J. Chem. Phys. 139, 174109 (2013); https://doi.org/10.1063/1.4828863 139, 174109

© 2013 AIP Publishing LLC.

Quantized Hamiltonian dynamics capturesthe low-temperature regime of chargetransport in molecular crystalsCite as: J. Chem. Phys. 139, 174109 (2013); https://doi.org/10.1063/1.4828863Submitted: 30 August 2013 . Accepted: 22 October 2013 . Published Online: 07 November 2013

Linjun Wang, Alexey V. Akimov, Liping Chen, and Oleg V. Prezhdo

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THE JOURNAL OF CHEMICAL PHYSICS 139, 174109 (2013)

Quantized Hamiltonian dynamics captures the low-temperature regimeof charge transport in molecular crystals

Linjun Wang,1,a) Alexey V. Akimov,1,2 Liping Chen,1 and Oleg V. Prezhdo1,a)

1Department of Chemistry, University of Rochester, Rochester, New York 14627, USA2Chemistry Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA

(Received 30 August 2013; accepted 22 October 2013; published online 7 November 2013)

The quantized Hamiltonian dynamics (QHD) theory provides a hierarchy of approximations to quan-tum dynamics in the Heisenberg representation. We apply the first-order QHD to study charge trans-port in molecular crystals and find that the obtained equations of motion coincide with the Ehrenfesttheory, which is the most widely used mixed quantum-classical approach. Quantum initial condi-tions required for the QHD variables make the dynamics surpass Ehrenfest. Most importantly, thefirst-order QHD already captures the low-temperature regime of charge transport, as observed ex-perimentally. We expect that simple extensions to higher-order QHDs can efficiently represent otherquantum effects, such as phonon zero-point energy and loss of coherence in the electronic subsystemcaused by phonons. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4828863]

I. INTRODUCTION

Charge transport is fundamental to all organic elec-tronic devices, including organic field-effect transistors(OFETs), light-emitting diodes (OLEDs), and photovoltaiccells (OPVs).1–4 The carrier mobility, characterizing thecharge transport efficiency of organic semiconducting func-tional layers, strongly influences the overall device perfor-mance. The temperature dependence of mobility marks thecharge transport mechanism and has been extensively stud-ied in literature in the past decades.5–11 Two distinct picturesare usually evoked when modeling charge transport in molec-ular crystals, that is, band and hopping models.8, 9 The sig-nature of each regime is conveyed by a characteristic tem-perature dependence of mobility. The band model applieswhen the electron-phonon coupling can be regarded as a per-turbation. Under such circumstances, the charge is delocal-ized over the whole crystal, and the mobility decreases withtemperature, since higher temperature results in more dis-order that destroys the translational symmetry of the elec-tronic Hamiltonian.12–15 In contrast, the hopping model workswhen the electronic couplings, also called transfer integrals,are treated as a perturbation. Then, the charge is self-trappedon individual molecules due to polaronic effects, and the mo-bility increases with temperature since there exists a barrierfor charge transfer due to nuclear reorganization.16–20 Bothcharge carrier mobility regimes can be described analyticallyat the perturbation level. The analytic expressions have beenapplied to different classes of organic materials.12–20

In principle, the band and hopping regimes of chargetransport may coexist in the same material, because the ef-fective electron-phonon coupling strength relies on the mag-nitude of nuclear vibrations, which is temperature depen-dent. At low temperatures, nuclear coordinate fluctuations are

a)Electronic addresses: linjun.wang@rochester.edu and oleg.prezhdo@rochester.edu

rather weak, and thus can be considered a perturbation, asin the band model. On the contrary, at high temperatures,large amplitude vibrations break the electronic symmetry,the charge becomes localized, and the hopping model takesover. This phenomenon is known as the band-to-hoppingtransition.21–25 The transition between the two regimes is notinstantaneous and covers a finite temperature range, due toinvolvement of phonons with different frequencies, disorder,and other effects. At extremely low temperatures, quantumrather than thermal effects govern nuclear fluctuations, lead-ing to a special charge transport behavior. Thus, accordingto the nature of the nuclear vibrations coupled to the chargecarriers, charge transport can be classified into three regimes,as schematically shown in Fig. 1. The band-to-hopping tran-sitions in molecular crystals have been extensively studiedin the past decades under the framework of the polarontheory,22–24, 26–30 where the polaronic couplings are generallytreated in a perturbative manner.

