Dynamics of Polarons in Organic Conjugated Polymers with Side

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Dynamics of Polarons in Organic Conjugated Polymers with Side Radicals Junjuan Liu, Zengjiang Wei, Yalin Zhang, Yan Meng, and Bing Di J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b00501 • Publication Date (Web): 20 Feb 2017 Downloaded from http://pubs.acs.org on February 25, 2017

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The Journal of Physical Chemistry

Dynamics of Polarons in Organic Conjugated Polymers with Side Radicals J. J. Liu,† Z. J. Wei,‡ Y. L. Zhang,¶ Y. Meng,∗,§ and B. Di∗,∥ Institute for Nationalities Attached to Hebei Normal University, Shijiazhuang 050091,China, Shijiazhuang Institute of Technology, Shijiazhuang 050200, China, College of Physics, Hebei Normal University, Shijiazhuang 050024, China, Department of Physics, Xingtai University, Xingtai 054001, China, and College of Physics, and Hebei Advanced Thin Films Laboratory, Hebei Normal University, Shijiazhuang 050024, China E-mail: [email protected]; [email protected]

∗ To

whom correspondence should be addressed Normal University ‡ Shijiazhuang Institute of Technology ¶ Hebei Normal University § Xingtai University ∥ Hebei Normal University † Hebei

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Abstract Based on the one-dimensional tight-binding Su-Schrieffer-Heeger (SSH) model, and using the molecular dynamics method, we discuss the dynamics of electron and hole polarons propagating along a polymer chain, as a function of the distance between side radicals and the magnitude of the transfer integrals between the main chain and the side radicals. We first discuss the average velocities of electron and hole polarons as a function of the distance between side radicals. It is found that the average velocities of the electron polarons remain almost unchanged, while the average velocities of hole polarons decrease significantly when the radical distance is comparable to the polaron width. Second, we have found that the average velocities of electron polarons decrease with increasing transfer integral, but the average velocities of hole polarons increase. These results may provide a theoretical basis for understanding carriers transport properties in polymers chain with side radicals.


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Introduction Due to the strong electron-lattice interactions in organic polymers, electrons and holes injected from the electrodes can induce lattice distortions to form self-trapped elementary excitations, such as solitons polarons and bipolarons. 1–4 The dynamic formation process of these elementary excitations can help us to understand transport in organic semiconductor as well as their luminescence properties. 5,6 Because of their high conductivity and light-emitting efficiency, 7 organic polymers have attracted considerable attention for their commercial applications, e.g., organic light emitting diodes, 8 field effect transistors, 9 memory devices, 10 organic solar cells, 11,12 etc. These materials are usually organic conjugated polymers with side radicals. For example, in organic solar cells, the mobility of electron and hole polarons plays a decisive role in the conduction efficiency of conjugated polymers, and the presence of side radicals is an effective means to improve the properties of these materials. In particular, side radicals allow control of hole transport and electron transmission in semiconductor materials through the influence of the functional side group, 13–15 the side radicals therefore significantly affect the photoelectric properties of conjugated polymers. 16,17 Baradwaj et al. demonstrated that charge transport in polymers with side radicals is inherently different than that in semicrystalline, conjugated polymers. 18 Casey et al. found that replacing the fluorine substituents of polymers with cyano groups in field effect transistors changed the charge transport from unipolar p-type to unipolar n-type. 19 Chan et al. discussed how refinement of pendant group chemistry and careful addition of intramolecular dopants can enhance solid-state transport in a radical polymer system, 20 and Tomlinson et al. discussed the application of polymers with side radicals in the field of organic electronics. 21 These results have been obtained mostly from the experiments designed to explore the effects of introducing side radicals. In the present paper, however, we implement a theoretical analysis, and discuss the dynamics of electron and hole polarons under the influence of side radicals. We have found that the average speeds of the hole and electron polarons are different in the presence of more than one side radical and that the side radicals directly affect carrier mobility in organic conjugated polymers. The mechanism of side radicals controlling charge carrier transport is due to 3 ACS Paragon Plus Environment


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the adjustment of the organic molecular structure and charge distribution through the interactions between the side radicals and the polymer chain, the interactions between the charge carriers and the side radicals play a significant role on the charge carrier transport, which is equivalent to the effects of a potential well or a potential barrier. It is therefore very important to study in depth the influence of multiple side radicals on organic conjugated polymer materials, and these conclusions are expected to provide guidance for the experiment designed.

