Growth and physical properties of crystalline rubrene - BOA Bicocca ...

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4 Organic semiconductors 1.3.1 Optical properties of organic semiconductors Among the most interesting properties of organic molecular crystals there are their peculiar optical properties, in particular their ability to conduct optical excitation energy within the crystal, and to transfer it. Moreover, unlike what happens for inorganic solids, the electronic excited states of organic molecular crystals strongly resemble those of the corresponding isolated molecules. This, like most organic molecular crystals properties, comes from the weakness of the intermolecular interactions in such materials, lead- ing only to a small perturbation of the electronic structure of the molecules when they are packed in the crystal. Nonetheless, the presence of the Van der Waals interactions leads to some difference between the electronic and vibrational spectra of the isolated molecule and of the relative molecular crystal, namely[36]: • a general shift, called solvent shift, toward lower energies, due to the van der Waals interactions between adjacent molecules; • a splitting of each peak in a maximum of Z different peaks, where Z is the number of molecules not symmetrically equivalent in the unit cell. This effect is called Davydov splitting. The different peaks are also often differently polarized; • the delocalization of the excitation between all the molecules in the crystal leads to the formation of the so-called excitonic bands, and thus to a general broading of the peaks; • a total, or partial, removal of the degeneracy of the excited levels and a change in the selection rules for the optical transitions. This is due to the fact that in crystals the selection rules are determined not only by the symmetry of the single molecule, but also by that of the whole crystal; • for transitions with a particularly high oscillator strength, the effects of the solvent shift and of the Davydov splitting can be strong enough to completely remove any resemblance of the crystal spectrum with that of the single molecule. All these effects find an explanation in the molecular exciton theory, de- veloped by Davydov[39], that permits to build the eigenstates of a perfect,
1.3 Small-molecule organic semiconductors 5 rigid crystal of Nσ identical weakly interacting molecules, N being the number of unit cells in the crystal and σ the number of molecules per unit cell. Considering the simpler case, in which the molecules are considered as two level systems with the wave function and energy of the ground and excited states given by ϕ 0 , e0, ϕ ∗ and e ∗ , respectively, the Hamiltonian H of the system is given by: H = N nα Hnα + ′ Vnαmβ nαmβ (1.3) where Hnα is the Hamiltonian of the isolated molecule in the lattice position nα and Vnαmβ is the van der Waals interaction potential between the two molecules in the nα and mβ lattice sites. If Vnαmβ is considered small with respect to Hnα it can be described as a perturbation within a tight-binding approximation. The wave function Ψ of the crystal ground state is given by the antisymmetrized product of the ground state wave functions of the single molecules and the corresponding energy at the first order is given by: E0 = Ne0 + ′ nαmβ 0 ϕnαϕ 0 mβ |Vnαmβ|ϕ 0 nαϕ 0 mβ = Ne0 + D (1.4) where the term D accounts for the total energy of the van der Waals interactions holding together the crystal. The wave functions Ψ ∗ of the crystal excited states (or excitons) can be built by considering a single molecule in its excited state, the excitation being completely shared among all the molecules constituting the crystal. Crystal periodicity has then to be included and a wave vector k is introduced, which takes N values in the first Brillouin zone. The corresponding eigenenergies E(k) form the electronic bands (or excitonic bands) of the crystal and are given by: E(k) = E0 + e ∗ − e0 + ∆D ± I(k) (1.5) Here, ∆D is the solvent shift, representing the difference between the interaction energy D of a single molecule in its ground state with all the other molecules comprised in the crystal and the interaction energy D’ of the same molecule in its excited state with all the other molecules (see also figure 1.2). I(k) is instead the so-called resonance term (or exciton shift), and repre- sents the transfer interaction between a molecule in a specific lattice site and all the other molecules and thus accounting for the fact that the molecular
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62 Rubrene thin films for a rubrene
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74 Rubrene thin films present. Sinc
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78 Rubrene thin films
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80 Oxidation dynamics of rubrene ep
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118 Conclusions and Perspectives On
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120 Conclusions and Perspectives
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122 Bibliography [13] D. A. Bernard
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124 Bibliography [41] V. Coropceanu
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126 Bibliography [71] A. J. Maliaka
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128 Bibliography [101] C. Kittel, I
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130 Bibliography [130] D. Beljonne,
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132 Bibliography
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134 PhD activity Attended courses
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