Electronic localization and optical activity of strain-engineered transition-metal dichalcogenide nanobubbles

Abstract

Strain-engineered transition-metal dichalcogenide nanobubbles are promising platforms for quantum emission, as revealed by recent experimental observations. In this work, we present an \textit{ab initio} investigation of MoS$_2$, WS$_2$, MoSe$_2$, and WSe$_2$ nanobubbles, linking their structural and electronic properties to predictions of their optical activity. Inflating forces yield tunable geometries with non-uniform, apex-concentrated strain, which is sensitive to material rigidity. Strain modifies band gaps and universally induces non-dispersive valence states, exhibiting composition-dependent wave-function character, as revealed by an in-depth analysis of band structures and orbital contributions. Crucially, transitions from these apex-localized valence states are predominantly dark. This characteristic is attributed to their localization at the $Γ$-point, inhibiting transitions to the lowest unoccupied states that reside at the K-valley. While revealing that the herein considered sub-10-nm nanobubbles fall short as single-photon emitters, our findings provide essential understanding of the structure-property relations in emerging quantum materials, providing robust design rules to optimize their characteristics for novel quantum applications.

Figures (6)

• Figure 1: a) Ball-and-stick representation, produced by the visualization software XCrySDenkokalj1999xcrysden, of the TMDC nanobubble supercell, with metal atoms in gray and chalcogen atoms in yellow. The lower-layer chalcogen atoms inside the orange circle with radius $r$ experience the inflating force. The metal atoms outside of the circle are held fixed. All other atoms are free to relax. b) Nanobubble profile, with orange bars defining the region of applied strain, while the black bars mark the supercell boundaries. c) Local strain map of a MoSe2 nanobubble subject to $F = 0.0175$ a.u./atom given by the relative change in interatomic distance between neighbouring metal atoms. Red and blue segments represent regions of compressive and tensile local strain, respectively, with values capped at +5% for visualization. Gray dots indicate metal atoms under the inflating force, with lighter shades indicating larger elevation. Green dots mark the positions of the metal atoms held fixed during relaxation. d) Maximum height of the nanobubbles as a function of the inflating force.
• Figure 2: a) Band gap and b) energy difference between the flat state and the highest occupied dispersive state as a function of the inflating force.
• Figure 3: Band structures of selected TMDC nanobubbles with the flat states highlighted by cyan dashed lines. The VBM is set to 0.0 eV. The heat maps show the wave function distribution (square modulus) of the states marked by red and green dots in the band structures, with the red and green circles marking the apex region (20% of the total supercell area) for the uppermost valence band and the lowest conduction band, respectively. The orange circles denote the area in which the force is applied. The composition and inflating force (in a.u./atom) of each nanobubble is reported as an inset in the corresponding band structure: a) and b) MoS2 at $F=0.0025$ a.u./atom and $F=0.01$ a.u./atom; c) and d) WS2 at $F=0.015$ a.u./atom and $F=0.02$ a.u./atom; e) and f) MoSe2 at $F=0.0175$ a.u./atom and $F=0.02$ a.u./atom; g) and h) WSe2 at $F=0.001$ a.u./atom and $F=0.0175$ a.u./atom.
• Figure 4: Wave-function distribution (WFD) at $\Gamma$ for the VBM and CBm localized at the apex encompassing 20% of the total supercell area. Empty symbols indicate the localized states within the valence band or nearly degenerate with the VBM at $\Gamma$ if the VBM is a flat state.
• Figure 5: Transition energy, defined as the energy difference between the initial and final state, of the average in-plane optical matrix elements for transitions at $\Gamma$ from the flat states of a) WSe2, b) WS2, c) MoSe2, and d) MoS2 nanobubbles and from the uppermost occupied dispersive state of e) WSe2, f) WS2, g) MoSe2, and h) MoS2 nanobubbles subject to a force $F \geq 0.01$ a.u./atom (see insets on the right). The oscillator strength is indicated by the color bar on top where transitions weaker than $<10^{-4}$ are considered dark and depicted in black. The width and thickness of the plotted bars increase (nonlinearly) with the transition energy, purely for visibility and clarity.