In recent years, mixed quantum-classical dynamics(MQCD) has developed into a powerful tool to study chargetransport in organic materials.31–38 As a nonperturbativeapproach, MQCD treats nuclear vibrations classically andelectron dynamics quantum mechanically. It has two majorflavors: mean field (MF), also known as Ehrenfest,39–41 andsurface hopping (SH).42–44 In both cases, the electronic stateis driven by the classical external field created by the nu-clear trajectory. The Schrödinger representation of quantumdynamics is usually used. The main difference between theEhrenfest and SH approaches lies in the description of theelectronic back-reaction – how the classical nuclei respond tothe evolution of electronic degrees of freedom. The nuclearevolution in the Ehrenfest approach is governed by a singlepotential energy surface (PES) averaged over all states ac-cording to the electronic probability amplitudes. In the SHstrategy, nuclei always move on a distinct, typically adiabaticPES, but incorporate nonadiabatic transitions, or hops, be-tween different PESs. The probability to stay in each PES is

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174109-2 Wang et al. J. Chem. Phys. 139, 174109 (2013)

FIG. 1. Schematic representation of the three regimes for charge transport:quantum regime, classical perturbative regime, and regime beyond perturba-tion. Typical temperature dependence of charge mobility in the log-log plotand applicable temperature range are indicated. ω is the phonon frequency,J represents the intermolecualr electronic coupling, K is the force constantof nuclear vibrations, and α (β) is the local (nonlocal) electron-phononcoupling.

determined from the corresponding quantum mechanical am-plitude. By treating nuclear motions classically, MQCD ne-glects such quantum effects as zero-point energy, tunnelingand loss of coherence. The latter in particular greatly af-fects quantum transition rates in condensed phase systemsand has to be incorporated as a semiclassical correction.45

In the Ehrenfest approach, nuclear vibrations do not inducedestruction of the electronic superposition states, indicatingthat the coherence time is infinite. Thus, Ehrenfest should beoptimal in the limit of weak electron-phonon coupling, as re-quested by the band transport. The SH approach implies a fi-nite decoherence time,44–49 corresponding to a relatively largeelectron-phonon coupling, and thus more likely falls into thehopping regime. Using analytical results as a benchmark, ithas been recently confirmed by numerical calculations thatSH is able to reproduce the hopping transport, while MF givescompletely wrong trends in this regime.38 A proper imple-mentation of SH works very well in the intermediate regime,where the transport is already band-like,37 showing a signifi-cantly wider applicability range compared to Ehrenfest.

In spite of the great successes of SH, the Ehrenfest tech-nique remains the most widely used approach, due to its sim-ple formulation. Maybe more importantly, the most activelystudied molecular crystals, such as pentacene and rubrene,generally show large mobility with a band-like temperaturedependence,50–52 which favors the MF description.31, 33 TheEhrenfest studies have shown that the feedback from the elec-tron dynamics to the nuclear motion can be safely neglectedwhen room-temperature mobility is higher than a criticalvalue. The value is ∼4.8 cm2/Vs and ∼0.14 cm2/Vs for one-and two-dimensional crystals, respectively.36 Such negligibleimportance of the feedback, which is a second-order effectof electron-phonon coupling, implies that a weak electron-phonon coupling regime and thus band-like transport can berealized up to a certain mobility. Similar observations havebeen made in the SH studies, where charge transport becomesband-like when carrier mobility exceeds ∼1 cm2/Vs.37 There-

fore, when the charge mobility is higher than these criticalvalues, which is the case for many organic systems, such asoligoacene crystals and graphene-based carbon materials,50–55

MF should be a reliable approach to study charge transport.Instead of the Schrödinger representation, charge trans-

port dynamics can be also studied within the Heisenberg rep-resentation. By treating quantum mechanically not only theelectron creation and annihilation operators, but also the nu-clear position and momentum variables, the time derivativeof the expectation value for any system observable of interestcan be achieved from the Heisenberg equation. An interestingphenomenon exists in this method: The original operator be-comes coupled to higher-order operators, resulting in a chainof equations. The resulting quantized Hamiltonian dynamics(QHD) approach can provide a hierarchy of approximationsto quantum dynamics.56–59 The approximations are achievedthrough termination of the chain with a closure that expressesthe expectation values of the higher-order operators in termsof products of the expectations of the lower-order operators.At this point, it would be very interesting to obtain the ex-act physical interpretation of different levels of QHD approx-imations, and to link QHD to the conventional MQCD ap-proaches, especially the Ehrenfest theory. Besides, followingthe Heisenberg formulation of quantum mechanics, the ini-tial conditions for the QHD variables, including nuclear po-sitions and momenta, should fulfill the quantum distributions.In this way, some quantum effects of nuclear vibrations60, 61

can be naturally incorporated. This issue is particularly im-portant when studying the temperature dependence of mobil-ity, especially the low temperature behavior of systems withhigh frequency modes.18, 62

