Table Of ContentCharge mobility of discotic mesophases: a multiscale quantum/classical study
James Kirkpatrick,1 Valentina Marcon,2 Jenny Nelson,1 Kurt Kremer,2 and Denis Andrienko2
1Department of Physics, Imperial College London,
Prince Consort Road, London SW7 2BW, United Kingdom
2Max-Planck-Institut fu¨r Polymerforschung, Ackermannweg 10, 55128 Mainz, Germany
7 (Dated: February 5, 2008)
0
A correlation is established between the molecular structure and charge mobility of discotic
0
mesophasesofhexabenzocoronenederivativesbycombiningelectronicstructurecalculations,Molec-
2
ular Dynamics, and kinetic Monte Carlo simulations. It is demonstrated that this multiscale ap-
n proach can providean accurate ab-initio description of charge transport in organic materials.
a
J
6 Organic photovoltaic (OPV) and other organic opto- Akeyquestionis the influence oflocalorderingin dis-
1
electronic devices rely upon blend films of two material cotic mesophases on charge transport. Computer sim-
phases, combining the attributes of a large interfacial ulation is a powerful tool for probing the influence of
]
i area for charge separation and recombination and effi- molecular structure on charge transport. However, the
c
s cientverticaltransportpaths. Transportinorganicelec- task is challenging because different theoretical methods
- tronic materials proceeds by incoherent hopping and de- are needed to describe different length- and time-scale
l
r pends upon local molecular ordering as well as on the phenomena. Atomistic simulations are needed for local
t
m presence of percolation paths, and therefore is an intrin- molecular arrangements [4], quantum chemical calcula-
. sically multiscale property. Improved mobilities are cru- tions are needed for the electron transfer mechanisms
t
a cial to improved OPV device performance, and in turn (and interaction with the electrodes) [5], and stochas-
m
require control of structure on both the intermolecular tic or rate-equation dynamic methods for simulation of
- and macroscopic length scales. charge dynamics. In this paper, we present the first sta-
d
tisticalmechanicsevaluationofthechargemobilitybased
n
Discotic liquid crystals could be ideal materials for on quantum chemical and atomistic MD methods. We
o
c OPV as they offer optimal design possibilities for both employ such a three level approach to explain the side
[ the charge mobility and the necessary underlying meso- chain dependence of charge mobility for differently sub-
1 scopic morphology. Discotic thermotropicliquid crystals stituted HBC derivatives.
v areformedbyflatmoleculeswithacentralaromaticcore Previous studies of liquid crystalline triphenylene
2 and aliphatic side chains [1]. They can form columnar derivatives [6] have established that polaron hopping is
8 phases, where the molecules stack on top of each other an appropriate description of microscopic charge trans-
3 into columns, which then arrange in a regular lattice. portinthesematerials. Tousechargetransporttheoryas
1
0 Along the stacks of aromatic cores in the column one a predictive, rather than descriptive tool ab initio calcu-
7 observes one dimensional charge transport [2, 3]. Per- lationsoftheratesforchargehoppingbetweenmolecules
0 pendiculartothe columnaxisthechargeshavetotunnel areneeded. Tothis endweuse the Marcus-Hushformal-
t/ through the insulating side chains, resulting in a much ism [7] for the charge transfer rate ωij
a reduced mobility. Therefore the discotics in a columnar
m phase act as nano-wires and are a promising alternative |J |2 π (∆G −λ)2
ij ij
ω = exp − , (1)
d- toconjugatedpolymers,wherechargemobilityindevices ij ¯h rλkT (cid:20) 4λkT (cid:21)
isoftenlimitedbyinefficientinter-chainhops. Columnar
n
o mesophases could, additionally, provide for laterally seg- where Jij is the transfer integral for electron or hole
c regatedphasesofdonorandacceptormoleculeswithhigh transfer, ∆Gij the difference in free energy between the
v: interface areas and efficient percolation pathways. The initial and final states, λ the reorganization energy, h¯
i reported high charge mobilities [2] would guarantee ef- Planck’s constant, k Boltzmann’s constant and T the
X
