and KMC simulations demonstrated that the dynamic disorder caused by intermolecular vibration has
negligible effect on the carrier mobility. We further apply the approach for molecular design of n-type
materials and for rationalization of experimental results. The charge reorganization energy is analyzed
through decomposition into internal coordinates relaxation, so that chemical structure contributions to
the intramolecular electron–phonon interaction are revealed and give helpful indication to reduce the
charge reorganization energy.
1. Introduction
Organic materials play an important role in next-generationelectronic applications, due to their processability and flexibility.However, low charge mobility and poor stability have restricted theirdevelopment. Although some new molecular materials emergedwith room-temperature mobilities up to tens cm2 V�1 s�1,1 acomplete understanding of the charge transport mechanism isstill not yet reached.2–5 Electron–local phonon interaction modelproposed by Holstein6,7 depicted a general scheme for chargetransport in organic solids. Non-local electron–phonon couplinghad been included in Hostein–Peierls model by Munn–Silbey8
and Hannewald–Bobbert.9 Mixed quantum-classical (quantumfor electrons and classical for atoms) non-adiabatic dynamicshave been proposed with local electron–phonon coupling byHultell and Stafstrom10 and non-local electron–phonon coupling
by Troisi and Orlandi.11 However, mapping a complex molecularsystem onto a simple model even with both local and non-localcouplings may lose important aspects of the structure features.4
From the point of view of material design, a microscopic whilerealistic model for the intrinsic property is highly desirable.Some polyacenes single crystals indicate band-like chargetransport.12 According to the magnitude of electron–phononinteraction, different charge transport models have beenadopted: (i) band model based on delocalized charge picture,where the intermolecular electron coupling (V) is much largerthan the molecular reorganization energy (l); (ii) intermediateregime, where V is comparable to l. None of these interactionscan be treated as a perturbation, and all relevant interactionsshould be treated on the same footing.13–18 (iii) V is much lessthan l, and the electron interacts strongly with intramolecularvibrations which eventually lead to self-localization: hoppingmodel applies in this case. Although the localized charge modelproposed by Bredas et al.19 has achieved great successes inmolecular design to achieve high charge mobility,20–26 there arestill some issues unsolved,26 such as the involvement of mole-cular excited states of the ions in evaluating the electroniccoupling term, the treatment of site variability, especially foramorphous organic films,28 and the involvement of the nucleartunneling effect in the charge transfer process.29
a Key Laboratory of Organic OptoElectronics and Molecular Engineering,
Department of Chemistry, Tsinghua University, 100084 Beijing, China.
Marcus semi-classical theory30 relied on classical treatment ofnuclear motions in organic semiconductors.19,31,32 The charge islocalized in one molecule and the transport is carried outthrough the hoping process from one molecule to another viathe charge transfer.33,34 We recall that in studying the chargetransfer process of the ferrous–ferric electron exchange reac-tion, Marcus originally assumed that electron transfer takesplace after solvent reorganization, the charge solvation processbeing mainly associated with low frequency polarizationmotion (o { kT).29 Therefore, the solvent polarization andfluctuation of the environment is described classicallyand vanishes with decreasing temperature. Hence, the semi-classical Marcus electron transfer theory predicts that themacroscopic current vanishes if the temperature approachesabsolute zero. However, in organic semiconductors, the
environment of the electrons consists of phonons, both intra-molecular and intermolecular, which often possess highfrequency o c kT. Thus, the classical treatment for theenvironment is not appropriate for organic materials. In fact,a number of experiments have shown that the conductivity oforganic materials at low temperatures is finite,35–38 whichdemonstrates obvious quantum nuclear tunneling effects inorganic semiconductors and the semi-classical approachesbreak down at low temperature. Quantum effect for chargetransfer process have been considered by Jortner with a singleeffective high frequency vibration mode,39 and by Emin40,41
for polaron conductivity. Even for the original classical ferrous–ferric electron exchange reaction, quantum Monte-Carlosimulation treating water solvent quantum mechanicallyalready demonstrated significant nuclear tunneling effects
Zhigang Shuai
Zhigang Shuai, PhD 1989 FudanUniversity. 1990–2001, postdocand research scientist the Univer-sity of Mons, Belgium. 2002–2008,‘‘Hundre-Talent’’ professor at theInstitute of Chemistry of theChinese Academy of Sciences inBeijing. 2008–date, ChangjiangScholar Chair professor, TsinghuaUniversity, Beijing. His researchinterests are theoretical chemistryand modeling of the organicfunctional materials for opto-electronic properties. 278 publica-
tions with h-index 48. Outstanding Young Investigator’s Fund (2004)and the Chinese Chemical Society–AkzoNobel Chemical SciencesAward (2012). Member of the International Academy of QuantumMolecular Science, the Royal Academy of Belgium, and Fellow of theRoyal Society of Chemistry.
Hua Geng
Hua Geng graduated with a MScdegree in physics from TianjinUniversity. Since 2003, she becamea research assistant with ProfessorZhigang Shuai. She received herPhD degree supervised by Prof.Zhigang Shuai from the Institute ofChemistry of the Chinese Academyof Sciences in 2012. Currently, herresearch interests are theoreticalevaluation of electron–phonon scat-terings and charge transport inorganic semiconductors.
Wei Xu
Wei Xu, received his PhD degreefrom the Chemistry Department,Wuhan University in 1997. Afterworking as a Postdoc in ProfessorDaoben Zhu’s group in theInstitute of Chemistry of theChinese Academy of Sciences, heobtained a permanent researchscientist position there.Currently, he holds a researchprofessor position. His researchinterests focus on design,synthesis, and device fabricationof organic semiconductors andthermoelectric materials.
Yi Liao
Yi Liao obtained her BSc degreeand MSc degree in the Depart-ment of Chemistry, NortheastNormal University in 1994 and2003, respectively. She receivedher PhD from Jilin University in2006 on computational studiesof organic electroluminescentmaterials under the supervisionof Prof. Jikang Feng. Afterpostdoctoral work at NanjingUniversity in 2009, she joinedthe Department of Chemistry inthe Capital Normal University in
Beijing as a full professor. Her major research interests focus on thedesign and synthesis of organic optoelectronic materials with theaid of computational chemistry.
from solvent polarization fluctuations,42 which later have beenconfirmed by Marcus himself.43,44 For organic semiconductors,Nan et al.29 have proposed the all-mode quantum chargetransfer rate formalism in line with the non-radiative rateformalism45 to investigate quantum nuclear tunneling effects,while several authors have also considered this effect by takingone effective quantum mode, keeping a classical treatment ofthe other modes.46,47 All these efforts pointed out the impor-tance of quantum effects. Especially, under such a local picturewith nuclear tunneling, the dynamic disorder was found tohave negligible effect on the mobility,48 in sharp contrast to theprevious investigations.49 Preliminary ultrafast spectroscopyexperiments indeed confirmed that the dynamic disorderdoes not play an appreciable role in charge transport at thenanosecond scale,50 rendering consolidated support for sucha picture.
