J. Phys. Chem. B 2006, 110, 14659-14666
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Efficient Charge Transport along Phenylene-Vinylene Molecular Wires Paulette Prins,* Ferdinand C. Grozema, and Laurens D. A. Siebbeles Opto-Electronic Materials Section, DelftChemTech, Delft UniVersity of Technology, Julianalaan 136, 2629BL, Delft, The Netherlands ReceiVed: April 28, 2006; In Final Form: June 14, 2006
We have studied the motion of charge carriers along isolated phenylene-vinylene (PV) chains using a combination of experimental and theoretical methods. The conductive properties of positive charges along PV chains in dilute solution were studied by using the pulse-radiolysis time-resolved microwave conductivity (TRMC) technique. This technique enables the measurement of high-frequency (tens of GHz) charge carrier mobilities along isolated PV chains without the use of electrodes. The charge carrier mobility along PV chains with finite and infinite length was studied theoretically by charge transport simulations with parameters from density functional theory (DFT) calculations. The high-frequency charge carrier mobility is found to depend strongly on the conjugation length of the PV chains and is found to increase both with increasing length of the PV chain and with increasing conjugation fraction. The experimental results are in good agreement with the calculated results. On the basis of this combined experimental and theoretical study an intrachain charge carrier mobility of a few tens of cm2/Vs is expected for an infinitely long PV chain without conjugation breaks.
1. Introduction Poly(p-phenylene-vinylene) (PPV) derivatives are prototype conjugated polymers that have been extensively studied for application in plastic electronic devices because of their light emitting1 and conductive properties.2 In addition, individual chains of oligo PV’s may be applied as charge transporting wires in nanoscale molecular electronic circuits.3,4 There has been a growing interest in molecular electronics because conventional chip manufacturing methods are rapidly approaching the fundamental size limits that are dictated by the wavelength of the light used for photolithography. Moore’s law predicts that the dimensions of the interconnecting wires will decrease from about 90 nm presently used in computer chips to approximately 10 nm within the next 10 years.5 At this small length scale the use of single molecules is inherently attractive because of their size. An additional advantage of using single molecules is the full control over the molecular structure through organic synthetic methods, which ensures that the physical properties can be tuned for specific applications. Charge transport through single molecules has been studied by positioning single molecules between electrodes6 or by studying self-assembled monolayers of conjugated molecules containing thiol groups.7 The optoelectronic properties of single molecules in a direct current (DC) setup are often determined to a large extent by the contact between the electrodes and the molecule. Although a vast number of these studies have been performed, there is relatively little systematic information about the relationship between molecular structure and the charge transport properties, mostly because of the variety of device setups that have been used, making it difficult to compare results. To date, no absolute values for the DC charge carrier mobility measured through single molecular wires between electrodes have been published. * Address correspondence tnw.tudelft.nl.
to
this
author.
E-mail:
p.prins@
Alternatively, charge transport along conjugated molecular wires can be studied by pulse-radiolysis time-resolved microwave conductivity (TRMC) measurements.8 With this technique an oscillating microwave field is used to measure the mobility of charge carriers along isolated polymer chains, without the need to apply electrodes. This circumvents some of the inherent problems encountered in DC experiments. TRMC measurements have been performed for a variety of conjugated polymers, including derivatives of PPV, poly(thienylene-vinylene), poly(thiophene), poly(fluorene), and a ladder-type polymer.9-11 The microwave mobility (at 34 GHz) is found to depend strongly on the molecular structure of the polymer. In conjugated oligomers and polymers, chain ends, conjugation breaks or torsional disorder, can act as barriers to charge transport. The presence of these barriers leads to an increase of the charge carrier mobility with probing frequency since the charge carrier motion is probed on a smaller length scale, i.e., between these barriers, at higher frequencies.12-14 In this work, we use the sensitivity of the high-frequency mobility to the presence of barriers to charge transport to gain a unique fundamental insight into the nature of charge transport along isolated chains of PPV derivatives. We present mobilities obtained from TRMC measurements for oligomers of PV derivatives with varying chain length and for PPV derivatives with varying conjugation fraction. In addition, simulations of charge transport along PV chains with dynamic torsional disorder have been performed with use of parameters from density functional theory (DFT) calculations. The calculated charge carrier mobilities are found to be in good agreement with the experimental data, which makes a reliable prediction of the intrachain (DC) mobility of charge along PPV chains possible. 2. Experimental Section The phenylene-vinylene (PV) derivatives studied in this work are depicted in Figure 1. The PV oligomers were synthesized by means of Wittig-Horner condensation reactions and have a
10.1021/jp0626115 CCC: $33.50 © 2006 American Chemical Society Published on Web 07/13/2006
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Figure 1. Chemical structure of the phenylene-vinylene derivatives used in this study. p denotes the conjugation fraction.
