Decoherence and Energy Relaxation in the Quantum-Classical Dynamics for Charge Transport in Organic Semiconducting Crystals: an Instantaneous Decoherence Correction Approach
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Decoherence and Energy Relaxation in the Quantum-Classical Dynamics forCharge Transport in Organic Semiconducting Crystals: an InstantaneousDecoherence Correction Approach
Wei Si and Chang-Qin Wu
1, 2, a) State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433,China Collaborative Innovation Center of Advanced Microstructures, Fudan University, Shanghai 200433,China (Dated: 12 September 2018; Revised)
We explore an instantaneous decoherence correction (IDC) approach for the decoherence and energy relaxationin the quantum-classical dynamics of charge transport in organic semiconducting crystals. These effects,originating from environmental fluctuations, are essential ingredients of the carrier dynamics. The IDC iscarried out by measurement-like operations in the adiabatic representation. While decoherence is inherent inthe IDC, energy relaxation is taken into account by considering the detailed balance through the introductionof energy-dependent reweighing factors, which could be either Boltzmann (IDC-BM) or Miller-Abrahams(IDC-MA) type. For a non-diagonal electron-phonon coupling model, it is shown that the IDC tends toenhance diffusion while energy relaxation weakens this enhancement. As expected, both the IDC-BM andIDC-MA achieve a near-equilibrium distribution at finite temperatures in the diffusion process, while theEhrenfest dynamics renders system tending to infinite temperature limit. The resulting energy relaxationtimes with the two kinds of factors lie in different regimes and exhibit different dependence on temperature,decoherence time and electron-phonon coupling strength, due to different dominant relaxation process.
I. INTRODUCTION
The advances of organic field-effect devices in re-cent years have inspired renewed research interest inthe charge transport problems in organic semiconductingcrystals.
Recent studies focus on the intrinsic regimeof charge transport, featured by observation of bothband-like and hopping-like properties in the same de-vice under different operating conditions.
Besides, theunderlying electronic states are proved to be localizedboth theoretically and experimentally. It is realizedthat the coupling between carriers, which is denoted aselectrons for simplicity throughout the paper, and lat-tice vibrations (phonons) plays an essential role in thecharge transport process. Further, in many cases, the en-ergy scales of the electron-phonon coupling, temperatureand intermolecular transfer integral overlap, which ren-ders the problem in the intermediated parameter regime,where the traditional pictures may not be effective. Forexample, Troisi and Orlandi proposed that the carriersin such materials are no longer localized by the self-trapping mechanism in polaron problems. Rather, itis due to dynamic disorder brought by phonons at fi-nite temperature. Many theoretical works have beencontributed to this topic.
Among them, the se-ries of studies using quantum-classical dynamics, whichtreat the electronic system quantum-mechanically andthe phonons classically, are promising ones.
They of-fer a time-dependent view of the transport process, en-abling the calculation of quantities such as the transient a) Email: [email protected] conductivity. Computationally, they are more efficientthan the full quantum-mechanical methods and are con-venient to be tuned for realistic materials. For example,the detailed crystal and electronic structures extractedby experiments and first-principle calculations can be in-put as model parameters for multi-scale simulations. Besides, existing studies on chemical dynamics providevaluable resources for the improvement of those meth-ods. For example, Wang et al. successfully generalizedthe surface hopping approach to such systems, reveal-ing an interesting crossover between band-like and hop-ping behavior.Despite the successes of the quantum-classical dynam-ics, there are still some inconsistencies in the resultingcarrier dynamics. For example, in the pioneering work ofTroisi et al. by Ehrenfest dynamics, the form of evo-lution of the electronic system is quantum-mechanicallycoherent. The electron wave functions keep expand-ing without a well-defined localization length, whichdoes not coincide with experimental findings. . Thiscould be ascribed to the lack of explicit treatment ofdecoherence. In the charge transport process, theelectronic system (electron) can be seen as the centralsystem. In realistic situations, it would interact withmany other species (environment), especially the domi-nant phonon degrees of freedom that are treated classi-cally here. Together, they form a system-environmentarchitecture. One direct consequence for the systemfrom coupling with environment is the destruction ofphase coherence within the system, i.e. , decoherence, which is essential for the consistency of quantum-classicaldynamics. The incorporation of decoherence is indi-cated or proposed in several further studies. Fratiniet al. introduced a relaxation time approximation, inwhich the information of initial dynamics are used andthe long-time dynamics are corrected. Yao et al. intro-duced the instantaneous decoherence correction (IDC) bymeasurement-like operations in the lattice site represen-tation, through which band-like temperature dependenceof mobility and localized electronic states coexist. Also,in the surface hopping studies of Wang et al., decoher-ence corrections are included by collapsing the electronicstates to the active states at the surface hopping events. Besides decoherence, another prominent process due tothe system-environment coupling is the energy relaxation(dissipation), by which the system absorbs/dissipatesenergy from/to the environment and relaxes to the ther-mal equilibrium with the environmental temperature. This process is essential for the interpretation of tran-sient experiments, such as the pump-probe ones.
