Probing spatially resolved spin density correlations with trapped excitons

TL;DR

The paper introduces an optical probe based on excitons trapped in a moiré lattice to measure spatial spin density correlations in an adjacent 2D electron system. Via virtual interlayer tunneling and spin-selective electron–exciton scattering, a spin-dependent static potential arises, causing exciton energy shifts that encode the two-point spin density-density correlator. Second-order energy shifts between two excitons separated in space reveal both magnetic spin correlations and the pairing symmetry of superconducting states, enabling spatially resolved access to electronic quantum phases. The approach is adaptable to detect critical behavior at magnetic transitions and to map gap symmetry in superconductors, providing a new spectroscopic handle on strongly correlated 2D materials. Practical realizations can explore higher-order correlation functions by using more excitons, with tunable interlayer coupling and detuning shaping the signal.

Abstract

The rapidly growing class of atomically thin and tunable van der Waals materials is intensely investigated both in the context of fundamental science and for new technologies. There is in this connection a widespread need for new ways to probe the electronic properties of these layered materials, since their two-dimensional (2D) character make conventional probes less efficient. Here, we show how excitons trapped in a moiré lattice can be used as an optical probe for spatially resolved electron spin density correlations in such materials. The electrons in the material of interest virtually tunnel to the moiré lattice where they scatter on the excitons after which they tunnel back. This gives rise to an effective spin-dependent and spatially localised potential felt by the electrons, which in turn leads to energy shifts that can be measured spectroscopically in the exciton spectrum. Using second order perturbation theory combined with a solution to the exciton-electron scattering problem, we show that the electrons mediate an interaction between two excitons resulting in an energy shift proportional to their two-point spin density-density correlation function evaluated at the exciton positions. We then discuss two specific applications of our setup. First, we show that quantum phase transitions between different in-plane anti-ferromagnetic orders in a 2D lattice give rise to large and measurable shifts in the exciton spectrum in the critical regions. Second, we analyse how different pairing symmetries of superconducting phases can be probed. This demonstrates that our scheme opens up new ways to probe electron spin density correlations, which is a key property of many quantum phases predicted to exist in the new 2D materials.

Figures (10)

• Figure 1: Electrons (blue balls) in the lower material can tunnel to the upper material, where they scatter on excitons with opposite spin trapped by a deep moiré potential. The exciton-electron interaction potential supports a bound state, i.e. a trion. The energy levels of the electron in the lower material as well as of the electron and trion in the upper material are shown on the right.
• Figure 2: (a) Diagrams for electron-exciton scattering matrix in the upper layer. (b) Self-energy of the electrons in the lower layer due to tunneling to the upper layer, scattering on the exciton, and tunneling back. (c) The interaction between two excitons mediated by electrons in the lower layer. Red stars (grey crosses) represent the static exciton (tunneling between two layers), single (double) wavy lines are the bare electron-exciton interaction (electron-exciton scattering matrix), and the blue lines are the propagator for the electron.
• Figure 3: The second order energy shift of the spin model given by Eq. \ref{['Eq-H']} due to the excitons in the upper layer. The blue line represents the result for one exciton and the red line shows the interaction term between two nearest neighbour excitons with the same spin.
• Figure 4: The second order energy shift for two excitons with parallel spins in the upper layer as a function of their separation $r$ in units of the lattice constant with $\phi=0.34\pi$.
• Figure 5: The energy shift of two excitons with anti-parallel spins interacting with superconducting samples having three different kinds of Cooper pairing as a function of their separation. We have subtracted the energy shift when the excitons are infinitely apart.