distinct length of 12 and 16 PV repeat units.15 In addition the partially conjugated polymers poly(2-methoxy-5-(3,7-dimethyloctyloxy)-1,4-phenylenevinylene)s (MDMO-PPV) and poly(2-methoxy-5-(2′-ethylhexyloxy)-1,4-phenylenevinylene)s (MEHPPV) are studied. The fraction of conjugation (p) was limited by incorporation of a specific fraction (1 - p) of “saturated vinylene” units.16 These “saturated vinylene” units are located at random positions in the polymer chain and break the conjugated pathway along the polymer backbone. Hence they act as conjugation breaks and limit the conjugation length. In this study polymers with a conjugation fraction of 0.70, 0.85, and 0.90 are used. The polymer denoted with p ∼ 1 in Figure 1 is free of intentionally incorporated saturated vinylene units. However, the presence of a small fraction of defects in the polymer chains is inevitable due to the method of synthesis.17,18 Therefore, this polymer (MEH-PPV(p∼1)) is likely to contain a few percent of defects that reduce the conjugation for this polymer. According to gel permeation chromatography (GPC) with a polystyrene calibration standard the estimated weight averaged molecular weight (Mw) for MEH-PPV with p ) 0.70 and 0.85 is in the range 200-260 kDa.16 For the MDMOPPV with p ) 0.90 the value of Mw is at least 200 kDa with a polydispersity index between 2 and 3.19,20 The MEH-PPV with p ∼ 1.00 was purchased from Sigma-Aldrich and has a Mw of 200 ( 100 kDa. Hence all polymers used in this study have a length of about 800 PV repeat units. The experiments discussed here were performed on dilute solutions of chains of the PV derivatives in benzene at a PV repeat unit concentration between 0.1 and 1 mM. At this low concentration the PV chains are isolated from each other by the solvent molecules, and interchain effects do not affect the charge carrier motion.9 Charge carriers were generated by irradiation of the dilute solution with a 5-20 ns pulse of 3 MeV electrons from a Van de Graaff electron accelerator. The high-energy electrons scatter on the solvent molecules and produce a close to uniform distribution of excess electrons and benzene cations with a known concentration. These excess electrons and benzene cations can diffuse toward the PV chains, where they undergo charge transfer, thus yielding a charge on the backbone of the PV chain.9 The change in conductivity after the generation of charges was monitored by time-resolved microwave conductivity (TRMC) measurements at a microwave frequency of 10 or 34 GHz.8 With this technique, the absorbance of microwave power as a result of the presence of mobile charges is monitored. In this way, the TRMC method enables determination of the high-frequency mobility of charges on isolated polymer chains in solution, without the use of electrodes.9-11,21 Note that the high-frequency mobility deduced from the TRMC measurements described here cannot be compared directly to the device mobility as determined by space charge limited current (SCLC), in a field effect transistor (FET) or with TOF measurements. In a device setup, the charges not only have
to overcome the injection barriers at the contacts, but the charges also have to migrate over the entire thickness of the sample (which is often in the order of 100 nm) to contribute to the current. Therefore, the mobility of charges determined by using a device setup is sensitive to interchain disorder and limited by charge transport from chain to chain, transport over grain boundaries, and transport over electrode-polymer interfaces. In contrast, in our microwave conductivity measurements we probe the motion of charges along isolated polymer chains and the transient conductivity observed is dominated by the most mobile charge carriers. 3. Theoretical Basis According to the work of Kubo, the frequency dependent (one-dimensional) mobility of charge carriers is given by22-25
µac(ω) ) -
eω2 2kBT
∫0∞ 〈∆x2(t)〉 cos(ωt) dt
(1)
