Zero-point motion and direct-indirect band-gap crossover in layered transition-metal dichalcogenides

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Zero-point motion and direct-indirect band-gap crossover in layered transition-metal dichalcogenides

Luciano Ortenzi, Luciano Pietronero, and Emmanuele Cappelluti

Phys. Rev. B 98, 195313 – Published 26 November 2018

Abstract

Two-dimensional transition-metal dichalcogenides $M X_{2} ({MoS}_{2}, {WS}_{2}, {MoSe}_{2}$ ,...) are among the most promising materials for band-gap engineering. Widely studied in these compounds, by means of ab initio techniques, is the possibility of tuning the direct-indirect gap character by means of in-plane strain. In such kind of calculations however the lattice degrees of freedom are assumed to be classical and frozen. In this paper we investigate in details the dependence of the band-gap character (direct vs indirect) on the out-of-plane distance $h$ between the two chalcogen planes in each $M X_{2}$ unit. Using DFT calculations, we show that the band-gap character is indeed highly sensitive on the parameter $h$ , in monolayer as well as in bilayer and bulk compounds, permitting for instance the switching from indirect to direct gap and from indirect to direct gap in monolayer systems. This scenario is furthermore analyzed in the presence of quantum lattice fluctuation induced by the zero-point motion. On the basis of a quantum analysis, we argue that the direct-indirect band-gap transitions induced by the out-of-plane strain as well as by the in-plane strain can be regarded more as continuous crossovers rather than as real sharp transitions. The consequences on the physical observables are discussed.

Received 17 September 2018
Revised 16 October 2018

DOI:https://doi.org/10.1103/PhysRevB.98.195313

Physics Subject Headings (PhySH)

Band gap

Functional materials Transition metal dichalcogenides

Density functional calculations Optical spectroscopy

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Luciano Ortenzi^1,2, Luciano Pietronero^2,1, and Emmanuele Cappelluti³

¹Istituto dei Sistemi Complessi, CNR, 00185 Roma, Italy
²Dipartimento di Fisica, Università La Sapienza, P.le A. Moro 2, 00185 Roma, Italy
³Istituto di Struttura della Materia, CNR, Division of Ultrafast Processes in Materials (FLASHit), 34149 Trieste, Italy

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Vol. 98, Iss. 19 — 15 November 2018

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Images

Figure 1
(a) Top view and (b) side view of monolayer transition-metal dichalcogenides $M X_{2}$ . The unit cell (marked by dashed lines) contain two $X$ atoms and one $M$ atom. In the top view, the bottom $X$ atom is hidden by the top one. The structural parameters $a$ and $h$ are here also defined. (c) Side view of bilayer $M X_{2}$ in the 2H- $M X_{2}$ structure. The unit cell is here specified. (d) Side view of bulk $M X_{2}$ in the 2H- $M X_{2}$ structure. (e)-(f) Lattice displacements of the $A_{1}^{'}$ and $A_{1 g}$ modes in the single layer and bilayer/bulk structure, respectively.
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Figure 2
Characteristic band structure of monolayer ${MoS}_{2}$ , here evaluated within GGA calculations, with $a = 3.197$ Å and $h_{\exp} = 3.172$ Å. The electronic dispersion is characterized by three gaps: the direct one $E_{KK}$ at the K point, the indirect one $E_{Γ K}$ between the band edge of the valence band at the $Γ$ point and the band edge of the conduction band at the K point, and the indirect one $E_{KQ}$ between the band edge of the valence band at the K point and the band edge of the conduction band at the Q point.
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Figure 3
Direct/indirect band-gap crossover for monolayer ${MoS}_{2}$ as a function of the interplane $S − S$ structural parameter $h$ : (a) evolution of the band structure for three representative values of $h$ . (b) evolution of the band gaps $E_{α} (h)$ as a function of $h$ . An indirect gap $E_{Γ K}$ is predicted for $h \leq 3.148$ Å, a direct gap $E_{KK}$ is predicted for $3.148 Å \leq h \leq 3.244$ Å, and an indirect gap $E_{KQ}$ is predicted for $h \geq 3.244$ Å. (c) Frozen phonon energy $V (h)$ as a function of $h$ (open symbols). The shape and color of the symbols show the character of the gap as evaluated by panel (b). The horizontal line denotes the zero point energy $ℏ ω_{A_{1}^{'}}$ , whose intercept with $V (h)$ gives the zero point motion lattice displacement. The dashed Gaussian-like line is the probability distribution $P (h)$ defined in Eq. (1), weighting the available phase space for each direct/indirect gap case.
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Figure 4
Effects of the quantum lattice fluctuations on the band gap for different families of TMDs. For each compound, in the lower panel we show the evolution of the different band gaps $E_{α}$ as a function of the interplane $X − X$ distance; in the upper panel the frozen phonon DFT total energy (open symbols), the quadratic fit (solid line), the energy $ω_{A_{1}^{'} / A_{1 g}} / 2$ relevant for lattice quantum fluctuations (horizontal line), and the probability distribution function $P (h)$ (dashed line).
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Figure 5
(a) Band gaps $E_{α}$ at the classical level (i.e., using $h_{cl})$ for different in-plane lattice constants $a$ of ${MoS}_{2}$ monolayer. The values of $h_{cl}$ for each $a$ are shown in Table 2. (b),(c) Evolution of the band gaps $E_{α}$ and of the frozen phonon energy with $h$ for 2% compressive in-plane strain [panel (b)] and for 0.5% tensile strain [panel (c)].
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