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Modeling of electron injection and transport in conjugated polymers
Synthetic Metals 141 (2004) 123–128
Modeling of electron injection and transport in conjugated polymers D.L. Smith∗ , A. Saxena Theoretical Division, Los Alamos National Laboratory, T-11 B262, Los Alamos, NM 87545, USA Received 14 July 2003; accepted 6 October 2003
Abstract Conjugated polymers are an important class of electronic and electro-optic materials. Nonlinear excitations are critical in determining the electrical and optical properties of these materials. Michael J. Rice was a pioneer in elucidating the nature and importance of nonlinear excitations in conjugated polymers. Here, we describe modeling of electron injection and transport in conjugated polymers that was strongly influenced by the seminal work of Michael Rice. We consider (i) charge transport in conjugated polymers based on a molecular fluctuation model and (ii) charge and spin injection at conjugated polymer/metal interfaces. © 2003 Elsevier B.V. All rights reserved. PACS: 72.25.-b; 73.61.Ph; 75.30.Vn; 71.38.-k Keywords: Electron injection; Transport; Conjugated polymers
1. Introduction Electronic and electro-optic devices based on conjugated polymers are attracting a great deal of attention because of their performance and fabrication advantages. Nonlinear excitations such as solitons, polarons, bipolarons and excitons are critical in determining the electrical, optical and transport properties of these materials. Rice [1–10] was a pioneer in elucidating the nature and importance of nonlinear excitations in conjugated polymers. His seminal work has had a great influence on this field. Here we describe the modeling of: (i) charge transport in conjugated polymers based on a molecular fluctuation model and (ii) charge and spin injection at conjugated polymer/metal interfaces that owes much to the insight provided by Michael Rice.
2. Mobility of conjugated polymers The carrier transport properties of conjugated polymers is important in the operation of device structures that utilize these materials. Time-of-flight (TOF) measurements show that the electric field (E) dependence of the mobility in many conjugated polymers has approximately the Poole–Frenkel
∗ Corresponding author. Tel.: +1-505-667-2056; fax: +1-505-665-4063. E-mail address: [email protected] (D.L. Smith).
0379-6779/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.synthmet.2003.10.031
form, i.e. the √ mobility increases approximately exponentially with E [11]. Study of mobility in these materials using Monte Carlo simulations based on a Gaussian disorder model explained many features of the observed mobility [12]. However, a spatially correlated potential for the carriers is needed to describe the observed Poole–Frenkel behavior for the range of electric fields over which this behavior is observed [13]. The Poole–Frenkel behavior was first seen in molecularly doped polymers in which the dopant molecules have permanent dipole moments. A mobility model invoking the long-range interaction between the charged carriers and the dipole moments of the molecular dopants successfully described the mobilities observed in the molecularly doped polymers [14]. An essential ingredient of this model is the long-range correlation of carrier energies at different spatial positions that results from the long-range charge–dipole interaction. However, the mechanism leading to the Poole–Frenkel behavior in conjugated polymers cannot be due to charge–dipole interactions, as it is in molecularly doped materials, because most conjugated polymers do not have permanent dipole moments. An alternative mechanism is needed to explain the field dependence of the mobility in conjugated polymers. There are systematic differences in the mobility parameters of various conjugated polymers that provide a clue to the mechanism leading to the field dependence. For example, TOF measurements for holes in poly[2-methoxy,5-(2 ethyl-hexyloxy)-1,4-phenylene vinylene] (MEH-PPV) [15] and poly(9,9-dioctylfluorene) (PFO) [16] have recently been
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published. The mobility in PFO is about two orders higher than that for MEH-PPV, and the field √ dependence is much weaker; that is, the coefficient of E in the exponential is much smaller for PFO than for MEH-PPV. In conjugated polymers like PPV and its derivatives, the orientation of the phenylene rings can easily fluctuate. In PFO, however, this ring-torsion motion is suppressed by chemical bonding between phenylene rings. This difference suggests that fluctuations in molecular geometry, specifically in the ring-torsion, determines the mobility in these materials. The restoring force for ring-torsion fluctuations can be inter-molecular or intra-molecular. For the PPV family of polymers the inter-molecular restoring force dominates in dense films, because the molecules are closely packed. Fluctuations of the adjacent molecular orientations give rise to a large steric energy. By contrast, the intra-molecular restoring force for these materials is small: the characteristic energy for the ring-torsion mode of an isolated PPV molecule is small. To describe the role of fluctuations in molecular geometry, we used a model for the mobility in dense films of conjugated polymers [17] based on a tight-binding Hamiltonian, which includes on-site carrier (polaron) energy, intra-molecular and inter-molecular elastic energy due to
ring-torsion, coupling energy between the torsion angle and the polaronic charge, and polaron hopping energy between sites. For a one-dimensional (1D) system we calculated the field-dependent mobility analytically in the continuum limit and found the Poole–Frenkel behavior at large electric fields. For dense three-dimensional (3D) films we studied the mobility numerically in a 3D lattice by solving a steady state master equation. In 3D systems, a carrier can optimize its path to avoid high energy barriers and achieve a higher mobility. Fig. 1 illustrates the current patterns in the low-field and the high-field regions. In the figure, we projected the 3D lattice onto the x–y plane by summing over the currents in different planes. The width of each bond in the figure is proportional to the current across the bond. Darker bonds indicate that the current is opposite to the standard directions (from left to right and from down to up). In the low-field regime, we see that the carriers take complex paths involving many chains. When such irregular paths occur, a 1D model, where the path is always along the field, is not appropriate. In the high-field regime, where the field is strong enough to overcome the energy barriers, the carrier paths are essentially one-dimensional. Therefore, the 3D and 1D numerical results merged in the high-field regime.