Our previous review51 summarized our progress in develop-ing computational tools for the assessment of charge mobilitystarting from Marcus theory. Here we summarize our recentdevelopments in three aspects, namely, quantum nuclear tun-neling effect, dynamic disorder, and an internal coordinateprojection scheme for charge reorganization energy as well asits application for molecular design. This article is organized inthe following way: Section 2 describes the quantum nucleareffect and its application in organic semiconductors. Section 3introduces electron–phonon coupling from intra and inter-molecular vibrations evaluated from first principle quantumchemistry calculations. Section 4 presents the application inn-type semiconductors and gives some strategies towards highmobility charge transport materials. The summary and theoutlook are presented in Section 5.
2. Quantum nuclear tunneling effectfor electron transfer in organicsemiconductors
As for charge transfer in organic semiconductors, the localizedcharge strongly interacts with the high frequency intra-molecular vibration, thus demanding quantum treatments ofthe nuclear vibrations. Even at absolute-zero temperature, thesystem processes a quantum mechanical ‘‘zero-point energy’’,which will reduce the barrier between initial state to final state,quantum effect of nuclear vibration, also called nuclear tunnel-ing effect, eventually will play an important role in the chargetransfer processes. In this section, we introduce the derivation ofthe full quantum charge transfer rate based on a perturbationalapproach. In addition, we introduce our implementation of non-local electron phonon coupling by kinetic Monte-Carlo simulationunder the localized charge picture.
2.1 Methodologies for quantum nuclear tunneling anddynamic disorder effects in the hopping model
Full-mode quantum charge transfer rate can be derived fromFermi golden rule. Within Condon approximation, the contribu-tions of the electronic state and of the nuclear vibrational statescan be written separately and the charge transfer rate is:
k ¼ 2p
�h2V2Xn;n0
Pin Yfn0 Yinjh ij j2d ofn0;in� �
(1)
Here, V = |hcf|H0|cii| is the electronic coupling between thefinal (initial) electronic states, Pin is the Boltzmann distributionfunction of n-th vibrational quanta in the initial state, Yin(fn0)
is the wave function of the initial (final) nuclear vibration,and ofn0,in is the energy difference between the final and initialvibronic states.
ofn0;in ¼ ofi þXnj
Xn0j
nj0 þ 1
2
� �oj0 � nj þ
1
2
� �oj
� �(2)
Under the harmonic approximation, the nuclear vibration wavefunction can be written as a product of independent harmonic
oscillators, Yin ¼Qj
winj ðQjÞ, Yfn0 ¼Qj
wfnj0 ðQj0 Þ, and the distri-
bution function of initial state is Pin ¼Qj
Pinj . The wave func-
tion of a harmonic oscillator is
wnj ðQjÞ ¼ bj=ffiffiffipp
2njnj !� �1=2
Hnj ðbjQjÞ expð�bj2Qj2=2Þ (3)
Where bj and Hnjare the Hermite polynomials. The distribution
function for the collection of quanta {Vj} of the initial state isgiven by:
Piu ¼Xn
exp�Ein
kBT
� �" #�1exp
�Ein
kBT
� �
¼YNj
2 sinh�hoj
2kBTexp ��hoj nj þ
1
2
� �kBT
� � (4)
Jean-Marie Andre
Jean-Marie Andre received hisPhD in 1968 from the CatholicUniversity of Louvain, Belgium.After a postdoctoral stay withProf. E. Clementi at IBM SanJose, he joined the University ofNamur in 1971 as associateprofessor and then he waspromoted to be a full professorin 1974. His research interestsare methodological developmentfor the electronic structure ofpolymers and nonlinear opticalproperties. He received the
International Academy of Quantum Molecular Science AnnualMedal in 1984 for his contribution in understanding theproperties of polymers. He was awarded the Francqui Prize onExact Sciences in 1991, the highest distinction for scientists inBelgium. He is the honorary President of the Royal Academy ofBelgium, and the honorary fellow of the Chinese Chemical Society.He received the Doctor Honoris Causa from the University ofWarsaw, Poland.
Expressing the d function as a Fourier integral of time, eqn (1)becomes
k ¼ V2
�h2
ð1�1
dteitofi
Yj
GjðtÞ (5)
GjðtÞ ¼Xnj
Xn0j
Pinj wf n0jwinj
D E 2
� exp it nj0 þ 1
2
� �oj0 � nj þ
1
2
� �oj
� �� �
Under the displaced harmonic oscillator approximation, thephonon frequency and the modes of the initial (final) vibrationstate obey the following relationships: oj = oj
0, DQj = Qj0 � Qj.
Then the charge transfer rate can be expressed as:
k¼ V2
�h2
ð1�1
dt exp itofi�Xj
Sj ð2nj þ 1Þ�nje�itoj�ðnj þ 1Þeitoj �( )
(6)
Where, the intermolecular coupling V between molecules mand n can be obtained with the site-energy overlap correctionmethod:52
Vmn ¼V0
mn �1
2ðem þ enÞOmn
1�Omn2
(7)
Here, em = hFm|H|Fmi, V0mn = hFm|H|Fni, and Omn = hFm|O|Fni,
where Fm(n) is the frontier molecular orbital of the isolatedmolecule m(n) in the dimer structure. Namely, for hole (electron)transport, HOMO (LUMO) should be plugged in. H and O arerespectively the dimer Hamiltonian and the overlap matrices. %nj =1/(ehoj/kBT � 1) is the occupation number for the j-th phonon modewith frequency oj, the intra-molecular vibration modes andfrequencies can be determined by quantum chemical calculation,Sj is the Huang–Rhys factor measuring the coupling strengthbetween the carrier on-site energy and the j-th intramolecularvibration mode, which can be obtained from a normal modeanalysis approach with DUSHIN program.53 In this way, quantumeffect arising from local electron–phonon coupling (Holstein-type)naturally enters into the electron transfer process.
In the limit of strong coupling S c 1, the short timeapproximation can be applied (eiot E 1 + iot + (iot)2/2). Inthe case of �hoj/kBT { 1, i.e. high temperature approximation,the occupation number of phonons turn to %nj E kBT/�hoj, so thateqn (6) goes back to Marcus formula with l ¼
Pj
lj ¼Pj
Sj�hoj ,
where li is the charge reorganization energy through thedominating i-th intramolecular vibration mode, by its electro-nic coupling upon going from the neutral to the charged stateand vice versa.