In this study, we apply the first-order QHD to chargetransport in molecular crystals. Higher-order QHDs will beconsidered in further investigations. We derive the equationsof motion for both electrons and nuclei based on modelHamiltonians with nuclear vibrations coupled to electron siteenergies and intermolecular electronic couplings, and com-pare with the Ehrenfest expressions. We implement bothclassical and quantum initial conditions, and reveal the im-portance of the nuclear quantum effects in describing the low-temperature transport behavior. We consider variations of theparameters of the Hamiltonians in order to establish their in-fluence on the temperature dependence of mobility. Finally,we investigate the dependence of the results on the number ofvibrations per molecular site.

II. METHODOLOGY

A. Local and nonlocal Hamiltonians

In order to investigate charge transport in molecular crys-tals, we adopt a one-dimensional model system containingN molecular sites with periodic boundary conditions. Allnearest neighbors are equally spaced by a distance L. Eachmolecule k has one harmonic intramolecular vibrational de-gree of freedom with position xk and momentum pk, and theforce constant K and effective mass m independent of k. Al-though real molecular crystals are generally assembled layerby layer and the molecules are arranged in a herringbone

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pattern within each layer,63, 64 the chosen one-dimensionalmodel with one effective vibrational mode is still widelyused in literature.31–38 Despite its simplicity, the model re-flects most characteristics of charge transport in bulk mate-rials. Note that the methodologies described throughout thiscontribution can be easily extended to higher dimensions.

For each molecule k, there is one associated molecular or-bital |k〉, representing the highest occupied molecular orbitalfor hole transport and the lowest unoccupied molecular orbitalfor electron transport. At equilibrium geometry, the electroniccoupling between the nearest neighbors is J, and the onsiteenergies of all molecules are the same and are therefore setto zero for simplicity. The electron-phonon couplings havetwo flavors: (1) the local electron-phonon coupling, wherethe onsite energy of any molecule k is linearly modulated byxk with the local electron-phonon coupling constant α and(2) the nonlocal electron-phonon coupling, where the elec-tronic coupling between molecule k and its neighbor k + 1 islinearly modulated by xk + 1 − xk with the nonlocal electron-phonon coupling constant β. We consider both Hamiltoniansas follows:36

Hlocal =∑

kJ (Pk,k+1 + Pk+1,k) +

∑kαxkPk,k

+∑

k

1

2

(Kx2

k + p2k

m

), (1)

Hnonlocal =∑

k[J + β(xk+1 − xk)](Pk,k+1 + Pk+1,k)

+∑

k

1

2

(Kx2

k + p2k

m

), (2)

where Pk, l ≡ |k〉〈l| are the electronic projection operators.

B. QHD equations of motion

Within the QHD framework, the time derivative of asystem observable 〈A〉 of interest can be achieved from theHeisenberg equation of motion (EOM),56–59

i¯d

dt〈A〉 = 〈[A, H ]〉, (3)

where H is the system Hamiltonian and the brackets de-note a quantum average over the initial state. In order to de-rive the first-order QHD (QHD-1) EOMs for charge trans-port, we need to regard the nuclear positions and momenta inEqs. (1) and (2) as quantum operators, and take into ac-count the canonical commutation relations between positionand momentum, namely, [xk, pl] = i¯δkl. Applying the localHamiltonian in Eqs. (1)–(3), we can get

d〈Pij 〉dt

= J

i¯(〈Pi,j+1〉 + 〈Pi,j−1〉 − 〈Pi+1,j 〉 − 〈Pi−1,j 〉)

+ α

i¯(〈xjPij 〉 − 〈xiPij 〉), (4)

d〈xi〉dt

= 1

m〈pi〉, (5)

d〈pi〉dt

= −K〈xi〉 − α〈Pii〉. (6)

