Dynamical simulations of polaron transport in conjugated polymers with the inclusion of electron-electron interactions
aa r X i v : . [ c ond - m a t . m t r l - s c i ] O c t Dynamical simulations of polaron transport in conjugatedpolymers with the inclusion of electron-electron interactions
Haibo Ma ∗ and Ulrich Schollw¨ock † Institute for Theoretical Physics C,RWTH Aachen University, D-52056 Aachen, Germany (Dated: Latest revised on October 25, 2018)
Abstract
Dynamical simulations of polaron transport in conjugated polymers in the presence of an exter-nal time-dependent electric field have been performed within a combined extended Hubbard model(EHM) and Su-Schrieffer-Heeger (SSH) model. Nearly all relevant electron-phonon and electron-electron interactions are fully taken into account by solving the time-dependent Schr¨odinger equa-tion for the π -electrons and the Newton’s equation of motion for the backbone monomer displace-ments by virtue of the combination of the adaptive time-dependent density matrix renormalizationgroup (TDDMRG) and classical molecular dynamics (MD). We find that after a smooth turn-on ofthe external electric field the polaron is accelerated at first and then moves with a nearly constantvelocity as one entity consisting of both the charge and the lattice deformation. An ohmic region(3 mV/˚A ≤ E ≤ U =2.0 eV and V =1.0 eV. The maximal velocity is well abovethe speed of sound. Below 3 mV/˚A the polaron velocity increases nonlinearly and in high electricfields with strength E ≥ U will suppress the polaron transportand small nearest-neighbor interactions V values are also not beneficial to the polaronic motionwhile large V values favor the polaron transport. ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION Charge transport properties in conjugated polymers have attracted sustained attentionfrom both academic and industrial researchers, since it was discovered in the 1970s thatthe electrical conductivity of trans-polyacetylene (PA) can be improved significantly bydoping with strong electron acceptors or donors [1, 2, 3]. Many electronic devices have beenfabricated based on these conjugated polymers.[4, 5, 6, 7]All the conducting polymers can generally be sorted into two classes. The first classis trans-polyacetylene, in which there exists a twofold degeneracy of ground state energydistinguished by the positions of the double and single bonds. The degenerate ground state oftrans-polyacetylene leads to solitons [8] as the important excitations and the dominant chargestorage species, which take the form of domain walls separating different districts of oppositesingle and double bonds alternation patterns. The second class of conducting polymersis given by all the other systems except trans-polyacetylene, in which the ground statedegeneracy is weakly lifted. So polarons and confined soliton pairs (bipolarons) [8] insteadof solitons are the important excitations and the dominant charge storage configurationsfor this class of conducting polymers. Nowadays it has been widely accepted that thesequasiparticles (solitons, polarons and bipolarons) are the fundamental charge carriers inconducting polymers. The studies of the dynamics of these charge carriers are therefore ofgreat interest for the purpose of modulating or devising new organic electronic materialsbased on conjugated polymers.So far, there have been many extensive theoretical studies on polaron dynamics in conju-gated polymers [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] based on the Su-Schrieffer-Heeger(SSH) model [21, 22], an improved H¨uckel molecular orbital model, in which π -electronsare coupled to distortions in the polymer backbone by the electron-phonon interaction.Rakhmanova and Conwell [9, 10] have considered the motion of a polaron under low andhigh electric fields respectively by using an adiabatic simulation in which the electronic en-ergy is treated within the Born-Oppenheimer approximation. Their results show that thepolaron can form and move in its entity in low electric fields but dissociates in high fields.The highest electric field strength under which the polaron can sustain is about 6 mV/˚A.Basko and Conwell [11] have also analyzed the dynamics of a polaron and found that thespeed of a steadily moving polaron cannot exceed the sound speed ( V S ). These conclusions2re consistent with those from the numerical results presented by Arikabe et al [12] butinconsistent with those given by Johansson and Stafstr¨om [13, 14], who performed a nona-diabatic simulation in which transitions between instantaneous eigenstates are allowed. InJohansson and Stafstr¨om’s work, the maximum velocity (about 4 V S ) is reached at a highelectric field strength ( ∼ et al haveconsidered extended Hubbard model (EHM) combined with the SSH model and investigatedthe dynamics of polarons within this model at the level of an unrestricted Hartree-Fock(UHF) approximation.