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Phase-breaking effect on polaron transport in organic conjugated polymers
Organic Electronics 49 (2017) 33e38
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Organic Electronics journal homepage: www.elsevier.com/locate/orgel
Phase-breaking effect on polaron transport in organic conjugated polymers Ruixuan Meng a, Sun Yin a, Yujun Zheng a, Liu Yang a, Shijie Xie a, *, Avadh Saxena b a b
School of Physics, State Key Laboratory of Crystal Materials, Shandong University, Jinan, 250100, China Los Alamos National Laboratory, NM 87545, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 April 2017 Received in revised form 7 June 2017 Accepted 11 June 2017 Available online 15 June 2017
Despite intense investigations and many accepted viewpoints on theory and experiment, the coherent and incoherent carrier transport in organic semiconductors remains an unsettled topic due to the strong electron-phonon coupling. Based on the tight-binding Su-Schrieffer-Heeger (SSH) model combined with a non-adiabatic dynamics method, we study the effect of phase-breaking on polaron transport by introducing a group of phase-breaking factors into p-electron wave-functions in organic conjugated polymers. Two approaches are applied: the modification of the transfer integral and the phase-breaking addition to the wave-function. Within the former, it is found that a single site phase-breaking can trap a polaron. However, with a larger regular phase-breaking a polaron becomes more delocalized and lighter. Additionally, a group of disordered phase-breaking factors can make the polaron disperse in transport process. Within the latter approach, we show that the phase-breaking can render the delocalized state in valence band discrete and the state in the gap more localized. Consequently, the phase-breaking frequency and intensity can reduce the stability of a polaron. Overall, the phase-breaking in organic systems is the main factor that degrades the coherent transport and destroys the carrier stability. © 2017 Elsevier B.V. All rights reserved.
Keywords: Polaron Electronic structures Phase Localized states Organic semiconductor
1. Introduction Energy migration and carrier transport are important current topics in organic semiconductors and devices. Although the organic conductance has been effectively improved and the organic devices, such as organic solar cells (OSC) and organic light emitting diodes (OLED), have been successfully fabricated, some basic physical phenomena in organic electronics are still controversial. For example, there is no consistent conclusion on the organic transport mechanism whether band-like transport or hopping is the dominant mechanism. This controversy is recently shown to relate closely to the coherence or incoherence of the transport mechanism. The specific nature of organic semiconductors on physical and structural basis determines their transport performance. In traditional inorganic semiconductors and metals, the rigid periodic lattice structure makes the electronic state behave in the form of a Bloch wave. The band-like transport mechanism dominates, which leads to a high mobility. Simple lattice structure and the absence of
* Corresponding author. E-mail address: [email protected] (S. Xie). http://dx.doi.org/10.1016/j.orgel.2017.06.026 1566-1199/© 2017 Elsevier B.V. All rights reserved.
electron-phonon coupling make the effective transport easily understood. While in organic semiconductors, the strong electronphonon coupling causes the carriers or excited states to couple with the soft lattice resulting in the formation of spatially localized self-trapped states, such as polarons, bipolarons or excitons. These quasi-particles are heavier and move slower than the usual electrons and holes. As a result, the mobility is adversely affected even in organic crystals. Moreover, the actual material microstructure is not as perfect as one would expect. Besides electron-lattice interaction, there also exist the structural diversity, impurities and thermal effects. All these factors complicate the transport mechanism. Up to now, much work has been done to study carrier transport in organic molecular crystals or polymers [1]. In 2006, Troisi and Orlandi used a one-dimensional semiclassical model to compute charge carrier mobility in the presence of thermal fluctuations of the electronic Hamiltonian. This transport mechanism explains several contrasting experimental observations pointing sometimes to a delocalized ‘‘band-like’’ transport and sometimes to the existence of strongly localized charge carriers [2]. The effects of structural diversity [3] and impurities are also considered as the diagonal or non-diagonal disorder, and the thermal effect has been modeled by dynamic disorder exerting a random force on lattice
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R. Meng et al. / Organic Electronics 49 (2017) 33e38
sites [4,5]. The comprehensive effect of these factors leads to the inferior transport performance of organic semiconductors. From what is stated above, we can infer that the poor mobility should be intrinsic to organic materials. However, two promising facts are worth considering. One is the high mobility at low temperature. The negative temperature coefficient of mobility is the fingerprint of the band-like transport mechanism. In one experiment, Sakanoue and Sirringhaus found that in the crystalline pentacene or polymer at low temperature, the mobility is high and “thermally deactivated”, e.g. it decreases with rising temperature [6]. Early investigation by Palstra and coworkers also showed that the mobility of crystal pentacene decreased with temperature and impurity [7]. This phenomenon should be attributed to degeneration of band-like or coherent transport caused by thermal lattice fluctuations. The other fact is the observation of room-temperature coherence. In 2009, Scholes observed efficient coherent transport in organic semiconductors at room temperature [8,9], which offered the possibility of efficient transport of charge and energy in the organic material. It is considered that coherent transport dominated at short times if the polymer chains were protracted, while incoherent transport dominated if the chains were folded seriously. Cho et al. experimentally measured electron transport properties during an ultrashort time and found the transition from coherent transport to incoherent transport [10]. Recently, Takeya et al. reduced molecular thermal vibrations by pressurizing the material and also experimentally obtained the corresponding result [11]. As the intermolecular spacing decreases the coupling is enhanced, which leads to the electron transport from incoherent to band transport in the organic crystal. These observations indicate that coherence supports effective transport and the structural disorder and thus the phase disorder should be