However, there are signs that it is not treated properlyin some forms of quantum-classical dynamics. For ex-ample, if the effect of phonon motion is taken to be astochastic potential for the electronic system, all the adi-abatic states tend to be populated with equal probabil-ity in the long time limit.
This means the effectivetemperature of the electronic system is infinitely large.Thus these methods are rigourous only when the ther-mal energy is much larger than the bandwidth. In theEhrenfest dynamics the electronic system could influencethe phonons by the mean-field Hellmann-Feynman force.However, it does not prevent the electronic system todeviate from the near-equilibrium distribution in the dif-fusion dynamics, which is also shown in the follow-ing. Besides, in the study using Kubo formalism withadiabatic approximation, the dc conductivity turns outto be zero. The surface hopping approach is promis-ing in treating the energy relaxation through the frus-trated hops in the switching procedure.
However,these hops also lead to inconsistency problems, the solu-tion of which is under active studies. Thus the problemof properly accounting for both decoherence and energyrelaxation in the quantum-classical dynamics of the dy-namic disordered systems remains an open question.In this paper, we extend the IDC approach to theadiabatic representation to investigate the decoherenceand energy relaxation processes. Similar approaches havebeen employed to the study of spin dynamics of excitedstates and carrier dynamics of the Anderson model. The evolution of the system in short time regime (com-parable to the phonon frequency) are governed by thequantum-classical dynamics. The long time dynamics,which is recognized to be problematic, are modifiedby decoherence corrections. The IDC are carried outby measurement-like operations with different schemes.For energy relaxation, the detailed balance is consideredby introducing the IDC with energy-dependent reweigh-ing factors, such as the Boltzmann (IDC-BM) and theMiller-Abrahams (IDC-MA) factors. The IDC with onlydestruction of phase coherence (IDC-DP) is also stud-ied for comparison. Based on the off-diagonal electron- phonon coupling model, the physical consequences of theIDC approach are explored. This paper is organized asfollows. In Section II, the model and the IDC approachare presented in detail. In Section III, the results of theIDC are shown, including both diffusion and electronicenergy. The paper is summarized briefly in Section IV. II. MODEL AND METHOD
In this section, we first present the off-diagonalelectron-phonon coupling model and the equations of mo-tion. We then move on to introduce the IDC approachwith a couple of schemes for energy relaxation, includingthat with Boltzmann and Miller-Abrahams factors.