Fig. 1. Current patterns of carrier paths for two different applied field. The width of each bond is proportional to the current across the bond. Upper and lower panels are for E = 0.5 × 105 and 2 × 106 V/cm, respectively.
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Fig. 3. Schematic diagram showing an oligomer chain with 2N atoms sandwiched between two semi-infinite metal leads and the tight-binding model used for the structure.
Fig. 2. Field-dependent mobility plotted on a logarithmic scale as a function of E1/2 . In panel (a), dots are experimental data for MEH-PPV [15], and the solid line is our theoretical result with model parameters chosen to represent a PPV based polymer from quantum chemical calculations for a model system. In panel (b), dots are experimental data for PFO [16], and the solid line is our theoretical result with model parameters chosen to represent PFO.
To interpret TOF mobility measurements in MEH-PPV and PFO, we fit the mobility data of MEH-PPV by adjusting the inter-molecular force constant and the polaron–torsion coupling coefficient around the values estimated from quantum chemical AM1 calculations for biphenyl and a three-benzene system used as model systems. We found that a good fit, as shown in Fig. 2(a), can be obtained. Torsion fluctuations are strongly suppressed by chemical bonding in PFO and we used a larger inter-molecular force constant to describe this situation. We found that the theoretical results are in good agreement with the experimental data for PFO, as shown in Fig. 2(b). We proposed a tight-binding model to describe electrical transport in dense films of conjugated polymers in which thermal fluctuations in the molecular geometry modify the energy levels of localized (polaronic) electronic states. The primary restoring force for the fluctuations is steric in origin. Because the restoring force is inter-molecular there is a spatial correlation in the molecular distortions which leads to a spatial correlation in the energies of the localized states. The model describes the experimentally observed field and carrier density dependence of the mobility and provides a framework to understand the transport in conjugated polymers.
3. Effects of lattice fluctuations on electronic transmission Electron transmission at metal/conjugated–oligomer/metal structures in the presence of lattice fluctuations was studied
for short oligomer chains [18]. An oligomer chain with 2 N atoms sandwiched between two semi-infinite metal leads is schematically illustrated in Fig. 3. Lattice fluctuations were first approximated by static white noise disorder. Resonant transmission occurs when the energy of an incoming electron coincides with a discrete electronic level of the oligomer. The corresponding transmission peak diminishes in intensity with increasing disorder strength. Because of disorder there is an enhancement of electronic transmission for energies that lie within the electronic gap of the oligomer. If fluctuations are sufficiently strong, a transmission peak within the gap is found at the mid-gap energy E = 0 for degenerate conjugated oligomers (e.g. trans-polyacetylene [1]) and E = 0 for AB-type degenerate oligomers (e.g. polycarbonitrile [4]. These results can be interpreted in terms of soliton–antisoliton states created by lattice fluctuations. Fig. 4 shows the transmission coefficient for a 2N = 8 oligomer chain versus the energy of the incoming electron for different disorder strengths. Here the oligomer is of trans-polyacetylene-type. When fluctuations are absent (no disorder), four resonant tunneling peaks are observed and their energies correspond to the four discrete levels above the gap in the oligomer. As the disorder strength increases,
Fig. 4. The probability of electron transmission (t) for an eight-atom oligomer structure as a function of the energy of the incoming electron (E). The solid, dashed, and long-dashed lines correspond to increasing disorder strengths.