responsible for the reduction of mobility. Impurities or defects are easily created in the organic samples; thermal effect is apparent for organic materials with light atoms. Moreover, the electron-electron interaction is also an important source of phase-randomizing collisions. For effective carrier transport and energy migration, investigations on the coherent mechanism and keeping track of phase information are necessary. Rapid progress has been made in studying coherent transport. Scholes and coworkers proposed some methods to define coherence in theory [12], such as the relative coupling strength theory. In experiment, they also observed the cross peak in two dimensional electronic spectroscopy [13], implying the existence of a coherent process. These theoretical descriptions can well present the coherence phenomenon in organic semiconductors and have played a guiding role in experiments, but the analysis of intrinsic phase information of the wave-function which is the root cause of all the coherence phenomena has been obviously lacking. In 2012, Yao et al. considered the decoherence process in organic amorphous semiconductors by introducing a decoherence time, and how the type of transport is changed from coherent to incoherent [14]. With Monte Carlo simulation, they obtained the dependence of diffusion coefficient on decoherence time, which is consistent with some experimental observations [15]. In many previous works on p-conjugated molecules or polymers, an atomistic simulation dynamic approach has been widely used to describe the dynamic behaviors of carriers in organic semiconductors, such as polaron transport and dissociation, polaron collisions and exciton production [16e20]. In these investigations, the phases of the electronic states are strictly determined by the Hamiltonian. From the perspective of coherence, the electronic phase is supposed to be reserved, which means a coherent transport. Obviously, this kind of treatment is deficient when involving phase-breaking processes. In this paper, we consider this question in the context of phase conservation. In a highly conjugated polymer, as the main carrier, a
polaron is a combination of an electron (hole) and the lattice distortion. How the phase of the p-electron wave-function affects the polaron should be considered by invoking the electron-phonon coupling. Consequently, this paper is organized as follows. The model and method are described in the next section. A qualitative analysis is also given in this section. Then in section three we present calculations on a polaron and its dynamics based on two phase-breaking mechanisms we proposed. Finally in section four, we discuss the phase-breaking effect and conclude that phasebreaking is unfavorable to carrier stability and its transport in the materials with band-like transport mechanism. 2. Model and method In our previous works, we have studied polaron motion, dissociation and collisions. The behavior of a polaron (bipolaron or exciton) is closely related to the electron-phonon coupling. For a conjugated polymer chain described by the SSH model, the evolution of the electronic wave-functions and the lattice configuration is determined by Refs. [3,21,22].
iZ
v 4 ðt Þ ¼ tn1 4m;n1 tnþ1 4m;nþ1 eðna þ un Þ4m;n vt m;n
Mu€n ðtÞ ¼ fl þ fe þ fE
(1) (2)
with the elastic force, electron-phonon coupling force and external electric field force separately given by
fl ¼ Kðunþ1 2un þ un1 Þ
(2a)
fe ¼ a rnþ1;n þ rn;nþ1 rn;n1 rn1;n
(2b)
fE ¼ eE rn;n 1
(2c)
where tn ¼ t0 aðunþ1 un Þ. All the notations have the normal meaning as in previous reference [23]. Normally, the non-adiabatic dynamic process is expressed by the solution of the coupled de€ dinger rivative equations consisting of the time-dependent Schro equation and the Newton's equation of motion. We first prescribe an initial system state, of which the band structure and wavefunction are determined. Starting with the initial state, the electron wave-function evolves following the time-dependent €dinger's equation. Meanwhile, due to the electron-phonon Schro coupling, the lattice motion is controlled by the driving force in Eq. (2), which is partially determined by the electron state at previous time step. Therefore, the updated lattice structure determines the electron state of the next time step. With increasing time, the evolving electron wave-function and the lattice motion condition at every time step are determined. P * In Eq. (2b), the density matrix is given by rm;n ¼ OCC m 4m;m 4m;n, from which we can discern the importance of phase. For the diagonal terms, rn;n means the electronic density at site n. But for the P * off-diagonal terms, 4m;n ¼ 4m;n eiqm;n , rm;n ¼ OCC m 4m;m 4m;n eiðqm;n qm;m Þ , the effect of phase is important. From the expression of the electron-phonon coupling term, we see that this force is sensitive to the phase difference between the neighboring sites. Therefore, if the phase is altered by some physical factors, it is expected that the lattice configuration or the carrier would be changed. It should be noticed that for usual rigid materials, the electron-phonon coupling a is very weak, and in this case the electron-phonon coupling force has been neglected.
R. Meng et al. / Organic Electronics 49 (2017) 33e38
35
We use two approaches to introduce the phase-breaking. One is 0 to modify the transfer integral tn into tn þ itn . This modification is equivalent to applying a spatially varying magnetic field to the electronic states [24]. It will affect the phase of the electronic states [25]. Another approach is to modify the phase of the electronic states at each step while solving the coupled equations (1) and (2) self-consistently. For step k, electronic state 4m;n ðtk Þ is obtained by
solving Eq. (1), then we substitute 4m;n ðtk Þ for 4m;n ðtk Þeiqm;n into Eq. (2) to give the next-step lattice configuration un ðtkþ1 Þ. Here, qm;n means the phase-breaking on the mth state at site n. We repeat this process until an expected result is achieved. In our calculations, we choose polyacetylene as the representative of organic conjugated polymers. Note that all the parameters have been obtained via DFT and from experiment. They are 2
t0 ¼ 2:5eV, a ¼ 4:1eV=Å, K ¼ 21:0eV=Å , M ¼ 1349:14eV$fs =Å , and a ¼ 1:22Å. 3. Results and discussions For the first kind of modification, the electronic Hamiltonian becomes,
He ¼
X n
0 þ tn þ itn Cnþ1 Cn þ h:c:
(3)
tn where the spin index s is omitted. Let cos qn ¼ pffiffiffiffiffiffiffiffiffiffi , we have 02 2 t n þt n
ffi X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 þ He ¼ Cn þ h:c: t 2n þ t n2 eiqn Cnþ1 n
¼
X n
þ tn eiqn Cnþ1 Cn þ h:c:
(4)
Pn1 i q We take the transformation for the operator, Cn ¼ an e j¼1 j and Pn Pn1 i j¼1 qj i q þ iqn . Cnþ1 Cn ¼ aþ an e j¼1 j ¼ aþ Then Eq. (4) nþ1 e nþ1 an e becomes
He ¼
X n
tn aþ nþ1 an þ h:c:
0
Fig. 1. Dependence of polaron width on the phase-breaking parameter tn.
