Tunneling signatures of interband coherence in dilute exciton condensates

TL;DR

This work develops a mean-field, low-density framework to study exciton condensates in two-band systems and how scanning tunneling microscopy can reveal interband coherence signatures in the dilute BEC regime. For monolayers, it predicts lattice-symmetry-breaking spatial oscillations in the tunneling conductance at the coherence wavevector, and demonstrates how combining spatially averaged and oscillatory signals can reconstruct the exciton wavefunction $\phi^{1s}_{\boldsymbol{k}}$; for bilayers, it identifies a characteristic peak in the averaged tunneling spectrum signaling exciton formation and enabling local extraction of the exciton density $n_{ex}$. The analysis covers spin and valley degrees of freedom, showing how spin-resolved tunneling can probe interband spin textures and distinguish SDW from CDW orders. Overall, the results position STM as a powerful local probe to characterize exciton insulators, their coherence, and density, with implications for material candidates and inhomogeneous samples.

Abstract

We theoretically investigate signatures of exciton condensation and the underlying interband coherence in scanning tunneling microscopy. We consider both monolayer and bilayer condensates in the regime of a dilute condensate of tightly bound excitons. For monolayer condensates, interband coherence is directly encoded in spatially oscillating contributions to the tunneling conductance, which break the underlying lattice symmetry. We show how scanning tunneling microscopy allows one to extract the exciton wavefunction. For bilayer condensates, we show that the formation of the exciton insulator is signaled by the emergence of a characteristic peak in the tunneling conductance, which can be used to extract the (local) exciton density. Our results are based on analytical considerations using a systematic solution of the mean-field equations in powers of the exciton density as well as numerical calculations.

Figures (4)

• Figure 1: (a) Schematic rendering of STM measurements on a monolayer exciton insulator. The tip tunnel couples to both bands. A momentum offset between the band edges of the conduction and valence bands by a wavevector $\mathbf w$ leads to spatial oscillations in the STM signal with wavelength ${2\pi}/{|\mathbf w|}$. These are a direct signature of interband coherence. Moreover, in combination with the spatially uniform contribution to tunneling, these allow the exciton wavefunction to be extracted. (b) Schematic rendering of a STM experiment on a bilayer exciton condensate. The tip electrons are tunnel coupled only to the top layer. Exciton condensation enables a process in which a conduction electron bound in an exciton tunnels out into the tip, leaving behind a valence hole. This leads to a characteristic peak in the tunneling spectrum, indicative of the formation of excitons.
• Figure 2: Spatially averaged tunneling conductance for a monolayer, see Eq. \ref{['eq:spatiallyaveragedspinlessdos']}. Numerical results (blue) are compared to analytical results from Eqs. \ref{['eq:analyticaldosfinalupper']} and \ref{['eq:analyticaldosfinallower']}, working at $n_{\text{ex}}=0.025\, (a_B^*)^{-2}$. The panels show results for (a) equal masses, $m_c=m_v$ and (b) unequal masses, $m_v=3m_c$.
• Figure 3: (a) Fourier component of differential conductance at wavevector $\mathbf w$ for $n_{\text{ex}} (a_B^*)^2 = 0.025$ (orange) and $n_{\text{ex}} (a_B^*)^2 = 0.05$ (blue). (b) Exciton wavefunction extracted by combining numerical results for the spatially oscillating and averaged differential conductance, plotted as a function of $k^2 = \frac{e \overline I}{G_0\,\mathrm{Ry}^*} (a_B^*)^{-2}$, cf. Eq. \ref{['eq:currentkconversion']}. The dashed red line is the analytical wavefunction, Eq. \ref{['eq:excitonwfmonolayer']}, for comparison. Only a constant prefactor is fitted. Both panels: $m_c=m_v$.
• Figure 4: (a) Spatially averaged differential conductance into a bilayer exciton condensate for $|z_a-z_b|=0.2\, a_B^*$ for two values of $n_{\text{ex}}$. The appearance of a peak at negative bias is characteristic for exciton condensation. Its integrated weight provides a measure of $n_{\text{ex}}$, cf. Eq. \ref{['eq:peakintegralnxconversion']}. (b) Like (a) but with $|z_a-z_b|=0.5\,a_B^*$. Both panels: $m_c=m_v$.