Exciton-based sensing of remote electron correlations in 2D heterostructures

TL;DR

The paper addresses how exciton spectroscopy in 2D TMDs can serve as a quantitative probe of remote correlated electron states in adjacent layers. It develops a compact theory within the $G_0W_0$ framework that links the proximity-induced bandgap shift $\delta E_g$ in a TMD to the full dynamic charge susceptibility $\chi_q^{cf}(\omega)$ of the neighboring layer, including retardation corrections via $\delta E_g^{dyn}$. Applying the formalism to rhombohedral trilayer graphene and Bernal bilayer graphene demonstrates that dynamic screening substantially reduces static estimates and produces characteristic even–odd signatures tied to flavor polarization and Landau-level filling, in line with experiments. The framework provides a quantitative tool for interpreting exciton-sensing measurements and can be extended to finite thickness, spin–orbit coupling, and other correlated states in 2D heterostructures, enabling optical access to complex many-body phenomena.

Abstract

Many monolayer transition metal dichalcogenides, including MoS$_2$, MoSe$_2$, WS$_2$, and WSe$_2$, are direct bandgap two-dimensional (2D) semiconductors with sharp optical resonances at excitonic bound state frequencies. Recent experiments have demonstrated that excitonic resonance frequencies in multilayer van der Waals stacks are altered by long-range Coulomb interactions with electrons in nearby but electrically isolated 2D materials. These modulations have been successfully used to detect transitions between distinct states of remote strongly correlated 2D electron fluids. In this Letter we provide a theory of these frequency shifts, enabling a more quantitative interpretation of excitonic-sensing experiments, and apply it as an example to WSe$_2$ that is proximate to graphene bilayers and multilayers.

Figures (9)

• Figure 1: Optical sensing of proximate layers using semiconductor bandgap renormalization. (a) Side view of an exciton (ellipse) in semiconductor layer $\ell=1$, in the absence (left) and presence (right) of electrons in a proximate $\ell=2$ layer separated by distance $d$. The arrows indicate electronic intra- and interlayer interactions. (b) Schematic band structure and exciton energies for each case. For weakly bound excitons, the change in binding energy ($\delta_{2s}$) is negligible. The resonance shift ($\delta E_X$) then measures the bandgap renormalization ($\delta E_g$).
• Figure 2: Feynman diagrams for the $G_0W$ approximation. (a) Dyson equations for the full Green's function $G$ with self-energy $\Sigma$, and the dressed interaction $W$ with irreducible polarizability $\Pi$. (b) Diagonal self-energy $\Sigma_b$ for band $b$ and $\mathsf{k}=(\bm{k},i\omega_n)$, where $\bm{k}$ is the momentum and $\omega_n$ is a (fermionic) Matsubara frequency, in the $G_0W$ approximation. We implicitly integrate/sum over internal variables, such as $\mathsf{q}=(\bm{q},i\Omega_m)$, where $\Omega_m$ is a (bosonic) Matsubara frequency. (c) In the $G_0W_0$ approximation the irreducible polarizability is replaced by the free-particle susceptibility $\chi_0$.
• Figure 3: Rhombohedral trilayer graphene (RTG) at large displacement field $U_D = 30$ meV and density $n_e = -0.97\times10^{12}$ cm$^{-2}$, and the induced bandgap shift in proximate WSe$_2$. (a) Mean-field electronic bands of the symmetric (paramagnetic; PM) state (red: valley $K$, blue: valley $K'$) and Fermi surface (inset). (b) Real part of the momentum-dependent static RPA polarizability along $q_x$ for competing flavor orders—PM, spin-polarized with valley polarization (SP-VP), spin-polarized with inter-valley coherence (SP-IVC), valley-polarized (VP), and inter-valley coherent (IVC). (c) Induced WSe$_2$ bandgap shift versus interlayer spacing $d$ for the same set of metastable Hartree–Fock flavor states; values are absolute (i.e., not referenced to the PM state). Dashed curves show the static approximation [cf. \ref{['eq:finalenergygap3']}], and solid curves include the retardation correction [cf. \ref{['eq:finalenergygap_corr']}]. (d) WSe$_2$ bandgap change relative to PM, i.e. $\delta E_g-\delta E_g^{\mathrm{PM}}$. We used a six-band-per-flavor continuum model, $300\times300\times3000$ momentum–frequency grids, and dual-gate screening (see SM for details), and $a_G$ is the graphene lattice constant.
• Figure 4: Landau levels and susceptibility of bernal bilayer graphene, and the resulting even–odd bandgap shift of a proximate monolayer WSe$_2$. (a) Landau‐level (LL) spectrum versus magnetic field $B$. (b) Static susceptibility $\mathrm{Re}\,\chi_0$ at $B=3$ T versus $q_x$ at $q_y=0$ for fillings $\nu=0,1,2,3,4$; shading highlights the extra ${0\!\to\!1}$ contribution present at odd filling. Gray dotted lines indicate how the finite interlayer spacing $d$ suppresses contributions to the sensing‐layer band shift. The corresponding dynamic susceptibility is shown in \ref{['fig:bilayer_graphene_chi0_filling0', 'fig:bilayer_graphene_chi0_filling1']}. (c–d) Even–odd bandgap shift in monolayer WSe$_2$ when the BBG filling changes from even $(\nu=0)$ to odd $(\nu=1)$, plotted versus the LL separation $\Delta_{01}$, for (c) $B=3$ T and (d) $B=6$ T. Blue lines: static‐screening contribution only; black lines: static+dynamic. Unless stated otherwise we use a Landau‐level cutoff $N_{\rm cut}=70$; see SM for details.
• Figure S1: Non-interacting, trigonally-warped band structure of bernal bilayer graphene (sketched in inset) at large displacement fields. (a) Electronic bands near valley $K$ at displacement potential $U_D=60$ meV. (b) Same bands but zoomed in on the valence band to highlight the strong trigonal warping and (c) Fermi surfaces at different hole doping potential, illustrating Lifshitz transitions from pockets, to annular to single pocket. (b) Density of states (DOS) at the Fermi surface as function of electronic density $n_e$ and displacement potential $U_D$. Insets indicate the Fermi sea (FS) shape and topology per valley--spin found in different regions of the parameter space. The Lifshitz transition produce distinct features in the DOS.