Plasmons dispersion and nonvertical interband transitions in single crystal Bi${}_2$Se${}_3$ investigated by electron energy-loss spectroscopy

Plasmons dispersion and nonvertical interband transitions in Bi${}_2$Se${}_3$ single crystals were investigated by electron energy-loss spectroscopy in conjunction with (scanning) transmission electron microscopy [(S)TEM-EELS]. Both volume plasmons (π plasmon at 7 eV and π + σ plasmon at 17 eV) and surface plasmons (∼5.5 and 10 eV) were demonstrated in STEM-EELS spectra and the corresponding spectral imaging in real space. In further EELS experiments in reciprocal space, the momentum-dependent spectra reveal very different dispersion behavior between π and π + σ plasmons, with π + σ plasmons showing a typical quadratic dependence, whereas the π plasmon exhibited a linear dispersion analogous to what was reported for graphene. Furthermore, a low energy excitation 0.7–1.6 eV was also observed which is attributed to direct nonvertical interband transitions along the Γ$F$ direction.

Figure 1

(a) STEM-EELS spectra acquired on single-crystal Bi${}_2$Se${}_3$ with the electron probe positioned at various locations of the material (inset HAADF image; probe step, 2 nm). The color circles (inset) denote probe locations and the corresponding EELS spectra are shown by the same colors. The intensities of the EELS spectra were normalized to ZLP and then ZLP was removed. Solid black arrow indicates the spectrum acquired at the sample edge. (b) The complex dielectric function of Bi${}_2$Se${}_3$ derived from the black spectrum in (a). The complex dielectric function below 2.5 eV was reproduced from Ref. 23. The dark arrow indicates the broad feature around 8–12 eV in ${\varepsilon }_2$ curve. The pink, sky blue, blue, and green spectra are the blowups of those in (a) with removed ZLP for clarity. Inset is the calculation of effective number $n_{\mathrm{eff}}$(ω) of electrons per atom in Bi${}_2$Se${}_3$. (c)–(e) The STEM-EELS excitation-intensity maps acquired from the HAADF image [inset, (a)] at different energy: (c) ∼5.5 eV, (d) 10 eV, and (e) 17 eV. The dashed black lines in (c)–(e) indicate the interface between material (upper part) and vacuum (bottom part). Color scale bar represents the linearly normalized image intensity.

Figure 2

Momentum-dependent EELS spectra obtained along the momentum transfer ($q$) parallel to the (a) Γ$M$, (b) Γ$K$, and (c) Γ$Z$ directions. The spectra are all normalized to the intensity of peak at 17 eV and then displaced vertically to improve readability. Inset in (a)–(c) are the selected-area electron diffraction (SAED) patterns of Bi${}_2$Se${}_3$ acquired with the incident electron beam parallel to the $c$ axis along the [001] direction [inset, (a) and (b)], and perpendicular to the $c$ axis along the $\left[\overline{1}10\right]$ direction [inset, (c)]. The (hexagonal) dash lines in (a) and (b) illustrates the Brillouin zone. The arrows with white/red color in (a), (b), and (c) indicate the momentum transfer range for momentum-dependent EELS experiments.

Figure 3

(a) Lorentz curve fitting of EELS spectra for $q$ = 0, 0.8, and 1.28 Å${}^{-1}$, respectively. (b) The energy of π and π + σ plasmons along Γ$M$, Γ$K$, and Γ$Z$ directions as a function of momentum transfer ($q$) with black square (Γ$M$ direction), gray circle (Γ$K$ direction), and red triangle (Γ$Z$ direction), respectively.

Figure 4

(a) Momentum-dependent EELS spectra of additional peak at 0.7–1.6 eV observed with sample tilting. (b) The dispersion curve of the nonvertical interband transition peaks as a function of momentum transfer with sample tilting (marked with black square, respectively). The dashed line is the linear curve fitting. The dispersion of the lowest conduction band along the Γ$F$ direction in Ref. 12 with energy normalized to the experimental value of energy gap ($E_g$ = 0.32 eV) at $q$ = 0 is also plotted for comparison.