[23] They found that the localization of the polaron is enhanced andthe stationary velocity of the polaron is decreased by both the on-site repulsions U and thenearest-neighbor interactions V . Furthermore, they found that the local extremum of thestationary velocity of the polaron appears at U ≈ V . However, in recent years a lot of theo-retical calculations of the static properties of conjugated polymers have shown that the elec-tron correlation effect plays a very important role in determining the behavior of the chargecarriers in conjugated polymers.[24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]For example, electron correlation is essential in obtaining correct pictures of the spin polar-izations in trans-polyacetylene.[38] Therefore, theoretical simulations for polaron dynamicswith electron correlations considered, which step beyond the SCF level, are highly desirable.However, the calculations for large polymer systems with traditional advanced electron-correlation methods such as configuration interaction method (CI), multi-configuration self-consistent field method (MCSCF), many-body Moller-Plesset perturbation theory (MPn)and coupled cluster method (CC) are currently still not feasible due to the huge compu-tational costs. Fortunately, the adaptive time-dependent density-matrix renormalizationgroup (TDDMRG) method [41, 42, 43] can be used instead. In the context of 1D cor-related electronic and bosonic systems, the adaptive TDDMRG has been found to be ahighly reliable real-time simulation method at economic computational cost, for example inthe context of magnetization dynamics [44], of spin-charge separation [45, 46], or far-fromequilibrium dynamics of ultracold bosonic atoms [47]. Recently, Zhao et al have performedadaptive TDDMRG simulations for polaronic transport within a combined model of SSHand Hubbard model to take the on-site Coulomb interactions into account and found that3hat the velocity of the polaron is suppressed by the on-site Coulomb interaction U .[48]In this paper, combining the SSH and extended Hubbard model to take both the on-site Coulomb interactions and nearest-neighbor electron-electron interactions into account,we simulate the motion of a polaron in conjugated polymers under an applied externalelectric field by using the adaptive TDDMRG for the π -electron part and classical moleculardynamics (MD) for the lattice backbone part. The aim of this paper is to give an exhaustivepicture of polaron transport in conducting polymers at a theoretical level with all relevantelectron-electron interactions and correlations included and to show how the on-site Coulombinteractions U and nearest-neighbor electron-electron interactions V influence the behaviorof polaron transport in conducting polymers. II. MODEL AND METHODOLOGY
We use the well-known and widely used SSH Hamiltonian [21, 22] combined with theextended Hubbard model (EHM), and include the external electric field by an additionalterm: H ( t ) = H el + H E ( t ) + H latt (1)This Hamiltonian is time-dependent, because the electric field E ( t ) is explicitly time-dependent.The π -electron part includes both the electron-phonon and the electron-electron interac-tions, H el = − X n,σ t n,n +1 ( c + n +1 ,σ c n,σ + h.c. )+ U X n,σ ( c + n,σ c n,σ −
12 )( c + n, − σ c n, − σ −
12 )+ V X n,σ,σ ′ ( c + n,σ c n,σ −
12 )( c + n +1 ,σ ′ c n +1 ,σ ′ −
12 ) (2)where t n,n +1 is the hopping integral between the n -th site and the ( n + 1)-th site, while U is the on-site Coulomb interaction and V denotes the nearest-neighbor electron-electroninteraction. Because the distortions of the lattice backbone are always within a certainlimited extent, one can adopt a linear relationship between the hopping integral and thelattice displacements as t n,n +1 = t − α ( u n +1 − u n ) [21, 22], where t is the hopping integral4or zero displacement, u n the lattice displacement of the n th site, and α is the electron-phonon coupling.Because the atoms move much slower than the electrons, we treat the lattice backboneclassically with the Hamiltonian H latt = K X n ( u n +1 − u n ) + M X n ˙ u n , (3)where K is the elastic constant and M is the mass of a site, such as that of a CH monomerfor trans-polyacetylene.The electric field E ( t ) directed along the backbone chain is uniform over the entire system.The field which is constant after a smooth turn-on is chosen to be E ( t ) = E exp[ − ( t − T C ) /T W ] , for t < T C , E , for t ≥ T C , (4)where T C , T W and E are the center, width and strength of the half Gaussian pulse. Thisfield gives the following contribution to the Hamiltonian: H E ( t ) = | e | X n,σ ( na + u n )( c + n,σ c n,σ −