Non-local electron–phonon coupling (Peierls type) arisingfrom intermolecular low-frequency vibration has been consideredas the major scattering sources for the charge transport in organicsemiconductors, or the dynamic disorder limited transportmodel.11 It is thus intriguing to see how dynamic disorder
influences the charge transport in the hopping model. Namely,in eqn (1), V was assumed to be constant under the Condonapproximation. In practice, V is fluctuating with time. Since theperiod of thermal fluctuation of the transfer integral is muchlarger than the time of a single charge transfer, therefore, theintermolecular motion is treated classically, as usually made inthe literature. We incorporated the non-local electron–phononcouplings in the hopping picture through a two-step approach,namely, the transfer integrals are kept constant during thecharge transfer processes and they are updated after eachhopping step.54 Classical MD simulations were performed toobtain nuclear vibration trajectories at different temperatures.For each MD snapshot, electronic coupling can be obtainedfrom quantum chemical calculation based on frontier orbitalsof a dimer. Through the discrete Fourier transformation, thetime-varying electronic coupling V can be expressed as:
VmnðtÞ ¼ Vmnh i þXN=2k¼0
ReVk � cosðoktþ j0Þ
þXN=2k¼0
ImVk � sinðoktþ j0Þ
(8)
where N is the total number of MD snapshots, Re V and Im Vare the amplitudes; on this basis, the contributions of inter-molecular vibration modes to the transfer integral fluctuationcan be obtained. The phase factor j0 for each Fourier componentis randomly set as rokt, where r is a random number uniformlydistributed in [0,1], ok is the Fourier frequency, and t is thetotal simulation time.
Only one single hopping step is especially insufficient todescribe charge transport in inhomogeneous systems, since themobility is a bulk parameter and so is related to the long rangemolecular packing. Therefore, random walk schemes to simu-late the charge diffusion using the kinetic Monte-Carlo (KMC)technique has been proposed. The flow chart of the simulationis shown in Fig. 1. First, charge transport parameters localelectron phonon coupling l, and intermolecular electroniccoupling V was evaluated from a quantum chemistry level,starting from the XRD crystal structure experimentally deter-mined and choosing an arbitrary molecule within the bulk asthe starting point for the carrier, the charge is only allowed tohop to its nearest neighbor molecules; based on charge transferparameters, full quantum charge transfer rates are computedfor all the possible neighbors, and the next position of thecharge carrier is chosen randomly from the neighbors with a
probability Pa ¼ kmn
, Pn0ðanÞ
kmn0 , the residence time of carrier
at site m is 1
Pn
kmn; combined classical MD and quantum
chemistry method, the time-varying electronic coupling Vcan be obtained, the evolution of electronic coupling obeyseqn (8); The simulation will be stopped when the diffusiondistance exceeds the intermolecular spacing by 2–3 ordersof magnitude.
Thousands of KMC simulations must be performed to get aconverged diffusion constant:
D ¼ 1
2nd lðtÞ2D E.
dt; (9)
where n is the spatial dimension, l(t) is the distance of carrier’sdiffusion. Finally, charge mobility can be obtained throughEinstein’s relation m = eD/kBT. Here, e is the electron charge,and kB is the Boltzmann constant. It has been verified that bothKMC and the Pauli master equation approach give essentiallythe same results and are found to be necessary for correctdescription for the mobility anisotropy.27
2.2 Tunneling enabled hopping model applied to pentacenederivatives
Polyacenes are typical organic semiconductors. However, theseare not soluble nor air-stable materials. A variety of chemicalmodifications based on polyacenes has been carried out inorder to achieve high mobility, solubility, and stability.6,13-Bis(triisopropylsilyl ethynyl)-pentacene (TIPS-P) is a example
of pentacene with good solubility and its charge transportproperties have attracted extensive interest.55 From electricalmeasurement, Sakanoue and Sirringhaus have recently reported a‘‘bandlike’’ transport in TIPS-P. However, when they performed acharge modulation spectroscopy on the FET device, they observeda charge induced absorption peak at 1.3 eV, the same position asin the electrochemical doping in solution, indicating that thecharge is localized on one single molecule in the FET device.56
Compared with pentacene, TIPS-P reveals a much smaller inter-molecular electronic coupling (about V B 20 meV for dimer A),57
and a much larger intramolecular reorganization energy l(around 220 meV), namely, V { l. Therefore, the localizedcharge picture is especially suitable to describe the chargetransport properties in this case.
Based on their optimized molecular geometry for the neutraland singly charged molecules, the vibration frequencies andmodes are calculated with the BhandHLYP DFT functional andwith a 6-31G(d) basis set. The corresponding Huang–Rhys factorsare obtained through normal mode analysis under the displacedharmonic oscillator approximation using the Dushin program.As can be seen in Fig. 2a, we found that the high frequencyvibration plays a dominant role in the charge reorganizationenergy, indicating that quantum effect of intramolecular vibra-tion should be considered.
Quantum charge transfer rates compared to the classical ones aregiven in Fig. 2(b). The classical Marcus CT rate increases with T, thecharacteristic of a thermal activation process. However, large chargetransfer rates which originate from the quantum mechanical zero-point vibration energy have been found at low temperature. There-fore, the quantum rate is rather insensitive and slightly decreaseswhen the temperature increases. However, the semi-classical Marcusrate only approaches the quantum rate around T B 1000 K, whichmeans that the high temperature approximation is not applicablefor organic semi-conductors at room temperature.
The dynamic disorder has been shown to play a dominantrole within the small polaron model applied with the thermalaverage approximation.55 Troisi et al. proposed a disorder-limitedtransport mechanism based on the Su–Schrieffer–Heeger model.11
However, experimentally, no improvement in charge carriermobility was observed at nanosecond time scales after pulsed
Fig. 1 Flow chart for multiscale charge diffusion simulation in organicsemiconductors.
Fig. 2 (a) Contribution of the normal modes to the charge reorganization energy l(1) from the singly positively charged molecule to the equilibriumstructure of the neutral molecule. (b) Comparison of the hole transfer rates (k): black line for full quantum rate, and red line for semi-classical Marcustheory. (c) Hole mobility: dashed line for dynamic disorder and solid line for dynamically averaged static V’s. Classical Marcus rate result is shown in theinset. Adapted with permission from ref. 56. Copyright r 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.
photoexcitation when decreasing the dynamic disorder at lowtemperatures.50 Therefore, the impact of the dynamic disorderon charge transport seems to support our approach (Fig. 2(c)).To generate many trajectories for a dimer, we adopt MolecularDynamics (MD). Then, for each snapshot, we perform quantumchemical calculations, and the thermal fluctuation or dynamicsdisorder of the transfer integral can be obtained by the proce-dure outlined before and discussed in more detail in Section 3.In order to simulate the random diffusion of the carrier, a 30 �30 � 1 crystal lattice is constructed according to the parametersof unit cell. The probability of hopping from site m to the
nearest neighbor site n is determined by Pmn ¼knP
n
kmn. By
averaging over 2000 trajectories, the diffusion constant isobtained. Finally, charge mobilities dependent on temperatureare evaluated according to the Einstein formula. As depicted inFig. 2(c), it should be noted that the tunneling enabled hopping
model gives admdT
o 0 ‘‘bandlike’’ behavior for TIPS-pentacene,
in sharp contrast to the classical Marcus theory (see insetfigure). Therefore, we estimate that the band like charge trans-port is due to the quantum effect of the intramolecular highfrequency vibration within the localized charge picture. Inaddition, the dynamic disorder has little effect on chargemobilities at lower temperature; however, it will enhance themobility at room temperature.