A. Model
The Hamiltonian we consider in this work is composedof the electronic and the phonon part, which is H = H el + H ph . The electronic part is H el = X j J [ − α ( u j +1 − u j )]( c † j c j +1 + c † j +1 c j ) , (1)where J is the transfer integral, α the electron-phononcoupling constant, u j the displacement of phonon on the j -th site, c † j ( c j ) the creation (annihilation) operators ofelectron. The Hamiltonian for the phonons is H ph = X j (cid:20) m ˙ u j + 12 mω u j (cid:21) , (2)where m is the effective mass of the phonon and ω isthe frequency. The electron state is described by thewave function | ψ ( t ) i that is governed by the Schr¨odingerequation, which isi ~ ∂ | ψ ( t ) i ∂t = H el | ψ ( t ) i . (3)The electronic system back-reacts on the phononsthrough the mean-field Hellmann-Feynman force. TheLangevin heat bath is included to account for the fluctu-ations of the phonons due to the thermal environment.The equations of motion for the phonons is m ¨ u j = − mω u j − ∂E el ∂u j − γm ˙ u j + ξ j , (4)where E el = h ψ ( t ) | H el | ψ ( t ) i is the expectation value ofenergy at time t , γ the friction constant of the Langevinheat bath and ξ j the stochastic forces. ξ j satisfy the cor-relation function h ξ i ( t ) ξ j (0) i = p γmk B T δ ( t ) δ ij , where k B is the Boltzmann constant and T is the tempera-ture. The initial displacements u j (0) and velocities ˙ u j (0)are drawn from the Maxwell distributions with variance k B T /mω and k B T /m respectively. The initial state ofthe electron is chosen among the adiabatic states (energyeigenstates) at t = 0 according to the Boltzmann distri-bution P ν = exp( − E ν /k B T ) / P µ exp( − E µ /k B T ). Theevolution is carried out by the 4th order Runge-Kuttamethod and the stochastic forces are incorporated by themethod of Wang et al. The final result is averagedover enough realizations for the convergence of relevantphysical quantities.
B. Instantaneous Decoherence Correction
We now move on to describe the instantaneous de-coherence corrections (IDC), which is carried out byrepeated measurement-like operations on the electronicwave function in the adiabatic representation. A de-coherence time t d is introduced as the time interval be-tween successive IDC. It reflects the decoherence rate dueto the system-environment interaction. This approach isshown to be effective to incorporate decoherence in thedynamics. Although discrete, it is shown in the siterepresentation that the results resemble those by actuallyadding an attenuation term in the equation of motion ofthe non-diagonal elements of the density matrix. Forthe following results, a fixed t d is used for convenience.Actually, more elaborate forms for t d can be taken toreflect the situation in realistic materials. For example,proper distributions, such as the Poisson distribution, can be used, together with specific models to determine t d from other parameters. In detail, the dynamics are sliced into segments of timeperiod t d . Within each t d , the carrier and phonons areevolved by the quantum-classical dynamics. After each t d , a measurement-like operation is carried out for theelectronic system by collapsing the electron state to acertain adiabatic state according to a chosen distribu-tion, which can be set by different schemes. Suppose theadiabatic states are denoted as | E µ i with energy E µ andthe distribution is P µ , where µ is the index. A randomnumber χ is chosen uniformly in [0 , | E ν i if ν − X µ =0 P µ ≤ χ < ν X µ =0 P µ . (5)For the distribution, the direct choice is that with thewave function in the adiabatic representation, which is P µ = | h E µ | ψ ( t ) i | , (6)where | ψ ( t ) i is the wave function prior to the decoherencecorrection at time t . This scheme removes the phaserelation among adiabatic states and is termed as the IDCwith destruction of phase coherence (IDC-DP). However,as is shown below, this scheme only includes decoherenceand energy relaxation is still not properly treated. C. Energy Relaxation
For the purpose of energy relaxation in the IDC, weconsider the detailed balance through the introductionof energy-dependent reweighing factors. We propose acouple of schemes in this work. The first one is to imple-ment a Boltzmann factor, which is P BM µ = P µ · exp( − E µ /k B T ) /C BM , (7)where C BM is the normalization factor. This schemecould reflect the ability of the environment to recoverthe thermal distribution for the electronic system andis denoted as the IDC-BM in the following. It is moti-vated by Einstein’s theory of spontaneous emission ofexcited states, which is the result of quantum fluc-tuations of radiation fields. Further, motivated by theMiller-Abrahams formula, an alternative scheme is im-plemented. If the