Polaron spin filtering in an organic ferromagnetic polymer: a dynamics simulation

Abstract

We present a model study of the dynamic properties of a polaron in an organic ferromagnetic polymer by focusing on the spin correlation between the polymer backbone and the side radicals. The simulations are performed by using a tight-binding description coupled with a nonadiabatic dynamics method. We find that, in the presence of an external electric field, the polarons with both up and down spins can get trapped near the side radicals of the polymer chain unless the electric field is stronger than a critical field. However, the magnitudes of the critical electric field vary quite differently for the spin-up and spin-down polarons as a function of the number of side radicals in the polymer, leading to the exponential change of the range of the electric field within which the spin-filtering takes place. The range of the electric field increases nearly in a linear manner with the strength of the electron–lattice coupling as a result of the increase of the polaron binding energy. The impact of the strength of the spin correlation between the backbone and the side radicals on the polaron spin filtering is also discussed. These findings are expected to be useful for the design of organic-based spin filters.

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Introduction

Spintronics, a field aiming at the development of the next generation of functional devices by exploiting electron spin, has become increasingly active in recent years. Organic materials are currently seen as compelling candidates for spintronic applications because of their unique advantages such as flexibility, low-cost and large-area fabrication, and weak spin–orbit and hyperfine interactions. In particular, organic ferromagnets (OFMs), a class of materials possessing both organic and ferromagnetic properties, have been widely investigated in both chemical synthesis and device fabrication.^1–4 In general, OFMs can be classified into two categories. One is organic magnetic macromolecules containing transition-metal ions, such as Mn, Fe, Cr and V;^5–7 these types of OFMs usually have stable magnetism and high Curie temperature.⁷ The other is OFMs with only purely organic elements (i.e., C, H, O, and N) in the molecule.^2,8,9 Many of these materials have been realized by using spin radicals which are usually heterocycles containing an unpaired electron.^8,10,11

As functional materials, OFMs have been recently studied for potential device applications. Yoo et al. measured the spin-dependent electronic transport properties of V[TCNE]_x through a Au/OFM/Au structure, and a magnetoresistance with a maximum value of about 2.5% was obtained.³ Li et al. also investigated the magnetoresistance properties of such a material by bridging it to magnetic electrodes, and a room-temperature magnetoresistance was observed when the Al₂O₃ barrier was used.¹² Sugawara et al. investigated the spin-dependent transport properties of spin-polarized wire molecules (ESBN)ClO₄ connected to gold nanoparticles, they found that the transport mechanisms were different when the temperature was above and below 30 K, and the phenomenon of magnetoresistance was also observed.^4,13 In addition, several theoretical studies were also performed to investigate the magnetism and electronic structure of OFMs. For instance, Ovchinnikov et al. first discussed the ferromagnetic ordering in a low-dimensional organic substance to explain the magnetism in poly-BIPO by considering the ferromagnetic exchange in carbon systems.¹⁴ Fang et al. described the ground state energy level and spin configuration of the quasi-one-dimensional organic ferromagnetic polymers with the extended Su–Schrieffer–Heeger (SSH) model by considering the itineracy of π electrons, the strong electron–lattice interaction, and the electron-electron correlation; the phenomena of spin-splitting and spin density wave were obtained according to their calculations.^15,16 Xie et al. found that the spin density wave and charge density wave are closely related to the sizes and dimensions of the organic ferromagnets.¹⁷ Furthermore, OFMs were also used to design functional devices. For instance, Wang¹⁸ and Hu et al.¹⁹ respectively designed a spin filter based on organic ferromagnetic polymers, and large spin polarization currents are obtained according to their calculation. Zhu et al. reported a molecular spin valve with large abnormal magnetoresistance by combining spin density functional theory with nonequilibrium Green's function techniques.²⁰ Hu et al. reported the phenomena of both charge rectification and spin rectification through an organic spin diode based on organic magnetic/nonmagnetic co-oligomers.²¹