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the transmission above the gap is reduced and the peaks diminish in intensity. This is easily understood since the incoming electron loses its coherence due to scattering resulting from the disorder. When the energy of the incoming electron lies within the gap, the tunneling is enhanced. This enhancement becomes more pronounced as we increase the fluctuations. For sufficiently strong fluctuations we find a peak in the middle of the gap, i.e. E = 0 which arises from the lattice-assisted tunneling. We developed a Green’s function formalism to study dynamics in the transport of organic tunneling devices [19]. We found a crossover behavior in the transport from free electron-like to polaron-like behavior as the ratio between the electronic and organic lattice vibration time scales is varied. If the electronic time scale is fast compared to the lattice vibration time scale, the lattice motion lags behind the incoming electron wavepacket and the transmission is similar to that in the static case (described above where the lattice is frozen). In the opposite limit, the lattice follows the electron and the first transmission peak shifts from the conduction band edge toward the self-trapped polaron level. We investigated the transmission coefficient, the transfer of energy between the incident electron and the lattice, and the time evolution of the electron energy distribution function as the ratio of these time scales is changed. To simulate lattice fluctuations we studied a pre-existing lattice distortion and found enhanced sub-gap transmission. Fig. 5 is a snapshot of an incoming electron wavepacket at different times for a high transmission (resonance) case. Before the wavepacket enters the oligomer, the wavepacket has a Gaussian profile. While the wavepacket is in the oligomer, the wavepacket profile is severely distorted due to the scattering by the interface and the oligomer. Eventually part of the wavepacket is reflected and part is transmitted, and the profile of the wavepacket splits into several subwavepackets. In Fig. 6, we show transmission as a function of the energy of the incoming electron wavepacket with a pre-existing polaron-like lattice distortion. The dashed and dot–dashed lines correspond to ω = 0 and ω = ω0 where ω is the lattice vibration frequency and ω0 , the corresponding value for trans-polyacetylene. For ω = 0, because the lattice does not move, the first resonant tunneling peak arises from the polaron level in the oligomer. For ω = ω0 , although the first transmission peak shifts from the polaron level toward higher energy compared to the case of ω = 0, it is still much lower than the energy gap (first peak of the solid line). This indicates that the wavepacket can use a “partially formed” polaron level produced by lattice fluctuations to tunnel through the oligomer, although the carrier behavior here is more akin to a free electron. Thus, polaron effects may be important even in the low-frequency regime due to the presence of strong lattice fluctuations in oligomers. The sub-gap transmission is enhanced in both cases, although at somewhat different energies.
Fig. 5. Snapshots of an electron wavepacket at different times for a case of resonant transmission. The five panels correspond to varying times τ = 1, 4, 8, 12, 16 fs, respectively. The oligomer is between the two dashed lines.
Fig. 6. Electron transmission as a function of incoming energy with a pre-existing polaron-like lattice distortion. Dashed and dot–dashed lines correspond to ω = 0 and ω = ω0 where, ω is the lattice vibration frequency and ω0 , corresponding value for trans-polyacetylene. The solid line, for reference purpose, is obtained by using the equilibrium lattice configuration and ω = ω0 .
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4. Ferromagnetic metal/conjugated polymer interfaces We investigated the ground state properties of ferromagnetic metal/conjugated polymer interfaces [20]. The work was motivated by recent experiments in which injection of spin-polarized electrons from ferromagnetic contacts into thin films of conjugated polymers was reported [21]. We used a one-dimensional nondegenerate Su–Schrieffer–Heeger (SSH) Hamiltonian to describe the conjugated polymer and one-dimensional tight-binding models to describe the ferromagnetic metal. We considered both a model for a conventional ferromagnetic metal, in which there are no explicit structural degrees of freedom, and a model for a half-metallic ferromagnetic colossal magnetoresistance (CMR) manganite, which has explicit structural degrees of freedom. We investigated electron charge and