Observing the spatial and temporal evolution of exciton wave functions

TL;DR

This work addresses the challenge of accessing exciton wave functions by introducing femtosecond time-resolved photoemission orbital tomography (trPOT) and a quantitative real-space reconstruction that captures both amplitude and phase. By applying trPOT to aligned α-sexithiophene films, the authors image the momentum-space distribution of excitons and, with GW/BSE calculations, connect these signatures to a delocalized, phase-coherent wave function spanning about three molecules. Time-resolved measurements reveal a roughly $\sim 20\%$ contraction of the exciton radius within $400\ \text{fs}$, signaling self-trapping driven by exciton–phonon coupling and providing direct insight into exciton dynamics. The results establish trPOT as a general, experimentally accessible framework for visualizing exciton wave functions in molecular and low-dimensional materials, enabling direct comparisons with advanced many-body theories.

Abstract

Excitons, the correlated electron-hole pairs governing optical and transport properties in organic semiconductors, have long resisted direct experimental access to their full quantum-mechanical wave functions. Here, we use femtosecond time-resolved photoemission orbital tomography (trPOT), combining high-harmonic probe pulses with time- and momentum-resolved photoelectron spectroscopy, to directly image the momentum-space distribution and ultrafast dynamics of excitons in $α$-sexithiophene thin films. We introduce a quantitative model that enables reconstruction of the exciton wave function in real space, including both its spatial extent and its internal phase structure. The reconstructed wave function reveals coherent delocalization across approximately three molecular units and exhibits a characteristic phase modulation, consistent with ab initio calculations within the framework of many-body perturbation theory. Time-resolved measurements further show a $\sim 20$\% contraction of the exciton radius within 400 fs, providing direct evidence of self-trapping driven by exciton-phonon coupling. These results establish trPOT as a general and experimentally accessible approach for resolving exciton wave functions -- with spatial, phase, and temporal sensitivity -- in a broad class of molecular and low-dimensional materials.

Figures (12)

• Figure 1: (a) Structural model of four monolayers of 6T (yellow) on the Cu(110)-$(2 \times 1)$O substrate in which molecular planes are tilted by $\pm 32^\circ$ with respect to the sample surface (gray: copper atoms, red: oxygen atoms). The normal incidence pump (blue) and p-polarized probe at 55$^{\circ}$ incidence (red) pulses as well as the angular distribution of the emitted electrons are illustrated. (b) Chemical structure of 6T. (c) Illustration of the lowest optically allowed exciton state $S_1$ with the primary HOMO to LUMO and the secondary HOMO-1 to LUMO+1 contributions.
• Figure 2: (a) Momentum-integrated photoemission spectrum with molecular emission from the HOMO and HOMO-1 below the Fermi energy $E_\mathrm{F}$ and the excitonic signature above $E_\mathrm{F}$ at $t=0$ fs. (b) Momentum maps averaged over the blue-shaded energy regions indicated in panel (a) corresponding to the HOMO (lower row) and exciton (upper row). The experimental data (left column) are compared to simulations for the HOMO and LUMO, respectively, computed in the independent particle approximation (IPA) for a free-standing monolayer of 6T molecules. (c) Momentum-integrated intensity, averaged over energies between 0.6 and 1.4 eV, as a function of delay time indicating an exponential decay of the exciton with a lifetime of $\tau=(422 \pm 15)$ fs.
• Figure 3: (a) Illustration of the exciton model in real space (upper row) and momentum space (lower row) according to Eqs. \ref{['eq:excitonmodel1']} and \ref{['eq:excitonmodel2']}, respectively. The four columns (i--iv) correspond to different choices for the envelope function $\alpha(\bm{R})$, indicated by the black arrow, the phase modulation $e^{\mathrm{i} \beta(\bm{R})}$, and the unit cell shape as detailed in the text. (b) Electron distribution $\Psi(\bm{r}_e, \bm{r}_h)$ of the ab initio exciton $S_1$ with respect to a fixed hole position $\bm{r}_h$ indicated by the red sphere located at the central molecule. For the visualization of the isosurface, we have chosen a value of 10% (3.5%) of the maximum value of the absolute value $|\Psi|$ for the central three (outer two) molecules, while the phase $\arg (\Psi)$ is color-coded (see color-wheel in the inset). (c) One-dimensional line cut of the complex wave function $\Psi(\bm{r}_e, \bm{r}_h=\mathrm{fixed})$ along crystalline $\bm{a}_2$ direction (black line in panel (b)). The solid line represents the absolute value and the filled colors the phase $\beta$, respectively. The gray line shows a Gaussian envelope extracted from the experimental map. (d) Momentum map corresponding to exciton $S_1$ according to Eq. \ref{['eq:exPOTperiodicsimple']}. The solid (dashed) gray lines are isolines of the experimental exciton map (see Fig. \ref{['fig2']}b) at 50% (5%) of its intensity maximum.
• Figure 4: (a) Experimental momentum maps at five different pump/probe delay times $t$ (upper row) together with intensity linescans (lower row) along $k_x$ in the red-boxed regions. The blue-shaded areas correspond to the fits to the data (black line) using two independent Gaussians with a common $\mathrm{FWHM}_k$. (b) Position (red symbols) and full width at half maximum (black symbols) of the intensity peaks as a function of delay time resulting from the fits shown in panel (a).
• Figure 5: (a) Experimental momentum maps at five different pump-probe delays extracted in the energy range of $E - E_\mathrm{F} \in [+0.6, +1.5]$ eV, as well as the momentum map of the Fermi edge taken between $E - E_\mathrm{F} \in [-0.4, +0.4]$ eV. (b) Intensity linescans (black) along $k_{x}$ for each delay time extracted within the red boxes in panel (a) together with the fitted linescan of the Fermi edge (gray). (c) Background-subtracted linescans (black) fitted by two independent Gaussians with a common $\mathrm{FWHM}_k$ (blue-shaded area). (d) Resulting fit parameters of the $\mathrm{FWHM}_k$ and mean peak position as a function of delay time, together with their statistical uncertainties. (e) Residues between background- subtracted linescans and fitted Gaussians.