Bloch oscillations in a one-dimensional organic lattice

PHYSICAL REVIEW B 74, 184303 共2006兲

Bloch oscillations in a one-dimensional organic lattice Yuan Li, Xiao-jing Liu, Ji-yong Fu, De-sheng Liu,* Shi-jie Xie, and Liang-mo Mei School of Physics and Microelectronics, Shandong University, Jinan 250100, China and State Key Laboratory of Crystal Materials, Shandong University, Jinan 250100, China 共Received 6 April 2006; revised manuscript received 3 September 2006; published 3 November 2006兲 We present a model study on the dynamics of electron transport of a dissociated polaron under a strong uniform electric field in a one-dimensional organic lattice. The simulations are performed within the framework of the Su-Schrieffer-Heeger model coupled by a Newtonian equation of motion with a nonadiabatic evolution method. It is found that the dissociated polaron propagates in the form of a freelike electron and performs spatial Bloch oscillations 共BO’s兲. In contrast to normal BO’s in a rigid lattice, the mean displacement of the oscillating electron will have a net forward movement in the direction of the electric field, which is dependent on the strong electron-lattice couplings in the organic lattice. It is also found that a transient polaron forms at the end of each period of BO’s, accompanying the appearance of discrete polaron levels in the gap of the band. The effects of electron-electron interactions and bond disorder in the organic lattice are briefly discussed. DOI: 10.1103/PhysRevB.74.184303

PACS number共s兲: 71.38.⫺k, 72.80.Le, 72.90.⫹y

I. INTRODUCTION

The intriguing properties and wide variety of applications of organic conducting materials, from optoelectronic devices to molecular electronics and organic spin electronics, have attracted the increasing interest of many researchers. One of the most important characteristics of organic conducting system is the strong electron-lattice coupling.1 It has been well known that the predominant carriers in the nonmetallic phase of conjugated polymers are self-trapping excitations, for example solitons 关only in trans-polyacetylene 共tPA兲兴1 and polarons,2 resulting from the strong electron-lattice couplings. There have been many detailed studies on the dynamics of soliton3–5 and polaron6–10 transport in polymer chains under the influence of moderate or weak external uniform electric fields. For the polaron, it is shown that there exists an upper critical electric field over which a preexisting polaron will dissociate due to the charge moving faster than the lattice atoms, e.g., 106 V / cm in Ref. 6 by Rakhmanova and Conwell, or about 4 ⫻ 105 V / cm in Ref. 8 by Johansson and Stafström. In the study of polaron relaxation in Ref. 6, an even lower value for the critical field is presented in which injected charge in the presence of fields over 6 ⫻ 104 V / cm does not induce the creation of a polaron because the charge is moving too fast for the lattice distortion to occur. A similar value is also reported by Wu et al.10 in a study of charges injection in a metal/polymer junction, where they found that the injected charges could not form polarons under electric fields 艌104 V / cm. Therefore, it can be expected that the carriers in organic conducting polymers under strong electric fields are, at least theoretically, free charges other than selftrapping excitations. Experimental studies also demonstrate that the extreme enhanced conductance in conducting polymers at strong electric fields of about 106 – 107 V / cm is due to the contribution from the generation of free carriers.11,12 The recently discovered negative differential resistance 共NDR兲 in polymeric and molecular electronic devices13–15 also shows that free charges play an important role in this intriguing phenomenon. Therefore, it is important to under1098-0121/2006/74共18兲/184303共9兲

stand the role of free charges, especially the microcosmic dynamic behavior. To our knowledge, however, the behavior of free charges, especially ones derived from the dissociation of polarons under strong electric fields, has scarcely been mentioned to date in such organic conducting systems. Bloch oscillations 共BO’s兲, a quantum phenomenon described as temporal and spatial oscillations of electrons in a periodic lattice potential subject to a uniform electric field, were first theoretically predicted by Bloch16 and Zener.17 The period ␶B and total spatial amplitude L of the oscillation can be tuned by the applied electric field according to