spin transfer from the ferromagnetic metal to the organic polymer, and structural relaxation near the interface. By adjusting the relative chemical potential of the ferromagnetic contact and the polymer, electrons can be transferred into the polymer from the magnetic layer through the interfacial coupling. We calculated the density of states (DOS) before and after coupling. The DOS has important consequences for spin injection under electrical bias: polarized spin injection is possible when the Fermi level of the ferromagnet lies below the bipolaron level of the polymer. However, if the Fermi level of the CMR manganite lies above the bipolaron level of the polymer, the transferred electrons form bipolarons, which have no spin, and there is no spin density in the bulk of the polymer. We considered the polymer chain in contact with a half-metallic ferromagnetic Re1−x Akx MnO3 chain with average electron concentration of 0.32, where Re represents a rare earth atom such as La and Ak represents an alkaline atom such as Ca. By adjusting the relative chemical potential, electrons are transferred between the CMR manganite and polymer. For a relative chemical potential of 2.15 eV, there is essentially no electron transfer between the segments. In this case if we compare the displacements of the atoms with their uniform bulk positions we find that within the CMR manganite segment, both the manganese and oxygen atoms are only slightly displaced (not shown). These small displacements are due to the finite length of the segment and disappear as the segment length is increased. The carbon atoms have a displacement of 0.05 Å, corresponding to the bulk dimerization of the polymer chain. The interfacial atoms have a deviation from the bulk dimerization, which results in a small expansion of the end bonds of the CMR manganite segment and a contraction of the first few polymer bonds. Because the CMR material is completely spin-polarized at the Fermi surface, the charge and spin densities coincide in this segment. The distributions of charge and spin density in each segment (not shown) are uniform except for a small modulation near the interface. The modulation in the CMR manganite is a finite size effect. There
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is neither a net charge nor spin distribution within the bulk polymer. There is a gap in the spin-down states at the Fermi energy, and the occupied states near the Fermi surface are strongly spin-polarized. Polarized spin injection is possible in this case because a bias voltage will draw spin-polarized electrons from the spin-polarized Fermi level of the CMR manganite. When we increase the chemical potential of the CMR manganite, electrons are transferred into the polymer. Results for a relative chemical potential of 2.90 eV are shown in Fig. 7. At this value for the chemical potential, 6.11 electrons transfer to the polymer segment. The CMR manganite segment keeps a nearly uniform lattice structure except for a small deviation at the interface. In the polymer, bipolaron states form as seen from the displacements shown in Fig. 7(a). The localized electronic density is shown in Fig. 7(b). The transferred electrons form spin-less bipolarons, and there is no spin amplitude within the polymer segment. The corresponding DOS is depicted in Fig. 8(a) before coupling and in Fig. 8(b) after coupling of the two segments. In this case the gap in spin-down states at the Fermi energy is filled because of the bipolaron states. Therefore, polarized spin injection is unlikely because a bias voltage will draw electrons from this spin-unpolarized source.
Fig. 7. For a ferromagnetic Re1−x Akx MnO (FM CMR manganite)/polymer chain: (a) site displacements of Mn (left dotted), O (left solid) and C (right solid) atoms; (b) charge (dotted line) and spin (solid line) density distributions. The charge and spin densities coincide in the CMR manganite. Electrons are transferred from the FM CMR manganite segment to the polymer through the interface by increasing the chemical potential of the FM CMR manganite, resulting in bipolarons forming in the polymer. The interface is between sites 1 0 0 and 1 0 1.
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spin transport in organics build upon the pioneering work by Michael Rice in this field. Acknowledgements We are pleased to acknowledge collaboration and fruitful discussions with K.H. Ahn, A.R. Bishop, R.L. Martin, S.J. Xie, and Z.G. Yu. This research is supported by the Los Alamos LDRD program. References
