at RTs. Higher mobilities can be expected in molecular architectures that effectively reduce fluctuations in relative molecular positions and orientations and associated fluctuations in the transfer integrals. Molecular modeling of dynamic disorder should provide a simulation tool at the molecular level to guide the design and development of new higher-mobility organic semiconductors.
Henning Sirringhaus, FRS, is the Hitachi Professor of Electron Device Physics at the Cavendish Laboratory, University of Cambridge. He has an undergraduate and PhD degree in physics from ETH Zürich (CH). His current research interests include the charge, spin transport, and photophysics of solution-processible polymer, molecular, and oxide semiconductors. He is a co-founder and Chief Scientist of Plastic Logic Ltd., a technology start-up company commercializing printed organic transistor and flexible display technology for applications in electronic readers, and of Eight 19, a spin-off company commercializing organic solar cell technology.
Tomo Sakanoue received his Ph.D. in 2006 from the Graduate University for Advanced Study, Japan under the supervision of Prof. H. Tada. He then joined Prof. C. Adachi's group at Kyushu University. From 2008 to 2011 he worked as a research associate at the University of Cambridge. After postdoctoral research fellow at Central Research Institute of Electric Power Industry (CRIEPI), Japan (2011–2012), he became an assistant professor at Yamagata University. His current interests are the charge transport physics and the electrochemical properties of organic semiconductors and polymers.
Jui-Fen Chang received her Ph.D. degree in physics at the University of Cambridge in 2007; the focus was on the transport mechanisms of microcrystalline polymer semiconductors. From 2007 to 2010 she worked as postdoctoral research assistant on the Hall effect of organic thin film transistors. In 2011 she took her present position as assistant professor in the Department of Optics and Photonics in the National Central University in Taiwan, where she is working in the field of organic light-emitting devices and solar cells.