2
2
(5)
Equation (5) has a similar form similar to that without the modification. But the transfer integral changes from tn to tn . If we consider the displacement unþ1 un as perturbation, we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 tn ¼ t 2n þ t n2 ztn0 ðt0 =tn0 Þaðunþ1 un Þ, where tn0 ¼ t 20 þ t n2 . Compared to the original SSH model, we can assess that phasebreaking will alter the effect of electron-phonon coupling. As t0 =tn0 1, a polaron tends to become more delocalized. We define 2 4 P P the width of a polaron as, w ¼ ð 4p;n Þ2 = 4p;n (in units of
polaron. Supposing there is a phase-breaking only at one site, 0 denoted as tc , we find that the polaron tends to locate at this site and its creation energy decreases with the phase-breaking strength (see Fig. 2). It seems that phase-breaking acts as an energy well and is not favorable for the polaron motion. Through systematic dynamical calculations, the single phase-breaking is found to have the trapping influence on polaron transport, similar to that of an impurity or a conjugation defect. However, if we calculate the dynamics of a polaron under a driving electric field, it is found that the continuous phase-breaking 0 may give a different result. Here we set tn as a constant, that is, there exists the same phase-breaking between every site. As shown in Fig. 3, the polaron moves faster with a larger continuous phasebreaking. This phenomenon may result from the adopted model of equation (3). To introduce a desired phase-breaking, we modify the transfer integral. Although this method indeed breaks phase conservation, at the same time, the transition rate increases from tn to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 tn ¼ t 2n þ t n2 . We also find that a sufficiently strong disordered phasebreaking will result in the dispersion of a polaron. To this end, a group of random phase-breaking, following the standard normal 0 distribution t Nð0; sÞ, is applied to the polymer chain, where s represents the degree of disorder. Fig. 4 depicts the evolution of the lattice displacement yn ¼ ð1Þn ðunþ1 þ un1 2un Þ=4 and charge 2 P density dn ¼ occ 4m;n with different intensity of the disordered m
phase-breaking. At the beginning, there is a charged polaron centered at site n ¼ 60. With an external driving electric field gradually applied, the polaron starts to move along the polymer chain. If there is no phase-breaking along the polaron transport routine, the electron wavefunction phases can keep regular. The
n
lattice constant a), where jp ¼
P n
n 4p;n n > is the highest occupied
state of the polymer with a polaron. If the state is confined to only one site, 4p;n is a delta function and we have w ¼ 1, which means that the polaron is strictly localized; if the state is extended over the pffiffiffiffi whole chain, 4p;n ¼ 1= N and we have w ¼ N, which means that the polaron is not localized. It also implies that the lattice cannot confine the p-electron. We show a result in Fig. 1, where it is found that the polaron width increases with the phase-breaking param0 eter tn. It proves what we have expected so far. One aspect is that the transfer integral tn0 broadens the energy band width, rendering the polaron delocalized. On the other hand, the effect of electronlattice coupling decreases, and the ability that lattice distortion confines the p-electron subsides. Next, we consider the trapping effect of the phase on the
Fig. 2. Dependence of polaron creation energy on the single phase-breaking param0 eter tc.
36
R. Meng et al. / Organic Electronics 49 (2017) 33e38
0
Fig. 3. Dependence of polaron velocity on the phase-breaking parameter tn. The driving field E ¼ 6 104 V=Å.
polaron moves perfectly with localized charge and lattice coupled together (Fig. 4(a)). If the phase-breaking intensity is s ¼ 0:2, a group of weak phase-breaking, it is found that the polaron is still robust during the transport process, although a little dispersion appears (Fig. 4(b)). It means that the phase-breaking is detrimental for the polaron stability. As the phase-breaking intensity is increased to s ¼ 0:4, as shown in Fig. 4(c), it is found that the distribution of charge density get disperse, and the lattice distortion become delocalized. In other words, the polaron dissociates quickly. In this case, the electronic states rapidly lose their initial phase information and cannot remain in a stable state. As a result, the random phase is not able to meet the requirement of coherent transport, and we have to describe the transport with incoherent hopping. Now let us consider the second approach to break the phase. In a real system, p-electrons and the lattice are strongly coupled together. To determine the most stable system state, we normally start from the Hamiltonian with a given lattice configuration to
Fig. 5. Dependence of the polaron localization on the phase-breaking intensity q0 .
obtain the electronic states. These electronic states in turn act on the lattice through the electron-phonon coupling. If the difference between successive lattice configurations in the iteration process is smaller than or within a given precision, we define it as the most stable system state. After taking phase-breaking into consideration, this self-consistent iteration is modified with a group of additional phase for each state. Therefore, the final state is the most stable state of this group of additional phase. Based on it, we next study the localization of a stationary polaron. The phase-breaking intensity on the wave-function is supposed to be distributed randomly in an interval qm;n 2½q0 ; q0 . The localization of the electronic state on the highest occupied state is defined as 4 P 2 P Pp ¼ 4p;n =ð 4p;n Þ2 , which is the reciprocal of the polaron n
n
width. Analogously, the localization of charge density and lattice 4 Pocc 2 2 P displacement are given by Рe ¼ occ m;n 4m;n =ð m;n 4m;n Þ and P P Рl ¼ jun j4 =ð jun j2 Þ2 . In a well extended polymer crystal, since n
n
the states in the valence band are completely delocalized, the local
Fig. 4. Dynamics of a polaron with phase-breaking intensity s ¼ 0 (a), s ¼ 0:2 (b) and s ¼ 0:4 (c). The driving field E ¼ 6 104 V=Å.
