ARTICLE pubs.acs.org/JPCC
Delocalization and Mobility of Charge Carriers in Covalent Organic Frameworks Sameer Patwardhan, Aleksey A. Kocherzhenko, Ferdinand C. Grozema, and Laurens D. A. Siebbeles* Optoelectronic Materials Section, Department of Chemical Engineering, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands ABSTRACT: A new generally applicable method to calculate the relative energetic stability of localized and delocalized charges in a system of two molecules is presented. The relative stability of localized and delocalized charges was calculated for π-stacked triphenylenes at varying twist angles and intermolecular distances. The reliability of the new method was validated by comparison with results from HartreeFock calculations on particular configurations. According to the calculations, charges are localized for larger twist angles that are typical for triphenylene derivatives in the liquid crystalline phase. In contrast, significant charge delocalization is expected for eclipsed stacking of triphenylene units in a covalent organic framework. This can give rise to band-like motion of delocalized charges with a mobility of the order of 10 cm2 V1 s1 or more.
1. INTRODUCTION Covalent organic frameworks (COFs) are new materials that have been reported for the first time only five years ago.1 In twodimensional COFs, π-conjugated aromatic molecular units form sheets with a periodic structure.17 These sheets can stack on top of each other in an eclipsed fashion with an intersheet distance of ∼0.34 nm; see Figure 1. The (close to) eclipsed stacking leads to large electronic coupling between the π-orbitals in the molecular units.810 The strong coupling is expected to give rise to efficient pathways for motion of excess charge carriers and photoexcited states (excitons). The prospect of efficient charge and exciton transport makes COFs very promising for (opto)electronic applications in, for example, thin film transistors, light-emitting diodes, solar cells, photodiodes, or photocatalysis.3,4,6,7 Very recently, it has been shown experimentally that mobile charge carriers can be produced by photoexcitation of COFs, containing columns of π-stacked triphenylene and pyrene units (TP-COF),3 pyrene units (PPy-COF),4 or nickel-phthalocyanine units (NiPc COF).7 Until now, no accurate value of the charge carrier mobility has been reported. The only quantitative information involves a lower limit to the mobility of positive charges (holes) in NiPc COF of 1.3 cm2 V1 s1.7 This value significantly exceeds typical values for more disordered columns of π-stacked phthalocyanine or triphenylene derivatives in the liquid crystalline (LC) phase.8,11,12 The high charge carrier mobility in NiPc COF and the ordered structure with strongly coupled molecular units suggest band-like motion of delocalized charges rather than hopping of localized charges.10 The present work aims to provide theoretical insights into the extent of delocalization and the transport mechanism of holes in triphenylene-based COFs1,13 (T-COF) and LC triphenylenes. A new method to calculate the relative energetic stability of localized and delocalized charges is presented and applied to a r 2011 American Chemical Society
stack of two triphenylene molecules with varying distance and mutual twist angle; see Figure 2.
