optical and transport properties of single crystal rubrene ... · optical and charge transport...

Chemical Physics xxx (2016) xxx–xxx

Contents lists available at ScienceDirect

Chemical Physics

journal homepage: www.elsevier .com/locate /chemphys

Optical and transport properties of single crystal rubrene: A theoreticalstudy

http://dx.doi.org/10.1016/j.chemphys.2016.05.0120301-0104/� 2016 Elsevier B.V. All rights reserved.

⇑ Corresponding author.E-mail address: [email protected] (Y. Zhao).

Please cite this article in press as: L. Chen et al., Chem. Phys. (2016), http://dx.doi.org/10.1016/j.chemphys.2016.05.012

Lipeng Chen a, Jing Lu a,b, Guankui Long a, Fulu Zheng a, Jingping Zhang b, Yang Zhao a,⇑aDivision of Materials Science, Nanyang Technological University, Singapore 639798, Singaporeb Faculty of Chemistry, Northeast Normal University, Changchun, China

a r t i c l e i n f o

Article history:Received 31 March 2016In final form 11 May 2016Available online xxxx

a b s t r a c t

Optical and charge transport properties of single crystal rubrene are investigated using the multi-modeBrownian oscillator (MBO) model, the charge hopping model with quantum nuclear tunneling, and theMunn–Silbey approach. The MBO model is adopted to calculate absorption and photoluminescence spec-tra, yielding results in excellent agreement with measurements. In addition, temperature dependence ofzero phonon lines (ZPL) and phonon sidebands (PSBs) of absorption spectra is also examined using theMBO model, revealing a nearly linear dependence of line widths of the ZPL and the PSBs on temperature.Model parameters obtained fromMBO fitting and TD-DFT computation are then utilized for hole mobilitycalculations. It is found that temperature dependence of the calculated mobility is in general agreementwith measurements, exhibiting ‘‘band-like” transport behavior.

� 2016 Elsevier B.V. All rights reserved.

1. Introduction

Organic semiconductors based on p-conjugated polymers andsmall molecule materials have attracted increasing attention inrecent years owing to their mechanical flexibility and inexpensivemass production among a variety of promising properties. Decadesof efforts have been dedicated to the research of organic semicon-ducting materials, leading to significant progress in the design andfabrication of devices, such as high performance organic light emit-ting diodes (OLEDs) [1,2], organic field effect transistors (OFETs)[3,4], and organic photovoltaics (OPVs) [5]. However, commercial-ization of organic materials is largely limited by its relatively lowercharge carrier mobility (typically 1–10 cm2=Vs or even lower).Recent successful synthesis of single crystalline rubrene providesattractive application potentials thanks to its high hole mobility,which, in the range of 10–40 cm2=Vs [6–8], is among the highestof acenes, comparable to that of amorphous silicon. Rubrene hasalso been considered as an important yellow fluorescent dye inthe field of OLED with improved electroluminescent efficiencyand enhanced device stability and lifetime [9–12].

Despite the high hole mobility and electroluminescence yield ofsingle crystal rubrene, a comprehensive understanding of theirunderlying mechanisms remains elusive. Absorption spectra ofrubrene in amorphous thin films and solutions have been analyzedusing the independent mode displaced harmonic oscillator model

(IMDHO), and results are supported by the time-dependent (TD)density functional theory (DFT) calculations [13]. Two forms ofrubrene isomers were considered, i.e., planner and twisted, and itis found that the thin film spectrum of rubrene is dominated bythe twisted isomer. Using a combination of the DFT-GGA level cal-culations, the GW-Bether-Salpeter equation and quantum MonteCarlo calculations, optical and electronic properties of crystallinerubrene have been investigated taking into account many bodyeffects [14]. Results show that observed yellow-green photolumi-nescence (PL) in rubrene can be attributed to the formation ofintermolecular charge transfer singlet excitons, and the exchangeenergy plays a decisive role in raising the energy of intra-molecularexcitons above the intermolecular ones. A detailed study has beenperformed on dissolved rubrene molecules by analyzing their opti-cal spectra (absorption and emission) [15]. In this study, normalcoordinate analysis and frequency calculations using DFT in conju-gation with linear response TD-DFT have been applied to unveil thephysical origin of the excited state difference between rubrene andtetracene, and it is found that the difference is mainly caused bythe inductive effect of the phenyl group that leads to the HOMOdestabilization. The optical response of rubrene crystals has alsobeen studied by providing the full UV–vis dielectric tensor ofrubrene crystals [16], and results show that the emission belongsto a series with different origins with respect to transitions, andthe intermolecular interactions are weak for the lowest opticaltransitions. Temperature-dependent absorption and photolumi-nescence spectroscopy of rubrene multichromophores has beenexamined by combining the time-dependent long-range-corrected

http://dx.doi.org/10.1016/j.chemphys.2016.05.012

mailto:[email protected]


http://www.sciencedirect.com/science/journal/03010104

http://www.elsevier.com/locate/chemphys


2 L. Chen et al. / Chemical Physics xxx (2016) xxx–xxx

DFT with the exciton model [17]. Spectra of rubrene multichro-mophores up to heptamers are calculated, and the effects of exci-ton–phonon coupling and temperature on photophysicalproperties of both H- and J-aggregated oligomers are addressed.It is found that the spectral behavior of rubrene aggregates is verymuch dependent on aggregation details. PL spectroscopy has beenapplied to study mechanisms of conductance in rubrene crystals[18,19], revealing that oxygen-related impurities are responsiblefor the increase of conductivity. As a result of defects and impuri-ties in the crystalline phase, the detailed theoretical analysis of therubrene PL spectra is nontrivial [20]. Kloc and coworkers comparedobserved Raman spectra of single rubrene crystals with calculatedspectra of isolated molecules [21]. It is found that unlike manyother oligoacenes, rubrene has very weak intermolecular couplingand no observable intermolecular Raman vibrational modes. It fol-lows that intermolecular forces in rubrene are too small to be mea-surable by Raman spectroscopy, a conclusion that is consistentwith those in Ref. [16].