R. Meng et al. / Organic Electronics 49 (2017) 33e38
states in the gap can dominantly determine the polaron formation. In other words, the behavior of polaron is consistent with that of the localized state. However, with disorder considered, the situation may be different. Fig. 5 shows the dependence of these quantities on phase-breaking intensity. We find that with the phase-breaking intensity, the highest occupied state becomes localized, but the polaron charge density and lattice distortion tend to be delocalized. The behavior of the polaron is inconsonant with that of the localized states. This indicates that the states in the valence band are not completely delocalized and also contribute to localization degree. Further, let us consider the dynamic process with phasebreaking. We assume that the phase-breaking happens after every time interval tq . During one time interval, the phase information is normally evolved without any destruction and keeps its initial phase relation. But at the end of every interval, all the states will suffer a random phase-breaking with an intensity q0. Thus, tq can be defined as the phase relaxation time or phase-coherence time. Fig. 6 shows the evolution of a polaron with different phase-relaxation time tq and phase-breaking intensity q0 within a timescale of 2000 fs. The colors stand for the distribution of localized charge density and the position of the lattice distortion. Fig. 6(a) describes a perfect polaron without any phase-breaking. Note that the polaron keeps its electron and lattice distortion localized and coupled together. With the phase-breaking effect applied, the localization of the polaron changes. Fig. 6(b) describes the polaron evolution with weak phase-breaking. The polaron is still essentially localized but with a little dispersion. By fixing the phase relaxation time and increasing the intensity, we find that the polaron is severely affected. Relative to Fig. 6(b) and (c) is a comparison that a polaron evolution with tq ¼ 200fs but a stronger phase-breaking q0 ¼ p=8. The polaron nearly extends over the whole polymer chain. Moreover, with time going on, the dispersion get more serious. This indicates that the phase-breaking intensity can weaken the polaron stability. Stepping further, the phasebreaking intensity is fixed but the phase-relaxation time is reduced. Apparently, as seen from the comparison of Fig. 6(c) and
37
(d), with a shorter phase-relaxation time tq ¼ 100fs, the polaron becomes more extended within a shorter evolution time. Therefore, both the phase-breaking frequency and the phase-breaking intensity are unfavorable to the stability of a polaron. Although it has been known that a polaron is a localized object in organics, the complexity in actual materials renders the mechanism causing the localization still being controversial [26]. Anderson effect and electron-phonon interaction are the two widely accepted factors that localize polaron. The theoretical methods have tackled Anderson effect with either diagonal or nondiagonal disorder. Dynamical disorder caused by temperature are also expressed by the random force on lattice sites, or electron Fermi distribution. These theoretical results are in well agreement with experimental observations, but the understanding is still incomplete. Electron phase information in organic semiconductors is speculated to be a key factor that influences the carrier itself and its transport, in which the electron-phonon coupling links the electron phase with the deformable lattice. We deduce that the experimental observation that mobility of an organic crystal decreases with impurity or temperature is partially determined by the electron phase-breaking originating from collisions with impurities and thermal activated phonons. Therefore, coherence transport generally appears in materials with high crystallization and low temperature environment. Material purification is an effective method to decrease phase-breaking to enhance coherent time and scattering length. From the adopted theoretical method, most present investigations treat lattice sites classically, such as the SSH model, TLM model and PPP model. There are also investigations based on the all-quantum description, such as the Holstein model [26e28]. The description of the electronic phase would be improved in allquantum description, in which the electronic wave phase and the acoustic wave phase would be considered simultaneously. But in this work, the behavior of electron is expressed adequately enough with the semi-classical method. Additionally, it should be noticed that beside the phase-breaking, complex environment and various collisions also cause other effects. For instance, different molecular vibration modes in real space have a direct and significant impact
Fig. 6. Dynamics of polaron dispersion with different intensities of random phase-breaking q0 and different phase-relaxation times tq .
38
R. Meng et al. / Organic Electronics 49 (2017) 33e38
on the interactions in reciprocal space [29,30]. This molecular vibration modes can be expressed by symmetric and antisymmetric electron-phonon coupling patterns, which reveal the different mechanisms on charge-transport properties of organic semiconductors. Therefore, more factors and modifications should be taken into consideration in the future work to complete the physics in organic electronics. 4. Conclusions In this paper, based on the tight binding SSH model, we theoretically studied the phase-breaking effect on a polaron and its transport in a conjugated polymer. Two kinds of phase-breaking approaches are considered. One is the modification on transfer integral. In this case, a single phase-breaking can trap a polaron. While the continuous regular phase-breaking leads to polaron delocalization, it also increases the velocity of transport. This phenomenon may originate from the increase of transfer integral and the reduced effect of electron-lattice coupling. Additionally, the disordered phase-breaking can make electron and lattice to be dispersive. The other approach is the addition of phase-breaking on wave-function. In this case, the disordered phase-breaking is also found to disperse the polaron, which proves the detrimental effect on the stability of polaron. We use different phase-relaxation times and phase-breaking intensities to describe the polaron evolution process, and find that a smaller phase-relaxation time and larger intensity are unfavorable to the stability of a polaron. Finally, we deduce that the decrease of mobility in experiments on organics is partially determined by the electron phase-breaking and provide a possible method to enhance coherence time and length. Acknowledgments This work is supported by the National Natural Science Foundation of the People's Republic of China (Grant No. 11574180) and 111 Project B13029. References das, [1] V. Coropceanu, J. Cornil, D.A. da Silva Filho, Y. Olivier, R. Silbey, J.L. Bre Charge transport in organic semiconductors, Chem. Rev. 107 (4) (2007) 926e952. [2] A. Troisi, G. Orlandi, Charge-transport regime of crystalline organic semiconductors: diffusion limited by thermal off-diagonal electronic disorder, Phys. Rev. Lett. 96 (8) (2006) 086601. [3] R. Meng, K. Gao, G. Zhang, S. Han, F. Yang, Y. Li, S. Xie, Exciton intrachain transport induced by interchain packing configurations in conjugated polymers, Phys. Chem. Chem. Phys. PCCP 17 (28) (2015) 18600e18605. [4] Liu Yang, Ding Yi, Shixuan Han, Shijie Xie, Theoretical investigation on thermal effect in organic semiconductors, Org. Electron. 23 (2015) 39e43. [5] Y.L. Zhang, X.J. Liu, Z. An, Temperature effects on the dynamics of photoexcitations in conjugated polymers, J. Phys. Chem. C 118 (2014) 2963e2969.