2. THERMODYNAMIC CONDITION FOR CHARGE LOCALIZATION AND DELOCALIZATION The thermodynamic condition for charge carrier localization can be quantified using the relative stability of a localized and a delocalized excess charge in a molecular dimer. The extension to the case of a large polaron in which the charge is delocalized over more than two molecules is discussed in the Appendix. For specificity, all calculations were performed for positive charges (holes); the calculations for negative charges (electrons) are similar. The relative stability of a localized and a delocalized hole in a dimer is defined as14 η ¼ Eþ1, 0 ðg þ1 ;g 0 Þ Eþ0:5,
þ 0:5
ðg þ0:5 ;g þ0:5 Þ
ð1Þ
Here the energy E þ1,0 (g þ1 ,g 0 ) corresponds to the state ψþ1,0(gþ1,g0) with an elementary charge localized on one molecule in the dimer with geometry gþ1 optimized for the charged (þ1) state; the geometry g0 of the other molecule is optimized for the neutral (0) state. The energy Eþ0.5,þ0.5(gþ0.5,gþ0.5) corresponds to the state ψþ0.5,þ0.5(gþ0.5,gþ0.5) with half an elementary charge on each molecule, both molecules having the geometry gþ0.5 optimized to the charge state þ0.5. If the relative stability η < 0, it is thermodynamically favorable for the charge to localize on a single molecule. If η > 0, it is thermodynamically favorable for the charge to delocalize over two or more molecules. Received: March 14, 2011 Revised: May 11, 2011 Published: May 13, 2011 11768
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of removing an electron from the highest occupied molecular orbital (HOMO) localized on one molecule in the dimer. The energy expectation value of this unrelaxed localized state is Eþ1;0 ðg 0 ;g 0 Þ ¼ Æψþ1;0 ðg 0 ;g 0 ÞjHdimer ðg 0 ;g 0 Þjψþ1;0 ðg 0 ;g 0 Þæ ð2Þ 0 0
where Hdimer(g ,g ) is the electronic Hamiltonian of the dimer with the intramolecular geometries of both molecules optimized for the neutral state. The state ψþ1,0(g0,g0) will undergo energy relaxation by adjustment of the intramolecular geometry and the electronic orbitals. Because the charge resides on one molecule, the geometry relaxation energy can to a good approximation be taken equal to that of an isolated molecule with charge þ1, which is equal to Figure 1. Schematic representation of (a) disordered columnar liquid crystalline triphenylenes, where R represents a long aliphatic side chain; (b) eclipsed π-stacking of triphenylene units in the covalent organic framework T-COF.1
þ1 þ1 þ1 0 Eþ1;0 georelax ¼ E ðg Þ E ðg Þ
ð3Þ
Using the same argument, the molecular orbital relaxation energy is approximated by that of an isolated charged molecule: þ1 0 0 0 HOMO 0 Eþ1;0 ðg Þ orbrelax ¼ E ðg Þ ½E ðg Þ E
ð4Þ
where Eþ1(g0) = Æψþ1(g0)|Hmonomer(g0)|ψþ1(g0)æ and ψþ1(g0) is the relaxed electronic state of a charged molecule with the geometry of the neutral state. The second term in eq 4 is the unrelaxed energy of the charged molecule, according to Koopmans’s theorem. The energy of the localized state ψþ1,0(gþ1,g0) is obtained by adding the corrections given by eqs 3 and 4 to eq 2, resulting in þ1;0 Eþ1;0 ðg þ1 ;g 0 Þ ¼ Eþ1;0 ðg 0 ;g 0 Þ þ Eþ1;0 georelax þ Eorbrelax
Figure 2. Model system of two parallel stacked triphenylene molecules with twist angle θ and intermolecular distance δ.
