Exfoliation energy, quasiparticle band structure, and excitonic properties of selenium and tellurium atomic chains

Effects that are not captured by the generalized-gradient density-functional theory have a prominent effect on the structural binding and on the electronic and optical properties of reduced-dimensional and weakly bound materials. Here, we report the exfoliation energy of selenium and tellurium atomic chains with nonempirical van der Waals corrections, and their electronic and optical properties with the $GW$ and Bethe-Salpeter formalisms. The exfoliation energy is found to be within 0.547–0.719 eV/u.c. for the selenium atomic chain, and 0.737–0.926 eV/u.c. for the tellurium atomic chain (where u.c. stands for unit cell), depending on the approximation for the van der Waals interaction and the numerical tool chosen. The $GW$ electronic band gap turned out to be 5.22–5.47 (4.44–4.59) eV for the Se (Te) atomic chains, with the lowest bound obtained with the Godby-Needs (GN), and the upper bound to the Hybertsen-Louie (HL) plasmon-pole models (PPMs). The binding energy of the ground-state excitonic state ranges between 2.69 and 2.72 eV for selenium chains within the HL and GN PPM, respectively, and it turned out to be 2.35 eV for tellurium chains with both approximations. The ground-state excitonic wave function is localized within 50 Å along the axis for both types of atomic chains, and its energy lies within the visible spectrum: blue [2.50(GN)–2.78(HL) eV] for selenium and yellow-green [2.09(GN)–2.28(HL) eV] for tellurium, which could be useful for LED applications in the visible spectrum.

Figure 1

Energy $E$ vs in-plane and out-of-plane lattice constants $a$ and $c$ for (a,b) bulk selenium and (c,d) bulk tellurium, as obtained from the qe code. The energy is reported with respect to the minimum for each calculation, labeled $E_{\text{bulk}}$.

Figure 2

(a,b) Convergence of the total energy vs $a$ on the tetragonal unit cell. (c,d) Total amount of charge $q$ within a cylinder centered about the chain axis with radius $a/4$ vs $a$ on the tetragonal unit cell. The vertical axes, named charge confinement, display $\frac{q}{18\text{electrons}}\times 100$. (e) Atomistic structure of the selenium atomic chain (the structure of the tellurium chain is similar, and not shown for that reason). (f,g) Total energy for the selenium atomic chain without and with vdW corrections. (h,i) Total energy for the tellurium atomic chain without and with vdW corrections, respectively. The minima in subplots (f)–(i) are at coordinates ($c_{\text{chain}},E_{\text{exf}}$) listed in Table 1.

Figure 3

Band structures for (a) selenium and (b) tellurium atomic chains with vdW exchange-correlation functionals. The golden lines correspond to a calculation with the qe code with NC pseudopotentials, the dashed line to a calculation with the vasp code without spin-orbit coupling, and the solid black lines to a vasp calculation with spin-orbit coupling. The inset in (b) shows bands above 3 eV.

Figure 4

Orbital-resolved density of states (DOS) for (a,b) selenium and (c,d) tellurium atomic chains with and without SOC (vasp code employed). Overall, the height of the van Hove singularities does not change drastically upon inclusion of SOC corrections, but SOC does produce a considerable split near the valence band, especially for the tellurium chain, and an increase in the DOS at about 2.0 (1.5) eV for the Se (Te) chain.

Figure 5

Convergence of the electronic band gap [subplots (a) and (c)] and of the energy difference at the end of the first Brillouin zone, point $\pi /c_{\text{chain}}$ [subplots (d) and (e)], vs the energy cutoff for the $q$ points within the berkeleyGW code, for the selenium and tellurium atomic chains, with the HL approximation to the plasmon-pole model (PPM).

Figure 6

$GW$-corrected band structure of (a) selenium and (b) tellurium atomic chains with the HL (full dots) and GN (open diamonds) approximations. The band gap is direct.

Figure 7

Imaginary part of the dielectric function (absorption spectrum) with and without electron-hole interactions. The position of the first peaks is highlighted. The blue curve was drawn semitransparently, which leads to a gray color at energies where the two curves overlap. The insets show the extent ($\sim 50\AA$) of the excitonic ground-state wave function.