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Quantum-dot-like states in molybdenum disulfide nanostructures due to the interplay of local surface wrinkling, strain, and dielectric confinement Christian Carmesin, Michael Lorke, Matthias Florian, Daniel Erben, Alexander Schulz, Tim O. Wehling, and Frank Jahnke Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.9b00641 • Publication Date (Web): 04 Apr 2019 Downloaded from http://pubs.acs.org on April 4, 2019
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Nano Letters
Quantum-dot-like states in molybdenum disulfide nanostructures due to the interplay of local surface wrinkling, strain, and dielectric confinement Christian Carmesin,∗,† Michael Lorke,†,‡ Matthias Florian,† Daniel Erben,† Alexander Schulz,† Tim O. Wehling,†,‡ and Frank Jahnke† Institute for Theoretical Physics, University of Bremen, P.O. Box 330440, 28334 Bremen, Germany, and BCCMS, University of Bremen, P.O. Box 330440, 28334 Bremen, Germany E-mail:
[email protected] Abstract
Keywords
The observation of quantum light emission from atomically thin transition metal dichalcogenides has opened a new field of applications for these material systems. The corresponding excited charge-carrier localization has been linked to defects and strain, while open questions remain regarding the microscopic origin. We demonstrate that the bending rigidity of these materials leads to wrinkling of the twodimensional layer. The resulting strain field facilitates strong carrier localization due to its pronounced influence on the band gap. Additionally, we consider charge carrier confinement due to local changes of the dielectric environment and show that both effects contribute to modified electronic states and optical properties. The interplay of surface wrinkling, straininduced confinement, and local changes of the dielectric environment is demonstrated for the example of nanobubbles that form when monolayers are deposited on substrates or other twodimensional materials.
2D materials, transition-metal dichalcogenides, microscopic modeling, strain, single-photon source, nanobubbles In the past, single-photon emission has been realized using trapped-ion systems, 1 NV centers in diamond, 2 and epitaxially grown quantum dots 3 with important applications in quantum information. Recently, single-photon emission from spatially localized centers in transition metal dichalcogenides (TMDs) has been demonstrated. 4–14 For this purpose, TMD monolayers have been placed over gold nanostructures, 8,9 substrates with etched holes, 10 and arrays of dielectric micropillars. 11,12 In the latter case, up to 96% active single-photon emitters have been obtained. Quantum light emission has been also achieved with nanobubbles, which are formed in vertically stacked TMD structures. 13,14 Single-photon emission from atomically thin TMDs can originate either from structural defects resulting in electronic trap states, which are known to be a source of photoluminescence centers, 15,16 or due to strain-induced potentials with three-dimensional carrier confinements. In this paper, we show that the strain-induced oc-
∗
To whom correspondence should be addressed Institute for Theoretical Physics, University of Bremen, P.O. Box 330440, 28334 Bremen, Germany ‡ BCCMS, University of Bremen, P.O. Box 330440, 28334 Bremen, Germany †
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currence of localized electronic states is intrinsically linked to a non-continuous deformation of the TMD surface. We find that bending of the atomically thin sheet leads to formation of local surface wrinkles and bond deformations are identified as their microscopic origin. Due to the strong dependence of the TMD band gap on local strain fields 17 this wrinkling induces a spatially localized carrier confinement. We show that this wrinkling takes place on length scales of 1 nm, providing efficient threedimensional localized states and thereby enabling single-photon emission. This supports recent experimental results, 9 which first established the connection between surface wrinkling and localized emission, although the resolution of the used confocal microscopy is limited to much larger µm length scales. The local wrinkling effect is different from surface corrugations on a 10 − 20 nm scale, that is known for many 2D materials, as studied for graphene in Ref. 18 To investigate the interplay between TMD layer deformations and carrier confinement, we model a TMD layer with atomic resolution using a million-atom supercell. Based on valence force field simulations, new equilibrium positions of the individual atoms in the bended material are determined. The information about the displaced atoms is used in a tightbinding electronic-state calculation for the supercell structure. For the example of nanobubbles, we demonstrate that surface wrinkling acts as the main cause for quantum confinement with quantum-dot-like electronic states. The considered geometry is depicted in Fig. 1(a). Nanobubbles are formed when placing a monolayer of MoS2 on another MoS2 sheet used as a flat substrate, which is comparable to recent experimental investigations. 13 Starting from a paraboloid profile of the nanobubble, the relaxation of atoms in the upper TMD layer is performed while keeping the atoms in the lower layer fixed. For this part, we utilize a REAX potential 19 with the parametrization from Ref. 20 within a valence force field calculation, which is capable of accurately describing bond deformations under strain as well as continuous bond formation and breaking dynamics. We