3. Electron–phonon couplings withboth intra- and inter-molecularvibrations
Organic molecules interact via weak van der Waals forces. As aconsequence, the electronic states available for the chargecarriers are spatially localized. Intermolecular electronic couplings(V) have been evaluated quantum chemically for many organicsemiconductors.58–60 Electron–phonon couplings are anotherimportant parameter in charge transfer processes, especially,in flexible organic semiconductors composed of light weightedatoms (C, S, N, H). The Hamiltonian including electron–phononcoupling is written as:
H0 = H0 + H1 (10)
H0 ¼Xm
emaþmam þXmn
tmnaþman þ
Xl
�hol bþl bl þ1
2
� �(11)
H1 = V localel-phonon + V non-local
el-phonon (12)
Here a+m and am denote the creation and annihilation
operators for an electron at site (molecule) m; tmn representsthe transfer integral between adjacent molecules as abovementioned; ol denotes the vibration frequency and b+
l and bl
represent the creation and annihilation operators for a phononat site l. Electron–phonon coupling is decomposed into:the local (Holstein-type) couplings and non-local (Peierls-type)
couplings, according to the modulation of the site energies andelectronic couplings:
V localel-phonon ¼
Xl
Xm
�holgl;mmðbþl þ b�lÞaþmam (13)
V localel-phonon ¼
Xl
Xman
�holgl;mmðbþl þ b�lÞaþmam (14)
Under the linear coupling approximation, the dimension-less gl,mm can be written as:
gl;mm ¼ ð�hol3Þ�1=2@em
@Ql(15)
In the case of non-polar molecules, the polarization of thesurrounding molecules can be neglected in first approximation.The local electron–phonon coupling is dominated by the individualintra-molecular relaxation, or, in the context of Marcus electron-transfer theory, by the reorganization energy l,
l ¼Xl
�holg2l;mm (16)
In the same way, the dimensionless non-local electron phononcoupling constant gl,mm reads as:
gl;mm ¼ �hol3
� ��1=2@tmm
@Ql(17)
In the limit of low and high temperature,61
s2 ¼Xl
olg2l;mm; �ho� kBT (18)
s2 ¼ 2kBTXl
g2l;mm; �ho� kBT (19)
Therefore, the strength of non-local electron–phonon couplingcan be expressed from the thermal fluctuation of the transferintegral. In view of the low frequency character of the intermolecularvibration, the condition of �ho { kBT is fulfilled, and the dynamicdisorder will follow a linear relationship with temperature.
3.1 Reorganization energy decomposition into internalcoordinates
In order to clarify the contribution of the local structurerelaxation to the reorganization energy, we have developed aninternal-coordinate decomposition approach,62 which identifiesthe segments that contribute the most to the reorganization energy.In conjunction with the frontier molecular orbital analysis, wecan design functional groups at appropriate positions so as toreduce the charge reorganization energy.
3.1.1 Methodology. Charge reorganization energies are tradi-tionally evaluated by the adiabatic potential energy surfaces (AP)method. The normal mode (NM) analysis is an alternative wayunder the harmonic oscillator approximation to obtain thevibronic relaxation contributions upon charge transfer, which,in many cases, are usually in good agreement with thoseobtained by the AP.63,64 The NM analysis can also be used tosimulate the UPS spectra. The agreement between the simulatedand experimental spectra confirms the validity of DFT estimations
of reorganization energies.63 Within the harmonic approxi-mation, the total reorganization energy can be written as:
l ¼Xi
li ¼Xi
1
2oi
2DQi2 (20)
Here, li is the reorganization energy from mode i withfrequency oi, DQi represents the displacement along the i-thnormal mode coordinate between the equilibrium positions ofthe charged and neutral states. The total reorganization energycan be obtained by the summations over all vibrational modes.It is difficult to visualize the variation of the reorganization energywith respect to local structural modifications, since the normalmodes are linear combination of all atomic displacements. Inorder to better understand the contribution of the local structureto the total reorganization energy, we proposed to decomposethe total reorganization energy into internal coordinates, whichallows the normal mode coordinates to be written as linearcombinations of internal coordinates:
DQi ¼Xj
aijDSj; (21)
Substituting eqn (21) into eqn (20), the reorganization energyin terms of internal coordinates reads as:
l ¼Xj
lðSjÞ ¼Xj
Xi
oi2
2aij2DSj
2 þXmðajÞ
aijaimDSjDSm
0@
1A
(22)
Here, lj ¼Pi
oi2
2aij2DSj
2 and ljm ¼Pi
oi2
2
PmðajÞ
aijaimDSjDSm
are the diagonal and non-diagonal terms respectively. Whenthe non-diagonal terms in eqn (22) are small enough, theinternal coordinate projection works appropriately. Assumingthat there is no Duschinsky mixing and thus that the molecularcharged state involved in electron transfer is characterized bythe same force constants as the neutral state, the chargereorganization energy can be written as a function of thevibronic coupling constants:3
l ¼Xi
li ¼Xi
Vi2
2oi2
(23)
The vibronic coupling constant Vi for vibrational mode ibecomes:
Vi ¼ cþðr;R0Þ@Hðr;RÞ@Qi
� �R0
cþðr;R0Þ
* +(24)
where, c+(r, R0) is the wavefunction of the charged state at theoptimized geometry (R0) of the neutral state. By the conversionfrom normal coordinates to internal coordinates, the reorganiza-tion energy can be expressed as a function of the vibronic couplingconstants in terms of internal coordinates:
~Vi ¼ cþðr;R0Þ@Hðr;RÞ@Sj
� �R0
cþðr;R0Þ
* +(25)
l ¼Xj
lð ~VjÞ ¼Xj
Xi
1
2oi2
aij2 ~Vj2 þ
XmðajÞ
aijaim ~Vj~Vm
0@
1A (26)
In contrast to expressions based on normal coordinates, eqn (26)provides a direct link between the reorganization energies andthe local molecular properties. It contains also the non-diagonal contributions from the internal coordinates (and therelated coupling constants). According to Koopmans’ theorem,the hole vibronic coupling constant can be written in terms ofnormal coordinates:
Vi ¼@EHOMO
@Qi(27)
In the tight-binding model, the energy of the HOMO can beexpressed:
EHOMO = C12b1,1 + C1
2b2,2 + 2C1C2b12 + 2C1C3b13 + . . .
(28)
The variation of the vibronic coupling constants uponsubstitution can be related to modifications of the atomiccharge densities:
~Vi ¼@EHOMO
@Si¼Xj;k
CjCk
@bj;k@Si
(29)
bi,j = hfi|F|fji (30)
Here, the fi terms form the atomic orbital basis in the LCAOtight-binding model, bi,i, being the site energies, and bi,j, thetransfer integrals (resonance integral) between atoms i and j. Inpractice, the derivative of the resonance integral (bi,j) between atomsi and j can be deduced from the dependence of the DFT Fockmatrixelements on the inter-atomic bond lengths. In this way, wehave established a link between the distribution of molecular orbitaland the reorganization energy of corresponding segment.