12 ) E ( t ) (5)where e is the electron charge and a is the unit distance constant of the lattice. The modelparameters are those generally chosen for polyacetylene: t =2.5 eV, α =4.1 eV/˚A, K =21eV/˚A , M =1349.14 eVfs /˚A , a =1.22 ˚A.[22] The results are expected to be qualitativelyvalid for the other conjugated polymers. The values of T C and T W ( T C =30 fs and T W =25fs) are taken from the paper of Fu et al .[18]For the purpose of performing real-time simulation of both the evolution of quantum π -electron part and the classical movement of the chain backbone, we adopt a newly developedreal-time simulation method in which classical molecular dynamics is combined with theadaptive TDDMRG. The main idea of this method is to evolve the π -electron part by theadaptive TDDMRG and move the backbone part by classical MD iteratively. Details aboutthis method can be found in recent papers [48, 49]. III. RESULTS AND DISCUSSION
To investigate the dynamic properties of polarons, we simulate the polaron transportprocess in a single model conjugated polymer chain under a uniform external electric field.5 ne t c ha r ge site index n t=0 fst=40 fst=80 fst=120 fst=160 fs FIG. 1: Time evolution of net charge distribution in a polymer chain with the transport of apolaron under an external electric field. (The polaron moves from the left to the right.) ( E =3.0mV/˚A, U=2.0 eV, V=1.0 eV) A polaron is different from a charged soliton in that it involves an unpaired electron andthus a nonzero S = 1 /
2. In our calculations, we simulate a model chain containing N = 100monomers and N e = 99 π -electrons to present a polaron defect without degenerate groundstates. The polaron is initially centered around site 30 and site 31 through imposing aconstraint of reflection symmetry around the center between site 30 and site 31. Then, thedynamics of polaron transport in 160 femtoseconds (fs) is simulated by virtue of classicalMD combined with the adaptive TDDMRG. A. General picture of polaron transport
Firstly let us show the general time evolution picture of polaron transport in a conductingpolymer. The time evolution of charge density and the staggered bond order parameter r n = ( − n (2 u n − u n +1 − u n − ) / r n / A ng s t r o m site index n t=0 fst=40 fst=80 fst=120 fst=160 fs FIG. 2: Time evolution of the staggered bond order parameter r n in a polymer chain with thetransport of a polaron under an external electric field. (The polaron moves from the left to theright.) ( E =3.0 mV/˚A, U=2.0 eV, V=1.0 eV) distortions don’t localize at one monomer. On the contrary, the polaron defect spreads overa delocalized region with a length of tens of sites. One can also clearly see that during theentire time evolution process the geometrical distortion curve and net charge distributionshape for the polaron defect always stay coupled and show no obvious dispersion. Thisimplies that the polaron defect is an inherent feature and the fundamental charge carrier inconducting polymer. In Fig. 2, we can also see that a long lasting oscillatory “tail” appearsbehind the polaron defect center. This “tail” is generated by the inertia of those monomersto fulfill energy and momentum conservation; they absorb the additional energy, preventingthe further increase of the polaron velocity after a stationary value is reached.In order to evaluate the velocity of polaron transport, it is necessary to know the centerpostions of a polaron at different times. The center position of a defect can be derived from7
30 35 40 45 50 55 60 65 0 20 40 60 80 100 120 P o l a r on c en t e r po s i t i on Time(fs) chargelattice order
FIG. 3: Temporal evolution of the center position of a polaron under an external electric fieldderived from both the charge density and the bond order parameter. ( E =3.0 mV/˚A, U=2.0 eV,V=1.0 eV) the charge density distribution picture or the bond length alternation pattern as x c,g ( t ) = N θ c,g ( t ) / π, if cos θ c,g ≥ θ c,g ≥ N ( π + θ c,g ( t )) / π, if cos θ c,g < N (2 π + θ c,g ( t )) / π, otherwise, (6)where θ c,g is defined according to the charge density ρ n or the staggered bond order parameter r n as θ c ( t ) = arctan P n ρ n ( t )sin(2 πn/N ) P n ρ n ( t )cos(2 πn/N ) ,θ g ( t ) = arctan P n ( r n ( t ) − r n )sin(2 πn/N ) P n ( r n ( t ) − r n )cos(2 πn/N ) , (7)in which r is the dimerized lattice displacement of the pristine chain without defects.From Fig. 1 and Fig. 2, one can also find that, after the external electric field is smoothlyturned on, the polaron is accelerated at first and then moves with a nearly constant velocityas one entity consisting of both the charge