Ultraviolet optical conductivity, exciton fine-structure and dispersion of freestanding monolayer h-BN

Abstract

Excitons govern the light-matter interaction in 2D gapped materials with intrinsically large binding energies. In spite of plentiful optical measurements in the visible for semiconducting transition-metal dichalcogenides, we still lack optical-absorption studies of the exciton structure of insulating 2D materials that requires UV light. Moreover, measurements of the momentum dispersion of excitons in the vicinity of optical limit are rare owing to low resolutions but hold the key to reveal quasiparticle interactions. To close this gap, we employ high momentum resolution electron energy loss spectroscopy ($q$-EELS) to explore exciton dispersions of mono- and few-layer hexagonal boron nitride. Surprisingly, we reveal a fine structure of the first bright exciton dispersion band composed by two features (A and A$'$), visible only at small momentum, not predicted by Bethe-Salpeter calculations. Introducing an optical conductivity approximation (OCA), we extract from the experimental $q$-EELS spectra the ultraviolet (UV) optical conductivity at zero momentum, $σ(ω)$, and discuss the exciton fine structure in $σ(ω)$, consistent with previous photoluminescence observations. Our findings establish a general methodology to probe the fine structure of exciton dispersions, providing new insights into exciton-phonon sidebands and eventually polarons in low-dimensional materials.

Figures (13)

• Figure 1: The q-E diagrams of monolayer h-BN by $q$-EELS. (a): Atomically resolved ADF-STEM image of monolayer h-BN. Scale bar: $1$ nm. Pink balls are N atoms and blue ones correspond to B atoms. (b): EEL spectrum at a finite momentum ($q=0.036$ Å$^{-1}$) in black. Peak positions and area are identified via a Four Lorentzian fitting model where each Lorentzian is plotted with dashed orange lines. The continuous orange line indicates the sum of the four Lorentzians, smoothing the experimental signal. (c): Experimental $q$-$E$ diagram for the valence electron energy-loss (EEL) spectra in monolayer h-BN along the $\Gamma$M direction. (d): Second order derivative of the spectrum intensity of A, A$'$ excitations. White (c) and cyan (d) dots correspond to the peak positions at each $q$.
• Figure 1: Experimental $q$-E diagrams of monolayer h-BN. a, $q$-E diagram along the $\Gamma$K direction. b, along the $\Gamma$M direction. Fine structure A, A' and B peaks are marked on the raw data without any smoothing.c-d, Zoom-in fine structure of the A and A' structure in a and b, respectively. White dots are extracted using 4LM fitting. e-f, Second derivative of the q-EEL spectrum intensity of a and b, respectively.
• Figure 2: Exciton structure of monolayers in the low q range and optical conductivity. (a)-(b): $q$-EELS experimental spectra (black) and obtained from the optical conductivity approximation [red lines of panels (c) and (d)]. Red momentum transfers correspond to the data used to extract the optical conductivity. We note the optical conductivity approximation slowly deteriorates over the increasing $q$. Real (c) and imaginary (d) parts of the 2D optical conductivity in units of the quantum conductance ($G_0 = e^2/\pi\hbar$) are extracted from experiments through the optical conductivity approximation (red) and calculated with the Bethe-Salpeter equation at fixed nuclei (blue). The presence of the A$'$ peak in the experimental optical conductivity while missing in the BSE one is an indication of the phonon nature of the A$'$ excitation.
• Figure 2: Exciton peak evolution in the low q range towards the optical limit for h-BN with different thicknesses. (a)-(b): $q$-EELS experimental spectra (black) and obtained from the optical conductivity approximation [red lines of panels (a) and (b)]. Red momentum transfers correspond to the data used to extrapolate the optical conductivity. We note the optical conductivity approximation slowly deteriorates over the increasing $q$. Real (c) and imaginary (d) parts of the 2D optical conductivity over the number of layers in units of the quantum conductance ($G_0 = e^2/\pi\hbar$) are extracted from experiments through the optical conductivity approximation (red) and for the monolayer calculated with the Bethe-Salpeter equation at fixed nuclei (blue). We report here data for monolayer h-BN of the $\Gamma$M and $\Gamma$K directions, along with $\Gamma$M data of the bilayer (2$\Gamma$M) and trilayer (3$\Gamma$M)
• Figure 3: Energy and oscillator strength dispersions of monolayer h-BN along $\Gamma$M direction. (a): Exciton dispersions of the A and A$'$ peaks in black and its weighted average (($E_AI_A+E_{A'}I_{A'})/(I_A+I_{A'})$) in green, and OCA (red) and BSE (blue) calculated dispersions. The OCA has been restricted to its range of validity up to $q_{\mathrm{max}} \approx 0.12$ Å$^{-1}$ (b): The dispersion of the B peak obtained by experiment (black), OCA (red) and BSE (blue). (c): Fraction of the intensity of A$'$ peak with respect to the A+A$'$ peaks. The same analysis is performed along the $\Gamma$K direction and for bi- and three-layer along the $\Gamma$M direction. Results can be found in the Extended Data Fig. \ref{['fig:dispersion_extended']}.