Coordinate-dependent electronic structures underlie chargetransport in organic crystals. A computational protocol incorporat-ing numerical evaluations of electronic and electron–phonon cou-pling was obtained by a combination of molecular dynamics (MD)simulation and semi-empirical calculation at the INDO/S level topredict charge carrier mobility of rubrene [22], in whichtime-dependent fluctuations of transfer integrals based on MD tra-jectories are utilized to compute the off-diagonal carrier-phononcoupling strength. Girlando and coworkers [23] calculated thediagonal and off-diagonal carrier-phonon coupling of orthorhom-bic rubrene from derivatives of the site energy and transfer integralwith respect to the normal mode coordinate [24,25], respectively.Li et al. calculatedthe electronic structure of rubrene crystals [26] and investigatedthe influence of off-diagonal carrier-phonon coupling on infraredoptical properties and charge transport of the rubrene crystal [27].

Charge transport in organic crystals is often handled by twosimplified treatments: the band-like transport model and the hop-ping-like approach. The former describes coherent charge trans-port, which is characterized by increased mobility at lowtemperatures. The latter captures thermally activated charge hop-ping over energy barriers exhibiting positive temperature depen-dence of mobility (i.e., the mobility increases with temperature).In pristine organic crystals, however, neither the band-like northe hopping-type models can correctly predict mobility becausecharge carrier motion in these systems is between these two limits[28–31]. Following the work of Munn and Silbey, Zhao and co-workers derived temperature-dependent mobility of a molecularcrystal incorporating both the diagonal and off-diagonal carrier-phonon coupling [32–36], which is capable to describe the twoaforementioned limits. This approach has been successfullyapplied to study anisotropic transport of holes in two-dimensionalperylene single crystals [37]. Vehoff et al. analyzed the relationshipamong the molecular structure, morphology, percolation network,and charge carrier mobility for rubrene and three other moleculesin the framework of the high-temperature non-adiabatic Marcustheory [38]. It was found that close packing and cofacially align-ment is preferable to achieve high mobilities. Nan et al. applied aquantum version of charge transfer theory coupled with random-walk simulation to evaluate the temperature dependence of theaveraged hole mobility of rubrene and tetracene [39].

In this paper, we present a detailed theoretical study on the opti-cal and charge transport properties of single rubrene crystal. Thepaper is organized as follows. In Sec. II, we present themethodologyincluding the DFT calculation and the Munn–Silbey method. Theformer is applied to calculate the electronic structure of rubrene,and the latter, to investigate the charge transport behavior inrubrene crystal. In Sec. III, detailed results on the optical and charge

Please cite this article in press as: L. Chen et al., Chem. Phys. (2016), http://dx

transport properties of rubrene are presented and compared withexperimental results. In particular, electronic structures of rubrenemonomer and dimer are obtained with the ZINDO and TD-DFTmethods, and the multimode Brownian oscillator (MBO) model isutilized to study temperature-dependent optical properties ofrubrene. The Munn–Silbey approach is then applied to calculatethemobility in rubrenewith parameters obtained from the DFT cal-culations. Conclusion are drawn in Sec. IV.

2. Methodology

2.1. DFT-based methods

Ground-state electronic structures of molecular systems areoften obtained by the DFT calculations with the B3LYP hybridexchange–correlation functional at the level of 6-31G (d) splitvalence polarized basis set. The B3LYP functional adopts theVMN functional III and the LYP expression to describe local andnon-local correlations, respectively, and has been widely used tomodel and investigate the electronic properties in polymeric andsmall molecular materials [29,40]. Molecular excited states, onthe other hand, are calculated using the TD-DFT and the semiem-pirical Zerner Intermediate Neglect of Differential Orbital (ZINDO)method parameterized to fit spectroscopic properties [41,42]. Tocompare calculated and measured spectra in a solution of toluene,solvent effects are taken into account by applying the self-consis-tent reaction field approach in the framework of polarizable con-tinuum model (PCM) [43]. Besides the information of the excitedstates, transfer integrals and reorganization energies are obtainedfrom the DFT calculations which are utilized to study the chargetransport properties in rubrene. All quantum chemical calculationsin this work are carried out using the Gaussian 09 program [44],and a frequency analysis follows to assure that the optimizedstructures are the most stable ones.

To gain insight into the measured spectra, the MBO model[45–48] is applied to simulate room-temperature absorption spec-trum of rubrene in a toluene solution. The MBO model describes atwo level system coupled to some primary oscillators which are, inturn, coupled linearly to abathof secondaryoscillators. This approachhas been successfully employed to interpret the spectra of chro-mophores in liquids [49–51] and conjugated polymers [52]. Adetailed description of theMBOmethod can be found in Appendix A.

Benefiting from the high mobility, rubrene has been applied tofabricate various devices [6–8,53–55] and in this work we alsoinvestigate the charge transport in rubrene from a theoretical per-spective. To study charge transport properties of organic crystals,we utilize the hopping-like model with the charge transfer (CT)rate between the donor and the acceptor as the most essentialparameter [56]. The well-known Marcus expression for semi-clas-sical electron-transfer rates is often used to predict the charge hop-ping rates in organic semiconductors [56–58]. It is noted that theMarcus theory treats nuclear motion classically and assumes weakelectronic coupling between donor and acceptor states. In organicsemiconductors, environmental phonons include both intramolec-ular and intermolecular modes, often with high frequencies. Thus,theory beyond the semiclassical treatment is needed in order toadequately account for the quantum aspects of the charge transferprocess. In this paper, we applied a quantummultimode formula tocalculate the charge transfer rate using the displaced harmonicoscillator approximation, which can be obtained from the FermiGolden Rule (FGR) as [59]:

k¼ J2

�h2

Z 1

�1dt�exp iDG0=�h�

Xj

Sj ð2njþ1Þ�nje�ixj t �ðnjþ1Þeixj t� �( )