3.1.2 Case study of the substitution effect on indolo[3,2-b]-carbazole derivatives. Substitution is an effective way both toimprove molecular stacking65 and to alter carrier type.66 However,the reorganization energies have been found to increase significantlyafter chemical substitutions,67,68 thus hindering charge transport.Here, we take typical hole transport materials like indolo[3,2-b]-carbazole derivatives as an example to analyze the chemicalorigin of the reorganization energy. The parent indolo[3,2-b]-carbazole molecule and its derivatives with two chlorine substituentsat symmetrically equivalent positions are depicted in Fig. 3(a).The molecular geometries and the normal modes for the groundthe charged states have been calculated using the B3LYP hybriddensity functional in conjunction with the 6-31g(d) basis set asimplemented in the Gaussian09 package.69 The normal displace-ments DQi between the neutral state and the charged state areacquired through DUSHIN program.53
We found that the substitution position has a prominenteffect on the reorganization energy, as shown in Fig. 3(b). Thepositions of the substituent largely influences the reorganizationenergy. Indeed, compared with the parent molecule (red line),
the various substitutions can enhance reorganization energy orreduce it according to the position of the substituents. According toour internal coordinates’ decomposition approach, the total reor-ganization energy can be related to the relaxation of bond lengths.We found that the relaxation processes along the C–Cl bondssignificantly contribute to the reorganization energy, while thecontributions from other coordinates are reduced (especially forN–C1) for 1a; thus, the overall reorganization energy turns out to besmaller than that observed for the parent molecule 1. In contrast,the contributions from these bonds increase for 1b, giving a muchlarger reorganization energy than the parent molecule 1 and 1a.Namely, the pattern of chlorination significantly influences therelaxation along N–C1 bonds.
Under Koopmans’ approximation, the hole (electron) vibro-nic coupling along a specific bond is related to the product ofthe HOMO (LUMO) coefficients of the two atoms forming thebond, which is approximately equal to the bond order in theHOMO orbital if one neglects the atomic orbital overlap.The modification of the resonant integral derivatives uponchlorination is so small that it can be neglected. Therefore,the changes in relaxation energy along the N–C1 bond originatefrom the modifications of the wavefunction amplitudes on theN and C1 atoms upon substitution. As shown in Table 1, thesmaller the charge density on the C1 and N atoms, the smallerthe reorganization energy of the bond N–C1 will be. The large
contribution of the bond length of N–C1 can be understoodfrom the polarization effect of nitrogen atoms.
Upon substitution, the charge distribution at some atoms ofthe conjugated system can be modified by long-range conjugationeffects, or hyperconjugation. Indeed, if some proper substituent ata given position can influence the charge distribution at thenearby C1 atoms so as to cancel the polarization effect of nitrogenatoms, the reorganization energy will be reduced, a result which isconsistent with our molecular orbital analysis.
In summary, the reorganization energy decomposition approachmakes it possible to establish a direct link between the localgeometric structure and the reorganization energy. Accordingto the modification of charge density distribution along aparticular bond of the neutral state, we can predict the trendsthat will be observed in the modification of the charge reorga-nization energy. It allows us to choose appropriate substituentsand proper positions to design organic semiconductors withlow reorganization energy.
3.1.3 Design of charge transport materials with smallreorganization energy. N-Heteropentacene (N-PEN) derivativeshave attracted extensive interest as n-type semiconductors,70–72
which can be classified into two kinds, the hydrogenated anddehydrogenated forms, corresponding to the ‘‘CH’’/NH and‘‘CH’’/N substitutions on the PEN core, respectively. An inter-esting phenomenon concerning these N-PEN derivatives,noticed by us, is that the dehydrogenated forms always havebetter electron transport capability than the correspondinghydrogenated ones. Meanwhile, it has been found that theposition of the N atoms plays an important role in tuning thestructures and properties of organic semiconductors based onN-heteropentacenes.73 Here we will analyze the role of theposition of nitrogen atoms and of the dehydrogenation effecton the value of the reorganization energy.
To address the above issues, a series of N-heteropentacenes(TIPS-N-PENs) derivatives, as depicted in Fig. 4, have been investi-gated at the single molecular level. We applied our reorganizationenergy decomposition approach to get a better insight into theeffect of chemical modifications on the reorganization processassociated with carrier transport. the equilibrium structuresof neutral and ionic states were fully optimized for the aboveN-PENs derivatives at the B3LYP/6-31G* level.
Based on adiabatic potential-energy surfaces, the reorganiza-tion energies due to the intramolecular vibrations have beenevaluated. Upon going from hydrogenated DHD(T)APs to thearomatic D(T)APs, the hole and electron reorganization energiesdecrease from 35–106 meV and 66–265 meV, respectively. Theprominent reduction in the reorganization energies for the
Fig. 3 (a) Molecular structures of 1, 1a, 1b, c, d, and e. (b) Dependence of thecalculated reorganization energy on substitution. The red line correspondsto the reorganization energy of unsubstituted indolo[3,2-b]carbazole.(c) Reorganization energy projected into internal bond-length coordinatesfor 1a, 1b, and 1. Reprinted with permission from ref. 62. Copyright by AIPPublishing LLC.
Table 1 HOMO molecular orbital coefficients on various atoms for compounds 1, 1a, and 1b. Reprinted with permission from ref. 62. Copyright by AIPPublishing LLC
aromatic D(T)APs can be partially ascribed to the larger delo-calization degree of their frontier molecular orbitals comparedto those of the formally anti-aromatic DHD(T)APs.
According to our reorganization energy decompositionapproach, the local bonding characters of nitrogen atoms areclosely connected with the reorganization energy. Therefore,the Natural Bond Orbital (NBO) method has been used toanalyze the local bonding characters of the orbitals. As illu-strated in Fig. 5, the contributions of the lone pair of N atoms tothe HOMO and LUMO are 39% and 6% for the DHDAP-1p,respectively. Upon dehydrogenation, DAP-1p displays anincreased sp2 character on the nitrogen atoms. The C–N bondof DAP-1p contains 5% bonding characters for the HOMO and27% anti-bonding one for the LUMO. Therefore, extracting anelectron or a hole will have more pronounced effects on theDHDAP-1p than on the DAP-1p, since a larger contribution tothe geometrical relaxation is expected from the delocalized LPN
than from the bonding/anti-bonding C–N. However, a cyanosubstitution with non-bonding character of C–N has little effecton the reorganization energy.74 Therefore, it can be understoodthat DCP-2p has the smallest l among all N-PENs studied here.We conclude that the contribution of the nitrogen part to the
reorganization energy roughly follows the order: delocalizedLPN > bonding or anti-bonding N > non-bonding N. The moreintense is the delocalized LPN, the more important the reorga-nization energy will be for the hydrogenated DHDAP systems;therefore, the reorganization energies of the DHDAPs could bereadily modulated by judicious modification of the LPN com-ponents in the Froutier Molecular Orbitals. An effective strategyis to alter the position of nitrogen atoms from end rings(DHDAP-1p) to center rings (DHDAP-3p). As indicated inFig. 5, the hole reorganization energies are reduced remarkablywith the decreasing LPN components of the HOMOs fromDHDAP-1p to DHDAP-2p and to DHDAP-3p. However, consider-ing that bonding and anti-bonding interactions dominateinstead of LPN in the FMOs of DAPs, the reorganizationenergies of dehydrogenated DAPs are almost unaffected bythe position of nitrogen atoms.