measurement is to collapse the state toan adiabatic state with higher energy, the probability isreduced by a factor from actually absorbing the energyfrom the environment. In this sense, the distribution is P MA µ = (cid:26) P µ exp[ − ( E µ − ¯ E ) /k B T ] /C MA , E µ > ¯ E,P µ /C MA , E µ ≤ ¯ E, (8)where C MA is a normalization constant. This schemeis termed as the IDC-MA in the following. Similar ex-pressions are widely used for the hopping rates in thesimulation of charge transport in amorphous organicsemiconductors. They are shown to capture the tran-sient mobility behavior and charge extraction tran-sients in organic solar cells, which depend on the correcttreatment of energy relaxation.It is noted that with the IDC, the electronic statesremain fairly localized. It is not necessary to solve theHamiltonian of the whole lattice and a specific regionis chosen instead, within which the electronic state ispopulated, which is more efficient. The region is deter-mined as follows. The index of the left/right bound-ary of the region is denoted as j l/r . The quantity p l/r = P µ P µ | (cid:10) j l/r | µ i | is calculated and is ensured tobe smaller than a critical value. Otherwise the regionshould be expanded. Actually, p l/r reflect the expecta-tion value of populations on the boundary sites after thedecoherence correction. In the following calculations, thecritical value is chosen to be 10 − , which is small enoughto not influence the final result. III. RESULTS
For the following results, the parameters are taken tobe typical for pentacene. The transfer integral J is0 . e V, the electron-phonon coupling constant α = 3 . − , the phonon frequency ω = 7 . − , the phononeffective mass m = 250 amu; the lattice constant a = 4˚A; the temperature T = 150 K. The friction constant γ is taken to be 1 ps − . A lattice with 600 sites is takento avoid the boundary effects. The decoherence time istaken to be t d = 10 ~ /J ≈
180 fs. This set of parametersis taken in the following unless stated otherwise.
A. Diffusion
We first consider the diffusion with the IDC, which isreflected by the time-dependent averaged population ofelectron among lattice sites P j ( t ) = 1 N s X s (cid:12)(cid:12) ψ sj ( t ) (cid:12)(cid:12) , (9)where ψ sj ( t ) the coefficient of the electron wave functionfor the j -th site in the s -th realization. N s is the to-tal number of realizations, which is taken to be at least10000. The mean squared displacement (MSD) is calcu-lated from the distribution P j ( t ), which isMSD( t ) = a X j j P j ( t ) − X j jP j ( t ) . (10)The diffusion constant is the time derivative of MSD inthe long time limit, which is D = 12 lim t →∞ dMSD( t )d t . (11)The evolution time is taken to be at least 10 ps for theconvergence of D .The typical MSD with the above IDC schemes areshown in Fig. 1(a), together with that from the Ehren-fest dynamics of Troisi et al. The prominent effect ofthe IDC in adiabatic representation is that the diffusionis enhanced, in contrast to the suppression of diffusionwith the IDC in site representation, which is a result ofthe quantum Zeno effect. The enhancement is reducedby inclusion of energy relaxation with the IDC-BM andIDC-MA schemes. For comparison, the results after av-eraging over 10 and 100 realizations with the IDC-BMare also shown. It can be seen that with 100 realizations,the result is already close to the final one.The enhancement of diffusion originates from the waythe electron diffuses in the quantum-classical dynamics.Take the IDC-DP as an example. The average populationon the j -th site before the decoherence correction is P bj = | X µ h j | E µ i h E µ | ψ i | . (12)The decoherence correction destroys the phase coherencebetween different adiabatic states and the correspondingpopulation after the decoherence correction is P aj = X µ | h j | E µ i | | h E µ | ψ i | . (13) M S D ( a ) t (ps)IDC-BMIDC-MAIDC-DPEhrenfest D ( c m · s − ) T (K) IDC-BMIDC-MAIDC-DPEhrenfest FIG. 1. (a) Evolution of the mean squared displacement(MSD) with different instantaneous decoherence correction(IDC) schemes applied to an off-diagonal electron-phononcoupling model, in comparison with the result of Troisi’smethod with Ehrenfest dynamics (green dot-dot-dash): IDCwith Boltzmann factor (IDC-BM, red solid), IDC with Miller-Abrahams factor (IDC-MA, blue dash) and IDC with destruc-tion of phase coherence (IDC-DP, purple dot-dash). All re-sults are got by averaging over more than 10000 realizations.The results with the IDC-BM with 10 and 100 realizationsare also shown (grey solid). (b) Temperature dependenceof diffusion constant with the above IDC schemes: IDC-BM(red square), IDC-MA (blue circle), IDC-DP (purple lower-triangle) and Ehrenfest dynamics (green upper triangle).
