Quantum Dots in Transition Metal Dichalcogenides Induced by Atomic-Scale Deformations

Single-photon emission from monolayer transition metal dichalcogenides requires the existence of localized, atom-like states within the extended material. Here, we predict from first-principles the existence of quantum dots around atomic-scale protrusions, which result from substrate roughness or particles trapped between layers. Using density functional theory, we find such deformations to give rise to local membrane stretching and curvature, which lead to the emergence of gap states. Having enhanced outer-surface localization, they are prone to mixing with states pertaining to chalcogen vacancies and adsorbates. If the deformation is sharp, the conduction band minimum furthermore assumes atomic and valley-mixed character, potentially enabling quantum light emission. When such structural defects are arranged in an array, the new states couple to form energetically separated sub-bands, holding promise for intriguing superlattice dynamics. All of the observed features are shown to be closely linked to elastic, deformation-induced intra- and intervalley scattering processes.

The small displacement of K with respect to supercell Γ is applied to break zone-center-specific symmetries.The feint circles indicate the integration domains used to compute the populations P nK (K), P nK (−K) and P nK (Γ), which enable the determination of the valley polarization VP nK : all tiles touched by the circles are included, amounting to 9 tiles each for K and −K, 21 for Γ.Note that the absolute-valued A nK are considered for this procedure, not the spin-polarized version S nK on display here.

S1 Effect of Hybrid Functional
To check whether hybrid functionals give rise to differences in the electronic structure beyond a rigid gap opening in inhomogeneously strained 1L-TMDCs, we calculate and compare the band structure of a 1L-MoS 2 wrinkle with PBE and HSE 1,2 [Fig.S2b)-d)].It is found that HSE opens the gap by an additional amount at the M point, which influences the slope of the bands between the high-symmetry points.Thus, the higher-level result brings about more than a mere rigid shift.However, the difference is only quantitative, not qualitative, and the M point is irrelevant in our discussions.Thus, we employ the computationally more efficient PBE.Comparing the band structure of pristine 1L-MoS 2 [Fig.2b)] with the unfolded ones of the bubbles (Fig. 3), the most salient differences are the emergence of satellite bands and band splittings in the latter.As it turns out, these two are closely related.To see this, it is instructive to investigate CL bubbles with increasing heights z = h of the central Mo above the base plane (z = 0) (Fig. S6).

S2 Side Bands and Localized States
When h = 0.5 Å, the perturbation is rather weak and the main band prevails [panel a)].Some first-order side bands can clearly be spotted, but they appear superposed without interactions.These side bands are not entirely new states, but represent perturbative components obtained by the main wave functions around Γ that have the same energy.The main band is separated from its associated side bands by multiples of 2π/l, where l is the distance between bubble replica in neighboring supercells, for reasons laid out in the next section.Wave functions not having a well-defined wave vector is a manifestation of the loss of unit-cell translational symmetry in the deformed structure.Once the perturbative components become substantial, the supercell wave functions can turn into wave packets.Thus, electrons and holes are no longer necessarily delocalized as in the flat material, but may prefer certain subregions of the supercell, e.g., the top of the dome.For h = 0.5 Å, the side bands are still feeble and cannot influence the main bands to a significant degree.This the upper state straightens out to become completely dispersionless, turning it into the gap state strongly localized at the top of the dome.While spread significantly in reciprocal space, it hosts just two electrons with opposite spin.The round probability distribution in real space is reflected in a similarly shaped k-distribution around Γ.