and the lattice deformation. We calculate thepostions of the center of a polaron during the transport process according to Eq. 6 and showthe temporal evolution of the polaron center position in Fig. 3. It can be clearly seen thatthe center position of the polaron increases nearly linearly with the increase of time after theinitial short-time acceleration. This implies the polaron defect is transported with a nearlyconstant velocity and supports our observation from Fig. 1 and Fig. 2. Therefore, only8 v p ( A ng s t r o m /f s ) E (mV/Angstrom) FIG. 4: The stationary velocity of a polaron v p as a function of the electric field strength. (U=2.0eV, V=1.0 eV) the stationary velocity will be considered for polaron transport in the following discussions.Meanwhile, one also finds that the lattice distortion center always stays well together with thecharge density center. But it should also be noticed that, under high electric field strengththe deviation of the lattice distortion center position from the charge density center positionmay become much larger because the charge density moves much faster under high fieldand the lattice distortion cannot catch up with the charge density. This separation betweenthe charge center and lattice distortion center may lead to the dissociation of a polaron.The dissociation of a polaron under high electric field strength will be discussed in the nextsection. B. Relationship between polaron transport velocity and electric field strength
The dependence of the stationary velocity of a polaron v p on the electric field strength isshown in Fig. 4. We find that v p increases with increasing electric field strength. Apparently,the polaron velocity can easily exceed the sound speed ( V S = p K/M a/ V S ) is reached at a high electric field strength ( ∼ v p as a function of the electric field strength isobserved. An ohmic region where v p increases nearly linearly with the electric field strengthis found. This ohmic region extends approximately from 3 mV/˚A to 9 mV/˚A for the caseof U =2.0 eV and V =1.0 eV. Below 3 mV/˚A the polaron velocity increases nonlinearly andwhen E ≥
10 mV/˚A the polaron becomes unstable and the charge will be decoupled from9
30 40 50 60 70 80 90 100 110 0 20 40 60 80 100 120 P o l a r on c en t e r po s i t i on Time(fs) charge(E =1 mV/Angstrom)lattice order(E =1 mV/Angstrom)charge(E =3 mV/Angstrom)lattice order(E =3 mV/Angstrom)charge(E =9 mV/Angstrom)lattice order(E =9 mV/Angstrom)charge(E =10 mV/Angstrom)lattice order(E =10 mV/Angstrom) FIG. 5: Temporal evolution of the center position of a polaron under different external electricfield strengths. (U=2.0 eV, V=1.0 eV) the lattice distortion.We illustrate the temporal evolutions of the polaron center postion under different exter-nal electric field strengths. We can find that the lattice distortion center stays well togetherwith the charge center under low electric field strengths. However, the deviation of thelattice distortion center from the charge density center becomes much larger under higherelectric field strengths because the charge density moves much faster under high field andthe lattice distortion cannot catch up with the charge density. Under the electric field of10 mV/˚A, the distance between the two centers becomes even larger than several unit dis-tances, implying that the polaron has been completely dissociated. This dissociation canbe clearly seen from the time evolution pictures of charge density and the staggered bondorder parameter in a polymer chain with a polaron defect under a external electric field of10 mV/˚A in Fig. 6 and Fig. 7. While the charge density accelerates to a high speed as shownin Fig. 6, the original lattice distortion stays far behind the charge density with a distanceof more than tens of unit distances. The charge density will induce new lattice distortionin the nearby region around the charge density center, while the original lattice distortioncan hardly move further, as illustrated in Fig. 7. Therefore, the polaron defect has becomecompletely dissociated under the high electric field. Apparently, our calculated threshold10 ne t c ha r ge site index n t=0 fst=40 fst=80 fst=120 fs FIG. 6: Time evolution of net charge distribution in a polymer chain with the transport of apolaron under a high external electric field. (The polaron moves from the left to the right.)