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Organic Electronics journal homepage: www.elsevier.com/locate/orgel
Phase-breaking effect on polaron transport in organic conjugated polymers Ruixuan Meng a, Sun Yin a, Yujun Zheng a, Liu Yang a, Shijie Xie a, *, Avadh Saxena b a b
School of Physics, State Key Laboratory of Crystal Materials, Shandong University, Jinan, 250100, China Los Alamos National Laboratory, NM 87545, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 April 2017 Received in revised form 7 June 2017 Accepted 11 June 2017 Available online 15 June 2017
Despite intense investigations and many accepted viewpoints on theory and experiment, the coherent and incoherent carrier transport in organic semiconductors remains an unsettled topic due to the strong electron-phonon coupling. Based on the tight-binding Su-Schrieffer-Heeger (SSH) model combined with a non-adiabatic dynamics method, we study the effect of phase-breaking on polaron transport by introducing a group of phase-breaking factors into p-electron wave-functions in organic conjugated polymers. Two approaches are applied: the modification of the transfer integral and the phase-breaking addition to the wave-function. Within the former, it is found that a single site phase-breaking can trap a polaron. However, with a larger regular phase-breaking a polaron becomes more delocalized and lighter. Additionally, a group of disordered phase-breaking factors can make the polaron disperse in transport process. Within the latter approach, we show that the phase-breaking can render the delocalized state in valence band discrete and the state in the gap more localized. Consequently, the phase-breaking frequency and intensity can reduce the stability of a polaron. Overall, the phase-breaking in organic systems is the main factor that degrades the coherent transport and destroys the carrier stability. © 2017 Elsevier B.V. All rights reserved.
Keywords: Polaron Electronic structures Phase Localized states Organic semiconductor
1. Introduction Energy migration and carrier transport are important current topics in organic semiconductors and devices. Although the organic conductance has been effectively improved and the organic devices, such as organic solar cells (OSC) and organic light emitting diodes (OLED), have been successfully fabricated, some basic physical phenomena in organic electronics are still controversial. For example, there is no consistent conclusion on the organic transport mechanism whether band-like transport or hopping is the dominant mechanism. This controversy is recently shown to relate closely to the coherence or incoherence of the transport mechanism. The specific nature of organic semiconductors on physical and structural basis determines their transport performance. In traditional inorganic semiconductors and metals, the rigid periodic lattice structure makes the electronic state behave in the form of a Bloch wave. The band-like transport mechanism dominates, which leads to a high mobility. Simple lattice structure and the absence of
* Corresponding author. E-mail address: [email protected] (S. Xie). http://dx.doi.org/10.1016/j.orgel.2017.06.026 1566-1199/© 2017 Elsevier B.V. All rights reserved.
electron-phonon coupling make the effective transport easily understood. While in organic semiconductors, the strong electronphonon coupling causes the carriers or excited states to couple with the soft lattice resulting in the formation of spatially localized self-trapped states, such as polarons, bipolarons or excitons. These quasi-particles are heavier and move slower than the usual electrons and holes. As a result, the mobility is adversely affected even in organic crystals. Moreover, the actual material microstructure is not as perfect as one would expect. Besides electron-lattice interaction, there also exist the structural diversity, impurities and thermal effects. All these factors complicate the transport mechanism. Up to now, much work has been done to study carrier transport in organic molecular crystals or polymers [1]. In 2006, Troisi and Orlandi used a one-dimensional semiclassical model to compute charge carrier mobility in the presence of thermal fluctuations of the electronic Hamiltonian. This transport mechanism explains several contrasting experimental observations pointing sometimes to a delocalized ‘‘band-like’’ transport and sometimes to the existence of strongly localized charge carriers [2]. The effects of structural diversity [3] and impurities are also considered as the diagonal or non-diagonal disorder, and the thermal effect has been modeled by dynamic disorder exerting a random force on lattice
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R. Meng et al. / Organic Electronics 49 (2017) 33e38
sites [4,5]. The comprehensive effect of these factors leads to the inferior transport performance of organic semiconductors. From what is stated above, we can infer that the poor mobility should be intrinsic to organic materials. However, two promising facts are worth considering. One is the high mobility at low temperature. The negative temperature coefficient of mobility is the fingerprint of the band-like transport mechanism. In one experiment, Sakanoue and Sirringhaus found that in the crystalline pentacene or polymer at low temperature, the mobility is high and “thermally deactivated”, e.g. it decreases with rising temperature [6]. Early investigation by Palstra and coworkers also showed that the mobility of crystal pentacene decreased with temperature and impurity [7]. This phenomenon should be attributed to degeneration of band-like or coherent transport caused by thermal lattice fluctuations. The other fact is the observation of room-temperature coherence. In 2009, Scholes observed efficient coherent transport in organic semiconductors at room temperature [8,9], which offered the possibility of efficient transport of charge and energy in the organic material. It is considered that coherent transport dominated at short times if the polymer chains were protracted, while incoherent transport dominated if the chains were folded seriously. Cho et al. experimentally measured electron transport properties during an ultrashort time and found the transition from coherent transport to incoherent transport [10]. Recently, Takeya et al. reduced molecular thermal vibrations by pressurizing the material and also experimentally obtained the corresponding result [11]. As the intermolecular spacing decreases the coupling is enhanced, which leads to the electron transport from incoherent to band transport in the organic crystal. These observations indicate that coherence supports effective transport and the structural disorder and thus the phase disorder should be responsible for the reduction of mobility. Impurities or defects are easily created