Dimensionality of electronic excitations in organic semiconductors: A dielectric function approach

We present a detailed investigation on the effective dimensionality (associated with the degree of delocalization) of electronic excitations in thin organic films using the dielectric function as obtained from ellipsometry. To this end, we study first the best analytical representation of the optical dielectric function of these materials and compare different approaches found in the literature: (i) the harmonic oscillator approximation, (ii) the standard critical-point model (SCP), (iii) the model dielectric function (MDF), and (iv) the Forouhi-Bloomer model. We use these models to analyze variable angle spectroscopic ellipsometry raw data for a thin poly(9,9-dioctylfluorene) (PFO) film deposited on quartz (taken as an archetypal sample). The superiority of the SCP model for PFO films and a wide range of other spin-coated conjugated polymers (and guest-molecules in polymers) is demonstrated. Moreover, we show how the SCP model can be used to gain physical information on the microscopic structure. As an example, we show that the delocalization of excitons decreases for nonconjugated polymers, such as polymethylmethacrylate and polyimide, while the conjugation length and exciton delocalization are, respectively, enhanced in cases where a planar conformation (e.g., $\beta$ phase of PFO) or a high degree of crystallinity [e.g., poly(3-hexylthiophene)] is achieved. As an additional example, we employ the SCP excitonic model to investigate the temperature dependence of the dielectric function of crystalline and glassy PFO films. We propose that the SCP excitonic model should be adopted as the standard choice to model the optical properties of polymer thin films from ellipsometry data.

Figure 1

Chemical structures of (a) PFO, (b) PMMA, (c) PI, (d) PCBM, and (e) P3HT.

Figure 2

Extinction coefficient of a $93\mathrm{nm}$ thick spin-coated PFO film deduced from transmission data using the Lambert law for absorption, film thickness from profilometry, and without correcting for reflections at the air/polymer and polymer/substrate interfaces.

Figure 3

Experimental (squares) and modeled (solid lines) $\sin \left(2\Psi \right)\cdot \cos \left(\Delta \right)$ vs photon energy for one representative angle of incidence (67°): (a) FB, (b) HOA, (c) SCP, and (d) MDF.

Figure 4

Extinction coefficient of a $128\mathrm{nm}$ thick spin-coated PFO film deduced from the ellipsometry data using (a) bulk calculation (solid line) and FB model (dashed line), (b) HOA with a Cauchy law plus three Lorentzian peaks, (c) SCP with three critical points of zero order (excitons), and (d) MDF with three ($M_0+3\mathrm{D}$ exciton) transitions.

Figure 5

Refractive index (solid squares) and extinction coefficient (open squares) of a $40\mathrm{nm}$ thick PCBM film deduced by fitting ellipsometry data with the SCP model using four exciton line shapes. For clarity, only a few data points are shown with symbols (120 measured at wavelength intervals with $\Delta \lambda =5\mathrm{nm}$).

Figure 6

Refractive index (solid symbols) and extinction coefficient (open symbols) for PMMA (squares) and PI (triangles) using one Lorentzian peak in the near UV and a Cauchy tail (first approximation of a Lorentzian peak with deep UV central energy). For clarity, only a few data points are shown with symbols. The two open circle data points for PI are taken from Ref. 51 for comparison.

Figure 7

Refractive index (solid squares) and extinction coefficient (open squares) of a $40\mathrm{nm}$ thick $\beta$-phase PFO film, deduced using the SCP model with three 1D critical points for the UV-vis absorption bands and one 0D critical point accounting for an average of far UV transitions. For clarity, only a few points are shown explicitly with symbols.

Figure 8

Refractive index (solid squares) and extinction coefficient (open squares) of a $30\mathrm{nm}$ thick P3HT film on spectrosil B. For clarity, only a few data points are explicitly shown with symbols.

Figure 9

Refractive index at $466\mathrm{nm}$ of an $\sim 185\mathrm{nm}$ thick crystalline PFO film (solid squares) and an $\sim 82\mathrm{nm}$ thick glassy nematic PFO film (open squares) as a function of film temperature. The cold crystalline phase transition can be seen as an abrupt change in the $dn∕dT$ slope.

Figure 10

Central energies of the two visible excitons of an $\sim 185\mathrm{nm}$ thick crystalline PFO film as a function of film temperature. The solid lines are fits of the data to Eq. (8).

Figure 11

Widths $\left(\Gamma \right)$ of the two visible excitons of an $\sim 185\mathrm{nm}$ thick crystalline PFO film as a function of film temperature. The solid lines are fits of the data to Eq. (9).