The energy of the delocalized state ψþ0.5,þ0.5(gþ0.5,gþ0.5) is calculated in a similar way. The delocalized state is written as a superposition of localized states, 1 ψþ0:5; þ 0:5 ðg 0 ;g 0 Þ ¼ pffiffiffiðψþ1;0 ðg 0 ;g 0 Þ ( ψ0; þ 1 ðg 0 ;g 0 ÞÞ 2 ð6Þ
2.1. Methods for Calculating the Relative Stability. The
relative stability can be computed using the following two methods. Method 1. This method was previously applied to porphyrin molecular wires.14 The state ψþ1,0(gþ1,g0) is obtained by taking one molecule with a geometry optimized for the charged (þ1) state and a second molecule with the geometry optimized for the neutral (0) state. The energy of this dimer with total charge þ1 is taken to correspond to state ψþ1,0(gþ1,g0). Calculating the energy of the state ψþ0.5,þ0.5(gþ0.5,gþ0.5) requires two steps. First, the optimum geometry of a single molecule in the þ0.5 charge state is found by optimizing the geometry of a charged (þ1) system of two molecules at a large fixed intermolecular distance (r = 10 nm). The interaction between the molecules is then negligible, and the charge is distributed equally over the two molecules. Next, the molecules are brought close together to form a dimer, and the energy of the interacting molecules, each having charge þ0.5, is calculated without changing the intramolecular geometry. The relative stability is then the energy difference of the ψþ1,0(gþ1,g0) and the ψþ0.5,þ0.5(gþ0.5,gþ0.5) states; see eq 1. Method 2. This method proceeds from the neutral state of the dimer ψ0,0(g0,g0). A hypothetical unrelaxed state ψþ1,0(g0,g0) with the charge localized on one molecule is defined as the result
ð5Þ
where the þ () sign corresponds to a negative (positive) value of the charge transfer integral, J ¼ Æψ0; þ 1 ðg 0 ;g 0 ÞjHdimer ðg 0 ;g 0 Þjψþ1;0 ðg 0 ;g 0 Þæ
ð7Þ
The energy expectation value of the unrelaxed delocalized state in eq 6 is Eþ0:5; þ 0:5 ðg 0 ;g 0 Þ ¼ Æψþ0:5; þ 0:5 ðg 0 ;g 0 ÞjHdimer ðg 0 ;g 0 Þjψþ0:5; þ 0:5 ðg 0 ;g 0 Þæ
¼ Eþ1;0 ðg 0 ;g 0 Þ jJj
ð8Þ
The first term on the right-hand side of eq 8 is the energy of the unrelaxed localized state in eq 2. The energy of the relaxed delocalized state is obtained by adding the energies associated with relaxation of the intramolecular geometry and the electronic orbitals to eq 8. These relaxation energies were obtained from calculations on dimers with the molecules at large distance (r = 10 nm), so that each molecule carries þ0.5 charge and the intermolecular interaction is negligible. The relaxation energies 11769
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Figure 3. The relative stability η calculated according to method 1 (empty squares) and method 2 (filled circles) as a function of the charge transfer integral, J, for combinations of twist angles θ and intermolecular distances δ listed in section 2.2.
are then obtained in the form þ 0:5 Eþ0:5; ¼ fEþ0:5; þ 0:5 ðg þ0:5 ;g þ0:5 Þ Eþ0:5; þ 0:5 ðg 0 ;g 0 Þgr ¼ 10nm georelax
ð9Þ þ0:5; þ 0:5 Eorbrelax
¼ fE
þ0:5; þ 0:5
ðg ;g Þ ðE ðg ;g Þ E 0
0
0;0
0
0
HOMO
ðg ;g ÞÞgr ¼ 10nm 0
0
ð10Þ The energy of the delocalized state is obtained by adding the relaxation energies given by eqs 9 and 10 to eq 8, resulting in þ0:5; þ 0:5 þ 0:5 Eþ0:5; þ 0:5 ðg þ0:5 ;g þ0:5 Þ ¼ Eþ1;0 ðg 0 ;g 0 Þ jJj þ Egeorelax þ Eþ0:5; orbrelax
Figure 4. The charge transfer integral J (filled squares) and the relative stability calculated according to method 2 (open circles) as a function of the twist angle θ at intermolecular distance δ = 0.35 nm.