Model and Numerical Method Based on the one-dimensional tight-binding SSH model, and using the molecular dynamics method, 22,23 the Hamiltonian can be written as

H = Hel + Hlattice + HSR + Him ,

(1)

The first term describes the electronic energy and the contribution of an external electric field, Hel = − ∑ tn (e−iγ A(t) c†n+1 cn + eiγ A(t) c†n cn+1 ),

(2)

n

where, tn = [t0 − α (un+1 − un ) + (−1)nte ] is the hopping integral of π electrons between sites n and n+1; t0 is the transfer integral between nearest-neighbor sites; α is the electron-phonon coupling constant; 24 un is the displacement coordinate of the n-th site, and te is introduced to lift the ground-state degeneracy for a nondegenerate polymer. c†n (cn ) is the electron creation (annihilation) operator on site n of the main chain, and the electric field is introduced through the vector potential A(t) appearing in a complex phase factor in the transfer integral, to which periodic boundary conditions are applied. The relation between the potential A(t) and the uniform electric field E(t) is given by E(t) = −∂t A(t)/c, and the coefficient γ in the exponent is defined as γ = ea/¯hc with e, a and c being the electron charge, lattice constant and speed of light, respectively. The vector


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potential A(t) is taken to have the form:    −cE0t 2 /2Tc , 0 < t < Tc A(t) =   −cE0 (t − Tc /2), Tc ≤ t < To f f

(3)

The second term in Eq.(1) includes the lattice potential and kinetic energy, 1 1 Hlattice = K ∑(un+1 − un )2 + M ∑ u˙2n , 2 n 2 n

(4)

where K is the elastic constant and M is the mass of a CH group. The third term describes the Hamiltonian related to the side radicals, ( ) HSR = − ∑ t1 δn,SR c†SR cn + h.c. ,

(5)

n

where t1 is the hopping integral between the π -electrons of the main chain and the unpaired electrons of the side radicals, and c†SR (cSR ) denotes the creation (annihilation) operator of the side radical, which is bonded to the n-th carbon atom of the main chain. The fourth term describes the contribution of the side radical potential, Him = ∑ VSR c†SR cSR ,

(6)

SR

where VSR is potential energy of a side radical. The time evolution of the lattice is determined by the equation of motion M u¨n (t) = −K[2un (t) − un+1 (t) − un−1 (t)] +α e−iγ A(t) [ρn+1,n (t) − ρn,n−1 (t)] +α eiγ A(t) [ρn,n+1 (t) − ρn−1,n (t)] ,

(7)

where ρn,n′ (t) = ∑ Φ∗µ ,n (t) f µ Φµ ,n′ (t), and f µ (= 0, 1, 2) is the time-independent distribution funcµ



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tion determined by the initial state. The electronic wave functions are the solutions of the timedependent Schrödinger equation, ˙ µ ,n (t) = ∑ hn,n′ (t)Φµ ,n′ (t), i¯hΦ

(8)

n′

where hn,n′ is the Hamiltonian matrix,

hn,n′

   −[t0 − α (un′ − un ) + (−1)nte ],       −t1 δn,SR1 +VSR1 δn,SR1 , =   −t1 δn,SR2 +VSR2 δn,SR2 ,       0,

n′ = n ± 1 n = SR1

(9)

n = SR2 otherwise

The coupled differential equations (7) and (8) are solved using the Runge-Kutta method of order 8 with step-size control, 25 which has proven to be an effective approach for investigating polaron dynamics in conjugated polymers. 26–28 The applied electric field is turned on smoothly over a time interval Tc and then keep fixed, i.e.    E0t/Tc , 0 < t < Tc E(t) =   E0 , Tc ≤ t < To f f