ficient operation. However, the spatial arrangement of temperature. J andλcanbe computedusingquantum
ij
r
stacks is never perfect: the columns can be misaligned, chemical methods [5]. Note that J is highly sensitive
a ij
tilted,orformvarioustypesoftopologicaldefects,which to the relative position and orientation of the molecules
are metastable or even stable at ambient conditions. In involved,whichwillbedetermined,inturn,usingMolec-
a recentmoleculardynamics (MD) study we haveshown ular Dynamics. If entropic effects and energetic disorder
that the degree of order in discotic stacks of hexaben- are ignored, ∆G = F ·d , where d is the vector be-
ij ij ij
zocoronene (HBC) derivatives is sensitive to the type tweenthecenterofthemoleculesinvolvedinchargehop-
of attached side chains [4]. The local alignment of the ping and F the electric field. In this study we choose to
molecules in columns and the global arrangement of the neglect site energy differences due to electrostatic inter-
columns in the mesophase, are thus both sensitive to actions between the molecules, because such differences
molecular architecture. vanish when the conjugated cores are parallel. Further-
2
R C12 shot, the overlap integrals Jij between intracolumnar
nearest neighbors i and j were calculated. To calculate
R R
J , the aromaticcoreofeachmolecule,asoutputby the
PhC12 ij
MD simulation, is replacedby a rigid copy of the energy
minimized configuration, with the same axial and tor-
C10-6 sional orientation as the MD result. Because of molec-
R R
ular symmetry, the highest occupied molecular orbital
(HOMO) and the lowest unoccupied molecular orbital
R
(LUMO) of HBC are doubly occupied. Thus we have
to calculate four transfer integrals for hole and four for
FIG. 1: Stick diagram of HBC derivatives with different side electron transport [10]. For each component we used an
chains. We studied alkyl chains of different lengths, Cn, adaptation of the ZINDO method, because of its speed
with n= 10,12,14,16; branched side chains, C10−6; and the anditsinsensitivitytopolarizationeffects[19]. Theeffec-
dodecylphenyl-substitutedside chains PhC12.
tive J values are then taken as the root-mean-square of
thefourHOMOtransferintegralsforholesandofthefour
LUMO transfer integrals for electrons [20]. The inner-
more, the exclusion of site energy disorder allows us to
sphere reorganization energy for cation and anion radi-
focus exclusivelyonthe effect ofconfigurationaldisorder
calswascalculatedusingunrestrictedwavefunctions,the
onchargetransport,throughdisorderinducedvariations
B3LYPfunctionaland6-31g**basis set. We foundreor-
in J .
ij ganization energies λ of 0.13eV for cations and 0.11eV
Once allω are known,chargedynamics canbe simu-
ij foranions. The outer-spherecontributionwasneglected.
lated by Monte Carlo (MC) methods [8] or by a Master
Kinetic MC and Master Equation (ME) simulation.
Equation (ME) approach [9]. Though several aspects of
For kinetic MC, columns of molecules produced by MD
the problem are treated in the literature [10, 11], a com-
arestackedperiodicallytoproduceastack1µmthick. A
prehensivestudyonlargecolumnarsystemsdoesnotyet
uniform field is applied along the stack, a charge carrier
exist. In organic molecular crystals, Deng and cowork-
representing either an electron or a hole is introduced
ers[12]showedthatmobilitiesinpentacenecanberepro-
near one end, and the charge drift is simulated by a
ducedbymixedquantumchemicalandmoleculardynam-
continuous-time random walk algorithm. The method
icsstudies. Hereweuseasimilarcombinationofmethods
used for MC simulation was similar to that in Ref. [21]
to achieve a truly ab-initio description of charge mobil-
with adjustments for a disordered lattice. The charge
ity in HBC: (i) molecular dynamics simulations (MD)
mobility µ is obtained from the transit time ttr of the
are performed on discotic mesophases of different HBC
simulated transient via µ=L/Fttr where L is the stack
derivatives, (ii) positions and orientations of molecules
thickness. ttr is taken as the point of intersection of the
areusedtocalculatethe transferintegralJ forallneigh-
twoasymptotestothesimulatedphotocurrentplottedon
boring pairs of molecules in a column, and (iii) charge
a log-log plot, as would be done in a ToF experiment.