Fig. 8. Density of states of the FM CMR manganite and the polymer: (a) before coupling and (b) after coupling. The thick solid line in (a) is for both spin-up and spin-down electrons in the polymer, the thin solid (dashed) line in (a) is for spin-up (spin-down) electrons in the CMR manganite. The solid (dashed) line in (b) is for spin-up (spin-down) electrons. A Lorentz line width of 0.15 eV is used. The Fermi level of the CMR manganite lies above the bipolaron energy of the polymer, so that electrons transfer from the CMR manganite segment to the polymer after coupling.
5. Conclusion We studied the role of disorder (such as ring-torsion) and lattice fluctuations on the charge transport properties of conjugated oligomers and polymers. In addition, we studied the ground state properties of ferromagnetic metal/conjugated polymer interfaces in order to understand polarized spin injection across the interface. These results for charge and
[1] M.J. Rice, Phys. Lett. A 71 (1979) 152. [2] M.J. Rice, A.R. Bishop, J.A. Krumhansl, S.E. Trullinger, Phys. Rev. Lett. 36 (1976) 432. [3] M.J. Rice, V.M. Yartsev, C.S. Jacobson, Phys. Rev. B 21 (1980) 3437. [4] M.J. Rice, E.J. Mele, Phys. Rev. Lett. 49 (1982) 1455. [5] M.J. Rice, N.O. Lipari, S. Strassler, Phys. Rev. Lett. 39 (1977) 1359. [6] M.J. Rice, C.B. Duke, N.O. Lipari, Solid State Commun. 17 (1975) 1089. [7] M.J. Rice, H.Y. Choi, Phys. Rev. B 45 (1992) 10173. [8] M.J. Rice, Y.N. Gartstein, Phys. Rev. Lett. 73 (1994) 2504. [9] M.J. Rice, J. Timonen, Phys. Lett. A 73 (1979) 368. [10] M.J. Rice, N.O. Lipari, Phys. Rev. Lett. 38 (1977) 437. [11] W.D. Gill, J. Appl. Phys. 43 (1972) 5033. [12] H. Bässler, Phys. Stat. Sol. (b) 175 (1993) 15. [13] Y.N. Gartstein, E.M. Conwell, Chem. Phys. Lett. 245 (1995) 351. [14] D.H. Dunlap, P.E. Parris, V.M. Kenkre, Phys. Rev. Lett. 77 (1996) 542. [15] I.H. Campbell, D.L. Smith, C.J. Neef, J.P. Ferraris, Appl. Phys. Lett. 74 (1999) 2809. [16] M. Redecker, D.D.C. Bradley, M. Inbasekaran, E.P. Woo, Appl. Phys. Lett. 73 (1998) 1565. [17] Z.G. Yu, D.L. Smith, A. Saxena, R.L. Martin, A.R. Bishop, Phys. Rev. Lett. 84 (2000) 721; Z.G. Yu, D.L. Smith, A. Saxena, R.L. Martin, A.R. Bishop, Phys. Rev. B 63 (2001) 85202. [18] Z.G. Yu, D.L. Smith, A. Saxena, A.R. Bishop, Phys. Rev. B 56 (1997) 6494. [19] Z.G. Yu, D.L. Smith, A. Saxena, A.R. Bishop, Phys. Rev. B 59 (1999) 16001. [20] S.J. Xie, K.H. Ahn, D.L. Smith, A.R. Bishop, A. Saxena, Phys. Rev. B 67 (2003) 125202. [21] V. Dediu, M. Murgia, F.C. Matacotta, C. Taliani, S. Barbanera, Solid State Commun. 122 (2002) 181.
Modeling of electron injection and transport in conjugated polymers D.L. Smith∗ , A. Saxena Theoretical Division, Los Alamos National Laboratory, T-11 B262, Los Alamos, NM 87545, USA Received 14 July 2003; accepted 6 October 2003
Abstract Conjugated polymers are an important class of electronic and electro-optic materials. Nonlinear excitations are critical in determining the electrical and optical properties of these materials. Michael J. Rice was a pioneer in elucidating the nature and importance of nonlinear excitations in conjugated polymers. Here, we describe modeling of electron injection and transport in conjugated polymers that was strongly influenced by the seminal work of Michael Rice. We consider (i) charge transport in conjugated polymers based on a molecular fluctuation model and (ii) charge and spin injection at conjugated polymer/metal interfaces. © 2003 Elsevier B.V. All rights reserved. PACS: 72.25.-b; 73.61.Ph; 75.30.Vn; 71.38.-k Keywords: Electron injection; Transport; Conjugated polymers
1. Introduction Electronic and electro-optic devices based on conjugated polymers are attracting a great deal of attention because of their performance and fabrication advantages. Nonlinear excitations such as solitons, polarons, bipolarons and excitons are critical in determining the electrical, optical and transport properties of these materials. Rice [1–10] was a pioneer in elucidating the nature and importance of nonlinear excitations in conjugated polymers. His seminal work has had a great influence on this field. Here we describe the modeling of: (i) charge transport in conjugated polymers based on a molecular fluctuation model and (ii) charge and spin injection at conjugated polymer/metal interfaces that owes much to the insight provided by Michael Rice.
2. Mobility of conjugated polymers The carrier transport properties of conjugated polymers is important in the operation of device structures that utilize these materials. Time-of-flight (TOF) measurements show that the electric field (E) dependence of the mobility in many conjugated polymers has approximately the Poole–Frenkel
∗ Corresponding author. Tel.: +1-505-667-2056; fax: +1-505-665-4063. E-mail address: [email protected] (D.L. Smith).