Different from the case of inorganic materials, the charge carriers in organic polymers are polarons consisting of a charge (electron or hole) coupled with a localized distortion of the polymer geometry as a result of the strong electron–lattice couplings. Typically, depending on the strength of the electron–lattice coupling, the size of a polaron can span from a few to dozens of lattice sites. The dynamic properties of polarons in organic polymers have been widely investigated by considering their formation, movement and dissociation processes.^22,23 When an external electric field is applied, the polaron moves as a whole along the polymer chain, and the velocity will not exceed the sound velocity according to Basko's calculation with adiabatic dynamics simulations.²⁴ Moreover, as pointed out by Johansson and Stafström in a nonadiabatic dynamics simulation, a step-like sonic to supersonic transition of the polaron velocity can be explicitly observed in a conjugated polymer chain.²⁵ By using the time-dependent density matrix renormalization group method, Wu et al. investigated the dynamic properties of polarons in conjugated polymers and found that the velocity of polarons can be suppressed by the on-site Coulomb interaction.²⁶ Furthermore, Wei and Xie et al. discussed the effect of the electric field mode on the dynamic process of polarons, and they concluded that a polaron can remain stable under a very strong electric field if the field was switched on slowly enough.²⁷ More investigations by focusing on the effects of interchain couplings, impurities and disorders on the dynamic properties of polarons have also been recently reported in the literature.^28–30

Most recently, a model describing the polaron properties in OFMs was proposed by Hu et al.³¹ It was found that the charge distribution of a polaron is highly asymmetric under spatial reflection while the spin distribution is nearly symmetric. It is noted that all these findings are based on only a static picture of polarons in OFMs. However, to better understand the charge and spin transport properties of these materials, the knowledge of the dynamic properties of polarons in the presence of external electric field is highly required. On the other hand, while polaron dynamics in organic polymers have been widely investigated, our understanding of the effects of the spin side radicals on polaron spin, in particular their dynamic behaviors, is actually very limited. Here, we present a simulation of the polaron dynamics in OFMs by using a model that well accounts for the spin degree of freedom of OFMs and the induced polarons. As will be shown, we obtain and analyze a spin-filtering behavior of the polaron dynamics in the OFM system, which has potential applications in organic ferromagnetic devices.

Model and methods

In our simulations, we consider a one-dimensional organic ferromagnetic polymer chain with side radicals such as poly-BIPO,^2,8 as shown schematically in Fig. 1. The backbone of the system can be described by the well-known SSH model, and the spin correlation between the backbone and the side radicals can be formulated by a Heisenberg-like Hamiltonian. Thus, the total Hamiltonian of the system is written as:

H = H_e + H_R-s + H_L. (1)

Fig. 1 The schematic diagram of the organic ferromagnetic polymer.

Here, H_e represents the Hamiltonian of the π-electrons on the backbone and takes the form of:³²

where t_n,n+1 = t₀ − α(u_n+1 − u_n) denotes the electron transfer integral between sites n and n + 1, and t₀ the average transfer integral at the equilibrium geometry. α is the electron–lattice coupling parameter and u_n the displacement of the nth site. c_n,s⁺ and c_n,s are respectively the electron creation and annihilation operators at the nth site with spin s. The exponential term describes an external electric field applied along the polymer chain in the form of E₀ = −∂A(t)/∂t, and A is a time-dependent vector potential and γ a coefficient defined as γ = ea/ℏ with a the lattice constant and e the electron charge.

For poly-BIPO-like organic ferromagnetic polymers, the ferromagnetic properties were inferred from the π electrons on the backbone and the nonbonding electrons on the side radicals.^15–17,31 The second term in eqn (1) denotes the correlation between the spin of electrons on the backbone and that on the side radicals (see Fig. 1)

where J_f denotes the strength of the spin correlation, and a positive value of J_f is chosen in our simulations to describe the antiferromagnetic correlation that widely exists in poly-BIPO-like organic ferromagnetic polymers. S⃑ represents the spin operator on the corresponding sites, δ_n,i denotes a Kronecker delta function and the subscript i represents the sequence number of the site to which the side radical connected.