( E =10.0 mV/˚A, U=2.0 eV, V=1.0 eV) field strength of around 10 mV/˚A for polaron dissociation is in agreement with the value of9.5 mV/˚A obtained by SSH-EHM/UHF calculations by Di et al [23] and much closer to theexperimental determined value of about 15 mV/˚A [50] compared to the value of 6 mV/˚Acalculated by Rakhmanova and Conwell [9, 10] and 3.5 mV/˚A calculated by Johansson andStafstr¨om [13, 14]. The large deviations of the calculated threshold field strength from theexperimental value by the latter two groups are due to the fact that they considered onlythe SSH model and considered no electron-electron interactions in their studies, and theagreement of SSH-EHM calculations with experimental results shows again that the inclu-sion of electron-electron interactions is vital for the theoretical studies of charge carriers inconducting polymers. Another thing that should be noticed is that our U value (2.0 eV)and V value (1.0eV) are not the standard values for any special kind of polymers. Herewe use these two values only for the purpose to show how a polaronic transport will gen-erally evolve and whether the inclusion of electron-electron interactions is important for areasonable theoretical simulation of the polaronic transport process. The detailed discussion11 r n / A ng s t r o m site index n t=0 fst=40 fst=80 fst=120 fs FIG. 7: Time evolution of the staggered bond order parameter r n in a polymer chain with thetransport of a polaron under a high external electric field. (The polaron moves from the left to theright.) ( E =10.0 mV/˚A, U=2.0 eV, V=1.0 eV) of how the on-site repulsions U and the nearest-neighbor interactions V will influence thepolaronic transport will be shown in the next sections. C. Influence of on-site repulsions U on polaron transport velocity In order to study the influence of electron-electron interactions on polaron transport, wefocus on the stationary velocity of a polaron v p calculated with different electron-electroninteraction strengths under the constraint that the other parameters are fixed. As shouldbe noticed that, the real conjugated polymer is with weak interactions. Strong interactionswill lead to too strong charge polarizations which are unrealistic. Considering that we areonly focusing on the study of real conjugated polymer system, in this work we adopt onlythe weak-interaction parameters ( U < t, V < t ).Firstly, we study the Hubbard model with only on-site Coulomb interactions U , i.e. , V = 0 in the EHM. The dependence of the polaron stationary velocity v p on the on-site12 v p ( A ng s t r o m /f s ) U (eV)
FIG. 8: The stationary transport velocity of a polaron as a function of different U values. ( E =3.0mV/˚A, V =0.0 eV) r n / A ng s t r o m site index n U=0.0 eVU=1.5 eVU=4.0 eV FIG. 9: The staggered bond order parameter r n of a static polaron for several different U values.( V =0.0 eV) Coulomb interactions U is displayed in Fig. 8. We find that v p decreases monotonically withthe increasing U value and its maximum value is achieved at U =0.0 eV. This can be easilyunderstood because the variation of the charge carrier transport velocity is strongly relatedto the delocalization level of the charge carrier defect. The lattice tends to be occupiedby one electron per site when the on-site Coulomb repulsion U increases. Therefore anincreasing U will lead to more localized charge density and a smaller defect width of the13olaron and accordingly restrain the polaron transport. This analysis is supported by ourcalculated charge density width values of a static polaron W c . W c is calculated according tothe following formula: W c = ( X n [ n − X c ] ρ n ) / . (8)Through calculations we find the polaron width W c also decreases monotonically with anincreasing U value, from 10.10 ( U =0.0 eV) to 9.68 ( U =1.5 eV) and then to 7.62 ( U =4.0 eV).This sequence is in agreement with that of the polaron stationary velocity v p and supportsour analysis of the relationship between the polaron transport velocity and the delocalizationlevel of a polaron. Actually, the change of the polaron delocalization can also be directlyviewed through the geometrical picture of the static polaron defect. In Fig. 9, the staggeredbond order parameter r n of a static polaron calculated by different U values is shown. Itcan be clearly seen that the lattice dimerization is enhanced and the polaron width becomesnarrower while the U value increases. All these pictures show that the polaron transport issuppressed by the on-site Coulomb repulsions U . The monotonic decrease picture of polarontransport velocity with on-site Coulomb repulsions U observed by us is in agreement withDi et al ’s UHF calculations [23] and Zhao et al ’s recent adaptive TDDMRG results [48], butnot in accordance to our recent studies on charged soliton transport, in which we found thatcharged soliton transport velocity is non-monotonic in U [49]. The different behaviors ofcharge carrier transport with U increasing is due to the different characteristics of chargedsolitons and polarons. It was found that the polaron defect is more delocalized than thecharged soliton defect and the height of the charge density peaks in polarons is only roughlyone half that of the charge density peaks in charged solitons.