␶B = h/eFd

共1兲

L = ⌬/eF,

共2兲

and

where d is the period of lattice, F is the applied electric field, e is the electron charge, ⌬ is the width of the band in which the electron propagates, and h is Planck’s constant. BO’s have been intensively studied both experimentally18–23 and theoretically24–28 during the past decade after its first experimental observation in a semiconductor superlattice.29 Up to now, nearly all of the experimental studies of electronic BO’s have been performed in semiconductor superlattice materials, because the repetitive unit of the superlattice is large enough to make the Bloch period shorter than the electron dephasing time, within which the coherence of the electron is not easily broken by the scattering of defects and phonons.30 In addition, motivated by the concepts introduced for electrons in crystals, many other analogues of electronic BO’s are also reported, e.g., BO’s of Bose-Einstein condensates 共BEC兲 in an optical potential,31,32 BO’s of optical waves in transport through periodic dielectric systems,33 and magnetic BO’s in nanowire superlattice rings subject to a timedependent magnetic flux threading the ring.34 Up to now, however, there have not been any reports on BO’s in organic conducting systems. It is well known that organic conducting materials, usually as one-dimensional organic lattice systems, are excellent candidates for use as

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LI et al.

semiconductors in electronics. Many of their electronic properties are extremely similar to those of traditional inorganic semiconductors.35 We conjecture that many intriguing electronic phenomena of inorganic semiconductors, e.g., BO’s, may also exist in organic conducting materials. In this paper, we focus on a one-dimensional organic lattice with strong electron-lattice couplings and have a model study on the propagation of a freelike electron derived from the dissociation of a preexisting polaron under a high uniform electric field. The dynamics is described by a time-dependent Schrödinger equation coupled by a Newtonian equation of motion. The paper is organized as follows. The methodology is introduced in Sec. II and the main results and discussions are presented in Sec. III. Finally, a summary and concluding remarks are given in Sec. IV.

Mu¨n = − K共2un − un+1 − un−1兲 + e−i␥A共t兲␣关␳n+1,n共t兲 − ␳n,n−1共t兲兴 + ei␥A共t兲␣关␳n,n+1共t兲 − ␳n−1,n共t兲兴 − ␭Mu˙n , where the density matrix ␳n,m共t兲 is given by

␳n,m共t兲 = 兺 ⌽*v共n,t兲f v⌽v共m,t兲

H = He + Hlatt

共3兲

† cn + ei␥A共t兲c†ncn+1兴, He = − 兺 tn关e−i␥A共t兲cn+1

共4兲

n

K M Hlatt = 兺共un+1 − un兲2 + 兺 u˙2n , 2 n 2 n

共5兲

where un is the lattice displacement from its undimerized equilibrium position of site n, c†n 共cn兲 is the creation 共annihilation兲 operator of an electron at site n, K is the elastic constant between neighboring sites, M is the mass of a site, and the coefficient ␥ is taken as ␥ = ea / បc, with e being the electron charge, a the average lattice spacing, c the velocity of light, and the electric field F共t兲 is proportional to the time derivative of the time-dependent vector potential A共t兲 as F共t兲 = −⳵tA共t兲 / c. The transfer integral tn between sites n and n + 1 reads tn = t0 − ␣共un+1 − un兲,

共6兲

where t0 represents the nearest-neighbor transfer integral for an undimerized lattice, and ␣ is the electron-lattice coupling constant. Here the electron-electron 共e-e兲 interactions are neglected. The time evolution of electronic state 兩⌽v共t兲典 depends upon the time-dependent Schrödinger equation iប⳵t兩⌽v共t兲典 = He兩⌽v共t兲典,

共7兲

and the development of lattice displacements is classically described by the following Newtonian equation of motion in the mean-field approximation:36

共9兲

v

and ⌽v共n , t兲 = 具n兩⌽v共t兲典 is the projection of electronic state 兩⌽v共t兲典 on the Wannier state of site n, and f v denotes the time-independent occupation function of state 兩⌽v共t兲典共f v being 0,1,2 and solely determined by the initial occupation兲. In Eq. 共8兲, a damping term is introduced to dissipate the extra kinetic energy of the lattice by a tuning parameter ␭.9 For any given time, the electronic state 兩⌽v共t兲典 can be expanded on the basis of instantaneous eigenstates

II. METHODOLOGY

We consider a one-dimensional organic lattice with strong electron-lattice couplings, which can be well described by the well-known Su-Schrieffer-Heeger 共SSH兲 Hamiltonian.1 By introducing an external uniform electric field in the vector potential gauge, which is available for periodic boundary conditions,30 we express the Hamiltonian in the following form:5