3.2 Intermolecular dynamic disorder effects throughQM/MM study
There have been increasing interests on the non-local (Peierls-type) coupling in the charge-transport properties of manyorganic semiconductors.3,75 Assuming that the vibration
Fig. 4 (a) Molecular structures of the systems investigated, (b) the reorganization energies evaluated from the adiabatic potential-energy surface.Reprinted from ref. 60 with permission from The Royal Society of Chemistry.
Fig. 5 (a) The frontier molecular orbitals and nitrogen NBO lone pair (LPN) assignments of DHDAP-1p, DAP-1p and DCP-2p. (b) The LPN components inthe HOMOs of DHDAP-1p, DHDAP-2p and DHDAP-3p. Reprinted from ref. 60 with permission from The Royal Society of Chemistry.
frequency does not change with temperature, the non-localelectron–phonon coupling can be obtained through numericalderivation combined with a phonon modes calculationwithin the harmonic oscillator approximation.76,77 Classicalmolecular-dynamics simulation is an alternative way to obtaintime-dependent molecular geometries. Using these moleculargeometries, the time evolution of the electronic coupling can beobtained from quantum chemical calculation. The strength ofthe non-local electron–phonon coupling can be evaluated bythe thermal fluctuation of the transfer integral:
s2 = h(Vmn � hVmni)2i (31)
In order to better understand the non-local electron–phononcoupling dependence on molecular structure, we have chosenpristine pentacene and its derivative tips-pentacene as examples.We have adopted a molecular dynamics (MD) approach to estimatethe nuclear vibrations at different temperatures. A 5 � 5 � 3and 3� 3� 3 super cell has been chosen for tips-pentacene andpristine pentacene respectively; therefore, the dimer that we areinterested in is surrounded by many environmental molecules,so that artificial boundary effects can be prevented. The super-cell structure can be seen in Fig. 6. The MD simulations havebeen run at constant temperatures every 50 K to 300 K using theBerendsen thermostat with the COMPASS force field. Thesystem was first equilibrated for 60 ps in the NVT ensemblewith a time step of 1 fs; after equilibration, a simulation of120 ps has been run and 2000 frames were extracted by taking asnapshot every 60 fs along the trajectory.
Here, we choose the dimer A as an example. The intermole-cular electronic couplings V were calculated for each snapshot andthe thermal fluctuation can be seen in Fig. 7(a). Through discreteFourier transformation, the vibration frequency dependence onthe amplitudes of periodic function is illustrated in Fig. 7(b). Wefound that TIPS-pentacene reveals a higher vibration frequencythan that of pentacene. Both of them satisfy a Gaussian distribu-tion, although the average transfer integral of pentacene is largerthan that of tips-pentacene. The latter reveals a larger broadeningthan that of pentacene. The thermal fluctuation and dynamics
Fig. 6 (a) A 3 � 3 � 3 supercell structure of pentacene crystal, and an a–b plane projection from the supercell. Reproduced from ref. 48 with permissionfrom the Royal Society of Chemistry. (b) A 5 � 5 � 3 supercell structure of tips-pentacene crystal, and a a–b plane projection from the supercell.Reprinted with permission from ref. 56. Copyright r 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.
Fig. 7 (a) Thermal fluctuation of the transfer integral (dimer A) for pentacene(left) and tips-pentacene (right) at 300 K; (b) Fourier transformation of thermaldeviation amplitude (ReV and ImV) at 300 K; (c) probability distribution ofthe transfer integral V of dimer A. The solid line represents a Gaussian fit.(d) Square of the standard deviation of transfer integral as a function oftemperature. (left) Reproduced from ref. 48 with permission from the RoyalSociety of Chemistry. (right) Reprinted with permission from ref. 56. Copyrightr 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.
disorder increase linearly with the temperature for both systems.When �ho { kBT, the non-local electronic phonon coupling isthe slope of the standard deviation, therefore, tips-pentacenedisplays a larger non-local electron–phonon coupling than thatof pentacene. It maybe suggests that herringbone packingstructures will have a smaller thermal fluctuation than that ofbrick p�p stacking structures, further investigation will beneeded in this field.
4. Application to the molecular designof organic transport materials
Recently, tremendous progress has been achieved in molecular andpolymeric semiconducting materials. Some p-type organic materialswith a charge mobility comparable or even larger than amorphoussilicon have been experimentally found.78–81 However, the develop-ment of n-type semiconductors lags behind in both charge mobilityand air stability. Quantum chemical calculations can provide usefulinformation for molecular design in terms of the charge injectionlevels, air stability and molecular reorganization energy and inter-molecular electronic couplings. Examples have been shown to beuseful for the computational design of molecules of interest for highmobility organic semiconductors.82 We have shown that a multi-scale approach starting from a molecule to charge diffusion inthe entire bulk material can describe adequately the transportbehaviors. Such an approach has been widely applied to therationalization of experimental results on the novel n-type materialsor on the donor–acceptor mixed co-crystal for achieving ambi-polar materials. Some examples are discussed below.
4.1 Application to n-type organic semiconductors
To check the applicability of our theoretical simulation approach,we have applied it to a series of naphthalene, perylene diimide(NDI, PDI) derivatives and perfluoroalkyl modified oligothiophenewhere charge mobilities range from 0.1 to 6 cm2 V�1 s�1.79,83–88
The molecular structures are presented in Fig. 8. Experimentallymeasured charge mobility could depend on the device fabricationconditions, or on materials processing and impurities presented.Nevertheless, the theoretical results can be regarded as an intrinsicproperty of the materials, are helpful to rationalize the measure-ments and can serve as some standard to guide experiment.
The electron transfer integrals Ve between the center moleculeand all its neighbors are evaluated. The important electroniccouplings from the main hopping pathways are illustrated inFig. 8 for NDI derivatives. Among different substitutions, 1 exhibitsa 2D brick stacking configuration, showing the largest transferintegral among the eight molecules (see left bottom of Fig. 8).2 is a 1-D slipped-stacking structure possessing the second largesttransfer integral. The other molecules have smaller electronictransfer integrals, less than 50 meV.
The geometric structure and the total energy of neutral andcharge states have been calculated at the B3LYP/6-31G(d) level.The charge reorganization energies for electrons (le) fromnormal mode analysis are in good agreement with the adiabaticpotential approach for these systems, implying the adequacy ofthe harmonic oscilltor model. The magnitudes of le are all muchlarger than the intermolecular electron coupling V’s. Accordingto a normal mode analysis, we found that the high frequencymodes (1350–1700 cm�1) play a dominant role in all the systems.For example, CQO and (single and double) C–C bond stretching
Fig. 8 (Top) molecular structures of the three types of n-type semiconductors, (left bottom) crystal packing for NDI derivatives with the electronictransfer integrals V (meV) for main carrier hopping pathways P1–P3. The dashed line refers to the position of minimum reorganization energy. (Rightbottom) contribution of the normal modes to the electron reorganization energy le. The blue and red parts in the pie graph represent contributions fromC–C single and double bond stretching modes and CQO stretching modes, respectively. Reprinted with permission from ref. 54. Copyright r 2012Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.
modes contribute to 17.4% and 58.7% for molecule 1; othersystems have similar contributions from high frequency vibration.