S6
In our simulations, we enforce periodic boundary conditions.Thus, we effectively simulate not an isolated bubble, but an two-dimensional superlattice of them, which is spanned by the supercell lattice vectors A 1 and A 2 .In the following, we assume |A 1 | = |A 2 |, although this is not strictly true, and refer to these lenghts as l.As pointed out in the previous section, the perturbative side bands in the unfolded band structures are displaced by n × 2π/l with respect to the main bands, where n is an integer.This can be verified by inserting the same mCL bubbles into two different supercells (Fig. S7), thus varying l but leaving all other parameters unchanged.This gives rise to tangible differences in the electronic properties, e.g., in the energy gap between the CBm and the first subband above it.Consequently, such properties have to be regarded as features of the superlattice rather than of isolated bubbles.
We consider the interactions taking place at π/l away from the CBm, which lead to a small split of the lowest conduction band into an upper and a lower branch.These correspond to standing waves localized in complementary parts of the supercell.The dependence on the spacing l can be explained through the superlattice's Bragg condition, which is met for electrons with different k due to the difference in l.However, it is clear that a periodic array of centers can only effectively reflect waves if these centers have a large scattering cross section.Consequently, the splits are indicators of intravalley scattering, since the interactions occur between different states within the same valley.

S4 Real-Space Localization
We define an indicator for the real-space localization (RSL) of Bloch states as RSL nK = 2

Figure S1 :
Figure S1: Spin-resolved spectral function S nK for state n = 3108 of the central-load structure, which lies 200 meV below the gap state (n = 3120).Every tile represents a k-point mapping onto the supercell wavevector K= 0.01π(1/|A 1 |, 1/|A 2 |).The small displacement of K with respect to supercell Γ is applied to break zone-center-specific symmetries.The feint circles indicate the integration domains used to compute the populations P nK (K), P nK (−K) and P nK (Γ), which enable the determination of the valley polarization VP nK : all tiles touched by the circles are included, amounting to 9 tiles each for K and −K, 21 for Γ.Note that the absolute-valued A nK are considered for this procedure, not the spin-polarized version S nK on display here.

Figure
Figure S2: a) Primitive-cell Brillouin zone (delimited by black lines) and path along which the band structure for the bubbles is calculated.The small zone outlined in red pertains to the superlattice, which is spanned by the real-space vectors A 1 and A 2 .b) Side view of a 10% compressively strained MoS 2 wrinkle in 20 × √ 3 cell, which is used as a test system to compare PBE (c) and HSE (d) results.

Figure S3 :
Figure S3: Optimized geometry of a deformation arising as a result of strong forces being applied to the three S atoms below the central Mo atom, which is elevated to a height h = 3.9 Å above the base plane.The zoom into the central region shows intact Mo-S bonds and no reconstruction.

Figure
Figure S4: a) Strain distribution in the uniform-load bubble.For each pair of Mo atoms, the strain is calculated as ε = (d − d 0 )/d 0 , where d is the distance between them and d 0 is the lattice constant.A radial (r) and a circumferential path (φ) are drawn.b) Strain along r and φ directions for uniform (UL) and central load (CL) bubbles.For each value r/r 0 , where r 0 is the radius of the bubble, the r strain is calculated by averaging the six values obtained by a C 6 rotation of the path segments shown in panel a); the φ strain is calculating by averaging over all values in the hexagonal φ path drawn in panel a).

Figure S5 :
Figure S5: Local valence band maximum at Γ in a central-load bubble with compressive (-2.5%) and tensile (+2.5%) global strain applied, compared to the unstrained case (±0%).The same Mo are fixed at the same heights in all cases.

Figure S6 :
Figure S6: Evolution of the band edges in a central-load protrusion upon increasing the height h of the central Mo atom.
changes once h is increased to 1 Å [panel b)].The intensity of the first side band increases, and even a faint second-order satellite emerges at twice the k-distance.At this point, interactions start to take place at the intersection between the main band and the first side band, resulting in a mixing of the |k| = ±π/l states.Further elevating the central Mo to h = 1.5 Å [panel c)], the interactions are so strong that they cause a band split, giving rise to standing waves localized in different parts of the supercell.Standing waves have forwardand backward-propagating components, implying a large intravalley scattering cross section for corresponding charge carriers.Finally, reaching heights of h = 2 − 2.5 Å [panels d)-e)],

Figure S10 :
Figure S10: Valence band maximum at Γ for bilayer MoS 2 , where a r = 12 Å, h = 1.85 Å central-load structure is situated in the upper layer.Spin-orbit coupling is not taken into account.The system is contained in a 8 × 5 √ 3 supercell.