ð1Þ

.doi.org/10.1016/j.chemphys.2016.05.012


L. Chen et al. / Chemical Physics xxx (2016) xxx–xxx 3

where J is the transfer integral between the initial state (donor) andthe final state (acceptor), DG0 is the variation of the Gibbs freeenergy during the charge transfer process (DG0 equals zero in theself-exchange reaction [60]), Sj is the Huang–Rhys factor character-izing the coupling strength between the charge carrier and the jthphonon mode, and nj ¼ 1=ðexp �hxj=kBT � 1Þ is the occupation num-ber for jth phonon mode with frequencyxj. In the high temperaturelimit, Eq. (1) recovers the Marcus formula for the charge transferrate. In the above FGR formula, the Huang–Rhys factor for eachmode is calculated by the normal mode (NM) analysis methodimplemented in the Dushin program [61]. In this approach, we par-tition the total reorganization energy into the contributions from

each normal mode as given by k ¼Pjkj ¼P

jMjxjDQ

2j

2 , where DQj

denotes the displacement between the equilibrium geometries ofthe neutral and the charged molecules along normal mode Qj, andMj and xj are the mass and the frequency of mode j, respectively[29]. Correspondingly, the Huang–Rhys factor Sj can be easilyobtained by the relation kj ¼ �hxjSj. The transfer integral betweenthe donor and the acceptor molecule is evaluated directly as thecoupling element between the frontier orbitals using the unper-turbed density matrix of the dimer Fock operator, which reads [62]

J12 ¼ hW1jFjW2i ð2Þwhere W1 and W2 are the frontier orbitals of the two isolated mole-cules in the dimer. F ¼ SC�C�1 is the Fock matrix, where S is thebasis set overlap matrix, and C and � represents the Kohn–Shamorbital coefficients and energies, respectively, which can beobtained by the Gaussian 09.

Given the quantum CT rates, the charge mobility can beobtained by the Einstein relation l ¼ eD=kBT , where e is the elec-tron charge and D is the diffusion coefficient. For simplicity, theisotropic diffusion coefficient can be approximately calculated byD ¼ 1

2

Pir

2i kiPi [63], where ri; ki; Pi are the center-to-center hopping

distance, the CT rate and the hopping probability (Pi ¼ ki=P

iki) forall the nearest neighboring molecules, respectively.

2.2. The Munn–Silbey approach

Intra- and inter-molecular phonons are among the crucial fac-tors that determine the charge transfer behavior in organic mole-cules. The Munn–Silbey approach [32–36] was proposed toevaluate charge transport in organic molecules in the simultaneouspresence of diagonal and off-diagonal charge-phonon coupling.Temperature-dependent mobility was only recently worked outby Zhao et al. [35,36], and application was made to investigatethe anisotropy of hole transfer in two-dimensional perylene singlecrystals [37]. In this paper we also apply the Munn–Silbeyapproach to study the charge transfer in rubrene. A generalizedHolstein Hamiltonian taking into account simultaneous diagonaland off-diagonal exciton–phonon coupling, often used to describecharge transport in organic crystals, can be written as [32–36]

H ¼Xn

eaynan þXn;m

Jnmaynam þ

Xq

xqðbyqbq þ 1=2Þ

þ N�1=2Xnmq

xqfqnma

ynamðbq þ by

�qÞ; ð3Þ

where aynðanÞ is the creation (annihilation) operator of an excitation

(i.e., an exciton or a charge carrier) with energy e, and byqðbqÞ is the

creation (annihilation) operator of a phonon with frequency xq anda wavevector q. The electronic transfer integral coupling two mole-cules n and m is given by Jnm. The last term of Eq. (3) describes theexciton–phonon coupling with f qnm denoting the linear couplingstrength.


It is more convenient to recast Hamiltonian (3) in the momen-tum representation due to its translational symmetry. Eq. (3) canbe written in the momentum representation as

H ¼Xk

ekaykak þXq

xqðbyqbq þ 1=2Þ þ N�1=2

Xkq

xqfq�ka

ykþqakðbq

þ by�qÞ ð4Þ

where ek ¼ eþ Jk, with Jk ¼P

meik�ðRn�RmÞJnm and f qk ¼

Pme

�ik�ðRn�RmÞ

f qn�m. Here, Jnm ¼ Jðdn;mþ1 þ dn;m�1Þ and f qk ¼ g � i/½sinðkÞ�sinðk� qÞ�, where J; g and / are the transfer integral, the diagonaland the off-diagonal charge phonon coupling strength, respectively.A unitary transformation can be applied to Hamiltonian (4) asfollows

H ! eH ¼ UyHU ð5Þ

with U ¼ eN�1=2P

kqAq�k

ðby�q�bqÞaykþqak . The transformation can be alterna-

tively written as

U ¼ e

Xkk0

aykSkk0 ak0

ð6Þ

where Skk0 ¼ N�1=2Ak�k0

�k ðbyk0�k � bk�k0 Þ with by

k0�k being the operator

creating a net phonon with momentum k0 � k. Thus, the excitationoperators can be written as

ak !Xk0hkk0ak0

ayk !Xk0hykk0a

yk0 ð7Þ

with

hkk0 ¼ expð�SÞ½ �kk0hykk0 ¼ expðSÞ½ �kk0 ð8ÞThe transformed Hamiltonian can be partitioned into the zer-

oth-order term (eH0) and the first-order part (V). The zeroth-order

Hamiltonian eH0 can be written as

eH0 ¼Xk

eþeJk � N�1Xq

jAqkj2xq

!aykak þ

Xq

xq byqbq þ 1=2

� �ð9Þ

with the renormalized transfer integral eJk ¼Pk0 Jk0 hhyk0khk0ki, wherethe bracket denotes the thermal average. The perturbative part Vcan be written as

V ¼Xkk0k00

fJkTkk0 ;kk00 � 2N�1 �Xq

xqfq�kA

�q�k00Tkþq;k0 ;k;k00�q þ N�1=2

�Xq

xqfq�kTkþq;k0 ;kk00 ðbq þ by

�qÞayk0ak00 Þ ð10Þ

where Tkk0 ;uu0 ¼ hykk0huu0 � hhy

kk0huu0 i. In this study, a self-consistentprocedure [35] will be utilized to obtain the transformation coeffi-cients Aq

k (the so-called A matrix) prior to the calculations of the dif-fusion coefficient. The diffusion coefficient can then be expressed as

D ¼ a2hht2k=Ckk þ ckkii ð11Þ

where a is the nearest neighbor distance, k is the wave vector, andtk ¼ rkEk with Ek denotes the polaron energy [32,34]. The doublebracket in Eq. (11) denotes the thermal average of polaron statesEk, with Ckk and ckk are the scattering and hopping rates, respec-tively. With the diffusion coefficient D, the mobility l can be easilyevaluated via the Einstein relation l ¼ eD=kBT.



