An exciton interacting with the phonons of an electronic Wigner crystal

TL;DR

This work analyzes an exciton in the top layer of a bilayer system coupled to electrons forming a two-dimensional Wigner crystal in the bottom layer, explicitly including coupling to the WC phonons. By formulating a Fröhlich-type exciton–phonon Hamiltonian and solving it with a non-perturbative self-consistent Born approximation, the authors show that Bloch-band polarons form as the exciton dresses itself with WC phonons, with their energy and damping strongly dependent on electron density. Resonant interband scattering can cause strong damping at intermediate densities, while detuning at higher densities can restore well-defined quasiparticles; these dynamics crucially affect the observed exciton spectra, including umklapp features. The results connect the microscopic exciton–phonon coupling to measurable optical spectra in van der Waals heterostructures and suggest using exciton spectroscopy as a probe of strongly correlated electronic states. Potential extensions include richer exciton band structures, in-plane excitations, transport properties, and induced interactions arising from Bloch polarons.

Abstract

With the advent of atomically thin and tunable van der Waals materials, a two-dimensional electronic Wigner crystal has recently been observed. The smoking gun signal was the appearance of an umklapp branch in optical exciton spectroscopy coming from the periodic potential generated by the Wigner crystal assumed to be static. Vibrations of the Wigner crystal however leads to a gapless phonon spectrum, which may affect the exciton spectrum. To explore this, we develop a field theoretical description of an exciton interacting with electrons forming a Wigner crystal including the coupling to the phonons. We show that importance of the exciton-phonon coupling scales with the exciton-electron interaction strength relative to the typical phonon energy squared. The motion of the exciton leads to two kinds of scattering processes, where the exciton emits a phonon either staying within the same Bloch band (intraband scattering) or changing its band (interband scattering). Using a non-perturbative self-consistent Born approximation, we demonstrate that these scattering processes lead to the formation of quasiparticles (polarons) consisting of the exciton in Bloch states dressed by Wigner crystal phonons. The energy shift and damping of these polarons depend on the electron density in a non-trivial way since it affects both the exciton-phonon interaction strength, as well as the phonon and exciton spectra. In particular, the damping is strongly affected by whether the polaron energy is inside the gapless phonon scattering continuum or not. Using these results, we finally analyse their effects on the observed spectral properties of the exciton.

Figures (6)

• Figure 1: (a) An exciton in the top layer interacts with the electrons forming a Wigner crystal in the lower layer. (b) This leads to the formation of lattice vibrations (phonons).
• Figure 2: (a) Contour plot of the lowest ($m=0$) Bloch band for $d/a=0.05$, $m_{x}a^2\kappa/d^4=217.5$, and a system size of 11x11 with energies in units of $1/m_xa^2$. (b) The two lowest Bloch bands following the orange path in the BZ shown in panel (a). The grey/black filled circles indicate inter- and intra-band scattering via the emmison of phonons (blue wavy lines).
• Figure 3: (a) Contour plot of the lowest (transverse) phonon mode for $a=23.5$nm and calculated using a system size of 11x11. (b) Dispersion of the transverse and longitudinal phonon modes along the orange path shown in Fig. \ref{['fig.DispTLB']}(a) with their dispersion around the $\mathbf{\Gamma}$-point indicated.
• Figure 4: Feynman diagrams for the exciton self-energies associated with intralayer $\Sigma_{00}$ and interlayer $\Sigma_{10}$ scattering. Wavy lines indicate phonons while the double lines are the dressed exciton propagator with the green/yellow color representing the $m=0$/$m=1$ Bloch bands. The red vertex is associated with scatterings where the exciton changes band, while the green and yellow vertices represent scatterings where the exciton stays in the same band.
• Figure 5: Diagonal exciton spectral functions in the Bloch band basis calculated with (full) and without (dashed) interband scattering for ${\bf p} = \bf \Gamma$ and the indicated densities. The spectral functions are normalized such that maximum value is $1$.