The last term in eqn (1) represents the lattice part of the Hamiltonian that contains both the potential and kinetic energies of the lattice sites.

where K denotes the elastic constant and M the mass of a CH group.

Based on the Hamiltonian of eqn (1), we simulate the dynamics of the system by using a nonadiabatic mixed quantum-classical method. In our calculations, the electronic wave function is expanded on the spin-dependent Wannier basis , where μ describes the μth energy level and Z_μ,n,s the probability amplitude of the state |ψ_μ〉 at the nth site with spin s. In this context, the evolution of the electronic states is obtained by solving the time-dependent Schrödinger equation:

while the evolution of the lattice coordinates is described by the classical Newtonian equation of motion:

Mü(t) = K(u_n+1(t) + u_n−1(t) − 2u_n(t)) + α{e^iγA[ρ_n,n+1(t) − ρ_n−1,n(t)] + c·c}. (6)

The symbol denotes the charge density matrix. The charge distribution of a polaron on the nth site at time t can be calculated by To better describe the geometry of the polymer chain and the polaron movement, we use a smoothed atomic displacement order parameter:as well as a parameter p_c describing the center of the motional polaron: Here, N is the number of lattice sites, , , and . The polaron velocity at time t is given by v(t) = [p_c(t + Δt) − p_c(t)]/Δt.

The values of parameters in the simulations are chosen as follows: t₀ = 2.5 eV, α = 41 eV nm⁻¹, K = 2100 eV nm⁻¹², and M = 1349.14 eV fs² A⁻², which are relevant to poly-BIPO.¹⁹ A polymer chain with total lattice sites of N = 150 is used. While the model parameters are chosen to specifically describe poly-BIPO organic ferromagnetic polymers, the conclusions obtained in our study are expected to be qualitatively applicable to other organic polymers with similar structures where the antiferromagnetic correlations between the spin of electrons on the backbone and that on the side radicals are important.

Results and discussion

In our simulations, the external electric field is added smoothly to the system in terms of E(t) = E₀·t/t_c for 0 < t < t_c and E(t) = E₀ for t ≥ t_c, where the critical time is chosen as t_c = 75 fs. A polaron with specific spin (up or down) is initially set to center on the 30th site of the chain, and the spin of side radical electrons is set to spin up.

Fig. 2 plots the results of the time evolution of the charge distribution and the smoothed lattice configuration of both spin-up and spin-down polarons moving in a polymer chain with only one side radical. It is seen that, due to the spin correlation between the backbone and the side radical, the motional behaviors of the polarons strongly depend on the type of their spin (up or down). When a weak electric field (E₀ = 1.0 × 10⁻³ V nm⁻¹) is applied, both spin-up and spin-down polarons are driven to move along the polymer chain (with a velocity of about 0.012 nm fs⁻¹) and, when approaching the side radical (the 120th site), the polarons get trapped and no longer pass through the molecule which is in accordance with the threshold voltage in oligothiophene-type spin-polarized wire molecule based devices.^4,13 Interestingly, the trapped polarons for both spin up and spin down vibrate around the side radical, as shown in Fig. 2(a), (b), (e), and (f). The difference for the two cases is that, during the vibrations, the spin-down polaron distributes on both sides of the 120th site while the spin-up polaron locates only on the left side, and the frequency of the vibration for the spin-down polaron is much higher than that of the spin-up polaron. However, when the applied electric field gets strong (E₀ = 4.0 × 10⁻³ V nm⁻¹), as shown in Fig. 2(c), (d), (g) and (h), the polarons for both cases can easily transport along the molecule without sign of obstacle by the side radical. In this situation, the polaron moves circularly under the periodic boundary condition with a constant velocity of about 0.041 nm fs⁻¹. It is noted that, for both weak and strong electric fields, the polaron charge is always coupled to its lattice distortion during the movements, as shown in Fig. 2(e), (f), (g) and (h), indicating that the polarons remain stable throughout the simulations.