Using similar procedures for the nonlocal Hamiltonian inEq. (2), one obtains

d〈Pij 〉dt

= J

i¯(〈Pi,j+1〉 + 〈Pi,j−1〉 − 〈Pi+1,j 〉 − 〈Pi−1,j 〉)

+ β

i¯(〈xj+1Pi,j+1〉 − 〈xjPi,j+1〉

+ 〈xjPi,j−1〉 − 〈xj−1Pi,j−1〉)

− β

i¯(〈xi+1Pi+1,j 〉 − 〈xiPi+1,j 〉

+ 〈xiPi−1,j 〉 − 〈xi−1Pi−1,j 〉), (7)

d〈xi〉dt

= 1

m〈pi〉, (8)

d〈pi〉dt

= −K〈xi〉 + β(〈Pi,i+1〉 + 〈Pi+1,i〉− 〈Pi,i−1〉 − 〈Pi−1,i〉). (9)

Hereby, we can find that Eqs. (5), (6), (8), and (9) dependonly on first-order variables, while Eqs. (4) and (7) rely alsoon second-order correlations between nuclear positions andelectronic projections. Principally, one needs to move on toinclude the second-order correlations 〈xiPkl〉 inside the EOMs.However, the simplest treatment is to neglect these correla-tions and make a closure at the first-order using the approxi-mation, 〈xiPkl〉 ≈ 〈xi〉〈Pkl〉. As a result, Eq. (4) goes to

d〈Pij 〉dt

= J

i¯(〈Pi,j+1〉 + 〈Pi,j−1〉 − 〈Pi+1,j 〉 − 〈Pi−1,j 〉)

+ α

i¯(〈xj 〉 − 〈xi〉)〈Pij 〉, (10)

and Eq. (7) becomes

d〈Pij 〉dt

= J

i¯(〈Pi,j+1〉 + 〈Pi,j−1〉 − 〈Pi+1,j 〉 − 〈Pi−1,j 〉)

+ β

i¯[(〈xj+1〉 − 〈xj 〉)〈Pi,j+1〉

+ (〈xj 〉 − 〈xj−1〉)〈Pi,j−1〉]

− β

i¯[(〈xi+1〉 − 〈xi〉)〈Pi+1,j 〉

+ (〈xi〉 − 〈xi−1〉)〈Pi−1,j 〉]. (11)

If one compares the final EOMs for the local Hamiltonian,i.e., Eqs. (5), (6), and (10), with the Ehrenfest formulations,36

one can immediately find that the equations for QHD-1 areexactly identical to that of the Ehrenfest approach. Similarobservations can be also made for the nonlocal Hamiltonian.Thus, we can conclude that the QHD reduces to Ehrenfest atthe first-order in case of no higher-order correlations betweenelectron and nuclei.65

C. Classical and quantum initial conditions

The initial settings of nuclear positions and momenta inthe EOMs are critical for the charge transport dynamics. Theclassical initial conditions, where both nuclear positions and

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FIG. 2. Schematic representation of the nuclear coordinate and momentumvariances as functions of temperature. The trends at low and high temper-atures are indicated as blue and red dashed lines, respectively, to guide theeye.

momenta are randomly chosen from Boltzmann distributions,are widely used in literature when implementing the Ehren-fest dynamics.31–38 This approach works well for high tem-perature investigations but fails completely to describe lowtemperature behaviors because of the absence of quantum ef-fect. Due to the quantum nature of QHD, one needs to makeuse of the quantum distribution of nuclear positions and mo-menta. For a harmonic oscillator, it is well-known that nuclearposition follows a Gaussian distribution,66

ρ(x) = 1

ξ√

πe−x2/ξ 2

. (12)

Here, ξ =√¯

mωcoth( ¯ω2kBT

), where kB is the Boltzmann con-

stant, T is the temperature, and the vibrational frequencyω = √

K/m. Thus the variance of the nuclear position reads

σ 2x = ¯

2√

Kmcoth

(¯ω

2kBT

). (13)

In Fig. 2, we show a schematic representation of the varianceas a function of temperature. In the low temperature limit,Eq. (13) goes to ¯

2√

Km, which is independent of temperature,

representing the coordinate uncertainty of ground state nu-clear vibrations. And in the high temperature limit, Eq. (13)becomes kBT