Because the relative stability is strongly dependent on the triphenylene dimer conformation, it was calculated for various twist angles (θ = 0, 15, 30, 45, 60) and intermolecular distances (δ = 0.32, 0.34, 0.35, 0.36, 0.5, 1.0, 10.0 nm). The mutual interaction of two triphenylene molecules is implicitly accounted for in the relative stability calculation according to method 1, and is accounted for by the charge transfer integral, J, in method 2, see eq 15. Because triphenylene dimers have C3 symmetry, the charge transfer integral, J, is to a good approximation equal to half the energy difference between the two-fold degenerate HOMO and HOMO 1 of a triphenylene dimer.810,14 Note that calculating the charge transfer integral in this way accounts for effects of spatial orbital overlap.810,14
ð11Þ Substitution of eqs 5 and 11 into eq 1 gives for the relative stability η ¼ jJj þ ΔEgeorelax þ ΔEorbrelax
ð12Þ
þ0:5; þ 0:5 Þ ΔEgeorelax ¼ ðEþ1;0 georelax Egeorelax
ð13Þ
þ0:5; þ 0:5 ΔEorbrelax ¼ ðEþ1;0 Þ orbrelax Eorbrelax
ð14Þ
where
Note that the charge transfer integral, J, depends on the mutual orientation of the two molecules,8 while the relaxation energies do not. For large intermolecular distances, J becomes negligible and the relative stability η¥ is given by the relaxation energies only; i.e., η ¼ jJj þ η¥
ð15Þ
η¥ ¼ ΔEgeorelax þ ΔEorbrelax
ð16Þ
where
2.2. Computational Methodology. To verify the validity of the new method 2, the values of the relative stability of localized and delocalized charges calculated using this method (eq 15) have to be compared to the values obtained using method 1. To this end, a system of two π-stacked triphenylenes with molecular planes perpendicular to the axis connecting their centers of mass was considered (see Figure 2). All calculations on this system were performed with the Gaussian 03 package employing the restricted open-shell HartreeFock (ROHF) method and the 6-311G(d,p) basis set.15
3. RESULTS AND DISCUSSION The relative stability calculated according to methods 1 and 2 are shown in Figure 3 as a function of the charge transfer integral. The charge transfer integral values correspond to the different combinations of θ and δ listed in section 2.2. The relative stability of a charge on two triphenylene molecules at a large distance is η¥ = 0.43 eV for both methods. Using method 2 reveals an interesting detail: the term ΔEorbrelax = 0.34 eV in eq 16 significantly exceeds ΔEgeorelax = 0.09 eV. Thus, the time scale of charge localization is predominantly determined by fast orbital relaxation, rather than by slower geometry relaxation. The relative stability values calculated with both methods show good agreement over a wide range of charge transfer integral, J, values. The relative stability calculated according to method 2 varies linearly with the value of the charge transfer integral, as expected from eq 15. The differences in the relative stability values calculated with the two methods can be attributed to several factors. First, in method 1, intermolecular interaction leads to partial delocalization of the charge in the state ψþ1,0(gþ1,g0), causing fractional charges on the two molecules to deviate from þ1 and 0. In Figure 3, only the results for θ and δ combinations where the fractional charge of the cation was þ0.9 or more were included. If the fractional charges significantly deviate from þ1 and 0, method 1 becomes unreliable and method 2 should be used instead. Second, approximations made in method 2, such as expressing the delocalized state in eq 6 as a superposition of two localized states only, are not employed in method 1. Triphenylene derivatives in the LC phase have a twist angle θ ≈ 45 and an average intermolecular distance δ = 0.35 nm.1618 11770