Fig. 2 Time evolution of the polaron charge distribution [(a)–(d)] and the smoothed lattice configuration [(e)–(h)]. The driven electric field is E₀ = 1.0 × 10⁻³ V nm⁻¹ for (a), (b), (e) and (f), and E₀ = 4.0 × 10⁻³ V nm⁻¹ for (c), (d), (g) and (h). The value of the spin coupling parameter is J_f = 0.3t₀, and only one side radical, which is connected to the 120th site (red dashed line), is considered.

The polaron behaviors displayed in Fig. 2 are essentially as a result of the spin correlation between the backbone and the side radical, which changes the electron on-site energy of the site to which the side radical is connected. For the given values of parameters, the on-site energy rises for the spin-up electron and lowers for the spin-down one by about 0.2 eV, respectively, as shown in Fig. 3. As a result, the 120th site serves as a quantum well for the spin-down polaron but in contrast as a quantum barrier for the spin-up polaron. In this context, the spin-down polaron gets trapped by the quantum well when it reaches to the 120th site while the spin-up polaron gets blocked by the quantum barrier. It is noted that the quantum well has a size of only one site, which is too narrow to trap the whole spin-down polaron such that it distributes on both sides of the 120th site. When the electric field is weak, the trapped spin-down polaron does not have enough energy to escape from the quantum well and it thereby oscillates as a result of the competition between the trapping of the site and the driving of the electric field; the blocked spin-up polaron, however, always moves back and forth on the left side of the quantum barrier because of the rebound in the presence of the electric field. When the electric field is strong enough, the polarons in both cases achieve sufficient energy to overcome the quantum well/barrier and thereby transport smoothly along the polymer chain.

Fig. 3 The on-site energy of electrons on each site. The value of the spin coupling parameter is J_f = 0.3t₀, and the side radical is connected to the 120th site.

Although the dynamics of the polarons for both spins are significantly impacted by the side radical when the electric field is weak, the values of the critical electric field above which the polarons can move freely through the OFM molecule are different. For instance, as shown in Fig. 4, the spin-down polaron can pass through the OFM molecule when the external electric field is 2.0 × 10⁻³ V nm⁻¹, while the spin-up polaron still remains trapped even when the driven electric field is increased to 3.0 × 10⁻³ V nm⁻¹. Given the present values of parameters, we obtain that the critical electric field is about E_c = 1.4 × 10⁻³ V nm⁻¹ for spin-down polarons and E_c = 3.5 × 10⁻³ V nm⁻¹ for spin-up polarons, respectively. The results are excepted to explain the nonlinear I–V characteristics measured by Sugawara et al. in oligothiophene-type spin-polarized wire molecule based devices. When the driven electric field is weaker than the critical field, the center of the spin-up polaron is closer to the side radical position for a stronger electric field. Moreover, the vibration frequency of the spin-up polaron also increases with the electric field, as shown in Fig. 4(e), (f) and (g). It is important to note that, when the strength of the driven electric field falls into the range between that of the critical fields for the spin-up and spin-down polarons, only the spin-down polaron can pass through the OFM molecule, leading to a phenomenon of spin filtering in such a system. According to our simulations, the range of the electric field for spin filtering spans about ΔE = 2.1 × 10⁻³ V nm⁻¹, which is consistent with the results obtained by using a static picture in ref. 19.

Fig. 4 Time evolution of polaron-charge distributions in the presence of different strengths of electric field. The driven electric field is E₀ = 1.0 × 10⁻³ V nm⁻¹ for (a) and (e), E₀ = 2.0 × 10⁻³ V nm⁻¹ for (b) and (f), E₀ = 3.0 × 10⁻³ V nm⁻¹ for (c) and (g), E₀ = 4.0 × 10⁻³ V nm⁻¹ for (d) and (h). The value of the spin coupling parameter is J_f = 0.3t₀, and the side radical is connected to the 120th site.