0379-6779/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.synthmet.2003.10.031
form, i.e. the √ mobility increases approximately exponentially with E [11]. Study of mobility in these materials using Monte Carlo simulations based on a Gaussian disorder model explained many features of the observed mobility [12]. However, a spatially correlated potential for the carriers is needed to describe the observed Poole–Frenkel behavior for the range of electric fields over which this behavior is observed [13]. The Poole–Frenkel behavior was first seen in molecularly doped polymers in which the dopant molecules have permanent dipole moments. A mobility model invoking the long-range interaction between the charged carriers and the dipole moments of the molecular dopants successfully described the mobilities observed in the molecularly doped polymers [14]. An essential ingredient of this model is the long-range correlation of carrier energies at different spatial positions that results from the long-range charge–dipole interaction. However, the mechanism leading to the Poole–Frenkel behavior in conjugated polymers cannot be due to charge–dipole interactions, as it is in molecularly doped materials, because most conjugated polymers do not have permanent dipole moments. An alternative mechanism is needed to explain the field dependence of the mobility in conjugated polymers. There are systematic differences in the mobility parameters of various conjugated polymers that provide a clue to the mechanism leading to the field dependence. For example, TOF measurements for holes in poly[2-methoxy,5-(2 ethyl-hexyloxy)-1,4-phenylene vinylene] (MEH-PPV) [15] and poly(9,9-dioctylfluorene) (PFO) [16] have recently been
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published. The mobility in PFO is about two orders higher than that for MEH-PPV, and the field √ dependence is much weaker; that is, the coefficient of E in the exponential is much smaller for PFO than for MEH-PPV. In conjugated polymers like PPV and its derivatives, the orientation of the phenylene rings can easily fluctuate. In PFO, however, this ring-torsion motion is suppressed by chemical bonding between phenylene rings. This difference suggests that fluctuations in molecular geometry, specifically in the ring-torsion, determines the mobility in these materials. The restoring force for ring-torsion fluctuations can be inter-molecular or intra-molecular. For the PPV family of polymers the inter-molecular restoring force dominates in dense films, because the molecules are closely packed. Fluctuations of the adjacent molecular orientations give rise to a large steric energy. By contrast, the intra-molecular restoring force for these materials is small: the characteristic energy for the ring-torsion mode of an isolated PPV molecule is small. To describe the role of fluctuations in molecular geometry, we used a model for the mobility in dense films of conjugated polymers [17] based on a tight-binding Hamiltonian, which includes on-site carrier (polaron) energy, intra-molecular and inter-molecular elastic energy due to
ring-torsion, coupling energy between the torsion angle and the polaronic charge, and polaron hopping energy between sites. For a one-dimensional (1D) system we calculated the field-dependent mobility analytically in the continuum limit and found the Poole–Frenkel behavior at large electric fields. For dense three-dimensional (3D) films we studied the mobility numerically in a 3D lattice by solving a steady state master equation. In 3D systems, a carrier can optimize its path to avoid high energy barriers and achieve a higher mobility. Fig. 1 illustrates the current patterns in the low-field and the high-field regions. In the figure, we projected the 3D lattice onto the x–y plane by summing over the currents in different planes. The width of each bond in the figure is proportional to the current across the bond. Darker bonds indicate that the current is opposite to the standard directions (from left to right and from down to up). In the low-field regime, we see that the carriers take complex paths involving many chains. When such irregular paths occur, a 1D model, where the path is always along the field, is not appropriate. In the high-field regime, where the field is strong enough to overcome the energy barriers, the carrier paths are essentially one-dimensional. Therefore, the 3D and 1D numerical results merged in the high-field regime.
Fig. 1. Current patterns of carrier paths for two different applied field. The width of each bond is proportional to the current across the bond. Upper and lower panels are for E = 0.5 × 105 and 2 × 106 V/cm, respectively.
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Fig. 3. Schematic diagram showing an oligomer chain with 2N atoms sandwiched between two semi-infinite metal leads and the tight-binding model used for the structure.
Fig. 2. Field-dependent mobility plotted on a logarithmic scale as a function of E1/2 . In panel (a), dots are experimental data for MEH-PPV [15], and the solid line is our theoretical result with model parameters chosen to represent a PPV based polymer from quantum chemical calculations for a model system. In panel (b), dots are experimental data for PFO [16], and the solid line is our theoretical result with model parameters chosen to represent PFO.
To interpret TOF mobility measurements in MEH-PPV and PFO, we fit the mobility data of MEH-PPV by adjusting the inter-molecular force constant and the polaron–torsion coupling coefficient around the values estimated from quantum chemical AM1 calculations for biphenyl and a three-benzene system used as model systems. We found that a good fit, as shown in Fig. 2(a), can be obtained. Torsion fluctuations are strongly suppressed by chemical bonding in PFO and we used a larger inter-molecular force constant to describe this situation. We found that the theoretical results are in good agreement with the experimental data for PFO, as shown in Fig. 2(b). We proposed a tight-binding model to describe electrical transport in dense films of conjugated polymers in which thermal fluctuations in the molecular geometry modify the energy levels of localized (polaronic) electronic states. The primary restoring force for the fluctuations is steric in origin. Because the restoring force is inter-molecular there is a spatial correlation in the molecular distortions which leads to a spatial correlation in the energies of the localized states. The model describes the experimentally observed field and carrier density dependence of the mobility and provides a framework to understand the transport in conjugated polymers.