K, which is actually the classical nuclear coordi-

nate distribution. Similarly, the variance for nuclear momen-tum is expressed as

σ 2p = ¯

√Km

2coth

(¯ω

2kBT

). (14)

Therefore, using Eqs. (13) and (14) as initial conditions,we should have a better description of charge transport atlow temperatures by QHD-1. Note that similar quantuminitial conditions based on Wigner distributions have beenwidely used in nonequilibrium, nonadiabatic dynamics of fi-nite systems.67–69

D. Charge carrier mobility

Using classical or quantum initial conditions, the expec-tation values for the electronic projections can be obtainedthrough solving the EOMs numerically with the standardfourth-order Runge-Kutta (RK4) algorithm.70 The mean-square displacement (MSD) can then be obtained through

MSD(t) =∑

k〈Pkk〉(t)(kL)2 −

(∑k〈Pkk〉(t)kL

)2.

(15)In the normal diffusive regime, there is a linear relationshipbetween MSD and the simulation time.7 Then the diffusioncoefficient can be evaluated through the time derivative of theMSD as

D = 1

2limt→∞

dMSD

dt. (16)

Finally, the charge carrier mobility can be calculated bymeans of the Einstein relation,71, 72

μ = e

kBTD. (17)

In the log μ vs. log T plot, an inverse power-law tempera-ture dependence, μ ∝ T−n, becomes a straight line, where theslope n ≡ −d(log μ)/d(log T). This slope varies along differ-ent lattice directions for different molecular crystals.73 Thus,we will focus on the temperature dependence of mobility,μ(T), and this slope, n(T), in the following discussions.

E. Computational details

Unless otherwise specified, we adopt the following setof reference parameters: J = 300 cm−1, α = 3500 cm−1/Å, β

= 995 cm−1/Å, K = 14500 amu/ps2, and m = 250 amu, basedon extensive earlier studies of charge transport in molecu-lar crystals.31, 36–38 The investigated temperatures fall into therange of 5–500 K. The intermolecular distance, L, is set tobe 5 Å. The time interval is fixed to be 0.025 fs. The totalsimulation time (at least 1 ps) and the system size (at least100 sites) are adjusted to ensure that equilibrium diffusion canbe obtained within the chosen simulation time without biasedby boundary effects. For each calculation, 2000 realizationsare carried out to get a smooth time dependence of the mean-square displacement.

III. RESULTS AND DISCUSSIONS

A. Critical temperatures for charge transport

As the only source of disorder to the electronic Hamil-tonian, the nuclear coordinate fluctuations determine entirelythe temperature dependence of mobility. As shown in Fig. 2,the nuclear coordinate variance (σ 2

x ) at extreme low tem-peratures is a constant value due to the quantum effect ofnuclear vibrations. Then the time-dependent electron dynam-ics remains the same and the diffusion coefficient is invari-ant of temperature. In this regime, the temperature contribu-tion to mobility solely comes from the Einstein relation (seeEq. (17)), resulting in a μ ∝ T−1 power-law dependence (seeFig. 1). When temperature is higher than a critical value,

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Tc1 = ¯ω/kB, nuclear coordinates start to follow classical dis-tributions, and the coordinate variance can be written as a lin-ear relationship with temperature, i.e., σ 2

x = kBT /K . If themagnitude of the electronic Hamiltonian fluctuations, that is,ασ x for the local Hamiltonian and

√2βσx for the nonlocal

Hamiltonian, is small in comparison to the intermoleculartransfer integral, J, the electron dynamics can be dealt with theperturbation theory. In this regime, the diffusion coefficientdecreases with temperature for a linear response, and then thetemperature dependence of mobility follows μ ∝ T −n1 withn1 > 1. When temperature exceeds the critical value, Tc2

= J2K/α2kB and Tc2 = J2K/2β2kB for local and nonlocalHamiltonians, respectively, the charge transport should bedescribed beyond the perturbation theory. There, the chargebecomes more localized, and thus the charge transport canbecome thermal activated or possess a weaker power-law tem-perature dependence of mobility. As a result, we can makea more quantitative description of the three charge transportregimes shown in Fig. 1. Note that if the classical instead ofthe quantum initial conditions are used, the first regime is dis-missed and the temperature range, T < Tc2, is dominant purelyby the second regime. Besides, if Tc2 < Tc1, the second regimedoes not apply. Analyzing the two critical temperatures, Tc1

and Tc2, is very helpful for understanding the overall temper-ature dependence of mobility.