Table 1Singlet excitation energies (eV) and their corresponding oscillator strength andtransition dipole moments (Debye) from ZINDO, TD-DFT calculations for a monomerand dimer along the c direction of rubrenea

System States Eex Transition dipole momentb

TD-DFT L N M osc. str.

Monomer 1 2.2382 0.0000 0.0000 �5.1895 0.22862 3.2479 0.6811 0.2300 0.0000 0.0064

Dimer 1 2.2177 0.0393 �0.1044 �7.4946 0.47256 3.2515 0.0930 �0.0776 �0.0090 0.00028 3.3009 �0.6280 �0.3354 �0.0025 0.006312 3.4909 �1.4023 �0.1146 0.0348 0.026214 3.6633 0.0881 0.0090 0.0417 0.0001

System States Eex Transition dipole moment

ZINDO L N M osc. str.

Monomer 1 2.3053 0.0000 0.0000 �7.8893 0.54412 2.8181 �6.1536 0.1600 0.0000 0.4050

Dimer 1 2.2832 �0.0163 0.0390 �11.2786 1.10144 2.8789 �8.8619 �0.1673 0.0144 0.85778 3.5565 �8.8068 �0.0843 0.0299 1.04619 3.5908 0.7283 �0.0084 �0.0043 0.007210 3.5908 0.1984 �0.0023 �0.0012 0.000512 3.8658 0.1834 �0.0116 �0.0688 0.0006

a The solvent effects have been included in TD-DFT and ZINDO calculations andthe dielectric constant � is 4.71.

b M, L and N denote the short, long and normal axes of the tetracene backbone ofrubrene respectively.


3. Results and discussions

3.1. Electronic structure and Optical properties

We adopt the structural parameters of rubrene crystal asreported in Ref. [64]. The rubrene crystal forms an orthorhombicunit cell with space group Bbcm D18

2hð64Þ and lattice parametersa ¼ 7:187 Å; b ¼ 14:430 Å, and c ¼ 26:901 Å. The molecular andcrystal structures of rubrene are shown in Fig. 1. It is clear thatthe packing of the ab plane is most beneficial energetically duringthe growth of the rubrene crystal, thus this plane should have thelargest surface area in the macroscopic crystal, as demonstratedexperimentally [64,65].

The lowest absorption peak of rubrene measured in chloroformis about 2.35 eV [66,67], and the 0–0 absorption band (or the zerophonon line (ZPL)) in rubrene single crystal is located at 2.32 eV[19]. In Table 1, we present the singlet excitation energies and theircorresponding oscillator strengths for a monomer and a dimeralong the c direction of rubrene calculated by the ZINDO and theTD-DFT methods. The results show that ZINDO underestimatesthe excitation energies for the pp� optical transition by 0.04 eVas compared with experimental data (2.35 eV), while TD-DFTunderestimates by 0.11 eV. The rubrene monomer is studied first.In the ZINDO calculation, the first excited state (S1) has the largestoscillator strength, and its transition dipole moment lies along theM direction in agreement with Tavazzi et al. [16]. Similar to theZINDO approach, the TD-DFT method also produces an S1 statewith the strongest oscillator strength for the rubrene monomerand the transition dipole moment for this state is also along theM direction. Rubrene dimers along the c direction are studied aswell. In the ZINDO method, the lowest dipole-allowed transitionsare from the S1 and S4 states with oscillator strengths of 1.1014and 0.8577, respectively, while they are replaced by the S1 statein the TD-DFT method. A close inspection on the transition orbitalshow that the S1 transition is mainly composed of HOMO! LUMO(47%) and HOMO-1! LUMO+1(47%) transition in the ZINDO calcu-lation, while it is mainly composed of HOMO! LUMO(51%) andHOMO-1! LUMO+1(49%) transition in the TD-DFT calculation.

Fig. 1. Molecular and crystal structure of rubrene. The ab, ac and bc surfaces of the crydirection of two nearest neighbor molecules along a, b and c directions.


We also note that the two adjacent transition dipoles are approxi-mately in a shoulder-to-shoulder configuration (H aggregation)along the a and b directions, and in a head-to-tail configuration (Jaggregation) along the c direction. This adjacent transition dipolearrangement can explain the red-shifted ZPL in the absorptionspectrum of single crystal rubrene compared to that measured insolution. If the absorption spectra of the rubrene crystal is mea-sured along the c direction [66,67,19], it will lead to J-aggregatebehavior in the absorption spectra of single crystal rubrene whichhas a red-shifted ZPL.

We then apply the MBO model to probe the underlying physicsof the rubrene absorption and photoluminescence (PL). In the MBO

stal are shown as above. The three arrows on the ab and ac surfaces indicate the



Fig. 2. The experimental and MBO absorption and PL spectra of rubrene attemperature T = 300 K in toluene solution. The MBO parameters used for theabsorption and PL spectra simulation are taken from Table 2.

Fig. 3. Effect of temperature on the MBO absorption spectra of rubrene in toluenesolution. The MBO parameters used for the absorption simulation are:x1 = 1375 cm�1;x2 = 70 cm�1; c1 = 200 cm�1; c2 = 200 cm�1; S1 = 1.1, S2 = 4.5. Theinset is the FWHM vs T plot for various peaks.


