Universal Scalings in Two-Dimensional Anisotropic Dipolar Excitonic Systems

Low-dimensional excitonic materials have inspired much interest owing to their novel physical and technological prospects. In particular, those with strong in-plane anisotropy are among the most intriguing but short of general analyses. We establish the universal functional form of the anisotropic dispersion in the small $k$ limit for 2D dipolar excitonic systems. While the energy is linearly dispersed in the direction parallel to the dipole in plane, the perpendicular direction is dispersionless up to linear order, which can be explained by the quantum interference effect of the interaction among the constituents of 1D subsystems. The anisotropic dispersion results in a $E^{\sim 0.5}$ scaling of the system density of states and predicts unique spectroscopic signatures including: (1) disorder-induced absorption linewidth, $W(\sigma )\sim {\sigma }^{2.8}$, with $\sigma$ the disorder strength, (2) temperature dependent absorption linewidth, $W(T)\sim T^{s+1.5}$, with $s$ the exponent of the environment spectral density, and (3) the out-of-plane angular $\theta$ dependence of the peak splittings in absorption spectra, $\mathrm{\Delta }E(\theta )\propto {\sin }^2\theta$. These predictions are confirmed quantitatively with numerical simulations of molecular thin films and tubules.

Figure 1

(a) A 2D square lattice of in-plane dipoles. (b) Two parallel chains of dipoles. (c) Left to right: square lattices with different dipole orientations (${\varphi }_{\mu }=0$, $\pi /8$), their 2D dispersion contours ($k\le (\pi /a)$), and those zoomed in ($k\le (\pi /10a)$). The dashed lines indicate the (reciprocal) lattice vectors. (d) Dispersion of the continuum model predicted by Eq. (3) with $k\le (\pi /10r_c)$ and $r_c=a$.

Figure 2

(a) The DOS from numerically evaluating Eq. (2) (circles), that of a ${\varphi }_{\mu }=\pi /4$ square lattice (squares), and a $E^{0.50}$ power-law fit (line). (b) Power-law exponents fitted for the low-energy DOS of square lattices with varying ${\varphi }_{\mu }$. Only cases with the bright ($k=0$) state close to band edge are included [40]. The dashed line indicates the theoretical value 0.5.

Figure 3

Disorder- and thermally-induced absorption linewidths. (a) Absorption linewidth as a function of site disorder strength for a ${\varphi }_{\mu }=\pi /4$ square lattice with 1600 sites. The solid and dashed lines indicate the strong and the weak disorder limits and the dotted line is a ${\sigma }^{2.8}$ fit to the intermediate regime. (b) Absorption linewidth as a function of temperature of the same system as in (a) coupled to a cubic super-Ohmic bath (dots) and its power-law fitting (line). (c) Power-law exponents of disorder (squares) and thermally (circles) induced absorption linewidths as functions of ${\varphi }_{\mu }$ for square lattices. The horizontal lines indicate theoretically predicted ${\sigma }^3$ (dotted) and $T^{4.5}$ (dashed) scalings.

Figure 4

(a) The transient absorption gap of 2D molecular films and (b) the energy splitting between the perpendicular and the parallel-polarized absorption peaks of molecular tubules, compared to the theoretical predictions Eqs. (5) and (6), respectively. The structure of the C8S3 dye molecule is shown in (a) and the arrow indicates the transition dipole direction of its lowest excited state. Here $r_c=5\AA$ and $k_b=2\times {10}^{-4}{\AA }^{-1}$, corresponding to peak absorption at 500 nm. Schematics of planar and tubular aggregates with their structural parameters (lattice offset $l$ for planes and helical pitch angle $\beta$ for tubes). We compare three different models of the excitonic coupling between two C8S3 molecules: simple dipole, extended dipole, and transition atomic charges. These models and the construction of planar and tubular lattices are detailed in S6 and S7 of the SM.