共8兲

兩⌽v共t兲典 = 兺 Cv,␮共t兲兩␾␮共t兲典, ␮

共10兲

where Cv,␮共t兲 = 具␾␮共t兲兩 ⌽v共t兲典, and 兵兩␾␮共t兲典其 are solutions to the time-independent Schrödinger equation at that moment, He兩␾␮共t兲典 = ␧␮共t兲兩␾␮共t兲典,

共11兲

where the Hamiltonian is determined by the instantaneous lattice configuration. For our study, it is convenient to present the electronic level, i.e., the expectation value of Hamiltonian He at electronic state 兩⌽v共t兲典, which is given by 具␧v共t兲典 = 具⌽v共t兲兩He兩⌽v共t兲典 = 兺 C2v,␮共t兲␧␮共t兲. ␮

共12兲

Equation 共12兲 also contains information concerning the distribution of electronic state 兩⌽v共t兲典 among the instantaneous eigenstates, that is, nv共␮兲 = C2v,␮ ,

共13兲

where nv共␮兲 is defined as the distribution function of 兩⌽v共t兲典. In contrast to the “adiabatic dynamics,”3,4 the electronic distribution on the instantaneous eigenstates is allowed to change in the “nonadiabatic dynamics,”36 which plays an important role in our studies. The method used in solving the coupled differential equations 共7兲 and 共8兲 is the Runge-Kutta method of order 8 with step-size control,37 which has been widely used and proven to be an effective approach in the study of dynamics in organic conducting systems.8–10,36 The chain contains 300 sites and a periodic boundary condition is proposed. The dynamic evolution starts from a static organic lattice with 300 ␲ electrons doubly occupying the 150 levels of the valence band. An extra electron, corresponding to an excitation of a polaron, is inserted to occupy the 151st level at the bottom of the conduction band. For convenience in studying the transport properties, the polaron is purposely centered on the 75th site in the initial configuration. The lattice configuration is determined by minimizing the total energy of the system and is given by

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BLOCH OSCILLATIONS IN A ONE-DIMENSIONAL…

un+1 − un = −

2␣ 兺⬘␾␮共n兲␾␮共n + 1兲 K ␮ N

+

2␣ 兺兺⬘␾␮共m兲␾␮共m + 1兲, KN m=1 ␮

共14兲

where ␾␮共n兲 is the wave function of the ␮th eigenstate on site n, and N is the number of total sites. Hereafter, we will focus on the lattice configuration in a smoothed form, yn =

共− 1兲n 共2un − un+1 − un−1兲. 4

共15兲

The spatial motion of the electron from the dissociated polaron, which always occupies the electronic state 兩⌽151共t兲典, can be monitored by the electronic state distribution ␳n共t兲 and the centroid xc共t兲,5 which is easily calculated for the periodic boundary condition using the following equation:

冦

if 具cos ␪n典艌 0 and 具sin ␪n典艌 0,

N˜␪/2␲

xc共t兲 = N共␲ + ˜␪兲/2␲

if 具cos ␪n典 ⬍ 0,

N共2␲ + ˜␪兲/2␲ otherwise,

冧

共16兲

where ˜␪ = arctan 具sin ␪n典具cos ␪n典

共17兲

and 具cos ␪n典 = 兺 ␳n共t兲cos ␪n, n

具sin ␪n典 = 兺 ␳n共t兲sin ␪n , n

共18兲 where ␪n = 2␲n / N and the electronic state distribution is defined as

␳n共t兲 = 兩⌽151共n,t兲兩2 .

共19兲

To avoid numerical errors, the applied electric field is turned on smoothly in the form of a half Gaussian function, that is, F共t兲 = F0 exp关−共t − tc兲2 / tw2 兴 when 0 ⬍ t ⬍ tc and F共t兲 = F0 when t 艌 tc.10 For all the results presented below, we choose tc = 50 fs, tw = 25 fs, and a time step of ⌬t = 1 fs. III. RESULTS AND DISCUSSIONS