in the organic samples; thermal effect is apparent for organic materials with light atoms. Moreover, the electron-electron interaction is also an important source of phase-randomizing collisions. For effective carrier transport and energy migration, investigations on the coherent mechanism and keeping track of phase information are necessary. Rapid progress has been made in studying coherent transport. Scholes and coworkers proposed some methods to define coherence in theory [12], such as the relative coupling strength theory. In experiment, they also observed the cross peak in two dimensional electronic spectroscopy [13], implying the existence of a coherent process. These theoretical descriptions can well present the coherence phenomenon in organic semiconductors and have played a guiding role in experiments, but the analysis of intrinsic phase information of the wave-function which is the root cause of all the coherence phenomena has been obviously lacking. In 2012, Yao et al. considered the decoherence process in organic amorphous semiconductors by introducing a decoherence time, and how the type of transport is changed from coherent to incoherent [14]. With Monte Carlo simulation, they obtained the dependence of diffusion coefficient on decoherence time, which is consistent with some experimental observations [15]. In many previous works on p-conjugated molecules or polymers, an atomistic simulation dynamic approach has been widely used to describe the dynamic behaviors of carriers in organic semiconductors, such as polaron transport and dissociation, polaron collisions and exciton production [16e20]. In these investigations, the phases of the electronic states are strictly determined by the Hamiltonian. From the perspective of coherence, the electronic phase is supposed to be reserved, which means a coherent transport. Obviously, this kind of treatment is deficient when involving phase-breaking processes. In this paper, we consider this question in the context of phase conservation. In a highly conjugated polymer, as the main carrier, a
polaron is a combination of an electron (hole) and the lattice distortion. How the phase of the p-electron wave-function affects the polaron should be considered by invoking the electron-phonon coupling. Consequently, this paper is organized as follows. The model and method are described in the next section. A qualitative analysis is also given in this section. Then in section three we present calculations on a polaron and its dynamics based on two phase-breaking mechanisms we proposed. Finally in section four, we discuss the phase-breaking effect and conclude that phasebreaking is unfavorable to carrier stability and its transport in the materials with band-like transport mechanism. 2. Model and method In our previous works, we have studied polaron motion, dissociation and collisions. The behavior of a polaron (bipolaron or exciton) is closely related to the electron-phonon coupling. For a conjugated polymer chain described by the SSH model, the evolution of the electronic wave-functions and the lattice configuration is determined by Refs. [3,21,22].
iZ
v 4 ðt Þ ¼ tn1 4m;n1 tnþ1 4m;nþ1 eðna þ un Þ4m;n vt m;n
Mu€n ðtÞ ¼ fl þ fe þ fE
(1) (2)
with the elastic force, electron-phonon coupling force and external electric field force separately given by
fl ¼ Kðunþ1 2un þ un1 Þ
(2a)
fe ¼ a rnþ1;n þ rn;nþ1 rn;n1 rn1;n
(2b)
fE ¼ eE rn;n 1
(2c)
where tn ¼ t0 aðunþ1 un Þ. All the notations have the normal meaning as in previous reference [23]. Normally, the non-adiabatic dynamic process is expressed by the solution of the coupled de€ dinger rivative equations consisting of the time-dependent Schro equation and the Newton's equation of motion. We first prescribe an initial system state, of which the band structure and wavefunction are determined. Starting with the initial state, the electron wave-function evolves following the time-dependent €dinger's equation. Meanwhile, due to the electron-phonon Schro coupling, the lattice motion is controlled by the driving force in Eq. (2), which is partially determined by the electron state at previous time step. Therefore, the updated lattice structure determines the electron state of the next time step. With increasing time, the evolving electron wave-function and the lattice motion condition at every time step are determined. P * In Eq. (2b), the density matrix is given by rm;n ¼ OCC m 4m;m 4m;n, from which we can discern the importance of phase. For the diagonal terms, rn;n means the electronic density at site n. But for the P * off-diagonal terms, 4m;n ¼ 4m;n eiqm;n , rm;n ¼ OCC m 4m;m 4m;n eiðqm;n qm;m Þ , the effect of phase is important. From the expression of the electron-phonon coupling term, we see that this force is sensitive to the phase difference between the neighboring sites. Therefore, if the phase is altered by some physical factors, it is expected that the lattice configuration or the carrier would be changed. It should be noticed that for usual rigid materials, the electron-phonon coupling a is very weak, and in this case the electron-phonon coupling force has been neglected.
R. Meng et al. / Organic Electronics 49 (2017) 33e38
35
We use two approaches to introduce the phase-breaking. One is 0 to modify the transfer integral tn into tn þ itn . This modification is equivalent to applying a spatially varying magnetic field to the electronic states [24]. It will affect the phase of the electronic states [25]. Another approach is to modify the phase of the electronic states at each step while solving the coupled equations (1) and (2) self-consistently. For step k, electronic state 4m;n ðtk Þ is obtained by
solving Eq. (1), then we substitute 4m;n ðtk Þ for 4m;n ðtk Þeiqm;n into Eq. (2) to give the next-step lattice configuration un ðtkþ1 Þ. Here, qm;n means the phase-breaking on the mth state at site n. We repeat this process until an expected result is achieved. In our calculations, we choose polyacetylene as the representative of organic conjugated polymers. Note that all the parameters have been obtained via DFT and from experiment. They are 2
t0 ¼ 2:5eV, a ¼ 4:1eV=Å, K ¼ 21:0eV=Å , M ¼ 1349:14eV$fs =Å , and a ¼ 1:22Å. 3. Results and discussions For the first kind of modification, the electronic Hamiltonian becomes,
He ¼
X n
0 þ tn þ itn Cnþ1 Cn þ h:c:
(3)
tn where the spin index s is omitted. Let cos qn ¼ pffiffiffiffiffiffiffiffiffiffi , we have 02 2 t n þt n
ffi X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 þ He ¼ Cn þ h:c: t 2n þ t n2 eiqn Cnþ1 n
¼
X n
þ tn eiqn Cnþ1 Cn þ h:c:
(4)
Pn1 i q We take the transformation for the operator, Cn ¼ an e j¼1 j and Pn Pn1 i j¼1 qj i q þ iqn . Cnþ1 Cn ¼ aþ an e j¼1 j ¼ aþ Then Eq. (4) nþ1 e nþ1 an e becomes
He ¼
X n
tn aþ nþ1 an þ h:c:
0
Fig. 1. Dependence of polaron width on the phase-breaking parameter tn.