B. Role of quantum initial conditions

In Fig. 3(a), we show μ(T) based on the Hamiltoniangiven by Eq. (1) with only local electron-phonon couplings.Using classical initial conditions, the Ehrenfest dynamicsgives an almost perfect linear relationship of μ(T) in the log-

log plot, which implies that the carrier mobility follows anoverall power-law dependence. With quantum initial condi-tions instead, the temperature dependence obtained by theQHD-1 remains the same above 50 K, but the mobility isstrongly reduced at lower temperatures. According to the ref-erence parameters used in this study, the vibrational frequencyis calculated to be about 40 cm−1, corresponding to a criti-cal temperature Tc1 of 58 K. The numerical calculations herethus confirmed the interpretations discussed above: a strongchange in the transport behavior happens around Tc1 becauseof the nuclear quantum effect. Due to the same reason, forthe Hamiltonian given by Eq. (2) with only nonlocal electron-phonon couplings, the role of quantum initial conditions re-mains basically the same (see Fig. 3(b)).

The above observations can be viewed more clearly whenwe extract n(T) from μ(T). As shown in Figs. 3(c) and 3(d),for both the local and nonlocal Hamiltonians, the n(T) pre-dicted by the Ehrenfest approach using classical initial condi-tions are more or less constant at low temperatures. This is ba-sically the perturbation regime shown in Fig. 1. However, theyexperience a reduction at temperatures higher than 150 K.This is related to the transition around Tc2, which is calcu-lated to be 128 K for the local Hamiltonian. The decrease ofn is even more significant for the nonlocal Hamiltonian dueto the difference in nature of the local and nonlocal electron-phonon couplings. For the local Hamiltonian, since the siteenergies at equilibrium geometries are zero, the fluctuationof site energies due to nuclear vibrations always result in aenhancement of electronic disorder and thus a reduction oftransport efficiency. However, this is not always the case fornonlocal Hamiltonians. There, the charge transport dependsstrongly on the absolute values of transfer integrals. When the

FIG. 3. Temperature dependence of (a) and (b) mobility, μ, and (c) and (d) its derivative, −d(log μ)/d(log T), for the Ehrenfest dynamics with classical initialconditions and the QHD-1 dynamics with quantum initial conditions. (a) and (c) are based on the local Hamiltonian, Eq. (1), while (b) and (d) are based on thenonlocal Hamiltonian, Eq. (2). In all cases, the reference parameter values are used: J = 300 cm−1, α = 3500 cm−1/Å, β = 995 cm−1/Å, K = 14500 amu/ps2,and m = 250 amu.

174109-6 Wang et al. J. Chem. Phys. 139, 174109 (2013)

FIG. 4. Temperature dependence of mobility for QHD-1 calculations based on the local Hamiltonian, Eq. (1). For each calculation, either (a) the electroniccoupling, J, (b) the local electron-phonon coupling, α, (c) the force constant, K, or (d) the effective mass, m, is changed, while keeping the other parameters totheir reference values: J = 300 cm−1, α = 3500 cm−1/Å, K = 14 500 amu/ps2, and m = 250 amu.

temperature is high enough, the fluctuation of the transferintegral can be of opposite sign of the transfer integral atequilibrium geometries, and thus transport can be relativelyenhanced,19, 20 resulting in a weaker temperature dependence.For the QHD-1 approach using quantum initial conditions,n(T) is close to unity at extreme low temperatures, indicatingthe signatures of quantum regime shown in Fig. 1. The hightemperature behaviors are similar to the Ehrenfest results us-ing classical initial conditions.

C. Role of parameters in the Hamiltonian

We start with the local Hamiltonian in Eq. (1). Based onthe reference parameters, we vary each fundamental param-eter within the Hamiltonian one by one, namely, the inter-molecular transfer integral, J, the local electron-phonon cou-pling, α, the force constant, K, and the effective mass, m.The temperature dependences, μ(T) and n(T), obtained by theQHD-1 approach using quantum initial conditions are shownin Figs. 4 and 5, respectively. At temperatures below 50 K,μ(T) is simply shifted when using different J and α (seeFigs. 4(a) and 4(b)), and thus n(T) keeps the same (seeFigs. 5(a) and 5(b)). This is because the transition betweenthe quantum regime and the classical perturbative regime inFig. 1 is unrelated to J and α. However, at high temperatures,we can observe a lower n for smaller J or larger α. As we men-tioned above, beyond the perturbation regime, the stronger thedynamic disorder strength the weaker the temperature depen-dence of mobility. When reducing J or enhancing α, the criti-cal temperature Tc2 gets smaller, resulting in a lower n.