In this section, we present the results of the electron motion in a one-dimensional organic lattice subject to a high electric field. Without any loss of generality, the value of the parameters in the Hamiltonian is chosen as those generally used for tPA in the SSH model: t0 = 2.5 eV, ␣ = 4.1 eV/ Å, K = 21.0 eV/ Å2, a = 1.22 Å, and M = 1349.14 eV fs2 / Å2. However, the results obtained below are also expected to be qualitatively available for all one-dimensional organic conducting systems. The damping tuning factor is chosen as ␭ = 0.05 fs−1 and the extra energies of the lattice can be dissipated as soon as possible. First of all, we present in Fig. 1 the time evolution of the electronic state distribution ␳n共t兲 and the centroid xc共t兲 in

FIG. 1. 共Color online兲 Time evolution of 共a兲 electronic state distribution ␳n and 共b兲 its centroid xc at an electric field of F0 = 8.0⫻ 105 V / cm.

coordinate space under an electric field of F0 = 8 ⫻ 105 V / cm. It is clearly shown that the electronic state 兩⌽151共t兲典 propagates regularly in the form of a periodic oscillation in the real space. To clarify the electronic motion in detail, we present the lattice configuration and electronic state distribution at a few typical times in Fig. 2, and in Fig. 3 we show the time dependence of the instantaneous velocity v of the electron scaled by the sound velocity vs = a冑K / M 共Ref. 7兲 within the first period of the oscillation. It can be seen from Fig. 2共a兲 that at the beginning of the evolution, the electronic state 兩⌽151共t兲典 couples with the lattice distortion to form a polaron. But by t = 50 fs, the polaron has already completely dissociated, which clearly shows the decoupling of electronic state 兩⌽151共t兲典 from the lattice distortion. A relevant study has shown that the maximum velocity of a polaron is only about four times the sound velocity.8 Thus, the great increase in the instantaneous velocity of the electron at about t = 25 fs 共see Fig. 3兲 also manifests the dissociation of the polaron. In the direction of the electric field, the freelike electron propagates in the form of a wave packet, which is more extended than the polaron state. An optical localized lattice vibration from the dissociation of the polaron is left behind 关see Fig. 2共b兲兴 and is soon removed by the damping, and the lattice configuration ultimately resumes a dimerized pattern. After about one circle motion in the periodic chain, the electron decelerates to zero 关see Fig. 2共c兲兴 and then continues its motion in the reverse direction 关see Fig. 2共d兲兴. At t = 222 fs 关see Fig. 2共e兲兴, the first period of the oscillation

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FIG. 2. Snapshots of the electronic state distribution ␳n 共open circles兲 and the smoothed lattice configuration y n 共solid circles兲 of the organic lattice at different times under an electric field of F0 = 8.0⫻ 105 V / cm. The dotted arrows denote the direction of the electron acceleration 共instantaneous velocity remains zero兲. The solid arrows denote the direction of the electron instantaneous velocity.

ends, and the electron continues to enter the next one. Hereafter, the electron repeats the same motion process and the oscillations are induced. It is noticed that in Fig. 2共e兲, due to the electron-lattice couplings 共which will be discussed in greater detail below兲, the envelope of the electronic wave packet is not so smoothe, and it directly affects the shape of the wave packet at the time of the next half period of the oscillation 关see Fig. 2共f兲兴. A similar oscillation also exists in the time evolution of the electronic level. In Fig. 4共a兲, we depict the time dependence of electronic levels 具␧151共t兲典 and 具␧150共t兲典, which are known as gap levels induced by the static polaron at the initial time. The figure shows that 具␧151共t兲典 and 具␧150共t兲典 periodically scan through the entire conduction band and valence band, respectively, which reveals the electron transition among instantaneous eigenstates. This indicates that the oscillation of the electron exists both in real space and momentum space. We expect that the electron from the dissociated polaron may perform BO’s. More oscillations of the electron at different strength of electric fields, which are selected according to related theoretical and experimental studies,6,12 are shown in Fig. 5. The oscillation period and total spatial amplitude 共i.e., the maximum displacement of the electron in one direction兲 defined by Eqs. 共1兲 and 共2兲 indicate a propor-

FIG. 3. The time dependence of the instantaneous velocity of the electron within the first period of the oscillation under an electric field of F0 = 8.0⫻ 105 V / cm.