3. Effects of lattice fluctuations on electronic transmission Electron transmission at metal/conjugated–oligomer/metal structures in the presence of lattice fluctuations was studied
for short oligomer chains [18]. An oligomer chain with 2 N atoms sandwiched between two semi-infinite metal leads is schematically illustrated in Fig. 3. Lattice fluctuations were first approximated by static white noise disorder. Resonant transmission occurs when the energy of an incoming electron coincides with a discrete electronic level of the oligomer. The corresponding transmission peak diminishes in intensity with increasing disorder strength. Because of disorder there is an enhancement of electronic transmission for energies that lie within the electronic gap of the oligomer. If fluctuations are sufficiently strong, a transmission peak within the gap is found at the mid-gap energy E = 0 for degenerate conjugated oligomers (e.g. trans-polyacetylene [1]) and E = 0 for AB-type degenerate oligomers (e.g. polycarbonitrile [4]. These results can be interpreted in terms of soliton–antisoliton states created by lattice fluctuations. Fig. 4 shows the transmission coefficient for a 2N = 8 oligomer chain versus the energy of the incoming electron for different disorder strengths. Here the oligomer is of trans-polyacetylene-type. When fluctuations are absent (no disorder), four resonant tunneling peaks are observed and their energies correspond to the four discrete levels above the gap in the oligomer. As the disorder strength increases,
Fig. 4. The probability of electron transmission (t) for an eight-atom oligomer structure as a function of the energy of the incoming electron (E). The solid, dashed, and long-dashed lines correspond to increasing disorder strengths.
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the transmission above the gap is reduced and the peaks diminish in intensity. This is easily understood since the incoming electron loses its coherence due to scattering resulting from the disorder. When the energy of the incoming electron lies within the gap, the tunneling is enhanced. This enhancement becomes more pronounced as we increase the fluctuations. For sufficiently strong fluctuations we find a peak in the middle of the gap, i.e. E = 0 which arises from the lattice-assisted tunneling. We developed a Green’s function formalism to study dynamics in the transport of organic tunneling devices [19]. We found a crossover behavior in the transport from free electron-like to polaron-like behavior as the ratio between the electronic and organic lattice vibration time scales is varied. If the electronic time scale is fast compared to the lattice vibration time scale, the lattice motion lags behind the incoming electron wavepacket and the transmission is similar to that in the static case (described above where the lattice is frozen). In the opposite limit, the lattice follows the electron and the first transmission peak shifts from the conduction band edge toward the self-trapped polaron level. We investigated the transmission coefficient, the transfer of energy between the incident electron and the lattice, and the time evolution of the electron energy distribution function as the ratio of these time scales is changed. To simulate lattice fluctuations we studied a pre-existing lattice distortion and found enhanced sub-gap transmission. Fig. 5 is a snapshot of an incoming electron wavepacket at different times for a high transmission (resonance) case. Before the wavepacket enters the oligomer, the wavepacket has a Gaussian profile. While the wavepacket is in the oligomer, the wavepacket profile is severely distorted due to the scattering by the interface and the oligomer. Eventually part of the wavepacket is reflected and part is transmitted, and the profile of the wavepacket splits into several subwavepackets. In Fig. 6, we show transmission as a function of the energy of the incoming electron wavepacket with a pre-existing polaron-like lattice distortion. The dashed and dot–dashed lines correspond to ω = 0 and ω = ω0 where ω is the lattice vibration frequency and ω0 , the corresponding value for trans-polyacetylene. For ω = 0, because the lattice does not move, the first resonant tunneling peak arises from the polaron level in the oligomer. For ω = ω0 , although the first transmission peak shifts from the polaron level toward higher energy compared to the case of ω = 0, it is still much lower than the energy gap (first peak of the solid line). This indicates that the wavepacket can use a “partially formed” polaron level produced by lattice fluctuations to tunnel through the oligomer, although the carrier behavior here is more akin to a free electron. Thus, polaron effects may be important even in the low-frequency regime due to the presence of strong lattice fluctuations in oligomers. The sub-gap transmission is enhanced in both cases, although at somewhat different energies.
Fig. 5. Snapshots of an electron wavepacket at different times for a case of resonant transmission. The five panels correspond to varying times τ = 1, 4, 8, 12, 16 fs, respectively. The oligomer is between the two dashed lines.
Fig. 6. Electron transmission as a function of incoming energy with a pre-existing polaron-like lattice distortion. Dashed and dot–dashed lines correspond to ω = 0 and ω = ω0 where, ω is the lattice vibration frequency and ω0 , corresponding value for trans-polyacetylene. The solid line, for reference purpose, is obtained by using the equilibrium lattice configuration and ω = ω0 .