model, the control parameters are the primary phonon mode fre-quency x, the Huang–Rhys factor S, the damping factor c and thetemperature T. The effects of these control parameters on the line-widths of the ZPL and the phonon side bands (PSBs) have been wellstudied by Ye et al. [48]. In the fitting procedure, we use a high fre-quency mode x1 with a small Huang–Rhys factor S1 and a low fre-quency mode x2 with a large Huang–Rhys factor S2. The initialvalues of S1 and x1 are usually taken as the intensity ratio andspacing between the first and zeroth phonon lines of experimentalspectra, respectively. The initial values of S2 and x2 are usuallytaken as 1 (strong coupling) and 100 cm�1, respectively [52]. Thepurpose of introducing a relatively strongly coupled low frequencymode is to account for the broadening of ZPL and PSBs, while theoverall pattern of spectra is controlled by the high frequency modestrongly coupled with the electronic transition. The damping factorc quantifies the dissipative effect of the Markovian bath. Theparameters for the MBO fitting is listed in Table 2, and the corre-sponding room temperature absorption spectra and the PL ofrubrene in the toluene solution are shown in Fig. 2. As can be seenfrom Fig. 2 that the fitted spectra match the experimental onesextremely well except for the high energy regime. Because ofnon-radiative decay after photo-excitation, the emitting statescan differ from the absorbing states, leading to different MBOparameters for PL and absorption spectra. The larger Huang–Rhysfactor S obtained for absorbing process as compared with that foremission process suggests that the charge couples to the vibra-tional modes more strongly in the excitation process than that inthe emission process. The primary phonon energy (1375 cm�1),which is responsible for the in plane stretching mode of the ben-zanthracene backbone in the rubrene molecule or the stretchingmode along the L direction, is in good agreement with the experi-mental absorption spectra (1346 cm�1).

We also studied the temperature effect on the rubrene absorp-tion spectra. Fig. 3 depicts the absorption spectra of rubrene in thetoluene solution at temperatures ranging from 50 to 300 K with theinset being the full width at half-maximum (FWHM) versus tem-perature plot for various absorption peaks. In general, increasingtemperature from 50 to 300 K tends to broaden both ZPL (peak1) and PSBs (peaks 2 and 3) and reduce their intensities. This canbe understood as follows: At higher temperatures, more bathmodes will be excited, and accordingly the stronger interactionswith the dissipative bath reduce the excited state lifetime, leadingto broadening on the ZPL and PSBs. Ye et al. [48] investigated theinfluence of temperature on the absorption spectra from 25 K to100 K, and found that the linewidths of the ZPL and PSBs are lin-early dependent on the temperature for an underdamped Brown-ian oscillator model with an Ohmic dissipative bath. Knox andcoworkers [68] studied the optical spectra of chromophores in hostsolids at low temperatures. They found that the ZPL linewidth van-ishes in the low temperature limit due to the suppression of fluc-tuations of the bath. In the Ohmic damping case, the linewidth ofZPL and the one-phonon peak change linearly with temperatureat low temperatures, which can be directly related to the lineardependence of real part of the line shape function on temperatureand agrees with the results in Ref. [48]. As shown in the inset ofFig. 3, we found that the linewidth of the peaks increases almostlinearly with temperature for low temperatures region, consistentwith [48,68]. However, the relation of FWHM and T deviates sub-stantially from a linear one when the temperature increase further.

Table 2Fitted MBO Parameters for the absorption and PL spectra of rubrene in toluene solution (T

x1ðcm�1Þ x2ðcm�1Þ c1ðcm�1ÞAbs 1375 70 200PL 1000 70 350


Through an accurate approximation for a hyperbolic cotangentfunction, Jang et al. [69] derived simple expressions of the line-shape function for a two level system linearly coupled with har-monic oscillator baths at arbitrary temperature. Their resultsshowed that in the strong-coupling limit, the line-shape functiontakes a Gaussian form, and the temperature dependence of thelinewidth seems almost linear. While in the weak-coupling limit,the line-shape function depends critically on the dimensionalityof the spectral density.

3.2. Mobility

According to Eqs. (1) and (3), the transfer integral and the elec-tron–phonon coupling strength are the key parameters that controlthe charge transport properties in organic crystals. First we use Eq.(2) to calculate the hole and electron transfer integrals of rubrene.Along the a direction of rubrene (see Fig. 1), the calculated hole and

= 300 K).

c2ðcm�1Þ S1 S2 �hxegðeVÞ200 1.100 4.500 2.31200 0.615 2.900 2.26



Fig. 4. Contribution of the neutral (a) and cationic state (b) vibrational modes tohole reorganization energy of rubrene as calculated at the DFT/B3LYP level with the6-31G(d) basis set.

Fig. 5. Temperature-dependent absolute mobilities calculated by FGR and modelHamiltonian methods as well as experimentally measured OFET mobility [28].


electron transfer integral are 0.105 and 0.0568 eV, respectively,while in other directions transfer integrals are negligible, consis-tent with values calculated from HOMO and LUMO bandwidths[58]. This is also in agreement with the experimental findings[28] which show that rubrene exhibits high hole mobility alongthe a direction.

To get a more complete understanding of charge transportproperties, we need to evaluate the corresponding intramolecularreorganization energy for hole transport in rubrene. The reorgani-zation energy can be calculated from the adiabatic potential (AP)energy surface method [29], which is written as:

k ¼ Ecn � Ecc þ Enc � Enn ð12Þ

where Ecn is the energy of the charged molecule (anion or cation) atthe optimal geometry of the neutral molecule, Enc is the energy ofthe neutral molecule at the optimal charged molecule geometry,and Enn and Ecc are the ground state energy of the neutral stateand the energy of the charged molecular state, respectively. The cal-culated total internal reorganization energy for hole transport ofrubrene is 0.1524 eV, implying that rubrene is in the intermediatehole-phonon coupling regime. In order to investigate contributionsof each individual vibrational mode to the hole reorganizationenergy, we apply NM analysis method [61]. Within the harmonicapproximation, the total reorganization energy can be decomposedinto the contributions of each vibrational mode as:k ¼Piki ¼

Pi�hxiSi, where ki denotes the reorganization energy

from vibrational mode i with frequency xi. The reorganizationenergy for holes calculated by the NM analysis method is in goodagreement with that from the AP method, implying the adequacyof the harmonic approximation. Fig. 4 shows contributions of indi-vidual vibrational modes to the total hole reorganization energy.According to the normal mode analysis, we found that most modesthat contribute to the reorganization energy are high frequencyones in the range of 1200–1600 cm�1, which can be attributed toC@C and CAC bond stretching modes. In addition to the high-fre-quency modes, we also found that a low-frequency vibrationalmode (at 25 cm�1), corresponding to a bending of the phenyl sidegroups around the tetracene backbond, has a large hole-phononcoupling constant (Sj = 2.64).

