Localization of excitons by molecular layer formation in a polymer film

Spin coated films of atactic polystyrene of two different molecular weights have been studied with uv spectroscopy and x-ray reflectivity, the film thickness $\left(d\right)$ varying from $\sim 2R_g$ to $\sim 12R_g$ where $R_g$ is the unperturbed radius of gyration of the polymer. uv extinction due to the pure electronic singlet ${}^{1}A_{1g}\to {}^{1}E_{1u}$ is seen to increase with $d^{-1}$ for $4R_g⩽d⩽12R_g$ (region 1). This suggests excitonic interaction along $d$. The variation of total exciton energy $\left(E\right)$ of the $A_{1g}\to E_{1u}$ singlet with $d$ in region 1 can be well explained by formation of linear $J$-aggregates of polystyrene molecules, in a lattice with spacing “$a$” (in Å) $R_g<a<2R_g$, along $d$. Atomic force microscopic images of the films show the presence of “spheres” distributed randomly on film surfaces with in-plane dimensions matching $a$. From the variation of $E$ with $d^{-2}$ the effective mass $\left(m_{\mathit{eff}}\right)$ of the exciton is also determined. For $R_g<d<4R_g$ (region 2) the extinction and $E$ become essentially independent of $d$, indicating exciton localization along $d$, and the value of $m_{\mathit{eff}}$ becomes very large. This enhancement in the effective mass maybe used to quantify localization. The variations of electron density $\left(\rho \right)$ with $d$, i.e., the electron density profiles (EDPs) of the films extracted from x-ray reflectivity studies, indicate formation of layers with period “$b$” (in Å), $R_g<b<2R_g$ parallel to substrate surface in region 2 and a constant $\rho$ film in region 1. On raising the temperature of a typical film to $60°\mathrm{C}$, the layering was seen to almost vanish, as obtained from both the EDP and the Patterson function of the reflectivity profile. The close correspondence between “$a$” and “$b$” indicates that the molecules forming the $J$-aggregates form the layers, too. The average difference in $\rho$ between successive extrema in the EDPs in region 2, denoted by $\delta$, can be used as the order parameter for the layering transition. For PS-5, $\delta >0$ at $d\simeq 4R_g$, where the exciton is still delocalized. Layering reduces the Hamaker constant $\left(A_H\right)$, deciding the cohesive force, between the layers and this reduction, $\Delta A_H$, is found to be less than $A_e$ at $d⩾4R_g$, where $i{A}_e∕\hslash$ is the amplitude for exciton transfer between neighboring molecules in the excitonic lattice of region 1. On the other hand, $\Delta A_H$ in region 2 starts from a value larger than $A_e$. This indicates that $\Delta A_H$ acts as a barrier between the layer, which localizes the exciton within the layers.

Figure 1

uv spectroscopic data for the pure electronic singlet band ${}^{1}A_{1g}\to {}^{1}E_{1u}$ for polystyrene films of molecular weights $560900\mathrm{g}{\mathrm{mol}}^{-1}$ (PS-5, closed circles) and $212400\mathrm{g}{\mathrm{mol}}^{-1}$ (PS-2, open squares), against film thickness ($d$, in Å) on fused quartz. Two regions, region 1 $(4R_g⩽d⩽12R_g)$ and region 2 $(R_g<d<4R_g)$ are marked, where $R_g=\text{unperturbed}$ radius of gyration of PS; $R_g=204\mathrm{\AA }$ for PS-5 and $R_g=125\mathrm{\AA }$ for PS-2. (a) Extinction coefficient with respect to thinnest film $\left({\kappa }_{\mathit{ext}}\right)$ vs $d$; (b) exciton energy ($E$, in eV) vs $d$, solid line for PS-5 and dashed line for PS-2 denote best fits with Eq. (3) (see text for details); (c) $E$ vs $d^{-2}$ in ${\mathrm{\AA }}^{-2}$, solid line for PS-5 and dashed line for PS-2 denote best fits with $E=E_b+h∕8\pi m_{\mathit{eff}}{d}^2$, $E_b=\text{bulk}$ PS energy, $m_{\mathit{eff}}=\text{effective}$ mass of exciton.

Figure 2

(Color online) Atomic force microscopic topographies of PS-5 film of thickness $\sim 518\mathrm{\AA }$. Top left: $1\mu \mathrm{m}\times 1\mu \mathrm{m}$ scan, bottom left: $500\mathrm{nm}\times 500\mathrm{nm}$ scan. Line profiles along two near-orthogonal directions $c$ and $d$, marked in bottom left scan, are shown in top and bottom right, respectively.

Figure 3

(Color online) X-ray reflectivity data for the PS-5 films. (a) Reflectivity profiles, i.e., specular reflectivity normalized with Fresnel reflectivity $(R∕R_F)$ vs normal momentum transfer $q_z$ (in ${\mathrm{\AA }}^{-1}$) in open circles and fits using DWBA analysis as lines of different colors for films of different thickness; thickness of film indicated beside each curve. Curves have been upshifted for clarity. (b) EDPs, i.e., electron density $\rho$ (in $e{\mathrm{\AA }}^{-3}$) vs film thickness $d$ (in Å), of the respective films extracted from these fits in the same color code. (c) Reflectivity profiles (open circles) and fits using DWBA analysis (black line) and constant density film (dark yellow line) of a $518\mathrm{\AA }$ thick film at $60°\mathrm{C}$ (top, red circles) and at room temperature ($24°\mathrm{C}$, bottom, blue circles). Top curve set upshifted for clarity. Inset I: Patterson function obtained from reflectivity profiles at $60°\mathrm{C}$ (red, top panel) and at $24°\mathrm{C}$ (room temperature, blue, bottom panel). Arrows indicate secondary peaks due to layering. Inset II: EDP extracted from films at $60°\mathrm{C}$ (black, top panel) and at room temperature (black, bottom panel). The constant density film EDP is shown in dark yellow in both panels.

Figure 4

(a) The order parameter for the layering transition $\delta$ $\left(e{\mathrm{\AA }}^{-3}\right)$ given by the average difference between successive extrema in each film for PS-5 (closed circles) and PS-2 (open squares); the solid (for PS-5) and dotted (for PS-2) lines are the best fits. (b) The consequent reduction in Hamaker constant from layer center to layer-layer interface, $\Delta A_H$ (meV) for PS-5 (closed circles) and PS-2 (open squares); the dotted (for PS-5) and dashed (for PS-2) lines show the corresponding values, 26 and $30\mathrm{meV}$, of $A_e$, the coefficient giving the probability amplitude for exciton transfer.