tional relationship to the reciprocal of the electric field, which agrees remarkably with the behavior of the electrons displayed in Fig. 5. For the dimerized lattice with a period of d = 2a, the period of BO’s obtained from Eq. 共1兲 at a field strength of F0 = 8.0⫻ 105 V / cm is ␶B = 212 fs, in good agreement with the average oscillation period of ¯␶ = 215 fs from the numerical solution. In addition, the instantaneous veloc-

FIG. 4. 共Color online兲 Time dependence of 共a兲 the electronic levels 具␧151共t兲典共red solid circles兲, 具␧150共t兲典共black open circles兲; and 共b兲 the instantaneous eigenlevels ␧151共t兲共red solid circles兲, ␧150共t兲共black open circles兲, and the occupation number of the instantaneous eigenstate 兩␾151共t兲典共blue solid squares兲 under an electric field of F0 = 8.0⫻ 105 V / cm.

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FIG. 5. Bloch oscillations of the electron under different strength of electric field. 共a兲 F0 = 5.0⫻ 105 V / cm, 共b兲 F0 = 1.0⫻ 106 V / cm, 共c兲 F0 = 1.5⫻ 106 V / cm, and 共d兲 F0 = 2.0⫻ 106 V / cm.

ity of the electron also follows the definition for the velocity of an electronic wave packet moving in the conduction band with dispersion E共k兲, i.e., ␷ = ⳵kE共k兲 / ប. Thus the electron indeed propagates as an electronic wave packet. Therefore, we arrive at the conclusion that an electron from the dissociated polaron in a one-dimensional organic lattice subject to a strong electric field can regularly perform BO’s. For a rigid lattice, the mean displacement of an electron through a period of BO’s can be obtained as follows: 具SB典 =

冕

␶B

0

␷dt = −

1 关E共k␶B兲 − E共k0兲兴 = 0. eF

tain the mean net forward displacement of the electronic wave packet centroid, i.e., the mean value of the difference between the forward displacement and the reverse one in each period, as about 15a. This raises the question of the source of this net forward displacement of oscillations. In Fig. 4共b兲, we present the time dependence of instantaneous eigenlevels ␧151共t兲 and ␧150共t兲, which are solutions to the instantaneous time-independent Schrödinger equation of the system 关Eq. 共11兲兴, and the occupation number of the eigenstate 兩␾151共t兲典. As mentioned above, only the electronic

共20兲

The zero displacement of electrons in a period of BO’s leads to the dynamical localization of electronic states in the electric field. However, the “softness” of the organic lattice has a different feature. We can see from Figs. 1 and 5 that the mean displacement of the electron in a period of BO’s is nonzero, typically different from BO’s in rigid materials.22 The net forward movement of the electron only occurs when the forward displacement is larger than the reverse one within a complete oscillation period. From Fig. 1共b兲, we ob-

FIG. 6. Time evolution of the smoothed lattice configuration y n under an electric field of F0 = 8.0⫻ 105 V / cm.

FIG. 7. Time dependence of 共a兲 the occupation number of the instantaneous eigenstates 兩␾151共t兲典, 兩␾152共t兲典, 兩␾153共t兲典, and 兩␾154共t兲典, and 共b兲 the instantaneous eigenlevel ␧151, ␧152, ␧153, and ␧154 at the end of the first period of BO’s. The electric field is F0 = 8 ⫻ 105 V / cm.

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FIG. 9. Time dependence of the instantaneous velocity of the electron in an organic and a rigid lattice, respectively, at the end of the first period of BO’s under an electric field of F0 = 8 ⫻ 105 V / cm.

FIG. 8. 共Color online兲 Time evolution of the smoothed lattice configuration y n in the formation and dissociation of the first transient self-trapping state. The unit of y n is angstroms and the electric field is F0 = 8.0⫻ 105 V / cm.