To further understand the phenomenon of spin filtering, we also calculated the total energy of the system during the time evolution of the polaron movement, and the results are shown in Fig. 5. In our simulations, the energy is calculated by E_t(t) = E_e(t) + E_l(t), where E_e(t) = 〈φ(t)|H|ψ(t)〉 and denote the energy of the electrons and lattice sites, respectively. When the polaron transports along the magnetic polymer chain, the system gets energy form the electric field. As a result, the total energy of the system increases during the polaron movement. On the other hand, the total energy of the system fluctuates as a result of the vibrations of the lattice sites. When a driven electric field of E₀ = 1.5 × 10⁻³ V nm⁻¹ is applied, the spin-down polaron can easily pass through the magnetic polymer molecule and the total energy of the system rises slightly throughout the transport. In the case of the spin-up polaron, the total energy of the system first increases in a way similar to the case of the spin-down polaron. When the spin-up polaron is closer to the side radical, the polaron level is lifted due to the spin correlation between electrons on the backbone and the side radicals. Furthermore, as the lattice sites vibrate more intensively, and the total energy of the system increases rapidly. As such, the spin-up polaron can no longer get enough energy from the electric field to move forward in this situation. For instance, when the center of the spin-up polaron moves to the 105th site, the spin-up polaron is blocked and the total energy of the system is about 10 eV higher than the case of the spin-down polaron.

Fig. 5 Total energy of the system as a function of the polaron center position. The driven electric field is E₀ = 1.5 × 10⁻³ V nm⁻¹.

We now turn to explore the effects of the width of the quantum well/barrier on the spin-filtering properties. The results are shown in Fig. 6. In our simulations, the width of the quantum well/barrier is tuned by changing the number of the successively arranged side radicals in the polymer chain. As seen from Fig. 6(a), the values of the critical electric field are found to increase slowly and almost linearly for the spin-up polaron but nearly exponentially for the spin-down polaron as a function of n_R. This is due to the fact that, in the case of the spin-up polaron, the width of the quantum barrier becomes larger with increasing n_R such that it is more difficult for the polaron to get across; in the case of the spin-down polaron, more parts of the polaron get trapped by the quantum well such that the critical electric field increases exponentially with n_R. As a result, the range of the electric field for spin filtering decreases almost exponentially with n_R, as shown in Fig. 6(b). Interestingly, it is noted that only the spin-up polaron can pass through the OFM molecule when the value of n_R is large enough.

Fig. 6 (a) The values of critical electric field for both spin-up and spin-down polarons and (b) the spin-filtering range of the electric field as a function of the number of successive sites to which the side radicals are connected. The value of the spin coupling parameter is J_f = 0.3t₀.

We also consider the effects of the electron–lattice coupling strength on the polaron transport, and the results are shown in Fig. 7. It is found that the values of the critical electric field for both the spin-up and spin-down polarons increase almost in a linear manner as a function of α, as seen from Fig. 7(a). However, as E_c for the spin-up polaron increases faster than that for the spin-down polaron, the range of the electric field for spin filtering also increases as a function of the strength of the electron–lattice coupling. As seen from Fig. 7(b), the range of the electric field for spin filtering is about ΔE = 0.2 × 10⁻³ V nm⁻¹ for a relatively weak electron–lattice coupling of α = 39 eV nm⁻¹, and it turns to ΔE = 1.7 × 10⁻³ V nm⁻¹ when the value of α increases to 43 eV nm⁻¹. These results suggest that OFMs with stronger electron–lattice coupling are more conducive to polaron spin filtering.

Fig. 7 (a) The value of the critical electric field for both spin-up and spin-down polarons and (b) the range of the electric field for spin filtering as a function of α. The value of the spin coupling parameter is J_f = 0.3t₀, and the number of side radicals is n_R = 4.