Comparing with P bj , the missing terms in P aj are the in-terference ones between adiabatic states. In the Ehren-fest dynamics, the wave function starts to change by themixing of adiabatic states due to phonon motion. Ifthe wave function is expanded in the evolved adiabaticstates, the interference terms tend to be constructive forthose sites that are initially populated and destructivefor those sites that are initially unpopulated. With fur-ther evolution, this kind of phase relation is disrupted bythe stochastic phonon motion and population emergeson those initially unpopulated sites. The IDC helps thisprocess and thus enhances diffusion. A specific exam-ple of this enhancement is shown in the supplementarymaterial. We further calculate the temperature dependence ofdiffusion constants. The results are shown in Fig. 1(b).It can be seen that all the IDC schemes, like the Ehrenfestdynamics, give a band-like dependence of diffusion con-stant. In the lower temperature regime around 100 K, the -3-2-10123 (a)-3-2-10123 0 2 4 6 8 10(b) E e l ( J ) LowestHighestEqual E e l ( J ) t (ps) IDC-BMIDC-MAIDC-DP FIG. 2. (a) Evolution of the electronic energy E el with theEhrenfest dynamics starting from the adiabatic states withlowest (red dash) and highest (blue dot-dash) energies, andenergies that are closest to ± . J , ± J , ± . J (grey solid).The result starting from random adiabatic states with equalprobability (green solid) is also shown. (b) Evolution of E el for the IDC-BM (red solid), IDC-MA (blue dash) and IDC-DP (purple dot-dash) schemes starting from adiabatic stateswith the lowest and the highest energies. The average valueof the Boltzmann distribution is shown by the arrow on theleft. diffusion constant with the IDC are substantially largerthan the ones given by the Ehrenfest dynamics. Withincreasing temperature, the diffusion constants from theIDC-DP scheme remain higher than the Ehrenfest ones,while those from the IDC-BM and IDC-MA schemesgradually decrease and nearly coincide with the Ehrenfestones. Besides, the diffusion constants with the IDC showsharper dependence on temperature. Beyond 150 K, theybegin to deviate from a power-law behavior. This can berelated to the corresponding deviation of the localizationlength of adiabatic states. The temperature dependenceof the averaged localization length is provided in the sup-plementary material. B. Energy Distribution of Electronic States
We now turn our attention to the electronic energy E el = h ψ ( t ) | H el | ψ ( t ) i for the energy relaxation process.Firstly, the evolution of E el in the Ehrenfest dynamicsare shown in Fig. 2(a). The initial condition of the elec-tronic system is chosen to be adiabatic states with boththe highest and the lowest energies, and those closest to ± . J , ± J , ± . J . They all tend to relax to energy values near the center of the density of states ( E = 0).To check the observation, the evolution of E el startingfrom random adiabatic states with equal probability isalso shown. An initial small decrease away from zero isobserved, which persists beyond 10 ps. The small devia-tion is caused by the polaron effect due to the Hellmann-Feynman force, which is a feedback from the electronicsystem to the phonons. It is confirmed by the evolutionof averaged displacements and phonon potential energiesthat are shown in the supplementary material. Theseresults show that the near-equilibrium Boltzmann distri-bution is violated in the Ehrenfest dynamics. All theadiabatic states tend to be populated with equal prob-ability, which corresponds to a large effective tempera-ture. This violation is from the inability of the coherentquantum dynamics to distinguish between the processesthat go upward and downward in energy. Similar prob-lems are also pointed out in the studies of the spin-bosondynamics by