dynamics are determined using both Master Equation
To compare these to steady-state mobilities along the
and kinetic MC methods.
column, we solvedthe linearizedMaster Equation[9] for
All studied HBC derivatives are shown in Fig. 1. The continuousboundaryconditions. The probabilityP ofa
i
molecules consist of a flat aromatic core and six side charge being on site i obeys the rate equation:
chains. Experimental data on their structural proper-
ties [2, 13, 14, 15, 16], as well as microwaveconductivity ∂P /∂t= [ω P −ω P ], (2)
i ij j ji i
(PR-TRMC) data [2, 17] are available. In some cases X
time-of-flight(ToF) mobility data are also available[18]. where the sum is over neighbors j 6= i. µ is
Molecular dynamics. In our simulation we adopt the then obtained from the charge velocity v = µF =
united atom approach for the side chains, and consider (ω P −ω P )(r −r ),wherer isthecoordinate
i6=j ji i ij j j i i
explicitly only the hydrogenatoms belonging to the aro- Pof site i in the direction of F.
matic rings of the central core. Details of the force Representative snapshots of two of the systems are
field and the MD setup can be found in Ref. 4. The shown in Fig. 2. For C12, the aromatic cores are well
molecules were arranged in 16 columns with 10, 20, 60, aligned and the columns are evenly spaced whilst for
or 100 molecules in a column. Production runs of 100ns C10−6 the branched side chains lead to a more disor-
wereperformedatconstantpressureof0.1MPaandtem- dered columnar structure, as studied in detail in Ref. 4.
perature T = 300K fixed using the Berendsen method Herewementiononlythe parametersrelevantforcharge
with anisotropic pressure coupling. System configura- transport, namely, the intracolumnar molecular separa-
4
tions were saved every 10 MD steps (the timestep was tion h and nematic order parameter S. The ordering for
2fs), yielding 200 snapshots for each run. C10 - C16 side chains is almost perfect, while PhC12 and
Quantum Chemical Calculations. For each MD snap- C10−6areslightlymoredisordered. ForthebulkierC10−6
3
compound µPR−TRMC µeToF µhToF µeME µhME S h
C10 0.5 [23] 0.22 0.75 0.14 0.49 0.98 0.36
C12 0.9 [23] 0.23 0.76 0.14 0.49 0.98 0.36
C14 1 [23] 0.27 0.80 0.17 0.59 0.98 0.36
(a) (b)
C16 - 0.29 0.91 0.17 0.58 0.98 0.36
C10−6 0.08 [17] - 0.01 6e-4 0.003 0.96 0.37
FIG. 2: MD simulation snapshots of columns of 10 HBC PhC12 0.2 [23] 0.036 0.13 0.012 0.045 0.95 0.41
molecules with (a) C12 and (b) C10−6 side chains at 300K.
Columns are pre-arranged on a rectangular lattice.
TABLE I: Electron (e) and hole (h) mobilities (cm2V−1s−1)
of different compounds calculated using time-of-flight (ToF)
and master equation (ME) methods, in comparison with ex-
we find numerous defects in the columnar arrangement. perimentallymeasuredPR-TRMCmobilities. Alsoshownare
The intracolumnar separations are practically the same the nematic order parameter S and the average vertical sep-
for all derivatives, as given in table I. aration between cores of HBC molecules h (nm).