Upon obtaining transfer integrals and hole-phonon couplingconstants based on the DFT and the NM analysis, we can calculatethe mobility of holes in rubrene using the FGR and the Munn–Sil-bey method as described above. The input parameters for the FGRmethod are the transfer integral J and the Huang–Rhys factor Sj foreach phonon mode with frequency xj (see Eq. (1)). While the for-mer is calculated by Eq. (2), the latter is obtained through NM anal-ysis method. For the Munn–Silbey approach, the input parametersare the transfer integral J, the characteristic phonon frequencyxLO,the phonon bandwidth Dx and the diagonal and off-diagonal hole-phonon coupling g and /. We set the characteristic phonon fre-quency xLO to 1450 cm�1 which is obtained from the MBO fittingof experimental spectra of rubrene single crystal, and Dx as 1/10ofxLO (i.e., a nearly dispersionless phonon spectrum). The diagonalhole-phonon coupling strength g is obtained from the relationS ¼ g2, where the Huang–Rhys factor S is calculated to be 0.849(S ¼PjSjxj=xLO). The off-diagonal hole-phonon coupling constantis 0.0479 as extracted from Ref. [27]. In Fig. 5 we report the tem-perature dependent absolute mobilities along the a axis calculatedby the FGR and the Munn–Silbey method, and compare them to themeasured OFET mobility [28]. Calculated temperature dependentmobilities by the Munn–Silbey method are in general agreementwith the measured ones, while the FGRmethod yields mobility val-ues 2–3 times larger. It is noted that other complex factors such asthe presence of interface, oxide layer and disorder caused by grainboundaries may also play important roles in the charge transport


process. In Ref. [28], the authors argue that the intrinsic transport(i.e., not limited by static disorder) at high temperatures (�150–300 K) does not necessary imply that the trapping is completelyeliminated. At high temperatures, the time a polaron spends resid-




ing in a shallow trap is much shorter than that it takes to hopbetween traps, and the effective drift mobility in the multiple trap-ping and release (MTR) model [70,28] is reduced to the intrinsicmobility. When the temperature is lower than 150 K, multipletrapping dominates charge transport and there is a crossover fromintrinsic to thermally activated hopping transport [28]. The micro-scopic model used in this work captures intrinsic transport proper-ties, and provides a clear physical picture for the complex interplaybetween the charge carrier and the vibrational modes and theireffects on charge transport. Using the Munn–Silbey approach, Chenet al. have studied the charge transport in a broad temperaturerange in the weak and intermediate charge-coupling regimes[36]. They found that the ‘‘band-like” transport is dominant atlow temperatures and the mobility decreases with the increasedtemperature. When the temperature is sufficiently high, the mobil-ity is mainly contributed by ‘‘hopping-like” transport. The temper-ature range of 160–300 K investigated in this work is relatively lowcompared to the characteristic phonon frequency. As shown inFig. 5 our calculated mobilities decrease with the increase of thetemperature, indicating that hole mobilities have ‘‘band-like”character in the temperature regime studied.

4. Conclusion

We have investigated the electronic structure of the rubrenecrystal with quantum chemistry techniques, and obtained transferintegrals and reorganization energies are utilized to study thecharge transport behavior in rubrene crystal. The MBO model isapplied to calculate the absorption and the PL spectra, with resultsin excellent agreement with measurements. The dependence of thelinewidths of ZPL and PSBs of absorption spectra on the tempera-ture are also studied with the MBO model. The results reveal lineardependence of linewidths of the ZPL and PSBs on the temperature.We have also studied charge transport in rubrene crystal makinguse of the Munn–Silbey approach and incorporating simultaneousdiagonal and off-diagonal coupling. In the temperature range of160–300 K, the hole mobility in rubrene single crystal as computedby the Munn–Silbey approach exhibits the ‘‘band-like” transportbehavior, which decreases with the increase of temperature. Thetemperature dependence of the hole mobility is generally in agree-ment with measurements. Deviation of the Munn–Silbey mobilityfrom the measurements may originate in the negligence of defectsand impurities which cannot be completely eliminated in purifica-tion procedures.

Acknowledgments

The authors thank Yao Yao and Christian Kloc for useful discus-sion. Support from the Singapore National Research Foundationthrough the Competitive Research Programme (CRP) under ProjectNo. NRF-CRP5-2009-04 is gratefully acknowledged.

Appendix A. MBO

The MBO model provides a convenient channel for incorporat-ing the coupling of vibrational motions to optical transitions. Inthe MBOmodel, we treat the nuclear motion in two different ways.We first consider some primary nuclear modes, which are stronglycoupled to the optical transitions. These nuclear motions can bemodeled as independent harmonic oscillators, which can bedescribed as [47,48]

H ¼ giHghgj þ jeiHehe�� ðA:1Þ

with


Hg ¼Xj

p2j

2mjþ 12mjx2

j q2j

" #ðA:2Þ

He ¼ �hxeg þXj

p2j

2mjþ 12mjx2

j ðqj þ djÞ2" #

ðA:3Þ

where xj;pj;mj and qj are the angular frequency, the momentum,the coordinate and the mass of the jth primary oscillator. dj is thedisplacement of the jth nuclear mode in the electronic excited state.�hxeg is the energy difference between the two-level system. Therest of nuclear degrees of freedom can be treated as a bath of sec-ondary phonons. The Hamiltonian describing the coupling betweenthe primary oscillators and the bath modes is given by

H0 ¼Xn

p2n

2mnþ 12mnx2

n qn �Xj

cnjqj

mnx2n

!224 35 ðA:4Þ

where xn; pn;mn; qn are the angular frequency, the momentum, thecoordinate and the mass of the nth bath mode. cnj is the couplingstrength between nth bath mode and jth primary mode.

The energy gap operator may be defined as

U ¼ He � Hg � �hxeg ðA:5ÞThen, the correlation function in the time domain for the jth

mode can be written as

CjðtÞ ¼ � 1

2�h2 hUðtÞUqgi � hUUðtÞqgih i

ðA:6Þ

where the operator UðtÞ is the interaction representation of theoperator U. qg is the equilibrium ground state vibrational densitymatrix defined as

qg ¼jgihgj expð�bbHgÞTr½expð�bbHgÞ�

ðA:7Þ

where b ¼ 1=kBT . The correlation function can be converted to fre-quency domain by the Fourier transformation, and its imaginarypart is known as the spectral density

eC 00j ðxÞ ¼ 2kjx2

j xcjðxÞx2c2j ðxÞ þ x2

j þxRjðxÞ �x2h i2 ðA:8Þ

where RjðxÞ is the real part of the self-energy, 2kj is the Stokes shiftof the jth mode

kj ¼mjx2

j d2j

2�h¼ Sj�hxj ðA:9Þ

where Sj is the dimensionless Huang–Rhys factor describing theexciton–phonon interaction strength.