state 兩⌽151共t兲典 moves in the conduction band. Thus the contribution to the occupation of the eigenstate 兩␾151共t兲典 nearly completely comes from the state 兩⌽151共t兲典共the contribution from electrons in the valence band by Landau-Zener tunneling is extremely small, less than 10−3兲. Therefore the instantaneous occupation number of eigenstate 兩␾151共t兲典 equals the projection of 兩⌽151共t兲典 onto 兩␾151共t兲典, i.e., n151共151兲. The eigenlevels ␧151共0兲 and ␧150共0兲 at the initial time correspond to the gap levels induced by the generation of a polaron. The complete occupation of state 兩␾151共0兲典 indicates the presence of a static polaron state. However, the time dependence of ␧151共t兲 and ␧150共t兲 displays that in addition to the static polaron at the initial time, gap levels are also induced periodically in the course of BO’s. We can see from the evolution of 具␧151共t兲典 in Fig. 4共a兲 that the generation of gap levels occurs

when 具␧151共t兲典 is at the bottom of the conduction band, and simultaneously the occupation number of eigenstate 兩␾151共t兲典 is markedly enhanced. One may expect that self-trapping states may be created in these cases. In Fig. 6, we present the time development of the smoothed lattice configuration y n. The dark gray in the graph indicates the generation of evident lattice distortions deviating from other light gray in a dimerized pattern of about 0.04 Å. By comparing these distortions with the configuration of the polaron at the initial time, we conjecture that a series of polaronic self-trapping states are induced in the course of BO’s. To study these self-trapping states further, we focus on the first one that arrives at the end of the first period of the BO’s. The time dependence of instantaneous eigenlevels near the bottom of the conduction band, i.e., ␧151, ␧152, ␧153, and ␧154, and the occupation number of instantaneous eigenstates 兩␾151共t兲典, 兩␾152共t兲典, 兩␾153共t兲典, and 兩␾154共t兲典 are depicted in Fig. 7. At about t = 210 fs, when the electronic level 具␧151共t兲典 develops near the bottom of a conduction band, the occupation number of the four eigenstates, especially that of 兩␾151共t兲典, starts to increase markedly. As a result of the symmetry of eigenstate 兩␾151共t兲典, the occupation of 兩␾151共t兲典 will lead to a lattice distortion due to the strong electron-lattice couplings

FIG. 10. Snapshots of the distribution n151共␮兲 of electronic state 兩⌽151共t兲典 among the instantaneous eigenstates at different times. 共a兲, 共b兲 Organic lattice; 共c兲, 共d兲 rigid lattice. The electric field is F0 = 8 ⫻ 105 V / cm.

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FIG. 11. 共Color online兲共a兲 Time evolution of the electronic state distribution centroid xc共t兲 with the effects of e-e interactions at F0 = 8.0⫻ 105 V / cm; 共b兲 a partial enlarged drawing of panel 共a兲 near the end of the first half-period of BO’s; 共c兲 a partial enlarged drawing of panel 共a兲 near the middle of the eighth period of BO’s.

in the organic lattice. Simultaneously, the eigenlevel ␧151 starts to split off from the conduction-band edge and goes deep into the gap. This lowering of the eigenlevel ␧151 makes it more favorable for the occupation of electrons. Thus, after the time t = 225 fs, the occupation number of the other three eigenstates decreases and turns to favor 兩␾151共t兲典. The depth of ␧151 into the gap is closely correlated with the occupation number of the eigenstate 兩␾151共t兲典. However, for an electron in the form of an electronic wave packet, the projection of electronic state 兩⌽151共t兲典 onto eigenstate 兩␾151共t兲典 cannot be a unit, but instead is about half of an electron charge. It is easy to understand that the electron velocity at the bottom of the conduction band is so slow that the electron will perform a similar behavior to that of an injected static electron relaxing to a polaron. Figure 8 depicts the time evolution of the lattice configuration corresponding to the process above. It is shown