To understand the findings presented in Fig. 7, we calculated the binding energy of the spin dependent polarons with different strengths of the electron–lattice coupling. The polaron binding energy is defined as ε^p_s = ε⁰_s − ε¹_s,³³ where ε¹_s and ε⁰_s denote the total energy of the system before and after the dissociation of the polaron, respectively, and the subscript s represents the spin index. As shown in Fig. 8, the binding energy of both spin-up and spin-down polarons increases almost linearly with the strength of the electron–lattice coupling. The binding energy of the spin-up polaron increases from 0.06 eV to 0.1 eV while it increases from 0.05 eV to about 0.09 eV for the spin-down polaron when α increases from 39 eV nm⁻¹ to 43 eV nm⁻¹. As a more stable polaron is harder to be driven by the electric field, it is more difficult for the polaron to overcome the quantum well/barrier when increasing the strength of the electron–lattice coupling. We also find that the velocity of the motional polaron decreases from 0.052 nm fs⁻¹ to nearly 0.008 nm fs⁻¹ when the value of α increases from 39 eV nm⁻¹ to 43 eV nm⁻¹. It is noted that, given the same values of parameters, the binding energy for the spin-up polaron is always higher than that for the spin-down polaron. As a result, the value of the critical electric field for the spin-up polaron increases faster than that for the spin-down polaron.

Fig. 8 Binding energy of spin-up and spin-down polarons as a function of α. The value of the spin coupling parameter is J_f = 0.3t₀, and the number of side radicals is n_R = 4.

We finally discuss how the strength of the spin correlation between the backbone and the side radical impacts the spin filtering, and the results are shown in Fig. 9. It is seen that the values of the critical electric field for both spin-up and spin-down polarons first increase rapidly and then turn to almost saturate as a function of J_f. For instance, the value of the critical electric field increases from 0 to about 1.4 × 10⁻³ V nm⁻¹ for the spin-down polaron while it increases from 0 to nearly 3.5 × 10⁻³ V nm⁻¹ for the spin-up polaron as a function of J_f increasing from 0 to 0.3t₀. When J_f is larger than 0.3t₀, the values of the critical field for both the spin-up and spin-down polarons increase very slightly as a function of J_f. As a result, the range of the electric field for spin filtering first increases with J_f and then it increases slightly when the value of J_f is larger than 0.3t₀, as shown in Fig. 9(b). These results indicate that the polaron spin filtering can be improved when improving the strength of the spin correlation between the backbone and the side radicals.

Fig. 9 (a) Value of the critical electric field for both spin-up and spin-down polarons and (b) the range of the electric field for spin filtering as a function of J_f. Only one side radical connected to the 120th site is considered.

Conclusion

In summary, we have simulated the dynamic properties of polarons in an organic ferromagnetic polymer chain by considering the spin correlation between the polymer backbone and the ferromagnetic side radicals. We find that a quantum barrier (quantum well) for spin-up (spin-down) polarons near the side radical can be created as a result of the lifting (lowering) of the electron on-site energy caused by the spin correlation. As a result, the motion of the polarons for both spins is impeded near the side radical when the applied electric field is weaker than a critical field. The value of the critical electric field is found to be markedly influenced by the width of the quantum barrier (or the quantum well). Importantly, at some ranges of the electric field, we find that only the polarons with one specific spin orientation (up or down) can pass through the ferromagnet molecule, which leads to a polaron spin-filtering phenomenon in the studied system. Our findings are expected to be useful for guiding the design of organic-based spin filter devices.

Acknowledgements

This work was supported by the Natural Science Foundation of China (Grant No. 11374183, 11574118, 21473102, 11404188).