(10)

In our simulations, we chose Tc = 25fs and To f f = 800fs, to observe the polaron motion for a long period of time, a periodic boundary condition is applied in our calculations. The model parameters are those generally chosen for polyacetylene: 29 t0 = 2.5 eV, α = 4.1 eV/Å, te = 0.05 eV, K = 21.0 eV/Å2 , M = 1349.14 eVfs2 /Å2 , a = 1.22 Å. The results are expected to be qualitatively valid for the other conjugated polymers.


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Results and discussions A polymer main chain consisting of 200 Carbon atoms is used in all simulations, and two side radicals are added to the chain. In what follows, we discuss the dynamics of electron and hole polarons as a function of both the distance between two side radicals (d) and the magnitude of the transfer integrals between the main chain and side radicals (t1 ) under the mediate electric field E = 0.6 mV/Å. Initially, a polaron is set up at site 160, with two side radicals at either sites 41 and 51 or at sites 35 and 51. The polaron is initially too far from the side radicals so that they have no interaction between the polaron and side radicals, and the polaron is accelerated and begins to move along the polymer chain towards the side radicals. Take two cases as an example, figure 1 shows the time evolution of the charge centers of the electron and hole polarons for the two cases: Two side radicals are separated by distances of d = 10 times the lattice constant (a) and d = 16 times the lattice constant (b). As noted above, in order to mitigate end effects of the finite chain length, periodic boundary conditions are imposed on the systems. We define the charge center xc of a polaron in a polymer chain as    N θ /2π , ⟨cos θn ⟩ ≥ 0, ⟨sin θn ⟩ ≥ 0,    xc = N(θ + π )/2π , ⟨cos θn ⟩ < 0,      N(θ + 2π )/2π , otherwise

(11)

where θ = arctan(⟨sin θn ⟩/⟨cos θn ⟩), and the average of sin θn and cos θn are defined as ⟨sin θn ⟩ = ∑ ρn sin θn , ⟨cos θn ⟩ = ∑ ρn cos θn , with the probability weight ρn = ρn,n − 1 and θn = 2nπ /N, and n

n

N is the number of sites in the polymer chain. In the simulation, the side radical potential strengths are VSR1 = VSR2 = 0.5eV, and the transfer integrals between the main chain and side radicals are t1 = 0.3eV. From figure 1(a), it may be seen that the electron or hole polaron can move along the polymer chain with a steady velocity under an external electric field. When the polaron meets with two side radicals, the speed of the electron polaron first decreases and then increases, the speed of the



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hole polaron first increases and then decreases. For both values of the side radical separation, the charge center for the electron polaron lies far ahead of the hole polaron. For instance, figure 1(a) shows that the electron polaron moves to the 71-th site at 800fs, while the hole polaron reaches only the 83-th site (the polaron moves from right to left), and there is a difference of 12 times the lattice constant between the hole polaron and electron polaron. Likewise in figure 1(b), the electron polaron moves to the 71-th site at the same 800fs, while the hole polaron arrives at only the 154-th site, a difference of 83 lattice constants. From figure 1(b), it can be seen that the velocity of the hole polaron is clearly affected by the distance between side radicals, the speed of electron polaron is greater than that of the hole polaron. In other words, the side radicals have a weaker influence on electron polaron speed. Above results are easily understood. This is due to the Coulomb repulsion interactions between the electron polaron and the side radicals, which acts in a manner similar to the potential barriers. However, the behavior of the hole polaron is due to the Coulomb attraction interactions between the hole polaron and the side radicals, which now acts like a potential well. In order to help analyze the different dynamical behaviors of the electron and hole polarons in the presence of side radicals, figure 2 shows the energy level diagrams for the electron and hole polarons in the presence of side radicals. When the side radical potential VSR > 0, a local energy level εSR forms in the upper gap. As a result, there are three energy levels in the gap, i.e. εup , εSR and εdown . Figure 2(a) represents the electronic configuration for an electron polaron with a side radical. The upper level is occupied by one spin-up electron, the middle one is occupied by the same electron and the lower one is occupied by two electrons. Figure 2(b) represents the electronic configuration for a hole polaron with a side radical, here the middle level is occupied by one spinup electron and the lower one occupied by the same electron. According to the Pauli exclusion principle, in the case of two electrons occupying the same orbital, the Pauli principle demands that they have opposite spins. For the electron polaron levels εup and εSR , there is no transition between levels with electrons of the same spin, so the interaction between the two levels is very small. In the case of the hole polaron levels, εSR level is occupied by one spin-up electron, while the εup