that an evident polaronic lattice distortion is induced and then breaks down quickly. An optical lattice vibration is also left and then removed by the damping, similar to the process of the dissociation of a static polaron 关see Fig. 2共b兲兴. As can be seen from the contour line in the bottom panel of Fig. 8, the lattice configuration y n of the chain is almost in a dimerized form of about 0.04 Å except for the polaronic lattice distortion. From Fig. 7共b兲, we may obtain that the time ␧151 takes to relax deepest into the gap is about 25 fs. According to our study, the time it takes for a static electron inserted into the lowest unoccupied molecular orbital of a neutral polymer tPA to relax to a polaron is about 50 fs. Therefore, although the maximum occupation number of the eigenstate 兩␾151共t兲典 is just about 0.5, which is only half of the value for an integrated polaron, it is purely a quantum effect and we can still regard the self-trapping state as a transient polaron. The presence of transient polaron states plays an important role in the net forward movement of the oscillation electron. Figure 9 displays the time dependence of the instantaneous velocity of the electron in the process of formation and dissociation of the first transient polaron state. For comparison, we also introduce a rigid lattice by freezing the evolution of site displacements and maintaining the lattice configuration in a dimerized pattern of 0.04 Å. As can be seen from Fig. 9, before t = 215 fs the electronic velocity in the two kinds of lattice systems accords well with each other. Thereafter, the occupation of eigenstate 兩␾151共t兲典 and the effect of electron-lattice couplings 关see Figs. 7共a兲 and 7共b兲兴 lead to the electronic velocity in the organic lattice deviating from that of the rigid one. It should be noticed that there exist two different phases for the electronic velocity above zero in the organic lattice: one is from about t = 225 to 245 fs in which the electronic velocity is lower than that in the rigid lattice, and the other is the reverse case from t = 245 to 260 fs. We now turn back to Fig. 7共b兲 and Fig. 8. Between t = 225 and 245 fs, the process of the formation and dissociation of the transient polaron state occurs. Although the transient polaron state is short-lived, the velocity of the transient polaron is lower compared to the freelike electron in the rigid lattice. After t = 245 fs, the charge of the transient polaron decouples from the lattice distortion and propagates nearly freely. Thus the electronic velocity after the dissociation of the transient polaron increases markedly. However, although the instantaneous electronic velocity is different for organic and rigid lattices, the total electronic displacement between t = 225 and 260 fs is nearly the same due to the compensation from each other in the two phases, which can be seen in the area below the two velocity curves. It seems that the displacement of electrons in an organic lattice is not affected by the electron-lattice couplings. In Fig. 10, we depict the distribution n151共␮兲 of the electronic state 兩⌽151共t兲典 among the instantaneous eigenstates. As mentioned above, the distribution n151共␮兲 actually corresponds to the occupation number of eigenstates in the conduction band. We can see from Figs. 10共a兲 and 10共c兲 that at t = 210 fs, just before the electronic wave packet develops into the bottom of the conduction band, the distribution n151共␮兲 is nearly the same for both organic and rigid lattices. However, after the complete dissociation of the transient polaron state at t = 260 fs,

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FIG. 12. Time evolution of the electronic state distribution centroid xc共t兲 at different bonddisordered chains under an electric field of F0 = 8.0⫻ 105 V / cm. 共a兲 te = 0.01 eV, 共b兲 te = 0.02 eV, 共c兲 te = 0.03 eV, and 共d兲 te = 0.04 eV.

the distribution n151共␮兲 in the organic lattice is markedly broadened due to the electron-lattice couplings 关see Figs. 10共b兲 and 10共d兲兴. This means that the distance between the centroid of n151共␮兲 and the eigenstate 兩␾300共t兲典, which can be seen as an effective bandwidth ⌬rig ⬘ , in the rigid lattice is smaller than ⌬org ⬘ in the organic lattice, that is, ⌬rig ⬘ ⬍ ⌬org ⬘ . According to Eq. 共2兲, the displacement of an electron in an organic lattice within the next half period of BO’s will be larger than that in a rigid lattice. The system considered here is one with strong electron-lattice interaction, and the purely electronic bandwidth can be strongly narrowed due to polaronic effects, as was pointed out by Hannewald et al.38 Thus, a net forward movement of the oscillation electron in the organic lattice is induced in the process of BO’s. It is well known that in most organic systems e-e interactions are important, and even as important as the electronlattice couplings to form the band gap. We give a simple calculation on the e-e interactions within the Hartree-Fock approximation by using the Hubbard model, He-e =

U V † † † † cn,s cn,scn,−s cn,−s + 兺 cn,s cn,scn+1,s⬘cn+1,s⬘ . 兺 2 n,s 2 n,s,s ⬘

共21兲 By setting the on-site repulsion U = 0 – 4.0 eV and the nearest-neighbor repultion V = U / 3, we present the simulation at F0 = 8.0⫻ 105 V / cm and find that there are almost no evident changes in the behavior of BO’s with the e-e interactions, as shown in Fig. 11共a兲. For the first period of the BO’s, a little decrease of the spatial oscillation amplitude, e.g., of the order of about three times the lattice constant for U = 4.0 eV, is induced, as shown in Fig. 11共b兲. This effect is directly induced by the slight narrowing of the width of the conduction band with the influence of e-e interactions. However, it is not the case for the next several periods of the BO’s. The spatial oscillation amplitude for U = 4.0 eV gradually increases and ultimately becomes larger, of the order of about four times the lattice constant, than that for U = 0 eV, as shown in Fig. 11共c兲. It may be induced by the effects of