with e the elementary charge, ω the (radial) frequency of the probing electric field, kB Boltzmann’s constant, T the temperature, and 〈∆x2(t)〉 the mean squared displacement of the charge. An implicit convergence factor exp(-t) (lim f 0) is understood in the integral.23,24 For normal Gaussian diffusion the mean squared displacement of charge carriers moving along an infinitely long polymer chain increases linearly with time
〈∆x2(t)〉 ) 2Dt
(2)
where D is the diffusion constant. In this special case, the mobility is frequency independent and eq 1 reduces to the Einstein relationship:
µ)
e D kBT
(3)
In the presence of structural disorder eqs 2 and 3 are not valid. In that case the charge carrier mobility can be obtained from the mean squared displacement by using eq 1. In the present work, the mean squared displacement is obtained from a numerical simulation of the motion of charge carriers on the PV chains. The polymer and the oligomers are modeled as a chain of sites that correspond to the molecular units of the PV chain, i.e., an alternating sequence of vinylene- and 2,5methoxy-substituted phenylene units. Structural fluctuations in the polymer chain and the surrounding environment led to incoherent charge transport.26-28 As a consequence the mean squared displacement can be calculated by considering charge carriers that are initially localized on a molecular unit with a probability corresponding to the charge distribution along the polymer chains.23,29 Hence the mean squared displacement can be obtained from the time-dependent wave function
Charge Transport along PV Molecular Wires
Ψ(t) )
J. Phys. Chem. B, Vol. 110, No. 30, 2006 14661
∑n cn(t,n0)Φn
(4)
with the initial condition cn(t)0,n0) ) δn,n0. In eq 4 Φn is an orbital localized at a molecular unit. The time-dependent expansion coefficients of the basis functions on the molecular units in the PV chain (cn(t)) are obtained by propagation of the wave function according to the time-dependent Schro¨dinger equation:
ip
∂Ψ(t) )H ˆ Ψ(t) ∂t
(5)
In the tight-binding approximation the diagonal matrix elements of the Hamiltonian (H ˆ ) correspond to the site energies (i,i), i.e., the energy of the charge localized on a single molecular unit in the PV chain. When only nearest neighbor interactions are taken into account, the off-diagonal matrix elements of the Hamiltonian bi,i(1 are equal to the electronic couplings between adjacent molecular units. The other off-diagonal matrix elements are zero in this approximation and the Hamiltonian matrix is given by
(
11 b12 b21 22
H) 0 ·· · 0
0 ··
···
0
· ··
· NN
)
(6)
The mean squared displacement of the charge as a function of time can be expressed in terms of the time-dependent site coefficients and the positions of the molecular units:
〈∆2(t)〉 )
f(n0)|cn(t,n0)|2(n - n0)2a2 ∑ n,n
(7)
0
where f(n0) describes the initial distribution of the charge and (n - n0)a is the distance between the orbitals at sites n and n0. cn(t,n0) is the coefficient of the orbital at site n at time t for a state that was initially localized at n0. The electronic coupling between adjacent molecular units in the PV chain strongly depends on the dihedral (torsional) angle between the molecular units. Hence, charge transport along a PV chain is determined by the molecular conformation. This structural dependence is taken into account through the angular dependence of the electronic coupling (bi,i(1(ϑ)). The dihedral angles (ϑ) between adjacent molecular units are distributed according to a Boltzman distribution:
P(ϑ) )
e-Etor(ϑ)/kBT
∫0
2π
e-Etor(ϑ)/kBT dϑ
(8)
where Etor(ϑ) is the torsion potential between adjacent molecular units. Changes in the molecular conformation on the time scale of the charge transport are taken into account by rotation of the phenyl rings with respect to the adjacent (static) vinyls. The change of the dihedral angles in time is determined by a combination of rotational drift and rotational diffusion. During a small time step (∆t) the change of the angles is given by30
∆ϑ ) -
Drot ∂Etor ∆t + ∆ϑdiff kBT ∂ϑ
(9)
The first term is the rotational drift due to the torsion potential.