In this work, we adopt a simple form of the MBO model, thespectral distribution function cjðxÞ is assumed to be in its Ohmiclimit (i.e., cjðxÞ = constant). For simplifications, we set the self-energy term RjðxÞ = 0. Under this condition, the spectral densityfor the jth primary oscillator has the following form

C 00j ðxÞ ¼ 2kjx2

j xcjx2c2j þ ðx2

j �x2Þ2ðA:10Þ

The PL and absorption line shape can be calculated as

IPLðxÞ ¼ 1pReZ 1

0exp½iðx�xeg þ kÞt � g�ðtÞ�dt ðA:11Þ

IabðxÞ ¼ 1pRe

Z 1

0exp½iðx�xeg � kÞt � gðtÞ�dt ðA:12Þ




with k ¼Pjkj. gðtÞ is the lineshape function, defined as

gðtÞ ¼ 12p

Z 1

�1dx

C 00ðxÞx2 1þ cothðb�hx=2Þ½ �½e�ixt þ ixt � 1�

ðA:13Þwhere function C00ðxÞ is the summation of C00

j ðxÞ(C00ðxÞ ¼PjC00j ðxÞ).

References

[1] F. Huang, P.I. Shih, C.F. Shu, Y. Chi, A.K.Y. Jen, Adv. Mater. 21 (2009) 361.[2] S.J. Su, T. Chiba, T. Takeda, J.J. Kido, Adv. Mater. 20 (2008) 2125.[3] H. Yan, Z.H. Chen, Y. Zheng, C. Newman, J.R. Quinn, F. Dötz, M. Kastler, A.

Facchetti, Nature 457 (2009) 679.[4] H.N. Tsao, D. Cho, J.W. Andreasen, A. Rouhanipour, D.W. Breiby, W. Pisula, K.

Müllen, Adv. Mater. 21 (2009) 209.[5] J.H. Hou, H.Y. Chen, S.Q. Zhang, G. Li, Y. Yang, J. Am. Chem. Soc. 130 (2008)

16144.[6] V. Podzorov, E. Menard, A. Borissov, V. Kiryukhin, J.A. Rogers, M.E. Gershenson,

Phys. Rev. Lett. 93 (2004) 086602.[7] I.N. Hulea, S. Fratini, H. Xie, C.L. Mulder, N.N. Iossad, G. Rastelli, S. Ciuchi, A.F.

Morpurgo, Nat. Mater. 5 (2006) 982.[8] J. Takeya, M. Yamagishi, Y. Tominari, R. Hirahara, Y. Nikazawa, T. Nishikawa, T.

Kawase, T. Shimoda, S. Ogawa, Appl. Phys. Lett. 90 (2007) 102120.[9] T.H. Liu, C.Y. Iou, C.H. Chen, Appl. Phys. Lett. 83 (2003) 5241.[10] D.J. Fatemi, H. Murata, C.D. Merritt, Z.H. Kafafi, Synth. Met. 85 (1997) 1225.[11] F. Li, J.L. Lin, J. Feng, G. Cheng, H.Y. Liu, S.Y. Liu, L.G. Zhang, X.F. Zhang, S.T. Lee,

Synth. Met. 139 (2003) 341.[12] Z.L. Zhang, X.Y. Jiang, S.H. Xu, T. Nagatomo, O. Omoto, Synth. Met. 91 (1997)

131.[13] M. Kytka, L. Gisslen, A. Gerlach, U. Heinemeyer, J. Kovac, R. Scholz, F. Schreiber,

J. Chem. Phys. 130 (2009) 214507.[14] N. Sai, M.L. Tiago, J.R. Chelikowsky, F.A. Reboredo, Phys. Rev. B. 77 (2008)

161306.[15] T. Petrenko, O. Krylova, F. Neese, M. Sokolowski, New. J. Phys. 11 (2009)

015001.[16] S. Tavazzi, A. Borghesi, A. Papagni, P. Spearman, L. Silvestri, A. Yassar, A.

Camposeo, M. Polo, D. Pisignano, Phys. Rev. B. 75 (2007) 245416.[17] F. Gao, W.Z. Liang, Y. Zhao, J. Phys. Chem. A 113 (2009) 12847.[18] O. Mitrofanov, C. Kloc, et al., Appl. Phys. Lett. 91 (2007) 212106.[19] O. Mitrofanov, D.V. Lang, C. Kloc, J.M. Wikberg, T. Siegrist, W.Y. So, M.A.

Sergent, A.P. Ramirez, Phys. Rev. Lett. 97 (2006) 166601.[20] Y. Chen, B. Lee, D. Fu, V. Podzorov, Adv. Mater. 23 (2011) 5370.[21] J.R. Weinberg-Wolf, L.E. McNeil, et al., J. Phys.: Condens. Matter 19 (2007)

276204.[22] A. Troisi, Adv. Mater. 19 (2007) 2000.[23] A. Girlando, L. Grisanti, M. Masino, et al., Phys. Rev. B 82 (2010) 035208.[24] V. Coropceanu, R.S. Sánchez-Carrera, P. Paramonov, G.M. Day, J.L. Brédas, J.

Phys. Chem. C 113 (2009) 4679.[25] R.S. Sánchez-Carrera, P. Paramonov, G.M. Day, V. Coropceanu, J.L. Brédas, J. Am.

Chem. Soc. 132 (2010) 14437.[26] Y. Li, V. Coropceanu, Jean-Luc Brédas, J. Phys. Chem. Lett. 3 (2012) 3325.[27] Y. Li, Y.P. Yi, V. Coropceanu, Jean-Luc Brédas, Phys. Rev. B 90 (2014) 245112.[28] V. Podzorov, E. Menard, A. Borissov, V. Kiryukhin, J.A. Rogers, M.E. Gershenson,

Phys. Rev. Lett. 93 (2004) 086602.