[39] In a more localized casewith large charge densities, as in a charged soliton, small U may favor the charge carriertransport because the increasing on-site repulsion for large charge densities can make theelectron (or hole) to hop more easily to the next site. Therefore the different behaviors ofcharge carrier transport with U increasing for charged solitons and polarons are reasonable. D. Influence of nearest-neighbor interactions V on polaron transport velocity Secondly, we also study the influence of nearest neighbor electron-electron interactions V on the polaron transport process. The dependence of the stationary velocity of a polaron v p on the V values is displayed in Fig. 10, where U values are supposed to be frozen to be zero.14 v p ( A ng s t r o m /f s ) V (eV)
FIG. 10: The stationary transport velocity of a polaron as a function of different V values. ( E =3.0mV/˚A, U =0.0 eV) -0.05 0 0.05 0.1 0.15 10 20 30 40 50 c ha r ge den s i t y site index n V=0.0 eVV=1.2 eVV=1.4 eV FIG. 11: The charge density of a static polaron as a function of the site index calculated withdifferent V values. (U=0.0 eV) This assumption is not realistic because the on-site Coulomb interactions U are normallymuch stronger than the neighbor electron-electron interactions V ; we make this assumptiononly for the purpose of studying of the influence of V on the polaronic transport processwithout the effect of U . As can be seen in Fig. 10, the situation is completely differentfrom the case of an on-site Coulomb interaction. Interestingly, v p is non-monotonic in V : itdecreases to a shallow minimum at V ≈ . U , the variation of v p with V is alsostrongly related to changing the delocalization of the polaron defect. For small V , larger15 r n / A ng s t r o m site index n V=0.0 eVV=1.2 eVV=1.4 eV FIG. 12: The staggered bond order parameter r n of a static polaron for several different V values.( U =0.0 eV) positive charge densities will be induced in the central part of a polaron defect while theinduced negative charge densities at nearest neighbor sites are still relatively small, as shownin Fig. 11. Therefore the polaron defect tends to be more localized and consequently thepolaron transport becomes more difficult. So, v p decreases while the value of V increasesfrom 0.0 eV to 1.2 eV as displayed in Fig. 10. However, when nearest neighbor electron-electron interactions V increase further and become dominant, very large charge polarizationwill be induced. Therefore, the electron-hole attraction between opposite charge densitiesat nearest-neighbor sites will contribute much more significantly to the polaron system andfavor the hoppings of the accumulated electrons (or holes) in the central part of a polarondefect to the neighbor sites. As shown in in Fig. 11, the two positive charge density peaksare separated further when V increases from 1.2 eV to 1.4 eV. This change leads to amore delocalized polaron defect and accordingly the increase of v p . Our calculated polaroncharge density width results, namely that W c decreases gradually from 10.10 ( V =0.0 eV)to 9.37 ( V =1.2 eV) and then increase gradually to 9.89 ( V =1.4 eV), support our analysiswell. In order to directly view the change of delocalization level of the polaron defect throughgeometrical pictures, we also show the staggered bond order parameter r n of a static polaroncalculated with different V values in Fig. 12. It is clearly shown that the lattice dimerization16 v p ( A ng s t r o m /f s ) U(eV) V=0.2 eVV=0.5 eVV=1.0 eVV=1.5 eV
FIG. 13: The stationary transport velocity of a polaron as a function of U for fixed values of V .( E = 3 . is enhanced and the polaron width becomes narrower with V for small V . We also find thatthe polaron becomes more delocalized with a smaller r n minimum while V increases furtherfrom 1.2 eV to 1.4 eV. The change of polaron delocalization illustrated from the geometricalpicture is in accordance to that has been shown in the charge density picture, verifyingthat small V suppresses the polaronic transport while large V favors the polaron transport.This non-monotonic picture of polaron transport velocity in V is similar to that has beenobserved in charged soliton transport [49]. E. Influence of both the on-site Coulomb interactions U and the nearest-neighborinteractions V on polaron transport velocity Furthermore, we consider the realistic case of conducting polymers in which both the on-site Coulomb interactions U and the nearest-neighbor interactions V are taken into account.Fig. 13 shows the stationary transport velocity of a polaron v p