find that both in-plane strain and bending contribute to the shape of the nanobubble (see Supporting Information for more details). Subsequent electronic-state calculations use this information about the positions of the individual atoms. For a supercell with an in-plane extension of 130 nm and up to 1.2×106 atoms, a 6-band tight-binding Hamiltonian is solved using the parametrization given in the Supporting Information. Strain-induced band gap changes arising from the displaced atomic positions are included via a generalized Harrison rule. 21 In our real space tight binding representation the energy gap varies spatially due to local changes in the strain. Additionally, when locally detaching the upper layer from the substrate underneath and changing from a commensurate bilayer to monolayer-like structures across the nanobubble, a modified dielectric environment and electronic hybridization is expected. 22–25 We include both effects into our model by changing individual tight-binding parameters based on GW calculations, as explained in the Supporting Information. Based on the calculated tight-binding states, optical emission spectra are obtained using Fermi’s golden rule. In our analysis, we consider nanobubbles of different aspect ratios h/r and sizes. The degree of localization is controlled by the aspect ratio, as shown in Fig. 1(b). In contrast, the nanobubble size does not change the qualitative behavior, as presented in the Supporting Information. For h/r = 0.1 the single-particle states are delocalized along the edge of the bubble, whereas for an increasing aspect ratio of h/r = 0.15 distinct maxima are found. Due to the increased strain, these maxima carry the C3v symmetry of the underlying crystal lattice. The trend of stronger localization with increasing aspect ratio continues for h/r = 0.175, which corresponds to the experimentally reported value of Ref. 13 In this case, the probability densities are localized at three distinct positions at the edge of the nanobubble. Figure 1(c) provides the emission spectra corresponding to the three different aspect ratios in Fig. 1(b), obtained from all confined states of the TB model and Fermi’s golden rule. For
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Figure 1: (a) MoS2 air nanobubble with radius r and height h. (b) Top view of the probability densities for electrons and holes, which are responsible for the strongest optical transition marked in (c). Results are shown for a bubble radius of r = 18 nm and various aspect ratios h/r. (c) Emission spectra calculated from Fermi’s golden rule based on the tight-binding states for the aspect ratios in (b). Results are plotted with respect to the bilayer band gap EBL . h/r = 0.1, the oscillator strength is very weak due to a small dipole interaction matrix element, which is the result of a much larger radial broadening of the probability density for the electrons than for the holes. More optical transitions are present for larger aspect ratios h/r. In the case of h/r = 0.175 strong emission from the electron and hole states, shown in the right panel of Fig. 1(b), as well as a broad background emission from other states is obtained. A similar behavior has been observed in recent experiments with WSe2 nanobubbles 14 and can be explained by increasing strain, which leads to a deeper confinement, stronger localization of the single-particle states, as seen in Fig. 1(b), and a redshift of the optical spectra. To analyze the mechanism behind the strong carrier localization, we quantify the contributions to the confinement potential due to strain as well as local changes of the screening and electronic hybridization for the upper TMD layer. As a result of the nanobubble geometry, shown in Fig. 2(c), the dielectric and electronic environment outside of the bubble corresponds
to a bilayer, while, in the center of the bubble, the large distance between top and bottom layer resembles the situation of a monolayer. This effect increases with increasing height h of the bubble, as illustrated in Fig. 2(c). The resulting changes of band gap are depicted in Fig. 2(d). Effectively, it leads to a repulsive potential in the center of the bubble. From our calculations we obtain band gap changes on the order of 100 meV, consistent with recent measurements. 24 The strain-induced changes of band gap for three different aspect ratios are displayed in Fig. 2(a). By comparing these strain effects with the probability densities in Fig. 1(b), we can conclude that the strain-induced changes of the band gap are responsible for the increasing localization, as the dielectric effect is rotationally symmetric (see Fig. 2(d)). Depending on the material, the magnitude of the respective shift of band gap varies. In our calculations for MoS2 bubbles, we found that dielectric effects are of minor amplitude compared to the strain effects. For aspect
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(a) Strain effect: Change of band gap h/r = 0.15 h/r = 0.1 nm
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Figure 2: (a) Strain-induced band gap shifts for different aspect ratios h/r. (b) Average angle α (in radians) between normal vectors of neighboring unit cells reveal wrinkling that corresponds to the spatially localized strain pockets. (c) Cross-section through the nanobubble of height h. The geometric situation implies a change of the dielectric and electronic environment for the MoS2 layer that is forming the nanobubble. The arrows indicating an angle α between normal vectors of neighboring unit cells on the discretized surface as used in (b). (d) Repulsive potential inside the nanobubble of height h = 34 Å and aspect ratio h/r = 0.175 as a result of the modified dielectric and electronic environment for the top layer. (e) Schematic representation of the orbital character of localized hole states. For shallow confinement, the states are composed of dz orbitals, while for deeper confinement dxy orbitals also contribute. ratios of h/r = 0.15 and h/r = 0.175, strain pockets are found. To illuminate their origin, the average angle between normal vectors at neighboring unit cells (schematically depicted in Fig. 2(c)) is displayed in Fig. 2(b), as described by the normal-normal correlation function (see Supporting Information). These average angles reveal a local wrinkling of the surface for aspect ratios of h/r = 0.15 and h/r = 0.175, with the angle of the normal vectors between neighboring unit cell varying up to 6◦ . A comparison between Figs. 2(a) and (b) shows that the strain becomes non-uniform and peaks in the wrinkles. The physical origin of this behavior can be traced back to the high Young’s modulus of MoS2 (see e.g. Ref. 20,26 ). The higher the modulus, the better a material resists against compression/elongation. Especially for a high modulus, the material tends to minimize its total change of length. In particular, this is possible by the formation of
