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WIREs Comput Mol Sci
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Comparison of different rate constant expressions for the prediction of charge and energy transport in oligoacenes

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Charge and exciton transport in organic semiconductors is crucial for a variety of optoelectronic applications. The prediction of the material‐dependent exciton diffusion length and the charge carrier mobility is a prerequisite for tailoring new materials for organic electronics. At room temperature often a hopping process can be assumed. In this work, three hopping models based on Fermi's Golden rule but with different levels of approximation—the spectral overlap approach, the Marcus theory, and the Levich–Jortner theory—are compared for the calculation of charge carrier mobilities and exciton diffusion lengths for oligoacenes, using the master equation approach and Monte Carlo simulations. WIREs Comput Mol Sci 2016, 6:694–720. doi: 10.1002/wcms.1273

Hole mobility in the pentacene crystal, calculated with the Marcus and the Levich–Jortner hopping rate. The experimental values are from Ref .
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Hole mobility in the tetracene crystal, calculated with the Marcus and the Levich–Jortner hopping rate. The experimental values are from Ref .
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Hole mobility in the pentacene crystal with the Marcus theory, calculated with and without external reorganization energy. The experimental values are from Ref .
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Hole mobility in the tetracene crystal with the Marcus theory, calculated with and without external reorganization energy. The experimental values are from Ref .
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The internal and external reorganization energies for the acenes depending on the number of benzene rings. The line is a fit, see Eq. . Note that the crystal symmetry changes when going from three to four benzene rings: naphthalene and anthracene belong to the space group P21/a, tetracene, pentacene, and hexacene belong to the space group P 1 ¯ .
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The exciton diffusion length in the naphthalene crystal in the (ab) plane (τ = 78 ns). The couplings were calculated with SCS‐ADC(2), SCS‐CC2, and TDHF, respectively. In all cases, the cc‐pVTZ basis sets were used and the reorganization energy was calculated with SCS‐CC2/cc‐pVTZ.
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The diffusion constant D of naphthalene along the unit cell vectors depending on the reorganization energy λ. The λ values from Table are tagged by arrows. (V ji was calculated with SCS‐ADC(2)/cc‐pVTZ.)
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The exciton diffusion length in the naphthalene crystal in the (ab), (bc), and (ac) plane. The dashed lines are calculated with the spectral overlap approach and the solid lines are calculated with the Marcus equation, both using the master equation approach. The dots are calculated with the Marcus equation using the Monte Carlo method. (V ji was calculated with SCS‐ADC(2)/cc‐pVTZ, λ and J were calculated with SCS‐CC2/cc‐pVTZ, τ = 78 ns.)
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The Davydov splitting between the first two excited states of the dimer with the energies E D1 and E D2. The splitting is twice the exciton coupling V ji . E i and E j are the monomer excitation energies.
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(a) HOMO orbital ϕ of ethene, (b) LUMO orbital ϕ #, (c) transition density ϱ, cf. Eq. . HOMO, highest occupied molecular orbital; LUMO, lowest unoccupied molecular orbital.
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The potential curves for the neutral/ground state and the charged/excited state (indicated by #) of the donor (index D, left) and the acceptor (index A, right) molecule, depending on the geometry coordinate q. λ indicates the reorganization energy.
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The potential curves for one vibrational mode in the neutral/ground state and the charged/excited state depending on the normal mode coordinate q, based on the harmonic approximation. Δq is the difference vector between the two equilibrium structures, λ indicates the relaxation energies.
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The hole mobilities for tetracene in the ab plane, calculated with the spectral overlap approach (solid lines) and the Marcus theory (dashed line). The experimental values (dots) are taken from Ref . (Calculated with DFT B3‐LYP/cc‐pVDZ.)
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The hole mobilities for pentacene in the ab plane, calculated with the spectral overlap approach (solid lines) and the Marcus theory (dashed line). The experimental values (dots) are taken from Ref . (Calculated with DFT B3‐LYP/cc‐pVDZ.)
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The Franck–Condon‐weighted density of states J ˜ E j i depending of the energetic shift between the molecules i and j caused by the external electric field for hole transport in tetracene. The red line is calculated with the spectral overlap approach, Eq. , the green solid line is the linear approximation used in Eq. , the green dashed lines are secants through the points ± E 1 | J ˜ ± E 1 , and the blue line is calculated with the Marcus theory. The lower graph is a magnification of the upper one, which shows the energy interval which is relevant for charge transport. (Calculated with DFT B3‐LYP/cc‐pVDZ.)
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The hole mobility for tetracene calculated with λ ext = 500 meV, which is chosen so that the mobility approximately fits to the experimental values.
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The hole mobility for pentacene calculated with λ ext = 29 meV (dashed), λext = 52 meV (dotted, for both values see Table ) and λ ext = 200 meV (solid), which is chosen so that the mobility fits to the experimental values (green).
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The probability distribution of the sum of the final vibrational numbers p = m D + n A for the regarded acenes.
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