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level is unoccupied. The electron in the εSR level may jump to the above εup level, and interaction between the two energy levels occurs easily. As a result, the speed of the hole polaron is affected more strongly by the side radicals than the speed of the electron polarons. To clarify this point further, we discuss the dynamics of the electron and hole polarons as a function of the distance between the side radicals. Figure 3 shows the average velocities of the electron and hole polarons for different values of d. With fixed side radical potentials (VSR1 = VSR2 = 0.5eV) and transfer integrals t1 = 0.3eV, the average velocities of the electron and hole polarons are obtained as a function of distance for values of d in the range 2-34 times the lattice constant. The results show that the average velocities of the electron polarons remain almost unchanged. The average velocities of the hole polaron decrease significantly when the distance between side radicals is comparable to the polaron width. When the distance between two side radicals is in the range 2 ≤ d < 14 or 22 < d ≤ 34 times the lattice constant, the average velocities of hole polaron remain almost unchanged, presenting just a few small oscillations. This can be explained as follows: This is due to the repulsive Coulomb interactions between the electron polaron and the side radicals, which means that the electron polaron is not easily bound by the side radicals. For hole polaron, there is Coulomb attraction between the hole polaron and side radicals, and arising from the transfer interaction between the main chain and the side radicals. Due to this reason, when the hole polaron meets the side radicals, electrons from the side radicals tend to transfer to the main chain. Therefore, the speed of the hole polaron is affected more strongly by the side radicals. When the distance between side radicals is comparable to the polaron width, the hole polaron can be easily trapped by the side radicals. This is due to the two side radicals generating different superposition effects for the potential well. Figure 4 shows the average velocities of the electron and hole polarons with different transfer integrals t1 . With fixed VSR1 = VSR2 = 0.4eV and d = 10 times the lattice constant, we find that the average velocities of the electron polaron decrease with increasing t1 . This is due to the fact that as the transfer integrals increase, more and more charges can transfer from the side radical to the main chain (see figure 5). The result is that the Coulomb repulsion between the electron



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polaron and side radical increases, and the average velocities of the electron polaron decrease. Correspondingly, the Coulomb attraction between the hole polaron and side radical is also larger, so the average velocities of the hole polaron increase. In addition, it should be mentioned that electron-electron (e-e) interactions are important in the 1D conjugated polymer chain. 30,31 Combining the one-dimensional tight-binding SSH model and the extended Hubbard model, we have investigated the effects of the e-e interactions on the dynamics of a charged polaron in a conjugated polymer chain, and shown that the dynamics of the polaron is intensively affected by both the on-site repulsions and the nearest-neighbor e-e interactions. 32 Further improvement of the works, e-e interactions will therefore be important and certainly have an impact on dynamics of polarons in organic conjugated polymers with side radicals in our future researches.