The second term accounts for the random diffusive rotation, such that 〈∆ϑdiff2〉 ) 2Drot∆t where Drot ) 1/2τrot and τrot is the diffusional rotation time of a dialkoxy-substituted phenyl. The mobility of positive charges on PV chains can be calculated by taking the HOMO orbitals on the vinylene and 2,5-methoxy-substituted phenylene units as the basis functions Φn in eq 4. The site energies for the HOMO orbitals are approximated by experimental values for the ionization energy for vinyl- (10.5 eV)31 and dialkoxy-substituted phenyl (7.9 eV).32 The electronic coupling between adjacent vinyl- and dimethoxysubstituted phenyl units in a PV chain is calculated by using density functional theory (DFT) in the program ADF,33,34 using fragment orbitals, as described previously.35,36 The calculations are performed by using the Becke-Perdew gradient-corrected exchange correlation functional and a double-ζ plus polarization (DZP) basis set consisting of Slater-type functions. Note that the electronic coupling used in this work is the “effective” coupling that incorporates the effects of the spatial overlap between the fragment orbitals.35,36 The geometry of the PV unit was optimized for each angle by using second-order MØller Plesset perturbation theory (MP2) with a cc-pVDZ basis set in the program Gaussian.37 The wave function of the charge and the conformation of the chain are propagated in time with a time step of one atomic unit (2.4189 × 10-17 s), using eqs 5 and 9, respectively. The rotation time of a dialkoxy-substituted phenylene used in eq 9 is taken to be τrot ) 200 ps, which is of the order of values typically found experimentally for molecules of comparable size.38 The calculated mobility increases by a factor of 1.5 upon a decrease of the rotation time from 400 to 100 ps. When the rotational diffusion of the phenylene units is not taken into account (τrot ) ∞), the PV chains exhibit only static disorder.39 In this case the mobility of charge carriers is limited by large angles between PV units and is reduced by a factor of 10 to 50. Such low mobility values are inconsistent with the experimental data (see section 4). Besides the torsional motion of the phenyl rings other vibrational modes will also contribute to fluctuations of the charge-transfer integrals and site-energies during time. To obtain insight into the effect of the latter on the charge carrier mobility, fluctuations in the charge-transfer integrals, and site-energies were introduced as uncorrelated stochastic processes, analogous to the work of Palenberg et al.27 The amplitude of the fluctuations in the chargetransfer integrals and site-energies was sampled from a uniform distribution centered around zero (δbi,i(1(t) ∈ [-A,A] and δi,i(t) ∈ [-B,B]), while the time correlation of the fluctuations was taken to decay exponentially with time, so that 〈δbi,i(1(t)δbi,i(1(0)〉 ) 1/3A2e-t/τb and 〈δi,i(t)δi,i(0)〉 ) 1/3B2e-t/τ. Introduction of these fluctuations with amplitudes A and B equal to 0.1 eV and average fluctuation times τb ) τ ) 50 fs results in changes of the charge carrier mobility by only a factor of about two. Hence, inclusion of details of fluctuations other than those due to torsional motion of the phenyl rings will not significantly affect the simulated charge carrier mobility and are therefore neglected. For PV chains with finite length, the charge is assumed to be initially equally distributed over the molecular units, i.e., in eq 7 f(n0) ) 1/n. The squared displacement of the charge carrier is averaged over a few hundred realizations of the initial conditions (i.e., angles between molecular units and initial position of the charge). For infinitely long PV chains the charge is initially localized in the middle of a chain with sufficient length such that the chain ends do not affect the mean squared displacement on the time scale of the simulation (25 ps, 1250 PV units). In this case, the squared displacement of the charge
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Figure 2. Conductivity due to positive charges on PV16 for an irradiation dose (Dirr) of 10, 20, and 40 Gy from bottom to top. The measurements were performed at a PV repeat unit concentration of 0.1 mM and a microwave frequency of 34 GHz.
Figure 3. Dose normalized conductivity due to positive charges on PV12 (bottom) and PV16 (top). The measurements were performed at a PV repeat unit concentration of 0.1 mM, a microwave frequency of 34 GHz, and an irradiation dose (Dirr) of 20 Gy. The smooth lines are the result of kinetic fits to the data.
carrier is averaged over a few hundred realizations of the chain conformation.