e-e interactions on the electron-lattice couplings in the formation of transient polaron states at the end of each period of BO’s. Similar results are also obtained at other strengths of the electric field. It is expected that, at least in the Hubbard model with the Hartree-Fock approximation, the e-e interactions do not qualitatively affect the behaviors of BO’s in an organic lattice. In addition, it is generally accepted that disorder widely exists in many organic conducting materials. We introduce an off-diagonal disorder into the hopping integrals to simulate bond disorder in the organic lattice,36 tn = t0 − ␣共un+1 − un兲 + ␤nte ,

共22兲

where ␤n is a random variable of standard normal distribution, i.e., the variance ␴2 = 1 and the mean value of ␤n is zero. te ranges from 0 to 0.04 eV. With one group of random numbers in a given strength te, we draw the time evolution of the electron in Fig. 12 under an electric field of F0 = 8.0 ⫻ 105 V / cm. The behaviors of BO’s depend markedly on the degree of bond disorder in the organic lattice. It is found that, for te 艌 0.02 eV, the centroid xc appears to oscillate randomly after some time, which means that the coherence of the electron in the process of BO’s is broken by the bond disorder. The BO’s breaks down quickly with the increase of the disorder, as shown in Figs. 12共c兲 and 12共d兲. Therefore, it can be concluded that bond disorder is not favorable to the formation of electronic BO’s in an organic lattice. IV. SUMMARY AND CONCLUDING REMARKS

In summary, we presented a model study on the dynamics of an electron derived from the dissociation of a preexisting polaron in a one-dimensional organic lattice at a high uniform electric field. The simulations were performed by employing a time-dependent Schrödinger equation to describe the evolution of electrons and a Newtonian equation of motion to describe the development of lattice sites with a nonadiabatic dynamic approach. Under the high electric field, a preexisting polaron in the organic lattice cannot survive and

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breaks down quickly. The freelike electron from the dissociated polaron propagates in the form of an electronic wave packet. BO’s of the electron are found in the onedimensional organic lattice. Unlike the BO’s in rigid materials, the strong electron-lattice couplings in the organic lattice induce a net forward movement of the oscillation electron. It was found that, even under a high electric field in which a preexisting polaron cannot survive, there still exist transient polaron states in the organic lattice. The formation and dissociation of the transient polaron states broaden the distribution of the electronic wave packet among the instantaneous eigenstates, which plays an important role in the net forward movement of the electron in the process of BO’s. A brief discussion on the effects of e-e interactions and bond disorder on the BO’s was presented. It was found that the behaviors of BO’s are not sensitive to the e-e interactions. As the bond disorder destroys the period of the lattice potential, it is found the BO’s will vanish if the disorder is strong. Therefore, a bond disorder is not favorable to form electronic BO’s in an organic lattice. It should be mentioned that we treat the lattice only classically within the mean-field approximation,

*Corresponding author. Electronic address: [email protected] 1

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and the quantum phonon effect is not included. In this study, the dynamics starts from a preexisting polaronic wave packet, and the electronic BO’s are excited from the dissociation of the polaron. As the excitation energy of BO’s referring to the binding energy of the polaron, about 0.21 eV,10 is larger than the zero-point fluctuation energy of the optical modes ប␼ = 0.16 eV, where ␼ = 冑4K / M is the bare optical phonon frequency, the classical treatment of the dynamics of lattice modes is suitable to some extent. The study of BO’s with phonons treated fully quantum mechanically is left as an open problem in this paper and still awaits further work. ACKNOWLEDGMENTS

Financial support from the National Natural Science Foundation of the People’s Republic of China 共Grants No. 90403110, No. 10574082, No. 10474056, and No. 50323006兲 and the Natural Science Foundation of Shandong Province 共Grant No. Z2005A01兲 is gratefully acknowledged. We are grateful to T. D. Corrigan for editing the manuscript.

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