References

1. J. S. Miller, A. J. Epstein and W. M. Reiff, Science, 1988, 240, 40–47.
2. Y. V. Korshak, T. V. Medvedeva, A. A. Ovchinnikov and V. N. Spector, Nature, 1987, 326, 370–372.
3. J. W. Yoo, R. S. Edelstein, D. M. Lincoln, N. P. Raju, C. Xia, K. I. Pokhodnya, J. S. Miller and A. J. Epstein, Phys. Rev. Lett., 2006, 97, 247205.
4. P. Nickels, M. M. Matsushita, M. Minamoto, S. Komiyama and T. Sugawara, Small, 2008, 4, 471–475.
5. K. Inoue, T. Hayamizu, H. Iwamura, D. Hashizume and Y. Ohashi, J. Am. Chem. Soc., 1996, 118, 1803–1804.
6. B. B. Kaul and G. T. Yee, Polyhedron, 2001, 20, 1757–1759.
7. J. M. Manriquez, T. Y. Gordon, R. S. McLean, A. J. Epstein and J. S. Miller, Science, 1991, 252, 1415–1417.
8. Y. Cao, P. Wang, Z. Hu, S. Li, L. Zhang and J. Zhao, Synth. Met., 1988, 27, 625–630.
9. A. Rajca, J. Wongsriratanakul and S. Rajca, Science, 2001, 294, 1503–1505.
10. Y. A. Katulevskii, M. A. Magrupov and A. A. Muminov, Phys. Status Solidi A, 1991, 127, 223–229.
11. T. Sugano, S. J. Blundell, T. Lancaster, F. L. Pratt and H. Mori, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 82, 180401.
12. B. Li, M. Zhou, Y. Lu, C.-Y. Kao, J.-W. Yoo, V. N. Prigodin and A. J. Epstein, Org. Electron., 2012, 13, 1261–1265.
13. T. Sugawara, M. Minamoto, M. M. Matsushita, P. Nickels and S. Komiyama, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 77, 235316.
14. A. A. Ovchinnikov and V. N. Spector, Synth. Met., 1988, 27, 615–624.
15. Z. Fang, Z. L. Liu and K. L. Yao, Phys. Rev. B: Condens. Matter Mater. Phys., 1994, 49, 3916–3919.
16. Z. Fang, Z. L. Liu, K. L. Yao and Z. G. Li, Phys. Rev. B: Condens. Matter Mater. Phys., 1995, 51, 1304–1307.
17. S. J. Xie, J. Q. Zhao, J. H. Wei, S. G. Wang, L. M. Mei and S. H. Han, Europhys. Lett., 2000, 50, 635.
18. W. Z. Wang, Phys. Rev. B: Condens. Matter Mater. Phys., 2006, 73, 235325.
19. G. Hu, Y. Guo, J. Wei and S. Xie, Phys. Rev. B: Condens. Matter Mater. Phys., 2007, 75, 165321.
20. L. Zhu, K. L. Yao and Z. L. Liu, Appl. Phys. Lett., 2010, 96, 082115.
21. G. Hu, K. He, S. Xie and A. Saxena, J. Chem. Phys., 2008, 129, 234708.
22. C. Q. Wu, Y. Qiu, Z. An and K. Nasu, Phys. Rev. B: Condens. Matter Mater. Phys., 2003, 68, 125416.
23. S. V. Rakhmanova and E. M. Conwell, Synth. Met., 2000, 110, 37–45.
24. D. M. Basko and E. M. Conwell, Phys. Rev. Lett., 2002, 88, 056401.
25. A. A. Johansson and S. Stafström, Phys. Rev. B: Condens. Matter Mater. Phys., 2004, 69, 235205.
26. H. Zhao, Y. Yao, Z. An and C. Q. Wu, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 78, 035209.
27. X. Liu, K. Gao, J. Fu, Y. Li, J. Wei and S. Xie, Phys. Rev. B: Condens. Matter Mater. Phys., 2006, 74, 172301.
28. Å. Johansson and S. Stafström, Phys. Rev. Lett., 2001, 86, 3602–3605.
29. Y. H. Yan, Z. An and C. Q. Wu, Eur. Phys. J. B, 2004, 42, 157–163.
30. X. J. Yuan, D. M. Li, S. Yin, K. Gao, B. Cui and D. S. Liu, Org. Electron., 2012, 13, 2094–2103.
31. G. C. Hu, H. Wang, J. F. Ren, S. J. Xie and C. Timm, Org. Electron., 2014, 15, 118–125.
32. Y. Li, X. J. Liu, J. Y. Fu, D. S. Liu, S. J. Xie and L. M. Mei, Phys. Rev. B: Condens. Matter Mater. Phys., 2006, 74, 184303.
33. Y. Li, K. Gao, Z. Sun, S. Yin, D. S. Liu and S. J. Xie, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 78, 014304.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c5cp05789b

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