as a function of U for fixedvalues of V = 0 . V = 0 . V =1.0 eV and V = 1 . v p behaviorwith variated U and fixed V values is quite similar to that has been illustrated in Fig. 8 inwhich V is frozen to be 0.0 eV. It can be found that v p decreases monotonically with U .17eanwhile, we can also find that, v p seems to decrease more rapidly in large V cases than insmall V cases for small U . Larger nearest-neighbor electron-electron interactions V will leadto larger charge polarizations and of course the increasing on-site Coulomb repulsions U willsuppress the transport of a polaron associated with larger charge polarizations more signif-icantly comparing to the case with smaller charge polarizations. Therefore the suppressionof polaron transport by the on-site Coulomb repulsions U is more remarkable in large V cases than in small V cases. The monotonic decrease picture of polaron transport velocitywith on-site Coulomb repulsions U observed by us is opposed to the non-monotonic pictureobtained by Di et al through UHF calculations [23]. They found a local extremum of thestationary velocity of the polaron can be achieved at U ≈ V . The difference between theresults of the adaptive TDDMRG and UHF calculations is due to the fact that the lattermethod ignores the important electron correlation effect. Previous theoretical static studieshave found that neglecting electron correlation effects will lead to an underestimation ofthe defect width as well as an overestimation of charge polarization.[34, 35, 36, 37] In amore localized case with large charge densities, presented by UHF calculations without theinclusion of electron correlation effect, introducing a small U may favor the charge carriertransport because the increasing on-site repulsion for large charge densities can make theelectron (or hole) to hop more easily to the next site. Therefore, the difference between TD-DMRG calculations and UHF calculations shows again that the electron correlation effectplays an important role in describing the charge carrier properties. IV. SUMMARY AND CONCLUSION
For a model conjugated polymer chain initially holding a polaron defect, described usingthe combined SSH-EHM model extended with an additional part for the influence of anexternal electric field, we have studied the dynamics of polaron transport through this chainby virtue of simulating the backbone monomer displacements with classical MD and evolvingthe wavefunction for the π -electrons with the adaptive TDDMRG.It is found that after a smooth turn-on of the external electric field the polaron is ac-celerated at first up to a stationary constant velocity as one entity consisting of both thecharge and the lattice deformation. During the entire time evolution process the geomet-rical distortion curve and charge distribution shape for the polaron defect always stay well18oupled and show no dispersion under low external electric fields, implying that the polarondefect is an inherent feature and the fundamental charge carrier in conducting polymer. Thedependence of the stationary velocity of a polaron v p on the external electric field strengthis also studied, and an ohmic region where v p increases linearly with the field strength isfound. Values beyond the speed of sound are achievable. This ohmic region extends ap-proximately from 3 mV/˚A to 9 mV/˚A for the case of U =2.0 eV and V =1.0 eV. Below 3mV/˚A the polaron velocity increases nonlinearly and in high external electric field strengthsthe polaron will become unstable and dissociate. Our calculated threshold field strength forpolaron dissociation is around 10 mV/˚A for the case of U =2.0 eV and V =1.0 eV. It is ingood agreement with experimental results and more physically intuitive than previous SSHcalculations which take no electron-electron interactions into account.The influence of electron-electron interactions (both the on-site Coulomb interactions U and the nearest-neighbor interactions V ) on polaron transport are investigated in detail.In general, the increase of the on-site Coulomb interactions U makes the lattice tend to beoccupied by one electron per site and accordingly suppress the polaron transport. Therefore, v p decreases monotonically with U . Meanwhile, small V values are not beneficial to thepolaronic motion because they induce a more localized defect distribution, and due to theinduced large charge polarization accompanied with large nearest-neighbor attractions large V values favor the polaron transport. When U and V are considered at the same time, v p also decreases monotonically with the increasing U value and v p decreases more rapidly forsmall U in large V cases than in small V cases. Acknowledgment
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