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The Journal of Physical Chemistry C Figure 4 shows the relative stability η as a function of the twist angle at this intermolecular distance. For twist angles below 20, the relative stability is close to zero. For larger twist angles, it is negative, and the localized state thus appears to be most stable. In the equilibrium conformation, θ = 45, the relative stability has an absolute value of 0.34 eV, which significantly exceeds the thermal energy of 0.024 eV at room temperature. Charges in triphenylenebased liquid crystals are thus expected to be localized on single triphenylene molecules and move by incoherent hopping from one molecule to another. This result is in agreement with the often applied criterion that localization occurs when the polaron formation energy Ep exceeds the charge transfer integral, J.19 Indeed the polaron formation energy Ep= 0.19 eV (calculated as half the reorganization energy)9 exceeds the charge transfer integral J = 0.1 eV at θ = 45, see Figure 4. In T-COF the triphenylene units are stacked at zero twist angle with intermolecular distances δ in the range between 0.34 and 0.36 nm.1,13 At δ = 0.34 nm and zero twist angle, the relative stability calculated according to method 2 is positive, η = 0.12 eV. Because η significantly exceeds the thermal energy at room temperature, charges are expected to be delocalized over at least two triphenylenes. Using the methodology described in the Appendix, it is found that the relative stability of a hole localized on one molecule and a hole delocalized over an infinitely long stack of triphenylenes (at θ = 0) is equal to η(N f ¥) = 0.28 eV. The fact that the relative stability is positive implies that the charge will delocalize completely over all triphenylenes in a stack with zero twist angle and δ = 0.34 nm. Upon increase of the intermolecular distance to δ = 0.36 nm, the relative stability for a dimer decreases to η = 0.02 eV. This value is negative, which implies a driving force for charge localization in a dimer. However, the relative stability for an infinitely long stack calculated by the methodology described in the Appendix amounts to 7 104 eV. Hence, upon going to longer stacks, the increase of the electronic coupling energy (eq A.2) compensates for the reduction of the relaxation energy (A.3). The very small positive relative stability for infinitely long stacks with δ = 0.36 nm slightly favors charge delocalization. The above results demonstrate the sensitivity of charge delocalization in T-COF to the intermolecular distance. This is because the charge transfer integral between two triphenylene molecules at zero twist angle decreases from J = 0.55 eV at δ = 0.34 nm to 0.41 eV at δ = 0.36 nm. Note, that these values are ∼30% higher than those obtained previously with density functional theory.8 Interestingly, both the J values from this work and from ref 8 significantly exceed the intramolecuar polaron formation energy (0.19 eV) associated with chargeinduced relaxation of the geometry of a triphenylene molecule. Therefore, intramolecular polaronic effects are weak. In the presence of a charge, the energy of a triphenylene stack can decrease by reduction of the intermolecuar distance δ as long as the energy gain due to the increase of |J| exceeds the enhanced nuclear and electronic repulsion. A reduction of δ by intermolecular electronphonon coupling increases |J| and will favor charge delocalization. On the basis of the effects of charge-lattice interactions discussed above, it is to be expected that charges in T-COF are to a significant extent delocalized, with the mechanism of charge transport having more band-like than hopping character. Disorder due to structural defects in the material will have a negative effect on charge delocalization and mobility. In particular, lateral slide of adjacent triphenylenes reduces the charge transfer integral8 and consequently the degree of
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delocalization and magnitude of mobility. Realization of COF materials with an ordered structure is essential to exploit the prospects for band-like transport of highly mobile charges. The charge carrier mobilities in T-COF can be estimated by comparison to materials for which the charge transport properties are known. The charge transfer integrals for T-COF significantly exceed values of less than 0.1 eV for oligoacenes. Taking into consideration the band-like charge transport mechanism in T-COF, the mobility is expected to be comparable to, or higher than, the values of the order of 10 cm2 V1 s1 for some oligoacene crystals.20,21 Fast band-like motion of charges is thermally deactivated, in contrast to hopping of localized polaronic charges. Hence, experimental studies of the temperature dependence of the charge mobility in COFs can provide further insights into the mechanism of charge transport.