Tostudy the effect ofdifferentside chainsonmobility,
weperformedToFsimulationsofphotocurrenttransients
oncolumnsconstructedfromcopiesofeachMDsnapshot small to include topological defects. ToF mobility data
and then averaged over results for different snapshots. would, in contrast, be heavily dependent on defects and
This sampling method is analogous to the experimental onfluctuationswithwavelengthslongerthanthecolumn
ToFsituationwherechargesarephotogeneratedinparal- sizewehaveused. Thesimulatedmobilitiesarecompared
lel columns of different configuration, and the measured with experimental data in table I and in figure 3c. The
transient is the sum of displacement currents from all systematicoverestimateofµbytheToFcomparedtothe
columns. For the same F, ttr varies significantly from MEsimulationisdue to the conventionusedhere forttr.
snapshottosnapshot,i.e. themobilityissensitivetothe Even though the differences in experimental PR-TRMC
particular orientational and positional configurations of mobilities are relatively small, we are able to reproduce
molecules in the column. Therefore, detailed tests were the experimental results not only qualitatively but also
performedtodeterminethenumberofdifferentcolumnar quantitatively. This suggests that our simulation pro-
configurations as well as the length of the stacks needed cedure and the underlying morphologies are appropriate
toobtainreproducibleresults. Testsshowedthat100dif- and that the calculated transport parameters are all in
ferent configurations are sufficient, and for 10 molecules the correct order of magnitude. Of particular interest is
per column the finite size error is less than 10%. the strong sensitivity of µ to the degree of order in the
The results of ToF simulations are shown in Fig. 3a. columns,whichisitselfcontrolledbythesidechaintype.
Comparison of the shape of the simulated transients for Thehighestµisobtained,inexperimentandsimulation,
C10-C16 and C10−6 at the same F shows that the disor- forthelinearside-chainderivativeswhichalsopresentthe
deredC10−6 structures leadto relativelydispersivetran- highest degree of intracolumnarorder while the lowestµ
sients while those for C10-C16 lead to the non-dispersive is obtained for the branched side-chain derivative with
transients typical of ordered materials [22]. In our ap- the least degree of order. The good correspondence be-
proach, where energetic disorder is neglected, disorder tween simulated and PR-TRMC mobilities arises partly
in charge transport can only arise from disorder in the fromthefactthatbothsimulationandexperimentprobe
nearest-neighbour transfer integrals J, which in turn transportinrelativelywellalignedcolumns. To simulate
arisesfromdisorderinintermolecularseparationandori- ToF experiments, large systems must be studied which
entation. Figure3bcomparesdistributionsoflog|J|2. As will require a multiscale ansatz [24].
the data clearly show,the disorderedcompoundleads to In summary, we have determined charge mobilities of
a much wider distribution of J leading to a wider distri- severalderivativesof the discotic liquid crystalhexaben-
bution of rates. In particular, the low J peak represents zocoronene. For the first time, three methods were com-
bottlenecks at discontinuities in the stacks, which act as binedintoonescheme: (i)quantumchemicalmethodsfor
traps in the one-dimensional transport. the calculationof molecularelectronic structures andre-
Finally, we compare our simulations to PR-TRMC organizationenergies(ii)moleculardynamicsforsimula-
measurements. PR-TRMC probes the sum of high fre- tionoftherelativepositionsandorientationsofmolecules
quencyphotoinducedconductivitiesduetobothtypesof in a columnar mesophase, and (iii) kinetic Monte Carlo
charge carriers, and therefore probes the fastest contri- simulations and Master Equation approach to simulate
butions to charge transport, sensitive only to the local charge transport. Applying this scheme to differently
disorderwithinwellordereddomains. PR-TRMCmobil- substituted HBC derivatives we are able to reproduce
ities are therefore appropriate for comparison with our the trends and magnitudes of mobilities as measured by
simulation since the column sizes we are using are too PR-TRMC.Thisstudyshowsthatitispossibletounder-
4
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Mu¨llen and J. Cornil are gratefully acknowledged.