The semiclassical Marcus theory is not adequate when thelocalized charge is strongly coupled with the high frequencyintramolecular vibration. Our tunneling-enabled hopping modelis more suitable in these cases. Combined with kinetic MonteCarlo simulation, a full quantum charge transfer rate formula isadopted to evaluate the charge mobility of these systems, asshown in Fig. 9. It should be noted that our quantum chargetransfer method correlates well with the experimental results.Marcus theory always underestimates charge mobility and eventhe trend does not correlate well with the experiment.
Fullerene (C60), naphthalene diimides (NDI), and perylenetetracarboxylic diimides (PTCDIs) are typical n-type semi-conductors. The electron mobility has reached as high as6.0 cm2 V�1 s�1 for the fullerene (C60) manufactured OFETwith a polymer as the dielectric layer. The electron mobilityof cyclohexyl substituted NDI has reached 7.5 cm2 V�1 s�1.Many derivatives of the PTCDI have been designed, synthesizedand characterized with good conducting properties.79,84 Somesubstitution groups have been introduced to improve thesolubility and air stability, with the long alkane chain to preventH2O and O2 contamination to some extent. Cyano-substitutedPTCDI derivatives have been found to have ‘‘band-like’’ char-acter from 230 K to the room temperature87 with the highestelectron mobility up to 6.0 cm2 V�1 s�1.88 Therefore, takingPTCDI as an example (Fig. 10), we have investigated the relation-ship between charge mobility and the substitution groups,and from the temperature dependence, we discuss the chargetransport mechanism.
Based on B3LYP functional and the 6-31G* basis set, thegeometric structure of the neutral state has been optimized.Upon substitution at the bay positions of the PTCDI molecule,the conjugated core keeps almost its full planarity for systems9, 10 and 11. However, the planarity of 12 is totally destroyeddue to the steric hindrance of the substitution group, which isdetrimental to the intermolecular stacking. Frontier molecularorbital profiles and energy levels are shown in Fig. 11, helpful forunderstanding the charge injection and air stability properties.The LUMO’s of system 10 and 12 have been reduced remarkably,which suggests that high air stability could be achieved as n-typetransport materials. The electron density of LUMO is mainlydistributed at the perylene core for the four systems, while moredelocalized electron distributions have been found in compounds10 and 11. Due to the non-bonding character of the cyanogroup, a lower reorganization energy is found, as expected froma model proposed by Chao et al.89,90
A normal mode analysis has been performed to investigatethe dependence of reorganization energy on intramolecularvibration. The relaxation from neutral state to charged state andvice versa reveals similar trends for the contributions from the lowfrequency region (around 500 cm�1) and the high frequency part(around 1500 cm�1). The high frequency vibration mainly comesfrom the CQC in-plane stretching vibration. Similar distributionsof the contributions to the reorganization energy of vibrationmodes have been found in the other three systems. Therefore,the quantum effect of nuclear vibration must be included in thecharge transfer processes.
When decomposing the reorganization energy into the internalcoordinates to reveal the substitution effect, as shown in Table 2,we observe that compound 12 has the largest reorganizationenergy, which is due to the C4F9 group causing additional con-tribution (about 18.2%), while the core part is almost not affectedby the substitution. Compound 10 reveals less reorganizationenergy when compared with the other systems. The contributionof the core part is reduced upon cyano substitution, since thecharge density distribution of the molecular orbital of the core partis well delocalized. Very little reorganization energy is introducedfrom the core substitution part (2.5%), which is consistent with thenon-bonding character of the molecular orbital.
Fig. 9 Correlation between experimental and calculated electron mobi-lity with full quantum and semiclassical Marcus methods. Reprinted withpermission from ref. 56. Copyright r 2012 Wiley-VCH Verlag GmbH & Co.KGaA, Weinheim.
Fig. 10 Structure scheme of systems investigated. R = CH2C3F7, X = H (9);X = CN (10); X = F (11); X = C4F9 (12).
The intermolecular stacking configurations and correspondingelectronic couplings as observed in Fig. 12 and Table 3. System 9reveals one-dimensional p–p stacking configurations. However,system 10 displays a brick stacking configuration with closerintermolecular distances, and thus larger intermolecular electroniccouplings. System 11 displays a herringbone packing character,and a large orbital overlap is found in the p–p stacking direction,while for the dimers with face to edge stacking, since the distance islarge, the intermolecular electronic couplings have been found
to be small. Due to the deformation of the core part of system12, the intermolecular distance is such that only the pathwaysP1 and P2 have noticeable electronic couplings. Therefore,system 10 possesses the best charge transport propertiesamong the four compounds, with theoretical charge mobilityas high as 16.96 cm2 V�1 s�1, in comparison with the experi-mentally measured 6.0 cm2 V�1 s�1.87
The temperature dependence for the four compounds isdepicted in Fig. 13. As expected, the nuclear tunneling modelalways presents a decreasing behavior with temperature, as itwould be in a ‘‘band-like’’ situation. Minder et al. have alsofound an experimental band like behaviour for compound 10,87
except for the thermally activated process below 210 K, sincecharge traps commonly existed in organic materials. When thecarrier concentration is large enough to fill up all the defectlevels, the nuclear tunneling effect can be found.38
Now we comment on the applicability of the Fermi golden rulehere, since it is a first-order perturbation. Often, one is surprisedby the success of the quantitative predictions by such a simpleapproximation. Recently, a more sophisticated electron wave-packet dynamics study has presented nearly identical results as
Fig. 11 Energy levels as well as charge density distribution of the HOMOs and LUMOs of compound 9–12. Reprinted with permission from ref. 90.Copyright r 2012 Elsevier.
Table 2 Decomposition of the reorganization energy of four systems onthe core (except the substitute groups), the end substitution groups andthe core substitution groups. Reprinted with permission from ref. 91.Copyright r 2012 Elsevier
Fig. 12 Electron transfer pathways for system 9–12 corresponding to (a)–(d) respectively. along short-axis view. All the alkyl chain substitutions at theend of molecules are removed for clarity. Reprinted with permission from ref. 90. Copyright r 2012 Elsevier.
the one obtained by the Fermi golden rule.91,92 This is very encoura-ging to extend the application scopes of the present procedure.