TABLE 1: Experimental and Calculated Values for the Mobility of a Positive Charge along Phenylene-Vinylene Chains at a frequency of 34 and 10 GHz (italic)a
4. Experimental Results
PV chain
In Figure 2 we present the change in conductivity after the generation of charge carriers for PV16 for a microwave frequency of 34 GHz. To selectively study the motion of positive charges along the PV chains, the solution was saturated with oxygen. Due to the relatively high concentration (12 mM) and electron affinity of oxygen, the excess electrons generated during the electron pulse (ebz-) rapidly react with the oxygen molecules (O2) forming the oxygen anion (O2-):
ebz- + O2 f O2-
(A)
In this way, the transfer of negative charges to the PV chains is prevented. Since the mobility of the oxygen anion in benzene is low (10-3 cm2/Vs)40 as compared to the mobility of the positive charges along the PV chain (see below), the contribution of the oxygen anion to the observed change in conductivity is negligible. The benzene cations generated during the 10 ns electron pulse (bz+) react with the PV chains by a diffusioncontrolled reaction, yielding positively charged PV chains (PV+ chain) +
bz + + PVchain f PVchain + bz
(B)
As this reaction proceeds, an increase in the transient conductivity is observed on a time scale of hundreds of nanoseconds. This increase directly indicates that the positive charge on the PV chain is more mobile than the benzene cation in benzene solution (1.2 × 10-3 cm2/Vs).41 On a time scale of ten to hundreds of microseconds a decrease in the conductivity signal is observed. The time scale of this decay decreases with increasing irradiation dose, or equivalently, with increasing initial concentration of charge carriers. The decay in the conductivity can therefore be attributed to bimolecular charge recombination between the oxygen anion and the positive charge on the PV chain +
O2- + PVchain f PVchain + O2
(C)
A more extensive description of the reactions upon irradiation with high-energy electrons and the resulting transient conductivity of dilute polymer solutions can be found elsewhere.9,11
2 µhole exp (cm /V‚s)
hole (cm2/V‚s) µcalc
0.018 0.029 600 units) PV chains, as expected. The calculated values for the mobility are in good agreement with the experimental results (see Figure 9 and Table 1). The mobility found experimentally for MEH-PPV(p∼1) can be reproduced by taking a conjugation fraction of p ) 0.955. This conjugation fraction is reasonable since a few percent of defects is inevitably induced during polymerization.17,18 As discussed above, the
Charge Transport along PV Molecular Wires
J. Phys. Chem. B, Vol. 110, No. 30, 2006 14665 MEH-PPV with p ) 0.70, 0.85, E. Staring, Philips, Eindhoven, The Netherlands, for providing the MDMO-PPV with p ) 0.9, and J. Wildeman, University of Groningen, Groningen, The Netherlands, for providing the p-phenylene-vinylene oligomers. This work is part of the research program of the “Stichting voor Fundamenteel Onderzoek der Materie (FOM)”, which is financially supported by the “Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)”. F.C.G. acknowledges NWO for support in the form of a VENI grant. References and Notes
Figure 9. Charge carrier mobility as a function of conjugation fraction calculated for a positive charge. The solid dots denote the experimental data.
mobility for an infinitely long PV chain is higher at 34 GHz than at 10 GHz (see Figure 7b). The limiting effect of large dihedral angles on the mobility of charge carriers is more pronounced at low frequency, when the motion of charges is probed over a larger distance. For MEH-PPV with a conjugation fraction of p ) 0.955 the frequency dependence is expected to be even larger due to the presence of conjugation breaks that act as barriers to charge transport. Analogously to the results in Figure 9, the mobility was calculated as a function of conjugation fraction for a frequency of 10 GHz. A mobility of 0.15 cm2/(V‚s) was found for a conjugation fraction of 0.955, which is significantly lower than the mobility calculated for 34 GHz. This frequency dependence of the mobility is in agreement with that found experimentally for the mobility for MEH-PPV(p∼1). The intrachain dc mobility of 59 cm2/(V‚s) found for PPV is several orders of magnitude higher than the mobility values obtained typically from device measurements on PPV.45,46 The value for the intrachain mobility deduced here is more than 2 orders of magnitude higher than the mobility deduced (with a kinetic model) from photocurrent measurements on highly ordered PPV films.47 This large difference between the device mobility and the intrachain mobility illustrates the that the device mobility is determined to a large extent by contact effects and interchain effects, as discussed in the Experimental Section. 6. Conclusions We have studied the motion of charge carriers along isolated phenylene-vinylene (PV) chains using a combination of experimental and theoretical methods. The mobilities calculated by using a tight binding model for a PV chain with dynamic torsional disorder are in good agreement with the experimental results, indicating that our theoretical model gives an adequate description of the charge transport along PV chains. Our results clearly show that the mobility of positive charges along PV chains is frequency dependent as a result of hindrance of charge carrier motion by large dihedral angles between adjacent PV units that act as barriers to charge transport. Furthermore, the mobility is found to depend strongly on the conjugation length and is found to increase both with increasing length of the PV chain and with increasing conjugation fraction. On the basis of this combined experimental and theoretical study an intrachain charge carrier mobility of a few tens of cm2/(V‚s) is expected for an infinitely long PV chain without conjugation breaks. Acknowledgment. The authors thank G. Badmanaban, Indian Institute of Science, Bangalore, India, for providing the
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