2
2
(5)
Equation (5) has a similar form similar to that without the modification. But the transfer integral changes from tn to tn . If we consider the displacement unþ1 un as perturbation, we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 tn ¼ t 2n þ t n2 ztn0 ðt0 =tn0 Þaðunþ1 un Þ, where tn0 ¼ t 20 þ t n2 . Compared to the original SSH model, we can assess that phasebreaking will alter the effect of electron-phonon coupling. As t0 =tn0 1, a polaron tends to become more delocalized. We define 2 4 P P the width of a polaron as, w ¼ ð 4p;n Þ2 = 4p;n (in units of
polaron. Supposing there is a phase-breaking only at one site, 0 denoted as tc , we find that the polaron tends to locate at this site and its creation energy decreases with the phase-breaking strength (see Fig. 2). It seems that phase-breaking acts as an energy well and is not favorable for the polaron motion. Through systematic dynamical calculations, the single phase-breaking is found to have the trapping influence on polaron transport, similar to that of an impurity or a conjugation defect. However, if we calculate the dynamics of a polaron under a driving electric field, it is found that the continuous phase-breaking 0 may give a different result. Here we set tn as a constant, that is, there exists the same phase-breaking between every site. As shown in Fig. 3, the polaron moves faster with a larger continuous phasebreaking. This phenomenon may result from the adopted model of equation (3). To introduce a desired phase-breaking, we modify the transfer integral. Although this method indeed breaks phase conservation, at the same time, the transition rate increases from tn to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 tn ¼ t 2n þ t n2 . We also find that a sufficiently strong disordered phasebreaking will result in the dispersion of a polaron. To this end, a group of random phase-breaking, following the standard normal 0 distribution t Nð0; sÞ, is applied to the polymer chain, where s represents the degree of disorder. Fig. 4 depicts the evolution of the lattice displacement yn ¼ ð1Þn ðunþ1 þ un1 2un Þ=4 and charge 2 P density dn ¼ occ 4m;n with different intensity of the disordered m
phase-breaking. At the beginning, there is a charged polaron centered at site n ¼ 60. With an external driving electric field gradually applied, the polaron starts to move along the polymer chain. If there is no phase-breaking along the polaron transport routine, the electron wavefunction phases can keep regular. The
n
lattice constant a), where jp ¼
P n
n 4p;n n > is the highest occupied
state of the polymer with a polaron. If the state is confined to only one site, 4p;n is a delta function and we have w ¼ 1, which means that the polaron is strictly localized; if the state is extended over the pffiffiffiffi whole chain, 4p;n ¼ 1= N and we have w ¼ N, which means that the polaron is not localized. It also implies that the lattice cannot confine the p-electron. We show a result in Fig. 1, where it is found that the polaron width increases with the phase-breaking param0 eter tn. It proves what we have expected so far. One aspect is that the transfer integral tn0 broadens the energy band width, rendering the polaron delocalized. On the other hand, the effect of electronlattice coupling decreases, and the ability that lattice distortion confines the p-electron subsides. Next, we consider the trapping effect of the phase on the
Fig. 2. Dependence of polaron creation energy on the single phase-breaking param0 eter tc.
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R. Meng et al. / Organic Electronics 49 (2017) 33e38
0
Fig. 3. Dependence of polaron velocity on the phase-breaking parameter tn. The driving field E ¼ 6 104 V=Å.
polaron moves perfectly with localized charge and lattice coupled together (Fig. 4(a)). If the phase-breaking intensity is s ¼ 0:2, a group of weak phase-breaking, it is found that the polaron is still robust during the transport process, although a little dispersion appears (Fig. 4(b)). It means that the phase-breaking is detrimental for the polaron stability. As the phase-breaking intensity is increased to s ¼ 0:4, as shown in Fig. 4(c), it is found that the distribution of charge density get disperse, and the lattice distortion become delocalized. In other words, the polaron dissociates quickly. In this case, the electronic states rapidly lose their initial phase information and cannot remain in a stable state. As a result, the random phase is not able to meet the requirement of coherent transport, and we have to describe the transport with incoherent hopping. Now let us consider the second approach to break the phase. In a real system, p-electrons and the lattice are strongly coupled together. To determine the most stable system state, we normally start from the Hamiltonian with a given lattice configuration to
Fig. 5. Dependence of the polaron localization on the phase-breaking intensity q0 .
obtain the electronic states. These electronic states in turn act on the lattice through the electron-phonon coupling. If the difference between successive lattice configurations in the iteration process is smaller than or within a given precision, we define it as the most stable system state. After taking phase-breaking into consideration, this self-consistent iteration is modified with a group of additional phase for each state. Therefore, the final state is the most stable state of this group of additional phase. Based on it, we next study the localization of a stationary polaron. The phase-breaking intensity on the wave-function is supposed to be distributed randomly in an interval qm;n 2½q0 ; q0 . The localization of the electronic state on the highest occupied state is defined as 4 P 2 P Pp ¼ 4p;n =ð 4p;n Þ2 , which is the reciprocal of the polaron n
n
width. Analogously, the localization of charge density and lattice 4 Pocc 2 2 P displacement are given by Рe ¼ occ m;n 4m;n =ð m;n 4m;n Þ and P P Рl ¼ jun j4 =ð jun j2 Þ2 . In a well extended polymer crystal, since n
n
the states in the valence band are completely delocalized, the local
Fig. 4. Dynamics of a polaron with phase-breaking intensity s ¼ 0 (a), s ¼ 0:2 (b) and s ¼ 0:4 (c). The driving field E ¼ 6 104 V=Å.