4.2 Ambipolar charge transport in fullerene–sulfur-bridgedannulene cocrystals
Ambipolar transport materials have attracted extensive interest,since (i) the fabrication of complementary-like circuits could besignificantly simplified and (ii) the integration of light-emitting
and field-effect transistors can be enabled, even achieving anelectrically pumped lasing application. Recently, it has beenfound that the organic donor–acceptor (D–A) dyads constitutean efficient way to realize ambipolar operations. Wakaharaet al.93 reported an ambipolar transistor based on C60–cobaltporphyrin (1 : 1) co-crystal with a balance electron and holemobilities in the range of 10�5–10�6 cm2 V�1 s�1. Park et al.94
have designed a donor–acceptor cocrystal with ambipolar fieldeffect mobility up to 10�3 cm2 V�1 s�1.
We have successfully synthesized a fullerene–sulfur-bridgedannulene (DPTTA) cocrystal,95 where both C60 and C70 havebeen employed as electron acceptors. The electronic structureis presented in Fig. 14. Large misalignments between frontiermolecular orbital energy levels of donor and acceptor mole-cules exclude any super-exchange interaction between donorsor between acceptors. C60 and C70 present similar adiabaticelectron affinities (2.07, 2.15 eV respectively).
Since the cocrystal presents a two-dimensional segregatedalternating layer structure, the electron transport mainly takesplace in acceptor molecular layers, while holes are in the donormolecular (DPTTA) layers. Electron reorganization energies forC60 and C70 molecules are calculated to be 135 and 142 meV,respectively. While the hole reorganization energy calculatedfor the DPTTA donor molecule is 201 meV.
Electron and hole transport pathways in C60–DPTTA are shownin Fig. 15. C70–DPTTA presents a similar transport network exceptwith different intermolecular distance and electronic coupling.The corresponding electronic coupling is listed in Table 4. UnderKoopmans’ approximation, transfer integrals for electron andhole are calculated from LUMO and HOMO, respectively. In thecase of C60 (C70) molecules, LUMOs have three-fold (two-fold)degeneracy. The electronic couplings between any two degeneratemolecular orbitals have been calculated. Supposing that thecarriers initially satisfy the Boltzmann distribution, and can hopto any one degenerate energy level, the effective transfer integralcan be derived as Veff
2 ¼Pij
PiVij2, where Pi is the Boltzmann
distribution function for the initial state.C60–DPTTA shows isotropic electronic couplings for electron
and hole. Obvious anisotropic electronic couplings can be found
Table 3 Electronic coupling (meV) along different pathways for foursystems. Reprinted with permission from ref. 90. Copyright r 2012Elsevier
for electrons in co-crystal C70–DPTTA. Similar transfer integralsfor the hole are found in both C60–DPTTA and C70–DPTTAcocrystals. Based on the above charge transfer parameters,our tunneling enabled hopping model coupled with kineticMonte Carlo simulation has been performed to obtain two-dimensional average diffusion mobilities, given in Table 5. It isseen that intrinsically, the electron mobility is much larger thanthe hole mobility. However, the experimental results showlower electron mobilities especially in the C60–DPTTA crystal.95
This might be due to the fact that there are more traps for
electrons than for holes in organic semiconductors and thesemiconductor/dielectric interface.
5. Conclusions and outlook
To summarize, we have presented a multiscale computationalapproach combining quantum charge transfer theory, moleculardynamics, and kinetic Monte Carlo simulations to assess thecharge mobility in organic semiconductors. This model featuresintramolecular high frequency vibration relaxation upon molecularcharging. Intermolecular vibration relaxation or dynamic disorderis modeled by Molecular Dynamics followed by quantum chemistrycalculations for intermolecular coupling at each snapshotas well as kinetic Monte Carlo simulation. We found that (i)mobility decreases with increasing temperature resulting froma nuclear tunneling effect instead of the claimed ‘‘bandlike’’,which can also explain the contradictory optical measurementsin TIPS-pentacene. Such an approach can also quantitativelypredict charge mobilities for a series of n-type materials,including naphthalene and perylene diimide derivatives andperfluoroalkyl modified oligothiophenes.
Chemical substitution is often used to improve the performanceand the stability of materials. But doing this often increases thereorganization energy thus lowering the mobility. We have presenteda computational scheme to decompose the reorganization energyinto molecular internal coordinates relaxations. Both substitutiontypes and positions have strong influences on the reorganizationenergy. Our approach presents a quantitative analysis on the role ofsubstitution, helpful for molecular design.
It should be noted that from both mechanism and computa-tional chemistry points of views, there are still important challengesin modeling the charge transport phenomena towards quantitativedescription and prediction.96 Organic materials cover both hoppingand bandlike behaviors and the criterion is still arguable.97
The role of both static and dynamic disorders in both chargeand exciton transport has been regarded as essential andcontinuously explorable.98 The work presented here is limited
Fig. 15 Crystal structures of C60–DPTTA and C70–DPTTA. ORTEP drawingsof the asymmetric unit of C60–DPTTA (a) and C70–DPTTA (c) showingthermal ellipsoids set at 50% probability level; stacking pattern in C60–DPTTA(b) and C70–DPTTA (d) view along the b-axis. Lower penal for illustrationof charge hopping pathways schemes in the C60 layer and DPTTA layer inco-crystal C60–DPTTA. Reprinted with permission from ref. 95. Copyright r2013 American Chemical Society.
Table 4 Transfer integrals and distances between center of mass of two molecules along different transfer pathways in the same layer in crystalC60–DPTTA and C70–DPTTA, respectively. Reprinted with permission from ref. 95. Copyright r 2013 American Chemical Society
to the dynamic disorder under the umbrella of tunnelingenabled hopping model, which needs further extension toinclude static disorder as well as charge delocalization effects.For the bandlike description, even though knowledge from tradi-tional inorganic semiconductor theory can be directly employed,the challenges arise from the description of electron–phononscattering/relaxation with dispersion effect.99 The computationalmethodology developments should go beyond the frontierorbital approximation. For example, the valence bond theoryprovided a natural way to construct the charge block state forthe charge transfer processes in a complex environment.100
Electron dynamics seems adequate to be applied to charge andexciton motion in organic systems as shown by Tretiak et al.101
Further development to include quantum nuclear effect isdesirable, since as shown in this review, the quantum nucleartunneling is important in both organic semiconductors andconducting polymers. In one word, we are still a long wayaway from a full understanding of the transport behavior inorganic materials.
Acknowledgements
This work is supported by National Natural Science Foundationof China (Grant Nos. 21290191, 21303213, 91333202) and theMinistry of Science and Technology of China through 973 program(Grant Nos. 2011CB932304, 2011CB808405, and 2013CB933503).The following collaborators have made important contributionsto the original work: Dr Linjun Wang, Dr Guangjun Nan,Dr Liping Chen, Dr Lili Lin, Prof. Yi Zhao, and Prof. Qiang Shi.JMA’s scientific stay in Tsinghua is supported by the ‘‘Oversea TopAcademic Visitor’’ (OTAV) Program of the Ministry of Education ofChina. H. Geng wishes to thank Prof. Yuanping Yi for valuablediscussions. The numerical calculations have been done in theCNIC supercomputer center of the Chinese Academy of Sciencesand the Tsinghua University Supercomputer Center.
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