Comparatively, K and m have a much stronger impact onthe temperature behavior of charge transport (see Figs. 4(c)

and 4(d)), because they together determine the frequency ofthe nuclear vibrations, ω = √

K/m, which plays the centralrole in the quantum effect. With the increase of K or decreaseof m, ω is increased, then the quantum effect plays a role upto a higher critical temperature Tc1 (see Figs. 5(c) and 5(d)).The major difference between the roles of K and m happensat high temperatures. An increase of K results in a larger Tc2,while changing m has no impact on Tc2. Thereby, different Kcan end up with similar n at high temperatures, because bothTc1 and Tc2 changes with K, and these high temperature calcu-lations fall in the same classical perturbative regime. Smallerm results in smaller n at high temperatures since the perturba-tion regime is compressed, and its characteristic temperaturedependence cannot be fully reflected.

The results with only nonlocal electron-phonon cou-plings (as shown in Figs. 6 and 7) are very similar to thoseobtained using only local electron-phonon couplings. Re-cent surface hopping studies have shown that the bandlikecharge carrier mobility decreases with increasing nonlocalelectron-phonon coupling strength in the presence of localelectron-phonon couplings.37 Thereby, we expect that com-bined investigations with both local and nonlocal electron-phonon couplings should yield similar conclusions in thepresent Ehrenfest-like dynamics, and thus are not carried out.In line with the observations in Fig. 3, the major differencebetween the Hamiltonians with local and nonlocal electron-phonon couplings happens at high temperatures. A muchstronger reduction in n can be found, because strong nonlo-cal electron-phonon couplings at high temperatures tend toincrease the effective electronic couplings between neighbor-ing molecules, counteracting the decrease of mobility due tolarger disorder.19, 20

174109-7 Wang et al. J. Chem. Phys. 139, 174109 (2013)

FIG. 5. Temperature dependence of −d(log μ)/d(log T) for the same calculations as in Figure 4.

D. Role of multiple phonons per molecule

In real molecular crystals, there exist a lot of vibra-tional modes with different frequencies reacting on the samemolecule.19, 23 Thereby, we study further the role of multiplephonons, using the local Hamiltonian with two uncorrelatedvibrations per molecule as an example. We keep J and K to beunchanged. The electron-phonon coupling strengths are re-

duced to 1/√

2 times the reference value to conserve the to-tal reorganization energy λ = ∑

i α2i /Ki ,36 where i covers all

considered vibrational modes. We choose m1 = 250 amu withthe vibrational frequency as 40 cm−1, and m2 = m1/16 so thatthe frequency is 4-fold larger. As shown in Fig. 8, the temper-ature dependence of mobility using two phonons per moleculefalls exactly between the results based on one single phonon

FIG. 6. Temperature dependence of mobility for QHD-1 calculations based on the nonlocal Hamiltonian, Eq. (2). For each calculation, either (a) the electroniccoupling, J, (b) the nonlocal electron-phonon coupling, β, (c) the force constant, K, or (d) the effective mass, m, is changed, while keeping the other parametersto their reference values: J = 300 cm−1, β = 995 cm−1/Å, K = 14500 amu/ps2, and m = 250 amu.

174109-8 Wang et al. J. Chem. Phys. 139, 174109 (2013)

FIG. 7. Temperature dependence of −d(log μ)/d(log T) for the same calculations as in Figure 6.

per molecule and proper electron-phonon couplings yieldingthe same reorganization energy. Thus, the major charge trans-port mechanism under multiple phonons per molecule can befully understood with one single vibrational mode as system-atically investigated above.

