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, 8 (1), 95

Probing Low-Energy Hyperbolic Polaritons in Van Der Waals Crystals With an Electron Microscope

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Probing Low-Energy Hyperbolic Polaritons in Van Der Waals Crystals With an Electron Microscope

Alexander A Govyadinov et al. Nat Commun.

Abstract

Van der Waals materials exhibit intriguing structural, electronic, and photonic properties. Electron energy loss spectroscopy within scanning transmission electron microscopy allows for nanoscale mapping of such properties. However, its detection is typically limited to energy losses in the eV range-too large for probing low-energy excitations such as phonons or mid-infrared plasmons. Here, we adapt a conventional instrument to probe energy loss down to 100 meV, and map phononic states in hexagonal boron nitride, a representative van der Waals material. The boron nitride spectra depend on the flake thickness and on the distance of the electron beam to the flake edges. To explain these observations, we developed a classical response theory that describes the interaction of fast electrons with (anisotropic) van der Waals slabs, revealing that the electron energy loss is dominated by excitation of hyperbolic phonon polaritons, and not of bulk phonons as often reported. Thus, our work is of fundamental importance for interpreting future low-energy loss spectra of van der Waals materials.Here the authors adapt a STEM-EELS system to probe energy loss down to 100 meV, and apply it to map phononic states in hexagonal boron nitride, revealing that the electron loss is dominated by hyperbolic phonon polaritons.

Conflict of interest statement

The authors declare no competing financial interests.

Figures

Fig. 1
Fig. 1
STEM-EELS map of a hexagonal boron nitride flake. a STEM HAADF image of the h-BN flake on a TEM membrane. Brighter areas correspond to a larger h-BN thickness, black is the supporting TEM membrane. Scale bar, 500 nm. White dashed lines are guides marking steps on the flake surface, dash-dotted line marks the flake edge. Blue, red, and green rectangles mark the area from which the spectra in Fig. 4d were collected. Orange line shows where the data for Fig. 6c were taken from. b Typical spectra acquired on h-BN (open circles) and on an empty TEM membrane (magenta); the corresponding locations are marked, respectively, by magenta and blue crosses in a. Each spectrum is normalized to its ZLP maximum. Inset shows the 3D sketch of the sample topography from a. c Close-up view of the same spectrum as in b (open circles); the thick magenta curve shows the ZLP averaged along the horizontal magenta line in a. Closed circles show the spectrum after ZLP subtraction and the red curve shows a Gaussian fit. d Map of the peak position ω 0. Black color marks regions of insufficient signal level (below 2.5x10−4 threshold) and of otherwise irregular fits (see Methods for details). Black dashed and dash-dotted lines are the same eye guides as marked in a
Fig. 2
Fig. 2
Dielectric permittivity tensor of h-BN in the upper Reststrahlen band. Real (solid red) and imaginary (dashed red) parts of the principal components of ϵ^ perpendicular to the h-BN optical axis. For the parallel component, the imaginary part is vanishingly small and only the real part is shown (blue). Inset illustrates the orientation of the tensor's principal axes with respect to the atomic sheets of h-BN. Left and right vertical dashed lines mark the TO and LO phonon frequencies, respectively
Fig. 3
Fig. 3
Theoretical EEL probability spectrum of a homogeneous h-BN slab. a Schematic representation of the electron energy loss due to the excitation of the bulk phonon (left) and guided modes (right) in a dielectric slab. b Intensity profiles of three guided HPhP modes (M0, M1, and M2). c Energy-momentum map P bulk(q,ω) calculated for a 30-nm-thick h-BN slab. d Same as in c, but showing P guid(q,ω). e Same as in c, but showing P begr(q,ω). The dashed curves in d and e depict the dispersion of M0 mode of the hyperbolic phonon polariton (calculated according to the Supplementary Eq. (27)). Inset in d is a close up on P guid(q,ω) at low momenta; dashed purple arrow shows the energy ω g at which the line q=3ωv (solid purple line, corresponding to the maximum of momentum transfer from electron to the guided mode) intersects with the M0-HPhP mode dispersion. f EEL probabilities, Γ(ω), corresponding to bulk (blue), guided-mode (red), Begrenzungseffekt (gray) losses and their sum (green). The vertical purple arrow marks the same energy as in the inset of d. In all plots, the vertical dashed lines mark the TO and LO phonon frequencies; vertical dot-dashed line marks the location of the SO energy (corresponds to ϵ  = −1)
Fig. 4
Fig. 4
Thickness dependence of h-BN EEL spectra. a Dispersions (low energy part) of the fundamental M0-HPhP mode in h-BN films of 2 nm (blue), 15 nm (red), and 30 nm (green) thicknesses. Blue, red, and green vertical dashed lines mark the energies at which these dispersion curves intersect the line q=3ωv (purple line), and which determine the positions of spectral maxima of Γ guid(ω). b Theoretically calculated EEL probability for the corresponding film thicknesses. c A Gaussian with FWHM = 46 meV representing the experimental ZLP. d Experimental spectra (dots) of h-BN averaged over areas marked by blue, red, and green boxes in Fig. 1a. The shaded curves are theoretical EEL spectra obtained by convolving spectra of b with the ZLP in c. All calculated spectra were scaled by the same factor to correspond with the experiment. In all plots, the left and right vertical dashed lines mark the TO and LO phonon energies, respectively
Fig. 5
Fig. 5
Volume vs. edge-localized hyperbolic polaritons. a EEL probability spectrum of 30-nm-thick semi-infinite slab for an on-edge electron trajectory (blue) and for a trajectory 5 nm outside the slab (thin purple) obtained using full-wave simulation (Comsol). b EEL probability spectrum for the beam passing through the h-BN slab away from the edge (off-edge trajectory) obtained using Comsol (solid green). Dashed green spectra in a and b are retarded theoretical calculations (see Supplementary Note 5) using Eq. (1). c Cross section of the induced electric fields (z-component) taken at ω = 1570 cm−1 for the on-edge electron trajectory. d Same as in c but for the off-edge electron trajectory. e Momentum dependent probability П(q y,ω) for the on-edge electron trajectory. f Retarded calculation of P(q y,ω) for a 30-nm-thick, laterally infinite h-BN layer (see Supplementary Note 5), i.e. the sum of contributions depicted in Fig. 2a–c, but with retardation included. Dashed blue curve marks the dispersion of M0-HPhP mode. Insets in e and f illustrate the electron trajectories and the polariton propagation directions. In all plots, the vertical dashed lines mark the TO, SO, and LO phonon energies
Fig. 6
Fig. 6
Electron energy loss spectra near flake edges. a Numerically calculated (Comsol) EEL probability spectra for 30-nm-thick semi-infinite h-BN slab as a function of distance to its edge. Black dashed curves trace the interference fringe maxima appearing due to reflection of the guided volume polaritons from the slab edge (schematically shown at the bottom) calculated using Eq. (6) with reflection phase ϕ refl = π/2. b Spectra extracted from a, but after convolution according to Eq. (5). The spectra are offset vertically for better visibility. Red line marks the position of the peak maximum. Blue spectrum corresponds to the convolution of the solid blue spectrum in Fig. 5a. Vertical dashed lines mark the TO and LO phonon energies. c Gaussian fits to the experimental STEM-EELS spectra for different beam positions near the edge of the 30 nm thick part h-BN flake. The spectra are collected along the orange line marked in Fig. 1a (see Methods). Red line marks the position of the peak maximum, ω 0

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