Summary Based on the one-dimensional tight-binding SSH model, and using the molecular dynamics method, we discuss the dynamics of polarons as influenced by the distance between side radicals and by the magnitude of transfer integrals between the main chain and side radicals. It is find that the average velocities of electron polarons remain almost unchanged with increasing d, this case of the electron polaron not easily bound by the side radicals is due to the Coulomb repulsion interactions between the electron polaron and the side radicals. On the other hand, the average velocity of the hole polaron can be affected by the distance between side radicals. When the distance between side radicals is comparable to the polaron width, the hole polaron can be easily trapped by side radicals, and the average velocities of hole polaron decrease significantly. The behavior of the hole polarons depends on the Coulomb interactions and the transfer interactions. With fixed side radical potentials and distance between side radicals, we find that the average velocities of the electron polarons decrease with increasing transfer integral, but the average velocities of the hole polaron increase. This is due to the fact that when the transfer integrals are large,


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the Coulomb interactions between the polaron and side radical are also large. This study can be of significance for the optimisation of the electron/hole transport properties of polymer derivatives of relevance in photovoltaics and optoelectronics.

Acknowledgement Project supported by the National Natural Science Foundation of China (Grant No.11074064), the Natural Science Fund of Hebei Province of China (Grant No.A2016205271) and the Educational Commission of Hebei Province of China (Grant No.ZD2014052). We thank Professor N. E. Davison for helpful discussions.

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Figure Captions Figure 1 Temporal evolution of the charge centers of the electron and hole polarons for d = 10 times the lattice constant (a) and d = 16 times the lattice constant (b). Figure 2 Energy level diagram with side radicals for the electron polaron (a) and hole polaron (b). Figure 3 Average velocities of the electron and hole polarons for different values of d. Figure 4 Average velocities of the electron and hole polarons for different values of t1 . Figure 5 Excess charge density on the main chain and the side radicals for different values of t1 . Figure 6 Table of Contents Graphic.



200 charge center x (n) c

(a)

d=10 times the lattice constant

150

electron hole

100

t1=0.3eV

50

VSR1=0.5eV VSR2=0.5eV

0

200

400

600

800

600

800

time (fs)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 21

(b)


150

electron hole

100

t1=0.3eV

50

VSR1=0.5eV VSR2=0.5eV

0

200

400 time (fs)

Figure 1: Temporal evolution of the charge centers of the electron and hole polarons for d = 10 times the lattice constant (a) and d = 16 times the lattice constant (b).


Page 17 of 21

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


conduction band

conduction band

up

up

SR

SR

down

down

valence band

valence band

(a) electron polaron

(b) hole polaron

Figure 2: Energy level diagram with side radicals for the electron polaron (a) and hole polaron (b).



0.50

A

O

average velocity ( /fs)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 21

0.45

electron 0.40

hole

t1=0.3eV

0.35

VSR1=0.5eV

0.30

VSR2=0.5eV

0.25 0

5

10

15

20

25

30

35

d

Figure 3: Average velocities of the electron and hole polarons for different values of d.


Page 19 of 21

0.50 electron hole 0.48

A

average velocity ( /fs)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


O

0.46


VSR1=0.4eV

0.44

VSR2=0.4eV 0.42 0.1

0.2

0.3

t

1

0.4

0.5

(eV)

Figure 4: Average velocities of the electron and hole polarons for different values of t1 .



0.30

excess charge density

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 21

VSR1=VSR2=0.05eV

0.15

side radical

VSR1=VSR2=0.3eV VSR1=VSR2=0.5eV

0.00

-0.15

VSR1=VSR2=0.05eV VSR1=VSR2=0.3eV VSR1=VSR2=0.5eV

-0.30

0.2

0.4

main chain

0.6 t

1

0.8

1.0

1.2

(eV)

Figure 5: Excess charge density on the main chain and the side radicals for different values of t1 .


Page 21 of 21


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



150

electron hole

100

t1=0.3eV

50

VSR1=0.5eV VSR2=0.5eV 0

200

400

600

time (fs)

Figure 6: Table of Contents Graphic


800

Dynamics of Polarons in Organic Conjugated Polymers with Side

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