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4. Ferromagnetic metal/conjugated polymer interfaces We investigated the ground state properties of ferromagnetic metal/conjugated polymer interfaces [20]. The work was motivated by recent experiments in which injection of spin-polarized electrons from ferromagnetic contacts into thin films of conjugated polymers was reported [21]. We used a one-dimensional nondegenerate Su–Schrieffer–Heeger (SSH) Hamiltonian to describe the conjugated polymer and one-dimensional tight-binding models to describe the ferromagnetic metal. We considered both a model for a conventional ferromagnetic metal, in which there are no explicit structural degrees of freedom, and a model for a half-metallic ferromagnetic colossal magnetoresistance (CMR) manganite, which has explicit structural degrees of freedom. We investigated electron charge and spin transfer from the ferromagnetic metal to the organic polymer, and structural relaxation near the interface. By adjusting the relative chemical potential of the ferromagnetic contact and the polymer, electrons can be transferred into the polymer from the magnetic layer through the interfacial coupling. We calculated the density of states (DOS) before and after coupling. The DOS has important consequences for spin injection under electrical bias: polarized spin injection is possible when the Fermi level of the ferromagnet lies below the bipolaron level of the polymer. However, if the Fermi level of the CMR manganite lies above the bipolaron level of the polymer, the transferred electrons form bipolarons, which have no spin, and there is no spin density in the bulk of the polymer. We considered the polymer chain in contact with a half-metallic ferromagnetic Re1−x Akx MnO3 chain with average electron concentration of 0.32, where Re represents a rare earth atom such as La and Ak represents an alkaline atom such as Ca. By adjusting the relative chemical potential, electrons are transferred between the CMR manganite and polymer. For a relative chemical potential of 2.15 eV, there is essentially no electron transfer between the segments. In this case if we compare the displacements of the atoms with their uniform bulk positions we find that within the CMR manganite segment, both the manganese and oxygen atoms are only slightly displaced (not shown). These small displacements are due to the finite length of the segment and disappear as the segment length is increased. The carbon atoms have a displacement of 0.05 Å, corresponding to the bulk dimerization of the polymer chain. The interfacial atoms have a deviation from the bulk dimerization, which results in a small expansion of the end bonds of the CMR manganite segment and a contraction of the first few polymer bonds. Because the CMR material is completely spin-polarized at the Fermi surface, the charge and spin densities coincide in this segment. The distributions of charge and spin density in each segment (not shown) are uniform except for a small modulation near the interface. The modulation in the CMR manganite is a finite size effect. There
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is neither a net charge nor spin distribution within the bulk polymer. There is a gap in the spin-down states at the Fermi energy, and the occupied states near the Fermi surface are strongly spin-polarized. Polarized spin injection is possible in this case because a bias voltage will draw spin-polarized electrons from the spin-polarized Fermi level of the CMR manganite. When we increase the chemical potential of the CMR manganite, electrons are transferred into the polymer. Results for a relative chemical potential of 2.90 eV are shown in Fig. 7. At this value for the chemical potential, 6.11 electrons transfer to the polymer segment. The CMR manganite segment keeps a nearly uniform lattice structure except for a small deviation at the interface. In the polymer, bipolaron states form as seen from the displacements shown in Fig. 7(a). The localized electronic density is shown in Fig. 7(b). The transferred electrons form spin-less bipolarons, and there is no spin amplitude within the polymer segment. The corresponding DOS is depicted in Fig. 8(a) before coupling and in Fig. 8(b) after coupling of the two segments. In this case the gap in spin-down states at the Fermi energy is filled because of the bipolaron states. Therefore, polarized spin injection is unlikely because a bias voltage will draw electrons from this spin-unpolarized source.
Fig. 7. For a ferromagnetic Re1−x Akx MnO (FM CMR manganite)/polymer chain: (a) site displacements of Mn (left dotted), O (left solid) and C (right solid) atoms; (b) charge (dotted line) and spin (solid line) density distributions. The charge and spin densities coincide in the CMR manganite. Electrons are transferred from the FM CMR manganite segment to the polymer through the interface by increasing the chemical potential of the FM CMR manganite, resulting in bipolarons forming in the polymer. The interface is between sites 1 0 0 and 1 0 1.
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spin transport in organics build upon the pioneering work by Michael Rice in this field. Acknowledgements We are pleased to acknowledge collaboration and fruitful discussions with K.H. Ahn, A.R. Bishop, R.L. Martin, S.J. Xie, and Z.G. Yu. This research is supported by the Los Alamos LDRD program. References
Fig. 8. Density of states of the FM CMR manganite and the polymer: (a) before coupling and (b) after coupling. The thick solid line in (a) is for both spin-up and spin-down electrons in the polymer, the thin solid (dashed) line in (a) is for spin-up (spin-down) electrons in the CMR manganite. The solid (dashed) line in (b) is for spin-up (spin-down) electrons. A Lorentz line width of 0.15 eV is used. The Fermi level of the CMR manganite lies above the bipolaron energy of the polymer, so that electrons transfer from the CMR manganite segment to the polymer after coupling.
5. Conclusion We studied the role of disorder (such as ring-torsion) and lattice fluctuations on the charge transport properties of conjugated oligomers and polymers. In addition, we studied the ground state properties of ferromagnetic metal/conjugated polymer interfaces in order to understand polarized spin injection across the interface. These results for charge and
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