Watching Polarons Form in Real Time

Abstract

Polaron formation in pump-probe experiments is an inherently non-equilibrium phenomenon, driven by the ultrafast coupled dynamics of electrons and phonons, and culminating in the emergence of a localized quasiparticle state. In this work, we present a first-principles quantum-kinetic theory of polaron formation that captures the real-time evolution of electronic and lattice degrees of freedom in presence of electron-phonon coupling. We implement this framework to investigate the ultrafast polaron formation in the prototypical polar insulator MgO. This approach allows us to determine the characteristic timescales of polaron localization and to identify its distinctive dynamical fingerprint. Our results establish clear and experimentally accessible criteria for identifying polaron formation in pump-probe experiments.

Figures (4)

• Figure 1: Schematic illustration of the ground-state potential energy surface $E$ for an $N$- (light) and $N-1$-electron systems (dark) as a function of $Q_p$ -- a generalized coordinate connecting the high-symmetry and the distorted polaronic structures. In the $N-1$-electron system, polaron formation proceed via a dynamical process in which a structural distortion emerges in concomitance with carrier localization, resulting in an energy lowering by the polaron formation energy $\varepsilon_p$.
• Figure 2: (a) Time dependence of the polaron formation energy throughout the dynamical formation of a hole polaron in MgO. The static polaron formation energy obtained from the solution of the static polaron equation or Ref. sio_ab_2019 is reported as dashed horizontal line. (b-d) Hole envelope function $A_{n{\bf k}}(t)$ obtained from the time propagation of the time-dependent polaron equations for time delays of $t=10$, 40, and 500 fs, respectively (marked by black dots in panel (a)). (e-g) Density of an extra hole injected at the top of conduction band for the same time steps of panels (b-d).
• Figure 3: (a) Ultrafast modulation of the phonon polaron envelope function $B_{{\bf q} \nu}(t)$ during the formation of a hole polaron in MgO. The phonon mode index $\nu$ is set to the longitudinal optical phonon, and the momentum ${\bf q}$ runs along the W-L-$\Gamma$-X high-symmetry path in the Brillouin zone. (b-c) Enlarged views of the regions highlighted by the blue and red rectangles in panel (a). (d) Time dependence of the phonon envelope function $B_{{\bf q} \nu}(t)$ for the momenta $q_1,q_2,q_3$ marked by vertical lines in panel (a). (e) Fourier transform $B_{{\bf q}\nu}(\omega)$ of the data in (d).
• Figure 4: (a) Polaron spectral function $\mathcal{B}^2_{{\bf q}} (\omega) = \sum_\nu |B_{{\bf q} \nu}(\omega)|^2$ for momenta along the W-L-$\Gamma$-X high-symmetry path. The phonon dispersion obtained from density-functional perturbation theory (black line) is included for comparison. (b) Comparison between the polaron (blue) and the phonon density of state (black). (c-f) Polaron spectral function $\mathcal{B}^2_{{\bf q}} (\omega)$ obtained by restricting the Fourier transforms to time intervals of 0.5 ps for different time delays after the onset of the dynamics.