4. CONCLUSIONS A new computational method to gain insight into the extent of charge delocalization in molecular dimers was presented. The method was applied to π-stacked triphenylenes at varying twist angles and intermolecular distances. The eclipsed stacking of triphenylene units, typical for COFs, leads to a large electronic coupling and charge delocalization. In contrast for larger twist angles between triphenylene units, typical for the LC phase, the coupling is small and the charge localizes. According to the calculations, holes in triphenylene-based COFs are delocalized and move via a band-like mechanism with a mobility that is potentially higher than the largest mobilities for oligoacene crystals of the order of 10 cm2 V1 s1.20,21 ’ APPENDIX: LARGE POLARON FORMATION BY DELOCALIZATION OVER MORE THAN TWO MOLECULES This appendix describes how the formalism developed in method 2 can be extended to a stack of N > 2 molecules. The resulting equation of the relative stability η(N) of a charge localized on one molecule and delocalized over N molecules is given by ηðNÞ ¼ jΔEel:coupling ðNÞj þ η¥ ðNÞ
ðA.1Þ
In eq A.1 the energy gain due to electronic coupling between N molecules is given by22 π ΔEel:coupling ðNÞ ¼ 2jJjcos ðA.2Þ N þ1 which increases from |J| for a dimer (N = 2) to 2|J| for an infinitely long stack (N f ¥). The second term at the right-hand side of eq A.1 is the relaxation energy for the case of infinite distance between the molecules in the stack, which analogously to eq 16, is given by η¥ ðNÞ ¼ ΔEgeorelax ðNÞ þ ΔEorbrelax ðNÞ
ðA.3Þ
The geometry relaxation energy ΔEgeorelax(N) and orbital relaxation energy, ΔEorbrelax(N) are similar to eqs 13 and 14; i.e. þ1=N; þ 1=N; þ 1=N:::
ðA.4Þ
þ1=N; þ 1=N; þ 1=N:::
ðA.5Þ
ΔEgeorelax ðNÞ ¼ Eþ1;0 georelax Egeorelax
ΔEorbrelax ðNÞ ¼ Eþ1;0 orbrelax Eorbrelax
for a charge that The energy relaxation Eþ1/N,þ1/N,þ1/N... orbrelax is delocalized over N molecules becomes less negative for 11771
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larger N and becomes zero as N f ¥. Because Eþ1,0 georelax and Eþ1,0 orbrelax are independent of N, the effect of delocalization is a decrease of ΔEgeorelax(N) and ΔEorbrelax(N) to more negative values. For infinite delocalization, η¥(N f ¥) = Eþ1,0 georelax þ and the relative stability becomes Eþ1,0 orbrelax þ1;0 ηðN f ¥Þ ¼ 2jJj þ η¥ ðN f ¥Þ ¼ 2jJj þ Eþ1;0 georelax þ Eorbrelax
ðA.6Þ Upon increasing the number of molecules, N, the electronic coupling term, |ΔEel.coupling(N)|, in eq A.1 increases monotonically, while the second term, η¥(N), decreases monotonically to become more negative. If the relative stability for both a dimer (N = 2) and an infinitely long stack are positive (negative), the charge will completely delocalize (localize). In case the relative stability is positive for a dimer (N = 2) and negative for an infinite stack, the charge will delocalize over a finite number of molecules Ndel for which η(N) changes sign.
’ AUTHOR INFORMATION
Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03; Gaussian, Inc.: Wallingford, CT, 2004. (16) Adam, D.; Schuhmacher, P.; Simmerer, J.; Haussling, L.; Siemensmeyer, K.; Etzbach, K. H.; Ringsdorf, H.; Haarer, D. Nature 1994, 371, 141. (17) Fontes, E.; Heiney, P. A.; Dejeu, W. H. Phys. Rev. Lett. 1988, 61, 1202. (18) Heiney, P. A.; Fontes, E.; Dejeu, W. H.; Riera, A.; Carroll, P.; Smith, A. B. J. Phys. 1989, 50, 461. (19) Stafstr€om, S. Chem. Soc. Rev. 2010, 39, 2484. (20) Coropceanu, V.; Sanchez-Carrera, R. S.; Paramonov, P.; Day, G. M.; Bredas, J. L. J. Phys. Chem. C 2009, 113, 4679. (21) Anthony, J. E. Chem. Rev. 2006, 106, 5028. (22) Andre, J. M.; Delhalle, J.; Bredas, J. L. Quantum Chemistry Aided Design of Organic Polymers, World Scientific Lecture and Course Notes in Chemistry; World Scientific: Singapore, 1991; Vol 2, p44, ISBN 98102-0004-8.
Corresponding Author
*E-mail:
[email protected].
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