R. Meng et al. / Organic Electronics 49 (2017) 33e38
states in the gap can dominantly determine the polaron formation. In other words, the behavior of polaron is consistent with that of the localized state. However, with disorder considered, the situation may be different. Fig. 5 shows the dependence of these quantities on phase-breaking intensity. We find that with the phase-breaking intensity, the highest occupied state becomes localized, but the polaron charge density and lattice distortion tend to be delocalized. The behavior of the polaron is inconsonant with that of the localized states. This indicates that the states in the valence band are not completely delocalized and also contribute to localization degree. Further, let us consider the dynamic process with phasebreaking. We assume that the phase-breaking happens after every time interval tq . During one time interval, the phase information is normally evolved without any destruction and keeps its initial phase relation. But at the end of every interval, all the states will suffer a random phase-breaking with an intensity q0. Thus, tq can be defined as the phase relaxation time or phase-coherence time. Fig. 6 shows the evolution of a polaron with different phase-relaxation time tq and phase-breaking intensity q0 within a timescale of 2000 fs. The colors stand for the distribution of localized charge density and the position of the lattice distortion. Fig. 6(a) describes a perfect polaron without any phase-breaking. Note that the polaron keeps its electron and lattice distortion localized and coupled together. With the phase-breaking effect applied, the localization of the polaron changes. Fig. 6(b) describes the polaron evolution with weak phase-breaking. The polaron is still essentially localized but with a little dispersion. By fixing the phase relaxation time and increasing the intensity, we find that the polaron is severely affected. Relative to Fig. 6(b) and (c) is a comparison that a polaron evolution with tq ¼ 200fs but a stronger phase-breaking q0 ¼ p=8. The polaron nearly extends over the whole polymer chain. Moreover, with time going on, the dispersion get more serious. This indicates that the phase-breaking intensity can weaken the polaron stability. Stepping further, the phasebreaking intensity is fixed but the phase-relaxation time is reduced. Apparently, as seen from the comparison of Fig. 6(c) and
37
(d), with a shorter phase-relaxation time tq ¼ 100fs, the polaron becomes more extended within a shorter evolution time. Therefore, both the phase-breaking frequency and the phase-breaking intensity are unfavorable to the stability of a polaron. Although it has been known that a polaron is a localized object in organics, the complexity in actual materials renders the mechanism causing the localization still being controversial [26]. Anderson effect and electron-phonon interaction are the two widely accepted factors that localize polaron. The theoretical methods have tackled Anderson effect with either diagonal or nondiagonal disorder. Dynamical disorder caused by temperature are also expressed by the random force on lattice sites, or electron Fermi distribution. These theoretical results are in well agreement with experimental observations, but the understanding is still incomplete. Electron phase information in organic semiconductors is speculated to be a key factor that influences the carrier itself and its transport, in which the electron-phonon coupling links the electron phase with the deformable lattice. We deduce that the experimental observation that mobility of an organic crystal decreases with impurity or temperature is partially determined by the electron phase-breaking originating from collisions with impurities and thermal activated phonons. Therefore, coherence transport generally appears in materials with high crystallization and low temperature environment. Material purification is an effective method to decrease phase-breaking to enhance coherent time and scattering length. From the adopted theoretical method, most present investigations treat lattice sites classically, such as the SSH model, TLM model and PPP model. There are also investigations based on the all-quantum description, such as the Holstein model [26e28]. The description of the electronic phase would be improved in allquantum description, in which the electronic wave phase and the acoustic wave phase would be considered simultaneously. But in this work, the behavior of electron is expressed adequately enough with the semi-classical method. Additionally, it should be noticed that beside the phase-breaking, complex environment and various collisions also cause other effects. For instance, different molecular vibration modes in real space have a direct and significant impact
Fig. 6. Dynamics of polaron dispersion with different intensities of random phase-breaking q0 and different phase-relaxation times tq .
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R. Meng et al. / Organic Electronics 49 (2017) 33e38
on the interactions in reciprocal space [29,30]. This molecular vibration modes can be expressed by symmetric and antisymmetric electron-phonon coupling patterns, which reveal the different mechanisms on charge-transport properties of organic semiconductors. Therefore, more factors and modifications should be taken into consideration in the future work to complete the physics in organic electronics. 4. Conclusions In this paper, based on the tight binding SSH model, we theoretically studied the phase-breaking effect on a polaron and its transport in a conjugated polymer. Two kinds of phase-breaking approaches are considered. One is the modification on transfer integral. In this case, a single phase-breaking can trap a polaron. While the continuous regular phase-breaking leads to polaron delocalization, it also increases the velocity of transport. This phenomenon may originate from the increase of transfer integral and the reduced effect of electron-lattice coupling. Additionally, the disordered phase-breaking can make electron and lattice to be dispersive. The other approach is the addition of phase-breaking on wave-function. In this case, the disordered phase-breaking is also found to disperse the polaron, which proves the detrimental effect on the stability of polaron. We use different phase-relaxation times and phase-breaking intensities to describe the polaron evolution process, and find that a smaller phase-relaxation time and larger intensity are unfavorable to the stability of a polaron. Finally, we deduce that the decrease of mobility in experiments on organics is partially determined by the electron phase-breaking and provide a possible method to enhance coherence time and length. Acknowledgments This work is supported by the National Natural Science Foundation of the People's Republic of China (Grant No. 11574180) and 111 Project B13029. References das, [1] V. Coropceanu, J. Cornil, D.A. da Silva Filho, Y. Olivier, R. Silbey, J.L. Bre Charge transport in organic semiconductors, Chem. Rev. 107 (4) (2007) 926e952. [2] A. Troisi, G. Orlandi, Charge-transport regime of crystalline organic semiconductors: diffusion limited by thermal off-diagonal electronic disorder, Phys. Rev. Lett. 96 (8) (2006) 086601. [3] R. Meng, K. Gao, G. Zhang, S. Han, F. Yang, Y. Li, S. Xie, Exciton intrachain transport induced by interchain packing configurations in conjugated polymers, Phys. Chem. Chem. Phys. PCCP 17 (28) (2015) 18600e18605. [4] Liu Yang, Ding Yi, Shixuan Han, Shijie Xie, Theoretical investigation on thermal effect in organic semiconductors, Org. Electron. 23 (2015) 39e43. [5] Y.L. Zhang, X.J. Liu, Z. An, Temperature effects on the dynamics of photoexcitations in conjugated polymers, J. Phys. Chem. C 118 (2014) 2963e2969.
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