[29] V. Coropceanu, J. Cornil, D.A. da Silva Filho, Y. Olivier, R. Silbey, J.L. Brédas,Chem. Rev. 107 (2007) 926.

[30] W. Zhang, W. Liang, Y. Zhao, J. Chem. Phys. 133 (2010) 024501.[31] Y. Yao, W. Si, X.Y. Hou, C.Q. Wu, J. Chem. Phys. 136 (2012) 234106.[32] R. Silbey, R.W. Munn, J. Chem. Phys. 72 (1980) 2763.[33] R.W. Munn, R.J. Silbey, J. Chem. Phys. 83 (1985) 1843.[34] R. Silbey, R.W. Munn, J. Chem. Phys. 83 (1985) 1854.[35] Y. Zhao, D.W. Brown, K. Lindenberg, J. Chem. Phys. 100 (1994) 2335.[36] D.M. Chen, J. Ye, H.J. Zhang, Y. Zhao, J. Phys. Chem. B. 115 (2011) 5312.[37] H. Jiang, K.K. Zhang, J. Ye, F. Wei, P. Hu, J. Guo, C. Liang, X. Chen, Y. Zhao, L.E.

McNeil, W. Hu, C. Kloc, Small 9 (2012) 990.[38] T. Vehoff, B. Baumeier, A. Troisi, D. Andrienko, J. Am. Chem. Soc. 132 (2010)

11702.[39] G.J. Nan, X.D. Yang, L.J. Wang, Z.G. Shuai, Y. Zhao, Phys. Rev. B. 79 (2009)

115203.[40] N. Kuritz, T. Stein, R. baer, L. Kronik, J. Chem. Theory. Comput 7 (2011) 2408.[41] M.C. Zerner, G.H. Lowe, R.F. Kirchner, U.T. Muller-Westerhoff, J. Am. Chem. Soc.

102 (1980) 589.[42] M.C. Zerner, P. Correa de Mello, M. Hehenberger, Int. J. Quantum Chem. 21

(1982) 251.[43] M. Cossi, V. Barone, R. Cammi, J. Tomasi. Chem. Phys. Lett. 255 (1996) 327.[44] M.J. Frisch. et al. Gaussian 09, revision A.1; Gaussian Inc: Wallingford, CT,

2009.[45] S. Mukamel, Principles of Nonlinear Optical Spectroscopy, Oxford University

Press, New York, 1995.[46] Y. Zhao, V. Chernyak, S. Mukamel, J. Phys. Chem. A. 102 (1998) 6614.[47] Y. Zhao, R.S. Knox, J. Phys. Chem. A. 104 (2000) 7751.[48] J. Ye, Y. Zhao, N. Ng, J.S. Cao, J. Phys. Chem. B. 113 (2009) 5897.[49] L.T. Su, A.I.Y. Tok, Y. Zhao, N. Ng, F.Y.C. Boey, J.L. Woodhead, C.J. Summers, J.

Phys. Chem. B. 112 (2008) 10830.[50] S.L. Shi, G.Q. Li, S.J. Xu, Y. Zhao, G.H. Chen, J. Phys. Chem. B. 110 (2006) 10475.[51] S.J. Xu, G.Q. Li, Y.J. Wang, Y. Zhao, G.H. Chen, D.G. Zhao, J.J. Zhu, H. Yang, D.P.

Yu, J.N. Wang, Appl. Phys. Lett. 88 (2006) 083123.[52] J. Ye, A.C. Grimsdale, Y. Zhao, J. Phys. Chem. A. 114 (2010) 504.[53] V.C. Sundar, J. Zaumseil, V. Podzorov, E. Menard, R.L. Willett, T. Someya, M.E.

Gershenson, J.A. Rogers, Science 303 (2004) 1644.[54] H. Najafov, B. Lee, Q. Zhou, L.C. Feldman, V. Podzorov, Nat. Mater. 9 (2010) 938.[55] H. Alves, R.M. Pinto, E.S. Maçôas, Nat. Commun. 4 (2013) 1842.[56] R.A. Marcus, N. Sutin, Biochim. Biophys. Acta. 811 (1985) 265.[57] J.L. Brédas, J.P. Calbert, D.A. da Silva Filho, J. Cornil, Proc. Natl. Acad. Sci. U.S.A.

99 (2002) 5804.[58] D.A. da Silva, E.G. Kim, J.L. Brédas, Adv. Mater. 17 (2005) 1072.[59] S.H. Lin, C.H. Chang, K.K. Liang, R. Chang, Y.J. Shiu, J.M. Zhang, T.-S. Yang, M.

Hayashi, F.C. Hsu, Adv. Chem. Phys. 121 (2002) 1.[60] Y. Zhao, W.Z. Liang, Chem. Soc. Rev. 41 (2012) 1075.[61] J. Reimers, J. Chem. Phys. 115 (2001) 9103.[62] A. Troisi, G. Orlandi, J. Phys. Chem. B. 106 (2002) 2093.[63] W.Q. Deng, W.A. Goddard III, J. Phys. Chem. B. 108 (2004) 8614.[64] O.D. Jurchescu, A. Meetsma, T.T.M. Palstra, Acta Cryst. Sec. B 62 (2006) 330.[65] B.D. Chapman, A. Checco, R. Pindak, T. Siegrist, C. Kloc, J. Cryst. Growth 290

(2006) 479.[66] Y. Hamada, H. Kanno, T. Tsujioka, H. Takahashi, T. Usuki, Appl. Phys. Lett. 75

(1999) 1682.[67] A. Otomo, S. Otomo, S. Yokoyama, S. Mashiko, Opt. Lett. 27 (2002) 891.[68] R.S. Knox, G.J. Small, S. Mukamel, Chem. Phys. 281 (2002) 1.[69] S. Jang, J.S. Cao, R.J. Silbey, J. Phys. Chem. B. 106 (2002) 8313.[70] G. Horowitz, AdV. Mater. 10 (1998) 365.


http://refhub.elsevier.com/S0301-0104(16)30281-6/h0005






































































































optical and transport properties of single crystal rubrene ... · optical and charge transport...

Documents