stochastic Sch¨odinger equation with real-valued stochastic terms and the Anderson model withHaken-Strobl-Reineker method. We then study the evolution of the electronic energywith the IDC schemes starting from adiabatic states withboth the highest and the lowest energies, which are shownin Fig. 2(b). For the IDC-DP scheme, the final energytends to a finite negative value different from the valueof the Boltzmann distribution, which is indicated by thearrow on the left. This negative value has the same originwith the negative dip in the Ehrenfest dynamics as dis-cussed above. Here, the effect is made more pronouncedby the decoherence corrections to adiabatic states. Fur-ther, it is clear that for the IDC-BM scheme, the elec-tronic energies starting from both the lowest and highestadiabatic states tend to the same limit, which is onlyslightly lower ( ∼ . J ) than the Boltzmann value. Forthe IDC-MA scheme, the energies also relax to a finalvalue, which is slightly higher than the Boltzmann one.Besides, the IDC-BM scheme gives a much faster relax-ation process than the IDC-MA. In all, both schemes helprecover the near-equilibrium Boltzmann distribution inthe dynamics, which is not observed in the Ehrenfest dy-namics and the IDC-DP scheme. This property is presentin the temperature regime considered in this paper, whichis shown in the supplementary material. We further calculated the population distributions P ( E ) of the the electronic states with respect to energyafter evolving the system for 10 ps, which are shown inFig. 3(a). In the calculation of P ( E ), the contributionfrom each realization is broadened by a Lorentzian line-shape with width Γ, which is P ( E ) = C P X s,µ |h E sµ | ψ s i| Γ ( E − E sµ ) + Γ , (14)where C P is the normalization constant to ensure R P ( E )d E = 1. Here Γ is chosen to be 0 . J . The den-sity of states D ( E ) can be defined similarly. A quantity f ( E ) is further defined to depict the probability distri-bution of the electronic system as f ( E ) = P ( E ) /D ( E ), P ( E ) IDC-BMIDC-MAIDC-DPEhrenfestBoltzmannDOS f ( E ) E ( J ) FIG. 3. (a) Population distribution P ( E ) of the electronicstates with respect to energy E after evolution of 10 ps start-ing from adiabatic states with the Boltzmann distribution:IDC-BM (red solid), IDC-MA (blue dash), IDC-DP (purpledot-dash) and Ehrenfest dynamics (green dot-dot-dash). Theresult of the Boltzmann distribution (black triple-dot-dash)and the density of states D ( E ) (black dot) are also shownfor comparison. (b) Corresponding probability distributiondefined as f ( E ) = P ( E ) /D ( E ). which is shown in Fig. 3(b). In both figures, the energyis from − . J to − J , in which the population is pro-nounced. A plot over the full energy regime is providedin the supplementary material for clarity. For compar-ison, the results of the Boltzmann distribution are alsoshown. The deviation of the Boltzmann result from lin-ear behavior around E = − . J in Fig. 3(b) is an ar-tifact from the fast variation of D ( E ). In Fig. 3(a), D ( E ) is also shown as the case when all the adiabaticstates are populated with equal probability. It can beseen that the results of the Ehrenfest dynamics tend to D ( E ) and the results of the IDC-DP scheme also de-viate from the near-equilibrium Boltzmann distribution.In contrast, the results of both the IDC-BM and IDC-MA are not far away from equilibrium ones. Besides,in the lower energy regime below the peak of the pop-ulation, the IDC-BM gives a distribution that is closerto the Boltzmann one; in the higher energy regime, thecorrespondence is better with the IDC-MA scheme. C. Energy Relaxation Time
Furthermore, we note that an energy relaxation time t r can be extracted from the the evolution of electronicenergy, which is a quantity commonly used for describ- . . . . . .