wrinkles. These results are supported by nanoindentation experiments, 27 which showed that monolayer MoS2 is a material with a high Young’s modulus with intrinsic strength of the Mo-S chemical bond. Moreover, corresponding calculations of the nanoindentation 28 predicted strong local changes of band gap . The bond deformations responsible for the wrinkling are more prominent in TMDs than in graphene, as the former consist of a tri-atomic structure, in which the transition metal atoms are sandwiched between chalcogen atoms. Additionally, the spherical deformation of the material that is imposed by the nanobubble geometry favors wrinkle formation. We expect the wrinkles to be a general feature of spherical deformations, in contrast to depositing the material on a cylindrical surface or over a ridge. Mathematically, the difference between these situations lies in the Gaussian curvature 29 of the geometry that is imposed upon the TMD monolayer.
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The strong band gap variations, induced by strain and the dielectric environment, also influence the oscillator strength of the optical transitions in Fig. 1(c). The main reason for this change is illustrated in Fig. 2(e). The localized electron states, originating from the conduction band, inherit the dz2 orbital character of the K-point. In the strained bubble geometry, the Γ valley lies energetically higher than the K-valley for holes. Depending on the depth of the confinement potential, the orbital character of the localized hole states is than either determined by states originating from the Γ-point (dz2 ), or from both the Γ- and the K-point (dxy ). In the former case, the emission is weak, as the electron and hole wave functions carry the same orbital character (dz2 ). In the latter case, the strong dipole matrix element of the K-point transition dominates, as known from the 2D monolayer. 17 Comparing our calculated dipole matrix elements for the nanobubble (D ∼ 102 nm ps pA) with those for the monolayer 17 (DML ∼ 104 nm ps pA), we can estimate the lifetimes (τ ∼ |D|2 ) of the localized states to be τ ∼ 10 ns in contrast to τML ∼ 1 ps in the monolayer. 30,31 The good agreement between these estimated lifetimes and the experimentally measured spontaneous emission times in TMD nanostructures 4,14 point towards the strain-induced confinement mechanism as a main source of single-photon emission. In summary, our calculations provide insight into the mechanism of quantum-dot-like carrier localization in TMD nanostructures for the example of MoS2 nanobubbles. We find a strong localization on the nanometer scale inside strain pockets, which carry the C3v symmetry of the underlying crystal lattice. It is demonstrated that the formation of such strain pockets is caused by a non-uniform strain, which peaks in surface wrinkles, an effect that is inaccessible to continuum theory. Our method is based on the requirements of covering strain as well as local changes of the screening and electronic hybridization on the nanometer scale. We quantify the contributions of these effects to the confinement potential in a combined atomistic valence force-field and million-atom supercell
tight-binding approach. We expect that our results for TMD nanobubbles can be transferred to different nanostructures, including nanoindentation 27,28 and pillar arrays, 11,12 where a specific pressure profile is exerted instead of isotropic pressure inside the nanobubble. All systems share the characteristic of spherical deformation, that can lead to wrinkle formation if the induced strain is sufficiently strong. These properties and mechanisms are expected to be present also in other TMD materials, including the experimentally frequently used WSe2 . These materials share similar mechanical stiffness characteristics, which in connection with the strong dependence of the optical band gap on local strain facilitates local surface wrinkling and the formation of a quantum-dot-like confinement potential. We find that the orbital character of the localized states changes compared to the 2D monolayer, which explains the measured recombination times of a few nanoseconds 4,14 in TMD nanostructures. In contrast, spontaneous emission times from defects are below 100 ps, 32 which points towards the strain-induced confinement mechanism as a main source of singlephoton emission in this nanostructures. We expect that our results stimulate new and alternative realizations of strong carrier confinement in atomically thin TMD semiconductors. Acknowledgements. The authors gratefully acknowledge computational resources from HLRN Berlin and funding from the DFG ("Deutsche Forschungsgemeinschaft") via the graduate school "Quantum-Mechanical Material Modeling". Supporting Information. The Supporting Information contains a description of the tightbinding model, its parameters, as well as the implementation of strain and dielectric effects.
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Graphical TOC Entry MoS2 Nanobubble
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