E. Comparisons with experiment

Due to the great difficulty in fabricating large size and ul-tra pure molecular crystals for time of flight measurements,52

reliable experimental data on the temperature dependence ofintrinsic mobility over a wide temperature range is not widelyavailable in literature. The work done by Karl on naphthalene

FIG. 8. Temperature dependence of mobility with either one nuclear vibra-tion (α = 3500 cm−1/Å, m = 250 amu or m = 15.625 amu) or two nuclearvibrations (α1 = α2 = 2475 cm−1/Å, m1 = 250 amu, and m2 = 15.625 amu)coupled to each molecule. The QHD-1 approach based on the local Hamilto-nian, Eq. (1), is used.

crystals73 is widely used in comparison with theories.22–24

The reference parameters adopted in the present study arebased on pentacene crystal,31 which has a slightly larger in-termolecular transfer integral and a smaller electron-phononcoupling strength than naphthalene crystal.74 By varyingthese reference parameters (i.e., Figs. 4–7) we are able to rep-resent the major transport behavior of other related systems,including naphthalene crystal. The transition in the slope ofμ(T) observed at about 30 K, arises from the quantum effectof nuclear vibrations discussed above. A power-law tempera-ture dependence of mobility is generally observed for highertemperatures. The slope n is around 1.5 for most transport di-rections, with the exception of hole transport along the a-axis,exhibiting n = 2.9.73 As shown in Figs. 4–7, the value of ncalculated at the QHD-1 level falls in the range between 1and 2, agreeing with these measurements on naphthalene.

Recently, it has been found that thermal expansion ofmolecular crystals has a strong impact on the temperature de-pendence of mobility.24 With the increase of temperature, thelarger spacing between neighboring molecules induce smallerintermolecular transfer integrals and also larger electron-phonon couplings. When these temperature-dependent struc-ture modifications are taken into account, a larger n is ex-pected and thus the large n in experiment can be explained.Note that acoustic phonons, which have not been consideredin the present models, may also have a strong impact on thetemperature dependence of mobility at low temperatures.22

Besides, we notice that there is another transition in the exper-imental temperature dependence of mobility along the b-axisof naphthalene crystal at around 100 K. It can be attributed toour second transition from the perturbative regime to the non-perturbative regime (see Fig. 1). In our calculations, such tran-sition happens at around 100–200 K, depending strongly onthe intermolecular transfer integrals (see Figs. 5(a) and 7(a)).

174109-9 Wang et al. J. Chem. Phys. 139, 174109 (2013)

Finally, it is important to emphasize the application rangeof QHD-1 for charge transport. We have proved that theEhrenfest approach is a special case of QHD, in the absenceof higher-order correlations and quantum effects. Thus, simi-larly to Ehrenfest, QHD-1 does not work well in the hoppingregime of charge transport.38 In the regime beyond perturba-tion shown in Fig. 1, higher-order QHD or other advancedapproaches like the flexible surface hopping37 should be usedto yield more accurate descriptions. As a result, QHD-1 worksproperly for charge transport in a relatively low temperaturerange, where quantum initial conditions should be consideredto achieve correct description of the nuclear quantum effects.

IV. CONCLUSIONS

In summary, we have derived the equations of motionfor charge transport in molecular crystals using the first-orderquantized Hamiltonian dynamics theory, and found that theQHD equations of motion are identical to the Ehrenfest for-malism. The difference between the Ehrenfest and QHD-1methods arises in the initial conditions. Ehrenfest uses clas-sical initial conditions, while the initial conditions of QHDare quantum mechanical. Neglect of correlations between dif-ferent degrees of freedom constitutes the main approximationof QHD-1. The use of quantum initial conditions for the nu-clear position and momentum variables allowed us to repro-duce the low-temperature regime of charge transport, as ob-served experimentally. We suggested a three-regime pictureof charge transport in order to understand the temperature de-pendence of mobility at the QHD-1 level. We investigated theeffect of all parameters in the Hamiltonian on charge trans-port for cases including both single and multiple phonons permolecular site. The developed description allowed us to ex-plain the experimental results on naphthalene crystal.

ACKNOWLEDGMENTS

This work is supported by the US National Science Foun-dation Grant No. CHE-1300118. A.V.A. was funded by theComputational Materials and Chemical Sciences Network(CMCSN) project at Brookhaven National Laboratory underContract No. DE-AC02-98CH10886 with the U.S. Depart-ment of Energy and supported by its Division of ChemicalSciences, Geosciences & Biosciences, Office of Basic EnergySciences. O.V.P. acknowledges financial support of the U.S.Department of Energy, Grant No. DE-SC0006527.

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