53 100 150 200 250 300 350 0 . . . . . . . . t − r ( p s − ) T (K)IDC-BMIDC-MA t − r ( p s − ) t d (fs) t − r ( p s − ) α ( α ) FIG. 4. (a) Dependence of the inverse energy relaxation time t − r on temperature T for the IDC-BM (red square) and theIDC-MA (blue circle) schemes. The decoherence time is cho-sen to be t d = 180 fs. (b) Dependence of t − r with decoherencetime t d . The temperature is fixed at T = 150 K. (c) Depen-dence of t − r with electron-phonon couplings α , with α = 3 . − . ing the associated transient properties. For this pur-pose, we calculate the evolution of E el from randomlychosen adiabatic states with equal probability. The evo-lution can be well fitted by an exponential dampingfunction ∼ exp( − t/t r ) in the temperature regions con-sidered. The dependence of the inverse energy relax-ation time t − r on temperature, decoherence time andelectron-phonon couplings are shown in Fig. 4, wherelarge t − r means faster energy relaxation. It can be seenthat the behaviors of t − r with the IDC-BM and IDC-BM schemes are very different. For the temperature de-pendence, t − r from the IDC-BM scheme decreases withtemperature while that from the IDC-MA scheme in-creases. For the decoherence-time dependence, t − r fromthe IDC-BM scheme decreases with t d while that fromthe IDC-MA scheme is insensitive. Both schemes give in-verse t − r that increase with increasing electron-phononcoupling α . These differences come from the differentdominant energy relaxation processes. In the IDC-BM,energy relaxation is dominated by the decoherence cor-rection. With lower temperature, the Boltzmann factor ∼ exp( − E/k B T ) has greater ability to redistribute thepopulation toward lower adiabatic states. For smaller t d , the redistribution is more frequent. In both cases,the relaxation is faster. In contrast, for the IDC-MAscheme, energy relaxation is through the energy fluctu-ation of adiabatic states. The decoherence correctionsbring about the imbalance between the processes thatgo upward and downward in energy. With lower tem-perature, the fluctuation is slower, leading to slower en-ergy relaxation. Besides, different t d do not influencethe fluctuation substantially, thus leading to the insen-sitivity. For both schemes, increasing electron-phononcoupling leads to faster mixing of adiabatic states andfaster energy relaxation. Besides, it is noted that t r withthe IDC-MA (1 ps) are generally ne order of magnitudelarger than the implemented decoherence time t d (100 fs),which follow the same relationship revealed in the stud-ies of some elementary models, such as the spin-bosonmodel. In contrast, t r with the IDC-BM are compara-ble to t d , which may correspond to a faster relaxationmechanism. IV. SUMMARY
In summary, we have studied the decoherence and en-ergy relaxation, which are essential processes inducedby the (quantum) system-environment interaction in thequantum-classical dynamics of charge transport in or-ganic semiconducting crystals. They can be incorpo-rated by the IDC with either the Boltzmann or theMiller-Abrahams factors. We find that, while decoher-ence enhances diffusion, the energy relaxation weakensthis enhancement. With both the IDC-BM and IDC-MA,the distributions of electronic states tend to the near-equilibrium one, which is a sign of proper treatment ofenergy relaxation. The energy relaxation time from thetwo schemes show different behaviors in its dependenceon temperature, decoherence time and electron-phononcoupling strength, which result from different relaxationprocesses. Furthermore, it can be shown that the deco-herence and energy relaxation are crucial for observinga direct current response with quantum-classical dynam-ics, which make a direct calculation of mobility under afinite external electric field possible. ACKNOWLEDGMENTS
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